U 38. Understanding the Southern U. vs MVSU Football Game: A Comprehensive Analysis

How did Southern U. defeat MVSU in their recent football matchup. What were the key statistics and standout performances that led to the final score of 38-25. How did the game unfold quarter by quarter, and what does this result mean for both teams.

Game Overview and Final Score

The Southern University Jaguars emerged victorious in a high-scoring affair against the Mississippi Valley State Delta Devils, with a final score of 38-25. This exciting matchup showcased offensive prowess from both teams, but Southern U. ultimately proved to be the stronger side.

Scoring Breakdown by Quarter

Let’s examine how the game unfolded, quarter by quarter:

First Quarter

• Southern U.: Jackson 43-yard field goal (7:46)
• MVSU: Clayton 30-yard pass from Eason (kick blocked) (2:39)

Second Quarter

• Southern U.: Howard 18-yard pass from McDaniel (Jackson kick) (14:16)
• Southern U.: Washington 32-yard pass from McDaniel (Jackson kick) (7:54)
• Southern U.: McClain 12-yard pass from McDaniel (Jackson kick) (0:30)

Third Quarter

• MVSU: Johnson III 21-yard pass from Eason (Fernandez kick) (10:51)
• Southern U.: Kirkwood 46-yard pass from McDaniel (Jackson kick) (7:11)
• MVSU: Johnson 4-yard run (pass failed) (3:29)
• Southern U.: O’Connor 13-yard pass from McDaniel (Jackson kick) (0:34)

Fourth Quarter

• MVSU: Rankin 11-yard pass from Eason (pass failed) (10:16)

The scoring pattern reveals that Southern U. dominated the second quarter, which proved to be crucial in securing their victory.

Key Statistics Comparison

To understand the dynamics of the game, it’s essential to analyze the key statistics:

These statistics highlight Southern U.’s superiority in several key areas, particularly in rushing yards and time of possession.

Standout Offensive Performances

Several players delivered impressive performances that contributed significantly to the game’s outcome:

Southern U. Standouts

• Glendon McDaniel (QB): 18-26, 235 yards, 5 TDs
• Kobe Dillon (RB): 13 rushes, 109 yards
• Jarod Sims (WR): 7 receptions, 90 yards, 1 TD

MVSU Standouts

• Jalani Eason (QB): 18-27, 210 yards, 3 TDs, 1 INT
• Caleb Johnson (RB): 19 rushes, 87 yards, 1 TD
• Jarius Clayton (WR): 6 receptions, 109 yards, 1 TD

McDaniel’s five touchdown passes were a key factor in Southern U.’s victory, while Eason’s three touchdown passes kept MVSU competitive throughout the game.

Defensive Highlights and Turnovers

While the game was primarily an offensive showcase, defensive plays and turnovers still played a role in shaping the outcome:

• Southern U. recorded one interception against MVSU’s quarterback
• MVSU fumbled twice but managed to recover both times
• Southern U. committed no turnovers, demonstrating better ball security

The lack of turnovers by Southern U. contributed to their ability to maintain control of the game and secure the win.

Impact of Special Teams

Special teams play often goes unnoticed, but it can significantly influence a game’s outcome. In this matchup:

• Southern U.’s kicker, Jackson, successfully converted all five extra point attempts and a 43-yard field goal
• MVSU had one blocked extra point attempt, which proved costly in a game decided by 13 points
• Southern U. averaged 44.5 yards per punt compared to MVSU’s 38.0, giving them an edge in field position battles

The consistency of Southern U.’s kicking game and their superior punting average provided them with valuable advantages throughout the contest.

Coaching Strategies and Game Management

The coaches’ decisions and game management strategies played a crucial role in the final outcome:

• Southern U.’s coach effectively utilized their rushing attack, keeping the offense balanced and unpredictable
• MVSU’s coach made aggressive decisions, attempting two-point conversions after touchdowns in an effort to close the gap
• Southern U.’s time of possession advantage (33:00 vs. 27:00) suggests effective clock management, particularly in the second half

The strategic choices made by both coaching staffs influenced the flow of the game and ultimately contributed to Southern U.’s victory.

Implications for Both Teams

This game has several implications for both Southern U. and MVSU moving forward:

For Southern U.

• The win boosts their confidence and potentially improves their standing in their conference
• The offense’s performance, particularly in the passing game, demonstrates their ability to score consistently
• Areas for improvement include reducing penalties and enhancing their run defense

For MVSU

• Despite the loss, their offensive output shows potential for future success
• Improving ball security and special teams play will be key focus areas in practice
• The defense will need to work on limiting big plays, particularly in the passing game

Both teams can use this game as a learning experience to refine their strategies and improve their performance in future matchups.

In conclusion, the Southern U. vs MVSU football game was a high-scoring affair that showcased offensive firepower from both teams. Southern U.’s balanced attack, led by quarterback Glendon McDaniel’s five touchdown passes, proved to be the deciding factor. While MVSU put up a strong fight, they were ultimately unable to overcome Southern U.’s consistent scoring and time of possession advantage. This game provides valuable insights for both teams as they move forward in their respective seasons, highlighting areas of strength to build upon and weaknesses to address in future matchups.

Southern U. 38, MVSU 25

SOU_FG Jackson 43, 07:46

MVSU_Clayton 30 pass from Eason (kick blocked), 02:39

Second Quarter

SOU_Howard 18 pass from McDaniel (Jackson kick), 14:16

SOU_Washington 32 pass from McDaniel (Jackson kick), 07:54

SOU_McClain 12 pass from McDaniel (Jackson kick), 00:30

Third Quarter

MVSU_Johnson III 21 pass from Eason (Fernandez kick), 10:51

SOU_Kirkwood 46 pass from McDaniel (Jackson kick), 07:11

MVSU_Johnson 4 run (pass failed), 03:29

SOU_O’Connor 13 pass from McDaniel (Jackson kick), 00:34

Fourth Quarter

MVSU_Rankin 11 pass from Eason (pass failed), 10:16

	SOU	MVSU
First downs	23	18
Rushes-yards	40-223	35-125
Passing	235	210
Comp-Att-Int	18-26-0	18-27-1
Return Yards	36	15
Punts-Avg.	2-44.5	4-38.0
Fumbles-Lost	0-0	2-0
Penalty-Yards	8-46	6-44
Time of Possession	33:00	27:00

___

INDIVIDUAL STATISTICS

RUSHING_Southern U., Ko. Dillon 13-109, De. Benn 11-41, Br. Hinton 5-37, Gl. McDaniel 7-14, Cr. Nelson 2-13, Ja. Washington 1-7, Ma. Fontenot 1-2. MVSU, Ca. Johnson 19-87, Ja. Eason 10-25, Ja. Thomas 3-8, Ja. Lindsey 1-5, Ja. Rankin 1-2, Qu. Ivory 1-(minus 2).

PASSING_Southern U., Gl. McDaniel 18-26-0-235. MVSU, Ja. Eason 18-27-1-210.

RECEIVING_Southern U., Ja. Washington 7-90, Et. Howard 4-61, Ty. Kirkwood 3-52, Tr. O’Connor 3-20, Ma. McClain 1-12. MVSU, Ja. Clayton 6-109, Ja. Rankin 5-35, Do. Johnson III 2-26, Ma. Myers 2-21, Tr. Mumphrey 2-10, Ma. Gladney 1-9.

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Southern U.

38, MVSU 25

SOU_FG Jackson 43, 07:46

MVSU_Clayton 30 pass from Eason (kick blocked), 02:39

Second Quarter

SOU_Howard 18 pass from McDaniel (Jackson kick), 14:16

SOU_Washington 32 pass from McDaniel (Jackson kick), 07:54

SOU_McClain 12 pass from McDaniel (Jackson kick), 00:30

Third Quarter

MVSU_Johnson III 21 pass from Eason (Fernandez kick), 10:51

SOU_Kirkwood 46 pass from McDaniel (Jackson kick), 07:11

MVSU_Johnson 4 run (pass failed), 03:29

SOU_O’Connor 13 pass from McDaniel (Jackson kick), 00:34

Fourth Quarter

MVSU_Rankin 11 pass from Eason (pass failed), 10:16

	SOU	MVSU
First downs	23	18
Rushes-yards	40-223	35-125
Passing	235	210
Comp-Att-Int	18-26-0	18-27-1
Return Yards	36	15
Punts-Avg.	2-44.5	4-38.0
Fumbles-Lost	0-0	2-0
Penalty-Yards	8-46	6-44
Time of Possession	33:00	27:00

___

INDIVIDUAL STATISTICS

RUSHING_Southern U., Ko. Dillon 13-109, De. Benn 11-41, Br. Hinton 5-37, Gl. McDaniel 7-14, Cr. Nelson 2-13, Ja. Washington 1-7, Ma. Fontenot 1-2. MVSU, Ca. Johnson 19-87, Ja. Eason 10-25, Ja. Thomas 3-8, Ja. Lindsey 1-5, Ja. Rankin 1-2, Qu. Ivory 1-(minus 2).

PASSING_Southern U., Gl. McDaniel 18-26-0-235. MVSU, Ja. Eason 18-27-1-210.

RECEIVING_Southern U., Ja. Washington 7-90, Et. Howard 4-61, Ty. Kirkwood 3-52, Tr. O’Connor 3-20, Ma. McClain 1-12. MVSU, Ja. Clayton 6-109, Ja. Rankin 5-35, Do. Johnson III 2-26, Ma. Myers 2-21, Tr. Mumphrey 2-10, Ma. Gladney 1-9.

Film Review: Michigan State 38 – Miami 17

The Miami Hurricanes were blown out by the Michigan State Spartans on Saturday afternoon by a final score of 38-17. MSU scored 21 points in the 4th quarter, one where many felt The U would prevail due to the heat and humidity of Hard Rock Stadium. The pundits were wrong. Sparty dominated The U in the 4th quarter heat and put the ‘Canes away with ground and pound in true Big Ten fashion.

Coach Diaz is now 1-2 on the season and 1-4 over his last five games. If it wasn’t for a safety on a punt long snap gone bad, Miami might’ve started 0-3 in 2021. This is following an 8-3 season where Miami made bad teams look better and got slapped by UNC on senior day.

The stats

Want to lose to a Big Ten team? Run for only 2.2 yards per carry. While running back Cam’Ron Harris ran for four yards per carry, quarterback D’Eriq King ran for only 0.6. King was beat up on the afternoon with shoulder and lower leg issues but Miami ran him out there, even down 21 with only minutes to go in the game. Tyler Van Dyke and Jake Garcia have still failed to take meaningful snaps in orange and green. That doesn’t bode well with King hobbled and the ACC Coastal slate fast approaching.

Photo by Joel Auerbach/Getty Images

Miami lacked the explosive plays that an allegedly fast program should make. The ‘Canes, with an obvious focus on body building and power lifting, looked slow and lacked the wiggle or breakaway speed needed for explosive plays. Harris is easy to take down in space, while Charleston Rambo had a fantastic day, he too lacks wiggle in the open field. Miami had no receivers hit 16 or more yards per catch, and the U’s longest rush was 11 yards.

On the other side of the ball, MSU QB Payton Thorne had a 22 yard long, RB Kenneth Walker III had a 26 yarder, and two MSU receivers had 20+ yards per grab. Sparty looked more explosive in pads in the heat than the ‘Canes have in years.

Miami was 6-of-17 on 3rd downs while Sparty was 7-of-15, and The U turned the ball over four times on Saturday to MSU’s ZERO. The Hurricanes committed 10 penalties for 93 yards while MSU committed nine penalties for 80 yards. Overall the game was a slog that took almost four hours to complete.

The ‘Canes offense

After 3 games the #canes are averaging 18.3 points per game which is dead last in the ACC. They are 116/130 nationally.

— rOmaN (@Romancane) September 19, 2021

I could probably just stop now and save us all the misery. Roman summed up the ‘Canes offense in one tweet, but, the show goes on. Miami was a paltry 35% on third downs against the Spartans, but did finish 3-of-4 on 4th down which shows some guts by Manny Diaz. I like going for it on 4th down and not taking safe field goals or punting from midfield and in.

While King threw for 388 yards that’s a Jon Kitna “in a losing effort” (Solid Verbal) type of stat. King averaged only 6.6 yards per attempt while throwing two TD’s and two interceptions. Charleston Rambo had a big afternoon as a possession receiver hauling in 12 balls for 156 yards and two TD’s but Mike Harley and Will Mallory suffered through the drops once again. MSU also had three sacks, six tackles for loss, six pass breakups and two hurries.

Above– The weird run play where King fumbles made no sense to me. It’s not inverted veer (power read) and it’s not outside zone with the way the back side guard and tackle fan block out on their DT and DE. It’s not a play-action and I can’t see an RPO tag.

Above– This is just a play where I’m assuming King is faking it to the RB and has carte blanche to run around.

Above– I absolutely hate the tight splits Miami is using. The angle to the QB is way too narrow. Defensive linemen line up on the OL by shoulder. You can either give the DE 5-yards to cover or 3-yards. Lashlee is choosing 3-yards. I choose 5-yards. The 5-tech (outside defensive end) isn’t giving up his leverage on the OT’s outside shoulder, so why not force that angle further out? Lashlee has abandoned all things learned from Sonny Dykes, much like Dan Enos abandoned all things learned from Nick Saban in 2019.

Above– Then you have the technique used by Jarid Williams, the right tackle. You can clearly see why Williams has struggled to start in this offense, even with an atrocious O-Line ran onto the field by Garin Justice. Williams never keeps knee-to-crotch relationship and it costs him. His feet are together, and then as the end uses a hand slap move, Williams has to lunge at him instead of moving his feet and keeping his outside knee in the DE’s crotch.

Above– Here’s the difference between King and some other QB’s, like Chase Brice. Brice would’ve fired this thing in at 100 MPH and made it a much harder ball to catch. King puts the perfect touch on it and it’s in a spot where only Rambo can grab it. King doesn’t always throw the most catchable ball (loose spirals) but this is that veteran touch Miami needs.

Above– I don’t understand what Zion Nelson is doing here. With Harris protecting the other side, Nelson is big-on-big with the DE. He can’t step inside first. The let guard has to have his DT and Nelson has to have the DE. Now, why Harris isn’t protecting inside-out on that side is beyond me as the right side looks to be in slide protection.

The ‘Canes defense

Miami’s defense came away with only three sacks, seven tackles for loss, only two PBU’s and a single hurry on the afternoon. After the first quarter, the pass rush dipped significantly once MSU figured out Miami’s striker-will blitz.

