Turbulence (T0, T1, T2, T3)

#119 measures what the scheme does to turbulence before anything is built on it. The diagnostics, the keys and the models are in input and numerics.

Contract legs

src/tests/flume/verification/turbulence/check.sh, CPU and FNL.

Leg	What it verifies	Measured
tt1	first history row of taylor-green against the exact integrals, M 0.1, 1.25	K ${10}^{-16}$; Z and $\epsilon$ order 3.98–4.00; ${\epsilon }_d$, $p_d$ ${10}^{-30}$
tt2	one rank against 2	$6\cdot {10}^{-14}$
tt3	energy budget to $t=2$, Re 1600, M 0.1: what the integrals leave of the $K$ drop	70 %, 8.8 %, 0.28 % at ${32}^3$, ${64}^3$, ${128}^3$
tt6	pressure work, M 0.8: the budget closes only with $p_d$	8.5 % left, 203 % without, 414 % flipped
ht1	isotropic-turbulence field: every shell, $K$, dilatation	shells $7\cdot {10}^{-15}$, $K$ $1.5\cdot {10}^{-14}$
ht2	one rank against 2 on the random field	$7\cdot {10}^{-15}$
sg1	the formulas against numpy on 200 tensors; the zeros of each model	$3.6\cdot {10}^{-15}$, $1.2\cdot {10}^{-15}$, $1.8\cdot {10}^{-11}$
sg2	budget with the model, M 0.4: Vreman 36 %, 5.7 %, 0.27 % left at ${32}^3$, ${64}^3$, ${128}^3$	at ${128}^3$: 21.6 % without ${\epsilon }_{\mathrm{sgs}}$, 43 % flipped
sg3	no viscosity with each model	$\epsilon =0$, mass and energy $3.3\cdot {10}^{-16}$
sg4	one rank against 2 with a model	${10}^{-14}$
hy1	inviscid Taylor-Green: dissipated $K$, hybrid against WENO	47 times less on FNL
wm1	laminar channel, isothermal walls, Re 100: force against $3/Re$; one rank against 2	order 1.99, $4.8\cdot {10}^{-4}$ at 64 across; $4\cdot {10}^{-11}$
wm2	test_wall_model; laminar channel on modelled walls	see numerics; order 1.84, 1.9 % at 64
wm3	half the wall refined, 2:1: force between the coarse and the fine uniform runs	0.03 % from their mean, mass ${10}^{-16}$
wm4	hot wall, modelled against resolved: energy change; one rank against 2 on modelled walls	7.8 % (bound 10 %); $4\cdot {10}^{-11}$
wm5	matching height 0.3, laminar channel at Re 5 against the exact slip solution $3/Re/(1+3h^2/2)$; semi-local against classical damping	$6.4\cdot {10}^{-3}$, $2.2\cdot {10}^{-3}$, $3.0\cdot {10}^{-5}$ at 16, 32, 64; $4\cdot {10}^{-7}$
wm6	semi-local against classical damping on a hot wall, Re 100	force 0.7 % apart (min 0.01 %)
wm7	input filter: $T_f\to 0$ against no filter; $T_f=0.05$ near the steady state; $T_f=1$ over a transient; one rank against 2	0 (bit for bit); $3.0\cdot {10}^{-4}$ (bound ${10}^{-3}$); $1.4\cdot {10}^{-2}$ (min ${10}^{-3}$); $4\cdot {10}^{-11}$ (CPU), ${10}^{-13}$ (FNL); CPU against FNL $5\cdot {10}^{-11}$
wm8	inverse-transformation model: laminar channel against the equilibrium model; hot wall; one rank against 2 with the filter	$1.0\cdot {10}^{-4}$, the heat flux of the Reynolds analogy moving the energy by $2.3\cdot {10}^{-3}$ (bound ${10}^{-4}$); $9.8\cdot {10}^{-3}$ (min ${10}^{-3}$); $4\cdot {10}^{-11}$ (CPU), 0 (FNL); CPU against FNL $4\cdot {10}^{-11}$
wm9	restart with the filter averages (.wmf, one per rank): 10 steps + restart to 20 against 20 straight, inverse-transformation model, $T_f=0.01$; control with the .wmf removed	fields and histories bit for bit (CPU and FNL); the control differs and warns
wm10	dimensional units on a Vreman channel with the wall model and the forcing, gas cp, cv: input bit for bit (NV-5), output against scaling.py dimensionalize (NV-9), forcing and turbulence histories included	bit for bit (CPU and FNL); bit for bit, 4 histories
tb2	wall-model history: ${\tau }_w$ of wm5's laminar channel against the driving force; one rank against 2 on the Mach 2.5 box; CPU against GPU	$3.9\cdot {10}^{-4}$ (bound ${10}^{-3}$); $4\cdot {10}^{-16}$; $1.1\cdot {10}^{-13}$
tb1	digital-filter inflow: the generator's stresses, temporal and spanwise correlations, periodic span, plane restart (test_inflow_df); a Mach 2.5 box: the inflow ghosts carry the table; one rank against 2; restart with the .dff against straight, control without it	stresses 0.4–0.7 % (bound 3 %), correlation exact, integral length 5 % short (bound 10 %); $⟨u⟩$ at $\delta /2$ 0.877 against 0.891; $2\cdot {10}^{-16}$; fields and histories bit for bit (CPU and FNL), the control differs; FNL against CPU ${10}^{-15}$
hy3	the central flux: order, kinetic energy identity, pressure equilibrium	order 5.96–5.99; $4\cdot {10}^{-16}$; $dp/dt$ $9\cdot {10}^{-15}$ across 100:1
hy4	contact of density ratio 8 and 100	$u$, $p$ within $9\cdot {10}^{-14}$
hy5	2:1 box, conservation; passes with any skin	mass ${10}^{-16}$
hy6	Sod against WENO	L1 $8.3\cdot {10}^{-5}$
hy7	runtime regridding; a forest with open ends	${10}^{-16}$; $5.7\cdot {10}^{-10}$, the single realm's drift
hy8	density wave, amplitude 0.2; WENO holds it too, so this does not separate them	$p$, $u$ ${10}^{-14}$
hy9	model only off flagged cells; one rank against 2; conservation (does not prove the mask on ghost cells)	${\epsilon }_{\mathrm{sgs}}$ exactly 0 with all flagged; $7\cdot {10}^{-14}$; $2.3\cdot {10}^{-16}$ with a model on a 2:1 box; whether a flagged cell faces a free one at an interface is not checked
hy11	relaxing the skin changes the density: seams, open ends (does not catch a shifted band)	$1.6\cdot {10}^{-4}$, $6.4\cdot {10}^{-5}$
hy12	restart on a refined grid, every cell	0
hy13	against WENO, L1: a vortex through a shock (2-D), Shu-Osher and a shock through 2:1 seams (1-D, no vorticity, so the Ducros ratio flags any varying flow): robustness, no accuracy claim	$1.0\cdot {10}^{-4}$, $1.7\cdot {10}^{-5}$, $3.4\cdot {10}^{-5}$

Taylor-Green vortex: the ILES baseline

Re 1600, $[0,2\pi {]}^3$, runge-kutta-ssp-54, CFL 0.5, weights = si, 4 ranks on A30, from turbulence/tgv_campaign.sh. weno is the flux splitting, HLLCweno-riemann with the 6th-order correction and the WENO sensor. All values per unit volume.

