Linearly Retarded Flow of Howarth

In this example case, the external flow is linearly decelerating on a flat plate, as rep­resented by:

Ue(x) = U0(1 – x/L).

02(*) v

Then, the Thwaites equation is integrated easily to obtain:

Then, X is given by:

From this, we can find the variation of the wall shear stress and displacement thick­ness along the plate from the previous correlation expressions. For example, the skin friction is given by:

C = pS(X) = 2 v(X + 0.09)062

f 1pU2 1pUB ^(1- x / L)0 .

In addition, separation is predicted at X = -0.09, which corresponds to:

leading to:

xsep

= 0.123,

L

which is within 3 percent of the exact result of 0.120.

Flow over a Circular Cylinder

Here, the potential flow is given by:

U

—- = 2sin ф.

U0

This is transformed into the distance along the cylinder, ф = x/R, where R is the cylinder radius and x is the distance along the cylinder. The expression then is expanded in a Taylor series to give:

Ue	f x >	f x 13	f 55 x
= 2		– 0.3331 — + 0.0167
U0	1R J	1R J	1R J

The presence of the boundary layer on such a bluff body, however, significantly alters the potential flow and the polynomial:

Ue	f x ^		f x ^	3	f x 1
-*■ = 1.814		– 0.271		– 0.0471
U0	v R J		v R J		1R J

+

which is a curve fit of the data found by Hiemenz in 1911. It represents a closer fit (White, 1974). Using this second polynomial results in a significantly more accu­rate solution. Separation is predicted at ф = 78.5°, which is close to the experimen­tally observed value of 80.5°. The first polynomial produces a separation point of ф = 104.5°.

Direct Numerical Solution of the Boundary-Layer Equations

In solving the boundary-layer equations, there are numerical alternatives to the integral methods using fully coupled, finite-difference methods. Integral methods tend to be fast, but they have limited range of applicability because they rely on the velocity profiles being specified—sometimes by guesswork—and being of the same geometric family at each axial location, which may not always be the case. Hence, they lack generality. Direct solutions of the governing equations by finite-difference methods are signifi­cantly slower due to the coupled nature of the equations. However, it is possible to take advantage of the rapid convergence of a novel, uncoupled procedure to give a signifi­cantly faster boundary-layer solution, which is then coupled to the inviscid-flow model.

‘ij+1 іі-

where:

The continuity equation is solved for a new value of w by a trapezoidal-rule integration:

(zij zij—1)

i wij—1———– 2—

The equations can be solved in an uncoupled manner, as described by Gad-El-Hak, 1989. The procedure is as follows:

1. The Initial values of и and w are obtained from the г-1 station.

2. The nonlinear coefficients are evaluated.

3. The momentum equation then is solved for u1j.

4. The continuity equation then is used to obtain a new value of wij.

5. The convergence of u1j. is checked.

6. If not converged, then the procedure is repeated from Step 2.

The method described is fast compared with other finite-difference calculations, even though iterations are used at each station. The iterative updating of the nonlinear coefficients, instead of simply using the value at г-1, improves accuracy and allows coarser spacing to be used. The computer code BL uses this solution procedure and is set up to demonstrate a number of flows for various freestream pressure distri­butions, as well as to accept pressure distributions from data files.