Above– I put this on MSU’s QB being too slow to get rid of the ball, and their bad play calling in the first half, as much as Miami’s defense. MSU was calling plays like you play Madden versus the CPU. You’ll get away with all of the slow drop backs against AI but once your Uncle Roscoe takes the sticks you have to change your game plan a little and get rid of the football faster.

Above– Diaz was dialing up that extra aggressive pressure and at this point I was getting a little excited. Chantz Williams is a player and should be on the first-team defense moving forward along with Jahfari Harvey.

Above– Live by the blitz, die by the blitz. Diaz loves to bring pressure, but on 3rd and 1 Gurvan Hall doesn’t understand the difference between a run blitz and a pass rush. Hall winds up 5-yards deep behind the line of scrimmage (LOS) while he needs to play at the LOS. Tyrique Stevenson, the CB, has NFL ability but he’ll need better coaching in order to be the “Day 2” pick he’s capable of being.

Above– Again, Miami loses contain and takes bad pursuit angles. This team STILL hasn’t learned how to finish (pursue and tackle) in six years under Manny Diaz.

Above– When Barstool picks up Hall pooping the bed, well, yeah.

Above– Kenneth Walker III wasn’t happy with his playing time at Wake Forest, a bad ACC team. So he transfers to Mel Tucker’s MSU squad and all of a sudden becomes Jerome Bettis.

The wrap

The Canyonero keys to victory for Miami against MSU were simple: 1- don’t give up big plays to the Spartans wide receivers, 2- don’t ‘force it’ on offense, and 3- beat someone with special teams.

1- Miami allowed two MSU receivers to go over 20 yards per catch.

2- Rhett Lashlee continued his futile attempt at running D’Eriq King on QB runs even while he was hobbled, and using the slow and stiff Cam Harris on obvious running situations.

3- Miami’s kicking game failed again. They were penalized, they caught a punt inside the 5-yard line, and they missed a field goal.

Manny Diaz’s time in Miami should be over, however, with there still being a chance to win the wide open Coastal, I’m sure he’ll be given time to see this season through. Miami won’t pull a USC and start the search early, they’ll wait until there’s a debacle against Virginia Tech or UNC.

The ‘Canes are facing Central Connecticut on Saturday, September 25th at Hard Rock in what could be an empty stadium. I’m not sure why Miami books these really bad FCS teams (see: Wagner) because it’s not like the win is worth anything, and having the watchful eyes of the state of Connecticut’s nine whole ranked players, none above a three-star ranking, seems moot. Here’s to seeing Tyler Van Dyke and Jake Garcia handoff three times a piece halfway through the 4th quarter.

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The MEMOS-U code description of macroscopic melt dynamics in fusion devices

For next generation tokamak devices such as ITER (currently under construction) and DEMO (currently in the planning phase), the plasma-facing component (PFC) lifetime and power handling capabilities constitute major design challenges. ITER features a beryllium (Be) first wall and a full tungsten (W) divertor [1]. In the divertor, intense steady state heat loads will elevate surface temperatures close to or above the W recrystallization range, raising concerns for edge melting and full surface melting during downward vertical displacement events (VDEs), edge-localized modes (ELMs) or loss of detachment control [2–4]. On the Be main chamber PFCs, the principal concern is the thermal and current quench phases of unmitigated VDEs or major disruptions.

Metallic tokamak melts are subject to strong electromagnetic forces which inevitably displace molten material, deforming the armour surface topology which could harm the power-handling capabilities by altering the incident magnetic field angle. Severe melt motion could also bridge castellations if melt fills the gaps between PFC elements. The main concern is the accumulated material excavation which may ultimately necessitate replacement of damaged components. Moreover, strong surface melting will drive immediate and deep recrystallization, leading to the propagation of macro-cracks. In order to minimize the risk of PFC damage, ITER aims to achieve complete ELM suppression and to deploy an effective disruption mitigation system [3, 5]. Nevertheless, some PFC melting appears to be inevitable; the Be wall is already at risk during early plasmas at low plasma current [6], while melting of W PFCs in the divertor becomes a risk later in the operational stages, when the stored plasma energies are higher [3].

In order to assess the PFC lifetime and power handling capabilities, it is necessary to acquire reliable predictions for the onset of melting, the macroscopic melt motion and the final surface deformation. This is a complex, multifaceted problem which involves the temperature, momentum and electromagnetic response of the metal to the incident plasma fluxes. Moreover, the fluid domain is bounded by two dynamically evolving interfaces, namely the plasma-facing free interface and the liquid–solid interface. These not only delimit the fluid motion domain, but also exert dynamic influence on the liquid—in particular the free surface spatial alignment and local curvature.

Thus far, fully self-consistent simulations of this class of problems have been carried out in two spatial dimensions, within the assumption of rotational symmetry or local invariance in a spatial dimension. Applications concerning gas-metal arc welding [7, 8], vacuum arcs [9–11] and unipolar arcs [12] have been modelled successfully. Under fusion-relevant conditions, 2D simulations of W melt layer instabilities have been performed that neglect phase transitions [13–16]. Such simulations require prescribed melt pools and are valid only for short timescales.

In the context of fusion reactor walls, the main objective is to predict the mass displacement and the gross surface damage inflicted over large spatial and long temporal scales. The potentially molten PFC area can range from ∼cm² to tens of cm² while relevant time scales vary from tens of milliseconds up to seconds [6, 17–21]. Naturally, numerical modelling of the complete problem that features the entangled heat transfer including phase transitions and the three-dimensional liquid dynamics is a formidable computational task given the above spatiotemporal scales. However, the problem can be simplified by taking advantage of key melt flow characteristics, identifiable from experiments in contemporary machines.

It has been recognized from the early experiments in the TEXTOR tokamak that the shallow water approximation is applicable, courtesy of the very large ratio between the molten surface extent and the melt depth [22]. The shallow water approximation makes the long length and timescale problem computationally feasible by reducing the dimensions of fluid flow by one [23, 24]. This feature has been implemented in the original macroscopic melt motion code MEMOS-3D [22, 25] which solves 3D heat transfer in combination with the shallow water equations.

More recent evidence obtained in tokamak experiments dedicated to ELM-induced W melting in the divertor (JET [17], ASDEX Upgrade [18, 19]) and VDE-induced Be melting in the main chamber (JET [20]) not only confirmed the shallow aspect of the pool dimensionality, but also brought up another crucial aspect of melt dynamics. The post-mortem analysis of these experiments revealed that frozen displaced material piles-up at distances of the order of the liquid pool extent, thus challenging the tacit assumption of MEMOS-3D [17, 26, 27] that the heat diffusion is much faster than the characteristic fluid motion. In scenarios where melt spills onto a progressively colder solid surface, the resolidification rate determines the viscous shear force and virtually governs melt dynamics [28]. Resolidification simultaneously stabilizes the liquid flow, as confirmed by experiments where a predominant flow direction can be discerned in the frozen material without signatures of liquid disintegration [17–20]. A new model has been developed and implemented in the MEMOS-U code, a significantly upgraded version of MEMOS-3D, aiming to address this novel fluid dynamics regime that is ubiquitous in contemporary fusion devices [28] and is also expected to be realized in melting scenarios on ITER [6].

The MEMOS-U code has been demonstrated to accurately describe the overarching physical processes behind shallow melting and surface deformation. Comprehensive benchmarking activities with four state-of-the-art tokamak experiments in JET and ASDEX Upgrade revealed that MEMOS-U is capable of quantitative predictions [28, 29]. The objective of the present work is to provide a complete detailed description of the physics model and the numerical schemes of MEMOS-U, especially since such a description has always been absent for the original MEMOS series of codes and is often sought after by the fusion plasma boundary research community. The shallow water fluid equations are established in the literature [23, 24] and have been widely utilized in studies of oceanic flows and atmospheric flows [30, 31]. Here, discussion will focus on peculiarities in the shallow water treatment of metal melts induced by hot magnetized plasmas, where gravity is negligible while phase transitions and electromagnetic responses are pivotal. Emphasis will be placed on the physics of boundary conditions, the underlying assumptions as well as the sensitivity to experimental uncertainties. Finally, aiming to highlight the most important aspects of melt dynamics under tokamak-relevant conditions, the JET ELM-driven W melting experiment [17] will be analysed in detail, demonstrating the shortcomings in the original MEMOS-3D analysis of experimental observations.

This section introduces the partitioning of domain dimensions, the key aspect of the physics approach behind MEMOS-U. The code features 3D heat transfer and 3D current density pathways, but only 2D fluid motion, the latter realized by employing the shallow water approximation where the Navier–Stokes equations are integrated across the vertical column depth, yielding two horizontal momentum equations and a continuity equation for the liquid column height h. The continuity equation then describes the surface deformation by providing the evolution of the sample surface position. The description of the free surface interface with a 2D vector field makes for extraordinary ease of interpretation and efficient numerical implementation [32, 33]. Thus, this approach facilitates MEMOS-U simulations of large domains and over long times within feasible time frames.

The shallow water equations are valid when the horizontal liquid length scales are much larger than the vertical length scale. It is not possible to set a strict maximum allowable ratio between the vertical and horizontal scales, but one can consider the value 0.05 as a soft validity limit for the shallow water approximation [24]. This threshold is typically not surpassed for molten PFCs in tokamaks, where the melt pool commonly extends several centimetres while the melt layer and surface build-up generally remain at around or below a millimetre [17, 19, 20].

Since the shallow water equations have been rather extensively described in the literature [23, 24], their complete derivation will not be repeated here. Instead, the discussion will focus on the applied physical boundary conditions, the neglected terms and the treatment of the transient boundary at the bottom of the liquid pool, i. e. the melting or re-solidification front.

2.1. The thermoelectric magnetohydrodynamic equations

The self-consistent description of the liquid metal is based on the incompressible resistive thermoelectric magnetohydrodynamic (TEMHD) equations, together with the temperature convection–diffusion equation [34]. The temperature equation is derived from conservation of energy; it includes the volumetric energy source terms from Joule and Thomson heating [34], but the negligible viscous heating term has been omitted. The fluid description is given by the two incompressible Navier–Stokes equations and includes the Lorentz force density.

The electromagnetic description is limited to the irrotational condition for the electric field and the solenoidal condition for the current density. This is a consequence of the magnetostatic approximation which allows important simplifications of Faraday’s and Ohm’s law, as well as of the strong tokamak magnetic fields that essentially allow Ampere’s and Gauss’ magnetic law to be bypassed [35]. Note that the thermoelectric current does not explicitly appear in the current density expressions, since it can be shown to be an irrotational vector for uniform material composition [34].

The emerging fluid, temperature and current density equations are the following:

with the fluid velocity, current density, magnetic flux density, fluid pressure, temperature and the mass density, specific isobaric heat capacity, thermal conductivity, dynamic viscosity, electrical resistivity, absolute thermoelectric power.

2.2. The shallow water equations

In what follows, the shallow water equations will be derived from the Navier–Stokes equations supplemented by appropriate boundary conditions corresponding to the free surface as well as the liquid–solid interface. In order to formulate boundary conditions on the free surface, not only the liquid but also the plasma-plus-vapour ambient medium need to be implicitly treated as a (immiscible) TEMHD fluid. Ultimately, the plasma and vapour effects appear exclusively through the pressure and drag force in the stress balance conditions as well as the incident plasma heat flux and the vapour cooling flux in the boundary condition for the temperature equation. Those plasma properties are physical quantities that can be extracted from external (kinetic) models or experimental observations.

The notations b₁ and b₂ are employed for the surface-normal positions of the liquid–solid and the free surface boundaries, respectively. Figure 1 offers an illustration of their relative placement with respect to the undeformed (pristine) surface. The melt layer thickness h is defined as the distance between the two interfaces, . Throughout the text, the subscript n is reserved for the inward surface normal (vertical, directed into the domain) components and the subscript t is reserved for the surface tangent (parallel) components of all vector quantities.

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Figure 1. Sketch of the 2D cross-section of a flat sample featuring a horizontally displaced induced melt layer. The liquid–solid interface (b₁), the free surface (b₂) and the melt thickness (h) are indicated. Note that the relative dimensions have been exaggerated for illustrative purposes.

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On the free interface, the kinematic boundary condition is applied with the surface normal velocity given by the time derivative of the surface deformation in the Lagrangian frame. Accounting for the rate of change of the interface position due to vaporization (), we obtain

In the application of the dynamic boundary condition, the Cauchy stress tensor is considered but the magnetic component of the Maxwell stress tensor (that results from the force density) is discarded due to the negligible induced magnetic field and the non-ferromagnetic nature of both media. Stress balance normal to the surface leads to a pressure jump at the free interface that is equal to the surface tension force:

where γ is the surface tension, κ is the free surface curvature and P is total ambient pressure (plasma plus metal vapour). In the shallow water applications of interest, the surfaces remain relatively flat and the surface curvature is generally very small so that surface normal tension can be neglected. Stress balance tangential to the surface leads to the boundary condition:

Here the viscous plasma contribution has been converted to the external drag force density . Assuming uniform material composition, the surface tension gradient leads to the thermo-capillary term .

On the liquid–solid interface, the no-slip condition is applied at the bottom of the melt pool. The relation:

forces a vertical velocity gradient which induces a shear force that counters the flow velocity. In the appendix, we discuss a more general kinematic boundary condition.

The first step in the derivation of the shallow water equations involves integration of the incompressibility equation, equation (1), across the shallow melt layer between the free surface and liquid–solid interface that introduces the depth-averaged velocity . Consideration of the kinematic boundary condition at the free surface and the no-slip condition at the liquid–solid interface yields a continuity equation for the surface-normal column height, or melt layer thickness h. The equation features a source or sink term accounting for melting or re-solidification () and a sink term accounting for vaporization (). The height equation reads as:

The derivation of the shallow water equations proceeds with inspection of the surface-normal momentum equation. Order-of-magnitude estimates reveal that the inertial terms are commonly negligible, and hydrostatic balance of the liquid column can then be assumed, as if the column were at rest at all times [24]. For the regimes that are encountered in tokamak melting scenarios, the vertical force components are typically negligible, resulting in from normal momentum balance. Accounting for the dynamic boundary condition at the free surface, we acquire:

A non-negligible normal force will be counteracted by a surface-normal pressure gradient, and the average pressure in the liquid column will depend on h through hydrostatic balance.

