On The Dynamic Response of Constant Temperature Hot-Wire ... · P. Freymuth, (1977), Further...
Transcript of On The Dynamic Response of Constant Temperature Hot-Wire ... · P. Freymuth, (1977), Further...
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On The Dynamic Response of Constant Temperature Hot-Wire Anemometers (CTHWA)
Elissavet Boufidi
Supervisor: Fabrizio Fontaneto (VKI)
University supervisor: Tony Arts (UCL/VKI)
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CFM LEAP engineHP turbineheat transfer
Turbulence effect in aero-engines
Combustion chamber
turbulence-chemistry interaction
Context
LP turbine
bl transition - separation
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CFM LEAP engine
High speed flow Unsteady phenomenaHigh turbulence level
Context
Turbulence measurements in turbomachinery flows:
BPF: 1~15 𝑘𝐻𝑧
M: 0.1~1.2
Tu: > 10% ?
Range of conditions through an aero-engine:
High frequency responseHigh spatial resolution
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Why CTHWA?
Basic research tool in turbulence
High spatial resolution:𝑑𝑤𝑖𝑟𝑒 ~5 − 10 𝜇𝑚
High frequency response:up to 60 𝑘𝐻𝑧
Constant Temperature Hot-Wire Anemometry
(CTHWA)
Investigation of the dynamic response of a CTHWA
system
TODAY’S PRESENTATION:
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Why CTHWA?
Basic research tool in turbulence
High spatial resolution:𝑑𝑤𝑖𝑟𝑒 ~5 − 10 𝜇𝑚
High frequency response:up to 60 𝑘𝐻𝑧
Constant Temperature Hot-Wire Anemometry
(CTHWA)
Investigation of the dynamic response of a CTHWA
system
TODAY’S PRESENTATION:
HWA is an established technique. What is left to
investigate?
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CTHWA
CTHWA: probe + electronics
Feedback amplifier
Wheatstone bridge
C
A𝑬𝒐𝒖𝒕
𝑹𝟏 𝑹𝟐
𝑹𝒘
𝑹𝑳
+
-
G
𝑹𝒅𝒆𝒄
B
+𝑹𝒘: HW resistance𝑹𝑳: probe stem+ prongs +cable resistance
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Square wave test
Square wave test signal
CTHWA: probe + electronics
“square wave test”Common method for dynamic
response optimization:
Feedback amplifier
Wheatstone bridge
C
A𝑬𝒐𝒖𝒕
𝑹𝟏 𝑹𝟐
𝑹𝒘
𝑹𝑳
+
-
G
𝑹𝒅𝒆𝒄
B
+𝑹𝒘: HW resistance𝑹𝑳: probe stem+ prongs +cable resistance
Input voltage at A
Output voltage
Square wave voltage perturbation
Velocity impulse perturbation
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CTHWA typical use
Static calibration
U = f (𝑬𝟐)
Optimized response to square wave
: -3 dB attenuation𝒇𝒄𝒖𝒕 =𝟏
𝟏. 𝟑𝝉
τ
0.97 +
Response assumed flat up to 𝑓𝑐𝑢𝑡
Static calibration used Dynamic response effects neglected!
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In this work:
Numerical model Experimental square
wave test
Highlight often neglected effects of the dynamic behavior of CTHWA on the measurement of turbulence
GOAL:
USING: +
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In this work:
Numerical model
Non-linear effects important for large fluctuations
Is the square wave test equivalent to the response to a velocity perturbation?
