A High- order Compact Finite Difference Solver for the Two-Dimensional Euler And Navier-Stroker Equations - Mawlodd, Mahmood Khalid (2004)
The objective of this study was to develop a high-order compact (HOC) finite difference solver for the two-dimensional Euler and Navier-Stokes equations. Before developing the solver, a detailed investigation was conducted for assessing the performance of the basic fourth-order compact central discretization schemes that are known as Hermitian or Pade schemes. Exact solutions of simple scalar model problems, including the one-dimensional viscous Burgers equation and two dimensional convection-diffusion equation were used to quantitatively establish the spatial convergence rate of these schemes. Examples of two-dimensional incompressible flow including the driven cavity and the flow past a backward facing step were used for qualitatively evaluating the accuracy of the discretizations. Resolution properties of the HOC and conventional schemes were demonstrated through Fourier analysis. Stability criteria for explicit integration of the convection-diffusion equation were derived using the on-Neumann method and validated.Due to aliasing errors associated with the central HOC schemes investigated. these were only used for the discretization of the viscous terms of the Navier-Stokes equations in developing the aimed solver. Dealiasing HOC methods were developed for the discretization of the Euler equations and the convective terms of the Navier- Stokes equations. Spatial discretization of the Euler equations was based on flux-vector splitting. A fifth-order compact upwind method with consistent boundary closures was developed for the Euler equations. Shock-capturing properties of the method were based on the idea of total variation diminishing (TVD). The accuracy, stability and shock capturing issues of the developed method were investigated through the solution of one-dimensional scalar conservation laws. Discretization of the convective flux terms of the Navier-Stokes equations was based on a hybrid flux-vector splitting, known as the advection upstream splitting method (AUSM), which combines the accuracy of flux-difference splitting and the robustness of flux-vector splitting. High-order accurate approximation to the derivatives was obtained by a fourth-order cell-centered compact scheme. The midpoint values of the staggered mesh were constructed using a fourth-order MUSCL (monotone upstream-centered scheme for conservation law) polynomial. Two temporal discretization methods were built into the developed solver. Explicit integration was performed using a multistage strong stability preserving (SSP) Runge-Kutta method for unsteady time-accurate flow problems. For steady state flows an implicit method using the lower-upper (LU) factorization scheme with local time stepping convergence accelerator was employed. An advanced two-equation turbulence model, known as k-o shear-stress-transport (SST), model has also been incorporated in the solver for computing turbulent flows. A wide variety of test problems in unsteady and steady state were solved to demonstrate the accuracy, robustness and the capability to preserve positivity of the developed solver. Although the main solver was developed for two-dimensional problems, a one-dimensional version of it has been used to solve some interesting and challenging one-dimensional test problems as well. The test problems considered contain various types of discontinuities such as shock waves, rarefaction waves and contact surfaces and complicated wave interaction phenomena. Quantitative and qualitative comparisons with exact solutions, other numerical results or experimental data, whichever is available, are presented. The tests and comparisons conducted have shown that the developed HOC methods and the solver are high-order accurate and reliable as an application CFD code for two-dimensional compressible flows and conducting further research. A number of avenues for further research are identified and proposed for future extension and improvement of the solver