## Abstract

In the present work, hot-wire anemometry was employed for the characterization of the turbulent field at the inlet of a high-speed low-pressure turbine cascade, in terms of turbulence intensity and integral length scales. This work addresses two major topics relevant to the turbomachinery field: the application of hot-wire anemometry in transonic and rarefied flow regimes and the decoupling of the deterministic and the stochastic fluctuations when measuring unsteady phenomena. In compressible and rarefied flows, a hot-wire is strongly sensitive to both density and velocity fluctuations, and the commonly used Nusselt–Reynolds correlations are not valid. In this article, a nondimensional calibration methodology, based on Nusselt, Reynolds, and Knudsen numbers, was coupled with a sensitivity analysis and employed to postprocess the experimental dataset, allowing to decouple the fluctuations of density and velocity and to compute the turbulence parameters. In the presence of unsteady wakes generated upstream of the cascade, two different phase-locked averaging techniques were employed to distinguish the wake deterministic fluctuations from the background turbulence intensity.

## 1 Introduction

*M*≤ 1), the hot-wire output voltage

*E*

_{b}is strongly sensitive to velocity, density, and total temperature fluctuations present in the flow:

The most common calibration methodology, the “King’s law” [4], directly links the bridge voltage *E*_{b} to the flow velocity *u*. Because of its simplicity, this technique can be applied only in simplified scenarios (isothermal and incompressible flows, and with a wire perpendicular to the stream). A further development of the King’s law is the mass flux calibration, linking bridge voltage and the mass flux *ρ* · *u*, where *ρ* is the flow density. An application of this calibration approach to low-density compressible flows was reported by Chemnitz and Niehuis [5]. To decouple density from velocity, this approach would need a time series of density fluctuations. The authors, however, exploited the average static density measured by 5-hole pressure probes to retrieve the velocity fluctuations. As a consequence, this method only provides reliable results in the case of negligible density fluctuations (*ρ*′ ∼ 0).

_{w}and the Nusselt number Nu [2,3,6,7]. The wire’s Reynolds number is calculated as follows:

*μ*is the dynamic viscosity and

*d*

_{w}is the wire diameter. The Nusselt number is calculated from the output bridge voltage as follows:

*R*

_{w},

*R*

_{s},

*R*

_{t},

*l*

_{w},

*k*,

*η*, and

*T*

_{w}are the wire resistance, support resistance, total resistance, effective wire length, thermal conductivity, recovery factor, and effective wire temperature, respectively. In these works, the recovery factor was determined as

*η*=

*f*(Re

_{w},

*M*) from semi-empirical correlations proposed by Dewey [8]. In incompressible and continuum flows, the Nusselt number is a unique function of the Reynolds number, so Nu =

*f*(Re

_{w}) [3,6]. Thus, from a Nu − Re

_{w}calibration curve, it is possible to compute Re

_{w}, and if the value of density

*ρ*is known, one can retrieve the velocity fluctuations. In compressible and/or rarefied flow regimes, the Nusselt number is also a function of the wire’s Knudsen number Kn [6,9]. The Knudsen number, defined as the ratio of the molecular mean free path length to a characteristic length (the wire diameter in this case), is used to express the deviation from the continuum flow state. Based on its value, three flow regimes can be identified, providing an indication of the degree of rarefaction of the gas: continuum flows for Kn < 0.01, slip flows for 0.01 < Kn < 0.1, and transitional to free-molecular flows for 0.1 < Kn < 10 [3]. The Knudsen number can also be expressed as a function of M and Re

_{w}[8]:

_{w}, M, and Kn, the third one is unequivocally defined. For hot-wire anemometry in compressible flows, the following equations are thus valid:

_{w}but also on Kn, i.e., on the Mach number. In Refs. [3,6], to tackle this problem and obtain a continuous curve, sets of semi-empirical correlations proposed by Dewey [8] and Klopfer [10] were exploited. These correlations allowed to obtain a Mach-independent corrected Nusselt number, Nu

_{corr}, solely function of Re

_{w}. Dewey’s formulation is given as follows:

*ϕ*(Re

_{w}, M) relates the actual Nusselt number to the one that would be obtained for a highly supersonic flow (Nu

_{corr}) that has been found to be Mach-independent. This correlation is valid for 0.3 ≤

*M*≤ 1. The correlation of Klopfer is an extension down to

*M*= 0 of Dewey’s equation, valid in the range 0 ≤

*M*≤ 0.4. In Refs. [3,6], the $Nucorr\u2212Rew$ approach eliminated the dependency on the Mach number. However, the correction term

*ϕ*is unknown a priori if M and Re

_{w}are unknown. For this reason, Cukurel et al. [6] developed an iterative procedure to compute a time series of Re

_{w}, i.e., $(\rho u)\u2032$. Again, in order to isolate the time series of $u\u2032$, an unknown time series of density fluctuations $\rho \u2032$ would be necessary.

