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 , System-I has moved in the flow direction at uniform speed 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

As the system coincides with the control volume at time ‘t’, a relation between the system and the control volume is

At time,

Using the definition of derivative, we can write,

or


The influx rate of N through the control surface can be computed as

Finally, Equation ( ) becomes

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 , 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. The time rate of change of ‘N’ for the system can be given by


Rearranging the above equation we have

As region II occupies the same space of control volume and the first term on the right side of the above equation becomes . The integral for region III 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, , the interface has moved a distance along a direction which is tangential to streamline at that point.
The volume of the fluid swept across the area dA is

Using the dot product we can define

So, the integral for the region III, is expressed by substituting. Efflux rate through control surface ABC is

Similarly, the influx rate through control surface ADC can be expressed
Influx rate:
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
Net efflux rate
Collecting the terms of equation ( ) gives the compact from of RTT as

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 it is necessary to compute relative velocity of fluid crossing the control surface. The relative velocity becomes . The flux terms are to be defined in terms of , but the volume integral remains unchanged when the control volume does not deform.
The RTT for a uniformly moving control volume is


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 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

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 during time interval . 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, _{particle at x, y, z}
At time , _{particle at}

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:

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

Applying chain rule, we get

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 x-component of velocity vector, u. Therefore

Similarly,

As discussed earlier the position vector of the fluid particle (x_particle, 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 in the Eularian description.
Therefore, the acceleration is defined by


In vector form

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 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 non-uniform 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


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

Neglecting the second and higher order terms in the above expression we obtain

Similarly the velocity components at point B and can be derived.

The pure rotation of the element is resulted from v-velocity component at point and the u-velocity component at point B.
The angle rotated during time

similarly,
The negative sign has been introduced because of clockwise rotation.

The average rotation angle is

Thus, the rate of rotation in the x and y planes becomes

In three dimension we can express rate of rotation or angular velocity in vector form as

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 pint P and Q at time t are u and respectively. During time , P moves to and Q to . The rate of linear deformation is


Thus
Similarly linear strain rate in other directions are

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 to , as demonstrated in Figure#. The rate of angle , already derived in section() is

The angular deformation in the xy plane is

Note that is in the clockwise direction. Extending to three dimensions the shear strain rate is given by