PROBLEM 1
Problem Statement:
For water flow at 30 deg. C. in a ½” pipe at Re = 500, plot the development of the axial velocity profile. A plot of dimensionless streamwise velocity (u x
/u average
) versus dimensionless radial position (r/R) should be constructed. Show this velocity profile for at least 5 dimensionless positions downstream (5 different x/D values). One of the profiles should be very close to the pipe entrance (x/D ~ 3) and one profile should be extremely close to the parabolic velocity profile characteristic of laminar flow in a pipe.
Schematic: r
D = 0.5" x
X = 5 m
X = 0
Assumptions:

Laminar flow

Incompressible flow

Constant properties
Properties 1 :
Water @ T = 30 o
C
  kg
995.18
m
3
 
7.7733

10

4 kg m
 s

Analysis:
Determine average velocity for Re = 500 and insert into FlowLab to generate plot for the axial development of the velocity profiles:
Re


VD


V

Re

D

V (Re

500 )

( 500 )( 7 .
7733

10

4 kg m
 s
)
( 995 .
18 kg
)( 0 .
01270 m ) m 3

0 .
030752 m s
Axial Development of Velocity Profiles at Re = 500
2
1.5
1
0.5
0
0 0.2
0.4
0.6
0.8
1 r/R
[1] Incroprera, F. and DeWitt, D. Fundamentals of Heat and Mass Transfer,5 th Ed . Table A-6. x=1*d x=3*d x=10*d x=15*d x=25*d

PROBLEM 2
Problem Statement:
For water flow at 30 deg. C. in a ½” pipe at Re = 10,000, plot the ratio of the maximum streamwise (x-direction) flow velocity to the average (x-direction) streamwise flow velocity as a function of dimensionless position downstream (as a function of x/D). Use at least 5 different x/D values to generate this plot. What is the limiting value for the ratio of the maximum flow velocity to the average flow velocity as x/D becomes very large? What is the flow development length (the value of x/D) when the ratio of the maximum streamwise flow velocity to the average flow velocity is equal to 98% of its limiting value?
Schematic: r
D = 0.5" x
X = 5 m
Properties
1
:
Water @ T = 30 o
C
  kg
995.18
m
3
 
7.7733

10

4 kg m
 s
X = 0

Analysis:
Create a plot of the centerline velocity distribution and non-dimensionalize the quantities:
Ratio of Streamwise Velocity to Average Velocity vs. Pipe Length
1.25
1.2
1.15
1.1
1.05
1
0.95
0 5 10 15 20 25 30 x/D
The maximum ratio of streamwise velocity to average velocity is 1.2.
To find the flow development length, find x/D where u x
is 98% of the ratio of streamwise velocity to average velocity:
(0.98)*(1.20) = 1.176
This occurs at x/D = 7.64
CHECK:
Confirm that the ratio approaches 1.2 by observing the axial development plot:
Axial Development of Velocity Profiles at Re = 10,000
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 0.2
0.4
0.6
r/R
0.8
1 1.2
From the axial development plot, we see that the fully-developed flow region at past our calculated x/D = 7.64 and beyond has an x-velocity to average velocity ratio of 1.2.
[1] Incroprera, F. and DeWitt, D. Fundamentals of Heat and Mass Transfer, 5 th Ed . Table A-2. x=1*d x=3*d x=10*d x=15*d x=25*d

PROBLEM 3
Problem Statement:
For air flow at 20 o
C over a 1.4 m flat plate at u

= 5 .
28 m
, determine the development of s the thermal boundary layer (99% of the free stream temperature) when the plate is maintained at 100 o
C (use at least 3 points along the plate). Make a plot of the thermal boundary thickness and hydrodynamic boundary layer thickness as a function of distance along the plate. Compare the boundary layer thickness predicted by FlowLab with the analytical solution at various Re x
.
Schematic: y
U
 x
X = 0
Assumptions:

