fluid mechanics lecture notes ii
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8/11/2019 Fluid Mechanics Lecture Notes II
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Mechanics of Fluids MECH2008 (2008 – 2009)
Assessment: In-course continuous assessment 10%
• a mid-term test to be announced later
Examination in December 90%
Topics Covered:
1. Flow kinematics with differential vector calculus
2. Differential equations of motion
3. Unidirectional viscous flow and hydrodynamic lubrication
4. Potential flow and stream function
5. Boundary layer and drag
6. Open-channel flow and fluid machines
Prerequisites:-
This is a Level-II Mechanics of Fluids course demanding the knowledge you acquired in thefirst-year fluid and mathematics courses. In particular, the following topics are relevant and
should be reviewed if you have already forgotten the stuffs:-
1) Properties of fluid (density, viscosity);
2) Principles of fluid statics;
3) Fluid dynamics by control volume approach
continuity equation
energy equation (or Bernoulli equation)
momentum equation
head, head loss
4) Differentiation and integration;
5) Vector differential calculus (grad, div, curl, Gauss theorem, etc);
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Lecture Notes and Worked Examples
(The corresponding section numbers in the textbook or other references are noted wherever
appropriate.)
(I) DIFFERENTIAL ANALYSIS OF FLUID FLOW
A. Description of Fluid Motion (Section 4.1.1)
• Lagrangian description: fluid particles are “tagged” or identified; rate of change of
flow properties as observed by following a fixed particle; variables are functions of
the initial position of particles and time.
• Eulerian description: fluid properties and variables are field variables, which are
functions of position in space (with respect to a fixed frame of reference) and time.
The Eulerian description, which is comparable to the data recorded by a measuring
device fixed in position, is more convenient to use in fluid mechanics.
Eulerian and Lagrangian descriptions
of temperature of a fluid discharging
from a smoke stack ual particles
In the Lagrangian description, one
must keep track of the position and
velocity of individ
•
Rectangular (Cartesian) coordinates:
( )
( )
( )
1 2 3
1 2 3
1 2 3
( , , ) ( , , ) 1, 2,3
( , , ) ( , , ) 1, 2,3
, , , , 1, 2,3
e.g.,
, , ( , , )
i
i
i
i
i
x y z x x x x i
u v w u u u u i
i x y z x x x x
uu v wu v w
x y z x y z x
= = = =
= = = =
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂= = = =⎜ ⎟⎜ ⎟
∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠
⎛ ⎞ ∂∂ ∂ ∂ ∂ ∂ ∂= + + =⎜ ⎟
∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
x
V
V =i i
z
V y
O
x
• Primitive variables: pressure ( , ) - scalar (0th order tensor)
velocity ( , ) - vector (1st order tensor)
p t
t
x
V x
Deduced variable stre ss ( , ) - 2nd order tensor t τ x
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In the Eulerian description, one
defines field variables, such as the
pressure field and the velocity field,
at any location and instant in time.
B. Kinematics (Sections 4.2 and 6.1)
• Total (a.k.a. material, substantial) derivative = local rate of change + convective (or
advective) rate of change = the rate of change as observed following a particle of
fixed identity. It is an operator that can be applied to any scalar or vector quantity.
( ) ( )( )( )
( )
( ) ( ) ( )
( )
local rate convective rate of changeof change
e.g., local acceleration =
convective acceleration =
d
dt t
u v wt x y z
t
u v w x y z
∂
= +∂
∂ ∂ ∂ ∂= + + +
∂ ∂ ∂ ∂
∂
∂∂ ∂ ∂
+ +∂ ∂ ∂
i
i
V
V
V V
V V =
V
-
The local rate of change, also called the unsteady term, vanishes identically for
a steady flow. Therefore a flow is steady if and only if / 0t ∂ ∂ ≡ .
-
The quantity ( )iV
is a scalar convective operator that determines the time
rate of change of any property (e.g., velocity, density, concentration,
temperature) of a particle by reason of the fact that the particle moves from a
place where the property has one value to another place where it has a
different value.
The total derivative is defined by
following a fluid particle as it moves
throughout the flow field. In this
illustration, the fluid particle is
accelerating to the right as it moves
up and to the right. A velocity field with respect to a fixedframe of reference ( x, y). A point fixed
in space is occupied by different fluid
particles at different time.
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•
Translation
+ (rigid body motion)
Rotation
General motion = +
Dilatation (change in volume)
+
Angular deformation (change in shape)
⎧ ⎫⎪⎪⎬⎪⎪⎪ ⎭
⎪⎨⎪⎪⎪⎪⎩
As illustrated by:-
The various modes of deformation can be expressed in terms of the velocity gradients.
• Divergence of velocity is the volumetric strain/dilatation rate (rate of change of
volume per unit volume)
u v w
x y z
∂ ∂ ∂≡ + +
∂ ∂ ∂V i
where u x∂ ∂ , v y∂ ∂ and w z∂ ∂ are the components of the volumetric strain rate due
to elongation of a fluid element in the x-, y-, and z-directions, respectively.
Consider a small element of dimensions x y zδ δ δ × × :
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Because of the velocity differential uδ over a distance xδ , the element is lengthened
in the x-direction by u t δ δ ⋅ over a small period of time t δ . The corresponding
change in volume is therefore
xV u t y zδ δ δ δ δ = ⋅ ⋅ ⋅ ,
and the volume strain rate (change in volume per volume per time) is
as , 0 xV u t y z u u x t
V t x y z t x x
δ δ δ δ δ δ δ δ
δ δ δ δ δ δ
⋅ ⋅ ⋅ ∂= = =
⋅ ⋅ ⋅ ⋅ ∂ → .
Similarly, for the lengthening of the element in the y- and z-directions
, as , , 0 y z
V V v w y z t
V t y V t z
δ δ δ δ δ
δ δ
∂ ∂= =
⋅ ∂ ⋅ ∂ →
The total volume strain rate is hence given by the divergence of the velocity.
as , , , 0V u v w x y z t V t x y zδ δ δ δ δ
δ
∂ ∂ ∂= + + →⋅ ∂ ∂ ∂
In this incompressible flow, in which the
velocity divergence is identically zero, an
initially square parcel of marked fluid will
deform into a long thin shape (stretch in x-
direction, but shrink in the y-direction) in the
course of movement shown in the figure. Theflow is irrotational is this case.
• Any shear deformation can be decomposed into rigid body rotation and angular
deformation. Consider a small element undergoing shear deformation
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Because of the velocity differential vδ over a distance xδ , the face OA rotates
counterclockwise by an angle /v t xδα δ δ δ = ⋅ over a small period of time t δ .
Therefore the angular velocity of OA is
as , 0d v
x t
dt x
α α δ δ
∂= = →
∂
Similarly, the face OB rotates clockwise at an angular velocity given by
as , 0d u
y t dt y
β β δ δ
∂= = →
∂
x
y
Oα
β
The deformation can be decomposed into a rigid body rotation at an angular velocity
( )1 1
2 2
v u
x yω α β
⎛ ∂ ∂= − = −⎜ ∂ ∂⎝ ⎠
⎞
⎟ , where counterclockwise rotation is taken to be positive,
x
y
O
( )1
2α β −
( )1
2α β −
and an angular deformation, where the corner angle decreases at a rate given by
v u x y
γ α β ∂ ∂= + = +∂ ∂ , where a positive rate means a decreasing angle,
x
y
O
( )1
2
( )1
2α β +
α β +
γ
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•
Rate of angular deformation of a 2-D fluid element moving in the x- y plane (angular
deformation is considered to be positive if it is to decrease the original right angle) is
hence defined to be
0lim
xyt
v u
t xδ y
δα δβ γ
δ →
+ ∂ ∂= = +
∂ ∂ .
For a 3-D element in general, the rate of change of the corner angle that is initially a
right angle between the i- j axes
( ) ji
ij
j i
uui j
x xγ
∂∂= + ≠
∂ ∂ ,
which is symmetric, i.e.,ij jiγ γ = .
• Rotation of a fluid element (about an axis which is perpendicular to the plane of the
fluid motion) is the average of the angular velocities of the two mutually
perpendicular sides of the element, where counterclockwise rotation is considered to
be positive:
1rotation about -axis:
2
1rotation about -axis:
2
1rotation about -axis:
2
z
x
y
v u z
x y
w v x
y z
u w y
z x
ω
ω
ω
⎛ ⎞∂ ∂= −⎜ ⎟
∂ ∂⎝ ⎠
⎛ ⎞∂ ∂= −⎜ ⎟
∂ ∂⎝ ⎠
∂ ∂⎛ ⎞= −⎜ ⎟
∂ ∂⎝ ⎠
Rotation (or angular velocity) vector ( ) x y z , ,ω ω ω ω =
Note that for a 2-D flow in the x- y plane, xω and
yω vanish identically; hence the
rotation vector is always perpendicular to the x- y plane.
• To generalize, we may define
1 1 1shear rate tensor angular deformation rate , and
2 2 2
ji
ij ij
j i
uue
x xγ
⎛ ⎞∂∂= + = =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠
1rate of rotation vorticity, where
2
vorticity (curl of velocity)
22 y x
i j k
x y z
u v w
w v u wi j
y z z x
ω ω
=
= ×
∂ ∂ ∂=
∂ ∂ ∂
⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞= − + − +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠
ω
ζ V
2 z
v uk
x y
ω
⎛ ⎞∂−⎜ ⎟∂ ∂⎝ ⎠
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2=
ζ ω
The direction of a vector cross product is
determined by the right-hand rule.
The vorticity vector is equal to twice the
angular velocity vector of a rotating fluid
particle.
The difference between a rotational and irrotational flow: fluid elements in a rotational
region of the flow rotate about their own axis, but those in an irrotational region of the
flow do not.
In this incompressible and rotational flow, an
initially square fluid parcel will not onlyelongate, but also rotate about its axis as it moves
over the time periods shown in the figure.
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C. The Reynolds Transport Theorem (Section 4.4)
Define:
The Reynolds transport theorem
(RTT) provides a link between
the system approach and thecontrol volume approach.
Two approaches of analyzing a problem.
(a) System approach: follow the fluid as
it moves and deforms; no mass crossesthe boundary. (b) Control volume
approach: consider the changes in a
certain fixed volume; mass crosses the
boundary.
- Material Volume: a volume that contains the same fluid as it moves and deforms
following the motion of the fluid- Material Surface: enclosing surface of a material volume; by definition no fluid
particles can cross it.
-
Control Volume: a volume of fluid in a flow field, usually fixed in space, to be
occupied by different fluid particles at different times.- Control Surface: imaginary or physical enclosing surface of a control volume.
- Flux: amount of property (e.g., mass, momentum, energy) crossing a unit area of asurface per unit time.
We state without proof the Reynolds transport theorem, which provides a basis for
developing differential equations for the various conservation laws:
rate of change of the local rate of change of the property within property within the fixed the material volume control volume that happens
to coincide with the materialvolum
MV CV
d bdV bdV
dt t ρ ρ
∂=
∂∫∫∫ ∫∫∫net out-flux of the
property across theentire control surface
e at that instant
CS b d A ρ +
∫∫ i
V n
where
density of fluid
an intensive property of fluid (property per unit mass)material volume that happens to coincide with at time t
b B MV CV
ρ =
==
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control volume (fixed in space)
control surface
unit outward normal to
CV
CS
CS
=
=
=n
The integral of over the
control surface gives the net amount
of the property B flowing out of the
control volume (into the control
volume if it is negative) per unit time.
