Numerical solution of Navier-Stokes equations on unstructured grids

The new direct action scheme is elaborated for approximation of two-dimensional Navier-Stokes equations on high cell aspect ratio (CAR) grids. For convective and viscous terms this scheme produced the conventional cross-type scheme on triangulated rectangular grids which is well known to be the best for high CAR grids. The scheme approximates the additional integral representing the conservation of angular momentum. An accuracy of approximation of direct action scheme can be estimated by error functional specified at each grid point and obtained by Taylor expansion analysis. This functional estimates the quality of unstructured grid for direct action scheme. The matrix dissipation is used for dumping odd-even nonstability of centered scheme. It combine the accuracy of upwind schemes with robustness and efficiency of centered ones. The implicit Newton algorithms is preferable for highly stretched grids. The algorithm of automatic choice of Courant number was developed. This algorithm utilized the small values of Courant number at initial iteration steps, followed by infinite increase of this number at final iteration steps. The calculation of both laminar and turbulent flows around single and multielement airfoil were performed by the elaborated method with Baldwin-Lomax algebraic turbulence model incorporated in it. INTRODUCTION In the present work a consideration is given to a number of problems connected with numerical modeling of viscous flows in complex twodimensional regions. A choice of computational grid is the principal item arising in discretization of governing equations. For domains of simple geometry this problem is usually solved by employment of one-block structured grid. If the computational domain has a complex geometry then the domain may be divided into a number of simple subdomains followed by a generation of structured grids in each of them. Automation of such a routine is a rather difficult task. An alternative to this is an application of unstructured grid. The highly automated unstructured grid generation algorithms can be implemented for domains of any complexity and an adaptation of the grid to the numerical solution can be realized more easily. It must be noted that the problems of approximation on unstructured grids are not completely investigated yet. The crucial issue for numerical solution of Navier-Stokes equations is connected with an approximation on the grids clustered in one direction, with their CAR (cell aspect ratio) reaching the values of 10. Such grid CAR values are needed in the boundary layers and wakes, where the gradients of flow parameters in one direction exceed in many times the gradients values in the other direction, and the accuracy of numerical solution essentially depends on the accuracy attained in these regions. Most of the approximation methods impose constrains on the maximal angle in Copyright © 19% by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. triangular elements and this results in nearly structured grid in regions with high CAR. The centered cross-type scheme is well known to be the best for approximation of both the convective terms and Laplace-type terms on high CAR structured grids. Generally the conventional scheme based on median-dual control volumes is utilized in unstructured grid computations (see Earth [1]). This scheme provides the accuracy of at least the first order for approximation of Euler equations on any grids, but significantly differs from cross-type scheme on structured grids with high CAR. Consequently this work employs the so called direct action scheme, elaborated in [2]. The control volume in this scheme is similar to the one used in [3]. The direct action scheme produces the cross-type scheme for triangulated rectangular grids and approximates the additional integral, representing the law of conservation of angular momentum. Meanwhile, this scheme approximates the governing equations only on some particular class of grids. An approximation accuracy is evaluated by the error functional specified at every grid node. This functional is obtained by Tailor expansions analysis. The error functional estimates the approximation accuracy both for convective and viscous terms. It is shown in the present work that the error functional takes on zero values for broad classes of grids, for example, for triangulated structured orthogonal grids, grids consisted of equilateral triangles. The main requirements of developed computational technique to computational grid are the Delaunay property of American Institute of Aeronautics ans Astronautics tessellation and the sufficiently small values of error functional. These requirements become too stiff for grids with high CAR and even slight nodes movements significantly changes the values of error functional added by great change of grid structure produced by edge swapping. That's why the grid generation method must strictly control the grid quality in the regions with high CAR. Consequently in the present work the grid is constructed in two steps. At the first step the nearly orthogonal structured grid is constructed by advancing layers method near wakes and boundary layers. This method strictly controls the value of error functional but can't be applied in the whole region. At the second step the grid is smoothly continued inside the region. This method treats the domains of complex geometry but does not control the values of error functional, that is not of primary importance because the grid in these regions has low CAR. The approximation of convective terms utilizes the centered scheme. This scheme needs the incorporation of artificial dissipation for damping odd-even instability and oscillations near shock waves. Generally the scalar dissipation constructed as a combination of second and fourth derivatives is employed. The dissipation of this type as was shown in [4] needs very refined grids in boundary layer regions for sufficiently accurate representation on Navier-Stokes numerical solutions. The numerical scheme can be significantly improved by employment of the matrix dissipation instead of scalar one. The construction of matrix dissipation emanates from the ideas applied in upwind schemes. The matrix dissipation was developed for structured grid case in [5], and was generalized for unstructured grids in the present work. The implicit scheme? are preferable on grids with high CAR. This is connected with the fact that the stability requirements for explicit schemes constrain the time step by the value of a grid step. This results in a great number of time steps to attain the convergence. On the contrary, the Newton's method realized in implicit scheme provides the quadratic convergence. But quadratic convergence can be obtained only for initial guess close enough to the solution due to essential nonlinearity of the NavierStokes equations governing the flows with shock waves and/or well developed separation. Therefore, the starting procedure with not large CFL numbers at initial iterations is utilized, followed by increase of CFL numbers, thus tending to Newton iterations. The presented results of numerical modeling of flows with strong shock waves and developed separation demonstrate the robustness and effectiveness of elaborated algorithm. The solution of system of linear equations in this algorithms is performed by GMRES method (see [6]). This work demonstrates the numerical solution of Euler equations on isotropic and highly stretched grids. The laminar flows governed by Navier-Stokes equations for low Reynolds numbers are calculated over airfoil. Numerical modeling of flows over single and multielement airfoils by solution of Reynolds averaged Navier-Stokes equations with Baldwin-Lomax algebraic turbulence model [7] is presented as well. The boundary-value problem for Navier-Stokes equations. The parameters of two-dimensional flow of viscous gas meet the conservation laws of mass, energy and momentum that can be written in undimensioned form in Cartesian coordinate system x,y as