final ppt m

8/7/2019 Final ppt m

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Stream Function

Velocity Potential Function


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The Stream Function

The steam function basically describes the velocity field in2D plane (x,y) in the absence of Vorticity (Irrotaional Flow).

xv

yu

=

=

;

The stream function is

defined as:


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The concept of stream function is based on the principle ofcontinuity and the properties of a stream line.

The simplest type of flow in fluid mechanics application is a"steady, incompressible, plane, two-dimensional flow". Thecontinuity equation for such type of flow can be written as,


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By definition,Streamlines are the lines in the flow field such that the

tangent at any point gives the direction of velocities. Sincestreamlines are tangent to the velocity vector of the flow.


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One way is to define the stream function () for a twodimensional flow such that the flow velocity can be expressed

as:

Now, "stream function" which relates the velocitycomponents in X and Y coordinate system such that,


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,

Fig: Velocity components along a streamline


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Using the components from above equation it can beshown that the continuity equation gets satisfied.

Hence, we have only one function that describes a steady,incompressible, plane, two-dimensional flow instead oftwo functions.


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To understand concept of stream function , Consider thetwo points P1 and P2, P1 lying on the streamline D while P2lies on another streamline C in Figure:From the definition of a streamline, it is known that no flowcrosses it and, therefore, the quantity of flow between any

two streamlines must remain the same in accordance with thecontinuity equation.If the points P1 and P2 have stream function values ofand + respectively, then the flow across P1P2 isIn triangle AP1 P2 the flow leaving the triangle (acrossP1P2) is whereas the flow entering it is in whichdx and dy are the lengths of AP1 and AP2 respectively.


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Now;As we know that,

By eliminating the time variable we obtain:

Along the streamline the value of stream function is

constant so

Which is the definition of streamline.


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So it can be written as:

Thus, if "stream function" is known, infinite number of

streamlines can be drawn to define a particular flow fieldpattern.


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i.e


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15.05.0),( == msuyyx

x0 1 2 3 4 5 6 7 8 9 10

y0

1

2

3

-1

-2

-3

-4

= 0

= 1

= 1

= 2

= 2

V=u0=0.5ms-1


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0.5m grid v0=0.5ms-1.

15.05.0),( == msvxyx

x0 1 2 3 4 5 6 7 8 9 10

y0

1

2

3

-1

-2

-3

-4

=0

=1

=2

=3

=4

=5

V=

v0

=0.5

ms-

1


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Adding two stream functions xvyuyx 00),( =

x0 1 2 3 4 5 6 7 8 9 10

y0

1

2

3

-1

-2

-3

-4

= 0

= 1

= 1

= 2

= 2

=0

=1

=2

=3

=4

=5

=0

=1

=1

=2

=3

=4

V

V=(u 0

2 +v 0

2 )1/2

=1.41m

s-1


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Generalised linear flow

xvyu 00 =

cos0 =Vu

sin0 =Vv

x

y V

u0

v0


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Now;In cylindrical coordinate systems the stream function for a

plane incompressible, two-dimensional flow can be written as,

Here and are the velocity components in radial andtangential directions. And is the stream function. Itmay be noted that this also satisfy the continuity equation incylindrical coordinate system i.e.


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Visualization


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19x-5 -4 -3 -2 -1 0 1 2 3 4 5

y0

1

2

3

-1

-2

-3

-4

= 0

=1

=2

=3


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20x-5 -4 -3 -2 -1 0 1 2 3 4 5

y0

1

2

3

-1

-2

-3

-4

= 0

=1

=2

=3

Source and horizontal flow

= 0

= 1

= 2

= 3

= 1

= 2

= 3

= 4

=

3

=4

=

5 =7

=5

=6

=6

=

6


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Velocity Potential

In Case of irrotational flows or three dimensional flow thereis another function that is use to describe the flow field,which is called velocity potential function.

The velocity potential function is also a scalar functionwhich is denoted by Greek symbol.


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So for the irrotational flow velocity potential function can bedefined as,


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So another way to express the velocity potential function isgiven by,

For incompressible flows, the conservation of mass equationis,


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Hence, for incompressible and irrotational flow, the aboveequation becomes,

This equation is known as Laplace equation". Thus, in viscid,incompressible, irrotationalflow fields are governed by Laplaceequation and these flows are characterized by "potential flow".

Lines of constant are called "equipotential lines" of the flow.

final ppt m

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