文档介绍：Problem Set 11B Name: ____________________
Assigned: Thursday, November 15
Due: Wednesday, November 21 (11:00 AM at 5-303)
1. Consider an approximate velocity profile within a laminar boundary layer over a wall
given by:
uyx(;) ax()ηη+< bx ()2 y δ() x
= (1)
Ux() 1()yx≥δ
where η≡ yx/δ( ) , and ax ( ) and bx( ) are constants at any given x, and δ(x ) is a
measure of the boundary layer thickness at x.
(a) By applying one boundary condition at y = δ which matches the boundary layer flow
with the outer flow Ux ( ) , obtain an algebraic relationship between a and b.
(b) Now write both a(x) and b(x) in terms of a new parameter Λ(x ) such that the
relationship between a and b found in (a) is maintained. In other words, find a()Λ and
b()Λ.
(c) Write the velocity profile (1) in terms of Λ instead of a and b.
(d) Using the assumed velocity profile and the x-momentum boundary layer equation
∂∂u u dU ∂2 u
uvU+= +ν
∂∂x y dx ∂ y2
evaluated at y = 0 , obtain an expression for Λ in terms of U, δ, and ν. Also obtain an
expression for the pressure gradient dp / dx in terms of ρ,U, Λ, δ, and ν.
(Hint: Polhausen approach, but a different profile than the Lecture 17 notes - be careful)
(e) For what value of Λ is dp/ dx zero? The pressure gradient is favorable when Λ is
[greater than, less than]