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ME 408
Fluid Mechanics II
Chapter 9
Flow Over Immersed Bodies
Content
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Classification of External Viscous Flow
Fluid Dynamic Forces: Lift and Drag
Reynolds Number Effect
Boundary Layer: Laminar and Turbulent
Flow Separation
Experimental Drag Data
Airfoil and Wing Characteristics
Flow
Classification
2-Dimensional:
3-Dimenstional:
09_01
Axi-symmetric:
Reynolds Number Effect
The Reynolds number, Re = U l/n ,
is the ratio between the inertial
force and the viscous force.
Low Re: Mostly viscous flow
High Re: Viscous Boundary Layer
near surface
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Moderate Re: Partial viscous flow
around body
Flow Past Cylinder
Low Re: Mostly viscous
flow
High
Re:
Viscous
Boundary
Layer
near
surface till separation and
wake
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Moderate
Re:
Partial
viscous flow around body
with separation and recirculation flow in wake
Surface Forces
Pressure:
Normal to surface
Shear Stress: Tangent to surface
09_03
Lift and Drag
The sum of forces
due
to
pressure
distribution and skin
friction (shear stress)
is the resultant force
on a 2-D object.
Lift:
Component
normal to the flow
Drag: Component in
the flow direction
09_02
This net force can be
represented by its two
components:
Example 9.1 (p. 329)
Flow parallel to flat plate
Skin Friction Drag only:
D = 0.0992 lbf, L = 0
Flow normal to flat plate:
Flow at an angle with plate:
Both Lift and Drag are present.
Drag consists of both pressure
drag and skin friction drag.
E_09_01
Pressure Drag only
D = 55.6 lbf,
L=0
Boundary Layer Flow Along a Smooth Flat Plate
Experimental observation:
At local Reynolds number (Rex = U x/n) around 5x105, transition
from Laminar to Turbulent Boundary Layer Flow occurs. This Rex
of 5x105 is known as the critical or transitional Reynolds number.
Velocity Profiles
The gradient (du/dy) of the
turbulent velocity profile at the
wall (y=0) is higher than that of
the laminar velocity profile.
Hence skin friction drag of
turbulent boundary layer is
higher than that of laminar one.
Boundary layer thickness d (x): The location normal to surface at
which the velocity reaches 0.99 of the velocity U in the inviscid
free-stream. It increases in the x-direction along the plate.
Displacement thickness d*(x): The distance normal to the surface
that the streamline passing d(x) is displaced from its original
distance (h) at the leading edge of the plate. Hence,
d*(x) = d(x) – h
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Laminar Boundary Layer on Flat Plate
• Blasius Solution
• Momentum Integral Method
Experimental Skin Friction Drag Data
Curve fit formula for
turbulent boundary
layer (Re > 500,000):
Drag Coefficient
of Flat Plate
with Roughness
Curve fitting of
Experimental Data
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Drag Coefficient of Flat Plate
Empirical Formulas
Boundary Layer Flow Separation
When flow separation occurs,
there is also pressure drag.
Pressure (Form) Drag due to Flow Separation
100% Pressure Drag
Total Profile Drag
= Skin Friction Drag
+ Form Drag
Development of
velocity profile in
the boundary layer
on curved surface:
09_12
Flow separation
occurs when the
gradient of the
velocity profile at
the wall is zero,
forming a recirculating wake
downstream.
Wind Tunnel Tests
Force transducer behind model senses lift, drag and pitching moment directly.
Motor-controlled mechanism adjusts the model’s angle of attack.
Typical Experimental Data
Notice the sudden drop at the transition Re of 5x105 (Point E)
For Re > 5x105, the boundary is turbulent, which
has a fuller velocity profile.
Flow separation is delayed, resulting in a smaller
wake, and hence the pressure drag.
Adding surface roughness on circular and spherical shapes triggers
turbulence at lower Re, and hence helps to reduce the drag coefficient
Benefit of Streamlining
Pressure drag is greatly reduced by preventing flow separation using a
gradually tapering tail. Though skin friction increases with larger area, the
total drag is much less. Hence streamlined bodies are made of smooth
surfaces to reduce skin friction.
These objects have approximately the same drag:
Test Data of 2D Objects
Test Data of
Axi-Symmetric Objects
Test Data of 3D Objects
Recommended films:
http://web.mit.edu/hml/ncfmf.html
Fluid Dynamics of Drag Part I-IV
Airfoil Characteristics
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