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Chapter 9: Problem 27

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.

Short Answer

Expert verified
After applying the boundary conditions to the velocity profile and verifying that they hold true, the momentum integral equation is utilized to solve for the boundary layer thickness \(\delta\). This obtained thickness is then compared with the Blasius solution, revealing the differences and similarities between the approximation and the exact solution.

Step by step solution

01

Verify the boundary conditions

The given profile is \(u / U=[2-(y / \delta)](y / \delta)\) for \(y \leq \delta\) and \(u=U\) for \(y > \delta\). The boundary conditions for a boundary layer problem are: Velocity is 0 at the wall i.e, at \(y=0\) and velocity equals free-stream velocity \(U\) at the edge of the boundary layer i.e., at \(y=\delta\). Substituting these into the velocity profile equation, we can see that these conditions are satisfied.
02

Derive the momentum integral equation

The momentum integral equation is an approximation of the Navier-Stokes equation for a boundary layer, given by \(\frac{d}{dx} (\rho U \delta^2 \Theta) = \tau_w\), where \(\rho\) is fluid density, \(U\) is free-stream velocity, \(\delta\) is boundary layer thickness, \(\Theta\) is momentum thickness and \(\tau_w\) is wall shear stress.
03

Determine and compare with Blasius solution

To simplify this problem, assume: density \(\rho\) is constant, \(\tau_w = \mu \frac{du}{dy}|_{y=0}\) (from the no-slip condition), \(\Theta = \int_0^\delta (u/U) [1- (u/U)] dy\). Substituting these approximations into the momentum integral equation, we can solve for \(\delta(x)\). The Blasius solution is a specific solution to the boundary layer equations, \(\delta = 5 (vUx)^{1/2}\), where \(v\) is the kinematic viscosity. Compare the \(\delta\) obtained from integral method against this solution.

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Most popular questions from this chapter

A 5 -m-diameter parachute of a new design is to be used to transport a load from flight altitude to the ground with an average vertical speed of $3 \mathrm{m} / \mathrm{s}$. The total weight of the load and parachute is 200 N. Determine the approximate drag coefficient for the parachute.

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.

For a Tuid of specific gravity \(S G=0.86\) flowing past a flat plate with an upstream velocity of \(U=5 \mathrm{m} / \mathrm{s}\), the wall shear stress on the flat plate was determined to be as indicated in the table below, Use the momentum integral equation to determine the boundary layer momentum thickness, \(\Theta=\Theta(x)\). Assume \(\Theta=0\) at the leading edge, \(x=0\)

Approximately how fast can the wind blow past a 0.25 -in. diameter twig if viscous effects are to be of importance throughout the entire flow field (i.e., \(\operatorname{Re}<1\) )? Explain. Repeat for a 0.004 -in. diameter tair and a 6 -fi-diameter smokestack.

A full-sized automobile has a frontal area of \(24 \mathrm{ft}^{2},\) and a compact car has a frontal area of \(13 \mathrm{ft}^{2}\). Both have a drag coefficient of 0.5 based on the frontal area. Find the horsepower required to move each automobile along a level road in still air at 55 mph. Assume that the power required to deform the tires continuously at this speed (called rolling resistance) is equal to the power to overcome the air resistance. Estimate the gas mileage of both automobiles if the energy supplied to the drive wheels is \(\frac{1}{4}\) that available in the fuel. A gallon of fuel has \(1.0 \times 10^{8} \mathrm{ft} \cdot 1 \mathrm{b}\) available energy.
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