All the calibration approaches presented in this introduction suffer from inaccuracies in compressible and rarefied flow regimes due to the difficulty of decoupling the instantaneous fluctuations of density and velocity and due to the Nusselt number dependency on both Mach and Knudsen numbers. The traditional calibration and data reduction techniques are thus oversimplifying the flow, neglecting the contribution of the density fluctuations, and they often rely on semi-empirical correlations in order to eliminate the Mach and Knudsen numbers dependencies.

## 2 The Sensitivity Equation

*S*

_{ρ},

*S*

_{u}, and $ST0$ are the hot-wire sensitivities to density, velocity, and total temperature, respectively. To the authors knowledge, this is the only equation isolating the simultaneous fluctuations of velocity, density, and total temperature. The sensitivities

*S*

_{ρ},

*S*

_{u}, and $ST0$ can be computed experimentally [3] from Eqs. (8)–(10):

_{w}, M, and recovery factor

*η*[9]:

*K*

_{1}= [1 + M

^{2}· (

*γ*− 1)/2]

^{−1}, $K2=\u2202logk/\u2202logT0$, $K3=\u2202log\mu /\u2202logT0$,

*θ*=

*T*

_{w}/

*T*

_{0}and

*τ*

_{w}= (

*T*

_{w}−

*ηT*

_{0})/(

*ηT*

_{0}). If a Nu

_{corr}–Re

_{w}calibration curve is employed, Boufidi and Fontaneto [3] proposed the following formulations to compute the partial derivatives of Nu with respect to Re

_{w}and M:

*ϕ*is the Dewey’s corrective factor of Eq. (6). Different authors who analyzed the hot-wire response in compressible flows [3,6,9,13] reported that typical values of

*S*

_{ρ}are on the order of 0.2–0.25. By contrast, the values of

*S*

_{u}were found to be scattered, ranging between 0.01 and 0.25. All the authors agreed that

*S*

_{u}is smaller than

*S*

_{ρ}, making extremely challenging to isolate $u\u2032$ in compressible flows. The total temperature sensitivity $ST0$ was always found to be negative and higher in absolute value compared to

*S*

_{u}and

*S*

_{ρ}. Its value was strongly dependent on the difference between the wire temperature

*T*

_{w}and the flow total temperature

*T*

_{0}, and varied between −0.5 and −1.2 [3]. In Eq. (7), the three fluctuations $u\u2032$, $\rho \u2032,$ and $T0\u2032$ are the unknowns. Multiwire probes can be exploited in order to isolate these unknowns: for a three-wire probe, provided that the sensitivities of each wire are different, a system with three sensitivity equations can be solved, decoupling $u\u2032$, $\rho \u2032,$ and $T0\u2032$. Yablochkin and Cukurel [14] proposed a guideline for the selection of an optimal multiwire probe, featuring wires with different sensitivities. The authors concluded that, in order to achieve different

*S*

_{u}and

*S*

_{ρ}, wires with different diameters must be selected, while different

*T*

_{w}values are necessary to vary $ST0$.

*S*

_{α}and

*S*

_{β}can be determined experimentally by systematically varying yaw and pitch angles in a dedicated angular calibration facility. The formulation is expressed in Eqs. (16) and (17):

_{corr}− Re

_{w}calibration curves. Consequently, the term $S\alpha \u22c5\alpha \u2032$ was computed and inserted in the left-hand side of Eq. (7).

