Chap_1_May_12

UEME3112

Fluid Mechanics II

1

Chapter 1 Inviscid Flow

2

Outline

Minor Riview

History of Potential Flow and Boundary Layer

Types of Motion or Deformation of Fluid Elements

Rotationality

3

Rotationality

Irrotational Flow Approximation

Continuity Equation

Stream Function

Velocity Potential

Elementary Flows

Complex Flows

Classification of Fluid Mechanics

Gas Liquids Statics Dynamics

0=∑ iF 0>∑ iF , Flows

Fluid Mechanics

Air, He, Ar,

N2, etc.

Water, Oils,

Alcohols, etc.

∑ i

Viscous/Inviscid

Steady/Unsteady

Compressible/

Incompressible

∑ i

Laminar/

Turbulent

Compressibility ViscosityVapor

Pressure

Density

PressureBuoyancy

Stability

Surface

Tension

Classification of Fluid Flows

� Viscous vs Inviscid Flow

� Internal vs External Flow

� Compressible vs Incompressible Flow� Compressible vs Incompressible Flow

� Laminar vs Turbulent Flow

� Steady vs Unsteady Flow

� One-, Two- and Three-Dimensional Flows

The Reynolds number describes

the degree of turbulence.

The Reynolds number describes

the degree of turbulence.

uDRe =

ρ

Reynolds Number

forceviscous

forceinertia

Du

Du

]/[

]/[

Re

2

2

==

=

ρµ

µ

Turbulent fluids have viscosity, but we usually measure

viscosity in laminar flow.

Wat er Dye

St ream line f low ing dye

� Laminar Flow: “Streamline” flow.

� Fluid moves in a

straight line with the

applied force.

� Layers slide by with

Wat er Dye

Turbulent f lowQuickTime?and a

TIFF (Uncompressed) decompressorare needed to see this picture.

� Layers slide by with

no swirls.

� Turbulent Flow: “Disorderly” flow.

� Small packets of fluid

moving in all directions

and all angles to

normal line of flow.

Viscosity

Some fluids move slowly.

They have a high viscosity.

Syrup

pours

It is very viscous.

pours

slowly.

Some fluids move quickly.

They have a low viscosity.

Water

It is less viscous.

pours

quickly.

Viscosity

� A fluid’s thickness or resistance to flow is called its viscosity.

� A fluid is said to be more viscous if it does not flow as readily as another fluid. not flow as readily as another fluid.

� Gases are less viscous than liquids,

so they flow very easily through

pipes.

Gases vs Liquids

� The particle theory helps us to understand that this resistance is due to the stronger attraction among particles.

Why?

� Different substances are composed of different particles and this is why fluids can have different viscosities.

� Viscosity is a useful and important fluid property.

� Industries produce liquids, such as lubricants, with

special viscosities. If motor oil is not viscous enough,

it will not protect the engine parts from friction; if it is

too viscous, it will not flow to all parts of the engine

that need protection. [Viscosity causes fluid to adhere

Importance of Viscosity

that need protection. [Viscosity causes fluid to adhere

to a surface – known as no-slip condition]

� Salad dressing must be thin enough to pour out of a

bottle, yet thick enough to coat lettuce properly.

� It influences the power needed to move an airfoil

through the atmosphere.

� It accounts for the energy losses associated with the

transport of fluids in ducts, channels, and pipes.

� It plays a primary role in the generation of turbulence.

� An object moving

through or on a fluid

meets resistance.F

vx

Fluid Resistance

� Force causes the fluid

to move.

� The velocity is

proportional to the

force.

Fvx ∝

� The resistance tends to

keep the fluid in place.

� Law of inertia

yvx

F

Velocity Gradient

� The fluid moves most

near the object and least

farther away.

� This is a velocity gradient

or strain rate or rate of

deformation.

yvx ∝

y

� Newton combined these

two proportionalities.

� This is the law of viscosity.yvx

F

Law of Viscosity

� A is the area of the

solid sliding on the fluid

� The constant µ is the

dynamic viscosity and

depends on the type of

fluid. dy

du

y

vAF x

µτ

µ

=

=

y

Viscous Behavior of Various Materials

18

Newtonian fluid:

the shear stress of

the fluid is directly

proportional to the

velocity gradient.

� Inviscid flow: The flow in which viscous effects do not significantly influence the flow and are thus neglected.

Inviscid Flow vs Viscous Flow

� Viscous flow: The flow in which viscous effects are important and cannot be ignored.

� The primary class of flows, which can be modeled as inviscid flow, is

external flows.

� Any viscous effects that may exist are confined to a thin layer, called a

boundary layer which is attached to the boundary. The inviscid flow

outside the boundary layer in an external flow is called free stream.

� For many flows, the boundary layers are so thin that the can simply be

ignored when studying the gross features (e.g., lift and drag and locate

possible separation points) of a flow around a streamlined body (e.g.,

Inviscid Flow vs Viscous Flow

possible separation points) of a flow around a streamlined body (e.g.,

airfoil).

