Some anomalies of numerical simulation of shock waves. I: Inviscid flows. II: Effect of artificial and real viscosity.

*(English)*Zbl 0970.76079Summary: In part I, we discuss nonunique solutions of potential and Euler equations. Previous work related to airfoil problem is reviewed, and some new results are presented. Simple models based on Burgers equation and quasi-one-dimensional nozzle flow are examined. We also include an example for a three-dimensional wing admitting nonunique solutions of potential and Euler equations.

Part II deals with shock structure and its limit for conservative and nonconservative formulations. In particular, we calculate the entropy jumps across inviscid shocks using different artificial viscosity forms. The effects of real viscosity for two-dimensional airfoil problems are studied, and nonunique solutions of Navier-Stokes equations for transonic flows are presented for simple symmetric configurations.

Part II deals with shock structure and its limit for conservative and nonconservative formulations. In particular, we calculate the entropy jumps across inviscid shocks using different artificial viscosity forms. The effects of real viscosity for two-dimensional airfoil problems are studied, and nonunique solutions of Navier-Stokes equations for transonic flows are presented for simple symmetric configurations.

##### MSC:

76M99 | Basic methods in fluid mechanics |

76L05 | Shock waves and blast waves in fluid mechanics |

76H05 | Transonic flows |

##### Keywords:

potential equations; nonunique solutions; airfoil problem; Burgers equation; quasi-one-dimensional nozzle flow; three-dimensional wing; Euler equations; shock structure; entropy jumps; artificial viscosity; Navier-Stokes equations; transonic flows##### Software:

OVERFLOW
PDF
BibTeX
XML
Cite

\textit{M. M. Hafez} and \textit{W. H. Guo}, Comput. Fluids 28, No. 4--5, 701--739 (1999; Zbl 0970.76079)

Full Text:
DOI

##### References:

[1] | Steinhoff, J.; Jameson, A., Multiple solutions for the transonic potential flow past an airfoil, AIAA journal, 20, 5-1521, (1982) |

[2] | Salas M, Jameson A, Melnik R. A comparative study of the nonuniqueness problem of the potential equation. AIAA Paper 83-1988, 1983 |

[3] | Bristeau, M.O.; Pironneau, O.; Glowinski, R.; Periaux, J.; Perrier, P.; Poirier, G., On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element method (II) application to transonic flow simulations, Computer methods in applied mechanics and engineering, 51, 94-363, (1985) · Zbl 0555.76046 |

[4] | Jameson A. Nonunique solutions to the Euler equations. AIAA Paper 91-1625, 1991 |

[5] | Hafez, M.; Dimanlig, A., Simulations of compressible inviscid flows over stationary and rotating cylinders, Acta mechanica, Suppl 4, 9-241, (1994) |

[6] | Hafez, M.; Dimanlig, A., Some anomalies of the numerical solutions of the Euler equations, Acta mechanica, 119, 14, 40-131, (1996) · Zbl 0886.76052 |

[7] | Garabedian, P., Nonparametric solutions of the Euler equations for steady flows, Communications in pure and applied mathematics, 36, 35-529, (1983) · Zbl 0506.35082 |

[8] | Buning PG, Steger JL. Solution of the two-dimensional Euler equations with generalized coordinate transformation using flux vector-splitting. AIAA Paper 82-0971, 1982 |

[9] | Pandolfi, M.; Larocca, F., Transonic flow about a circular cylinder, Computers and fluids, 17, 1, 20-205, (1989) |

[10] | Deconinck H, Hirsch Ch. Boundary conditions for the potential equation in transonic flow calculation. SME Paper 83-gt-135, 1983 |

[11] | Bauer F, Garabedian P, Korn D, Jameson A. Supercritical wing sections II. In: Lecture notes in Economics and Mathematical Systems. vol. 108. New York: Springer, 1975 · Zbl 0304.76026 |

[12] | Murman, E., Analysis of embedded shock waves calculated by relaxation methods, AIAA journal, 12, 5, 33-626, (1974) · Zbl 0282.76064 |

[13] | Engquist, B.; Osher, S., Stable and entropy satisfying approximations for transonic flow calculations, Mathematics of computing, 34, 149, 45-75, (1980) · Zbl 0438.76051 |

[14] | Embid, P.; Coodman, J.; Majda, A., Multiple steady states for 1-D transonic flow, SIAM journal of scientific and statistical computations, 5, 1, 21-41, (1984) · Zbl 0573.76055 |

[15] | Roe P, van Leer B. Nonexistence, nonuniqueness and slow convergence in discrete conservation laws. In: Numerical methods for fluid dynamics. Oxford: Clarendon Press, 1988 |

[16] | Hafez M, Palaniswamy S, Mariani P. Calculations of transonic flows with shocks using Newton’s method and direct solver. Part II, AIAA paper 88-0226, 1988 |

[17] | Kantrowitz A. The formation and stability of normal shock waves in channel flow. NACA TN 1225, 1947 |

[18] | Salas M. Local stability for a planar shock wave. NASA Technical Paper 2387, 1984 |

[19] | Liu, T.P., Nonlinear stability and instability of transonic flow through a nozzle, Communications in mathematical physics, 83, 60-243, (1982) · Zbl 0576.76053 |

[20] | Liu TP. Transonic gas flows along a duct of varying area. Archives of Rational Mechanics and Analysis, 1982. p. 1-18 |

[21] | Nikfettrat K, Hafez M. Nonunique solution of inviscid flow in two dimensional nozzle. AIAA paper 90-0443, 1990 |

[22] | Beauchamp, P.; Murman, E., Wavy wall solutions of Euler equations, AIAA journal, 24, 12, 5-2042, (1986) |

[23] | Overflow User’s Manual. Version 1.6bd. NASA Ames Research Center, 1995 |

[24] | Hafez M, Guo WH. Simulation of steady compressible flows based on Cauchy/Riemann equations and Crocco’s relations. In: Proceedings of the Ninth International Conference on Finite Elements in Fluids. Venezia, 1995 · Zbl 1073.76614 |

[25] | Cheng HK. The transonic flow theories of high- and low-aspect ratio wings. In: Numerical and physical aspects of aerodynamic flows. Berlin: Springer, 1982 |

[26] | Cole J, Cook P. Transonic aerodynamics. Amsterdam: North-Holland, 1986 · Zbl 0622.76067 |

[27] | Hafez M. Perturbation of transonic flow with shocks. In: Numerical and physical aspects of aerodynamic flow. Berlin: Springer, 1982. p. 421-38 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.