Point Velocity Measurements

P M V SubbaraoProfessor

Mechanical Engineering Department

Velocity Distribution is a Fundamental Symptom of Solid –

Fluid Interactions….

Point Velocity Measurement

• Pitot Probe Anemometry : Potential Flow Theory .

• Thermal Anemometry : Newton’s Law of Cooling.

• Laser Anemometry: Doppler Theory.

POTENTIAL FLOW THEORY

• Ideal flow past any unknown object can be represented as a complex potential.

• In particular we define the complex potential

iW

In the complex (Argand-Gauss) plane every point is associated with a complex number

ireiyxz

In general we can then write

zfzizW

Now, if the function f is analytic, this implies that it is also differentiable, meaning that the limit

so that the derivative of the complex potential W in the complex z plane gives the complex conjugate of the velocity. Thus, knowledge of the complex potential as a complex function of z leads to the velocity field through a simple derivative.

z

zfzzf

dz

dfz

0lim

ivudz

dW

ELEMENTARY IRROTATIONAL PLANE FLOWS

• The uniform flow

• The source and the sink

THE UNIFORM FLOW The first and simplest example is that of a uniform flow with velocity U directed along the x axis.

and the streamlines are all parallel to the velocity direction (which is the x axis).

Equi-potential lines are obviously parallel to the y axis.

In this case the complex potential is

UziW

THE SOURCE OR SINK • source (or sink), the complex potential of which is

• This is a pure radial flow, in which all the streamlines converge at the origin, where there is a singularity due to the fact that continuity can not be satisfied.

• At the origin there is a source, m > 0 or sink, m < 0 of fluid.

• Traversing any closed line that does not include the origin, the mass flux (and then the discharge) is always zero.

• On the contrary, following any closed line that includes the origin the discharge is always nonzero and equal to m.

zm

iW ln2

The flow field is uniquely determined upon deriving the complex potential W with respect to z.

Iso lines

Iso lines

zm

iW ln2

Stream and source: Rankine half-body

• It is the superposition of a uniform stream of constant speed U and a source of strength m.

2D Rankine half-body:

zm

UziW ln2

zm

UziW ln2

ireiyxZ

sin

cos

ry

rx

x

y1tan

ii re

mUreiW ln

2

im

rm

iUyUxi2

ln2

2ln

2

mUyir

mUxi

2

mUy

Shape of Zero Value Stream line

2sin0

mUr

2D Rankine half-body:

x

ymUy 1tan

20

y

3D Rankine half-body:

2224 zyx

mUx

Pitot Probe Anemometry : Henri Pitot in 1732

• Theory• A constant-density fluid flowing steadily without friction

through the simple device. • No heat being added and no shaft work being produced by the

fluid.• A simple expression can be developed to describe this flow:

2

22

21

21

1 22gz

upgz

up

Apply Bernoulli’s equation along the central streamline from a point upstream where the velocity is u1 and the pressure p1 to the stagnation point of the blunt body where the velocity is zero, u2 = 0. Also z1 = z2.

This increase in pressure which bring the fluid to rest is called the dynamic pressure.

Dynamic pressure =

or converting this to head

Dynamic head = g

p

g

u

g

p

2

211

2

2

21

12

upp

1 2

The total pressure is know as the stagnation pressure (or total pressure)

or in terms of head Stagnation head =

Stagnation pressure =

The blunt body stopping the fluid does not have to be a solid. It could be a static column of fluid. Two piezometers, one as normal and one as a Pitot tube within the pipe can be used in an arrangement to measure velocity of flow.

2

21

12

upp

g

u

g

p

2

211

Using the above theory, we have the equation for p2 ,

We now have an expression for velocity obtained from two pressure measurements and the application of the Bernoulli equation. g

p

g

u

g

p

2

211

2

2

21

12

upp 122 hhgu

Pitot Static Tube

• The necessity of two piezometers and thus two readings make this arrangement is a little awkward.

• Connecting the piezometers to a manometer would simplify things but there are still two tubes.

• The Pitot static tube combines the tubes and they can then be easily connected to a manometer.

• A Pitot static tube is shown below. • The holes on the side of the tube

connect to one side of a manometer and register the static head, (h1), while the central hole is connected to the other side of the manometer to register, as before, the stagnation head (h2).

A Pitot-static tube

Consider the pressures on the level of the centre line of the Pitot tube and using the theory of the manometer,

We know that

gXppA 2

ghhXgpp manB 1

BA pp

ghhXgpgXp man 12

2

21

12

uppp stag

AB

Xh

The Pitot/Pitot-static tubes give velocities at points in the flow. It does not give the overall discharge of the stream, which is often what is wanted. It also has the drawback that it is liable to block easily, particularly if there is significant debris in the flow.

2

21

11

upghp man

mangh

u2

1

Compressible-Flow Pitot Tube

0

2

2h

Vh

For an ideal compressible flow coming to rest from finite velocity:

For perfect gas :

0

2

2Tc

VTc pp

TTcV

p 0

2

2

This process of ideal compressible flow coming to rest is regarded as isentropic process.

0 vdpdhTds

vdpdTcp For perfect gas :

dpp

RTdTcp For ideal gas :

p

dp

T

dT

1

100

T

T

p

p

Compressible-Flow Pitot Tube

Subsonic pitot tube :A pitot tube in subsonic flow measures the local total pressure po together with a measurement of the static pressure p

120

21

Tc

V

p

p

p

12

00

2

p

pTcTTc

Vpp

RT

VM

120

2

11

M

p

p

11

21

02

p

pM

The pitot-static combination therefore constitutes a Mach meter .

With M2 known, we can then also determine the dynamic pressure.

11

21

02

2

p

ppV

M

The velocity can be determined from

11

21

02

p

ppV

Supersonic pitot tube

• A pitot probe in a supersonic stream will have a bow shock ahead of it.

• This complicates the flow measurement, since the bow shock will cause a drop in the total pressure, from po1 to po2 , the latter being sensed by the pitot port.

• It’s useful to note that the shock will also cause a drop in o, but ho will not change.

• The pressures and Mach number immediately behind the shock are related by

122

2

02

2

11

M

p

p

Normal Shock Relations

• Mach number relation

• Static pressure jump relation 1M2

21)M(M 2

1

212

2

1

1M2

p

p 21

1

2

122

2

02

2

11

M

p

p

Mach Number Range