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NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD of 56

Module 3 : Lecture 1 INVISCID INCOMPRESSIBLE FLOW

(Fundamental Aspects)

In general, fluids have a well-known tendency to move or flow. The slight change in

shear stress or appropriate imbalance in normal stresses will cause fluid motion. Fluid

kinematics deals with various aspects of fluid motion without concerning the actual

force that causes the fluid motion. In this particular section, we shall consider the

‘field’ concept to define velocity/ acceleration of fluid by virtue of its motion. In the

later part, some ‘visualization’ concepts are introduced to define the motion of the

fluid qualitatively as well as quantitatively.

There are two general approaches in analyzing the fluid motion. In the first

method (Lagrangian approach), the individual fluid particles are considered and their

properties are studied as a function of time. In the second method (Eulerian

approach), the ‘field’ concept is introduced and the properties are completely

prescribed as the functions of space and time. In other words, the attention is focused

at fixed points in space as the fluid passes those points.

Velocity and Acceleration Field

Since the ‘continuum’ assumption holds well for fluids, the description of any fluid

property (such as density, pressure, velocity, acceleration etc.) can be expressed as a

function of location. These representation as a function of spatial coordinates is called

as “field representation” of the flow. One of the most important fluid variables is the

velocity field. It is a vector function of position and time with components , andu v w

as scalar variables i.e.

( ) ( ) ( ) ˆˆ ˆ, , , , , , , , ,V u x y z t i v x y z t j w x y z t k= + +

(3.1.1)

The magnitude of the velocity vector i.e. ( )2 2 2V u v w= + +

, is the speed of fluid.

The total time derivative of the velocity vector is the acceleration vector field ( )a of

the flow which is given as,

( ){ } ( ){ } ( ){ }, , , , , , , , , ˆˆ ˆd u x y z t d v x y z t d w x y z tdVa i j kdt dt dt dt

= = + +

(3.1.2)



For instance, the scalar time derivative of u is expressed as,

( ){ }

( )

, , ,

.

d u x y z t u u dx u dy u dzdt t t dt t dt t dt

u u u uu v wt t t tu V ut

∂ ∂ ∂ ∂= + + +∂ ∂ ∂ ∂∂ ∂ ∂ ∂

= + + +∂ ∂ ∂ ∂∂

= + ∇∂

(3.1.3)

When u is replaced with andv w in the above equation, then the corresponding

expressions would be,

( ){ } ( )( ){ } ( )

, , ,.

, , ,.

d v x y z t v V vdt t

d w x y z t w V wdt t

∂= + ∇∂

∂= + ∇∂

(3.1.4)

Now, summing them into a vector quantity, one may write Eq. (3.1.2) in compact

form as,

( ).

ˆˆ ˆwhere, and

dV V V V V Va u v w V Vdt t x y z t

V u v w i j kx y z x y z

∂ ∂ ∂ ∂ ∂= = + + + = + ∇ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂= + + ∇ = + +

∂ ∂ ∂ ∂ ∂ ∂

(3.1.5)

In the above equation, Vt

∂∂

is called as “local acceleration” and the second part i.e.

V V Vu v wx y z

∂ ∂ ∂+ + ∂ ∂ ∂

is called a “convective acceleration”. The total time derivative

is called as “substantial/material” derivative. This field concept can be applied to any

variable (vector or scalar). For example, one may write the total derivative for

pressure and temperature field as,

( )

( )

.

.

dp p p p p pu v w V pdt t x y z t

dT T T T T Tu v w V Tdt t x y z t

∂ ∂ ∂ ∂ ∂= + + + = + ∇ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂= + + + = + ∇ ∂ ∂ ∂ ∂ ∂

(3.1.6)



Visualization of Fluid Flow

The quantitative and qualitative information of fluid flow can be obtained through

sketches, photographs, graphical representation and mathematical analysis. However,

the visual representation of flow fields is very important in modeling the flow

phenomena. In general, there are four basic types of line patterns used to visualize the

flow such as timeline, pathline, streakline and streamlines. Regardless of how the

results are obtained, (i.e. analytically/experimentally/computationally) it is necessary

to plot the data to get the feel of flow parameters that vary in time and/or shape (such

as profile plots, vector plots and contour plots).

(a) Timeline: A ‘timeline’ is a set of fluid particles that form a line at a given instant

(Fig. 3.1.1-a). Thus, it is marked at same instant of time. Subsequent observations of

the line provide the information of the flow field. They are particularly useful in

situations where uniformity of flow is to be examined.

(b) Pathline: It is the actual path traversed by a given fluid particle as it flows from

one point to another (Fig. 3.1.1-b). Thus, the pathline is a Lagrangian concept that can

be produced in the laboratory by marking the fluid particle and taking time exposure

photograph of its motion. Pathlines can be calculated numerically for a known

velocity field ( )V

i.e.

end

start

t

startt

x x V dt= + ∫ (3.1.7)

(c) Streakline: A streakline consists of all particles in a flow that has previously

passed through a common point (Fig. 3.1.1-c). Here, the attention is focused to a fixed

point in space (i.e. Eulerian approach) and identifying all fluid particles passing

through that point. These lines are laboratory tool rather than analytical tool. They are

obtained by taking instantaneous photographs of selected particles that have passed

through a given location in the flow field.



(d) Streamline: These are the lines drawn in the flow field so that at a given instant,

they are tangent to the direction of flow at every point in the flow field (Fig. 3.1.1-d).

Since the streamlines are tangent to the velocity vector at every point in the flow field,

there can be no flow across a streamline. Mathematically, these lines are obtained

analytically by integrating the equations defining lines tangent to the velocity field. In

a two dimensional flow field as shown in the figure, the slope of the streamline is

equal to the tangent of the angle that velocity vector makes with x-axis i.e.

dy vdx u

= (3.1.8)

This equation can be integrated to obtain the equation of streamlines.

When bundles of streamlines are considered in a flow field, it constitutes a

‘stream tube’ (Fig. 3.1.1-e). Since streamlines are everywhere parallel to the local

velocity, fluid cannot cross a streamline, so fluids within a stream tube remain there

and cannot cross the boundary at stream tube.

Fig. 3.1.1: Basic line patterns in fluid flow: (a) Timelines; (b) Pathline; (c) Streakline; (d) Streamline; (e) Streamtube.



The following observations can be made about the fundamental line patterns;

1. Mathematically, it is convenient to calculate a streamline while other three are

easier to generate experimentally.

2. The streamlines and timelines are instantaneous lines while pathlines and

streakline are generated by passage of time.

3. In a steady flow, all the four basic line patterns are identical. Since, the

velocity at each point in the flow field remains constant with time,

consequently streamline shapes do not vary. It implies that the particle located

on a given streamline will always move along the same streamline. Further,

the consecutive particles passing through a fixed point in space will be on the

same streamline. Hence, all the lines are identical in a steady flow. They do

not coincide for unsteady flows.

(f) Graphical data analysis techniques: Profile plots, vector plots and contour plots

are few important techniques in which fluid flow properties can be analyzed. The

profile plot (Fig. 3.1.2-a) indicates the variation of any scalar property (such as

pressure, temperature and density) along some desired direction in a flow field. Using

this plot, it is possible to examine the relative behavior of all variables in a

multivariate data set.

Fig. 3.1.2: Graphical representation of data analysis technique: (a) profile plot, (b) vector plot, (c) contour plot.



A vector plot (Fig. 3.1.2-b) is an array of arrows indicating the magnitude and

direction of a vector property at an instant of time. Although, streamlines indicate the

direction of instantaneous velocity field, but does not directly indicate the magnitude

of velocity. A useful flow pattern for both experimental and computational fluid flow

is the ‘vector plot’ that indicates the magnitude and direction of instantaneous vector

property.

A contour plot (Fig. 3.1.2-c) is a two-dimensional plot of a three-dimensional

surface showing lines where the surface intersects planes of constant elevation. Thus,

they are curves with constant values of scalar property (or magnitude of vector

property) at an instant of time. They can be filled in with either colors or sheds of gray

representing the magnitude of the property.



Module 3 : Lecture 2 INVISCID INCOMPRESSIBLE FLOW (Kinematic Description of Fluid Flow)

In order to illustrate this concept, we consider a typical fluid element of certain

volume at any arbitrary time as shown in Fig. 3.2.1. After certain time interval, it has

moved and changed its shape as well as orientation drastically. However, when we

limit our attention to an infinitesimal particle of volume ( ). .dV x y zδ δ δ= at time t

and ( )t tδ+ within the fluid element, it may be observed that the change of its shape

is limited to only stretching/shrinking and rotation with its sides remaining straight

even though there is a drastic change in the finite fluid element. Thus, the particle

motion in a fluid flow can be decomposed into four fundamental components i.e.

translation, rotation, linear strain and shear strain as shown in Fig. 3.2.2. When the

fluid particle moves in space from one point to another, it is referred as translation.

