Application of a Self-Adaptive Grid Method to Complex Flows

A directional-split, modular, user-friendly grid point distribution code is applied to several test problems. The code is self-adaptive in the sense that grid point spacing is determined by user-specified constants denoting maximum and minimum grid spacings and constants relating the relative influence of smoothness and orthogonality. Estimates of truncation error, in terms of flow-field gradients and/or geometric features, are used to determine the point distribution. Points are redistributed along grid lines in a specified direction in an elliptic manner over a user-specified subdomain, while orthogonality and smoothness are controlled in a parabolic (marching) manner in the remaining directions. Multidlrectional adaption is achieved by sequential application of the method in each coordinate direction. The flow-field solution is redistributed onto the newly distributed grid points after each unidirectional adaption by a simple one-dimensional interpolation scheme. For time-accurate schemes such interpolation is not necessary and time-dependent metrics are carried in the fluid dynamic equations to account for grid movement. THE METHOD The grid adaption method used herein is a pointwise redistribution of node points based on an approximation of the three-dimensional variational method of Brackbill and Saltzman (1982). Their variational method is analogous to minimizing the energy of a set of grid points which can be imagined as being connected by tension springs that are coaligned with the grid lines. The strengths of the springs are proportional to the gradients of the flow-field solution and are, in some sense, related to the discretization error of the solution. The Brackbill and Saltzman scheme is rigorous and robust but is computationally intensive for multidimensional flow fields. To make the method more efficient, Nakahashi and Deiwert (1984,1985) introduced the concept of directional splitting proposed by Gnoffo (1982) and Dwyer (1982). To keep the method truly multidimensional, torsion springs along the grid lines were introduced at each grid point to bring in the influence of the other directions. The strengths of these torsion springs were adjusted to control both the smoothness and the degree of orthogonality of the grid. To eliminate the problems of excessive computer time associated with multidimensional ellipticity, the torsion springs are constrained to have onesided influence. Hence, the grid points are redistributed elliptically in a single split direction according to tension springs while grid smoothness and orthogonality control are maintained in a one-sided manner in which spacewise marching techniques can be used. The redistribution of the flow-field solution is also simplified in that one-dimensional interpolation can be used to determine the solution on the redistributed I Eloret Institute. 2 Sterling Software. 3 NRC Fellow. 4 Boeing Military Airplane Co. set of points. Multidirectional adaption can be achieved by a sequence of split one-dimensional adaptions. The only significant penalty to using unidirectional splitting to achieve computational efficiency is the loss of direct control of computational cell volumes as a constraint on the variational scheme. The effect of constraining the influence of torsion springs to a single side should not be considered a serious penalty, since the additional precision realized by two-sided influence is within the uncertainty of optimal grid determination. The advantages of computational efficiency and ease of use outweigh this loss of precision. It should be noted, however, that the redistribution of grid points with a directionally split method and one-sided torsion control does not result in a unique solution. Rather the solution is dependent both on the order of directional splitting and on the direction of the one-sided torsion control. To make the method user friendly and robust, Nakahashi and Deiwert (1985) introduced the concept of "self adaption." In this approach the principal control parameters specified by the user are: 1) the minimum and maximum grid spacing in the split coordinate direction, 2) the magnitude of the torsional influence, and 3) the relative importance of grid smoothness and orthogonality. The effect of each of these parameters will be illustrated in this paper. Other control parameters, such as the adaption domain, the sequence of splitting for multidirectional adaption, and the direction for the one-sided torsion springs, are also user-specified but are less critical in the determination of an acceptable grid. CONTROL PARAMETERS The parameters A s_i, and A s,,,ax control the relative density of the redistributed points and are the maximum and minimum allowable grid spacings. They are input proportioned to the average spacing over the adapted domain, e.g., A 8,ha= 2.0 will constrain a converged redistribution to maintain a maximum grid spacing equal to twice the average step size. Note that because of additional constraints, such as smoothness and orthogonality, this control of grid spacing need not be precisely maintained. A torsion control parameter, ),, determines the torsion spring coefficient and controls the relative influence of the torsion parameter with respect to the tension parameter. Values for this parameter are required for each direction other than the principal split direction. A value of zero will turn off the torsion control, the effect of which may result in oscillatory grid lines which have been adapted to the "noise" in the solution. Large values of ), will diminish the influence of the flow-field gradients (tension spring control) and minimize alignment to high gradient regions. The direction of the torsion control is determined by C,t (whereas)_ determines magnitude). Values of Ct at or near zero favor orthogonality and values at or near unity favor smoothness. The default values of the principal control parameters set in the computer code are as follows: 1. As,hi, = 0.5 and A s,,,_ = 2.0 (0.01 < As,hi, < 1.0 and 1.0 < A s,na_ < 10.0 are typical ranges) 2. ), = 0.001 (where 0.00001 < ), < 0.5 is a typical range) 3. Ca=O.5(whereO<Ca< 1.0)