57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
1
Chapter 4: Fluids Kinematics
4.1 Velocity and Description Methods
Primary dependent variable is fluid velocity vector
V = V ( r ); where r is the position vector
If V is known then pressure
and forces can be
determined using
techniques to be discussed
in subsequent chapters.
r
r  xî  yĵ  zk̂
x
V  (r, t )  uî  vĵ  wk̂
Consideration of the velocity field alone is referred to as
flow field kinematics in distinction from flow field
dynamics (force considerations).
Fluid mechanics and especially flow kinematics is a
geometric subject and if one has a good understanding of
the flow geometry then one knows a great deal about the
solution to a fluid mechanics problem.
Consider a simple flow situation, such as an airfoil in a
wind tunnel:
U = constant
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
2
Velocity: Lagrangian and Eulerian Viewpoints
There are two approaches to analyzing the velocity field:
Lagrangian and Eulerian
Lagrangian: keep track of individual fluids particles (i.e.,
solve F = Ma for each particle)
Say particle p is at position r1(t1) and at position r2(t2) then,
r2  r1 dx dy dz
Vp  lim
 î  ĵ  k̂
t 0 t  t
dt
dt
dt
2
1
= u p î  v p ĵ  w p k̂
Of course the motion of one particle is insufficient to
describe the flow field, so the motion of all particles must
be considered simultaneously which would be a very
difficult task. Also, spatial gradients are not given directly.
Thus, the Lagrangian approach is only used in special
circumstances.
Eulerian: focus attention on a fixed point in space
x  xî  yĵ  zk̂
In general,
V  V( x, t )  uî  vĵ  wk̂
velocity components
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
3
where,
u = u(x,y,z,t), v = v(x,y,z,t), w = w(x,y,z,t)
This approach is by far the most useful since we are usually
interested in the flow field in some region and not the
history of individual particles.
However, must transform F = Ma
from system to CV (recall
Reynolds Transport Theorem
(RTT) & CV analysis from
thermodynamics)
Ex. Flow around a car
V can be expressed in any coordinate system; e.g., polar or
spherical coordinates. Recall that such coordinates are
called orthogonal curvilinear coordinates. The coordinate
system is selected such that it is convenient for describing
the problem at hand (boundary geometry or streamlines).
V  v r ê r  v ê 
x  r cos 
y  r sin 
ê r  cos  î  sin  ĵ
ê    sin  î  cos  ĵ
Undoubtedly, the most convenient coordinate system is
streamline coordinates:
V(s, t )  vs (s, t )ês (s, t )
However, usually V not known a priori and even if known
streamlines maybe difficult to generate/determine.
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
4
4.2 Flow Visualization and Plots of Fluid Flow Data
See textbook for:
Streamlines and Streamtubes
Pathlines
Streaklines
Timelines
Refractive flow visualization techniques
Surface flow visualization techniques
Profile plots
Vector plots
Contour plots
4.3 Acceleration Field and Material Derivative
The acceleration of a fluid particle is the rate of change of
its velocity.
In the Lagrangian approach the velocity of a fluid particle
is a function of time only since we have described its
motion in terms of its position vector.
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
5
r p  x p ( t )î  y p ( t ) ĵ  z p ( t )k̂

d rp
Vp 
 u p î  v p ĵ  w p k̂
dt


dv p d 2 rp
ap 
 2  a x î  a y ĵ  a z k̂
dt
dt
du p
dv p
dw p
ax 
ay 
az 
dt
dt
dt
In the Eulerian approach the velocity is a function of both
space and time; consequently,
x,y,z are f(t)
V  u(x, y, z, t )î  v(x, y, z, t ) ĵ  w(x, y, z, t )k̂

dV du dv dw
a
 î 
ĵ 
k̂  a x î  a y ĵ  a z k̂
dt dt
dt
dt
ax 
since we must
follow the
particle in
evaluating
du/dt
du u u x u y u z u
u
u
u





u v w
dt t x t y t z t t
x
y
z
called substantial derivative
Similarly for ay & az,
Dv v
v
v
v
 u v w
Dt t
x
y
z
Dw w
w
w
w
az 

u
v
w
Dt
t
x
y
z
ay 
Du
Dt
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
6
In vector notation this can be written concisely
DV  V

