# Computation of Low Mach Number Flow

## Background

The numerical solution of the compressible Euler and Navier-Stokes equations at low Mach numbers has important applications in aerodynamics, aeroacoustics and combustion.

## Asymptotic Expansion

A multiple time scale, single space scale asymptotic analysis of the compressible Navier-Stokes equations reveals the role of the zeroth-order global thermodynamic pressure, the first-order acoustic pressure and the second-order 'incompressible' pressure. My asymptotic analysis identifies the acoustic time change of the heat release rate as the dominant source of sound in low Mach number flow, e.g. in flames. If the flow equations are averaged over an acoustic wave period, the averaged velocity tensor describes the net acoustic effect on the averaged flow field.

## Perturbation Formulation

The perturbation form of the compressible flow equations allows to retain conservativity and all nonlinearities. Solving for the small changes of the conservative variables avoids the cancellation problem with the conservative variables at low Mach numbers at the expense of a more elaborate formulation. If the changes of the conservative variables with respect to a constant state are used, the extra effort with the perturbation form is small.

## Boundary Conditions

The boundary conditions have proved to be decisive for the accuracy of unsteady flow and the convergence of steady flow at low Mach numbers, because if the fast acoustic waves are reflected at a boundary, they will very quickly corrupt the interior flow field and thereby impair accuracy and convergence, respectively. Therefore, I introduced a simple implementation of non-reflecting boundary conditions and tested it for slow internal flow.

## Convergence Acceleration

In low Mach number flow, the Mach number can be considerably increased by artificially lowering the pressure and thereby decreasing the speed of sound. Since the time step of an explicit method in slow flow is proportional to the Mach number, the time step can be chosen much larger for the modified flow with higher Mach number and the steady state can be reached much faster. With that Mach transformation, Patrick Jenny could considerably reduce the long computing times for premixed laminar flames. If the algorithm is not implemented as a predictor-corrector method but in one step, I could show that the present convergence acceleration for steady state constitutes an efficient preconditioning of the time derivatives in low Mach number flow.

## References

Müller,B.: ``Computation of Compressible Low Mach Number Flow'', Habilitation Thesis, ETH Zürich, September 1996.
Müller,B.: "Low Mach Number Asymptotics of the Navier-Stokes Equations", Journal of Engineering Mathematics, Vol. 34, Nos. 1-2, July 1998, pp. 97-109.
Müller,B.: "Low Mach Number Asymptotics of the Navier-Stokes Equations and Numerical Implications", 30th Computational Fluid Dynamics Lecture Series, von Karman Institute for Fluid Dynamics, 8-12 March 1999. http://folk.ntnu.no/bmuller/vki_ls_cfd_1999_1.pdf
Sesterhenn,J., Müller,B., Thomann,H.: "On the Cancellation Problem in Calculating Compressible Low Mach Number Flows", Journal of Computational Physics, Vol. 151, No. 2, 20 May 1999, pp. 597-615.
Müller,B.: ``Influence of Inlet and Outlet Boundary Conditions in Internal Flow Computations at Low Mach Numbers'', in: Desideri,J.-A., Hirsch,C., Le Tallec,P., Pandolfi,M., Periaux,J. (Eds.), "Computational Fluid Dynamics `96", Proc. 3rd ECCOMAS CFD Conference, Paris, 9-13 Sept. 1996, John Wiley, Chichester, 1996, pp. 637-643.
Jenny,P., Müller,B.: ``Convergence Acceleration for Computing Steady-State Compressible Flow at Low Mach Numbers'', Computers & Fluids, Vol. 28, No. 8, pp. 951-972, November 1999.
Müller,B., Jenny, P.: "Rankine-Hugoniot-Riemann Solver and Convergence Acceleration for Low Speed Combustion", 30th Computational Fluid Dynamics Lecture Series, von Karman Institute for Fluid Dynamics, 8-12 March 1999. http://folk.ntnu.no/bmuller/vki_ls_cfd_1999_2.pdf
Müller,B., Jenny,P.: ``Improving the Reliability of Low Mach Number Flow Computations'', Proceedings of 8th International Symposium on Computational Fluid Dynamics, Bremen, 5-10 Sept. 1999, published in Computational Fluid Dynamics Journal, Vol. 9, No. 1-I, pp. 515-528, April 2001.

## Contact

Bernhard Müller
Department of Energy and Process Engineering,
NTNU, Kolbjĝrn Hejes vei 2, NO-7491 Trondheim, Norway
http://folk.ntnu.no/bmuller/