selection of optimal multi-hotwire probe in constant

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Experiments in Fluids (2019) 60:27 https://doi.org/10.1007/s00348-018-2675-0

RESEARCH ARTICLE

Selection of optimal multi-hotwire probe in constant temperature anemometry (CTA) for transonic flows

E. Yablochkin1 · B. Cukurel1

Received: 10 July 2018 / Revised: 25 December 2018 / Accepted: 26 December 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2019

AbstractTo decouple dimensional flow perturbation quantities, a probe optimization method is proposed for multi-wire constant temperature anemometry in transonic flow conditions. Historically, hotwire measurements of density, velocity, and total temperature fluctua-tions in transonic flows are challenging due to the complexity of calibration and difficulties in obtaining a probe with favorably conditioned sensitivity matrix. Based on universal empirical correlation for heated cylinders in compressible flow, the current method relies on evaluation of wire-voltage sensitivities to density, velocity, and total temperature perturbations. To maximize the signal-to-noise ratio (SNR) of the decoupling, a two-step optimization-sifting procedure is developed. The first-step optimization maximizes a value function geared towards lower sensitivity-matrix condition numbers, better robustness against wire temperature-setting errors, and large sensitivities. From the resulting Pareto front, the second step sifts further through the candidate probes by decoupling simulated noisy voltage and ranking the probes by their decoupling quality. The performance of the optimal wire temperatures and diameters combination is contrasted against probes stemming from a naive selection from the parameter space. According to the artificial data, only the optimal probes enable decoupling with reasonable SNR. Finally, the research effort pro-poses guidelines to define a probe with desirable properties, applicable across a wide range of transonic flow conditions.

Graphical abstract

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List of symbols� (–) Angle between the measured voltages

vector Eme

and the vector A ⋅ F

aref (1/°C) Wire resistance temperature coefficient� (–) Ratio of specific heats for air� (–) Measure of how much ||||A ⋅ F|||| falls

short of its maximum possible value� (–) Recovery factor, Tr∕T0� (–) Overheat ratio Tw∕T0�(A) (–) Condition number of matrix A

μ (N s/m2) Viscosity of air� (kg/m3) Freestream flow density�2 (–) Variance of condition number with

respect to different flow combinations�wr (–) Overheating parameter, (Tw − Tr)∕Tr� (–) Compressibility correction function� (Ω m) Wire resistivity�u (rad/s) Frequency of perturbation for artificial

signal of u�� (rad/s) Frequency of perturbation for artificial

signal of ��T0

(rad/s) Frequency of perturbation for artificial signal of T0

A (–) Probe sensitivity matrixAmprel-noise (%) Max amplitude for relative noiseAmpDC-noise (V) Max amplitude for DC noisedw (m) Wire diameterE (V) Wires voltagesEme

(V) Vector of measured normalized wire-voltage perturbations

F− (–) Vector of decoupled normalized flow

perturbationsh [W/(m2 K)] Convection heat transfer coefficientkf [W/(m2 K)] Thermal conductivity of the fluidKn∞ (–) Knudsen number of free streaml (m) Wire lengthM (–) Freestream Mach numberm (–)

[1 + (� − 1)∕2 ⋅M2

]−1mt [N s/(m2 K)] � log �∕� log T0nt [W/(m2 K)] � log kf∕� logT0nflows (–) Number of flowsNu (–) Nusselt numberQw (W) Convective heat transfer rate from a wireQualu (–) Measure of velocity decoupling qualityQual� (–) Measure of density decoupling qualityQualT0 (–) Measure of T0 decoupling qualityQRMS (–) RMS decoupling qualityRe (–) Reynolds numberReT0 (–) Reynolds number based on wire diam-

eter, with viscosity evaluated at T0Rl (Ω) Resistance of lead wiresRt (Ω) Top-of-bridge resistanceRw (Ω) Wire resistance

Rref (Ω) Reference wire resistance at ref temperature

Su (–) Sensitivity to velocity perturbationS� (–) Sensitivity to density perturbationST0 (–) Sensitivity to total temperature

perturbationsmax, smin (–) Largest and smallest singular values of

A

t (s) TimeT0 (K) Freestream flow total temperatureTw (K) Wire temperatureTref (K) Reference wire temperature for

resistanceΔTpert (K) Wire temperature perturbation applied

for estimate of sensitivity matrix pertur-bation (ΔA)

u (m/s) Freestream velocityW1,W2 (–) Objective weights for value function(⋅)� (–) Perturbation of a quantity(⋅) (–) Mean of a quantity(⋅)i (–) Index of flow combination i

(–) Averaged quantity normalized by min value in flow

�1

nflows

∑nflowsi

�(⋅)i

min(⋅)i

��

(–) Averaged quantity normalized by max value in flow

�1

nflows

∑nflowsi

�(⋅)i

max(⋅)i

��

(⋅)inst (–) Instantaneous property(⋅)inp (–) Input perturbation amplitude for artifi-

cial signal(⋅)clean (–) Artificial signal without noise(⋅)noisy (–) Artificial signal with noise||⋅|| (–) Frobenius normΔ (–) Small change in quantity

1 Introduction

Constant temperature anemometry (CTA) measurements for transonic flow conditions are typically very demanding, as there are simultaneous perturbations of velocity, density, and total temperatures. Therefore, defining the attributes of a probe to operate under these conditions is non-trivial.

In general, sensitivity-based compressible-flow turbu-lence measurements can be described by the individual contribution of the velocity, density, and T0 perturbations to the voltage perturbation relative to the mean values, as presented in the seminal work of Horstman and Rose (1977):

where E is the wire voltage, (⋅)� is for perturbation, and (⋅) is for mean quantities. The sensitivities to velocity, density, and

(1)E�

E= S� ⋅

��

�+ Su ⋅

u�

u+ ST0 ⋅

T �0

T0

,


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total temperature perturbations Su, S�, ST0 are logarithmic derivatives defined as follows:

It is worth mentioning that the full voltage-sensitivity equation includes sensitivities to angle at which the flow impinges upon the probe. According to Spangenberg (1955), only the flow component normal-to-the-wire affects the heat loss. Therefore, these angular effects depend only on the slant angle of the wires and not on wire temperature or diameter. Hence, their evaluation is possible based on the effective velocity concept (Bruun 1995). Then, the heat transfer–voltage relation of the slanted wires can be also reduced to the general form presented in Eq. (1).

