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.

Convected density jump
Density profile of a convected density jump on a Cartesian grid with 8 x 40 cells and polynomial degrees of 3, 3 and 2 for velocity, level set and pressure, respectively.
Convected density jump convergence
Spatial convergence study for level set function

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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.

low_mach_couette_u
low_mach_couette_v
low_mach_couette_p
low_mach_couette_T

Klein, Benedikt ; Müller, Björn ; Kummer, Florian ; Oberlack, Martin (2015):
A high-order Discontinuous Galerkin solver for low-Mach number flows.
In: International Journal for Numerical Methods in Fluids, ISSN 02712091,
[Online-Edition: http://dx.doi.org/10.1002/fld.4193],
[Article]

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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.

Klein, Benedikt ; Müller, Björn ; Kummer, Florian ; Oberlack, Martin (2015):
A high-order Discontinuous Galerkin solver for low-Mach number flows.
In: International Journal for Numerical Methods in Fluids, ISSN 02712091,
[Online-Edition: http://dx.doi.org/10.1002/fld.4193],
[Article]

go to TU-biblio search on ULB website