Comparison of NACA 0012 Laminar Flow Solutions: Structured and Unstructured Grid Methods

In this paper we consider the solution of the compressible Navier-Stokes equations for a class of laminar airfoil flows. The principal objective of this paper is to demonstrate that members of this class of laminar flows have steady-state solutions. These laminar airfoil flow cases are often used to evaluate accuracy, stability and convergence of numerical solution algorithms for the Navier-Stokes equations. In recent years, such flows have also been used as test cases for high-order numerical schemes. While generally consistent steady-state solutions have been obtained for these flows using higher order schemes, a number of results have been published with various solutions, including unsteady ones. We demonstrate with two different numerical methods and a range of meshes with a maximum density that exceeds 8 106 grid points that steady-state solutions are obtained. Furthermore, numerical evidence is presented that even when solving the equations with an unsteady algorithm, one obtains steady-state solutions

A Discontinuous Galerkin Method for Two-Dimensional Shock Wave Modeling

A numerical scheme based on discontinuous Galerkin method is proposed for the two-dimensional shallow water flows. The scheme is applied to model flows with shock waves. The form of shallow water equations that can eliminate numerical imbalance between flux term and source term and simplify computation is adopted here. The HLL approximate Riemann solver is employed to calculate the mass and momentum flux. A slope limiting procedure that is suitable for incompressible two-dimensional flows is presented. A simple method is adapted for flow over initially dry bed. A new formulation is introduced for modeling the net pressure force and gravity terms in discontinuous Galerkin method. To validate the scheme, numerical tests are performed to model steady and unsteady shock waves. Applications include circular dam break with shock, shock waves in channel contraction, and dam break in channel with 45∘ bend. Numerical results show that the scheme is accurate and efficient to model two-dimensional shallow water flows with shock waves

MODELING ONE- AND TWO-DIMENSIONAL SHALLOW WATER FLOWS WITH DISCONTINUOUS GALERKIN METHOD

Numerical models for one- and two-dimensional shallow water flows are developed using discontinuous Galerkin method. Formulation and characteristics of shallow water equations are discussed. The well-balanced property and wetting/drying treatment are provided in the numerical models. The shock-capturing property is achieved by the approximate Riemann solvers in the schemes. Effects of different approximate Riemann solvers are also investigated. The Total Variation Diminishing property is achieved by adoption of slope limiters. Different slope limiters and their effects are compared through numerical tests. Numerical tests are performed to validate the models. These tests include dam-break flows, hydraulic jump and shocks in channels, and flows in natural rivers. Results show that the numerical models developed in present work are robust, accurate, and efficient for modeling shallow water flows. The one-dimensional model shows that the area based slope limiter provided the best solution in natural channels. The slope limiter based on the water depth or water surface elevation performs progressively poorer as the cross-section shape deviates from rectangular. In the approximate Riemann solver, the wave speeds are based on the original form of the equations, although the pressure force and the gravity force terms are combined for solving the shallow water equations with discontinuous Galerkin method. The combined term is discretized, in one- and two-dimensional models, such that the stationarity property is preserved. Different wetting and drying procedures are evaluated for the one- and two-dimensional models. Analytical, laboratory, and field tests are conducted to verify the accuracy of the wetting and drying procedures