Point Velocity Measurements. P M V Subbarao Professor Mechanical Engineering Department. Velocity Distribution is a Fundamental Symptom of Solid â€“Fluid Interactionsâ€¦. Point Velocity Measurement. Pitot Probe Anemometry : Potential Flow Theory . Thermal Anemometry : Newtonâ€™s Law of Cooling.
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P M V Subbarao
Mechanical Engineering Department
Velocity Distribution is a Fundamental Symptom of Solid –Fluid Interactions….
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 source and the sink
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.
source (or sink), the complex potential of which is
Iso y lines
It is the superposition of a uniform stream of constant speed U and a source of strength m.
2D Rankine half-body:
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:
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.
We now have an expression for velocity obtained from two pressure measurements and the application of the Bernoulli equation.
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
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.
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 :
Subsonic pitot tube :
A pitot tube in subsonic flow measures the local total pressure potogether with a measurement of the static pressure p
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
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.
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.
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.