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Mark Zhang
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Add 3 new examples. (2D_kelvin_helmholtz, 2D_richtmyer_meshkov, 2D_viscous_shock_tube)
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‎examples/2D_kelvin_helmholtz/README.md‎

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# Kelvin-Helmholtz Instability (2D)
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Reference:
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> A.S. Chamarthi, S.H. Frankel, A. Chintagunta, Implicit gradients based novel finite volume scheme for compressible single and multi-component flows, arXiv preprint arXiv:2106.01738 (2021).
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**Key Parameters**
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* $\gamma = 5/3$
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**Initial Conditions**
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* **Spatial Domain:** $(x, y) \in [0, 1] \times [0, 1]$
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$$
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p = 2.5, \rho(x, y) = \begin{cases}
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2, \text{if } 0.25 < y \leq 0.75\\
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1, \text{else, }\\
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\end{cases}
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$$
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$$
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u(x, y) = \begin{cases}
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0.5, \text{if } 0.25 < y \leq 0.75\\
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-0.5, \text{else, }
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\end{cases}
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$$
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$$
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v(x, y) = 0.1 \sin(4\pi x) \left[ \exp\left( -\frac{(y-0.75)^2}{2\sigma^2} \right) + \exp\left( -\frac{(y-0.25)^2}{2\sigma^2} \right) \right]
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$$
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where $\sigma = \frac{0.05}{\sqrt{2}}$.
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### Initial State
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<img src="figure0.png" height="MAX_HEIGHT"/>
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### Evolved State
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<img src="figure1.png" height="MAX_HEIGHT"/>

‎examples/2D_kelvin_helmholtz/case.py‎

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#!/usr/bin/env python3
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import json
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import math
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eps = 1e-6
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time_end = 1.0
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time_save = time_end / 100.0
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# Configuring case dictionary
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print(
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json.dumps(
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{
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# Logistics
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"run_time_info": "T",
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# Computational Domain Parameters
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"x_domain%beg": 0,
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"x_domain%end": 1.0,
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"y_domain%beg": 0,
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"y_domain%end": 1.0,
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"m": 512,
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"n": 512,
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"p": 0,
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"cfl_adap_dt": "T",
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"cfl_target": 0.2,
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"n_start": 0,
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"t_stop": time_end,
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"t_save": time_save,
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# Simulation Algorithm Parameters
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"num_patches": 2,
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"model_eqns": 2,
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"alt_soundspeed": "F",
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"num_fluids": 2,
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"mpp_lim": "T",
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"mixture_err": "T",
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"time_stepper": 3,
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"weno_order": 5,
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"weno_eps": 1.0e-10,
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"null_weights": "F",
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"mp_weno": "F",
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"riemann_solver": 1,
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"wave_speeds": 1,
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"avg_state": 2,
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"bc_x%beg": -1,
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"bc_x%end": -1,
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"bc_y%beg": -1,
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"bc_y%end": -1,
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# Formatted Database Files Structure Parameters
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"format": 1,
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"precision": 2,
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"prim_vars_wrt": "T",
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"parallel_io": "T",
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# Background
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"patch_icpp(1)%geometry": 3,
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"patch_icpp(1)%hcid": 207,
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"patch_icpp(1)%x_centroid": 0.5,
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"patch_icpp(1)%y_centroid": 0.5,
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"patch_icpp(1)%length_x": 1.0,
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"patch_icpp(1)%length_y": 1.0,
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"patch_icpp(1)%vel(1)": -0.5,
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"patch_icpp(1)%vel(2)": 0.0,
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"patch_icpp(1)%pres": 2.5,
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"patch_icpp(1)%alpha_rho(1)": 1.0 - eps,
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"patch_icpp(1)%alpha(1)": 1.0 - eps,
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"patch_icpp(1)%alpha_rho(2)": eps,
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"patch_icpp(1)%alpha(2)": eps,
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# Center Strip (0.25 < y <= 0.75)
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"patch_icpp(2)%geometry": 3,
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"patch_icpp(2)%alter_patch(1)": "T",
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"patch_icpp(2)%hcid": 207,
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"patch_icpp(2)%x_centroid": 0.5,
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"patch_icpp(2)%y_centroid": 0.5,
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"patch_icpp(2)%length_x": 1.0,
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"patch_icpp(2)%length_y": 0.5,
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"patch_icpp(2)%vel(1)": 0.5,
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"patch_icpp(2)%vel(2)": 0.0,
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"patch_icpp(2)%pres": 2.5,
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"patch_icpp(2)%alpha_rho(1)": 2.0 * eps,
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"patch_icpp(2)%alpha(1)": eps,
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"patch_icpp(2)%alpha_rho(2)": 2.0 * (1.0 - eps),
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"patch_icpp(2)%alpha(2)": 1.0 - eps,
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# Fluids Physical Parameters
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"fluid_pp(1)%gamma": 1.0e00 / (5.0 / 3.0 - 1.0e00),
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"fluid_pp(1)%pi_inf": 0.0e00,
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"fluid_pp(2)%gamma": 1.0e00 / (5.0 / 3.0 - 1.0e00),
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"fluid_pp(2)%pi_inf": 0.0e00,
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}
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)
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)
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‎examples/2D_kelvin_helmholtz/figure0.png‎

