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A second order finite element method for the solution of the transonic Euler and Navier-Stokes equations. (English) Zbl 0840.76030
A finite element approach is proposed in which the fourth-order dissipation is recast as the difference of two Laplacian operators, allowing the use of bilinear elements. The Laplacians of the primitive variables of the first-order scheme are thus balanced by additional terms obtained from the governing equations themselves, tensor identities or other forms of nodal averaging. To demonstrate formally the accuracy of this scheme, an exact solution is introduced which satisfies the continuity equation identically and the momentum equations through forcing functions. The solutions of several transonic and supersonic inviscid and laminar viscous test cases are also presented and compared to other available numerical data.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76H05 Transonic flows
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##### References:
 [1] and , ’Euler Solvers as an Analysis Tool for Aircraft Aerodynamics’, in (ed.), Advances in Computational Transonics, Pineridgs Press, Swansea, 1985, pp. 371-404. [2] ’Implicit finite-difference methods for the Euler equations’, in (ed.), Advances in Computational Transonics, Pineridge Press, Swansea, 1985, pp. 503-542. [3] Roe, Ann. Rev. Fluid Mech. 18 pp 337– (1986) [4] , and , ’A comparison of numerical flux formulas for the Euler and Navier-Stokes equations’, AIAA Paper 87-1104, 1987. [5] and , ’A theoretical framework for Petrov-Galerkin methods with discontinuous weight functions: application to the streamline upwind procedure’, in et al. (eds.), Finite Element in Fluids, Vol. 4, Wiley, New York, 1982, pp. 47-65. [6] and , ’Finite element formulation for convection dominated flows with particular emphasis on the compressible Euler equations’, AIAA paper 83-0125, 1983. [7] and , ’An implicit finite element method for high speed flows’, AIAA Paper 90-0402, 1990. [8] and , ’Non-unique solutions of the Euler equations’, in and (eds.), Advances in Fluid Dynamics, Springer, New York, 1989, pp. 1-10. [9] Baruzzi, AIAA J. 29 pp 1886– (1991) [10] and , ’Numerical solution of the incompressible Navier-Stokes equations in primitive variables on unstructured grids’, AIAA Paper 91-1561, 1991. [11] and , ’Finite element simulation of compressible flow with shocks’, AIAA Paper 91-1551, 1991. [12] and , ’A second order method for the finite element solution of the Euler and Navier-Stokes equations’, Proc. 13th Int. Conf. on Numerical Methods in Fluid Dynamics, Springer, Roma, Italy. July 1992, pp. 509-513. [13] and , ’An improved finite element method for the solution of the compressible Euler and Navier-Stokes equations’, Proc. 1st European Computational Fluid Dynamics Conf. (ECCOMAS), Elsevier, Brussels, Belgium, September 1992, Vol. 2, pp. 643-650. [14] Pulliam, AIAA J. 28 pp 1703– (1990) [15] Peeters, Int. j. numer. methods fluids 13 pp 135– (1991) [16] Peeters, AIAA J. Propulsion and Power 8 pp 192– (1992) [17] Hinton, Int. j. numer. methods eng. 9 pp 235– (1975) [18] Langtanen, Commun. appl. numer. methods 5 pp 275– (1989) [19] , and , ’Parallelizable block-diagonal preconditiones for 3D viscous compressible flow calculations’, AIAA Paper 93-3309, Proc. 11th AIAA Computational Fluid Dynamics Conf., Orlando, Florida, Vol. 2, July 1993, pp. 135-143. [20] ’Structured mesh grid adapting based on a spring analogy’, Proc. CFD’93 Conf., CERCA (Centre for Research on Computation and its Applications), Montreal, June 1993, pp. 425-436. [21] and , ’Euler computations of AGARD Working Group 07 airfoil test cases’, AIAA Paper 85-0018, 1985. [22] ’Test cases for inviscid flow field methods’, AGARD Advisory Report No. 211, 1985, pp. 6-21, 22. [23] and , ’Résolution des équations de Navier-Stokes en fluide compressible par méthode implicite’, La echerche Aérospatiale, No. 1, Jan-Feb. 1986, pp. 23-46.
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