Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction
5-2 Boundary Layer Equations
5-3 Similarity Solution
5-4 Integral Method Approximation
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (1)
All experimental observations indicate that a fluid in
motion comes to a complete stop at the surface and
assumes a zero velocity relative to the surface (no-slip).
The no-slip condition is responsible for the development
of the velocity profile.
The flow region adjacent
to the wall in which the
viscous effects (and thus
the velocity gradients) are
significant is called the boundary layer.
5-1
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (2)
An implication of the no-slip condition is that heat transfer from
the solid surface to the fluid layer adjacent to the surface is by
pure conduction, and can be expressed as
q&conv = q&cond = −k fluid
∂T
∂y
(W/m 2 )
y =0
q&conv = h(Ts − T∞ )
Heat transfer coefficient
−k fluid ( ∂T ∂y ) y =0
h=
(W/m 2 ⋅ o C)
Ts − T∞
The convection heat transfer coefficient, in general, varies along
the flow direction.
5-2
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (3)
The Nusselt Number
It is common practice to nondimensionalize the heat transfer coefficient
h with the Nusselt number
hL
Nu =
c
k
Heat flux through the fluid layer by convection and by conduction can
be expressed as, respectively:
q&conv = hΔT
q&cond
ΔT
=k
L
Taking their ratio gives
q&conv
hΔT
hL
=
=
= Nu
q&cond k ΔT / L k
The Nusselt number represents the enhancement of heat transfer
through a fluid layer as a result of convection relative to conduction
across the same fluid layer.
Nu=1 Æpure conduction.
5-3
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (4)
Viscous versus inviscid regions of flow
Internal versus external flow
Compressible versus incompressible flow
Laminar versus turbulent flow
Natural (or unforced) versus forced flow
Steady versus unsteady flow
One-, two-, and three-dimensional flows
5-4
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (5)
Velocity Boundary Layer:
Consider the parallel flow of a fluid over a flat plate.
x-coordinate: along the plate surface
y-coordinate: from the surface in the normal direction.
The fluid approaches the plate in the x-direction with a uniform velocity V.
Because of the no-slip condition V(y=0)=0.
The presence of the plate is felt up to d.
Beyond d the free-stream velocity remains essentially unchanged.
The fluid velocity, u, varies from 0 at y=0 to nearly V at y=d.
5-5
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (6)
Velocity Boundary Layer:
The region of the flow above the plate bounded by d is called the velocity
boundary layer.
d is typically defined as
the distance y from the
surface at which
u=0.99V.
The hypothetical line of
u=0.99V divides the flow over a plate into two regions:
the boundary layer region, and
the irrotational flow region.
5-6
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (7)
Surface Shear Stress:
Consider the flow of a fluid over the surface of a plate.
The fluid layer in contact with the surface tries to drag the plate along via
friction, exerting a friction force on it.
Friction force per unit area is called shear stress, and is denoted by t.
Experimental studies indicate that the shear stress for most fluids is
proportional to the velocity gradient.
The shear stress at the wall surface for these fluids is expressed as
τs = μ
∂u
∂y
(N/m2 )
y =0
The fluids that that obey the linear relationship above are called Newtonian
fluids.
The viscosity of a fluid is a measure of its resistance to deformation.
5-7
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (8)
The viscosities of liquids decrease with temperature, whereas the
viscosities of gases increase with temperature.
In many cases the flow velocity profile is
unknown and the surface shear stress ts
from Eq. 6–9 can not be obtained.
A more practical approach in external flow
is to relate ts to the upstream velocity V as
τs = Cf
ρV 2
(N/m 2 )
2
Cf is the dimensionless friction coefficient (most cases is determined
experimentally).
The friction force over the entire surface is determined from
Ff = C f As
ρV 2
2
Advanced Heat Transfer
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Spring 2009
(N)
5-8
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (9)
Thermal Boundary Layer:
Like the velocity a thermal boundary layer develops when a fluid at a
specified temperature flows over a surface that is at a different
temperature.
Consider the flow of a fluid
at a uniform temperature of
T∞ over an isothermal flat
plate at temperature Ts.
The fluid particles in the
layer adjacent assume the surface temperature Ts.
A temperature profile develops that ranges from Ts at the surface to T∞
sufficiently far from the surface.