A moving system (hatched region)
and a fixed control volume (shaded
region) in a diverging portion of a
flow field at times t and
bV ndA
i ρ
t t + .
D. Conservation of Mass (Section 6.2)
If the property is mass, then b = 1, and
( )L.H.S. mass in 0
(by definition of , which always contains the same fluid)
MV d d dV MV dt dt
MV
ρ = =∫∫∫
( )
by Gauss theorem is stationary
R.H.S.
=
CV CS
CV CV
CV
dV d At
dV dV t
ρ ρ
ρ ρ
∂+
∂∂
+∂
∫∫∫ ∫∫
∫∫∫ ∫∫∫
V n
V
i
i
Equating L.H.S. and R.H.S., and removing the volume integral since CV is arbitrary, we
get the differential form of Continuity Equation
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( )
( )
( )
0
or, using the identity
0
or in index form 0 , 1,2,3, summation over repeated indexi
i
t
d
dt ud
i jdt x
ρ ρ
ρ ρ ρ
ρ ρ
ρ ρ
∂+ =
∂
=
+ =
∂+ = =
∂
V
V V + V,
V
i
i i i
i
INCOMPRESSIBLE FLOW is defined as one in which the density of a fluid particle
is invariant with time 0d
dt
ρ ⇔ = , which implies
0
(ie, divergence of velocity is zero for incompressible flow)
In Cartesian coordinates, the continuity equation for inc
V =i
ompressible flow reads
0u v w
x y z
∂ ∂ ∂+ + =
∂ ∂ ∂
Note that a flow with constant density is always incompressible, but an incompressible
flow does not necessarily have a constant density (e.g., flow in a stratified sea).
E. Applied Forces
• Body force due to gravity on a small fluid element = dV ρ g
•
Surface stress =s τ ni , where n is the unit outward normal vector to the surface, and
( ) , 1, 2, 3
xx xy xz
yx yy yz ij
zx zy zz
i j
τ τ τ
τ τ τ τ
τ τ τ
⎡ ⎤⎢ ⎥
= = =⎢ ⎥⎢ ⎥⎣ ⎦
τ
are the stress components on an infinitesimal cubic fluid element.
ijτ is a second order tensor, where
the f irst index i denotes the f ace (on which the stress acts) being normal to i x ,
and the second index j denotes the stress component being in the j x direction.
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In the textbook, the normal stress is denoted byiiσ in order to distinguish it from the
shear stress ( )ij i jτ ≠ .
It can be shown that ijτ is symmetric, ie, ij jiτ τ = . Therefore there are only 6
independent stress components.
F. Conservation of Linear Momentum (Section 6.3)
Apply Newton’s second law of motion to a material volume of fluid:
= +
rate of change surface bodyof momentum stress force
MV MS MV
d dV dA dV
dt ρ ρ ∫∫∫ ∫∫ ∫∫∫V s
g
The L.H.S. can be converted, using the transport theorem and the continuity equation,
into MV MV
d d dV dV
dt dt ρ ρ =∫∫∫ ∫∫∫
V V .
The first term on the R.H.S. is MS MS MV
dA dA dV = =∫∫ ∫∫ ∫∫∫s τ n τ i i on using Gauss theorem.
Plugging these terms back, and removing the volume integral since the volume is
arbitrary, we get the differential form of momentum equation
d
dt
ρ ρ = +V
τ g i
The left hand term is a total derivative, which can be expanded into the Eulerian form:
oriji i
j i
j j
u uu g
t t
τ
x x ρ ρ ρ
⎛ ⎞ ∂∂ ∂∂⎛ ⎞+ = + + = +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
V V V τ g i i ρ
∂ ∂
By now, there are more unknowns than equations. To close the problem, we need to
introduce CONSTITUTIVE (stress vs strain-rate) relations to relate the stress and the
kinematics.
If the fluid is Newtonian, a linear relationship is followed
1 forwhere , dynamic viscosity coefficient
0 for
jiij ij
j i
ij
uu p
x x
i j
i j
τ δ μ
δ μ
⎛ ⎞∂∂= − + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠
=⎧= =⎨
≠⎩
Finally, on substituting the above relationship, we obtain the Navier-Stokes equations
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( )
2
2
2
2
2
or in index form,
, 1,2,3, summation over repeated index
1or
(I) (II)
i i i j i
j i j
i i i j i
j i j
pt
u u u pu g i j
t x x x
u u u pu g
t x x x
ρ ρ μ
ρ ρ μ
ν ρ
∂⎛ ⎞+ = − + +⎜ ⎟∂⎝ ⎠
⎛ ⎞∂ ∂ ∂∂+ = − + + =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
∂ ∂ ∂∂+ = − + +
∂ ∂ ∂ ∂
V V V g V i
(III) (IV) (V)
where is the kinematic viscosityν μ ρ = /
Meanings of the five terms:-
(I) – local acceleration;
(II) – convective acceleration (inertia), nonlinear term of the equation;
(III) – pressure gradient;
(IV) – gravity;
(V)
– viscous diffusion of momentum owing to molecular viscosity of the fluid.
Now, we have 4 equations (1 continuity + 3 components of momentum) for the four
variables as functions of space and time, , , and x y zu u u p ( ), , , x y z t . Note that it is the
pressure gradient, rather than the pressure itself that drives the flow.
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The Equations of Motion for an Incompressible Newtonian Fluid
In Rectangular Coordinates ( x , y, z )
2 2 2
2 2 2
2 2 2
2 2
Continuity: 0
1-component:
1-component:
y x z
x x x x x x x x y z
y y y y y y y
x y z
uu u
x y z
u u u u u u u p x x u u u g
t x y z x x y z
u u u u u u u p y u u u
t x y z y x y z
ν ρ
ν ρ
∂∂ ∂
+ + =∂ ∂ ∂
⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂∂+ + + = − + + + +⎜ ⎟
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
∂ ∂ ∂ ∂ ∂ ∂ ∂∂+ + + = − + + +
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2
2 2 2
2 2 2
1-component:
y
z z z z z z z x y z
g
u u u u p u u u z u u u
t x y z z x y zν
ρ
⎛ ⎞+⎜ ⎟⎜ ⎟
⎝ ⎠
⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + + +⎜ ⎟
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ zg
( , , ) are the components of the acceleration due to gravity in the , , and directions.
If, say, and are horizontal axes and is positive upward, then 0, and .
Also, the gravity
x y z
x y z
g g g x y z
x y z g g= = = −
( )can be combined implicitly with the pressure term by introducing
* . x y z p p p g x g y g z ρ ρ ≡ − − + + g x =i
g g
In Cylindrical Coordinates (r , , z )
( )
( )
2
2 2
2 2 2 2
1 1Continuity: 0
1-component:
1 1 2
r z
r r r r r z
r r r
ru u u
r r r z
u uu u u u pr u u
t r r r z r
uu uru g
r r r r r z
θ
θ θ
θ
θ
θ ρ
ν θ θ
∂ ∂ ∂+ + =
∂ ∂ ∂
∂ ∂ ∂ ∂ ∂+ + − + = −
∂ ∂ ∂ ∂ ∂
⎡ ⎤∂∂ ∂ ∂ ∂⎛ ⎞+ + − +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦
+
( )2 2
2 2 2 2
1-component:
1 1 2
-component:
r
r r z
r
z z zr
u u u u u u u pu u
t r r r z r
u uuru g
r r r r r z
uu u u z u u
t r r
θ θ θ θ θ θ
θ θ θ θ
θ
θ θ ρ θ
ν θ θ
θ
∂ ∂ ∂ ∂ ∂+ + + + = −
∂ ∂ ∂ ∂ ∂
⎡ ⎤∂ ∂∂ ∂ ∂⎛ ⎞+ + + +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦
∂ ∂ ∂+ + +
∂ ∂ ∂
+
2 2
2 2 2
1
1 1
z z
z z z z
u p
z z
u u ur gr r r r z
ρ
ν θ
∂ ∂= −
∂ ∂
⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞+ + + +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦
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G. Scaling and Approximation
• Because of the inertia terms (convective acceleration), the Navier-Stokes (NS)
equations are non-linear equations.
•
Except for simple flow geometry, analytical solutions do not exist in general.
• Fortunately, for many practical applications, not all terms in the equations are equally
important, and therefore some subdominant terms can be dropped in favor of a firstapproximation of the problem. The approximate equations can then be solved
(analytically or numerically) with much greater ease than the full-blown ones.
•
It is important to judge, for a particular problem, the relative significance of the
individual terms in the NS equations, which can be reflected from the magnitude of
the corresponding non-dimensional parameters.
For illustration, consider incompressible unsteady flow past a body:
Body
L
U
Characteristic scales:
Length ( L); Time scale of unsteadiness (T ); Velocity (U ); Pressure (P)
Introduce dimensionless variables (distinguished by *):
/ , * / , / , * / , /U t t T L p p P g= = = = =V* V x* x g* g
the normalized NS can be expressed as
2
2 2*
*
L P gL p
UT t U U UL
ν
ρ
∗ ∗⎛ ⎞∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ = − + +⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
V * V * V*
∗ g* V i
*
The scales have been chosen to be representative of the variables so that all the
dimensionless terms are order unity. Now, the importance of each term (relative to
the inertia) is carried by its bracketed coefficient.
2
2
temporal accelerationStrouhal number (St)
convective accelertion
pressure forceEuler number (E)
inertia
ineritaReynolds number (Re)
viscous force
ineritaFroude number (Fr)gravi
L
UT
P
U
UL
U gL
ρ
ν
= =
= =
= =
= =ty force
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Possible Cases of Simplification:-
Large Re
Re 1 negligible viscous effect,
1 NS reduces to Euler equations p
t ρ
⇒
∂+ = − +
∂
V V V g
i
Small StSt 1 negligible unsteady effect quasi-steady flow,
The local (temporal) acceleration term can be dropped.
⇒ ⇒
∴
Small Re
Re 1 negligible inertia effect (good news!)
Viscous force is significant, and is to be balanced by pressure gradient.For slow and viscous flow and negligible gravity, the flow is
called
⇒
Creeping Flow
21 0
Nonlinear inertia terms are now gone, analytical solutions are possible
if the flow geometry is simple enough.
p ν ρ
= − + V
Spatial DimensionAlso, it is often the case that the flow varies only in one or two spatial dimensions,
and therefore the problem can be reduced to a one- or two-dimensional problem, for
which only one or two velocity components need to be solved. Some common cases
of one-dimensional flow:
• fully developed pipe or channel flow: axial velocity as a function of radial
distance from center of pipe ( )u u r = , or longitudinal velocity as a function of
distance from the bottom of channel ( )u u y= ;
• axi-symmetrical flow: velocity is symmetrical about an axis (e.g., point
source/sink, vortex).