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‎examples/2D_kelvin_helmholtz/figure1.png‎

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‎examples/2D_richtmyer_meshkov/README.md‎

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# Richtmyer-Meshkov Instability (2D)
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Reference:
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> A.S. Chamarthi, S.H. Frankel, A. Chintagunta, Implicit gradients based novel finite volume scheme for compressible single and multi-component flows, arXiv preprint arXiv:2106.01738 (2021)., see Example 4.18
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**Key Parameters**
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* $\lambda = 1.0$
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* $Re = 1e4$
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**Initial Conditions**
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* **Spatial Domain:** $(x, y) \in [0, 16\lambda] \times [0, \lambda]$
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The domain is divided into three regions: Post-shock Air, Pre-shock Air, and SF6. The shock position is given by $x_{shock}=0.7\lambda$. The interface position $x_{int}(y)$ is defined by the sinusoidal perturbation:
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$$
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x_{int}(y) = \lambda \left[ 0.4 - 0.1 \sin\left( 2\pi \left( \frac{y}{\lambda} + 0.25 \right) \right) \right]
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$$
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The initial state is given by:
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$$
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(\rho, u, v, p, \gamma) =
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\begin{cases}
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(1.4112, 0.8787, 0, 1.6272/1.4, 1.4) & \text{if } x > x_{shock} \quad \text{(Post-shock Air)} \\
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(1.0, 1.24, 0, 1/1.4, 1.4) & \text{if } x_{int}(y) < x \leq x_{shock} \quad \text{(Pre-shock Air)} \\
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(5.04, 1.24, 0, 1/1.4, 1.093) & \text{if } x \leq x_{int}(y) \quad \text{(SF6)} \\
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\end{cases}
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$$
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The initial perturbation is then smoothed by
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$$f_{sm} = \frac{1}{2}\left(1 + \text{erf}\left(\frac{\Delta D}{E_i\sqrt{\Delta x \Delta y}}\right)\right)$$
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$$u = u_L(1 - f_{sm}) + u_Rf_{sm}$$
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where $u$ is each primitive variable. $E_i$ is a thickness constant, and $\Delta D$ is the signed horizontal distance from $x_{int}(y)$. $u_L$ and $u_R$ are left and right interface conditions.
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### Initial State
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<img src="figure0.png" height="MAX_HEIGHT"/>
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### Evolved State
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<img src="figure1.png" height="MAX_HEIGHT"/>

