Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

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The remaining dissipative terms, present in the expression (4.21), are determined in a similar way. Quantities ε ( 2 ) i + 1 / 2 , j , k and ε ( 4 ) i + 1 / 2 , j , k in relations (4.21.2) and (4.21.4) are defined by the expres sions (4.9.7) and (4.9.8). After the dissipative terms introduction, the system the equation (4.20) receives the final form I + β ∆ t ( δ − ξ A + + δ − η B + + δ − ζ C + ) n ∗ ∗ I + β ∆ t ( δ + ξ A − + δ + η B − + δ + ζ C − ) n ∆ q n + ∆ t [ R n − R n 1 ] = 0 . (4.22) The integration step ∆ t for cell ( i , j , k ) is determined by the relation (4.12), in the same way as in the case of explicit numerical scheme. Stability analysis of exposed two-factor implicit LU scheme [2] showed the insensitivity of the scheme to relatively large values of the Courant number. Also, in the described implicit approach the application of the variable integration steps lead to a significant acceleration of solution convergence, so the conclusions from the chapter 4.2.1 can be applied without any restriction. The mentioned discussion certainly relates to stationary flow computation, while in the non-stationary flow field analysis a constant integration step ∆ t = min ( ∆ t i , j , k ) has to be used. Figures 4.3 and 4.4 show results of numerical calculation by the described procedure. Station ary transonic flow around rectangular wing and non-stationary flow around the rectangular wing in oscillatory translational motion are analyzed. In both cases three-dimensional algebraic non orthogonal “C-H” computational grid is used. The presented results refer to a convergent solution that is realized at the moment when the value of the residual term is reduced to the size of the fourth order relative to the initial.

1.2

1

0.8

0.6

0.4

0.2

-C p

0

-0.2

-0.4

upper lower upper lower

-0.6

and (SWANSON TURKEL) and (SWANSON TURKEL)

-0.8

-1

-1.2

0

0.2

0.4

0.6

0.8

1

x/l

Figure 4.3: Pressure distribution on a rectangular wing – plane of symmetry.

The wing in non-stationary flow field, at Mach number M ∞ = 0 . 8 of undisturbed flow and angle of attack α = 1 . 25 ◦ was observed. Pressure distribution on the airfoil in the plane of symmetry is