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(II) SIMPLE (EXACTLY OR NEARLY ONE-DIMENSIONAL)
VISCOUS FLOW (Section 6.9)
A. Mathematical Formulation for a Fluid Dynamics Problem
Assumptions:
• constant fluid properties (density ρ , viscosity μ )
• Newtonian fluid (linear, isotropic and purely viscous material)
Basic Variables:
( )
( )
Velocity ( , , ) , , , (3 variables)
Pressure , , , (1 variable)
u v w x y z t
p p x y z t
= =
=
V V
Basic Governing Equations:
2
Continuity 0 (1 equation)
1 Navier-Stokes (3 equations) p
t ν
ρ
∂+ = − + +
∂
V =
V V V g V
i
i
Other derived variables:
( )Stress ( , , , ) , isotropic tensor
with stress components (see the definition on page 11):
2 , (no
T
xx
x y z t p
u p
x
μ
τ μ
⎡ ⎤= − ∇ ∇ =⎣ ⎦
∂= − +
∂
τ I + V + V I
rmal stress)
2 , (normal stress)
(shear stress)
yy
xy yx
v p
y
u v
y x
τ μ
τ τ μ
∂= − +
∂
⎛ ⎞∂ ∂= = +⎜ ⎟
∂ ∂⎝ ⎠
etc.
Vorticity w v u w v ui j k y z z x x y
⎛ ⎞ ⎛ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞= × − + − + −⎜ ⎟ ⎜⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ζ V = ⎞⎟
⎠
Boundary Conditions:
• No-slip boundary condition: the velocity of a fluid in contact with a solid
impermeable wall must equal that of the wall
fluid solid along a fluid-solid interface=V V
If in particular the wall is stationary, the fluid adjacent to the wall must have zero
velocity.
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The development of velocity profiles due to the no-slip condition as a fluid flows past
a blunt nose and a flat plate.
• Interface boundary condition between two fluids: when fluid A and fluid B meets at
an interface, the velocity and stress must match between the two fluids at the interface
A B A B, along a fluid-fluid interface= =V V τ τ
If, say, the interface is flat (along x-direction) and the fluids are moving parallel to the
interface, the continuity of stress implies the continuity of pressure and shear stress at
the interface
A B A B
A B
,du du
p pdy dy
μ μ = =
• Free-surface boundary condition: a degenerate form of the above interface boundary
condition occurs at the free-surface of a liquid, meaning that fluid A is a liquid (say,
water, oil) and fluid B is a gas (usually air). By virtue of the factair liquid
μ μ , the
shear stress at the air-liquid interface is negligibly small, and it is reasonable to
approximate the shear stress to be at the interface, which is hence called a free surface,
liquid atmosphere liquid
liquid
, 0 along the free surfacedu p p
dyμ = =
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•
Other boundary conditions, such as inlet condition, outlet condition, periodic
condition and symmetry, may also apply to certain types of boundaries, depending on
the problem.
Boundary conditions along a plane of symmetry are defined so as to ensure that the
flow field on one side of the symmetry plane is a mirror image of that on the other
side, as shown above for a horizontal symmetry plane.
Initial Condition If the problem is time dependent (i.e., unsteady), an initial condition alsoneeds to be specified.
*************************************************************************
Let us consider in the following sections a few applications of the Navier-Stokes equations,
in which the flow configuration is simple enough for analytical solutions (exact or
approximate) to be deduced. The assumptions are that the flow is steady ( / 0t ∴ ∂ ∂ = ),
laminar, and incompressible and the fluid is Newtonian.
B. Plane Poiseuille-Couette Flow
Note that this is a unidirectional flow ( ), 0u u y v= = . Therefore there is no dependence
on x for all variables: ./ 0 x∂ ∂ =
The flow is driven by three forcings: (1) motion of the upper plate; (2) pressure gradient
in the x-direction, / a constant p x∂ ∂ = ; (3) gravity, if x is not in a horizontal direction.
Recall the momentum equations:
x
Upper plate moving at a constant speed U
Lower fixed plate
y
u( y)
y = h
y = 0
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-component:u
xt
∂
∂
uu
x
∂+
∂
uv
y
∂+
∂
2
2
1 p u
x xν
ρ
∂ ∂= − +
∂ ∂
2
2
2
2
1 ....................... (1)
-component:
x
x
ug
y
u pg
y x
v y
t
ν ρ
⎛ ⎞∂+ +⎜ ⎟⎜ ⎟∂⎝ ⎠
∂ ∂⇒ = −
∂ ∂
∂
∂
vu
x
∂+
∂
vv
y
∂+
∂
2
2
1 p v
y xν
ρ
∂ ∂= − +
∂ ∂
2
2
v
y
∂+
∂
1 0 ....................... (2)
y
y
g
pg
y ρ
⎛ ⎞+⎜ ⎟⎜ ⎟
⎝ ⎠
∂⇒ − + =
∂
Note that the inertia terms are identically zero, which is true for all unidirectional flows
irrespective of the Reynolds number.
Equation (2) simply gives that the pressure ( ) y p p x g y ρ = + .
The R.H.S. of equation (1) is constant, so the equation can be integrated twice with
respect to y, giving2
1 2( )2
x
p yu y g C y C
x ρ
μ
∂⎛ ⎞= − + +⎜ ⎟∂⎝ ⎠
where and are integration constants that can be determined using the boundary
conditions that1
C 2
C
( 0) 0 (no slip at the lower plate), and
( ) (speed of the upper plate).
u y
u y h U
= =
= =
Solving for these constants, we obtain the solution for the velocity profile (see Fig. 6.31
below):
22
( )2
Couette FlowPoiseuille Flow
x
p h y y yu y g U
x h h h ρ
μ
⎡ ⎤∂⎛ ⎞ ⎛ ⎞= − + − +⎢ ⎥⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
Couette flow is caused by the motion of a boundary wall moving in its own plane, while
Poiseuille flow is caused by axial pressure gradient or gravity in the direction of flow.
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The shear stress in the flow is
constant stresslinear stress distribution due to due to Couette flowPoiseuille flow,zero stress at the centerline
( )
2
xy x
du y p h U g y
dy x h
τ μ ρ μ ∂⎛ ⎞ ⎡ ⎤
= = − + − +⎜ ⎟ ⎢ ⎥∂⎝ ⎠ ⎣ ⎦
The discharge (flow-rate) per unit width of channel is given by
3
0 12 2
h
x
p h hU Q udy g
x ρ
μ
∂⎛ ⎞= = − + +⎜ ⎟
∂⎝ ⎠∫
The volume flow averaged (mean) velocity2
/12 2
x p h U u Q h g x
ρ μ
∂⎛ ⎞= = − + +⎜ ⎟∂⎝ ⎠
It is left as an exercise for you to show the following
Given that / p x−∂ ∂ is a positive constant and 0 xg = , determine the location of the
maximum velocity. It is also the point where the shear stress vanishes (why?).
Hence, find the minimum value of U such that the shear stress will not vanish
throughout the flow.
C.
Circular Poiseuille Flow
We now consider laminar flow through a circular tube:
•
The objective to find the relationship between volumetric flow rate and pressure
change along a pipe of circular section.
• Examples include blood flow in capillaries, air flow in lung alveoli, where the
Reynolds number is not high enough for the flow to become turbulent.
• Navier-Stokes equations in cylindrical coordinates are to be used, where
/ 0θ ∂ ∂ = , since the flow is axially-symmetric (i.e., no dependence on angular
position in a cross-section of the flow).
• We have seen that the gravity can be combined with the pressure gradient in a
trivial manner, so let us ignore gravity in the following analysis.
Again, this is a unidirectional flow: 0, 0r u u uθ
r
( ) zu r
Circular pipe of radius R
z
z= = ≠
is driven by a constant andsteady pressure gradient dp/dz in the axial direction.
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( )
The continuity equation reduces to
1
r ru
r r
∂
∂
1 u
r
θ
θ
∂+
∂0 must not depend on , ( )
The -component momentum equation is simplified to
z z z
z
uu z u
z
z
u
t
∂+ = ⇒ ∴ =
∂
∂
∂
zu r
zr
uu
r
∂+
∂ zu u
r
θ
θ
∂+
∂ z
z
uu
z
∂+
∂
2
2 2
1
1 1 z z
p
z
u ur
r r r r
ρ
ν θ
∂= −
∂
∂ ∂ ∂⎛ ⎞+ +⎜ ⎟
∂ ∂ ∂⎝ ⎠
2
2
zu
z
∂+
∂ zg
⎡ ⎤+⎢ ⎥
⎢ ⎥⎣ ⎦
2
1 2
, which can be integrated twice with respect to to give
( ) ln4
z
z
d du r dpr r
dr dr dz
r dpu r C r C
dz
μ
μ
⎛ ⎞⇒ =⎜ ⎟
⎝ ⎠
= + +
The two integration constants C and can be determined using the boundary
conditions:1 2C
1
2
2
( 0) is finite 0
( ) 0 (no slip at boundary wall)4
z
z
u r C
R dpu r R C
dzμ
= ⇒ =
= = ⇒ = −
Plugging back, we get the expression for the velocity profile
( )
2 2
214 z
dp R r
u r dz Rμ
⎡ ⎤
= − −⎢ ⎥⎣ ⎦ which is a parabolic distribution with the maximum at the center:
2
max ( 0)4
z
R dpu u r
dzμ = = = −
The flow-rate is4
02
8
R
z z A
R dpQ u dA u rdr
dz= = = −∫ ∫
π π
μ
The mean velocity is half the maximum velocity4
2max
2
8/ ..................... (1)
8 2
R dp
u R dpdzu Q A
R dz
π
μ π μ
−= = = − =
The shear stress at wall is given by
42
zw
r R
du R dp u
dr dz Rτ μ μ
=
= − = − =
For a given length L of the pipe, the pressure drop is ( )/ p dp dzΔ = − L
g
and the head loss
due to friction is h p / f ρ = Δ . Hence we may obtain from equation (1) the Darcy-
Weisbach equation
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D. Nearly One-Dimensional Flow by Lubrication Approximation
Lubrication in a Slider Bearing
A slider bearing is designed as a thrust bearing to support very large loads. To carry these
loads, the fluid film between the solid surfaces must develop normal stresses, so we are
interested in predicting the pressure distribution and thus the load-carrying capacity of the
bearing. Typical examples of slider bearings are found in the shafts of screw-propelledships and in the high-speed turbines of electricity-generating stations. For example, the
thrust of the ship’s propellers may be transmitted through a series of pads (see the figure
below) to the hull of the ship. Each pad (slider) may be tilted slightly to account for the
relative effects of pressure, speed and viscosity, and thus maintain the fluid film between
the two surfaces (the slider and its guide) which are in relative motion, and thereby reduce
friction.
In most lubrication problems the relevant Reynolds number is so small that viscous terms
in the Navier-Stokes equation dominate completely. The reason is not necessarily that the
coefficient of viscosity is large; it is more due to the fact that the thickness of the film is
extremely small compared to the lateral dimensions of the bearing. The Reynolds number
may be defined asslider speed film thickness
1 Re ρ
μ
× ×=
Let us now formulate a model of the slider bearing with a planar face, with the further
assumptions that
1.
The lubricant is an incompressible Newtonian viscous fluid with constant viscosity.
2. The bearing has infinite length into the paper, and the bearing guide is flat. The gap
height h( x) between the slider and its guide varies so gently that the flow is nearly
one-dimensional through a section of the bearing.
3.
