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Paper describes a numerical calculation method using eddy viscosity/mixing length concepts for tangential slot injection (wall-wake) flows; application of the method over a wide range of flow conditions indicates increased accuracy compared to previous work. Predictions from the numerical code were in good agreement with experiment (velocity profile, skin friction, and effectiveness data) for low and high speed flows. To achieve improved accuracy, improvements in the turbulence modeling, compared to previous research, were necessary for the imbedded shear layer region in the near field and for the wall region near shear layer impingement. Anomalous behavior was noted for far field experimental velocity profiles in low speed flow when the slot-to-free stream velocity ratio was near one.

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When x<0 we must close the contour in the upper half plane and when;t>0 in the lower half plane. Hence K 0 (x) = K + (x) x<0 -(x) x>0 (18) rs^. , ŵ here K ± (x) r* • /upper ) half plane. Res m (lower Then upon evaluating the residues we find that K + (x)= -r Hm2i y^o dy (a+-k/M) and sinh (cos) e /w 2 cosh (cos 1 ) -cps (ar --T hm (20) These series are only conditionally convergent and will not converge at all if we take the derivative term by term. To obtain convergent series, notice that since a* ~T"/ (s^±/3s) + 0(n ~]) as n^oo the nth term of these sums behave like ]]hThe series composed of these terms will converge to a row of step functions. Hence, its derivative will converge to a row of delta functions. We can evaluate the latter series by using the theory of distributions to show that 3 = J_ Hence where --..] e'^+^^/i) 6(^^185/1) (21) 2s and co sinh (cos) e'" x 2[cosh(co5<) -cos(a -s f co) e iMkx rll' -a;r--are now convergent series. The kernel function is given by Eqs. (18) and (21). Only a finite number of the infinite row of delta functions in Eq. (21) will contribute to the integral in Eq. (13). However when this kernel is substituted into Eq.(13), we obtain a functional integral equation (and not an ordinary integral equation) due to the introduction of terms of the form [P(x n + ns@) ] caused by the integration over the delta functions. The series which appear in K ± are only conditionally convergent. But the same device that was used to make the original series converge can also be used to render these latter series absolutely convergent. This removal of the slowly convergent part of the series results in a row of step functions which explicitly exhibit the discontinuities of K ± (and occur at the points x n = ±f$sn). The remaining series will represent continuous functions and will be quite suitable for numerical computation. Nomenclature= skin friction coefficient, ij\/1p e u e 2 = mixing length =. Mach number = power law velocity exponent, Eq. (1) .= Reynolds number based on momentum thickness = temperature = velocity = friction velocity, (r/p) 1/2 = normal coordinate = Reynolds stress = velocity boundary-layer thickness = shear stress = density = ratio of specific heats IJL= viscosity Subscripts m = maximum value, evaluated herein at y/ 5 = 0.5 e = edge w = wall t -stagnation

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