_{in}∼ 0.4 − 0.5. Due to the low-density environment, the wires were operating in the slip flow regime, with a Knudsen number around ∼ 0.1. The maximum fluctuations of total temperature $T0\u2032$ expected in the facility were in the order of 0.05–0.1 K. It was estimated that the contribution of $T0\u2032$ in Eq. (7) was at least one order of magnitude smaller compared to the contribution brought by $\rho \u2032$ and $u\u2032$ terms. Moreover, the pitch angle fluctuations were expected to be smaller than 0.5 deg, and for such range, according to Bruun [1],

*S*

_{β}∼ 0. For these reasons, the total temperature and pitch angle fluctuating terms were neglected, and the following sensitivity equation was employed:

## 3 Measurements Setup and Apparatus

The CTA measurements presented in this work were performed at the inlet of a low-pressure turbine linear cascade, in the high-speed low-Reynolds facility S-1/C of the von Karman Institute for fluid dynamics.

### 3.1 Turbine Rig.

The S-1/C rig (Fig. 1) is a closed-loop wind tunnel driven by a 615 kW axial compressor, which controls the amount of mass flow by adjusting its rotational speed, while the pressure is regulated by a vacuum pump, allowing to reach minimum absolute pressures of the order of 5000 Pa. Past the compressor, the flow goes across a cooler, employed to lower the temperature to near-ambient one. Subsequently, the flow passes through a divergent section, functioning as settling chamber and equipped with wire meshes and honeycombs. A rectangular duct finally delivers the flow to the test section. A more detailed description of the facility can be found in Ref. [16]. In the present analysis, the test section was equipped with a linear cascade of 23 airfoils, with a span of 165 mm and a pitch of 32.95 mm. The measuring plane of the present investigation is indicated as “Plane 02” in Fig. 2(a). This plane, located 0.5 axial chords (*C*_{ax}) upstream of the blade leading edge plane, featured an inclination of 40.73 deg with respect to the rectangular duct cross section. In order to obtain the desired level of turbulence, the test section was equipped with a passive turbulence grid (TG), presented in Fig. 2(b), placed perpendicularly to the test section inlet, at a distance of 400 mm to the central blade leading edge. It consisted of 41 cylindrical rods of 3 mm in diameter, with a spacing between their centerlines of 12 mm. According to the correlation proposed by Roach [17], the expected value of turbulence intensity Tu generated by the grid at the measuring plane was around $2.5%$. A spoked-wheel type wake generator (WG), consisting of a brass disk of 625 mm in diameter and with 96 bars of 1 mm in diameter, was employed to recreate the effect of unsteady incoming wakes, as shown in Figs. 2(b) and 2(c). The wake reduced frequency was on the order of ∼ 0.95, resulting in a flow coefficient, computed as the ratio of the inlet axial velocity to the peripheral velocity of the bars, of 0.80. When the WG was employed, the inclination of Plane 02 with respect to the duct was changed to 46.9 deg. Geometrical details about the linear cascade and the flow conditions are reported in Table 1. Further details of the test case were provided by Simonassi et al. [18].

