Prove that the streamlines $\psi(r, \theta)$ in polar coordinates from Eqs. (8.10) are orthogonal to the potential lines $\phi(r, \theta)$

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The steady plane flow in Fig. P8.2 has the polar velocity components $v_{\theta}=\Omega r$ and $v_{r}=0 .$ Determine the circulation $\Gamma$ around the path shown.

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Using cartesian coordinates, show that each velocity component $(u, v, w)$ of a potential flow satisfies Laplace's equation separately.

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Is the function $1 / r$ a legitimate velocity potential in plane polar coordinates? If so, what is the associated stream function $\psi(r, \theta) ?$

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Consider the two-dimensional velocity distribution $u=$ $-B y, v=B x,$ where $B$ is a constant. If this flow possesses a stream function, find its form. If it has a velocity potential, find that also. Compute the local angular velocity of the flow if any, and describe what the flow might represent.

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An incompressible flow has the velocity potential $\phi=2 B x y$ where $B$ is a constant. Find the stream function of this flow, sketch a few streamlines, and interpret the pattern.

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Consider a flow with constant density and viscosity. If the flow possesses a velocity potential as defined by Eq. (8.1) show that it exactly satisfies the full Navier-Stokes equations $(4.38) .$ If this is so, why for inviscid theory do we back away from the full Navier-Stokes equations?

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For the velocity distribution of Prob. $8.5,$ evaluate the cir culation $\Gamma$ around the rectangular closed curve defined by $(x, y)=(1,1),(3,1),(3,2),$ and $(1,2) .$ Interpret your result, especially vis-à-vis the velocity potential.

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Consider the two-dimensional flow $u=-A x, \quad v=A y$ where $A$ is a constant. Evaluate the circulation $\Gamma$ around the rectangular closed curve defined by $(x, y)=(1,1)$ $(4,1),(4,3),$ and $(1,3) .$ Interpret your result, especially vis-à-vis the velocity potential.

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A mathematical relation sometimes used in fluid mechanics is the theorem of Stokes [1]

$$\oint_{C} \mathbf{V} \cdot d \mathbf{s}=\int_{A} \int(\nabla \times \mathbf{V}) \cdot \mathbf{n} d A$$

where $A$ is any surface and $C$ is the curve enclosing that surface. The vector $d$ s is the differential arc length along $C$ and $\mathbf{n}$ is the unit outward normal vector to $A .$ How does this relation simplify for irrotational flow, and how does the resulting line integral relate to velocity potential?

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A power plant discharges cooling water through the manifold in Fig. P8.11, which is 55 $\mathrm{cm}$ in diameter and 8 $\mathrm{m}$ high and is perforated with 25,000 holes $1 \mathrm{cm}$ in diameter. Does this manifold simulate a line source? If so, what is the equivalent source strength $m ?$

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Consider the flow due to a vortex of strength $K$ at the origin. Evaluate the circulation from Eq. (8.15) about the clockwise path from $(r, \theta)=(a, 0)$ to $(2 a, 0)$ to $(2 a, 3 \pi / 2)$ to $(a, 3 \pi / 2)$ and back to $(a, 0) .$ Interpret the result.

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A well-known exact solution to the Navier-Stokes equations (4.38) is the unsteady circulating motion [15]

\[

v_{\theta}=\frac{K}{2 \pi r}\left[1-\exp \left(-\frac{r^{2}}{4 \nu t}\right)\right] \quad v_{r}=v_{z}=0

\]

where $K$ is a constant and $\nu$ is the kinematic viscosity. Does this flow have a polar-coordinate stream function and/or velocity potential? Explain. Evaluate the circulation $\Gamma$ for this motion, plot it versus $r$ for a given finite time, and interpret compared to ordinary line vortex motion.

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A tornado may be modeled as the circulating flow shown in Fig. $\mathrm{P} 8.14,$ with $v_{r}=v_{z}=0$ and $v_{\theta}(r)$ such that

$$v_{\theta}=\left\{\begin{array}{ll}

\omega r & r \leq R \\

\frac{\omega R^{2}}{r} & r>R

\end{array}\right.$$

Determine whether this flow pattern is irrotational in either the inner or outer region. Using the $r$ -momentum equation (D.5) of App. D, determine the pressure distribution $p(r)$ in the tornado, assuming $p=p_{\infty}$ as $r \rightarrow \infty$ Find the location and magnitude of the lowest pressure.

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Evaluate Prob. 8.14 for the particular case of a small-scale tornado, $R=100 \mathrm{m}, v_{8, \max }=65 \mathrm{m} / \mathrm{s},$ with sea-level conditions at $r=\infty$. Plot $p(r)$ out to $r=400 \mathrm{m}$

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Consider inviscid stagnation flow, $\psi=K x y$ (see Fig. $8.15 b$ ) superimposed with a source at the origin of strength $m$. Plot the resulting streamlines in the upper half plane, using the length scale $a=(m / K)^{1 / 2}$. Give a physical interpretation of the flow pattern.

