Airflow over Flat Plate-Boundary Layer
Numerical Simulation and Experimental Design
Introduction
The objective of this numerical experiment includes two aspects: 1) simulate airflow
over a flat plate in COMSOL (a commercial finite element package, available on
vlabs/Mechanical Engineering or General Engineering 2/COMSOL Multiphysics
5.5.); 2) use the simulation results to guide the design of an experiment, in which
you will measure the thickness of the velocity boundary layer over the surface of a
flat plate.
Reading: Heat transfer from surface of flat plates has been investigated extensively.
Thermal boundary layers for fully heated or partially heated flat plates are presented
in your heat transfer textbook by Incropera, DeWitt, Bergman, and Lavine entitled
“Fundamentals of Heat and Mass Transfer”, Chapter 7; Theories of velocity
boundary layer over a flat surface has been elaborated in detail in Fundamentals of
Fluid Mechanics textbook by Munson, Chapter 9.
The Velocity Boundary Layer
Consider flow over a flat plate, as shown in Figure 6.1 (Heat transfer textbook).
When fluid particles contact the surface, they assume zero velocity. These particles
then act to retard the motion of particles in the adjoining fluid layer, which act to
retard the motion of particles in the next layer, and so on until, at a distance y=δ from
the surface, the effect becomes negligible. This retardation of fluid motion is
associated with shear stresses τ acting in planes that are parallel to the fluid velocity.
With increasing distance y from the surface, the x velocity component of the fluid,
u, must then increase until it approaches the free stream value uꝏ. The quantity δ is
(0,0)
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termed the velocity boundary layer thickness, and it is typically defined as the value
of y for which,
u u 0.99 = ∞ (1)
For laminar flow over an isothermal plate, based on the similarity solution from the
heat transfer textbook, the velocity boundary layer thickness could be expressed as:
5 5
Re / x
x
u vx
δ
∞
= = (2)
For turbulent flow over an isothermal plate, to a reasonable approximation, the
velocity boundary thickness may be shown as:
1/5 0.37 Rex δ x − = (3)
Comparing equation (2) and (3), we see that turbulent boundary layer growth is
much more rapid (δ varies as x4/5 in contrast to x1/2 for laminar flow).
Transition from Laminar to Turbulent Flow
Boundary layer development on a flat plate is illustrated in Figure 6.6. In many cases,
laminar and turbulent flow conditions both occur, with the laminar section preceding
the turbulent section. The transition from laminar to turbulent flow is ultimately due
to triggering mechanisms, such as the interaction of unsteady flow structures that
develop naturally within the fluid or small disturbances that exist within many
typical boundary layers. It is reasonable to assume that transition begins at some
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location xc. The location is determined by the critical Reynolds number Rex,c. For
flow over a flat plate, Rex,c is known to be:
5 Re , 5 10 c
x c
ρu x
µ
∞ = = × (4)
For a fixed location x, there is a critical velocity associated with the critical Reynolds
number Rex,c, it can be calculated by:
,
,
Rex c
c u
x
µ
ρ ∞ = (4’)
The Thermal Boundary Layer
Just as a velocity boundary layer develops when there is fluid flow over a surface, a
thermal boundary layer must develop if the fluid free stream and surface
temperatures differ. Consider flow over an isothermal flate plate (Figure 6.2). At the
leading edge, the temperature profile is uniform, with T(y)=Tꝏ. However, fluid
particles that come into contact with the plate achieve thermal equilibrium at the
plate’s surface temperature. In turn, these particles exchange energy with those in
the adjoining fluid layer, and temperature gradients develop in the fluid. The region
of the fluid in which these temperature gradients exist is the thermal boundary layer,
and its thickness δt is typically defined as the value of y for which:
0.99 s
s
T T
T T∞
− = − (5)
With increasing distance from the leading edge, the effects of heat transfer penetrate
further into the free stream and the thermal boundary layer grows.
For laminar flow, the ratio of the velocity to thermal boundary layer thickness is:
1/3 Pr
t
δ
δ ≈ (6)
For turbulent flow, boundary layer development is influenced strongly by random
fluctuations in the fluid and not by molecular diffusion. Hence relative boundary
layer growth does not depend on the value of Pr. Therefore, for turbulent flow,
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t δ δ ≈ (7)
Flat Plat in parallel flow with unheated starting length
When there is an unheated starting length upstream of a heat section, as shown in
Figure 7.4. Velocity boundary layer growth begins at x=0, while thermal boundary
layer development begins at x=ξ. Hence there is no heat transfer for 0 ≤ ≤ x ξ . Through
the use of an integral boundary layer solution, you can calculate the average heat
transfer coefficient:
,
1
theory x theory
L
h h dx
L = ∫ (8)
Here, hx,theory can be obtained by the heat flux balance at y=0,
( )
( )
0,
0,
x xs f
y x
f
x
s y x
T q hT T k
y
k T h
TT y
∞
=
∞ =
∂ = − =− ∂
∂ = − − ∂
(9)
On the other hand, if you know the power generated in the heating element, you
could calculate the average heat transfer coefficient using:
( ) t
t s
Q q hT T L w = = − ∞
⋅ (10)
Here, w is the width of the plate; Qt is the power in W; qt is the power density in
W/m2
.
