For fully-developed turbulent flow in a pipe:

u u rR

n

max 1

1

where n varies from 6 to 10 depending on the flow Reynolds number and umax is the velocity on the pipe axis.

Let s rR

ds drR

r R s 1 1;

Q R u nn n

n nR u n

n n

22 1 1

1 2 12

1 2 12

2 2

maxmax

u n

n nuave

21 2 1

2

max

- 61 -

u

d

r

n = 7 in most cases (1/7th power-law velocity distribution in fully-developed turbulent pipe flow)

with n = 7

u uave

2 498 15 max

u u uave 98

1204960max max

Flow along curved streamlines

Fluid is accelerated both along the streamline (in the s-direction) and normal to the streamlines (in the n-direction)

Velocity field:

V V s t ,

Accleration:

a a DV

DtV

dtV

Vss

(along streamline)

Acceleration normal to the streamline:

adt

Vs

dsdt

VVs

ds dnn

1 1

sin

d is small sin d dds =Rd; ds is small

adt

VVs

s ddt

VVs

sdsR

VR

dsdt Rdt

Vs

dsn

1 1 1 2

(second-order term)

- 62 -

s

R

Radius of

d

s = st = t streamlin

e

Centre of curvature

C

dsd

s = s + ds

6.34 Water flows steadily up a vertical 0.1 m diameter pipe and out the nozzle, which is 0.05m in diameter, discharging to atmospheric pressure. The stream velocity at the nozzle exit must be 20m/sec. Calculate the gage pressure required at section (1), assuming frictionless flow.

Given: Flow is steady

Frictionless flow can be assumed

Working fluid is water Flow is incompressible

From the foregoing, Bernoulli’s equation can be applied along a streamline

Central streamline is chosen

- 63 -

1

2V2

Datum

4 m

Flow

Continuity:

6.36 Water may be considered to flow without friction through the siphon. The water flow rate is 0.03m3/sec, its temperature is 20C, and the pipe diameter is 75 mm. Compute the maximum allowable height, h, so that the pressure at point A is above the vapor pressure of the water.

Given: Flow without friction frictionless flow

Flow of water incompressible flow

Assume: Flow is steady

Bernoulli’s equation can be applied along a streamline from (1) to (2)

(since tank >> delivery pipe)

(see handout)

- 64 -

hz

D = 75

flo

A

1

Continuity: Q V A V Q

AmA A A

A

0 030 075

4

6 7923.

.. / sec

h mmax .7 73

6.53 The flow system of parallel disks shown contains water. As a first approximation, friction may be neglected. Determine the volume flow rate and the pressure at point (C). (R = 300mm and rc = 150mm.)

Given: Flow of water incompressible flow

Friction may be neglected frictionless flow

Assume: Container much larger than space between the disks V1 0 flow is steady

Bernoulli’s equation can be applied along a streamline from (1) to (2)

- 65 -

C z

1

R

rC

r1

H = 1.5 mm

2

H = 1 m

V

g V m22

221 2 9 81 1 4 43. . / sec

Volume flow rate: Q A V RhV m 2 2 232 2 0 3 0 0015 4 43 0 0125 . . . . / sec

Continuity: A V A V Q V

QA

Qr h

mc c cc c

2 2 20 0125

2 015 0 00158 86

.

. .. / sec

Bernoulli’s equation applied along a streamline from (C) to (2)

6.45 An Indianapolis racing car travels at a maximum speed of 350 km/hr. A pressure gage attached to the airfoil reads -50mm of water (gage). Estimate the air speed relative to the car at that location.

Compressibility effects in air become significant when the local speed is more than 30% of the speed of sound at the prevailing temperature c k RT c m s : /340 .

Flow is incompressible

Let flow observer be located on the wing Flow appears steady to the observer

Flow outside the boundary layer can be treated as inviscid, i.e., frictionless

Bernoulli’s equation can be applied along a streamline

- 66 -

But

V m13503 6

97 2 .