Rotation of the fluid particle can occur in any of the orthogonal axis. In the case of

linear strain, the particle’s side can stretch or shrink. When the angle between the

sides of the particle changes, it is called as shear strain.

Fig. 3.2.1: Schematic representation of motion of finite fluid element and infinitesimal particle mass at two different time

steps.



Fig. 3.2.2: Basic deformations of fluid mass: (a) Linear deformation; (b) Angular deformation.

Linear Motion and Deformation

Translation is the simplest type of fluid motion in which all the points in the fluid

element have same velocity. As shown in Fig. 3.2.3-a, the particle located as point O

will move to O’ during a small time interval tδ . When there is a presence of velocity

gradient, the element will tend to deform as it moves. Now, consider the effect of

single velocity gradient ( )u x∂ ∂ on a small cube having sides , andx y zδ δ δ and

volume V x y zδ δ δ δ= . As shown in Fig. 3.2.3-b, the x-component of velocity of O

and B is u . Then, x-component of velocity of points A and C would be,

( )u u x xδ+ ∂ ∂ , which causes stretching of AA’ by an amount ( )u x x tδ δ∂ ∂ as

shown in Fig.3.2.3-c. So, there is a change in the volume element

( )uV x y z tx

δ δ δ δ δ∂ = ∂



Fig. 3.2.3: Linear deformation of a fluid element.

Rate at which the volume Vδ changes per unit volume due to the velocity gradient

( )u x∂ ∂ is

( ) ( )0

1 limt

d V u x t uV dt t xδ

δ δδ δ→

∂ ∂ ∂= = ∂

(3.2.1)

In the presence of other velocity gradients ( ) ( )andv y w z∂ ∂ ∂ ∂ , Eq. (3.2.1) becomes,

( )1 .d V u v w V

V dt x y zδ

δ∂ ∂ ∂

= + + = ∇∂ ∂ ∂

(3.2.2)

If we look closely to the unit of velocity gradients ( ) ( ) ( ), andu x v y w z∂ ∂ ∂ ∂ ∂ ∂ , then

they resemble to unit of strain rate and the deformation is associated in the respective

directions of orthogonal coordinates in which the components of the velocity lie.

Thus, the linear strain (Fig. 3.2.2-a) is defined as the rate of increase in length to

original length and the linear strain rates are expressed as,

, ,xx yy zzu v wx y z

ε ε ε∂ ∂ ∂= = =∂ ∂ ∂

(3.2.3)

The volumetric strain rate/volumetric dilatation rate is defined as the rate of increase

of volume of a fluid element per unit volume.

1xx yy zz

dV u v wV dt x y z

ε ε ε ∂ ∂ ∂= + + = + +

∂ ∂ ∂ (3.2.4)

In an incompressible fluid, the volumetric dilatation rate is zero because the fluid

element volume cannot change without change in fluid density.



Angular Motion and Deformation

The variations of velocity in the direction of velocity is represented by the partial

derivatives ( ) ( ) ( ), andu x v y w z∂ ∂ ∂ ∂ ∂ ∂ , which causes linear deformation in the

sense that shape of the fluid element does not change. However, cross variations of

derivatives such as ( ) ( ) ( ), ,u y v z w x∂ ∂ ∂ ∂ ∂ ∂ will cause the fluid element to rotate.

These motions lead to angular deformation which generally changes the shape of the

element.

Fig. 3.2.4: Angular deformation of a fluid element.

Let us consider the angular motion in x-y plane in which the initial shape is

given by OACB, as shown in Fig. 3.2.4-a. The velocity variations cause the rotation

and angular deformation so that the new positions become OA’ and OB’ after a time

interval tδ . Then the angles AOA’ and BOB’ are given by andδα δβ , respectively

as shown in Fig. 3.2.4-b. Thus, the angular velocities of line OA and OB are,

( )

( )

0 0lim ; lim

For smallangles, tan

and tan

OA OBt t

v ut x t y

v x x t v tx x

u y y t u ty y

δ δ

δα δβω ωδ δ

δ δδα δα δ

δδ δ

δβ δβ δδ

→ →

∂ ∂ = = = = ∂ ∂ ∂ ∂ ∂

≈ = =∂

∂ ∂ ∂≈ = =

∂

(3.2.5)



When, both ( ) ( )andv x u y∂ ∂ ∂ ∂ are positive, then both andOA OBω ω

will be in

counterclockwise direction. Now, the rotation of the fluid element about z-direction

(i.e x-y plane) zω can be defined as the average of andOA OBω ω . If counterclockwise

rotation is considered as positive, then

12z

v ux y

ω ∂ ∂

= − ∂ ∂ (3.2.6)

In a similar manner, the rotation of the fluid element about x and y axes are denoted as

andx yω ω , respectively.

1 1;2 2x y

w v u wy z z x

ω ω ∂ ∂ ∂ ∂ = − = − ∂ ∂ ∂ ∂

(3.2.7)

These three components can be combined to define the rotation vector ( )ω in the form

as,

ˆˆ ˆ

1 1 1 ˆˆ ˆor,2 2 2

ˆˆ ˆ

1 1or,2 2

x y zi j k

w v u w v ui j ky z z x x y

i j k

Vx y z

u v w

ω ω ω ω

ω

ω

= + +

∂ ∂ ∂ ∂ ∂ ∂ = − + − + − ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂= = ∇×

∂ ∂ ∂

(3.2.8)

It is observed from Eq.(3.2.6) that the fluid element will rotate about z-axis, as an

undeformed block, only when, ( ) ( )i.e.OA OB v x u yω ω= − ∂ ∂ = − ∂ ∂ . Otherwise it will

be associated with angular deformation which is characterized by shear strain rate.

When the fluid element undergoes shear deformation (Fig. 3.2.2-b), the average shear

strain rates expressed in different cartesian planes as,

1 1 1; ;2 2 2xy yz zx

v u w v w ux y y z z x

ε ε ε ∂ ∂ ∂ ∂ ∂ ∂ = + = + = + ∂ ∂ ∂ ∂ ∂ ∂

(3.2.9)



Strain rate as a whole constitute a symmetric second order tensor i.e.

xx xy xz

ij yx yy yz

zx zy zz

ε ε εε ε ε ε

ε ε ε

=

(3.2.10)

Vorticity

In a flow field, vorticity is related to fluid particle velocity which is defined as twice

of rotation vector i.e.

ˆˆ ˆ2 w v u w v uV i j ky z z x x y

ζ ω ∂ ∂ ∂ ∂ ∂ ∂ = = ∇× = − + − + − ∂ ∂ ∂ ∂ ∂ ∂

(3.2.11)

Thus, the curl of the velocity vector is equal to the vorticity. It leads to two important

definitions:

If 0V∇× ≠

at every point in the flow, the flow is called as rotational. It

implies that the fluid elements have a finite angular velocity.

If 0V∇× =

at every point in the flow, the flow is called as irrotational. It

implies that the fluid elements have no angular velocity rather the motion is

purely translational.

Irrotational Flow

In Eq.(3.2.11) , if 0V∇× =

is zero, then the rotation and vorticity are zero. The flow

fields for which the above condition is applicable is known as irrotational flow. The

condition of irrotationality imposes specific relationship among the velocity gradients

which is applicable for inviscid flow. If the rotations about the respective orthogonal

axes are to be zero, then, one can write Eq. (3.2.11) as,

0 ; 0 ; 0z y xv u u w w vx y z x y z

ω ω ω∂ ∂ ∂ ∂ ∂ ∂= ⇒ = = ⇒ = = ⇒ =

∂ ∂ ∂ ∂ ∂ ∂ (3.2.12)



A general flow field would never satisfy all the above conditions. However, a uniform

flow field defined in a fashion, for which (a constant); 0; 0u U v w= = = , is

certainly an example of an irrotational flow because there are no velocity gradients. A

fluid flow which is initially irrotational may become rotational if viscous effects

caused by solid boundaries, entropy gradients and density gradients become

significant.

Circulation

It is defined as the line integral of the tangential velocity component about any closed

curve fixed in the flow i.e.