Gravity can be ignored.4. The flow has settled down and we need consider only the steady problem.
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The local film thickness is
2 11
( ) h h
h x h x L
−⎛ ⎞= + ⎜ ⎟
⎝ ⎠ (1)
where so that the film is extremely thin.1 2h h L<
The flow is quasi-one-dimensional, and we may recall the equation for discharge for
combined Poiseuille-Couette flow (on page 31):
3
12 2
dp h hU Q
dx μ = − −
where the gravity has been ignored and U is now in the negative direction. By
conservation of mass, Q must be a constant. If there were Couette flow alone, the
discharge would decrease down the slider as the gap height decreases from to .
Therefore in order to balance the flow, the pressure gradient must not be zero.
Rearranging the terms the above equation gives
2h 1h
3
12
2
dp hU Q
dx h
μ ⎛ = − +⎜
⎝ ⎠
⎞⎟ (2)
We further suppose that both ends of the bearing are exposed to surrounding lubricant,or to the atmosphere. Then we have
0( ) (0) p L p p= =
as the boundary condition for the pressure. It follows that integrating (2) from 0 to L is
equal to zero
0
2 30
( ) (0) 0
02
L
L
dpdx p L p
dx
U Q
dxh h
= − =
⎛ ⎞⇒ − + =
⎜ ⎟⎝ ⎠
∫
∫
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Now, on substituting (1) for h( x), the above integral can be carried out to give
( )1 2
1 2
h hQ U
h h
−=
+
We may put Q back into (2), which is integrated again, but the upper limit is now a
general position “ x”
0 2 30 0 ( ) 12
2
x xdp U Qdx p x p dx
dx h hμ
⎛ ⎞= − = − +⎜ ⎟
⎝ ⎠∫ ∫
After some algebra, we get
( )(0 12 2 2
2 1
6( )
( )
UL)2 p x p h h h h
h h h
μ = − − −
−
Note that sinc h 0e ,1 ( )h h x≤ ≤2 ( ) p x p≥ or a positive pressure distribution is
established within the bearing fluid to support the normal load. It can be readily shown
that, by setting dp/dx = 0 in (2), the pressure reaches a maximum
( )( ) ( )
2 1 1max 0
1 2 1 2 1 2
3 at
2
UL h h h L p p x
h h h h h h
μ −− = =
+ +
It may be shown that in the left-hand section of the bearing, the pressure gradient is
positive so that it drives fluid in the flow direction, and in the right-hand section the
pressure gradient is negative so that it drives fluid against the flow direction. In this way,
the Poiseuille flow will balance the Couette flow to result in a constant dischargethroughout the pad.
Lubrication performance is found to be favorable, since by using orders of magnitude
we may estimate that
0
2
0
Drag shear stress /1
Bearing load pressure /
L
L
dx U h h
UL h L pdx
τ μ
μ
∫
∫∼ ∼ ∼ ∼
By virtue of the small film thickness, the bearing can support a large load with only
small frictional resistance.
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(III) INVISCID AND POTENTIAL FLOWS (Sections 6.4-6.6)
Analysis can be considerably simplified if the flow under consideration can be regarded as
INVISCID and IRROTATIONAL.
A.
Inviscid (Nonviscous) Flow (Section 6.4)• Flow of an ideal fluid with zero viscosity ( )0μ = would be inviscid exactly.
• In practice, flow is approximately inviscid when the effects of shear stresses on the
motion are small as compared to other influences. One guiding condition is that the
Reynolds number Re must be very large:
viscous force 11 , where Re
inertia force Re
VL ρ
μ =∼
• Many flows involving water or air, whose viscosity is small, can practically be
considered as inviscid as long as the viscous effects are not dominant (e.g., far from a
wall).• When the viscous force becomes negligible, the Navier-Stokes equations reduce to
Euler’s equations
nonlinear termis still here
1 (viscous term is missing here) p
t ρ
∂+ = − +
∂
V V V g
∇
i
• For incompressible flow, Euler’s equations of motion can be integrated along a
streamline to yield the Bernoulli equation (which you learnt already in Year I; read
Section 6.4.2 for a review)
2
constant along a streamline2
p z
g g ρ + + =V
• It is remarkable that the Bernoulli equation provides an algebraic (rather than vector
differential) relationship between pressure, velocity and position in the earth’s
gravitational field.
B. Irrotational (Potential) Flow (Section 6.4.3)
• Recall that vorticity (curl of velocity) is twice the rotation (angular velocity) of a fluid
element.
• A fluid element will acquire vorticity when acted upon by a couple to cause it to
rotate. One source of rotation is unbalanced shear stresses acting on its periphery.
When shear stresses are absent, it is possible that the flow is irrotational.
• A flow field is irrotational if, at every point, the vorticity vanishes or
0∇ ×V = .
• It can be shown that the flow of an inviscid fluid which is irrotational at a particular
instant of time remains irrotational for all subsequent times. That means, the motion
of an inviscid fluid which is started from rest is always irrotational (provided the flow
lies outside a boundary layer).
•
This result is known as the Persistence of Irrotational Motion of an inviscid fluid. It
is because the setting up of a rotation would require forces tangential to the boundary;
and such forces, which arise through the viscous properties of the fluid, are non-existent in the inviscid fluid model.
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•
The constant in the Bernoulli equation becomes universal (i.e., not specific to a
streamline) when the flow is irrotational (Section 6.4.4). Therefore, for
incompressible irrotational flow, the Bernoulli equation can be applied between any
two points in the flow field:
2 2
1 1 2 21 2
2 2 p V p V z z
g g g g ρ ρ + + = + +
•
The procedures of finding a solution for an irrotational flow field are typically:
o Firstly, solve for the kinematics (velocity components) from an equation
derived from the condition of zero vorticity, which is the subject matter of the
following sections;
o Secondly, find the pressure from the Bernoulli equation.
You should appreciate that solving irrotational flow equations is usually much simpler
than solving the full Navier-Stokes equations.
• You are cautioned that irrotationality fails to apply to a boundary layer, which is a
thin layer that develops next to a solid wall owing to no-slip of the flow at the wall. No matter how small its viscosity is, a real fluid cannot “slide” past a solid boundary.
The flow in a boundary layer is always viscous and highly rotational (a rapid change
in velocity from zero at wall to the free stream value over a short distance); real fluid
behavior must be accounted for in a boundary layer (Chapter 9).
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C. The Velocity Potential (Section 6.4.5)
• For any scalar field φ , curl(grad φ ) = 0 is an identity. See a proof below.
•
Alternatively speaking, a velocity field V is irrotational or curl V = 0 if and only if
there exists a scalar field φ such that V = grad φ .
• The scalar function is called velocity potential
Cartesian coordinates: , ,
1Cylindrical coordinates: , ,r z
u v w x y z
u u ur r
θ
φ
φ φ φ
φ φ
θ
≡ ∇
∂ ∂= = =
∂ ∂
∂ ∂= =
∂ ∂
V
∂
∂
z
φ ∂=
∂
Irrotational flow is therefore also called potential flow.
•
The velocity potential satisfies Laplace’s equation on substituting the above relation
into the continuity equation:
2 2 2
2 2 2
2 2
2 2 2
0 0, or 0
Cartesian coordinates: 0
1 1Cylindrical coordinates: 0
x y z
r r r r r z
φ φ
φ φ φ
φ φ φ
θ
= ⇒ = =
∂ ∂ ∂+ + =
∂ ∂ ∂
∂ ∂ ∂ ∂⎛ ⎞+ + =⎜ ⎟
∂ ∂ ∂ ∂⎝ ⎠
i iV
∇ ∇ ∇ ∇
• The immediate upshot is: for irrotational flow, one only needs to solve for a scalar
function (instead of a vector with 2 or 3 components) from one single equation in
order to determine the kinematics (good news!). However, the differential equation
for the scalar function is one order higher than that for the vector function (no free
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lunch!). Once the potential is found, its spatial gradients will give the velocity
components.
D. Equipotential Lines and Streamlines (Sections 6.2.3 and 6.5)
• A two-dimensional potential flow field can be graphically represented using a flow
net composed of equipotential lines and streamlines.
• Equipotential lines are (contour) lines of constant velocity potential, while streamlines
are lines in the flow field that are everywhere tangent to the velocity. It can be shown
that these two sets of lines are orthogonal (i.e., they intersect each other at right
angles).
• You may recall the following mathematical statement:
It follows that:
equipotential lines streamlines equipotential lines
φ ≡ ⊥⇒ ⊥
V
The grad of a scalar function, say φ , gives the maximum rate of
spatial change of the function, and is in a direction normal to the local
line along which the function is constant..
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Figure 6.15 shows a flow net for a 90o bend. A
flow net is useful in the visualization of a flow
pattern. To further understand what information a
flow net can provide, we need to knowsomething about stream function.
Stream Function
• For 2-D incompressible flow, another scalar function, viz stream function can be
introduced to identically satisfy the continuity equation.
( ) ( )A stream function , or , is defined such that
, for 2-D flow in Cartesian coordinates,
which satisfie
x y r
u v y x
ψ ψ θ
ψ ψ ∂ ∂= = −
∂ ∂
( )
s 0 identically.
1
, for 2-D flow in Polar coordinates,
which satisfies 0 identical
r
r
u v
x y
u u
r r uru
r
θ
θ
ψ ψ
θ
θ
∂ ∂+ =
∂ ∂
∂ ∂= = −
∂ ∂ ∂∂+ =
∂ ∂ly.
Note that the stream function is introduced based on kinematics consideration only. It
is definable for any two-dimensional incompressible flow fields, irrespective of the
flow being inviscid or not.
•
Physically, ψ is constant along a streamline since
V
ψ is constant
d dx dy vdx udy
x y
ψ ψ ψ
∂ ∂= + = − +
∂ ∂
That means, a line of constant ψ (along which 0d ψ = ) will have its slope in the same
direction of flow: . This is nothing but the defining property for a
streamline.
/ /dy dx v u=
Note that a solid boundary is always a
streamline. At a particular instant of time,
there is no fluid crossing any streamline,
and distinct streamlines cannot cross.
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•
Given any two points in space whose stream function values are known, then the
volume flow rate across any line joining these two points is equal to the difference in
values of their stream functions.
2
12 1
One can readiy see from the above figure that
dq udy vdx dy dx d y x
q d ψ
ψ
ψ ψ ψ
ψ ψ ψ
∂ ∂= − = + =
∂ ∂
∴ = = −∫
• If the 2-D flow is irrotational, the stream function also satisfies Laplace’s equation,
since
2 2
2 2
0 0
0
0
v u
x y
x x y y
x y
ψ ψ
ψ ψ
∂ ∂× = ⇒ − =
∂ ∂
⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞⇒ − − =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠
∂ ∂
⇒ + =∂ ∂
V
Therefore, for two-dimensional irrotational flow, both the velocity potential φ and the
stream function ψ satisfy Laplace’s equation. They are called harmonic functions,
and they are harmonic conjugates of each other. These functions are related, but
their origins are different:
– The stream function is defined by continuity; the Laplace equation for ψ results
from irrotationality.
– The velocity potential is defined by irrotationality; the Laplace equation for φ
results from continuity.
