PART II
INVISCID INCOMPRESSIBLE FLOW
In part II,we deal with the flow of a fluid which has
constant density– incompressible flow,This applies to the
flow of liquids,such as water flow,and to low-speed flow
of gases,The material covered here is applicable to low-
speed flight through the atmosphere—flight at a Mach
number of about 0.3
Chapter 3
Fundamentals of Inviscid,
incompressible Flow
Theoretical fluid dynamics,being a difficult subject,is for
convenience,commonly divided into two branches,one
treating frictionless or perfect fluids,the other treating of
viscous or imperfect fluids,The frictionless fluid has no
existence in nature,but is hypothesized by mathematicians in
order to facilitate the investigation of important laws and
principles that may be approximately true of viscous or natural
fluids,
3.1 Introduction and Road Map
From an aerodynamic point of view,at
air velocities between 0 to 360km/h the
air density remains essentially constant,
varying only a few percent,
Purpose of this chapter is to establish
some fundamental relations applicable to
inviscid,incompressible flows and to
discuss some simple but important flow
fields and applications,
3.2 Bernoulli’s Equation
c o n s tVp ?? 2
2
1 ?
Derivation of Bernoulli’s equation
Step 1,x component momentum equation
without viscous effect and body forces
x
p
Dt
Du
?
????
or
x
p
z
uw
y
uv
x
uu
t
u
?
???
?
??
?
??
?
??
?
? ????
For steady flow,,then 0??? tu
x
p
z
uw
y
uv
x
uu
?
???
?
??
?
??
?
?
?
1
Multiply both sides with dx
dx
x
pdx
z
uwdx
y
uvdx
x
uu
?
???
?
??
?
??
?
? ???
Along a streamline in 3D space,there are
0
0
??
??
udyv d x
w d xudz
Substituting the differential equations of the
streamline into x component momentum equation
dx
x
pdz
z
uudy
y
uudx
x
uu
?
???
?
??
?
??
?
?
?
1
or
dx
x
p
dz
z
u
dy
y
u
dx
x
u
u
?
?
????
?
?
??
?
?
?
?
?
?
?
?
?
?
?
1
as
dz
z
udy
y
udx
x
udu
?
??
?
??
?
??
We have
dx
x
pudu
?
???
?
1
or
dx
x
pdu
?
???
?
1
2
1 2
In the same way,we can get
dy
y
pdv
?
???
?
1
2
1 2 dz
z
pdw
?
???
?
1
2
1 2
??
?
?
??
?
?
?
?
?
?
?
?
?
?
???? dz
z
p
dy
y
p
dx
x
p
wvud
?
1
)(
2
1 222
2222 Vwvu ???
as
and
dpdz
z
pdy
y
pdx
x
p ?
?
??
?
??
?
?
Then,we have
dpdV
?
1
2
1 2 ?? or V d Vdp ???
V d Vdp ???
The above equation is called Euler’s Equation,
precondition,inviscid,without body force,along a
streamline,
usage,setup the relation between dpa n ddV
Step 2,integration of Euler’s equation
For incompressible flows,
The integration from point 1 to point 2 along a
streamline is
c o n s t??
?? ??
2
1
2
1
V
V
p
p
V d Vdp ?
or
??
?
?
??
?
?
????
22
2
1
2
2
12
VV
pp ?
or
2
11
2
22 2
1
2
1 VpVp ?? ???
The above equation is called Bernoulli’s Equation,
precondition,steady,inviscid,incompressible,without
body force,along a streamline,
usage,setup the relation between at point
1 on a streamline to at another point 2 on
the same streamline,
11 pa n dV
22 pa n dV
As point 1 and point 2 are arbitrarily chosen on a
streamline,so
c o n s tVp ?? 2
2
1 ?
Along a streamline
1,As there are no stipulation being made to whether
the flow is rotational or irrotational,it satisfies for
both irrotational flows and rotational flows as well,
2,The value of the constant will change from one
streamline to another,
3,For irrotational flows
c o n s tVp ?? 2
2
1 ? Through out the flow
Bernoulli’s equation is also a relation for mechanical
energy in an incompressible flow
c o n s tVp ?? 2
2
1 ?
where is the kinetic energy per unit volume 22V?
As Bernoulli’s equation can also be derived from the
energy equation,the energy equation is redundant
for the analysis of inviscid,incompressible flow,
The way for solving inviscid,incompressible flows,
1,Obtain the velocity field from the governing
equations,
2,Obtain the pressure field from Bernoulli’s equation
3.3 Incompressible flow in a duct,the
Venturi tube and low-speed wind tunnel
Assumption of quasi-one-dimensional flows
1,Flow-field properties are uniform across any cross
section,
2,All the flow-field properties are assumed to be
functions of x only,
Respect to continuity equation,and for steady case
0222111 ??? VAVA ??
or
222111 VAVA ?? ?
For incompressible flow
2211 VAVA ?
Applications of venturi tube
Measurements of flow speed in a duct,
2
212
2
1 )(
2 VppV ???
?
