×

zbMATH — the first resource for mathematics

Stream vectors in three dimensional aerodynamics. (English) Zbl 0625.76009
This paper deals with the decomposition of a velocity field in inviscid aerodynamics into a potential part and a stream function part in three- dimensional flows. Stream function has a long history in two-dimensional and axisymmetrical flows. It is convenient to find a stream function instead of finding two velocity components directly. However, in order to form a well posed problem in the mathematical sense, more attention must be paid on the boundary conditions. It has been treated carefully in numerical calculations. In dealing with three-dimensional flows in this paper, a stream vector is a function whose curl is a part of the velocity vector. It serves as a correction to isentropic potential flow, e.g., in transonic case when strong shocks develop. However, the physical meaning of three-dimensional stream vector is no longer as obvious as the stream function in two-dimensional ones.
C. Bernardi [Thèse de 3ème cycle. Université Paris VI (1979)] and later J. M. Dominguez [A. Bendali, J. M. Dominguez and S. Gallic, A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three dimensional problems. Rapport interne No.95, Ecole Polytechnique, CMA (1983)] have proved the uniqueness of decomposition of a velocity vector into a potential part and a stream function part in the case when stream vector is solenoidal and when boundary conditions are homogeneous. This result has been extended to non-simply-connected domains.
The authors extended the above result to steady inviscid flows with homogeneous and non-homogeneous boundary conditions. They proved that the uniqueness of decomposition of velocity field is possible by solving first a Laplace-Beltrami problem on the boundary. In summary, this decomposition transfers a three-dimensional problem into a Neumann problem for velocity potential, a mixed problem for stream vector when stream vector is solenoidal and a Laplace-Beltrami problem for a boundary function. Some applications to aerodynamics are also given.
An interesting example is entropy correction in transonic flow by iterating from the potential flow (zero stream vector) to obtain non- homoenergetic and shock-developing flows where both entropy and stagnation enthalpy are not constant. Other examples include wings and nozzles at small Mach numbers. Finally, in applying the decomposition to finite element calculations, element discretization is given; existence, uniqueness and error estimation of approximation of stream vector and boundary function are proved and presented. The authors claim that except for non-simply-connected domains calculations are not more expensive than that in potential flows even in the case of simple incompressible flow. The Kutta-Joukowski condition is also easier to apply.
The present scheme is a worthwhile alternative mathematical method especially in numerical calculations of three-dimensional external flows. Since for such a large-scale computation, the present scheme may significantly reduce the storage space, the initial potential flow results will provide the initial values of iteration. However, how serious is the limitation on the stream vector to be solenoidal is not clear so far.
Reviewer: A.Ting

MSC:
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
76H05 Transonic flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] De Rham, G.: Varietes differentiables. Paris: Ed. Hermann 1960
[2] Bernardi, C.: These de 3 eme cycle. Universite Paris VI, 1979
[3] Bendali, A., Dominguez, J.M., Gallic, S.: A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three dimensional problems. Rapport interne No 95, Ecole Polytechnique, CMA (1983) · Zbl 0591.35053
[4] Lacor, C., Hirsch, C.: Rotational flow calculations in three dimensional blade passage. ASME report, 82-GT-316 · Zbl 0487.76017
[5] Sokhey, J.S.: Transonic flow around axisymetric inlets including rotational flow effects. AIAA 18th meeting, Pasadena (1980)
[6] Papaillon, N., Chaviaropoulos, P., Gianakoglou, K.: Numerical computation of 2-D rotational inviscid transonie flows, using the decomposition of the flow field into a potential and a rotational part, 7th ISABE Conf. Bejung, China, 1985
[7] Amara, M.: Analyse de methodes d’elements finis pour des ecoulements transsoniques. These d’Etat, Universite Paris VI, Analyse Numerique (1983)
[8] Dominguez, J.M.: Etude des equations de la Magnetohydrodynamique stationnaires et leur approximation par elements finis. These de Docteur Ingenieur, Universite Paris VI (1982)
[9] Lions, J.L., Magenes, E.: Problemes aux limites non homogenes et applications, Vol. 1. Paris: Dunod 1968 · Zbl 0165.10801
[10] Foias, C., Temam, R.: Remarques sur les equations de Navier-Stokes et phenomenes successifs de bifurcation. Ann. Sc. Norm. Super. Pisa, IV. Ser.V, 29-63 (1978) · Zbl 0384.35047
[11] El Dabaghi, F., Periaux, J., Pironneau, O., Poirier, G.: 3-D finite element solution of steady Euler transonic flow by stream vector correction, AIAA 7th computational fluid dynamics conference, AIAA-85-1532-CP, Cincinnati, Ohio, USA, 1985 · Zbl 0645.76070
[12] Jameson, A.: A multi-grid scheme for transonic potential calculations on arbitrary grids. Proceedings of the AIAA 4th computational fluid dynamics conference, Williamsburg, Va., pp. 122-146, 1979
[13] Jameson, A.: Numerical solutions of the euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA paper 81-1259, AIAA 14th fluid and plasma dynamics conference, Palo Alto, CA 1981
[14] Bristeau, M.O., Glowinski, R., Pironneau, O., Periaux, J., Perrier, P.: On the numerical solution of nonlinear problems by least squares and finite element methods (I) Least square formulation and conjugate gradient solution of the continuous problems, pp. 619-657, 17/18. Amsterdam: North-Holland 1979 · Zbl 0423.76047
[15] Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Mathe.8, 309-332 (1982) · Zbl 0505.76100 · doi:10.1007/BF01396435
[16] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland 1978 · Zbl 0383.65058
[17] Verfurth, R.: Error estimates for a mixed finite element approximation of the Stokes equations. R.A.I.R.O. (To appear) · Zbl 1189.76394
[18] Nedelec, J.C.: Mixed finite elements in R3. Numer. Math.35, 315-341 (1980) · Zbl 0419.65069 · doi:10.1007/BF01396415
[19] Peetre, J.: Another approach to elliptic boundary problems. Pure Appl. Math. Sci.14, 711-731 (1961) · Zbl 0104.07303 · doi:10.1002/cpa.3160140404
[20] Dominguez, J.: Formulation en potentiel vecteur du système de Stokes dans un domaine deR 3. Rapport 83015 LAN, Université Paris 6, 1983
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.