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The multigrid method is applied to the calculation of transonic, potential flow fields about arbitrary three dimensional wing-body combinations. Numerical results for iterative convergence rate are in good agreement with those predicted by a local mode analysis and show substantial improvement over conventional relaxation algorithms.
A method for obtaining the fine detail of a transonic flowfield is presented. The technique employs the multigrid method to embed very dense meshes in regions of interest. Accurate results are obtained on meshes of a heretofore unobtainable density with reasonable computer expenditures. Comparisons of results with data reveal accurate predictions in the supersonic bubble of a transonic inlet, an area which is incorrectly predicted by existing techniques. More accurate results are also obtained with the new method on a mesh of a density comparable to existing codes and at a lower cost.
Relaxation schemes based on approximate and incomplete factorization technique (AF) are described. The AF schemes allow construction of a fast multigrid method for solving integral equations of the second and first kind. The smoothing factors for integral equations of the first kind, and comparison with similar results from the second kind of equations are a novel item. Application of the MD algorithm shows convergence to the level of truncation error of a second order accurate panel method.
A detailed description of spectral multigrid methods is provided. This includes the interpolation and coarse-grid operators for both periodic and Dirichlet problems. The spectral methods for periodic problems use Fourier series and those for Dirichlet problems are based upon Chebyshev polynomials. An improved preconditioning for Dirichlet problems is given. Numerical examples and practical advice are included.
The systems of algebraic equations which arise from spectral discretizations of elliptic equations are full and direct solutions of them are rarely feasible. Iterative methods are an attractive alternative because Fourier transform techniques enable the discrete matrix-vector products to be computed with nearly the same efficiency as is possible for corresponding but sparse finite difference discretizations. For realistic Dirichlet problems preconditioning is essential for acceptable convergence rates. A brief description of Chebyshev spectral approximations and spectral multigrid methods for elliptic problems is given. A survey of preconditioners for Dirichlet problems based on second-order finite difference methods is made. New preconditioning techniques based on higher order finite differences and on the spectral matrix itself are presented. The preconditioners are analyzed in terms of their spectra and numerical examples are presented.
Papers presented in this volume provide an overview of recent work on numerical boundary condition procedures and multigrid methods. The topics discussed include implicit boundary conditions for the solution of the parabolized Navier-Stokes equations for supersonic flows; far field boundary conditions for compressible flows; and influence of boundary approximations and conditions on finite-difference solutions. Papers are also presented on fully implicit shock tracking and on the stability of two-dimensional hyperbolic initial boundary value problems for explicit and implicit schemes.
The solution by multigrid techniques of the steady inviscid compressible equations of gas dynamics, the Euler equations is investigated. Steady two dimensional transonic flow over an airfoil section is studied intensively. Most of the material is applicable to three dimensional flow problems of aerodynamic interest.
An iterative algorithm for the Helmholtz equation is presented. This scheme was based on the preconditioned conjugate gradient method for the normal equations. The preconditioning is one cycle of a multigrid method for the discrete Laplacian. The smoothing algorithm is red-black Gauss-Seidel and is constructed so it is a symmetric operator. The total number of iterations needed by the algorithm is independent of h. By varying the number of grids, the number of iterations depends only weakly on k when k(3)h(2) is constant. Comparisons with a SSOR preconditioner are presented.
A multigrid scheme for solving the Euler equations is presented. The method has been successfully applied to two-dimensional airfoil calculations on both O-type and C-type meshes. In three dimensions the scheme has proved equally effective and calclations of flows over wing/body combinations are possible with convergence achieved in less than 100 cycles.
A detailed description of spectral multigrid methods is provided. This includes the interpolation and coarse-grid operators for both periodic and Dirichlet problems. The spectral methods for periodic problems use Fourier series and those for Dirichlet problems are based upon Chebyshev polynomials. An improved preconditioning for Dirichlet problems is given. Numerical examples and practical advice are included.
