mech 221 fluid mechanics (fall 06/07) chapter 7: inviscid flows
DESCRIPTION
MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 7: INVISCID FLOWS. Instructor: Professor C. T. HSU. 7.1 Inviscid Flow. Inviscid flow implies that the viscous effect is negligible. This occurs in the flow domain away from a solid boundary outside the boundary layer at Re . - PowerPoint PPT PresentationTRANSCRIPT
1
MECH 221 FLUID MECHANICS(Fall 06/07)
Chapter 7: INVISCID FLOWS
Instructor: Professor C. T. HSU
2
MECH 221 – Chapter 7
7.1 Inviscid Flow Inviscid flow implies that the viscous effect is
negligible. This occurs in the flow domain away from a solid boundary outside the boundary layer at Re.
The flows are governed by Euler Equations
where , v, and p can be functions of r and t .
gvvv
pt
)(
0)(
vt
3
MECH 221 – Chapter 7
7.1 Inviscid Flow On the other hand, if flows are steady but
compressible, the governing equation becomes
where can be a function of r
For compressible flows, the state equation is needed; then, we will require the equation for temperature T also.
gvv p)(
0)( v
RTp
4
MECH 221 – Chapter 7
7.1 Inviscid Flow
Compressible inviscid flows usually belong to the scope of aerodynamics of high speed flight of aircraft. Here we consider only incompressible inviscid flows.
For incompressible flow, the governing equations reduce to
where = constant.
gvvv
pt
)(
0 v
5
MECH 221 – Chapter 7
7.1 Inviscid Flow For steady incompressible flow, the governing eqt reduce further to
where = constant. The equation of motion can be rewrited into
Take the scalar products with dr and integrate from a reference at along an arbitrary streamline =C , leads to
since
)(2
)v( 2
vv
zgp
gvv p)(
0 v
constant22
v 22
gz
vpgz
p
Calong0)d( rvvr
6
MECH 221 – Chapter 7
7.1 Inviscid Flow If the constant (total energy per unit mass) is the same for
all streamlines, the path of the integral can be arbitrary, and in the flow domain except inside boundary
layers.
Finally, the governing equations for inviscid, irrotational steady flow are
Since is the vorticity , flows with are called irrotational flows.
0 v
0 v
0 v
vω 0 v
constant2
v2
gzp
7
MECH 221 – Chapter 7
7.1 Inviscid Flow
Note that the velocity and pressure fields are decoupled. Hence, we can solve the velocity field from the continuity and vorticity equations. Then the pressure field is determined by Bernoulli equation.
A velocity potential exists for irrotational flow, such that,
and irrotationality is automatically satisfied.
v
0 v
8
MECH 221 – Chapter 7
7.1 Inviscid Flow
The continuity equation becomes
which is also known as the Laplace equation.
Every potential satisfy this equation. Flows with the existence of potential functions satisfying the Laplace equation are called potential flow.
02
2
2
2
2
22
zyx
9
MECH 221 – Chapter 7
7.1 Inviscid Flow The linearity of the governing equation for the flow fields
implies that different potential flows can be superposed.
If 1 and 2 are two potential flows, the sum =(1+2) also constitutes a potential flow. We have
However, the pressure cannot be superposed due to the nonlinearity in the Bernoulli equation, i.e.
21 vvv 2121 )(
21 ppp
10
MECH 221 – Chapter 7
If restricted to steady two dimensional potential flow, then the governing equations become
E.g. potential flow past a circular cylinder with D/L <<1 is a 2D potential flow near the middle of the cylinder, where both w component and /z0.
7.2 2D Potential Flows
L
D
U
x
y
z
0
y
v
x
uv
0
ky
u
x
vv
11
MECH 221 – Chapter 7
The 2-D velocity potential function gives
and then the continuity equation becomes
The pressure distribution can be determined by the Bernoulli equation,
where p is the dynamic pressure
7.2 2D Potential Flows
xu
yv
02
2
2
22
yx
constant)( 2221 vup
12
MECH 221 – Chapter 7
7.2 2D Potential Flows For 2D potential flows, a stream function (x,y) can also be
defined together with (x,y). In Cartisian coordinates,
where continuity equation is automatically satisfied, and irrotationality leads to the Laplace equation,
Both Laplace equations are satisfied for a 2D potential flow
02
2
2
22
yx
02
2
2
2
yx
02
2
2
2
yx
yu
xv
13
MECH 221 – Chapter 7
7.2 Two-Dimensional Potential Flows For two-dimensional flows, become:
In a Cartesian coordinate system
In a Cylindrical coordinate system
v
y
,x
v,ujiv vu and
r,
rv,ur iiv vu rr and
14
MECH 221 – Chapter 7
7.2 Two-Dimensional Potential Flows Therefore, there exists a stream function such that in the Cartesian coordinate system and
in the cylindrical coordinate system.
