Ask Question
13 September, 06:01

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.

+3
Answers (1)
  1. 13 September, 06:17
    0
    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
Know the Answer?
Not Sure About the Answer?
Find an answer to your question 👍 “A crude approximation for the x component of velocity in an incompressible laminar boundary layer is a linear variation from u = 0 at the ...” in 📗 Physics if the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers