Download The Reynolds transport Theorem

The Reynolds Transport Theorem
The basic equations given in section (), involving the time derivative of extensive
properties (mass, linear momentum, angular momentum, energy) are required to analyse
any fluid problem. In solid mechanics, we often use a system representing a quantity of
mass of fixed identity. The basic equations are therefore directly applied to determine the
time derivatives of extensive properties. However, in fluid mechanics it is convenient to
work with control volume, representing a region in space considered for study. The basic
equations based on system approach can not directly applied to control volume approach.
Fig. illustrates different types of control volume: fixed control volume, control volume
moving at a constant speed and deforming control volume. In this section, it is aimed to
derive a relationship between the time derivative of system property and the rate of
change of that property within a control volume. This relationship is expressed by the
Reynolds Transport Theorem (RTT) which establishes a link between the system and
control volume approaches.
Before deriving the general form of the RTT, a derivation for one dimensional fixed
control volume is explained in the next section.
One- dimensional fixed control volume:
Consider a diverging (expanding) flow field bounded by a stream tube. The chosen
control volume is to be fixed between section ‘a’ and section ’b’. Note that both the
sections are normal to the direction of flow. At initial time t, System I exactly coincides
with the chosen control volume. This assumption says that the system and control volume
are identical at that time. At time t   t , System-I has moved in the flow direction at
uniform speed v and a part of system II has entered into the control volume.
Let ‘N’ represent any properties of the fluid (mass, momentum, energy) and then 
represent the amount of ‘N’ per unit mass (known as intensive property) in a small
proportion of the fluid. The total amount of ‘N’ in a control volume is expressed as
N    dm    dv where  
cv
cv
dN
dm
As the system coincides with the control volume at time ‘t’, a relation between the system
and the control volume is
N system,t  Ncv ,t
At time t  d t ,
N system,t t  N cv ,t  dt  N II , t  dt  N I ,t  dt
Using the definition of derivative, we can write,
dN sys
dt
 lim
t  o
 lim
N sys ,t  dt  N sys ,t
t
N cv,t t  N cv,t
t
t  o
 lim
N II ,t t
t
t o
 lim
t o
N I ,t t
t
or
dN sys
dt

 the time rate of
change of N of

 the system
dN cv
 Nin ,t t  N out ,t t
dt
  the rate of change   the flux of N   the flux of N 
  of N within the    passing into the    passing out the 
 
 
 

  Control volume   control surface  control surface 
The influx rate of N through the control surface can be computed as
N in ,t t  lim
N II ,t t
t
  Av  II
t  0
where N II ,t t   Av  II  t
Finally, Equation ( ) becomes
dN sys
dt

dNcv
  Av  II   Av  I
dt
This equation implies that the time rate of change of any extensive property for a system
is equal to the rate of change of that extensive property inside the control volume plus the
net efflux of the property through the control surface. This is known as Reynolds
Transport Equation which relates the change of a property of a system to the change of
that property for a control volume.
Arbitrary Fixed Control Volume
As similar to the previous derivation, consider a fixed control volume with an arbitrary
flow pattern passing through. At time t the system coincides with the control volume
which is fixed relative to the x, y, and z axes. At time t  t , the system has moved and
occupies the region II and III as shown in Fig.#. Note that the region II is common to the
system at both times t and t  t . The time rate of change of ‘N’ for the system can be
given by



1 
 dN 
 lim   dv    dv      dv    dv  


 dt system t 0 t  III
II
II
t t  I
t 
Rearranging the above equation we have
 dN 


 dt  system





 
 
 
    dv      dv  
    dv  
    dv  
 III
t t  II
t 
 III
t t 
 I
t 
 lim 
  lim
  lim

t 0
t 0 
t 0 
t
t
t












As t  0 region II occupies the same space of control volume and the first term on the
d
right side of the above equation becomes
 dv . The integral for region III
dt 
cv
approximates the amount of ‘N’ that has crossed the control surface ABCD shown in
Figure.#.
α
Let an area dA on the control surface where a steady flow velocity v is attained during
time interval, t , the interface has moved a distance V dt along a direction which is
tangential to streamline at that point.
The volume of the fluid swept across the area dA is dv  V .dt  dA cos  
Using the dot product we can define
dv  V . nˆ  dA dt 
So, the integral for the region III, is expressed by substituting dv . Efflux rate through
control surface ABC is
  V .nˆ  dA
ABC
Similarly, the influx rate through control surface ADC can be expressed


