Examples

The application cases are in exec/, grouped below by model. Each is set up and tested in 2D. The header of its prob/inputs describes the problem and its units, and the inputs set the methods, so the commands on these pages show only what a run changes. A case builds and runs like the first simulation, from its own directory:

cd exec/DropletSplashing-5Eq
make -j4
mpirun -n 4 ./main2d.gnu.MPI.ex prob/inputs

Every case except IsentropicVortex-5Eq writes in-situ diagnostics to a CSV file in its plot directory through its UserOutputFunction (Setting up a case).

Cases compared with a reference

These cases are compared with exact solutions, theory or experiments on the Verification and validation page:

Case	Model	Problem	Also shows
DoubleMachReflection-5Eq	FIVEEQS	Mach 10 shock reflecting off a wedge	a BoundaryFill_User face that follows the exact moving shock
GuderleyImplosion-5Eq	FIVEEQS	Converging shock on a perturbed heavy gas	the similarity solution as initial condition and exact inflow
VortexWindup-5Eq	FIVEEQS	Interface wound up by a steady vortex	THINC in a sheared flow, with an exact solution
ThreeMaterialRM-5Eq-N	FIVEEQS_NPHASE, 3 phases	Shock through two successive interfaces	SI units; make NPHASE=4 adds an auxiliary material for a reduction-consistency test
ShockBubble-AirHelium-5Eq	FIVEEQS	Shock and helium cylinder	six levels of refinement on the interface and the shocks
ShockRefraction-AirWater-5Eq	FIVEEQS	Shock refraction at a water surface	a stiffened gas, a sharp initial interface and a post-shock inflow
KelvinHelmholtz-5Eq	FIVEEQS	Compressible shear layer	refined bands from spatio_temporal_tag; the run histories in data/
WaterHammer-6Eq	SIXEQS	Water column stopped by a wall	the exact solution in Parm::DynamicInit

The test cases in exec/_Tests/ that appear there are listed under Test cases.

Two-phase five-equation model

DropletSplashing-5Eq

PHYSICS=FIVEEQS, DIFFUSION=true, 2D. A heavy droplet of radius 0.25 falls under gravity through a light gas into a pool of the same heavy material, in a closed box, with density ratio 100. The case shows the viscous five-equation model with gravity added through user_source_term, and writes the area, centroid height and mean vertical velocity of the heavy fluid above \(y = R/2\) to droplet.csv. The figures come from two runs with one more level than the inputs, 512 by 768 finest cells, with MUSCL-MC alone and with THINC:

cd exec/DropletSplashing-5Eq && make -j4
mpirun -n 4 ./main2d.gnu.MPI.ex prob/inputs amr.max_level=3 amr.t_write_interval=0.125
mpirun -n 4 ./main2d.gnu.MPI.ex prob/inputs amr.max_level=3 amr.t_write_interval=0.125 \
    FiniteVolume.THINC=1 amr.case_name=./plot/DropletSplashing-5Eq-THINC

exec/DropletSplashing-5Eq at impact (\(t = 2\)), in the crater (\(t = 2.5\)) and late in the splash (\(t = 3\)), with MUSCL-MC (top row) and MUSCL-MC + THINC (bottom row). Right half of each panel: light-gas volume fraction \(\alpha_1\). Left half: pressure schlieren \(|\nabla P|\), the interface \(\alpha_1 = 0.5\) (black) and the outlines of AMR levels 1 to 3.

By impact the droplet has flattened and the gas flowing around it has rolled its edges into thin rims. It traps a thin gas film against the pool, merges with the pool and opens a crater, whose rim throws up a splash sheet as the pool climbs the side walls. With THINC the droplet and pool surfaces stay about two cells thick until impact and about three afterwards, against 6 to 12 cells with MUSCL-MC alone. The difference is largest in the gas film, which MUSCL-MC spreads into a diffuse band and THINC keeps sharp until it breaks into a chain of small gas pockets. With THINC the splash is a tall sheet whose rim rolls up and sheds small droplets; without it, it forms thicker, rolled-up fingers. THINC leaves small roll-ups on the sides of the falling droplet, where the gas flows past, and the thinnest late sheets, below a cell or two, as faint smeared trails.

