This document discusses fluid mechanics and heat transfer concepts related to forced convection. It covers boundary layer development over flat plates, including definitions of boundary layer thickness and wall shear stress. It discusses the governing equations for laminar boundary layer flow and provides the exact solution known as the Blasius solution. It also discusses heat transfer coefficients, turbulent boundary layers, and forced convection over external bodies like cylinders. For internal flows like pipe flow, it covers thermal entrance regions and definitions of mean temperature used in the energy balance.

1. Fluid Mechanics
Prepared By:
Md Ateeque Khan
Assistant Professor
JIT
5/26/2017 JIT 1

2. les
x q
Ts
T
h=f(Fluid, Vel ,Distance,Temp)
•Fluid particle adjacent to the
solid surface is at rest
•These particles act to retard the
motion of adjoining layers
• boundary layer effect
Momentum balance: inertia forces, pressure gradient, viscous forces,
body forces
Energy balance: convective flux, diffusive flux, heat generation, energy
storage
FORCED CONVECTION:
external flow (over flat plate)
An internal flow is surrounded by solid boundaries that can restrict the
development of its boundary layer, for example, a pipe flow. An external flow, on
the other hand, are flows over bodies immersed in an unbounded fluid so that the
flow boundary layer can grow freely in one direction. Examples include the flows
over airfoils, ship hulls, turbine blades, etc.
5/26/2017 JIT 2

3. One of the most important concepts in understanding the external flows is the
boundary layer development. For simplicity, we are going to analyze a boundary
layer flow over a flat plate with no curvature and no external pressure variation.
laminar turbulent
transition
Dye streak
U U U
U
Hydrodynamic boundary layer
Boundary layer definition
Boundary layer thickness (d): defined as the distance away from the surface
where the local velocity reaches to 99% of the free-stream velocity, that is
u(y=d)=0.99U. Somewhat an easy to understand but arbitrary definition.
Boundary layer is usually very thin: d/x usually << 1.5/26/2017 JIT 3

4. Hydrodynamic and Thermal
boundary layers
• As we have seen earlier,the hydrodynamic boundary layer is a
region of a fluid flow, near a solid surface, where the flow
patterns are directly influenced by viscous drag from the surface
wall.
• 0<u<U, 0<y<d
• The Thermal Boundary Layer is a region of a fluid flow, near a
solid surface, where the fluid temperatures are directly
influenced by heating or cooling from the surface wall.
• 0<t<T, 0<y<dt
• The two boundary layers may be expected to have similar
characteristics but do not normally coincide. Liquid metals
tend to conduct heat from the wall easily and temperature
changes are observed well outside the dynamic boundary layer.
Other materials tend to show velocity changes well outside the
thermal layer.5/26/2017 JIT 4

5. Effects of Prandtl number, Pr
d
dT
Pr >>1
 >> 
e.g., oils
d, dT
Pr = 1
 = 
e.g., air and gases
have Pr ~ 1
(0.7 - 0.9)
W
W
TT
TT
U
u



tosimilar
(Reynold’s analogy)
dT
d
Pr <<1
 << 
e.g., liquid metals
5/26/2017 JIT 5

6. Boundary layer equations (laminar flow)
• Simpler than general equations because boundary layer is thin
d
dT
U
WT
T
U
x
y
• Equations for 2D, laminar, steady boundary layer flow








































y
T
yy
T
v
x
T
u
y
u
ydx
dU
U
y
u
v
x
u
u
y
v
x
u


:energyofonConservati
:momentum-xofonConservati
0:massofonConservati
• Note: for a flat plate, 0hence,constantis 

dx
dU
U5/26/2017 JIT 6

7. Exact solutions: Blasius
3
1
2
1
3
1
2
1
PrRe678.0numberNusseltAverage
PrRe339.0numberNusseltLocal
Re
Re
328.11
tcoefficiendragAverage
,Re
Re
664.0
tcoefficienfrictionSkin
Re
99.4
x
knesslayer thicBoundary
0
0
2
2
1
L
xx
L
L
L
fD
y
wx
x
w
f
x
uN
Nu
LU
dxC
L
C
y
uxU
U
C


























 




d
5/26/2017 JIT 7

8. Heat transfer coefficient
• Local heat transfer coefficient:
x
k
x
kNu
h xx
x
3
1
2
1
PrRe339.0

• Average heat transfer coefficient:
L
k
L
kuN
h L
3
1
2
1
PrRe678.0

• Recall:   wallfromrateflowheat, TTAhq ww• Recall:   wallfromrateflowheat, TTAhq ww
• Film temperature, Tfilm
For heated or cooled surfaces, the thermophysical properties within
the boundary layer should be selected based on the average
temperature of the wall and the free stream;   TTT wfilm 2
1
5/26/2017 JIT 8

9. Heat transfer coefficient
U
x Hydrodynamic
Boundary Layer, d
Convection
Coefficient, h.
Thermal Boundary
Layer, dt
Laminar Region Turbulent Region
Laminar and turbulent b.l.
5/26/2017 JIT 9

