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Abstract
Glimm's (1965) random choice method has been successfully applied to compute steady two-dimensional supersonic/hypersonic flow using a new Lagrangian formulation. The method is easy to program, fast to execute, yet it is very accurate and robust. It requires no grid generation, resolves slipline and shock discontinuities crisply, can handle boundary conditions most easily, and is applicable to hypersonic as well as supersonic flow. It represents an accurate and fast alternative to the existing Eulerian methods. Many computed examples are given.
... In this case, an arbitrary value of the integration step, Dl n (chosen to satisfy the stability condition), usually leads to significant numerical errors and to the inability of capturing sharp shocks without numerical distortions, as shown in Section 4.1 (Fig. 4). This type of loss of accuracy has also been observed by Loh and Hui (1993) who used a local analytical solution at the location of the slope change, similar to that used by Glatz and Wardlaw (1984) for a Godunov scheme. In the following, we will use a different procedure by conveniently adjusting the Lagrangian-distance increment Dl n : First, the integration step Dl nÀ1 is adjusted such that the Lagrangian-distance line l ¼ l n coincides with the location of the wall slope change. ...
This paper presents solutions for several 2-D aerodynamic problems with geometrically unspecified boundaries. These solutions are obtained with an enhanced Lagrangian method based on stream function and Lagrangian-distance coordinates, which includes special procedures to substantially improve the numerical resolution of the shock waves and for the numerical implementation of the aerodynamic conditions defining the geometrically unspecified solid or fluid boundaries. The method is first validated for flows with specified solid boundaries by comparison with exact analytical solutions and with previous computational results obtained by numerical methods using Eulerian formulations. In all cases, this enhanced Lagrangian method displayed a very good accuracy and computational efficiency and a sharp numerical resolution of shock waves. Then this method is used to obtain solutions for problems with geometrically unspecified boundaries, such as: (i) indirect problems of determining the geometrical shape of airfoils and nozzle walls for a specified pressure distribution; (ii) supersonic nozzle design problem for a specified uniform flow at the nozzle outlet based on reflection-suppression condition; (iii) analysis of flexible-membrane airfoils; and (iv) analysis of jet-flapped airfoils.
The random choice method has now been shown to be successfully extendible from the original one-dimensional unsteady formulation to inert high-speed flow fields which are steady and two-dimensional using Cartesian, axisymmetric and Lagrangian formulations. This paper deals with the description of a new implementation of the random choice method formulated for natural co-ordinates based on streamlines and normals. Comparisons between theoretical and computed results for several different physical configurations are presented.
The exact real gas Riemann solution and the random choice method (RCM) are briefly reviewed and the derivation of the Lagrangian geometrical quantities, which represent the deformation of fluid particles in the motion, are described in detail. Extensive calculations were made to test the accuracy against the exact solution and the robustness of the Lagrangian RCM approach for real gas supersonic flows, including complex wave interactions of different types. The real gas effect is also presented by comparison with the perfect gas solution. The inherent parallelism in the Lagrangian approach lends a natural application in the massively parallel computation.
A second-order Godunov-type shock-capturing scheme for solving the
steady Euler equations in generalized Lagrangian coordinates has been
developed and applied to compute steady supersonic and hypersonic flow
problems. Following Hui and Zhao, the Lagrangian distance and a stream
function are used as the coordinate lines that not only simplify the
Riemann solution procedure but also have an intrinsic flow adaptive
property embedded. Numerical examples for various supersonic flows
involving strong flow discontinuities are given. Good agreement is
obtained between computed results and shock expansion theory or
available experimental data. It was found that the resolution of the
slip line is almost exactly without smearing, the resolution of shock is
always crisp even at increasing Mach number, and the Prandtl-Meyer
expansion is adequately resolved with the second-order-accurate scheme.