Numerical Simulation
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Figure 1 Schematic Illustration of the flat plate in parallel flow with unheated
length
As shown in Figure 1, we are tying to simulate the laminar flow passing through a
flat plate, which is made of resin in the unheated length (1.2 inch) and copper in the
heated length (4.8 inch). The width of the flat plat is 1.25 inch. From x=0 to x=ξ, the
flat plate surface temperature Ts equals the air temperature Tꝏ; with x> ξ, a piece of
copper has been heated with an electric heater by 4 Watts, which will raise the flat
plate surface temperature. A short area has been added before the leading edge of
the flat plate to avoid the effect of boundary condition brought to the edge.
According to the introduction, there will be two boundary layers developed along
the flat plate surface: velocity boundary layer and thermal boundary layer.
A COMSOL model has been built to simulate the phenomenon. Please follow the
steps below to analyze the simulation results.
1. Analyze how the velocity boundary layer thickness has been developed in the
x direction. Use a 1D graph to demonstrate it (x axis is the x coordinate, y axis
is the boundary layer thickness).
2. Analyze how the thermal boundary layer thickness has been developed in the
x direction. Use a 1D graph to demonstrate it (x axis is the x coordinate, y axis
is the boundary layer thickness).
3. Select a x coordinate, perform a parametric study of velocity. Use a 1D graph
to demonstrate the correlation of velocity and boundary layer thickness (x axis
is the velocity, y axis is the velocity boundary layer thickness). Use Eq. 4’ to
back calculate the critical velocity to ensure that Rex is smaller than Rex,c, so
the flow is laminar.
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4. Plot δ as a function of Reynolds number with a fixed x (same x you picked in
question 3). Compare the results with δ from equation (2).
5. (Bonus) Plot δt as a function of Reynolds number with a fixed x (same x you
picked in question 3). Compare the results with δtfrom equation (6).
6. (Bonus) Calculate the average heat transfer coefficient based on equation (10).
Note: Ts has been calculated by the average of temperature across the copper
surface in the Comsol model as a global variable.
Design of Experiment
Objective: Measure the velocity profiles in the velocity boundary layer of laminar
flow over a flat plate in a wind tunnel.
Recommended Devices and Instruments:
A Flat Plate: the flat plate should have a specified designed leading edge for the
smooth development of the velocity boundary layer. It is recommended that the flat
plate being placed in a wind tunnel to provide a uniform and stable laminar flow.
Funnel with pressure measurement: The funnel is connected with a Betz micro
manometer, which can measure the absolute pressure of the air before existing the
funnel. With this you can carefully set the speed of the flow from the funnel and
calibrate the hot wire.
Flow velocity measurements: Hot Wire Anemometry (HWA) of Dantec is capable
to measure flow speeds in and around the velocity boundary layer. It is capable to
measure velocity from a few cm/s to 200 m/s. Miniature wire probe should be used.
The probe is 1.25mm long plated tungsten wire with a diameter of 5μm, which
results in a short response time. The height of the HWA should be able to be varied
by a small increment (depending on the thickness of the boundary layer).
Figure 2 left: Betz manometer Middle: Funnel with probe for calibration Right:
Flat plate
Task 1:
Describe how you would scope your experiment and select appropriate
apparatus. Be sure to specify: the dimension of the flat plate; the air velocity range
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you would like to test; the critical velocity associated with the critical Reynolds
number; the thickness of the velocity layer; what are the operating range and
accuracy requirements for each apparatus.
Task 2:
Describe how you would calibrate the HWA. Be sure to specify: how to
calibrate the air velocity of HWA by the pressure from Betz manometer; the velocity
range you should calibrate.
Task 3:
Describe how you would set up the experiment. Be sure to specify: where
should you set up the HWA (x coordinate); how to vary the height of HWA probe
to measure velocity at different y (how much is your increment for each
measurement).
Task 4:
Describe how you would postprocess your data. Be sure to specify: what
theoretical calculation are you going to use as a comparison for your experimental
data; how are you going to make comparison.