The transformation between the two coordinate systems
r
,r
v,ur
x
,y
v,u
v
u
v
u ,
v
u
v
u rr
cossin
sin-cos
cossin-
sincos
15
MECH 221 – Chapter 7
7.2 Two-Dimensional Potential Flows The potential function and the stream function are
conjugate pair of an analytical function in complex variable analysis. The conditions:
These are the Cauchy-Riemann conditions. The analytical property implies that the constant potential line and the constant streamline are orthogonal, i.e.,
and to imply that .
xyyx
and
u,v- v,u 0
16
MECH 221 – Chapter 7
7.3 Simple 2-D Potential Flows
Uniform Flow
Stagnation Flow
Source (Sink)
Free Vortex
17
MECH 221 – Chapter 7
7.3.1 Uniform Flow
)( VU,v
yU
x
xV
y
VyUx VxUy
and
and
For a uniform flow given by , we have
Therefore,
Where the arbitrary integration constants are taken to be zero at the origin.
18
MECH 221 – Chapter 7
7.3.1 Uniform Flow This is a simple uniform flow along a single direction.
22 vuU
19
MECH 221 – Chapter 7
7.3.2 Stagnation Flow For a stagnation flow, . Hence,
Therefore,
2 cos22
222 rB
)yx(B
xBy
y ,
yBx
x
By,Bx v
2sin 2
2rB
Bxy
20
MECH 221 – Chapter 7
The flow an incoming far field flow which is perpendicular to the wall, and then turn its direction near the wall
The origin is the stagnation point of the flow. The velocity is zero there.
7.3.2 Stagnation Flow
x
y
21
MECH 221 – Chapter 7
7.3.3 Source (Sink) Consider a line source at the origin along the z-
direction. The fluid flows radially outward from (or inward toward) the origin. If m denotes the flowrate per unit length, we have (source if m is positive and sink if negative).
Therefore,
mur r 2
ru
r
rr
mu
r r
02
and
22
MECH 221 – Chapter 7
7.3.3 Source (Sink) The integration leads to
Where again the arbitrary integration constants are taken to be zero at .
rm
ln2
2
m
1,0,r
and
23
MECH 221 – Chapter 7
A pure radial flow either away from source or into a sink A +ve m indicates a source, and –ve m indicates a sink The magnitude of the flow decrease as 1/r z direction = into the paper. (change graphics)
7.3.3 Source (Sink)
24
MECH 221 – Chapter 7
7.3.4 Free Vortex Consider the flow circulating around the
origin with a constant circulation . We have: where fluid moves counter clockwise if is positive and clockwise if negative.
Therefore,
ur2
rru
r
ru
r r
20 and
25
MECH 221 – Chapter 7
7.3.4 Free Vortex
2
rln2
1,0,r
and
The integration leads to
where again the arbitrary integration constants are taken to be zero at
26
MECH 221 – Chapter 7
The potential represents a flow swirling around origin with a constant circulation .
The magnitude of the flow decrease as 1/r.
7.3.4 Free Vortex
27
MECH 221 – Chapter 7
7.4. Superposition of 2-D Potential Flows Because the potential and stream functions
satisfy the linear Laplace equation, the superposition of two potential flow is also a potential flow.
From this, it is possible to construct potential flows of more complex geometry.
Source and Sink Doublet Source in Uniform Stream 2-D Rankine Ovals Flows Around a Circular Cylinder
28
MECH 221 – Chapter 7
7.4.1 Source and Sink Consider a source of m at (-a, 0) and a sink of m at (a, 0)
For a point P with polar coordinate of (r, ). If the polar coordinate from (-a,0) to P is and from (a, 0) to P is
Then the stream function and potential function obtained by superposition are given by:
11 ,r
22 ,r
1212 lnln22
rrm
, m
29
MECH 221 – Chapter 7
7.4.1 Source and Sink
30
MECH 221 – Chapter 7
7.4.1 Source and Sink Hence,
Since
We have
12
1212 tantan1
tantantan
2tan
m
22
sin22tan
ar
ar
m
ar
r
ar
r
cos
sintan
cos
sintan 12 and
31
MECH 221 – Chapter 7
7.4.1 Source and Sink We have
By
Therefore,
22
1- sin2tan
2 ar
arm
cos2cossin
cos2cossin22222
1
222222
arararrr
arararrr
cos2
cos2ln
2 22
22
arar
ararm
32
MECH 221 – Chapter 7
7.4.1 Source and Sink The velocity component are:
sin2
sin
sin2
sin
2
cos2
cos
cos2
cos
2
2222
2222
arar
r
arar
rmv
arar
ar
arar
armu
33
MECH 221 – Chapter 7
7.4.1 Source and Sink
34
MECH 221 – Chapter 7
7.4.2 Doublet The doublet occurs when a source and a sink
of the same strength are collocated the same location, say at the origin.