A space marching Godunov method using control volumes consisted of streamlines and the coordinate lines in the marching direction is described for the computation of two-dimensional and axisymmetric steady supersonic flows. These streamline meshes are solution-generated. They follow the fluid particles closely and also simplify the implementation of the Godunov method. A second order extension to the Godunov scheme using essentially nonoscillatory interpolation is also presented. Computations of various supersonic and hypersonic flow problems indicate that the method accurately represents smooth flow and crisply resolves all flow discontinuities such as shocks and slip lines even at high Mach numbers.
Thesis (M. Eng.)--McGill University, 1997. Includes bibliographical references. Includes abstract in French.
In the aeronautical applications, many problems involving boundaries of unspecified geometry are of interest, such as the indirect problems of determining the shape of an airfoil to generate a specified pressure distribution, supersonic nozzle design based on reflection-suppression condition, flexible-membrane airfoils or wings, jet-flapped airfoils, and others. In these cases, the utilization of the Eulerian formulations of the Euler equations of motion, may lead to long iterative computations for successive shapes of the boundaries, until the final geometry is reached. For this type of problems, the Lagrangian formulations using the streamline coordinates are more suitable, since the geometrically-unspecified boundaries are always represented by streamlines. Computational methods based on Lagrangian formulations have been recently developed for supersonic flows for the solution of the system of Euler equations. These Lagrangian formulations use the stream-function and the Lagrangian-time or -distance coordinates. In our present study, the numerical method applied to solve aerodynamic problems of unspecified geometry, is based on a finite volume discretization in the Lagrangian coordinates (stream-function and Lagrangian-distance), in which the flux values are determined by using the Riemann problem solution. Improvements leading to a better resolution of the shock waves are included. The method has been first validated for nozzles and airfoils of specified geometry, by comparison with analytical results and previous solutions obtained using Eulerian formulations. Then, the method has been applied for the solution of the above mentioned problems of unspecified geometry. In the cases of the flexible-membrane airfoils and the jet-flapped airfoils, an analytical solution has been developed in addition to the numerical solution.
We investigate Glimm's method, a method for constructing approximate solutions to systems of hyperbolic conservation laws in one space variable by sampling explicit wave solutions. It is extended to several space variables by operator splitting. We consider two problems: (1) we propose a highly accurate form of the sampling procedure, in one space variable, based on the van der Corput sampling sequence. We test the improved sampling procedure numerically in the case of inviscid compressible flow in one space dimension and find that it gives high resolution results both in the smooth parts of the solution, as well as at discontinuities; (2) we investigate the operator splitting procedure by means of which the multidimensional method is constructed. An O(1) error stemming from the use of this procedure near shocks oblique to the spatial grid is analyzed numerically in the case of the equations for inviscid compressible flow in two space dimensions. We present a hybrid method which eliminates this error, consisting of Glimm's method, used in continuous parts of the flow, and the nonlinear Godunov method, used in regions where large pressure jumps are generated. The resulting method is seen to be a substantial improvement over either of the component methods for multidimensional calculations.
This paper studies the problem of steady two-dimensional supersonic flow of an inviscid compressible fluid using the new Lagrangian formulation of Hui and Van Roessel, in which the stream function and the Lagrangian time are used as independent variables. A shock capturing method is developed by applying the first-order Godunov scheme to the conservation form equations of this formulation. The method is fast and robust. Furthermore, extensive comparisons with exact solutions and with the second-order Godunov scheme of Glaz and Wardlaw based on the Eulerian formulation show that the first-order Lagrangian method generally attains the same level of accuracy as the second-order Eulerian method and is even better in resolving slip line discontinuities.
A random choice method for solving nonlinear hyperbolic systems of conservation laws is presented. The method is rooted in Glimm's constructive proof that such systems have solutions. The solution is advanced in time by a sequence of operations which includes the solution of Riemann problems and a sampling procedure. The method can describe a complex pattern of shock wave and slip line interactions without introducing numerical viscosity and without a special handling of discontinuities. Examples are given of applications to one- and two-dimensional gas flow problems.