.V dsΓ = ∫

(3.2.13)

where, ds is an elemental vector tangent to the curve and with length ds with

counterclockwise path of integration considered as positive. For the closed curve path

OACB as shown in Fig. 3.2.4-a, we can develop the relationship between circulation

and vorticity as follows;

( )

or, 2

Then, . 2

z

z zA A

v uu x v x y u y x v yx y

v u x y x yx y

V ds dA V dA

δ δ δ δ δ δ δ

δ δ δ ω δ δ

ω

∂ ∂ Γ = + + − + − ∂ ∂ ∂ ∂

Γ = − = ∂ ∂

Γ = = = ∇×∫ ∫ ∫

(3.2.14)

Hence, circulation around a closed contour is equal to total vorticity enclosed within

it. It is known as Stokes theorem in two dimensions.



Module 3 : Lecture 3 INVISCID INCOMPRESSIBLE FLOW (Stream Function and Velocity Potential)

Basic Equations of Fluid Motion

The differential relations for fluid particle can be written for conservation of mass,

momentum and energy. In addition, there are two state relations for thermodynamic

properties. They can be summarized as;

( )

( ) ( )

( ) ( )

Continuity: . 0

Momentum: .

Energy: . .

Thermodynamicstate relations: , ; ,

ij

Vt

dV g pdt

e p V k Tt

p T e e p T

ρ ρ

ρ ρ τ

ρ

ρ ρ

∂+∇ =

∂

= −∇ +∇

∂+ ∇ = ∇ ∇ +Φ

∂= =

(3.3.1)

Here, Φ is the viscous-dissipation function, e is the internal energy and k is the

thermal conductivity of the fluid. In general, the density is a variable and all these

equations have 5-unknown parameters i.e. , , , andV p e Tρ

. In an incompressible flow

with constant viscosity, the momentum equation can be decoupled from energy

equation. Thus, continuity and momentum equations are solved simultaneously for

pressure and velocity. However, there are certain flow situations, which can wipe out

continuity equation by defining a suitable variable (called as stream function) and

thereby solving the momentum equation with a single variable.



Stream Function

The idea of introducing stream function works only if the continuity equation is

reduced to two terms. There are 4-terms in the continuity equation that one can get by

expanding the vector equation (3.3.1) i.e.

( ) ( ) ( ) 0u v w

t x y zρ ρ ρρ ∂ ∂ ∂∂

+ + + =∂ ∂ ∂ ∂

(3.3.2)

For a steady, incompressible, plane, two-dimensional flow, this equation reduces to,

0u vx y∂ ∂

+ =∂ ∂

(3.3.3)

Here, the striking idea of stream function works that will eliminate two velocity

components andu v into a single variable (Fig. 3.3.1-a). So, the stream function

( ){ },x yψ relates to the velocity components in such a way that continuity equation

(3.3.3) is satisfied.

;

ˆ ˆor,

u vy x

V i jy x

ψ ψ

ψ ψ

∂ ∂= = −∂ ∂

∂ ∂= −∂ ∂

(3.3.4)

Fig. 3.3.1: Velocity components along a streamline.



Fig. 3.3.2: Flow between two streamlines.

The following important points can be noted for stream functions;

1. The lines along which ψ is constant are called as streamlines. In a flow field, the

tangent drawn at every point along a streamline shows the direction of velocity (Fig.

3.3.1-b). So, the slope at any point along a streamline is given by,

dy vdx u

= (3.3.5)

Referring to the Fig. 3.3.2-a, if we move from one point ( ),x y to a nearby point

( ),x dx y dy+ + , then the corresponding change in the value of stream function is dψ

which is given by,

d dx dy v dx u dyx yψ ψψ ∂ ∂

= + = − +∂ ∂

(3.3.6)

Along a line of constant ψ ,

0

or,

d v dx u dydy vdx u

ψ = − + =

= (3.3.7)

The Eq. (3.3.5) is same as that of Eq. (3.3.7). Hence, it is the defining equation for the

streamline. Thus, infinite number streamlines can be drawn with constant ψ . This

family of streamlines will be useful in visualizing the flow patterns. It may also be

noted that streamlines are always parallel to each other.



2. The numerical constant associated toψ , represents the volume rate of flow.

Consider two closely spaced streamlines ( )and dψ ψ ψ+ as shown in Fig. 3.3.2-a.

Let dq represents the volume rate of flow per unit width perpendicular to x-y plane,

passing between the streamlines. At any arbitrary surface AC, this volume flow must

be equal to net outflow through surfaces AB and BC. Thus,

or,

dq v dx u dy dx dy dx y

dq d

ψ ψ ψ

ψ

∂ ∂= − + = + =

∂ ∂=

(3.3.8)

Hence, the volume flow rate ( )q can be determined by integrating Eq. (3.3.8) between

streamlines 1 2andψ ψ as follows;

2

1

2 1q dψ

ψ

ψ ψ ψ= = −∫ (3.3.9)

So, the change in the value of stream function is equal to volume rate of flow. If the

upper streamline 2ψ has a value greater than the lower one 1ψ , then the volume flow

rate is positive i.e. flow takes place from left to right (Fig. 3.3.2-b).

3. In cylindrical coordinates, the continuity equation for a steady, incompressible,

plane, two-dimensional flow, reduces to

( )1 1 0rr v vr r r

θ

θ∂ ∂

+ =∂ ∂

(3.3.10)

The respective velocity components andrv vθ are shown in Fig. 3.3.1-c. The stream

function ( ){ },rψ θ that satisfies Eq. (3.3.10), can then be defined as,

1 ;rv vr rθ

ψ ψθ

∂ ∂= = −

∂ ∂ (3.3.11)



4. In a steady, plane compressible flow, the stream function can be defined by

including the density of the fluid. But, the change in the stream function is equal to

mass flow rate ( )m .

;

or,

u vy x

dm v dx u dy dx dy dx y

dm d

ψ ψρ ρ

ψ ψρ ρ ψ

ψ

∂ ∂= = −∂ ∂

∂ ∂= − + = + =

∂ ∂=

(3.3.12)

5. One important application in a two-dimensional plane is the inviscid and

irroational flow where, there is no velocity gradient and 0zω = . Then, the vorticity

vector becomes,

2 2

2 2

2

ˆ ˆ2 0

ˆor, 0

or, 0

or, 0

zv uk kx y

kx x y y

x y

ζ ω

ψ ψ

ψ ψ

ψ

∂ ∂= = − = ∂ ∂ ∂ ∂ ∂ ∂ − − = ∂ ∂ ∂ ∂ ∂ ∂

− =∂ ∂

∇ =

(3.3.13)

This is a second order equation and is quite popular in mathematics and is known as

Laplace equation in a two-dimensional plane.

Velocity Potential

An irrotational flow is defined as the flow where the vorticity is zero at every point. It

gives rise to a scalar function ( )φ which is similar and complementary to the stream

function ( )ψ . Let us consider the equations of irrortional flow and scalar function ( )φ

. In an irrotational flow, there is no vorticity ( )ξ

0Vξ = ∇× =

(3.3.14)

Now, take the vector identity of the scalar function ( )φ ,

( ) 0φ∇× ∇ = (3.3.15)



i.e. a vector with zero curl must be the gradient of a scalar function or, curl of the

gradient of a scalar function is identically zero. Comparing, Eqs. (3.3.14) and (3.3.15),

we see that,

V φ= ∇

(3.3.16)

Here, φ is called as velocity potential function and its gradient gives rise to velocity

vector. The knowledge φ immediately gives the velocity components. In cartesian

coordinates, the velocity potential function can be defined as, ( ), ,x y zφ φ= so that

Eq. (3.3.16) can be written as,

ˆ ˆˆ ˆ ˆ ˆu i v j wk i j kx y zφ φ φ∂ ∂ ∂

+ + = + +∂ ∂ ∂

(3.3.17)

So, the velocity components can be written as,

; ;u v wx y zφ φ φ∂ ∂ ∂

= = =∂ ∂ ∂

(3.3.18)

In cylindrical coordinates, if ( ), ,r zφ φ θ= , then

; ;r zV V Vr zθφ φ φ

θ∂ ∂ ∂

= = =∂ ∂ ∂

(3.3.19)

Further, if the flow is incompressible i.e. ( )constant and 0tρ ρ= ∂ ∂ = , then

continuity equation can be written as,

( )( )

. 0

or, . 0

or, . 0

Vt

V

V

ρ ρ

ρ

∂+∇ =

∂

∇ =

∇ =

(3.3.20)



Therefore, for a flow which is incompressible and irrotational, Eqs. (3.3.16) and

(3.3.20) can be combined to yield a second order Laplace equation in a three-

dimensional plane.

( )2

2 2 2

2 2 2

. 0

or, 0

or, 0x y z

φ

φ

φ φ φ

∇ ∇ =

∇ =

∂ ∂ ∂+ + =

∂ ∂ ∂

(3.3.21)

Thus, any irrotational, incompressible flow has a velocity potential and stream

function (for two-dimensional flow) that both satisfy Laplace equation. Conversely,

any solution of Laplace equation represents both velocity potential and stream

function (two-dimensional) for an irrotational, incompressible flow.