The next step of the shallow water equation derivation involves the depth averaging of the tangential momentum equation across the shallow melt layer, the application of the kinematic boundary condition and combination with the column height continuity equation. Neglecting, for simplicity, the variation of the velocity from the average, we get:

Explicit momentum transfer across the evolving liquid–solid boundary has been simplified, see the appendix for details. The last viscosity term can be evaluated by prescribing a parabolic profile for the depth-dependence of . The three unknown polynomial coefficients are then computed with the (a) tangential stress condition equation (8), (b) no-slip condition equation (9), (c) equality of the depth-averaged polynomial to the depth average velocity field. Denoting the depth average operation with brackets and introducing the surface temperature , we have:

Thus, the shallow water approximation reduces the unknowns from four () to three (), i.e. to the melt layer thickness and depth averaged tangential velocity.

2.3. The model equations

The depth-averaged melt velocity should also be used in the advective term of the temperature equation. The present treatment of the convective derivative is based on two simplifying assumptions that are aligned with the shallow water approximation: (a) the surface-tangential velocity is constant along the melt column depth; (b) the surface-normal velocity component is zero. Moreover, the irrotational electric field allows the definition of an auxiliary potential ψ and conversion of the current continuity condition into a differential equation for this scalar field. Overall, the MEMOS-U system of equations reads as:

with the electrical conductivity. Since P, as well as are external quantities and b₁, are fully determined by the temperature, there are essentially 5 unknowns ().

The boundary condition for the height equation is formulated by specifying the inflow at the edge of the fluid domain. Since no material can flow into the domain from outside, a trivial condition emerges when the vectors , are anti-parallel, where the latter is the outward pointing normal to the 2D fluid domain. On the other hand, there is a free outflow, that is no boundary condition is enforced, when , are parallel. Hence, we have:

The boundary condition for the momentum equation is:

for the advective term and shear stress due to horizontal gradients. The procedure of removing or adding occupied fluid cells is slightly involved; details of the numerical implementation will be discussed in section 3. The boundary condition for the temperature equation stems from prescribing a heat flux across the plasma-exposed surface. The modification of Fourier’s law by thermoelectricity has also been included [34] to obtain:

In the above; is the inward directed unit normal to the 3D domain, is the incident heat flux from the plasma-facing side, is the surface cooling flux. Finally, the boundary condition for the auxiliary potential stems from current continuity and reads as [35]:

where denotes the total current density on the sample surface. The description and the physics behind all the so far unspecified terms will be thoroughly discussed in section 4.

2.4. The coupling between equations

The shallow water equations for incompressible 3D flows are fully analogous to a compressible flow in 2D, where the column height plays the role of the mass density [24]. Typically, in shallow water applications gravity acts as a restoring force with the corresponding equation of state given by the hydrostatic balance of the liquid column. In the present case, no such explicit relation exists between equations (10) and (11). The evolution of the column height h is determined by the 2D velocity field, but there are no restoring forces in the momentum equation. Only the vertical shear stress, which is the third rhs term of equation (11), is a strong decelerating term depending on h.

The spatiotemporal evolution of the solidification front b₁ is determined by heat transfer. Thus, equation (12) implicitly affects equation (10). Temperature diffusion dictates surface cooling and in this manner affects the momentum source terms which depend on surface temperature, for instance the thermo-capillary and Lorentz force densities. Transversely, the velocity field of the molten material determines the convective term of the temperature equation and the continuity equation for h determines filled grid points to be added or removed in the process of surface deformation, determining the heat conduction domain.

Equations (10) and (11) describing liquid metal motion are two-dimensional, while equations (12) and (13) describing heat transfer and the current pathways are three-dimensional. Therefore, there is a respective partition of the numerical domains in the MEMOS-U code; the quantities (T, ψ), or equivalently , are defined on a 3D Cartesian grid, whereas the quantities are defined on a 2D grid parallel to the exposed side, with correspondingly overlapping grid point positions.

The MEMOS-U code solves the system of equations (10)–(13) on a rectilinear grid with a finite difference approach [36]. The solution is propagated one time step via a series of incremental sub-steps, each updating and transferring an intermediate solution to the other. These sub-steps are:

• (a)  

  The diffusive part of the temperature equation is solved with an implicit scheme.

• (b)  

  The convective part of the temperature equation is solved with a first order upwind scheme.

• (c)  

  The steady state current distribution is obtained with an iterative scheme.

• (d)  

  The column height continuity equation is solved with a first order upwind scheme.

• (e)  

  The momentum equation is solved with an explicit scheme.

• (f)  

  The heat conduction domain is re-evaluated.

The sub-steps (a)–(f) are discussed in detail below.

3.1. Temperature diffusion

The temperature equation is solved on the 3D domain with an implicit scheme, splitting the treatment into one tri-diagonal matrix inversion for each dimension so that the solution requires operations with N the number of grid points. Figure 2 illustrates a sample geometry at two exposure phases—before and after some surface deformation by fluid motion. The height function b₂ determines the surface index that constitutes the boundary of the heat conduction domain where condition equation (16) is applied by utilizing ghost point temperatures. As demonstrated in the figure, the heat conduction domain approximates the deformed surface by a step function.

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Figure 2.  3D configuration of the MEMOS-U domain (a) prior to and (b) during melt layer deformation. The melt is displayed in red. Here the domain is cross-sectioned along the n, t₁-plane on a point of symmetry of the t₂-direction where . Note that the relative dimensions have been exaggerated and that a low resolution has been selected for illustrative purposes.

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The transformation between liquid and solid phases is treated with the heat integration method [37], where an algorithm keeps track of the latent heat of fusion for each individual grid point, and the temperature is kept at the melting point until phase transformation is completed. This casts the melt interface problem in a weak formulation, where each grid point can be considered as fully molten, fully solid, or undergoing a phase transition. The interface position within the grid point is unknown, thus the melt layer thickness accuracy is directly related to the grid size.

3.2. Temperature advection

Fluid motion displaces the sample’s temperature profile in the fixed grid reference frame. The treatment of advective temperature derivatives is standard in computational fluid dynamics and is implemented in MEMOS-U via the finite difference discretization of the advective equation:

by employing a first order upwind scheme.

Owing to the dimensionality mismatch, the melt velocity is applied to the whole molten material column at the corresponding surface point. As a consequence of the step function approximation of the melt layer thickness and the deformed surface, some fluid elements will be index-adjacent to solid or non-material elements and the advective derivative is set to zero for the corresponding direction. This can be discerned in the illustrative melt pool example of figure 2, where molten elements are adjacent to empty or solid grid points due to the step-wise approximation. In a highly resolved simulation, there will be few such grid points compared to the fully surrounded ones, thus introducing negligible errors.

3.3. Current density distribution

The finite difference approximation of equation (13) leads to the linear system of equations AΨ = B where Ψ is the auxiliary potential array, B the boundary value array and A the matrix operator for the spatial derivatives [35]. The system is solved with the iterative generalized minimum residual method (GMRES) [38].

Figure 3 features 2D cross-section sketches for two sample geometries which have been used on tokamaks in recent years in order to investigate melt dynamics: a sloped lamella (a) and a leading edge (b). The typical shape of the melt layer is included and simplified, but realistic current paths are drawn, see [35]. for the exact numerical solutions. The sketches illustrate the important differences in the near-surface current pathways between the leading edge and sloped samples as well as in the current density component that mainly induces melt motion through the force. In the sloped lamella, the normal current density component is sought, implying a negligible difference between the current density within the melt layer and the surface current density . On the other hand, in the leading edge, the tangential current density component is sought which can be significantly smaller than depending on the bending of the current paths [35].

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Figure 3. 2D sketch of the (a) sloped surface and (b) leading edge. The typical melt layer shape is indicated in red and the characteristic replacement current pathways are depicted in a simplified manner, see [35] for numerical solutions. The alignment of the local Cartesian coordinate system is also displayed (different for each exposure geometry). In both scenarios, the melt layer displacement is directed out of the page, i.e. perpendicular to .

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3.

4. Column height continuity

The column height h is determined by equation (10) and is propagated with the aid of an explicit first order upwind scheme. The liquid–solid interface b₁ is determined solely by the temperature equation and incremented in the heat diffusion sub-step, described above. Thus, surface deformation is also determined in this sub-step.

3.5. Momentum equation

The velocity field is updated by employing a first order upwind scheme for the advective term and an explicit representation of the momentum source terms. When a liquid metal element re-solidifies, the corresponding melt velocity is set to zero. On the other hand, when a solid element melts, a distinction should be made. In case the surface element melted under the effect of the surface heat flux only, then the melt velocity begins from zero. In case the surface element melted due to material outflow from an adjacent boundary, then the melt velocity of the corresponding upstream element is assigned.

3.6. Re-evaluation of the simulation domain

The re-evaluation of the simulation domain constitutes the last sub-step of each time step propagation. For each grid point on the surface-parallel plane, the surface position b₂ is used to detect the corresponding surface-normal grid point index that constitutes the domain boundary.

The updated surface indices delimit the heat conduction domain during the next computational cycle (time step), when the heat flux boundary condition will be enforced. This procedure results in the removal or the addition of grid points for each surface-parallel index and the temperature treatment at these points needs to be considered. When elements are being discarded, no action is taken and their removal is assumed not to impact the adjacent filled grid points. When elements are added, a distinction should be made similar to that in the momentum equation. In the case in which a molten grid point is added on top of a molten column, the temperature of the previous surface element at the same position is assigned. On the other hand, in the case in which a molten grid point is added on top of a previously solid element, the surface temperature of the upstream element is assigned.

4.1. The surface current density

The only non-trivial boundary condition in the replacement current module of MEMOS-U is the surface current density , whose nature depends on the plasma scenario and the sample material composition. In case of steady-state driven, ELM driven and disruption thermal quench driven melting in tokamaks, the surface current is given by the local escaping thermionic current that is dictated by the temperature field and the incident plasma fluxes. In contrast, in case of melting due to disruption current quench heat fluxes, the surface current is given by the halo current which strongly depends on the plasma conditions and in general requires global modelling.

4.1.1. Thermionic surface current.

Within the assumption of plasma ambipolarity, the release of thermionic electrons from the surface of a grounded bulk metallic sample generates a replacement current which flows through its volume [4, 35]. The associated current density is computed from the boundary value problem of equations (13) and (17) and is responsible for the dominant force density acting on the molten metal [35]. In the absence of external electromagnetic fields, the nominal thermionic current is described by the Richardson–Dushman formula that reflects the classical over-the-barrier physical mechanism [39]:

with the room temperature work function and the effective, material-dependent Richardson constant. It is evident that thermionic emission drastically increases for elevated surface temperatures and/or low work functions. As a result, considering the main PFC materials on ITER, it is very important for W melting (K, eV), but it is totally negligible when considering Be melting (K, eV).

The unimpeded thermionic current from W is known to greatly exceed the incident plasma current at the strike point position of typical Type-I ELMing H-mode AUG and JET discharges already at the melting point [40, 41]. However, complications arise due to the self-consistent sheath electrostatic fields and strong external magnetic fields. Such high emission intensities are incompatible with the classical Bohm pre-sheath structure and eventually space charge accumulation leads to the formation of a virtual cathode limiting the escaping current density to a maximum value [42, 43]. This is referred to as a space charge limited sheath. In the case of oblique magnetic field lines, a fraction of the low energy thermo-electrons is recaptured by the surface during the first Larmor gyration [44, 45]. This is referred to as prompt re-deposition.

Both suppression mechanisms need to be properly considered, since thermionic emission has a rather complex impact on the melt dynamics, directly influencing both liquid motion and heat transfer. The emitted current density determines the volumetric force exerted on the melt layer and thus the displaced liquid volume as well as the displacement itself. Furthermore, thermionic emission constitutes an important cooling mechanism (see section 4. 3) which affects heat balance and thus the quantity of liquid metal volume produced. The intricate coupling between the momentum and energy transfer aspects of thermionic emission is so important that any miscalculation of the escaping current density would result in large errors in the melt displacement estimates. The central role of thermionic emission has been recognized in MEMOS-U and led to its rigorous treatment via dedicated particle-in-cell (PIC) simulations [40, 41, 46].

2D3V PIC simulations of magnetized plasma sheaths with intense thermionic emission revealed that the escaping thermionic current saturation is a global characteristic of the space-charge limited regime, irrespective of the inclination angle of the magnetic field [46]. Extensive simulations led to the identification of an accurate and physically transparent expression that describes the dependence of the limited thermionic current on the plasma conditions, magnetic field strength and inclination angle. The semi-empirical expression was constructed on the basis of the Takamura model of unmagnetized space charge limited sheaths [43], the Chodura picture of inclined magnetized sheaths [47] and a single particle field-free approximation of prompt re-deposition. It reads as [46]:

with the plasma density, the electron thermal velocity, α the magnetic field inclination angle with respect to the surface normal. The above expression is valid for the inter- as well as intra-ELM plasmas of contemporary devices. It is accurate within 25% for non-grazing incidence () and within 60% for α = 2.5° [46]. Note though that , which suggests a rather negligible value of .

Overall, due to the strict current limitation in the space charge limited regime, the thermionic current is given by:

The following remarks are pertinent to the MEMOS-U implementation. (a) The escaping thermionic current expression stems from PIC simulations which assume an infinite emitting wall with a homogeneous prescribed temperature [46]. Since equation (20) is applied to all surface elements, it is implicitly assumed that the escaping current density contributions from each surface element are independent. PIC simulations of hot spots should be carried out to confirm the validity of this convenient assumption. (b) The low temperature branch of equation (20), which is selected prior to current limitation, follows the unimpeded Richardson–Dushman expression. This means that prompt re-deposition is only considered after the virtual cathode has formed. This simplification does not lead to errors since, for oblique inclination angles, the transition to the space-charge limited regime takes place at temperatures well below the W melting point. (c) As a result of surface deformation by melt motion, the surface normal is spatiotemporally varying which modifies the escaping current density [40, 41, 46]. The overall sin²(α) angular dependence implies a rapid escaping current decrease for shallow inclinations. Thus, depending on the geometry, it may be important that the inclination angle is not assigned its static value defined for a pristine unexposed sample but its local dynamic value [29]. (d) The Child–Langmuir type expression for the escaping current density depends on the plasma density and electron temperature profiles. Since such profiles are generally not available during ELMs, this expression can be transformed into a dependence on the incident field line parallel heat flux by assuming that the heat flux spatiotemporal variations can be attributed to temperature variations, that the plasma is isothermal, and by adopting a standard value for the sheath heat transmission coefficient [48].

