Lecture 7
Flow over immersed bodies.
Boundary layer.
Analysis of inviscid flow.
Flow over immersed bodies
• flow classification: 2D,
axisymmetric, 3D
• bodies: streamlined
and blunt
Shuttle landing: examples of various body types
Lift and Drag
• shear stress and pressure
integrated over the surface of a
body create force
• drag: force component in the
direction of upstream velocity
• lift: force normal to upstream
velocity (might have 2
components in general case)
D   dFx   p cos  dA    w sin  dA
CD 
D
2
1
2 U A
L   dFy   p sin  dA    w cos  dA
CL 
L
2
1
2 U A
Flow past an object
• Dimensionless numbers involved
Re 
Ul

Ma 
U
c
• for external flow: Re>100 dominated by inertia, Re<1 – by viscosity
Fr 
U
gl
Flow past an object
Character of the steady, viscous
flow past a circular cylinder:
(a) low Reynolds number flow, (b)
moderate Reynolds number flow,
(c) large Reynolds number flow.
Boundary layer characteristics
• for large enough Reynolds number flow can be divided into
boundary region where viscous effect are important and
outside region where liquid can be treated as inviscid
Re x 
Ux

Re xcr
2 105  3 106
Laminar/Turbulent transition
• Near the leading edge of a flat
plate, the boundary layer flow is
laminar.
• f the plate is long enough, the flow
becomes turbulent, with random,
irregular mixing. A similar
phenomenon occurs at the
interface of two fluids moving with
different speeds.
Boundary layer characteristics
• Boundary layer thickness

 u
Boundary layer displacement thickness:     1 
U
0
*

dy

Boundary layer momentum thickness (defined in terms of momentum flux):

 bU    b  u U  u dy
2
0

u
  b 
U
0
u

1  
 U
Prandtl/Blasius boundary layer solution
Let’s consider flow over large thin plate:
  2u  2u 
u
u
1 p
u
v

  2  2 
x
y
 x
y 
 x
  2v  2v 
v
v
1 p
u v  
  2  2 
x
y
 y
 x y 

x
• approximations:
v
• than:
u v

0
x y
u

y
u
u
 2u
u
v
 2
x
y
y
• boundary conditions:
u  v  0 on y  0
u  U as y  
u v

0
x y
Prandtl/Blasius boundary layer solution
• as dimensionless velocity profile should
be similar regardless of location:
u
 y
 g 
U
 

½
• dimensionless similarity variable
U 
  y
 x 
  ( xU )½ f ( )
• stream function
u


 Uf ( )
y
   U 
u

  f ( )  f ( ) 
x  4 x 
½
2 f   f  f   0
f  f   0 at   0; f   1 as   
 x 
 
U 
½
Drag on a flat plate
• Drag on a flat plate is related to the momentum deficit within
the boundary layer
D   bU 
2

u
  b
U
0
 u
1  
 U
dD

 b w   bU 2
dx
x
• Drag and shear stress can be calculated just by assuming
some velocity profile in the boundary layer
Transition from Laminar to Turbulent flow
• The boundary layer will become turbulent if a plate is long enough
turbulent profiles are flatter and produce
larger boundary layer
Inviscid flow
• no shearing stress in inviscid flow, so
 p   xx   yy   zz
• equation of motion is reduced to Euler equations
 xx
u
u
u
u
 gx 
 (  u  v  w )
x
t
x
y
z
 yy
v
v
v
v
gy 
 (  u  v  w )
y
t
x
y
z

w
w
w
w
 g z  zz   (  u
v
w )
x
t
x
y
z
 V

 g  p   
 (V  V 
 t

Bernoulli equation
• let’s write Euler equation for a steady flow along a streamline
 g  p   (V  V
(V  )V  12 (V V )  V  ( V )
g   g z
  gz  p 

2
 (V V )  V  (  V )
• now we multiply it by ds along the streamline
  gz  ds  p  ds 
dp
1
 d (V 2 )  gdz  0
 2

2
 (V V )  ds  V  ( V )  ds
or
p V2

 gz  const
 2
Irrotational Flow
• Analysis of inviscide flow can be further simplified if we
assume if the flow is irrotational:
1  v
u 
z      0
2  x y 
v u
 ;
x y
w v
 ;
y z
u w

z x
• Example: uniform flow in x-direction:
Bernoulli equation for irrotational flow
  gz  p 

2
 (V V )  V  (  V )
always =0, not only along a stream line
• Thus, Bernoulli equation can be applied between any two
points in the flow field
p V2

 gz  const
 2
Velocity potential
• equations for irrotational flow will be satisfied automatically if
we introduce a scalar function called velocity potential such
that:

u
x

v
y

w
z
V  
• As for incompressible flow conservation of mass leads to:
∇ V  0,
 2  0
  