An irrotational flow allows a velocity potential to be defined and leads to

simplification of fundamental equations. Instead of dealing with the velocity

components , andu v w as unknowns, one can deal with only one parameterφ , for a

given problem. Since, the irrotational flows are best described by velocity potential,

such flows are called as potential flows. In these flows, the lines with constant φ , is

known as equipotential lines. In addition, a line drawn in space such that φ∇ is the

tangent at every point is defined as a gradient line and thus can be called as

streamline.

Stream Function vs Velocity Potential

The velocity potential is analogous to stream function in a sense that the derivatives of

both andφ ψ yield the flow field velocities. However, there are distinct differences

between andφ ψ :

The flow field velocities are obtained by differentiating φ in the same

direction as the velocities, whereas, ψ is differentiated normal to the velocity

direction.

The velocity potential is defined for irrotational flows only. In contrast, stream

function can be used in either rotational or irrotational flows.

The velocity potential applies to three-dimensional flows, whereas the stream

function is defined for two dimensional flows only.



It is seen that the stream lines are defined as lines of constant ψ which are same

as gradient lines and perpendicular to lines of constant φ . So, the equipotential lines

and stream lines are mutually perpendicular. In order to illustrate the results more

clearly, let us consider a two-dimensional, irrotational, incompressible flow in

Cartesian coordinates.

For a streamline, ( ), constantx yψ = , and the differential of ψ is zero.

constant

0

or, 0

or,

d dx dyx y

d v dx u dydy vdx uψ

ψ ψψ

ψ

=

∂ ∂= + =∂ ∂= − + =

=

(3.3.22)

Similarly, for an equipotential line, ( ), constantx yφ = , and the differential of φ is

zero.

constant

0

or, 0

or,

d dx dyx y

d u dx v dydy udx vφ

φ φφ

φ

=

∂ ∂= + =∂ ∂= + =

= −

(3.3.23)

Combining Eqs. (3.3.22) and (3.3.23), we can write,

( )constant constant

1dydx dy dxψ φ= =

= −

(3.3.24)

Hence, the streamlines and equipotential lines are mutually perpendicular.




(Basic Potential Flows - I)

Potential Theory

In a plane irrotaional flow, one can use either velocity potential or stream function to

define the flow field and both must satisfy Laplace equation. Moreover, the analysis

of this equation is much easier than direct approach of fully viscous Navier-Stokes

equations. Since the Laplace equation is linear, various solutions can be added to

obtain other solutions. Thus, if we have certain basic solutions, then they can be

combined to obtain complicated and interesting solutions. The analysis of such flow

field solutions of Laplace equation is termed as potential theory. The potential theory

has a lot of practical implications defining complicated flows. Here, we shall discuss

the stream function and velocity potential for few elementary flow fields such as

uniform flow, source/sink flow and vortex flow. Subsequently, they can be

superimposed to obtain complicated flow fields of practical relevance.

Governing equations for irrotational and incompressible flow

The analysis of potential flow is dealt with combination of potential lines and

streamlines. In a planner flow, the velocities of the flow field can be defined in terms

of stream functions ( ){ },x yψ and potential functions ( ){ },x yφ as,

( ){ } ( ){ }

( ){ } ( ){ }

, ,;

, ,;

x y x yu v

y xx y x y

u vx y

ψ ψ

φ φ

∂ ∂= = −

∂ ∂

∂ ∂= =

∂ ∂

(3.4.1)

The stream function ψ is defined such that continuity equation is satisfied whereas,

for low speed irrotational flows ( )0V∇× =

, if the viscous effects are neglected, the

continuity equation ( )0V•∇ =

, reduces to Laplace equation for φ . Both the functions

satisfy the Laplace equations i.e.

2 2 2 2

2 2 2 20; 0x y x yφ φ ψ ψ∂ ∂ ∂ ∂+ = + =

∂ ∂ ∂ ∂ (3.4.2)



Thus, the following obvious and important conclusions can be drawn from Eq.

(3.4.2);

• Any irrotational, incompressible and planner flow (two-dimensional) has a

velocity potential and stream function and both the functions satisfy Laplace

equation.

• Conversely, any solution of Laplace equation represents the velocity potential

or stream function for an irrotational, incompressible and two-dimensional

flow.

Note that Eq. (3.4.2) is a second-order linear partial differential equation. If there are

n separate solutions such as, ( ) ( ) ( )1 2, , , ,........, ,nx y x y x yφ φ φ then the sum (Eq. 3.4.3)

is also a solution of Eq. (3.4.2).

( ) ( ) ( ) ( )1 2, , , ........ ,nx y x y x y x yφ φ φ φ= + + + (3.4.3)

It leads to an important conclusion that a complicated flow pattern for an irrotational,

incompressible flow can be synthesized by adding together a number of elementary

flows which are also irrotational and incompressible. However, different values of

orφ ψ represent the different streamline patterns of the body and at the same time

they satisfy the Laplace equation. In order to differentiate the streamline patterns of

different bodies, it is necessary to impose suitable boundary conditions as shown in

Fig. 3.4.1. The most common boundary conditions include far-field and wall

boundary conditions. on the surface of the body (i.e. wall).

Fig. 3.4.1: Boundary conditions of a streamline body.



Far away from the body, the flow approaches uniform free stream conditions in

all directions. The velocity field is then specified in terms of stream function and

potential function as,

0

u Vy x

vx y

ψ φ

ψ φ

∞

∂ ∂= = =∂ ∂∂ ∂

= − = =∂ ∂

(3.4.4)

On the solid surface, there is no velocity normal to the body surface while the tangent

at any point on the surface defines the surface velocity. So, the boundary conditions

can be written in terms of stream and potential functions as,

0; 0n sφ ψ∂ ∂= =

∂ ∂ (3.4.5)

Here, s is the distance measured along the body surface and n is perpendicular to the

body. Thus, any line of constant ψ in the flow may be interpreted as body shape for

which there is no velocity normal to the surface. If the shape of the body is given by

( )by f x= , then =constantbbody y yψ ψ == is alternate boundary condition of Eq. (3.4.5).

If we deal with wall boundary conditions in terms of andu v , then the equation of

streamline evaluated at body surface is given by,

( )constant constant

1b

surfaceb

dy vdx dy dx uψ φ= =

= − =

(3.4.6)

It is seen that lines of constant φ (equi-potential lines) are orthogonal to lines of

constant ψ (streamlines) at all points where they intersect. Thus, for a potential flow

field, a flow-net consisting of family of streamlines and potential lines can be drawn,

which are orthogonal to each other. Both the set of lines are laplacian and they are

useful tools to visualize the flow field graphically.



Referring to the above discussion, the general approach to the solution of

irrotational, incompressible flows can be summarized as follows;

• Solve the Laplace equation for orφ ψ along with proper boundary

conditions.

• Obtain the flow velocity from Eq. (3.4.1)

• Obtain the pressure on the surface of the body using Bernoulli’s equation.

2 21 12 2

p V p Vρ ρ ∞∞+ = + (3.4.7)

In the subsequent section, the above solution procedure will be applied to some basic

elementary incompressible flows and later they will be superimposed to synthesize

more complex flow problems.

Uniform Flow

The simplest type of elementary flow for which the streamlines are straight, parallel

with constant velocity, is known as uniform flow. Consider a uniform flow in positive

x-direction as shown in Fig. 3.4.2. This flow can be represented as,

; 0u V v∞= = (3.4.8)

Fig. 3.4.2: Schematic representation of a uniform flow.



The uniform flow is a physically possible incompressible flow that satisfies continuity

equation ( )0V•∇ =

and the flow is irrotational ( )0V∇× =

. Hence, the velocity

potential can be written as,

; 0

; 0

u V vx y

u V vy y

φ φ

ψ ψ

∞

∞

∂ ∂= = = =

∂ ∂∂ ∂

= = = − =∂ ∂

(3.4.9)

Integrating Eq.(3.4.9) with respect to x,

( ) ( )( ) ( )

( ) ( )( ) ( )

1 1 1

1 1 1

1

2 2 2

2 2 2

2

;

and

;

and

V x f y g x C

g x V x f y CV x C

V y f x g y C

g y V y f x CV y C

φ φ

φψ ψ

ψ

∞

∞

∞

∞

∞

∞

= + = +

⇒ = =

⇒ = +

= + = +

⇒ = =

⇒ = +

(3.4.10)

In practical flow problems, the actual values of andφ ψ are not important, rather it is

always used as differentiation to obtain the velocity vector. Hence, the constant

appearing in Eq. (3.4.10) can be set to zero. Thus, for a uniform flow, the stream

functions and potential function can be written as,

;V x V yφ ψ∞ ∞= = (3.4.11)



Fig. 3.4.3: Flow nets drawn for uniform flow.