From Bernoulli’s equation
From continuity equation,we have
1
2
1
2 VA
AV ?
Then
2
1
2
2
1
12
2
1 )(
2
V
A
A
ppV ??
?
?
??
?
?
???
?
? ?1)(
)(2
2
21
12
1 ?
?
?
AA
pp
V
?
3.4 Pitot tube,measurement of airspeed
Definition of static pressure,the measurement of
the static pressure,
Definition of stagnation pressure,the measurement
of the stagnation pressure,
The usage of Pitot tube,
The pressure at point C is the stagnation pressure
0p
The pressure at point A is the static pressure
1p
The speed at point C is equal to zero
The speed at point A is equal to free stream velocity
1V
The speed and pressure and at point A and B are the
same,apply Bernoulli’s equation,we have
22
2
1
2
1
BBAA VpVp ?? ???
0
2
1
0
2
11 ??? pVp ?
or
?
)(2 10
1
pp
V
?
?
Combined instrument for the measurement of both
static and total(stagnation) pressure,
Definition of dynamic pressure,the relationship
between of the static pressure,total pressure and
dynamic pressure,
2
2
1 Vq ??
For incompressible flows
0
2
11 2
1 pVp ?? ?
static
pressure
dynamic
pressure
total
pressure
Keep in mind !!!!!
Any result derived from Bernoulli’s equation holds for
incompressible flow only
3.5 Pressure coefficient
Definition of the pressure coefficient
?
???
q
ppC
p
where
2
2
1
??? ? Vq ?
The pressure coefficient is a nondimensional value,
it can be used throughout aerodynamics,for both
compressible flow and incompressible flow,
For incompressible flow,the pressure coefficient
can be expressed in terms of velocity only
If and are defined as the freestream velocity
and pressure,And and are the velocity and
pressure at an arbitrary point in the flow,Then,with
Bernoulli’s equation,
?V ?
p
V p
22
2
1
2
1 VpVp ?? ???
??
or
)(
2
1 22 VVpp ???
?? ?
?
???
q
ppC
p
? )(
2
1 22 VVpp ???
?? ?
2
22
2
1
)(
2
1
?
?
?
?
?
?
?
?
V
VV
q
pp
C
p
?
?
or
2
1 ??
?
?
??
?
?
??
?
?
??
?
V
V
q
pp
C p
Valid for
incompressible
flow only
??
??
?
?
?
??
??
??
??
??
??
ppVV
Cqpp
CVV
CVV
CVV
CV
p
p
p
p
p
0
0
0
10
3.6 Condition on velocity for
incompressible flow
Contents in the center branch
1,Laplace’s equation
2,General philosophy and use in solving problems
3,Synthesis of complex flows from a superposition of
elementary flows
Basic condition on velocity in an incompressible
flow,
From the definition for the divergence of velocity
? ?
Dt
VD
V
V ?
?
1??? ?
If the density of the fluid is equal to a constant,then
? ? 0?
Dt
VD ?
That means,for incompressible flow,the velocity
must satisfies
0??? V?
If we consider the continuity equation
? ? 0????
?
? V
t
?
??
And apply it for the incompressible flows,then,the
continuity equation becomes
00 ???? V??
?
0??? V?
3.7 Governing equation for irrotational,
incompressible flow,Laplace’s Equation
The continuity equation for incompressible flow is
0??? V?
???V?
Therefore,for incompressible and irrotational flows
0)( 2 ?????? ?? Laplace’s equation
If the flow is irrotational,then,the velocity can be
expressed with velocity potential
Solutions of Laplace’s Equation are called harmonic
functions,
Solutions of Laplace’s Equation
in Cartesian coordinates,
02
2
2
2
2
2
2 ?
?
?
?
?
?
?
?
?
??
zyx
???
?
in cylindrical coordinates,
0
11
2
2
2
2
2
2 ?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
??
zrr
r
rr
?
?
??
?
For 2D incompressible flow,a stream function can
be defined,and the relation between velocity and
stream function is
x
v
y
u
?
???
?
?? ??,
The continuity equation,expressed in Cartesian
coordinates is,
0?
?
??
?
????
y
v
x
uV?
0
22
?
??
?
?
??
?
??
?
?
?
?
?
?
?
?
?
?
???
?
?
??
?
?
?
?
?
?
xyyxxyyx
????
The very definition and use of stream function is a
statement of the continuity equation of mass,and
therefore Equations bellow
x
v
y
u
?
???
?
?? ??,
Can be used in place of the continuity equation itself,
If the flow is irrotational,then
0?
?
??
?
?
y
u
x
v
? xvyu ?
???
?
?? ??,
0???
?
?
??
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
yyxx
??
?
02
2
2
2
?
?
?
?
?
?
yx
??
Laplace’s equation
Important conclusions
1,Any irrotational,incompressible flow has a velocity
potential and stream function(for 2D flow) that
both satisfy Laplace’s Equation
2,Conversely,any solution of Laplace’s Equation
represents the velocity potential or stream