Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

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Figure 4.2: The algebraic “C-H” computational grid.

Pressure distribution on a rectangular wing with an airfoil NACA 0012, constant along the wing span, is shown for cross section in the plane of symmetry of the wing at Figure 4.1. In numerical calculation an algebraic non-orthogonal “C-H” grid was used in three-dimensional space, whose appearance in the wing symmetry plane is illustrated in Figure 4.2. At Mach number of undisturbed flow M ∞ = 0 . 8 and angle of attack α = 1 . 25 ◦ the results obtained show very good agreement with the results [45], despite coarse grid discretization (65 × 12 × 15). Described procedure in the analysis of non-viscous flow in transonic speed range made possible the calculation of the aerodynamic load in presence of strong shock waves, when the application of potential theory becomes unsustainable. Numerical stability and fast convergence of system of differential equations solution by the Runge-Kutta method make exposed procedure very acceptable in the analysis of three-dimensional flow. The great flexibility of the applied approach is reflected in the fact that it allows very easy introduction of real viscous terms in the Euler equations, thus leading to the Navier-Stokes equations, which represents a significant improvement in the aerodynamic load analysis in the cases of large wing attack angles with pronounced effects of flow separation. 4.2.2 Implicit numerical scheme Application of an explicit scheme in Euler equations solving, explained in chapter 4.2.1 of this book and literature [18], [27] and [19], enables a very precise analysis of the transonic three-dimensional flows, eliminating irregularities related to results of potential theory. The main disadvantage of the explicit scheme is reflected, however, in limited value of the Courant number ( CFL ), related to value of integration step. The introduction of an implicit scheme will have the task of increasing the allowable value of the Courant number and thus the calculation of the transonic flows makes more acceptable, even in solving problem related to non-stationary flow. The procedure of two-factor LU implicit factorization, based on finite volume method with the application of “flux decomposition” approach, is described in this chapter. Special attention will be paid to the correct implementation of boundary conditions related to features of “flux decomposition".