‎examples/2D_richtmyer_meshkov/case.py‎

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#!/usr/bin/env python3
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import json
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import math
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mu = 1.0e-4
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lambd = 1.0
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time_end = 15.0
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time_save = time_end / 20.0
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eps = 1.0e-6
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# Configuring case dictionary
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print(
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json.dumps(
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{
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# Logistics
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"run_time_info": "T",
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# Computational Domain Parameters
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"x_domain%beg": 0,
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"x_domain%end": 16.0 * lambd,
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"y_domain%beg": 0,
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"y_domain%end": lambd,
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"m": 4096,
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"n": 256,
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"p": 0,
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"cfl_adap_dt": "T",
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"cfl_target": 0.1,
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"n_start": 0,
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"t_stop": time_end,
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"t_save": time_save,
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# Simulation Algorithm Parameters
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"num_patches": 2,
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"model_eqns": 2,
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"alt_soundspeed": "F",
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"num_fluids": 2,
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"mpp_lim": "T",
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"mixture_err": "T",
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"recon_type": 1,
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"time_stepper": 3,
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"weno_order": 5,
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"weno_eps": 1.0e-9,
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"null_weights": "F",
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"mapped_weno": "T",
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"mp_weno": "T",
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"riemann_solver": 2,
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"wave_speeds": 1,
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"avg_state": 2,
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"bc_x%beg": -17,
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"bc_x%end": -17,
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"bc_y%beg": -15,
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"bc_y%end": -15,
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# Formatted Database Files Structure Parameters
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"format": 1,
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"precision": 2,
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"prim_vars_wrt": "T",
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"parallel_io": "T",
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# Fluid #1 = Heavier Fluid
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# Fluid #2 = Lighter Fluid
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# Pre Shock
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"patch_icpp(1)%geometry": 3,
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"patch_icpp(1)%hcid": 208,
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"patch_icpp(1)%x_centroid": 0.5 * 0.7 * lambd,
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"patch_icpp(1)%y_centroid": 0.5 * lambd,
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"patch_icpp(1)%length_x": 0.7 * lambd,
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"patch_icpp(1)%length_y": lambd,
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"patch_icpp(1)%vel(1)": 1.24,
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"patch_icpp(1)%vel(2)": 0.0,
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"patch_icpp(1)%pres": 1.0 / 1.4,
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"patch_icpp(1)%alpha_rho(1)": 1.0 * eps,
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"patch_icpp(1)%alpha(1)": eps,
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"patch_icpp(1)%alpha_rho(2)": 1.0 * (1.0 - eps),
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"patch_icpp(1)%alpha(2)": (1.0 - eps),
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# Post Shock
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"patch_icpp(2)%geometry": 3,
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"patch_icpp(2)%hcid": 208,
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"patch_icpp(2)%x_centroid": 0.7 * lambd + 0.5 * 15.3 * lambd,
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"patch_icpp(2)%y_centroid": 0.5 * lambd,
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"patch_icpp(2)%length_x": 15.3 * lambd,
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"patch_icpp(2)%length_y": lambd,
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"patch_icpp(2)%vel(1)": 0.8787,
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"patch_icpp(2)%vel(2)": 0.0,
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"patch_icpp(2)%pres": 1.6272 / 1.4,
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"patch_icpp(2)%alpha_rho(1)": 1.4112 * eps,
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"patch_icpp(2)%alpha(1)": eps,
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"patch_icpp(2)%alpha_rho(2)": 1.4112 * (1.0 - eps),
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"patch_icpp(2)%alpha(2)": (1.0 - eps),
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# Fluids Physical Parameters
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"fluid_pp(1)%gamma": 1.0e00 / (1.093 - 1.0e00),
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"fluid_pp(1)%pi_inf": 0.0e00,
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"fluid_pp(2)%gamma": 1.0e00 / (1.4 - 1.0e00),
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"fluid_pp(2)%pi_inf": 0.0e00,
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"viscous": "T",
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"fluid_pp(1)%Re(1)": 1 / mu,
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"fluid_pp(2)%Re(1)": 1 / mu,
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}
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)
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)
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‎examples/2D_richtmyer_meshkov/figure0.png‎

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‎examples/2D_richtmyer_meshkov/figure1.png‎

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‎examples/2D_viscous_shock_tube/README.md‎

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# Viscous Shock Tube (2D)
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Reference:
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> A.S. Chamarthi, S.H. Frankel, A. Chintagunta, Implicit gradients based novel finite volume scheme for compressible single and multi-component flows, arXiv preprint arXiv:2106.01738 (2021)., see Example 4.13
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**Key Parameters**
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* $\gamma = 7/5$
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* $Re = 1000$
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**Initial Conditions**
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* **Spatial Domain:** $(x, y) \in [0, 1] \times [0, 0.5]$
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$$
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(\rho, u, v, p) =
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\begin{cases}
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(120, 0, 0, \frac{120}{\gamma}) & \text{if } 0 < x < 0.5 \\
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(1.2, 0, 0, \frac{1.2}{\gamma}) & \text{if } 0.5 \leq x < 1
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\end{cases}
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$$
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### Initial State
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<img src="figure0.png" height="MAX_HEIGHT"/>
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### Evolved State
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<img src="figure1.png" height="MAX_HEIGHT"/>

