Development and validation of a SUPG finite element scheme for the compressible Navier–Stokes equations using a modified inviscid flux discretization

Kirk, Benjamin S.; Carey, G. F.

doi:10.1002/fld.1635

Cited by 40 publications

(17 citation statements)

References 31 publications

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“…The grouped variable discretization of F i,h is nonstandard, but has shown to improve some aspects of the numerical scheme [21]. This issue is also discussed in Reference [22], where it is pointed out that this approximation of the fluxes is compatible with a discrete Gauss divergence theorem and therefore conservative.…”

Section: Finite Element Formulationmentioning

confidence: 95%

“…The no-penetration boundary condition u·n = 0 holds on the cylinder surface and is enforced as a natural boundary condition that results upon integrating the convective term in (36) by parts. (This approach is described further in [21].) The importance of a continuous field was discussed in Section 4.…”

Section: Inviscid Flow Over a Cylindermentioning

confidence: 99%

“…Additional details regarding the important issues of boundary condition enforcement, transient discretization, and nonlinear treatment can be found in References [11,21].…”

Section: Finite Element Formulationmentioning

confidence: 99%

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Adiabatic shock capturing in perfect gas hypersonic flows

Kirk¹

2009

Numerical Methods in Fluids

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SUMMARYThis paper considers the streamline-upwind Petrov/Galerkin (SUPG) method applied to the compressible Euler and Navier-Stokes equations in conservation-variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, is briefly reviewed. Of particular interest is the behavior of the shock-capturing operator, which is required to regularize the scheme in the presence of strong, shock-induced gradients. A standard shock-capturing operator that has been widely used in previous studies by several authors is presented and discussed. Specific modifications are then made to this standard operator that is designed to produce a more physically consistent discretization in the presence of strong shock waves. The actual implementation of the term in a finite-dimensional approximation is also discussed. The behavior of the standard and modified scheme is then compared for several supersonic/hypersonic flows. The modified shock-capturing operator is found to preserve enthalpy in the inviscid portion of the flowfield substantially better than the standard operator.

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Section: Finite Element Formulationmentioning

confidence: 95%

Section: Inviscid Flow Over a Cylindermentioning

confidence: 99%

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Adiabatic shock capturing in perfect gas hypersonic flows

Kirk¹

2009

Numerical Methods in Fluids

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“…It involves the linearization of all the terms contributing to the residual vector R(Ū) (See Equation (27)). In the current work the linearization of the inviscid and viscous flux terms is handled through the implementation an inviscid flux Jacobian matrix A i and the diffusivity matrix K ij , presented in [23], whereas a hand-coded linearization is implemented for the streamline and transverse stabilization terms. Following this idea, the general form of the Jacobian matrix can be expressed as…”

Section: Solution Methodsmentioning

confidence: 99%

“…In Equation (23), τ = β |v|+vc is the the stabilization parameter and the stabilization matrices S and B i in Eq. (22) The Galerkin form of the discretized equations is obtained by making the weighting functions equal to the interpolation ones (W = N).…”

Section: Spatial Discretizationmentioning

confidence: 99%

An implicit stabilized finite element method for the compressible Navier–Stokes equations using finite calculus

Kouhi

Oñate

2015

Comput Mech

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A new implicit stabilized formulation for the numerical solution of the compressible NavierStokes equations is presented. The method is based on the Finite Calculus (FIC) scheme using the Galerkin finite element method (FEM) on triangular grids. Via the FIC formulation, two stabilization terms, called streamline term and transverse term, are added to the original conservation equations in the space-time domain. The non-linear system of equations resulting from the spatial discretization is solved implicitly using a damped Newton method benefiting from the exact Jacobian matrix. The matrix system is solved at each iteration with a preconditioned GMRES method. The efficiency of the proposed stabilization technique is checked out in the solution of 2D inviscid and laminar viscous flow problems where appropriate solutions are obtained especially near the boundary layer and shock waves. The work presented here can be considered as a follow up of a previous work of the authors [24]. In that paper, the stabilized Galerkin FEM based on the FIC formulation was derived for the Euler equations together with an explicit scheme. In the present paper, the extension of this work to the Navier-Stokes equations using an implicit scheme is presented.

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Fluid‐structure interaction simulations of venous valves: A monolithic ALE method for large structural displacements

Calandrini

Aulisa

2018

Numer Methods Biomed Eng

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Venous valves are bicuspidal valves that ensure that blood in veins only flows back to the heart. To prevent retrograde blood flow, the two intraluminal leaflets meet in the center of the vein and occlude the vessel. In fluid-structure interaction (FSI) simulations of venous valves, the large structural displacements may lead to mesh deteriorations and entanglements, causing instabilities of the solver and, consequently, the numerical solution to diverge. In this paper, we propose an arbitrary Lagrangian-Eulerian (ALE) scheme for FSI simulations designed to solve these instabilities. A monolithic formulation for the FSI problem is considered, and due to the complexity of the operators, the exact Jacobian matrix is evaluated using automatic differentiation. The method relies on the introduction of a staggered in time velocity to improve stability, and on fictitious springs to model the contact force of the valve leaflets. Because the large structural displacements may compromise the quality of the fluid mesh as well, a smoother fluid displacement, obtained with the introduction of a scaling factor that measures the distance of a fluid element from the valve leaflet tip, guarantees that there are no mesh entanglements in the fluid domain. To further improve stability, a streamline upwind Petrov-Galerkin (SUPG) method is employed. The proposed ALE scheme is applied to a two-dimensional (2D) model of a venous valve. The presented simulations show that the proposed method deals well with the large structural displacements of the problem, allowing a reconstruction of the valve behavior in both the opening and closing phase.

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Development and validation of a SUPG finite element scheme for the compressible Navier–Stokes equations using a modified inviscid flux discretization

Cited by 40 publications

References 31 publications

Adiabatic shock capturing in perfect gas hypersonic flows

Adiabatic shock capturing in perfect gas hypersonic flows

An implicit stabilized finite element method for the compressible Navier–Stokes equations using finite calculus

Fluid‐structure interaction simulations of venous valves: A monolithic ALE method for large structural displacements

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