Information regarding the derivation of the sensitivities and references to historical sources are conveniently found in Tropea et al. (2007). The analytical sensitivities can be derived utilizing logarithmic derivatives of Nu with respect to its dependencies. Conventionally, the well-accepted rela-tion of heat transfer over a specific wire is given by the fol-lowing equation (Nagabushana and Stainback 1992):

where ReT0 is the Reynolds number based on wire diameter with viscosity evaluated at T0 , and the overheat ratio is � =

Tw

T0 . Considering that convective heat transfer rate from

the wire is as follows:

For moderately transonic flows (0.5 < M < 0.8) , and as long as the wire is hotter than flow by a temperature differ-ence of at least

(Tw − T0

)>∼ 70K , the Nu dependence on

temperature is very small, Baldwin (1958). Furthermore, for small T0 perturbations, which result in small (Tw − T0) variations, the change in Nu is insignificant across all M . Therefore, it is implied that h is invariant for small changes in surface or gas temperatures. Hence, it is a good practi-cal approximation to assume the reduction of the overheat dependence:

(2)S�(�, u, T0

)=

� logE

� log �

||||u,T0=const,

(3)Su(�, u, T0

)=

� logE

� log u

||||�,T0=const,

(4)ST0

(�, u, T0

)=

� logE

� log T0

||||u,�=const.

(5)Nu =hdw

kf= f (ReT0 ,M, �),

(6)Qw = �dwlh(Tw − �T0

).

(7)Nu = f (ReT0 ,M).

This representation is consistent with scientific commu-nity on slip flows over heated wires (Dewey 1965; Behrens 1971).

Having established the dependencies of Nu as Re and M, it is possible to reformulate Eq. (6) from power to voltage supplied to the wire and plug it in Eqs. (2)–(4). Pioneering derivation of sensitivities is conducted by Morkovin (1956) for constant current anemometers (CCA); for a CTA appli-cation, the results of the logarithmic differentiation are con-veniently presented in Nagabushana and Stainback (1992):

To bring utility through this formulation, deterministic Nu − Re −M and � −M − Re relations are necessary. An example of a convenient semi-empirical formulation for infi-nite length wires is provided in Dewey (1965):

This formulation states in a disjoint manner the Mach independent behavior of Nu , Nu

(ReT0 ,∞

) , and the com-

pressibility correction, Φ(ReT0 ,M

) . For convenience, the

empiric relations are repeated here:

(8)S� = 0.5

(� logNu

� logReT0−

1

�wr

� log �

� logReT0

),

(9)Su = S� +1

2m

(� logNu

� logM−

1

�wr

� log �

� logM

),

(10)

ST0 = 0.5

[nt + 1 − mt

� logNu

� logReT0−

�

� − �

+1

�wr

(1

2m

� log �

� logM+ mt

� log �

� logReT0

)−

1

2m

� logNu

� logM

].

(11)Nu(ReT0 ,M

)= Nu∞

(ReT0 ,∞

)⋅Φ

(ReT0 ,M

).

(12)

Nu∞(ReT0 ,∞

)= Ren

T0

[0.14 + 0.2302 ⋅

(Re0.7114

T0

15.44 + Re0.7114T0

)

+

(0.01596

0.3077 + Re0.7378T0

)⋅

(15

15 + Re3T0

)],

(13)n = 1 − 0.5 ⋅

(Re0.6713

T0

2.571 + Re0.6713T0

),

(14)

Φ(ReT0 ,M

)= 1 + (M) ⋅

[1.834 − 1.634

(Re1.109

T0

2.765 + Re1.109T0

)]

⋅

[1 +

(0.3 −

0.065

M1.67

)⋅

(ReT0

4 + ReT0

)],


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27 of 19

For completeness, the empiric relations for the recovery ratio are also repeated here:

Thus, implementing the assumption of Eq. (7), along with the logarithmic derivation of Nu and � relations Eqs. (11)–(17), estimation of individual wire sensitivities is possible. This is inline with the conceptualized universal calibration procedure described in Klopfer (1974).

In general, the use of the universal relations mentioned above must be applied with caution. This correlation is a rea-sonable estimator for comparison of the decoupling ability of different probes. However, one must keep in mind that the validity of the correlations is only for infinitely long wires with defined exact diameters. Accounting for the end loss correction improves the sensitivity estimate (Dewey 1961; Lord 1974); however, wire-specific calibration is still neces-sary for a reasonable accuracy (Horstman and Rose 1977; Cukurel et al. 2012).

Regardless, there have been only a few attempts in decou-pling independent instantaneous transonic flow quantities, with varying levels of success (Jones et al. 1989; Walker et al. 1987; Rose and McDaid 1977; Cukurel et al. 2012). The main issue has been associated with the uncertainties introduced through ill-posing of the sensitivities matrix, requiring over-determined methods through addition of more wires (Walker et al. 1987). However, to the authors’ knowledge, there has been no investigation attempting to optimize probe wire parameters, to improve the performance of the decoupling.

2 Motivation

Numerous probe design guidelines suggest good working practices towards admissible ranges of wire length-to-diam-eter ratio, wire material and coating, prong geometry, and system damping (Bruun 1995). Moreover, for low-speed applications, guidelines relation of the system frequency response to wire diameter and overheat ratio exist. In con-trast, for transonic applications, where the wire diameter and its temperature are critical factors, guidelines are limited. In transonic flow regimes, the selection process of wire

(15) (M) =0.6039

M+ 0.5701

[(M1.222

1 +M1.222

)1.569

− 1

].

(16)

� = 1 + 0.2167

(Kn1.193

∞

0.493 + Kn1.193∞

)(M2.8

0.8512 +M2.8

)

− 0.05

(M3.5

1.175 +M3.5

),

(17)Kn∞ =√0.5�� ⋅M∕(Re∞).

diameter and its temperature should be driven by density, velocity, and temperature sensitivities. This work conducts an optimization of diameters and temperatures for a multi-wire-probe over a range of transonic flow conditions. The novelty of the effort is utilization of linear algebra tools to better condition the sensitivity matrix, as well as minimiz-ing noise amplification in the decoupled flow quantities. To achieve this flow decoupling, a matrix being invertible (non-singular) is necessary but insufficient, and the error propaga-tion needs to be considered additionally.

3 Methodology

3.1 Probe sensitivity matrix

For compressible transonic flow with work addition (com-mon in turbomachinery applications), perturbations in local velocity, density, and stagnation temperature are expected. To deduce the measured voltage fluctuations into different flow perturbations, a minimum of three wires with different sensitivities are required. Then, each probe (based on the number of wires—three, four, or more) will have its own sensitivity matrix.

For example, a sensitivity matrix A for a four-wire probe would satisfy the following equation:

where:

Over-determined systems as such can allow greater reso-lution of individual flow quantities (instantaneous density, velocity, and temperature), along with decreased noise amplification. To characterize the trends associated with selected wire diameters and temperatures, as a first-order approximation, the use of full empirical relations is suffi-cient. Therefore, this investigation utilizes the formulations outlined in Eqs. (11)–(17).

