Abstract
The Euler equations describe compressible inviscid gas flows with rotation. They are widely used in the aerospace industry. The Euler equations are derived by considering the laws of conservation of mass, momentum and energy for an inviscid gas. The result is a nonlinear hyperbolic system of conservation laws. Only for very simple flow problems, analytical solutions exist. For almost all engineering problems solutions must be found numerically. Several discretization methods have been developed which yield solutions of good quality (good resolution of shock waves, slip lines, etc.). However, generally the computational cost is high. In 1983 a project was started at the Centre for Mathematics and Computer Science (CWI) in Amsterdam for the development of more efficient methods. So far, a multigrid method for the solution of the 2D steady Euler equations has been developed, implemented and tested.
Key Words and Phrases
An efficient iterative solution method for second-order accurate discretizations of the 2D Steady Euler equations is described and results are shown. The method is based on a nonlinear multigrid method and on the defect correction principle. Both first- and second-order accurate finite-volume upwind discretizations are considered. In the second-order discretization a limiter is used.
An Iterative Defect Correction process is used to approximately solve the system of second-order discretized equations. In each iteration of this process, a solution is computed of the first-order system with an appropriate right-hand side. This solution is computed by a nonlinear multigrid method, where Symmetric Gauss-Seidel relaxation is used as the smoothing procedure.
The computational method does not require any tuning of parameters. Flow solutions are presented for an airfoil and a bi-airfoil with propeller disk. The solutions show good resolution of all flow phenomena and are obtained at low computational cost. Particularly with respect to efficiency, the method contributes to the state of the art in computing steady Euler flows with discontinuities.
1980 Mathematics Subject Classification: 35L65, 35L67, 65N30, 76G15, 76H05.
This work was supported in part by the Netherlands Technology Foundation (STW).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
G.D. van Albada, B. van Leer & W.W. Roberts (1982): A Comparative Study of Computational Methods in Cosmic Gasdynamics. Astron. Astrophys. 108, 76–84.
K. Bohmer, P.W. Hemker & H.J. Stetter (1984): The Defect Correction Approach. Computing, Suppl. 5, 1–32.
A. Brandt (1982): Guide to Multigrid Development. Lecture Notes in Mathematics 960, Springer Verlag.
S.K. Godunov (1959): Finite Difference Method for Numerical Computation of Discontinuous Solutions of the Equations of Fluid Dynamics (in Russian, also Cornell Aeronautical Lab. Transl.). Math. Sbornik 47, 272–306.
W. Hackbusch (1985): Multi-Grid Methods and Applications. Springer Verlag.
A. Harten, P. Lax & B. van Leer (1983). On Upstream Differencing and Godunov-type Schemes for Hyperbolic Conservation Laws. SIAM Review 25, 35–61.
P.W. Hemker (1985): Defect Correction and Higher-Order Schemes for the Multigrid Solution of the Steady Euler Equations. Report NM-R8523, Centre for Mathematics and Computer Science, Amsterdam. To appear in Proceedings 2nd European Multigrid Conference, Cologne, 1985. Lecture Notes in Mathematics, Springer Verlag.
P.W. Hemker & B. Koren (1986): A Non-linear Multigrid Method for the Steady Euler Equations. Report NM-R8621, Centre for Mathematics and Computer Science, Amsterdam. To appear in Proceedings GAMM-Workshop on The Numerical Simulation of Compressible Euler Flows, Rocquencourt, 1986. Vieweg Verlag Series Notes on Numerical Fluid Mechanics.
P.W. Hemker & S.P. Spekreijse (1985): Multiple Grid and Osher’s Scheme for the Efficient Solution of the Steady Euler Equations. Report NM-R8507, Centre for Mathematics and Computer Science, Amsterdam. To appear in Appl. Num. Math., 1986.
B. Koren (1986): Euler Flow Solutions for a Transonic Windtunnel Section. Report NM-R8601, Centre for Mathematics and Computer Science, Amsterdam.
B. Koren (1986): Evaluation of Second Order Schemes and Defect Correction for the Multigrid Computation of Airfoil Flows with the Steady Euler Equations. Report NM-R8616, Centre for Mathematics and Computer Science, Amsterdam.
D. Kuchemann (1978): The Aerodynamic Design of Aircraft. Pergamon Press.
B. van Leer (1982): Flux-Vector Splitting for the Euler Equations. Proceedings 8 th International Conference on Numerical Methods in Fluid Dynamics, Aachen, 1982. Lecture Notes in Physics 170, Springer Verlag.
B. van Leer (1985): Upwind-Difference Methods for Aerodynamic Problems governed by the Euler Equations. Lectures in Applied Mathematics 22, AMS.
H.W. Liepmann & A. Roshko (1966): Elements of Gasdynamics. Wiley.
S. Osher & F. Solomon (1982): Upwind Difference Schemes for Hyperbolic Systems of Conservation Laws. Math. Comp. 38, 339–374.
S. Osher & S. Chakravarthy (1983): Upwind Schemes and Boundary Conditions with Applications to Euler Equations in General Geometries. J. Comp. Phys. 50, 447–481.
P.L. Roe (1981): Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes. J. Comp. Phys. 43, 357–372.
S.P. Spekreijse (1985): Second-Order Accurate Upwind Solutions of the 2D Steady Euler Equations by the Use of a Defect Correction Method. Report NM-R8520, Centre for Mathematics and Computer Science, Amsterdam. To appear in Proceedings 2nd European Multigrid Conference, Cologne, 1985. Lecture Notes in Mathematics, Springer Verlag.
S.P. Spekreijse (1986): Multigrid Solution of Monotone Second-Order Discretizations of Hyperbolic Conservation Laws. Report NM-R8611, Centre for Mathematics and Computer Science, Amsterdam. To appear in Math. Comp.
S.P. Spekreijse (1986): A Comparison of Several Multigrid Methods for the Solution of Second- Order Upwind Discretizations of the Steady Euler Equations. Report NM-R86xx, Centre for Mathematics and Computer Science, Amsterdam.
J.L. Steger & R.F. Warming (1981): Flux-Vector Splitting of the Inviscid Gas Dynamic Equations with Applications to Finite-Difference Methods. J. Comp. Phys. 40, 263–293.
P.K. Sweby (1984): High Resolution Schemes using Flux Limiters for Hyperbolic Conservation Laws. SIAM J. Num. Anal. 21, 995–1011.
H. Viviand (1985): Numerical Solutions of Two-dimensional Reference Test Cases. In: Test Cases for Inviscid Flow Field Methods. H. Yoshihara, et al. (eds.). AGARD Advisory Report 211.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1987 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig
About this chapter
Cite this chapter
Koren, B., Spekreijse, S. (1987). Multigrid and Defect Correction for the Efficient Solution of the Steady Euler Equations. In: Wesseling, P. (eds) Research in Numerical Fluid mechanics. Notes on Numerical Fluid Mechanics, vol 17. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-89729-9_7
Download citation
DOI: https://doi.org/10.1007/978-3-322-89729-9_7
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-08090-7
Online ISBN: 978-3-322-89729-9
eBook Packages: Springer Book Archive