Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

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computational methods in Euler equations solving, as well as the results obtained experimentally. Exceptional accuracy of the solution in the vicinity of shock waves, in the case of high Mach numbers of undisturbed flow, represents a significant advantage of the described approach in aerodynamic load calculation over methods based on potential theory. 4.2.3 Godunov scheme Tendency in the formation of such a mathematical flow model which would show universality in solving a wide spectrum of problems is present in the modern numerical fluid mechanics. This universality consists in achieving satisfactory accuracy of the numerical solution, stability and rapid convergence of solutions in all flow problems, without introducing subsequent adjustment of mathematical model to a specific calculation task. Most recently developed methods of mathematical flow problems modeling are based on the introduction of appropriate adjustment constants, often empirically derived. The presence of the mentioned constants has for aim to obtain results by numerical simulation of physical problems that are nearly identical with experimentally determined ones. In this chapter, starting from known gasdynamic laws and some approaches performed so far in the field of numerical fluid mechanics [33], [31], a universal modeling procedure will be recom mended in modern computational fluid dynamics. The main goal will be forming a simple scheme that would be used, due to its own structure, in very complex computation of three-dimensional ( 3D ) flow without request for powerful supercomputer usage. The name “ Godunov scheme ” will refer to the whole a class of methods based on the wave properties of hyperbolic partial differential equations. Analysis of wave components propagation, caused by decomposition of complex nonlinear waves, led to the emergence of this numerical schemes class [52]. Godunov scheme – one-dimensional Euler equations Euler equations in the case of one-dimensional ( 1D ), non-stationary flow can be written in integral form in the following way: I [ ρ dx − ρ udt ] = 0 , I h ρ udx − ( p + ρ u 2 ) dt i = 0 , I ρ ( e u + 1 2 u 2 ) dx − ρ u ( e u + p ρ + 1 2 u 2 ) dt = 0 , (4.23) where the first equation (4.23) represents continuity equation, while the second and third represent momentum and energy equation, respectively. Integrals in expressions (4.23) are calculated on a closed contour in the ( x , t ) plane, where the x coordinate is in the flow direction, and t is time. Closed contours can also contain lines (surfaces) of discontinuity of flow quantities ρ , u , p and e u which denote density, velocity, pressure and internal energy per unit fluid mass, respectively. With the adopted assumption that the fluid moves parallel to the axis x at the moment t = t 0 the area occupied by that gas will be divided on segments of equal lengths h . The mentioned segment boundaries will be marked with x m , x m + 1 , . . . , while the values of the flow variables in the segment with the boundaries x m and x m + 1 will be denoted by the lower index m + 1 / 2. Calculated values of the flow quantities at time t = t 0 + τ in the mentioned segment will be denoted by the upper index m + 1 / 2. With the known flow variables values at time t = t 0 , an approximation that the quantities ρ m + 1 / 2 , u m + 1 / 2 , p m + 1 / 2 and e um + 1 / 2 are constant within each cell is introduced, bearing in mind that the cell dimensions are small enough.