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Chapter 6: Problem 121

\(\mathrm{A}\) crude model of a tornado is formed by combining a \(\operatorname{sink},\) of strength \(q=2800 \mathrm{m}^{2} / \mathrm{s},\) and a free vortex, of strength \(K=5600 \mathrm{m}^{2} / \mathrm{s} .\) Obtain the stream function and velocity potential for this flow field. Estimate the radius beyond which the flow may be treated as incompressible. Find the gage pressure at that radius.

Short Answer

Expert verified
The stream function for this flow field is \(1400/ \pi r\) m虏/s and the velocity potential is \(-1400 cos\theta/\pi\) m虏/s. The specific values of the radius for the flow to be treated as incompressible and the gage pressure at that radius depend on the values of the speed of sound and the fluid density.

Step by step solution

01

Calculate the Stream Function

Stream function (\( \psi \)) for a flow comprising a sink and a free vortex can be given by superposing the stream functions for each component. The stream function for a sink is \(-q/\(2\pi r\), and for a free vortex it's \(K/\(2\pi r\). So, the combined stream function is \((K-q)/2\pi r\). Given that \(q=2800 m^2/s\) and \(K=5600 m^2/s\), we get \( \psi = (5600-2800)/2\pi r = 1400/ \pi r \)
02

Calculate the Velocity Potential

The velocity potential (\(\phi\)) is null for a free vortex and \(-q/2\pi cos\theta\) for a sink. By adding both terms \(\phi = -q/2\pi cos\theta\), given that \(q=2800 m^2/s\), we get \(\phi = -2800/2\pi cos\theta = -1400 cos\theta/\pi \)
03

Calculate Radius for Incompressible Flow

A flow field may be treated as incompressible if the speed of the flow is much less than the speed of sound. For our tornado model, the radial velocity (\(u_r\)) is \( -q/2\pi r\), and the tangential velocity (\(u_\theta\)) is \( K/2\pi r\). The speed \(U\) \(= sqrt(u_r^2 + u_\theta^2)\). Let's equate \(U/c\) (c is speed of sound) to a very small number (like 0.03), and solve this equation for \(r\).
04

Calculate the Gage Pressure at the Radius

The gage pressure at a point in a fluid is given by \(P = P_0 + 1/2 * rho * U^2\), where \(P_0\) is reference pressure (we can take it as atmospheric pressure), rho is fluid density, and \(U\) is speed of the flow. Substitute \(U\) from step 3 into this equation to calculate the gage pressure. The value of \(rho\) should be known or assumed.

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Most popular questions from this chapter

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.

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 \(x\) component of velocity in an incompressible flow field is given by \(u=A x,\) where \(A=2 \mathrm{s}^{-1}\) and the coordinates are measured in meters. The pressure at point \((x, y)=(0,0)\) is \(p_{0}=190 \mathrm{kPa}\) (gage). The density is \(\rho=1.50 \mathrm{kg} / \mathrm{m}^{3}\) and the \(z\) axis is vertical. Evaluate the simplest possible \(y\) component of velocity. Calculate the fluid acceleration and determine the pressure gradient at point \((x, y)=(2,1) .\) Find the pressure distribution along the positive \(x\) axis.

The inlet contraction and test section of a laboratory wind tunnel are shown. The air speed in the test section is \(U=50 \mathrm{m} / \mathrm{s} .\) A total- head tube pointed upstream indicates that the stagnation pressure on the test section centerline is \(10 \mathrm{mm}\) of water below atmospheric. The laboratory is maintained at atmospheric pressure and a temperature of \(-5^{\circ} \mathrm{C}\) Evaluate the dynamic pressure on the centerline of the wind tunnel test section. Compute the static pressure at the same point. Qualitatively compare the static pressure at the tunnel wall with that at the centerline. Explain why the two may not be identical.

Consider the flow field represented by the potential function $\phi=A x^{2}+B x y-A y^{2} .$ Verify that this is an incompressible flow and determine the corresponding stream function.
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