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Chapter 6: Problem 58

Consider frictionless, incompressible flow of air over the wing of an airplane flying at \(200 \mathrm{km} / \mathrm{hr}\). The air approaching the wing is at 65 kPa and \(-10^{\circ} \mathrm{C}\). At a certain point in the flow, the pressure is 60 kPa. Calculate the speed of the air relative to the wing at this point and the absolute air speed.

Short Answer

Expert verified
The speed of the air relative to the wing is around 289.15 m/s and the absolute air speed is approximately 233.59 m/s.

Step by step solution

01

Conversion into consistent units

First, convert the speed of the airplane from km/hr into m/s. This is done by multiplying the given speed \(200 \mathrm{km/hr}\) by \(\frac{5}{18}\) to obtain \(55.56 \mathrm{m/s}\).
02

Application of Bernoulli's Principle

Next, apply Bernoulli鈥檚 principle to solve for the speed of air relative to the wing. Bernoulli's Principle \( P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \), where \(P_1\) and \(P_2\) are the pressure at the two points, \(\rho\) is the fluid density, \(v_1\) and \(v_2\) are the velocities at the two points, \(g\) is the acceleration due to gravity, and \(h_1\) and \(h_2\) are the heights at the two points. Ignoring the heights (since height change is negligible for flow over airplane wing) and the initial velocity (which is zero), simplifying the equation gives \(v_2 = \sqrt { \frac{2 (P_1 - P_2)}{\rho}} \). The air density \(\rho\) can be found using the Ideal Gas Law \( \rho= \frac {P_{air}}{RT}\), where \(P_{air}\) is the pressure of the air, \(R\) is the specific gas constant, and \(T\) is the temperature in Kelvin. Convert -10掳C to Kelvin to get 263.15 K. The specific gas constant for air is 287 J/(kg K). Using this information, calculate \(\rho\) to get roughly 0.816 kg/m\(^3\).
03

Calculate the Relative Speed

Use the simplified Bernoulli's equation and the found density value to calculate \(v_2\), thus determining the relative speed of the air to the wing. Use \(P_1=65 KPa\) and \(P_2=60 KPa\). Be sure to convert these pressures into Pascal (Pa) to maintain consistent units. This results in a relative speed of approximately 289.15 m/s.
04

Calculate the Absolute Speed

Subtract the speed of the airplane from the calculated relative speed to obtain the actual speed of the air or absolute speed. This results in an absolute speed of approximately 233.59 m/s.

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Most popular questions from this chapter

For the flow of Problem 4.123 show that the uniform radial velocity is \(V_{r}=Q / 2 \pi r h .\) Obtain expressions for the \(r\) component of acceleration of a fluid particle in the gap and for the pressure variation as a function of radial distance from the central holes.

A horizontal axisymmetric jet of air with 0.4 in. diameter strikes a stationary vertical disk of 7.5 in. diameter. The jet speed is \(225 \mathrm{ft} / \mathrm{s}\) at the nozzle exit. A manometer is connected to the center of the disk. Calculate (a) the deflection, if the manometer liquid has \(\mathrm{SG}=1.75,(\mathrm{b})\) the force exerted by the jet on the disk, and (c) the force exerted on the disk if it is assumed that the stagnation pressure acts on the entire forward surface of the disk. Sketch the streamline pattern and plot the distribution of pressure on the face of the disk.

The velocity field in a two-dimensional, steady, inviscid flow field in the horizontal \(x y\) plane is given by \(\vec{V}=(A x+B) \hat{i}-A y \hat{j},\) where \(A=1 \mathrm{s}^{-1}\) and \(B=2 \mathrm{m} / \mathrm{s} ; x\) and \(y\) are measured in meters. Show that streamlines for this flow are given by \((x+B / A) y=\) constant. Plot streamlines passing through points \((x, y)=(1,1),(1,2),\) and \((2,2) .\) At point \((x, y)=(1,2),\) evaluate and plot the acceleration vector and the velocity vector. Find the component of acceleration along the streamline at the same point; express it as a vector. Evaluate the pressure gradient along the streamline at the same point if the fluid is air. What statement, if any, can you make about the relative value of the pressure at points (1,1) and (2,2)\(?\)

The flow field for a plane source at a distance \(h\) above an infinite wall aligned along the \(x\) axis is given by \\[ \begin{aligned} \vec{V} &=\frac{q}{2 \pi\left[x^{2}+(y-h)^{2}\right]}[x \hat{i}+(y-h) \hat{j}] \\\ &+\frac{q}{2 \pi\left[x^{2}+(y+h)^{2}\right]}[x \hat{i}+(y+h) \hat{j}] \end{aligned} \\] where \(q\) is the strength of the source. The flow is irrotational and incompressible. Derive the stream function and velocity potential. By choosing suitable values for \(q\) and \(h,\) plot the streamlines and lines of constant velocity potential.

An incompressible liquid with negligible viscosity and density \(\rho=1.75\) slug/ft \(^{3}\) flows steadily through a horizontal pipe. The pipe cross- section area linearly varies from 15 in \(^{2}\) to 2.5 in \(^{2}\) over a length of 10 feet. Develop an expression for and plot the pressure gradient and pressure versus position along the pipe, if the inlet centerline velocity is \(5 \mathrm{ft} / \mathrm{s}\) and inlet pressure is 35 psi. What is the exit pressure? Hint: Use relation \\[ u \frac{\partial u}{\partial x}=\frac{1}{2} \frac{\partial}{\partial x}\left(u^{2}\right) \\]
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