By now, referring back to Figure 6.15, you should understand that in a flow net the
velocity is roughly given by
V n s
φ ψ Δ Δ≈ ≈
Δ Δ
where is the spacing between two adjacent equipotential lines, and is the
spacing between two adjacent streamlines. Therefore, the velocity is higher in a
region where the mesh is finer, and lower where the mesh is coarser.
nΔ sΔ
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E. Some Simple Plane Potential Flows (Sections 6.5.1-6.5.4)
1) Uniform Flow with constant velocity U
For case (a) where the flow is purely in the x-direction:
( )
( )
Velocity potential cos
equipotential lines are parallel to the -axis
Stream function sin
streamlines are parallel to the -axis
Ux Ur
y
Uy Ur
x
φ θ
ψ θ
= =
∴
= =
∴
Can you write down the corresponding φ and ψ for case (b) where the flow is at an
angle α with the x-axis?
2)
Source and Sink
A 2-D source is a line (from a mathematical perspective)
that runs perpendicular to the plane of flow and injects
fluid equally in all directions. The figure shows the flow
field of a source at the origin, from which fluid particles
emerge and follow radial pathlines. The strength of a
source, denoted by m, is the volume rate of flow
emanating from unit length of the line.
/m V L=
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By conservation of mass, 2r m ruπ = for any radial distance r from the source located at the
origin. Hence,1
2
mu
r r
1, 0
r ur r r
θ
φ ψ φ ψ
θ
∂ ∂= = = = − =
∂ ∂θ π
∂ ∂=
∂ ∂. On integrating,
( )
( )
Velocity potential ln2
equipotential lines are concentric circles centered on the origin
Stream function2
streamlines are radial lines
mr
m
φ π
ψ θ π
=
∴
=
∴
The radial and tangential velocities are:
02r
mu ur
θ π = =
o when 0m > , the flow is radially outward, the origin is a SOURCE
o when 0m < , the flow is radially inward, the origin is a SINK
o the origin is a singularity where r u → ∞
o conservation of mass is satisfied everywhere except the origin
3) Vortex
In contrast to a source, a vortex has the pathlines
being circles centered on the origin, and fluid
particles move along these circles. The vortex can be used to model the flow round the plughole in a
bathtub. An irrotational vortex is called a free
vortex. The strength of a vortex is measured by the
circulation around a closed curve C
that encloses the center of the vortex. Hence,
C
d Γ = ∫V si
1 1
2r
r r r r r θ
0,u uφ ψ φ ψ Γ
θ θ π
∂ ∂ ∂ ∂= = = = = − =
∂ ∂ ∂ ∂.
On integrating,
( )
( )
Velocity potential2
equipotential lines are radial lines
Stream function ln2
streamlines are concentric circles centered on the origin
r
φ θ π
ψ π
Γ=
∴
Γ= −
∴
The radial and tangential velocities for a free vortex are
02
r u ur
θ π Γ= =
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o
the flow is not defined at the origin
o the vorticity curl V = 0, except at r = 0, where V is not defined
o free vortex (a) is irrotational flow, tangential velocity decreases radially 1u r θ
−∝
o forced vortex (b) is rotational flow, tangential velocity increases radially u r θ ∝
4)
DoubletConsider a combination of a source and a sink of equal strength m and separated at a
distance 2a (left figure):
If the source and sink are moved indefinitely closer together ( )0a → in such a way
that the product 2am (distance apart × strength) is kept finite and constant, then we
obtain a doublet. The streamline pattern for a doublet is shown in the right figure
above. The line joining the source to the sink is called the axis of the doublet, and is
taken to be positive in the direction from sink to source. The strength of the doubletis /K ma π = .
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examples, you will see how ideal flows can be described by a combination of basic
solutions. The key thing is to locate the dividing streamline. The general procedures are
as follows. (1) Sketch some streamlines for the combined flow. (2) Find the location of a
stagnation point where the velocity vanishes (you may expect that when flow past a body,
there is a point somewhere on the body surface where the flow velocity is zero). (3)
Evaluate the stream function at the stagnation point stagnationψ . (4) The dividing streamline,
which passes through the stagnation point, can be determined by letting the stream
function be equal to the stagnation stream function ( ),r ψ θ = stagnationψ .
1) Source + Uniform Flow = Flow Past a Half Body
It is more convenient to use polar coordinates ( ),r θ where r is the radial distance
from the source. It is along the negative x-axis where the flow due the uniform flow is
directly opposite to that due the source. At a point x b= − the velocities due to the two
flows cancel each other, and this is identified as the stagnation point.
( )uniform flow velocity radial outward flow due to source
2 2
r U um m
U bb U π π
r b∴ = =⇒ = ⇒ =
( ) uniform flow sourceCombined stream function ,
2
r
mUy
ψ θ ψ ψ
θ π
= +
= +
sin2
mUr θ θ
π = +
The stream function at the stagnation point has the value( )stagnation ,
2
mr b bU ψ ψ θ π π = = = = =
Therefore the dividing streamline that passes through the stagnation point is given by
( )
( )
( )
stagnation ,
or sin2 2
orsin
or
r
m mUr
br
y b
ψ θ ψ
θ θ π
π θ
θ
π θ
=
+ =
−=
= −
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This streamline, which has the shape shown above, can be considered as a solid
boundary of a half-body that extends from x b= − to x → +∞ . The flow exterior to
this streamline represents the flow past a half-body, whose thickness at large x can be
estimated to be 2 bπ , since
0 as
2
b y
b
π θ
π π
⎧ ⎧→ →⎨ ⎨
−⎩ ⎩
Note that the every fluid particle emanated from the source is completely enclosed
within the dividing streamline. The flow pattern around the half-body is described by
streamlines stagnationψ ψ > . The velocity components and the pressure can then be
determined as described in earlier sections.
2) Source + Sink + Uniform Flow = Flow Past a Rankine Oval
The source and the sink are of the same strength: any mass of fluid injected by the
source is eventually drawn into the sink. The dividing streamline is now a closed
curve. This finite body, called Rankine Oval, has two stagnation points, one at the
front end and the other at the rear end of its boundary.
3) Doublet + Uniform Flow = Flow Past a Circular Cylinder
As the source and the sink combine to become a doublet, the Rankine Oval becomes a
circular cylinder. As the flow past a circular cylinder is of fundamental interest, let us
examine the flow in some detail.
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The combined stream function is
sinsin
( strength of doublet)
K Ur
r
K
θ ψ θ = −
=
The radial velocity is2
1cosr
K u U
r r
ψ θ
θ
∂ ⎛ ⎞= = −⎜ ⎟∂ ⎝ ⎠
Obviously, the radial velocity vanishes at a circular surface with the radius
( )1/ 2
/r a K U = =
This defines the dividing streamline, which represents the surface of a circular
cylinder of radius a. Substituting a for K , the stream function can be written as2
2
1 sina
Ur r
ψ θ ⎛ ⎞
= −⎜ ⎟⎝ ⎠
from which we obtain the velocity components2
2
2
2
11 cos
1 sin
r
au U
r r
au U
r r θ
ψ θ
θ
ψ θ
⎛ ⎞∂= = −⎜ ⎟
∂ ⎝ ⎠
⎛ ⎞∂= − = − +⎜ ⎟
∂ ⎝ ⎠
On the cylinder surface the tangential velocity is,r a= 2 sinsu U θ θ = − . As expected,
there are 2 stagnation points, at 0,θ π = .
The pressure distribution on the cylinder surface can be found from the Bernoulli
equation
( )
2 2
0
2 2
0
1 1
2 2
1 1 4sin
2
s s
s
p U p u
p p U
θ ρ ρ
ρ θ
+ = +
⇒ − = −
where 0 p is the far upstream pressure. It is
remarkable that the pressure distribution is
symmetrical about the horizontal and thevertical diameters. Therefore there is no net
force arising from the pressure distribution
around the cylinder in both streamwise and
lateral directions. In other words, both drag
and lift forces are exactly zero, as predicted
from the potential flow theory.
This zero drag prediction is contrary to what has been observed in reality. There is
always a significant drag developed on a cylinder when it is placed in a stream of
moving fluid. This discrepancy is called d’Alembert’s Paradox, which was not
explained until the concepts of boundary layer and flow separation were developed.A comparison between the inviscid and the real pressure distributions is shown above.
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4) Free Vortex + Doublet + Uniform Flow = Flow Past a Rotating Circular Cylinder
The effect of adding a vortex is to upset
the symmetry of flow about the
horizontal diameter. Therefore, the
pressure in the upper half of the cylinderis not balanced by the pressure in the
lower half. This results in a net lift
force acting laterally on the cylinder.
5) Sink + Free Vortex = Spiral Flow
6) Two separated sources of equal strength = source flow with a neighboring wall
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(IV) FLOW PAST A BODY AND BOUNDARY LAYER THEORY
(Chapter 9)
A. Introduction (Section 9.1.2)
In 1904, Prandtl developed the concept of the boundary layer, which provides an
important link between ideal-fluid flow (inviscid irrotational flow) and real-fluid flow(viscous rotational flow). It was accepted that for fluids with relatively small viscosity (or
more exactly, flow with a high Reynolds number), the effect of internal friction in the
fluid is appreciable only in a narrow region surrounding the fluid boundaries. Therefore
the flow sufficiently far away from the solid boundaries may be considered as ideal flow
(in which effects of viscosity are neglected). However, flow near the boundaries suffers
retardation by the boundary shear forces and at the boundaries the velocity is zero (no-
slip condition). A steep velocity gradient is therefore resulted in a thin layer adjacent to
the boundaries, which is known as the boundary layer. It is of great significance when
behavior of real fluid is considered. For example, it explains the d’Alembert’s paradox –
the drag force experienced by a cylinder in stream that cannot be predicted with a
potential theory.
Flow of a uniform stream parallel to
a flat plate. The larger the Reynolds
number, the thinner the boundary
layer along the plate at a given x-
location.
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Flow past a circular cylinder; the boundary layer separates from the surface of the body in
the wake for large Reynolds number.
B. Description of the Boundary Layer (Section 9.2.1)
(1) Development of the Boundary Layer
nominal limit
of boundarylayer
u = 0.99 U y
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U
• On-coming flow is irrotational and has a uniform velocity U .
• The boundary layer starts out as a laminar boundary layer, in which fluid particles
move in smooth layers and the velocity distribution is approximately parabolic. As the
flow moves on, the continual action of shear stress tends to slow down additional
fluid particles, causing the boundary layer thickness to increase with distance
downstream from the leading edge. See below for a definition of the boundary layer
thickness.
• The flow within the boundary layer is subject to wall shear, and dominated by viscous
forces. The velocity gradient (hence the rotation of fluid particles) is the largest at the
wall, and decreases with distance away from the wall, and tends to zero on matching
with the main stream flow. Roughly speaking, the flow is said to be rotational within
the boundary layer, but is irrotational outside the boundary layer.
•
As the thickness of laminar boundary layer increases, it becomes unstable and some
eddying commences. These changes take place over a short length known as thetransition zone.
• It finally transforms into a turbulent boundary layer, in which particles move in
haphazard paths. Due to the turbulent mixing, the velocity distribution is much more
uniform than that in the laminar boundary layer. The increase of thickness along the
plate continues indefinitely but with a diminishing rate. If the plate is smooth (i.e.,
negligible roughness size), laminar flow persists in a very thin film called the viscous
sub-layer in immediate contact with the plate and it is in this sub-layer that the greater
part of the velocity change occurs.
