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Fluid Mechanics

Frank M. White

Chapter 8

Potential Flow and Computational Fluid Dynamics - all with Video Answers

Educators

Chapter Questions

08:58

Problem 1

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

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
01:49

Problem 2

The steady plane flow in Fig. $\mathrm{P} 8.2$ has the polar velocity components $v_{\theta}=\Omega r$ and $v_{r}=0 .$ Determine the circulation $\Gamma$ around the path shown.

James Kiss
James Kiss
Numerade Educator
01:32

Problem 3

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

Penny Riley
Penny Riley
Numerade Educator
02:35

Problem 4

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

James Kiss
James Kiss
Numerade Educator
02:16

Problem 5

A proposed harmonic function $F(x, y, z)$ is given by
\[
F=2 x^{2}+y^{3}-4 x z+f(y)
\]
(a) If possible, find a function $f(y)$ for which the laplacian of $F$ is zero. If you do indeed solve part $(a),$ can your final function $F$ serve as $(b)$ a velocity potential or $(c)$ a stream function?

Penny Riley
Penny Riley
Numerade Educator
07:43

Problem 6

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

Susan Hallstrom
Susan Hallstrom
Numerade Educator
02:07

Problem 7

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?

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
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Problem 8

For the velocity distribution $u=-B y, v=+B x, w=0$ evaluate the circulation $\Gamma$ about the rectangular closed curve defined by $(x, y)=(1,1),(3,1),(3,2),$ and $(1,2) .$ Interpret your result, especially vis-ã -vis the velocity potential.

Victor Salazar
Victor Salazar
Numerade Educator
03:25

Problem 9

Consider the two-dimensional flow $u=-A x, v=A y$ 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-ã -vis the velocity potential.

Chai Santi
Chai Santi
Numerade Educator
03:03

Problem 10

A two-dimensional Rankine half-body, $8 \mathrm{cm}$ thick, is placed in a water tunnel at $20^{\circ} \mathrm{C}$. The water pressure far upstream along the body center line is 105 kPa. What is the nose radius of the half-body? At what tunnel flow velocity will cavitation bubbles begin to form on the surface of the body?

Amit Srivastava
Amit Srivastava
Numerade Educator
07:58

Problem 11

A power plant discharges cooling water through the manifold in Fig. $\mathrm{P} 8.11$, 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 $m ?$

Ivan Kochetkov
Ivan Kochetkov
Numerade Educator
01:52

Problem 12

Consider the flow due to a vortex of strength $K$ at the origin. Evaluate the circulation from Eq. (8.23) 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.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
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Problem 13

Starting at the stagnation point in Fig. $8.6,$ the fluid acceleration along the half-body surface rises to a maximum
and eventually drops off to zero far downstream. $(a)$ Does this maximum occur at the point in Fig. 8.6 where $U_{\max }=$ $1.26 U ?(b)$ If not, does the maximum acceleration occur before or after that point? Explain.

Victor Salazar
Victor Salazar
Numerade Educator
02:35

Problem 14

A tornado may be modeled as the circulating flow shown in Fig. P8.14, 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.

Narayan Hari
Narayan Hari
Numerade Educator
03:40

Problem 15

Hurricane Sandy, which hit the New Jersey coast on Oct. $29,2012,$ was extremely broad, with wind velocities of $40 \mathrm{mi} / \mathrm{h}$ at $400 \mathrm{miles}$ from its center. Its maximum velocity was $90 \mathrm{mi} / \mathrm{h}$. Using the model of Fig. $\mathrm{P} 8.14$, at $20^{\circ} \mathrm{C}$ with a pressure of $100 \mathrm{kPa}$ far from the center, estimate ( $a$ ) the radius $R$ of maximum velocity, in $\mathrm{mi} ;$ and $(b)$ the pressure at $r=R$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
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Problem 16

Air flows at $1.2 \mathrm{m} / \mathrm{s}$ along a flat surface when it encounters a jet of air issuing from the horizontal wall at point $A,$ as in Fig. $P 8.16 .$ The jet volume flow is $0.4 \mathrm{m}^{3} / \mathrm{s}$ per unit depth into the paper. If the jet is approximated as an inviscid line source, (a) locate the stagnation point $S$ on the wall. (b) How far vertically will the jet flow extend into the stream?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
08:12

Problem 17

Find the position $(x, y)$ on the upper surface of the half-body in Fig. $8.9 a$ for which the local velocity equals the uniform stream velocity. What should be the pressure at this point?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:40

Problem 18

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 afar?