Experimental square wave test
Highlight often neglected effects of the dynamic behavior of CTHWA on the measurement of turbulence
GOAL:
USING: +
Thermal transient effects due to heat conductions to the wire supports
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Experimental method
HWA
• Phase averaging to eliminate effect of flow fluctuations and
obtain a clean signal
• Using the embedded square wave voltage generator of the Dantec Streamline system,
acquire the signal with 𝑓𝑠 = 1.25 𝑀𝐻𝑧
• HW probe placed in the flow (Response depends on the flow conditions)
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Experimental method
• Compute Fourier transform of averaged response
𝑓𝑐𝑢𝑡
• System is a “black-box”, unknown amplitude of input voltage
• To compute transfer function: Non-dimensionalization with
amplitude at low frequency, where amplitude can assumed to be 1
Amplitude transfer function
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Numerical modelProposed by Freymuth (1977) & used by Weiss et al. (2013):
1. Wheatstone bridge equation:
2. Amplifier equation:
3. Wire equation:
𝛿 =R1(𝑅
∗ − 𝑅𝑤)
(𝑅1+𝑟𝐿 + 𝑅∗)(𝑅1+𝑟𝐿 + 𝑅𝑤)
𝐸 + 𝑀𝐵𝑑𝐸
𝑑𝑡+R1(𝑟𝐿 + 𝑅𝑤)
(𝑅1+𝑟𝐿 + 𝑅𝑤)
𝐸𝑡𝑅𝑡
𝑀′′𝑑2𝐸
𝑑𝑡+ 𝑀′
𝑑𝐸
𝑑𝑡+ 𝐸 = 𝐺𝛿 + ΕΒ
𝑚𝑤𝑐𝑤𝜕𝑇𝑤𝜕𝑡=
𝐸2𝑅𝑤(𝑅1+𝑟𝐿 + 𝑅𝑤)
2 − 𝑁𝑢𝜋𝑙𝑤𝑘𝑓 𝑇𝑤 − 𝜂Τ0 − k𝑤𝐴𝑐𝜕2𝑇𝑤𝜕𝑥2
Convection losses term for non-dimensional calibration
𝑁𝑢 = 𝑓(𝑅𝑒)
heat conduction
term
Term only present for the square wave
test
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Numerical modelProposed by Freymuth (1977) & used by Weiss et al. (2013):
1. Wheatstone bridge equation:
2. Amplifier equation:
3. Wire equation:
𝛿 =R1(𝑅
∗ − 𝑅𝑤)
(𝑅1+𝑟𝐿 + 𝑅∗)(𝑅1+𝑟𝐿 + 𝑅𝑤)
𝐸 + 𝑀𝐵𝑑𝐸
𝑑𝑡+R1(𝑟𝐿 + 𝑅𝑤)
(𝑅1+𝑟𝐿 + 𝑅𝑤)
𝐸𝑡𝑅𝑡
𝑀′′𝑑2𝐸
𝑑𝑡+ 𝑀′
𝑑𝐸
𝑑𝑡+ 𝐸 = 𝐺𝛿 + ΕΒ
𝑚𝑤𝑐𝑤𝜕𝑇𝑤𝜕𝑡=
𝐸2𝑅𝑤(𝑅1+𝑟𝐿 + 𝑅𝑤)
2 − 𝑁𝑢𝜋𝑙𝑤𝑘𝑓 𝑇𝑤 − 𝜂Τ0 − k𝑤𝐴𝑐𝜕2𝑇𝑤𝜕𝑥2
System of 2 ODEs and 1 PDE
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Numerical solution
∙∙
x
x=0 x=L∙∙ ∙ ∙ ∙i i+1i-1
⋯
∙
𝐝𝐱𝐢
𝑚𝑤,𝑖𝑐𝑤,𝑖𝜕𝑇𝑤,𝑖𝜕𝑡
=𝐸2𝑅𝑤,𝑖
(𝑅1+𝑟𝐿 + 𝑅𝑤,𝑡𝑜𝑡)2 − 𝑁𝑢𝑖𝜋𝑙𝑖𝑘𝑓 𝑇𝑤,𝑖 − 𝜂Τ0 − k𝑤𝐴𝑐,𝑖
𝑇𝑤,𝑖+1 − 2𝑇𝑤,𝑖 + 𝑇𝑤,𝑖−1𝑑𝑥𝑖
The wire is discretized in n-1 elements and n nodes
𝑞𝑐𝑜𝑛𝑑
𝑞𝑗𝑜𝑢𝑙𝑒
𝑞𝑐𝑜𝑛𝑣
i
The spatial derivative is discretized and the wire equation is written for each node:
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Numerical solution
∙∙
x
x=0 x=L∙∙ ∙ ∙ ∙i i+1i-1
⋯
∙
𝐝𝐱𝐢
𝑚𝑤,𝑖𝑐𝑤,𝑖𝜕𝑇𝑤,𝑖𝜕𝑡
=𝐸2𝑅𝑤,𝑖
(𝑅1+𝑟𝐿 + 𝑅𝑤,𝑡𝑜𝑡)2 − 𝑁𝑢𝑖𝜋𝑙𝑖𝑘𝑓 𝑇𝑤,𝑖 − 𝜂Τ0 − k𝑤𝐴𝑐,𝑖
𝑇𝑤,𝑖+1 − 2𝑇𝑤,𝑖 + 𝑇𝑤,𝑖−1𝑑𝑥𝑖
The wire is discretized in n-1 elements and n nodes
𝑞𝑐𝑜𝑛𝑑
𝑞𝑗𝑜𝑢𝑙𝑒
𝑞𝑐𝑜𝑛𝑣
i
The spatial derivative is discretized and the wire equation is written for each node:
System of n+2 ODEs
𝑻𝒘,𝟎 = 𝑻𝟎 𝑻𝒘,𝑳 = 𝑻𝟎
Solve with Runge-Kutta method
B.C. B.C.