. / sec

(relative to the wing)

4.58 A jet of water issuing from a stationary nozzle at 15m/sec (Aj =0.05m2) strikes a turning vane mounted on a cart as shown. The vane turns the jet through angle =50. Determine the value of M required to hold the cart stationary.

- 67 -

V1

p1p2

V22

1

Assumptions

Flow is steadyFlow at (i) and (e) is uniformNo drag forces at the vane

Momentum equation for the C.V.:

(Steady flow) x-component of the momentum equation:

T 1000 0 05 15 15 15 50 1000 015 15 502. cos . cos

T N Mg 4020 since cart stays stationary

4.78 A turning vane, which deflects the water through 60, is attached to the cart under the conditions of Problem 4.57. Determine the tension in the wire holding the cart stationary and the force of the vane on the cart.

- 68 -

M

Ry

= 50o

Ty

x

i

Mg

CV

T

e

Assumptions

No drag forces on the vaneFlow is uniform at (i) and (e)

Given

Flow is steady, level in tank is held constantFlow of water = constantMomentum equation for the C.V.

(Steady flow)

x-component of the momentum equation for C.V.a

F Nxa 1000 600 10 10 10 60 306 cos

Fxa = Tension in wire holding cart = 30N

x-component of the momentum equation for C.V.b:

Force ( Px ) of vane on cart in the x-direction = -(Force of cart on vane=Fxb)

P F Nx xb 30

- 69 -

Fy

b

= 60o

e a

i b

CVa

e bCVb

Fxa

Fx

b

x

y

y-component of the momentum equation for C.V.b:

Force (Py) of vane on cart in the y-direction = -(Force of cart on vane=Fyb)

4.80 A conical spray head is shown. The fluid is water and the exit stream is uniform. Evaluate (a) the thickness of the spray sheet at 400mm radius and (b) the axial force exerted by the spray head on the supply pipe.

Assumptions

Flow is uniform at (i) and (e)Flow is steady

Given

Flow of water =const

(a) Conservation of mass for steady flow:

Q V A V A Rti i e e ( ) .10 2 0 03

- 70 -

x

y

= 30o

= 30o

CVRx

pig

D = 300

Q = 0.03

V = 10

i

e

p1 = 150 kPa

V Ai i 0 03.

t m mm 0 03

10 2 0 41194 10 11943.

.. .

(b) Momentum equation for the C.V.

(Steady flow)

x-component of the momentum equation for C.V.:

R Nx 1000 150 10134

0 3 1000 0 03 0 424 10 30 3605. . . . sin

Axial force (Ps) of spray head on supply pipe = -(axial force of supply pipe on spray head = Rx) P R N Ns x 3605 3605

4.106 Experimental measurements are made in a low-speed air jet to determine the drag force on a circular cylinder. Velocity measurements at two sections, where the pressure is uniform and equal, give the results shown. Evaluate the drag force on the cylinder, per unit width.

- 71 -

U = 50 m/sec = 1.2 kg/m3

D = 30 mma = 2.2D

u Uya

y a

sin ;

2

0

u = U; y > a

Assumptions

Flow is steadyFlow is incompressible since flow is low

Conservation of mass for steady flow

m m m m1 2 3 4 0

U w a m uwdy ma

2 2 02 0 4

m m U w a uwdy U u wdya a

2 4 0 02 2 2

Momentum equation for the c.v.:

F F

tV dV V V dAs B

V A

(Steady flow)

- 72 -

x

y

U D

1

2

3

4

u

U

a

CV

Rx

x-component of the momentum equation: R F U m m U w a U u wdyx Bx

a ( )2 4

2

02 2

R U m m u U wdyx

a 2 4

2 2

02

R U Uu wdy u U wdy u u U wdyx

a a a 2 2 22

0

2 2

0 0

Drag force on cylinder (FD) = -(force of cylinder on the c.v. = Rx)

FD = -Rx

F u U u wdy u U u wdyD

a a 2 2

0 0

Fw

U uU

uU

dyD a

2 12

0 u

Uya

sin

2

Fw

Uya

ya

dyD a

2

21

22

0

sin sin

y

ad

ady

2 2

Fw

U a dD 2 1 22

02

sin sin

Fw

D Nm541.