3.2 Selection of optimization objectives

Based on a set of wire diameters and temperatures con-strains, the optimization method determines the optimal combination of wire properties in a given probe over a range of flow conditions for a given cost function. The choice of the cost function for the optimization is arbitrary. There are

(18)A ⋅ F = Eme,

(19)A =

⎡⎢⎢⎢⎢⎣

S�1 Su1 ST01S�2 Su2 ST02S�3 Su3 ST03S�4 Su4 ST04

⎤⎥⎥⎥⎥⎦; F =

⎛⎜⎜⎜⎝

��

�u�

uT �0

T0

⎞⎟⎟⎟⎠; E

me=

⎡⎢⎢⎢⎢⎢⎢⎣

E�1

E1E�2

E2E�3

E3E�4

E4

⎤⎥⎥⎥⎥⎥⎥⎦

.


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no clear demands that can define a cost function in a unique manner.

The optimization evaluates the sensitivity matrix for all the combinations of wire diameters and temperatures at dif-ferent flow conditions. In following, according to the cost function selected, a “grade” is assigned to each probe based on all the calculated sensitivity matrixes stemming from various flow parameter combinations.

Towards formulating the cost function, the condition number of matrix �

(A) can be a useful parameter. Since

the ultimate goal is to decouple the separate perturbations by solving Eq. (18), the quality of the solution depends on the well-posedness of the sensitivity matrix A . Higher con-dition number indicates a larger manifestation of the error from the input wire-voltage vector in the output set of flow perturbations. There is a multitude of ways in computing the condition number of a matrix; in this investigation, singular value decomposition (SVD) is used:

where smax, smin correspond to the largest and smallest singu-lar values. �

(A) represents the accuracy of pseudo-inverting

a non-square over-defined matrix via least square fitting.Another relevant parameter may be the robustness of the

matrix condition number across a wide range of flow condi-tions. Such robustness can be represented by the variance of the condition numbers of all matrixes related to a single probe:

Here, �i is a short notation for �(Ai

) and index i indicates the

different flow conditions evaluated.Looking for other possible cost function objectives, it is

possible to directly consider the voltage noise propagation in the solution vector F , described in Treferhen (1997) at Chap. 18:

where � is the angle between the measured voltages vector Eme

and the vector A ⋅ F (range of A ). The deviation is due to the inability of the columns of A to map the entire domain of E

me . Small � would result in a better fit of the decoupled

flow perturbations F . � is the measure of how much ||||A ⋅ F|||| falls short of its maximum possible value:

(20)�(A)= smax∕smin,

(21)�2 =(�i − �i

)2.

(22)||||ΔF||||||||F||||

≤

�(A)

� ⋅ cos (�)

||||||ΔEme

||||||

||||||Eme

||||||,

(23)� =||||A||||||||F||||||||A ⋅ F||||

; 1 ≤ � ≤ �(A).

Alternatively, another source of noise arises from the inaccurate determination of sensitivity matrix A in an experimental setting. Typically, this would manifest itself from an error in the evaluated wire temperature; therefore, a probe, which is less sensitive to slight deviations in wire temperature, may be desirable. The errors of the solution vector F due to sensitivity matrix noise can be written as Treferhen (1997):

where ΔA is the sensitivity matrix error estimate due to the uncertainty in wire temperature estimates. This value is computed by the maximal possible difference in sensitivity matrix stemming from wire temperatures varying by ΔTpert individually. In this investigation, the matrix norm ||||A|||| and the perturbation norm ||||ΔA|||| are calculated using the Frobe-nius norms.

Addressing the angle � and � is not in the scope of this paper, as, in reality, the voltage signal is uncontrolled by the user and it is not possible to minimize α and maximize ζ without information about the real signal. Therefore, ignor-ing the contribution of � and � to the problem, Eqs. (22) and (24) reduce to:

Thence, to improve invertibility and minimize the ampli-fication of the errors into the decoupled flow perturbations, we are interested in minimizing �

(A) ||ΔA||||A|| and �

(A) ||ΔEme||||Eme|| .

However, as the normalized error in voltage, ||ΔEme||||Eme|| , cannot be accurately predicted, then the latter objective reduces to minimizing �

(A) alone.

From another perspective, we are interested in detecting small perturbations, and, therefore, need the sensitivity val-ues to be larger; this translates into maximization of ||||A|||| . This is inline with minimizing �

(A) ||ΔA||||A|| ; however, �

(A) is

the dominant factor, and for a single-value objective func-tion, minimization of �

(A) ||ΔA||||A|| is drawn towards the

smaller condition number, without any control of the matrix norm.

Thus, a value function of two objectives is proposed:

(24)||||ΔF||||||||F||||

≤

(�(A)2

tan (�)

�+ �

(A))||||ΔA||||

||||A||||,

(25)||||ΔF||||||||F||||

≤ �(A) ||||||ΔEme

||||||

||||||Eme

||||||,

(26)||||ΔF||||||||F||||

≤ �(A) ||||ΔA||||||||A||||

.


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where (�(A)||||ΔA||||

)min

, ||||A||||max are the minimal and maxi-

mal values of the considered probe collection. With this normalization, the range of the optimization objectives indi-vidually spans between 0 and 1. A max–min formulation is used in the case of convex Pareto front shape. This value function maximizes ||||A|||| and 1

�(A)||ΔA|| , and, thus, minimiz-

ing �(A)||ΔA||||A|| as desired. The weights W1 and W2 are the

choice of the user and it should be noted that the larger weight draws the result to the other objective.

3.3 Synthetic data generation

In cases where selection of weights for the cost function is not trivial, an additional method is needed to assist with sift-ing through probes found on the Pareto front. A virtual data-reduction check is proposed, which will allow assigning an additional score to the candidate probes. Creating artificial voltage signal consists of superpositioning the mean flow and the perturbation-induced signals.

The mean voltage is evaluated through the following (Nagabushana and Stainback 1992):

The Nu and � can be estimated using the relationships given in Eqs. (11)–(17). After rearranging the equation, the voltage is extracted at each time instance and then aver-aged. Additional assumptions for the evaluation are the exposed length of the wire l = 1 mm, and lead cable resist-ance Rl = 1 Ω . Wire resistance is evaluated through the resistivity:

where aref = 0.0036 1/°C, Tref = 20 °C, and � = 5.5 μΩ cm, adapted from Bruun (1995). For operating the anemometer in fast flows where the convection is large, the top-of-bridge resistance for high-power option is Rt = 25 Ω.