Chai Santi
Chai Santi
Numerade Educator
03:40

Problem 19

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?

Chai Santi
Chai Santi
Numerade Educator
02:14

Problem 20

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?

Chai Santi
Chai Santi
Numerade Educator
02:12

Problem 21

At point $A$ in Fig. $\mathrm{P} 8.21$ is a clockwise line vortex of strength $K=12 \mathrm{m}^{2} / \mathrm{s}$. At point $B$ is a line source of strength $m=25 \mathrm{m}^{2} / \mathrm{s} .$ Determine the resultant velocity induced by these two at point $C$.

Supratim Pal
Supratim Pal
Numerade Educator
05:32

Problem 22

Consider inviscid stagnation flow, $\psi=K x y(\text { see Fig. } 8.19 b)$ 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.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:35

Problem 23

Sources of strength $m=10 \mathrm{m}^{2} / \mathrm{s}$ are placed at points A and B in Fig. $P 8.23 .$ At what height $h$ should source $B$ be placed so that the net induced horizontal velocity component at the origin is $8 \mathrm{m} / \mathrm{s}$ to the left?

Alan Gavel
Alan Gavel
Numerade Educator
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Problem 24

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.

Victor Salazar
Victor Salazar
Numerade Educator
03:44

Problem 25

Let the vortex/sink flow of Eq. (8.16) simulate a tornado as in Fig. $P 8.25 .$ 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 $2200 \mathrm{Pa}$ less than the far-field pressure. Assuming inviscid flow at sealevel 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. $\mathrm{P} 8.25$ ).

Chai Santi
Chai Santi
Numerade Educator
01:37

Problem 26

A coastal power plant takes in cooling water through a vertical perforated manifold, as in Fig. $\mathrm{P} 8.26 .$ The total volume flow intake is $110 \mathrm{m}^{3} / \mathrm{s}$. Currents of $25 \mathrm{cm} / \mathrm{s}$ flow past the manifold, as shown. Estimate (a) how far downstream and ( $b$ ) how far normal to the paper the effects of the intake are felt in the ambient 8 -m-deep waters.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:31

Problem 27

Water at $20^{\circ} \mathrm{C}$ flows past a half-body as shown in Fig. P8.27. Measured pressures at points $A$ and $B$ are $160 \mathrm{kPa}$ and $90 \mathrm{kPa}$, respectively, with uncertainties of $3 \mathrm{kPa}$ each. Estimate the stream velocity and its uncertainty.

Dominador Tan
Dominador Tan
Numerade Educator
03:32

Problem 28

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?

Arwa Ali
Arwa Ali
Numerade Educator
04:26

Problem 29

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 $p_{\infty} .(a) \mathrm{Is}$ this point outside the body? Estimate (b) the appropriate source strength $m$ and $(c)$ the pressure at the nose of the body.

James Kiss
James Kiss
Numerade Educator
01:42

Problem 30

A tornado is simulated by a line sink $m=-1000 \mathrm{m}^{2} / \mathrm{s}$ plus a line vortex $K=1600 \mathrm{m}^{2} / \mathrm{s}$. Find the angle between any streamline and a radial line, and show that it is independent of both $r$ and $\theta$. If this tornado forms in sea-level standard air, at what radius will the local pressure be equivalent to 29 inHg?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
03:41

Problem 31

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 $A$.

