A high order Spectral Volume method for moving boundary problems (original) (raw)

Accurate and efficient solutions of unsteady viscous flows

International Journal of Numerical Methods for Heat & Fluid Flow, 2001

This paper describes two accurate and efficient numerical methods for computing unsteady viscous flows. The first one solves the incompressible Navier‐Stokes equations in their vorticity‐velocity formulation, using a staggered‐grid finite‐volume spatial discretization to provide second‐order accuracy on arbitrary grids, and combines effectively an alternating direction implicit scheme for the vorticity transport equation and a multigrid line‐Gauss‐Seidel relaxation for the velocity equations. The second method solves the compressible Reynolds‐averaged Navier‐Stokes equations in strong conservation form, with a k−ω turbulence closure model. The equations are discretized in time using an implicit three‐time‐level scheme, combined with a dual time stepping approach, so that the residual at every physical time step is annihilated using an efficient multigrid Runge‐Kutta iteration with variable time stepping and implicit residual smoothing. The space discretization uses a Roe’s flux diff...

Numerical simulation of the turbulent boundary layer equations via a Runge‐Kutta algorithm

Numerical Methods for Partial Differential Equations, 1991

This paper addresses a hybrid computational procedure for the step‐by‐step calculation of momentum transfer in turbulent boundary layer flows along flat plates. The proposed procedure relies on a modified method of lines wherein transversal discretizations are carried out by a “control volume” being infinitesimal in the streamwise direction and finite in the transversal direction of the fluid flow. Using mixing length theory and coarse intervals in the transversal direction, the resulting system of ordinary differential equations of first order may be readily integrated on a personal computer utilizing a fourth‐order Runge‐Kutta algorithm. In general, a maximum number of sixteen lines is necessary at the trailing edge of the flat plate for a typical calculation. As a consequence, computing time and storage for each run were very small when compared to other finite‐difference methods. Furthermore, to validate the hybrid procedure involving the method of lines and control volumes (MOL...

A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow

Journal of Computational Physics, 1983

A new spectral method for solving the incompressible Navier-Stokes equations in a plane channel and between concentric cylinders is presented. The method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step. In addition, the number of dependent variables is reduced, resulting in a reduction in computer memory requirements. Several test problems have been computed for the channel flow and for flow between concentric cylinders, including Taylor-Couette flow with axisymmetric Taylor vortices and wavy vortices. In all cases, agreement with available experimental and theoretical results is very good.

Assessment of the Spectral Volume Method on Inviscid and Viscous Flows

Computational Fluid Dynamics 2010, 2011

The compact high-order 'Spectral Volume Method' (SVM, Wang ) designed for conservation laws on unstructured grids is presented. Its spectral reconstruction is exposed briefly and its applications to the Euler equations are presented through several test cases to assess its accuracy and stability. Comparisons with classical methods such as MUSCL show the superiority of SVM. The SVM method arises as a high-order accurate scheme, geometrically flexible and computationally efficient.

Spectral element approximation of the incompressible Navier-Stokes equations in a moving domain and applications

2009

In this thesis we address the numerical approximation of the incompressible Navier-Stokes equations evolving in a moving domain with the spectral element method and high order time integrators. First, we present the spectral element method and the basic tools to perform spectral discretizations of the Galerkin or Galerkin with Numerical Integration (G-NI) type. We cover a large range of possibilities regarding the reference elements, basis functions, interpolation points and quadrature points. In this approach, the integration and differentiation of the polynomial functions is done numerically through the help of suitable point sets. Regarding the differentiation, we present a detailed numerical study of which points should be used to attain better stability (among the choices we present). Second, we introduce the incompressible steady/unsteady Stokes and Navier-Stokes equations and their spectral approximation. In the unsteady case, we introduce a combination of Backward Differentiation Formulas and an extrapolation formula of the same order for the time integration. Once the equations are discretized, a linear system must be solved to obtain the approximate solution. In this context, we consider the solution of the whole system of equations combined with a block type preconditioner. The preconditioner is shown to be optimal in terms of number of iterations used by the GMRES method in the steady case, but not in the unsteady one. Another alternative presented is to use algebraic factorization methods of the Yosida type and decouple the calculation of velocity and pressure. A benchmark is also presented to access the numerical convergence properties of this type of methods in our context. Third, we extend the algorithms developed in the fixed domain case to the Arbitrary Lagrangian Eulerian framework. The issue of defining a high order ALE map is addressed. This allows to construct a computational domain that is described with curved elements. A benchmark using a direct method to solve the linear system or the Yosida-q methods is presented to show the convergence orders of the method proposed. Finally, we apply the developed method with an implicit fully coupled and semi-implicit approach, to solve a fluid-structure interaction problem for a simple 2D hemodynamics example.

An Implicit Solution of the Unsteady Navier-Stokes Equations on Unstructured Moving Grids

2004

An efficient dual-time implicit approach combined with the unstructured moving grids is presented for solution of the unsteady turbulent flows. Unstructured grids suitable for both inviscid and turbulent viscous flow regions are generated using a successive refinement method and the grid is moved adaptively based on the boundary movements. Special care is taken to maintain the quality of the grid near the surface. The unsteady two-dimensional compressible Navier-Stokes equations are discretised by an implicit approach in a real time basis. This approach allows the real time step to be chosen on the basis of accuracy rather than stability thus, enabling the use of large CFL numbers for computational efficiency. The resulting set of implicit non-linear equations is then solved iteratively in a pseudo-time using a RungeKutta scheme. The ε − k turbulence model equations are solved together with the main flow equations in a fully coupled manner. Results are presented for two unsteady tes...