1
MECH 221 FLUID MECHANICS(Fall 06/07)
Chapter 4: FLUID KINETMATICS
Instructor: Professor C. T. HSU
2
MECH 221 – Chapter 4
4. FLUID KINETMATICS Fluid kinematics concerns the motion of fluid
element. As the fluid flows, a fluid particle (element) can translate, rotate, and deform linearly and angularly
Translation Rotation
Linear deformation Circulation
Dilation Viscous stress
Angular deformation
3
MECH 221 – Chapter 4
4.1. Translation The translation considers mainly the velocity and
acceleration along the trajectory of fluid element in linear motion
z
y
x
0
r
d
r
4
MECH 221 – Chapter 4
4.1. Translation For the fluid element moving along the
trajectory r(t), the velocity is simply given by v =dr/dt = (u,v,w). As the description is basically Lagrangian, the acceleration a is given by
which, for steady flows, reduces to
zw
yv
xu
tDt
D
vvvvv
a
zw
yv
xu
Dt
D
vvvv
a
5
MECH 221 – Chapter 4
4.2. Linear Deformation (Strain) Deformation: change of shape of fluid element
For easily understanding, we illustrate here in two-dimensions. The results then can be easily extended to 3-dimensions. Consider the rectangular fluid element at the initial time instant given in the following picture
6
MECH 221 – Chapter 4
4.2. Linear Deformation (Strain) The initial distance between points A and B is ∆x
and between A and C is ∆y. After a short time of ∆t, the distances then become ∆x+∆Lx and ∆y+∆Ly due to different velocities at B and C from A
A u( x) ; B u( x)+uxx
A v(y) ; C v( y)+vy y
Lx=(u B–uA)t=u
xxt
L x
t=uxx
Ly=(vC–vA)t=vyyt
L y
t=uyy
7
MECH 221 – Chapter 4
4.2. Linear Deformation (Strain) The linear strain rate in x and y directions are
then given by
Similarly, for 3-D flows we have in the z-direction,
xu
tx
L
tx
xx
lim
0
yv
ty
L
ty
yy
lim
0
zw
tz
L
tz
zz
lim
0
8
MECH 221 – Chapter 4
4.3. Dilation Volumetric expansion & contraction
The fluid dilation is defined as the change of volume per unit volume. We are more interested in the rate of dilation that determines the compressibility of fluids. For 2-D flows,
yx)y
Ly)(x
LxV1yxV ( ;
t
yx)tyyv
y)(txxu
x(
yx
1
t
V
V
1
9
MECH 221 – Chapter 4
4.3. Dilation Then, the rate of dilatation becomes,
It is easy to generalize this dilation rate for 3-D flows and to reach
For incompressible flow, the rate of dilation is zero,
v
y
v
x
u
txyAxyA
tV
V
lim0Δt
lim0Δt
z
w
y
v
x
u
v
0 v
for 2-D flows
10
MECH 221 – Chapter 4
4.4. Angular Deformation (Strain) Now consider the deformation between A and B
caused by the change in velocity v, and the deformation between A and C by change in u
A v ( x ) , B v ( x )+ v
x x ; =
v
x x t
txv
xη
αα ΔΔΔ
)tan(ΔΔ
A u ( y ) , C u ( y )+ u
y
y ; = u
y
y t
tyu
yΔ
ΔΔ
)tan(ΔΔ
11
MECH 221 – Chapter 4
4.4. Angular Deformation (Strain) For , the counter clockwise rotation of AB is
equal to clockwise rotation of AC; therefore, the fluid element is in pure angular strain without net rotation and the angular strain is equal to either or . However, if ≠ , the strain then is equal to
. The rate of angular strain is then given by
2/)(
)(21
2)(1
0lim
yu
xv
ttyxxy
12
MECH 221 – Chapter 4
4.4. Angular Deformation (Strain) Similarly, we can extend to other planes y-z and z-
x to obtain:
)(21
zv
yw
zyyz
)(21
xw
zu
xzzx
13
MECH 221 – Chapter 4
4.5. Rotation If then the fluid element is under rigid
body rotation on the x-y plane. No angular strain is experienced, i.e.,
0 yxxy with yu
xv
14
MECH 221 – Chapter 4
4.5. Rotation When ≠ , the rotation of fluid element in x-y
plane is the average rotation of the two mutually perpendicular lines AB and AC; therefore,
where a counter clockwise rotation is chosen as positive and the rotation axis is in the z direction