James Kiss
James Kiss
Numerade Educator
06:05

Problem 32

Line sources $m_{1}$ and $m_{2}$ are near point $A,$ as in Fig. $P 8.32$ If $m_{1}=30 \mathrm{m}^{2} / 2,$ find the value of $m_{2}$ for which the resultant velocity at point A is exactly vertical.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
04:33

Problem 33

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

KS
Keyan Sheppard
Numerade Educator
01:17

Problem 34

Consider three equally spaced sources of strength $m$ placed at $(x, y)=(+a, 0),(0,0),$ and $(-a, 0) .$ Sketch the resulting streamlines, noting the position of any stagnation points. What would the pattern look like from afar?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
08:09

Problem 35

A uniform stream, $U_{\infty}=4 \mathrm{m} / \mathrm{s},$ approaches a Rankine oval as in Fig. $8.13,$ with $a=50 \mathrm{cm} .$ Find the strength $m$ of the source-sink pair, in $\mathrm{m}^{2} / \mathrm{s}$, which will cause the total length of the oval to be $250 \mathrm{cm} .$ What is the maximum width of this oval?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
04:45

Problem 36

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?

Manish Kumar
Manish Kumar
Numerade Educator
03:00

Problem 37

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.

Chai Santi
Chai Santi
Numerade Educator
03:40

Problem 38

Consider potential flow of a uniform stream in the $x$ direction plus two equal sources, one at $(x, y)=(0,+a)$ and the other at $(x, y)=(0,-a) .$ Sketch your ideas of the body contours that would arise if the sources were ( $a$ ) very weak and (b) very strong.

Chai Santi
Chai Santi
Numerade Educator
03:40

Problem 39

A large Rankine oval, with $a=1 \mathrm{m}$ and $h=1 \mathrm{m},$ is immersed in $20^{\circ} \mathrm{C}$ water flowing at $10 \mathrm{m} / \mathrm{s}$. The upstream pressure on the oval centerline is 200 kPa. Calculate $(a)$ the value of $m ;$ and $(b)$ the pressure on the top of the oval (analogous to point A in Fig. P8.37)

VS
Vivek Singh
Numerade Educator
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Problem 40

Modify the Rankine oval in Fig. $\mathrm{P} 8.37$ so that the stream velocity and body length are the same but the thickness is unknown (not $1 \mathrm{m}$ ). The fluid is water at $30^{\circ} \mathrm{C}$ and the pressure far upstream along the body centerline is 108 kPa. Find the body thickness for which cavitation will occur at point $A$.

Victor Salazar
Victor Salazar
Numerade Educator
01:57

Problem 41

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 Bana
Prashant Bana
Numerade Educator
01:33

Problem 42

The vertical keel of a sailboat approximates a Rankine oval $125 \mathrm{cm}$ long and $30 \mathrm{cm}$ thick. The boat sails in seawater in standard atmosphere at 14 knots, parallel to the keel. At a section $2 \mathrm{m}$ below the surface, estimate the lowest pressure on the surface of the keel.

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
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Problem 43

Water at $20^{\circ} \mathrm{C}$ flows past a 1 -m-diameter circular cylinder. The upstream centerline pressure is 128,500 Pa. If the lowest pressure on the cylinder surface is exactly the vapor pressure, estimate, by potential theory, the stream velocity.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 44

Suppose that circulation is added to the cylinder flow of Prob. $\mathrm{P} 8.43$ sufficient to place the stagnation points at $\theta$ equal to $35^{\circ}$ and $145^{\circ} .$ 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?

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 45

If circulation $K$ is added to the cylinder flow in Prob. $\mathrm{P} 8.43$ $(a)$ for what value of $K$ will the flow begin to cavitate at the surface? $(b)$ Where on the surface will cavitation begin? $(c)$ For this condition, where will the stagnation points lie?

Victor Salazar
Victor Salazar
Numerade Educator
05:24

Problem 46

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.

Sophie S
Sophie S
Numerade Educator
05:30

Problem 47

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 $p_{a}, p_{b}, \rho,$ and the cylinder radius $a$.

