Euler equations - Implicit schemes and boundary conditions

In the present work an algorithm for the numerical integration of the three-dimensional unsteady Euler equations in vibrating transonic compressor cascades is described. The equations are discretized in finite-volume formulation in a mobile grid using isoparametric brick elements. They are integrated in time using Runge-Kutta schemes. A thorough discussion of the boundary conditions used and of their influence on results is undertaken. The influence of grid refinement on computational results is examined. Unsteady convergence of results is discussed.

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The Limit $\alpha \to 0$ of the $\alpha$-Euler Equations in the Half-Plane with No-Slip Boundary Conditions and Vortex Sheet Initial Data

SIAM Journal on Mathematical Analysis ◽

10.1137/19m1303721 ◽

2020 ◽

Vol 52 (5) ◽

pp. 5257-5286

Author(s):

Adriana V. Busuioc ◽

Dragos Iftimie ◽

Milton D. Lopes Filho ◽

Helena J. Nussenzveig Lopes

Keyword(s):

Initial Data ◽

Boundary Conditions ◽

Euler Equations ◽

Half Plane ◽

Vortex Sheet ◽

Slip Boundary Conditions ◽

Slip Boundary

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Preconditioned characteristic boundary conditions for solution of the preconditioned Euler equations at low Mach number flows

Journal of Computational Physics ◽

10.1016/j.jcp.2012.01.040 ◽

2012 ◽

Vol 231 (12) ◽

pp. 4384-4402 ◽

Cited By ~ 11

Author(s):

Kazem Hejranfar ◽

Ramin Kamali-Moghadam

Keyword(s):

Mach Number ◽

Boundary Conditions ◽

Euler Equations ◽

Low Mach Number ◽

Characteristic Boundary

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Stability of Leapfrog Constant-Coefficients Semi-Implicit Schemes for the Fully Elastic System of Euler Equations: Case with Orography

Monthly Weather Review ◽

10.1175/mwr2907.1 ◽

2005 ◽

Vol 133 (5) ◽

pp. 1065-1075 ◽

Cited By ~ 11

Author(s):

P. Bénard ◽

J. Mašek ◽

P. Smolíková

Keyword(s):

Euler Equations ◽

Evolution Equations ◽

Momentum Equation ◽

Primitive Equations ◽

Prognostic Variables ◽

Flat Terrain ◽

Implicit Schemes ◽

Constant Coefficients ◽

The Stability ◽

The One

Abstract The stability of constant-coefficients semi-implicit schemes for the hydrostatic primitive equations and the fully elastic Euler equations in the presence of explicitly treated thermal residuals has been theoretically examined in the earlier literature, but only for the case of a flat terrain. This paper extends these analyses to a case in which an orography is present, in the shape of a uniform slope. It is shown, with mass-based coordinates, that for the Euler equations, the presence of a slope reduces furthermore the set of the prognostic variables that can be used in the vertical momentum equation to maintain the robustness of the scheme, compared to the case of a flat terrain. The situation appears to be similar for systems cast in mass-based and height-based vertical coordinates. Still for mass-based vertical coordinates, an optimal prognostic variable is proposed and is shown to result in a robustness similar to the one observed for the hydrostatic primitive equations system. The prognostic variables that lead to robust semi-implicit schemes share the property of having cumbersome evolution equations, and an alternative time treatment of some terms is then required to keep the evolution equation reasonably simple. This treatment is shown not to modify substantially the stability of the time scheme. This study finally indicates that with a pertinent choice for the prognostic variables, mass-based vertical coordinates are equally suitable as height-based coordinates for efficiently solving the compressible Euler equations system.

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On numerical errors in the boundary conditions of the Euler equations

Applied Mathematics and Computation ◽

10.1016/s0096-3003(01)00032-7 ◽

2002 ◽

Vol 128 (1) ◽

pp. 129-140 ◽

Cited By ~ 1

Author(s):

Lars Ferm ◽

Per Lötstedt

Keyword(s):

Boundary Conditions ◽

Euler Equations ◽

Numerical Errors

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One-Dimensional Unsteady Periodic Flow Model with Boundary Conditions Constrained by Differential Equations

Journal of Fluids Engineering ◽

10.1115/1.3130244 ◽

2009 ◽

Vol 131 (6) ◽

Cited By ~ 1

Author(s):

Nhan T. Nguyen

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Euler Equations ◽

Periodic Boundary ◽

Closed Loop ◽

Fluid Transport ◽

Periodic Boundary Conditions ◽

Transport Systems ◽

Variable Approach ◽

Upwind Method

This paper describes a modeling method for closed-loop unsteady fluid transport systems based on 1D unsteady Euler equations with nonlinear forced periodic boundary conditions. A significant feature of this model is the incorporation of dynamic constraints on the variables that control the transport process at the system boundaries as they often exist in many transport systems. These constraints result in a coupling of the Euler equations with a system of ordinary differential equations that model the dynamics of auxiliary processes connected to the transport system. Another important feature of the transport model is the use of a quasilinear form instead of the flux-conserved form. This form lends itself to modeling with measurable conserved fluid transport variables and represents an intermediate model between the primitive variable approach and the conserved variable approach. A wave-splitting finite-difference upwind method is presented as a numerical solution of the model. An iterative procedure is implemented to solve the nonlinear forced periodic boundary conditions prior to the time-marching procedure for the upwind method. A shock fitting method to handle transonic flow for the quasilinear form of the Euler equations is presented. A closed-loop wind tunnel is used for demonstration of the accuracy of this modeling method.

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