This can be obtained by placing a source at (-a,0) and a sink of equal strength at (a,0) and then letting a 0, and m , with ma keeping constant, say 2am=M
35
MECH 221 – Chapter 7
7.4.2 Doublet For source of m at (-a,0) and sink of m at (a,0)
Under these limiting conditions of a0, m , we have
cos2
cos2
cos2lnlim
sin2sin2tanlim
22
22
0a
221-
0a
r
a
arar
arar
r
a
ar
ar
and ar
arm
22
1- sin2tan
2
cos2
cos2ln
2 22
22
arar
ararm
36
MECH 221 – Chapter 7
7.4.2 Doublet Therefore, as a0 and m with 2am=M
The corresponding velocity components are
r
M
2
cosand
r
M
2
sin
22
sin
r
Mu
and 22
cos
r
Mun
37
MECH 221 – Chapter 7
7.4.2 Doublet
38
MECH 221 – Chapter 7
7.4.3 Source in Uniform Stream
Assuming the uniform flow U is in x-direction and the source of m at(0,0), the velocity potential and stream function of the superposed potential flow become:
2sin
2
ln2
cosln2
mUr
mUy
rm
Urrm
Ux
39
MECH 221 – Chapter 7
7.4.3 Source in Uniform Stream
2m
ψ 2m
ψ
0ψ
40
MECH 221 – Chapter 7
7.4.3 Source in Uniform Stream The velocity components are:
A stagnation point occurs at
Therefore, the streamline passing through the
stagnation point when .
The maximum height of the curve is
sin2
cos Ur
ur
mU
rur
and
U
mrs
2
and
Urm
ss 2
2
ms
and as r U
mrh 0
2sin
41
MECH 221 – Chapter 7
7.4.3 Source in Uniform Stream
For underground flows in an aquifer of constant thickness, the flow through porous media are potential flows.
An injection well at the origin than act as a point source and the underground flow can be regarded as a uniform flow.
42
MECH 221 – Chapter 7
7.4.4 2-D Rankine Ovals The 2D Rankine ovals are the results of the
superposition of equal strength sink and source at x=a and –a with a uniform flow in x-direction.
Hence,
12
12
2sin
lnln2
cos
mUr
rrm
Ur
43
MECH 221 – Chapter 7
7.4.4 2-D Rankine Ovals Equivalently,
22
1
22
22
sin2tan
2sin
cos2
cos2ln
2cos
ar
armUr
arar
ararmUr
sin2
sin
sin2
sin
2
cos2
cos
cos2
cos
2
2222
2222
arar
r
arar
rmv
arar
ar
arar
armu
44
MECH 221 – Chapter 7
7.4.4 2-D Rankine Ovals The stagnation points occur at
where with corresponding .
0
12
1
2
1
2
s
ss
y
Ua
m
a
xa
U
max
i.e., ,
0v 0s
45
MECH 221 – Chapter 7
7.4.4 2-D Rankine Ovals The maximum height of the Rankine oval is
located at when ,i.e.,
which can only be solved numerically.
20
,r 0 s
a
r
m
Ua
a
r
a
r
ar
armUr
o0
2tan1
2
1
02
tan2
2
0
220
010
or
46
MECH 221 – Chapter 7
7.4.4 2-D Rankine Ovals
rs
ro
ro
rs
47
MECH 221 – Chapter 7
7.4.5 Flows Around a Circular Cylinder
Steady Cylinder
Rotating Cylinder
Lift Force
48
MECH 221 – Chapter 7
7.4.5.1 Steady Cylinder Flow around a steady circular cylinder is the
limiting case of a Rankine oval when a0.
This becomes the superposition of a uniform parallel flow with a doublet in x-direction.