We emphasize that the present description of the escaping emitted current density applies to both inter- and intra-ELM periods in contemporary fusion devices as well as to inter-ELM periods in ITER. A number of complications emerge during the ITER intra-ELM periods. (a) Owing to the combination of high plasma densities with elevated electron temperatures (eV), the contributions of secondary electron emission [49] and electron backscattering [50] start to become important along with thermionic emission. (b) Due to the elevated plasma temperatures and densities, the wall electrostatic field becomes high enough ( V m⁻¹) so that the Schottky effect becomes important leading to thermionic emission enhancement [39]. (c) Due to the elevated electron thermal current, a virtual cathode is formed at unrealistically high—even for melting scenarios—surface temperatures which implies that the emitted currents are only suppressed by prompt re-deposition. PIC simulations of the magnetic pre-sheath and electrostatic sheath in the presence of all emitted electron populations should be carried out in order to acquire reliable expressions for the escaping emitted current during ITER intra-ELM periods.

4.1.2. Halo surface current.

In case of PFC melting that is induced during disruption current quench phases, the halo current density determines the volumetric Lorentz force. The charge continuity equation inside the metal implies that the current density gradient scales should have the same magnitude in each spatial direction in the proximity of the surface. Thus, depth-variations of the current density inside the molten material [35] can be neglected provided that the horizontal gradient scales are very large compared to the melt thickness. Then, and the current module can be circumvented with providing the Lorentz force density. Given the overlap between the plasma-wetted area and the halo region, the horizontal extent typically reaches several centimetres during disruptions [6, 20], while the melt layer thickness remains below a millimetre. Hence, there is no need to accurately model the complicated current paths through the vessel that is a topic of considerable complexity.

In contrast to thermionic emission where is calculated self-consistently by the surface temperature, incident heat flux and local inclination angle, constitutes an external input for the MEMOS-U code. In disruption current quench-driven Be melting in JET [28], 100 kA m⁻² was employed based on empirical input [21]. In predictive Be melt simulations of a 5 MA VDE ITER scenario, 20 kA m⁻² was used based on the total halo current and the wetted area [6].

4.2. The surface heat flux

In ELM-driven and disruption-driven melting events, the incident energies of the plasma particles are always less than few keV. The corresponding depth range, i.e. the total material thickness traversed by the impinging particles prior to thermalization, lies well in the nanometre range and is thus much smaller than the characteristic length scales of interest and the typical MEMOS-U mesh sizes. To be more specific; for keV the electron, proton and helium depth ranges are ∼300, ∼150, ∼100 nm in Be and ∼80, ∼40, ∼20 nm in W [50–52]. As a result, plasma incidence does not result in volumetric heating, but in surface heating which can be accounted for only by the boundary condition equation (16).

For completeness, we note that runaway electron (RE) driven melting events are subject to volumetric heating, since when MeV the electron depth ranges are ∼3 mm in Be and ∼0.4 mm in W [53]. For the relativistic initial electron energies, electronic stopping as well as nuclear stopping are important, but also internal photon and secondary electron transport are central in the determination of the energy deposition profile [54]. Heavy RE-driven melting events cannot be simulated with MEMOS-U not because of volumetric plasma heating, but because the shallow water approximation breaks down.

The incident plasma heat flux is an external input to the MEMOS-U code and is inferred either by experimental measurements or theoretical considerations. In the ELM-driven W melt experiments on JET and AUG using dedicated leading edge or sloped samples [17, 19], spatiotemporal heat flux profiles are deduced from infrared diagnostic measurements of the temperature response of adjacent flat W samples. Employing the optical approximation and in accordance with the local magnetic field line inclination, the extracted heat flux is mapped to the exposed surface of interest. In some cases, 3D diffusion effects and isotropic background contributions need to be accounted for [55]. The inaccuracies of such heat flux profiles are expected to be within ∼25% [19, 28, 29]. Since the reference sample temperature is low, the total is extracted.

In disruption driven (current quench) Be melting in JET, direct time resolved measurements are unavailable. The thermocouple measurements in the upper dump plates yield the total energy deposited onto the Be plates during the whole discharge. The heat flux is evaluated by estimating the elapsed deposition time and the wetted area [20, 56]. Given the thermocouple measurements, in this case only the conducted heat flux, , is extracted. Finally, in predictive disruption current quench-driven Be melt simulations for ITER, the adopted heat flux is based on theoretical estimations and simulations of the conducted power across the separatrix and of the deposition profile on the first wall [6].

The sensitivity of the MEMOS-U output to the heat flux input depends on the melting scenario. In repetitive-ELM melting of W samples, most of the incident energy is expended in raising the base sample temperature close enough to the melting point, so that superimposed ELM temperature excursions will suffice to initiate transient melting near the end of the exposure. This is a threshold phenomenon, where a minuscule () quantity of input energy is consumed for the phase transformation. Thus, small—well within experimental uncertainties—heat flux variations will significantly affect the amount of melt generated. A quantitative discussion will follow in section 7.4. On the other hand, in disruption current quench-induced melting of Be samples, a larger fraction of the input energy is expended for melting. Therefore, the amount of metallic melt produced is less sensitive to heat flux variations.

4.3. The surface cooling flux

Surface cooling consists of electron emission cooling, vapour cooling and thermal radiation cooling [57, 58]:

Their implementation in MEMOS-U is detailed below.

4.3.1. Thermionic cooling.

For any uncompensated emitted electron, not only its kinetic energy but also its binding energy (given by the work function) is released from the PFC surface. Local thermodynamic equilibrium implies that thermionic electrons follow a Maxwellian distribution with a temperature equal to the local surface temperature [39]. Hence, the cooling flux is given by [39, 59]:

where e denotes the elementary charge. Since holds even beyond the normal W boiling point, the binding energy contribution is dominant. This cooling flux plays an important role in bulk W heat balance where it can reach values in excess of , significantly limiting the surface temperatures attained by an exposed W surface.

4.3.2. Vapour cooling.

Within the assumption of thermodynamic equilibrium between the exposed surface and the emitted vapour, the evaporated mass flux is described by the Knudsen formula [60]:

where is the atomic mass and is the vapour pressure whose dependence on the surface temperature is described by the exponential Antoine correlation. Evaporation plays a role in both surface deformation and cooling. As mentioned earlier, surface mass loss is translated into an interface velocity which acts as an independent sink term in equation (10). The respective cooling flux has a similar form to the thermionic cooling flux, but with the binding energy equal to the vapourization enthalpy , i. e.

Since holds even beyond the normal boiling point of the materials of most current interest in tokamaks and for ITER, the binding energy contribution is dominant. For completeness, we note that exhibits a very weak temperature dependence with ∼3% and 5% differences between its room temperature and normal boiling point values for W and Be, respectively [61, 62].

At this point, we discuss the relevant issue of vapour shielding of the incident plasma heat flux. For high surface temperatures, saturated vapour pressure can exceed the plasma pressure ( for tokamak conditions). In this case, the vapour species is expected to strongly interact with the incident plasma particles, leading to a reduction of the absorbed plasma heat flux. Atomic processes such as ionization, recombination as well as scattering govern the local plasma-vapour energy transfer, while the vapour transport characteristics are also influenced by the plasma and surface parameters. In spite of the fact that the ionized vapour-plasma interplay is highly complicated, it has been clearly demonstrated that the choice of vapour shielding model has a minor effect on the absorbed heat flux [63, 64]. This is a direct consequence of the exponential temperature dependence of , which effectively masks any variable interaction strength. One must therefore distinguish between the anticipated accuracy in the evaluation of the absorbed heat flux and of the evaporated material loss. For the latter, inaccurate evaporation rates are expected in scenarios involving very high surface temperatures. On the other hand, the absorbed heat flux is expected to be relatively insensitive to the vapour shielding description, with the beneficial consequence of robust melt production predictions.

4.3.3. Thermal radiation cooling.

Surface cooling due to the emission of thermal radiation is described by the Stefan–Boltzmann law [65]:

where is the Stefan–Boltzmann constant and is the total hemispherical emissivity of the PFC which depends strongly on the surface temperature. This cooling term is significant at surface temperatures below melting and for longer time scales, being overpowered at higher temperatures by the exponential nature of thermionic emission and/or evaporation. This contribution is rather negligible for typical tokamak transient melting experiments.

The relevant significance of these surface cooling terms can be better understood by plotting their magnitude as a function of the surface temperature for W and Be, see figure 4. In the case of W, thermionic cooling dominates in the rather wide neighbourhood of the liquid–solid phase transition. The emergence of current limitation close to the melting point, here calculated for a typical heat flux of , switches the monotonic character of the increase from exponential (courtesy of the Richardson–Dushman formula) to weak linear (courtesy of the kinetic cooling contribution). For W, vapour cooling remains negligible even beyond melting, since W possesses one of the lowest known vapour pressures, but due to its continuous exponential dependence, ultimately surpasses thermionic cooling for higher surface temperatures (not shown here). As mentioned above, radiative cooling provides an important contribution only prior to melting. In the case of Be, thermionic and radiative cooling are always negligible due to the low melting point, while vapour cooling becomes significant after melting.

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Figure 4. The W and Be surface cooling fluxes as a function of the surface temperature (normalized by the respective melting point). The limited thermionic current , necessary for the calculation of the themionic cooling flux, has been calculated for a representative heat flux of .

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4.4. The surface acceleration

Here we discuss the physics and importance of the secondary momentum terms, i.e. all terms other than the main volumetric Lorentz force and viscous deceleration.

4.4.1. Surface pressure gradient.

4.4.2. Thermo-capillary convection or the Marangoni effect.

Thermodynamic arguments predict a linear decrease of the surface tension beyond the melting point , as confirmed by experimental results for most liquid metals [66, 67]. The resulting thermo-capillary flows are introduced through the tangential stress balance boundary condition equation (8) on the free surface and, after depth averaging, yield the fifth momentum source term on the rhs of equation (11). By simple inspection, it is evident that intense surface temperature gradients lead to an effective force on the melt layer which pulls the liquid metal from the hot zones toward the colder areas. This is the so-called Marangoni term that is calculated self-consistently through the known temperature field. Given the spatial extent of the metal melt in the case of tokamak experiments, strong tangential gradients are rather atypical for leading edge ELM-induced melting or disruption current quench-induced melting [6, 28], thus thermo-capillary accelerations are generally not important compared to the main acceleration mechanism. Nevertheless, the effect can be important in some exposure geometries, such as in the sloped W lamella experiments in AUG [29].

4.4.3. Plasma surface drag.

Plasma flows oriented along the surface lead to a drag force per unit area . After depth averaging, the respective force density is given by the sixth momentum source term in equation (11), formally also introduced through the tangential stress balance boundary condition equation (8). In contrast to the Marangoni term, the acceleration explicitly depends on the incident plasma parameters and is an external input to MEMOS-U.

The typical melt velocities found in contemporary tokamak experiments are of the order of a few m s⁻¹, that is orders of magnitude lower than the ∼ km s⁻¹ thermodynamic sound speed in liquid metals at their melting point [68]. These extremely low Mach numbers allow us to circumvent an equation of state and safely assume the validity of the incompressibility condition. Moreover, given the kPa plasma pressures; the ∼100 GPa inverse compressibilities and the ∼0.1 K MPa⁻¹ initial slopes of the melting curves of the liquid metals of interest [68–71] suggest that the thermophysical and electrochemical material properties are independent of the pressure and depend solely on the temperature.

MEMOS-U employs analytical expressions for the temperature dependence of valid from room temperature up to the atmospheric pressure boiling point, and also state-of-the-art values for the quantities . All expressions are based on reliable experimental data rather than theoretical calculations and great effort has been taken to ensure that extrapolations are minimized.

Most of the adopted W material properties follow the recommendations of a recent comprehensive literature review [72]. A similar—yet unpublished—investigation has been carried out for the Be material properties by one of the authors [73]. In contrast to W, the properties of polycrystalline Be are much less explored, particularly for the liquid phase. As a consequence, few data sources are available for most properties beyond the melting point. Hence, one should remain critical of the adopted values, but keep their accuracy in reasonable relation to the overall accuracy and the sensitivity of any specific modelling problem. The discussion below exemplifies this remark.

Under atmospheric pressure, pure Be undergoes a polymorphic phase transition from an hcp (alpha-phase) to a bcc solid (beta phase), that takes place at 1543 ± 5 K and is naturally accompanied by discontinuities in most material properties [74]. Since the melting phase transition from a bcc solid to a liquid takes place at 1560 ± 5 K, there is a narrow temperature window of 17 ± 10 K for the beta phase of Be. Such desired accuracy would lead to unnecessary computational cost given the uncertainty in the plasma-facing boundary conditions. Therefore, in MEMOS-U, the alpha-beta transition is ignored and the enthalpy of hcp-to-bcc transformation (6.855 kJ mol⁻¹) is combined with the enthalpy of fusion (7. 959 kJ mol⁻¹) in a single latent heat contribution that is absorbed or released only at the melting point.

In addition, only pure polycrystalline W and Be are considered. In fact, experience with JET tokamak operation in a Be/W environment has revealed that the thickness of deposited Be layers and the extent of Be-W intermetallic compound formation in the divertor as well as the depth of beryllium oxide formation in the chamber are of the order of few micron [20, 75–77] and thus much thinner than the typical melt layers. There are no significant benefits in attempting to incorporate the mostly unknown BeO or Be_x
W_y
material properties.

Finally, it is also worth pointing out that, in view of planned transient and steady state melt experiments with proxy materials in AUG and DIII-D, the MEMOS-U material library has been recently expanded to include niobium, iridium, and aluminium.

The initial computational domain should match the 3D geometry of the pristine sample. As the computation/exposure progresses, the sample is heated and eventually a melt pool is produced. The 2D outline of the melt delimits the fluid domain and the exposed surface of the 3D domain is enclosed by the dynamic scalar field b₂. As surface deformation progresses driven by the Lorentz force, the 3D heat conduction domain should be re-evaluated for consistency. When molten material is displaced along the surface and excavates or amasses in different parts of the sample, both the sensible and the latent heat are displaced with the fluid. The numerical difficulties arising in the treatment of the transient 3D domain along with the energy displacement constitute the main complications in the implementation of the full model, equations (10)–(13).

These complexities can be circumvented on the basis of two assumptions for the molten surface characteristics:

These assumptions drastically simplify the coupling between the fluid equations and the temperature equation, see equations (10)–(12). The fluid solver does not provide any input to the heat solver and the continuity equation for the surface deformation becomes linear. The surface deformation is stored, but it does not constitute input for the progressive heat transfer calculations, or for any other parts of the code. In this manner, surface deformation is not considered in the heat transfer domain, but the deformed surface shape can still be predicted after several accumulated events (for example in repetitive ELM scenarios) or continuous displacements (for example in longer timescale disruption current quench scenarios). Moreover, since the wetted PFC area is typically much smaller than their total area, surface melting is localized so that regions of freshly produced melt are surrounded by pristine solid or by re-solidified regions. The above simplifications lead to liquid pools which thin out toward the edges being a direct reflection of the incoming heat flux spatial profile. Liquid motion is therefore essentially arrested past the pool edges. Section 7 provides a detailed discussion of this particular aspect.

