Applications of Von Karman's Integral Momentum Equation for Boundary Layers

Applications of von Karman's Integral Momentum Equation for Boundar...

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Applications of von Karman's Integral Momentum Equation for Boundary Layers 1. An absolute estimate of flat-plate laminar boundary layer properties for Application of method A Self-similarity of boundary layer velocity profiles signifies that

where

, the boundary layer thickness, = f(x), the distance from the leading edge.

The boundary conditions to be satisfied for a boundary layer over a flat plate are : At y = 0 : u = 0 ,

(since

),

(The last condition is imposed by the B L equation

since LHS = 0 at y = 0.) and at

: u = uo ,

Simplest polynomial form of

(since

at

)

satisfying these is (see Tutorial example)

Now it is found that this form of velocity profile does indeed correspond quite well with measurements for laminar B.L.s over flat plates: so what follows can represent the behaviour of such a laminar B.L. Then, by definition, Displacement Thickness

since for all

, u = uo. Dividing by

is

(

) gives

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Similarly, for Momentum Thickness

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:

Further,

From the velocity profile assumed

Therefore

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Substituting in Momentum Equation for zero pressure gradient

Integrating

or

Hence

and

Local friction coefficient,

or

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( Cf., eg. this last result with the exact Blasius result for the laminar BL

This close agreement is, however, fortuitous.) Tutorial 6 Question 2 and 3 2. Deduction of the Characteristics of a Turbulent Boundary Layer over a Flat Plate from Pressure Drop Data for Turbulent Pipe Flow. Application of method B The velocity profile in that part of the turbulent BL accessible to measurement is

(1)

where

is the local boundary layer thickness.

We could find

if we knew

But, in fact, we do not know additional data to find

or

as a function of x and if equation (1) applied at the wall.

and Eq (1) clearly does not apply very close to the wall; so we require .

From measurements of the pressure drop in turbulent flow in pipes

Assuming that the velocity profile in the turbulent boundary layer in a pipe is described by equation (1), with pipe radius, it is easy to show that um = 0.817 uo, where uo is the velocity on the pipe axis.

(2)

Evaluating

directly from equation (1) by substitution in the defining equation, ignoring the small effect

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on the total momentum flux in the boundary layer of the departure of the velocity profile from eq (1) near the wall. (3)

Substituting equations (2) and (3) into the momentum equation, remembering that

here:

Rearranging and integrating from 0 to x,

Averaged over the length x, integration gives

Tutorial 6 Question 1

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3. Turbulent Boundary Layers on Curved Surfaces. Pressure Gradient Effects a) Favourable Pressure Gradient

Over portion of surface inclined to the flow as shown (+ve angle of incidence)

Consider first the implications for the overall behaviour of the BL, using the von Karman equation

As we have seen, LHS is only a weak function of uo, particularly for turbulent boundary layers; so LHS will change little as uo increases. Hence, as

changes from 0 to > 0,

Consider the relative magnitudes of these effects. For the turbulent BL, over a flat plate at

must diminish.

, for instance, our previous analysis shows that

is of the order of the BL thickness. The 2nd term on the RHS is thus of the order of the fractional change in uooccurring over an x-distance equivalent to the BL thickness. It is clearly possible for this to exceed

or 0.3%, a typical value of

, without violating the BL assumption that

; and at entry to a converging duct, for instance, where the BL thickness is comparable to the

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half width, such axial velocity gradients are easily achieved and exceeded. In such cases, if the 2nd term on RHS

becomes negative and the BL becomes thinner in the direction of flow - over

upwind surfaces of an obstruction or in the converging portion of a channel, for instance. b) Adverse Pressure Gradient

More interesting and complicated is the case where

ie flow with an adverse pressure gradient. Using the same arguments as before

may readily exceed

numerically; in which case if

, von Karman's equation requires

to

become more positive than in a straight parallel flow and the BL thickness increases abnormally rapidly in the flow direction. This result, that the BL thickness increases abnormally rapidly over the downstream (rear) face of an obstacle or at the wall of a diverging pipe section (diffuser) has very far reaching practical implications. These are best introduced by returning to the detailed equation of motion for a boundary layer. Next : Separation of Boundary Layer David Balmer Last modified: Wed Dec 2 15:48:29 GMT

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