Rescaling of the Roe scheme in low Mach-number flow regions

Computing and Visualization in Science, 2000

The low Mach number regime is characterized by a large discrepancy between the flow velocity and the speed of sound, leading to physical effects on different length scales and of different orders of magnitude. A single time scale, multiple space scale asymptotic analysis provides detailed insight into the limit behavior of solutions of the compressible Euler equations as the Mach number tends to zero. This analysis shows that "the pressure" splits up into three parts with different physical meanings. This knowledge is then used to develop a numerical scheme including multiple pressure variables to account for the different effects. The numerical method is a semi-implicit predictor-corrector algorithm. In the predictor step, the asymptotic equations are used to guess the global and large scale effects. Then the corrector step can be viewed as an incompressible solver with compressibility effects acting as source terms.

Journal of Computational Physics, 2013

For low Mach number flow calculation, when acoustic waves have to be captured, semiimplicit methods allow to avoid the time-step limitation that arises when explicit schemes are used. A method is suggested to solve the boundary equations so that the semi-implicitness of the algorithm is maintained, as well as its pressure-velocity coupling. This method is studied theoretically and numerically, in the low Mach number regime. Partially nonreflective characteristic-based boundary conditions, with the linear relaxation form suggested by Rudy and Strikwerda, [J. Comput. Phys. 36:55-70, 1980], are considered. It is shown that their properties, well known in the framework of explicit schemes, are recovered with the proposed semi-implicit treatment and an acoustic CFL number significantly larger than unity.

Journal of Computational and Applied Mathematics, 2018

The method presented below focuses on the numerical approximation of the Euler compressible system. It pursues a twofold objective: being able to accurately follow slow material waves as well as strong shock waves in the context of low Mach number flows. The resulting implicit-explicit fractional step approach leans on a dynamic splitting designed to react to the time fluctuations of the maximal flow Mach number. When the latter rises suddenly, the IMEX scheme, so far driven by a material-wave Courant number, turn into a time-explicit approximate Riemann solver constrained by an acoustic-wave Courant number. It is also possible to enrich the dynamic splitting in order to capture high pressure jumps even when the flow Mach number is low. One-dimensional low Mach number test cases involving single or multiple waves confirm that the present approach is as accurate and efficient as an IMEX Lagrange-Projection method. Besides, numerical results suggest that the stability of the present method holds for any Mach number if the Courant number related to the convective subsystem arising from the splitting is of order unity.

The aim of this paper is to simulate low Mach number unsteady viscous flows using a density-based finite volumes solver. This kind of solver is known to encounter some difficulties to simulate low Mach number flows in which the density is almost constant. Convergence fails or accuracy decreases when the speed of sound is much greater than the fluid velocity. Local preconditioning methods intend to solve this problem by altering the time derivative terms of the Navier-Stokes equations in order to artificially modify the speed of sound, improving the convergence and accuracy in the case of low Mach number steady flows [1], [4].

Journal of Computational Physics, 2007

When the Mach number tends to zero the compressible Navier-Stokes equations converge to the incompressible Navier-Stokes equations, under the restrictions of constant density, constant temperature and no compression from the boundary. This is a singular limit in which the pressure of the compressible equations converges at leading order to a constant thermodynamic background pressure, while a hydrodynamic pressure term appears in the incompressible equations as a Lagrangian multiplier to establish the divergence-free condition for the velocity. In this paper we consider the more general case in which variable density, variable temperature and heat transfer are present, while the Mach number is small. We discuss first the limit equations for this case, when the Mach number tends to zero. The introduction of a pressure splitting into a thermodynamic and a hydrodynamic part allows the extension of numerical methods to the zero Mach number equations in these non-standard situations. The solution of these equations is then used as the state of expansion extending the expansion about incompressible flow proposed by Hardin and Pope [J.C. Hardin, D.S. Pope, An acoustic/ viscous splitting technique for computational aeroacoustics, Theor. Comput. Fluid Dyn. 6 (1995) 323-340]. The resulting linearized equations state a mathematical model for the generation and propagation of acoustic waves in this more general low Mach number regime and may be used within a hybrid aeroacoustic approach.

Inertia terms are inffoduced in a Godunoy-type scheme. The resulting scheme, called AUSM-IT, is designed as an extension of the AUSM+-up scheme, in order to allow full Mach number range calculations of unsteady flows. In line with the continuous asymptotic analysis, the AUSM-IT scheme satisfies the exact conservation of the disuete linear acoustic energy in the low Mach number limit. Its capability to propedy handle low Mach number unsteady flows, that may include acoustic wayes or discontinuities, is uumerically illustrated.

2002

A preconditioned solution scheme for the computation of compressible flow in turboma- chinery at arbitrary Mach numbers is presented. The preconditioning technique used is applied to a state-of-the-art explicit, time-marching Navier-Stokes code which originally was developed for compressible, high-speed turbomachinery applications. It combines the ideas of low Mach number preconditioning and artificial compressibility method into a unified approach where principally fluids with arbitrary equations of state can be simulated. As shown by the test cases presented, it allows the code to simulate flows eciently and accurately independent of the Mach number. A description of the Navier-Stokes equations for rotating coordinate systems, along with the solution scheme and the details of the preconditioning method is given. Since turbomachinery computations are often performed on truncated domains, the solution scheme should be used in conjunction with non-reflecting boundary conditions. A ch...

geraldine.fjfi.cvut.cz

The work deals with numerical solution of two compressible flows problems. Firstly authors considered steady transonic flows through DCA 8% cascade (Double Circular Arc symmetrical) for increasing upstream Mach numbers M ∞ ∈ (0.813; 1.13). The cascade flows were suggested in Institute of Thermomechanics by Mr. Dvořák and flows were investigated experimentally. The structure of flow seems to be very complicated. It is possible to observe subsonic and supersonic part, shock wave structure, interaction of shock wave and boundary layer, wake etc. We investigated these flows numerically using composite scheme in the form of finite volume method for governing system of Euler equations. These numerical results are compared to experimental data of IT CAS CZ using comparison of several regimes with increasing upstream Mach numbers. The second problem is an unsteady viscous flow with very low upstream Mach number M∞ ≈ 0.02 in a 2D channel with a moving part of solid wall as a function of time. The flow is described by the system of Navier-Stokes equations for compressible laminar flows. The problem is numerically solved by MacCormack finite volume scheme. Moved grid of quadrilateral cells is considered in the form of conservation laws using Arbitrary Lagrangian-Eulerian method.

2014

Inertia terms are inffoduced in a Godunoy-type scheme. The resulting scheme, called AUSM-IT, is designed as an extension of the AUSM+-up scheme, in order to allow full Mach number range calculations of unsteady flows. In line with the continuous asymptotic analysis, the AUSM-IT scheme satisfies the exact conservation of the disuete linear acoustic energy in the low Mach number limit. Its capability to propedy handle low Mach number unsteady flows, that may include acoustic wayes or discontinuities, is uumerically illustrated.

The problem of accurately computing low Mach number flows, with the specific intent of studying the interaction of sound waves with incompressible flow structures, such as concentrations of vorticity is considered. This is a multiple time (and/or space) scales problem, leading to various difficulties in the design of numerical methods. Concentration is on one of these difficulties - the development of boundary conditions at artificial boundaries which allow sound waves and vortices to radiate to the far field. Nonlinear model equations are derived based on assumptions about the scaling of the variables. Then these are linearized about a uniform flow and exact boundary conditions are systematically derived using transform methods. Finally, useful approximations to the exact conditions which are valid for small Mach number and small viscosity are computed.