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 Navier-Stokes equations, using order of magnitude analysis which is based on the rigorous technique of asymptotic expansion. The м-component 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

Laminar Boundary Layer Theory (Prandtl 1904)

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:

Laminar Boundary Layer Theory (Prandtl 1904)

d2 d2

dx2 « dy2

 

(8.28)

 

Laminar Boundary Layer Theory (Prandtl 1904)

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

Laminar Boundary Layer Theory (Prandtl 1904)

(8.29)

 

The flow is almost aligned with the body.

(b) From the x-momentum equation: the acceleration terms in the left-hand-side are both of order O (pu^j and must balance the viscous term р,^ = о (pjU^j, hence

Laminar Boundary Layer Theory (Prandtl 1904)

(8.30)

 

Laminar Boundary Layer Theory (Prandtl 1904)

where the Reynolds number is defined as Rel = pU. In fact, this is true for any x sufficiently large, i. e.

Laminar Boundary Layer Theory (Prandtl 1904)

where Rex = pUx. Note that if there is a pressure gradient in the x-direction, its order will match the other terms, as dp = о (p ufsj.

(c) Consider the y-momentum equation. First we multiply it by |: the acceleration terms in the left-hand-side 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term of order O (p U^ as a consequence of the previous result (indeed, one term и has been replaced by v in both sides).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 Pdy-2

As a result, to first order, the y-momentum equation reduces to one term and the pressure gradient in the y-direction 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)

Подпись: dp _ dpi (x) dx dx Laminar Boundary Layer Theory (Prandtl 1904) Подпись: (8.35)

Taking the derivative in the x-direction yields

d2 u

Laminar Boundary Layer Theory (Prandtl 1904) Подпись: (8.36)

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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