Laminar flow

Incompressible flow

Constant properties
Properties
3
:
Air @ T = 20 o C
 
1 .
21 kg m
3

C p

1.79

10

5 kg m
 s

1005 .
20
J kg

K k
Pr

0 .
02566 m
W

K

0 .
7013
T= 100 o
C
X = 1.4 m

Analysis:
See attached plots.
The result of the von Karman balances gives:
4
3
10
 2 
3
10
 
2
15

3
140
1
 2

1
180
 3

37
315
Pr

1
The Prandtl number is known and the value of

is expressed as:
 


T
H
(
( x ) x )
We will obtain values for both the thermal and hydrodynamic boundary layer thicknesses using FlowLab: x/L Re x
Pr 37
Pr

1
315

T
( x ) from
FlowLab

H
( x ) from
FlowLab
 

T

H
( x )
( x )
L.H.S. of equation
0.25 125,000 0.7013 0.1675 0.005828 0.005418
0.50 250,000 0.7013 0.1675 0.008045 0.007334
1 500,000 0.7013 0.1675 0.01133 0.01020
1.076
1.097
1.111
0.1437
0.1516
0.1570
Generate a plot of the thermal boundary layer thickness and hydrodynamic boundary layer thickness as a function of distance along the plate:
Thermal and Hydrodynamic Boundary Layer vs. Distance Along the
Plate
%
Error
14.2
9.50
6.27
0.01
0.008
0.006
0.004
0.002
0
0 0.2
0.4
0.6
0.8
1 hydrodynamic thermal x/L
[3] SFPE Handbook of Fire Protection Engineering 2 nd Edition, Table B-2.
[4]Bird, Stewart, and Lightfoot. Transport Phenomena , 2 nd Edition. Equation 12D.8-1. pp 406.

PROBLEM 4
Problem Statement:
Determine the axial development of the temperature profile and the Nusselt number of the water flow in a 1” pipe with a heated wall at 100 o
C for Re = 1500. Plot the fluid temperature as a function of radial position for at least 8 locations along the pipe length
(x-direction). Plot the Nusselt Number (Nu) as a function of distance along the pipe length (x-direction). What is the limiting value of Nu? Compare with the literature value.
Schematic:
T wall
= 100 o
C
Water at T =
20 o
C r
D = 1" z
Z = L
Z = 0
Assumptions:

Uniform surface temperature

Constant properties
Properties
1
:
Water (T = 20
o
C):
  kg
997 .
98 m 3

C p

0 .
982

10

3

J
4182 kg

K kg m
 s k
Pr


W
0 .
60304 m

K
6 .
97

Analysis:
Use FlowLab to generate the axial development of the temperature profile and Nusselt number distribution:
Fully-developed Nusselt number results:
Nu FlowLab Value: 3.67
Nu Literature Value
4
: 3.66
% error: 0.273%
[1] Incropera, F. and DeWitt, D. Fundamentals of Heat and Mass Transfer, 5 th Ed . Table A-6.
[5] Incropera, F. and DeWitt, D. Fundamentals of Heat and Mass Transfer, 5 th Ed . pp 487.

PROBLEM 5
Problem Statement:
Determine the drag coefficient for flow past a cylinder of radius 0.5m at and Re

10
6
. Compare FlowLab results to literature values.
Re

100
Schematic:
R = 0.5 m
Properties:
 
1000 kg m
3
 
0 .
001 kg m
 s
Analysis:
Reynolds Number Literature Value 6
100
6
10
1.6
0.38 of C
D
FlowLab Value of
C
D
1.52
0.40
The % error is within reasonable boundaries for error.
5.0
5.2
% Error
[6] Incropera, F. and DeWitt, D. Fundamentals of Heat and Mass Transfer, 5 th Ed . pp 409.

