Validation of Implicit Algorithms for Unsteady Flows Including Moving and Deforming Grids

Abstract

Implicit subiteration algorithms for time-accurate Navier-Stokes flow solvers for moving body and deforming applications have been investigated. Several example calculations are computed to demonstrate the performance of these algorithms for solving unsteady Navier-Stokes problems. A set of relatively simple two-dimensional validation cases has been identified to assess the performance of unsteady CFD solvers. These cases demonstrate the advantages of second order time derivatives and subiterations for unsteady flow simulations. This investigation also indicates that the same level or a reduced level of numerical error relative to a baseline calculation using a smaller time step can be achieved with large time steps using subiterations when a highly convergent inner algorithm is used. here currently exists a great interest in performing calculations using Computational Fluid Dynamics (CFD) for high Reynolds number unsteady flows. Researchers are beginning to apply the new hybrid RANS/LES class of turbulence models to a host of unsteady high Reynolds flows containing large-scale coherent turbulent structures. Applications involving moving and deforming bodies are also becoming more common. Validation and verification of CFD codes becomes much more difficult for these unsteady flows than for traditional steady state CFD problems. Grid convergence studies do not address all of the relevant dimensions of a problem. Often refining the results in resolution of smaller scale structure in the unsteady flow, and hence a grid resolved solution does not exist except in the Direct Numerical Simulation (DNS) limit. In many cases convergence can only be judged in a statistical sense. The choice of time step can have a tremendous effect on the time-accuracy of a solution. The high frequency spectral regime will be under-resolved if the chosen time step is too large. Large time steps can introduce error in the solution if no means are provided to locally converge (i.e. convergence in both time and space at each time step) the solution. If the time step is too small a tremendous number of iterations will be required in order to adequately resolve the low frequency spectral regime. Hence the selection of a time step for a typical CFD problem is an exercise in the art of compromise and often requires some a priori knowledge of the unsteady nature of the flow. Much of the numerical technology for the solution of the Navier-Stokes equations over the last three decades has been focused on obtaining steady-state solutions. These algorithms generally provide large amounts of numerical dissipation in order to damp out spurious numerical fluctuations rapidly. Excessive numerical dissipation can cause the unsteady structures in the flow to be over-damped. This study focuses on subiteration strategies that allow for large time steps and local convergence at each step. Large time steps are useful in problems that require assembly or remeshing at each time step since it minimizes the number of these operations required for a simulation. This effort outlines the basic implicit numerical algorithms required for unsteady flow applications using large time steps including moving and deforming body applications. Simple two-dimensional example cases are provided that allow numerical algorithms to be assessed for unsteady flow applications. The test cases were chosen based on the availability of analytical solutions to the Navier-Stokes equations and/or a regular periodic behavior of the flow. This allows the user to evaluate the ability of a flow solver to provide a locally converged solution in time and to