The unsteady voltage is generated by the use of Eq. (18) in reverse manner. The desired u, �, T0 fluctuations are gen-erated, its mean and perturbations are evaluated along with the probe’s sensitivities Su, S�, ST0 , for each wire at mean flow values. The result is artificial E

me absent of noise. For

small disturbances, this signal is equivalent to directly using

(27)

Val = max

⎛⎜⎜⎜⎝min

⎛⎜⎜⎜⎝W1 ⋅

��A��ΔA��

�min

��A��ΔA��

,W2 ⋅

��A��A��max

⎞⎟⎟⎟⎠

⎞⎟⎟⎟⎠,

(28)Nu =E2

kf(Tw − �T0

)(

Rw

�l(Rt + Rl + Rw

)2),

(29)Rref = � ⋅ l∕(0.25� ⋅ d2w),

(30)Rw = Rref ⋅

[1 + aref

(Tw − Tref

)],

Eq. (28) with the instantaneous flow properties. The choice of using the linear derivative-based approach intends to decouple the contribution of non-linearity effects from the propagation of noise in the linear matrix algebra.

To evaluate probe’s performance for a real signal, an electrical noise must be added to the voltage signal. The noise can be split into two components, noise relative to the amplitude of the perturbation and parasitic DC noise with random amplitude. At each instance for each wire, the rela-tive noise is added as a uniformly distributed random noise signal with a maximum amplitude of Amprel-noise , relative to the instantaneous voltage perturbation. The parasitic volt-age is randomized up to a maximal amplitude AmpDC-noise at each instance.

3.4 Decoupling quality quantification

The noisy voltages are decoupled at every instance to the corresponding flow properties through the solution of Eq. (18). To choose a more robust probe within the bounds of wire temperature uncertainty ΔTpert , the sensitivity matrix permutation with the highest condition number is assumed for each considered probe.

A decoupling quality parameter can be defined as follows:

Here

Finally, the QRMS of each probe is normalized by the mini-mum observed QRMS among all probes for a given flow, and averaged across a broad range of flow conditions:

(31)QRMS =√

Qual2u+ Qual2

�+ Qual2

T0.

(32)Qualu =

∑�u�

u noisy−

u�

u clean

�2

∑�u�

u clean

�2,

(33)Qual� =

∑��

� noisy−

��

� clean

�2

∑��

� clean

�2,

(34)QualT0 =

∑�T �0

T0 noisy−

T �0

T0 clean

�2

∑�T �0

T0 clean

�2.

(35)Q̂RMS =1

nflows

nflows∑iflow =1

QRMSiflow

min(QRMSiflow

) .


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of 19 27

3.5 Input parameters

In the scope of the current investigation, the following mean flow conditions, wire parameters, artificial voltage inputs, and noise parameters are considered.

3.5.1 Mean flow conditions

The maximal stagnation pressure ratio of 1.7 is chosen con-sidering a typical highly loaded fan stage in a turbomachin-ery application. The upper limit on the total flow temperature is calculated using the isentropic flow relations correspond-ing to the compression. The density is deduced from the ideal gas state equation. Encompassing a wide range of mean flow conditions, a total of 40 combinations are examined from the following parameter space (combinations of high P0 with T0 lower than 320 K are ignored): 0.5 ≤ M ≤ 0.9 , 290K ≤ T0 ≤ 350K, and 1 atm ≤ P0 ≤ 1.7 atm. Although it may be possible to extend this work to supersonic condi-tions, it is unclear whether the generality of the formulation can be retained. Exemplary experiments on evaluation of single-wire sensitivities were successfully conducted, Rong et al. (1985), yet the treatment of shock waves would require special attention as the actual flow conditions around the wire are not trivial.

3.5.2 Wire parameters

Wire diameters are limited in the lower range by structural, manufacturing, and operability considerations (described in detail by Sandborn 1972), and in the upper limit by the decreased circuit resistance, and reduced frequency response (Bruun 1995). Moreover, to prevent the oxidation of a tung-sten wire, the upper temperature is not to exceed 350 °C (Bruun 1995), for generality, a slightly wider range is exam-ined. Thence, the ranges of wire properties for the optimiza-tion are as follows:

In addition, based on the previous methods of wire tem-perature evaluation (Cukurel et al. 2012), it is reasonable to assume that the uncertainty in the evaluation of wire tem-perature is bound by ΔTpert = ±2.5K.

3.5.3 Artificial voltage and noise input parameters

Sinusoidal flow fluctuations are used as a benchmark example:

5 μm ≤ dw ≤ 10 μm, 380K ≤ Tw ≤ 650K.

(36)uinst = u

[1 +

(u�

u

)

inp

sin(�u ⋅ t

)],

The exact values of the frequencies are not critical. How-ever, they should merely be different from one another, to ensure that the scheme successfully decouples the different flow quantities which are not correlated by frequency. The arbitrary-chosen frequencies are �u = 7 ⋅ 2� rad∕s , �� = 5 ⋅ 2� rad∕s , and �T0

= 2 ⋅ 2� rad∕s , respectively. The time vector t should be long enough to minimize the change in probe performance evaluation between different simula-tion runs due to randomization. In this case, a time duration of 10 s is considered with 1KHz sampling. The perturbation amplitudes are typical to turbomachinery applications, cho-sen as

(u�

u

)inp

= 0.2,(

��

�

)inp

= 0.1, and(

T �0

T0

)inp

= 0.01 ,

respectively.On the arising clean signal, a random noise, with ampli-

tude proportional to the instantaneous voltage fluctuations ( Amprel-noise = 1% ), is added. Moreover, related to the anemometry system, a constant random noise is also super-posed. Based on a typical figure from an anemometry system datasheet (AN-1003 Anemometry Systyem Data sheet), its 3-sigma peak to peak is AmpDC-noise = 0.5mV . In reality, the typical noise levels of the anemometry system are much lower.

4 Results

To identify the probe characteristics most suitable for decou-pling flow quantities, an optimization is performed, forming a Pareto front for a min–max formulation. In following, all the candidate Pareto probes are ranked by their decoupling of synthetic data, yielding a clear guideline for choice of probe wire diameter and temperatures.

4.1 Value function‑based optimization

The input parameters for the optimization are the mean flow quantities of Mach number, total temperature, and total pres-sure. Probes comprising of three and four wires are exam-ined, with possible wire diameters of 5, 7, 9, 10 μm. The wire temperatures examined vary between 380 and 650 K, with 10K increments. The overall number of probe combinations with repetitions for three- and four-wire probes are 227,920 and 6,894,580, respectively.

(37)�inst = �

[1 +

(��

�

)

inp

sin(�� ⋅ t

)],

(38)T0inst = T0

⎡⎢⎢⎣1 +

�T0

�

T0

�

inp

sin��T0

⋅ t�⎤⎥⎥⎦.


1 3

27 of 19

For a four-wire probe, Fig. 1 presents the inverse of the sensitivity matrix condition number with respect to a broad range of flow conditions, and probe wire diameter and tem-perature combinations. Evidently, some wire diameter and temperature combinations result in the more desirable lower condition numbers.