y
u
x
v
2
1
2t
1Ω
ΔΔlim
0Δtz
15
MECH 221 – Chapter 4
4.5. Rotation Rotation is a vector quantity for fluid elements in
3-D motion. A fluid particle moving in a general 3-D flow field may rotate about all three coordinate axes, thus:
kjiΩ zΩyΩxΩ and so,
yu
xv
Ω
xw
zu
Ω
zv
yw
Ω
z
y
x
21
,21
,21
16
MECH 221 – Chapter 4
4.5. Rotation The vorticity of a flow field is defined as
wvuzyx
kji
ω v
kji
yu
xv
xw
zu
zv
yw
17
MECH 221 – Chapter 4
4.5. Rotation Therefore,
The flow vorticity is twice the rotation
In 2-D flow, ∂/∂z=0 and w=0 (or const.), so there is only one component of vorticity,
Irrotational flow is defined as having
ω
Ωω 2 v
0 v
Ω
zΩ
18
MECH 221 – Chapter 4
4.5. Rotation A fluid particle moving, without rotation, in a flow
field cannot develop a rotation under the action of a body force or normal surface force. If fluid is initially without rotation, the development of rotation requires the action of shear stresses. The presence of viscous forces implies the flow is rotational
The condition of irrotationality can be a valid assumption only when the viscous forces are negligible. (as example, for flow at very high Reynolds number, Re, but not near a solid boundary)
19
MECH 221 – Chapter 4
4.6. Circulation Consider the flow field as shown below
The circulation, , is defined as the line integral of the tangential velocity about a closed curve fixed in the flow,
Γ
V
ds
sdΓ v
where ds is the tangential vector along the integration loop. i.e. with being the unit tangential vector
Tdcd is Ti
20
MECH 221 – Chapter 4
4.6. Circulation Where is the line-element vector tangent to the
closed loop C of the integral. It is possible to decompose the integral loop C into the sum of small sub-loops, i.e.,
Without loss of generality, each sub-loop can be a rectangular grid as illustrated below.
C
s)d(dΓΓ v
21
MECH 221 – Chapter 4
4.6. Circulation Therefore,
As a result, we have
yvxyyu
uyxxv
vxud ΔΔΔΔΔΔ
Ayxyxyu
xv
dΓ zz ΔΔΔΔΔ
dAdAdΓAA z ks )( vv
where A is the area enclosed the contour
22
MECH 221 – Chapter 4
4.6. Circulation
Stokes' theorem in 2-D:
The circulation around a closed contour (loop) is the sum of the vorticity (flux) passing through the loop
This is an expression to illustrate the Green’s Theorem. In fact, the surface A can be a curved surface
23
MECH 221 – Chapter 4
4.6. Circulation Then for each sub-loop on the surface, we
have locally
where is the vorticity normal to the surface enclosed by the small increment loop C
nA
d
0A
Csv
lim
n
24
MECH 221 – Chapter 4
4.7. Viscous Stresses The strain rate tensor S is a symmetric tensor
that measures the rate of linear and angular deformations of fluid element. The strain rate tensor is expressed as:
where the superscript “T” represents the transpose
T)( vv S
25
MECH 221 – Chapter 4
4.7. Viscous Stresses In term of a Cartesian coordinate system,
they are expressed as:
z/wz/vz/u
y/wy/vy/u
x/wx/vx/u
v
zwywzvxwzu
ywzvyvxvyu
xwzuxvyuxu
/2)//()//(
)//(/2)//(
)//()//(/2
S
and
26
MECH 221 – Chapter 4
4.7. Viscous Stresses Following the Stokes’ hypothesis, the viscous
stress tensor is linearly related to the rate of dilation and the strain rate tensor by
where I represents the unit tensor, i.e.,
SIσ )(3
2v v)(
100
010
001
I
vσ
27
MECH 221 – Chapter 4
4.7. Viscous Stresses The proportional constants of the above linear
relation are the volume viscosity and shear viscosity of the fluid respectively. It is seen that the fluid viscosity leads to additional normal stresses, as well as shear stresses. Note that is a symmetric tensor, i.e.,
Total stress is given by
vσ
xzzxzyyzyxxy ττ,ττ,ττ
ISIσσσ ))(( v 3
2pvp
σ
and
Top Related