Heavy fluid above the cut \(y = R/2\), with MUSCL-MC and with MUSCL-MC + THINC. (a) Area \(V_2\), normalized by the droplet area \(\pi R^2\) (dashed); the rise after \(t \approx 2.7\) is the pool climbing the walls past the cut. (b) Centroid height \(y_2\), with free fall from rest (dashed) and the cut height (dash-dot). (c) Mass-weighted mean vertical velocity \(v_2\), with free fall (dashed). Dotted line: the free-fall contact time \(t_c\).

The diagnostics are nearly the same in both runs. Until impact the heavy fluid above the cut keeps the area of the droplet to within 1 percent, and its centroid and velocity follow free fall, trailing it slightly because of the drag of the gas. After impact its mean velocity turns upward with the splash rim, peaking at 0.72 with MUSCL-MC and 0.59 with THINC.

GasImpact-5Eq

PHYSICS=FIVEEQS, 2D. A dense, pressurized rectangular block moving at \(u = 0.2\) strikes a plate 15 times denser than the surrounding gas. The plate and the gas share one EOS and differ only in density, and symmetry at \(y = 1\) halves the block. The case shows THINC under a strong density contrast. GasImpact-5Eq-N is the three-material version.

CylindricalRM-5Eq

PHYSICS=FIVEEQS, 2D. A Mach 2 cylindrical shock in air converges on a heavy SF\(_6\)-like gas whose interface is \(R(\theta) = 0.5 + 0.02 \cos(8\theta)\), in one quarter of the plane with symmetry on the axes. The case shows THINC in a converging flow, AMR on the interface and the shocks, and in-situ diagnostics: it writes the equivalent radius of the heavy gas and the amplitude of the perturbation mode to interface.csv, and the scripts in analysis/ plot the contours, the time series and the tracked amplitude. The COMPAS tutorial builds this case step by step, and the version here is the finished one. CylindricalRM-5Eq-N is the three-material version.

N-phase five-equation model

CylindricalRM-5Eq-N

PHYSICS=FIVEEQS_NPHASE, NPHASE=3, 2D. The converging shock of CylindricalRM-5Eq meets two interfaces: a Mach 2 cylindrical shock starts at \(r = 0.92\) in air and converges on a heavy SF\(_6\)-like shell with outer interface \(r = 0.75 + 0.03 \cos(16\theta)\), around a light helium-like core with interface \(r = 0.5 + 0.02 \cos(12\theta)\). All three are ideal gases in nondimensional units, and one quarter of the plane is simulated. The case shows the N-material model with THINC on both interfaces and AMR that follows the interfaces and the shocks, and writes the equivalent radii of the two interfaces and the half widths of their mixing zones to interfaces.csv. The video on the home page and the figure use four levels above the 64 by 64 base grid, 1024 by 1024 cells at the finest level; the inputs use three:

cd exec/CylindricalRM-5Eq-N && make -j4
mpirun -n 8 ./main2d.gnu.MPI.ex prob/inputs amr.max_level=4 amr.t_write_interval=0.008

exec/CylindricalRM-5Eq-N at \(t = 0.2\), 0.3, 0.4, 0.56 and 1: air sand, SF\(_6\) shell teal and helium core dark blue, and in the lower half of each panel the pressure-gradient schlieren, the two interfaces (black) and the outline of the region each AMR level covers. The panels zoom in as the shell converges; each has its own scale bar, of length 0.1.

By \(t = 0.2\) to 0.3 the shock has crossed both interfaces; the shell perturbations grow into spikes while the core stays nearly round. By \(t = 0.4\) the shell has become long SF\(_6\) jets with rolled-up mushroom caps and small Kelvin-Helmholtz roll-ups along their stems, the helium core bulges out between their roots, and the shocks focus inside the core. Near \(t = 0.56\) the reflected shock comes back out and re-shocks the debris, and the imploded region is smallest near \(t = 0.57\). After reshock the jets break up, and SF\(_6\) and helium mix into a zone that grows back out to about \(r = 0.5\) by \(t = 1\), with parcels down to a few cells across. THINC keeps the interfaces about two cells thick through the jets and the roll-ups, and about two and a half in the mixing zone, and the finest level follows the interfaces and the shocks, so the schlieren shows the transmitted and reflected shocks inside the shell and the core.