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Numerical solution
𝑚𝑤,𝑖𝑐𝑤,𝑖𝜕𝑇𝑤,𝑖𝜕𝑡
=𝐸2𝑅𝑤,𝑖
(𝑅1+𝑟𝐿 + 𝑅𝑤)2 − 𝑁𝑢𝑖𝜋𝑙𝑖𝑘𝑓 𝑇𝑤,𝑖 − 𝜂Τ0 − k𝑤𝐴𝑐,𝑖
𝑇𝑤,𝑖+1 − 2𝑇𝑤,𝑖 + 𝑇𝑤,𝑖−1𝑑𝑥𝑖
Wire equation:
𝑇𝑤 =1
𝐿 0
𝐿
𝑇𝑤 𝑥 𝑑𝑥
𝑅𝑤 = 𝑅𝑟𝑒𝑓(1 + 𝑎𝑤(𝑇𝑤 − 𝑇𝑟𝑒𝑓))
Wheatstone bridge equation:
𝛿 =R1(𝑅
∗ − 𝑅𝑤)
(𝑅1+𝑟𝐿 + 𝑅∗)(𝑅1+𝑟𝐿 + 𝑅𝑤)
𝐸 + 𝑀𝐵𝑑𝐸
𝑑𝑡+R1(𝑟𝐿 + 𝑅𝑤)
(𝑅1+𝑟𝐿 + 𝑅𝑤)
𝐸𝑡𝑅𝑡
Temperature profile along the wire at each time instant
𝑇𝑤 spatial average and corresponding 𝑅𝑤
Input to
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Circuit parameters’ estimationCommercial system used, certain circuit parameters are not know:
𝑮,𝑴′, 𝑴′′, 𝑬𝒃, 𝑴𝑩
• 𝐸𝑏, offset voltage of the amplifier. No adjustment in our system. Increases the damping of the system, a constant high value is selected following Perry (1972).
C
A𝑬𝒐𝒖𝒕
𝑹𝟏 𝑹𝟐
𝑹𝒘
𝑹𝑳
+
-
G
𝑹𝒅𝒆𝒄
B
𝐸𝑏 = 5 𝑉
• 𝑀𝐵, bridge time constant takes into account high frequency components. It is adjusted by varying an inductance 𝜆𝐿 of a set of coils to compensate for the inductance of the cable 𝜆𝐶(Freymuth 1977),
• G, amplifier gain. Adjusted during square wave test.
𝑀𝐵 = 𝜆𝐿 R2 − 𝜆𝐶 R1(𝑛 + 1)2
= 10−10
• 𝑴′,𝑴′′time constants of the amplifier. Adjusted during square wave
𝝀𝑳𝝀𝑪
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Comparison with experimentsComparison of the model with experimental square wave test
Tungsten wire𝑑𝑤 = 9 𝜇𝑚l𝑤 = 1𝑚𝑚𝑇𝑤 = 455 𝐾
M=0.25
𝐺 = 230𝑀′ = 1.6 ∙ 10−5
𝑀′′ = 10−11To predict the same 𝑓𝑐𝑢𝑡
• Steeper attenuation at high frequencies
• Different behavior at low frequencies
Further investigation required!
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In this work:
Numerical model
Non-linear effects important for large fluctuations
Is the square wave test equivalent to the response to a velocity perturbation?
Experimental square wave test
Highlight often neglected effects of the dynamic behavior of CTHWA on the measurement of turbulence
GOAL:
USING: +
Thermal transient effects due to heat conductions to the wire supports
![Page 21: On The Dynamic Response of Constant Temperature Hot-Wire ... · P. Freymuth, (1977), Further Investigation of the Nonlinear Theory for Constant-Temperature Hot-Wire Anemometers, Journal](https://reader035.fdocuments.net/reader035/viewer/2022081407/5f1da886d516ea1dfa494867/html5/thumbnails/21.jpg)
Velocity vs voltage perturbation
Voltage perturbation “square wave test”
Electronic testing Common method to test dynamic
response
Velocity (Reynolds) perturbation
Direct testing Real case Difficult to perform
The square wave test should be always used to adjust the response and avoid a self-oscillating system
BUT can it be used as a reliable estimation of the system’s response?
