Chapter 3

A truck carries an open tank, that is \(6 \mathrm{~m}\) long, \(2 \mathrm{~m}\) wide, and \(3 \mathrm{~m}\) deep. Assuming that the driver will not accelerate or decelerate the truck at a rate greater than \(2 \mathrm{~m} / \mathrm{s}^{2}\), what is the maximum depth to which the tank may be filled so that the water will not be spilled?

A truck carries an open tank that is \(20 \mathrm{ft}\) long, \(6 \mathrm{ft}\) wide, and \(10 \mathrm{ft}\) deep. Assuming that the driver will not accelerate or decelerate the truck at a rate greater than \(6.3 \mathrm{ft} / \mathrm{s}^{2}\), what is the maximum depth to which the tank may be filled so that the water will not be spilled?

What conditions must be satisfied before we can use Bernoulli's equation to relate the flow characteristics between two points in the flow field?

A high-rise office building located in a city at sea level is exposed to a wind of \(75 \mathrm{~km} / \mathrm{h}\). What is the static pressure of the airstream away from the influence of the building? What is the maximum pressure acting on the building? Pressure measurements indicate that a value of \(C_{p}=-4\) occurs near the corner of the wall parallel to the wind direction. If the internal pressure equals to the free-stream static pressure, what is the total force on the pane of glass \(1 \mathrm{~m} \times 3 \mathrm{~m}\) located in this region?

(a) What conditions are necessary before you can use a stream function to solve for the flow field? (b) What conditions are necessary before you can use a potential function to solve for the flow field? (c) What conditions are necessary before you can apply Bernoulli's equation to relate two points in a flow field? (d) Under what conditions does the circulation around a closed fluid line remain constant with respect to time?

What is the circulation around a circle of constant radius \(R_{1}\) for the velocity field given as $$ \vec{V}=\frac{\Gamma}{2 \pi r} \hat{e}_{\theta} $$

Find the integral along the path \(\vec{s}\) between the points \((0,0)\) and \((1,2)\) of the component of \(\vec{V}\) in the direction of \(\vec{s}\) for the following three cases: (a) \(\vec{s}\) a straight line. (b) \(\vec{s}\) a parabola with vertex at the origin and opening to the right. (c) \(\vec{s}\) a portion of the \(x\) axis and a straight line perpendicular to it. The components of \(\vec{V}\) are given by the expressions $$ \begin{aligned} &u=x^{2}+y^{2} \\ &v=2 x y^{2} \end{aligned} $$

Consider the incompressible, irrotational two-dimensional flow where the potential function is $$ \phi=K \ln \sqrt{x^{2}+y^{2}} $$ where \(K\) is an arbitrary constant. (a) What is the velocity field for this flow? Verify that the flow is irrotational. What is the magnitude and direction of the velocity at \((2,0)\), at \((\sqrt{2}, \sqrt{2})\), and at \((0,2)\) ? (b) What is the stream function for this flow? Sketch the streamline pattern. (c) Sketch the lines of constant potential. How do the lines of equipotential relate to the streamlines?

The stream function of a two-dimensional, incompressible flow is given by $$ \psi=\frac{\Gamma}{2 \pi} \ln r $$ (a) Graph the streamlines. (b) What is the velocity field represented by this stream function? Does the resultant velocity field satisfy the continuity equation? (c) Find the circulation about a path enclosing the origin. For the path of integration, use a circle of radius 3 with a center at the origin. How does the circulation depend on the radius?

The absolute value of the velocity and the equation of the streamlines in a velocity field are given by $$ \begin{aligned} |\vec{V}| &=\sqrt{4 x^{2}-4 x y+5 y^{2}} \\ 4 x y-y^{2} &=y^{2}+2 x y=\text { constant } \end{aligned} $$ Find \(u\) and \(v\).

Consider the superposition of a uniform flow and a source of strength \(K\). If the distance from the source to the stagnation point is \(R\), calculate the strength of the source in terms of \(U_{\infty}\) and \(R\). (a) Determine the equation of the streamline that passes through the stagnation point. Let this streamline represent the surface of the configuration of interest. (b) Noting that $$ v_{r}=\frac{1}{r} \frac{\partial \psi}{\partial \theta} \quad v_{\theta}=-\frac{\partial \psi}{\partial r} $$ complete the following table for the surface of the configuration. \begin{tabular}{llll} \hline\(\theta\) & \(\frac{r}{R}\) & \(\frac{U}{U_{\infty}}\) & \(C_{\rho}\) \\ \hline \(30^{\circ}\) & & \\ \(45^{\circ}\) & & \\ \(90^{\circ}\) & & \\ \(135^{\circ}\) & & \\ \(150^{\circ}\) & & \\ \(180^{\circ}\) & & \\ \hline \end{tabular}

What is the stream function that represents the potential flow about a cylinder whose radius is \(1 \mathrm{~m}\) and which is located in an air stream where the free-stream velocity is \(50 \mathrm{~m} / \mathrm{s}\) ? What is the change in pressure from the free-stream value to the value at the top of the cylinder (i.e., \(\theta=90^{\circ}\) )? What is the change in pressure from the free-stream value to that at the stagnation point (i.e., \(\theta=180^{\circ}\) )? Assume that the free-stream conditions are those of the standard atmosphere at sea level.

Calculate the force and the overturning moment exerted by a 45 -mph wind on a cylindrical flagpole that has a diameter of 6 in. and a height of \(15 \mathrm{ft}\). Neglect variations in the velocity of the wind over the height of the flagpole. The temperature of the air is \(85^{\circ} \mathrm{F} ;\) its pressure is \(14.4\) psi. What is the Reynolds number of this flow?

There were early attempts in the development of the airplane to use rotating cylinders as airfoils, Consider such a cylinder having a diameter of \(1 \mathrm{~m}\) and a length of \(10 \mathrm{~m}\). If this cylinder is rotated at \(100 \mathrm{rpm}\) while the plane moves at a speed of \(100 \mathrm{~km} / \mathrm{h}\) through the air at \(2 \mathrm{~km}\) standard atmosphere, estimate the maximum lift that could be developed, disregarding end effects.

What condition(s) must prevail in order for a velocity potential to exist? For a stream function to exist?