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Examine the flow of Fig. 8.30 as an analytical (not a numerical) problem. Give the appropriate differential equation and the complete boundary conditions for both the stream function and the velocity potential. Is a Fourierseries solution possible?

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Plot the streamlines and potential lines of the flow due to a line source of strength $m$ at $(a, 0)$ plus a source $3 m$ at $(-a, 0) .$ What is the flow pattern viewed from $\operatorname{afar} ?$

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Plot the streamlines and potential lines of the flow due to a line source of strength $3 m$ at $(a, 0)$ plus a $\operatorname{sink}-m$ at $(-a, 0) .$ What is the pattern viewed from afar?

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Plot the streamlines of the flow due to a line vortex $+K$ at $(0,+a)$ and a vortex $-K$ at $(0,-a) .$ What is the pattern viewed from afar?

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Plot the streamlines of the flow due to a line vortex $+K$ at $(+a, 0)$ and a vortex $-2 K$ at $(-a, 0) .$ What is the pattern viewed from afar?

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Plot the streamlines of a uniform stream $\mathbf{V}=i U$ plus a clockwise line vortex $-K$ located at the origin. Are there any stagnation points?

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Find the resultant velocity vector induced at point $A$ in Fig. $\mathrm{P} 8.23$ by the uniform stream, vortex, and line source.

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Line sources of equal strength $m=U a,$ where $U$ is a reference velocity, are placed at $(x, y)=(0, a)$ and $(0,-a)$ Sketch the stream and potential lines in the upper half plane. Is $y=0$ a "wall"? If so, sketch the pressure coefficient

$$C_{p}=\frac{p-p_{0}}{\frac{1}{2} \rho U^{2}}$$

along the wall, where $p_{0}$ is the pressure at $(0,0) .$ Find the minimum pressure point and indicate where flow separation might occur in the boundary layer.

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Let the vortex/sink flow of Eq. (4.134) simulate a tornado as in Fig. P8.25. Suppose that the circulation about the tornado is $\Gamma=8500 \mathrm{m}^{2} / \mathrm{s}$ and that the pressure at $r=40 \mathrm{m}$ is 2200 Pa less than the far-field pressure. Assuming inviscid flow at sea-level density, estimate $(a)$ the appropriate sink strength $-m,(b)$ the pressure at $r=15 \mathrm{m},$ and $(c)$ the angle $\beta$ at which the streamlines cross the circle at $r=40 \mathrm{m}$ (see Fig. P8.25).

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Find the resultant velocity vector induced at point $A$ in Fig. P8.26 by the uniform stream, line source, line sink, and vortex.

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A counterclockwise line vortex of strength $3 K$ at $(x, y)=(0, a)$ is combined with a clockwise vortex $K$ at $(0,-a) .$ Plot the streamline and potential-line pattern and find the point of minimum velocity between the two vertices

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Sources of equal strength $m$ are placed at the four symmetric positions $(x, y)=(a, a),(-a, a),(-a,-a),$ and $(a,-a) .$ Sketch the streamline and potential-line patterns Do any plane "walls" appear?

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A uniform water stream, $U_{\infty}=20 \mathrm{m} / \mathrm{s}$ and $\rho=998 \mathrm{kg} / \mathrm{m}^{3}$ combines with a source at the origin to form a half-body. At $(x, y)=(0,1.2 \mathrm{m}),$ the pressure is $12.5 \mathrm{kPa}$ less than $p_{\infty} \cdot(a)$ Is this point outside the body? Estimate ( $b$ ) the appropriate source strength $m$ and $(c)$ the pressure at the nose of the body.

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Suppose that the total discharge from the manifold in Fig. P8.1 1 is $450 \mathrm{m}^{3} / \mathrm{s}$ and that there is a uniform ocean current of $60 \mathrm{cm} / \mathrm{s}$ to the right. Sketch the flow pattern from above, showing the dimensions and the region where the cooling-water discharge is confined.

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A Rankine half-body is formed as shown in Fig. P8.31. For the stream velocity and body dimension shown, compute ( $a$ ) the source strength $m$ in $\mathrm{m}^{2} / \mathrm{s},$ ( $b$ ) the distance $a$

$(c)$ the distance $h,$ and $(d)$ the total velocity at point $A$.

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Sketch the streamlines, especially the body shape, due to equal line sources $+m$ at $(-a, 0)$ and $(+a, 0)$ plus a uniform stream $U_{\infty}=m a$.

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Sketch the streamlines, especially the body shape, due to equal line sources $+m$ at $(0,+a)$ and $(0,-a)$ plus a uniform stream $U_{\infty}=m a$

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Consider three equal sources $m$ in a triangular configuration: one at $(a / 2,0),$ one at $(-a / 2,0),$ and one at $(0, a)$ Plot the streamlines for this flow. Are there any stagnation points? Hint: Try the MATLAB contour command [34].

