ME3560 – Fluid Mechanics
Summer 2016
ME 3560 Fluid Mechanics
Chapter IX. Flow over Immersed Bodies
1
Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
9.1 General External Flow Characteristics
•A body immersed in a moving fluid experiences a resultant force due to
the interaction between the body and the fluid surrounding it.
•In some instances the fluid far from the body is stationary and the body
moves through the fluid with velocity U.
•Or the body is stationary and the fluid flows past the body with
velocity U.
•In both cases, the coordinate system can be fixed to the body and treat
the situation as fluid flowing past a stationary body with velocity U,
the upstream velocity.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•The structure of an external flow and the description and analysis of the
flow depend on the nature of the body in the flow.
•Two-dimensional objects infinitely long and of constant crosssectional size and shape.
•Axisymmetric bodies formed by rotating their cross-sectional shape
about the axis of symmetry.
•Three-dimensional bodies that may or may not possess a line or plane of
symmetry.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
9.1.1 Lift and Drag
•When any body moves through a fluid, two types of forces occur as a
result of this motion:
•Wall shear stress (τw) forces, due to viscous effects
•Normal stress forces, due to the pressure, p.
•The resultant force in the direction of
the upstream velocity is termed
the drag, D, and the resultant force
normal to the upstream velocity is
termed the lift, L
•For some three-dimensional bodies
there may also be a side force that is
perpendicular to the plane containing D
and L.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•The resultant of the shear stress and pressure distributions can be
obtained by integrating the effect of these two quantities on the body
surface:
dFx = ( pdA) cosθ + (τ w dA) sin θ
dFy = −( pdA) sin θ + (τ w dA) cosθ
D = ∫ dFx = ∫ p cosθ dA + ∫ τ w sin θ dA
L = ∫ dFy = − ∫ p sin θ dA + ∫ τ w cosθ dA
•To carry out the integrations and determine the lift and drag, the body
shape, the distribution of τw and p along the surface need to be known.
• These distributions are often extremely difficult to obtain.
•The pressure distribution can be obtained experimentally by use of a
series of static pressure taps along the body surface.
• It is usually quite difficult to measure the wall shear stress distribution
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Chapter IX. Flow over Immersed Bodies
Summer 2016
ME3560 – Fluid Mechanics
The equations to determine drag and lift,
D = ∫ p cosθ dA + ∫ τ w sin θ dA
L = −∫ p sin θ dA + ∫ τ w cosθ dA
are valid for any body, however, it is necessary to know the appropriate
shear stress and pressure distributions on the body surface. This is not
easy, such information is available only for certain simple situations.
D=
θ dA
∫1p4cos
243
+
sin θ dA
∫1τ42
43
w
PressureDrag Friction (viscous)
Drag
L = − ∫ p sin θ dA +
cosθ dA
∫1τ42
43
w
Usually Negligible
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Chapter IX. Flow over Immersed Bodies
Summer 2016
ME3560 – Fluid Mechanics
Alternatively, dimensionless lift and drag coefficients are defined and
their approximate values are determined by means of either a simplified
analysis, some numerical technique, or an appropriate experiment.
The lift coefficient, CL , and drag coefficient, CD, are defined as
CD =
D
1
2
ρ
U
A
2
CL =
A is a characteristic area of the object.
L
1
2
ρ
U
A
2
U is the upstream velocity.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
Pressure-Sensitive Paint (PSP)
PSP is now gaining acceptance as an alternative to the static surface
pressure ports. The PSP material is typically a luminescent compound
that is sensitive to the pressure on it and can be excited by an appropriate
light which is captured by special video imaging equipment. Thus, it
provides a quantitative measure of the surface pressure. PSP is a global
measurement technique, measuring pressure over the entire surface, as
opposed to discrete points. PSP also has the advantage of being
nonintrusive to the flow field.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
Particular Cases:
•θ = 0.
•Friction Drag = 0. Drag is due to
pressure only (form drag).
•θ = π/2. Flat plate parallel to flow
•Pressure Drag = 0. Only viscous
drag is present.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
9.1.2 Characteristics of Flow Past an Object
Re =
Ul
ν
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Re =
Summer 2016
UD
ν
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Chapter IX. Flow over Immersed Bodies
Summer 2016
ME3560 – Fluid Mechanics
9.2 Boundary Layer Characteristics
•Consider the case in which the boundary layer is formed on an
infinitely long flat plate along which flows a viscous, incompressible
fluid.