� Inviscid flow is also encountered in contractions inside piping systems

(viscous effects require substantial areas in order to be significant), in

short regions of internal flows (called inviscid core length) where

viscous effects are negligible, and wake in high-Reynolds-number flow

around bodies

� Viscous flows include the broad class of internal flows, such as flows in

pipes, conduits and open channels. In such flows, viscous effects cause

substantial “losses” and account for the huge amounts of energy that

must be used to transport oil and gas in pipelines.

� In the 19th century, two schools of thought existed on fluid

mechanics.

� Hydraulicians: looked at experimental data and attempted to

generalize them into useful design equations. Their equations were

generally empirical, without much theoretical content.

� Hydrodynamicists: started with 2D- and 3D- equations and tried to

The History of Potential Flow and Boundary Layer

21

� Hydrodynamicists: started with 2D- and 3D- equations and tried to

apply them to practical problems. They ignored the viscous-friction

and density change term by hypothesizing a perfect fluid with zero

viscosity and constant density, in order to calculate the complete

behavior of many kinds of flows. These mathematical solutions

agreed to very well with some observed behavior (except involving

no solid surfaces, and etc).

� “Hydrodynamicists calculate that which cannot be observed;

hydraulicians observe that which cannot be calculated.”

� In 1904 Ludwig Prandtl introduced a new concept, called the

boundary layer: if a fluid flows past the leading edge of a flat

surface, there will generate a velocity profile. Inside the boundary

layer, the effects of viscosity are too large to be ignored. Outside the

boundary layer, the laws of perfect-fluid flow should be satisfactory.

� The calculations were still very difficult, and so only approximate

mathematical solutions were possible.

The History of Potential Flow and Boundary Layer

22

mathematical solutions were possible.

� But, this idea clarified numerous unexplained phenomena and

provided a much better intellectual basis for discussing complicated

flows.

� So, it is clear that the ideas of perfect-fluid flow and boundary layer

are intimately tied together.

� We will consider perfect-fluid or inviscid flows in this chapter, and

boundary layer in chapters 3 and 4.

Deformation of a Fluid Elements

General deformation of fluid element is rather complex; however, we can

break the different types of deformation or motion into a superposition of

each type.

23

Velocity Angular Velocity

Linear Strain Rate Shear Strain Rate

In order for these deformation rates to be useful in the calculation of fluid

flows, they must be expressed in terms of velocity and derivatives of velocity.

1. Linear Motion/Translation

� “Simplest” form of motion — the element moves as a solid

body.

� The rate of translation vector is described as the velocity

vector in Cartesian coordinates:

kwjviuV ˆˆˆˆ ++=

24

2. Linear Deformation

� Linear strain rate is defined as the rate of increase in length per unit

length. It is expressed in Cartesian coordinates as

� Solid objects such as wires, rods, and beams stretch while pulled. Then,

they usually shrink in direction(s) normal to that direction.

z

w

y

v

x

uzzyyxx

∂

∂=

∂

∂=

∂

∂= εεε ,,

25

they usually shrink in direction(s) normal to that direction.

� This is also true for fluid elements. Therefore, for an incompressible flow,

if the element stretches in one direction, it must shrink by an appropriate

amount in other direction(s) to compensate.

The shape does not change, “linear deformation”

2. Linear Deformation

Velocity gradients can cause deformation, “stretching” resulting in a change

in volume of the fluid element.

Rate of Change for one direction:

For all THREE directions: The volumetric strain rate is

26

For all THREE directions:

Vz

w

y

v

x

u

dt

dzzyyxx

ˆ)(1⋅∇=

∂

∂+

∂

∂+

∂

∂=++=

∀

∀εεε

δ

δ

� The rate of increase of volume of a fluid element per unit volume is

called as volumetric strain rate or volumetric dilatation rate or bulk

strain rate. It is positive if the volume increases.

�The linear deformation is zero for incompressible fluids. 0ˆ =⋅∇ V

The volumetric strain rate is

the sum of the linear strain

rates in three mutually

orthogonal directions.

3. Angular Motion/Rotation

� Angular velocity or rate of rotation at a point is defined as the average

rotation rate of two initially perpendicular lines that intersect at that point:

Angular motion results from

cross derivatives.

27

3. Angular Motion/Rotation

The rotation of the element about the z-axis is the average of the angular

velocities :

Likewise, about the y-axis, and the x-axis:

Counterclockwise rotation is considered positive

and

28

and

The three components gives the rotation vector:

Using vector identities, the rotation vector is one-half the curl of the

velocity vector:

Vorticity and Rotationality

The definition of the rotation vector operation is the following:

29

� If , then there is no rotation, and the flow is said to be

irrotational ; the vorticity of an irrotational flow is zero.

The vorticity is twice the angular rotation:

� These calculations were carried out for any rigid-body rotation.


For a flow to be irrotational, each of the rotation (and therefore vorticity)

vector components must be equal to zero.