10. Turbulent boundary layer
 
 
etc.:wayusualintcoefficienferheat transCalculate*
Re664.0Re036.0PrPrRe036.0
PrRe029.0
Re328.1Re072.0
Re
1
Re072.0
)105(ReRe059.0
:dataalexperimentonbasednscorrelatiousemainlywillWe*
solve.todifficultmore
infinitelybutones,laminarsimilar toareequationsb.l.Turbulent*
)105plate,flatoverflowFor(Re
.turbulenteventuallyandnaltransitiobecomesflowthe),ReRe(
numberReynoldsofvaluecriticalaBeyondwith x.increasesRe*
5.08.08.0
8.0
5.08.0
52.0
5
3
1
3
1
3
1
x
kNu
h
uN
Nu
C
C
x
xU
xcxcL
xx
xcxc
L
LD
xxf
c
c
xc
xcx
x










5/26/2017 JIT 10

11. d x( )
x
0 0.5 1
0
0.5
1
• Boundary layer growth: d  x
• Initial growth is fast
• Growth rate dd/dx  1/x,
decreasing downstream.
 w x( )
x
0 0.5 1
0
5
10
• Wall shear stress: w  1/x
• As the boundary layer grows, the
wall shear stress decreases as the
velocity gradient at the wall becomes
less steep.
Laminar Boundary Layer Development
5/26/2017 JIT 11

12. Determine the boundary layer thickness, the wall shear stress of a laminar water flow
over a flat plate. The free stream velocity is 1 m/s, the kinematic viscosity of the water
is 10-6 m2/s. The density of the water is 1,000 kg/m3. The transition Reynolds number
Re=Ux/=5105. Determine the distance downstream of the leading edge when the
boundary transitions to turbulent. Determine the total frictional drag produced by the
laminar and turbulent portions of the plate which is 1 m long. If the free stream and
plate temperatures are 100 C and 25 C, respectively, determine the heat transfer rate
from the plate.
3
2
w
( ) 5 5 10 ( ).
Therefore, for a 1m long plate, the boundary layer grows by 0.005(m),
or 5 mm, a very thin layer.
0.332 0.0105
The wall shear stress, 0.332 ( )
Re
The transition Reyn
x
x
x x m
U
U U
U Pa
x x

d
 



 

  
  
5U
olds number: Re 5 10 , 0.5( )tr
tr
x
x m


   
Example
5/26/2017 JIT 12

13. 2
D
0 0
f
2
The total frictional drag is equal to the integration of the wall shear stress:
0.664
F (1) 0.332 0.939( )
Re
Define skin friction coefficient: C
0.664
for a
1 Re
2
tr tr
tr
x x
w
x
w
f
x
U U
dx U dx N
x
C
U
 



 


   
 
 
laminar boundary layer.
Example (cont..)
5/26/2017 JIT 13

14. 
 DU
D

Re
Perimeter
Area4
hD
D
kN
h
k
Dh
N
AU
C u
u
normal
D 

;;
forceDrag
2
2
1

Forced convection over exterior bodies
• Much more complicated.
•Some boundary layer may exist, but it is likely
to be curved and U will not be constant.
• Boundary layer may also separate from the
wall.
• Correlations based on experimental data can
be used for flow and heat transfer calculations
• Reynolds number should now be based on a
characteristic diameter.
• If body is not circular, the equivalent
diameter Dh is used
5/26/2017 JIT 14

15. 0.70.0810-102
0.60.26102-10
0.50.5110-40
0.40.75401
mCRe
Pr
Pr
ReuN
65
53
3
D
25.
62.




s
m
DC
Flow over circular cylinders
All properties at free stream
temperature, Prs at cylinder
surface temperature
5/26/2017 JIT 15

16. • Thermal conditions
 Laminar or turbulent
 entrance flow and fully developed thermal condition
For laminar flows the thermal entrance length is a function of the
Reynolds number and the Prandtl number: xfd,t/D  0.05ReDPr,
where the Prandtl number is defined as Pr = / and  is the thermal
diffusitivity.
For turbulent flow, xfd,t  10D.
FORCED CONVECTION: Internal flow
Thermal entrance region, xfd,t
e.g. pipe flow
5/26/2017 JIT 16

17. Thermal Conditions
5/26/2017 JIT 17
• For a fully developed pipe flow, the convection
coefficient does not vary along the pipe length.
(provided all thermal and flow properties are constant)
x
h(x)
xfd,t
constant
• Newton’s law of cooling: q”S = hA(TS-Tm)
Question: since the temperature inside a pipe flow does not remain
constant, we should use a mean temperature Tm , which is defines
as follows:

18. Energy Transfer
5/26/2017 JIT 18
Consider the total thermal energy carried by the fluid as
(mass flux) (internal energy)v
A
VC TdA 
Now imagine this same amount of energy is carried by a body
of fluid with the same mass flow rate but at a uniform mean
temperature Tm. Therefore Tm can be defined as
v
A
m
v
VC TdA
T
mC



&
Consider Tm as the reference temperature of the fluid so that the
total heat transfer between the pipe and the fluid is governed by the
Newton’s cooling law as: qs”=h(Ts-Tm), where h is the local
convection coefficient, and Ts is the local surface temperature.
Note: usually Tm is not a constant and it varies along the pipe
depending on the condition of the heat transfer.

19. Energy Balance
5/26/2017 JIT 19
Example: We would like to design a solar water heater that can heat up the water
temperature from 20° C to 50° C at a water flow rate of 0.15 kg/s. The water is
flowing through a 0.05 m diameter pipe and is receiving a net solar radiation
flux of 200 W/m of pipe length. Determine the total pipe length required to
achieve the goal.

20. 5/26/2017 JIT 20