Re =
Ux
ν
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•Some distance downstream from the leading edge, the boundary layer
flow becomes turbulent.
•Turbulent flow is characterized by the occurrence of irregular mixing of
fluid particles that range in size from the smallest fluid particles up to
those comparable in size with the object of interest.
•For laminar flow, mixing occurs only on the molecular scale.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•The transition from a laminar boundary layer to a turbulent boundary
layer occurs at a critical value of the Reynolds number,
Rexcr= 2 × 105 to 3 × 106
•Depending on the surface roughness and the amount of turbulence in the
upstream flow.
•The location along the plate where the flow becomes turbulent, xcr,
moves towards the leading edge as the free-stream velocity increases.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•The boundary layer thickness, δ, is the distance from the plate at which
the fluid velocity is 0.99U:
y = δ whereu = 0.99U
• Fig. b shows two velocity profiles—one if there were no
viscosity (a uniform profile) and one considering the viscosity
and zero slip at the wall (the boundary layer profile).
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Chapter IX. Flow over Immersed Bodies
Summer 2016
ME3560 – Fluid Mechanics
Rexcr = 5×105
Laminar Boundary Layer: If Rex < 5×105
νx
δ =5
τ w = 0.332U
U
3/ 2
τw
Cf = 1
2
ρ
U
2
C Df
or
FD
=1
2
ρ
U
A
2
δ
Ux
5
=
; Re x =
x
ν
Re x
ρµ
x
0.664
Cf =
Re x
CDf
1.328
=
Re L
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
9.2.4Transition from Turbulent to Laminar Flow
•The analytical results shown in the previous section are restricted to
laminar boundary layer flows along a flat plate with zero pressure
gradient.
•They agree quite well with experimental results up to the point where
the boundary layer flow becomes turbulent.
•Any BL will become turbulent for any free-stream velocity and any
fluid provided the plate is long enough.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
9.2.4Transition from Turbulent to Laminar Flow
•Re at the transition location is a function of various parameters:
- Roughness of the surface.
- The curvature of the surface (for example, a flat plate or a sphere).
- The disturbances in the flow outside the boundary layer.
•On a flat plate with a sharp leading edge in a typical airstream, the
transition takes place at Rexcr = 2 × 105 to 3 × 106.
•For our Calculations we will use Rexcr = 5×105
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•The transition from laminar to turbulent flow involves the instability of
the flow field.
•Small disturbances imposed on the boundary layer flow: from a
vibration of the plate, a roughness of the surface will either grow
(instability) or decay (stability), depending on where the disturbance is
introduced into the flow.
•If these disturbances occur at a location with Rex < Rexcr they will die
out, and the boundary layer will return to laminar flow at that location.
•Disturbances imposed at a location with Rex > Rexcr will grow and
transform the boundary layer flow downstream of this location into
turbulence.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•The transition from laminar to
turbulent
flow
involves
a
noticeable change in the shape of
the boundary layer velocity
profile.
•The turbulent profiles are flatter,
have a larger velocity gradient at
the wall, and produce a larger
boundary layer thickness than do
the laminar profiles.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
9.2.5 Turbulent Boundary Layer Flow
•The structure of turbulent
boundary layer flow is very
complex, random, and irregular.
•The figure shows a laser-induced
fluorescence visualization of a
turbulent boundary layer on a flat
plate.
•There are no “exact” solutions
for turbulent boundary layer flow.
•Considerable effort has been
made to obtain numeric solutions
for turbulent flow by using
approximate
shear
stress
relationships.