The z-component:

The x-component lead to a similar result:

30

The y-component lead to a similar result:


� The vorticity vector in cylindrical coordinates (r, θ, z):

� For 2-D flow in the rθ-plane:

zrzr

rz e

u

r

ru

re

r

u

z

ue

z

uu

rˆ

)(1ˆˆ

1ˆ

∂

∂−

∂

∂+

∂

∂−

∂

∂+

∂

∂−

∂

∂=

θθζ θ

θθ

31

� For 2-D flow in the rθ-plane:

zr e

u

r

ru

rˆ

)(1ˆ

∂

∂−

∂

∂=

θζ θ

Example

Example: A velocity field in a particular flow is given by V = 20y2i – 20xyj

m/s. Calculate the angular velocity and the vorticity vector at the point (1,

-1, 2).

32

Example

Example: Determine whether the following 2-D flows are rotational or

irrotational:

(a) u = -2y, v = 3x;

(b) v = 0, w = 3yz;

(c) u = 2x, w = 2z.

33

The Irrotational Flow Approximation

� Irrotational approximation:

� We must keep in mind that the assumption of irrotationality

is an approximation, which may be appropriate in some

regions of a flow field, but not in other regions.

� In general, inviscid regions of flow far away from solid walls

0ˆˆ ≈×∇= Vζ

34

� In general, inviscid regions of flow far away from solid walls

and wakes of bodies are also irrotational.

� However, there are situations in which an inviscid region of

flow may not be irrotational (e.g., solid-body rotation).

Inviscid Flow: Irrotational Flow

Examples where inviscid flow theory can be used:

35

Viscous Region - RotationalInviscid Region – Irrotational

4. Angular Deformation/Shear Strain

� Shear strain rate at a point is defined as half of the rate of

decrease of the angle between two initially perpendicular lines that

intersect at that point.

� Consider a fluid element translating and deforming in 2-D xy-

plane, the shear strain rate, initially perpendicular lines in the x- and

y-directions: ∂∂ vud 11

36

� Consider 3-D, the shear strain rate in Cartesian coordinates:

∂

∂+

∂

∂=−=

x

v

y

u

dt

dxy

2

1

2

1αε

∂

∂+

∂

∂=

∂

∂+

∂

∂=

∂

∂+

∂

∂=

x

w

z

v

z

u

x

w

x

v

y

uyzzxxy

2

1

2

1

2

1εεε

Angular Deformation/Shear Strain

� Now, we combine linear strain rate and shear strain rate into

shear strain tensor:

∂

∂+

∂

∂

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂

∂

=

=y

w

z

v

y

v

y

u

x

v

x

w

z

u

x

v

y

u

x

u

yzyyyx

xzxyxx

ij2

1

2

1

2

1

2

1

εεε

εεε

ε

37

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂

∂∂∂

∂∂

z

w

z

v

y

w

z

u

x

w

yzyyxzzzyzx

2

1

2

1

22εεε

We will see it again in Chap 2We will see it again in Chap 2

Example

Example: A velocity field in a particular flow is given by V = 20y2i – 20xyj

m/s. Calculate the nonzero strain rate components at the point (1, -1, 2).

38

Conservation of Mass: Cartesian Coordinates

The differential form of the equation for Conservation of Mass:

In vector notation, the equation is the following:

If the flow is steady and compressible:

39

The Continuity Equation


If the flow is steady and incompressible:

Examples

Example: Assuming ρ to be constant, do the following flows satisfy continuity?

(a) u = -2y, v = 3x;

(b) u = 0, v = 3xy;

40

Streamlines

� A streamline is a line drawn

through the flow field in such a

manner that the local velocity vector

is tangent to the streamline at every

point along the line at that instant.

� The tangent of the streamline at a

41

� The tangent of the streamline at a

given time gives the direction of the

velocity vector. A streamline does

not indicate the magnitude of the

velocity.

� The flow pattern shown by the

streamlines is an instantaneous

visualization of the flow field.

Conservation of Mass: Cylindrical Coordinates

42


If the flow is steady and incompressible:

The Continuity Equation

Examples

Example: Check the following incompressible flows for continuity and

determine the vorticity of each:

(a) vθ = 6r, vr = 0;

(b) vθ = 0, vr = -5/r.

43

Stream Functions

Stream Functions are defined for steady, incompressible, 2D flow.

2-D Continuity Equation:

Then, we define the stream functions as follows:

44

Now, substitute the stream function into the continuity equation:

Any flow that satisfies stream

function automatically satisfies

the continuity condition.

Stream Functions

The slope at any point along a streamline:

Streamlines have constant ψ, thus dψ = 0:

45

Streamlines have constant ψ, thus dψ = 0:

w

dz

v

dy

u

dx==

The stream function is

constant along a streamline.

Stream Functions

Now, calculate the volume flow rate per unit depth between streamlines:

46

The change in the value of the stream function is related to

the volume flow rate per unit depth between two

streamlines.

� Incompressible, planar stream function in cylindrical coordinates:

� Incompressible, axisymmetric stream function in cylindrical

coordinates:

Stream Functions in Cylindrical Coordinates

47

coordinates:

0)()(1

=∂

∂+

∂

∂

z

u

r

ru

r

zr

rru

zru zr

∂

∂=

∂

∂−=

ψψ 11

Interpretation of Stream Function

� A single variable (ψ) replaces two variables (u and v); once ψ

is known, both u and v can be generated.

� The stream function satisfies the continuity equation.