Satpal Satpal
Satpal Satpal
Numerade Educator
01:29

Problem 48

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

Penny Riley
Penny Riley
Numerade Educator
07:35

Problem 49

In strong winds the force in Prob. $\mathrm{P} 8.48$ 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?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:07

Problem 50

It is desired to simulate flow past a two-dimensional ridge or bump by using a streamline that 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 $U ?$

Amit Srivastava
Amit Srivastava
Numerade Educator
03:19

Problem 51

A hole is placed in the front of a cylinder to measure the stream velocity of sea-level fresh water. The measured pressure at the hole is $2840 \mathrm{lbf} / \mathrm{ft}^{2} .$ If the hole is misaligned by $12^{\circ}$ from the stream, and misinterpreted as stagnation pressure, what is the error in velocity?

Ma Ednelyn Lim
Ma Ednelyn Lim
Numerade Educator
00:26

Problem 52

The Flettner rotor sailboat in Fig. E8.3 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 $15-\mathrm{mi} / \mathrm{h}$ wind. What is the optimum angle between the boat and the wind?

AG
Ankit Gupta
Numerade Educator
02:06

Problem 53

Modify Prob. $\mathrm{P} 8.52$ as follows. For the same sailboat data, find the wind velocity, in $\mathrm{mi} / \mathrm{h}$, that will drive the boat at an optimum speed of 8 kn parallel to its keel.

Simran Hiranandani
Simran Hiranandani
Numerade Educator
03:30

Problem 54

The original Flettner rotor ship was approximately $100 \mathrm{ft}$ long, 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 outside the range of Fig. $8.15 .$ 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 $25 \mathrm{ft} / \mathrm{s}$, as in Fig. $\mathrm{P} 8.54$, what will the wind force parallel and normal to the ship centerline be? Estimate the power required to drive the rotors.

Chai Santi
Chai Santi
Numerade Educator
03:07

Problem 55

Assume that the Flettner rotor ship of Fig. P8.54 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.]

Sachin Rao
Sachin Rao
Numerade Educator
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Problem 56

A proposed free-stream velocimeter would use a cylinder with pressure taps at $\theta=180^{\circ}$ and at $150^{\circ} .$ The pressure difference would be a measure of stream velocity $U_{\infty} .$ However, the cylinder must be aligned so that one tap exactly faces the free stream. Let the misalignment angle be $\delta$; that is, the two taps are at $\left(180^{\circ}+\delta\right)$ and $\left(150^{\circ}+\delta\right) .$ Make a plot of the percentage error in velocity measurement in the range $-20^{\circ}<\delta<+20^{\circ}$ and comment on the idea.

Victor Salazar
Victor Salazar
Numerade Educator
01:04

Problem 57

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.

Narayan Hari
Narayan Hari
Numerade Educator
08:09

Problem 58

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

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:37

Problem 59

The Transition® car-plane in Fig. 7.30 has a gross weight of 1430 lbf. Suppose we replace the wing with a 1 -ft-diameter rotating cylinder $20 \mathrm{ft}$ long. $(a)$ What rotation rate from Fig. $8.15,$ in $\mathrm{r} / \mathrm{min},$ would lift the plane at a take-off speed of $55 \mathrm{mi} / \mathrm{h} ?(b)$ Estimate the cylinder drag at this rotation rate. Neglect fuselage lift and cylinder end effects.

Penny Riley
Penny Riley
Numerade Educator
02:54

Problem 60

One of the corner flow patterns of Fig. 8.18 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. (8.53)$?$

Supratim Pal
Supratim Pal
Numerade Educator
01:07

Problem 61

Plot the streamlines of Eq. (8.53) 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 $x^{5} ?$

Carson Merrill
Carson Merrill
Numerade Educator
01:17

Problem 62

Combine stagnation flow, Fig. $8.19 b$, 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.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:36

Problem 63

The superposition in Prob. P8.62 leads to stagnation flow near a curved bump, in contrast to the flat wall of Fig. $8.19 b .$ Determine the maximum height $H$ of the bump as a function of the constants $A$ and $m$.