Influx rate:     V . nˆ  dA


ADC
The negative sign indicates influx rate of N pass through the control surface. The net
efflux rate of N through the whole control surface is
  Efflux rate on ABC   influx rate on ADC 
Net efflux rate 
   Vn  dAt   V .n  dA
ABC

ADC

 V .nˆ dA
Control surface
Collecting the terms of equation ( ) gives the compact from of RTT as
d
 dN 

    dv 
  V .nˆ dA
 dt  sys dt cv
control surface
 
The above equation states that the time rate of change of property N within a system is
equal to the time rate of change of property N within the arbitrary shaped control volume
plus the net rate of efflux of the property across the control surface.
Special Cases
Control volume moving at constant velocity:
In the case of fixed control volume the velocity field was measured with reference to x, y,
z axes. If the control volume moves at a uniform velocity Ve it is necessary to compute
relative velocity Vr of fluid crossing the control surface. The relative velocity becomes
Vr  V  Ve . The flux terms are to be defined in terms of Vr , but the volume integral
remains unchanged when the control volume does not deform.
The RTT for a uniformly moving control volume is
d
 dN 

    dv    p Vr nˆ dA
 dt  sys dt cv
contrl surface
 
Types of Fluid Flow
Laminar and Turbulent flow
In fluid flows, there are two distinct fluid behaviors experimentally observed. These
behaviors were first observed by Sir Osborne Reynolds. He carried out a simple
experiment in which water was discharged through a small glass tube from a large tank
(the schematic of the experiment shown in Fig.). A colour dye was injected at the
entrance of the tube and the rate of flow could be regulated by a valve at the out let.
When the water flowed at low velocity, it was found that the die moved in a straight line.
This clearly showed that the particles of water moved in parallel lines. This type of flow
is called laminar flow, in which the particles of fluid moves along smooth paths in layers.
There is no exchange of momentum from fluid particles of one layer to the fluid particles
of another layer. This type of flow mainly occurs in high viscous fluid flows at low
velocity, for example, oil flows at low velocity. Fig. shows the steady velocity profile for
a typical laminar flow.
When the water flowed at high velocity, it was found that the dye colour was diffused
over the whole cross section. This could be interpreted that the particles of fluid moved in
very irregular paths, causing an exchange of momentum from one fluid particle to
another. This type of flow is known as turbulent flow. The time variation of velocity at a
point for the turbulent flow is shown in Fig.
Turbulent
It means that the flow is characterized by continuous random fluctuations in the
magnitude and the direction of velocity of the fluid particles.
Velocity Field
Consider a uniform stream flow passing through a solid cylinder (Fig.). The typical
velocities at different locations within the fluid domain vary from position to position at a
particular time t. At different time instants this velocity distribution may change. Keeping
this observation in mind, the velocity within a flow domain can be represented as
function of position (x, y, z) and time t.
In the Cartesian co-ordinates the variation of velocity can be represented as a vector
V  uiˆ  vjˆ  wkˆ where u, v, w are the velocity scalar components in x, y and z directions
respectively. The scalar components u, v and w are dependent functions of position and
time. Mathematically we can express them as
u  u  x, y , z , t 
v  v  x, y , z , t 
w  w  x, y , z , t 
This type of continuous function distribution with position and time for velocity is known
as velocity field. It is based on the Eularian description of the flow. We also can represent
the Lagrangian description of velocity field.
Let a fluid particle exactly positioned at point A moving to another point A during time
interval t . The velocity of the fluid particle is the same as the local velocity at that point
as obtained from the Eulerian description
At time t,
V particle at x, y, z  t   V  x, y, z, t 
At time t  t ,
V
particle at x, y , z 
 t  t   V  x, y, z, t  t 
This means that instead of describing the motion of the fluid flow using the Lagrangian
description, the use of Eularian description makes the fluid flow problems quite easier to
solve. Besides this difficult, the complete description of a fluid flow using the Lagrangian
description requires to keep track over a large number of fluid particles and their
movements with time. Thus, more computation is required in the Lagrangian description.
The Acceleration field
At given position A, the acceleration of a fluid particle is the time derivative of the
particle’s velocity.
Acceleration of a fluid particle: a particle 
dVparticle
dt
Since the particle velocity is a function of four independent variables (x, y, z and t), we
can express the particle velocity in terms of the position of the particle as given below
a particle 
dVparticle
dt

dV  x particle , y particle , z particle 
dt
Applying chain rule, we get
a particle 
 V dt
 V dx particle  V dy particle  V dz particle
. 


