Nonlinear Steady Compressible Flow Problem for a Foil

Consider a moderately curved foil in a two-dimensional steady compressible flow near a flat ground. To the lowest order, the corresponding channel flow equation, formulated with respect to the relative motion velocity potential фх, can be derived from the more general three-dimensional equations (5.14)- (5.16) in the form

Подпись: (5.51)

Подпись: where ys(x) = ys(x)/h, u(x)

1 + ^(7 – 1)M,2(1 – w2) – ^s(*)M02u-jlu2 = 0,

Grouping with respect to d In ys and dp,

Подпись: dp. Подпись: (5.58)

(1 + pp){ 1 – p) dlnys – 1(1 + pp)dp + 1m02(1 – p)dp = 0,

Integrating (5.58),

In2/s = ~^ln(l – p)~ ^^-hi(l +pp)+]nC*. (5.59)

Applying the trailing edge condition ys(0) = l, p = 0, we find that С* = 1 so that

4 = (X ~pX1 + PP)M°/>1- (5-60)

Vs

My*)’

where 9 = 9/h, and 9 is pitch. For a flat plate


We introduce the pressure coefficient

ys(x) = l + 9x, y’s = e re Є [-1,1],

so that the lift coefficient is given by

Подпись: Cy(0,MoПодпись: V(p,e,M0)^dp,

Подпись: Уо Подпись: (5.64) Подпись: where pi = p(9) is determined by using formula (5.60). For a flat plate in incompressible flow, the expression for the lift coefficient to the leading order is

(5.63)

Some calculated results, corresponding to the case of a flat plate, are pre­sented versus the relative pitch angle 9 for the Mach numbers M0 = 0.5 and M0 = 0 (incompressible flow case) in Fig. 5.10.

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