M S
M S
Numerade Educator
02:06

Problem 64

Consider the polar-coordinate stream function $\psi=B r^{1.2}$ $\sin (1.2 \theta),$ with $B$ equal, for convenience, to $1.0 \mathrm{ft}^{0.8} / \mathrm{s}$ (a) Plot the streamline $\psi=0$ in the upper half plane. (b) Plot the streamline $\psi=1.0$ and interpret the flow pattern.(c) Find the locus of points above $\psi=0$ for which the resultant velocity $=1.2 \mathrm{ft} / \mathrm{s}$.

Manish Jain
Manish Jain
Numerade Educator
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Problem 65

Potential flow past a wedge of half-angle $\theta$ leads to an important application of laminar boundary layer theory called the Falkner-Skan flows $[15, \mathrm{pp} .239-245] .$ Let $x$ denote distance along the wedge wall, as in Fig. $\mathrm{P} 8.65,$ and let $\theta=10^{\circ}$ Use Eq. (8.53) to find the variation of surface velocity $U(x)$ along the wall. Is the pressure gradient adverse or favorable?

Victor Salazar
Victor Salazar
Numerade Educator
05:46

Problem 66

The inviscid velocity along the wedge in Prob. P8.65 has the analytic form $U(x)=C x^{m},$ where $m=n-1$ and $n$ is the exponent in Eq. $(8.53) .$ Show that, for any $C$ and $n$ computation of the boundary layer by Thwaites's method, Eqs. (7.53) and $(7.54),$ leads to a unique value of the Thwaites parameter $\lambda$. Thus wedge flows are called similar $[15, \mathrm{p} .241]$.

Satpal Satpal
Satpal Satpal
Numerade Educator
03:54

Problem 67

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

Chai Santi
Chai Santi
Numerade Educator
03:54

Problem 68

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

Chai Santi
Chai Santi
Numerade Educator
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Problem 69

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?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
00:57

Problem 70

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

Joseph Liao
Joseph Liao
Numerade Educator
09:52

Problem 71

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?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
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Problem 72

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?

Victor Salazar
Victor Salazar
Numerade Educator
01:02

Problem 73

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.

Raj Bala
Raj Bala
Numerade Educator
02:12

Problem 74

A positive line vortex $K$ is trapped in a corner, as in Fig. $\mathrm{P} 8.74 .$ 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.

Supratim Pal
Supratim Pal
Numerade Educator
View

Problem 75

Using the four-source image pattern needed to construct the flow near a corner in Fig. $\mathrm{P} 8.72,$ find the value of the source strength $m$ that will induce a wall velocity of $4.0 \mathrm{m} / \mathrm{s}$ at the point $(x, y)=(a, 0)$ just below the source shown, if $a=50 \mathrm{cm}$.

Victor Salazar
Victor Salazar
Numerade Educator
04:54

Problem 76

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 $A, B, C,$ and $D, \mathrm{com}-$ paring with a cylinder flow in an infinite expanse of fluid.

James Kiss
James Kiss
Numerade Educator
03:22

Problem 77

Discuss how the flow pattern of Prob. P8.58 might be interpreted to be an image system construction for circular walls. Why are there two images instead of one?

Kevin Corkran-Itagaki
Kevin Corkran-Itagaki
Numerade Educator
02:42

Problem 78

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 particular dimensions shown in this figure, estimate the position of the nose of the resulting half-body.

Narayan Hari
Narayan Hari
Numerade Educator
03:47

Problem 79

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?

Ajay Singhal
Ajay Singhal
Numerade Educator
03:59

Problem 80

The beautiful expression for lift of a two-dimensional airfoil, Eq. $(8.59),$ 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 $x=-\varepsilon \ll$ $1, y=0,$ and radius $(1+\varepsilon)$ will become an airfoil shape in the $\zeta$ plane. [Hint: The Excel spreadsheet is excellent for solving this problem.