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When a line source-sink pair with $m=2 \mathrm{m}^{2} / \mathrm{s}$ is combined with a uniform stream, it forms a Rankine oval whose minimum dimension is $40 \mathrm{cm} .$ If $a=15 \mathrm{cm},$ what are the stream velocity and the velocity at the shoulder? What is the maximum dimension?

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A Rankine oval $2 \mathrm{m}$ long and $1 \mathrm{m}$ high is immersed in a stream $U_{\infty}=10 \mathrm{m} / \mathrm{s},$ as in Fig. $\mathrm{P} 8.37 .$ Estimate $(a)$ the velocity at point $A$ and $(b)$ the location of point $B$ where a particle approaching the stagnation point achieves its maximum deceleration.

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A uniform stream $U$ in the $x$ direction combines with a source $m$ at $(a, 0)$ and a $\operatorname{sink}-m$ at $(-a, 0) .$ Plot the resulting streamlines and note any stagnation points.

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Sketch the streamlines of a uniform stream $U_{\infty}$ past a line source-sink pair aligned vertically with the source at $+a$ and the sink at $-a$ on the $y$ -axis. Does a closed-body shape form?

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Consider a uniform stream $U_{\infty}$ plus line sources $+m$ at $(x,$ $y)=(+a, 0)$ and $(-a, 0)$ and a single line $\operatorname{sink}-2 m$ at the origin. Does a closed-body shape appear? If so, plot its shape for $m /\left(U_{\infty} a\right)$ equal to $(a) 1.0$ and $(b) 5.0$

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A Kelvin oval is formed by a line-vortex pair with $K=9$ $\mathrm{m}^{2} / \mathrm{s}, a=1 \mathrm{m},$ and $U=10 \mathrm{m} / \mathrm{s} .$ What are the height, width, and shoulder velocity of this oval?

Prashant B.

Numerade Educator

For what value of $K /\left(U_{\infty} a\right)$ does the velocity at the shoulder of a Kelvin oval equal $4 U_{\infty} ?$ What is the height $h / a$ of this oval?

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Consider water at $20^{\circ} \mathrm{C}$ flowing at $6 \mathrm{m} / \mathrm{s}$ past a $1-\mathrm{m}-\mathrm{di}$ ameter circular cylinder. What doublet strength $\lambda$ in $\mathrm{m}^{3} / \mathrm{s}$ is required to simulate this flow? If the stream pressure is $200 \mathrm{kPa},$ use inviscid theory to estimate the surface pressure at $\theta$ equal to $(a) 180^{\circ},(b) 135^{\circ},$ and $(c) 90^{\circ}$.

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Suppose that circulation is added to the cylinder flow of Prob. 8.43 sufficient to place the stagnation points at $\theta$ equal to $50^{\circ}$ and $130^{\circ} .$ What is the required vortex strength $K$ in $\mathrm{m}^{2} / \mathrm{s} ?$ Compute the resulting pressure and surface velocity at (a) the stagnation points and ( $b$ ) the upper and lower shoulders. What will the lift per meter of cylinder width be?

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What circulation $K$ must be added to the cylinder flow in Prob. 8.43 to place the stagnation point exactly at the upper shoulder? What will the velocity and pressure at the lower shoulder be then? What value of $K$ causes the lower shoulder pressure to be $10 \mathrm{kPa} ?$

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A cylinder is formed by bolting two semicylindrical channels together on the inside, as shown in Fig. P8.46. There are 10 bolts per meter of width on each side, and the inside pressure is $50 \mathrm{kPa}$ (gage). Using potential theory for the outside pressure, compute the tension force in each bolt if the fluid outside is sea-level air.

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A circular cylinder is fitted with two surface-mounted pressure sensors, to measure $p_{a}$ at $\theta=180^{\circ}$ and $p_{b}$ at $\theta=105^{\circ}$ The intention is to use the cylinder as a stream velocimeter. Using inviscid theory, derive a formula for estimating $U_{\infty}$ in terms of $p_{a}, p_{b r}, \rho,$ and the cylinder radius a.

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Wind at $U_{\infty}$ and $p_{\infty}$ flows past a Quonset hut which is a half-cylinder of radius $a$ and length $L$ (Fig. $P 8.48$ ). The in ternal pressure is $p_{r}$ Using inviscid theory, derive an expression for the upward force on the hut due to the difference between $p_{i}$ and $p_{\mathrm{s}}$

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In strong winds the force in Prob. 8.48 can be quite large. Suppose that a hole is introduced in the hut roof at point $A$ to make $p_{i}$ equal to the surface pressure there. At what angle $\theta$ should hole $A$ be placed to make the net wind force zero?

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It is desired to simulate flow past a two-dimensional ridge or bump by using a streamline which passes above the flow over a cylinder, as in Fig. P8.50. The bump is to be $a / 2$ high, where $a$ is the cylinder radius. What is the elevation $h$ of this streamline? What is $U_{\max }$ on the bump compared with stream velocity $U ?$

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Modify Prob. 8.50 as follows. Let the bump be such that $U_{\max }=1.5 U .$ Find $(a)$ the upstream elevation $h$ and $(b)$ the height of the bump.

