Page 250 - Computational Fluid Dynamics for Engineers

Problems 239

To demonstrate the power of Richardson extrapolation, consider the solu-
tions of Eq. (7.3.11) and compare the computed f^ values with h = 0.1 with
those obtained by Richardson extrapolation. Use three different uniform spac-
1
ings, h(°) = 0.8, M ) = 0.4, and h^ = 0.2. Perform extrapolation with /^[/i (0) ]
and /^[/i (1) ] to get the results for f£[h(°\ h^]; with / ^ ( 1 ) ] and /^[/i (2) ] for

plicitly, in order to determine f!^[h^\h^ \ l use Eq. (P7.2.1a) to get

Since h^ = \h^ in this case, Eq. (P7.2.3a) can also be written as

(
/> °U (1) ] = f/> (1) ] - \f>^] (P7.2.3b)

7-3. Compute the boundary-layer development of a laminar flow over a circular
cylinder of radius ro placed normal to the freestream velocity UQQ. Take the
inviscid velocity distribution with £ = x/ro as

= 2UOQ sin £
u e
with 2, 4, 6, 8 degree intervals in £. Plot T W/ QU^^OOTQ/U) / as a function of
1 2
£. Identify the location of flow separation. Note that from the definition of ra,
£cos£
srn£

which indicates that the flow starts as a stagnation point flow (why?).
Take h = 0.2.

7-4. Apply the Richardson extrapolation to compute f!^ values obtained in
Problem 7.3 for streamwise spacings, k^ — 8°, k^ = 6° and k^ = 4° and
compare the results with those corresponding to k n = 2°. Take h = 0.2 in all
cases.

7-5. Consider a laminar flow past a flat plate with uniform suction. The bound-
ary conditions follow from Eq. (7.3.1) and can be written as

y = 0, u = 0, v = v w = const (P7.5.1a)
y = (5, U = U e = Uoo (P7.5.1b)

At a certain distance from the leading edge, the boundary-layer thickness <5,
which in general is a function of x, becomes constant and stays constant with
increasing x. A s a result the streamwise velocity component u varies only with