James Kiss
James Kiss
Numerade Educator
01:22

Problem 81

Given an airplane of weight $W,$ wing area $A,$ aspect ratio AR, and flying at an altitude where the density is $\rho .$ Assume all drag and lift is due to the wing, which has an infinite-span drag coefficient $C_{D \infty} .$ Further assume sufficient thrust to balance whatever drag is calculated. $(a)$ Find an algebraic expression for the best cruise velocity $V_{b},$ which occurs when the ratio of drag to speed is a minimum. $(b)$ Apply your formula to the data in Prob. P7.119 for which a laborious graphing procedure gave an answer $V_{b} \approx 180 \mathrm{m} / \mathrm{s}$.

Carson Merrill
Carson Merrill
Numerade Educator
06:05

Problem 82

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 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 kn.

Chai Santi
Chai Santi
Numerade Educator
02:00

Problem 83

The world's largest airplane, the Airbus $\mathrm{A} 380,$ has a maximum weight of 1,200,000 lbf, wing area of $9100 \mathrm{ft}^{2},$ wingspan of $262 \mathrm{ft},$ and $C_{D o}=0.026 .$ When cruising at maximum weight at $35,000 \mathrm{ft}$, the four engines each provide 70,000 lbf of thrust. Assuming all lift and drag are due to the wing, estimate the cruise velocity, in $\mathrm{mi} / \mathrm{h}$.

Anand Jangid
Anand Jangid
Numerade Educator
09:10

Problem 84

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:
$$\begin{array}{lcc}
\hline x / c & V / U_{\infty}(\text { upper }) & V / U_{\infty}(\text { lower }) \\
\hline 0.0 & 0.0 & 0.0 \\
0.025 & 0.97 & 0.82 \\
0.05 & 1.23 & 0.98 \\
0.1 & 1.28 & 1.05 \\
0.2 & 1.29 & 1.13 \\
0.3 & 1.29 & 1.16 \\
0.4 & 1.24 & 1.16 \\
0.6 & 1.14 & 1.08 \\
0.8 & 0.99 & 0.95 \\
1.0 & 0.82 & 0.82
\end{array}$$
Use these data, plus Bernoulli's equation, to estimate (a) the lift coefficient and ( $b$ ) the angle of attack if the airfoil is symmetric.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:29

Problem 85

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 sealevel standard air at $200 \mathrm{ft} / \mathrm{s}$ and is found to have lift of $30 \mathrm{lbf}$ 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.

Akshaya Rs
Akshaya Rs
Numerade Educator
10:24

Problem 86

An airplane has a mass of $20,000 \mathrm{kg}$ and flies at $175 \mathrm{m} / \mathrm{s}$ at 5000 -m standard altitude. Its rectangular wing has a 3 -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.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
04:37

Problem 87

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 $7 \mathrm{m} / \mathrm{s}$ and $\alpha=2.5^{\circ}$ 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 that delivers 20 hp to the water.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
03:51

Problem 88

The Boeing $787-8$ Dreamliner has a maximum weight of 502,500 lbf, a wingspan of $197 \mathrm{ft}$, a wing area of $3501 \mathrm{ft}^{2}$ and cruises at $567 \mathrm{mi} / \mathrm{h}$ at $35,000 \mathrm{ft}$ altitude. When cruising, its overall drag coefficient is about $0.027 .$ Estimate $(a)$ the aspect ratio (b) the lift coefficient, $(c)$ the cruise Mach number, and ( $d$ ) the engine thrust needed when cruising.

Shoukat Ali
Shoukat Ali
Other Schools
02:51

Problem 89

The Beechcraft $\mathrm{T}-34 \mathrm{C}$ aircraft has a gross weight of 5500 lbf 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 that delivers 300 hp to the air. Assume for this problem that its airfoil is the NACA 2412 section described in Figs. 8.23 and $8.24,$ and neglect all drag except the wing. What is the appropriate aspect ratio for the wing?