GasImpact-5Eq-N

PHYSICS=FIVEEQS_NPHASE, NPHASE=3, 2D. The block of GasImpact-5Eq strikes a plate of a second material surrounded by a light gas, a third material, each with its own EOS. The case shows the N-material model with THINC. The video compares the case as provided with the same inputs without THINC, both with three levels of refinement, 1152 by 512 finest cells:

cd exec/GasImpact-5Eq-N && make -j4
mpirun -n 3 ./main2d.gnu.MPI.ex prob/inputs amr.case_name=./plot/thinc amr.chk_int=-1
mpirun -n 3 ./main2d.gnu.MPI.ex prob/inputs FiniteVolume.THINC=0 amr.case_name=./plot/base amr.chk_int=-1

The block (dark blue) striking the plate (teal) in the gas (sand), from t = 0 to 10: the THINC run above the symmetry plane y = 1 and the run with MUSCL-MC alone mirrored below it, in colors blended by volume fraction. Download the MP4.

At first the two halves are nearly the same: the corners of the block roll up, and weak circular waves run through the gas. As the block pushes into the plate the runs differ behind it: without THINC the block material rolls up into smooth vortices and mixes diffusely with the gas, and with THINC the interfaces stay sharp and the block leaves a cloud of parcels a few cells across. The outer shape of the block, its leading edge and the height of its main body agree closely. Late in the run the block spreads into a thin cap wrapped in plate material, and the stem of the plate bends back toward the axis, smooth without THINC and broken into sharp fragments with it.

TriplePoint-5Eq-N

PHYSICS=FIVEEQS_NPHASE, NPHASE=3, 2D. Three gases at rest in a \(7 \times 3\) box with reflecting walls (Galera, Maire and Breil, J. Comput. Phys. 229, 2010): a high-pressure driver (\(\rho = 1\), \(P = 1\), \(\gamma = 1.5\)) fills \(x < 1\), and to its right lie a dense gas below \(y = 1.5\) (\(\rho = 1\), \(P = 0.1\), \(\gamma = 1.4\)) and a light gas above (\(\rho = 0.125\), \(P = 0.1\), \(\gamma = 1.5\)). The case shows the N-material model with sharp initial interfaces. The inputs run it to \(t = 5\) on 224 by 96 base cells with two levels, MUSCL-MC, HLLC and TVD-RK3, and the second run adds THINC:

cd exec/TriplePoint-5Eq-N && make -j4
mpirun -n 3 ./main2d.gnu.MPI.ex prob/inputs
mpirun -n 3 ./main2d.gnu.MPI.ex prob/inputs FiniteVolume.THINC=1 amr.case_name=./plot/TriplePoint-THINC

exec/TriplePoint-5Eq-N at \(t = 1\), 3 and 5, with MUSCL-MC (top row) and MUSCL-MC + THINC (bottom row): dense gas dark blue, driver teal and light gas sand, blended by volume fraction, with the outlines of AMR levels 1 and 2.

The shock runs faster through the light gas than through the dense gas, so its front tilts, and the shear along the dense-light interface puts vorticity at the triple point, where the dense gas rolls up into a large spiral that wraps driver and light gas around it, as in the reference results, and the shear layer behind it rolls up into a row of small Kelvin-Helmholtz billows. The vortex and the fronts are in the same places with both schemes. With MUSCL-MC alone the interfaces spread over nine to ten cells by \(t = 5\), and the core of the roll-up is a diffuse blend of all three gases. With THINC every interface stays under two cells thick, the billows stay sharp, and the wound-up layers stay separate and break into thin filaments and small pockets of light gas and driver.

