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Chapter 6: Problem 30

To model the velocity distribution in the curved inlet section of a water channel, the radius of curvature of the streamlines is expressed as \(R=L R_{0} / 2 y .\) As an approximation, assume the water speed along each streamline is \(V=10 \mathrm{m} / \mathrm{s}\) Find an expression for and plot the pressure distribution from \(y=0\) to the tunnel wall at \(y=L / 2,\) if the centerline pressure (gage) is \(50 \mathrm{kPa}, L=75 \mathrm{mm},\) and \(R_{0}=0.2 \mathrm{m} .\) Find the value of \(V\) for which the wall static pressure becomes 35 kPa.

Short Answer

Expert verified
By using Bernoulli's equation, it's possible to create an expression to represent the pressure distribution along the y-axis of the water flow. To get a visual representation, plot the curve of this function. When the speed is adjusted to make the pressure at the wall 35 kPa, we will find the required velocity.

Step by step solution

01

Pinpoint Known Variables

First identify and note down all the given variables in the problem. We have \(V=10 \mathrm{m}/\mathrm{s}\) as the water speed along each streamline, the centerline gage pressure is \(50 \mathrm{kPa}\), the length \(L=75 \mathrm{mm} = 0.075 \mathrm{m}\) , and \(R_{0}=0.2 \mathrm{m}\) as the radius of curvature.
02

Applying Bernoulli's Equation

The Bernoulli's equation will be a key in this problem solving. It suggests that some mechanical energy losses notwithstanding, the sum of pressure, kinetic and gravitational potential energies per unit volume in a flowing fluid stays constant. The fluid pressure \(p(y)\) at the height \(y\) can be defined as \(p(y) = p_0 + 0.5 \rho (V^2 - (V^2 \cdot (1-(2y/L))^2))\). Here, \(p_0 = 50 \mathrm{kPa}\) is the pressure when \(y=0\), \(V=10 \mathrm{m}/\mathrm{s}\) is the speed of the fluid along the streamline, \(\rho = 1000 \mathrm{kg}/\mathrm{m}^3\) is the density of water and \(L= 0.075 \mathrm{m}\) is the length of the streamline.
03

Plotting The Pressure

This step needs to plot the pressure distribution from \(y=0\) to \(y=L/2\) by using the expression obtained in step 2. This can be solved using a graph plotting software or online tool.
04

Finding the Speed for a Specific Pressure

The final step requires to find the speed \(V\) such that \(p(L/2) = 35 \mathrm{kPa}\). For this, first calculate the term inside the square root of the Bernoulli's equation obtained in step 2 for \(y=L/2\). It is \((1-(4y/L))^2\). Then, solve the equation \(35 = 50 + 0.5 \times 10^3 \times (V^2 - V^2(1-(4*0.0375/0.075))^2) / 10^3\) for \(V\).

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Most popular questions from this chapter

The velocity field for a two-dimensional flow is \(\vec{V}=(A x-B y) t \hat{\imath}-(B x+A y) t \hat{j},\) where \(A=1 \mathrm{s}^{-2} B=2 \mathrm{s}^{-2}\) \(t\) is in seconds, and the coordinates are measured in meters. Is this a possible incompressible flow? Is the flow steady or unsteady? Show that the flow is irrotational and derive an expression for the 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) \\]

Air flow over a stationary circular cylinder of radius \(a\) is modeled as a steady, frictionless, and incompressible flow from right to left, given by the velocity field \\[ \vec{V}=U\left[\left(\frac{a}{r}\right)^{2}-1\right] \cos \theta \hat{e}_{r}+U\left[\left(\frac{a}{r}\right)^{2}+1\right] \sin \theta \hat{e}_{\theta} \\] Consider flow along the streamline forming the cylinder surface, \(r=a .\) Express the components of the pressure gradient in terms of angle \(\theta .\) Obtain an expression for the variation of pressure (gage) on the surface of the cylinder. For \(U=75 \mathrm{m} / \mathrm{s}\) and \(a=150 \mathrm{mm},\) plot the pressure distribution (gage) and explain, and find the minimum pressure. Plot the speed \(V\) as a function of \(r\) along the radial line \(\theta=\pi / 2\) for \(r>a\) (that is, directly above the cylinder), and explain.

\(\mathrm{A}\) horizontal flow of water is described by the velocity field \(\vec{V}=(-A x+B t) \hat{i}+(A y+B t) \hat{j},\) where \(A=1 \mathrm{s}^{-1}\) and \(B=2 \mathrm{m} / \mathrm{s}^{2}, x\) and \(y\) are in meters, and \(t\) is in seconds. Find expressions for the local acceleration, the convective acceleration, and the total acceleration. Evaluate these at point (1,2) at \(t=5\) seconds. Evaluate \(\nabla p\) at the same point and time.

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