Top: M 0.1, WENO-5 by resolution; middle: M 0.1 at ${128}^3$ by scheme and order; bottom: $M_0=1.25$ by resolution. Left $-dK/dt$, right the enstrophy; dashed: the 512³ spectral DNS, at M 0.1.

Case	Peak $-dK/dt$	at $t$	Peak $Z$	at $t$	Numerical share to the peak	$K(20)$
M 0.1, WENO-5, ${64}^3$	0.01377	5.08	1.71	6.25	77 %	0.0245
M 0.1, WENO-5, ${128}^3$	0.01242	8.24	3.40	8.11	58 %	0.0219
M 0.1, HLLC, ${64}^3$	0.01305	5.42	1.83	6.56	76 %	0.0284
M 0.1, HLLC, ${128}^3$	0.01249	8.28	3.59	8.11	55 %	0.0227
M 0.1, WENO-7, ${128}^3$	0.01224	8.19	3.17	8.14	60 %	0.0181
M 0.1, WENO-5, ${256}^3$	0.01272	8.75	6.18	8.56	27 %	0.0223
M 0.1, WENO-9, ${128}^3$	0.01299	8.28	4.45	8.03	43 %	0.0213
$M_0$ 1.25, WENO-5, ${64}^3$	0.01383	7.12	1.81	7.62	80 %	0.0206
$M_0$ 1.25, WENO-5, ${128}^3$	0.01081	8.62	3.34	9.09	46 %	0.0243
$M_0$ 1.25, WENO-5, ${256}^3$	0.01101	11.12	5.95	11.94	20 %	0.0245
$M_0$ 1.25, HLLC, ${64}^3$	0.01246	7.22	2.29	7.72	69 %	0.0299
$M_0$ 1.25, HLLC, ${128}^3$	0.01120	8.56	3.86	11.44	36 %	0.0265
$M_0$ 1.25, HLLC, ${256}^3$	0.01122	11.11	6.31	11.99	14 %	0.0243

The share is that of the energy lost until the peak of $-dK/dt$ not accounted for by $\epsilon -p_d$.

Reference. The 512³ pseudo-spectral DNS of the High-Order CFD Workshop (case C3.5, turbulence/reference/spectral_Re1600_512.gdiag), incompressible, so compared at M 0.1 only: peak $-dK/dt$ 0.01286 and peak $Z$ 10.29, both at $t=8.97$. The table in Against the DNS gives the errors. Compressible Mach 1.25 is compared with nothing here.

What it shows.

• Numerical dissipation dominates: up to the peak of $-dK/dt$ it is 77 % of the energy lost at ${64}^3$ and 58 % at ${128}^3$ with WENO-5 at M 0.1. At ${128}^3$ the enstrophy peak is a third of the reference, so the resolved gradients are far from converged even where the total $-dK/dt$ looks right. At ${64}^3$ the peak of $-dK/dt$ comes at $t=5$.
• Scheme and order matter less than resolution at ${128}^3$: HLLC gains a few percent, WENO-9 reaches $Z$ 4.45. WENO-7 is below WENO-5 ($Z$ 3.17 against 3.40), which is not explained.
• At $M_0=1.25$ the numerical share falls with the resolution, 80 %, 46 %, 20 %.

Subgrid-scale models (T1) and the go/no-go

Top: M 0.1 at ${64}^3$; middle: at ${128}^3$; bottom: a 2:1 box on the centre of the ${64}^3$ grid against the uniform grids.

Case, M 0.1	Peak $-dK/dt$	at $t$	Peak $Z$	Numerical share to the peak	$K(20)$
no model, ${64}^3$	0.01377	5.08	1.71	77 %	0.0245
Vreman, ${64}^3$	0.01385	5.05	1.59	79 %	0.0236
WALE, ${64}^3$	0.01374	5.04	1.60	79 %	0.0239
Sigma, ${64}^3$	0.01380	5.17	1.59	79 %	0.0237
no model, ${128}^3$	0.01242	8.24	3.40	58 %	0.0219
Vreman, ${128}^3$	0.01210	8.29	3.18	61 %	0.0213
WALE, ${128}^3$	0.01199	8.29	3.16	61 %	0.0210
Sigma, ${128}^3$	0.01206	8.30	3.18	60 %	0.0213

Here the share leaves out the model's own dissipation, so it is not a pure numerical share: ${\epsilon }_{\mathrm{sgs}}$ is not subtracted. At $M_0$ 1.25, ${128}^3$, Vreman gives peak $-dK/dt$ 0.01085 and peak $Z$ 3.10, against 0.01081 and 3.34.

No model beats no model on this measure. On a dissipative scheme each one lowers the enstrophy peak further from the reference (1.59 against 1.71 at ${64}^3$, 3.18 against 3.40 at ${128}^3$), and the three are nearly the same. The eddy viscosity adds to a numerical dissipation that already dominates; in the laminar phase, to $t=2$ at M 0.4, it takes 21–53 % of the energy lost. Under the rule of #119 this makes the hybrid central/WENO scheme (T2) necessary before any model is recommended. The constants come from other discretizations and were not tuned. A spectral comparison with filtered DNS could still rank the models differently; on the integrals, the DNS confirms this.

The effect of a 2:1 box (T0.6)

M 0.1, base ${64}^3$, the box of side $\pi$ on the centre refined to the ${128}^3$ spacing, so one eighth of the volume. The first attempt never refined (amr_iterations = 0), found because the runs equal the uniform grid to every digit.

Case	Peak $-dK/dt$	at $t$	Peak $Z$	at $t$	$K(20)$	$max|p_d|$
uniform ${64}^3$	0.01377	5.08	1.71	6.25	0.0245	0.00026
box, tricubic fill, mean restriction	0.01312	6.39	1.87	6.39	0.0233	0.0070
box, quintic fill, lagrange6	0.01314	6.39	1.85	6.38	0.0234	0.0115
uniform ${128}^3$	0.01242	8.24	3.40	8.11	0.0219	0.00007

The box delays the peak of $-dK/dt$ from $t=5.1$ to $6.4$ and lowers it, and the two seam operators give nearly the same integrals: $-dK/dt$ and $Z$ do not tell them apart.

$p_d$ cannot be read on these grids. It is 27 to 44 times the uniform value on the box. The likely reason, not verified, is the diagnostic: at M 0.1, $p\approx 71$, and a central difference across a 2:1 face does not telescope, so $\int p\theta dV$ picks up the mean pressure times a nonzero discrete $\int \theta dV$. So the budget and the numerical share are not reported on a 2:1 grid.

Decaying compressible isotropic turbulence (T0.5)

$M_t=0.6$, $R{e}_{\lambda }=100$, $k_0=4$, $\tau ={\lambda }_0/u_{\mathrm{rms}}=0.5$, power-law viscosity 0.76, Pr 0.7, one seed. The DNS of Johnsen et al. (2010) is not available to this tree: only self-convergence is measured.

