# 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.

**Klein, Benedikt** (2015):

*A high-order Discontinuous Galerkin solver for incompressible and low-Mach number flows.*

Darmstadt, Technische Universität, [Ph.D. Thesis]

# Couette flow with temperature gradient

*by: Benedikt Klein*

For Couette flow with a cold temperature

applied to the lower wall and a hot temperature

to the upper wall, the analytical solution

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.

**Klein, Benedikt ; Müller, Björn ; Kummer, Florian ; Oberlack, Martin** (2016):

*A high-order Discontinuous Galerkin solver for low-Mach number flows.*

81, In: International Journal for Numerical Methods in Fluids, (8), S. 489-520, ISSN 02712091, [Online-Edition: http://dx.doi.org/10.1002/fld.4193],

[Article]

# 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=10^{2} to Ra=10^{7}.

**Klein, Benedikt ; Müller, Björn ; Kummer, Florian ; Oberlack, Martin** (2016):

*A high-order Discontinuous Galerkin solver for low-Mach number flows.*

81, In: International Journal for Numerical Methods in Fluids, (8), S. 489-520, ISSN 02712091, [Online-Edition: http://dx.doi.org/10.1002/fld.4193],

[Article]