Question

If a velocity profile of a boundary layer is u= a+by+cy^2, please calculate the value of a, b and c, using the boundary conditions.

Please show all steps. I am trying to understand the process. Thank you (:

Answer 1

velocity profile =

let's assume that the height of the plate = y = h

if we apply boundary conditions,

in the vicinity of the plate when y= 0, velocity u = 0.

at the free surface of the plate y= h, velocity u = U

at free surface velocity gradient = du/dy =0

where U = free stream velocity.

The velocity(u) gradually reaches to free stream velocity (U) as we move from plate surface to the free surface along the boundary layer. Thus velocity gradient (du/dy) is present only within the boundary layer, at the free surface, velocity gradient du/dy = 0.

we have three boundary conditions, and three unknowns namely a,b and c.

let us apply three boundary conditions to the velocity distribution equation of   in the boundary layer.

by applying first boundary condition, y=0 at u= o we get

0 = a+0+0 = a=0.

a=0.

by applying second boundary condition,

U=0+bh+ch²

by applying third boundary condition du/dy =0 at the free surface, y=h. we get

du/dy = b+2cy.

0 = b+2ch

b = -2ch

let us substitute equation 5 in equation 4.

U = -2ch²+ch² = ch²

c = U/h².

we know a=0 and c = U/h².

from eqution 5 we know that b = -2ch

so b = -2(U/h²)*h = -2U/h.

Constant values are

a = 0

b = -2U/h

c=U/h²