This test case solves the isentropic injection flow in a square cavity at \( v=-0.01 ms^{-1} \). The aim of this test case is to validate the compressible Stokes equations, coupled with the energy equation and a perfect gas equation of state

Configurations

A square cavity of length \( L\!=\!1 mm \) is filled with air considered as a perfect gas ( \(R=287 JK^{-1}kg^{-1} \), \( \gamma=c_p/c_v=1.4 \)).

At initial time, the following thermodynamic state is imposed \((T_0,p_0,\rho_0)=(300 K,101325 Pa,\frac{p_0}{R T_0})\). A fluid in the same thermodynamic state as the cavity is injected from the top with a vertical velocity \(v_0 = -0.01 ms^{-1}\). The dimensionless parameters of the problem are respectively the initial Reynolds, Mach and Prandtl numbers \( Re_0=\rho_0 v_0 L / \mu_0 = 6.36 \ 10^{-1}\), \(Ma_0=u_0 /c_0 = 5.37 \ 10^{-4}\), \(Pr_0=c_p\mu_0 /\lambda_0 = 7.04 \ 10^{-1}\).

The analytical solution of the problem can be found from [1,2]. Under the Stokes hypothesis ( \(Re\leq1\)), the test case exhibits a vertical linear velocity field \(v_y = -v_0 y\), with a constant velocity divergence \( \nabla \cdot \mathbf{u} = -v_0 \). Considering our hypothesis, using the law of reversible adiabatic process and the perfet gas EoS, the thermodynamic solution of the problem starting at \(t_0=0\) reads to

\[ p = p_0 \exp(\gamma t v_0/L), \]

\[ T = T_0 \exp((\gamma-1) t v_0/L), \]

\[ \rho = \rho_0 \exp(t v_0/L ). \]

For velocity boundary conditions, left and right boundaries have slip conditions, top has a Dirichlet condition for injection \(\mathbf{u}_{top}=[0,-v_0]^T\) and bottom has a no-slip condition. For temperature boundary conditions, all the boundaries have homogeneous Neumann conditions.

Runtime parameters

Main parameters to run this test case are:

• advection term of the Navier-Stokes equations not activated (Stokes flow)
• incremental pressure correction method for subsonic compressible flows
• implicit second order scheme
• HYPRE or MUMPS solvers

Comments

Thermodynamic variables do not vary in space (0D benchmark) allowing temporal convergence study without any effect of spatial error (linear velocity).

Results

Next tables presents the temporal convergence study. Temporal second-order is achieved for pressure, density and temperature, for both \(L^2\) and \(L^∞\) norms. We do not present velocity errors in the table because, whatever the time step, the exact velocity is reached as expected with errors close to the resolution tolerance of linear systems ( \(10^{−14}\)). As the problem is 1D for velocity and 0D for the other variables, conclusions do not change whatever be the mesh size form \(8^2\) to \(128^2\). Let us note the significant variations in pressure, temperature and density, final values at time \(t_f = 0.1\) being \(3.0955 \ 10^5 Pa\), \(4.4755 \ 10^2 K\) and \(3.1988 kg m^{−3}\) , respectively.

References

[1] J.-P. Caltagirone. An alternative model of euler equations based on conservation of acceleration. Submited to Journal of Computational Physics, 2024.

[2] J. Jansen, S. Glockner, D. Sharma, A. Erriguible, Incremental pressure correction method for subsonic compressible flows, Submited to Journal of Computational Physics, 2024.