Stream vectors in three dimensional aerodynamics.

*(English)*Zbl 0625.76009This 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.

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 |

##### Keywords:

inviscid aerodynamics; three-dimensional flows; well posed problem; boundary conditions; uniqueness of decomposition; Laplace-Beltrami problem; Neumann problem; mixed problem; entropy correction in transonic flow; potential flow; shock-developing flows; finite element calculations; element discretization; existence; error estimation of approximation of stream vector; Kutta-Joukowski condition; three- dimensional external flows.##### References:

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