The thermal boundary layer ─ the flow region over the surface in
which the temperature variation in the direction normal to the surface is
significant.
Advanced Heat Transfer
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Spring 2009
5-9
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (10)
The thickness of the thermal boundary layer dt at any
location along the surface is defined as the distance
from the surface at which the temperature difference
T(y=dt)-Ts= 0.99(T∞-Ts).
The thickness of the thermal boundary layer increases
in the flow direction.
The convection heat transfer rate anywhere along the
surface is directly related to the temperature gradient
at that location.
5-10
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (11)
Prandtl Number:
The relative thickness of the velocity and the thermal
boundary layers is best described by the dimensionless
parameter Prandtl number, defined as
Molecular diffusivity of momentum ν μcp
Pr =
= =
Molecular diffusivity of heat
α
k
Heat diffuses very quickly in liquid metals (Pr«1) and very
slowly in oils (Pr»1) relative to momentum.
Consequently the thermal boundary layer is much thicker for
liquid metals and much thinner for oils relative to the velocity
boundary layer.
5-11
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (12)
Laminar and Turbulent Flows:
Laminar flow ─ the flow is characterized by
smooth streamlines and highly-ordered motion.
Turbulent flow ─ the flow is
characterized by velocity
fluctuations and
highly-disordered motion.
The transition from laminar
to turbulent flow does not
occur suddenly.
5-12
Advanced Heat Transfer
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Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (13)
The velocity profile in turbulent flow is much fuller than that in laminar flow, with a
sharp drop near the surface.
The turbulent boundary layer can be considered to consist of four regions:
Viscous sublayer
Buffer layer
Overlap layer
Turbulent layer
The intense mixing in turbulent flow enhances heat and momentum transfer, which
increases the friction force on the surface and the convection heat transfer rate.
5-13
Advanced Heat Transfer
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Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (14)
Reynolds Number:
The transition from laminar to turbulent flow depends on the surface geometry, surface
roughness, flow velocity, surface temperature, and type of fluid.
The flow regime depends mainly on the ratio of the inertia forces to viscous forces in
the fluid.
This ratio is called the Reynolds number, which is expressed for external flow as
Inertia forces VLc ρVLc
=
=
Re =
μ
Viscous forces ν
At large Reynolds numbers (turbulent flow) the inertia forces are large relative to the
viscous forces.
At small or moderate Reynolds numbers (laminar flow), the viscous forces are large
enough to suppress these fluctuations and to keep the fluid “inline.”
Critical Reynolds number ─ the Reynolds number at which the flow becomes
turbulent.
5-14
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (15)
Heat and Momentum Transfer in Turbulent Flow:
Turbulent flow is a complex mechanism dominated by
fluctuations, and despite tremendous amounts of research the
theory of turbulent flow remains largely undeveloped.
Knowledge is based primarily on experiments and the empirical
or semi-empirical correlations developed for various situations.
Turbulent flow is characterized by random and rapid fluctuations
of swirling regions of fluid, called eddies.
The velocity can be expressed as the sum
of an average value u and a fluctuating
component u’
u = u +u'
Advanced Heat Transfer
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Spring 2009
5-15
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (16)
It is convenient to think of the turbulent shear stress as consisting of two parts:
the laminar component, and
the turbulent component.
The turbulent shear stress can be expressed as
τ turb = − ρ u ' v '
The rate of thermal energy transport by turbulent eddies is
q&turb = ρcp v 'T '
The turbulent wall shear stress and turbulent heat transfer
τ turb = −ρ u ' v ' = μt
∂u
∂y
;
q&turb = ρcp vT = −kt
mt ─ turbulent (or eddy) viscosity.
kt ─ turbulent (or eddy) thermal conductivity.
Advanced Heat Transfer
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Spring 2009
∂T
∂y
5-16
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (17)
The total shear stress and total heat flux can be expressed as
∂u
∂u
τ turb = ( μ + μt )
= ρ (ν +ν t )
∂y
∂y
and
∂T
∂T
q&turb = − ( k + kt )
= − ρ c p ( α + αt )
∂y
∂y
In the core region of a turbulent boundary layer ─ eddy motion
(and eddy diffusivities) are much larger than their molecular
counterparts.
Close to the wall ─ the eddy motion loses its intensity.
At the wall ─ the eddy motion diminishes because of the noslip condition.
