Procedure for Determining Turbulence

Length Scales Using Hotwire Anemometry

NASA/TM—2014-218403

December 2014

Lamyaa A. El-Gabry

The American University in Cairo, New Cairo, Egypt

Douglas R. Thurman

U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio

Philip E. Poinsatte

Glenn Research Center, Cleveland, Ohio
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Procedure for Determining Turbulence

Length Scales Using Hotwire Anemometry

NASA/TM—2014-218403

December 2014

Lamyaa A. El-Gabry

The American University in Cairo, New Cairo, Egypt

Douglas R. Thurman

U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio

Philip E. Poinsatte

Glenn Research Center, Cleveland, Ohio

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

The authors would like to acknowledge the support of the NASA Glenn Faculty Fellowship program and the input of 

Dr. Khairul Zaman on hotwire measurements.

Available from

NASA STI Information Desk

Mail Stop 148

NASA Langley Research Center

Hampton, VA 23681-2199

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identi¿ cation 

only. Their usage does not constitute an of¿ cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2014-218403 1 

Procedure for Determining Turbulence Length Scales 

Using Hotwire Anemometry 

Lamyaa A. El-Gabry 

The American University in Cairo 

New Cairo, 11835, Egypt 

Douglas R. Thurman 

U.S. Army Research Laboratory 

Glenn Research Center 

Cleveland, Ohio 44135 

Philip E. Poinsatte 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Hotwire anemometers are used to measure instantaneous velocity from which the mean velocity and 

the velocity fluctuation can be determined. Using a hotwire system, it is possible to deduce not only the 

velocity components and their fluctuation but to also analyze the energy spectra and from that the 

turbulence length scales. In this experiment, hotwire anemometry is used to measure the flow field 

turbulence for an array of film cooling holes. The objective of this paper is to document the procedure 

that is used to reduce the instantaneous velocity measurements to determine the turbulence length scales 

using data from the film-cooling experiments to illustrate the procedure. 

Introduction 

Hotwire systems have several advantages for turbulence research including lower cost than other 

systems such as laser Doppler anemometers, high frequency response, small size, and lower noise/signal 

ratio (Ref. 1). Hotwire anemometry relies on the principle of convective heat transfer to measure velocity. 

The heated wire or film element is placed in the flow field; changes in fluid velocity will change the 

resistance of the wire. The wire being very thin instantly responds to the change in fluid conditions. The 

output of the hotwire is a voltage signal that can be calibrated to velocity from which mean and 

turbulence quantities can be determined. 

The objective of the current study is to characterize the turbulence field for an array of film cooling 

holes at a high blowing ratio where the jet is detached. The detailed turbulence measurements are an 

enabler for the development and improvement of numerical methods and tools for predicting film cooling 

flow and heat transfer. Turbulence is a familiar notion but remains a complex physical phenomenon to 

model and predict. Bradshaw (Ref. 2) defines turbulence as “a three-dimensional time-dependent motion 

in which vortex stretching causes velocity fluctuations to spread to all wavelengths between a minimum 

determined by viscous forces and a maximum determined by the boundary conditions of the flow.” Some 

describe turbulence in terms of an energy cascade in which large scale vortices are dissipated into smaller 

vortices which in turn dissipate into yet smaller scales and so on until the viscous effects dominate and the 

turbulence energy is dissipated into heat. A detailed description of the associated length scales and 

mechanism of turbulent transport is beyond the scope of this technical memo. The authors refer readers to 

several seminal texts on the subject (Refs. 2 and 3). 
NASA/TM—2014-218403 2 

Experimental Apparatus 

The apparatus, shown in Figure 1, consists of an open-loop wind tunnel with a temperature-controlled 

coolant loop. The tunnel is a suction type tunnel that draws air from the room and passes it through flow 

conditioning screens prior to entering the test section. The freestream velocity is 30 ft/sec, and the 

Reynolds number based on freestream velocity and coolant hole diameter is 11,000. The test section is a 

square section 8.2 by 8.2 in. and 34 in. in length. 

The model consists of a three-hole array of film cooling holes that are fed by three long tubes. The 

model is inserted into the test section of the tunnel such that the heat transfer surface forms the wind 

tunnel floor. The cooling holes have a diameter D = 0.75 in., spacing P/D = 3, and are angled 30q from 

the streamwise direction. The coolant was fed through a manifold into three separate flow meters, then 

through 18 in. of hose and 12 in. of acrylic tube. The entire coolant path was insulated. 