4.134 A jet of water is directed against a vane, which could be a blade in a turbine or in any other piece of hydraulic machinery. The water leaves the stationary 50 mm diameter nozzle with a speed of 20 m/sec and enters the vane tangent to the surface at A. The inside surface of the vane at B makes angle = 150o with the x direction. Compute the force that must be applied to maintain the vane speed constant at U = 5 m/sec.

- 73 -

Assumptions

Flow is steadyFlow is uniform at ‘i’ and ‘e’No drag forces at the vane

Momentum equation for the c.v.:

F F

tV dV V V dAs B

V A

(Steady flow)

x-component of the momentum equation:

R F u V A Q V Vx B x xi xei

e

.

R A V U V U V Ux cos

Velocity is referenced to the C.V.

Rx

10004

501000

20 5 1 1502

2 cos

R Nx 824 4.

y-component of the momentum equation:

- 74 -

Ry

= V

y

x

i

CV

e

Rx

U = 5

R Mgy

10004

501000

20 5 0 1502

2 sin

R Mg Ny ( . )220 9

R i j N N ( . . ) .824 4 220 9 8535 165

4.144 Consider a single vane, with turning angle , moving horizontally at constant speed, U, under the influence of an impinging jet as in Problem 4.135. The absolute speed of the jet is V. Obtain general expressions for the resultant force and power that the vane could produce. Show that the power is maximized when U = V/3.

Assumptions:

Flow is steadyFlow is uniform at (i) and (e)No drag forces at the vane

Momentum equation for the c.v.:

F F

tV dV V V dAs B

V A

(Steady flow)

x-component of the momentum equation:

- 75 -

Rx

V

y

x

i

CV

U

e

R A V U V U V Ux cos

R A V Ux 2 1cos

Resultant force exerted by vane (Kx) = -(force of cart on vane = Rx)

K R A V Ux x 2 1 cos

Power produced by vane: cosW K U A V U Uout x 2 1

For maximum power:

A V VU U1 4 3 02 2 cos

V U V U U V or U V 3 0

3

FLOW OVER A VANE (Divided Flow)

Assumptions

No drag force at the vaneFlow is steadyFlow is incompressibleFlow is uniform at the inlet and exits

Momentum equation for the c.v.

- 76 -

CV

V1

A1

t

vane

Fn

A2V2

V0A0

V0

n

V0sinV0cos

(steady flow)

Momentum equation in the t-direction:

0 0 0 1 1 2 2Q V V Q V Qcos

V V V0 1 2 (no drag!)

Q Q Q0 1 2cos (1)

Conservation of mass for steady flow:

Q Q Q Q Q Q0 1 2 0 1 20 (2)

Solve (1) & (2) simultaneously

QQ

QQ

10

20

21

21cos ; cos

Line of action of the normal force

h = width

- 77 -

CV

V1

b1

t

Fn

b2V2

V0

n

n fb0

O

n h

b V b V

Ffn

1

21

22

22

2

2

n-component of the momentum equation:

n

hV b b

hb V

b b

bf

0

21

22

2

0 02

12

22

02 2sin sin

Momentum flux correction factor

Consider flow through a duct of constant area, A

flow is incompressible

- 78 -

c

Vave

M Mactual ideal

correction factor

12

Au

VdA

aveA

4.146 Water, in a 100mm diameter jet with speed of 30 m/sec to the right, is deflected by a cone that moves to the left at 15 m/sec. Determine (a) the thickness of the jet sheet at a radius of 200mm and (b) the external horizontal force needed to move the cone.

