Page 263 - Computational Fluid Dynamics for Engineers

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8.3 e -Method 253

To find the eigenvalues at a new chosen value of a r , we may again obtain
better estimates of UJ and R by writing

BR \
R = R 0 + ( _ _ ) ( 8a r (8.2.25a)
da r)

i BUJ ,
UJ = UJ 0+ ( ^r ) Sa r (8.2.25b)
da r
However, the additional work required to compute duj/8a r may not justify this
advantage since the calculations are being performed near the critical Reynolds
number region for small increments in a r which make the initial guess and the
eigenvalue problem easier.
The calculation of UJ and R for a specified value of a r is very similar to the
one in which UJ and a r are computed for a specified R. With initial estimates of
UJ and i?, Eq. (4.4.29) can be solved to see whether Eq. (8.2.11) is satisfied. If
not, new estimates are computed by using Newton's method. Equations similar
to (8.2.15b) and (8.2.20) are solved to get df /duo and df/dR, respectively, and
then new increments in R and UJ are obtained from the solution of Eqs. (8.2.24).
The procedure is repeated until convergence, when, for example, \6UJ\ and \6R\
become less than a specified tolerance parameter.

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8.3 e -Method

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The e -method, which utilizes linear stability theory, was first used by Smith [4]
and van Ingen [5] and is discussed in detail in [3]. The basic assumption is that
transition starts when a small disturbance is introduced at a critical Reynolds
number and is amplified by a factor of e n which, for a typical value of n equal to
9, is about 8000. The calculation of transition with this procedure is relatively
straight-forward in two-dimensional (and in axisymmetric) flows, requiring the
calculation of the amplification rates (—on) as a function of x (or R) for a
range of dimensional frequencies UJ* . The stability calculations are preceded
by boundary-layer calculations and, for a given external velocity distribution
u e(x) and freestream Reynolds number, the laminar boundary-layer equations
are solved to obtain the streamwise velocity profile u and its second derivative
u". The stability calculations begin at a Reynolds number, R$*, slightly larger
than the critical Reynolds number, R$* r, on the lower branch of the neutral
stability curve at an x-location, say x = x\ (see Fig. 8.3). At this point, since i/,
u' f and R are known, a r and UJ can be computed by the procedure of subsection
8.2.2 and the dimensional frequency co>* determined from Eq. (2.5.10), that is,

UJ* =UJ^± (8.3.1)