Velocity and turbulence are measured using a two-wire hotwire system for obtaining components. 

The hotwire is powered by a constant temperature anemometry system, and data is acquired at a rate of 

50 kHz for 3 sec. Two different X-wire probes are used; one is used to measure the U- and V-components 

of velocity, and the other is used to measure the U- and W-components. In both cases, the two wires of the 

probe are separated by some distance and crossed to make an “X” shape. Figure 2 shows the probe 

(TSI Model 1246) used to measure the U- and W-components, and Figure 3 shows the probe (TSI Model 

1240) used to measure the U- and V-components. 

Figure 1.—Wind tunnel test apparatus (left) and film cooling model (right). 

Figure 2.—TSI Probe Model 1246 for measuring U- and W-components of velocity. 
NASA/TM—2014-218403 3 

Figure 3.—TSI Probe Model 1240 for measuring U- and V-components of velocity. 

Procedure 

Calibration 

The probes are calibrated in-situ, with the probe located upstream of the film cooling holes at midheight of the tunnel and the coolant flow shut off. The freestream velocity, measured using total and static 

pressure probes, is then varied from 0 to 150 ft/s, and the wire voltage is measured and recorded at each 

freestream velocity. A calibration curve for one of the probes is shown in Figure 4. The points are curve 

fit using a 4th order polynomial that relates the delta voltage (the difference between the mainstream 

velocity voltage and the zero velocity voltage) for each wire to the freestream tunnel velocity. This 

velocity is assumed to be in the streamwise direction with no V- or W-components. The calibration is 

verified at the start of each run and repeated as needed. 

Velocity Trace 

During tunnel operation, the wire voltages are recorded to a data acquisition system and converted to 

velocity using the calibration curves to obtain the velocity components. As an example, Figure 5 shows 

the velocity trace at a location upstream of the cooling hole leading edge along the centerline of the tunnel 

and 1.0 in. above the floor of the tunnel. The mean velocity, Umean, is 32 ft/s. The standard deviation, nj, of 

the velocity is 0.32 ft/s. Therefore the turbulence intensity, nj/Umean, at this location is 1 percent. 

Turbulence Energy Density Spectra Analysis 

The velocity signal shown in Figure 5 is in the time domain. By using a fast Fourier transformation 

(FFT), the data can be transformed to the frequency domain. The Mathworks Inc. software package 

Matlab is used for the analysis. The Appendix contains a listing of the Matlab script, parts of which are 

repeated here for clarity. 

FFT Analysis Transforms U From Time to Frequency Domain 

The velocity trace in Figure 5 contains 150,000 points (sampling frequency of 50 kHz for 3 sec). Let 

U be the array consisting of the 150,000 velocity points sampled over the 3-sec duration. aFData is the 

FFT of the U-velocity (i.e., the transform into the frequency domain). L is the length of the signal. 

aFData = fft(U) 

L = length(aFData) 

The FFT of the U-velocity signal results in half the number of unique points, so let 

n = L/2 = 150,000/2 = 75,000 
NASA/TM—2014-218403 4 

Figure 4.—Sample hotwire calibration curve for two-wire probe. 

Figure 5.—U-velocity trace in the freestream. 

y =Ͳ0.0502x4 + 2.0995x3 Ͳ 1.9471x2 + 6.8845x Ͳ 0.0307

R2 = 0.9999

y =Ͳ0.0987x4 + 2.4105x3 Ͳ 3.2774x2 + 7.7103xͲ 0.0349

R2 = 0.9999

Ͳ20

0

20

40

60

80

100

120

140

160

0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5.000

Uveclocity ft/s

delta_Voltage

delV1

delV2

Poly. (delV1)

Poly. (delV2)

0 0.5 1 1.5 2 2.5 3

30

30.5

31

31.5

32

32.5

33

33.5

U-velocity trace

time (s)

U-velocity (ft/s)
NASA/TM—2014-218403 5 

Plot Amplitude and Power Spectra of U-Velocity 

To plot the velocity in the frequency domain, the absolute value of the FFT is used to calculate the 

magnitude, or amplitude, of the velocity array, i.e. 

aFMag = abs(aFData(1:n)/L) 

Here, L is used to normalize the data. The corresponding frequency array is 

aFreq = Fs*(1:n)/n

Figure 6 is the amplitude spectrum of the U-velocity component. The amplitude ranges from 0 to the 

mean velocity 32 ft/s. The first point in the FFT corresponds to the zero frequency and mean velocity. 

This first point has been removed from the plot to enable us to analyze the turbulence energy. 