Left: the enstrophy against $t/\tau$; right: the spectrum of $\sqrt{\rho }\mathbf{u}$ at $t/\tau =4$.

Case	Peak $Z$	at $t/\tau$	$K(t/\tau =4)$	$E(16)$ at $t/\tau =4$	$E(30)$
no model, ${64}^3$	34.5	0.66	0.303	$7.4\cdot {10}^{-4}$	$8.3\cdot {10}^{-6}$
Vreman, ${64}^3$	32.6	0.58	0.292	$5.9\cdot {10}^{-4}$	$5.8\cdot {10}^{-6}$
no model, ${128}^3$	51.0	1.14	0.384	$3.0\cdot {10}^{-3}$	$1.0\cdot {10}^{-4}$
Vreman, ${128}^3$	48.9	1.12	0.376	$2.7\cdot {10}^{-3}$	$8.1\cdot {10}^{-5}$
no model, ${256}^3$	60.5	1.36	0.413	$4.2\cdot {10}^{-3}$	$2.3\cdot {10}^{-4}$

$K$ keeps changing with the resolution by 7 % from ${128}^3$ to ${256}^3$, so ${256}^3$ with WENO-5 is not converged either. As on the Taylor-Green vortex, Vreman moves away from the finer runs at each resolution.

The hybrid central/WENO scheme (T2)

scheme_space = hybrid-kep-weno (numerics), the same inputs.

Top: M 0.1, the hybrid against WENO-5; middle: with Vreman; bottom: $M_0=1.25$. Left $-dK/dt$, right the enstrophy; dashed: the 512³ spectral DNS, on the M 0.1 rows.

Case, M 0.1	Peak $-dK/dt$, raw	Over $\mathrm{\Delta }t=0.5$	at $t$	Flicker, $7<t<10$	Peak $Z$	Numerical share to the peak
WENO-5, ${64}^3$	0.01377	0.01357	5.09	0.3 %	1.71	77 %
hybrid, ${64}^3$	0.01591	0.01334	8.09	7.2 %	3.82	50 %
hybrid + Vreman, ${64}^3$	0.01347	0.01173	8.21	3.5 %	2.92
WENO-5, ${128}^3$	0.01242	0.01216	8.29	0.5 %	3.40	58 %
hybrid, ${128}^3$	0.01389	0.01317	8.48	2.0 %	6.86	18 %
hybrid + Vreman, ${128}^3$	0.01258	0.01226	8.58	0.8 %	5.53
WENO-5, ${256}^3$	0.01272	0.01261	8.72	0.7 %	6.18	27 %

The flicker is the rms of $-dK/dt$ about its running mean, over the smoothed peak. The share with Vreman is left out: it does not separate the model from the scheme. The model acts off flagged cells only (T2.6); against the same runs with the model everywhere, $Z$ moves under 1 % at M 0.1, where the sensor rarely fires.

Case, $M_0=1.25$	Peak $-dK/dt$	Peak $Z$	Numerical share
WENO-5, ${64}^3$ / ${128}^3$ / ${256}^3$	0.01383 / 0.01081 / 0.01101	1.81 / 3.34 / 5.95	80 % / 46 % / 20 %
hybrid, ${64}^3$ / ${128}^3$ / ${256}^3$	0.01266 / 0.01057 / 0.01112	2.20 / 4.83 / 7.21	73 % / 36 % / 10 %

Isotropic turbulence: peak $Z$ 41.0 and 57.4 at ${64}^3$, ${128}^3$ against 34.5 and 51.0 on WENO-5 (60.5 at ${256}^3$), 37.8 and 54.1 with Vreman; $K(t/\tau =4)$ 0.367 and 0.408 against 0.303 and 0.384 (0.413 at ${256}^3$).

What it shows.

• At M 0.1 the hybrid reaches at ${128}^3$ a higher enstrophy peak than WENO-5 at ${256}^3$, and at ${64}^3$ more than WENO-5 at ${128}^3$, with the dissipation peak where the finer runs have it. $Z$ is still below the quoted 10.3.
• At $M_0=1.25$ the gain is smaller, as the sensor keeps WENO around the shocklets.
• The raw peak of $-dK/dt$ is not a measure on the hybrid: it flickers, 7 % at ${64}^3$, 2 % at ${128}^3$. The source is not established; the sensor switching on and off is the first candidate. Smoothed, the hybrid is 4–6 % above WENO-5 at ${256}^3$.
• Vreman on the hybrid removes half of the flicker and lowers $Z$ by 20 %. At ${64}^3$ it lands below the finer runs (0.0117). Against the DNS it makes the hybrid worse at both resolutions (below).
• The thresholds (0.2, 0.1) and the floor are not calibrated.

On a 2:1 box, and what the skin costs

M 0.1, a box of half the side refined once on ${64}^3$. Left $-dK/dt$, right the enstrophy.

Case	Smoothed peak $-dK/dt$	Peak $Z$
WENO-5, box	0.01304	1.87
hybrid, box, WENO skin (default)	0.01239	3.69
hybrid, box, central skin	0.01319	4.13
hybrid, uniform ${64}^3$	0.01333	3.82

With the WENO skin the box ends below the uniform run in $Z$: the skin takes more than the extra cells give. The central skin is stable to $t=20$ and above it. Against the DNS the central skin is the closer one (table below), on this smooth flow; the case with a shock has no reference. A vortex after Jiang & Shu (the code's profile, same strength and radius, unit square) through a Mach 1.1 shock crossing the box seams also runs with the central skin; it differs from the WENO skin by at most $2.8\cdot {10}^{-4}$ in density at $t=0.35$.

Against the DNS

M 0.1 against the 512³ DNS, from turbulence/tgv_dns.py. Peak $-dK/dt$ over a window of 0.5; L2 errors over $0.3<t<19.7$ relative to the DNS, $-dK/dt$ over the same window.

Case	Peak $-dK/dt$	at $t$	Peak $Z$	at $t$	L2 $-dK/dt$	L2 $Z$	L2 $K$
DNS	0.01286	8.97	10.29	8.97
WENO-5, ${64}^3$	0.01357	5.09	1.709	6.25	0.660	0.844	0.187
WENO-5, ${128}^3$	0.01216	8.29	3.400	8.11	0.278	0.643	0.083
WENO-5, ${256}^3$	0.01260	8.72	6.182	8.56	0.090	0.331	0.020
HLLC, ${128}^3$	0.01235	8.32	3.585	8.11	0.260	0.623	0.070
WENO-7, ${128}^3$	0.01215	8.11	3.171	8.14	0.291	0.688	0.101
WENO-9, ${128}^3$	0.01274	8.27	4.454	8.03	0.177	0.539	0.037
Vreman, ${64}^3$	0.01373	5.11	1.591	6.23	0.671	0.854	0.200
Vreman, ${128}^3$	0.01193	8.28	3.177	8.15	0.295	0.662	0.092
WALE, ${128}^3$	0.01183	8.26	3.164	8.16	0.296	0.660	0.091
Sigma, ${128}^3$	0.01192	8.29	3.184	8.14	0.293	0.660	0.091
WENO-5, 2:1 box	0.01304	6.41	1.865	6.39	0.617	0.829	0.178
hybrid, ${64}^3$	0.01333	8.09	3.822	7.86	0.237	0.615	0.057
hybrid, ${128}^3$	0.01317	8.47	6.856	8.25	0.067	0.271	0.014
hybrid + Vreman, ${64}^3$	0.01173	8.21	2.917	8.09	0.283	0.678	0.086
hybrid + Vreman, ${128}^3$	0.01226	8.58	5.529	8.33	0.114	0.400	0.029
hybrid, 2:1 box, WENO skin	0.01239	8.14	3.688	7.88	0.269	0.618	0.063
hybrid, 2:1 box, central skin	0.01319	8.11	4.134	7.91	0.236	0.580	0.059