 t dt  x particle dt
 y dt
 z dt
Where  and d are the partial derivative operator and total derivative operator
respectively. The time rate of change of the particle in the x-direction equals to the xcomponent of velocity vector, u. Therefore
dx particle
Similarly,
dt
dy particle
dt
dz particle
dt
u
v
w
As discussed earlier the position vector of the fluid particle (xparticle, y particle, z particle) in the
Lagranian description is the same as the position vector (x, y, z) in the Eulerian frame at
time t and the acceleration of the fluid particle, which occupied the position (x, y, z) is
equal to a( x, y, z, t ) in the Eularian description.
Therefore, the acceleration is defined by
V
V
V
V
a x, y , z ,t  
u
v
w
t
x
y
z
In vector form
a x , y , z ,t  
V
t

V . V
 local accelaration   convective accelartion 
where  is the gradient operator.
The first term of the right hand side of equation represents the time rate of change of
velocity field at the position of the fluid particle at time t. This acceleration component is
also independent to the change of the particle position and is referred as the local
acceleration. However the term V . V accounts for the affect of the change of the


velocity at various positions in this field. This rate of change of velocity because of
changing position in the field is called the convective acceleration.
Deformation of fluid particles
Fig. illustrates the deformation of three fluid particles originating from a uniform stream
flow. The fluid particle ‘A’ of a definite shape (square in this example) moves from its
initial position along the direction of stream flow. As there is no significant velocity
gradient, the particle has undergone only translation motion without any deformation.
But in the case of the particle B, it can easily be seen that the particle rotates in clockwise
direction near the obstruction. This results due to the presence of the velocity gradient at
that region. So, this type of motion of a fluid particle is known as rotation.
The particle C moves in the region of high velocity gradient. Therefore, the particle is
deformed volumetrically and is also undergoes angular deformation because of nonuniform distribution of velocity in the path x and y directions.
In short the types of primary motion of a fluid particle are described in four ways: (a)
translation (b) rotation (c) linear deformation and (d) angular deformation.
Rotation
Consider a two dimensional fluid particle motion in a fluid flow domain. The flow
velocity at point ‘A’ of the particle is expressed as
V  uiˆ  vjˆ
As per the continuum hypothesis the velocity components u and v are continuous
functions of space and time. The velocity at point A can be expressed using the Taylor
series
u
 2u  x 

u  x  x, y  at A  u  x, y   x  2

x
x
2
2
Neglecting the second and higher order terms in the above expression we obtain
u  x  x, y   u  x, y  
u
x
x
Similarly the velocity components at point B and A can be derived.
The pure rotation of the element is resulted from v-velocity component at point A and
the u-velocity component at point B.
The angle 1 rotated during time  t
v 

 v  x  t  vt v
x 
1  
 t
x
x


u 
 u  y  t  ut 
y 

   u t
similarly,  2  
y
y
The negative sign has been introduced because of clockwise rotation.
The average rotation angle is
1
  1   2 
2
Thus, the rate of rotation in the x and y planes becomes
w
d 1   v  u 
 


dt 2   x  y 
In three dimension we can express rate of rotation or angular velocity in vector form as
1   w v  ˆ 1  u  w  ˆ 1  v u  ˆ
w 
 i  


k
 j 
2 y z  2z x 
2x  y 
Linear deformation
In fluid mechanics the rate of linear deformation is emphasized instead of linear
deformation in solid mechanics. The rate of linear deformation or linear strain rate is the
rate of increased or decreased length per unit length.
Consider two points P and Q located on a fluid particle in the x-direction. The velocity at
u
x respectively. During time t , P moves to P 
pint P and Q at time t are u and u 
x
and Q to Q  . The rate of linear deformation xx is
 PQ  PQ  1
xx  lim 

t  0
PQ

 t

u 

 u  x  t  x  ut   x
x 


 lim 
t  0
x  t
xx 
Thus
u
x
Similarly linear strain rate in other directions are
v
y
w
zz 
z
yy 
Angular deformation:
As shown in figure#, angular deformation at point P is defined as the half of the rate of
the angle decreased between two mutually perpendicular axes.
The angle between these two axes decreases from  / 2 to  / 2  1  2  ,
as
demonstrated in Figure#. The rate of angle 1 , already derived in section() is
1 
d1  v

dt  x
The angular deformation in the xy plane is
1
1   2 
2
1  v u 
   
2  x y 
xy 
Note that  2 is in the clockwise direction. Extending to three dimensions the shear strain
rate is given by
1  v u 
xy  


2x  y 
1 v w
yz  


2z  y 
1  w u 
xz  


2 x z 