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The Flettner rotor sailboat in Fig. E8.2 has a water drag coefficient of 0.006 based on a wetted area of 45 $\mathrm{ft}^{2} .$ If the rotor spins at $220 \mathrm{r} / \mathrm{min},$ find the maximum boat velocity that can be achieved in a 15 -mi/h wind. What is the optimum angle between the boat and the wind?

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Modify Prob. 8.52 as follows. For the same sailboat data, find the wind velocity, in $\mathrm{mi} / \mathrm{h}$, which will drive the boat at an optimum speed of $10 \mathrm{kn}$ parallel to its keel.

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The original Flettner rotor ship was approximately $100 \mathrm{ft}$ Iong, displaced 800 tons, and had a wetted area of 3500 $\mathrm{ft}^{2} .$ As sketched in Fig. $\mathrm{P} 8.54,$ it had two rotors $50 \mathrm{ft}$ high and $9 \mathrm{ft}$ in diameter rotating at $750 \mathrm{r} / \mathrm{min},$ which is far out side the range of Fig. $8.11 .$ The measured lift and drag coefficients for each rotor were about 10 and $4,$ respectively. If the ship is moored and subjected to a crosswind of 25 ft/s, as in Fig. P8.54, what will the wind force parallel and normal to the ship centerline be? Estimate the power required to drive the rotors.

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Assume that the Flettner rotorship of Fig. $\mathrm{P} 8.54$ has a water resistance coefficient of $0.005 .$ How fast will the ship sail in seawater at $20^{\circ} \mathrm{C}$ in a 20 -ft/s wind if the keel aligns itself with the resultant force on the rotors? Hint: This is a problem in relative velocities.

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The measured drag coefficient of a cylinder in crossflow, based on frontal area $D L$, is approximately 1.0 for the laminar-boundary-layer range (see Fig. 7.16a). Boundarylayer separation occurs near the shoulder (see Fig. $7.13 a$ ). This suggests an analytical model: the standard inviscidflow solution on the front of the cylinder and constant pressure (equal to the shoulder value) on the rear. Use this model to predict the drag coefficient and comment on the results with reference to Fig. $7.13 c$

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In principle, it is possible to use rotating cylinders as aircraft wings. Consider a cylinder $30 \mathrm{cm}$ in diameter, rotating at $2400 \mathrm{r} / \mathrm{min}$. It is to lift a 55 -kN airplane cruising at $100 \mathrm{m} / \mathrm{s} .$ What should the cylinder length be? How much power is required to maintain this speed? Neglect end effects on the rotating wing.

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Plot the streamlines due to the combined flow of a line $\operatorname{sink}-m$ at the origin plus line sources $+m$ at $(a, 0)$ and $(4 a, 0) .$ Hint: $\mathrm{A}$ cylinder of radius $2 a$ will appear.

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By analogy with Prob. 8.58 plot the streamlines due to counterclockwise line vortices $+K$ at (0,0) and $(4 a, 0)$ plus a clockwise vortex $-K$ at $(a, 0)$. Again a cylinder appears.

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One of the comer-flow patterns of Fig. 8.15 is given by the cartesian stream function $\psi=A\left(3 y x^{2}-y^{3}\right) .$ Which one? Can the correspondence be proved from Eq. (8.49)$?$

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Plot the streamlines of Eq. (8.49) in the upper right quadrant for $n=4 .$ How does the velocity increase with $x$ outward along the $x$ -axis from the origin? For what corner angle and value of $n$ would this increase be linear in $x ?$ For what corner angle and $n$ would the increase be as $x^{5} ?$

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Combine stagnation flow, Fig. $8.14 b,$ with a source at the origin:

$$f(z)=A z^{2}+m \ln z$$

Plot the streamlines for $m=A L^{2}$, where $L$ is a length scale. Interpret.

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The superposition in Prob. 8.62 leads to stagnation flow near a curved bump, in contrast to the flat wall of Fig. $8.14 b .$ Determine the maximum height $H$ of the bump as a function of the constants $A$ and $m$.

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Determine qualitatively from boundary-layer theory (Chap. 7) whether any of the three stagnation-flow patterns of Fig. $8.15 \mathrm{can}$ suffer flow separation along the walls.

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Potential flow past a wedge of half-angle $\theta$ leads to an important application of laminar-boundary-layer theory called the Falkner-Skan flows $[15, \mathrm{pp} .242-247] .$ Let $x \mathrm{de}$ note distance along the wedge wall, as in Fig. $\mathrm{P} 8.65,$ and let $\theta=10^{\circ} .$ Use Eq. (8.49) to find the variation of surface velocity $U(x)$ along the wall. Is the pressure gradient adverse or favorable?

