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

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### Abstract

To 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 fluctuations 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 proposes guidelines to define a probe with desirable properties, applicable across a wide range of transonic flow conditions.

### Graphical abstract

## List of symbols

- \(\alpha\) (–)
Angle between the measured voltages vector \(\underline{E} _\text{me}\) and the vector \(\underline{A} \cdot \underline{F}\)

- \({a_{{\text{ref}}}}\) (1/°C)
Wire resistance temperature coefficient

- \(\gamma\) (–)
Ratio of specific heats for air

- \(\zeta\) (–)
Measure of how much \(\left| {\left| {\underline{A} \cdot \underline{F} } \right|} \right|\) falls short of its maximum possible value

- \(\eta\) (–)
Recovery factor, \({T_{\text{r}}}/{T_0}\)

- \(\theta\) (–)
Overheat ratio \({T_{\text{w}}}/{T_0}\)

- \(\kappa \left( \underline{A} \right)\) (–)
Condition number of matrix \(\underline{A}\)

- \(\upmu\) (N s/m
^{2}) Viscosity of air

- \(\rho\) (kg/m
^{3}) Freestream flow density

- \({\sigma ^2}\) (–)
Variance of condition number with respect to different flow combinations

- \({\tau _{{\text{wr}}}}\) (–)
Overheating parameter, \(({T_{\text{w}}} - {T_{\text{r}}})/{T_{\text{r}}}~\)

- \(\varPhi\) (–)
Compressibility correction function

- \(\chi\) (Ω m)
Wire resistivity

- \({\omega _u}\) (rad/s)
Frequency of perturbation for artificial signal of \(u\)

- \({\omega _\rho }\) (rad/s)
Frequency of perturbation for artificial signal of \(\rho\)

- \({\omega _{{T_0}}}\) (rad/s)
Frequency of perturbation for artificial signal of \({T_0}\)

- \(\underline{A}\) (–)
Probe sensitivity matrix

- \({\text{Am}}{{\text{p}}_{{\text{rel-noise}}}}\) (%)
Max amplitude for relative noise

- \({\text{Am}}{{\text{p}}_{{\text{DC-noise}}}}\) (V)
Max amplitude for DC noise

- \({d_{\text{w}}}\) (m)
Wire diameter

*E*(V)Wires voltages

- \(\underline{E} _{\text{me}}\) (V)
Vector of measured normalized wire-voltage perturbations

- \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{F}\) (–)
Vector of decoupled normalized flow perturbations

- \(h\) [W/(m
^{2}K)] Convection heat transfer coefficient

- \({k_{\text{f}}}\) [W/(m
^{2}K)] Thermal conductivity of the fluid

- \({Kn_\infty }\) (–)
Knudsen number of free stream

- \(l\) (m)
Wire length

- \(M\) (–)
Freestream Mach number

- \(m\) (–)
\({\left[ {1+(\gamma - 1)/2 \cdot {M^2}} \right]^{ - 1}}\)

- \({m_{\text{t}}}\) [N s/(m
^{2}K)] \(\partial \log ~\mu /\partial \log {T_0}\)

- \({n_{\text{t}}}\) [W/(m
^{2}K)] \(\partial \log {k_{\text{f}}}/\partial \log {T_0}~\)

- \({n_{{\text{flows}}}}\) (–)
Number of flows

- \(Nu\) (–)
Nusselt number

- \({Q_{\text{w}}}\) (W)
Convective heat transfer rate from a wire

- \({\text{Qua}}{{\text{l}}_u}\) (–)
Measure of velocity decoupling quality

- \({\text{Qua}}{{\text{l}}_\rho }\) (–)
Measure of density decoupling quality

- \({\text{Qua}}{{\text{l}}_{{T_0}}}\) (–)
Measure of \({T_0}\) decoupling quality

- \({Q_{{\text{RMS}}}}\) (–)
RMS decoupling quality

- \(Re\) (–)
Reynolds number

- \(R{e_{{T_0}}}\) (–)
Reynolds number based on wire diameter, with viscosity evaluated at \({T_0}\)

- \({R_{\text{l}}}\) (Ω)
Resistance of lead wires

- \({R_{\text{t}}}\) (Ω)
Top-of-bridge resistance

- \({R_{\text{w}}}\) (Ω)
Wire resistance

- \({R_{{\text{ref}}}}\) (Ω)
Reference wire resistance at ref temperature

- \({S_u}\) (–)
Sensitivity to velocity perturbation

- \({S_\rho }\) (–)
Sensitivity to density perturbation

- \({S_{{T_0}}}\) (–)
Sensitivity to total temperature perturbation

- \({s_{{\max} }},{s_{{\min} }}\) (–)
Largest and smallest singular values of \(\underline{A}\)

- \(t\) (s)
Time

- \({T_0}\) (K)
Freestream flow total temperature

- \({T_{\text{w}}}\) (K)
Wire temperature

- \({T_{{\text{ref}}}}\) (K)
Reference wire temperature for resistance

- \(\Delta {T_{{\text{pert}}}}\) (K)
Wire temperature perturbation applied for estimate of sensitivity matrix perturbation \((\underline {{\Delta A}} )\)

- \(u\) (m/s)
Freestream velocity

- \({W_1},{W_2}\) (–)
Objective weights for value function

- \({\left( \cdot \right)^\prime }\) (–)
Perturbation of a quantity

- \(\overline {{( \cdot )}}\) (–)
Mean of a quantity

- \({\left( \cdot \right)_i}\) (–)
Index of flow combination \(i\)

- Open image in new window (–)
Averaged quantity normalized by min value in flow \(\left( {\frac{1}{{{n_{{\text{flows}}}}}}\sum _{i}^{{{n_{{\text{flows}}}}}}\left[ {\frac{{{{\left( \cdot \right)}_i}}}{{{\min} {{\left( \cdot \right)}_i}}}} \right]} \right)\)

- Open image in new window(–)
Averaged quantity normalized by max value in flow \(\left( {\frac{1}{{{n_{{\text{flows}}}}}}\sum _{i}^{{{n_{{\text{flows}}}}}}\left[ {\frac{{{{\left( \cdot \right)}_i}}}{{{\max} {{\left( \cdot \right)}_i}}}} \right]} \right)\)

- \({\left( \cdot \right)_{{\text{inst}}}}\) (–)
Instantaneous property

- \({\left( \cdot \right)_{{\text{inp}}}}\) (–)
Input perturbation amplitude for artificial signal

- \({\left( \cdot \right)_{{\text{clean}}}}\) (–)
Artificial signal without noise

- \({\left( \cdot \right)_{{\text{noisy}}}}\) (–)
Artificial signal with noise

- \(\left| {\left| \cdot \right|} \right|\) (–)
Frobenius norm

- \(\Delta\) (–)
Small change in quantity

## Notes

### Acknowledgements

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

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