Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

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information is transported from the boundary to the interior of the flow domain. At the entry point P 0 the characteristics C 0 and C + have slopes u and u + c which are always positive for flow in positive direction of the axis x . The third characteristic C − has the slope u − c which sign depends on the Mach number on the inflow domain boundary. Characteristics in supersonic flow are shown in Figure 4.9.a for the case of one-dimensional flow in the nozzle, while their disposition in the subsonic flow field is illustrated in Figure 4.9.b. If the transformation matrix L − 1 of variables W and V , defined by the first rela tion (4.92), is written so that the set of variables that match physical boundary conditions is separated from the remaining variables that correspond to numerical boundary conditions, as described in [30] and [48], there must be enough information about the wave propaga tion along the characteristics to determine the required quantities inside the domain. The relation between the variables W and V can be written in the form ∆ W =   ∆ w 3 ∆ w 1 ∆ w 2   =   − 1 / ρ c 0 1 − 1 / c 2 1 0 1 / ρ c 0 1     ∆ p ∆ ρ ∆ u   . (4.96) Denoting by ∆ W F characteristic variables associated with physical boundary conditions and by ∆ W N remaining variables associated with numeric boundary conditions that provide information from the interior of the domain toward to boundaries, the equation (4.96) becomes ∆ W = ∆ W F ∆ W N = " ( L − 1 ) F I ( L − 1 ) F II ( L − 1 ) N I ( L − 1 ) N II #( ∆ V I ∆ V II ) . (4.97) The variables V I in the equation (4.97) represent imposed physical boundary conditions, while a group of variables V II denotes variables determined by numerical boundary conditions. In case of subsonic outlet W F = w 3 , while at subsonic inflow W F consists of the variables w 1 and w 2 . All combinations of conservative and nonconservative variables may determine physical boundary conditions, except couple ( u , p ) in the subsonic inflow [14]. In the case of supersonic boundaries, for the problem to be well posed, it is necessary to define all variables at the inflow, while at the outflow it is not allowed to specify any variable, as shown in Figure 4.9.a. Numeric variables ∆ W N at outflow are determined by extrapolating the values from interior points. For the zero order extrapolation, the following relation is valid: ∆ W N | M = ∆ W N | M − 1 , (4.98) where M is total number of grid points. At the inflow boundary conditions are determined by similar relations ∆ W N | 1 = ∆ W N | 2 . (4.99) In the expressions (4.98) and (4.99) the operator ∆ indicates the corresponding increment in time. Implicit difference scheme, represented by the system of equations (4.60), can now be supplemented by boundary conditions based on equation (4.98), which for the subsonic