As a desired potential objective for optimization, probe’s mean condition number and its variance with respect to dif-fering flow conditions are charted in Fig. 2. The results are normalized by the minimal mean condition number (�min) at and the minimum variance of condition numbers

(�2min

)

across all flows. Evidently, the �2 and �(A) are positively

correlated (improving one also benefits the other). Thus, minimizing either is sufficient for optimization of both vari-ables, moving towards the desired top right corner region in the figure. Therefore, it is sufficient to choose �

(A) as one of

the optimization objectives.In this light, the other important parameters for the probe

optimization are the condition number �(A) , as well as the

�(A)||||ΔA|||| term, in Eqs. (25) and (26), respectively. For all

four-wire probes considered, Fig. 3 charts a map of inverse normalized mean condition number with respect to inverse normalized mean norm of matrix perturbation. Evidently, the ||||ΔA|||| and �

(A) parameters are also positively corre-

lated; so minimizing only one of the variables is sufficient. Similarly, it is possible to select the multiplication of the two quantities �

(A)||||ΔA|||| as an objective for the optimization.

Considering this corollary along with the significance of matrix norm, the value function described in Eq. (27) is the natural choice for optimization. Along these lines, Fig. 4 por trays a map of the two objectives

Obj1 =

(�(A)||ΔA||)

min

�(A)|||ΔA||| , Obj2 =||A||||A||max

, for all possible four-

wire probe combinations. In this optimization, Obj1 and Obj2 are both maximized. The most desirable probe is in the top right corner of the chart, and hence, there are no

universally dominant probes; instead, they are scattered along a convex Pareto front.

Interestingly, the probes on the Pareto front are all with wire diameters of 5, 5, 10, 10 μm with differing temperatures, Fig. 5. This implies that the choice of wire diameters should be spread to pairs at the upper and lower constraint values for probe decoupling performance. Notably, although this

Fig. 1 Inverse condition number of all four-wire probe combinations for broad range of mean flow properties

Fig. 2 Map of inverse normalized mean condition number with respect to inverse normalized mean norm of matrix perturbation for all four-wire probe combinations

Fig. 3 Map of inverse normalized mean condition number with respect to inverse normalized mean norm of matrix perturbation for all four-wire probe combinations


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figure only considers potential wire diameters of 5, 7, 9, 10 μm, findings of finer 1 μm discretization yield similar results.

In the scope of this work, Table 1 presents the weighting combinations considered. The optimization results for best probes for the five different value functions are superim-posed by red symbols on Fig. 4. Selecting the desired probe from the Pareto front requires analyzing in detail the char-acteristic performance. Although it is possible to treat each objective as a criteria, the critical threshold to the decou-pling is associated with the invertibility of the matrix which is characterized by the condition number. Therefore, the condition numbers of the best four-wire probes are charted for the various value functions across the considered flow conditions, Fig. 6a–e. The particular traits of the resulting probes can be found in Table 2.

For value function 1, Fig. 6a presents a choice favor-ing large matrix norm over a minimal �

(A)||||ΔA|||| .

The ensuing probe consists of 5, 5, 10, 10 μm wires at 390, 400, 390, 400K , respectively. Although the sensitivi-ties to changes in flow properties are amplified (large ||||A|||| ), the resulting condition numbers for higher flow temperatures and higher pressures are approaching a value of a 1000. High condition numbers are not desirable and this deems the con-figuration inadmissible.

Figure 6b–d charts the condition number for the best probes resulting from value functions 2–4. These configu-rations all yield wire diameters as far apart as possible at the limits of the constraint, two 5 μm and two 10 μm wires. The corresponding wire temperatures are spread over the intermediate range of 470–560 K. Throughout the entire flow range, the corresponding condition numbers are less than ∼ 500 for all three probes, which suggests that all three

probes are acceptable from an invertibility perspective. Moreover, due to objective weighting that favors the lower condition number probes, value function 4 yields slightly lower �(A) . In general, the condition numbers rise for flows with higher Mach numbers and total pressures.

For value function 5 that strongly favors �(A) over the norm, the results are presented in Fig. 6e. The probe con-figuration associated still yields two 5 (µm) and two 10 (µm) wires, with temperatures of 640 K and 650 K pairs. Con-sidering that the optimization has reached the upper limit in the wire temperature constraint, and that the temperature values are very close to each other, this probe configuration is less desirable.

To assess the dependence of the optimization on the num-ber of probe wires, Fig. 7 is a representative example con-trasting the condition numbers of best three-wire and four-wire probes for value function 4 at T0 = 350K . For value function 4, the best three-wire probe diameters and tempera-ture are dw = 5, 10, 10 μm; Tw = 540, 530, 560K , respec-tively. It can be seen that the condition number for a

Fig. 4 Map of the two objectives for all four-wire probe combinations

Table 1 Weighting of value function objectives

Value func-tion #

W1 W2

1 0.98 0.022 0.7 0.33 0.5 0.54 0.4 0.65 0.1 0.9

Fig. 5 Map of the two objectives for all four-wire probe combina-tions, location of probes with dw = 5, 5, 10, 10 μm


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Fig. 6 Condition numbers of four-wire probes contrasted across different mean flow combinations: a best probe for value function 1, b best probe for value function 2, c best probe for value function 3, d best probe for value function 4, and e best probe for value function 5


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three-wire probe is slightly worse than a four-wire probe. To demonstrate the ramifications associated with noise propaga-tion, Fig. 8 directly charts the noise amplification term, �(A) ||ΔA||||A|| , for the same probes. Expectedly, the noise

amplification is lower for the four-wire probe, owing to the redundant measurement carried by the additional wire. Simi-lar trends can be observed for other value functions.

Concluding the initial optimization, the main significant result is that probes populating the Pareto front consist solely of 5, 10 μm wires. And for 4-wire probes, selecting wire diameter pairs at the extremes of the available range pro-duce the most significant beneficial impact on decoupling performance.

4.2 Analysis based on quality of decoupling

To verify that the considered value function objectives, indeed, correlate with the decoupling ability of the probe, synthetic data analysis is conducted. Along these lines, in the probe parameter space, randomly selected (via square

point picking) wire diameter and temperature combinations are examined, Fig. 9. In the following, each selected probe is evaluated based on the previously described Q̂RMS criterion and the 100 probes with the lowest values are charted on the scatter plot, Fig. 10. In general, the location of the top probes correlate well with the Pareto front. However, some of the specific low Q̂RMS probes are not strictly on the actual boundary. This is expected, considering that the optimiza-tion objectives are derived from universal upper bounds, whereas the decoupling quality of each probe is slightly dependent on the particular signal and noise combination. Nevertheless, due to robustness considerations, selection of probes from the Pareto front is recommended. Increasing the sensitivities as much as possible does not appear to be necessary if a reasonable value of ||||A|||| is achieved, indicated by the best probes being located in an intermediate region in terms of ||||A||||.

