Fluid Mechanics. Lecture 6 The boundary-layer equations. The need for the boundary-layer model.
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The boundary-layer equations
( A question for you)
x has a magnitude comparable to L
x* has an order of magnitude of 1.
y* is at least an order of magnitude smaller than 1.
[V*] has to be of order O(d) to satisfy continuity, i.e..
No term can be omitted hence the continuity equation remains as it is, i.e.
To make the above equation valid, we must have:
ReL has to be large and x-gradients in the viscous term can be dropped in comparison with y-gradients. The dimensional form of the equation thus becomes:
To do an order of magnitude analysis for each term and estimate the order of magnitude for
Hence at most
The pressure can be assumed to be constant across the boundary layer over a flat plate. Hence the pressure only varies in the x-direction and the pressure at the wall is equal to that at the edge of the layer, i.e.
it may not be adequate to
assume constant pressure across boundary layer. Then one needs to apply radial equilibrium to compute P (see Slide 16)
and then only velocity gradients normal to the wall are significant in the viscous term
At wall :
y = 0; U = V = 0
y = d; U = V U
Pressure gradient across boundary layer:
Assume a linear velocity distribution, i.e. integrating from y=0 to d gives
Hence pressure variations across the boundary layer are negligible when