A splitting scheme is proposed for the numerical solution of the time-dependent, incompressible Navier-Stokes equations by spectral methods. A staggered grid is used for the pressure, improved intermediate boundary conditions are employed in the split step for the velocity, and spectral multigrid techniques are used for the solution of the implicit equations.
A rapid quasi three-dimensional analysis was developed for blade-to-blade flows in turbomachinery. The analysis solves the unsteady Euler or thin layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream-surface thickness. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. The equations which are solved by a two-stage Runge-Kutta scheme made efficient by use of vectorization, a variable time-step, and a flux-based multigrid scheme, are described. A stability analysis is presented for the two-stage scheme. Results for a flat-plate model problem show the applicability of the method to axial, radial, and rotating geometries. Results for a centrifugal impeller and a radial diffuser show that the quasi three-dimensional viscous analysis can be a practical design tool.
A solution algorithm has been developed for the prediction of recirculating flows. Brandt's multilevel acceleration technique is used with Leonard's QUICK differencing scheme and a modified pressure implicit operator splitting scheme. Intermediate calculations enable a tau error distribution to be used for the identification of regions for local grid refinement, i.e., multigrid. The algorithm was tested for the prediction of laminar flow in a shear-driven and a buoyancy-driven cavity.
A rapid quasi three-dimensional analysis was developed for blade-to-blade flows in turbomachinery. The analysis solves the unsteady Euler or thin layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream-surface thickness. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. The equations which are solved b a two-stage Runge-Kutta scheme made efficient by use of vectorization, a variable time-step, and a flux-based multigrid scheme, are described. A stability analysis is presented for the two-stage. Results for a flat-plate model problem show the applicability of the method to axial, radial, and rotating geometries. Results for a centrifugal impeller and a radial diffuser show that the quasi three-dimensional viscous analysis can be a practical design tool.
The problem of estimating spatially varying coefficients of partial differential equations is considered from observation of the solution and of the right hand side of the equation. It is assumed that the observations are distributed in the domain and that enough observations are given. A method of discretization and an efficient multigrid method for solving the resulting discrete systems are described. Numerical results are presented for estimation of coefficients in an elliptic and a parabolic partial differential equation.
Three explicit multigrid methods, Ni's method, Jameson's finite-volume method, and a finite-difference method based on Brandt's work, are described and compared for two model problems. All three methods use an explicit multistage Runge-Kutta scheme on the fine grid, and this scheme is also described. Convergence histories for inviscid flow over a bump in a channel for the fine-grid scheme alone show that convergence rate is proportional to Courant number and that implicit residual smoothing can significantly accelerate the scheme. Ni's method was slightly slower than the implicitly-smoothed scheme alone. Brandt's and Jameson's methods are shown to be equivalent in form but differ in their node versus cell-centered implementations. They are about 8.5 times faster than Ni's method in terms of CPU time. Results for an oblique shock/boundary layer interaction problem verify the accuracy of the finite-difference code. All methods slowed considerably on the stretched viscous grid but Brandt's method was still 2.1 times faster than Ni's method.
A coarse-grid correction algorithm has been implemented into an implicit upwind Euler solver and tested for transonic airfoil problems. The Euler solver uses split-flux formulation and penta-diagonal scalar equations, respectively, for the explicit and implicit operators. The multigrid sequence starts at the fine grid level, then steps down to each coarse grid level to smooth error components using implicit operators. Estimate of residuals can be obtained by two approaches, which differ in the level where the residuals are collected. Both approaches will lead to a work reduction factor of 12 for a Mach 0.75 flow at 2 degrees incidence on a 65x26 grid. The work reduction factor is found to increase proportional to the number of grid levels.
A new efficient relaxation scheme in conjunction with a multigrid method is developed for the Euler equations. The LU SSOR scheme is based on a central difference scheme and does not need flux splitting for Newton iteration. Application to transonic flow shows that the new method surpasses the performance of the LU implicit scheme.