Under this limit and with M=2a. m=constant,
is the radius of the cylinder.
2
1
0 2
U
Mrr s
49
MECH 221 – Chapter 7
7.4.5.1 Steady Cylinder The stream function and velocity potential
become:
The radial and circumferential velocities are:
sin12
sinsin
cos12
coscos
2
2
2
2
r
rUr
r
MUr
r
rUr
r
MUr
o
o
and
sin1cos12
20
2
20
r
rUU
r
rUU r and
50
MECH 221 – Chapter 7
7.4.5.1 Steady Cylinder
ro
51
MECH 221 – Chapter 7
7.4.5.2 Rotating Cylinder The potential flows for a rotating cylinder is the free
vortex flow given in section 7.3.3. Therefore, the potential flow of a uniform parallel flow past a rotating cylinder at high Reynolds number is the superposition of a uniform parallel flow, a doublet and free vortex.
Hence, the stream function and the velocity potential are given by
rr
rUr
r
rUr
ln2
sin1
2cos1
2
20
2
20
52
MECH 221 – Chapter 7
7.4.5.2 Rotating Cylinder The radial and circumferential velocities are given by
rr
rU
ru
2sin1
2
20
cos1
2
20
r
rU
rur
53
MECH 221 – Chapter 7
7.4.5.2 Rotating Cylinder The stagnation points occur at
From
0 ssr uu
0cos
0cos12
20
s
os
s
rr
r
rU
:B Case
:A Case
0sru
54
MECH 221 – Chapter 7
7.4.5.2 Rotating Cylinder
2
12
22
41
4sin
14
02
sin2
Urryrx
Urry
Ur
rU u
rr
oosos
osos
o
oss
os
:A Case
when exits only Solution
55
MECH 221 – Chapter 7
7.4.5.2 Rotating Cylinder
sr real positivefor implies which
with sign :B Case
Ur
UrUrr
r
rr
rU u
s
ss
ss
1
04
2
12
000
2
20
144
02
10
1sin0cos
56
MECH 221 – Chapter 7
7.4.5.2 Rotating Cylinder The stagnation points occur at
Case 1:
Case 2:
Case 3:
14 0
Ur
14 0
Ur
14 0
Ur
57
MECH 221 – Chapter 7
7.4.5.2 Rotating Cylinder Case 1:
14 0
Ur
2
12
0000 41
4
Urr
x
Urr
y ss
and
58
MECH 221 – Chapter 7
7.4.5.2 Rotating Cylinder Case 2:
The two stagnation points merge to one at cylinder surface where .
14 0
Ur
00 r,y,x ss
59
MECH 221 – Chapter 7
7.4.5.2 Rotating Cylinder Case 3:
The stagnation point occurs outside the cylinder
when where . The condition of
leads to
Therefore, as , we have
14 0
Ur
ss ry 0u2
2
12
0000
144
UrUrr
r
r
y ss
14 0
Ur
12 00
Urr
ys
60
MECH 221 – Chapter 7
7.4.5.2 Rotating Cylinder Case 3: 1
4 0
Ur
61
MECH 221 – Chapter 7
7.4.5.3 Lift Force The force per unit length of cylinder due to
pressure on the cylinder surface can be obtained by integrating the surface pressure around the cylinder.
The tangential velocity along the cylinder surface is obtained by letting r=ro,
0
0 2sin2
0r
Ur
urr
62
MECH 221 – Chapter 7
7.4.5.3 Lift Force The surface pressure as obtained from Bernoulli
equation is
where is the pressure at far away from the cylinder.
0p
22
2sin2
00
2
2
Up
rU
p
p
63
MECH 221 – Chapter 7
7.4.5.3 Lift Force Hence,
The force due to pressure in x and y directions are then obtained by
22
02
2
0
20 4
sin2
sin412 UrUr
Up p
2
drp drp dpFFCCyx ]sincos[ 00000 jiji sF
UdrpF drpF yx 0
2
0 00
2
0 0 sin0cos and
ji drd o sincos swhere
64
MECH 221 – Chapter 7
7.4.5.3 Lift Force The development of the lift on rotating bodies is called
the Magnus effect. It is clear that the lift force is due to the development of circulation around the body.
An airfoil without rotation can develop a circulation around the airfoil when Kutta condition is satisfied at the rear tip of the air foil.
Therefore, The tangential velocity along the cylinder surface is obtained by letting r=ro:
This forms the base of aerodynamic theory of airplane.