Stream Function | Fluid Mechanics for Mechanical Engineering PDF Download

Stream Function
Let us consider a two-dimensional incompressible flow parallel to the x - y plane in a rectangular cartesian coordinate system. The flow field in this case is defined by
u = u(x, y, t)
v = v(x, y, t)
w = 0 

The equation of continuity is

If a function ψ(x, y, t) is defined in the manner

so that it automatically satisfies the equation of continuity (Eq. (10.1)), then the function is known as stream function. 
Note that for a steady flow, ψ is a function of two variables x and y only.

Constancy of ψ on a Streamline

Since ψ is a point function, it has a value at every point in the flow field. Thus a change in the stream function ψ can be written as

It follows that dψ = 0 on a streamline.This implies the value of ψ is constant along a streamline. Therefore, the equation of a streamline can be expressed in terms of stream function as

ψ(x, y) = constant   (10.3)

Once the function ψ is known, streamline can be drawn by joining the same values of ψ in the flow field.

Stream function for an irrotational flow
In case of a two-dimensional irrotational flow 

Conclusion drawn:For an irrotational flow, stream function satisfies the Laplace’s equation


Physical Significance of Stream Funtion ψ

Figure 10.1 illustrates a two dimensional flow.

Fig 10.1   Physical Interpretation of Stream Function

Let A be a fixed point, whereas P be any point in the plane of the flow. The points A and P are joined by the arbitrary lines ABP and ACP. For an incompressible steady flow, the volume flow rate across ABP into the space ABPCA (considering a unit width in a direction perpendicular to the plane of the flow) must be equal to that across ACP. A number of different paths connecting A and P (ADP, AEP,...) may be imagined but the volume flow rate across all the paths would be the same. This implies that the rate of flow across any curve between A and P depends only on the end points A and P.

Since A is fixed, the rate of flow across ABP, ACP, ADP, AEP (any path connecting A and P) is a function only of the position P. This function is known as the stream function ψ.

The value of ψ at P represents the volume flow rate across any line joining P to A. 
The value of ψ at A is made arbitrarily zero. If a point P’ is considered (Fig. 10.1b),PP’ being along a streamline, then the rate of flow across the curve joining A to P’ must be the same as across AP, since, by the definition of a streamline, there is no flow across PP'

The value of ψ thus remains same at P’ and P. Since P’ was taken as any point on the streamline through P, it follows that ψ is constant along a streamline. Thus the flow may be represented by a series of streamlines at equal increments of ψ.

In fig (10.1c) moving from A to B net flow going past the curve AB is


The stream function, in a polar coordinate system is defined as

The expressions for V_r and V_θ in terms of the stream function automatically satisfy the equation of continuity given by



Stream Function in Three Dimensional and Compressible Flow
Stream Function in Three Dimensional Flow

In case of a three dimensional flow, it is not possible to draw a streamline with a single stream function.

An axially symmetric three dimensional flow is similar to the two-dimensional case in a sense that the flow field is the same in every plane containing the axis of symmetry.

The equation of continuity in the cylindrical polar coordinate system for an incompressible flow is given by the following equation

For an axially symmetric flow (the axis r = 0 being the axis of symmetry),  the term    ,and simplified equation is satisfied by  functions defined as 
    (10.4)
 

The function ψ , defined by the Eq.(10.4) in case of a three dimensional flow with an axial symmetry, is called thestokes stream function.

Stream Function in Compressible Flow
For compressible flow, stream function is related to mass flow rate instead of volume flow rate because of the extra density term in the continuity equation (unlike incompressible flow)

The continuity equation for a steady two-dimensional compressible flow is given by

Hence a stream function ψ is defined which will satisfy the above equation of continuity as
     [where ρ₀ is a reference density]       (10.5)

ρ₀ is used  to retain the unit of ψ same as that in the case of an incompressible flow. Physically, the difference in stream function between any two streamlines multiplied by the reference density ρ₀ will give the mass flow rate through the passage of unit width formed by the streamlines.

 

Continuity Equation: Integral Form
Let us consider a control volume     bounded by the control surface S. The efflux of mass across the control surface S is given by
 

where  is the velocity vector at an elemental area( which is treated as a vector by considering its positive direction along the normal drawn outward from the surface).

Fig 10.2  A Control Volume for the Derivation of Continuity Equation (integral form)

The rate of mass accumulation within the control volume becomes

where d is an elemental volume, ρ is the density and  is the total volume bounded by the control surface S. Hence, the continuity equation becomes (according to the statement given by Eq. (9.1))
    (10.6)

The second term of the Eq. (10.6) can be converted into a volume integral by the use of the Gauss divergence theorem as

Since the volume  does not change with time, the sequence of differentiation and integration in the first term of Eq.(10.6) can be interchanged. 
Therefore Eq. (10.6) can be written as

    (10.7)

Equation (10.7) is valid for any arbitrary control volume irrespective of its shape and size. So we can write
   (10.8)

 

Conservation of Momentum:  Momentum Theorem
In Newtonian mechanics, the conservation of momentum is defined by Newton’s second law of motion.