Comparison of laminar andturbulent flat plate
boundary layer profiles
(left: non-dimensionalized
by the boundary layer
thickness; right: in physical
variables).
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(2) Thicknesses of the Boundary Layer
i) Boundary Layer Thickness δ
The velocity within the boundary layer increases to the velocity of the main stream
asymptotically. It is conventional to define the boundary layer thickness δ as the distance
from the boundary at which the velocity is 99% of the main stream velocity.
There are other ‘thicknesses’, precisely defined by mathematical expressions, which are
measures of the effect of the boundary layer on the flow.
ii) Displacement Thickness δ∗
It is defined by
*
01
udy
U δ
∞ = −
∫
δ∗ is the distance by which the boundary surface would have to be shifted outward if the
fluid were frictionless and carried at the same mass flowrate as the actual viscous flow. It
also represents the outward displacement of the streamlines caused by the viscous effects
on the plate. Conceptually one may ‘add’ this displacement thickness to the actual wall
and treat the flow over the ‘thickened’ body as an inviscid flow.
iii) Momentum Thickness θ It is defined by
01
u udy
U U θ
∞ = −
∫
θ is the thickness of a layer of the main stream whose flux of momentum equals the
deficiency in the boundary layer, equivalent to the loss of momentum flux per unit width
divided by 2U ρ due to the presence of the growing boundary layer. The momentum
thickness is often used when determining the drag on an object.
Note that when evaluating the above integrals for *δ and θ, the upper integration limit
can practically be replaced by δ.
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C. Laminar Boundary Layer Over a Flat Plate
• Heuristic Analysis
x
δ (x)
U
leading edge
of plate
Consider steady flow past a flat plate at zero incidence. The effect of viscosity is to
diffuse momentum normal to the plate. Consider a fluid element that is close enough
to the wall to be influenced by viscosity. In travelling a distance x, it has been
influenced by viscosity for a time . The influence of viscosity will have
spread laterally to a distance
/t x U ∼
( )
( )
1/ 21/ 2
1/ 21/ 2
or Re x
xt
U
x Ux
ν δ ν
δ ν −
≡
∼ ∼
∼
The above analysis is rather crude, and does not yield a full equation for the growth of
the boundary layer thickness. It however correctly describes one important
relationship for the laminar boundary layer: where( )1/ 2
/ Re x
xδ −
∝ Re / x
Ux ν ≡
x
is
the local Reynolds number in terms of the distance from the leading edge x. This
relationship is found to be valid at a distance far behind the leading edge: / 1.δ
The heuristic analysis can be further carried on to find relations for the wall stress:1/ 2
3
0
or ww
y
u U
y x
τ ν τ ρν ν
ρ δ =
∂=
∂ ∼ ∼
U
The wall shear stress wτ decreases with increase of x until the boundary layer turns
turbulent. The local friction coefficient, which is defined as follows, is given by
( )1/ 2
1 2
2
2 Rew f xC
U
τ
ρ
−≡ ∼
While the numerical factor of 2 is far from the true value, the functional dependence
of C f on Re x is correctly predicted.
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•
Exact Solution by Blasius (Section 9.2.2)
A more rigorous analysis, using the technique of similarity solution, was developed
by Blasius for the laminar boundary layer over a flat plate. While the details of the
analysis are beyond the scope of this course, it is important to note the following
results derived from Blasius’ solution.
( )
( )
( )
( )
1/ 2
1 12
2
1 12
2
5 boundary layer thickness
Re
0.664friction coefficient
Re
1.328drag coefficient
Re
x
w f
x
D
L
x
x
C U
C U L
δ
τ
ρ
ρ
=
≡ =
≡ =D
/ 2
/ 2
1
whereD
is the skin friction drag force on unit width of a plate of length L:
.0
L
wdxτ = ∫D
D. The Boundary Layer Momentum-Integral Equation (Section 9.2.3)
By virtue of the property that the boundary layer thickness δ is much smaller than the
streamwise length scale (say, L): / Lδ , one may simplify the Navier-Stokes equations
to obtain the boundary-layer approximation:
2
2
continuity 0
1-momentum
1-momentum 0
u v
x y
u u p x u v
u
x y x
p y
y
ν ρ
ρ
∂ ∂
+ = ∂ ∂
y
∂ ∂ ∂+ = − +
∂
∂ ∂ ∂ ∂
= −∂
∂
with the boundary conditions:
( , ) (0,0) at 0
, as
(where , are the velocity and pressure
of the inviscid flow just outside the boundary layer)
u v y
u U p P y
U P
= =
= = → ∞
From the y-momentum equation, it is clear that the pressure in the boundary layer is
constant laterally across the layer and equal to the near-wall pressure of the inviscid flow
outside the boundary layer.
On integrating the x-momentum equation with respect to y from y = 0 to y = δ, and after
some algebra including the use of the continuity equation, one may obtain the Karman
momentum integral equation
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( )2 *
where wall shear stress
density near-wall velocity of the outer inviscid flow
moment
w
w
d dU U U
dx dx
U
τ ρ θ ρ δ
τ
ρ
θ
= +
=
==
=0
*
0
um thickness 1
displacement thickness 1
boundary layer thickness
u udy
U U
udy
U
δ
δ
δ
δ
= −
= =
=
∫
∫ −
0
This momentum integral equation is applicable to laminar, transitional or turbulent boundary layer.
In particular, in the absence of pressure gradient (e.g., flow over a flat plate), the free stream velocity
U = constant and dU , and therefore the momentum integral equation reduces to/ dx =
2
w
d U
dx
θ τ ρ =
by which the skin friction drag and drag coefficient are simply given by
22 2
1 1 20 0
2 2
2, 2
L L L L
w L DU L
U d dx U dx U C
dx U L L ρ
ρ θ θ θ τ ρ ρ θ
ρ = = = = = =∫ ∫
DD
where Lθ is the momentum thickness at L= .
(1)
Laminar Boundary Layer Over a Flat Plate Revisited – approximate solution by
momentum integral equation
It is remarkable that approximate solutions, which are reasonably close to the exact
ones, can be obtained for the boundary layer thickness and drag coefficients from the
momentum integral equation on adopting an assumed velocity profile
( )
( )
where is the -coordinate normalized with respect to the local boundary
layer thickness.
u f
U
y y
x
η
η δ
=
=
The steps are as follows:-
a) Find the relation between θ and δ
( ) (1
0 0
1 1 , is a numerical constantu u
dy f f d a a
U U a
δ
θ δ η δ
= − = − =
∫ ∫
)
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b) Find the wall shear stress from Newton’s law of viscosity
( )
0 0
0
/
/
is another numerical constant
w
y y
b
du U u U
dy y
U df U
b bd η
τ ρν ρν δ δ
ρν
ρν δ η δ
= =
=
∂= =
∂
= =
c) Substitute θ and wτ into the momentum integral equation
2
2
1
2
U d b a U
dx
b d d
aU dx dx
ρν δ ρ
δ
ν δ δ δ
=
⇒ = =
Integrating the above equation with respect to x, assuming that δ = 0 at x = 0:
( )
2
1/ 2
2
2 /
Re x
b
aU
b a
x
ν δ
δ
=
⇒ =
Furthermore,
( )
( )
1 12
2
1 12
2
2friction coefficient
Re
2 2drag coefficientRe
w f
x
D
L
abC
U
abC U L
τ
ρ
ρ
≡ =
≡ =D
/ 2
/ 2
)
It turns out that the values of a and b are rather insensitive to the choice of the
approximate velocity profile u U / ( f η = as long as it is a reasonable one
satisfying the boundary conditions.
Some assumed velocity profiles are
( )
2
3
parabolic
cubic2 2
sine2
η η
η
πη
−
2
3
sin
f η η
= −
which satisfy
( )
( )
( )
0 0, (no-slip at 0)
1 1, ( at )
' 1 0 (no stress at )
f y
f u U y
f y
δ
δ
= =
= = =
= =
.
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(2) Turbulent Boundary Layer Over a Flat Plate (Section 9.2.5)
The one-seven-power law, suggested by Prandtl, is used for the velocity profile in the
turbulent boundary layer with zero pressure gradient:
( )1/ 7
1/ 7 oru y
f U
η η δ
= =
by which the momentum thickness is
( ) ( )1 1
1/ 7 1/ 7
0 0
71 1
72 f f d d θ δ η δ η η η = − = − =∫ ∫ δ
The one-seven-power law fails to describe the velocity profile at , where
. The following empirical formula obtained for pipe flow can be adoptedhere:
0 y =
/u y∂ ∂ → ∞
1/ 4
20.0225w U
U
ν τ ρ
δ
=
Substituting θ and wτ into the momentum integral equation, and integrating with
respect to x:
1/ 4
5/ 44 72 0.0225constant
5 7 x
U
ν δ
× = +
It is assumed that the turbulent boundary layer starts from 0. x = (This is a
contradiction to the fact that the boundary layer starts out as a laminar one, but this
assumption has given good results.) Therefore, the constant = 0. Further simplificationyields
( ) ( ) ( )1/5 1/5 1/52
0.370 0.0288 0.072, ,
Re Re Re
w D
x x
C x U
L
τ δ
ρ = = = .
These results are valid for smooth flat plates with 5 1 .5 7
0 Re 10 L× < <
Note that for the turbulent boundary layer flow the boundary layer thickness increases
with x as4 /5
xδ ∼ and the shear stress decreases as1/ 5
wτ −
∼ . For laminar flow thesedependencies are 1/ 2 and 1/ 2 x− , respectively.
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E. Effect of Pressure Gradient (Section 9.2.6)
The pressure in the streamwise direction (i.e., along the body surface) will not be constant
if the body is not a flat plate. Consequently, the free stream velocity at the edge of the boundary layer U is also not a constant but a function of x. Whether the free-stream flow
is accelerating or decelerating along the body surface will have dramatically different
effects on the development of the boundary layer.
Let us re-examine flow past a circular cylinder, and find out what causes d’Alembert’s paradox.
You may recall that inviscid flow past a circular cylinder has a symmetrical pressuredistribution around the surface of the cylinder about the vertical axis. This results in a
zero pressure drag, which is however not true in reality for any fluid with a finiteviscosity. Such discrepancy is now referred to as d’Alembert’s paradox.
Despite the discrepancy, the potential theory helps to reveal that the pressure and hence
the free-stream velocity U fs on the cylinder’s surface are not constant. From A to C , the pressure gradient is negative and the flow is accelerating, and from C to F , the opposite istrue.
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The real fluid flow past a circular cylinder is like this:
Flow Past A-B-C
the streamlines are converging, i.e., flow is accelerating, and the free-stream velocityU reaches a maximum at C .
•
• the pressure is decreasing along the cylinder surface, i.e., / 0 p x∂ ∂ < , net pressure
force is in forward direction, and the pressure gradient is said to be ‘favorable’.
the accelerating flow tends to offset the ‘slowing down’ effect of the boundary on thefluid. Therefore, the rate of boundary layer thickening decreases and flow remains
stable.