Chai Santi
Chai Santi
Numerade Educator
05:28

Problem 90

NASA is developing a swing-wing airplane called the Bird of Prey $[37] .$ As shown in Fig. $\mathrm{P} 8.90,$ the wings pivot like a pocketknife blade: forward ( $a$ ), straight ( $b$ ), or backward $(c) .$ Discuss a possible advantage for each of these wing positions. If you can't think of one, read the article [37] and report to the class.

Kristela Garcia
Kristela Garcia
Numerade Educator
00:55

Problem 91

If $\phi(r, \theta)$ in axisymmetric flow is defined by Eq. (8.72) and the coordinates are given in Fig. 8.28 , determine what partial differential equation is satisfied by $\phi$.

R M
R M
Numerade Educator
03:00

Problem 92

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 $4.5 \mathrm{m} / \mathrm{s}$.

Chai Santi
Chai Santi
Numerade Educator
01:31

Problem 93

The Rankine half-body of revolution (Fig. 8.30 ) could $\operatorname{sim}$ ulate 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.

Narayan Hari
Narayan Hari
Numerade Educator
07:42

Problem 94

Determine whether the Stokes streamlines from Eq. (8.73) are everywhere orthogonal to the Stokes potential lines from Eq. $(8.74),$ as is the case for Cartesian and plane polar coordinates.

John Gehad
John Gehad
Numerade Educator
08:09

Problem 95

Show that the axisymmetric potential flow formed by superposition of a point source $+m$ at $(x, y)=(-a, 0),$ a point $\sin \mathrm{k}-m$ 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 $a$.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:40

Problem 96

Consider inviscid flow along the streamline approaching the front stagnation point of a sphere, as in Fig. 8.31 . Find
(a) the maximum fluid deceleration along this streamline and $(b)$ its position.

Narayan Hari
Narayan Hari
Numerade Educator
02:41

Problem 97

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 lowpressure water tunnel as shown, cavitation may appear at point $A .$ Compute the stream velocity $U,$ neglecting surface wave formation, for which cavitation occurs.

Amit Srivastava
Amit Srivastava
Numerade Educator
08:09

Problem 98

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. $\mathrm{P} 8.98 ?$ 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 $m /\left(U_{\infty} L^{2}\right)$.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
02:36

Problem 99

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. P8.49, at what point $A$ on the hemisphere should a hole be cut so that the pressure force will be zero according to inviscid theory?

Guilherme Barros
Guilherme Barros
Numerade Educator
03:03

Problem 100

A 1 -m-diameter sphere is being towed at speed $V$ in fresh water at $20^{\circ} \mathrm{C}$ as shown in Fig. P8.100. 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?

Amit Srivastava
Amit Srivastava
Numerade Educator
05:33

Problem 101

Consider a steel sphere $(\mathrm{SG}=7.85)$ of diameter $2 \mathrm{cm}$ dropped from rest in water at $20^{\circ} \mathrm{C}$. Assume a constant drag coefficient $C_{D}=0.47 .$ Accounting for the sphere's hydrodynamic mass, estimate ( $a$ ) its terminal velocity and ( $b$ ) the time to reach 99 percent of terminal velocity. Compare these to the results when hydrodynamic mass is neglected, $V_{\text {terminal }} \approx 1.95 \mathrm{m} / \mathrm{s}$ and $t_{99 / 6} \approx 0.605 \mathrm{s},$ and discuss.

Vishal Gupta
Vishal Gupta
Numerade Educator
10:15

Problem 102

A golf ball weighs 0.102 Ibf 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. P7.108. 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 $7500 \mathrm{r} / \mathrm{min}$.

Meghan Miholics
Meghan Miholics
Numerade Educator
01:40

Problem 103

Consider inviscid flow past a sphere, as in Fig. 8.31 . Find (a) the point on the front surface where the fluid acceleration
$(c)$ If the $a_{\max }$ is maximum and $(b)$ the magnitude of $a_{\max }$ stream velocity is $1 \mathrm{m} / \mathrm{s},$ find the sphere diameter for which $a_{\max }$ is 10 times the acceleration of gravity. Comment.