To sift through and identify the best probes from the Pareto front, all probe combinations that reside on the bound-ary are evaluated for their decoupling quality. Table 3 pre-sents the properties of the top 20 probes sorted by increasing

Table 2 Best four-wire properties for value functions

Value function dwire μm Twire K

1 5 5 10 10 390 400 390 4102 5 5 10 10 470 500 470 5003 5 5 10 10 510 530 520 5204 5 5 10 10 540 540 530 5605 5 5 10 10 640 650 640 650

Fig. 7 Contrasting the condition numbers of best four-wire and three-wire probes for value function 4 at T0 = 350K

Fig. 8 Contrasting the error amplification due to sensitivity matrix perturbation of best four-wire and three-wire probes for value func-tion 4 at T0 = 350K


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Q̂RMS . It is important to note that the best performing probes do not contain wires near the limits of the temperature range considered. This implies that optimal decoupling conditions can not be achieved simply by maximizing or minimizing temperature spreads. Moreover, three different “families” of probes can be identified by wire temperature-diameter pairing options: the same two temperatures repeated over

different diameter pairs (Family A), a broad and a narrower temperature spread over repeated diameters (Family B), a temperature spread over a repeated diameter pair, and a redundant wire pair at intermediate temperature (Family C). In each family, the probe with the lowest overall Q̂RMS is selected, indicated by an asterisk in Table 3. For future reference, the actual sensitivity matrices of the three probes are presented in the “Appendix” for select flow conditions.

In hotwire anemometry, it is typically rather difficult to exactly prescribe the temperature value of each wire. To ensure that the recommended probes are robust selections, their wire temperatures are checked for deviation of ±5K from their nominal values. Figure 11 presents the map of the two objectives, with these resultant perturbation groups consisting in total 216 members. Evidently, each perturba-tion group is tightly packed and close to the Pareto front. Moreover, their respective Q̂RMS are among the 400 lowest of all considered probes.

Therefore, the findings of this decoupling study indicate that there exist three families of acceptable probe wire tem-perature combinations, and among them, the Family A-type probes (with same two temperatures repeated over different diameter pairs) perform slightly better.

4.3 Decoupling of synthetic data for select probes

Highlighting the value of this analysis with respect an “intuitive” approach, two other exemplary probes are indicated in Fig. 11. One with a reason-able “naïve” spread in wire temperatures and diam-eters (Probe 1: 5, 7, 9, 10 μm, 450, 500, 550, 600K), and the other with the same temperature spread but with the optimal diameters, as suggested before (Probe 2: 5, 5, 10, 10 μm, 450, 500, 550, 600K). Clearly, neither of the points lies on the Pareto front, and moreover, Probe 1 with diameter spread is even further from the boundary.

At this stage, it is interesting to examine the actual decoupling of flow properties for these “intuitive” probes (Probe 1 and 2) with respect to the recommended selection (Probe A*). Artificial voltages are created with the same perturbation quantities and noise parameters as described in section 5c. The decoupling is conducted for two mean flow conditions of M̄ = 0.9, T0 = 350K,P0 = 1.7Atm , and M̄ = 0.5, T0 = 290K,P0 = 1Atm , respectively, presented in Figs. 12 and 13.

To asses the performance of the decoupling, signal-to-noise ratio (SNR) of each quantity is calculated using squared clean signal (power of meaningful information) divided by the squared difference between the decoupled and clean flow perturbations (power of the noise). Evidently, for the high M flow (Fig. 12), probe 1 fails to adequately decou-ple the flow, evidenced by the large deviations from the clean signal; meanwhile, the SNR is significantly lower than

Fig. 9 An example of square point picking of probes in the objectives domain range

Fig. 10 Map of the two objectives for all four-wire probes combi-nations, indicating top 100 probes with lowest Q̂RMS from a random selection


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1. This is expected considering the non-optimal selection of both wire diameters and temperatures. In contrast, probe 2 decoupling is better and seems to follow the general trends of a sine signal. However, the SNR of the decoupling still appears very low in some of the perturbations. In this case, particularly for the density fluctuations, the decoupling SNR

is in the order of 1. The relative but insufficient improvement is associated with the selection of optimal wire diameters. At this high Mach number, to get adequate decoupling from the probes (SNR greater than 4), optimization of the wire tem-peratures is critical, as represented by probe A∗ . For lower Mach flow regimes (Fig. 13), the decoupling is relatively easier for all probes, and the overall SNRs are comparatively larger. Along these lines, the SNR for the recommended probe A∗ is exceptional, above 40.

It is also interesting to contrast the actual decoupling ability of optimal three-wire probes with respect to optimal four-wire probes. In order for a fair comparison, a similar multi-flow optimization and sifting procedure is conducted for three-wire probe combinations with the same constraints. The ensuing best three-wire probe (Probe 3) has the follow-ing properties dw = 5, 5, 10 μm and Tw = 440, 510, 460K . In general, this wire temperature combination seems to resemble four-wire Probe A∗ , with the fourth wire absent. As an additional comparison, another three-wire probe is conceived (Probe 4), also stemming from Probe A∗ in the absence of the third wire; dw = 5, 5, 10 μm and Tw = 460, 510, 510K . The decoupling results along with their respective SNRs are charted in Fig. 14 for the chal-lenging flow conditions (similar results can are found for other flows). Remarkably, the decoupled signals are visually quite similar; however, the SNR suggests that the optimal Probe 3 is only slightly better than Probe 4, and expectedly, the optimal 4 wire probe (Probe A∗ ) outperforms both. It is important to note that the SNR of the optimal three-wire

Table 3 Probes with lowest Q̂RMS , ranked and divided to three family types

# Probe family dwire μm Twire K Q̂RMS

1 A* 5 5 10 10 460 510 460 510 1.142 B* 5 5 10 10 500 570 520 540 1.303 B 5 5 10 10 510 580 530 550 1.364 B 5 5 10 10 470 530 480 510 1.385 A 5 5 10 10 410 450 410 450 1.386 B 5 5 10 10 480 510 470 530 1.407 B 5 5 10 10 490 520 480 540 1.408 C* 5 5 10 10 510 580 540 540 1.419 B 5 5 10 10 500 530 490 550 1.4110 B 5 5 10 10 510 540 500 560 1.4411 B 5 5 10 10 520 590 540 560 1.4412 C 5 5 10 10 490 560 520 520 1.4713 C 5 5 10 10 500 570 530 530 1.4714 C 5 5 10 10 520 590 550 550 1.4815 B 5 5 10 10 480 540 490 520 1.5116 B 5 5 10 10 520 550 510 570 1.5117 B 5 5 10 10 530 600 550 570 1.5718 A 5 5 10 10 420 460 420 460 1.5819 B 5 5 10 10 490 550 500 530 1.6320 B 5 5 10 10 500 560 520 530 1.68

Fig. 11 Map of the two objectives for all four-wire probe combina-tions, indicating ±5K Tw perturbation groups of the three recom-mended probes


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probe (Probe 3) with a determined decoupling matrix is much better than the intuitively selected four-wire Probes 1 and 2 with over-determined decoupling matrixes. Neverthe-less, the SNR of Probe 3 is not consistently above 4, and, therefore, not recommended.

