Convection Heat Transfer
LECTURE
Laws of the Wall
for turbulent flows
Turbulent boundary layer equations
Consider the 2-D steady turbulent flow and the corresponding reynold's averaged equations.
u v
0
x y
u '2 u ' v '
2u 2u
u
u
P
v
2 2
u
x
y
x

Convection Heat Transfer
LECTURE
Natural Convection
(concluded)
9.5 The effect of Turbulence
The Rayleigh number and convection correlations
Natural convection correlations appear much like those for forced convection
Nux = C Grm Prn
In most cases it is f

Convection Heat Transfer
LECTURE
Natural Convection
Forced convection fluid motion is due to a pump or fan
Free convection
A body force acts on the fluid
which is due to gravity and the
fluid motion is due to density gradient
(buoyancy)
which is due to te

Convection Heat Transfer
LECTURE
Natural Convection
(continued)
9.4 Laminar free convection on a vertical surface
similarity solution (variable surface temperature)
The governing equations
are to be solved with the
following BCs
Ts T Ax n
using
Dr. S. Z.

Convection Heat Transfer
LECTURE
Turbulent Internal
flows
Boundary layer equations for axi-symmetric flows
Axisymmetric flows occur e.g. in a circular pipe or circular jet
or the boundary layer on a body of circular cross section.
u 1 rv
0
z r r
u
u
1 dp

Convection Heat Transfer
LECTURE
Turbulent heat transfer
(External flows)
Law of the wall for the thermal boundary layer
Dr. S. Z. Shuja
2
Flat plate with unheated starting length
Dr. S. Z. Shuja
3
Turbulent flow over a flat plate (superposition method)
A

Convection Heat Transfer
LECTURE
Friction Laws
Turbulent boundary layer equations
Consider the 2-D steady turbulent flow and the corresponding reynold's averaged equations.
u v
0
x y
u '2 u ' v '
2u 2u
u
u
P
v
2 2
u
x
y
x
y
y
x
x
u ' v ' v

Convection Heat Transfer
LECTURE
Turbulent Internal
flow heat transfer
Momentum-heat transfer analogies
This analogy is developed for the case of constant heat flux boundary condtions.
Note: Although an analogy cannot be made for the case of constant surf

Convection Heat Transfer
LECTURE
Arbitrary surface
temperature and effect
of viscous dissipation
Laminar flow over a flat plate (superposition methods)
Arbitrary specified surface temperature
Consider the solution Nu x 0.332 Pr1/ 3
k
0.332 Pr1/3 Re1/ 2
x

Convection Heat Transfer
LECTURE
Turbulent heat
transfer
Analogy heat transfer solutions for turbulent BL flows
Recall the energy equation:
T
T t T
v
x
y y Pr Prt y
we shall consider the following Heat/Momentum transfer analogies for turbulent flow he

Convection Heat Transfer
LECTURE
Turbulent flows
Reynolds averaged conservation equations (Recall)
The time averaged continuity equation
no. of equations = 4
V 0
The time averaged N-S equation is written as
unknowns = u , v , w, P and
the 6 turbulent str

Convection Heat Transfer
LECTURE
Fully developed (FD)
flows in pipe
8.2.3 Fully developed (HFD & TFD) conditions
u
For HFD region
0
x
For TFD region the temperature is changing due to Heat Transfer
BUT for the special case of
Ts constant, or
q" const

Convection Heat Transfer
LECTURE
HFD and Thermally
developing (TD) flow
in pipe (constant qs)
HFD and Thermally developing flow in a pipe (constant qs)
The energy equation for, axisymmetric HFD ( Re Pr 10) flow is (neglecting viscous dissipation)
2
2um r

Convection Heat Transfer
LECTURE
Internal flows
Applications (internal flow)
flow in pipes of circular cross-section
with constant surface temperature (Ts)
with constant heat flux at the surface (q)
flow in ducts of non-circular cross-section
with constan

Convection Heat Transfer
LECTURE
HFD and Thermally
developing (TD) flow
in pipe (constant Ts)
HFD and Thermally developing flow in a pipe
the energy equation, axisymmetric HFD ( Re Pr 10) flow is (neglecting viscous dissipation)
2
2um r T 1 d dT
1
r
R

Convection Heat Transfer
LECTURE
External flows (More
Temperature solutions)
Laminar BL over a flat plate
(variable plate temperature)
Ts fn x
The energy equation is
T
T 2T
u
v
x
y Pr y 2
recall the
following
u Uf ' v
u
U
f' y
x
U
U
1 U
f ' f y
x
2 x

Convection Heat Transfer
LECTURE
External flows
Applications (external flow)
flow over flat plates
flow over cylinders and spheres
flow over airfoils and turbine blades
2
Dr. S. Z. Shuja
Examples of external flows
1) Flow over a flat plate
2) Flow over a

Convection Heat Transfer
LECTURE
External flows
Temperature solutions
The Pohlhausens solution for the energy equation
(constant plate temperature)
The energy equation is
T Ts
T
T 2T
introducing the dimensionless temperatute
u
v
T Ts
x
y Pr y 2
The equat

Convection Heat Transfer
LECTURE
Non-similar solutions of
the BL equations
Differential Vs. Integral formulation
Differential
u
u
2u
dU
v
U
2
Momentum u
x
y
y
dx
Energy
solution
T
T 2T
v
u
x
y Pr y 2
Exact
(however difficult )
Integral
Cf
2
d
dU
2 H

Convection Heat Transfer
LECTURE
The boundary layer
approximation
Velocity boundary layer thickness
3 measures of BL thickness are in common use.
1) Distrubance thickness
The physical thickness of the BL. It is defined
as the point where the velocity in

Convection Heat Transfer
LECTURE
Some exact solutions
Some exact solutions: Uni-directional / Parallel flows
V u x, y , z , t i
u
0
The continuity
x
equation implies
or u u y, z , t
V
i.e. the velocity component does not
depend on the flow direction.
su

Convection Heat Transfer
LECTURE
INTRODUCTION
(Revision)
Forms of Energy
As studied in Thermodynamics I (ME 203)
Transfers (enters or leaves)
at the system boundary as
t th
t
b
d
Heat or Work (in general)
and with mass flow
(
(for open systems)
p
y
)
Cont

Convection Heat Transfer
LECTURE
Conservation Equations
Alternate derivation
The three laws (Conservation of mass, momentum and energy)
The 3 laws are written for a control mass
(control mass is called a System, and is a
fixed collection of material parti

Convection Heat Transfer
LECTURE
Convection Heat Transfer
Introduction
Convection heat transfer
Energy transfer involving fluid motion, generally
between a surface (Ts) and a fluid (T) moving near it, when Ts T
Convection has 2 elements: Diffusion + Advec

Convection Heat Transfer
LECTURE
Conservation Equations
The three laws (Conservation of mass, momentum and energy)
The 3 laws are written for a control mass
(control mass is called a System, and is a
fixed collection of material particles)
Conservation la