Six-equation models

ShockBubble-WaterAir-6Eq

PHYSICS=SIXEQS, 2D. A Mach 1.2 shock in water (stiffened gas) hits a cylindrical air bubble (ideal gas) of diameter 1, with the upper half simulated. The case shows the six-equation model with instantaneous pressure relaxation, THINC and AMR at a liquid-gas interface.

ShockDroplet-AirWater-6Eq

PHYSICS=SIXEQS, 2D. A Mach 4 shock in air hits a cylindrical water droplet of diameter 1, with the upper half simulated. The case shows the six-equation model with pressure relaxation and AMR on the droplet and the shocks. The figure uses THINC on a coarser grid than the inputs, 96 by 48 base cells with two levels, run to half the case’s end time:

cd exec/ShockDroplet-AirWater-6Eq && make -j3
mpirun -n 3 ./main2d.gnu.MPI.ex prob/inputs FiniteVolume.THINC=1 stop_time=0.1 amr.n_cell="96 48 8" amr.max_level=2

exec/ShockDroplet-AirWater-6Eq with THINC at \(t u_0/D = 0.01\), 0.04, 0.07 and 0.10, mirrored about \(y = 0\). Upper half: air volume fraction \(\alpha_1\). Lower half: pressure schlieren \(|\nabla P|\), the interface \(\alpha_1 = 0.5\) and the patch boxes of AMR levels 1 and 2. The shock moves in \(+x\).

A curved shock reflects from the upstream face of the column and becomes nearly planar by \(t = 0.1\), and the two branches of the incident shock meet in Mach stems downstream. The water barely compresses, but the flow behind the shock flattens the column into a lens, with thin sheets starting to peel off near its widest points by \(t = 0.1\). THINC keeps the water-air interface one to two cells thick the whole time, even on these coarse cells.

ShockBubble-WaterAir-6Eq-Nphase

PHYSICS=SIXEQS_IE_NPHASE, NPHASE=2, 2D. The setup of ShockBubble-WaterAir-6Eq at Mach 1.1, with the N-phase six-equation model in internal-energy form and instantaneous pressure relaxation. The shock drives the upstream side of the bubble into a re-entrant jet that crosses it, and the air is compressed to under one percent of its initial volume before it rebounds. The figures compare the case as provided, with MUSCL, and the same inputs with THINC:

cd exec/ShockBubble-WaterAir-6Eq-Nphase && make -j4
mpirun -n 4 ./main2d.gnu.MPI.ex prob/inputs
mpirun -n 4 ./main2d.gnu.MPI.ex prob/inputs FiniteVolume.THINC=1 \
    amr.case_name=./plot/ShockBubble-WaterAir-6Eq-Nphase-THINC

exec/ShockBubble-WaterAir-6Eq-Nphase at \(t u_0/D = 0.010\), 0.018, 0.020 and 0.024, with MUSCL (top row) and MUSCL + THINC (bottom row). Upper half of each panel: air volume fraction \(\alpha_2\). Mirrored lower half: pressure schlieren \(|\nabla P|\) with the interface \(\alpha_2 = 0.5\).

The two runs, with times in units of \(D/u_0\), \(u_0 = \sqrt{P_0/\rho_0}\). (a) Air volume \(V\) over the simulated upper half, normalized by its initial value \(V_0\); the inset shows it on a log scale around the collapse, with the minima, 0.0063 (MUSCL) and 0.0059 (THINC), at \(t u_0/D = 0.020\) (open circles). (b) Positions on the axis of the upstream interface, which becomes the jet, and of the downstream interface, until the jet reaches the downstream side at \(t_j\) (open circles). Dashed line: \(t_a\), when the shock reaches the bubble.

With THINC the bubble surface stays about two cells wide for the whole run, while with MUSCL alone the upstream interface, which turns into the jet, spreads over many cells as the jet crosses the bubble. After the collapse the remaining air is a single diffuse pocket trailing a spiral of partly mixed fluid without THINC, and a few sharp pockets and small fragments with it. The air volume follows nearly the same history in both runs.