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The inviscid velocity along the wedge in Prob. 8.65 has the analytic form $U(x)=C x^{m},$ where $m=n-1$ and $n$ is the exponent in Eq. $(8.49) .$ Show that, for any $C$ and $n$ computation of the boundary layer by Thwaites' method, Eqs. (7.53) and $(7.54),$ leads to a unique value of the Thwaites parameter $\lambda$. Thus wedge flows are called simi$\operatorname{lar}[15, \mathrm{p} .244]$

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Investigate the complex potential function $f(z)=$ $U_{\infty}\left(z+a^{2} / z\right)$ and interpret the flow pattern.

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Investigate the complex potential function $f(z)=U_{\infty} z+$ $m \ln [(z+a) /(z-a)]$ and interpret the flow pattern.

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Investigate the complex potential $f(z)=A \cosh [\pi(z / a)]$ and plot the streamlines inside the region shown in Fig. P8.69. What hyphenated word (originally French) might describe such a flow pattern?

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Show that the complex potential $f=U_{\infty}\{z+\frac{1}{4} a$ coth. $[\pi(z / a)]\}$ represents flow past an oval shape placed midway between two parallel walls $y=\pm \frac{1}{2} a .$ What is a practical application?

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Figure $\mathrm{P} 8.71$ shows the streamlines and potential lines of flow over a thin-plate weir as computed by the complex potential method. Compare qualitatively with Fig. $10.16 a$ State the proper boundary conditions at all boundaries. The velocity potential has equally spaced values. Why do the flow-net "squares" become smaller in the overflow jet?

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Use the method of images to construct the flow pattern for a source $+m$ near two walls, as shown in Fig. P8.72. Sketch the velocity distribution along the lower wall (y = 0). Is there any danger of flow separation along this wall?

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Set up an image system to compute the flow of a source at unequal distances from two walls, as in Fig. P8.73. Find the point of maximum velocity on the $y$ -axis.

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A positive line vortex $K$ is trapped in a corner, as in Fig. P8.74. Compute the total induced velocity vector at point $B,(x, y)=(2 a, a),$ and compare with the induced velocity when no walls are present.

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The flow past a cylinder very near a wall might be simulated by doublet images, as in Fig. P8.75. Explain why the result is not very successful and the cylinder shape becomes badly distorted.

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Use the method of images to approximate the flow pattern past a cylinder a distance $4 a$ from a single wall, as in Fig. P8.76. To illustrate the effect of the wall, compute the velocities at corresponding points $A, B$ and $C, D,$ comparing with a cylinder flow in an infinite expanse of fluid.

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Discuss how the flow pattern of Prob. 8.58 might be interpreted to be an image-system construction for circular walls. Why are there two images instead of one?

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Indicate the system of images needed to construct the flow of a uniform stream past a Rankine half-body constrained between two parallel walls, as in Fig. P8.78. For the narticular dimensions shown in this figure, estimate the position of the nose of the resulting half-body.

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Explain the system of images needed to simulate the flow of a line source placed unsymmetrically between two parallel walls as in Fig. $\mathrm{P} 8.79$. Compute the velocity on the lower wall at $x=a .$ How many images are needed to estimate this velocity within 1 percent?

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The beautiful expression for lift of a two-dimensional airfoil, Eq. $(8.69),$ arose from applying the Joukowski transformation, $\zeta=z+a^{2} / z,$ where $z=x+i y$ and $\zeta=\eta+i \beta$ The constant $a$ is a length scale. The theory transforms a certain circle in the $z$ plane into an airfoil in the $\zeta$ plane. Taking $a=1$ unit for convenience, show that $(a)$ a circle with center at the origin and radius $>1$ will become an ellipse in the $\zeta$ plane and $(b)$ a circle with center at $x=$ $-\epsilon<1, y=0,$ and radius $(1+\epsilon)$ will become an airfoil shape in the $\zeta$ plane. Hint: The Excel spreadsheet is excellent for solving this problem.

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A two-dimensional airfoil has 2 percent camber and 10 percent thickness. If $C=1.75 \mathrm{m}$, estimate its lift per meter

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The ultralight plane Gossamer Condor in 1977 was the first to complete the Kremer Prize figure-eight course under human power. Its wingspan was $29 \mathrm{m},$ with $C_{\mathrm{av}}=2.3 \mathrm{m}$ and a total mass of $95 \mathrm{kg}$. The drag coefficient was approximately $0.05 .$ The pilot was able to deliver $\frac{1}{4}$ hp to propel the plane. Assuming two-dimensional flow at sea level, estimate ( $a$ ) the cruise speed attained, $(b)$ the lift coefficient, and $(c)$ the horsepower required to achieve a speed of $15 \mathrm{kn}$.

