Reference no: EM131711175
Problem 1:
Consider a flow field with constant density (ρ) and viscosity (ν) and assume the below equations represent the conservation equations for this flow field.
∂u/∂t + u.∂u/∂x + v.∂u/∂y = g + ν[∂2u/∂x2 + ∂2u/∂y2]
∂v/∂t + u.∂v/∂x + v.∂v/∂y = g + ν[∂2v/∂x2 + ∂2v/∂y2]
∂u/∂x + ∂v/∂y = 0
(a) List the assumptions made in order to simplify the Navier-Stokes and Mass Conservation Equations to the ones show above
(b) Differentiate the x-momentum equation with respect to y
(c) Differentiate the y-momentum equation with respect to x
(d) Now subtract the equation obtained in (c) from the equation obtained in (b)
(e) Rewrite the equation in terms of ωz. Assume that that ωz = (∂v/∂x - ∂u/∂y)
(f) Use the definition of ψ(v = ∂ψ/∂x, u = ∂ψ/∂y), the stream function, to rewrite ωz in terms of ψ
(g) Using the new definition of ωz rewrite the Equation obtained in (e) to show that it is now:
∂/∂t(∇2ψ) + ∂ψ/∂y.∂/∂x(∇2ψ) - ∂ψ/∂x.∂/∂y(∇2ψ) = υ(∇4ψ)
(h) Expand ∇4? What is it?
Problem 2:
For practical applications we can assume that an airplane wing is a flat plate. A plane wing flying at altitude of 10km (ρair = 0.4135 kg/m3 and μ = 1.458 x 10-5 kg/ (m-s)) can be approximated by a stationary flat plate with a uniform velocity fluid (U) passing over it. If you assume the below velocity distribution, use the momentum integral and determine the drag force per unit width on a 0.2m long wing flying at 100 m/s.
u/U = a + b(Y/δ) + c(y/δ)2 + d(y/δ)3
You will need to use boundary conditions below with the momentum integral to find the values of the coefficients.
At y = 0 u = 0 At y = δ ∂u/∂y = 0
At y =0 ∂2u/∂y2 = 0 At y = δ u = U
d/dx 0∫∞ (U∞ - u)u dy + dU∞/dx 0∫∞ (U∞ - u)dy = ν(∂u/∂y)y=0
You may estimate drag force on the wing using the below relationships (notice that dz = 1):
Τo = μdu/dy|y=0
Cf,x = Τo/(1/2ρU2)
C‾f,L = 1/L 0∫L Cf,x dx
Drag = 1/2C‾f,LρAsU2
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