BubbleFreeSurface-WaterHeAir

PHYSICS=SIXEQS_IE_NPHASE, NPHASE=3, 2D. A gas bubble at 1/30 of the ambient pressure collapses in water at 5 MPa, three bubble radii below a free surface with air above it. Half of the bubble is simulated, with symmetry at \(x = 0\). The case shows the N-phase six-equation model with pressure relaxation on three materials and two interfaces of large density ratio. The same inputs also run with PHYSICS=FIVEEQS_NPHASE, and run.slurm is a template for a cluster job.

Test cases

The cases in exec/_Tests/ are small. The test suite builds and runs all of them, and each is also a starting point for a new case. The directories with a BUILD in the table hold one problem (Diffusion several) with one build per model (per dimension in RayleighTaylor), chosen with BUILD on the make line, and several base runs, named by their labels; the others are one case each. The first base run of the default build reads prob/inputs, the others prob/inputs.<label> (Several builds and base runs in one case). For example, Sod-6Eq is

cd exec/_Tests/Sod
make -j4 BUILD=6Eq
mpirun -n 4 ./main3d.gnu.MPI.6Eq.ex prob/inputs.Sod-6Eq

Case (label)	Directory, BUILD	Model	Dim	Problem and what it tests
Advection-5Eq	Advection, 5Eq (default)	FIVEEQS	2	Four shapes advected in a periodic box (interface advection); base test of the model, with variants for every method
Advection-5Eq-N	Advection, 5Eq-N	FIVEEQS_NPHASE, 5 phases	2	Four shapes of four materials; base test of the model, with THINC
Advection-6Eq	Advection, 6Eq	SIXEQS	2	Four shapes; base test of the model, with pressure relaxation and ACDI
Advection-6Eq-N	Advection, 6Eq-N	SIXEQS_IE_NPHASE, 5 phases	2	Four shapes of four materials; base test of the model, with pressure relaxation and ACDI
Advection-THINC-5Eq	Advection, 5Eq (default)	FIVEEQS	2	Advection-5Eq with THINC
Advection-PF-5Eq	Advection, 5Eq (default)	FIVEEQS	2	Advection-5Eq with implicit CAC-Adv Phase-Field regularization
Couette2Layer-5Eq	Diffusion, 5Eq (default)	FIVEEQS	2	Two-layer Couette flow with a moving wall (user boundary condition), viscosity ratio 10; the arithmetic face viscosity of the five-equation models
Couette2Layer-6Eq	Diffusion, 6Eq	SIXEQS	2	As Couette2Layer-5Eq; harmonic face viscosity, exact steady profile across a sharp interface on a cell face; a variant with the interface in the middle of a cell (first-order error)
Couette2Layer-6Eq-N	Diffusion, 6Eq-N	SIXEQS_IE_NPHASE, 2 phases	2	As Couette2Layer-6Eq
HeatConduction2Mat-5Eq	Diffusion, 5Eq (default)	FIVEEQS	2	Conduction between two different ideal gases, against the exact interface temperature and erf profiles
HeatConduction2Mat-6Eq	Diffusion, 6Eq	SIXEQS	2	As HeatConduction2Mat-5Eq; mixture heat flux with pressure-temperature relaxation, also across a sharp interface
HeatConduction2Mat-6Eq-N	Diffusion, 6Eq-N	SIXEQS_IE_NPHASE, 2 phases	2	As HeatConduction2Mat-6Eq; no heat passes between the phases (known limitation)
ViscousShockTube-5Eq	Diffusion, 5Eq (default)	FIVEEQS	2	Viscous shock tube of Daru and Tenaud; WENO5 with viscous fluxes and no-slip walls