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Two-dimensional lift-drag data for the NACA 2412 airfoil with 2 percent camber (from Ref. 12 ) may be curve-fitted accurately as follows:

$$\begin{aligned}

C_{L} & \approx 0.178+0.109 \alpha-0.00109 \alpha^{2} \\

C_{D} \approx & 0.0089+1.97 \mathrm{E}-4 \alpha+8.45 \mathrm{E}-5 \alpha^{2} \\

&-1.35 \mathrm{E}-5 \alpha^{3}+9.92 \mathrm{E}-7 \alpha^{4}

\end{aligned}$$

with $\alpha$ in degrees in the range $-4^{\circ}<\alpha<+10^{\circ} .$ Compare ( $a$ ) the lift-curve slope and ( $b$ ) the angle of zero lift with theory, Eq. $(8.69) .(c)$ Prepare a polar lift-drag plot and compare with Fig. 7.26

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Reference 12 contains inviscid-theory calculations for the upper and lower surface velocity distributions $V(x)$ over an airfoil, where $x$ is the chordwise coordinate. A typical result for small angle of attack is as follows:

Use these data, plus Bernoulli's equation, to estimate $(a)$ the lift coefficient and $(b)$ the angle of attack if the airfoil is symmetric.

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A wing of 2 percent camber, 5 -in chord, and 30 -in span is tested at a certain angle of attack in a wind tunnel with sea-level standard air at $200 \mathrm{ft} / \mathrm{s}$ and is found to have lift

of 30 lbf and drag of 1.5 lbf. Estimate from wing theory

(a) the angle of attack,

( $b$ ) the minimum drag of the wing and the angle of attack at which it occurs, and

( $c$ ) the maximum lift-to-drag ratio.

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An airplane has a mass of $20,000 \mathrm{kg}$ and flies at $175 \mathrm{m} / \mathrm{s}$ at $5000-\mathrm{m}$ standard altitude. Its rectangular wing has a $3-$ $\mathrm{m}$ chord and a symmetric airfoil at $2.5^{\circ}$ angle of attack. Estimate ( $a$ ) the wing span, ( $b$ ) the aspect ratio, and (c) the induced drag.

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A freshwater boat of mass $400 \mathrm{kg}$ is supported by a rectangular hydrofoil of aspect ratio 8,2 percent camber, and 12 percent thickness. If the boat travels at $8 \mathrm{m} / \mathrm{s}$ and $\alpha=$ $3.5^{\circ},$ estimate $(a)$ the chord length, $(b)$ the power required if $C_{D_{\infty}}=0.01,$ and $(c)$ the top speed if the boat is refitted with an engine which delivers 50 hp to the water.

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The Boeing 727 airplane has a gross weight of 125,000 Ibf, a wing area of $1200 \mathrm{ft}^{2},$ and an aspect ratio of $6 .$ It is fitted with two turbofan engines and cruises at $532 \mathrm{mi} / \mathrm{h}$ at 30,000 ft standard altitude. Assume for this problem that its airfoil is the NACA 2412 section described in Prob. $8.83 .$ If we neglect all drag except the wing, what thrust is required from each engine for these conditions?

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The Beechcraft $\mathrm{T}-34 \mathrm{C}$ aircraft has a gross weight of 5500 Ibf and a wing area of $60 \mathrm{ft}^{2}$ and flies at $322 \mathrm{mi} / \mathrm{h}$ at $10,000-$ ft standard altitude. It is driven by a propeller which delivers 300 hp to the air. Assume for this problem that its airfoil is the NACA 2412 section described in Prob. 8.83 and neglect all drag except the wing. What is the appropriate aspect ratio for the wing?

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When moving at $15 \mathrm{m} / \mathrm{s}$ in seawater at its maximum liftto-drag ratio of $18: 1,$ a symmetric hydrofoil, of plan area $3 \mathrm{m}^{2},$ develops a lift of $120 \mathrm{kN}$. Estimate from wing theory $(a)$ the aspect ratio and $(b)$ the angle of attack in degrees.

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If $\phi(r, \theta)$ in axisymmetric flow is defined by Eq. (8.85) and the coordinates are given in Fig. 8.24 , determine what partial differential equation is satisfied by $\phi$

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A point source with volume flow $Q=30 \mathrm{m}^{3} / \mathrm{s}$ is immersed in a uniform stream of speed $4 \mathrm{m} / \mathrm{s}$. A Rankine half-body of revolution results. Compute ( $a$ ) the distance from source to the stagnation point and $(b)$ the two points $(r, \theta)$ on the body surface where the local velocity equals $4.5 \mathrm{m} / \mathrm{s}$

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The Rankine half-body of revolution (Fig. 8.26 ) could simulate the shape of a pitot-static tube (Fig. 6.30 ). According to inviscid theory, how far downstream from the nose should the static pressure holes be placed so that the local velocity is within ±0.5 percent of $U_{\infty} ?$ Compare your answer with the recommendation $x \approx 8 D$ in Fig. 6.30

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Determine whether the Stokes streamlines from Eq. (8.86) are everywhere orthogonal to the Stokes potential lines from Eq. $(8.87),$ as is the case for cartesian and plane polar coordinates.