When Eq. (3.4.11), is substituted in Eq. (3.4.2), Laplace equation is satisfied. Further,

if the uniform flow is at an angle θ with respect to x-axis as shown in Fig. 3.4.2, then

the generalized form of stream function and potential function is represented as

follows;

( ) ( )cos sin ; cos sinV x y V y xφ θ θ ψ θ θ∞ ∞= + = − (3.4.12)

The flow nets can be constructed by assuming different values of constants in Eq.

(3.4.11) and with different angle θ as shown in Fig. 3.4.3. The circulation in a

uniform flow along a closed curve is zero which gives the justification that the

uniform flow is irrotational in nature.

0C C

V ds V ds• ∞Γ = = =∫ ∫

(3.4.13)



Source/Sink Flow

Consider a two-dimensional incompressible flow where the streamlines are radially

outward from a central point ‘O’ (Fig. 3.4.4). The velocity of each streamlines varies

inversely with the distance from point ‘O’. Such a flow is known as source flow and

its opposite case is the sink flow, where the streamlines are directed towards origin.

Both the source and sink flow are purely radial. Referring to the Fig. 3.4.4, if

andrv vθ are the components of velocities along radial and tangential direction

respectively, then the equations of the streamlines that satisfy the continuity equation

( )0V•∇ =

are,

; 0rcv vr θ= = (3.4.14)

Here, the constant c can be related to the volume flow rate of the source. If we define

Λ as the volume flow rate per unit length perpendicular to the plane, then,

( )2

or, and2 2

r

r

r v

v cr

π

π π

Λ =

Λ Λ= =

(3.4.15)

The potential function can be obtained by writing the velocity field in terms of

cylindrical coordinates. They may be written as,

1; 02

1 ; 02

r

r

v vr r r

v vr r y

θ

θ

φ φπ θ

ψ ψθ π

∂ Λ ∂= = = =

∂ ∂∂ Λ ∂

= = − = =∂ ∂

(3.4.16)



Fig. 3.4.4: Schematic representation of a source and sink flow.

Integrating Eq. (3.4.16) with respect to andr θ , we can get the equation for potential

function and stream function for a source and sink flow.

( ) ( )

( ) ( )

( ) ( )

( ) ( )

3 3 3

3 3 3

3

4 4 4

4 4 4

4

ln ;2

and ln2

ln2

;2

and2

2

r f C g r

f C g r r

r C

f r C g

f r C g

C

φ θ φπ

θπ

φπ

ψ θ ψ θπ

θπ

ψ θπ

Λ= + = +

Λ⇒ = =

Λ⇒ = +

Λ= + = +

Λ⇒ = =

Λ⇒ = +

(3.4.17)

The constant appearing in Eqs (3.4.17) can be dropped to obtain the stream function

and potential function.

ln ;2 2

rφ ψ θπ πΛ Λ

= = (3.4.18)

This equation will also satisfy the Laplace equation in the polar coordinates. Also, it

represents the streamlines to be straight and radially outward/inward depending on the

source or sink flow while the potential lines are concentric circles shown as flow nets

in Fig. 3.4.5. Both the streamlines and equi-potential lines are mutually perpendicular.



It is to be noted from Eq. (3.4.15) that, the velocity becomes infinite at origin

( )0r = which is physically impossible. It represents a mathematical singularity where

the continuity equation ( )0V•∇ =

is not satisfied. We can interpret this point as

discrete source/sink of given strength with a corresponding induced flow field about

this point. Although the source and sink flows do not exist, but many real flows can

be approximated at points, away from the origin, using the concept of source and sink

flow.

Fig. 3.4.5: Flow nets drawn for of a source and sink flow.




(Basic Potential Flows - II)

Doublet Flow

The third type of basic elementary flow is the combination of source-sink pair of

equal strength in a special fashion. It leads to singularity called as doublet. Consider a

source and sink pair of equal strength Λ and separated by a distance 2a as shown in

Fig. 3.5.1. At a point ‘P’ in the flow field, the combined stream function of the pair

can be written as,

( )1 22ψ θ θ

πΛ

= − − (3.5.1)

Fig. 3.5.1: Combination of source and sink located along x-axis.

Eq. (3.5.1) can be re-written in the following form.

( )

( )

1 2

1 21 2

1 2

2

tan tan2tan tan1 tan tan

πψ θ θ

θ θπψ θ θθ θ

− = − Λ − ⇒ − = − = Λ +

(3.5.2)

Referring to Fig. 3.5.1 and using the concepts of trigonometry, the following relations

can be obtained i.e.

1 2sin sintan ; tan

cos cosr r

r a r aθ θθ θ

θ θ= =

− + (3.5.3)



Substituting these results in Eq. (3.5.2), we can obtain,

( ) ( )

2 2

12 2

2 2

2 2 sintan

2 sintan2

sin For small values of

arr a

arr a

a r ar a

πψ θ

θψπ

θψπ

−

− = Λ − Λ ⇒ = − − Λ

⇒ = −−

(3.5.4)

Now, a doublet can be formed by bringing the source-sink pair as close to each other

( )0a → while increasing its strength ( )Λ →∞ and keeping the product ( )a πΛ

constant. Then, ( )2 2 1r r a r− → and Eq. (3.5.4) can be simplified as,

sinr

κ θψ = − (3.5.5)

Here, ( )2aκ π= Λ is a constant and called as strength of the doublet. Since the

stream function and potential function is mutually perpendicular, we can write the

velocity potential for the doublet as,

cosr

κ θφ = (3.5.6)

Combining Eqs (3.5.5) and (3.5.6), the flow nets can be drawn as shown in Fig. 3.5.2.

Fig. 3.5.2: Flow nets drawn for doublet flow.



The equation of streamline can be obtained from Eq.(3.5.5) for a doublet flow i.e.

( )sin constant

sin

cr

rc

κ θψ

κ θ

= − =

⇒ =

(3.5.7)

Fig. 3.5.3: Streamlines for doublet flow.

It may be seen from the Fig. (3.5.3) and Eq. (3.5.7) that the streamlines for a doublet

flow are the family of circles with diameter cκ

if the source and sink are placed at

origin ( )0a → . The different circles correspond to the different values of the

parameter c . The doublet has associated with a direction with which the flow moves

around the circular streamlines. By convention, the direction of the doublet is shown

by an arrow drawn from sink to the source so that Eq. (3.5.5) is consistent with

respect to sign convention.

Vortex Flow

Three types of elementary flows (uniform flow, source/sink flow and doublet flow)

have been discussed earlier. Now, the last elementary flow will be introduced called

as vortex flow. Consider a flow field in which the streamlines are concentric circles

about a given point which is exactly opposite case when the velocity potential and



stream function for the source is interchanged. Here, the velocity along any given

circular streamline is constant, while it can vary inversely with distance from one

streamline to another from a common center. Referring to the Fig. 3.5.4, if andrv vθ

are the components of velocities along radial and tangential direction respectively,

then the flow field can be described as given below,

0;rcv vrθ= = (3.5.8)

Fig. 3.5.4: Schematic representation of a vortex flow.

It may be easily shown that streamlines satisfy the continuity equation i.e. 0V•∇ =

and the vortex flow is irrotational i.e. 0V∇× =

at every point except origin ( )0r = .

In order to evaluate the constant appearing in Eq. (3.5.8), let us take the circulation

around a given streamline of radius r :

( )2

2

CV ds v r

vr

θ

θ

π

π

Γ = = −

Γ⇒ = −

∫

(3.5.9)

It may be seen by comparing Eqs. (3.5.8) and (3.5.9) that

22

c cππΓ

= − ⇒ Γ = − (3.5.10)

Thus, the circulation taken about all the streamlines is the same value. So, it is called

as the strength of the vortex flow while the velocity field is given by Eq. (3.5.9). It

may be noted that vθ is negative when Γ is positive i.e. vortex of positive strength

rotates in clockwise direction. Now, let us obtain the velocity potential and stream

function for the vortex flow from the velocity field. By definition,



10;2

1 0;2

r

r

v vr r r

v vr r r

θ

θ

φ φθ π

ψ ψθ π

∂ ∂ Γ= = = = −

∂ ∂∂ ∂ Γ

= = − = = −∂ ∂

(3.5.11)

Integrating the above equations, the velocity potential and stream function are

obtained as,

; ln2 2

rφ θ ψπ πΓ Γ

= − = (3.5.12)

Once again it is clear from this equation that streamlines ( )constantψ = for a vortex

flow is given by concentric circles with fixed radius while equipotential lines

( )constantφ = are the straight radial lines from the origin with constant θ . Both

streamlines and equipotential lines are mutually perpendicular as shown in Fig. 3.5.5.