Quantity | Value | Units |
---|---|---|

Cascade chord C | 52.28 | mm |

Cascade axial chord C_{ax} | 47.61 | mm |

Cascade pitch g | 32.95 | mm |

Cascade span s | 165 | mm |

Number of blades N_{b} | 23 | mm |

Inlet Mach number M_{in} | ∼ 0.46 | – |

Inlet Reynolds number Re_{in,C} | ∼ 45k | – |

Inlet total temperature T_{0,in} | ∼ 300 | K |

Inlet total pressure p_{0,in} | ∼ 9000 | Pa |

Inlet velocity u_{in} | ∼ 160 | m · s^{−1} |

Inlet density ρ_{in} | ∼ 0.097 | kg · m^{−3} |

Knudsen number Kn (9 $\mu $m wire) | ∼ 0.1 | – |

Outlet Mach number M_{out} | ∼ 0.90 | – |

Outlet Reynolds number Re_{out,C} | ∼ 70k | – |

Wake reduced frequency f^{+} | ∼ 0.95 | – |

Quantity | Value | Units |
---|---|---|

Cascade chord C | 52.28 | mm |

Cascade axial chord C_{ax} | 47.61 | mm |

Cascade pitch g | 32.95 | mm |

Cascade span s | 165 | mm |

Number of blades N_{b} | 23 | mm |

Inlet Mach number M_{in} | ∼ 0.46 | – |

Inlet Reynolds number Re_{in,C} | ∼ 45k | – |

Inlet total temperature T_{0,in} | ∼ 300 | K |

Inlet total pressure p_{0,in} | ∼ 9000 | Pa |

Inlet velocity u_{in} | ∼ 160 | m · s^{−1} |

Inlet density ρ_{in} | ∼ 0.097 | kg · m^{−3} |

Knudsen number Kn (9 $\mu $m wire) | ∼ 0.1 | – |

Outlet Mach number M_{out} | ∼ 0.90 | – |

Outlet Reynolds number Re_{out,C} | ∼ 70k | – |

Wake reduced frequency f^{+} | ∼ 0.95 | – |

### 3.2 Instrumentation.

For the present work, two different in-house built hot-wire probes were employed (Fig. 3). The first probe was a cross-wire (XW) probe, featuring two slanted wires forming an angle of 120 deg. Following the guidelines provided by Yablochkin and Cukurel in Ref. [14], the two wires were chosen with different diameters, 5 *μ*m and 9 *μ*m, with the purpose of obtaining different sensitivities *S*_{u} and *S*_{ρ} for the two wires. The cutoff frequency of both wires was optimally adjusted with a square-wave test at the turbine operating conditions, and was found to be around 10 and 20 kHz for the 9 *μ*m and 5 *μ*m wires, respectively. Due to a failure of the XW probe, the tests with the WG were performed with a single-wire (SW) probe, equipped with a single 9 *μ*m sensor with active length of 0.7 mm and cutoff frequency around 10 kHz. A *Dantec Dynamics StreamLine*$\xae$*Pro* anemometer was employed during the measurements. The signals were filtered with a 100 kHz analog anti-aliasing low-pass filter prior to digital sampling. A 16-bit *NI* board was used to digitize the hot-wire signal, at 300 kHz for the tests without the WG, and 1200 kHz for the experiments with the WG. A *Dantec Dynamics* calibration nozzle was used to perform the ex situ calibration.

## 4 Calibration and Data Reduction Methodology

Being the hot wires operated in a compressible slip flow, a nondimensional calibration approach was adopted and a sensitivity-based postprocessing technique was implemented to decouple the velocity and the density fluctuations.

### 4.1 Cross-Wire Probe Calibration and Data Reduction.

The XW calibration can be essentially divided into two different steps: a low-speed ex situ calibration, to determine the wire temperatures *T*_{w} and the angular response, and an in situ calibration to obtain the Nu − Re_{w} − Kn correlations and compute the wire sensitivities.

#### 4.1.1 Determination of Wire Temperature.

The effective wire temperature *T*_{w} is the temperature that best represents the convective heat transfer between the hot wire and the flow. This value typically differs from the temperature imposed by the user during the hot-wire setup, since the temperature distribution is not uniform over the whole wire length. The temperature *T*_{w} is defined as the one collapsing a set of Nu-Re_{w} data, obtained at different flow temperatures, on a single fourth-order polynomial curve. To determine its value, a low-speed calibration was carried out in the *Dantec* calibration nozzle (*StreamLine ^{®}$\xae$ Pro Automatic Calibrator*), where flow velocity and total temperature were varied systematically in the range ∼0 – 50 m/s and ∼20 – 40 °C, as shown in Fig. 4(a). During the calibration, the probe head was kept in a fixed (reference) position, aligned with the incoming flow, as shown in Fig. 3. For low-speed continuum flows, the Nusselt number is a unique function of the Reynolds number and was computed with Eq. (5), while the Reynolds number was calculated with Eq. (2). The Nusselt number was computed for a set of expected

*T*

_{w}, and the value providing the best

*R*

^{2}of the polynomial interpolation was selected [6]. The results are reported in Fig. 4(b), and the wire temperatures were found to be $Tw1=400.4K$ and $Tw2=456.8K$.