PROBLEM 6
Problem Statement:
For flow through a flat orifice of 1.2 cm in a pipe of 6 cm ID transporting water at 20 deg. C and Re=50,000 (through the orifice), what is the volumetric flowrate of the recirculating fluid (in m 3 /s) at a location one-half orifice diameter downstream from the orifice?
Schematic:
R r
L
Assumptions:

Incompressible flow

Constant properties

Axisymmetric flow
Properties
1
:
Water @ T = 20 o
C
 
0 .
982

10

3 kg m
 s
  kg
997 .
984 m
3
Analysis:
Set all geometry, material, and boundary conditions in orifice plate template then iterate for the solution making sure that at least of the x/D positions is defined as 0.5. Generate axial velocity magnitude plot and integrate the following numerically to find volume of recirculating flow:
Total volumetric flow rate of recirculating flow is described by:
Q
 
V

A
= r r
2

1
2
  r

V ( r )
 dr
= 2
 r 2
 r 1
V ( r )
 r
 dr

Use Simpson’s 1/3 Rule to numerically integrate the total volumetric flow rate equation to obtain the following value:
Q = -0.0005 m^3/s
Appendix:
Data Table for Simpson’s Rule
R
(m)
V(R)
(m/s)
0 4.53785
0.000454 4.53184
0.000896 4.49715
0.001326 4.4419
0.001745 4.36807
0.002153 4.27766
0.002551 4.17273
0.002938 4.05541
0.003316 3.92781
0.003683 3.79204
0.004041 3.65013
0.004389 3.50399
0.004729 3.35538
0.005059 3.20585
0.005381 3.05677
0.005695 2.90933
0.006 2.7642
0.006239 2.6503
0.006496 2.5284
0.006773 2.39806
0.00707 2.2593
0.007389 2.11237
0.007732 1.95771
0.008101 1.79608
0.008497 1.62854
0.008923 1.45652
0.009381 1.28187
0.009874 1.10689
0.010403 0.934416
0.010971 0.767839
0.011583 0.611168
0.01224 0.469037
0.012946 0.34661
0.013705 0.249404
0.014521 0.182868
0.015398 0.149593
0.016275 0.144973
0.017139 0.155476
0.01799 0.171253
0.018828 0.188008
0.019654 0.204299

0.020468 0.219687
0.021269 0.234046
0.022058 0.247338
0.022836 0.259542
0.023602 0.270635
0.024356 0.280588
0.025099 0.28937
0.025832 0.29695
0.026553 0.303253
0.027263 0.308237
0.027963 0.311752
0.028652 0.314035
0.029331 0.31774
0.03 0
Limits of Integration:
R1 = 0.016275 m
R2 = 0.03 m
*The highlighted portion of the data table marks the radial location where recirculation begins. The radial location of recirculation can be roughly estimated by observing where the velocity (which has a decreasing trend) suddenly increases. This is due to the fact that FlowLab outputs a plot of velocity magnitude—all values of velocity are positive, even in the region of recirculation.
[1] Incroprera, F. and DeWitt, D. Fundamentals of Heat and Mass Transfer,5 th Ed . Table A-6.

PROBLEM 7
Problem Statement:
Compare the pressure drop in Pa for water at 20 deg. C predicted by FlowLab for flow through a sudden expansion when Re=10,000 and D
2
/D
1
= 5 to an estimated pressure drop in the same geometry using estimates of the viscous losses via pipe friction factors and a friction loss coefficient for flow sudden expansion. Values for the pipe friction factors and friction loss coefficients can be found in your textbook. Use L
1
= 5 m and
L
2
= 100 m as in the schematic below.
Schematic:
Let:
L
1
= 5 m
L
2
= 100 m
Assumptions:

Incompressible flow

Axisymmetric flow

Maximum recirculation occurs at one pipe diameter downstream of expansion

Constant properties
Properties
1
:
Water @ T = 20 o
C
 
0 .
982

10

3 kg m
 s
  kg
997 .
984 m
3
Analysis:
Experimental value of pressure difference is obtained by:

P
 g
 h m
 h f
, where h are the losses due to skin friction and the sudden expansion
(the

indicates the differences between the inlet and the outlet)

P
 g

K
 v

1
2
2 g
 f
1
L
1
D
1
 v

1
2
2 g
 f
2
L
2
D
2
 v
 2
2
2 g

The loss coefficient, K, is described by:
K




1



D
D
2
1
By continuity:


2



2
 v

2
A
2
 v

1
A
1
Sample calculation for the case
D
2
D
1
K
 v




1


1



D
1
D
2
Re
D
1




2



2
 



1

0 .
2

0 .
049199 m s
1
2



2

5 :

0 .
9216
 v

2
 v

1
Re
2


A
1
 v


2
A
2
D
2

( 0 .
049199
0 .
2
2
1
2


1 .
97

10

3
( 997 .
984 )( 1 .
97

10

3
)( 1 )
9.82
x 10
4 m s

2000
To find friction factor, f, refer to graph of experimental values 2 ; use Re at inlet and expansion, and assume pipe is smooth: f f
1
2


0 .
032
0 .
030
Insert all known values after rearranging the pressure difference equation:

P
 g

K
 v
2 g

1
2
 f
1
L
1
D
1
 v

1
2
2 g
 f
2
L
2
D
2
 v
2 g
 2
2
 
P
 
K
 v

1
2
2
 f
1
L
1
D
1
 v

1
2
2
 f
2
L
2
D
2
 v
 2
2
2

( 997 .
984 )
( 0 .
9216 )( 0 .
049199 )
2
2

5
( 0 .
032 )
0 .
2
( 0 .
049199 )
2
2

( 0 .
030 )
100
1
( 1 .
97 E

3 )
2
2

2 .
09 Pa
Ratio of
Diameter
(A
2
/A
1
)
Exp.
Value
(Pa)
FlowLab
Value
(Pa)
%
Error
Re1 Re2 f1
5 2.09 2.37 13.4 10000 2000 0.032
[1] Incroprera, F. and DeWitt, D. Fundamentals of Heat and Mass Transfer,5 th Ed . Table A-6.
[7] Whitaker, S. Introduction to Fluid Mechanics, 1 st Ed . pp 235. f2
0.030

PROBLEM 8
Problem Statement:
For air flow at 20 o
C over a 1.4 m flat plate determine the axial development of the hydrodynamic boundary layer. Generate a dimensionless plot of boundary layer thickness versus x-position along the plate for u


5 .
28 m s
. Determine how the boundary layer thickness varies with Re x
in the laminar flow regime (use at least three points along the plate). Compare the boundary layer thickness predicted by FlowLab at various Re x
with the analytical solution. Plot u as a function of y at the end of the plate. x
Schematic: y
U
 x
L
X = 0
Assumptions:

Laminar flow

Incompressible flow

Constant properties
Properties
3
:
Air @ T = 20 o
C
 
1 .
21 kg m
3
 
1.79

10

5 kg m
 s
NOTE: The limitations on the flat plate template are as follows:

For laminar boundary layer flow, 0 < Re x
< 5

10
5
 u


1 .
00 m

L

1 .
00 m s
Determine Rex at end of plate: Re x
=
 u x

 
 

1 .
79
5 .
28

10
 

5
 
5

10

5
X = 1.4 m

Analysis:
1

10
5
, 2 .
5

10
5
, and
Integral Method 8 :
2 .
5

10
4
, 5

10
4
Compare the boundary layer thickness calculated by FlowLab at Re x
=
5

10
5
with the analytical solution derived using the von Karman
,

( x )

1260
37


 x v



Flowlab BL
Thickness
Analytical BL
Thickness x/L Re x
0.05
0.100
0.199
25000
50000
100000
0.500
250000
1.00
500000
0.00183
0.00260
0.00342
0.00527
0.0102
0.00185
0.00260
0.00368
0.00585
0.0116
% Error
0.39
0.54
7.36
10.06
12.07
Axial Development of the Hydrodynamic Boundary Layer for
U_infinity = 5.28 m/s
0.008
0.006
0.004
0.002
0
0 0.1
0.2
0.3
0.4
0.5
x/L
0.6
0.7
0.8
0.9
1
[3] SFPE Handbook of Fire Protection Engineering 2 nd Edition, Table B-2.
[8] Bird, Stewart, and Lightfoot. Transport Phenomena , 2 nd Edition. Equation 12.4-12. pp 389.