Course I – Fluid Mechanics and Heat Transfer
Tutors – Prof. B. Uensal, Prof. F. Durst
By,
Sangram R. Patil
Third Year Undergraduate
Mechanical Department
IIT Bombay

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Introduction
Principle and Probe Construction
Configurations
Modes of Operation
General Equation
Hot Wire Anemometry Theory
Sensitivity Calculations
Temperature Measurements
Constant Voltage Anemometer
Measurement Accuracy
Advantages and Shortcomings
Advanced Topics

Flow Measuring Devices
Velocity Components :
•
Pitot tube
•
Thermal Anemometry
•
Laser Doppler Anemometry
•
Particle Image Velocimetry
•
Sonic Anemometers
Temperature :
•
Thermocouples
•
Resistive Sensors
•
Liquid Crystals
Pressure :
•
Manometer
•
Pressure Transducers
Density :
•
Shadow Graph
•
Schleieren
•
Interferometry
Concentration :
•
Laser Induced Fluorescence
 Intrusive and Non-Intrusive Methods

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The most widely used device for measuring turbulent fluctuations
Measures rapidly varying velocities with good spatial and time resolution
Consists of a tiny platinum or tungsten wire heated by an electric current which responds to changes in velocity and temperature of the fluid around the wire.

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Electrical power dissipation through the wire resistance and convection heat transfer to the fluid.
The normal wire configuration is usually placed perpendicular to the mean flow
It is capable of sensing the mean velocity U, the streamwise component of turbulent fluctuations u and also the temperature fluctuations under some circumstances.
Very sensitive to alignment with respect to the flow direction
Inflow cone in which error in velocity measurement is very low
Suitable for moderately turbulent flows with a distinct main flow direction

A hot-wire type sensor must have two characteristics to make it a useful device:
• A high temperature coefficient of resistance
• An electrical resistance such that it can be easily heated with an electrical current at practical voltage and current levels.

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Aspect Ratio > 300 is preferred
Following assumptions hold :
Hot wire is cooled by velocity component normal to the wire
Conduction heat losses to the supports are minimum
For fixed aspect ratio smaller diameter implies shorter wire : Localized measurements and a high frequency response

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Sensing element : round wire of tungsten, platinum or platinum-iridium alloy (10-20 percent rhodium).
Tungsten is available in sizes down to 2.5um, it is stronger, high temperature coefficient of resistance
Platinum is available in sizes down to 1um.
Both tungsten and platinum have poor oxidation resistance at high temperatures.
Platinum-Iridium : good oxidation resistance, better strength but low temperature coefficient of resistance
The sensor supports, or prongs, are made of stainless steel and tapered, providing an end surface of around 0.1 mm in diameter to which the wires are spot-welded.
The remaining immersed body is made of ceramic

Used in different configurations depending on the quantity to be measured
Ref : Springer Handbook of Experimental Fluid Mechanics

CONSTANT CURRENT
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Wheatstone bridge is fed by constant electric current
The series resistivity of the energy source is set large compared to the total resistivity of the bridge in order to keep current constant at all times.
Temperature and resistivity change of the hot wire induces a voltage difference along the diagonal which is manifested as flow velocity
CONSTANT TEMPERATURE
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Maintaining constant resistance
R of the wire implies that in steady state the temperature is also kept constant
The output voltage provides measure of the heat transfer from the probe
The heat transfer is a measure of the fluid parameter under consideration at that time

Steady State Solution
ρ c w w
∂
T w
∂ t
Non dimensional temperature distribution along wire length
δ = k A w
∂ 2
T w
∂ x 2
δ x
+
I
2
δ
R w
− π
( w
−
T o
)
δ x
+ σε
( T w
4 −
T
4 sur
)
π δ

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In dimensionless form,
Nu = Nu(Re , Pr , Kn , M , t , l / d )…………(Bruun)
Nu
= f (Re, Pr,
Tw
, Kn )
Tf
King’s Law – A simpler approach by grouping several of the dimensionless quantities into one or two coefficients which are generally constant over a wide range of conditions –
Nu A B (Re)
C
The coefficients A and B can be shown to be dependent on the mean fluid temperature. Thus change in Tf will invalidate the calibration and necessitate correction. In flows where both velocity and temperature fluctuate the variations in heat transfer will not be an index of the velocity fluctuations alone.
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Overheat ratio : Non dimensional measure of the rise in temperature of the hot wire

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Consider a wire that is immersed in a fluid flow. The electrical power dissipation is equal to the heat lost due to convective heat transfer :
I R
= hA T T w
( w
− f
) w
The wire resistance is also a function of temperature :
Rw
= h
R o
[1
+ α a bv
( T c w
−
T f
)]
Heat Transfer co-efficient h can be obtained from
King’s Law : f
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Combining the above three equations allows us to eliminate the heat transfer co-efficient h : v f
= 



