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### Point Velocity Measurements

P M V Subbarao

Professor

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.

THE COMPLEX POTENTIAL

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

In particular we define the complex potential

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

In general we can then write

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.

THE UNIFORM FLOW

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

In this case the complex potential is

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.

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.

Iso f lines

Iso y lines

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

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:

Pitot Probe Anemometry : Henri Pitot in 1732

Theory

Consider, for example, a constant-density fluid flowing steadily without friction through the simple device.

If it is assumed that there is no heat being added and no shaftwork being produced by the fluid, a simple expression can be developed to describe this flow:

2

1

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 =

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

Stagnation pressure =

or in terms of head Stagnation head =

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.

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.

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

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.

Compressible-Flow Pitot Tube

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

For perfect gas :

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

For perfect gas :

For ideal gas :

Compressible-Flow Pitot Tube

Subsonic pitot tube :

A pitot tube in subsonic flow measures the local total pressure potogether with a measurement of the static pressure p

The pitot-static combination therefore constitutes a Mach meter .

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

The velocity can be determined from

- but this requires knowing either the static speed of sound a, or the stagnation speed of sound ao.
- The latter can be obtained by measuring the stagnation temperature at the tip of the pitot probe.

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 ro, but ho will not change.

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

Normal Shock Losses

Stagnation pressure jump relation

The stagnation pressure ratio across the shock is

Combining normal shock and subsonic pitot tube relations produces the relation between the po2 measured by the pitot probe, the static p1, and the required flow Mach number M1.

- After some manipulation, the result is the Rayleigh Pitot tube formula.

Five Hole Probes

The five-hole probe is an instrument often used in low-speed wind tunnels to measure flow direction, static pressure, and total pressure in subsonic flows.

This adaptation permits extending the useful calibration range up to 85 ° .

A special calibration is to been done, and new, extended range calibration curves are to be provided.

Probe Description

- The probe consists of four direction-sensing ports plus a center port, precision bored into a conical brass tip.
- Four individual small diameter stainless steel tubes connect the four side sensing ports to individual pressure transducers.
- The outer 3.175 millimeter diameter tube serves as the pressure transmitting channel for the center tube, as well as housing for the four side-port tubes.
- This small 3.175 millimeter tube is fitted within a larger tube for increased stiffness away from the sensing tip.

Thermal Anemometry

A thermal anemometer measures the velocity at a point in a flowing fluid — a liquid or a gas.

A typical industrial thermal anemometer used to monitor velocity in gas flows has two sensors —

a velocity sensor and a temperature sensor —

that automatically correct for changes in gas temperature.

Both sensors are reference-grade platinum resistance temperature detectors (RTDs).

The electric resistance of RTDs increases as temperature increases.

For this reason, they are one of the most commonly used sensors for accurate temperature measurements.

The electronics circuit passes current through the velocity sensor, thereby heating it to a constant temperature differential (Tv – Ta) above the gas temperature Ta and measures the heat qc carried away by the cooler gas as it flows past the sensor.

- Hence, it is called a “constant-temperature thermal anemometer.”

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