The aforementioned assumptions further reduce the computational complexity and have been implemented in the original MEMOS code employed in [17, 26, 27]. However, as already pointed out in the Introduction, recent cross-machine experiments provided strong evidence that contradict their validity. In particular, post-mortem analysis after the repetitive ELM-driven melting of misaligned W samples in the JET and AUG divertors [17, 19], as well as the current quench-driven melting of the Be upper dump plates in JET [20], clearly revealed that the melt displacements are large—of the order of the pool extent. The metallic melt is accelerated out of the pool and displaced on top of the adjacent solid, where the strong temperature gradient facilitates heat conduction and rapid solidification near the contact. This regime necessitates a solution of the full model described in section 2. This fact, together with the unjustified circumvention of the field equations and the lack of rigorous boundary conditions in MEMOS-3D, provided the motivation for the development and implementation of the new physics model in MEMOS-U.

In this section, aiming to demonstrate the capabilities of the MEMOS-U code and the predictive power of the new physics model, we will present simulation results for the JET W leading edge exposures [17], the first attempt in a tokamak to provide a controlled, transient-induced melt for the purposes of model validation. In the experiment, a specially engineered lamella located in the outer, full W divertor, was deliberately misaligned, exposing a tapering leading edge of width 0.25–, intercepting the full heat flux flowing parallel to the magnetic field lines—see a schematic of the geometry in figure 5. The outer separatix strike point was swept across the sample for a few seconds at a time, during nine consecutive type I ELMing H-mode pulses. High resolution visible camera images taken after each pulse revealed that the majority of the surface deformation was inflicted during pulse #84779 [17]. Hence, the MEMOS-U simulations reported here will focus on this single discharge. For a complete description of the experiment and its results, see [17].

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Figure 5. Schematic of the leading edge geometry in the JET W melting experiments. A representative ELM heat flux profile is illustrated in red, and the plasma-wetted surface near the edge is indicated. The directions of the local toroidal magnetic field and induced force density are also indicated. The dashed white line corresponds to the uppermost part of the exposed sample, along which the surface deformation and melt thickness will be plotted in figures 7 and 8.

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An experimental spatiotemporal heat flux is available that was reconstructed from IR thermography data [55]. The typical ELM duration was ms, the average ELM frequency 30 Hz and the toroidal field 2.9 T. The maximum ELM heat flux varied within 0.5–1.5 and the inter-ELM heat flux varied within 50–200 . The experimentally extracted heat flux results in superficial melting that is incompatible with the observed surface deformation. To produce melting in accordance with the observations, the heat flux has been uniformly scaled up by a constant factor 1.10, an increase that lies well within the estimated 25% experimental uncertainty.

The replacement current module is coupled to the heat solver through the escaping thermionic current and the temperature-dependent electrical conductivity, but also coupled to the fluid solver owing to surface deformation. In the leading edge case, as a result of the exposure geometry and the dominant toroidal magnetic field, the main tangential force component lies along the poloidal direction and requires the determination of the radial current density [35]. Dedicated MEMOS-U simulations of current flow in an un-deformed leading edge domain have revealed that the average radial current component inside the melt layer is . This approximation is invoked in the present simulations instead of solving the aforementioned boundary value problem in the deformed domain for each time step. This circumvention of the replacement current module results in a drastic decrease of the computational cost and is deemed appropriate on the basis that uncertainties in the experimental heat flux should outweigh the marginal accuracy gained by a fully self-consistent bulk current.

In what follows, in order to demonstrate the inapplicability of the small melt displacement and deformation simplification to leading edge exposures, we shall compare simulation results for heat transfer independent of melt motion as employed in the original MEMOS-3D code analysis of the JET experiment in [17]. and for fully coupled heat transfer—melt motion as employed in the MEMOS-U code. Hereafter these are labelled as ‘the simplified model’ (see section 6) and ‘the full model’ (see section 2), respectively. Both models will be realized with the MEMOS-U code in order to quantify this specific model aspect without interference from (a) all other differences between the MEMOS-3D and MEMOS-U models and their numerical implementation, (b) differences in the employed heat fluxes (the reconstruction of the heat flux as in [55]. employed here was not available at the time of [17]).

7.1. Response to a single ELM

Let us first consider the effect a single JET ELM whose energy is deposited onto a W sample with an assumed uniform elevated temperature. Here a 2D sample geometry suffices for the illustration. The chosen initial temperature of 3200 K is high enough so that absorption of the ELM energy will increase the sample temperature beyond the W melting point. The temperature response and the surface deformation predictions of the simplified and full model are illustrated in figure 6. A 2D sample-cross section is considered at three time instants. In both cases, the surface is mainly cooled by thermal conduction, with respectable contributions from thermionic and radiative cooling.

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Figure 6. MEMOS-U simulations of an illustrative scenario where a single ELM from the JET pulse #84779 is deposited onto a W sample of initial homogeneous temperature of 3200 K. The temperature response and the surface deformation computed by the simplified model (top row) and the full model (bottom row) are plotted at 2, 8, 24 ms for a sample cross-section. The ELM duration is approximately 3 ms. The two models are described in sections 2 and 6. Note the horizontal scale is cm and the normal scale is mm.

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The top row in figure 6 features the temperature response and surface deformation resulting from the simplified model. As a result of the non-symmetric spatial profile of the ELM heat deposition, an irregular melt pool is produced that features irregular isotherms. The metallic melt reaches a depth of and a horizontal extent of . The ELM lasts ∼3 ms, and the induced melt pool resolidifies within 24 ms. Since the heat conduction domain is not updated to reflect the lateral motion of the molten material, the latent heat of fusion is always deposited back in the area where melting was originally induced. Consequently, after complete resolidification, the hottest point resides between 3.5 and 4.0 cm. As mentioned above, the spatiotemporal dependence of the solid–liquid interface b₁, obtained by the heat solver, is forwarded as input to the fluid solver which then predicts the surface deformation. The induced melt layer thickness tapers off toward the pool edges owing to the spatial profile of the heat flux pulse. This is significant because: (a) the thickest central molten region is subject to a stronger acceleration for a longer duration compared to the pool edge regions, (b) the depth-dependent viscous shear force () is relatively weak in the pool centre and gradually increases in magnitude toward the pool edge. These suggest that the depth-averaged fluid velocity develops a smooth gradient from its maximum value at the pool centre to a zero value at the pool edges. Since the heat conduction domain is undisturbed by melt motion, this gradient is present at all times. As a result, the linearized column height continuity described by equation (26) will lead to a surface deformation that builds up smoothly across the whole downwind region of the melt pool. The resulting surface deformation is illustrated by the dashed line that is superimposed on the temperature profile.

The bottom row shows the temperature response and the surface deformation resulting from the full model. Up to 2 ms, the temperature responses of the simplified and the full model are nearly identical owing to the validity of the small melt displacement approximation, since the short elapsed time does not allow for appreciable melt motion. However, at longer times, the predictions of the two models for the temperature profile and surface deformation begin to deviate strongly. Molten material that is produced in the central pool region eventually reaches the edge of the initially induced pool and spills over. The latent and sensible heat are transported with the melt, elevating the temperature of the colder surface underneath which provides an additional conduction channel for the deposited energy. Consequently, after complete resolidification, the hottest point resides around ∼1 cm.

7.2. Response to multiple consecutive ELMs

We shall now investigate the combined effect of multiple consecutive ELM pulses by simulating the full exposure during JET pulse #84779. The evolution of the induced melt pools will be easier understood on the basis of the single melting event discussion. The melt layer thickness at the top of the leading edge is illustrated in figure 7 as a function of poloidal coordinate and exposure time.

As for figure 6, the top row of plots illustrates the predictions of the simplified model. The ELMs repeatedly strike near the centre of the sample, where they locally induce melting. The omission of poloidal motion in the heat solver implies that there is no convective heat transport (latent and sensible). Thus, the corresponding region is efficiently heated. At the beginning of the exposure, the pools completely re-solidify in-between ELMs. As time progresses and the base temperature increases, melt layers begin to overlap. Melting becomes sustained and the surface temperature gradually rises well above the melting point. The surface deformation as described by equation (26) leads to a significant overestimation and the material pile-up (constrained to be uniquely within the melt pool extent) resides in a place along the lamella (∼2 cm) that is inconsistent with post-mortem observations of the lamella melt profile (see figure 9).

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Figure 7. MEMOS-U simulations of the JET W leading edge melting during pulse #84779 (1.25 s of steady strike point). The melt layer thickness as a function of the exposure time and the poloidal coordinate, near the end of the exposure. Results for (a) the simplified model (see section 6), (b) the full model (see section 2). The results are plotted at the uppermost edge of the exposed sample (where the deepest melting occurs), as illustrated by the dashed white line in figure 5.

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The bottom row gives the predictions of the full model. Due to the inclusion of deformed domains in the heat solver, the absorbed energy is transported together with the molten material toward the lower poloidal coordinates. As a consequence, the centre of the sample is less heated and the local induced melt pools re-solidify faster. Moreover, the transported melt gradually heats the lower coordinate regions, where no melting is originally induced (cm), allowing the subsequent melt layers to move further along the sample prior to re-solidification.

The direct comparison of the outcome of the two models with experimental evidence demonstrates how the violated assumptions of the simplified model distort the temperature response and surface deformation profile. In fact, in order to avoid disagreement with observations, early MEMOS-3D simulations [17] needed to invoke vapour shielding despite the modest heat and energy loads, as well as the lack of direct experimental evidence in favour of significant W vapour generation. The MEMOS-U model, within the experimental heat flux uncertainties, is capable of reproducing all main experimental features without introducing ad hoc factors or vapour shielding effects.

7.3. Build-up and final surface deformation profile

The final surface deformation profile is naturally determined by the accumulation of the material displaced from the individual transient melt pools depicted in figure 7. In what follows, we shall limit the discussion to the predictions of the full MEMOS-U model. It is worth pointing out that melting is induced only by the last few tens of ELMs in the discharge, since the absorbed energy is exclusively expended in raising the sample temperature during a large fraction of the ∼1 s exposure.

The gradual build-up of the surface deformation profile is illustrated in figure 8 at the onset of transient melting. The surface profile has been plotted at five instants that are evenly spaced over ten incident ELMs. The region of material pile-up is clearly observed to progressively move further toward the lower coordinates. In the course of the exposure, the deformation builds up after repeated cycles of melting, poloidal acceleration and displacement, surface excavation, material pile-up and resolidification.

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Figure 8. MEMOS-U simulations of the JET W leading edge melting during pulse #84779 (1.25 s of steady strike point). The evolution of the surface deformation profile near the onset of melting, at five evenly spaced consecutive time instants. Within this time interval, 10 ELMs impacted the sample and induced melting. The surface deformation is plotted along the poloidal coordinate at the uppermost edge of the exposed sample, as illustrated by the dashed white line in figure 5.

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The final MEMOS-U surface deformation profile is displayed in figure 9(a), together with a post-exposure image of the JET-exposed lamella from a similar angle. The results correspond to the constant heat flux pre-factor 1.10 that well reproduces the experimental surface profile of [17]. The specific deformation metrics of interest concern; the crater position (1.7 cm from the edge), its depth (0.6 mm) and extension (2 cm), but also the melt pile-up location (0.7 cm from the edge), its height (up to 1.8 mm) and extension (1 cm).

7.4. Sensitivity to the incident heat flux uncertainties

The MEMOS-U input that is required for the simulation of the JET ELM-driven W melting exposures consists of the sample geometry, local magnetic field and incident heat flux. Since only the latter quantity is characterized by non-negligible uncertainties, we shall investigate the sensitivity of the final deformation profile to the heat flux input. In principle, the plasma surface drag force per unit area and plasma pressure also constitute input, but their effect here is insignificant, see section 4.4.

In order to quantify the sensitivity to heat flux uncertainties, the final surface deformation profile is illustrated in figure 9(b) for a heat flux pre-factor of 1. 15. Compared to the reference case of a 1.10 pre-factor, the displaced volume doubles from ∼10 to . This seemingly disproportional melting enhancement can be understood by considering the partition between the latent and sensible heat during the exposure. The total energy incident on the W sample is ∼11 kJ and the total energy released by the W sample in the form of thermionic electrons, W atoms and thermal photons is ∼1 kJ. Hence, the total energy absorbed by the sample during the exposure is ∼10 kJ. In contrast, the energy required to melt the observed of W is ∼30 J, calculated assuming a constant mass density of and using for the enthalpy of fusion [72]. Therefore, as already mentioned in section 4.2, approximately 99% of the absorbed energy is expended to raise the base sample temperature, while only a very small fraction corresponds to latent heat, which explains the emerging threshold-like behaviour.

The strong sensitivity of the displaced volume to the heat flux input casts a doubt on the value of such benchmarking activities. However, careful comparison between figures 9(a) and (b) reveals that the surface morphologies are nearly identical and the deformation profiles are approximately scaled versions of each other. Furthermore, the key deformation characteristic (molten material strongly displaced from the transient pool locations which appear as a crater upon resolidification) is well captured by the full MEMOS-U model and cannot be predicted by the simplified model featuring stagnated melt pools. In general, it is crucial to recognize that benchmarking activities should not be limited to comparisons of the highly sensitive displaced volume, but should be focused on deformation profile comparisons. It is worth pointing out that in the similar leading edge exposures performed on AUG, molten material was actually ejected from the sample edge, a salient feature that was successfully reproduced only by the MEMOS-U full model, see [28]. for details. Finally, it is important to reiterate that such strong sensitivity is a peculiarity of leading edge exposures to type-I ELMs and that other exposure scenarios exhibit much more moderate sensitivity, see the discussion in section 4. 2.

7.5. Surface cooling and thermionic emission

Let us expand the discussion on heat flux sensitivity by including the effect of cooling fluxes. In figure 10, the integrated incident plasma energy and the integrated cooling energy are plotted as functions of the exposure time. Clearly, the total cooling energy released from the sample constitutes only ∼5% of the overall energy budget. As discussed in section 4.3 and observed in figure 4, the major cooling contribution stems from thermionic cooling (the maximum surface temperature reaches 4500 K in this simulation). Therefore, a high sensitivity to the heat flux input directly translates to a high sensitivity to thermionic cooling.