Disagreement in literature
Use the model to simulate both cases:
𝐸𝑡• Case1, Inputs: 𝑅𝑒 = 𝑐𝑜𝑛𝑠𝑡
• Case2, Inputs: 𝑅𝑒
𝐸𝑡 = 0
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Velocity vs voltage perturbation
Only difference at high frequencies
Compare response to square wave voltage and response to Reynolds
impulse
Square wave seems to simulate adequately Reynolds
impulse
Transfer function could be used to correct amplitude
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Thermal transient effects
Low frequency attenuation due to heat conduction to the wire supports from
𝑓𝑙~150 𝐻𝑧𝑓𝑙~150 𝐻𝑧
1.4 𝑘𝐻𝑧
From 𝑓 > 1.4 𝑘𝐻𝑧 amplification due to system dynamics, attenuation not
completed
Temperature profile along the wire
𝑇𝑤 = 450 𝐾
𝑇𝑤,𝑒𝑥𝑝 = 455 𝐾
Square wave response also features attenuation
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Thermal transient effects
𝐺 = 1000𝑀′ = 5 ∙ 10−6
𝑀′′ = 10−12
Adjust amplifier parameters to move cut-off to higher frequencies
Focus on attenuation
l/d ratio effect
High l/d
Flatter 𝑇𝑤 disribution
convection > conduction
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Thermal transient effects
High l/d
Final attenuation value decreases
High l/d should be used for less attenuation
Small l should be used for spatial resolution
Min d restricted by mechanical strength
Choice depends on application
Trade-off
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Thermal transient effects
Attenuation important for a wide range of frequencies even for l/d=330
Correction should be applied
𝑓𝑙~150 𝐻𝑧
1.4 𝑘𝐻𝑧
Attenuation also present at square wave test response,
negligible difference
Square wave transfer function could be used for
correction
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Non-linearity
+
Non-linear static responseheat transfer law
Non-linear dynamic responseBridge and wire equations of the
dynamic model
The response of CTHWA is non-linear
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Non-linearity
+
Non-linear static responseheat transfer law
Non-linear dynamic responseBridge and wire equations of the
dynamic model
The response of CTHWA is non-linear
Corrected with calibration
Valid for small fluctuations
Usually neglected
Errors for large fluctuations
Turbomachinery flows!
Errors mainly affect odd turbulence moments,e.g. skewness
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Non-linearity
Reynolds input to the model:
𝟐𝝅𝒇
𝟐𝑨
Output voltage
𝑹𝒆𝒊𝒏
Static calibrationOutput Reynolds 𝑹𝒆𝒐𝒖𝒕
Skewness of 𝑹𝒆𝒐𝒖𝒕
Increasing with increasing amplitude
Results agree with Weiss et al. (2013)
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Non-linearity
𝑻𝒖,𝒐𝒖𝒕 < 𝑻𝒖,𝒊𝒏
For increasing frequency:
Heat conduction attenuation +system damping
Range of input frequencies affected by attenuation
Significant attenuation
Correction should be applied!
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Conclusions
For small fluctuations, the first two could be corrected by applying a transfer function to the measured spectrum. According to this model, the transfer function obtained with the square wave test could be used but further validation is required.
• Attenuation due to heat conduction • Amplification or damping due to system dynamics• Non-linearity errors
For large fluctuations the effect of non-linearity can have a significant effect on the skewness. It can not be corrected by a transfer function. Correction method existing for Constant-Voltage-Anemometry (Berson et al. 2009) but not for CTHWA.
Identification of possible sources of errors in turbulence measurements due to the dynamic response of CTHWA :
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Future work
Further work on the validation of the dynamic model:
• Use a VKI anemometer instead of a Dantec system to better estimate the circuit parameters
• Consider direct testing (eg by laser heating) in addition to the electronic testing
Further work on the possibility of corrections.
Use of the model for the design of HW probes with optimized response
Investigate spatial resolution issues
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References
P. Freymuth, (1977), Further Investigation of the Nonlinear Theory for Constant-Temperature Hot-Wire Anemometers, Journal of Physics E: Scientific Instruments 10.
J. Weiss, A. Berson, G. Comte-Bellot, (2013) Investigation of Non- Linear Effects in Constant Temperature Anemometers, Measurement Science and Technology 24.
A. Berson, P. Blanc-Benon, G. Comte-Bellot, (2009), A Strategy to Eliminate All Nonlinear Effects in Constant-Voltage Hot-Wire Anemometry, Review of Scientific Instruments 80.
A. E. Perry, G. L. Morisson, (1971) A Study of the Constant- Temperature Hot-Wire Anemometer, Journal of Fluid Mechanics 47 (3).
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Thank you for your attention!