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The use of Richardson extrapolation for the numerical solution of low Mach number flow in confined regions

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Abstract

We use artificial compressibility together with Richardson extrapolation in the Mach numberM as a method for solving the time dependent Navier-Stokes equation for very low Mach number flow and for incompressible flow. The question of what boundary conditions one should use for low Mach number flow, especially at inflow and outflow boundaries, is investigated theoretically, and boundary layer suppressing boundary conditions are derived. For the case of linearization around a constant flow we show that the low Mach number solution will converge with the rateO(M2) to the true incompressible solution, provided that we choose the boundary conditions correctly. The results of numerical calculations for the time dependent, nonlinear equations and for flow situations with time dependent inflow velocity profiles are presented. The convergence rateM 2 to incompressible solution is numerically confirmed. It is also shown that using Richardson extrapolation toM 2= 0 in order to derive a solution with very small divergence can with good result be carried through withM 2 as large as 0.1 and 0.05. As the time step in numerical methods must be chosen approximately such thatΔt · (i/(M Δx)−v/Δx 2) is in the stability region of the time stepping method, and asM 2=0.05 is sufficiently small to yield good results, the restriction on the time step due to the Mach number is not serious. Therefore the equations can be integrated very fast by explicit time stepping methods. This method for solving very low Mach number flow and incompressible flow is well suited to parallel processing.

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References

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Christer, B., Johansson, V. The use of Richardson extrapolation for the numerical solution of low Mach number flow in confined regions. J Sci Comput 8, 307–340 (1993). https://doi.org/10.1007/BF01061142

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Key words

  • Navier-Stokes equations
  • incompressible
  • low Mach number
  • Richardson extrapolation
  • boundary layer
  • boundary condition
  • boundary layer suppressing boundary condition
  • inflow
  • outflow
  • open boundary