Narayan Hari
Narayan Hari
Numerade Educator
01:23

Problem 104

Consider a cylinder of radius $a$ moving at speed $U_{\infty}$ through a still fluid, as in Fig. P8.104. 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. (8.91).

Dominador Tan
Dominador Tan
Numerade Educator
01:14

Problem 105

A 22 -cm-diameter solid aluminum sphere $(\mathrm{SG}=2.7)$ is accelerating at $12 \mathrm{m} / \mathrm{s}^{2}$ in water at $20^{\circ} \mathrm{C}$. ( $a$ ) According to potential theory, what is the hydrodynamic mass of the sphere?
(b) Estimate the force being applied to the sphere at this instant.

Narayan Hari
Narayan Hari
Numerade Educator
02:08

Problem 106

Laplace's equation in plane polar coordinates, Eq. (8.11) is complicated by the variable radius. Consider the finite difference 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 (8.96).

Manik Pulyani
Manik Pulyani
Numerade Educator
01:43

Problem 107

SAE $10 \mathrm{W} 30$ oil at $20^{\circ} \mathrm{C}$ is at rest near a wall when the wall suddenly begins moving at a constant $1 \mathrm{m} / \mathrm{s}$. (a) Use $\Delta y=$ $1 \mathrm{cm}$ and $\Delta t=0.2 \mathrm{s}$ and check the stability criterion (8.101) (b) Carry out Eq. (8.100) to $t=2$ s and report the velocity $u$ at $y=4 \mathrm{cm}$.

Narayan Hari
Narayan Hari
Numerade Educator
03:40

Problem 108

Consider two-dimensional potential flow into a step contraction as in Fig. $\mathrm{P} 8.108 .$ 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 computer and plot the streamlines in the flow.

Chai Santi
Chai Santi
Numerade Educator
02:24

Problem 109

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.)

Breanna Ollech
Breanna Ollech
Numerade Educator
View

Problem 110

For fully developed laminar incompressible flow through a straight noncircular duct, as in Sec. $6.8,$ the Navier-Stokes 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 $b$

Victor Salazar
Victor Salazar
Numerade Educator
01:04

Problem 111

Solve Prob. P8.110 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
\[
Q \approx 0.1143 \frac{b^{4}}{\mu}\left(-\frac{d p}{d x}\right) \quad f \operatorname{Re}_{D_{n}} \approx 62.19
\]
where $D_{h}=4 A / P=4 b / 3$ for this case. Comment on the possible truncation errors of your model.

Narayan Hari
Narayan Hari
Numerade Educator
02:39

Problem 112

In CFD textbooks $[5,23-27],$ one often replaces the lefthand sides of Eqs. $(8.102 b \text { and } c)$ with the following two expressions, respectively:
\[
\frac{\partial}{\partial x}\left(u^{2}\right)+\frac{\partial}{\partial y}(v u) \text { and } \frac{\partial}{\partial x}(u v)+\frac{\partial}{\partial y}\left(v^{2}\right)
\]
Are these equivalent expressions, or are they merely simplified approximations? Either way, why might these forms be better for finite difference purposes?

Lottie Adams
Lottie Adams
Numerade Educator
01:41

Problem 113

Formulate a numerical model for Eq. $(8.99),$ which has no instability, by evaluating the second derivative at the next time step, $j+1 .$ Solve for the center velocity at the next time step and comment on the result. This is called an implicit model and requires iteration.

James Kiss
James Kiss
Numerade Educator
04:43

Problem 114

If your institution has an online potential flow boundary element 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]
\[
\begin{aligned}
\frac{2 y}{t_{\max }} \approx & 1.4845 \zeta^{1 / 2}-0.63 \zeta-1.758 \zeta^{2} \\
&+1.4215 \zeta^{3}-0.5075 \zeta^{4}
\end{aligned}
\]
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 $C$.

Chai Santi
Chai Santi
Numerade Educator
02:02

Problem 115

Use the explicit method of Eq. (8.100) to solve Prob. P4.85 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 $y=2 \mathrm{cm}$.

James Kiss
James Kiss
Numerade Educator