2
I R o
[1
+ α
( T w
−
T ref
( w w
−
T f
)
)]
− a

 / b
 1/ c

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For a constant temperature anemometer, the current is changed in such a way that the temperature and hence resistance of the wire remain constant a bv f c =
2
I R w
( w w
−
T f
)
We can measure the temperature of the flow and thus fluid velocity is a function of current only
The velocity can be obtained by solving the above equation
Note the electrical power dissipation can be expressed as VI and it is the potential difference V which is measured to find the fluctuation in fluid properties.
Most turbulence measurements depend implicitly on the existence of a mean velocity U which is considerably greater than the fluctuation u, so that variations in V are relatively small

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The wire sensitivity can be expressed as the rate of change of voltage with respect to velocity fluctuations.
For Constant Current mode : S
I
For Constant Temperature mode :
=
 dV

 u 
I
S
T
=
dV
u
T
( R w
− f
IR U f
1
− n
)
( R w
f
2 IU
1
− n
From this it may appear that the constant current sensitivity is higher but it must be
S
S
I
T
=
2( R w
R
− f
R f
) borne in mind that a higher over-heating ratio can be selected in the constant temperature mode, since the temperature control system automatically prevents wire burn-out following a large decrease in velocity
)

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Thermal inertia of the wire; its small reservoir of energy ensures that when the velocity changes rapidly, the temperature lags behind.
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Relation is found to be :
S
I
=
 dV

 u

I
M 2
ω
2 )
−
1
2
(where ω is the frequency of the fluctuation and M is a time constant)
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Typically the time constant is of the order of 1msec and the uncorrected response drops off rapidly above ω = 500 Hz
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Frequency Compensation : For measurements of turbulent fluctuations with very high frequencies – beyond 100 KHz
Pass the output from constant current wire through the compensator
Amplification has the inverse frequency dependence

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Ignoring the end losses the energy balance for the wire element can be written as :
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Express Tw in terms of Rw :
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Thus, the response of Rw to fluctuations in velocity, temperature or both is characterized by the following time constant :
CT Mode
CC Mode

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Small fluctuations in fluid temperature will influence the heat transfer from the wire in several ways. In general,
I R
BU ( R dV
R
=
)
S u u
+
S
θ
Consider constant current operation and assume that the velocity dependent term in the heat transfer law is a good deal larger than the other,
2 n w w f
Now if we find sensitivities wrt u and θ :
S u
=
S
I
≈ 
 nR w
2
1
+
BR U f n


I
3 S
θ
R w
R f
I

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S
S
0 operates as a resistance thermometer.
Sensitivity of the constant temperature mode thermometer drops rapidly with current, hence it does not make such a good thermometer.
One conclusion that can be drawn from the above results is that for high temperature operations, the contribution of the temperature fluctuations can be minimized by operating the wire at as high a temperature as possible

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The previous results can be used to measure both fluctuating velocities and temperature
Adjacent hot and cold wires :
S
L
= α
I
R w
R f Hot wire : dV
=
S u u
+
S
θ
These can be solved for θ and u dV
= −
S S u
L u

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Wire voltage is dependent upon the temperature difference
Measurement error due to temperature fluctuations
Solution : Operate wire at high temperatures and calibrate at mean flow temperature
A means of compensation will otherwise be required: there are two main practical ways, Bruun, 1995:
1. Automatic compensation : Use a temperature sensor in the
2. Analytical correction : Measure the flow temperature separately and compensate using the heat transfer equation.
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Since automatic compensation has a bandwidth of approximately 100 Hz, analytical correction is the only means of compensation at most experimental frequencies.

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Same wire is operated at different temperatures.
Measurements are made at different times and hence cannot be combined to give u and θ solely.
Statistical properties appearing in the mean square of the combined signal can be measured : dV
2 = u
2 2
S u
+
S
θ
2
2 +
2
 dV
S S found by calibration, then the remaining three statistical averages can be calculated. These quantities are presumes to remain same throughout the measuring process.

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The heat transfer from a fine wire is dependent primarily on the component of velocity normal to it
Specifying the turbulent flow at the probe by the components U+u, v and w, we find the normal component of velocity :
U n
=
U
+ u
+ v
Substitute in King’s Law with n=0.5 :
A
+
BU n
1
2
≈ +
(
1 cos ) (1
+
1
2 u
U
+
1
2 tan
v
U
)

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Suppose we take two wires oriented in opposite sense with respect to the mean flow direction then the signals obtained will be : s
= au
+
v s
= au
−
1 2
These can be combined to give u and v.
v
The third velocity component w can be found by rotating the probe through 90 degrees about its axis, or by similarly combining the signals from three slant wires.

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Used mainly to find out the constants A and B in the anemometer equations
General Method : Measurements performed at constant temperature for a number of values of flow velocity U, measured by some other device
Graph : a bv c f
=
(
2
I R w w
− w
T f
)
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Experimental Calibration vs. Computational Procedures :
Exponent n in King’s Law obtained experimentally
Physical properties of thin metal wire are different from that of the material in bulk
Cold length of wire due to presence of massive holders
Uncertainty in determination of Reynolds Number

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New technique
Specifically designed for high performance flow measurements
No need for tuning of frequency response
Almost constant bandwidth operation even when the flow and sensor conditions are varied
High frequency response and low noise characteristics
Suitable for turbulent flows with large frequency content and/or low turbulent intensity.

