Semi-implicit extension of a godunov-type scheme based on low mach number asymptotics I: One-dimensional flow

Klein, Rupert

doi:10.1016/s0021-9991(95)90034-9

Cited by 287 publications

(323 citation statements)

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“…When dependencies on both x and ξ are relevant and when there are leading order density variations on the small scale, the long wave acoustic and the small scale quasi-incompressible flow modes do not decouple as we will show in this section (see also [12]). …”

Section: Scale Interactions For Variable Density Flowsmentioning

confidence: 69%

“…This is motivated by the physics of thermoacoustic refrigeration and of accelerating premixed flames in the first part of this paper. Then we review the single time scale/multiple length scale low Mach number asymptotic analysis from [12,15], which explicitly reveals the mathematics of this interaction: in the resulting multiscale model the zero Mach number variable density flow equations describe the small scale motions, linear acoustics represents the long waves, and there are specific coupling terms involving correlations between the small scale density and velocity fluctuations. In turn it follows that in the absence of small-scale leading order density variations the large scale linear acoustics entirely decouple from the small scale flow, at least on the time scales considered here.…”

Section: R Kleinmentioning

confidence: 99%

“…The first part of this paper reviews the single time scale/multiple length scale low Mach number asymptotic analysis by Klein (1995Klein ( , 2004. This theory explicitly reveals the interaction of small scale, quasi-incompressible variable density flows with long wave linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes.…”

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Multiple spatial scales in engineering and atmospheric low Mach number flows

Klein

2005

ESAIM: M2AN

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Abstract. The first part of this paper reviews the single time scale/multiple length scale low Mach number asymptotic analysis by Klein (1995Klein ( , 2004. This theory explicitly reveals the interaction of small scale, quasi-incompressible variable density flows with long wave linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a single spatial scale reproduce automatically the zero Mach number variable density flow equations for the small scales, and the linear acoustic equations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number of wellknown simplified equations of theoretical meteorology can be derived in a unified fashion directly from the three-dimensional compressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are, however, characterized by several singular perturbation parameters that appear in addition to the Mach number, and that are defined independently of any particular length or time scale associated with some specific flow phenomenon. These are the ratio of the centripetal acceleration due to the earth's rotation vs. the acceleration of gravity, and the ratio of the sound speed vs. the rotational velocity of points on the equator. To systematically incorporate these parameters in an asymptotic approach, we couple them with the square root of the Mach number in a particular distinguished so that we are left with a single small asymptotic expansion parameter, ε. Of course, more familiar parameters, such as the Rossby and Froude numbers may then be expressed in terms of ε as well. Next we consider a very general asymptotic ansatz involving multiple horizontal and vertical as well as multiple time scales. Various restrictions of the general ansatz to only one horizontal, one vertical, and one time scale lead directly to the family of simplified model equations mentioned above. Of course, the main purpose of the general multiple scales ansatz is to provide the means to derive true multiscale models which describe interactions between the various phenomena described by the members of the simplified model family. In this context we will summarize a recent systematic development of multiscale models for the tropics (with Majda).Mathematics Subject Classification. 34E13, 76B99, 76Q05, 86A10.

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Section: Scale Interactions For Variable Density Flowsmentioning

confidence: 69%

Section: R Kleinmentioning

confidence: 99%

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confidence: 99%

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Multiple spatial scales in engineering and atmospheric low Mach number flows

Klein

2005

ESAIM: M2AN

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“…The non-dimensional quantities in (6) Since the Mach number occurs as parameter in the equations, the flow variables density, velocity and pressure depend on the Mach number as well. Therefore, an asymptotic expansion of the form…”

Section: Asymptotic Analysis For Low Mach Number Flowsmentioning

confidence: 99%

“…With this transformation, the spatial derivatives become ∇f = ∇ x f + M∇ ξ f . This kind of expansion was introduced by Klein [6] and Meister [8].…”

Section: Asymptotic Analysis For Low Mach Number Flowsmentioning

confidence: 99%

Calculation of low Mach number acoustics: a comparison of MPV, EIF and linearized Euler equations

Roller¹,

Schwartzkopff²,

Fortenbach³

et al. 2005

ESAIM: M2AN

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Abstract. The calculation of sound generation and propagation in low Mach number flows requires serious reflections on the characteristics of the underlying equations. Although the compressible Euler/Navier-Stokes equations cover all effects, an approximation via standard compressible solvers does not have the ability to represent acoustic waves correctly. Therefore, different methods have been developed to deal with the problem. In this paper, three of them are considered and compared to each other. They are the Multiple Pressure Variables Approach (MPV), the Expansion about Incompressible Flow (EIF) and a coupling method via heterogeneous domain decomposition. In the latter approach, the non-linear Euler equations are used in a domain as small as possible to cover the sound generation, and the locally linearized Euler equations approximated with a high-order scheme are used in a second domain to deal with the sound propagation. Comparisons will be given in construction principles as well as implementational effort and computational costs on actual numerical examples.Mathematics Subject Classification. 41A60, 65M55, 76Q05.

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Improvements to compressible Euler methods for low-Mach number flows

Sabanca

Brenner

Alemdaroğlu

2000

Int. J. Numer. Meth. Fluids

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In the present study improvements to numerical algorithms for the solution of the compressible Euler equations at low Mach numbers are investigated. To solve flow problems for a wide range of Mach numbers, from the incompressible limit to supersonic speeds, preconditioning techniques are frequently employed. On the other hand, one can achieve the same aim by using a suitably modified acoustic damping method. The solution algorithm presently under consideration is based on Roe's approximate Riemann solver [Roe PL. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics 1981; 43: 357–372] for non‐structured meshes. The numerical flux functions are modified by using Turkel's preconditioning technique proposed by Viozat [Implicit upwind schemes for low Mach number compressible flows. INRIA, Rapport de Recherche No. 3084, January 1997] for compressible Euler equations and by using a modified acoustic damping of the stabilization term proposed in the present study. These methods allow the compressible Euler equations at low‐Mach number flows to be solved, and they are consistent in time. The efficiency and accuracy of the proposed modifications have been assessed by comparison with experimental data and other numerical results in the literature. Copyright © 2000 John Wiley & Sons, Ltd.

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Semi-implicit extension of a godunov-type scheme based on low mach number asymptotics I: One-dimensional flow

Cited by 287 publications

References 31 publications

Multiple spatial scales in engineering and atmospheric low Mach number flows

Multiple spatial scales in engineering and atmospheric low Mach number flows

Calculation of low Mach number acoustics: a comparison of MPV, EIF and linearized Euler equations

Improvements to compressible Euler methods for low-Mach number flows

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