• The hybrid at ${128}^3$ is closer than WENO-5 at ${256}^3$ on all three errors, with one eighth of the cells. At ${64}^3$ it is closer than WENO-5 at ${128}^3$.
• Its peak of $-dK/dt$ is 2.4 % above the DNS and comes at $t=8.47$ rather than 8.97; $Z$ at ${128}^3$ reaches two thirds of the DNS peak, so the gradients are not converged.
• A model makes it worse: Vreman takes the $-dK/dt$ error from 0.067 to 0.114 and the $Z$ error from 0.27 to 0.40 at ${128}^3$. All three models also worsen WENO-5. This flow at Re 1600 is not a case where the subgrid scales are far from resolved, and the constants were not tuned; on these integrals no model is recommended here.
• WENO-9 at ${128}^3$ (0.177) is much closer than WENO-5 (0.278); WENO-7 is further (0.291), still unexplained.
• The skin: on the 2:1 box the WENO skin (0.269) is further than uniform ${64}^3$ (0.237); the central skin ties it on $-dK/dt$ (0.236) and beats it on $Z$ (0.580 against 0.615). Smooth flow only.

Cost

200 steps on ${128}^3$, two A30, three samples: WENO 24.1–24.6 s, every cell flagged 24.2–24.7 s, every cell central 26.7–26.9 s. The sensor costs under 1 %, the central flux about 10 % on top of WENO, startup included. A single kernel with a mask was not tried, so what skipping WENO would save is not measured.

Plane channel and the wall model (T3)

Coleman, Kim & Moser (1995): bulk Mach 1.5, bulk Re 3000 on the half height and the wall viscosity, isothermal walls, viscosity $(T/T_w{)}^{0.7}$, Pr 0.7; $4\pi \times 2\times \frac{4}{3}\pi$, mass-flow forcing; $R{e}_{\tau }$ about 220–222 for the DNS, as cited by Qi et al. (2022), Cao et al. (2024), Alberti et al. (2025b). The hybrid scheme throughout. From turbulence/channel_campaign.sh with channel_cases_t35.txt, 4 A30, after the no-slip leak fix. ${\tau }_w$ is the mean force per unit volume times $h$; $R{e}_{\tau }=3000\sqrt{{\tau }_w{\rho }_w}$ with ${\rho }_w=1.30$ for every row, so the rows compare among themselves; the reference run's own wall density, extrapolated from its last checkpoint, is 1.25, which lowers every $R{e}_{\tau }$ by 2 %. $u_c$ and $T_c/T_w$ are the Favre plane means at the centreline of the last checkpoint (one snapshot).

${\tau }_w$ against time; dashed: $R{e}_{\tau }$ 220 and 222; dotted: the laminar value.

Case	Cells	${\tau }_w$, $100<t\le 150$	$150<t\le 200$	$R{e}_{\tau }$	$u_c$, $T_c/T_w$
resolved walls	$256\times 192\times 128$, $\mathrm{\Delta }{y}^{+}\approx 2.3$	3.82e-3	–	211 ($50<t\le 100$: 218)	1.16, 1.26
resolved walls	$64\times 32\times 32$	2.35e-3	2.72e-3	166, 178	1.22, 1.26
equilibrium, 3rd cell	$64\times 32\times 32$	4.37e-3	4.49e-3	226, 229	1.21, 1.29
equilibrium, 3rd cell	$96\times 48\times 48$	4.36e-3	4.52e-3	226, 230	1.19, 1.30
equilibrium, 3rd cell	$128\times 64\times 64$	4.69e-3	4.66e-3	234, 234	1.18, 1.31
equilibrium, 3rd cell, semi-local	$128\times 64\times 64$	4.33e-3	4.33e-3	225, 225	1.18, 1.30
equilibrium, $h=0.15$	$96\times 48\times 48$	4.71e-3	4.77e-3	235, 236	1.19, 1.30
equilibrium, $h=0.15$	$128\times 64\times 64$	4.89e-3	4.90e-3	239, 239	1.19, 1.30
equilibrium, $h=0.15$, semi-local	$96\times 48\times 48$	4.15e-3	4.09e-3	220, 219	1.18, 1.29
equilibrium, $h=0.15$, semi-local	$128\times 64\times 64$	4.13e-3	4.21e-3	220, 222	1.17, 1.29
… and the filter $T_f=1$	$96\times 48\times 48$	4.15e-3	4.03e-3	220, 217	1.18, 1.29
… and the filter $T_f=1$	$128\times 64\times 64$	4.15e-3	4.16e-3	220, 221	1.19, 1.29
inverse, $h=0.15$, $u_e=1.18$	$128\times 64\times 64$	4.13e-3	4.08e-3	220, 218	1.16, 1.21
… and the filter $T_f=1$	$128\times 64\times 64$	4.04e-3	4.11e-3	217, 219	1.17, 1.21
inverse, $h=0.15$, $u_e=1.18$	$96\times 48\times 48$	2.03e-3	1.62e-3	154, 138: relaminarises	1.51, 1.11
inverse, 3rd cell, $u_e=1.18$	$96\times 48\times 48$	1.61e-3	1.40e-3	137, 128: relaminarises	1.44, 1.12
inverse, $h=0.15$, $u_e=1.10$	$96\times 48\times 48$	3.61e-3	4.15e-3	206, 220	1.18, 1.21

The inverse model's edge state is given ($u_e$, $T_e/T_w=1.29$): the centreline of the $h=0.15$ semi-local runs. Every run conserves mass to ${10}^{-16}$ (the wall-resolved ones gained 11 % and 1.3 % before the leak fix).

What it shows.