� Curves of constant ψ are streamlines of the flow.

� The difference in the value of ψ from one streamline to

48

� The difference in the value of ψ from one streamline to

another is equal to the volume flow rate per unit width

between the two streamlines.

� In steady flow, there is no flow across (perpendicular) to a

streamline.

Example

Example: The velocity components in a steady, incompressible, 2D flow field

are u = 2y and v = 4x. Determine the stream function and show on a

sketch with several streamlines.

49

Streamlines

� For real fluid flows, the fluid adjacent to the boundary of a solid body

does not move relative to body – it sticks to the wall. So, in real fluids

the wall is a streamline of zero velocity.

� But, the perfect fluid has no tendency to stick to walls because it has

no viscosity. So, the streamline adjacent to a solid body in perfect-fluid

flow is one with finite velocity.

� This leads to the idea that we may divide a perfect-fluid flow along a

50

� This leads to the idea that we may divide a perfect-fluid flow along a

streamline and substitute a solid body for the flow on one side of the

streamline.

Potential Flow

� In the region outside the boundary layer, where the fluid may be

assumed to have no viscosity, the mathematical solution takes on the

form known as irrotational flow (also known as potential flow).

� This form is analogous to the flow of heat in a temperature field or to

the flow of charge in an electrostatic field. All these flows obey

Laplace’s equation under certain restrictions (for example: steady-state

mass balance for a constant-density fluid).

51

mass balance for a constant-density fluid).

� Not every velocity potential satisfies Laplace’s equation, and so not

every velocity potential represents a potential flow. For example, φ = x2,

x2 + y2, ex, sin x do not satisfy Laplace’s equation, so they cannot

represent potential flows because they violate the mass balance for a

constant-density fluid.

Consider a velocity field that is given by the gradient of a scalar function φ(called velocity potential function):

Such velocity field is called a potential flow (or an irrotational flow) and

possesses the property that the vorticity ω, which is the curl of a velocity

vector, is zero:

Potential Flow: Velocity Potential

φ∇=V̂

52

vector, is zero:

With the velocity given by the gradient of a scalar function, the differential

continuity equation ( ), for an incompressible flow, gives

which is known as Laplace’s equation.

0ˆ =×∇= Vω

02 =∇=∇⋅∇ φφ

The flow must be irrotational if there is a velocity potential.

0ˆ =⋅∇ V


For irrotational flow, there exists a velocity potential:

Take one component of vorticity to show that the velocity potential is irrotational:

53

Substitute u and v components of velocity potential:

02

1 22

=

∂∂

∂−

∂∂

∂

xyyx

φφWe could do this to show all vorticity components are zero.

The flow must be irrotational if there is a velocity potential.

If the curl of a vector is zero, the vector can be expressed as the gradient of

a velocity potential.


Then, rewriting the u, v, and w components as a vector:

For irrotational, planar flow:

Now substitute the stream function:

54

Then for incompressible irrotational flow:

Now substitute the stream function:

Then, Laplace’s Equation

Pierre Laplace (1749-1827)


� Potential flows are

irrotational – vorticity is

zero.

� If the vorticity is present

(e.g., boundary layer, wake),

then the flow cannot be

55

then the flow cannot be

described by Laplace’s

equation.


Laplacian Operator in Cartesian coordinates:

Laplacian Operator in cylindrical coordinates:

If a Potential Flow exists,

with appropriate boundary

conditions, the entire velocity

and pressure field can be

specified.

56

where the gradient in cylindrical coordinates, the gradient operator,

Then,

May choose cylindrical

coordinates based on the

geometry of the flow problem,

i.e., pipe flow.


Lines of constant ψ are streamlines:

Now, the change of φ from one point (x, y) to a nearby point (x + dx, y + dy):

57

Along lines of constant φ, we have dφ = 0,

0

The equipotential lines are orthogonal to streamlines where they intersect.

Lines of constant φ are called equipotential lines.


The flow net consists of a family of streamlines and equipotential lines.

The combination of streamlines and equipotential lines are used to visualize a

graphical flow situation.

The velocity is inversely

proportional to the spacing

Velocity decreases

along this streamline.

58

between streamlines.

Velocity increases

along this streamline.

Streamlines and equipotential lines

intersect at right angles.

Stream Function and Velocity Potential

� The stream function is

defined by continuity; the

Laplace equation for ψresults from irrotationality.

59

� The velocity potential is

defined by irrotationality;

the Laplace equation for φresults from continuity. 0

2

1 22

=

∂∂

∂−

∂∂

∂

xyyx

φφ

Potential Flow: Plane Potential Flows

Velocity components for steady, incompressible, irrotational, 2D regions of

flow in terms of velocity potential and stream function in various

coordinate systems:

Planar, Cartesian:

Planar, Cylindrical:

60


Planar, Cartesian:


rru

zru zr

∂

∂=

∂

∂−=

ψψ 11

Axisymmetric, Cylindrical:

Axisymmetric, Cylindrical:

zu

ru zr

∂

∂=

∂

∂=

φφ

The Conversion between Velocity Field, Stream Function and Velocity Potential

φψ ⇔⇔ ),( vu

61

φ⇔),,( wvu

Incompressible? Continuity equation? Stream function exists? Irrotational? Velocity

potential exists?