#### 4.1.2 Angular Calibration.

*u*and

*T*

_{0}. The Nu − Re

_{w}curves computed to determine

*T*

_{w}(Fig. 4(b)) were exploited to compute $Reeff1$ and $Reeff2$, where the subscript “eff” indicates the effective Reynolds number, i.e., the one that would yield to the same Nusselt number if the probe were aligned at the reference position. These values were then normalized by the $Re\alpha 0,1$ and $Re\alpha 0,2$, where the subscript “

*α*

_{0}” denotes the values that would be computed at the reference position. The formulation is given as follows:

*f*

_{1}(

*α*) and

*f*

_{2}(

*α*) show an opposite and symmetric trend. When a XW probe is yawed in the jet, one of the two wires gets more orthogonal to the flow, experiencing a higher normal velocity, i.e., a higher cooling, while the opposite occurs on the other wire. A higher cooling corresponds to a higher bridge voltage

*E*

_{b}, which translates in a higher Nu and consequently in a higher Re

_{eff}. In the scope of the present work, and in agreement with the conclusions in Refs. [6,15], no variations in the angular response, expressed in the form of Eq. (19), were detected for different Reynolds and Mach numbers and flow total temperature values. Consequently, a single curve regression, performed on the low-speed ex situ data, was assumed to be representative of the angular response independently of the actual flow velocity, density, and total temperature. The yaw angle was then expressed as a unique function of the two effective Reynolds numbers, yielding to:

*S*

_{α}=

*f*(

*α*, Nu).

#### 4.1.3 In Situ Calibration.

Since the vacuum and transonic flow conditions encountered during the actual tests (Table 1) could not be replicated in an open jet facility, the nondimensional mass flux calibration was performed *in situ*. A total of 14 calibration points were collected for different levels of Re_{w} and M. During the in situ operations, due to the combination of high flow velocities and low densities, the flow around the wires was in the slip regime, with a Knudsen number on the order of 0.1. In compressible and rarefied flows, Eq. (5) applies: the dependency of Nu on both Re_{w} and M is highlighted in Fig. 6(a), where scattered and discontinuous points are visible around M = 0.4 for the 9 *μ*m wire dataset. The same trend was also detected for the 5 *μ*m wire. In order to apply the angular calibration (Eq. (20)), the Reynolds number time series of both wires were required, but scattered calibration points could not be used for this purpose. To obtain continuous curves, the Nusselt number was corrected exploiting the correlations of Dewey [8] (Eq. (6)) and Klopfer [10]. The resulting Nu_{corr} curves were assumed to be Mach independent, so Re_{w} was uniquely a function of Nu_{corr}, as shown in Fig. 6(b).

_{corr}−

*Re*

_{w}calibration curve is adopted, the link between the physical Nusselt number (Nu) and the corrected one Nu

_{corr}is the Dewey’s correcting factor

*ϕ*=

*f*(Re

_{w}, M). Therefore, the hot-wire sensitivities to density and velocity could be computed through the differentiation of

*ϕ*with respect to Re

_{w}and M, exploiting Eqs. (14) and (15). In the present case, this approach led to nonphysical negative values of velocity sensitivity

*S*

_{u}for both wires. This observation suggested that Dewey’s correlation, whose range of validity was defined by a range of M, but not of Kn, is inaccurate when used to compute the hot-wire sensitivities in slip flow regimes, for Kn ∼ 0.1. For this reason, the Nu

_{corr}− Re

_{w}curves were employed uniquely to obtain Re

_{w}time series, used then to compute the fluctuating yaw angle by means of Eq. (20), but not to calculate

*S*

_{ρ}and

*S*

_{u}. To accurately determine the density and velocity sensitivities directly applying Eqs. (11) and (12), Nu − Re

_{w}− Kn or Nu − Re

_{w}− M functions were thus necessary. A recent numerical work of Xie et al. [19] proposed a correlation to describe the heat transfer for infinite cylinders, valid in the slip flow regime (0.01 < Kn < 0.1) and for 0.1 < Re

_{w}< 20:

*μ*m wire is presented in Fig. 7(a). This offset was thought to be related to the heat conduction between the hot-wire sensor and the prongs, neglected in the Nusselt number formulation of Eq. (3) and leading to a Nusselt number overestimation. Although almost constant, this offset could not be ignored in the presence of a logarithmic differentiation, thus

*S*

_{ρ}and

*S*

_{u}could not be obtained directly differentiating the correlation. So, in order to apply Eqs. (11) and (12), a Nu − Re

_{w}− Kn interpolation on the calibration points was mandatory. Following the methodology of Xie et al. [19], a Levenberg–Marquardt nonlinear fitting was performed on the experimental dataset. Due to the small number of experimental points, a simpler function, with the following form, was adopted:

