Compressible Couette flow

ID	Group	Milestone	Backends	Figure	Regression
VV-3	Dissipative terms (M4)	M4 (#65)	CPU, FNL	yes	couette

Case and scope

A channel between a fixed isothermal wall at $y=0$ and an adiabatic no-slip wall moving at $U$ along $x$ at $y=H$. The exact steady solution has uniform $p$, the linear velocity $u=Uy/H$ and the viscous-heating parabola $T=T_w+\mu U^2/(kH)(y-y^2/(2H))$. It is an exact steady solution of the compressible Navier–Stokes equations with constant $\mu$ and $k$, so the run starts from it and the departure at the final time is the error of the scheme, with no relaxation transient.

What the case catches: the first test of the no-slip walls with the viscous fluxes, and of a nonlinear energy flux (the viscous heating $u\tau$): a wrong wall ghost, a wrong wall velocity or temperature, or a wrong viscous work term shows as a departure from the exact profile.

Models and numerics

Item	Choice	Code
Equations	compressible Navier–Stokes, $\gamma =1.4$, $R=1$, $\mu =0.01$, $Pr=0.75$ ($k=\mu c_p/Pr$)	models; load_from_file, adam_flume_dissipation_object.F90
Dissipative fluxes	dissipative_order = 4	add_dissipative_fluxes_o4, adam_flume_cpu_dissipation_kernels.F90; numerics
Inviscid fluxes	WENO5 flux splitting, characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90
Time	runge-kutta-ssp-54, CFL 0.2	adam_rk_object.F90
Wall ghosts	velocity mirrored about the wall's (exact for the linear $u$); the moving adiabatic wall mirrors $T$ symmetrically (exact for the symmetric $T$); the isothermal wall sets the geometric mirror $T_g=T_w^2/T$ and ${\rho }_g=p/(R{T}_g)$ (second order)	wall_noslip_ghost, adam_flume_bc_object.F90; set_boundary_conditions, adam_flume_cpu_object.F90; boundary conditions
FNL backend	the same fluxes and wall ghosts on the device	adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Domain	$[0,1]\times [0,H]$, $H=1$, $z$ null
Grid	quadtree, $4\times 4$ blocks of $6\times N/4$ cells: $x$ active and thin (a null $x$ would freeze $\rho u$), $N$ = 32 / 64 / 128 cells across the channel
Ranks	2
Initial condition	couette: the exact profile, $p=1$, $U=1$, $T_w=1$
bc_x_min, bc_x_max	periodic
bc_y_min ($y=0$)	wall-isothermal, wall_temperature = 1 (at rest)
bc_y_max ($y=1$)	wall-noslip, wall_u = 1 (adiabatic, moving)
bc_z_min, bc_z_max	extrapolation
Final time	$t=0.2$

Input files

File	Role
src/tests/flume/verification/viscous/profiles.py	make --case couette writes each input; oracle measures the RMS of $u/U$ and $T/T_w$ against the exact profile
src/tests/flume/verification/viscous/check.sh --legs vv3	runs the ladder (--np N; FLUME_EXE selects the backend)
src/tests/flume/regression/couette/input.ini	the frozen case, $t=0.05$
make_doc_figures.py (fig_couette, fig_viscous)	the figures, from the CPU work directories of viscous/check.sh

cd src/tests/flume/verification/viscous && ./check.sh --legs vv3             # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh --legs vv3       # FNL

Results

Top: $u/U$ and $T/T_w$ at 64 cells and $t=0.2$, and the 32-cell profiles on the exact ones; bottom: the pointwise errors of $u$ and $T$ at 64 cells, and the temperature error averaged along $x$ at every resolution.

The Couette temperature ladder is the "VV-3 Couette (T)" curve of the left panel.

Oracle: order $\ge 1.9$ on $u/U$ and $T/T_w$ (finest pair).

Quantity	N	RMS error (CPU)	Order	FNL
$u/U$	32 / 64 / 128	$1.71\cdot {10}^{-7}$ / $3.84\cdot {10}^{-8}$ / $8.72\cdot {10}^{-9}$	+2.15, +2.14	the same; the last digit at 128 cells differs
$T/T_w$	32 / 64 / 128	$3.01\cdot {10}^{-6}$ / $6.62\cdot {10}^{-7}$ / $1.49\cdot {10}^{-7}$	+2.19, +2.15	the same

Both backends ran on 2 ranks.

Related cases: the wall-free viscous waves are VV-1 and VV-2; the other exact steady profile is Becker's shock.

Findings

• The run is second order overall: the moving-wall mirror is exact for the linear $u$ and the symmetric $T$, the isothermal geometric mirror $T_w^2/T$ is second order.
• The fields show where that second order comes from. Velocity and temperature are uniform along the periodic $x$, and the profiles lie on the exact ones (the linear $u$, the viscous-heating parabola of $T$). The error enters at the isothermal wall and diffuses into the channel: at $t=0.2$ its front has reached about $y=0.5$, and beyond it the solution is exact to round-off at 128 cells. The moving adiabatic wall adds nothing.
• With the linear isothermal mirror of P1 ($2T_w-T$) the errors were 18% lower, at the same orders. The geometric mirror was adopted because the linear one made negative ghost densities beside hot gas; the two differ by $O(\mathrm{\Delta }{x}^2)$.