Low-Mach equations

Low-Mach number flows

The solver for non-isothermal flows is based on the low-Mach number equations, i.e. the density is a function of the thermodynamic pressure (which is constant in space) and the temperature but does not depend on the hydrodynamic pressure.

A segregated approach based on the SIMPLE algorithm is used to solve the equations for the primitive variables in each time step. A mixed-order formulation is used for the spatial DG discretization, i.e. the polynomial order for the pressure is reduced by one compared to the order for the velocity and the temperature.

Convected density jump

by: Benedikt Klein

The first test case for the variable density solver is the convection of a smooth density jump, which can also be seen as a multiphase problem with a smooth interface. In this case, the density is a linear function of a conservative level set which is advected with the velocity of the flow field. A fairly high density ratio of 1000 is applied. The h-convergence study for the level set function presented below shows a convergence rate of k+1.

Couette flow with temperature gradient

by: Benedikt Klein

For Couette flow with a cold temperature

\[T_c\]

applied to the lower wall and a hot temperature

\[T_h\]

to the upper wall, the analytical solution

\[\begin{align*}& u = x_2,\\& p = -\frac{p_0}{Fr^2(T_h – T_c)} \ln\left((T_h – T_c) x_2 + T_c\right) + C,\\& T = (T_h – T_c) x_2 + T_c\end{align*}\]

can be derived, where constant viscosity and heat conductivity have been assumed. The h-convergence study shown below on successively refined Cartesian grids shows the high-order accuracy of the method.

Natural convection in a heated cavity

by: Benedikt Klein

The natural convection in a square cavity is a common benchmark test for low-Mach number solvers. A hot temperature is applied to the left wall and a cold temperature to the right wall, while the upper and lower domain boundaries are adiabatic walls. The equations for low-Mach number flows are solved on a rather coarse uniform Cartesian grid with 26 x 26 cells. Polynomial orders are 5 for velocity and temperature and 4 for pressure. Below contour plots for the temperature and the streamlines are shown for various Rayleigh numbers ranging from Ra=10² to Ra=10⁷.

heated_cavity_temperatures — Temperature contours for natural convection in a heated cavity. Results are calculated on Cartesian grid with 26 x 26 cells and polynomial orders of 5 for velocity and temperature and 4 for pressure.

heated_cavity_streamlines — Streamlines for natural convection in a heated cavity. Results are calculated on Cartesian grid with 26 x 26 cells and polynomial orders of 5 for velocity and temperature and 4 for pressure.

Low-Mach number flows

Convected density jump

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Couette flow with temperature gradient

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Natural convection in a heated cavity

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