Wire | d_{w} (μm) | A | B | C | D | E |
---|---|---|---|---|---|---|

W1 | 5 | 1.045 | 0.601 | 0.314 | −2.177 | 0.900 |

W2 | 9 | 2.584 | 0.708 | 0.386 | −3.588 | 0.768 |

Wire | d_{w} (μm) | A | B | C | D | E |
---|---|---|---|---|---|---|

W1 | 5 | 1.045 | 0.601 | 0.314 | −2.177 | 0.900 |

W2 | 9 | 2.584 | 0.708 | 0.386 | −3.588 | 0.768 |

The resulting calibration map for the 9 *μ*m sensor is shown in Fig. 7(b) as a function of Re_{w} and M. The figure clearly shows that the dependency of the Nusselt number on the Mach number goes well below the “compressibility limit” of M = 0.3, which is also the lower bound of the correlation of Dewey. This observation suggested that the Nu − Re_{w} − Kn function of Eq. (22) is necessary to accurately describe the heat transfer physics in the slip flow regimes.

#### 4.1.4 XW Postprocessing Methodology.

The proposed postprocessing methodology for the XW probe follows:

The unsteady voltages of the two wires were digitally filtered at 30 kHz and resampled at 70 kHz.

The local flow total temperature $T0\xaf$ was known from thermocouple measurements. Similarly, the local time-averaged Mach ($M\xaf$), Reynolds ($Rew1,2\xaf$), and Knudsen ($Kn1,2\xaf$) numbers were retrieved from five-hole pressure probes measurements and from the knowledge of the inlet total pressure

*p*_{0,in}.Equation (3) was applied to compute time series of the two wires Nusselt number (Nu

_{1,2}). The time series of Nu_{1,2}were then converted into $Nucorr1,2=f(Rew1,2\xaf,M\xaf,Nu1,2)$, exploiting Dewey’s correlation.The time series of $Nucorr1,2$ were converted in time series of $Rew1,2$ through the Nu

_{corr}− Re_{w}calibration curves.Using the angular calibration curve of Eq. (20), the ratio of $Rew1$ and $Rew2$ provided the angular fluctuations $\alpha \u2032$.

The sensitivities

*S*_{ρ}and*S*_{u}were computed with Eqs. (11) and (12), from the calibration map of Eq. (22), while the angular sensitivity*S*_{α}was computed exploiting Eq. (16).

*E*(

*f*) is the power spectrum of the velocity fluctuations, extrapolated for a frequency value approaching zero. To compute this value, the power spectrum was averaged up to a frequency before the decay, as discussed later in Sec. 5.

### 4.2 Single-Wire Probe Calibration and Data Reduction.

The calibration procedure for the single-wire probe is very similar to that of the XW probe. The wire temperature was determined by means of the same low-speed calibration, but no angular calibration was required, since a single-wire probe is not suitable to measure instantaneous flow angles. The in situ calibration was obtained through the same Nu − Re_{w} − Kn mapping of Eq. (22), allowing to compute the wire sensitivities. The computed calibration coefficients are reported in Table 3.

*μ*m wire of the XW probe were also postprocessed by means of Eq. (26).

## 5 Results and Discussion

In this section, the results of the steady and unsteady measurements, i.e., without and with WG, are discussed. All the tests were conducted at the cascade inlet at blade midspan, traversing the hot-wire probes over two blade pitches, from *y* = −*g* to *y* = *g*. The position *y* = 0 indicates the pitchwise coordinate of the leading edge of the cascade’s central blade.

### 5.1 Hot-Wire Sensitivities.

Before discussing the resulting turbulence parameters, the computed hot-wire sensitivities are presented and compared with the available literature. The sensitivities to density and velocity for the two probes are plotted in Fig. 8. It is immediately apparent that the density sensitivities were systematically higher than those to velocity, and this result is in agreement with all values reported in literature for compressible flows [3,6,9,13]. In the present case, only the 5 *μ*m wire (XW1) presented a *S*_{ρ} higher than 0.2, while the two 9 *μ*m wires (XW2 and SW) are on the order of 0.15 and 0.175, respectively. The velocity sensitivities were found to be around 0.08 for both XW1 and XW2 and on the order of 0.055 for the SW. These trends are different from the ones reported by Cukurel et al. [6], where a 9 *μ*m wire presented higher sensitivities to density and velocity compared to a 5 *μ*m wire. To justify these differences, it is worth mentioning that the present investigation was performed in a compressible and rarefied flow, with Kn ∼ 0.1, while the work in Ref. [6] was performed at atmospheric pressure conditions. Moreover, in the work in Ref. [6], the sensitivities were computed starting from a Nu_{corr} − Re_{w} curve, thus performing a direct derivative of Dewey’s correlation using Eqs. (14) and (15). As previously discussed, this approach led to nonphysical values of velocity sensitivity when applied to the present dataset.

