Laminar Boundary Layer Theory (Prandtl 1904)
Let S be the boundary layer thickness which represents the limit beyond which viscous effects are negligible. Let l be the characteristic length of the obstacle. We assume that S ^ l, see Fig. 8.6. Inside the boundary layer one can simplify the NavierStokes equations, using order of magnitude analysis which is based on the rigorous technique of asymptotic expansion. The мcomponent of velocity is of order U, the incoming flow velocity. Pressure is assumed to be of order p = O (pU2), in other words, it can be written
Fig. 8.6 Geometric interpretation of displacement thickness S* (x) 
P(x, y) = pu2n(<8 27)
where П is a dimensionless function of x/1 and y/6(x) only. The coefficient in front of the dimensionless function, pU2, is called a “gage”. It gives the order of magnitude of the term. Furthermore, the partial derivatives are of order, respectively:




(a) From continuity, the two terms must balance, hence


The flow is almost aligned with the body.
(b) From the xmomentum equation: the acceleration terms in the lefthandside are both of order O (pu^j and must balance the viscous term р,^ = о (pjU^j, hence


where the Reynolds number is defined as Rel = pU. In fact, this is true for any x sufficiently large, i. e.
where Rex = pUx. Note that if there is a pressure gradient in the xdirection, its order will match the other terms, as dp = о (p ufsj.
(c) Consider the ymomentum equation. First we multiply it by : the acceleration terms in the lefthandside are both now of order O (pир^ and balance the viscous
term of order O (p U^ as a consequence of the previous result (indeed, one term и has been replaced by v in both sides). The pressure derivative term is now of order O (pир. Comparing this term’s magnitude with the other terms indicates that this term dominates the equation since
U2 
U262 
U2 
( 62 
l > p 13 
= pT 
U) * 

d p 
dv 
dv 
d2v 
dy y> pu dx 
+pv d 
0Г Pdy2 
As a result, to first order, the ymomentum equation reduces to one term and the pressure gradient in the ydirection must vanish
^ = 0 ^ p = p(x) = pi (x) (8.33)
d y
where the subscript i stands for “inviscid”, i. e. in the inviscid flow. Let ui (x) be the inviscid flow velocity at the edge of the boundary layer. From Bernoulli’s equation
p™ + [7] pU2 = pi (x) + 2p (u2(x) + v2(x)) ~ pi (x) + 2pu2(x) (8.34)
Taking the derivative in the xdirection yields
d2 u 
Hence the Prandtl boundary layer equations reduce to
This is a nonlinear system of two equations in two unknowns, (u, v) with a source term function of ui (x) provided by the inviscid flow. It is of parabolic type and can be marched in the flow direction as long as u(x, y) > 0.
The initial/boundary conditions are the following:
• at x = 0 u(0, y) = U, v(0, y) = 0, a uniform, undisturbed velocity profile is given,
• at y = 0, u(x, 0) = v(x, 0) = 0, along the solid wall,
• at y = S(x), u (x, S(x)) = ui (x), at the edge of the boundary layer.
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