Mathematical Physics - Volume II - Numerical Methods

4.3 Solution of Navier–Stokes equations

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while σ ± j , i + 1 / 2 = λ ± ( j ) i + 1 / 2 α j , i + 1 / 2 , bearing in mind that the quantities α , λ and r were pre viously defined by the relations (4.38). Compression parameter b , present in the rela tion (4.124), is defined by the expression b = ( 3 − k ) / ( 1 − k ) , as in [51].

4.3 Solution of Navier–Stokes equations 4.3.1 Implicit numerical scheme

The basic limitation in the application of Euler equations is reflected in flow calculation around the wing at large angles of attack, when the viscous effects of the flow cannot be ignored. In cases where occurs flow separation it is necessary to add viscous terms to Euler equations, ie. to model the flow by Navier-Stokes equations. The use of "full" Navier-Stokes equations is a problem due to high memory space requirements, on the one hand, and large CPU time, on the other hand. In this chapter is presented model based on Navier-Stokes approximate equations, in the literature known as the thin layer equations. Numeric solution of the equations is based on the application of the LU implicit factorization approach in finite volume method with "flux decomposition", described in chapter 4.2.2. LU implicit factorization method Three-dimensional non-stationary Navier-Stokes equations can be write in Cartesian coordinate system in conservative form ∂ t q + ∂ x ( F − F v )+ ∂ y ( G − G v )+ ∂ z ( H − H v ) = 0 , (4.126) where q is a vector of flow variables, defined by the relation (4.14), while F , G and H are flux vector projections on the three coordinate axes determined by the expressions (4.15). Viscous terms F v , G v and H v , present in the equation (4.126), are determined as follows:

    0 τ xx τ yx τ zx u τ xx + v τ xy + w τ xz − Q x 0 τ xy τ yy τ zy u τ yx + v τ yy + w τ yz − Q y 0 τ xz τ yz τ zz u τ zx + v τ zy + w τ zz − Q z        

    ,         .

F v =

G v =

and

(4.127)

H v =