### 5.2 Steady Measurements

#### 5.2.1 Power Spectra.

*pwelch*function, for the cases with and without the TG, are presented in Fig. 9. It is possible to identify the turbulent inertial subrange, characterized by the −5/3 slope, starting around 3 kHz for the freestream flow and around 5 kHz for the flow perturbed by the TG. However, these spectra were noisy and spurious peaks appeared in the PSD at frequencies of 6 kHz, 10.4 kHz, 12.5 kHz, 16.3 kHz, 20.5 kHz, and 23.7 kHz. Their presence was not related to any flow fluctuations, since the wake generator was not employed for steady measurements, and could not be ascribed to vortex shedding from the TG bars since they were present even in the absence of the TG. Consequently, the spurious peaks could only be related to the facility itself, perhaps due to vibrations generated by the compressor or the vacuum pump, or to resonances of the hot-wire probe. Moreover, a noisy spectrum makes it difficult to extrapolate at zero frequency to compute the integral length scales (Eq. (25)), so the peaks had to be eliminated to perform proper measurements. Median and Savitzky–Golay filters were hence applied to smooth the spectra, as suggested by Heckmeier et al. [20]. A filtered spectrum is presented in Fig. 10 by the red line: the fluctuations were significantly reduced, allowing a safer zero-frequency extrapolation, but the spurious peaks were still persistent. For this reason, a tailor-made smoothing spline fitting (black line in Fig. 10) was performed in few tests, in order to cancel out the peaks and determine their impact on the turbulence intensity computation. The standard deviation of the velocity fluctuations was then computed as follows:

#### 5.2.2 Turbulence Intensity.

The turbulence intensity values, computed with Eq. (24), are shown in Fig. 11. The caption “XW” refers to the results obtained from the sensitivity system postprocessing methodology (Eq. (23)). As previously explained, Eq. (26) was also used, treating the 9 *μ*m wire of the XW probe as a single wire, in order to understand the influence of neglecting the $\rho \u2032$ contribution. These results are indicated by the caption “1W.” According to a previous investigation performed in the same rig, Tu naturally achieved by the facility was expected to be around $0.9%$ [21]. In the present investigation, the results were slightly lower, and the Tu measured by the XW without the grid was around 0.55%. The results obtained with the single-wire method were even lower (Tu ∼ 0.3%). A bigger difference between the two processing methodologies was observed for the tests with the TG. The XW method provided a Tu bounded between 2.5% and 3%. As previously discussed, the estimated boost of Tu imputable to the spurious peaks in the spectra was around 0.25%. The corrected Tu values, obtained by removing this offset, are also presented in Fig. 11. These results are in fair agreement with Tu predicted for grid-generated turbulence by Roach’s model [17], shown by the black continuous line in Fig. 11. The single-wire method provided a significantly lower value of Tu, between 1.4% and 1.5%, proving that neglecting the density fluctuations in such a high-speed low-density flow is a questionable assumption, leading in this case to a severe underestimation of Tu by ∼1%. This evidence suggested that a sensitivity-based system (Eq. (23)) is mandatory to decouple density and velocity fluctuations in compressible slip flows applications.

