Edexcel FP3 by Keith Pledger, Dave Wilkins

By Keith Pledger, Dave Wilkins

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1 ) . 55) 32 JULIAN D. COLE since The boundary value problem for ( 2 . 4 9 ) together cp is continuous. with shock jumps and K-J (

0+) = 9 ^ ( 1 , 0 - ) ) presumably conditions x defines a unique solution whose perturbation velocities die off at infin­ ity. Direct analytic solutions of ( 2 . 5 3 ) for flow past realistic airfoils with K > 0 are non-existent due to the non-linearity and possible shock waves. However computational algorithms which capture shocks have been developed. These are based on implicit finite difference schemes with central differencing at elliptic points and backward (up­ wind) differencing at hyperbolic points.

37) 5 < 0 , \ Thus a fixed. 39^ If $y is matched a condition on u = 0^ is obtained directly. Comparing ( 2 . 3 3 ) and ( 2 . in) provides the initial condition for the solution of ( 2 . 3 6 ) in the wave-zone. However ( 2 . 3 6 ) is a non-linear equation, a version of the 28 JULIAN D. COLE continuity equation ( 2 . 1 ) , and as such admits shock waves. The solution to ( 2 . 1*2) u is constant on a characteristic value of u (t = const, af. Figure 2 . 3 ) . The for those characteristics originating ahead of the airfoil is zero, so that based on characteristics above the solution is multi­ valued.

A general theory for the problem (P^) can now be constructed, using these examples as guides. We consider first boundary layer behavior for solutions of (P^). Suppose then that the reduced problem |2 u + h(t,u) = 0, a < t < b, (R R) u(b) = B, SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS n has a smooth solution u = which is close to u'(t) : I order that (P 3) have a solution u R except near t = a (where, in general, u R(a) ^ A ) , we must first ask that positive constant 61 k u R be stable in the sense that there exists a for which < 0 on [a,b].

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