MAE 3130: Fluid Mechanics Lecture 10: Internal and External Flows Spring 2003

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MAE 3130 Fluid MechanicsLecture 10 Internal
and External FlowsSpring 2003

• Dr. Jason Roney
• Mechanical and Aerospace Engineering

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Outline

• Overview of Viscous Pipe Flow
• Laminar Pipe Flow
• Turbulent Pipe Flow
• Dimensional Analysis of Pipe Flow
• Overview of External Flows
• Boundary Layer Characteristics
• Pressure Gradients Effects
• Lift and Drag
• Some Example Problems

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Viscous Pipe Flow Overview
Pipe Flow is important in daily operations and is
described in general as flow in a closed conduit
(pipes and ducts). It is also known as an
internal flow.
Some common examples are oil and water pipelines,
flow in blood vessels, and HVAC ducts.
When real world effects such as viscous effects
are considered, it is often difficult to use only
theoretical methods. Often theoretical,
experimental data, and dimensional analysis is
used,
Some common pipe flow components are shown
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Viscous Pipe Flow Overview
Pipe flow versus Open-channel flow
Open-Channel Flow
Pipe Flow

• Pipe is not full of fluids
• Pressure gradient is constant
• Gravity is the driving force

• Pipe is completely filled with fluid
• Pressure Gradients drive the flow
• Gravity can also be important

i.e., flow down a concrete spill way.
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Viscous Pipe Flow Flow Regime
Osborne Reynolds Experiment to show the three
regimes Laminar, Transitional, or Turbulent
Laminar
Experiment
Transitional
Turbulent
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Viscous Pipe Flow Flow Regime
If we measure the velocity at any given point
with respect to time in the pipe
Re gt 4000
Reynolds Number Dependency
2100lt Re lt 4000
Re lt 2100

1. Turbulence is characterized by random
   fluctuations.
2. Transitional flows are relatively steady
   accompanied by occasional burst.
3. Laminar flow is relatively steady.

For laminar flow there is only flow direction
For turbulent flow, there is a predominate flow
direction, but there are random components normal
to the flow direction
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Viscous Pipe Flow Entrance and Fully Developed
The entrance region in a pipe flow is quite
complex (1) to (2)
The fluid enters the pipe with nearly uniform
flow. The viscous effects create a boundary layer
that merges. When they merge the flow is fully
developed.
There are estimates for determining the entrance
length for pipe flows
and
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Viscous Pipe Flow Entrance and Fully Developed
For very low Reynolds numbers (Re 10), the
entrance length is short
For large Reynolds number flow the entrance
length can be several pipe diameters
For many practical engineering problems
Bends and Ts affect Fully Developed Flow
Pipe is fully developed until the character of
the pipe changes.
It changes in the bend and becomes fully
developed again after some length after the bend.
Many disruptions can cause the flow to never be
fully developed.
In many flows, the fully developed region is
greater than the developing region.
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Viscous Pipe Flow Pressure and Shear Stress
The shear stress in laminar flow is a direct
result of momentum transfer along the randomly
moving molecules (microscopic).
The shear stress in turbulent flow is due to
momentum transfer among the randomly moving,
finite-sized bundles of fluid particles
(macroscopic).
The physical properties of shear stress are quite
different between the two.
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Fully Developed Laminar Flow Overview
Both turbulent and laminar flows become fully
developed in long enough straight pipes.
However, the details of the two flows are quite
different.
Some important quantities that we calculate
velocity profiles, pressure drop, head loss, and
flow rate.
Although most flows are turbulent rather than
laminar, and many pipes are not long enough to
allow the attainment of fully developed flow, a
fully understanding of fully developed laminar
flow is important.
This study is the basis for more complex
analysis, and there are some cases where these
assumption are good.
The equations or a description can be obtained in
three different ways (1) Momentum applied to a
fluid element, (2) Navier-Stokes equations, and
(3) dimensional analysis methods.
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Fully Developed Laminar Flow Fluid Element Method
Basic Pipe flow is governed by a balance of
viscous and pressure forces.
Consider and cylindrical fluid element within a
pipe
Free-Body Diagram
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Fully Developed Laminar Flow Fluid Element Method
Now since neither the pressure gradient nor the
length depend on r, the R.H.S. must also be
independent of r.
Then at r 0, t 0, and at r D/2, t is the
wall shear stress.
Then,
Now,
The shear profile is linear.
A small shear stress can produce a large pressure
difference if the pipe is relatively long.
The shear stress for laminar Newtonian Flow
Velocity decreases from the center-line.
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Fully Developed Laminar Flow Fluid Element Method
Now, recall
Substitute, the shear stress definition, and
rearrange
Integrate,
Apply the boundary conditions, no-slip, u 0 at r
D/2, and solve for C1
Vc centerline velocity
Also, we can write in terms of shear stress
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Fully Developed Laminar Flow Fluid Element Method
Find the Volumetric Flow Rate
The average velocity is V.
The average velocity is 1/2Vc
Hagen-Poiseuille Flow
or,
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Fully Developed Laminar Flow Fluid Element Method
Some general remarks

1. The flowrate is directly proportional to the
   pressure drop.
2. The flowrate is inversely proportional to the
   viscosity.
3. The flowrate is inversely proportional to the
   pipe length.
4. The flowrate is inversely proportional to the
   pipe diameter to the 4th power.