Figure 6.—Amplitude spectrum of U-velocity (omitting mean velocity at 0 Hz). 

 

100 101 102 103 104 105

0

0.005

0.01

0.015

0.02

0.025

Single-Sided Amplitude Spectrum of U(t)

Frequency (Hz)

|U(f)| (ft/s)
NASA/TM—2014-218403 6 

Power is defined as: 

Power = abs(fft(U)).^2/L 

Dividing by L ensures that the power is independent of the length of the signal. The energy E(f) can be 

determined by dividing power by the frequency. The energy density spectrum is shown in Figure 7. 

Integral Length Scale 

The integral length scale, Ȧ௫ (macro-scale), is a measure of the largest eddy size in the turbulent fluid 

and is calculated using the method shown by Roach (Ref. 4): 

Ȧ௫ ൌ ቈ

ܧሺ݂ሻܷ୫ୣୟ୬

Ͷݑതଶ ቉

௙՜଴

where nj is the standard deviation of the U-velocity. To get E(f) as the frequency approaches zero from the 

data, the first 100 points are averaged, where the data asymptotes to a fixed value at the low frequencies. 

The dashed yellow line in Figure 7 shows E(f) as f approaches zero, and is determined to be 1.3445e–4. 

Thus the length scale is calculated to be 

Ȧ௫ ൌ 

ሺͳǤ͵ͶͶͷ݁ିସ ݂ݐଶ

ݐ݂ ͳǤ ͅ͵ሻሺ ݏ

ሻ ݏ

Ͷ ൬ͲǤ͵ʹଶ ݂ݐଶ

ݏଶ ൰

 ൌ ͲǤͲͳͲͷ݂ݐ ൌ ͲǤͳʹͷ͸݅݊Ǥ

The length scale at this location (freestream inlet) is found to be about 1/8th in., which is roughly the size 

of the wire mesh upstream of the test section. 

Figure 7.—Energy density spectrum of U-velocity. 

10-1 100 101 102 103 104 105 10-12

10-10

10-8

10-6

10-4

10-2 Energy Density Spectrum

Frequency (Hz)

Energy/Hz (ft2/s) Energy, E(f) (ft2/s) 
NASA/TM—2014-218403 7 

Dissipation Length Scale 

The dissipation length scale, ߣ௫ (micro-scale), is a measure of the average dimension of the eddies 

responsible for the dissipation of turbulent energy. It is calculated as shown by Roach (Ref. 4): 

ͳ

௫ߣ

ଶ ൌ ʹߨଶ

ܷଶݑതതଶത න ݂ଶܧሺ݂ሻ݂݀

ஶ

଴

The trapezoid rule is used for the numerical integration; the integration yields 2.3067e6

 ft2/s4. 

ͳ

௫ߣ

ଶ ൌ ʹߨଶ

ሺ͵ͳǤ ͅଶ ݂ݐଶ

ݏଶ ሻሺͲǤ͵ʹଶ ݂ݐଶݏଶ ሻ

ቈʹǤ͵Ͳ͸͹݁଺ ݂ݐଶ

ݏସ ቉ ൌ ͶǤͶͲͲͳ݁ହ ͳ

ଶݐ݂

? ߣ௫ =0.0015 ft = 0.0181 in. 

Results 

The same procedure for calculating the length scale is repeated at each measurement location. 

Figure 8 shows the integral length scale along the centerline of the central cooling hole. Figure 9 shows 

the dissipation length scale along the jet centerline. The integral length scale is the size of the largest 

eddy. Note that at the hole location, the integral length scale is roughly 0.75 in. which is the diameter of 

the coolant hole. The dissipation length scale is smaller and is independent of the geometry. Whereas the 

integral length scale decreases in the flow direction as the coolant mixes and diffuses in the freestream, 

the micro-scale does not decrease or increase with the flow direction. The macro-scale decreases as the 

larger eddies break down into smaller eddies. 

Figure 8.—Integral length scale contour along jet centerline. 

x [inch]

z [inch]

0 2 4 6 0

1

2

3

4 MacroLS [inch]: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
NASA/TM—2014-218403 8 

Figure 9.—Dissipation length scale contour along jet centerline. 