Assumptions

Flow is steadyFlow is uniform at (i) and (e)No drag force at the cone

Given

Flow of water = const

Conservation of mass for steady flow:

- 79 -

Rx

= 60o

Vj

y

xi

CV

VC

e

cone

Momentum equation for the c.v.:

(steady flow)

x-component of the momentum equation:

Rx = 7952 N

4.153 A steady jet of water is used to propel a small cart along a horizontal track as shown. Total resistance to motion of the cart assembly is given by FD=kU2, where k = 0.92 N sec2/m2. Evaluate the acceleration of the cart at the instant when its speed is U=10m/sec.

Assumptions

No drag forces at the vaneFlow is uniform at (i) and (e)

Given

Flow is steady

Momentum equation for the c.v.

- 80 -

= V

y

x

i

CV

e

FDU = 10.0

V = 30.0 m/s

D = 25.0

M = 15.0

(flow is steady)

x-component of the momentum equation:

a m0

2 2 2

20 92 10 1000 30 10

40 025 15

15135

. . .. / sec

Example: Neglecting losses, determine the force needed to hold the Y shown in place. Assume the Y to be in a horizontal plane.

Assumptions

Flow is steady

- 81 -

CV

Rx

Ry

500 L/s

300 L/s

200 L/s

F1

F2

F3

1

2

3

p1 = 60 kPa

20o30o

30 cm

45 cm

15 cm

y

x

Flow is uniform at 1,2 & 3

Given

Water as working fluid = const

Conservation of mass

since = const

Flow is assumed uniform

VA Q Q Q Q Q 0 01 2 3

since (1) is an inlet

Bernoulli is used along a central streamline from (1) to (2) and from (1) to (3) to obtain the pressures p2 & p3

Note: flow is frictionless along the central streamlines since the rate of shear (du/dy) is zero there. Flow has been assumed steady and water is the working fluid which is incompressible

z z z1 2 3 0 (Y is in the horizontal plane)

- 82 -

Similarly,

x-component of the momentum equation for a c.v.

(V1x = 0)

F F R Q V Q Vx x x x x2 3 2 2 3 3

3954 2 30 152 20 1000 0 3 4 24 30 0 2 1132 20. cos . cos . . cos . . cos Rx

y-component of the momentum equation for a c.v.

F F F R Q V Q V Q Vy y y y y y y1 2 3 1 1 2 2 3 3

F F F R Q V Q V Q Vy y y y1 2 3 1 1 2 2 3 330 20 sin sin

Ry

9542 6 3954 2 30 15 2 20 10 05 314 10 0 3 4 24 30

0 2 10 1132 20

3 3

3

. . sin . sin . . . . sin

. . sin

R Ny 7720

Force needed to hold Y in place:

- 83 -

R i j N 2384 7720 or R N8080

Problem 4.196 (Fox & McDonald, 4th edition)

CVI for linear momentum equationCVII for conservation of mass

Given

Ac/Ae = 10, Vc(t) = V0-ktM0 is the initial massWater is the working fluid = const

Assumption

Flow is uniform at (e) & (c)

Conservation of mass for the CVI

- 84 -

VC = V0 - kt

O xy

CV

CVII

c

ex

y

(since flow is assumed uniform at (e) & (c) )

A V A V V VAA

V ktc c e e e cc

e

0 10 0 (a)

Conservation of mass for the CVII

d Mdt

V Ad Mdt

V Ae e e e 0

d Mdt

V kt A mc 0

M M A V t k tc

0 0

2

2 (b)

Momentum equation for the c.v.II Consider the y-component

Mg a Mt

MV V my y e y0

V V m dMdte y e ,

- 85 -

Mg a M MdVdt

V dMdt

V myy

y e0

aMgM

MM

dVdt

mM

V Vyy

y e0

V V ktdVdt

kyy 0

a g k mM

V V k

V kt A VVV

M V t k t Agy e y

c ye

y

c

0

0

0 02

1

12

(c)

- 86 -