We could adjust the equations for non-horizontal
pipes
Mean Velocity
Volumetric Flow
Lastly, we could develop these flows from
Navier-Stokes as in Lecture 8.
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Fully Developed Laminar Flow Dimensional Analysis
Important Variables
Density is not important because of fully
developed
Number of dimensionless groups
Two dimensional groups
If we state that the pressure drop has to be
directly proportional to l
Or, solving for V, and substituting in Q
C in this case must be determined by experimental
analysis. It is 32 for circular pipes.
Now, return to
and divide by dynamic pressure
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Fully Developed Laminar Flow Dimensional Analysis
Now, simplifying,
Now, rewriting,
f is the Darcy Friction Factor (dimensionless).
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Fully Developed Turbulent Flow Overview
Turbulent flow is the least understood of all
flow phenomena, yet is more likely to occur than
laminar flow, so we address ways of describing
the flow.
Transition from Laminar to Turbulent Flow in a
Pipe
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Fully Developed Turbulent Flow Overview
One see fluctuation or randomness on the
macroscopic scale.
fluctuating
mean
One of the few ways we can describe turbulent
flow is by describing it in terms of
time-averaged means and fluctuating parts.
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Fully Developed Turbulent Flow Overview
Now consider, the time average of the fluctuating
parts
The fluctuations are equally distributed on
either side of the average.
Now, consider the average of the square of the
fluctuations
Turbulence Intensity
Indication of the gustiness of the flow.
In good wind tunnel
in Atmosphere,
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Fully Developed Turbulent Flow Overview
Now, shear stress
However,
for turbulent flow.
Turbulent Flow
Laminar Flow
Experiment
Shear comes from eddy motion which have a more
random motion and transfer momentum.
Shear relates to random motion as particles glide
smoothly past each other.
For turbulent flow
Is the combination of laminar and turbulent
shear. If there are no fluctuations, the result
goes back to the laminar case. The turbulent
shear stresses are positive, thus turbulent flows
have more shear stress.
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Fully Developed Turbulent Flow Overview
The turbulent shear components are known as
Reynolds Stresses.
Shear Stress in Turbulent Flows
Turbulent Velocity Profile
In viscous sublayer tlaminar gt tturb 100 to 1000
times greater.
In the outer layer ttirb gt tlaminar 100 to 1000
time greater.
The viscous sublayer is extremely small.
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Fully Developed Turbulent Flow Velocity Profile
The velocity profile for turbulent flow is been
obtained through experimental analysis,
dimensional analysis, and semiempirical
theoretical efforts.
for a smooth wall, Law of the Wall
In the viscous sublayer
is the friction velocity, and
In the overlap region
From dimensional analysis arguments
Possible outer region approximation
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Fully Developed Turbulent Flow Velocity Profile
Some alternative, approach include the Power-Law
equation
n 7 for many practical flows.
n, chosen based on the Reynolds number.
Turbulent velocity profiles are relatively flat
in a pipe flow.
Profiles
The power-law equation is not valid near the
wall, since that would give an infinite velocity
gradient.
Also, the shear does not go to zero at the
center-line.
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Dimensional Analysis of Pipe Flow Moody Chart
Most turbulent pipe flow data is based on
experiments. In turbulent flow, in order to do
dimensional analysis we consider the roughness of
the pipe, as well as density which relates to
momentum.
Variables
roughness
Roughness is important in the viscous sub-layer
in turbulent flows, if it protrudes sufficiently
in this layer.
The viscous layer in laminar flow is so large,
that small roughness does not play a role.
Then range of roughness for validity of this
analysis is for
Then, the dimensionless groups are the following
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Dimensional Analysis of Pipe Flow Moody Chart
As for laminar flow, the pressure drop must be
proportional to the pipe length
Recalling the definition of the friction factor
Then the friction factor is one of our
dimensionless groups
Then using experiments, we can find the above
relationship with various manufactured pipe
roughness values
Moody Chart
Colebrook Relation for Non-Laminar part of the
Moody Chart (curve fit)
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Dimensional Analysis of Pipe Flow Moody Chart
Laminar
Marks Reynolds Number independence
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Dimensional Analysis of Pipe Flow Moody Chart
Energy Equation relation to Pipe Flow
as account for non-uniform velocity profiles.
For fully developed pipe flow in a horizontal
pipe
And,
Darcy-Weisbach Equation
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Some Example Problems