4.4 Comparison with real exemplary probe from literature

Based on the information available, the sensitivity matrix of the three-parallel-wire probe of Nagabushana and Stainback (1992) (wire diameters of 4, 8, 13 μm, rounded to the nearest micron, at temperatures of 550, 515, 450K1) is recalculated

according our methodology using the empirical Re–Nu rela-tion from Dewey’s paper (1965), at the same flow condition, noted in Nagabushana’s paper as “obs 30.” (M = 0.3, � = 1.133

kg

m3, T0 = 322K1

) . Comparing the in-

situ sensitivity calibration from the reference to the com-puted sensitivities from the universal relation for infinite wires, the differences are up to 17.5%, 28.5%, and 32% for Srho , Su , and ST0 respectively. The corresponding condition numbers are 166 for the sensitivities from the reference, and 127 for the Dewey-based matrix for the same probe wire diameters and temperatures. For completeness, the sensitiv-ity matrices along with their condition numbers and probe properties are given in Table 4. The discrepancies are expected, since in-situ calibration is always necessary for quantitative decoupling. By capturing the adequate trends, the suggested method provides a comparative decision-mak-ing tool for the most favorable probe in a given set of flow conditions. The location of the probe is identified with

Fig. 12 Selected four-wire probes decoupling performance comparison for M̄ = 0.9,T0 = 350K,P0 = 1.7Atm , Amprel-noise = 1% , and AmpDC-noise = 0.5mV in terms of a decoupled �0 perturbations, b decoupled U perturbations, and c decoupled T0 perturbations

1 It seems that there is a mistake in Nagabushana and Stainback 1992 regarding the units of flow total-temperature and wire tempera-tures. The reported units are ℉, if true the temperatures would be too high for the wires and tunnel capabilities described in the paper. It is assumed the correct units are °R.


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respect to the Pareto front for a three-wire optimization under the single-flow conditions with extended search-range for wire diameters of 4–13 μm (Fig. 15), and evidently, there is room for potential improvement in terms of SNR. Accord-ing to our methodology, the properties of an optimal three-wire probe for this single-flow condition are wire diameters 4, 4, 13 μm at temperatures of 400, 450, 410K . Counter-intuitively, spreading wire diameters to the extreme [turning the 8 μm wire to a 4 μm], and lowering the temperatures of the wires, appears to be a better selection. When the SNRs of the optimal and Nagabushana’s original probes are com-pared by decoupling a synthetic flow with added artificial noise, the optimal three-wire probe presents 2.3, 3.3, and 3.5 times better SNR for the test case decoupling of density, velocity, and total temperature, respectively. Notably, the condition numbers are similar, yet this is not enough for lower SNR and the optimal probe can offer a significant benefit.

5 Conclusions

In the scope of decoupling density, velocity, and tempera-ture perturbations in transonic flows, a two-step optimization methodology is proposed towards selecting the properties of a constant temperature anemometry probe. Based on empiri-cal Nu − Re −M and � − Re −M correlations, the wire sensitives to density, velocity, and total temperature per-turbations are evaluated synthetically. Thereby, sensitivity matrices can be calculated for different hypothetical probes. In this case, a broad range of flow conditions characteristic of a highly loaded fan stage are considered.

To identify the probes with better decoupling properties, the invertibility is considered to be the critical factor. The condition number is the dominant quantity, characterizing the well-posedness of the sensitivity matrix. It is demon-strated that the variance of the condition number across different flows and the robustness of the sensitivity matrix against wire temperature errors are both positively correlated

Fig. 13 Selected four-wire probes decoupling performance compare, for M̄ = 0.5,T0 = 290K,P0 = 1Atm , Amprel-noise = 1% , AmpDC-noise = 0.5mV in terms of a decoupled �0 perturbations, b decoupled U perturbations, and c decoupled T0 perturbations


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Fig. 14 Selected four-wire and three-wire probes decoupling performance compare, for M = 0.9,T0 = 350K,P0 = 1.7Atm , Amprel-noise = 1% , AmpDC-noise = 0.5mV in terms of a decoupled �0 perturbations, b decoupled U perturbations, and c decoupled T0 perturbations

Table 4 Properties and sensitivity matrices of Nagabushana’s probe and the optimal three-wire probe proposed by the optimization for M = 0.3, � = 1.133, and T0 = 322K

Probe properties Sensitivity matrix (Nagabushana) ⎡⎢⎢⎣

S�1 Su1 ST01S�2 Su2 ST02S�3 Su3 ST03

⎤⎥⎥⎦{condition number}

Sensitivity matrix (Re–Nu from Dewey) ⎡⎢⎢⎣

S�1 Su1 ST01S�2 Su2 ST02S�3 Su3 ST03

⎤⎥⎥⎦{condition number}

Nagabushana’s 3-parallel-wire probe

dw = 4, 8, 13 μm

Tw = 550, 515, 450K⎡⎢⎢⎣

0.232 0.140 −0.662

0.260 0.144 −0.796

0.211 0.155 −1.190

⎤⎥⎥⎦{166}

⎡⎢⎢⎣

0.237 0.166 −0.503

0.251 0.201 −0.652

0.255 0.214 −1.082

⎤⎥⎥⎦{127}

Optimal three-wire probe from single-flow optimi-zation

dw = 4, 4, 13 μm

Tw = 400, 450, 410K

NA ⎡⎢⎢⎣

0.238 0.169 −1.858

0.237 0.167 −1.054

0.255 0.216 −1.652

⎤⎥⎥⎦{123}


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with the condition number itself. Thus, the need for these additional independent objectives is obviated. However, the other considerations must include large sensitivities. Compromising between the two objectives associated with minimum error amplification and maximizing the sensitivity matrix, a value function in max–min formulation is chosen, resulting in a Pareto front. To sift through the probes on the boundary, artificial voltage is generated and the perturbation decoupling quality is evaluated in the presence of realis-tic upper bound noise. Three probe families are identified, describing the wire temperature combinations with high-est decoupling quality. It is important to note that none of the best performing probes contain wires near the limits of the temperature range considered. The best overall probe is contrasted against the other configurations which could have been naively selected. The signal-to-noise ratio of even the optimal three-wire probe (with a determined decoupling matrix) outperformed the intuitively selected four-wire probes (with over-determined decoupling matrixes).

