A Preconditioned implicit Roe's scheme for barotropic flows: towards simulation of cavitation phenomena

The discretisation of the Euler equations for a barotropic state law is considered. An upwind scheme based on the definition of a Roe's type matrix is first obtained for this particular hyperbolic problem. A low Mach number asymptotic study is performed both in the continuous and discrete case showing that the discrete solution admits pressure fluctuations in space much larger than those of the exact one. This is the same kind of behaviour observed for the case of a polytropic state law. A preconditioning is then applied such that the obtained discrete formulation has an asymptotic behaviour in agreement with the continuous case. A linearised implicit scheme is defined using the properties of the Roe matrix instead of the first-order homogeneity of the flux function which is not satisfied here. The implicit formulation is also extended to the preconditioned scheme. All the proposed ingredients are validated in the case of a quasi 1-D nozzle flow of a cavitating liquid

Upwind stabilization of Navier-Stokes solvers

We present a study of the effect of upwinding on stabilisation of both advective and pressure terms in a family of primitive-variable Navier-Stokes solvers. We consider two MUSCL schemes, the first one applies to compressible flow, the second one to incompressible flow. We illustrate the fact that both numerical models suffer oscillations if a minimal (but not large) amount of upwinding is not associated with acoustics, while advection can be stabilized by the physical diffusion terms when the mesh Reynolds number is small enough

Extract-gradient shape optimization of a 2D Euler flow

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Multilevel optimization : application to shape optimum design with a one-shot method

Gradient method is applied to the optimal control of a system for which each simulation is expensive. Instead of solving completely the flow equation as a state equation in the shape optimization, we use a "one-shot method" which solves simultaneously the system optimality. It is tested for the problem of shape optimization of a nozzle in a 2D Euler flow

A Hierarchical approach for shape optimisation

We consider the gradient method applied to the optimal control of a system for which each simulation is expensive. A method for increasing the number of uknowns, and relying on multilevel ideas is tested for the academic problem os shape optimization of a nozzle in a subsonic or transonic Euler flow

Linearised implicit time-advancing applied to sediment transport simulations

The numerical simulation of sediment transport problems is considered.The problem is modeled through the shallow-water equations coupled with the Exner equation to describe the time evolution of the bed prole. The Grass model is used for the sediment transport. The governing equations arediscretized by using two different finite-volume methods, the SRHN predictorcorrector scheme and a Modied Roe scheme for non conservative systems of equations. As for the time advancing, starting from the explicit versions, linearised implicit schemes are generated, in which the fl ux Jacobians are computed through automatic dierentiation. This allows the complexity of the analytical differentiation of the numerical schemes to be avoided. Second-order accuracy in space and time is obtained through MUSCL reconstruction together with a defect-correction approach. Finally the considered numerical ingredients are compared in terms of accuracy and computational time using different one dimensional and two dimensional sediment transport problems, characterised by different time scales for the evolution of the bed and of the water flow

A numerical study of non-cavitating and cavitating liquid flow around a hydrofoil

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

Upwind Stabilization Of Navier-Stokes Solvers

: We present a study of the effect of upwinding on stabilisation of both advective and pressure terms in a family of primitive-variable Navier-Stokes solvers. We consider two MUSCL Schemes, the first one applies to compressible flow, the second one to incompressible flow. We illustrate the fact that both numerical models suffer oscillations if a minimal (but not large) amount of upwinding is not associated with acoustics, while advection can be stabilized by the physical diffusion terms when the mesh Reynolds number is small enough. RESUME : On pr'esente une 'etude de l'effet du d'ecentrage sur la stabilisation des termes advectifs et de pression d'une famille de solveurs de Navier Stokes en variables primitives. On consid`ere deux sch'emas MUSCL, l'un s'appliquant `a des 'ecoulements compressibles, l'autre `a des 'ecoulements incompressibles. On illustre le fait que les deux sch'emas pr'esentent des oscillations si une quantit'e minimale de d'ecentrage n'est pas appliqu'ee aux terme..