Fig. 3.5.5: Flow nets drawn for of a free vortex flow.

The following general remarks may be made for the vortex flow;

• The vortex flow is irrotational everywhere in the flow field except at origin

where the vorticity is infinite. Hence the origin 0r = is a singular point in the

flow field which may be interpreted as point vortex and it induces circular

vortex flow as shown in the Fig. 3.5.4. The irrotational vortex is usually called

as free vortex. The swirling motion of water as it drains out of a bathtub is an

example of free vortex.



• When the fluid particles rotate as a rigid body such that

1 1; is a constantv c r cθ = , then the vortex motion is rotational and it cannot be

described with velocity potential. This type of rotational vortex is commonly

called as forced vortex. The motion of a liquid contained in a tank, when

rotated about its axis with angular velocity ω corresponds to a forced vortex.

• A combined vortex is the one with a forced vortex as the central core and

velocity distribution corresponding to that of free vortex outside the core.

( ) ( )20 0;r

cv r r v r r rr θ ω= > = ≤ (3.5.13)

Here, 2 andc ω are constants and 0r is corresponds to the radius of the central

core.




(Superposition of Potential Flows - I)

Method of Superposition

The potential flows are governed by the linear partial differential equation commonly

called as “Laplace Equation”. The elementary basic plane potential flows include

uniform flow, source/sink flow, doublet flow and free vortex flow. The details of

these flow fields have already been discussed and are summarized in the following

Table 3.6.1. A variety of interesting potential flow can be obtained by combination of

velocity potential and stream function of basic potential flows.

In an inviscid flow field, a streamline can be considered as a solid

boundary because there is no flow through it. Moreover, the conditions along the sold

boundary and the streamline are the same. Hence, the combinations of velocity

potential and stream functions of elementary flows will lead to a particular body

shape that can be interpreted as flow around that body. The method of solving such

potential flew problems is commonly called as, method of superposition.



Table 3.6.1: Summary of basic, plane potential flows

Description of Flow Field Velocity Potential Stream Function Velocity components

Uniform flow at an angle θ with x-axis

( )cos sinV x yφ θ θ∞= +

( )cos sinV y xψ θ θ∞= −

cossin

u Vv V

θθ

∞

∞

==

Source ( )0Λ > or Sink

( )0Λ < ln

2rφ

πΛ

= 2

ψ θπΛ

= 20

rvr

vθπΛ

=

=

Doublet cosr

κ θφ =

sinr

κ θψ = −

2

2

cos

sin

rvr

vrθ

κ θ

κ θ

= −

= −

Free Vortex

Counter-clockwise ( )0Γ <

Clockwise ( )0Γ >

2φ θ

πΓ

= − ln2

rψπΓ

= 0

2

rv

vrθ π

=Γ

=

Combination of a Uniform Flow with a Source

A source of strength Λ , located at origin is superimposed with a uniform stream with

velocity V∞ as shown in Fig. 3.6.1. The resulting stream function can be written as,

uniform source sin2

V rψ ψ ψ θ θπ∞

Λ= + = + (3.6.1)

Fig. 3.6.1: Superposition of uniform flow and a source.



The streamlines of the combined flow leads to the flow over a semi-infinite body and

are obtained as,

sin constant2

V rψ θ θπ∞

Λ= + = (3.6.2)

The velocity field is obtained from stream function through differentiation i.e.

1 cos ; sin2rv V v V

r r rθψ ψθ θθ π∞ ∞

∂ Λ ∂= = + = − = −

∂ ∂ (3.6.3)

From the Fig. 3.6.1, it is clear that the flow becomes stagnant at some point because

the velocity due to the source will cancel with that of uniform flow. Hence, a

stagnation point is created which is obtained by assigning the velocity components to

zero value in Eq. (3.6.3).

cos 0; sin 02

V Vr

θ θπ∞ ∞

Λ+ = = (3.6.4)

Solving for andr θ , the coordinates of stagnation point ‘B’ is found out i.e.

( ) ( ), ,r bθ π= and the distance 2

bVπ ∞

Λ= , directly upstream of the source. It may be

observed that the point ‘B’ will be blown further downstream, if source strength is

increased keeping V∞ same, and increasing V∞ , keeping source strength same. When

the coordinates of ‘B’ is substituted in Eq.(3.6.1), the streamline ‘ABC’ passing

through the stagnation point is obtained i.e.

stagnation sin constant2 2 2

VV

ψ π ππ π∞

∞

Λ Λ Λ= + = =

(3.6.5)



Since, 2bVπ ∞

Λ= , it follows that the equation of the streamline passing through the

stagnation point is obtained from Eq. (3.6.1) as follows;

( )

stagnation 2

sin

; varies from 0 to 2sin

bV

bVV r bV

br

ψ π

πθ θ ππ

π θθ π

θ

∞

∞∞ ∞

Λ= =

⇒ + =

−⇒ =

(3.6.6)

The streamline ‘ABC’ following the equation (3.6.6) is shown in Fig. 3.6.1. The

following important observations can be made from the figure;

• The streamline ‘ABC’ contains the stagnation point at ‘B’ and separates the

flow coming from the free stream and fluid emanating from source. All the

fluid outside ‘ABC’ are from the free stream while the fluid inside ‘ABC’ are

from the source. Hence, the singularity in the flow field (i.e. source) occurs

inside the body whereas there is no singularity in the free stream (outside

‘ABC’).

• In inviscid flow, the velocity at the surface of the solid body is tangent to the

body. So, any streamline of this combined flow field can be replaced by a

solid surface of same shape. Hence, with respect to free stream, the flow

would not feel the difference if the streamline ‘ABC’ is replaced with a solid

body. The streamline stagnation 2ψ Λ

= extends downstream to infinity, forming a

semi-infinite body and is called as ‘Rankine Half-Body”.

• Referring to Eq. (3.6.6), it is seen that the width of the half body ( )y b π θ= −

asymptotically approaches to 2 bπ while the half-width is given by bπ± when

0 or 2θ π→ .

• For the half-body shown in Fig. 3.6.1, the magnitude of velocity at any point is

given by the following equation; 2

2 2 2 2

22 2

2

cos and2 2

21 cos

rVV v v V b

r r V

b bV Vr r

θθ

π π π

θ

∞∞

∞

∞

Λ Λ Λ = + = + + =

⇒ = + +

(3.6.7)



Using the Bernoulli’s equation, the pressure any arbitrary point on the half-body

can be obtained with the knowledge of free stream pressure ( )p∞ and velocity

( )V∞ .

2 21 12 2

p V p Vρ ρ∞ ∞ ∞+ = + (3.6.8)

Here, the elevation change is neglected.

Combination of a Uniform Flow with a Source-Sink Pair

The superposition of a uniform flow with a source results in the flow over the semi-

infinite body ‘ABC’ as shown in Fig. 3.6.1. This is a half-body that stretches to

infinity in the downstream and the body is not closed. If a sink of equal strength is

added to this flow in the downstream and equally spaced from origin as that of source,

then the resulting shape will be closed. The resulting flow field is shown in Fig. 3.6.2.

Fig. 3.6.2: Superposition of uniform flow and a source-sink pair.



Let us consider a source and sink with strengths andΛ −Λ equally spaced at

a distance b from the origin. When a uniform stream of velocity V∞ is superimposed

on it, the combined flow at any point in the flow can be written as follow;

( )uniform source sink 1 2

12 2

12 2 2

sin2

2 sinor, sin tan2

2or, tan2

V r

brV rr bbyV y

x y b

ψ ψ ψ ψ θ θ θπ

θψ θπ

ψπ

∞

−∞

−∞

Λ= + + = + −

Λ = − − Λ

= − + −

(3.6.9)

From the geometry of Fig. 3.6.2, it is seen that 1 2andθ θ are the functions of

, andr bθ . Assigning the velocity components to zero value, two stagnation points

can be obtained in similar manner and they are located at points ‘A and B’ in the Fig.

3.6.2. Their distances from origin can also be calculated.

2

1

bOA OB l bV

lb V b

π

π

∞

∞

Λ= = = +

Λ⇒ = +

(3.6.10)

The equation of the streamline is given by,

( )1 2sin constant2

V rψ θ θ θπ∞

Λ= + − = (3.6.11)

The equation of specific streamlines passing through the stagnation points

( ) ( )1 2 1 2A and B 0θ θ θ π θ θ θ= = = = = = and B ( )1 2 0θ θ θ= = = ’ is obtained by

assigning the constant appearing in Eq. (3.6.11) as 0.