•
Flow Past C-D
the streamlines are diverging, and the flow is retarding.•
• the pressure is increasing along the cylinder surface, i.e., / 0 p x∂ ∂ > , net pressure
force opposes the flow, and the pressure gradient is said to be ‘adverse’ or
‘unfavorable’.
•
y
it reduces the energy and forward momentum of the fluid particles in proximity to thesurface, causing the thickness to increase sharply and fluid near the surface be brought
to a standstill ( /u∂ ∂ at the surface is zero) at D. See figure (b).
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Flow Past D-E-F
flow close to the cylinder surface starts to reverse at D (separate point), i.e., fluid nolonger to follow the contour of the surface. The phenomenon is termed separation.
•
•
•
large irregular eddies formed in the reverse flow (the wake), in which much energy islost to heat.
the pressure in the wake remains approximately the same as at the separation point D,and is therefore lower than that predicted by the inviscid theory (see figure c). Thislowering of pressure behind the cylinder resulting from flow separation leads to a net
pressure drag on the cylinder. This explain d’Alembert’s paradox. Note that the wider
the wake, the larger the pressure drag, and vice versa.
Influence of the pressure gradient
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F. Drag (Section 9.3)
Any object moving through a fluid (or a stationary object immersed in a viscous flow)
will experience a drag, – a net force in the direction of flow due to the pressure andshear forces on the surface of the object.
D
Drag = Pressure Drag + Skin Friction Drag
where
Pressure Drag = resultant force arising from the non-uniform and asymmetrical pressure
distribution around the surface the body. It is also called form drag as itdepends on the form or the shape of the body.
Skin Friction Drag = resultant force due to fluid shear stress on the surface of the object.
cos w p dA sin dAθ τ θ = +∫ ∫D
The drag coefficient C D is given by the ratio of the total drag force to the dynamic force
1 2
2
DC U A ρ
= D
where U = relative velocity of fluid far upstream of the object, A = frontal area – the projected area of the object when viewed from a direction
parallel to the oncoming flow if it is a blunt (or bluff) object (e.g., a cylinder);
or the planform area – the projected area of the object when viewed from
above it if it is a streamlined object (e.g., a flat plate).
drag
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Typically the drag coefficient depends on(i) the shape of the object,
(ii) orientation of the object with the flow (e.g., a flat plate normal to flow has adifferent C D than a flat plate parallel to flow),
(iii) the Reynolds number Re /UD ν = where D is a characteristic dimension of the
object,(iv) surface roughness if the drag is dominated by skin friction and the boundary
layer is turbulent.
Flow Past a Flat Plate
When a flat plate is held normal to flow, the flow is
separated upon past over the plate. A region of
eddying motion (wake) is formed at the rear of the plate, the pressure there being much reduced.
Therefore the pressure drag is dominant, and the
plate is a bluff body in this position. The drag showslittle dependence on the Reynolds number.
When a flat plate is held parallel to flow, formation of the boundary layer over the plate isappreciable and flow separation is
negligible. Therefore the skin frictiondrag is significant. The plate is a
streamlined body in this position.
The drag coefficient increases when the boundary layer becomes turbulent.
Flow Past a Circular Cylinder/Sphere
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• Re ≤ 1 – creeping flow
– no flow separation
– DC decreases with increasing Re ( 64/ Re D
C = for a sphere)
(Note that a decrease in the drag coefficient with Re does not necessarily imply acorresponding decrease in drag. The drag force is proportional to the square of the
velocity, and the increase in velocity at higher Re will usually more than offset thedecrease in the drag coefficient.)
• Re = 10 – separation starts occurring on the rear of the body forming a pair of vortex bubbles
there – vortex shedding begins at Re ≅ 90, leading to an oscillating Karman vortex street
wake (see next page)
– region of separation increases with increasing Re
– DC continues to decrease with increasing Re until Re = 10
3, at which pressure
drag dominates
•
3 5
10 Re 10< <
– DC remains relatively constant, which is a characteristic behavior of blunt bodies
– flow in the boundary layer is laminar, but the flow in the separated region is
highly turbulent, thereby a wide turbulent wake
•
5 610 Re 10< <
– a sudden drop in DC somewhere within this range of Re
– this large reduction in DC is due to the flow in the boundary layer becoming
turbulent, which moves the separation point further on the rear of the body,
reducing the size of the wake and hence the magnitude of the pressure drag. Thisis in sharp contrast to streamlined bodies, which experience an increase in the drag
coefficient (mostly due to skin friction drag) when the boundary layer turnsturbulent.
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Laminar boundary layer separation with
a turbulent wake for flow past a circular
cylinder at Re = 2000.
(a) (b)
low over (a) a smooth sphere at Re = 15,000, and (b) a sphere at Re = 30,000 with a tripire; the delay of boundary layer separation is clearly seen by comparing these two
hotographs. The delay of separation in turbulent flow is caused by the rapid fluctuations
f the fluid in the transverse direction, which enables the turbulent boundary layer toavel farther along the surface before separation occurs, resulting in a narrower wake andsmaller pressure drag. Recall also that turbulent flow has a fuller velocity profile as
ompared to the laminar case, and thus it requires a stronger adverse pressure gradient to
vercome the additional momentum close to the wall.
Fw
p
otr a
c
o
Karman Vortex StreetsT
91
he Karman vortex street is one of the best-known vortex patterns in fluid mechanics.
al to
res.
eriodic flow is that the forces on theecause the flow
ually
The vortex street is just a special type of unsteady separation over bluff bodies such as a
cylinder. The vortex street is highly periodic having a frequency which is proportionU/D, where D is the length of the bluff body measured transverse to the flow and U is the
incoming flow speed. This periodicity is responsible for the "singing" of telephone wi
In fact, vortex streets are almost always involved when the wind generates a fairly puretone as it blows over obstacles.
A practical consequence of the regular,
p body are also periodic. B
is asymmetric fore and aft as well as inthe direction transverse to the flow, the
body will experience both an oscillating
drag and lift. If the frequency of theshedding is close to a structural
frequency, resonance can occur, uswith unpleasant results.
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(V) OPEN-CHANNEL FLOW (Chapter 10)
A. Introduction
•
Studies of open channel flow are important for design and planning of river control,
inland navigation, surface drainage, irrigation, water supply and urban sanitation.
•
An open channel is a conduit in which the liquid flows with a free surface subjected toatmospheric pressure. It includes river channels (natural water courses), constructed
channels (e.g., canals, flumes) and enclosed conduits (e.g., sewers, culverts) operating
partially full.
• Open channels normally have a very small slope 0.01S < , and the pressure variation
with depth is nearly hydrostatic. The hydraulic grade lines for all the streamtubes are
the same and coincide with the free surface of the flow. Open channel flow is
essentially caused by slope of the channel and self-weight of the liquid.
B.
Types of Flow
1. Uniform and Non-Uniform FlowUniform flow – mean velocity does not change (in both magnitude and direction) from
one section to another along the channel. With a free surface this implies
a constant cross section and flow depth, which is called the normal
depth. Hence the liquid surface is parallel to the base of the channel.
Uniform flow results from an exact balance between the gravity and
frictional effects. (NB: uniform channel flow however does not require
uniformity of velocity across any one section of the flow.)
Non-uniform (varied) flow – the mean velocity V and the fluid depth y change with
distance x along the channel. Non-uniform channel flow can be classified
into gradually, or rapidly varying flow when the flow depth changesslowly ( )/dy dx1 , or rapidly ( )/dy dx ∼1 with distance along the channel.
2. Steady and Unsteady Flow
The flow is unsteady or steady depending on whether or not the depth at a given location
changes with time. Surface waves propagating on a channel is in fact unsteady, but may
appear steady to an observer who travels at the same speed as the waves.
3. Laminar and Turbulent Flow
The flow is laminar or turbulent depending on the magnitude of the Reynolds number,which for open channels can be defined as Re /hVR ν ≡ , where
h R is the hydraulic radius
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equal to the ratio of the cross section area to the wetted perimeter. Open-channel flows in
rivers, culverts, surface drainage, etc typically involve water as the fluid (with fairly small
viscosity) and have relatively large dimensions so that the flows are invariably turbulent.
For turbulent channel flow, the velocity profile is rather uniform (except close to the
bottom and lateral boundaries) across a section of the flow, and therefore it is a common
practice to use the section-averaged velocity V = V ( x) as the primary variable in openchannel flow.
4. Tranquil (Sub-critical) and Rapid (Super-critical) Flow
Using the momentum principle, one may show that a small-amplitude surface wave (which
is, say, caused by some small disturbance to the flow) will travel in a shallow pool of
liquid at the speed c g= y , where y is the local fluid depth. An open channel flow may
have sharply different behaviors when its flow velocity V is smaller or larger than this
wave speed. The ratio of these two velocities is known as the Froude number, the
magnitude of which corresponds to the following types of flow:
1 s⎧ ub-critical flow
Fr 1 critical flow
1 super-critical flow
V
gy
<⎪
≡ =⎨⎪>⎩
Briefly speaking, a sub-critical flow is so low in speed (thereby called tranquil) that a
disturbance to the flow may send waves both upstream and downstream of the channel. In
sharp contrast, a super-critical flow is a high speed flow (thereby called rapid) and a wave
cannot be transmitted upstream.
C.
Energy Considerations
For channels under consideration, the bottom slope is very small ( 0.001). Streamlines are
therefore virtually straight and parallel, and pressure variation is hydrostatic. This implies
∼
Free surface = hydraulic grade line
Datum
Streamline
Energy line
Channel bed
Lh
2 / 2V g
/ p g ρ
z
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D. Uniform Flow: Chezy-Manning Equation
Define:
Normal depth n y = depth of uniform flow
Hydraulic radius h R =
flow sectional area
wetted perimeter
A
P=
There is no change in momentum along the channel when the flow is uniform, and therefore
the net force acting on a control volume, as shown above, must be zero. By uniformity, the
two end forces are equal . Consequently, the downward gravity force is exactly balanced by the bottom friction:
1F F = 2
( ) 0
0
sinw
w h
Pl W gAl S
gR S
τ θ ρ
τ ρ
= =
⇒ =
By analogy with pipe flow, the wall stressw
τ can be expressed as
2
8w
f V τ ρ =
where f is the friction factor, which for complete turbulent flow depends only on the
roughness of channel. Combining these equations, we get the so-called Chezy equation for
uniform flow0h
V C R S =
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E. Specific Energy: Alternative Depth of Flow
Let us recall the specific energy that has been introduced in Section C. The specific energy,
E , is the sum of the pressure and velocity heads2
2
V
E y g= + .
Specific energy is also the energy head referred to the base of the channel. It is a very
important concept in open-channel flow. While the specific energy is constant along the
channel when the flow is uniform, it may increase or decrease down the channel when the
flow is non-uniform.
For the convenience of discussion, let us consider only rectangular cross section from here
onward. The results can be extended to an arbitrary cross section, but will not be considered
here.
1. Variation of Specific Energy with Flow DepthSince the velocity is related to the unit width discharge by /V q y= , we may write the specific
energy as
2
22
q E y
gy= +
Here the equation consists of three variables: E , y, q. It would be of interest to examine the
cases: i) q is constant, and E varies with y; and ii) E is constant, and q varies with y.
If q is kept constant, E varies with y in the following manner.
This is known as the specific energy diagram, in which the following points are notable.
