Dynamic Meteorology 2 Lecture 9
Sahraei
Physics Department Razi University http://www.razi.ac.ir/sahraei Stream Function Incompressible fluid
uv 0 xy
u ; v Definition of Stream Function y x
Substituting these in the irrotationality condition, we have
vu 0 xy
22 0 xy22 Velocity Potential Irrotational flow
vu Since the, V 0 V 0 0 the flow field is irrotational. xy
u ; v Definition of Velocity Potential x y
Velocity potential is a powerful tool in analysing irrotational flows.
Continuity Equation
22 uv 2 0 0 0 xy xy22 As with stream functions we can have lines along which potential is constant. These are called Equipotential Lines of the flow.
Thus along a potential line c Flow along a line l
Consider a fluid particle moving along a line l . dx For each small displacement dl dy
dl idxˆˆ jdy
Where iˆ and ˆj are unit vectors in the x and y directions, respectively.
Since dl is parallel to V , then the cross product must be zero. V iuˆˆ jv V dl iuˆ ˆjv idx ˆ ˆjdy udy vdx kˆ 0 dx dy uv Stream Function dx dy Since must be satisfied along a line l , such a line is called a uv streamline or flow line.
A mathematical construct called a stream function can describe flow associated with these lines.
The Stream Function xy, is defined as the function which is constant along a streamline, much as a potential function is constant along an equipotential line.
Since xy, is constant along a flow line, then for any dl , d dx dy 0 along the streamline xy Stream Functions
d dx dy 0 dx dy xy xy dx dy vdx ud y uv uv , from which we can see that yx
So that if one can find the stream function, one can get the discharge by differentiation. Interpreting Stream Functions
Any Line l is an arbitrary line tˆ is tangent to l ˆ t nˆ is normal to 2 l Flow Line 2 tnˆ.0ˆ nˆ Flow Line 1 1 dy dx What is the flow that crosses dQ V. ndlˆ uiˆ vj ˆ. i ˆ ˆj dl between flow Lines 1 and 2? dl dl
for each increment dl : udy vdx dy dx d dx dy yx tˆ iˆˆ j dl dl Discharge Between Lines 1, 2 dy dx nˆ iˆˆ j If we integrate along line , between flow lines 1 and 2 dl dl we will get the total flow across the line. 2 2 2 2 2 Q dQ d d 12 1 1 1 1 1
Properties of Stream Function
2 Q
1 Flow field variables are found by f and
We notice that velocity potential f and stream function are connected with velocity components.
It is necessary to bring out the similarities and differences between them.
Differentiating potential function f in the same direction as velocities Differentiating stream function in direction normal to velocities
Potential function f applies for irrotational flow only Stream function applies for rotational or irrotational flows
Potential function f applies for 2D flows [f (x,y) or f (r,q)] and 3D flows [f (x,y,z) or f (r,q, f)] Stream function applies for 2D (x,y) or (r, q) flows only Properties of Stream Function
2
1
f2 f1
Stream lines ( =constant) and equipotential lines (f =constant) are always perpendicular to each other Slope of a line with =constant is the negative reciprocal of the slope of a line with f =constant
Stream Function- Physical meaning
Statement: In 2D (viscous or inviscid) flow (incompressible flow OR steady state compressible flow), = constant represents the streamline. (Properties of Stream Function)
Proof
If = constant, then d 0
d dx dy v dx u dy 0 xy
14 dy v If = constant, then () dx u v
Along a streamline u
Along an equipotential line
d dx dy udx vdy 0 xy dy u dy dy () ( ) ( ) 1 dx v dx dx showing that equipotential lines and streamlines are orthogonal to each other. This enables one to calculate the stream function when the velocity potential is given and vice versa. Fig. shows the flow through a bend where the streamlines and the equipotential lines have been plotted.