The Full Approximation Scheme (FAS) multigrid method is applied to several implicit flux-split algorithms for solving the three-dimensional Euler equations in a body fitted coordinate system. Each of the splitting algorithms uses a variation of approximate factorization and is implemented in a finite volume formulation. The algorithms are all vectorizable with little or no scalar computation required. The flux vectors are split into upwind components using both the splittings of Steger-Warming and Van Leer. The stability and smoothing rate of each of the schemes are examined using a Fourier analysis of the complete system of equations. Results are presented for three-dimensional subsonic, transonic, and supersonic flows which demonstrate substantially improved convergence rates with the multigrid algorithm. The influence of using both a V-cycle and a W-cycle on the convergence is examined.

Three-dimensional multigrid algorithms for the flux-split Euler equations

The convergence characteristics of various approximate factorizations for the 3D Euler and Navier-Stokes equations are examined using the von-Neumann stability analysis method. Three upwind-difference based factorizations and several central-difference based factorizations are considered for the Euler equations. In the upwind factorizations both the flux-vector splitting methods of Steger and Warming and van Leer are considered. Analysis of the Navier-Stokes equations is performed only on the Beam and Warming central-difference scheme. The range of CFL numbers over which each factorization is stable is presented for one-, two-, and three-dimensional flow. Also presented for each factorization is the CFL number at which the maximum eigenvalue is minimized, for all Fourier components, as well as for the high frequency range only. The latter is useful for predicting the effectiveness of multigrid procedures with these schemes as smoothers. Further, local mode analysis is performed to test the suitability of using a uniform flow field in the stability analysis. Some inconsistencies in the results from previous analyses are resolved.

/pdf/stability-analysis-of-factorization-methods-5csezfhhsf.pdf

Stability analysis of factorization methods

A STUDY OF FLOW SEPARATION IN TRANSONIC FLOW USING INVISCID AND VISCOUS CFD SCHEMES J a n e s Andrew Rhodes Old Don in i on U n i v e r s i t y , 1987 D i r e c t o r : Dr . S u r e n d r a N. T iw ar i C o D i r e c t o r : D r . E r n s t von L a v a n te A com par i son of f lo w s e p a r a t i o n i n t r a n s o n i c f lows i s made u s in g v a r i o u s c o m p u t a t i o n a l schemes which s o l v e t h e E u l e r and t h e N a v i e r S t o k e s e q u a t i o n s o f f l u i d m e c h a n i c s . The f lows examined a r e computed u s i n g s e v e r a l s i m p l e t w o d i m e n s i o n a l c o n f i g u r a t i o n s i n c l u d i n g a backward f a c i n g s t e p and a bump in a c h a n n e l . Comparison o f t h e r e s u l t s o b t a i n e d u s i n g shock f i t t i n g and f l u x v e c t o r s p l i t t i n g methods a r e p r e s e n t e d and t h e r e s u l t s o b t a i n e d u s i n g t h e E u l e r codes a r e compared t o r e s u T t s on t h e same c o n f i g u r a t i o n s u s i n g a code which s o l v e s t h e N a v i e r S t o k e s e q u a t i o n s . R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS The w r i t e r w ish es t o acknowledge t h e en co u rag em en t , s u p p o r t and g u i d a n c e g iven by each member o f t h e gu id a n c e co m m it t ee . In p a r t i c u l a r , t h e w r i t e r would l i k e t o th an k Dr . Surendri . N. T iwar i f o r h i s f a i t h in t h e w r i t e r ' s a b i l i t i e s and w i l l i n g n e s s t o p ro v id e f i n a n c i a l s u p p o r t , Dr. E r n s t von La va n te f o r t h e many v a l u a b l e d i s c u s s i o n s which t h e w r i t e r had w i t h him and f o r t h e use of s e v e r a l o f h i s computer c o d e s , and t o Mr. Manual D. S a l a s , who was w i l l i n g t o p r o v id e t h e w r i t e r w i th t h e i n i t i a l c o n c e p t f o r t h e d i s s e r t a t i o n t o p i c and wi th NASA f a c i l i t i e s w i t h o u t which t h e work would have been i m p o s s i b l e .

/pdf/a-study-of-flow-separation-in-transonic-flow-using-inviscid-3a0nuqouwn.pdf

A Study of Flow Separation in Transonic Flow Using Inviscid and Viscous CFD Schemes

Three different implicit operators in a numerical method for solving the two-dimensional steady Euler equations using flux vector splitting are investigated. These include the implementation of the true Jacobian matrices in the implicit part, the use of their splitting approximate form, and a modification of the implicit part of the scheme that uses the scalar form of the implicit part based on the spectral radii of the split Jacobian matrices. The three versions of the basic algorithm are compared with the results given by two common numerical methods using several two-dimensional test cases. Special attention is paid to the quality of results as well as computational efficiency and convergence properties.

Effects of various implicit operators on a flux vector splitting method

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Effects of various implicit operators on a flux vector splitting method