Chapter IX. Flow over Immersed Bodies
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Summer 2016
ME3560 – Fluid Mechanics
Rexcr = 5×105
Turbulent Boundary Layer: If Rex > 5×105
 ν  4/5
δ = 0.370  x
or
U 
1/ 5
δ
x
Ux
ν
τw
Cf = 1
2
ρ
U
2
τ w = 0.0288ρ U 2 Re −x1/ 5
C f = 0.0576 Re −x 1/ 5
= 0.370 Re x ; Re x =
1/ 5
C Df
FD
0.072
=1
=
1/ 5
2
Re
ρ
U
A
L
2
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Chapter IX. Flow over Immersed Bodies
Summer 2016
ME3560 – Fluid Mechanics
Rexcr = 5×105
In Summary:
Laminar Boundary Layer: If Rex < 5×105
δ =5
νx
or
U
δ
5
Ux
=
; Re x =
x
ν
Re x
τ w = 0.332U
τw
Cf = 1
2
ρ
U
2
CDf
FD
=1
2
ρ
U
A
2
3/ 2
ρµ
x
0.664
Cf =
Re x
CDf
1.328
=
Re L
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Chapter IX. Flow over Immersed Bodies
Summer 2016
ME3560 – Fluid Mechanics
Rexcr = 5×105
Turbulent Boundary Layer: If Rex > 5×105
 ν  4/5
δ = 0.370  x
or
U 
1/ 5
δ
x
= 0.370 Re x ; Re x =
1/ 5
Ux
ν
τ w = 0.0288ρ U 2 Re −x1/ 5
τw
Cf = 1
2
ρ
U
2
C Df
C f = 0.0576 Re −x 1/ 5
FD
0.072
=1
= 1/ 5
2
ρ U A Re L
2
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
9.3 Drag
D
CD = 1
2
ρ
U
A
2
Chapter IX. Flow over Immersed Bodies
25
Summer 2016
ME3560 – Fluid Mechanics
9.4 Lift
Typically, the lift is given in terms of the lift coefficient
L
CL = 1
2
ρ
U
A
2
•Most common lift-generating devices (i.e., airfoils, fans, spoilers on
cars, etc.) operate in the large Re range in which the flow has a boundary
layer character, with viscous effects confined to the boundary layers and
wake regions.
• For such cases the wall shear stress, τw, contributes little to the lift.
•Most of the lift comes from the surface pressure distribution.
•The distribution, for the most part, is consistent with simple Bernoulli
equation analysis.
•Locations with high-speed flow have low pressure, while locations with
low-speed flow have high pressure.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•A typical device designed to produce lift does so by generating a
pressure distribution that is different on the top and bottom surfaces.
•For large Reynolds number flows these pressure distributions are
usually directly proportional to the dynamic pressure, ρU2/2, with
viscous effects being of secondary importance.
•Two airfoils used to produce lift are shown.
•The symmetrical one cannot produce
lift unless the angle of attack, α, is
nonzero.
•Because of the asymmetry of the
nonsymmetric airfoil, the pressure
distributions on the upper and lower
surfaces are different, and a lift is
produced even with α = 0.
Chapter IX. Flow over Immersed Bodies
27
ME3560 – Fluid Mechanics
Summer 2016
•Since most airfoils are thin, it is customary to use the planform
area, A = bc, in the definition of the lift coefficient.
• b is the airfoil span and c is the chord length—the length from the
leading edge to the trailing edge.
•Typical lift coefficients so defined are on the order of unity. That is, the
lift force is on the order of the dynamic pressure times the planform area
of the wing, L ≈ (ρU2/2)A.
•The wing loading, defined as the average lift per unit area of the
wing, L/A, therefore, increases with speed.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•Typical lift and drag coefficient data as a function of angle of attack, α,
and aspect ratio, AR, are shown.
•AR is the ratio of the square of the wing span to the planform area,
AR = b2/A.
•If the chord length, c, is constant along the length of the wing (a
rectangular planform wing), AR= b/c
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•In general, CL increases and CD decreases with an increase in AR.
• Long wings are more efficient because their wing tip losses are
relatively minor than for short wings.
•The increase in drag due to the finite length (AR < ∞) of the wing is
often termed induced drag.
•It is due to the interaction of the complex swirling flow structure near
the wing tips and the free stream.
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•Although viscous effects and the wall shear stress contribute little to the
direct generation of lift, they play an extremely important role in the
design and use of lifting devices.
•This is because of the
viscosity-induced
boundary layer
separation that can
occur on
nonstreamlined bodies
such as airfoils that
have too large an angle
of attack
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Chapter IX. Flow over Immersed Bodies
ME3560 – Fluid Mechanics
Summer 2016
•As is indicated in the figure, up to a certain point, CL increases rather
steadily with the angle of attack.
•If α is too large, the boundary layer on the upper surface separates, the
flow over the wing develops a wide, turbulent wake region, the lift
decreases, and the drag increases.
•This condition, as indicated by the
figures in the margin, is termed
stall.
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Chapter IX. Flow over Immersed Bodies