POTENTIAL FLOW THEORY
Definition: Potential Flow describes the velocity field as the gradient of a scalar function i.e. the velocity potential. As a result, a potential flow is characterized by an
irrotational velocity field. The irrotationality of a potential flow is due to the curl of a
gradient always been zero.
Important Cases of Potential Flow:
a) Uniform flow b) Source flow
c) Sink flow d) Doublet
e) Superimposed flow f)Flow over a cylinder
a) Uniform Flow:
In a uniform flow, the velocity remains constant. All the fluid particles are moving with same velocity. The uniform flow may be:
Let U=Velocity which is uniform or constant along x‐axis
u and v=Components of uniform velocity U along x and y axis.
For the uniform flow, parallel to x‐axis, the velocity components u and v are given as
u=U and v=0 ‐ (i)
But the velocity u in terms of stream function is given by,
u=
And in terms of velocity potential the velocity u is given by,
u=
u= = ‐ (ii)
Similarly it can be shown that v=‐ = ‐ (iii)
But u=U from eqn (i). Substituting u=U in eqn (ii), we have
∴ U= = (IV)
U= and also U=
First part gives whereas second part gives dϕ= Udx
Integration of these parts gives as
Ψ=Uy+C1 and ϕ=Ux+C2
Where C1 and C2 are constants of integration.
Now let us plot the stream lines and potential lines for uniform flow parallel to x‐axis
Plotting of Stream lines: For stream lines, the equation is
Ψ=U×ψ+C1
Let ψ=0, where y=0. Substituting these values in the above equations, we get
0=U×0+C1 or C1=0
Hence the equation of stream lines becomes as
Ψ=U.y ‐ (v)
The stream lines are straight lines parallel to x‐axis and at a distance y from the x‐axis
as shown.
In equation (v), U.y represents the volume flow rate (i.e. m3/s) between x‐axis and the
stream line at a distance y.
0
1
2
3
4
5
6
0 2 4 6 8
Ψ5
Ψ1
Ψ2
Ψ3
Ψ4
Plotting Of Potential Lines: For potential lines, the equation is
Φ=U.x+C2 (VI)
Let ϕ=0, where x=0. Substituting these values in the above equations, we get C2=0.
Hence equation of potential lines becomes as
Φ=U.x
The above equation shows that potential lines are straight lines parallel to y‐axis and at
a distance x from y‐axis as shown in fig
The fig 2
Shows the plot of stream lines and potential lines for uniform flow parallel to x‐axis. The
stream lines and potential lines intersect each other at right angles.
A matlab image for uniform flow
0
1
2
3
4
5
6
7
0 2 4 6
Φ3
Φ4
Φ5
Φ1
Φ2
0
1
2
3
4
5
6
7
0 2 4 6 8
Series1
Φ1
Φ2
Φ3
Φ4
Ψ1
Ψ2
Source moving o
in which
strength
strength
Let ur=r
The radi
decrease
approxim
uϴ=0.
Let us kn
source fl
function
and Sink Fout radially
h the point O
h of a source
h of source is
radial veloci
q=volum
r=radius
ius velocity
ur=
The a
es. And at a
mately equa
now find the
low. As in th
will be obt
Flow: The sy in all direct
O is the sour
e is defined
s m3/s.It is
ity of flow a
me flow rate
s
ur at any ra
above equat
large distan
al to zero. Th
e equation o
his case, uϴ=
ained from
ource flow i
tions of a pl
rce from wh
as the volum
represented
at a radius r
per unit de
adius r is giv
‐ (i)
ion shows t
nce away fro
he flow is in
of stream fu
=0, the equa
ur.
is the flow c
lane at unifo
hich the flui
me flow rate
d by q.
from the so
pth
ven by,
that with th
om the sour
n radial dire
unction and
ation of stre
coming from
orm rate. Fi
d moves rad
e per unit d
ource O
e increase o
rce, the velo
ection, henc
velocity po
am function
m a point (so
ig shows a s
dially outwa
depth. The u
of r, the radi
ocity will be
ce the tange
otential func
n and veloci
ource) and
source flow
ard. The
unit of
ial velocity
e
ntial velocit
ction for the
ity potentia
ty
e
l
Equation of Stream Function:
By definition, the radial velocity and tangential velocity components in terms of
stream function are given by
ur=1
and uϴ= ‐
But, ur=2
∴ = 2
Or dψ= r. 2 . dϴ =
2dϴ
Integrating the equation wrt , we get
Ψ=2 +C1 , where C1 is a constant of integration.