‎examples/2D_viscous_shock_tube/case.py‎

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#!/usr/bin/env python3
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import json
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import math
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mu = 1.0e-3
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time_end = 1.2
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time_save = time_end / 100.0
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eps = 1.0e-6
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# Configuring case dictionary
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print(
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json.dumps(
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{
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# Logistics
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"run_time_info": "T",
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# Computational Domain Parameters
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"x_domain%beg": 0,
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"x_domain%end": 1.0,
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"y_domain%beg": 0,
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"y_domain%end": 0.5,
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"m": 1280,
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"n": 640,
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"p": 0,
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"cfl_adap_dt": "T",
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"cfl_target": 0.1,
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"n_start": 0,
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"t_stop": time_end,
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"t_save": time_save,
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# Simulation Algorithm Parameters
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"num_patches": 2,
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"model_eqns": 2,
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"alt_soundspeed": "F",
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"num_fluids": 2,
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"mpp_lim": "T",
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"mixture_err": "T",
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"time_stepper": 3,
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"weno_order": 5,
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"weno_eps": 1.0e-10,
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"null_weights": "F",
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"mapped_weno": "T",
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"mp_weno": "T",
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"riemann_solver": 1,
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"wave_speeds": 1,
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"avg_state": 2,
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"bc_x%beg": -8,
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"bc_x%end": -15,
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"bc_y%beg": -16,
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"bc_y%end": -15,
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# Formatted Database Files Structure Parameters
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"format": 1,
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"precision": 2,
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"prim_vars_wrt": "T",
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"parallel_io": "T",
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# Left
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"patch_icpp(1)%geometry": 3,
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"patch_icpp(1)%x_centroid": 0.5 * 0.5,
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"patch_icpp(1)%y_centroid": 0.5 * 0.5,
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"patch_icpp(1)%length_x": 0.5,
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"patch_icpp(1)%length_y": 0.5,
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"patch_icpp(1)%vel(1)": 0.0,
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"patch_icpp(1)%vel(2)": 0.0,
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"patch_icpp(1)%pres": 120.0 / (7.0 / 5.0),
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"patch_icpp(1)%alpha_rho(1)": 120.0 * (1 - eps),
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"patch_icpp(1)%alpha(1)": 1.0 * (1 - eps),
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"patch_icpp(1)%alpha_rho(2)": 120.0 * eps,
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"patch_icpp(1)%alpha(2)": 1.0 * eps,
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# Right
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"patch_icpp(2)%geometry": 3,
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"patch_icpp(2)%x_centroid": 0.5 + 0.5 * 0.5,
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"patch_icpp(2)%y_centroid": 0.5 * 0.5,
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"patch_icpp(2)%length_x": 0.5,
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"patch_icpp(2)%length_y": 0.5,
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"patch_icpp(2)%vel(1)": 0.0,
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"patch_icpp(2)%vel(2)": 0.0,
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"patch_icpp(2)%pres": 1.2 / (7.0 / 5.0),
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"patch_icpp(2)%alpha_rho(1)": 1.2 * eps,
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"patch_icpp(2)%alpha(1)": 1.0 * eps,
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"patch_icpp(2)%alpha_rho(2)": 1.2 * (1 - eps),
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"patch_icpp(2)%alpha(2)": 1.0 * (1 - eps),
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# Fluids Physical Parameters
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"viscous": "T",
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"fluid_pp(1)%gamma": 1.0e00 / ((7.0 / 5.0) - 1.0e00),
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"fluid_pp(1)%Re(1)": 1 / mu,
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"fluid_pp(1)%pi_inf": 0.0,
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"fluid_pp(2)%gamma": 1.0e00 / ((7.0 / 5.0) - 1.0e00),
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"fluid_pp(2)%Re(1)": 1 / mu,
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"fluid_pp(2)%pi_inf": 0.0,
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}
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)
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)
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