Shock over a cylinder

ID	Group	Milestone	Backends	Figure	Regression
V6	Euler	M1 (#35), M3 (#47)	CPU, FNL	yes	shock-cylinder-ib

Case and scope

A Mach 2 planar shock runs into a cylinder imposed by the immersed boundary, in a channel with slip walls. The flow has no closed form, but the setup is mirror symmetric about $y=0.5$, and so is the exact solution. The oracle asserts what the setup guarantees: the refinement, the symmetry and the positivity of the fluid.

The case exercises every immersed-boundary piece together: the distance function, the eikonal extrapolation into the solid (Jacobi sweeps) and the Euler-wall inversion, the cut spacing of the fluid cells next to the surface, the solid masks of the Runge–Kutta stages, and the solid AMR marker with the coarse–fine reflux of the refined ring.

What the case catches:

• any defect that breaks the symmetry: indexing and ordering errors in the distance function, the eikonal extrapolation, the cut spacing or the solid marker, stencils reading the wrong ghost, data races. A wrong wall law that is itself symmetric is not caught;
• a solid marker that misses blocks: every block crossed by the surface must be refined;
• loss of positivity next to the surface.

Models and numerics

Item	Choice	Code
Equations	compressible Euler, ideal gas, $\gamma =1.4$	models
Space, weno	WENO5 (weno-u-5) flux splitting on characteristic variables	compute_residuals_weno, adam_flume_cpu_object.F90
Space, weno-riemann	WENO5 interpolation (characteristic), HLLC, the 6th-order flux correction with the WENO-weight sensor	compute_residuals_riemann, adam_flume_cpu_object.F90; compute_riemann_hllc, adam_flume_euler_library.F90
Time	SSP Runge–Kutta, 3 stages (runge-kutta-ssp-33), CFL 0.5	adam_rk_object.F90
Immersed boundary	analytical circle, Euler wall (bc_type = 2), n_eikonal = 2 Jacobi sweeps	compute_phi_analytical_circle, evolve_eikonal, invert_eikonal, adam_ib_object.F90; numerics
AMR	init-time solid marker (blocks crossed by the surface refined once), reflux at the 2:1 faces	mark_by_solid, adam_flume_common_object.F90; accumulate_coarse_flux, accumulate_fine_flux, adam_flux_register_object.F90; numerics
Boundaries	inflow at $x_{min}$, extrapolation at $x_{max}$, inviscid walls in $y$	set_boundary_conditions, adam_flume_cpu_object.F90; boundary conditions
FNL backend	the same kernels on the device	adam_flume_fnl_object.F90

Initial and boundary conditions

Item	Value
Domain	$[0,1{]}^2$, $z$ null
Grid	octree (the 2:1 reflux restriction needs two tangential axes), 64 blocks of $16\times 16\times 4$ cells (base spacing $1/64$), the blocks crossed by the cylinder surface refined once (delta_fine = 0.0078125); measured: 176 blocks, 96 crossed by the surface, 32 coarse–fine faces
Ranks	2
Cylinder	centre $(0.5,0.5)$, radius 0.13, axis $z$; its surface crosses the block faces $x=0.375$ and $0.625$ at a shallow angle
Post-shock state, $x\le 0.2$ and inflow	$(\rho ,u,p)=(3.7333,1.25,4.5)$ (Mach 2 shock)
Pre-shock state, $x>0.2$	$(\rho ,u,p)=(1.4,0,1)$ (sound speed 1)
Initial condition	riemann-problem, two regions, amr_iterations = 1
Boundaries	$x_{min}$ inflow (post-shock state), $x_{max}$ extrapolation, $y$ wall-inviscid, $z$ extrapolation
Final time	$t=0.25$ (the shock has passed the cylinder; 270 steps)

Input files

File	Role
src/tests/flume/verification/shock-cylinder/shock-cylinder.ini	the case ([solids], [solid_1] definition = analytical_circle, an [amr] solid marker)
src/tests/flume/verification/shock-cylinder/check.sh	V6 (--np N, --legs v6; FLUME_EXE selects the backend), mirror tolerance ${10}^{-10}$
src/tests/flume/verification/shock-cylinder/shock_cylinder_oracle.py	the refinement, mirror-symmetry and fluid-sanity checks; --compare for two runs
src/tests/flume/verification/riemann-flux/check.sh --legs cylinder	the same case on weno-riemann
src/tests/flume/verification/make_doc_figures.py (fig_cylinder)	the figure, from the HLLC work directory of riemann-flux/

cd src/tests/flume/verification/shock-cylinder && ./check.sh                  # CPU
FLUME_EXE=$PWD/../../../../../exe/adam_flume_fnl ./check.sh                   # FNL
cd ../riemann-flux && ./check.sh --legs cylinder                              # weno-riemann HLLC

Results

Density with the block outlines (the refined ring on the surface) and Mach number at $t=0.25$, weno-riemann HLLC.

Check	Bound	weno, CPU	weno, FNL	HLLC, CPU	HLLC, FNL
crossed blocks refined	all	pass	pass	pass	pass
relative asymmetry about $y=0.5$	${10}^{-10}$	$1.2\cdot {10}^{-11}$	$4.8\cdot {10}^{-12}$	$4.9\cdot {10}^{-12}$	$1.9\cdot {10}^{-12}$
min $\rho$, min $p$ in the fluid	positive, finite	positive	positive	1.365, 0.968	positive
CPU against FNL, relative, every field	—	$1.1\cdot {10}^{-11}$

Related cases: Conservation across AMR seams checks the reflux this case relies on; the regression case shock-cylinder-ib freezes the input at 120 steps.

Findings

• The radius 0.13 makes the surface cross block faces at a shallow angle, so fluid and solid cells face each other across block faces. The case therefore depends on the all-solids $\varphi$ summary in the ghost cells: with the former interior-only summary the solution moves by $3.9\cdot {10}^{-2}$.
• Both flux paths keep the symmetry to round-off and the fluid positive; HLLC is the more symmetric of the two.
• The symmetry check is blind to a wrong but symmetric wall law; the case verifies the immersed-boundary plumbing, not its accuracy.