# A pressure correction algorithm for unsteady low and high Mach number flows

## Description

The research is about development of finite-volume algorithms for unsteady flows of compressible fluids, functioning equally well at low and high Mach numbers, allowing precise representation of acoustic propagation and convective transport. Difficulties come mainly for low Mach number flows, where the speed of propagation of pressure perturbations is much larger than the convective speed. The studied algorithms are iterative pressure-based with segregation of velocity and pressure updates and unsteady Rhie-Chow type interpolation for face velocity. The current status is that algorithms can be formulated for low and high Mach number flows and for flows with smooth variations and discontinuities. But the algorithm cannot be identical for all cases. The current research effort is on the formulation of a unique algorithm for all cases.

## Publications

- MOGUEN Y., KOUSKSOU T., BRUEL P., VIERENDEELS J., DICK E.: Pressure-velocity coupling allowing acoustic calculation in low Mach number flow. J. of Computational Physics, 231 (2012), 5522-5541.

Finite-volume second-order TVD space discretisation is done with co-located variables and AUSM+ flux vector splitting. The time discretisation is second-order backward differencing. The face velocity for all equations is derived from a time-consistent momentum interpolation of Rhie-Chow type applied to preliminary discretised momentum and continuity equations, obtained with face velocity and face pressure by the low Mach number formulation of AUSM+, called AUSM+-up, but without velocity and pressure diffusion terms. In the final discretisation, this face velocity is used in all equations, together with the previous expression for face pressure. The solution is obtained by pressure-correction derived from the energy equation. The time-consistent momentum interpolation algorithm is suitable for steady and unsteady low Mach number flows with smooth variation of flow parameters. Good performance is demonstrated for propagation of acoustic waves in low Mach number flows.

- MOGUEN Y., DICK E., VIERENDEELS J., BRUEL P.: Pressure-velocity coupling for unsteady low Mach number flow simulations: An improvement of the AUSM+-up scheme. J. of Computational and Applied Mathematics, 246 (2013), 136-143.

A time-step dependent modification of the coefficient of the pressure diffusion term in the face velocity expression of the AUSM+-up scheme is derived based on the Mach number scaling of the time-consistent momentum interpolation method. The face pressure is still the AUSM+-up expression without the velocity diffusion term. The performance for propagation of acoustic waves in low Mach number flows is much improved but does not reach fully that of the momentum interpolation method.

- MOGUEN Y., BRUEL P., DICK E.: Semi-implicit characteristic-based boundary treatment for acoustics in low Mach number flows. J. of Computational Physics, 255 (2013), 339-361.

Time-consistent, partially non-reflecting, boundary conditions of the time-consistent momentum interpolation method are explained.

- MOGUEN Y., DELMAS S., PERRIER V., BRUEL P., DICK E.: Godunov-type schemes with an inertia term for unsteady full Mach number range flow calculations. J. of Computational Physics, 281 (2015), 556-590.

An inertia term is added to the face velocity expression of the AUSM+-up algorithm, similar to the inertia term in the time-consistent momentum interpolation method. This face velocity is combined with the AUSM+-up face pressure, but without velocity diffusion term. The obtained algorithm has the same low Mach number scaling as the momentum interpolation method and it functions equally well for propagation of acoustic waves in low Mach number flows. The algorithm functions also well for low Mach number Riemann problems, for which the momentum interpolation method produces strong oscillations around discontinuities. For such problems, the AUSM+-up algorithm functions also well, while the AUSM+ algorithm produces oscillations. For high Mach number Riemann problems, the algorithm produces sometimes a slight sonic glitch in expansion fans and oscillations around shocks. The same deficiencies are observed with the AUSM+-up scheme. So, none of the AUSM+, AUSM+-up and time-consistent momentum interpolation methods functions equally well for low and high Mach number flows and for flows with smooth and discontinuous flow parameter changes.

- MOGUEN Y., BRUEL P., DICK E.: Solving low Mach number Riemann problems by a momentum interpolation method. J. of Computational Physics, 289 (2015), 741-746.

It is demonstrated that the time-consistent momentum interpolation method may produce oscillations in low Mach number Riemann problems. A simple repair is replacing the interpolation of the momentum terms by interpolation of velocities. This makes the method similar to the AUSM+-up algorithm with an added inertia term. But, the resulting algorithm does not function well for high Mach number flows.