Conclusions 

The procedure for calculating the turbulence length scales from using hotwire anemometry is 

outlined, and a data point in the freestream inlet is calculated to illustrate the process. The instantaneous 

velocity signal obtained using a hotwire anemometer is processed using a fast Fourier transform to 

convert the signal from the time domain to the frequency domain. The magnitude is plotted against the 

frequency to yield the amplitude spectrum. The energy spectrum is obtained and plotted against 

frequency. It is from the energy spectrum and equations derived in Roach (Ref. 4) that the integral length 

scale (macro-scale) and dissipation length scale (micro-scale) are calculated. The results for the length 

scales along the centerline are plotted and discussed. The integral length scale is commensurate with the 

geometry. At the inlet, the integral length scale is approximately the size of the wire mesh upstream, 

while at the hole discharge, the integral length scale is roughly equal to the hole diameter. The integral 

length scale decreases in the flow direction as large eddies dissipate into smaller eddies. The micro-scale 

is at least one order of magnitude smaller than the integral length scale, and unlike the macro-scale, it 

appears to be independent of geometry and flow direction. 

References 

1. Bruun, H.H., 1995, “Hot-Wire Anemometry Principles and Signal Analysis,” Oxford University 

Press: Oxford, UK. 

2. Bradshaw, P., 1971, “An Introduction to Turbulence and its Measurement,” Pergamon Press: Oxford, 

UK. 

3. Hinze, J.O., 1957, “Turbulence,” McGraw-Hill: New York, USA. 

4. Roach, P.E., 1986, “The generation of nearly isotropic turbulence by means of grids,” Heat and Fluid 

Flow, vol. 8, No. 2, pp. 82-92. 

x [inch]

z [inch]

0 2 4 6 0

1

2

3

4

MicroLS [inch]: 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07
NASA/TM—2014-218403 9 

Appendix—MATLAB Script 

% Read in the two wire signal and process it to get velocity 

E1 = data(:,1); %let column 1 be voltage 1 (wire 1)

E2 = data(:,2); %let column 2 be voltage 2 (wire 2)

E0_1 = 4.667; %Zero voltage for wire 1

E0_2 = 4.336; %Zero voltage for wire 2

delE1 = E1 - E0_1; 

delE2 = E2 - E0_2;

a1=0.1839;

b1=-0.198;

c1=3.9795;

d1=1.8669;

a2=0.134;

b2=0.259;

c2=2.9707;

d2=1.8816;

vel1=a1*delE1.^4+b1*delE1.^3+c1*delE1.^2+d1*delE1;

vel2=a2*delE2.^4+b2*delE2.^3+c2*delE2.^2+d2*delE2;

U=0.5*(vel1+vel2);

W=0.5*(vel1-vel2);

% Plot the U-velocity trace

Fs=50000; % Sampling frequency of 50 kHz

L=length(U); % Number of samples

t=(0:L-1)/(Fs); % Time vector

plot(t,U); % Plot the velocity trace 

title('U-velocity trace');

ylabel('U-velocity (ft/s)');

xlabel('time (s)');

% Perform FFT analysis on the U-velocity signal and plot Amplitude spectrum

aFData=fft(U); % Take FFT of U signal

n=L/2; % FFT will yield half the number of unique points

aFreq=Fs*(1:n)/n; % Frequency array (half the length of signal)

aFMag=abs(aFData(1:n)/L); % Normalized Magnitude array (half the length of 

signal)

figure;

semilogx(aFreq(2:n),aFMag(2:n)) % Plot frequency against magnitude

title('Single-Sided Amplitude Spectrum of U(t)')

xlabel('Frequency (Hz)')

ylabel('|U(f)| (ft/s)')

% Plot the energy density spectrum

Power=abs(fft(U)).^2/L; %Power is the magnitude squared by L

Energy=Power/Fs;

figure

loglog(aFreq,Energy(2:L/2+1))

title('Energy Density Spectrum')

xlabel ('Frequency (Hz)')

ylabel ('Energy, E(f) (=Power/frequency) (ft2/s)')

 
NASA/TM—2014-218403 10 

% Calculate the integral length scale (macro scale) Roach Eqn 15

uprime=std(U)

uprimesquared=uprime^2;

Ubar=mean(U)

Tu=uprime/Ubar

Ef0=mean(Energy(2:100))

IntegralLengthScale=Ef0*mean(U)/(4*uprimesquared)

% Calculate the dissipation length scale (micro scale) Roach Eqn 8

for i=1:length(aFreq)

 Y(i)=aFreq(i)*aFreq(i)*Energy(i);

end

Z=trapz(aFreq,Y) % Use trapezoid rule for integration

Z=Z*2*pi^2/(Ubar^2*uprimesquared);

DissipationScale=sqrt(1/Z)