( )1 2sin 02

V rψ θ θ θπ∞

Λ= + − = (3.6.12)



The body-half width ( )h can be obtained by determining the value y where the y-axis

intersects streamline 0.ψ = , Thus, from Eq. (3.6.9), one can obtain the body half

width with y h= .

21 21 tan2

V bh h hb b b

π ∞ = − Λ

(3.6.13)

From the above mathematical analysis the following physical interpretation can be

made;

• The stagnation streamline described by Eq. (3.6.12), is the equation of an oval

and is the dividing streamline. This particular shape is called as “Rankine

Oval”. All the flow from the source is consumed by the sink and is contained

entirely inside the oval. The flow outside the oval is originated through

uniform flow only and can be interpreted as inviscid, irrotational and

incompressible flow over solid body. Also, the potential solution for the

Rankine oval gives the reasonable approximation of velocity outside the thin,

viscous boundary layer and pressure distribution on the front part of the body.

• Using Eqs. (3.6.10) and (3.6.13), a large variety of body shapes with different

length to width ratio can be obtained for different values of the parameter

V b∞ Λ

. As this parameter becomes large, the flow around a slender body is

described while the smaller values give the flow field around a blunt shape

body.




(Superposition of Potential Flows - II)

Non-Lifting Flow over a Circular Cylinder

It is seen earlier that flow over a semi-infinite body can be simulated by combination

of a uniform flow with a source and flow over an oval-shaped body can be

constructed by superimposing a uniform flow and a source-sink pair. A circular

cylinder is one of the basic geometrical shapes and the flow passing over it can be

simulated by combination of a uniform flow and doublet. When the distance between

source-sink pair approaches zero, the shape Rankine oval becomes more blunt and

approaches a circular shape.

Consider the superposition of a uniform flow of velocity V∞ and a

doublet of strength Λ as shown in Fig. 3.7.1. The direction of the doublet is upstream,

facing into uniform flow.

Fig. 3.7.1: Superposition of a uniform flow and doublet.

The stream function for the combined flow is,

uniform doublet 2

2

2

sinsin sin 12 2

sin 1 ;2

V r V rr V r

RV r Rr V

θψ ψ ψ θ θπ π

ψ θπ

∞ ∞∞

∞∞

Λ Λ = + = − = −

Λ⇒ = − =

(3.7.1)



The velocity field is obtained as,

2 2

2 2

1 1 cos ; 1 sinrR Rv V v V

r r r rθψ ψθ θθ ∞ ∞

∂ ∂= = − = − = − + ∂ ∂

(3.7.2)

In order to locate the stagnation point, assign the velocity components in Eq. (3.7.2) to

zero value and simultaneously solve for andr θ . There are two stagnation points,

located at ( ) ( ) ( ), ,0 and ,r R Rθ π= and denoted by points A and B, respectively. The

equation of streamlines that passes through the stagnation points A and B, is given by

the following expression;

2

2sin 1 0RV rr

ψ θ∞

= − =

(3.7.3)

This equation is satisfied by r R= for all values of θ . Since R is a constant, Eq.

(3.7.3) may be interpreted as the equation of a circle with radius R with center at the

origin. It is satisfied by 0 andθ π= for all values of R . Different values of R may

be obtained by varying the uniform velocity and/or doublet strength. Hence, entire

horizontal axis through the points A and B, extending infinitely far upstream and

downstream, is a part of stagnation streamline. The above discussions can be

summarized as follows;

• The dividing streamline 0ψ = that passes through the stagnation points A and

B as shown in Fig. 3.7.1.

• The dividing streamline is a circle of radius R . The family of circles can be

obtained by assigning different values of R with various doublet strength and

free stream velocity.

• The flow inside the circle is generated from the doublet whereas flow outside

the circle comes from the uniform flow. So, the flow inside the circle may be

replaced by solid body and the external flow will not feel the difference.

• Thus, the inviscid, irrotational, incompressible flow over a circular cylinder of

radius R can be simulated by adding a uniform flow of velocity V∞ and a

doublet of strength Λ and R is related to andV∞ Λ .

2R

Vπ ∞

Λ= (3.7.4)



Referring to the Fig. 3.7.1, it is seen that the entire flow field is symmetrical

about both horizontal and vertical axes through the center of the cylinder. It means the

pressure distribution is also symmetrical about both the axes. When the pressure

distribution over the top part of the cylinder is exactly balanced by the bottom part,

there is no lift. Similarly, when the pressure distribution on the front part of the

cylinder is exactly balanced by rear portion, then there is no drag. This is in contrast

to the realistic situation i.e. a generic body placed in a flow field will experience finite

drag and zero lift may be possible. This paradox between the theoretical result of zero

drag in an inviscid flow and the knowledge of finite drag in real flow situation is

known as d’ Alembert’s paradox.

Pressure Coefficient

In general, pressure is a dimensional quantity. Many a times, it is expressed in a non-

dimensional form with respect to free stream flow and the ‘pressure coefficient’ is

defined as follows;

212

pp p p pc

q Vρ

∞ ∞

∞∞ ∞

− −= =

(3.7.5)

Here, and Vρ∞ ∞ are the free stream density and free stream velocity, respectively. The

term q∞ is called as dynamic pressure. For incompressible flow, if a body is

immersed in the free stream, then Bernoulli’s equation can be written at any arbitrary

point in the flow field as,

( )

2 2

2 2

2

1 12 2

12

1p

p V p V

p p V V

p p Vcq V

ρ ρ

ρ

∞ ∞ ∞

∞ ∞

∞

∞ ∞

+ = +

⇒ − = −

−⇒ = = −

(3.7.6)

On the surface of the cylinder ( )r R= shown in Fig. 3.7.1, the velocity distribution

can be obtained from Eq. (3.7.2) i.e.

0; 2 sinrv v Vθ θ∞= = − (3.7.7)



Fig. 3.7.2: Maximum velocity in the flow over a circular cylinder.

Here, rv is geometrically normal to the surface and vθ

is tangential to the surface of

the cylinder as shown in Fig. 3.7.2(a). The negative sign signifies that vθ is positive

in the direction of increasing θ . It may be observed that the velocity at the surface

reaches to maximum value of 2V∞ at the top and bottom of the cylinder as shown in

Fig. 3.7.2(b). Eqs. (3.7.6) and (3.7.7) can be combined to obtain the surface pressure

coefficient as,

21 4sinpc θ= − (3.7.8)

The pressure distribution over the cylinder is plotted in Fig. 3.7.3. Here, pc varies

from 1 at the stagnation point to -3 at the points of maximum velocity. It is also clear

that the pressure distribution at the top half the cylinder is equal to the bottom half and

hence the lift is zero. Similarly, the pressure distribution on the front part of the

cylinder is exactly balanced by rear portion and there is no drag. Both, normal force

and axial force coefficients ( )andn ac c are same as lift and drag coefficients. They

are calculated from pc as given below;

( ) ( ), , , ,0

1 10; 0c TE

l n p l p u d a p u p lLE

c c c c dx c c c c dyc c

= = − = = = − =∫ ∫ (3.7.9)



Here, LE and TE stands for leading edge and trailing edge, respectively. The

subscripts u and l refers to upper and lower surface of the cylinder. The chord c is the

diameter of the cylinder ( )R .

Fig. 3.7.3: Surface pressure coefficient for a circular cylinder.

When the surface pressure matches with free stream pressure, then Eq. (3.7.8) reduces

to,

2 0 0 0 011 4sin 0 sin 30 , 150 , 210 , 3302pc θ θ θ= − = ⇒ = ± ⇒ = (3.7.10)



These points as well as the stagnation points and location of minimum pressure are

illustrated in Fig. 3.7.4.

Fig. 3.7.4: Pressure values at various locations on the surface of the cylinder.



Module 3 : Lecture 8 INVISCID INCOMPRESSIBLE FLOW (Superposition of Potential Flows - III)

Lifting Flow over a Circular Cylinder

When a doublet flow is superimposed on a uniform flow, the combined flow fields

can be visualized as possible flow pattern over a circular cylinder. In addition, both

lift and drag force are zero for such flows. However, there are other possible flow

patterns around a circular cylinder resulting non-zero lift. Such lifting flows are

discussed here.

Consider the flow synthesized by addition of the non-lifting flow over

a cylinder and a vortex of strength Γ as shown in Fig. 3.8.1. The stream function of

for a circular cylinder of radius R is given by the following equation.

2

1 2sin 1 ;2

RV r Rr V

ψ θπ∞

∞

Λ= − =

(3.8.1)

Fig. 3.8.1: Superposition of non-lifting flow over a cylinder and a vortex.