• Matching height 0.15 with semi-local damping puts the equilibrium model in the DNS band on both grids (219–222). Either change alone is worse: the 3rd-cell sample is 226–234 and rises with the grid, $h=0.15$ with the classical damping 235–239, the semi-local damping at the 3rd cell 225. The fixed height removes the grid trend; the semi-local damping removes the excess. This is one Mach and Reynolds number, single runs, and the band sits within the $\pm 2\mathrm{\%}$ of the wall density.
• The input filter does not move the means (within 1.5 %); on the coarser grid it slows the transient.
• The inverse model is in the band on the finer grid with a cooler centreline ($T_c/T_w$ 1.21 against 1.29 for the equilibrium model, 1.26 for the resolved run): its heat flux differs. On $96\times 48\times 48$ it relaminarises with $u_e=1.18$, at $h=0.15$ and at the 3rd cell, but not with $u_e=1.10$: the model is sensitive to the given edge velocity, which the flow does not set (the paper takes the edge from the flow; a face kernel cannot). Why 1.18 drains the turbulence on that grid is not explained.
• The wall-resolved reference did not change with the leak fix (211 against 211 before, $100<t\le 150$), so its 4–5 % shortfall from the DNS was not the leak: its resolution or the hybrid scheme's dissipation, not tested. The modelled rows in the band agree with the DNS, not with it.
• The coarse grid with resolved walls still falls 20 % short: without a model the grid cannot carry the wall layer.
• Only ${\tau }_w$ and the centreline are compared: no averaged profiles, and the DNS profiles are not in this tree.

Mach 3

Result first: with the wall model's eddy viscosity below $h$ (wall_model_eddy_viscosity, below) and run to a thermal steady state at $t=800$, $h=0.15$ with semi-local damping gives $R{e}_{\tau }=443$–$445$ against about 451 for the DNS; the classical damping overshoots by 13–21 %. The runs without it, to $t=200$ only, come first: they show why the key is needed, and their $R{e}_{\tau }$ are transient values, not means.

Without a flux path below $h$, to $t=200$

Coleman, Kim & Moser's second case: bulk Mach 3, bulk Re 4880, $R{e}_{\tau }$ about 450 for the DNS; the rest as above, the equilibrium model only, to $t=200$, channel_cases_t3_mach3.txt. The wall density changes with the run at this Mach number, so each row takes its own: ${\rho }_w$ is the first-cell mean pressure of the last checkpoint over $T_w$, and $R{e}_{\tau }=4880\sqrt{{\tau }_w{\rho }_w}$. $T_1/T_w$ is the first cell (Favre, both walls). $Q_w/W$ is the wall heat flux over the forcing work for $180<t\le 200$, from the total-energy budget (channel_wall_cells.py budget; $Q_w$ is the residual): 1 at a thermal steady state.

Left: ${\tau }_w$ against time (dotted: the laminar value). Middle and right: the first cell of the two $h=0.15$ runs on $128\times 64\times 64$ against time, from a checkpoint every 500 steps (channel_cases_t3_mach3_drain.txt, the same runs, bitwise).

Case	Cells	${\tau }_w$, $100<t\le 150$	$150<t\le 200$	${\rho }_w$	$R{e}_{\tau }$	$u_c$, $T_c/T_w$	$T_1/T_w$	$Q_w/W$
equilibrium, 3rd cell	$96\times 48\times 48$	1.77e-3	1.43e-3	1.65	264, 238: relaminarises	1.43, 1.63	0.86	0.71
equilibrium, $h=0.15$, semi-local	$96\times 48\times 48$	1.89e-3	1.59e-3	1.54	263, 241: relaminarises	1.50, 1.53	0.64	0.67
equilibrium, 3rd cell	$128\times 64\times 64$	4.29e-3	4.44e-3	2.12	465, 473	1.25, 2.22	1.03	0.87
equilibrium, 3rd cell, semi-local	$128\times 64\times 64$	3.20e-3	3.03e-3	2.02	392, 381	1.20, 2.04	1.28	0.85
equilibrium, $h=0.15$, semi-local	$128\times 64\times 64$	3.58e-3	3.46e-3	1.91	404, 397	1.28, 2.01	0.81	0.83
equilibrium, $h=0.15$	$128\times 64\times 64$	6.46e-3	7.49e-3	2.12	571, 615: still rising	1.48, 2.63	0.41	0.88

What it shows.

• No run is at a thermal steady state at $t=200$: the total energy still rises in every run ($Q_w/W$ 0.67–0.88), so these $R{e}_{\tau }$ are values on the way, not means.
• The Mach 1.5 setting does not carry over: $h=0.15$ with semi-local damping is 12 % below the DNS; the 3rd-cell sample is 5 % above. The coarse grid relaminarises with either setting.
• The first cell is drained. The wall face takes the modelled ${\tau }_w$ and $q_w$ from the sample at $h$; the first cell's own state never enters, and nothing below $h$ carries those fluxes (no subgrid model): the molecular plus resolved stress at the face between cells 1 and 2 is 0.54 ${\tau }_w$ with semi-local damping and 0.21–0.39 with the classical one (after $t=30$). With $h=0.15$ and the classical damping the first cell falls to $0.26T_w$, reverses its flow and holds 17 % of the mass at $t\approx 80$; ${\tau }_w$ follows it, also down ($t$ 80–110) and up again: the run does not settle. With semi-local damping the drain is slower and ${\tau }_w$ stays flat, but the first cell is still below $T_w$ from $t\approx 125$. A first cell colder than an isothermal wall that takes heat out of it is not a boundary layer: the treatment below $h$, not the model's law, is what fails here.

With the eddy viscosity below $h$, to $t=800$

The face key wall_model_eddy_viscosity = .true. sets ${\mu }_t=max({\mu }_{t,\mathrm{SGS}},\kappa y\sqrt{\rho {\tau }_w}{D}^2)$ in the cells whose centre is below $h$ (numerics): the flux path that the drain above lacked. The three runs on $128\times 64\times 64$ are the inputs above plus the key, to $t=200$ (channel_cases_t3_mach3_ev.txt), then continued from the restart to $t=800$ (channel_cases_t3_mach3_ev800.txt). ${\rho }_w$ and $T_1/T_w$ are taken at the last checkpoint as above; $Q_w/W$ over $700<t\le 800$.

Left: $R{e}_{\tau }$ against time, with the wall density of each checkpoint (dashed: the DNS). Middle: the first cell. Right: the wall heat flux over the forcing work in 50-unit windows, 1 at a thermal steady state.

Case	${\tau }_w$, $600<t\le 700$	$700<t\le 800$	${\rho }_w$	$R{e}_{\tau }$	$u_c$, $T_c/T_w$	$T_1/T_w$	$Q_w/W$
equilibrium, $h=0.15$, semi-local	3.52e-3	3.48e-3	2.36	445, 443	1.17, 2.44	1.79	0.99
equilibrium, 3rd cell	4.51e-3	4.35e-3	2.51	519, 510	1.21, 2.65	1.57	0.99
equilibrium, $h=0.15$	5.17e-3	5.15e-3	2.44	548, 547	1.21, 2.53	1.84	1.00

What it shows.