Let Ψ=0, when = 0, then C1=0.
Hence the equation of stream function becomes as
Ψ=2.
In the above equation, q is constant.
The above equation shows that stream function is a function of ϴ.
Equation of Velocity Potential:
Now consider a length in the radial direction
ds = dr
At radius r the velocity potential is defined as
dϕ=Vrdr
This becomes d= 2dr for source
d=‐2dr for sink
To find the expression for a length of one radius, we integrate with respect to r
=2 lnr for source
− 2 lnr for sink
Doublet:
A doublet is formed when an equal source and a sink are brought close
together. Consider a source and sink of equal strength placed at A and B respectively.
The stream function for point P relative to A and B are respectively.
B= 2 for the source
ΨA= ‐2 for the sink
ΨP= ψA+ψB = 2(2‐1)
Referring to the diagram,
tanϴ1= , tanϴ2=
Tan (ϴ2‐ ϴ1) =ϴ2 − tanϴ1
1 tanϴ1 ϴ2
Tan (ϴ2‐ ϴ1) =( )
1+x2− 2
Tan (ϴ2‐ ϴ1) =2b
x2+y2
As b tends to zero, b2 tends to zero and the tan of the angle becomes the same as the
angle itself in radians
(ϴ2‐ ϴ1)= 2b
x2+y2
ΨP=2 2b
x2+ 2
When the source and sink are brought close together we have DOUBLET but b remains
finite.
Let B= ψ= 2+ 2
Since y=rsinϴ and x2 + 2 =r2
Then = 2 = 1 ,
=0 is a streamline across which there is no flux circle so it can be used to
represent a cylinder.
Combination of Uniform flow and source or sink:
For this development, consider the case for the source at the origin of the x‐y
coordinates with a uniform flow of velocity u from left to right. The development for a
sink in uniform flow follows the same principle. The uniform flow encounters the flux
from the source producing a pattern as shown. At large values of x the flow has become
uniform again with velocity u .The flux from the source is Q this divides equally to the
top and bottom. At point s there is a stagnation point where the radial velocity from the
source is equal and opposite of the uniform velocity u.
The radial velocity is
= Q/2 r.
Equating to u we have
r=Q/2.u
and this is the distance from the origin to the stagnation point.
For uniform flow
1=‐uy.
For the source
2=Qϴ/2 .
The combined value is
=‐u.y+Q/2.
The flux between the origin and the stagnation point S is half the flow from the source
.Hence the flux and angle ϴ is radian. The dividing streamline emanating from S is the
zero streamline =0.Since no flux crosses this streamline, the dividing streamline could
be a solid boundary.
Rankine Oval
Flow around a long cylinder
When an ideal fluid flows around a long cylinder ,the stream lines and velocity
potentials from the same pattern as a doublet placed in a constant unifrom flow.It
follows that we may use a doublet to study the flow around a cylinder. The result of
combining a doublet with a uniform flow at velocity is as shown.
Consider a doublet at the origin with a uniform flow from left to right.The stream
function for point p is obtained by functions for a doublet and a uniform flow.
For a doublet is ψ=Bsin
For a uniform flow ψ=‐u.y
For the combined flow ψ= Bsin ‐u.y
Where B= (Qb/ ) from the diagram we have y=rsinϴ and substituting this into the
stream function gives
Ψ=Bsin
r – ursin =[
B
r− ]sin
Ψ=[
−B
r2− ]sin
Ψ=[ − ]
The equation is usually given in the form Ψ=[B
r− ] cos where A=u
The stream functions may be converted into velocity potentials by use of the above
equations as follows
Vr=Ψ =
d
dr VR=‐
Ψ=d
dr
Ψ=d
dr ‐ =r
d
dr
The equations can further be solved
At any given point in the flow with coordinate’s r, ϴ the velocity has a radial and
tangential component. The true velocity vϴ is the vector sum of both which, being at
right angle to each other, is found by Pythagoras as
Vϴ=2 + 2
From eqn we can show that
=‐u {2
2 − 1}
=‐u{2
2 + 1}
R is the radius of the cylinder.
Pressure Distribution around A cylinder:
The velocity of the main stream flow is u and the pressure is ‘p’.When it flows
over the surface of the cylinder the pressure is p because of the change in velocity .The
pressure change is p‐p’.
The dynamic pressure for a stream is defined as 2
2
The pressure distribution is usually shown in the dimensionless forms
=2(p‐p’)/ ( 2)
For an infinitely long cylinder placed in a stream of mean velocity u we apply.
