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Unstructured Grid Smoothing for Turbomachinery Applications

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Unstructured Grid Smoothing for Turbomachinery Applications International Journal of Fluid Machinery and Systems DOI: http://dx.doi.org/10.5293/IJFMS.2017.10.4.421 Vol. 10, No. 4, October-December 2017 ISSN (Online): 1882-9554 Original Paper Unstructured Grid Smoothing for Turbomachinery Applications Mehdi Falsafioon1, Sina Arabi1, Ricardo Camarero1 and François Guibault2 1Department of Mechanical Engineering, École Polytechnique de Montréal CP 6079, succ. Centre-ville, Montréal, QC, H3C 3A7, Canada, [email protected], [email protected], [email protected] 2Department of Computer Engineering, École Polytechnique de Montréal CP 6079, succ. Centre-ville, Montréal, QC, H3C 3A7, Canada, [email protected] Abstract In the present study, two mesh smoothing techniques, Laplace and Winslow smoothing techniques, for unstructured grids on turbomachinery application are investigated. These operators are based on the solution of elliptic equations. In the first case, Laplace`s equations are solved using a barycentric averaging procedure. Solution of Winslow`s equations has been a challenging work for unstructured grids because of existence of cross derivative terms in the equations. This issue is addressed devising a local control volume. Both methods are compared using different grid quality criteria. Finally, these operators have been applied to turbomachinery configurations and the advantages and disadvantages are discussed. Keywords: Grid Smoothing, Unstructured Grid, Winslow`s Equation, Grid quality 1. Introduction Grid smoothing is a post-processing procedure designed to improve the mesh quality. This can be done with various techniques where the nodal coordinates are modified using an operator such as Laplace, Winslow smoothing [1], or the use of functionals [2]. These have been successfully used for structured meshes. However, elliptic smoothing has been challenging for unstructured grid generation due to the non-conservative form of these equations, as well as the lack of an implied unique computational domain. Knupp [3] applied unstructured Winslow mesh smoothing on unstructured quadrilateral meshes using a locally defined computational domain. Karman [4] and Sahasrabudhe [5] successfully applied unstructured smoothing by introducing a local computational domains called virtual control volumes. Masters [6] extended this to stretched viscous grids. Arabi et al. [7] addressed two major problems concerning the application of elliptic smoothing methods to unstructured meshes. The first is the non-conservative aspect, and the second addresses the mapping procedure by employing an explicit mapping. However, the method is not unique because the physical space is mapped to a square which can be arbitrary for a general unstructured mesh. The solution of the Winslow`s equation on an unstructured mesh was achieved by introducing a 9-point finite difference stencil around each node of the mesh to correctly handle the mixed derivatives. Following the work of Winslow [1] and Sahasrabudhe [5], in the current study, local computational space along with a finite volume scheme introduced by [7] is used to handle cross derivatives using an averaging process instead of augmented cells. The present article, is an extension to the study presented at the 27th IAHR Symposium on Hydraulic Machinery and Systems [8]. 2. Elliptic Smoothing Elliptic smoothing consists in solving a partial differential equations, where the dependent variables are the coordinates in physical space, (x,y), in terms of independent variables in computational space (ξ,η). Two such techniques will be investigated and compared. Laplacian smoothing is the simplest and consists in solving a Laplace operator for (x, y) 2 x 0 (1) 2 y 0 In this work, this will be carried out simply by a barycentric averaging procedure of the coordinates connected to each node of the mesh. Received October 5 2016; accepted for publication August 7 2017: Review conducted by Yoshinobu Tsujimoto. (Paper number O16030J) Corresponding author: Mehdi Falsafioon, [email protected] 421 n xi x j1 i n (2) n yi y j1 i n where n is the number of nodes around the specific node i of the mesh. This yields smooth meshes for most regular geometries, but invalid meshes can result for configurations with concave corners where large changes of curvature occur. The smoothing procedure can be improved with Winslow`s equations (eq. (1)) in computational space (ξ,η) given as, L (x) g11x 2 g12x g11x 0 (2) L (y) g11y 2 g12y g11y 0 where 2 2 g11 x y g12 x x y y (3) 2 2 g22 x y While Winslow`s operator guarantees continuum global mapping [3, 9], truncation errors can lead to folded meshes. In such instances, additional control is needed to adapt the mesh around the boundaries to insure the validity of the results, especially around discontinuous parts of the physical boundary. A method based on Taylor series expansion to solve this operator on equilateral triangles, where all the angles are equal to π/3, has been proposed in [3]. Another method is presented in [4], based on generating a virtual control volume in the physical domain locally around each node, as a local computational space with the same number of neighboring nodes as in physical space. Arabi et al. [7] compared two techniques for the solution of the functional operator [2] using finite volume and finite difference methods. The latter based on inserting a 9-point finite difference stencil presented several sharp corners. In the current work, Winslow`s equation is solved using a local computational space [4] combined with the averaging method presented by [7] for evaluating of cross- derivatives. 