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In this survey paper,
we are concerned with the zero Mach number limit
for compressible viscous flows.
For the sake of (mathematical) simplicity,
we restrict ourselves to the case of barotropic
fluids and we
assume that the flow evolves in the whole space
or satisfies periodic boundary conditions. We focus on the case of ill-prepared data.
Hence highly oscillating acoustic waves
are likely to propagate through the fluid.
We nevertheless state
the convergence to the incompressible Navier-Stokes
equations when the Mach number ϵ goes to 0.
Besides, it is shown that the global existence for the limit equations
entails the global existence for the compressible model with
small ϵ. The reader is referred to [R. Danchin, Ann. Sci. Éc. Norm. Sup. (2002)] for the detailed proof in the whole space case,
and to [R. Danchin, Am. J. Math.124 (2002) 1153–1219] for the case of periodic boundary conditions.
The purpose of this work is to study an example of low Mach (Froude) number
compressible flows when the initial density (height) is almost equal to a
function depending on x.
This allows us to connect the viscous shallow water equation
and the viscous lake equations.
More precisely, we study this asymptotic with well prepared
data in a periodic domain looking at the influence of the variability of the
depth. The result concerns weak solutions.
In a second part, we discuss the general low Mach number limit for standard
compressible flows given in P.–L. Lions' book that means with constant
We propose a Diphasic Low Mach Number (DLMN) system for the modelling of diphasic flows without phase change at low Mach number, system which is an extension of the system proposed by Majda in [Center of Pure and Applied Mathematics, Berkeley, report No. 112] and [Combust. Sci. Tech.42 (1985) 185–205] for low Mach number combustion problems. This system is written for a priori any equations of state. Under minimal thermodynamic hypothesis which are satisfied by a large class of generalized van der Waals equations of state, we recover some natural properties related to the dilation and to the compression of bubbles. We also propose an entropic numerical scheme in Lagrangian coordinates when the geometry is monodimensional and when the two fluids are perfect gases. At last, we numerically show that the DLMN system may become ill-posed when the entropy of one of the two fluids is not a convex function.
Preconditioners for hyperbolic systems are numerical
artifacts to accelerate the convergence to a steady state.
In addition, the preconditioner should also be included in the
artificial viscosity or upwinding terms to improve the accuracy
of the steady state solution. For time dependent problems
we use a dual time stepping approach. The preconditioner
affects the convergence rate and the accuracy of the
subiterations within each physical time step. We consider
two types of local preconditioners:
Jacobi and low speed preconditioning.
We can express the algorithm in several sets of variables
while using only the conservation variables for the flux terms.
We compare the effect of these various variable sets
on the efficiency and accuracy of the scheme.
The first part of this paper reviews the single time scale/multiple
length scale low Mach number asymptotic analysis by Klein (1995, 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
well-known 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).
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.
The results of a workshop concerning the numerical
simulation of the liquid flow around a hydrofoil in non-cavitating and
cavitating conditions are presented. This workshop was part of the
conference “Mathematical and Numerical aspects of Low Mach Number
Flows” (2004) and was aimed to investigate the capabilities of
different compressible flow solvers for the low Mach number regime and for
flows in which incompressible and supersonic regions are
simultaneously present. Different physical models of cavitating
phenomena are also compared. The numerical results are validated
against experimental data.
This paper is devoted to the numerical simulation of wave
breaking. It presents the results of a numerical workshop that was
held during the conference LOMA04. The objective is to compare
several mathematical models (compressible or incompressible) and
associated numerical methods to compute the flow field during a
wave breaking over a reef. The methods will also be compared with
There are very few reference solutions in the literature on
non-Boussinesq natural convection flows. We propose here a test
case problem which extends the well-known De Vahl Davis
differentially heated square cavity problem to the case of large
temperature differences for which the Boussinesq approximation is
no longer valid. The paper is split in two parts: in this first
part, we propose as yet unpublished reference solutions for cases
characterized by a non-dimensional temperature difference of 0.6,
Ra = 106 (constant property and variable property cases) and
Ra = 107 (variable property case). These reference solutions were
produced after a first international workshop organized by CEA and
LIMSI in January 2000, in which the above authors volunteered to
produce accurate numerical solutions from which the present
reference solutions could be established.
In the second part of the paper, we compare the solutions produced
in the framework of the conference “Mathematical and numerical
aspects of low Mach number flows” organized by INRIA and MAB in
Porquerolles, June 2004, to the reference solutions described in
Part 1. We make some recommendations on how to produce good
quality solutions, and list a number of pitfalls to be avoided.