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Inverting amplifier circuit with hot wire resistance Rw connected within the feedback loop
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The choice of wire voltage Vw will depend on the type of measurements to be performed.
Ref : Product Datasheet Tao Systems

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CVA is particularly suited for flows with temperature drifts or non-isothermal flows
No need for auxiliary probe
New calibration scheme
PDR is the ratio of the power dissipated by the wire and the difference between heated and cold resistance.
PDR is shown to be only dependent on velocity even when temperature fluctuations are present
This is calibrated and related to velocity

the hot wire probes used here, all the different sources of error must be considered. These are:
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Disturbance Errors
High Frequency Errors
Spatial Resolution Errors
Calibration measurement Errors
Calibration Equation Errors
Calibration Drift Errors
Approximation Errors
Contamination of the sensing element

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The sensitive element is small enough not to introduce any disturbance into the flow
Suitable for both gases and liquids
Responds almost instantaneously to rapid fluctuations so that high frequency effects can be recorded without distortion
High sensitivity
Electrical output signal can be easily processed by analog and digital systems

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Does not sense flow direction
Intrusive method – leads to flow disturbances that are unacceptable in recirculating flows
Conduction losses to supports
Resistance of the sensor supplying cable affects the overheating ratio
Easily broken by mechanical impact or by a stream with high dynamic pressure
Not suitable for use in liquids with high electrical conductivity

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Time averaged Navier Stokes Equations : u
∂ u
∂ x
+ v
∂ u
∂ y
+ w
∂ u
∂ z
= −
1
ρ dp dx
+ v


∂ 2 u
∂ x
2
+
∂ 2 u
∂ y
2
+
∂ 2 u
∂ z
2


−
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Integral Method
Eddy Viscosity
Reynolds Stress Models
∂ u
' 2
∂ x
+
∂ u ' v '
∂ y
+
∂ u ' w '
∂ z uv
=
1
N
N ∑
1
( U i
−
U mean
)( V i
−
V mean
)
The Skewness is a measure of the lack of statistical symmetry in the flow.
3
Skewness
=
1
3 Nu rms
N
1
 u i
− u
2 u rms
=
1
N i
N
=
1 u u
2

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Three regions downstream of the grid :
Developing Region : Rod wakes merge, anisotropic and produces turbulent kinetic energy
Middle Region : Flow is nearly homogenous and isotropic
Last Region : Viscous Effects become significant
Decay law : Relates Reynolds number to starting position of Isotropic region and states that Turbulence decays with distance downstream
Theoretical Methods to validate decay law :
Skewness (Two Methods)
Dissipation rate of turbulent kinetic energy
Experimental Method : Use Hot Wire Anemometry to determine the turbulent intensities Ix, Iy and Iz
For isotropic turbulence
Ref: S.Meleschi (2004)

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Very Low Velocities :
At very low velocities the heat transfer from the wire is governed by natural convection.
Extrapolating the high velocity calibration curves gives error in the measured value
H.H.Bruun has proposed A swinging arm calibration method for low velocity hot-wire probe calibration
An integral technique is used for calibration

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High Velocities, compressible flow :
Effects from compressibility
Simultaneous measurement of pressure and velocity
The correction is quite complicated and is very often neglected.
Problem in Hypersonic flows : At higher Mach numbers for air the total temperature must be high enough to prevent liquefaction.
There is a maximum recommended overheat ratio for each wire material.
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Rarefied Flows (low pressure) :
Knudsen number increases.
If Kn>0.01 then heat transfer becomes a function of both velocity and Knudsen Number.
Fibre film probes instead of wire probes are used

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Wall Effects :
Heat conducted to the wall
Very high velocity is measured
The critical wall distance is typically 0.1 to 0.2 mm depending on free stream velocity (y+ < 3.5)
Heat loss increases with wire diameter and overheat ratio
For a fixed distance, effect increases with wall conductivity up to k/ko =
100 after which it becomes asymptotic.
Very high turbulence, revering flows :
 If the turbulence is very high or in reversing flows, where the velocity vector falls outside the opening angle of the sensor array
 Flying hot-wire systems
 Resulting velocity vector inside the opening angle.

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Wall shear stress :
Arrays of film probes mounted on the wall
Heat transfer can then be expressed as a function of the wall shear stress. The probes have to be calibrated in a known shear.
Secondary heat transfer to the wall changes the frequency response as compared to freely mounted probes
Two-phase flows :
Heat transfer is much higher in liquids than in gases
This can be utilized to measure the passage of for example air bubbles in water.
Binary mixtures :
Heat transfer to gases depends on the heat conductivity of the gas.
Can be used to measure the concentration in binary mixtures.

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HWA Construction
Physics of CT and CC mode
Sensitivity calculations and frequency response
Turbulence modeling
Applications in practical flows

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Anemometer
Prof. F. Durst, E. S. Zanoun : Experimental measurements
Shaghari B. Meleschi , Ultrasonic technique in grid generated turbulent flow , Masters Thesis
H.H.Bruun et al, A swinging arm calibration method for low velocity hot-wire probe calibration ,
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Calibration of Hot wire anemometers
Wikipedia