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Show that the-axisymmetric potential flow formed by superposition of a point source $+m$ at $(x, y)=(-a, 0), a$ point $\operatorname{sink}-m$ at $(+a, 0),$ and a stream $U_{\infty}$ in the $x$ direction forms a Rankine body of revolution as in Fig. P8.95. Find analytic expressions for determining the length $2 L$ and maximum diameter $2 R$ of the body in terms of $m$ $U_{\infty},$ and $a$.

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Suppose that a sphere with a single stagnation hole is to be used as a velocimeter. The pressure at this hole is used to compute the stream velocity, but there are errors if the hole is not perfectly aligned with the oncoming stream. Using inviscid incompressible theory, plot the percent error in stream velocity estimate as a function of misalignment angle $\phi .$ At what angle is the error 10 percent?

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The Rankine body of revolution in Fig. $\mathrm{P} 8.97$ is $60 \mathrm{cm}$ long and $30 \mathrm{cm}$ in diameter. When it is immersed in the low-pressure water tunnel as shown, cavitation may appear at point $A .$ Compute the stream velocity $U,$ neglecting surface wave formation, for which cavitation occurs.

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We have studied the point source (sink) and the line source (sink) of infinite depth into the paper. Does it make any sense to define a finite-length line sink (source) as in Fig. P8.98? If so, how would you establish the mathematical properties of such a finite line sink? When combined with a uniform stream and a point source of equivalent strength as in Fig. $\mathrm{P} 8.98$, should a closed-body shape be formed? Make a guess and sketch some of these possible shapes for various values of the dimensionless parameter $m /\left(U_{\infty} L^{2}\right)$.

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Consider air flowing past a hemisphere resting on a flat surface, as in Fig. $\mathrm{P} 8.99 .$ If the internal pressure is $p_{i}$ find an expression for the pressure force on the hemisphere. By analogy with Prob. 8.49 , at what point $A$ on the hemisphere should a hole be cut so that the pressure force will be zero according to inviscid theory?

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A $1-\mathrm{m}$ -diameter sphere is being towed at speed $V$ in fresh water at $20^{\circ} \mathrm{C}$ as shown in Fig. $\mathrm{P} 8.100 .$ Assuming inviscid theory with an undistorted free surface, estimate the speed $V$ in $\mathrm{m} / \mathrm{s}$ at which cavitation will first appear on the sphere surface. Where will cavitation appear? For this condition, what will be the pressure at point $A$ on the sphere which is $45^{\circ}$ up from the direction of travel?

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Normally by its very nature inviscid theory is incapable of predicting body drag, but by analogy with Fig. $8.16 \mathrm{c}$ we can analyze flow approaching a hemisphere, as in Fig. P8.101. Assume that the flow on the front follows inviscid sphere theory, Eq. $(8.96),$ and the pressure in the rear equals the shoulder pressure. Compute the drag coefficient and compare with experiment (Table 7.3 ). What are the defects and limitations of this analysis?

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A golf ball weighs 0.102 lbf and has a diameter of 1.7 in. A professional golfer strikes the ball at an initial velocity of $250 \mathrm{ft} / \mathrm{s},$ an upward angle of $20^{\circ},$ and a backspin (front of the ball rotating upward). Assume that the lift coefficient on the ball (based on frontal area) follows Fig. $P 7.108 .$ If the ground is level and drag is neglected, make a simple analysis to predict the impact point (a) without spin and $(b)$ with backspin of $7500 \mathrm{r} / \mathrm{min}$.

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Modify Prob. 8.102 as follows. Golf balls are dimpled, not smooth, and have higher lift and lower drag $\left(C_{\mathrm{L}} \approx 0.2$ and \right. $C_{D} \approx 0.3$ for typical backspin). Using these values, make a computer analysis of the ball trajectory for the initial conditions of Prob. $8.102 .$ If time permits, investigate the effect of initial angle for the range $10^{\circ}<\theta_{0}<50^{\circ}$Modify Prob. 8.102 as follows. Golf balls are dimpled, not smooth, and have higher lift and lower drag $(C_{\mathrm{L}} \approx 0.2$ and. $C_{D} \approx 0.3$ for typical backspin). Using these values, make a computer analysis of the ball trajectory for the initial conditions of Prob. $8.102 .$ If time permits, investigate the effect of initial angle for the range $10^{\circ}<\theta_{0}<50^{\circ}$

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Consider a cylinder of radius $a$ moving at speed $U_{x}$ through a still fluid, as in Fig. $\mathrm{P} 8.104$. Plot the streamlines relative to the cylinder by modifying Eq. (8.32) to give the relative flow with $K=0 .$ Integrate to find the total relative kinetic energy, and verify the hydrodynamic mass of a cylinder from Eq. (8.104)

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In Table 7.2 the drag coefficient of a 4: 1 elliptical cylinder in laminar-boundary-layer flow is 0.35 . According to Patton $[17],$ the hydrodynamic mass of this cylinder is $\pi \rho h b / 4,$ where $b$ is width into the paper and $h$ is the maximum thickness. Use these results to derive a formula from the time history $U(t)$ of the cylinder if it is accelerated from rest in a still fluid by the sudden application of a constant force $F$

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Laplace's equation in plane polar coordinates, Eq. (8.11) is complicated by the variable radius. Consider the finitedifference mesh in Fig. $\mathrm{P} 8.106,$ with nodes $(i, j)$ equally spaced $\Delta \theta$ and $\Delta r$ apart. Derive a finite-difference model for Eq. (8.11) similar to the cartesian expression (8.109)

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Set up the numerical problem of Fig. 8.30 for an expansion of $30^{\circ} .$ A new grid system and a nonsquare mesh may be needed. Give the proper nodal equation and boundary conditions. If possible, program this $30^{\circ}$ expansion and solve on a digital computer.