Bernoulli’s equation between a point well away from the stream and a point on the
surface. At the surface the velocity is entirely tangential so:
P’ + 2/2=p+ 2/2
From the previous work it becomes
P’+ u2/2=p+ (2usinϴ)2/2
p‐p’= u2/2‐ 4 ) = ( u2/2) (1‐4 )
p‐p’/ u2/2=1‐4
If this function is plotted against angle we find that the distribution has a
maximum value of 1.0 at the front and back and a min value of ‐3 at the sides.
Vortices
Circulation
Consider a stream line that forms a closed loop. The velocity of the streamline at
any point is tangential to the radius of curvature R, the radius is rotating at angular
velocity ω .Now consider a small length of that streamline ds.
The circulation is defined as K=∫VTds and the integration is around the entire loop.
Substituting VT=ωR ds=R dϴ
K=∫ 2 the limits are 0 and 2
K=2 R
In terms of VT K=2
Vorticity:
Vorticity is defined as G=∫ ds/A where A is the area of the rotating element.
The area of the element shown in the diagram is small sector of arc ds and angle
dϴ.
dA=d
2 R2= R2
d
2
A= ∫ R2 d
2
G= ∫2
2
2
= 2 at any point.
It should be remembered in this simplistic approach that to may vary with angle.
Vortices
Consider a cylindrical mass rotating disk about a vertical axis. The streamlines
form concentric circles. The streamlines are so close that the circumference of each is
the same and length 2 r.Let the depth is dh, a small part of actual depth.
The velocity of the outer streamline is u+du
and the inner streamline is u.
The pressure at inner streamline is p and the outer diameter is p+dp.
The mass of the element is p2 r dh dr
The centrifugal force acting on the mass is p2 r dh dr 2/r
It must be the centrifugal force acting on the element that gives rise to the
change in pressure dp.
It follows that
dp 2 r dh=p 2 r dh dr 2/r
and dp/p= 2/r
The kinetic head at the inner streamline is = 2/2g
Differentiating w.r.t radius we get u du/ (g dr)
The total energy may be represented as a head H where H=Total Energy/mg.
The rate of change of energy head with radius is dH/dr.It follows that this must
be the sum of the rate of change of pressure and kinetic heads so
dH/dr=u2/gr +u du/ (gdr)
Free vortex
A free vortex is one with no energy added nor removed so dH/dr=0. It is also
irrotational which means that although the streamlines are circle and the individual
molecules orbit the axis of the vortex.Thy do not spin. This may be demonstrated
practically with a vorticity meter that is a float with a cross on it. The cross can be seen
to orbit the axis but not spin as shown.
Since the total head H is the same at all radii it follows the dH/dr=0.The equation
reduces to
u/r+du/dr=0
dr/r+du/v=0
Integrating ln u+ ln r = constant
ln (ur)=constant
ur=C
Note that a vortex is positive for anticlockwise rotation’s is the strength of the
free vortex with units of m2/s
Stream function for a free vortex
The tangential velocity was shown to be linked to the stream function by
dψ = Vt dr
Substituting Vt = C/r
dψ= C dr/r
Suppose the vortex has an inner radius of a and outer radius of R
Ψ=C∫ = C ln(R/a)
Velocity potential for a free vortex
The velocity potential was defined in the equation
dϕ= Vt r dϴ
Substituting, Vt=C/r and integrating
Φ=∫ ( )
Over the limits 0 to ϴ we have
Φ=Cϴ
Surface Profile of a free vortex
It was shown that dh/dr=u2/gr where h is depth .Substituting u=C/r we
get
dh/dr=C2/gr3
dh = C2gr3dr/g
Integrating between a small radius r and a large radius R we get
H2‐h1= (C2/2g) (1/R2‐1/r2)
Plotting h against r produces a shape as shown
Forced Vortex
A forced vortex is one in which the whole cylindrical mass rotates at one
angular velocity ω. It was shown earlier that dH/dr=u2/gr+ u du/(g dr) where h is depth
Substituting u=ωr and noting du/dr=ω we have
dH/dr= (ωr2)/gr + ω2r/g
dH/dr=2ω2r/g
Integrating without limits yields
H= ω2r2/g+A
H was also given by
H= h+ u2/2g =h+ω2r2/2g
Equating we have
h= ω2r2/2g+A
At radius r h1= ω2r2/2g+A
At radius R h2= ω2R2/2g+A
H2‐h1= (2/2g) (R2‐r2)
This produces a parabolic surface