Incompressible? Satisfy

continuity equation?

Stream function

Irrotational? Velocity

potential exists?

62

exists?

0=•∇ V

02 =∇ φ 02 =∇ ψ

0or0 == ζω

Example

Example: A velocity potential in 2D flow is . Find the stream

function for this flow.

22yxy −+=φ

63

Example

Example: The 2D stream function for a flow is . Find

the velocity potential for this flow.

xyyx 7469 +−+=ψ

64

Example

Example: In a 2D incompressible flow, the fluid velocity components are

given by and . Show that the flow satisfied the

continuity equation and obtain the expression for the stream function.

If the flow is potential, obtain also the expression for the velocity

potential.

yxu 4−= xyv 4−−=

65

Example

Example: The 2D flow of a non-viscous, incompressible fluid in the vicinity of

the 90º corner is described by the stream function .

Determine the corresponding velocity potential. θψ 2sin2 2

r=

66α

πθφ απ cos/

Ar=

α

πθψ απ sin/

Ar=

� Before we discuss the elemental flows, let’s visit this

website:

http://simscience.org/fluid/green/potential.html

Elementary Flows

67

� Next we will learn how the velocity fields of some

elementary and complex flows can be expressed in

terms of stream function and velocity potential.

http://simscience.org/fluid/green/potential.html

1. Uniform Flow

Question: Do φ1 = Ax and φ2 = Ax + By satisfy Laplace’s equation?

For φ1 = Ax , u = ∂φ1/∂x = A, v = ∂φ1/∂y = 0

So, φ1 describes a uniform, steady flow of velocity A in the positive x

direction. This might be the description of a wind blowing over the ocean at

a steady, uniform velocity of A.

68

a steady, uniform velocity of A.

For φ2 = Ax + By, u = ∂φ2/∂x = A, v = ∂φ2/∂y = B

So, φ2 describes a uniform, constant-velocity flow with velocity (A2 + B2)1/2,

making the angle arctan (B/A) with the x axis.

These uniform flows are not of much practical interest

alone, but they can be combined with other flows to solve

more interesting problems.

1. Uniform Flow

For Uniform Flow in an arbitrary direction, α:

φ = Ux

ψ = Uy

u = U

v= 0

69

2. Source and Sink Flow

0

Now, obtain the stream function for the flow:

Integrating to obtain the solution: rm

ln2ππππ

φφφφ =

( ) mvr r =π2

vr vθ

70

The streamlines are radial lines and the equipotential

lines are concentric circles centered about the origin:

φ lines

ψ lines

2. Source and Sink Flow

� If m is positive, the flow is source; if m is negative, the flow is

sink.

is the volume rate of flow per unit depth issuing from the

source or sink, where Q is flow rate, and L is height.

� These flows are of practical significance in the petroleum

LQm /=

71

industry; it describes the flow into oil well in a thick horizontal

stratum.

� The equations vr = m/2πr, vθ = 0 show that the radial flow

velocity becomes infinite at r = 0 (mathematical singularity); thus,

this equation cannot describe any real flow at r = 0.

Examples

Example: A nonviscous, incompressible fluid flows between wedge-shaped

walls into a small opening. The velocity potential (in m2/s), which

approximately describes this flow is φ = -2 ln r. Determine the

volume rate of flow (per unit length) into the opening.

72

3. Vortex Flow

In vortex flow the streamlines are concentric circles, and the equipotential

lines are radial lines.

where K is a constant, namely the strength of the vortex.

Solution:

The sign of K determines whether the flow rotates

clockwise (-) or counterclockwise (+).

73

clockwise (-) or counterclockwise (+).

In this case, ,

The tangential velocity varies inversely with

the distance from the origin.

At the origin it encounters a singularity

becoming infinite.

φ lines

ψ lines

3. Vortex Flow

� Rotation refers to the orientation of a fluid element and not the path

followed by the element. The elements deform to maintain a constant

orientation.

� In general flow there is both deformation and rotation.

�An ideal flow is one that has no viscosity and is incompressible.

74

� If an ideal flow is initially irrotational, it will remain irrotational.

� Two vortices: free vortex and forced vortex.

� The swirling motion of the water as it drains from a bathtub is similar to

that of a free vortex, while the motion of a liquid contained in a tank is

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

Free Vortex and Forced Vortex

Irrotational Flow: Free Vortex Rotational Flow: Forced Vortex

Traveling from A to B, consider two sticks

Velocity

increases

inward.

Velocity

increases

outward.

i.e., water

draining from a

bathtub

i.e., a rotating tank

filled with fluid

75

Initially, sticks aligned, one in the flow direction,

and the other perpendicular to the flow.

As they move from A to B the perpendicular-

aligned stick rotates clockwise, while the flow-

aligned stick rotates counter clockwise.

The average angular velocities cancel each other,

thus, the flow is irrotational.

Irrotational Flow: Rotational Flow: Rigid Body Rotation

Initially, sticks aligned, one in the

flow direction, and the other

perpendicular to the flow.