2.1 Computational Space Elliptic equations enforce a smoothness condition in physical space (x y) through the use of a regular computational domain. In structured meshes, the computational domain has the same topology as the physical space, and ideal spacing i.e δx = δy, thus effectively enforcing an equal spacing around each node. For an unstructured mesh, there is no such a universal computational domain that matches every unstructured topology; therefore, for each unstructured mesh a computational domain must be devised to match the topology of the physical mesh. A solution to this problem was proposed by Sahasrabudhe with the introduction of computational domains that are only defined locally [10]. For a stencil of elements surrounding a single node, a regular mesh can be defined by equally spacing the connected points on a unit circle. These stencils, called virtual control volumes, are locally defined and can be used to drive the solution of the smoothing equations. Thus formulated, smoothing by the use of elliptic operators consists in solving a distinct boundary value problem with dirichlet boundary conditions for each node. 2.2 Finite Volume Scheme The Winslow`s operator is in non-conservative form and the three coefficients, g11, g12 and g22 in eq. 2 are functions of the gradients of the dependent variables in the computational space. The coefficients are frozen during each iteration. Using a linearization procedure, the various terms of eq. 3 can be integrated separately over a control volume defined around each point of the mesh in computational space. The integration path for the application of Green’s theorem is formed by joining the centroid of each triangular element to the midpoints of its sides, as shown by the dashed lines in Fig. 1. The cell edges divide each triangular element into three equal areas which form non-overlapping contiguous control volumes associated with a node in the mesh. The hashed region in Fig. 1 indicates a control volume with a centroid node which is the storage location of all dependent variables. Using integral form of eq. 2 and linearizing this equation results in the following form 422 (a) (b) Fig. 1 Local mapping from physical to virtual computational space g x d 2g x d g x d 0 11 12 22 (4) g y d 2g y d g y d 0 11 12 22 Applying the divergence theorem to the second order derivative terms, xξξ and xηη, gives x d .F d where the components of function F(xξ , 0). Similarly, for the cross derivative terms, for xξη we have x d .Q d While mathematically xξη = xηξ for continuous functions, numerically these integration procedures yield different results depending on the order of the integration. In [4] these terms were calculated using a set of augmented cells attached to the control volumes. Therefore, the shape of virtual control volumes is different for the cross derivative terms. In [7], it was found that taking an average by splitting the cross derivatives in two components and applying the Green’s theorem in both directions avoids the generation of folded cells in the physical domain. In addition, this averaging procedure maintains the symmetry of the final mesh when the deformation of the physical boundary is symmetric. Ref. [7] concluded that for the cross derivative terms, applying Green’s theorem only for one component on each control volume side around node (ξi, ηi) yields a degenerated final mesh in most cases of geometries with severe boundary curvature variations. In other words, for arbitrary deformations in the (x, y) plane, the values of the calculated fluxes in (ξi, ηi) are dominated by the values from the cross derivatives terms. Moreover, taking only one component of the cross derivative term after applying the Green’s theorem, deflects the final mesh in one direction. Based on this, the term Q is computed in two manners. Using Q1 = (0, xη) and Q2 = (xξ, 0), gives Q as an average of these two 1 Q (Q Q ) 2 1 2 Integrating over the control volume and applying the divergence theorem for each dependent variable, for example ξ, gives .Q d F.nˆdS (5) The term on the RHS integral represents the net flux through the surface of the volume and, for the Winslow`s operator, can be evaluated as 1 g11 x nxdS 2g12 ( x nxdS x nydS) g22 x nydS 0 2 (6) 1 g11 y nxdS 2g12 ( y nxdS y nydS) g22 y nydS 0 2 These line integrals are approximated by a summation over the sides of a polygon forming a co-volume around each node.
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