scholarly journals A 3D Chebyshev-Fourier algorithm for convection equations in low Mach number approximation

2009 ◽  
pp. 607-625
Author(s):  
Ouafa Bouloumou ◽  
Eric Serre ◽  
Jochen Fröhlich
Keyword(s):  
Mach Number ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Time Discretization ◽  
Navier Stokes ◽  
Working Fluid ◽  
Uzawa Algorithm ◽  
Low Mach Number ◽  
Preconditioned Iterative

A three-dimensional spectral method based on Chebyshev-Chebyshev-Fourier discretizations is presented in the framework of the low Mach number approximation of Navier-Stokes equations. The working fluid is assumed to be a perfect gas with constant thermodynamic properties. The generalized Stokes problem, which arises from the time discretization by a second-order semi-implicit scheme, is solved by a preconditioned iterative Uzawa algorithm. Several validation results are presented in the case of steady and unsteady flows. This model is also evaluated for natural convection flows with large density variations in the case of a tall differentially heated cavity of aspect ratio 8. It is found that on contrary to convection at small temperature differences (Boussinesq), the 2D unsteady solution at Ra = 3.4 x 105 is unstable to 3D perturbations.

scholarly journals Implicit MAC scheme for compressible Navier–Stokes equations: low Mach asymptotic error estimates

2020 ◽  
Author(s):  
David Maltese ◽  
Antonín Novotný
Keyword(s):  
Mach Number ◽  
Initial Data ◽  
Error Estimate ◽  
Strong Solution ◽  
Stokes Equations ◽  
Main Tool ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Mach Number ◽  
Mac Scheme

Abstract We investigate the error between any discrete solution of the implicit marker-and-cell (MAC) numerical scheme for compressible Navier–Stokes equations in the low Mach number regime and an exact strong solution of the incompressible Navier–Stokes equations. The main tool is the relative energy method suggested on the continuous level in Feireisl et al. (2012, Relative entropies, suitable weak solutions, and weak–strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech., 14, 717–730). Our approach highlights the fact that numerical and mathematical analyses are not two separate fields of mathematics. The result is achieved essentially by exploiting in detail the synergy of analytical and numerical methods. We get an unconditional error estimate in terms of explicitly determined positive powers of the space–time discretization parameters and Mach number in the case of well-prepared initial data and in terms of the boundedness of the error if the initial data are ill prepared. The multiplicative constant in the error estimate depends on a suitable norm of the strong solution but it is independent of the numerical solution itself (and of course, on the discretization parameters and the Mach number). This is the first proof that the MAC scheme is unconditionally and uniformly asymptotically stable in the low Mach number regime.

scholarly journals On the Low Mach Number Limit for Quantum Navier--Stokes Equations

2020 ◽  
Vol 52 (6) ◽  
pp. 6105-6139
Author(s):  
Paolo Antonelli ◽  
Lars Eric Hientzsch ◽  
Pierangelo Marcati
Keyword(s):  
Mach Number ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Mach Number ◽  
Low Mach Number Limit

scholarly journals Splitting methods for low Mach number Euler and Navier-Stokes equations

1989 ◽  
Vol 17 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Saul Abarbanel ◽  
Pravir Duth ◽  
David Gottlieb
Keyword(s):  
Mach Number ◽  
Stokes Equations ◽  
Splitting Methods ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Mach Number
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