ViscousShockTube-6Eq	Diffusion, 6Eq	SIXEQS	2	ViscousShockTube-5Eq with one gas in the six-equation model; WENO5 with viscous fluxes and no-slip walls
ViscousShockTube-6Eq-N	Diffusion, 6Eq-N	SIXEQS_IE_NPHASE, 2 phases	2	As ViscousShockTube-6Eq
Advection-Viscous-6Eq	Diffusion, 6Eq	SIXEQS	2	A liquid drop advected in gas with viscosity and heat conduction; uniform pressure, velocity and phase temperatures
Advection-Viscous-6Eq-N	Diffusion, 6Eq-N3	SIXEQS_IE_NPHASE, 3 phases	2	As Advection-Viscous-6Eq, with a third phase that stays exactly absent
Advection-Viscous-5Eq-N	Diffusion, 5Eq-N	FIVEEQS_NPHASE, 5 phases	2	Four shapes with viscosity and heat conduction in every material
Sod-5Eq	Sod, 5Eq (default)	FIVEEQS	3	Sod shock tube with AMR and outflow boundaries; variants along \(y\) and \(z\), between walls, and the water-air shock tube
Sod-5Eq-N	Sod, 5Eq-N	FIVEEQS_NPHASE, 2 phases	3	As Sod-5Eq
Sod-6Eq	Sod, 6Eq	SIXEQS	3	As Sod-5Eq, with pressure relaxation; the water-air tube also on 512 uniform cells
Sod-6Eq-N	Sod, 6Eq-N	SIXEQS_IE_NPHASE, 2 phases	3	As Sod-5Eq, with pressure relaxation
NASG-5Eq	NASG, 5Eq (default)	FIVEEQS	2	Noble-Abel stiffened gas: air-water and water shock tubes with liquid water of Le Métayer and Saurel (2016), a water drop advected at uniform pressure and velocity, and \(b = q = 0\) against the stiffened gas; conservation and covolume diagnostics in CSV
NASG-5Eq-N	NASG, 5Eq-N	FIVEEQS_NPHASE, 2 phases	2	As NASG-5Eq
NASG-6Eq	NASG, 6Eq	SIXEQS	2	As NASG-5Eq, with pressure and pressure-temperature relaxation
NASG-6Eq-N	NASG, 6Eq-N	SIXEQS_IE_NPHASE, 2 phases	2	As NASG-5Eq, with pressure relaxation
ShuOsher-5Eq	ShuOsher-5Eq	FIVEEQS	2	Shu-Osher shock-entropy wave interaction with AMR; variants compare the reconstruction schemes
ShockVortex-5Eq	ShockVortex-5Eq	FIVEEQS	2	Vortex through a stationary shock; quadrature WENO5 with AMR
IsentropicVortex-5Eq	IsentropicVortex-5Eq	FIVEEQS	2	Isentropic vortex over one period; WENO5; the order-of-accuracy case
COMPAS-STL-5Eq	COMPAS-STL-5Eq	FIVEEQS	2	A solid read from an STL file falls into a pool; STL reader, gravity source term, viscosity, THINC, on CPUs
RichtmyerMeshkov-5Eq	RichtmyerMeshkov-5Eq	FIVEEQS	2	Single-mode Richtmyer-Meshkov instability (against experiment); interface diagnostics with ghost cells, checkpoint and restart
RichtmyerMeshkov-Multimode-5Eq	RichtmyerMeshkov-Multimode-5Eq	FIVEEQS	2	Multimode Richtmyer-Meshkov instability; parameter arrays sized at run time and copied to the GPU
RayleighTaylor-5Eq	RayleighTaylor, 2D (default)	FIVEEQS	2	Single-mode Rayleigh-Taylor instability (against theory); gravity source term, viscosity, THINC
RayleighTaylor-3D-5Eq	RayleighTaylor, 3D	FIVEEQS	3	3D Rayleigh-Taylor instability; 3D build, gravity source term, viscosity
Jet-Inflow-5Eq	Jet-Inflow-5Eq	FIVEEQS	2	Gas jet injected into a box with an outflow face; Robin user boundary function on an inflow face
NonsphericalCollapse-6Eq	NonsphericalCollapse-6Eq	SIXEQS	3	Gas bubble collapsing near a wall (Johnsen and Colonius 2009); pressure relaxation, THINC, user refinement tagging, no subcycling