• The drain is gone. The first cell is $1.6$–$1.8T_w$ at every time after the transient, the two walls agree within 2 %, and the two first-cell layers hold 4–5 % of the mass (3.1 % if uniform), against 16 % at $0.41T_w$ without the key.
• The runs are thermally steady at $t=800$: $Q_w/W$ is 0.99–1.00 and $dE/dt$ about 60 times smaller than at $t=200$.
• $h=0.15$ with semi-local damping matches the DNS within 2 %. With the classical damping $R{e}_{\tau }$ is 13 % (3rd cell) and 21 % ($h=0.15$) above.
• $R{e}_{\tau }$ needs the thermal steady state at this Mach number. The bulk heats until $t\approx 600$, ${\rho }_w$ rises from about 2.1 to 2.4–2.5, and $R{e}_{\tau }$ rises with it: at $t=200$ the 3rd-cell run gave 438, 448, which looked like a match.
• Cost: from the same $t=800$ state of the semi-local run to $t=810$ (medooza, 4 A30), the key on takes 1.69 times the wall time of the key off: 1.28 times the steps (the eddy viscosity enters the viscous time-step limit) and 1.32 times the time per step.

Compressible boundary layer (T3.7)

The zero-pressure-gradient layer of the DNS of Zhang, Duan & Choudhari (2018, AIAA J 56(11)), case M2p5: Mach 2.5, adiabatic wall ($T_w/T_r=1$), $R{e}_{\theta }=2835$ and $R{e}_{\tau }=510$ at its station. The station profiles come from the NASA Turbulence Modeling Resource (turbulence/reference/). The layer is fed through the face inflow-digital-filter (numerics): the DNS station profile scaled to the inflow thickness ${\delta }_i$, with fluctuations that have the DNS Reynolds stresses. $C_f$ is the wall model's own ${\tau }_w$, written by [diagnostics] wall_history_save (leg tb2).

Setup
box, cells	$90\times 6\times 6{\delta }_i$, $1080\times 72\times 72$, cubic ${\delta }_i/12$; periodic span, extrapolated top and outflow
gas	$Re=U_{\mathrm{\infty }}{\delta }_i/{\nu }_{\mathrm{\infty }}=19370$ (the DNS's 4 mm ${\delta }_i$), $\mu \propto T^{0.76}$ (the DNS uses Sutherland), $Pr=0.71$
wall	wall-modelled, $T_w=2.104T_{\mathrm{\infty }}$, the 3rd cell, semi-local damping, its eddy viscosity below $h$ (the setting that matched the Mach 3 channel)
inflow	integral lengths $0.8$, $0.3$, $0.3{\delta }_i$ (streamwise as a time scale $L_x/U_{\mathrm{\infty }}$, wall-normal, spanwise)
scheme	hybrid-kep-weno, no subgrid model
run	tbl_cases_t37_l90.txt, to $t=360{\delta }_i/U_{\mathrm{\infty }}$ (4 flow-throughs); the wall history every 20 steps and 34 checkpoints of $180<t\le 360$, averaged over the span and in time (tbl_oracle.py compare); 0.40 s per step on 4 A30

Left: $C_f$ from the wall model's ${\tau }_w$ (blue) and from the momentum integral $2d\theta /dx$ (grey), and $R{e}_{\theta }$ (right axis) against $x$; the square is the DNS at its $R{e}_{\theta }$. Middle: $u_{VD}^{+}$ at the station where $R{e}_{\theta }$ is the DNS's, scaled with the run's $u_{\tau }$. Right: the rms over $u_{\tau }$ at the same station; dashed: the DNS.

At $R{e}_{\theta }=2836$ ($x=55{\delta }_i$)	LES	DNS
$C_f$, the wall model's ${\tau }_w$	2.21e-3 (the two halves of the window 2.19e-3, 2.22e-3)	2.31e-3
$C_f$, momentum integral	1.56e-3
$H$, against the edge state (the shape factor)	4.08	4.10 (its profile: 4.22)
$u_{VD}^{+}$ at $y^{+}=30$, $100$, $200$	13.1, 17.5, 19.8	13.6, 16.7, 18.8
$u_{rms}$, $v_{rms}$, $w_{rms}$ over $u_{\tau }$ at $y/\delta =0.2$	2.06, 0.81, 1.22	1.49, 0.93, 1.12
$-⟨u^{\prime }{v}^{\prime }⟩/u_{\tau }^2$ at $y/\delta =0.2$, $0.5$	0.80, 0.50	0.62, 0.37

What it shows.

• $C_f$ within 4.4 % of the DNS at its $R{e}_{\theta }$, with the setting that matched the Mach 3 channel. The two halves of the averaging window differ by 1.1 %. Most of the 4.4 % is the inflow's recovery, not the plateau: farther downstream, at $R{e}_{\theta }=3000$–$3350$, $C_f$ is within 1.6–0.4 % of the DNS's value scaled as $R{e}_{\theta }^{-1/4}$, and at ${\delta }_i/16$ it is 0.3 % at the station (resolution).
• The inflow recovers by $x\approx 50{\delta }_i$. The layer reaches the DNS's $R{e}_{\theta }$ at $x=55$, against 53 in the DNS. $C_f$ falls from $2.8\cdot {10}^{-3}$ at the inlet to $1.8\cdot {10}^{-3}$ at $x\approx 25$, the adaptation of the synthetic eddies, then rises to a plateau of $2.2\cdot {10}^{-3}$ from $x\approx 50$. That matches the $40{\delta }_i$ recovery length of a digital filter at this Mach number (Huang, Duan & Choudhari 2022).
• The momentum integral does not measure $C_f$ here. Fitted over $\pm 3{\delta }_i$, $2d\theta /dx$ swings between $0.9$ and $4\cdot {10}^{-3}$ along $x$ on the same averaged data, and is $-33$ % at the station. A first run on a $60{\delta }_i$ box had only this estimate, and it reported $C_f$ 16–28 % low.
• The log layer sits 0.9–1.0 above the DNS in $u_{VD}^{+}$, and $u_{rms}$ peaks too high, the near-wall streaks of a coarse LES. The resolved $-⟨u^{\prime }{v}^{\prime }⟩$ in the outer layer is above the DNS's. At ${\delta }_i/16$ the log layer, $u_{rms}$ and $-⟨u^{\prime }{v}^{\prime }⟩$ move toward the DNS. $H$, taken against the local free stream as the DNS takes it, is the DNS's within 1 % (4.08).

Resolution: Mach 2.5 at ${\delta }_i/16$

The Mach 2.5 case on the same $90{\delta }_i$ box at ${\delta }_i/16$: $1440\times 96\times 96$ cells, the matching point still the 3rd cell (so $h$ shrinks with the grid, as in Kawai & Larsson 2012), the filter's node spacing the new $\mathrm{\Delta }y$; everything else as at ${\delta }_i/12$. 30 checkpoints of $180<t\le 360$ (tbl_cases_t37_r16.txt, 35264 steps, 0.80 s per step on 4 A30).

Left: $C_f$ from the wall model's ${\tau }_w$ at ${\delta }_i/12$ (dashed) and ${\delta }_i/16$ (solid); the square is the DNS at its station. Middle: $u_{VD}^{+}$ at the station of each run. Right: the rms and $-⟨u^{\prime }{v}^{\prime }⟩$ over $u_{\tau }$; dotted: the DNS.