As discussed in the previous lecture, different values of R can be obtained by

assigning the various values of doublet strength ( )Λ and uniform free stream velocity

( )V∞ to synthesize the flow over a circular cylinder. Now, the stream function for a

vortex of strength Γ may be written as,

2 1ln ln ln ln2 2 2 2

rr c r RR

ψπ π π πΓ Γ Γ Γ = + = − =

(3.8.2)



Since 1c is any arbitrary constant, it can be replaced with ln2

RπΓ −

in the Eq.

(3.8.2). The resulting stream function for the flow pattern is given by the sum of the

stream functions, i.e.

2

1 2 2sin 1 ln2

R rV rr R

ψ ψ ψ θπ∞

Γ = + = − +

(3.8.3)

The streamlines expressed by Eq. (3.8.3), represents the equation of a circle of radius

R . A special case will arise that will represent the flow over a circular cylinder when

0Γ = . If r R= then 0ψ = for all values of θ . The velocity fields can be obtained

by differentiating Eq. (3.8.3) i.e.

2

2

2

2

1 1 cos

1 sin2

rRv V

r r

Rv Vr r rθ

ψ θθ

ψ θπ

∞

∞

∂= = − ∂

∂ Γ= − = − + − ∂

(3.8.4)

In order to locate the stagnation points, one can put 0rv vθ= = in Eq. (3.8.4) and

solve for resulting coordinates ( ),r θ :

sin4

r RV R

θπ ∞

Γ= ⇒ = −

(3.8.5)

Since Γ is a positive number, the value of θ must be lie in the range and 2π π and

there are three possibilities;

Case I: If 14 V Rπ ∞

Γ<

, then the two stagnation points are shown by the points ‘A

and B’ lies in the bottom half of the cylinder as shown in Fig. 3.8.2. The locations of

these points are given by the Eq. (3.8.5).



Case II: If 14 V Rπ ∞

Γ=

, then there is one stagnation point on the surface of the

cylinder at the point ‘C’ as shown in Fig. 3.8.2. It means that the point ‘A and B’

come closer meet at point ‘C’ on the surface at and2

r R πθ= = − .

Case III: When, 14 V Rπ ∞

Γ>

, no interpretation can be made from Eq. (3.8.5).

Referring to Eq. (3.8.4), the stagnation point 0rv = is satisfied for both

and or2 2

r R π πθ= = − . Now, substitute 2πθ = − in Eq. (3.8.4) and solve for r by

setting 0vθ = at the stagnation point.

22

4 4r R

V Vπ π∞ ∞

Γ Γ= ± −

(3.8.6)

Eq. (3.8.6) is a quadratic equation and the two possible solutions can be interpreted as

stagnation points: one lies inside the cylinder (point ‘D’) and other lies outside the

cylinder (point ‘E’) as shown in Fig. 3.8.2. Physically, the point ‘D’ is generated

within the cylinder r R< , when a doublet flow at origin is superimposed on a vortex

flow while the point ‘E’ lies on same vertical axis for r R> .

Fig. 3.8.2: Stagnation points for lifting flow over a cylinder.



Based on the results shown in Fig. 3.8.2, the following inferences can be made;

• The circular streamline 0 and r Rψ = = is one of the allowed streamline in

the synthesized flow field that divides the doublet flow and vortex flow. So,

one can replace it as a solid body i.e. circular cylinder and the external flow

will not feel the difference. The free stream can be considered as a vortex

flow.

• With reference to the solid body, the stagnation point ‘D’ has no meaning and

only point ‘E” is the meaningful stagnation point.

• Since, the parameter Γ can be chosen freely, there are infinite numbers of

possible potential flow solutions, for incompressible flow over a circular

cylinder. This is also true for incompressible potential flows over all smooth

two-dimensional bodies.

Lift and Drag Coefficients for Circular Cylinder

Intuitively, one can say that there is a finite normal force when a circular cylinder is

placed in a vortex flow while the drag is zero i.e. d’Alembert’s paradox still prevails.

Let us quantify the results;

First, the velocity on the surface of the cylinder ( )r R= can be written as,

2 sin2

V v VRθ θ

π∞

Γ= = − − (3.8.7)

The pressure coefficient is obtained as,

2 2 22 2 sin1 1 2sin 1 4sin

2 2pVcV RV RV RV

θθ θπ π π∞ ∞ ∞ ∞

Γ Γ Γ = − = − − − = − + +

(3.8.8)



The force coefficients can be obtained by integrating pressure coefficient and skin

friction coefficient over the surface. For the inviscid flow, there is no skin friction

coefficient. Hence, the drag coefficient is written as,

( ), , , ,1 1 1

2 2 2

TE TE TE

d p l p u p u p lLE LE LE

c c c dy c dy c dyR R R

= − = −∫ ∫ ∫ (3.8.9)

Converting Eq. (3.8.9) to polar coordinates by replacing sin ; cosy R dy R dθ θ θ= = ,

we can obtain,

0 2

, ,1 1cos cos2 2d p u p lc c d c d

π

π π

θ θ θ θ= −∫ ∫ (3.8.10)

Here, the first part of integration is performed from the leading edge (i.e. front point)

and moving over the top surface. In the second part, the integration is done from the

leading edge moving over the bottom portion of the cylinder. Finally, Eq. (3.8.10) can

be written as,

0 2 2

0

1 1 1cos cos cos2 2 2d p p pc c d c d c d

π π

π π

θ θ θ θ θ θ= − − = −∫ ∫ ∫ (3.8.11)

Substitute the value of pc from Eq.(3.8.8) in Eq.(3.8.11),

222

0

1 2 sin1 4sin cos2 2dc d

RV RV

π θθ θ θπ π∞ ∞

Γ Γ = − − + +

∫ (3.8.12)

Use the following trigonometric relations in Eq. (3.8.12);

2 2 22

0 0 0

cos 0; sin cos 0; sin cos 0d dπ π π

θ θ θ θ θ θ θ= = =∫ ∫ ∫ (3.8.13)

It leads to 0dc = , which implies that the drag on a cylinder in an inviscid,

incompressible flow is zero, regardless of whether or not the flow has circulation

about the cylinder. The lift can be evaluated in the similar manner from the first

principle i.e.

( )2 2 2

, , , ,0 0 0

1 1 12 2 2

R R R

l p l p u p l p uc c c dx c dx c dxR R R

= − = −∫ ∫ ∫ (3.8.14)



Converting Eq. (3.8.14) to polar coordinates by replacing

cos ; sinx R dx R dθ θ θ= = − , we can obtain,

2 0 2

, ,0

1 1 1sin sin sin2 2 2l p l p u pc c d c d c d

π π

π π

θ θ θ θ θ θ= − + = −∫ ∫ ∫ (3.8.15)

Substitute the value of pc from Eq.(3.8.8) in Eq.(3.8.15),

222

0

1 2 sin1 4sin sin2 2lc d

RV RV

π θθ θ θπ π∞ ∞

Γ Γ = − − + +

∫ (3.8.16)

Use the following trigonometric relations in Eq. (3.8.16);

2 2 22 3

0 0 0

sin 0; sin ; sin 0d dπ π π

θ θ θ π θ θ= = =∫ ∫ ∫ (3.8.17)

The lift coefficient can be obtained as,

212

lLc

RV RVV Sρ∞ ∞∞ ∞

′Γ Γ= ⇒ =

(3.8.18)

The value of lift per unit span ( )L′ can be obtained by considering the plan-form area

2S R= in Eq. (3.8.18) and after simplification, one can obtain,

L Vρ∞ ∞′ = Γ (3.8.19)

It is seen from Eq.(3.8.19) that the lift per unit span for a circular cylinder in a given

free stream flow is directly proportional to the circulation. This simple and powerful

relation is known as Kutta-Joukowski theorem. This result shows the importance of

the concept circulation and the same result can be extended for two-dimensional

bodies. The inviscid potential flow does not provide proper explanation for drag

calculation because zero drag in a flow field is quite un-realistic. Because of viscous

effects the flow separates from the rear part of the cylinder, creating recirculating flow

in the wake downstream of the body. This separated flow greatly contributes the finite

drag measured for the cylinder. However, the prediction of lift by Kutta-Joukowski

theorem is quite realistic.



Magnus Effect

The general idea of generation of lift for a spinning circular cylinder can be extended

to sphere. Here, non-symmetric flows are generated due to spinning of bodies in all

dimensions. It leads to the generation of aerodynamic force perpendicular to the

body’s angular velocity vector. This phenomenon is called as Magnus effect. The

typical examples include the spinning of three-dimensional object such as soccer,

tennis and golf balls where the side force is experienced.

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