Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

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determine the grid surfaces in physical space, time intervals ( ∆ t ξ ) i , j , k , ( ∆ t η ) i , j , k and ( ∆ t ζ ) i , j , k can be calculated: ( ∆ t ξ ) i , j , k = 1 | U | + c q ξ 2 x + ξ 2 y + ξ 2 z i , j , k , ( ∆ t η ) i , j , k = 1 | V | + c q η 2 x + η 2 y + η 2 z i , j , k , ( ∆ t ζ ) i , j , k = 1 | W | + c q ζ 2 x + ζ 2 y + ζ 2 z i , j , k . (4.12.1) In the expressions (4.12.1) U , V and W denote contravariant velocity vector coordinates, 1 while c represents the local speed of sound. Integration step ∆ t , determined by the expression (4.12), can be scaled by a constant, known in literature as the Courant number [ aga ], which in the case of the fourth order Runge-Kutta method equals to 2 √ 2. Taking into account that grid cell dimensions can vary drastically, depending on the type of chosen grid, application of the variable integration steps leads to a significant acceleration of solution convergence. As the values of the flow variables do not change significantly within of an iterative cycle, that the time interval ∆ t calculation does not need to be performed after each iteration, resulting in significant savings in computational (CPU) time.

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Figure 4.1: Pressure distribution on a rectangular wing – plane of symmetry.

1 Numeric determination of partial derivatives ξ x , ξ y , ξ z , η x , η y , η z , ζ x , ζ y and ζ z are explained in detail in literature [ hirsch ] (Hirsch, C. Numerical Computation of Internal and External Flows , Vol. 1, pp. 253–260).