	${\delta }_i/12$	${\delta }_i/16$	DNS
$C_f$ at $R{e}_{\theta }=2835$	2.208e-3 (halves 2.195e-3, 2.220e-3)	2.306e-3 (halves 2.299e-3, 2.312e-3)	2.310e-3
$C_f$ against the DNS scaled as $R{e}_{\theta }^{-1/4}$, at $R{e}_{\theta }=2600$, 2835, 3000, 3200, 3350	−12.0, −3.9, −1.6, −0.9, −0.4 %	−3.6, −0.3, +1.0, +0.7, −0.5 %
$H$, against the edge state	4.08	4.23	4.10 (its profile: 4.22)
$u_{VD}^{+}$ at $y^{+}=30$, $100$, $200$	13.1, 17.5, 19.8	12.8, 17.3, 19.2	13.6, 16.7, 18.8
$u_{rms}$, $v_{rms}$, $w_{rms}$ over $u_{\tau }$ at $y/\delta =0.2$	2.06, 0.81, 1.22	1.79, 0.83, 1.16	1.49, 0.93, 1.12
$-⟨u^{\prime }{v}^{\prime }⟩/u_{\tau }^2$ at $y/\delta =0.2$, $0.5$	0.80, 0.50	0.66, 0.43	0.62, 0.37

What it shows.

• $C_f$ is grid-converged once the layer has recovered. Downstream of $x\approx 60$ ($R{e}_{\theta }\ge 3000$) the two grids agree with each other and with the DNS's $C_f$ scaled as $R{e}_{\theta }^{-1/4}$ within 1–2 %. The scaling is an approximation (the DNS file holds one station), so these figures carry its error too.
• The finer grid shortens the inflow's recovery. The $C_f$ dip after the inlet ends by $x\approx 40$ at ${\delta }_i/16$, against $x\approx 55$–$60$ at ${\delta }_i/12$: the $-4.4$ % at the DNS station of the ${\delta }_i/12$ run is mostly the layer still recovering. At ${\delta }_i/16$ the station is past it, and $C_f$ is $-0.2$ %; the halves of the window differ by 0.6 %.
• The near-wall and log-layer statistics converge toward the DNS: the $u_{rms}$ peak at $y/\delta =0.2$ from 2.06 to 1.79, $-⟨u^{\prime }{v}^{\prime }⟩$ from 0.80 to 0.66, the log-layer offset in $u_{VD}^{+}$ at $y^{+}=200$ from +1.0 to +0.4.
• $H$ is the DNS's on both grids (4.08, 4.23, against 4.10) once taken against the local free stream (the shape factor).

The shape factor and the free stream

The DNS's profiles are normalised by its free stream at the station, and are 1 there. The run's free stream is not the nominal one: the layer thickens downstream and turns the supersonic stream over it, a weak compression of order $\gamma M^2(d{\delta }^{\ast }/dx)/\sqrt{M^2-1}$ (linear supersonic theory). tbl_oracle.py compare takes ${\delta }^{\ast }$ and $\theta$ against the local edge state: ${\rho }_e$ and the Favre $u_e$ the means over $1.5$–$2.5\delta$, the integrals to $1.5\delta$, the DNS's own profile integrated the same way (4.22 at Mach 2.5, against the header's rounded 4.1). Earlier versions of this page took them against the nominal ${\rho }_{\mathrm{\infty }}$, $U_{\mathrm{\infty }}$ and integrated to the top of the box. That reported $H$ 14 % low at Mach 2.5 and 30 % low at Mach 5.86, and the runs below set out to explain a deficit that was the measurement.

Left and right: $H$ of the Mach 2.5 baseline and the Mach 5.86 run against $x$, against the edge state (solid) and the nominal free stream (dashed); the DNS's header and its profile's at its station. Middle: the edge pressure and density over the nominal ones against $x$.

• The free stream is compressed. At Mach 2.5 $p_e/p_{\mathrm{\infty }}$ rises to 1.027 at $x\approx 15$ and is 1.013 at the station, ${\rho }_e/{\rho }_{\mathrm{\infty }}$ 1.009; at Mach 5.86 1.08 and 1.042. Integrated to the top of the box ($6{\delta }_i$, more than $3\delta$), the denser stream over the layer takes 0.5 off $H$ at Mach 2.5 and 4 at Mach 5.86. The dip of the nominal $H$ to 2.8 at $x\approx 15$ is the compression's peak, not the layer.
• Against the edge state $H$ is the DNS's: 4.08 at Mach 2.5 (4.10, its profile 4.22), 13.4 at Mach 5.86 (13.6, its profile 14.9). Taken over $2.5$–$3.5$, $3$–$5$ or $4$–$5.5{\delta }_i$, the edge moves it by 0.04 at Mach 2.5.

Five runs at ${\delta }_i/12$. Before the measurement was found, the Mach 2.5 case ran with one setting changed each, to look for the deficit: the filter's integral lengths raised from $0.8,0.3,0.3$ to $1.2,0.5,0.5{\delta }_i$ and the span doubled to $12{\delta }_i$ ($1080\times 72\times 144$ cells; tbl_cases_t37_hdef.txt), and an algebraic subgrid model, [turbulence] sgs_model = wale and vreman, their default constants (tbl_cases_t37_sgs.txt; WALE on dev-gn, Vreman on prod-gn). Same averaging, 33–35 checkpoints of $180<t\le 360$. They measure how much the other statistics depend on these settings.

Four of the runs (the span of $12{\delta }_i$ in the table only). Left: $H$ against the edge state, against $x$; the square is the DNS at its station. Middle: $u_{VD}^{+}$ at the station of each run. Right: $-⟨u^{\prime }{v}^{\prime }⟩/u_{\tau }^2$ against $y/\delta$; black, the DNS.

At $R{e}_{\theta }=2835$	baseline	lengths $1.2,0.5,0.5{\delta }_i$	span $12{\delta }_i$	WALE	Vreman	DNS
station $x/{\delta }_i$	55.2	51.4	54.6	54.7	57.5	53
$H$, against the edge state	4.08	4.08	4.11	4.05	4.11	4.10 (its profile: 4.22)
$C_f$, the wall model's ${\tau }_w$	2.208e-3 (−4.4 %)	2.248e-3 (−2.7 %)	2.205e-3 (−4.6 %)	2.235e-3 (−3.3 %)	2.206e-3 (−4.5 %)	2.310e-3
$u_{VD}^{+}$ at $y^{+}=30$, $100$, $200$	13.1, 17.5, 19.8	13.0, 17.7, 19.8	13.0, 17.5, 19.9	13.2, 17.4, 19.7	13.0, 17.5, 19.7	13.6, 16.7, 18.8
$u_{rms}$, $v_{rms}$, $w_{rms}$ over $u_{\tau }$ at $y/\delta =0.2$	2.06, 0.81, 1.22	1.97, 0.75, 1.15	2.08, 0.79, 1.22	1.97, 0.76, 1.19	1.94, 0.74, 1.27	1.49, 0.93, 1.12
$-⟨u^{\prime }{v}^{\prime }⟩/u_{\tau }^2$ at $y/\delta =0.2$, $0.5$	0.80, 0.50	0.66, 0.57	0.74, 0.56	0.72, 0.55	0.66, 0.49	0.62, 0.37

What it shows.