Towards future implementation in various turbomachin-ery and propulsion applications, below are some observa-tions and general guidelines as applied to four-wire hotwire probes:

• Based on the trends observed in the first step of the opti-mization, selecting wire diameter pairs at the extremes of the available range produces the most significant ben-eficial impact on decoupling performance (in this case, 5–5–10–10 μm wires).

• Among the three families of probes with highest decou-pling quality, the top performing candidate consists of the following respective wire diameters and temperatures dw = 5, 5, 10, 10 μm and Tw = 460, 510, 460, 510 K , and it is recommended to use across all flows in the range of the current optimization.

• The recommended probe in contrast to configurations with reasonable “naïve” spread in wire temperatures and/or diameters, is the only one with an adequate signal-to-noise ratio of above 4 for all the considered flow condi-tions.

It is important to note that the optimization procedure is a decision-making tool for the selection of wire diameters and temperatures to allow for a better decoupling of flow quantities for a practical probe. The actual in-situ calibra-tion is unavoidable, as the optimization can only produce recommendations based on comparative calculations stem-ming from empirical universal Re-Nu curves.

Acknowledgements The authors acknowledge the partial financial support of Minerva Research Center (Max Planck Society Contract No. AZ5746940764).

Appendix

Inspecting the resulting sensitivity matrices of the three rec-ommended probes A∗,B∗ and C∗ , in Table 5, it is evident that the M is a dominant factor, and the pressure does not change the sensitivities drastically. The Srho and Su values of the recommended probes are quite close for the high M = 0.9 with high T0 flow combination; however, they are sufficiently far to allow decoupling. Probe A∗ having lower wire temperatures has a better spread in sensitivities as the ST0 is increased, serving as a relative “cold wire”. As the flow temperature drops, the presence of a “cold wire” is no longer possible (due to the constraints of this optimization), but the spread is still sufficient. As the M decreases (presented by the case of M = 0.5 ), the density and velocity sensitivities spread naturally as Su decreases.

Fig. 15 Map of the two objectives, single-flow optimization for all three-wire probe combinations, indicating top 100 probes with lowest Q̂RMS from the Pareto front and an exemplary probe from Nagabusha-na’s paper dw = 4, 8, 13 μm;Tw = 550, 515, 450K


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Table 5 Sensitivity matrix of selected probes on different representative flows, all dw = 5, 5, 10, 10 μm

M T0K

P0

Atm

Probe A∗

Tw = 460, 510, 460, 510K

Probe B∗

Tw = 500, 570, 520, 540K

Probe C∗

Tw = 510, 580, 540, 540K

0.9 350 1.7 ⎡⎢⎢⎢⎣

0.2697 0.2650 −1.3330

0.2690 0.2446 −0.8765

0.2618 0.2647 −1.3268

0.2615 0.2437 −0.8720

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2691 0.2476 −0.9444

0.2686 0.2319 −0.5953

0.2614 0.2409 −0.8120

0.2614 0.2362 −0.7103

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2690 0.2446 −0.8765

0.2686 0.2305 −0.5623

0.2614 0.2362 −0.7103

0.2614 0.2362 −0.7103

⎤⎥⎥⎥⎦0.9 350 1 ⎡⎢⎢⎢⎣

0.2766 0.2609 −1.3389

0.2755 0.2412 −0.8798

0.2677 0.2654 −1.3314

0.2672 0.2447 −0.8755

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2756 0.2441 −0.9481

0.2747 0.2291 −0.5974

0.2671 0.2420 −0.8153

0.2670 0.2374 −0.7134

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2755 0.2412 −0.8798

0.2746 0.2277 −0.5643

0.2670 0.2374 −0.7134

0.2670 0.2374 −0.7134

⎤⎥⎥⎥⎦0.9 290 1 ⎡⎢⎢⎢⎣

0.2720 0.2334 −0.6614

0.2717 0.2252 −0.4760

0.2643 0.2342 −0.6575

0.2641 0.2257 −0.4727

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2717 0.2265 −0.5062

0.2714 0.2192 −0.3394

0.2641 0.2244 −0.4452

0.2641 0.2222 −0.3966

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2717 0.2252 −0.4760

0.2714 0.2185 −0.3221

0.2641 0.2222 −0.3966

0.2641 0.2222 −0.3966

⎤⎥⎥⎥⎦0.5 350 1.7 ⎡⎢⎢⎢⎣

0.2628 0.1782 −1.3759

0.2626 0.1727 −0.8860

0.2598 0.1846 −1.3773

0.2597 0.1789 −0.8879

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2626 0.1735 −0.9581

0.2625 0.1694 −0.5906

0.2597 0.1782 −0.8243

0.2597 0.1770 −0.7171

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2626 0.1727 −0.8860

0.2625 0.1690 −0.5562

0.2597 0.1770 −0.7171

0.2597 0.1770 −0.7171

⎤⎥⎥⎥⎦0.5 350 1 ⎡⎢⎢⎢⎣

0.2628 0.1649 −1.3709

0.2625 0.1597 −0.8804

0.2623 0.1806 −1.3767

0.2622 0.1750 −0.8869

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2625 0.1605 −0.9525

0.2623 0.1566 −0.5846

0.2622 0.1743 −0.8233

0.2621 0.1731 −0.7160

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2625 0.1597 −0.8804

0.2623 0.1562 −0.5503

0.2621 0.1731 −0.7160

0.2621 0.1731 −0.7160

⎤⎥⎥⎥⎦0.5 290 1 ⎡⎢⎢⎢⎣

0.2627 0.1641 −0.6548

0.2626 0.1620 −0.4622

0.2612 0.1747 −0.6591

0.2611 0.1725 −0.4667

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2626 0.1624 −0.4934

0.2626 0.1605 −0.3217

0.2611 0.1721 −0.4382

0.2611 0.1716 −0.3881

⎤⎥⎥⎥⎦

⎡⎢⎢⎢⎣

0.2626 0.1620 −0.4622

0.2625 0.1603 −0.3039

0.2611 0.1716 −0.3881

0.2611 0.1716 −0.3881

⎤⎥⎥⎥⎦

http://www.lab-systems.com/products/flow-mea/an1003/AN-1003%20Brochure.pdf

http://www.lab-systems.com/products/flow-mea/an1003/AN-1003%20Brochure.pdf


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Affiliations

E. Yablochkin1 · B. Cukurel1

* B. Cukurel [email protected]

E. Yablochkin [email protected]

1 Technion-Israel Institute of Technology, Technion City, 3200003 Haifa, Israel

http://orcid.org/0000-0002-9899-4693

selection of optimal multi-hotwire probe in constant

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