An accurate finite volume scheme for Euler and Navier-Stokes equations on unstructured adaptive grids

Delanaye, Michel; Essers, J. A.

doi:10.2514/6.1995-1710

Cited by 37 publications

(28 citation statements)

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“…The momentum equation, Equation (6), is often referred to as the stress-divergence form [28], which is the dominantly adopted form in FE-based flow solvers [5,26,27], due to the convenience of enforcing the natural boundary conditions in the weak form of Equation (6). Alternatively, the incompressible momentum equations can also be written as…”

Section: Governing Equationsmentioning

confidence: 99%

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A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

Wan

Aliabadi

Bigler

2009

Numerical Methods in Fluids

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SUMMARYWe have successfully extended our implicit hybrid finite element/volume (FE/FV) solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centered FV method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix-free FV method, as the one used for momentum equations, is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm that enforces the conservation of the mass for each fluid.

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Section: Governing Equationsmentioning

confidence: 99%

“…which is the simplified form by taking into account ∇ ·u = 0 in Equation (6). The momentum equation in the form of Equation (9) is more commonly used in non-FE-based flow solvers [2,3].…”

Section: Governing Equationsmentioning

confidence: 99%

A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

Wan

Aliabadi

Bigler

2009

Numerical Methods in Fluids

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“…One of the most common methods of gradient estimation is based on local Taylor series expansions and least squares. Such methods are not unique to meshless discretizations and have been applied in the context of finite volume gradient reconstruction [19,20]. The Taylor series method requires that for every point in the global domain to be solved, there exist a local cloud of points consisting of nearest neighbors.…”

Section: Traditional Weighted Least-squares Gradient Proceduresmentioning

confidence: 99%

Meshless Scheme Based on Alignment Constraints

Katz

Jameson

2010

AIAA Journal

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Development of a new meshless technique is described that presents certain advantages over conventional meshless schemes due to unique weights that force alignment of the derivative approximation. The method is completely meshless, yet retains the low storage and simple flux distribution technique that are characteristic of finite volume methods. The new method is based on the well-known Taylor series expansion method with least squares. The proposed weighting scheme significantly reduces storage and computational requirements compared to other meshless schemes. The scheme is shown to be analogous to finite volume schemes in many ways, but it still lacks a discrete telescoping property and is not discretely conservative. The method is applied to the Euler equations in two dimensions. Results are presented that agree well with established methods.

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“…Most of the unstructured adaptive schemes that have been proposed so far exploit h-reÿnement by directly splitting the elements (i.e. insertion of new points at the midpoints of edges) [1][2][3]. This approach has the advantage of being relatively straightforward to implement, but can often lead to an excessively large increase in the number of mesh points since each tetrahedron to be fully reÿned has new points inserted on each of its six edges.…”

Section: Introductionmentioning

confidence: 99%

Mesh deformation and modification for time dependent problems

Baker

2003

Numerical Methods in Fluids

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SUMMARYMesh coarsening and mesh reÿnement are combined to provide a exible approach for the adaptation of time dependent problems. When the shape of the domain remains ÿxed but the computed solution is unsteady (e.g. vortex shedding of laminar ow over a cylinder), successive applications of coarsening and reÿnement allow the computed solution to be tracked in an e cient manner. The addition of a mesh movement phase extends this technique to handle a domain whose shape is also evolving with time.

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An accurate finite volume scheme for Euler and Navier-Stokes equations on unstructured adaptive grids

Cited by 37 publications

References 0 publications

A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

Meshless Scheme Based on Alignment Constraints

Mesh deformation and modification for time dependent problems

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