• None of the settings changes the layer much. $H$ stays within 0.06 of the DNS's header value, $C_f$ within 2 % of the baseline's; longer integral lengths shorten the inflow's recovery a little (the layer reaches $R{e}_{\theta }=2835$ at $x=51$, $C_f$ −2.7 %), as ${\delta }_i/16$ did.
• The subgrid model is active, and does not matter at this grid. WALE's eddy viscosity, computed from a checkpoint, is about 1.5 times the molecular one over $0.3$–$0.75\delta$ and 0.4 at the edge.
• What is left is in the stresses, not in the mean: the outer $-⟨u^{\prime }{v}^{\prime }⟩$ is 0.49–0.57 at $0.5\delta$ against 0.37 in every run, and the near-wall $u_{rms}$ peak 1.9–2.1 against 1.5. At ${\delta }_i/16$ both move toward the DNS (resolution).

Mach 5.86, cold wall

Case M6Tw076 of the same DNS: Mach 5.86, a cold wall ($T_w/T_r=0.76$, $T_w=5.45T_{\mathrm{\infty }}$), $R{e}_{\theta }=9455$ and $R{e}_{\tau }=453$ at its station ($x=54{\delta }_i$), wall heat-flux parameter $B_q=q_w/({\rho }_w{c}_p{u}_{\tau }{T}_w)=-0.02$ (heat into the wall). The wall history gives $q_w$ as well as ${\tau }_w$.

Setup
box, cells	$120\times 6\times 6{\delta }_i$, $1440\times 72\times 72$, cubic ${\delta }_i/12$: the filter's recovery length grows with the Mach number (Huang, Duan & Choudhari 2022)
gas	$\mu \propto T^{0.76}$ with ${\mu }_w$ matched to the DNS's Sutherland value at the wall, where the model acts (${\mu }_w/{\mu }_{\mathrm{\infty }}=5.13$ against ${5.45}^{0.76}=3.63$): $Re=97120$, the DNS's $137350$ per ${\delta }_i$ over $1.414$; $R{e}_{\theta }$ is reported with $137350$ (tbl_oracle.py compare --re-inf)
wall, inflow, scheme	as at Mach 2.5
run	tbl_cases_t37_m6.txt, to $t=360$ (21604 steps); 29 checkpoints of $180<t\le 360$ and the wall history every 20 steps; 0.54 s per step on 4 A30

The panels of the Mach 2.5 figure; the title adds $B_q$ at the station.

At $R{e}_{\theta }=9453$ ($x=55{\delta }_i$)	LES	LES, Sutherland	DNS
$C_f$, the wall model's ${\tau }_w$	9.47e-4 (the two halves of the window 9.70e-4, 9.25e-4); $x=60$–$110$: 9.78e-4	9.51e-4; $x=60$–$110$: 9.80e-4	9.84e-4
$B_q$, the wall model's $q_w$	−0.0189 (halves −0.0191, −0.0187); $x=60$–$110$: −0.0194	−0.0190	−0.02 (one significant figure)
$\delta /{\delta }_i$	1.61	1.61	1.72
$H$, against the edge state	13.41	13.71	13.6 (its profile: 14.9)
$u_{VD}^{+}$ at $y^{+}=30$, $100$, $200$	12.4, 17.9, 20.8	12.4, 17.9, 20.6	13.9, 17.2, 19.5
$u_{rms}$, $v_{rms}$, $w_{rms}$ over $u_{\tau }$ at $y/\delta =0.2$	2.21, 0.56, 0.94	2.24, 0.55, 0.90	1.33, 0.82, 0.99
$-⟨u^{\prime }{v}^{\prime }⟩/u_{\tau }^2$ at $y/\delta =0.2$, $0.5$	0.64, 0.39	0.65, 0.35	0.49, 0.25

What it shows.

• $C_f$ within 3.8 % of the DNS at its $R{e}_{\theta }$, with no change of the wall-model setting from Mach 2.5; past $x=60$ the plateau is 0.7 % below. $B_q$ agrees to the precision of the reference: the DNS file gives $-B_q=0.02$, one significant figure, and the run gives 0.0189 at the station, 0.0194 on the plateau.
• $C_f$ drifts by 4.6 % between the two halves of the window; $q_w/{\tau }_w$ by 0.4 %. The layer itself changes slowly over the window, and the wall model's heat flux follows its friction. 180 time units are 1.5 flow-throughs of this box, so the $C_f$ figure carries about $\pm 2$ %.
• The inflow recovers by $x\approx 55$–$60{\delta }_i$, farther than at Mach 2.5 (50): $C_f$ falls from $1.4\cdot {10}^{-3}$ at the inlet to $0.8\cdot {10}^{-3}$ at $x\approx 33$, then rises to its plateau. The layer reaches the DNS's $R{e}_{\theta }$ at $x=55$, against 54 in the DNS.
• $H$ is the DNS's against the edge state (13.4; the header 13.6, its profile 14.9): the free stream over this layer is compressed by 4 % (the shape factor).
• The stresses have the Mach 2.5 faults, larger. $\delta$ is 6 % thin; $u_{rms}$ peaks at 1.7 times the DNS's at $y/\delta =0.2$, and $u_{VD}^{+}$ is 1.5 below the DNS at $y^{+}=30$ (the first cells under the matching point) and 0.7–1.3 above it at $y^{+}=100$–$200$. At ${\delta }_i/12$ the layer of $R{e}_{\tau }=453$ has about 20 cells across $\delta$; at Mach 2.5 a finer grid moves the log layer and the stresses toward the DNS (resolution).
• Sutherland's law with the DNS's Reynolds number changes little. The viscosity, matched at the wall, is 1.41 times the DNS's in the free stream. A run under Sutherland's law (viscosity_law = sutherland, $S=110.4$ K, the DNS's $Re=137350$ per ${\delta }_i$; tbl_cases_t37_hdef.txt, 29 checkpoints) removes that: $H$ 13.71 against 13.41, $C_f$ −3.4 % against −3.8 % at the station, the plateau 9.80e-4, $B_q$ −0.0190.
• The momentum integral is useless here too ($-62$ % at the station, negative in places).

Not done

• HLLC at ${256}^3$, M 0.1 was dropped by owner decision.
• Spectra against the DNS: the file holds integrals only. The DNS of Johnsen et al. for the isotropic case.
• Calibration of the sensor; a model masked by face rather than by cell; central-bcs on a forest seam.
• T3: why the classical damping overshoots at Mach 3; a cheaper flux path below $h$; the boundary layer: the matching height, the outer-layer and near-wall stresses at ${\delta }_i/12$, a longer window at Mach 5.86; $R{e}_{\theta }$ and the station against the edge state (they use the nominal free stream); SBLI; runtime regridding along a modelled wall; mean profiles against the DNS; the relaminarisation of the inverse model on $96\times 48\times 48$ with $u_e=1.18$; an edge state taken from the flow.
• A model on 2:1 grids beyond conservation; runtime AMR with a model; a wall.