As they move from A to B the sticks

move in a rigid body motion, and thus

the flow is rotational.

0ˆˆ =×∇= Vζζζζ V̂2ˆ ×∇== ωζ

Free Vortex and Forced Vortex

76

A simple analogy can be made

between flow A and a merry-go-

round or roundabout, and flow B

and a Ferris wheel.

As children revolve around a

roundabout, they also rotate at

the same angular velocity as that

of the ride itself. This is analogous

to a rotational flow.

In contrast, children on a Ferris

wheel always remain oriented in

77

A simple analogy: (a) rotational

circular flow is analogous to a

roundabout, while (b) irrotational

circular flow is analogous to a

Ferris wheel.

wheel always remain oriented in

an upright position as they trace

out their circular path. This is

analogous to an irrotational flow.

Tornadoes and Hurricanes

� A combined vortex flow is one in which there is a forced vortex at the core, and a free

vortex outside the core.

� The minimum pressure at the vortex center can give rise to a “secondary flow” which

is produced by the pressure gradient in the primary (vortex) flow.

� In the region near the ground, the wind velocity is decreased due to the friction

provided by the ground.

� However, the pressure difference in the radial direction causes a radially inward flow

adjacent to the ground, and upward flow at the vortex center.

78

Pressure difference between the vortex

center and outer edge:

p1 – p0 = –ρVmax2

Circulation

� Circulation (Γ) or vortex strength gives a measure of the average of

rate of rotation of fluid particles that are situated in an area that is bounded

by a closed curved.

� This concept is often useful when evaluating forces (such lift force)

developed on bodies immersed in moving fluids.

� It is defined as the line integral of the tangential component of the

velocity (V) around a closed curve fixed in the flow.

79

velocity (V) around a closed curve fixed in the flow.

� ΓΓΓΓ = 0 for irrotational flow.

� If there are singularities enclosed within the curve, Γ ≠ 0, for example:

free vortex.

Circulation: Free Vortex

For the free vortex:

(Integrate the entire circle)

Γ= 0

80

The circulation is non-zero and constant for the free vortex:

The velocity potential and the stream function for the free vortex can

be rewritten in terms of the circulation:

Examples

Example: The pressure far from an irrotational vortex (a simplified tornado) in

the atmosphere is zero gauge. If the velocity at r = 20 m is 20 m/s, estimate

the velocity and the pressure at r = 2 m.

81

Circulation

How is circulation calculated from rpm and radius?

82

Example

Example: A liquid drains from a large tank through a small opening. A vortex

forms whose velocity distribution away from the tank opening can be

approximated as that of a free vortex having a velocity potential

Determine an expression relating the surface shape to the strength of

the vortex as specified by the circulation.

θπφ )2/(Γ=

83

4. Doublet Flow

Combination of an equal Source and Sink pair.

Rearrange and take tangent,

84

Note, the following:

Substituting the above expressions,

and

Then,

If a is small, then tangent of angle is approximated by the angle:

4. Doublet Flow

Now, we obtain the doublet flow by letting the source and sink approach one

another (a→ 0), and letting the strength increase (m→∞).

85

K is the strength of the doublet, and is equal

to ma/π.

is then constant.

The corresponding velocity potential then is the following:

Streamlines of a Doublet:ψ lines Doublet strength is for a

double oriented in the

negative x-direction.

Summary of Basic Flows

u = U

v = 0

Γ = 0

Γ = 0

86

Γ = 0Origin is singular point

Origin is singular point

Γ = K around any closed curved enclosing origin

Γ = 0 around any closed curved NOT enclosing origin

Origin is singular point Γ = around any closed curved

Superposition of Basic Flows

� Because potential flows are governed by linear partial differential

equations, the solutions can be combined in superposition; we will

superimpose simple functions to create flows of interest.

� If φ1 and φ2 are each solutions of the Laplace equation, then Aφ1, (A +

φ1), (φ1 + φ2), and (Aφ1 + φ2) are also solutions,.

� Thus, some of the basic ψ and φ can be combined to yield a streamline

87

� Thus, some of the basic ψ and φ can be combined to yield a streamline

that represents a particular body shape, or complicated incompressible,

plane flow.

� The superposition representing a body can lead to describing the flow

around the body in detail.

� The superposition is only valid for irrotatioanal flow fields for which the

equations for φ and ψ are linear.

� The velocity at any point in the composite field is the vector sum of the

velocities of the individual flow fields.

1. Rankine Half-Body

The Rankine Half-Body is a combination of a source and a uniform flow.

88

Stream Function (cylindrical coordinates):

Velocity Potential (cylindrical coordinates):


There will be a stagnation point, somewhere along the negative x-axis where

the velocities due to the source and uniform flow are cancelled (θ = π).

For the source: For the uniform flow: θcosUvr =

For θ = π, Uv r =

Then, for a stagnation point, at some x = -b (r = b), θ = π:

m

89

r

mUvr

π2== and

Now, the stagnation streamline can be defined by evaluating y at r = b, and

θ = π . The value of ψ at the stagnation point:

θπ

θψ2

sinm

Ur +=


Since m/2 = πbU, it follows that the equation of the streamline passing

through the stagnation point, and gives the outline of the Rankine half-

body:

Then,

For inviscid flow, a streamline can be replaced by a solid boundary. So,

90

For inviscid flow, a streamline can be replaced by a solid boundary. So,

the source and uniform can be used to describe the flow around a

streamlined body placed in a uniform stream – half-body.