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• For 0, , (asymptotic to the axis)q
y V E E y
→ = → ∞ ∴ → ∞ .
• For , 0, (asymptotic to the line )q
y V E h y E = . y
→ ∞ = → ∴ →
• Between these two extremes, E declines to a minimum min E at the critical point:
o
depth and velocity at this point are called the critical depth c y and critical
velocityc
V ;
o critical depth represents the least possible specific energy with which the fixed
discharge q is able to flow in the channel of given slope.
• For each other value of E greater than the minimum, there are two possible values of y,
one greater and one less than c y . The two corresponding depths for a given value
min E E > are known as the alternative depths.
• Flow can be classified into
, flow is tranquil or sub-critical
, flow is critical, flow is rapid or super-critical
c c
c c
c c
y y V V
y y V V y y V V
> <
= =< >
2. Criterion for Minimum Specific Energy
Since the minimum E occurs when d E /d y = 0, we may readily obtain, using the relation for E
given above, the following condition for critical flow
2
3
2 22
3
1 0
Fr 1
dE q
dy gy
q V
gy gy
= − =
⇒ ≡ = =
Being consistent with our earlier discussion in Section B.4, the flow is critical, sub-critical or
super-critical depending on the value of the Froude number. Combining all of the above,
min
min
min
1 , sub-critical flow,
Fr 1 , critical flow,
1 , super-critical flow,
c c
c c
c c
y y V V E E V
y y V V E E gy
y y V V E E
< > < >⎧⎪
≡ = = = =⎨⎪> < > >⎩
Also, the critical flow depthc y is given by
1/ 32 2
2
3Fr 1
c
c
q q y
gy g⎛ ⎞≡ = ⇒ = ⎜ ⎟⎝ ⎠
,
and hence the minimum specific energy is
2
min 2
3
2 2c c
c
q E y
gy= + = y .
It is remarkable that, for a given discharge, the critical flow occurs when the specific energy
is the lowest possible value .min E
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F. Gradually Varied Flow
1. Introduction
• Uniform flow, in which a uniform flow depth is maintained, requires constancy of all
channel characteristics (i.e., cross-sectional shape, bed slope, roughness) throughout the
flow. In natural streams, this condition is hardly attained and flow is invariably non-uniform in nature. Even for artificial channels with uniform cross section, etc., uniform
flow is only a condition to be approached asymptotically. The surface of a varied (i.e.,
non-uniform) flow is not parallel to the bed and takes the form of a curve.
• Steady varied flow is broadly divided into two kinds:-
o Gradually varied flow, in which changes of depth and velocity take place over a
long distance and degree of non-uniformity is very slight. Boundary friction is
significant and is to be accounted for.
o Rapidly varied flow, in which the sectional area of flow changes abruptly within a
short distance. Turbulent eddying loss is more important than boundary friction in
this case. Hydraulic jump, which is a typical example of rapidly varied flow, will
be examined in the next section.
• Gradually varied flow may result from
o a change in cross-sectional shape, bed slope, boundary roughness of the channel.
o the installation of control structures (e.g., sluice gate, weirs, etc.).
2. General Equation for Gradually Varied Flow
When the flow is non-uniform, the specific energy is no longer a constant, but varies with
distance, x, along the channel. It has been derived in Section C that
( ) ( ) ( )0 f E x x E x S S + Δ − = − Δ x
where and0S f S are respectively the slopes of the channel bed and the energy grade line.
This relation implies that the specific energy gradient is equal to the difference between these
two slopes.
0 f
dE S S
dx= −
On the other hand, by the definition of the specific energy,
( )2 2 2
2
2 31 1 F
2 2
dE d V d q q dy dy y y
dx dx g dx gy gy dx dx
⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + = + = − = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠r
Combining the two equations above, one may get the general equation governing the free
surface gradient in a gradually varied flow:
.0
21 Fr
f S S −dy
dx=
−
Note that f S and Fr are functions of x, and the geometrical bed slope S may or may not
change with x depending on construction. The equation above may be integrated numerically
to yield the surface profile for any type of gradually varied flow. This kind of profile
evaluation is normally carried out by a commercial package nowadays and will not be
discussed here.
0
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3. Profile Classification
The general surface profile equation can be utilized in establishing the various forms of
varied flow profile. The equation may be rearranged into
0
0 2
1 /
1 Fr
f S S dyS
dx
−⎛ ⎞= ⎜ ⎟
−⎝ ⎠
.
Let us consider a very wide rectangular channel. As noted earlier in Section D, when the flow
is uniform,
2
0 5/3 f
n
qnS S
y
⎛ ⎞= = ⎜ ⎟
⎝ ⎠
wheren y is the normal depth.
An assumption is made here that for gradually varied flow the energy gradient S can be
related to the local flow depth y in the same manner as in the above formula for uniform
flow. Therefore, when the flow is gradually varied
( )2 2
10/3
0 05/3 5/3, / / f f
n
qn qnS S S S y
y y
⎛ ⎞ ⎛ ⎞= = ⇒ =⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠ n y .
On the other hand, the Froude number can be written as
32 22
3 3 3Fr 1c
c
yq q
gy y gy
⎛ ⎞= = =⎜ ⎟
⎝ ⎠∵ .
Putting these relations back into the surface profile equation
( )
( )
10/3
0 3
1 /
1 /
n
c
y ydyS
dx y y
−=
−.
With the equation above, we may outline the various forms of varied flow profiles, with a
classification based on the following features.
Type of Bed Slope
Mild Slope (M): 0 0 ,c nS S y y< > c
c
c
∞
∞
Steep Slope (S): 0 0 ,c nS S y y> <
Critical Slope (C): 0 0 ,c nS S y y= =Horizontal Slope (H): 0 0, nS y= =
Adverse Slope (A): 0 0, nS y< =
( ) y Relative to Normal Depth ( )n y and Critical Depth ( )c y Flow Depth
Type 1: , andn c y y y y> > 0 Backwater curvedy
dx> ⇒
Type 2: is between andn c y y y 0 Dropdown curvedy
dx< ⇒
Type 3: , andn c y y y y< < 0 Backwater curvedy
dx > ⇒
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Hence, a flow occurring on a mild slope has an M1 curve when its depth is greater than the
normal depth, and has an M3 curve when its depth is less than the critical depth, and so on.
Further notes:-
The critical and normal depth lines (CDL, NDL) and the channel bed form the boundaries of
3 zones. The curves of gradually varied flow surface profile approach each of these zone boundaries in a specific manner:
a) Upper limits of depth – the curves tend to become asymptotic to a horizontal water line.
b) Normal depth line – approached asymptotically (except for C curves).
c) Critical depth line – intersected at right angles (except for C curves).
d) Bed – intersected at right angles.
There are altogether 12 possible profiles of gradually varied flow, as depicted below.
Channel
slope
Depth
Relations
dy
dx
Type of
ProfileSymbol
Type of
FlowForm of Profile
n c y y y> > + Backwater 1 M Sub-
critical
n c y y y> > - Dropdown 2 M Sub-
criticalMild
n c y y> > y + Backwater 3 M Super-
critical
c n y y y> = + Backwater 1C Sub-critical
c n y y y= = Parallel to
bed2C
Uniform,
CriticalCritical
c n y y y= > + Backwater 3C Super-
critical
c n y y y> > + Backwater 1S Sub-
critical
c n y y y> > - Dropdown 2S Super-
criticalSteep
c n y y y> > + Backwater 3S Super-
critical
c y y> - Dropdown 2 H Sub-
criticalHorizontal
c y y> + Backwater 3 H Super-
critical
c y y> - Dropdown 2 A Sub-
criticalAdverse
c y y> + Backwater 3 A Super-
critical
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G. Hydraulic Jump
1. Introduction
When a change from rapid (super-critical) to tranquil (sub-critical) flow occurs in open
channel, a hydraulic jump appears, through which the depth increases abruptly in the
direction of flow.
In engineering practice, the hydraulic jump frequently appears downstream from overflow
structures (spillways) or underflow structures (sluice gated) where velocities are high. It may
be used as an effective dissipation of kinetic energy (and thus prevent scour) of channel
bottom) or as a mixing device in water or sewage treatment designs where chemicals are
added to the flow.
2. General Equation of Hydraulic Jump
In spite of the complex appearance of a hydraulic jump withits turbulence and air entrainment, it may be analyzed by
application of the momentum equation.
Consider flow in a rectangular channel, and apply
momentum equation for the control volume of unit width
between sections 1 and 2:
( )1 2 2 1F F q V V ρ − = −
Substituting the following relations
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If the head loss is , the balance of energy heads before and after the jump gives Lh
( )
( )
2 2
1 21 2
2 2
1 2 1 2
2
1 2 2 2
1 2
2 2 22 21 2 1 2
1 2 1 2 1 22 2
1 2
3
2 1
1 2
2 2
1
2
1 1
2
1 1 2 0
4
4
L
L
V V y y h
g g
h y y V V g
q y y
g y y
y y y y q y y y y y y
y y g
y y
y y
+ = + +
⇒ = − + −
⎛ ⎞= − + −⎜ ⎟
⎝ ⎠
⎛ ⎞⎛ ⎞ ⎛ ⎞+= − + − + − =⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠⎝ ⎠
−=
∵
which also equals the difference in specific energy across the jump.
To ensure a positive magnitude,2 1 y y> and not vice versa.
Therefore, a hydraulic jump is irreversible.
The power dissipated per width of channel is LP gqh ρ = .
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H. Some Examples of Composite-Flow Profiles
• Downstream of a sluice gate
• A change of bed slope from steep to mild: the hydraulic jump may be formed either
on the steep slope or on the mild slope depending on whether the downstream
conjugate depth2 y is smaller or greater than the normal depth on the mild slope.
• Other types of change of bed slope
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•
Free fall at the end of a channel
• Flow over a bump (bottom friction is ignored): the free surface over the bump is
depressed or elevated when the flow is sub-critical or super-critical, respectively.
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When the flow before the bump is sub-critical (state 1a), the flow depth y2 decreases
s the bump height is increased, point 2 continues shifting to the left on the
(state 2a). Since the decrease in flow depth is always greater than the bump height
(why?), the free surface will be suppressed. But if the flow before the bump is super-
critical (state 1b), the flow depth rises over the bump (state 2b), creating a bigger
bump over the free surface. The situation is reversed if the channel has a depression
in its bed: the flow depth increases if the approach flow is sub-critical and decreases ifit is super-critical.
Ab zΔ
m (specific energy diagra while point 1 remains unaffected), until finally reaching the
critical point at which the specific energy is the minimum, and the flow over the bump
is critical. This critical height of a bump is given by the difference between the
original and the minimum specific energy levels1 minbc z E E Δ = − . Since the specific
energy has already reached the minimum level at flow over the bump
will only remain critical even when the bump height is further increased. To
overcome a bump of height greater than bc z
this state, the
Δ , the approach flow must adapt itself(say, either to increase the upstream energy level
1 E or to reduce the flow rate so that
min E is decreased) so that the bump height1 min E E = − is always satisfied. When this
ens, the flow is said to be choked. happ
he fact that flow over a sufficiently high obstruction in an open channel is alwaysT
critical is the working principle of weirs (broad-crested or sharp-crested), which are
used to measure the volume flow rate in open channels.
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