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Chapter 9: Problem 31

A laminar boundary layer velocity profile is approximated by \(u / U=2(y / \delta)-2(y / \delta)^{3}+(y / \delta)^{4}\) for \(y \leq \delta,\) and \(u=U\) for \(y >\delta .\) (a) Show that this profile satisfies the appropriate boundary conditions. (b) Use the momentum integral equation to determine the boundary layer thickness, \(\delta=\delta(x)\). Compare the result with the exact Blasius solution.

Short Answer

Expert verified
The velocity profile function satisfactorily satisfies the laminar boundary layer conditions. Integration uses the momentum integral equation, followed by comparison with the Blasius solution. Note that differences between the obtained result and the Blasius solution might arise due to the used approximation for the velocity profile.

Step by step solution

01

Check Boundary Conditions

The boundary conditions for a laminar boundary layer are as follows: 1. At \(y=0\), the fluid velocity is zero due to the no-slip condition (\(u=0\)). 2. At a location further from the boundary (\(y=\delta\)), the fluid velocity equals the free stream velocity (\(u=U\)).So, plug these values into the velocity profile to check if it satisfies these conditions.
02

Application of Momentum Integral Equation

The momentum integral equation is given by: \(\frac{d}{dx} \int_0^{\delta} (U-u)u dy = \tau_w / \rho\)where \(\tau_w\) is the wall shear stress and \(\rho\) is the fluid density. Integrate the profile function over \(y\), multiply the result with \( (U-u)\) and differentiate with respect to \( x\). This is then set equal to \(\tau_w / \rho\) to solve for \(\delta(x)\).
03

Comparison with Blasius solution

The Blasius solution is a well-known solution for laminar boundary layer over flat plates. It is given by: \(\delta = \frac{5x}{\sqrt{Re_x}}\)where \(Re_x\) is the Reynolds number at the position \(x\). To compare the result from the momentum integral equation with the Blasius solution, insert the obtained numerical value for \(\delta\) into the Blasius equation.

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Most popular questions from this chapter

An airplane flies at \(150 \mathrm{km} / \mathrm{hr}\). (a) The airplane is towing a banner that is \(b=0.8 \mathrm{m}\) tall and \(\ell=25 \mathrm{m}\) long. If the drag coefficient based on area \(b \ell\) is \(C_{D}=0.06,\) estimate the power required to tow the banner. (b) For comparison, determine the power required if the airplane was instead able to tow a rigid flat plate of the same size. (c) Explain why one had a larger power requirement (and larger drag) than the other. (d) Finally, determine the power required if the airplane was towing a smooth spherical balloon with a diameter of \(2 \mathrm{m}\)

Consider the following cases. (a) A small 0.6 -in.-long fish swims with a speed of 0.8 in/s. Would a boundary layer type flow be developed along the sides of the fish? Explain. (b) A \(12-f t\) -long kayak moves with a speed of \(5 \mathrm{ft} / \mathrm{s}\). Would a boundary layer type flow be developed along the sides of the boat? Explain.

An airplane flies at a speed of \(400 \mathrm{mph}\) at an altitude of \(10,000 \mathrm{ft}\). If the boundary layers on the wing surfaces behave as those on a flat plate, estimate the extent of laminar boundary layer flow along the wing. Assume a transitional Reynelds number of \(\mathrm{Re}_{\mathrm{xcr}}=5 \times 10^{5} .\) If the airplane maintains its 400 -nph speed but descends to sea- level elevation, will the portion of the wing covered by a laminar boundary layer increase or decrease compared with its value at \(10,000 \mathrm{ft}\) ? Explain.

Commercial airliners normally cruise at relatively high altitudes \((30,000 \text { to } 35,000 \mathrm{ft}) .\) Discuss how flying at this high altitude (rather than \(10,000 \mathrm{ft}\), for example) can save fuel costs.

A laminar boundary layer velocity profile is approximated by \(u / U=[2-(y / \delta)](y / \delta)\) for \(y \leq \delta,\) and \(u=U\) for \(y>\delta\) (a) Show that this parabolic profile satisfies the appropriate boundary conditions. (b) Use the momentum integral equation to determine the boundary layer thickness, \(\delta=\delta(x)\). Compare the result with the exact Blasius solution.
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