The other streamlines can be obtained by setting ψ = constant.

Singularity (inside the body)


The width of the half-body:Total width = 2ππππb

The magnitude of the velocity (V) at any point in the flow:

and

( )r

b θπθ

−=sin

91

Noting,

Knowing the velocity we can now determine the pressure field using the

Bernoulli Equation:

po and U are at a point far away from the body and are known.


� We wish to find the flow pattern around some arbitrary body.

This is normally done by combining uniform flows, sources,

sinks, etc.

� When a combination is found that produces a streamline with

the shape of the body in question, the flow outside the

streamline is a representation of the flow around the body.

92

streamline is a representation of the flow around the body.

� The flow inside that line normally has no meaning and is

ignored.

� The singularity in the flow field (source) only occurs inside

the body.


� The velocity tangent to the surface of the body is not zero,

i.e., the fluid “slips” by the boundary (as neglecting viscosity).

So, all potential flows differ from the flow of real fluids

(considering viscosity) and do not accurately represent the

velocity very near the boundary. However, outside this layer,

the velocity distribution will generally correspond to that

93

the velocity distribution will generally correspond to that

predicted by potential flow theory if flow separation does not

occur.

� The pressure distribution along the surface will closely

approximate that predicted from the potential flow theory since

the boundary layer is thin, and there is little variation of

pressure through the boundary layer.

Example

Example: The shape of a hill arising from a plain can be approximated with the top

section of a half-body. The height of the hill approaches 60 m.

(a) When a 60 km/hr wind blows toward the hill, what is the magnitude

of the air velocity at a point on the hill directly above the origin (point 2)?

(b) What is the elevation of point (2) above the plain and what is the

difference in pressure between point (1) on the plain far from the hill and

point (2)? Assume an air density of 1.23 kg/m3.

94

2. Rankine Oval

Rankine Ovals are the combination a source, a sink and a uniform flow,

producing a closed body.

95

Stream function and velocity function describing the flow:

2. Rankine Oval

The streamline ψ = 0 forms the surface of a body of length 2l and width 2 h

placed in a uniform flow.

Ua/m is large � slender body

Ua/m is small � blunt shape body

96

The body half-length

The body half-width

“Iterative”

2/1

1

+=

Ua

m

a

l

π

−

=

a

h

m

Ua

a

h

a

h π2tan1

2

12

2. Rankine Oval

Ua/m is large � slender body


97


2. Rankine Oval

� Downstream from the point of maximum

body width, the surface pressure increases

with distance along the surface.

� This condition (called adverse pressure

gradient) typically leads to separation (not

predicted by potential theory) of the flow

from the surface, resulting in a large low

98

from the surface, resulting in a large low

pressure wake on the downstream side of

the body.

� The potential solution for the Rankine

ovals will only approximate the velocity

outside the thin, viscous boundary layer and

the pressure distribution on the front part of

the body.

3.1. Flow Around a Stationary Circular Cylinder

Combines a uniform flow and a doublet flow

and

For the ψ to represent flow around a cylinder, ψ = constant for r = a (a = the

radius of the circular cylinder):

K = Ua2

99

K = Ua2

Then, and

Then the velocity components:

Doublet strength


On the surface of the cylinder (r = a):

The maximum velocity occurs at the top and bottom of the cylinder,

magnitude of 2U (θ = ± π/2).

vrs = 0

The figure shows the pattern

of streamlines for this flow.

Why are the

streamlines so

close here?

100

of streamlines for this flow.

We disregard the doublet flow

on the inside of the circle r = a

and imagine that a solid

cylinder replaces this portion

of the flow. A remarkable

feature is the symmetry of the

flow upstream and

downstream of the cylinder.

Why are the streamlines so far here?

close here?

No slip or slip?


Pressure distribution on a circular cylinder found with the Bernoulli equation

Then substituting for the surface velocity:

101

Theoretical and experimental agree well

on the front portion of the cylinder. The

actual surface pressures and ideal values

agree for a distance up to β = 60°.

Flow separation on the back-half in the

real flow due to viscous effects causes

differences between the theory and

experiment. So, the ideal flow is no

longer valid.

� The pressure gradient influences flow patterns and pressure distributions acting on bodies create forces.

� A common dimensionless group for describing the pressure distribution is pressure coefficient (Cp):

where p is the local pressure, p0 and V0 are the free-stream pressure and velocity.

Pressure Coefficient

2

021

0

V

ppC p

ρ

−=

102

velocity.

� The points B and D are points of stagnation (Cp = +1.0), and the minimum pressure (Cp = – 3.0) occurs at the point C.


Inviscid flow past a circular

cylinder:

(a) streamlines for the flow if

there were no viscous effects.

103

(b) pressure distribution on

the cylinder’s surface,

(c) free-stream velocity on

the cylinder’s surface.

