An adaptive-gridding solution method for the 2D unsteady Euler equations

Adaptive grid refinement is a technique to speed up the numerical solution of partial differential equations by starting these calculations on a coarse basic grid and refining this grid only there where the solution requires this, e.g. in areas with large gradients. This technique has already been used often, for both steady and unsteady problems. Here, a simple and efficient adaptive grid technique is proposed for the solution of systems of 2D unsteady hyperbolic conservation laws. The technique is applied to the Euler equations of gasdynamics. Extension to other conservation laws or to 3D is expected to be straightforward. A solution algorithm is presented that refines a rectangular basic grid by splitting coarse cells into four, as often as required, and merging these cells again afterwards. The small cells have a shorter time step too, so the grid is refined in space and time. The grid is adapted to the solution several times per coarse time step, therefore the total number of cells is kept low and a fast solution is ensured. The grid is stored in a simple data structure. All grid data are stored in 1D arrays and the grid geometry is determined with, per cell, five pointers to other cells: one `mother' pointer to the cell from which the cell was split and four `neighbour' pointers. The latter are arranged so, that all cells around the considered cell can be quickly found. To determine where the grid is refined, a refinement criterion is used. Three different refinement criteria are studied: one based on the first spatial derivative of the density, one on the second spatial derivative of the density and one on an estimate of the local truncation error, comparable to Richardson extrapolation. Especially the first-derivative $ho$ criterion gives good results. The algorithm is combined with a simple first-order accurate discretisation of the Euler equations, based on Osher's flux function, and tested. A second-order accurate discretisation of the Euler equations is presented that combines a second-order limited discretisation of the fluxes with the time derivatives of the Richtmyer scheme. This scheme can be easily combined with the adaptive-gridding algorithm. Stability is proved for CFL numbers below 0.25. For cells with different sizes, several interpolation techniques are developed, like the use of virtual cells for flux calculation. The scheme is tested with two standard test cases, the 1D Sod problem and the forward-facing step problem, known from the work of Woodward and Colella. The results show that the second-order scheme is more efficient than the first-order scheme. An accuracy, comparable with solutions on uniform grids is obtained, but with at least five times lower computational costs. Results from a last test problem, the shedding of vortices from a flat plate that is suddenly set into motion, confirm that the method can be used for different flow regimes and that it is very useful in practice for analysis of unsteady flow

Industrial Application of Adaptive Grid Refinement - The Case of FINE/Marine

This paper studies the introduction of adaptive grid refinement in the FINE/Marine simulation suite, in order to analyse the aspects of a mesh adaptation method which can dissuade, or encourage, industry users to adopt it

Efficient computation of steady water flow with waves

A surface-capturing model for steady water flow is presented. This volume-of-fluid model without reconstruction consists only of conservation laws, hence, it can be solved very efficiently. The model contains a high-accuracy compressive water surface discretization and turbulence; it is solved with a linear multigrid technique and defect correction. Results show that the model is accurate and the solver gives fast convergence

A surface capturing method for the efficient computation of steady water waves

A surface capturing method is developed for the computation of steady water–air flow with gravity. Fluxes are based on artificial compressibility and the method is solved with a multigrid technique and line Gauss–Seidel smoother. A test on a channel flow with a bottom bump shows the accuracy of the method and the efficiency of the multigrid solver

Five-equation model for compressible two-fluid flow

An interface-capturing, five-equation model for compressible two-fluid flow is presented, that is based on a consistent, physical model for the flow in the numerical transition layer. The flow model is conservative and pressure-oscillation free. Due to the absence of an interface model in the capturing technique, the implementation of the model in existing flow solvers is very simple. The flow equations are the bulk-fluid equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term, to account for the energy exchange between the fluids. The physical flow model enables the derivation of an exact expression for this source term, both in continuous and in discontinuous flow. The system is solved numerically with a limited second-order accurate finite-volume technique. Linde's HLL Riemann solver is used. This solver is simplified here and its combination with the second-order scheme is studied. When the solver is adapted to two-fluid flow, the source term in the flow equations is incorporated in the Riemann solver. Further, the total source term in the cells is integrated over each cell. Numerical tests are performed on 1D shock-tube problems and on 2D shock-bubble interactions. The results confirm that the method is pressure-oscillation free and show that shocks are captured sharply. Good agreement with known solutions is obtained. Two appendices show an approximate model for shocks in physical two-phase media and a theoretical study of the interaction of shocks with plane interfaces, which is used to analyse the shock-bubble interaction

Raffinement automatique de maillage pour la simulation des écoulements à surface libre

Nous présenterons une méthode de raffinement automatique de maillage pour la simulation des écoulements visqueux incompressibles, en calcul parallèle, sur des maillages non-structurés. La méthode est appliquée aux écoulements à surface libre stationnaires et non-stationnaires. Pour ces simulations, une forte augmentation de la qualité des solutions est obtenue

Accurate and efficient computation of steady water flow with surface waves and turbulence

A surface capturing method is developed for steady water-air flow with gravity. Second-order accuracy is obtained with flux limiting and turbulence is modeled with Menter's model. The model is solved efficiently with a combination of multigrid and defect correction. Results for two test cases confirm the efficiency and accuracy of the method

A multigrid method for the computation of steady water waves

An efficient method for the computation of steady water-air flow with gravity is presented. The method is designed for fast solution using multigrid, combined with a line Gauss-Seidel smoother. A capturing model is used for the water surface, with fluxes based on artificial compressibility. A test on a channel flow with a bottom bump shows the accuracy of the method and the efficiency of the multigrid solver