A crude approximation for the x component of velocity in an incompressible laminar boundary layer is a linear variation from u = 0 at the surface (y = 0 ) to the freestream velocity, U, at the boundary-layer edge (y = δ). The equation for the profile is u = Uy/δ, where δ = cx1/2 and c is a constant. (a) What is the simplest expression for the y component of velocity (use x, y, u)? (b) Evaluate the maximum value of the ratio v/U, at a location where x = 0.5 m and δ = 6.0 mm.

2.5 * 10^-3

Explanation:

solution:

The simplest solution is obtained if we assume that this is a two-dimensional steady flow, since in that case there are no dependencies upon the z coordinate or time t. Also, we will assume that there are no additional arbitrary purely x dependent functions f (x) in the velocity component v. The continuity equation for a two-dimensional in compressible flow states:

δu/δx+δv/δy=0

so that:  

δv/δy= -δu/δx

Now, since u = Uy/δ, where δ = cx^1/2, we have that:

u=U*y/cx^1/2

and we obtain:  

δv/δy=U*y/2cx^3/2

The last equation can be integrated to obtain (while also using the condition of simplest solution - no z or t dependence, and no additional arbitrary functions of x):  

v=∫δv/δy(dy)=U*y/4cx^1/2

 =y/x*(U*y/4cx^1/2)

 =u*y/4x

which is exactly what we needed to demonstrate.  

Also, using u = U*y/δ in the last equation we can obtain:  

v/U=u*y/4*U*x

     =y^2/4*δ*x

which obviously attains its maximum value for the which is y = δ (boundary-layer edge). So, finally:

(v/U)_max=δ^2/4δx

                =δ/4x

                =2.5 * 10^-3

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