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Consider two-dimensional potential flow into a step contraction as in Fig. P8.108. The inlet velocity $U_{1}=7 \mathrm{m} / \mathrm{s}$ and the outlet velocity $U_{2}$ is uniform. The nodes $(i, j)$ are labeled in the figure. Set up the complete finite-difference algebraic relations for all nodes. Solve, if possible, on a digital computer and plot the streamlines in the flow.

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Consider inviscid flow through a two-dimensional $90^{\circ}$ bend with a contraction, as in Fig. P8.109. Assume uniform flow at the entrance and exit. Make a finite-difference computer analysis for small grid size (at least 150 nodes), determine the dimensionless pressure distribution along the walls, and sketch the streamlines. (You may use either square or rectangular grids.)

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For fully developed laminar incompressible flow through a straight noncircular duct, as in Sec. $6.6,$ the NavierStokes equations (4.38) reduce to

$$\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=\frac{1}{\mu} \frac{d p}{d x}=\mathrm{const}<0$$

where $(y, z)$ is the plane of the duct cross section and $x$ is along the duct-axis. Gravity is neglected. Using a nonsquare rectangular grid $(\Delta x, \Delta y),$ develop a finite-difference model for this equation, and indicate how it may be applied to solve for flow in a rectangular duct of side lengths $a$ and $b$

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Solve Prob. 8.110 numerically for a rectangular duct of side length $b$ by $2 b,$ using at least 100 nodal points. Evaluate the volume flow rate and the friction factor, and compare with the results in Table 6.4

\begin{equation}Q \approx 0.1143 \frac{b^{4}}{\mu}\left(-\frac{d p}{d x}\right) \quad f \operatorname{Re}_{D_{h}} \approx 62.19\end{equation}

where $D_{h}=4 A / P=4 b / 3$ for this case. Comment on the possible truncation errors of your model.

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In his CFD textbook, Patankar [5] replaces the left-hand sides of Eq. $(8.119 b \text { and } c)$ with the following two expressions, respectively:

$$\frac{\partial}{\partial x}\left(u^{2}\right)+\frac{\partial}{\partial y}(v u) \quad \text { and } \quad \frac{\partial}{\partial x}(u v)+\frac{\partial}{\partial y}\left(v^{2}\right)$$

Are these equivalent expressions, or are they merely simplified approximations? Either way, why might these forms be better for finite-difference purposes?

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Repeat Example 8.7 using the implicit method of Eq. $(8.118) .$ Take $\Delta t=0.2 \mathrm{s}$ and $\Delta y=0.01 \mathrm{m},$ which ensures that an explicit model would diverge. Compare your accuracy with Example 8.7

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If your institution has an online potential-flow boundaryelement computer code, consider flow past a symmetric airfoil, as in Fig. P8.114. The basic shape of an NACA symmetric airfoil is defined by the function [12]

$$\begin{aligned}

\frac{2 y}{t_{\max }} &=1.4845 \zeta^{1 / 2}-0.63 \zeta-1.758 \zeta^{2} \\

&+1.4215 \zeta^{3}-0.5075 \zeta^{4}

\end{aligned}$$

where $\zeta=x / C$ and the maximum thickness $t_{\max }$ occurs at $\zeta=0.3 .$ Use this shape as part of the lower boundary for zero angle of attack. Let the thickness be fairly large, say, $t_{\max }=0.12,0.15,$ or $0.18 .$ Choose a generous number of nodes $(\geq 60),$ and calculate and plot the velocity distribution $V / U_{\infty}$ along the airfoil surface. Compare with the theoretical results in Ref. 12 for NACA $0012,0015,$ or 0018 airfoils. If time permits, investigate the effect of the boundary lengths $L_{1}, L_{2},$ and $L_{3},$ which can initially be set equal to the chord length $C$

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Use the explicit method of Eq. (8.115) to solve Prob. 4.85 numerically for SAE 30 oil at $20^{\circ} \mathrm{C}$ with $U_{0}=1 \mathrm{m} / \mathrm{s}$ and $\omega=M \mathrm{rad} / \mathrm{s},$ where $M$ is the number of letters in your surname. (This author will solve the problem for $M=5$.) When steady oscillation is reached, plot the oil velocity versus time at $y=2 \mathrm{cm}$

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