Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

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equations (4.25), thus obtaining the new value of the contact discontinuity pressure p c . d . . The process is repeated to complete convergence of pressure p c . d . . The mentioned iterative scheme need only be applied in the discontinuity zone (break down), when the cell boundary coincides with the shock wave. In general case, when the changes in dependent flow variables across the cell boundary are small compared to cell length, approximate form of the equation (4.25) and (4.26) can be used successfully a m = b m = h γ 4 ( p m − 1 / 2 + p m + 1 / 2 )( ρ m − 1 / 2 + ρ m + 1 / 2 ) i 1 / 2 , p c . d . = p m + 1 / 2 + p m − 1 / 2 2 + a m u m − 1 / 2 − u m + 1 / 2 2 , u c . d . = u m + 1 / 2 + u m − 1 / 2 2 + p m − 1 / 2 − p m + 1 / 2 2 a m . (4.27) To determine the quantities R , U , P and E u at the cell boundaries it is required to define three characteristic wave propagation speeds. Propagations speeds of the left and right acoustic waves are calculated by relations D L = u m − 1 / 2 − a m ρ m − 1 / 2 and D D = u m + 1 / 2 + a m ρ m + 1 / 2 , (4.28) respectively, while the third characteristic speed is precisely the contact discontinuity speed u c . d . . For time t > t 0 the ( x , t ) plane near the boundary x = x m is divided into four zones, shown in Figure 4.5. ∆ F 2 ∆ F 3 ∆ F 1

t

II

III

I

IV

x = x m

x

Figure 4.5: Wave propagation near the cell boundary.

Values of flow variables in zones I and IV are equal to values corresponding to the indices m − 1 / 2 and m + 1 / 2, respectively. In the zones II and III values of contact discontinuity pressure p c . d . and velocity u c . d . are calculated on the basis of equation (4.25), or (4.27). The density on the left side is determined from ρ L ρ m − 1 / 2 = ( γ + 1 ) p c . d . +( γ − 1 ) p m − 1 / 2 ( γ − 1 ) p c . d . +( γ + 1 ) p m − 1 / 2 , (4.29) while the corresponding value on the right side ρ D is obtained by substitution index m − 1 / 2 by index m + 1 / 2 in the equation (4.29).