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Figure 9. MEMOS-U simulations of the JET W leading edge melting during pulse #84779 (1.25 s of outer steady strike point). The final surface deformation profile (viewed from above the sample) for the uniform heat flux pre-factors (a) 1. 10 and (b) 1.15. A post-mortem lamella image from a similar viewing angle has been included for comparison (the image is courtesy of EURATOM). See the schematic in figure 5 for an illustration of the plasma-wetted area, the directions of the local magnetic field and incident plasma heat flux, as well as of the force density.

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Figure 10. MEMOS-U simulations of the JET W leading edge melting during pulse #84779 (1.25 s of steady strike point). The integrated plasma energy and cooling energy as functions of the exposure time. The slope change in the plasma energy coincides with the positioning of the outer strike point (OSP) on the lamella. Significant melting events refer to pool depths m.

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The above statements highlight the intricate role of thermionic emission in the determination of the surface deformation profile of these W leading edge exposed samples. The escaping thermionic current density not only dictates the melt acceleration through the volumetric Lorentz force, but also affects melt production through thermionic cooling. In the case that the escaping emission is underestimated, the amount of melt will substantially increase, but melt acceleration will decrease resulting to smaller displacements for a given melt layer. On the other hand, an overestimation of the escaping emission will result in stronger melt acceleration but less melting due to the efficient cooling.

The MEMOS-U code simulates the melting, macroscopic liquid motion and surface deformation induced by incident plasma fluxes on metallic PFCs of magnetic confinement fusion devices. The macroscopic melt dynamics are self-consistently described by the set of incompressible resistive TEMHD equations and temperature evolution is governed by the convection–diffusion equation. The subset of fluid equations is simplified within the shallow water approximation which transforms the three momentum equations and incompressibility condition to two momentum equations and a continuity equation for the melt layer column height. The electromagnetic field equations are simplified with the magnetostatic approximation for strong external magnetic fields which reduces Maxwell’s equations and Ohm’s law to a single, Poisson-like equation for an auxiliary potential that describes the bulk current pathways.

The MEMOS-U code is not only based on a transparent physical description but is also void of adjustable coefficients that could be utilized as free parameters owing to its rigorous boundary conditions and robust numerical schemes. The simulation outcome depends solely on the incident heat flux, local magnetic field and problem geometry for ELM-driven melting scenarios and also on the halo current density for melting induced by thermal plasma impact during the current quench phases of disruptions.

The physics model has been demonstrated to offer a unified description of ELM-induced W divertor melting and disruption-induced Be first wall melting, in spite of the significant differences in terms of heat loading, bulk current source and material composition [28]. A quantitative agreement with state-of-the-art fusion experiments performed in two tokamaks (JET and AUG), concerning two plasma scenarios (ELMs and VDEs) and involving two PFC geometries (sloped and leading edge), as well as two metal PFCs (W and Be) has been achieved, with the only variations allowed concerning the incident heat flux and strictly within the experimental uncertainties [28, 29].

It has been demonstrated that the upgraded MEMOS-U code is equipped with the essential physical and numerical features that allow reliable simulations of significant displacements of metallic melts, i.e. of the order of the melt spatial extent, in fusion environments. These novel features have been discussed in detail in the present investigation and comprise the continuously deforming heat conduction domain, free-flow boundary condition at the melt pool edge and inclusion of convective heat transport. Concerning the latter, it has been revealed that the horizontal displacement of latent and sensible heat affects the temperature response in an intuitive, but important fashion, in which the colder adjacent material is heated by the displaced liquid and the initial induced melt area cools down. This process explains the gradual surface deformation and horizontal melt displacement under repeated transient melting scenarios (ELMy H-mode W exposures, discussed herein), or continuous melting scenarios (disrupting plasma Be exposures, discussed elsewhere [28]).

The shallow water assumption leads to a drastic reduction of the computational cost, making the MEMOS-U code a highly suitable simulation tool for parametric predictive studies and input sensitivity studies, such as investigations of the impact of heat flux input uncertainties and of the adopted non-ambipolar current description on surface deformation. The natural trade-off has to do with loss of high-order accuracy relevant to the surface pinch-off in droplet ejection, instability growth, wetting dynamics, deep melting and melting of small scale or complex geometries. In the scenarios addressed here, re-solidification stabilizes the flow by controlling the melt depth and thus the viscous damping [28]. However, to reliably address the general flow stability and melt splashing, the full Navier–Stokes system needs to be employed. Simulation tools beyond the shallow water approximation are computationally heavy and cumbersome to use in parametric predictive studies. Nevertheless, the methodology is well developed, as mentioned in the Introduction.

Concerning transient W melting ITER scenarios, we reiterate that the present MEMOS-U description of the escaping emitted current is reliable for inter-, intra-ELM periods of contemporary fusion devices, where thermionic emission is dominant and becomes suppressed by space charge effects and prompt re-deposition. This expression should also be valid during inter-ELM ITER periods but is expected to be inaccurate during intra-ELM ITER periods due to role of electron-induced electron emission, the Schottky effect and the elevated incident plasma currents. An investigation has been initiated with the objective of identifying an accurate boundary condition for the escaping emitted current in this complicated regime [78], within the framework of the earlier studies [40, 41, 46].

The R&D programme which led to the development and benchmarking of the new MEMOS-U code version was proposed and initiated by the ITER Organization (IO) and conducted within the framework of an Implementing Agreement (IO/IA/16/4300001384) established under the Cooperation Agreement between IO and KTH. SR, PT acknowledge the financial support of the Swedish Research Council under Grant No. 2018-05273. The work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 and 2019–2020 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission or the ITER Organization.

Here we shall consider the kinematic boundary condition at the bottom of the melt pool instead of the no-slip boundary condition employed in section 2.2. For an impermeable moving boundary b₁, the condition is identical to that of the free surface, see equation (6) without the vaporization term. However, the liquid–solid interface does not displace the liquid in its movement, but instead the material transitions from one state to the other. Therefore, it would be more correct that also the partial time derivative is removed from this equation. In other words, we assume that the liquid–solid interface acts as a stationary impermeable boundary, even though it clearly evolves. The boundary condition becomes:

After depth-averaging the momentum equation (2), and combining equation (A1) with equations (6) and (10), we acquire:

There are subtleties involved with the first rhs term which describes the change of mean velocity in the column as liquid is added or removed by melting or solidification.

For the no-slip condition , this term represents the velocity change resulting from the addition or removal of stationary fluid at the bottom of the pool, without explicitly changing the parcel momentum. Two possible consequences of this term need to be considered which depend on the sign of . When this quantity is positive, the melt layer thickness increases due to melting and there is a stationary liquid that is ‘added’ to an already moving column. The net effect is a reduction of the average velocity of the column. On the other hand, when it is negative, the melt layer thickness decreases due to solidification and the average velocity should increase since the material that is removed from the column is stationary.

It might appear that this term adequately captures the explicit effects of phase change on the melt column velocity. However, the term arises because of the prescribed parabolic velocity profile which neglects the finite viscous propagation speed in the liquid. Let us consider the case in infinitesimal time steps. A small portion of stationary liquid is removed from the parcel, which will increase the column velocity average. Since now there is a slightly higher average velocity and a thinner melt layer, the parabolic velocity profile is assumed to ‘equilibrate’, adjusting to the new average, which implies a steeper gradient with higher maximum velocity value. This process repeats for each infinitesimal step, providing a non-physical acceleration of the liquid. The opposite effect is true for the melting phases. It is evident that this process does not adequately describe the dynamics in the melt layer—the actual scenario needs a more detailed description of viscous propagation of melt velocity upwards in the melt layer as the liquid–solid boundary moves. In the absence of such a treatment, and in light of other model uncertainties, the term has been neglected. This is equivalent to stating that the velocity of the ‘added/removed’ fluid is identical to the average melt column velocity, .

90,000 U-38 (1938) – Global wiki. Wargaming.net

Service

Germany

Historical data

General data

Power plant

Crew

Armament

Artillery armament

• 1 – 105mm / 42 gun SK C / 32 ;
• 1 – 37mm / 80 gun SK C / 30 ;
• 1 – 20mm / 65 gun C / 38 .

Mine-torpedo armament

Ships of the same type

Service history

She made 11 military campaigns, sank 35 ships (188 967 brt), damaged 1 ship (3 670 brt). Flooded by the crew on May 5, 1945 near the town of Wesermünde.

  1st hike 
 

On August 19, 1939, U-38 left the port of Wilhelmshaven. The boat took part in operations off the coast of Lisbon and returned to port on 18 September.During this four-week voyage, she sank two ships.

On September 5, 1939, U-38 stopped the French freighter SS Pluvoise, checked the papers, and released it. SS Pluvoise reported this on the air, which alerted everyone in the area about the presence of a German submarine. Liebe was reprimanded.

The British steam cargo ship SS Manaar was sunk on 6 September 1939. U-38 opened fire on the ship, and it snapped back. This was the first time a cargo ship had fired at a German submarine.U-38 sank SS Manaar with torpedoes. Since SS Manaar was shooting at him, Liebe did not help the survivors. Radio operator James Turner remained in office until the last moment. While leaving the ship, he found two Laskars, one of whom was seriously wounded. Turner saved both of them under continuous fire from U-38, for which he was awarded the Imperial Medal of Valor.

On September 11, 1939, the SS Inverliffey under the Irish tricolor was sunk by artillery fire.Despite protests from Captain William Trowsdale and claims that they were Irish, Liebe said that they were “sorry” that SS Inverliffey would be sunk as it was carrying gasoline to England, which is smuggling. The crew plunged into boats. The SS Inverliffey burned so violently that the boats were threatened. At risk, the German boat approached, lines were thrown onto the boats and they were towed to a safer location. As Captain Throwsdale’s boat was damaged, its passengers were allowed to board the submarine. , where the captain was also given a life jacket, due to the lack of his own.U-38 stopped the American tanker R.G. Stewart and offloaded the SS Inverliffey team to them. Just two days earlier, Inver Tankers had transferred all of their vessels to the UK register.

  2nd hike 
 

On November 12, 1939, after spending almost two months in port, U-38 left Wilhelmshaven. This second trip took the boat into the waters northwest of Norway.

November 17, 1939, Headquarters for the leadership of the war at sea (eng.) Russian. (SKL) ordered U-38 and U-36 to find a site for the Bazis Nord, a secret German naval base for raids on the sea routes of the Allies in the Kola Peninsula region and supported by the USSR.This mission required coded messages sent to the Soviet naval forces patrolling the area in order to request an escort to the alleged base location.
U-36, sunk by the British submarine HMS Salmon, never left the Norwegian Sea. [9] U-38 quietly rounded the North Cape and arrived in Teriberki Bay on the afternoon of 26 November. Sneaking into the bay, U-38 had to avoid detection by merchant ships in order to maintain the appearance of USSR neutrality. The U-38 commander said that while in the North Cape and the Kola Peninsula, he observed between thirty and forty targets, but regretfully was forced to remain “harmless to all of them.”

Upon completion of the secret reconnaissance mission, U-38 returned to her raiding duties, and sank three ships: two British and one Greek. The British steam freighter SS Thomas Walton was sunk on 7 December; 11 December Greek steam freighter SS Garoufalia and 13 December British steam freighter SS Deptford. On December 16, after four and a half weeks of combat patrols, U-38 returned to Wilhelmshaven.

3rd trip

On February 26, 1940, U-38, after spending considerable time in the port again, set out on a cruise and headed for the western approaches.[eleven]

U-38 sank six ships. Irish steam trawler ST Leukos was the first to go to the bottom on May 9, sunk by a single shell hit, and taking away its entire crew near the island of Tori ST Leukos. fished with British trawlers and presumably stood between the surfaced German submarine and the fleeing British, in the hope that neutral markings would protect him. This event was followed by the sinking of the Danish freighter SS Argentina on 17 March and SS Algier and SS Christiansborg on 21 March.The Norwegian cargo ship MV Cometa was sunk on 26 March. On April 2, the sixth and last victim of the boat on this voyage was the Finnish cargo steamer SS Signe. On April 5, 1940, after six weeks at sea, U-38 returned to Wilhelmshaven.

4th trip

On April 8, 1940, U-38 under the command of Heinrich Liebe left its home port in Wilhelmshaven. She had orders to comb the waters of Norway, supporting her occupation by German forces. During this campaign, U-38 recorded torpedo problems after unsuccessfully firing at HMS Effingham.) Russian .. U-38 returned to port on 27 April.

During both Battles of Narvik on April 10 and April 13, 1940, U-38 and U-65 held positions near the entrance to the fjord. When the Royal Navy arrived, U-38 fired at HMS Valiant and HMS Southampton, missing both. In the second battle, U-38 fired at HMS Effingham, but the torpedoes refused and exploded without reaching the target.

  5th hike 
 

On June 6, 1940, U-38 launched its fifth campaign again under the command of Heinrich Liebe and again from Wilhelmshaven.Heading towards southern Ireland, Liebe achieved hits on six ships, two of which were at that time in convoys. On June 6, U-38 sank the Greek freighter SS Mount Myrto. The next day, two freighters from convoy HX-47 from Halifax to Britain were sunk. The first was the Canadian steam cargo ship SS Erik Boye, followed by the Norwegian tanker MV Italia. Five days later, on June 20, the Swedish steam freighter SS Tilia Gorthon was torpedoed and sunk.The Belgian steam cargo ship SS Luxembourg was destroyed on June 21, followed by the Greek steam cargo ship SS Neion on June 22. On July 2, after completing a three-week trip, U-38 returned to Wilhelmshaven.

On June 12, U-38 was able to land two Nazi agents at Dingle Bay in Ireland. One of them, Walter Simon, did not realize that passenger traffic on the Tralee and Dingle Light Railway. were stopped fourteen months earlier, asked when the next train to Dublin would be, as a result of which he was arrested and interned in Camp Kurrach until the end of the war.

6th hike

On August 1, 1940, U-38 and again under the command of Heinrich Liebe, the boat left on its last voyage from Wilhelmshaven. During this month’s duty off the west coast of Ireland, the submarine achieved three victories, all from convoys. On August 1, 1940, the Egyptian SS Mohamed Ali El-Kebir was sunk in convoy HX-61 from Halifax to Gibraltar, claiming 320 lives. As part of the SL-41 convoy from Sierra Leone to Britain, the British steam freighter SS Llanfair was sunk.The third victim of U-38 was the British steam cargo ship SS Har Zion from the OB-225 convoy from Liverpool to the United States of America. On September 3, 1940, after four weeks at sea, U-38 arrived at its new home port at Lorient.

