External Flow:
The Flat Plate in Parallel Flow
Empirical Correlation Development
Dimensionless representation of convective heat
transfer coefficient measurement
Laminar flow over flat plate (Similarity Solution)
New Dependent Variable
Laminar flow over flat plate (Similarity Solution)
Continuity equation
Momentum equation
Stream function
New Independent variable
New Dependent Variable
Laminar flow over flat plate (Similarity Solution)
Momentum equation
Laminar flow over flat plate (Similarity Solution)
Substituting in Momentum equation
We get 3rd order ODE
Laminar flow over flat plate (Similarity Solution)
Boundary Conditions
In Similarity coordinate
Usimg
Laminar flow over flat plate (Similarity Solution)
Laminar flow over flat plate (Similarity Solution)
Boundary Layer Thickness
Shear stress
Skin Friction Coefficient
Laminar flow over flat plate (Similarity Solution)
Average Value
Energy Equation
Species Equation
Physical Features
Physical Features
Rex ,c = 0
Rex ,c = 0
Rex ,c = 0
ReL < Rex ,c → laminar flow throughout
Rex ,c ≈ 5 × 105.
ReL > Rex ,c → transition to turbulent flow at xc / L ≡ Rex ,c / ReL
• If boundary layer is tripped at the leading edge
Re x,c = 0
and the flow is turbulent throughout.
Turbulent Flow
Turbulent Flow
• Local Parameters:
Empirical
Correlations
C f , x = 0.0592 Rex−1/5
Nu x = 0.0296 Rex4/5 Pr1/3
How do variations of the local shear stress and convection coefficient with
distance from the leading edge for turbulent flow differ from those for laminar flow?
• Average Parameters:
1 xc
L
=
hL
∫0 hlam dx + ∫xc hturb dx
L
(
)
Substituting expressions for the local coefficients and assuming Rex ,c = 5 × 105 ,
0.074 1742
General Case
=
C f ,L
−
1/5
ReL
ReL
=
Nu L
( 0.037 Re
4/5
L
)
− 871 Pr1/3
For =
Rex ,c 0 or L >> xc ( ReL >> Rex ,c ) ,
C f , L = 0.074 ReL−1/5
Nu L = 0.037 ReL4/5 Pr1/3
Unheated Starting Length
Integral boundary
layer approach
Laminar Flow
Integral boundary
layer approach
Turbulent Flow
P =2 for laminar and p =8 for turbulent flow
Flat plate with constant heat flux condition
Laminar Flow
Turbulent Flow
Local surface temperature distribution
(It is not necessary to introduce average heat transfer coefficient for calculating q.)
The Nusselt number is about 36 % and 4% higher than constant temperature
case for laminar and turbulent flow condition respectively.
External Convection:
Flow over Bluff Objects
(Cylinders, Spheres)
Cylinder in Cross Flow
The Cylinder in Cross Flow
• Conditions depend on special features of boundary layer development, including
onset at a stagnation point and separation, as well as transition to turbulence.
– Stagnation point: Location of zero velocity ( u∞ = 0 ) and maximum pressure.
– Followed by boundary layer development under a favorable pressure gradient
( dp / dx < 0 ) and hence acceleration of the free stream flow ( du∞ / dx > 0 ) .
– As the rear of the cylinder is approached, the pressure must begin to increase.
Hence, there is a minimum in the pressure distribution, p(x), after which boundary
layer development occurs under the influence of an adverse pressure gradient
( dp / dx > 0, du∞ / dx < 0 ) .
Cylinder in Cross Flow (cont.)
– Separation occurs when the velocity gradient du / dy
y =0
reduces to zero
and is accompanied by flow reversal and a downstream wake.
– Location of separation depends on boundary layer transition.
ReD ≡
ρVD VD
=
µ
ν
Pressure distribution over a circular cylinder
Angular Nusselt Number Distribution
Average Nusselt Number of Circular Cylinder
All properties are calculated at film temperature.
Average Nusselt Number of non-Circular Cylinder
All properties are calculated at film temperature.
Average Nusselt Number of Circular Cylinder
n = 0.37 for Pr < 10
n=0.36 for Pr >10
All properties are calculated at free stream temperature except Prs, which
Is calculated at surface temperature.
Spheres and Packed Beds
The Sphere
• Flow over a rigid sphere
– Boundary layer development is similar to that for flow over a cylinder,
involving transition and separation.
All properties except µs are calculated at free stream temperature.
Thank you