Newton’s Second Law of Motion

• The rate of change of momentum of a body is proportional to the impressed action and takes place in the direction of the impressed action.
• If a force acts on the body ,linear momentum is implied.
• If a torque (moment) acts on the body,angular momentum is implied.

Reynolds Transport Theorem
A study of fluid flow by the Eulerian approach requires a mathematical modeling for a control volume either in differential or in integral form. Therefore the physical statements of the principle of conservation of mass, momentum and energy with reference to a control volume become necessary.
This is done by invoking a theorem known as the Reynolds transport theorem which relates the control volume concept with that of a control mass system in terms of a general property of the system.

Statement of Reynolds Transport Theorem
The theorem states that "the time rate of increase of property N within a control mass system is equal to the time rate of increase of property N within the control volume plus the net rate of efflux of the property N across the control surface”.

Equation of Reynolds Transport Theorem
After deriving  Reynolds Transport Theorem according to the above statement we get
   (10.9)

 

In this equation
N - flow property which is transported
η - intensive value of the flow property
 

Application of the Reynolds Transport Theorem to Conservation of Mass and Momentum 
Conservation of mass :The constancy of mass is inherent in the definition of a control mass system and therefore we can write
     (10.13a)

To develop the analytical statement for the conservation of mass of a control volume, the Eq. (10.11) is used with N = m (mass) and η = 1 along with the Eq. (10.13a).
This gives
    (10.13b)

The Eq. (10.13b) is identical to Eq. (10.6) which is the integral form of the continuity equation derived in earlier section. At steady state, the first term on the left hand side of Eq. (10.13b) is zero. Hence, it becomes
    (10.13c) 

Conservation of Momentum or Momentum Theorem - The principle of conservation of momentum as applied to a control volume is usually referred to as the momentum theorem.
Linear momentum - The first step in deriving the analytical statement of linear momentum theorem is to write the Eq. (10.11) for the property N as the linear - momentum  and accordingly η as the velocity () Then it becomes
  (10.14)

The velocity () defining the linear momentum in Eq. (10.14) is described in an inertial frame of reference. Therefore we can substitute the left hand side of Eq. (10.14) by the external forces   on the control mass system or on the coinciding control volume by the direct application of Newton’s law of motion. This gives
   (10.15)

This Equation is the analytical statement of linear momentum theorem.
In the analysis of finite control volumes pertaining to practical problems, it is convenient to describe all fluid velocities in a frame of coordinates attached to the control volume.
Therefore, an equivalent form of Eq.(10.14) can be obtained, under the situation, by substituting N as and accordingly η as    , we get
   (10.16)

With the help of the Eq. (10.12) the left hand side of Eq. can be written as


where    is the rectilinear acceleration of the control volume (observed in a fixed coordinate system) which may or may not be a function of time. From Newton’s law of motion

Therefore,      (10.17)

The Eq. (10.16) can be written in consideration of Eq. (10.17) as
   (10.18a)

At steady state, it becomes 
    (10.18b)

In case of an inertial control volume (which is either fixed or moving with a constant rectilinear velocity),   and hence Eqs (10.18a) and (10.18b) becomes respectively
   (10.18c)

and     (10.18d)

The Eqs (10.18c) and (10.18d) are the useful forms of the linear momentum theorem as applied to an inertial control volume at unsteady and steady state respectively, while the Eqs (10.18a) and (10.18b) are the same for a non-inertial control volume having an arbitrary rectilinear acceleration.

In general, the external forces  in Eqs (10.14, 10.18a to 10.18c) have two components - the body force and the surface force. Therefore we can write

  (10.18e)


where    is the body force per unit volume and    is the area weighted surface force.

Angular Momentum
The angular momentum or moment of momentum theorem is also derived from Eq.(10.10) in consideration of the property N as the angular momentum and accordingly η as the angular momentum per unit mass. Thus,
   (10.19)

where A_{Control mass system} is the angular momentum of the control mass system. . It has to be noted that the origin for the angular momentum is the origin of the position vector

The term on the left hand side of Eq.(10.19) is the time rate of change of angular momentum of a control mass system, while the first and second terms on the right hand side of the equation are the time rate of increase of angular momentum within a control volume and rate of net efflux of angular momentum across the control surface.

The velocity () defining the angular momentum in Eq.(10.19) is described in an inertial frame of reference.Therefore, the term  d/ dt (A_CMS) can be substituted by the net moment ΣM applied to the system or to the coinciding control volume. Hence one can write Eq. (10.19) as
    (10.20a)

At steady state
   (10.20b)