7th hike

On September 25, 1940, U-38 under the command of Heinrich Liebe set out on its first campaign from Lorient towards the Northwest approaches. During this watch, she attacked five ships, sinking four of them.On October 1, the British freighter MV Highland Patriot was torpedoed. Two unlucky weeks later, on October 17, U-38 sank the Greek steam freighter SS Aenos on board the SC-7 convoy from Sydney, Nova Scotia. in Grimsby. The next day, the submarine managed to damage, but not sink, the British steam freighter SS Carsbreck, sailing in the same convoy SC-7. On October 19, the submarine sank two cargo ships of the convoy HX-79: the Danish SS Bilderdijk and the British steamer SS Matheran.On October 24, having completed another victorious campaign, U-38 returned to Lorient.

8th hike

On December 18, 1940, U-38 under the command of Liebe, on its eighth combat duty, again departed from Lorient towards the Northwest Approaches, during which it sank two cargo ships. On December 18, the submarine destroyed the British steamer SS Waiotira, and on December 31, she sank the Swedish SS Valparaiso, which was part of the HX-97 convoy from Halifax to Glasgow. On January 22, 1941, depth charges were dropped on the boat from the escort ships, causing minor damage.On January 22, U-38 returned to port.

9th hike

April 9, 1941, after spending two and a half months in the port, U-38 went on alert to the west coast of Africa. Thanks to eight ships sunk during this voyage, it was recognized as the most successful. On May 4, the Swedish steam cargo ship SS Japan from the OB-310 convoy from England to the United States of America was torpedoed. The British freighter MV Queen Maud was sunk the next day.

On May 12, U-38 had the worst day of their career.While hunting a freighter, she fired four torpedoes, missing them all.

On 23 May, the account was replenished by the Danish cargo ship SS Berhala from the OB-318 convoy from England to the United States of America. The British steam freighter SS Vulcain was torpedoed and sunk on 24 May. Six days later, on May 29, the British steam freighter SS Tabaristan became another victim. The next day, the destruction continued and the British steam cargo ship SS Empire Protector was sent to the bottom, and on May 31, the Norwegian steam cargo ship SS Rinda disappeared.The eighth and final casualty of this U-38 cruise on 8 June was the British steam freighter SS Kingston Hill. After that, on June 29, 1941, after spending eleven and a half weeks at sea, the boat returned to Lorient.

10th hike

For the first time during her service, U-38 went to sea under the command of a new commander – Captain Zee Zee Heinrich Shuch. After leaving port on August 6, the submarine headed for its five-week watch in the North Atlantic. During this entire period, she achieved only one victory, sinking the Panamanian steam cargo ship SS Longtanker on 18 August.On September 14, 1941, U-38 returned to Lorient.

11th and 12th hikes

On October 15, 1941, U-38 again left Lorient under the command of Heinrich Schuch. The eleventh cruise took place in the North Atlantic, but not a single vessel was attacked. On November 21, U-38 arrived at the submarine base in Bergen. Later, on November 29, she left Bergen and came to Szczecin.

Life after the end of combat duty
From December 1, 1941 to November 30, 1943, the U-38 was used as a training boat in the 24th training flotilla and the 21st training flotilla.After that, she was used as a test boat until the very moment of sinking by the crew on May 5, 1945.

Ship history
Launching on August 9, 1938
Large ship type DPL
IXA project designation
Speed ​​(surfaced) 18.2
Speed ​​(underwater) 7.7
Maximum immersion depth 230
Crew 4 officers 44 sailors

Dimensions
Displacement above water 1,032
Displacement underwater 1 153
Length overall (at design waterline) 76.5
Body width naib.6.51
Height 9.4
Average draft (at design waterline) 4.7

Power unit
9-cylinder 4-stroke “MAN” M9V40 / 46 2×2 200
Electric motor SSW GU345 / 34 2×370

Armament ‘
Artillery 1 x 10,5 cm L / 45, 1 x 3,7 cm Flak, 1 x 2 cm Flak
Torpedo-mine armament 6 TA 22 torpedoes or 66 mines

90,000 U-38 (1938) – Wikipedia

This term has other meanings, see U-38.

U-38 – a large ocean-going German submarine of the IX-A type, during the Second World War.The order for the construction was given on July 29, 1936. The boat was laid down at the shipyard of the shipbuilding company AG Weser in Bremen on April 15, 1937 under the serial number 943. Launched on August 9, 1938. On October 24, 1938, it was put into service and, under the command of Lieutenant-Commander Heinrich Liebe, entered the 6th flotilla.

Service history [edit]

She made 11 military campaigns, sank 35 ships (188 967 grt), damaged 1 ship (3 670 grt). Flooded by the crew on May 5, 1945 near the town of Wesermünde.

1st campaign [edit]

August 19, 1939 U-38 left the port of Wilhelmshaven. The boat took part in operations off the coast of Lisbon and returned to port on 18 September. During this four-week voyage, she sank two ships.

5 September 1939 U-38 stopped the French freighter SS Pluvoise , checked the papers and released it. SS Pluvoise reported this on the air, which alerted everyone in the area about the presence of a German submarine.Liebe was reprimanded.

British steam cargo ship SS Manaar was sunk on 6 September 1939. U-38 opened fire on the ship and it snapped back. This was the first time a cargo ship had fired at a German submarine. U-38 sank SS Manaar with torpedoes. Since SS Manaar fired at him, Liebe did not help the survivors of ^[1] . Radio operator James Turner ( James Turner ) remained at his post until the last moment.While leaving the ship, he found two Laskars, one of whom was seriously wounded.
Turner saved both of them under continuous fire from U-38 , for which he was awarded the Imperial Medal of Valor. ^[2] .

On September 11, 1939, an artillery fire was sunk under the Irish tricolor SS Inverliffey ^[3] . Despite the protests of Captain William Trowsdale ( William Trowsdale and claims that they are Irish, Liebe said that they are “very sorry”, but SS Inverliffey will be sunk, since gasoline is transported to England, and this is smuggling.The crew plunged into the boats. SS Inverliffey burned so violently that the boats were threatened. At the risk of itself, the German boat came closer, lines were thrown onto the boats and they were towed to a safer place ^[4] . Since Captain Throwsdale’s boat was damaged, her passengers were allowed to board the submarine, where the captain was also given a life jacket, due to the lack of his own. U-38 stopped American tanker R.G. Stewart and offloaded the SS Inverliffey team. Just two days earlier, Inver Tankers had transferred all their vessels to the British register ^{[5] [6]} .

2nd campaign [edit]

On November 12, 1939, after spending almost two months in port, U-38 left Wilhelmshaven. This second trip took the boat into the waters northwest of Norway ^[7] .

November 17, 1939, Headquarters for the leadership of the war at sea (eng.) Russian. (SKL) ordered U-38 and U-36 to find a place to house Basis Nord – a secret German naval base for raids on the sea routes of the Allies in the Kola Peninsula region and provided by the USSR ^[8] …This mission required coded messages sent to Soviet naval forces patrolling the area to request an escort to the alleged location of Base ^[9] .

U-36 sunk by British submarine HMS Salmon , never left the Norwegian Sea ^[9] . U-38 calmly rounded the North Cape and arrived at Teriberki Bay ^[10] in the afternoon of 26 November. Sneaking into the bay, U-38 had to avoid detection by merchant ships in order to maintain the appearance of USSR neutrality ^[10] .Commander U-38 said that while in the North Cape and Kola Peninsula, he observed thirty to forty targets, but regretfully was forced to remain “harmless to all of them” ^[10] .

Upon completion of the secret reconnaissance mission , U-38 returned to her raiding duties and sank three ships: two British and one Greek. The British steam freighter SS Thomas Walton was sunk on 7 December; 11 December Greek steam cargo ship SS Garoufalia and 13 December British steam cargo vessel SS Deptford .On December 16, after four and a half weeks of combat patrols, U-38 returned to Wilhelmshaven ^[7] .

3rd campaign [edit]

February 26, 1940 U-38 , having again spent considerable time in the port, went on a cruise and headed for the western approaches ^[11] .

U-38 sunk six ships. Irish steam trawler ST Leukos was the first to sail to the bottom on May 9. , sunk by a single shell hit, and taking with him his entire crew near the island of Tori ^[12] ST Leukos (eng.) Russian. fished with British trawlers and is believed to have stood between the surfaced German submarine and the fleeing British, hoping that neutral markings would protect him ^[13] . This event was followed on 17 March by the sinking of the Danish freighter SS Argentina and on 21 March SS Algier and SS Christiansborg . Norwegian freighter MV Cometa was sunk on 26 March. On April 2, the sixth and last victim of the boat on this voyage was the Finnish freighter SS Signe .On April 5, 1940, after six weeks at sea, U-38 returned to Wilhelmshaven ^[11] .

4th campaign [edit]

April 8, 1940 U-38 under the command of Heinrich Liebe left her home port in Wilhelmshaven. She had orders to comb the waters of Norway, supporting her occupation by German forces. During this campaign, U-38 recorded problems with torpedoes after the HMS Effingham (eng.) Russian. . U-38 returned to port on 27 April ^[14] .

During both Battles of Narvik on April 10 and April 13, 1940, U-38 and U-65 held positions near the entrance to the fjord. When the Royal Navy arrived, U-38 fired at HMS Valiant and HMS Southampton , missing both ^[15] . In the second battle, U-38 fired at HMS Effingham. , however the torpedoes failed ^[16] , and detonated without reaching the target.

5th campaign [edit]

June 6, 1940 U-38 set out on her fifth campaign again under the command of Heinrich Liebe and again from Wilhelmshaven. Heading towards southern Ireland, Liebe achieved hits on six ships, two of which were at that time in convoys. June 6 U-38 sank Greek freighter SS Mount Myrto . The next day, two freighters from convoy HX-47 from Halifax to Britain were sunk. The first was the Canadian steam cargo ship SS Erik Boye , followed by the Norwegian tanker MV Italia .Five days later, on June 20, the Swedish steam freighter SS Tilia Gorthon was torpedoed and sunk. The Belgian steam cargo ship SS Luxembourg was destroyed on June 21, followed by the Greek steam cargo ship SS Neion on June 22. On July 2, after completing a three-week trip, U-38 returned to Wilhelmshaven ^[17] .

On June 12, march U-38 was able to land two Nazi agents at Dingle Bay in Ireland.One of them, Walter Simon (eng. Walter Simon ), not realizing that passenger traffic Tralee and Dingle Light Railway (eng.) Russian. were stopped fourteen months earlier, asked when the next train to Dublin would be, as a result of which he was arrested and interned at Camp Kurrach. until the end of the war ^[18] .

6th campaign [edit]

August 1, 1940 U-38 and again under the command of Heinrich Liebe, the boat set out on its last voyage from Wilhelmshaven.During this month’s duty off the west coast of Ireland, the submarine achieved three victories – all from convoys. On August 1, 1940, the Egyptian liner SS Mohamed Ali El-Kebir was sunk in convoy HX-61 from Halifax to Gibraltar, claiming 320 lives ^[19] . As part of convoy SL-41 from Sierra Leone to Britain, the British steam freighter SS Llanfair was sunk. The third victim of U-38 was the British steam cargo ship SS Har Zion from the OB-225 convoy from Liverpool to the United States of America.On September 3, 1940, after four weeks at sea, U-38 arrived at its new home port at Lorient. ^[20] .

7th campaign [edit]

September 25, 1940 U-38 under the command of Heinrich Liebe set out on her first campaign from Lorient in the direction of the Northwest approaches. During this watch, she attacked five ships, sinking four of them. On October 1, the British freighter MV Highland Patriot was torpedoed. Two unlucky weeks later, on October 17, U-38 sank the Greek steam freighter SS Aenos on board the SC-7 convoy from Sydney, Nova Scotia.) Russian. in Grimsby (eng.) Russian .. The next day, the submarine managed to damage but not sink the British steam freighter SS Carsbreck , sailing in the same convoy SC-7. On October 19, the submarine sank two cargo ships of the convoy HX-79: the Danish SS Bilderdijk and the British steamer SS Matheran . October 24, completing another victorious campaign, U-38 returned to Lorient ^[21] .

8th campaign [edit]

December 18, 1940 U-38 under the command of Liebe, on her eighth combat duty, again departed from Lorient towards the Northwest approaches, during which she sank two cargo ships.On December 18, the submarine destroyed the British steamer SS Waiotira , and on December 31 sank the Swedish SS Valparaiso , which was sailing with convoy HX-97 from Halifax to Glasgow. On January 22, 1941, depth charges were dropped on the boat from the escort ships, causing minor damage. January 22nd U-38 returned to port ^[22] .

9th campaign [edit]

On April 9, 1941, after spending two and a half months in the port, U-38 went on alert to the west coast of Africa.Thanks to eight ships sunk during this voyage, it was recognized as the most successful. On May 4, the Swedish steam freighter SS Japan from the OB-310 convoy from England to the United States of America was torpedoed. The next day the British freighter MV Queen Maud was sunk.

On May 12th, U-38 had the worst day of his career. While hunting a freighter, she fired four torpedoes, missing them all.

On 23 May, the account was replenished by the Danish freighter SS Berhala from the OB-318 convoy from England to the United States of America.The British steam freighter SS Vulcain was torpedoed and sunk on 24 May. Six days later, on May 29, the British steam freighter SS Tabaristan was another casualty. The next day, the destruction continued and the British steam cargo ship SS Empire Protector was sent to the bottom, and on May 31, the Norwegian steam cargo ship SS Rinda disappeared into the list. The eighth and final casualty on this cruise U-38 on June 8 was the British steam freighter SS Kingston Hill .After that, on June 29, 1941, after spending eleven and a half weeks at sea, the boat returned to Lorient ^[23] .

10th Campaign [edit]

For the first time during the service U-38 went to sea under the command of a new commander – Captain zuz zee Heinrich Schuch. After leaving port on August 6, the submarine headed for its five-week watch in the North Atlantic. During this entire period, she achieved only one victory, sinking the Panamanian steam freighter SS Longtanker on August 18.14 September 1941 U-38 returned to Lorient ^[24] .

11th and 12th Patrols [edit]

October 15, 1941 U-38 again left Lorient under the command of Heinrich Schuch. The eleventh cruise took place in the North Atlantic, but not a single vessel was attacked. November 21 U-38 arrived at the submarine base in Bergen. Later, on November 29, she departed from Bergen and arrived at Szczecin ^[25] .