5-17
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-1 Introduction (18)
In the core region ─ the velocity and temperature profiles are very
moderate.
In the thin layer adjacent to the wall ─ the velocity and
temperature profiles are very steep.
Large velocity and temperature gradients at the wall
surface.
The wall shear stress
and wall heat flux are much larger
in turbulent flow than they
are in laminar
flow.
5-18
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-2 Boundary Layer Equations (1)
Consider the parallel flow of a fluid over a
surface.
Assumptions:
steady two-dimensional flow,
Newtonian fluid,
constant properties, and
laminar flow.
The fluid flows over the surface with a uniform
free-stream velocity V, but the velocity within
boundary layer is two-dimensional (u=u(x,y),
v=v(x,y)).
Three fundamental laws:
conservation of mass Æ continuity
equation
conservation of momentum Æ momentum
equation
conservation of energy Æ energy equation
Advanced Heat Transfer
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Spring 2009
5-19
Chapter 5: Boundary Layer Flow
-Forced Convection
5-2 Boundary Layer Equations (2)
Boundary Layer Approximation
1)
2)
3)
Assumptions:
Velocity components:
u>>v
Velocity gradients:
∂v/∂x≈0 and ∂v/∂y≈0
∂u/∂y >> ∂u/∂x
Temperature gradients:
∂T/∂y >> ∂T/∂x
When gravity effects and other body forces are negligible the y-momentum
equation
∂P
∂y
=0
5-20
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-2 Boundary Layer Equations (3)
Consider laminar flow of a fluid over
a flat plate.
Steady, incompressible, laminar flow
of a fluid with constant properties
Continuity equation
∂u ∂v
+
∂x ∂y
Momentum equation
∂u
∂u
∂ 2u
u +v
=ν 2
∂x
∂y
∂y
Energy equation
∂T
∂T
∂ 2T
u
+v
=α 2
∂x
∂y
∂y
Advanced Heat Transfer
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Spring 2009
5-21
Chapter 5: Boundary Layer Flow
-Forced Convection
5-2 Boundary Layer Equations (4)
Boundary conditions
At x=0
At y=0
As yÆ∞
u ( 0, y ) = V ,
u ( x, 0 ) = 0,
T ( 0, y ) = T∞
v ( x, 0 ) = 0, T ( x, 0 ) = Ts
u ( x, ∞ ) = V ,
T ( x, ∞ ) = T∞
When fluid properties are assumed to be constant, the first two equations
can be solved separately for the velocity components u and v.
knowing u and v, the temperature becomes the only unknown in the last
equation, and it can be solved for temperature distribution.
Advanced Heat Transfer
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Spring 2009
5-22
Chapter 5: Boundary Layer Flow
-Forced Convection
5-3 Similarity Solution (1)
The continuity and momentum equations are solved by
transforming the two partial differential equations into a single
ordinary differential equation by introducing a new
independent variable (similarity variable).
The argument ─ the nondimensional velocity profile u/V
should remain unchanged when plotted against the
nondimensional distance y/d.
d is proportional to (nx/V)1/2, therefore defining dimensionless
similarity variable as
η = y Vν x
might enable a similarity solution.
5-23
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-3 Similarity Solution (2)
Introducing a stream function y(x, y) as
∂ψ
u=
∂y
;
∂ψ
v=−
∂x
Defining a function f(h) as the dependent variable as
ψ
f (η ) =
V ν x /V
The velocity components become
df
∂ψ ∂ψ ∂η
ν x df V
u=
=
=V
=V
V dη ν x
dη
∂y ∂η ∂y
⎞
1 Vν ⎛ df
∂ψ
ν x df V ν
v=−
= −V
−
−f⎟
f =
⎜η
2 x ⎝ dη
∂x
V dη 2 Vx
⎠
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Advanced Heat Transfer
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Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-3 Similarity Solution (3)
d3 f
d2 f
2 3+f
=0
2
dη
dη
f ( 0 ) = 0,
df
= 0,
dη η =0
df
=1
dη η →∞
5-25
Advanced Heat Transfer
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Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection
5-4 Integral Method Approximation
Mathematical Simplification
• Number of independent variables are reduced
• Reduction in order of differential equation
5-26
Advanced Heat Transfer
Y.C. Shih
Spring 2009
Chapter 5: Boundary Layer Flow
-Forced Convection