 V  V
Dt
t




î  ĵ  k̂ gradient operator
x y z
V
, called local or temporal acceleration results
t
from velocity changes with respect to time at a given point.
Local acceleration results when the flow is unsteady.
First term,
Second term, V  V , called convective acceleration
because it is associated with spatial gradients of velocity in
the flow field. Convective acceleration results when the
flow is non-uniform, that is, if the velocity changes along a
streamline.
The convective acceleration terms are nonlinear which
causes mathematical difficulties in flow analysis; also, even
in steady flow the convective acceleration can be large if
spatial gradients of velocity are large.
Example: Flow through a converging nozzle can be
approximated by a one dimensional velocity distribution
u = u(x). For the nozzle shown, assume that the velocity
varies linearly from u = Vo at the entrance to u = 3Vo at the
exit. Compute the acceleration
y
DV
as a function of x.
Dt
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Evaluate
Chapter 4
7
DV
at the entrance and exit if Vo = 10 ft/s and
Dt
L =1 ft.
We have V  u ( x )î ,
Assume linear
variation
between inlet
and exit
u(x) 
Du
u
 u  ax
Dt
x
2Vo
x   Vo  Vo  2x  1
L
L

2Vo2  2x 
u 2V0

 ax 
  1
x
L
L L

@x=0
ax = 200 ft/s2
@x=L
ax = 600 ft/s2
u(x) = mx + b
u(0) = b = Vo
u 3Vo  Vo 2Vo
m=


x
L
L
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
8
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
9
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
10
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
11
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
12
Example problem: Deformation rate of fluid element
Consider the steady, two-dimensional velocity field given
by
V  (u, v)  (0.5  0.8x)i  (1.5  0.8 y) j
Calculate the kinematic properties such as;
(a)
Rate of translation
(b)
Rate of rotation
(c)
Rate of linear strain
(d)
Rate of shear strain
Solution:
(a) Rate of translation:
u  0.5  0.8x , v  1.5  0.8 y , w  0
(b) Rate of rotation:
1  v
u 
1
     k   0  0 k  0
2  x y 
2
(c) Rate of linear strain:
 xx 
v
u
 0.8s 1 ,  zz  0
 0.8s 1 ,  yy 
y
x
(d) Rate of shear strain:
1  u
v 
1
 xy       0  0   0
2  y x  2
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
13
Rotation and Vorticity
 = fluid vorticity = 2  angular velocity = 2
=V
î
=
ĵ
i.e., curl V
x  v  u
x y
k̂
 w v   u w   v u 
  
=
 î   
 ĵ    k̂
x y z  y z   z x   x y 
u
v
w
To show that this definition is correct consider two lines in
the fluid
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
14
Angular velocity about z axis = average rate of rotation +
1  d d 
z  
 
2  dt dt 
v
dxdt
d  tan 1 x
u
dx  dxdt
x
lim 
dt 0
v
dt
x
i.e.,
d v

dt x
u
dydt

y
d  tan 1
v
dy  dydt
y
u
d u
i.e.,

lim  dt
dt 0
dt y
y
1  v u 
z    
2  x y 
similarly,
1  w v 
x  
 
2  y z 
1  u w 
y   

2  z x 
i.e.,  = 2
57:020 Fluid Mechanics
Professor Fred Stern Fall 2005
Chapter 4
15
Example problem: Calculation of Vorticity
Consider the following steady, three-dimensional velocity
field
V  (u, v, w)  (3.0  2.0 x  y)i  (2.0 x  2.0 y) j   0.5xy  k
Calculate the vorticity vector as a function of space  x, y, z 
Solution:
Vorticity vector in Cartesian coordinates:
 w v 
 u
w 
 v
u 
    i     j     k
 y z   z x   x y 
For u  3.0  2.0 x  y , v  2.0 x  2.0 y , w  0.5 xy
   0.5 x  0  i   0  0.5 y  j   2.0   1  k
  0.5 x  i   0.5 y  j   3.0  k