Weighted Least Squares Kinetic Upwind Method Using Eigendirections (WLSKUM-ED)
Least Squares Kinetic Upwind Method (LSKUM), a grid free method based on kinetic schemes has been gaining popularity over the conventional CFD methods for computation of inviscid and viscous compressible ﬂows past complex conﬁgurations. The main reason for the growth of popularity of this method is its ability to work on any point distribution. The grid free methods do not require the grid for ﬂow simulation, which is an essential requirement for all other conventional CFD methods. However, they do require point distribution or a cloud of points. Point generation is relatively simple and less time consuming to generate as compared to grid generation. There are various methods for point generation like an advancing front method, a quadtree based point generation method, a structured grid generator, an unstructured grid generator or a combination of above, etc. One of the easiest ways of point generation around complex geometries is to overlap the simple point distributions generated around individual constituent parts of the complex geometry. The least squares grid free method has been successfully used to solve a large number of ﬂow problems over the years. However, it has been observed that some problems are still encountered while using this method on point distributions around complex conﬁgurations. Close analysis of the problems have revealed that bad connectivity of the nodes is the cause and this leads to bad connectivity related code divergence. The least squares (LS) grid free method called LSKUM involves discretization of the spatial derivatives using the least squares approach. The formulae for the spatial derivatives are obtained by minimizing the sum of the squares of the error, leading to a system of linear algebraic equations whose solution gives us the formulae for the spatial derivatives. The least squares matrix A for 1-D and 2-D cases respectively is given by (Refer PDF File for equation) The 1-D LS formula for the spatial derivatives is always well behaved in the sense that ∑∆xi2 can never become zero. In case of 2-D problems can arise. It is observed that the elements of the Ls matrix A are functions of the coordinate differentials of the nodes in the connectivity. The bad connectivity of a node thus can have an adverse effect on the nature of the LS matrices. There are various types of bad connectivities for a node like insufficient number of nodes in the connectivity, highly anisotropic distribution of nodes in the connectivity stencil, the nodes falling nearly on a line (or a plane in 3-D), etc. In case of multidimensions, the case of all nodes in a line will make the matrix A singular thereby making its inversion impossible. Also, an anisotropic distribution of nodes in the connectivity can make the matrix A highly illconditioned thus leading to either loss in accuracy or code divergence. To overcome this problem, the approach followed so far is to modify the connectivity by including more neighbours in the connectivity of the node. In this thesis, we have followed a diﬀerent approach of using weights to alter the nature of the LS matrix A. (Refer PDF File for equation) The weighted LS formulae for the spatial derivatives in 1-D and 2-D respectively are are all positive. So we ask a question : Can we reduce the multidimensional LS formula for the derivatives to the 1-D type formula and make use of the advantages of 1-D type formula in multidimensions? Taking a closer look at the LS matrices, we observe that these are real and symmetric matrices with real eigenvalues and a real and distinct set of eigenvectors. The eigenvectors of these matrices are orthogonal. Along the eigendirections, the corresponding LS formulae reduce to the 1-D type formulae. But a problem now arises in combining the eigendirections along with upwinding. Upwinding, which in LS is done by stencil splitting, is essential to provide stability to the numerical scheme. It involves choosing a direction for enforcing upwinding. The stencil is split along the chosen direction. But it is not necessary that the chosen direction is along one of the eigendirections of the split stencil. Thus in general we will not be able to use the 1-D type formulae along the chosen direction. This diﬃculty has been overcome by the use of weights leading to WLSKUM-ED (Weighted Least Squares Kinetic Upwind Method using Eigendirections). In WLSKUM-ED weights are suitably chosen so that a chosen direction becomes an eigendirection of A(w). As a result, the multi-dimensional LS formulae reduce to 1-D type formulae along the eigendirections. All the advantages of the 1-D LS formuale can thus be made use of even in multi-dimensions. A very simple and novel way to calculate the positive weights, utilizing the coordinate diﬀerentials of the neighbouring nodes in the connectivity in 2-D and 3-D, has been developed for the purpose. This method is based on the fact that the summations of the coordinate differentials are of diﬀerent signs (+ or -) in different quadrants or octants of the split stencil. It is shown that choice of suitable weights is equivalent to a suitable decomposition of vector space. The weights chosen either fully diagonalize the least squares matrix ie. decomposing the 3D vector space R3 as R3 = e1 + e2 + e3, where e1, e2and e3are the eigenvectors of A (w) or the weights make the chosen direction the eigendirection ie. decomposing the 3D vector space R3 as R3 = e1 + ( 2-D vector space R2). The positive weights not only prevent the denominator of the 1-D type LS formulae from going to zero, but also preserve the LED property of the least squares method. The WLSKUM-ED has been successfully applied to a large number of 2-D and 3-D test cases in various ﬂow regimes for a variety of point distributions ranging from a simple cloud generated from a structured grid generator (shock reﬂection problem in 2-D and the supersonic ﬂow past hemisphere in 3-D) to the multiple chimera clouds generated from multiple overlapping meshes (BI-NACA test case in 2-D and FAME cloud for M165 conﬁguration in 3-D) thus demonstrating the robustness of the WLSKUM-ED solver. It must be noted that the second order acccurate computations using this method have been performed without the use of the limiters in all the ﬂow regimes. No spurious oscillations and wiggles in the captured shocks have been observed, indicating the preservation of the LED property of the method even for 2ndorder accurate computations. The convergence acceleration of the WLSKUM-ED code has been achieved by the use of LUSGS method. The use of 1-D type formulae has simplified the application of LUSGS method in the grid-free framework. The advantage of the LUSGS method is that the evaluation and storage of the jacobian matrices can be eliminated by approximating the split flux jacobians in the implicit operator itself. Numerical results reveal the attainment of a speed up of four by using the LUSGS method as compared to the explicit time marching method. The 2-D WLSKUM-ED code has also been used to perform the internal ﬂow computations. The internal ﬂows are the ﬂows which are confined within the boundaries. The inflow and the outflow boundaries have a significant effect on these ﬂows. The accurate treatment of these boundary conditions is essential particularly if the ﬂow condition at the outflow boundary is subsonic or transonic. The Kinetic Periodic Boundary Condition (KPBC) which has been developed to enable the single-passage (SP) ﬂow computations to be performed in place of the multi-passage (MP) ﬂow computations, utilizes the moment method strategy. The state update formula for the points at the periodic boundaries is identical to the state update formula for the interior points and can be easily extended to second order accuracy like the interior points. Numerical results have shown the successful reproduction of the MP ﬂow computation results using the SP ﬂow computations by the use of KPBC. The inflow and the outflow boundary conditions at the respective boundaries have been enforced by the use of Kinetic Outer Boundary Condition (KOBC). These boundary conditions have been validated by performing the ﬂow computations for the 3rdtest case of the 4thstandard blade conﬁguration of the turbine blade. The numerical results show a good comparison with the experimental results.
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