Life after the end of combat duty [edit]

From December 1, 1941 to November 30, 1943 U-38 was used as a training boat in the 24th training flotilla and 21st training flotilla.After that, she was used as a test boat until the very moment of sinking by the crew on May 5, 1945.

Sunken ships [edit]

vehicles

90 011 862 people (96 killed and 766 rescued) 90 012

90 011 5700 tons of general cargo and frozen food, 172 people (3 killed and 169 rescued)

returned to service

91 627 90 101 ↑ Blair, page 82.

↑ Blair, page 150

↑ Blair, page 155

90,000 Chills in a child – causes and symptoms, first aid for treatment

01.10.2021

140 212

4 minutes

Contents:

Causes of chills in children
What to do if a child has chills?
What medications can you give a child for a chill associated with a cold?

Chills are one of the symptoms that can accompany various diseases and conditions in children.Shivering provides intense heat generation and an increase in body temperature ² . In this case, the feeling of coldness arises because the process of heat transfer is disturbed: the blood vessels narrow and the skin cools ² .

Causes of chills in children

Coldness and muscle tremors often indicate the onset of an infection or fever ¹ , but they may appear without a high fever.

A chill in a child at a temperature is often a manifestation of an infectious disease.In this case, the fever becomes a defensive reaction that activates the immune system and helps fight off infectious agents ³ . Chills associated with the “invasion” of viruses and bacteria can also be accompanied by various manifestations of the common cold, diarrheal syndrome and urinary tract infection. Observed:

• manifestations of intoxication – headache and muscle pain, feeling unwell ¹³ ;
• symptoms of respiratory tract damage – sore throat, runny nose, cough, shortness of breath ¹⁰ ;
• disorders of the digestive system – vomiting, diarrhea ¹⁰ ;
• dysuria – frequent and painful urination, insurmountable urge to urinate ¹¹ .

Sometimes the chills are not related to infection or inflammation, so it is important for parents to pay attention to other accompanying symptoms and tell their doctor about them.

There are also many reasons for the development of chills in children without fever. Some of them are:

• Subcooling . At low ambient temperatures, the body tries to prevent cooling of the internal organs. In response to cold, 2 reactions occur: the body reduces the loss of heat through the skin – a spasm of the superficial vessels of the skin occurs – and increases heat production by stimulating muscle contraction.In addition to the muscles of the trunk, the chewing muscles are also involved in the work – that is why, when the child is cold, he begins to “chatter his teeth.” As soon as the body warms up, the muscle tremors stop ² .
• Hypothyroidism – insufficient thyroid function. With this disease, heat production decreases, which is manifested by a feeling of coldness, chilliness. In children, hypothyroidism can be congenital or occur due to a lack of iodine in the body. Additional symptoms usually include swelling, dry skin, muscle weakness, constipation and hearing impairment ⁵ .
• Hypoglycemia is a low blood glucose level that occurs during fasting, intense muscle work or diabetes mellitus (if you enter the wrong dose of insulin or eat incorrectly). Chills may be accompanied by sweating, confusion, and other symptoms ⁶ .
• Vegetative crisis. More common in schoolchildren and adolescents and is associated with dysregulation by the nervous system, or, more simply, “neurosis”.Additional signs are usually anxiety, fear, nervous tension, increased blood pressure, headache ⁷ .

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What to do if a child has chills?

If the child is shivering, first wrap him up ¹² , and then use a thermometer and find out if he has a fever – it often accompanies muscle tremors ⁸ . If the thermometer shows high values, the most likely cause of the chills is fever ⁸ .In this case, it is impossible to keep warm under warm clothes or a blanket – you need to allow the body to lose heat through the skin ¹⁰ . Also, do not rush to bring down the temperature, because this is a protective reaction of the body ⁸ . There are the following criteria by which you can determine when it is time to take antipyretic drugs ⁸ :

• temperature rise above 38 ⁰ C in children under 3 months;
• the value of the thermometer is higher than 39 ⁰ C in initially healthy children older than 3 months;
• temperature 37.5-38 ⁰ C and higher in children with convulsions or diseases of the heart and central nervous system.

Do not blow or dry your child with chills. These procedures can cause vasospasm and increase tremors ⁸ . The fact is that for blowing it is necessary to ventilate the room where the child lies, while he must be opened, and not wrapped up. If you feel cold, it is also not recommended to open it ⁸ . Rubbing with warm water is allowed only in the absence of tremors ⁸ , not less than 30 minutes after taking antipyretics ⁴ . You cannot wipe your child with vinegar or alcohol.Rubbing them into the skin may cause irritation ⁸ .

Severe and prolonged tremors, fever, pale and cold skin of the feet and palms may indicate a severe course of the disease and require urgent medical attention ⁴ .

If chills without fever are not associated with hypothermia and are not accompanied by a high fever, and attempts to warm the child do not help to get rid of shivering and cold sensations, there is a possibility that the symptoms are caused by hypoglycemia, thyroid insufficiency or nervous overstrain ^5.6.7 .In this case, it is necessary to contact the pediatrician, who clarifies the diagnosis and, if necessary, refers to narrow specialists.

What medications can I give my child for a chill associated with a cold?

Not all medicines are suitable for the symptomatic treatment of fever and chills with colds. One of the approved and well researched is paracetamol ⁸ .

Paracetamol is a part of combination preparations, for example, it is one of the active ingredients of RINZASIP® for children, powder for preparing a hot drink ⁹ .

RINZASIP® for children reduces fever, sore throat, headache and muscle soreness that may accompany colds. In addition to the antipyretic component, RINZASIP® for children contains pheniramine with antiallergic effect and ascorbic acid, which increases the body’s resistance ⁹ .

The drug is recommended to be used from the age of 6, the dose is determined by the doctor, depending on the age of the child and his condition. To prepare a hot drink, dissolve the contents of the sachet in a glass of warm water ⁹ .

Recommendations for taking RINZASIP ® for children ⁹

The course of admission should be no more than 5 days ⁹ .The interval between doses should be at least 4 hours ⁹ .

It is important to remember that although antipyretic therapy helps to bring down the temperature and fight chills, it does not affect the cause of the disease and can “lubricate” its other manifestations, which often makes it difficult for a doctor to diagnose. Therefore, before giving children antipyretic drugs, be sure to consult a pediatrician to find the cause of the increase in body temperature ⁸ .

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Literature:

1. ICD-10 (R68.83), 2021, http://www.icd10data.com/ICD10CM/Codes/R00-R99/R50-R69/R68-/R68.83
2. Litvitsky P.F. Disturbances in the heat balance of the body. Fever. Questions of modern pediatrics. 2009, volume 8, no. 6, pp.55-63. https://cyberleninka.ru/article/n/narusheniya-teplovogo-balansa-organizma-lihoradka
3. Zakharova I.N. Fever in children: from symptom to diagnosis. Medical Council, 2017, No. 1, pp. 212-218. https://cyberleninka.ru/article/n/lihoradka-u-detey-ot-simptoma-k-diagnozu-1
4. Korovina N.A. Fever. The variety of reasons and the complexity of the solution. Medical Council, 2013, No. 2, pp. 82-90. https://cyberleninka.ru/article/n/lihoradka-mnogoobrazie-prichin-i-slozhnost-resheniya
5. Verbovoy A.F., Sharonova L.A. Hypothyroidism syndrome: lecture. Journal for Continuing Medical Education of Physicians. Endocrinology: news, opinions, training, 2015, No. 4, pp. 71-75. https://cyberleninka.ru/article/n/sindrom-gipotireoza-lektsiya
6. Starostina E.G. Hypoglycemia and hypoglycemic coma.Medical Council, 2013, No.-5-6, p. 6-10. https://cyberleninka.ru/article/n/gipoglikemiya-i-gipoglikemicheskaya-koma-1
7. Autonomic dysfunction syndrome in children and adolescents: clinical manifestations, diagnosis and treatment. Part 2. Guidelines No. 25, Moscow 2015, Moscow Government, Moscow City Health Department
8. Polyakova A.S. and other Significance of fever in children. Questions of modern pediatrics, 2015, volume 14, No. 2, pp.272-276. https://cyberleninka.ru/article/n/znachenie-lihoradki-u-detey
9. Instructions for the use of Rinzasip for children.https://grls.rosminzdrav.ru/Grls_View_v2.aspx?routingGuid=3fe093f9-82a1-468b-ad1b-1c52f75b3c7e&t=
10. Pikuza O.I., Zakirova A.M. Fever in children. Teaching aid for students of medical universities, Kazan, KSMU, 2013, 56 p. https://kazangmu.ru/files/prop_det_bol/Lichoradka.pdf
11. Urinary tract infection in children. Clinical guidelines 2018 Union of Pediatricians of Russia
12. Avkhimenko MM Cold trauma: emergency medical care, prevention // Nurse / №1.- 2014 .– S. 33-39.
13. Differential diagnosis of infectious diseases: a tutorial. At 2 o’clock, Part 1 / comp .: D. Kh. Khunafina, O. I. Kutuev, A. N. Burganova, A. T. Galieva, L. R. Shaikhullina, T. A. Khabelova, V. I Starostina, G. R. Syrtlanova, R. S. Sultanov. – Ufa: Publishing house of the State Budgetary Educational Institution of Higher Professional Education BSMU of the Ministry of Health of Russia, 2012 .– 153 p.

Administration of the South Ural City District

Civil defense and emergency summary

cold water supply cutoff The day before yesterday, 06:55

11.10.21 g from 09:00 to 15:00 cold water supply shutdown:

st. Mira No. 11,12,13, 19, 21, 23 (MKD), cottages: 18,20,22,24.
st. Lenin No. 13,14,15,17,18,19,20,21,22,24.
st. Moscow 25.
Shop “Pyaterochka” on Lenin 16.
D / garden on Lenin 16 “A” “Golden Key”
The water supply will be carried out at the site between the houses of Lenin 14 and Lenin 16.

power outages. 07/10/2021 07:27

07.10.2021 power outage from 9-00 to 17-00
st.Kirov 1-31; 2-38; st. Chapaeva, 1-11; 2-12; st. Pugacheva, 1-23; 2-22; st. 40 years of October, 1-11; 2-12 .; st. Troitskaya, 10-38; st. Troitskaya, 28 – Mail; st. Kirov, 21-Shop. (Installation of PU)
st. Sverdlov, 1-11; 2-12; st. Uralskaya, 1-17; 2-20; st. Birches 1-11; st. Troitskaya, 2-8; st. Trinity, 1-Temple. (Installation of PU)

08.10.2021 power outage from 9-00 to 17-00

st. Kirov, 33-51; 40-80; st. Troitskaya, 40-56; 56/1; st. Razin, 1-23; 2-24;
st.Kalinin, 41-59; 40; 46, 48; st. Nakhimova, 1-23; 2-24; st. Kashirina, 1; 3; 17; 2-24; st. Ushakova, 1-21; 2-24, Chapaeva 1-11, 2-12. (Installation of PU).

Chapaeva street, 40 years of October, Sverdlova, Uralskaya, Berezki, Yuzhnaya, Kalinin. (Installation of PU)

power outages. 02.10.2021 22:56

06.10.2021 power outage from 9-00 to 17-00
st. Kirov, 33-51; 40-80; st. Troitskaya, 40-56; 56/1; st. Razin, 1-23; 2-24; st. Kalinin, 41-59; 40; 46, 48; st. Nakhimova, 1-23; 2-24; st. Kashirina, 1; 3; 17; 2-24; st. Ushakova, 1-21; 2-24. (Installation of PU).

EMERGENCY WARNING No. 42 dated 27.09.202127.09.2021 15:31

According to the data of the Chelyabinsk CGMS (http: // www.chelpogoda.ru/):
At night and in the afternoon on September 28 in the Chelyabinsk region, precipitation is expected in the form of rain, freezing rain, sleet, snow, in places strong and very strong, in some areas there is a strong deposition of ice and wet snow on the wires, ice on the roads, in places strong, increased wind 15 -20 m / s, locally up to 25 m / s.
The likelihood of emergencies caused by accidents at facilities and lines of power systems increases. Possible damage to poorly reinforced structures, damage to the roof, breakage of power lines, tree shaft.An increase in the number of accidents and congestion on federal, municipal and local roads are predicted.

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HAS-U 5.8 Anchor rod – Anchor rods and elements

HAS-U 5.8 Anchor rod – Anchor rods and elements – Hilti Russia

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Article # r9864250

Anchor rod for use with injection and capsule adhesive anchors (5.8 CS)

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Benefits and Applications

Benefits and Applications

Benefits

• Stud face markings for easy checking of steel grade and stud length even after installation

• Anchor rod with a sharp tip that allows you to cut and mix capsules with the HVU2 adhesive anchor, and also simplifies the insertion of the rod when used with Hilti HIT injection anchors

• Replaces previous HIT-V and HAS / HAS-E studs

• Wide range of anchor rod diameters (M6-M39)

• Custom lengths and additional diameters up to M39 available on request

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• Fastenings in concrete and masonry in combination with the corresponding Hilti HIT adhesive anchoring systems

For information on technical approvals and certifications, click on the corresponding article.

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Magnetic drilling machine Fein KBM 65 U

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• Magnetic drilling machine Fein KBM 65 U
• Product code: 72704361000
• Select size, about to whom we must notify you:

KBM 65 U is a very powerful two-speed magnetic machine that easily maintains speed stability of during heavy work.It is possible to set the spindle exactly , change the direction of rotation and adjust the speed on the electronic panel. The equipment is equipped with conical Morse connection No. 3 . An additional benefit is the is equipped with a double motor guide , thanks to which the largest drills can be used.

The price includes:

Machine tool, case, coolant container, fastening strap, chip hook, centering pin, touch guard and wedge for knocking out the tool.

Technical specifications:

90 013

Max. Core drill Ø:	65 mm
Max. Core drill Ø HSS:	45 mm
Max. drilling depth of core drill:	50 (75 *) mm
Max. Twist drill Ø:	25 mm
Tapping up to:	M 20
Max.Reamer Ø:	50 mm , 25 mm
Rated power input:	1 300 W
Effective power:	730 W
Idling speed in first gear:	120- 240 rpm
Idle speed in second gear:	260-520 rpm
Core drill clamp:	QuickIN (3/4 in Weldon: AU, CA, US, UK)
Tool clamp:	MK 3
Stroke:	145 mm
Full Stroke Range:	315 mm
Adjustment range of drill stand:	± 6 mm
Rotation range of drill stand:	± 7 °
Magnet holding force:	14,000 N
Magnetic stand dimensions:	205 x 95 mm
Cable with plug:	4 m
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