×

On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. II: Application to transonic flow simulations. (English) Zbl 0555.76046

[For part I, see the first authors, ibid. 17/18, 619-657 (1979; Zbl 0423.76047).] - The main goal of this paper is to show how the general methods discussed in the first part apply to the solution of a class of rather complicated nonlinear problems of industrial interest, namely the numerical simulation of potential transonic flows for compressible inviscid fluids, in two and three dimensions. We recall in section 2 the governing equations and the various conditions that physical solutions have to verify, such as entropy and Kutta-Joukowsky conditions. We shall comment also on the possible existence of multiple physical solutions. In section 3, we recall the least squares formulation for the continuous problem discussed in the first part, and describe in section 4 the finite element approximation of the several above formulations of the continuous problem. In section 5, we discuss the finite element implementation of the entropy condition by upwinding of the density in the flow direction. In section 6 various numerical experiments illustrate the possibility of the above methods.

MSC:

76H05 Transonic flows
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76M99 Basic methods in fluid mechanics
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing

Citations:

Zbl 0423.76047

Software:

BRENT
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bristeau, M.O.; Glowinski, R.; Périaux, J.; Perrier, P.; Pironneau, O., On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods (I). least squares formulation and conjugate gradient solutions of the continuous problems, Comput. meths. appl. mech. engrg., 17/18, 619-657, (1979) · Zbl 0423.76047
[2] Bristeau, M.O.; Glowinski, R.; Périaux, J.; Perrier, P.; Pironneau, O., Transonic flow simulations by finite element and least squares-methods, (), 453-482 · Zbl 0423.76047
[3] Glowinski, R., Numerical methods for nonlinear variational problems, (1984), Springer Berlin · Zbl 0575.65123
[4] Landau, L.; Lifchitz, E., Mécanique des fluides, (1953), Mir Moscow · Zbl 0144.47605
[5] Steinhoff, J.; Jameson, A., Multiple solutions of the transonic flow equation, Aiaa j., 20, 11, 1521-1525, (1982) · Zbl 0496.76013
[6] Salas, M.D.; Jameson, A.; Melnik, R.E., A comparative study of the non-uniqueness problems of the potential equation, ()
[7] Bristeau, M.O.; Glowinski, R.; Périaux, J.; Poirier, G., Multiple solutions for the full potential transonic flow equation, (), 583-589
[8] Polak, E., Computational methods in optimization, (1971), Academic Press New York · Zbl 0257.90055
[9] Householder, A.S., The numerical treatment of a single nonlinear equation, (1970), McGraw-Hill New York · Zbl 0242.65047
[10] Brent, R., Algorithms for minimization without derivatives, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0245.65032
[11] Shanno, D.F., Conjugate gradient method with inexact line search, Math. oper. res., 13, 244-255, (1978)
[12] Buckley, A.; Lenir, A., QN-like variable storage conjugate gradients, Math. programming, 27, 155-175, (1983) · Zbl 0519.65038
[13] Jameson, A., Numerical solutions of nonlinear partial differential equations of mixed type, (), 275-320
[14] Jameson, A., Transonic flow calculations, (), 1-87
[15] Jameson, A., Numerical calculation of transonic flow past a swept wing by a finite volume method, (), 125-148
[16] Jameson, A., Remarks on the calculation of transonic potential flows by a finite volume method, (), 363-386
[17] Bristeau, M.O., Application of optimal control to transonic flow computations by finite element methods, (), 103-124 · Zbl 0411.76048
[18] Bristeau, M.O., Application of a finite element to transonic flow problems using an optimal control approach, (), 281-328 · Zbl 0469.76045
[19] Holst, T., An implicit algorithm for the transonic full potential equation in conservative form, (), 157-174
[20] Kogan, A.; Migemi, S., Transonic flow calculation by a finite element method, Israel J. technology, 20, 166-170, (1982) · Zbl 0529.76057
[21] Deconinck, H.; Hirsh, Ch., Subsonic and transonic computations of cascade flows, (), 175-195
[22] Eberle, A., Finite element method for the solution of the full potential equation in transonic steady and unsteady flow, (), 483-504
[23] Akay, H.U.; Ecer, A., Treatment of transonic flows with shocks using finite elements, (), 505-530 · Zbl 0465.76054
[24] Amara, M.; Joly, P.; Thomas, J.M., Mixed finite element methods for solving transonic flow equations, Comput. meths. appl. mech. engrg., 39, 1-19, (1983) · Zbl 0497.76066
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.