Page 81 - Computational Fluid Dynamics for Engineers

66 2. Conservation Equations

Here x, y, t are dimensionless quantities denned in Eq. (2.5.5), q(y) is the
f
amplitude function of a typical flow variable q (x,y,t), a is a dimensionless
wave number, and uu is the radian (circular) frequency of the disturbance. The
dimensionless forms of a and u are defined by

a = —— = a*L, UJ = (2.5.10)
X x uo
where X x denotes the wavelength in the x-direction. In general q, q , f a and u are
f
complex. With q r and q[ denoting the real and imaginary parts of q', the mag-
,
2
1
2 1//2
nitude of q f is [{q' r) + (<z[) ] and its relative phase angle is ta,n~ (q[/q T). The
real part of the exponential term represents a growth of disturbance amplitude
in x or t, while the imaginary part, exp(i#), can be rewritten as cos# + isin#,
which represents the sinusoidal oscillation in x or i.
The small-disturbance equations given by Eqs. (2.5.6), (2.5.7) and (2.5.8)
can also be expressed in other forms. Eliminating pressure and introducing a
stream function ip(x, y, t) such that

we can express the momentum equations (2.5.7) and (2.5.8) as a fourth-order
partial differential equation

2
2
where V 4 = V V 2 and —S/ ip is the fluctuating ^-component of vorticity,
f
dv' jdx — du /dy. Equation (2.5.12) represents the rate of change of fluctuating
vorticity following the fluid along a mean streamline.
Taking q f in Eq. (2.5.9) to represent the disturbance stream function ty with
q(y) replaced by 0(y), and introducing the resulting expression into Eq. (2.5.12),
we obtain the following fourth-order ordinary differential equation for the am-
plitude <f>(y)
2
2
(f) iv - 2a </>" + « V = iR{pLU - u)((t)" - a 0) - iRau"^) (2.5.13)
where a prime denotes differentiation with respect to y. This equation is known
as the Orr-Sommerfeld equation and is the fundamental equation for incom-
pressible stability theory.
The solutions of Eq. (2.5.13) correspond to small disturbance waves and are
sometimes called Tollmien-Schlichting waves. Despite the major assumptions
made to derive this equation, the solutions of Eq. (2.5.13) are encouragingly
close to the experimental results. The support to linear stability theory was
first provided by the experiments of Schubauer and Skramstad [10] who used
a specially designed low turbulence wind tunnel and generated small sinusoidal