The pressure distribution up to the

point of separation is very nearly

the same as that predicted by

potential flow.

Boundary layer

characteristics on a circular

cylinder:

(a) boundary layer

separation location.


Wake

104

(b) typical boundary layer

velocity profiles at various

locations on the cylinder,

(c) surface pressure

distributions for inviscid

flow and boundary layer

flow.

Turbulent or laminar data

matches better with

irrotational flow

approximation? Why?

� From Euler’s equation for pressure gradient and acceleration along a

pathline,

� The fluid particle accelerates (at > 0) if the pressure decreases with

distance along a pathline (∂p/�s < 0) – favorable pressure gradient.

Favorable and Adverse Pressure Gradient

s

pat

∂

∂−=ρ

105

� The fluid particle decelerates (at < 0) if the pressure increases with

distance along a pathline (∂p/�s > 0) – adverse pressure gradient.

� Flow separation occurs when the fluid pathlines adjacent to body deviates

from the contour of the body and produce a wake.

� It tends to increase drag, reduce lift and produce unsteady forces leading

to structural failure (e.g., Tavoma Narrows Bridge in 1904).

� The prediction and control of separation is continuing challenge for

engineers involved with the design of fluid systems.

Flow Separation

106


The resultant force per unit length acting on the cylinder can be determined

by integrating the pressure over the surface (equate to lift and drag).

(Drag)

(Lift)

107

Substituting

Evaluating the integrals:

Both drag and lift are predicted to be zero on fixed cylinder in a uniform

flow. The zero-life prediction is acceptable for a real flow, but the zero-

drag result is unacceptable.

Jean le Rond

d’Alembert

(1717-1783)


� Mathematically, this makes sense since the pressure distribution is

symmetric about cylinder (because of the symmetric pressure distribution, the

force on the front half cancels that one the rear half to produce zero drag).

� However, in practice/experiment, we see substantial drag on a circular

cylinder (and called as d’Alembert’s Paradox, 1717-1783).

Potential theory incorrectly predicts that the drag on a cylinder is zero.

108

3.2. Flow Around a Rotating Circular Cylinder

� The addition of the vortex changes the flow pattern everywhere, except at the

cylinder surface and at infinity.

� The streamlines that represents the cylinder is still a circle, but the values of the

surface velocity are changed.

Flow around a rotating cylinder is approximated by the combination of a

uniform flow past a cylinder and a free vortex.

109

surface velocity are changed.

� This flow is relevant to the flow about wings and airfoils.

rr

aUr ln

2sin1

2

2

πθψ

Γ−

−= θ

πθφ

2cos1

2

2 Γ+

+=

r

aUrand

aU

rv

ar

sπ

θψ

θ2

sin2Γ

+−=∂

∂−=

=

On the surface of the cylinder (r = a):

The additional vortex only affects vθs, but not vr.


� If Γ = 0, then θstag = 0 or π,

i.e., the stagnation points occur at

the front and rear of the cylinder.

� If -1 ≤ Γ/(4πUa) ≤1, then the

stagnation points occur at some

Uastag

πθ

4sin

Γ=The stagnation points occur at θ = θStag where (vθ = 0):

110

stagnation points occur at some

other location on the surface as

Figures (b) and (c).

� If ||||Γ/(4πUa)|||| > 1, then the

stagnation point is located away

from the cylinder. There is a

portion of fluid that is trapped

next to the surface and continually

rotates around the cylinder.


For the cylinder with circulation, the surface pressure is obtained from

the Bernoulli equation.

2

2

0

2sin2

2

1

2

1

Γ+−+=+

aUpUp

s

πθρρ

Γ

−Γ

+−+=2

22 sin2sin41

1Upp

θθρ

111

Γ−

Γ+−+=

222

22

0

4

sin2sin41

2

1

UaaUUpp

s

ππ

θθρ


112


0=x

F

Substituting this equation into Fy, for the lift, and integrated, yields

For the rotating cylinder, no force in the direction of the uniform flow is developed.

Substituting this equation into Fx, for the drag, and integrated, yields

Γ−= UFy

ρ

Magnus Effect Magnus Effect Magnus Effect Magnus Effect ––––

Lift on rotating bodiesLift on rotating bodiesLift on rotating bodiesLift on rotating bodies

113

The negative sign means that if U is positive in the positive x direction, and

circulation is positive (a free vortex with counterclockwise rotation), the

direction is downward.

Potential flow past a cylinder with circulation gives zero drag, but non-zero lift.

Lift on rotating bodiesLift on rotating bodiesLift on rotating bodiesLift on rotating bodies

http://www.grc.nasa.gov/WWW/K-12/airplane/cyl.html

The equation relating lift force on airfoils to ρ, U, and Γ is called Kutta-

Joukowski law.


� Uniform flow towards +ve x-direction (U =

+ve), counterclockwise (Γ = +ve)

� Low v and high P on top-half of cylinder

� Downward force (Fy = -ve)

114

� Downward force (Fy = -ve)

� Uniform flow towards +ve x-direction (U =

+ve), clockwise (Γ = -ve)

� Low v and high P on bottom-half of cylinder

� Upward lift force (Fy = +ve)

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