 2  2 0
2
x y
z
2
2
2
Laplace equation
Some basic potential flows
• As Laplace equation is a linear one,
the solutions can be added to each
other producing another solution;
• stream lines (y=const) and
equipotential lines (f=const) are
mutually perpendicular
dy
v

dx along streanline u
d 


dy
u
dx 
dy  udx  vdy 

x
y
dx along  const
v
Both f and y satisfy Laplace’s equation
u v
  



y x
y  y
    
  


x

x



Uniform flow
• constant velocity, all stream lines are straight and
parallel

U
x
  Ux

U
y
  Uy

0
y

0
x
  U ( x cos   y sin  )
  U ( y cos   x sin  )
Source and Sink
• Let’s consider fluid flowing radially outward from a line through
the origin perpendicular to x-y plane
from mass conservation:
(2 r )vr  m

m

r 2 r

m
ln r
2
1 
m

r  2 r

1 
0
r 
m

2

0
r
Vortex
• now we consider situation when ther
stream lines are concentric circles i.e.
we interchange potential and stream
functions:   K
   K ln r
• circulation
   V  ds     ds 
C
C
 d  0
C
• in case of vortex the circulation is zero
along any contour except ones
enclosing origin
2
K
   (rd )  2 K
r
0

 
2

   ln r
2
Shape of a free vortex



2
p V2

 gz  const
 2
at the free surface p=0:
V12 V2 2

z
2g 2g

z 2 2
8 r g
2
Doublet
• let’s consider the equal strength, source-sink pair:
m
   (1   2 )
2
m
2ar sin 
 
tan 1 ( 2
)
2
2
r a
if the source and sink are close to
each other:
K sin 
 
r
K cos 

r
K – strength of a doublet
Summary
Superposition of basic flows
• basic potential flows can be combined to form new
potentials and stream functions. This technique is
called the method of superpositions
• superposition of source and uniform flow
m
  Ur sin   
2
m
  Ur cos  
ln r
2
Superposition of basic flows
•
Streamlines created by injecting
dye in steadily flowing water show
a uniform flow. Source flow is
created by injecting water through
a small hole. It is observed that for
this combination the streamline
passing through the stagnation
point could be replaced by a solid
boundary which resembles a
streamlined body in a uniform flow.
The body is open at the
downstream end and is thus called
a halfbody.
Rankine Ovals
• a closed body can be modeled as a combination of a
uniform flow and source and a sink of equal strength
m
  Ur sin  
(1   2 )
2
m
  Ur cos  
(ln r1  ln r2 )
2
Flow around circular cylinder
• when the distance between source and sink approaches 0,
shape of Rankine oval approaches a circular shape
K sin 
  Ur sin  
r
K cos 
  Ur cos  
r
Potential flows
•
•
Flow fields for which an incompressible
fluid is assumed to be frictionless and
the motion to be irrotational are
commonly referred to as potential flows.
Paradoxically, potential flows can be
simulated by a slowly moving, viscous
flow between closely spaced parallel
plates. For such a system, dye injected
upstream reveals an approximate
potential flow pattern around a
streamlined airfoil shape. Similarly, the
potential flow pattern around a bluff
body is shown. Even at the rear of the
bluff body the streamlines closely follow
the body shape. Generally, however, the
flow would separate at the rear of the
body, an important phenomenon not
accounted for with potential theory.
Effect of pressure gradient
• d’Alemberts paradox: drug on an object in inviscid liquid is
zero, but not zero in any viscous liquid even with vanishingly
small viscosity
inviscid flow
viscous flow
Effect of pressure gradient
•
•
At high Reynolds numbers, nonstreamlined (blunt) objects have wide,
low speed wake regions behind them.
As shown in a computational fluid
dynamics simulation, the streamlines for
flow past a rectangular block cannot
follow the contour of the block. The flow
separates at the corners and forms a
wide wake. A similar phenomenon
occurs for flow past other blunt objects,
including bushes. The low velocity wind
in the wake region behind the bushes
allows the snow to settle out of the air.
The result is a large snowdrift behind the
object. This is the principle upon which
snow fences are designed
Drag on a flat plate
Drag coefficient diagram
CD 
D
2
1

U
A
2
Drag dependence
• Low Reynolds numbers Re<1:
D  f (U , l ,  )
D  C lU
2C
CD 
Re
Drag dependence
• Moderate Reynolds numbers. Drag coefficient on flat plate ~Re-½; on blunt
bodies relatively constant (and decreases as turbulent layer can travel
further along the surface resulting in a thinner wake
Examples
Examples
• The drag coefficient for an
object can be strongly
dependent on the shape of
the object. A slight change
in shape may produce a
considerable change in
drag.
Problems
• 6.60. An ideal fluid flows past an infinitely
long semicircular hump. Far from hump
flow is uniform and pressure is p0. Find
maximum and minimum pressure along
the hump. If the solid surface is y=0
streamline, find the equation for the
streamline passing through Q=p/2, r=2a.
• 9.2. The average pressure and
shear stress acting on the surface of
1 m-square plate are as indicated.
Determine the lift and drag
generated.
• 9.12. Water flows past a flat plate with an upstream velocity of
U=0.02 m/s. Determine the water velocity 10mm from a plate at
distances 1.5mm and 15m from the leading edge.