#### 5.2.3 Integral Length Scales.

The traditional approach to compute integral length scales is based on the autocorrelation function of the velocity signal [17]. However, when this approach is employed on experimental datasets, the noise can offset the integral length scale value, resulting in extremely variable test-to-test results. As an example, for the tests without the TG, this method provided integral length scales oscillating in the order of 300 < Λ_{int} < 700 mm. The same trend was highlighted by Michálek et al. [21], who showed that low-frequency large-scale patterns, of the same dimension of the connecting duct (∼0.5 m) developed in the test section if the TG was not employed. In order to eliminate the low-frequency scales and to reduce the test-to-test variability, a different approach, based on Eq. (25), was employed in the scope of this study. To apply Eq. (25), a zero-frequency extrapolation in the PSD was performed, averaging the spectra over a certain range of low-frequencies, as shown in Fig. 10. Looking at the spectra of Fig. 9, a high-energy content at low frequencies is discernible for the tests without the TG, in agreement with the conclusions of Michálek et al. [21]. The spectrum drops and stabilizes in a region around 100 Hz. For this reason, the zero-frequency extrapolation, for the tests without the TG, was performed averaging the spectra between 100 Hz and 400 Hz, canceling out the low-frequency content associated with the largest low-frequency scales. On the other hand, the spectrum for the test with the TG flattened out at low frequencies, indicating that the large-scale structures previously detected were no longer dominant. Accordingly, for the tests with the TG, the zero-frequency extrapolation was performed averaging the PSD between 0 Hz and 100 Hz. The results are presented in Fig. 12, where the integral length scales are shown to be between 14 mm and 19 mm for the tests without the TG, and between 10 mm and 15 mm for the tests with the TG. In particular, the results with the TG suggested that the dominant structures were associated to scales of the same order of the mesh size of the TG. The computed values were comparable to the ones reported by Michálek et al. [21], amounting to 30 mm and 12 mm for the tests, respectively, without and with the TG. For the sake of completeness, the results obtained averaging the PSD of the tests without the TG between 0 Hz and 100 Hz, as well as the ones with the TG if the averaging was performed from 100 Hz to 400 Hz, are shown in Fig. 12. As expected, without the grid, the integral length scales resulting from an averaging at low frequencies were significantly higher, between 50 mm and 80 mm. On the other hand, if the tests with the TG were averaged between 100 Hz and 400 Hz, no major differences could be identified, just a small reduction of ∼3 mm. This trend was again justified by the fact that, in the presence of the TG, the spectra appeared almost flat in the whole range from 0 to 400 Hz, as shown in Figs. 9 and 10.

The integral length scales were found to be of the same order of magnitude when using the processing methodology of the single wire. However, these results showed an opposite trend, with higher scales (Λ_{int} ∼ 18 mm) for the tests performed with the TG, and lower values (Λ_{int} ∼ 14 mm) for the tests in the absence of the TG. This trend, opposite to the experimental evidence previously discussed, and to the findings reported in Ref. [21], suggested that an accurate estimation of the velocity fluctuations $u\u2032$, decoupled from the density fluctuations $\rho \u2032$ through Eq. (23), is mandatory to correctly estimate Λ_{int} with Eq. (25).

The time-average pitchwise average measurements of turbulence intensity and integral length scales are summarized in Table 4.

#### 5.2.4 Angle and Density Fluctuations.

The proposed experimental methodology provided also the unsteady fluctuations of yaw angle and density. The intensity of the density fluctuations was determined to be around 0.25% and 1.1% for the tests performed without and with the TG, respectively. On the other hand, the standard deviation of the yaw angle fluctuations was found to be around 0.35 deg when the tests were performed in the presence of the TG, with the fluctuations bounded in the interval ± 1.5 deg. For the tests performed without the TG, the standard deviation of $\alpha \u2032$ reduced to approximately 0.1 deg, and all the fluctuations were bounded in the interval ± 0.5 deg.

#### 5.2.5 Uncertainty Analysis.

Due to the complexity of the XW postprocessing methodology, the uncertainty analysis was carried out implementing a Monte Carlo algorithm across the whole measurement chain, from the in situ calibration to the postprocessing of the experiments, following the methodology proposed by Boufidi et al. [7]. Two major sources of uncertainity were identified in this analysis, namely, the uncertainty of the accessory measurements and the fitting error in the calibration laws. For each of the *N*_{i} Monte Carlo iterations, the uncertainties of the accessory measurements of total pressure, static pressure and total temperature, used to define the operating point of the facility were divided in random and bias errors and randomly perturbed around their statistical distribution. In order to compute Nusselt, Mach, Reynolds, and Knudsen numbers, accessory thermocouple and five-hole pressure probe measurements were required, and this uncertainty contribution was accounted too. For all the input values, rectangular and Gaussian distributions were used to account for the bias and random errors, respectively. The complete breakdown in random and bias errors of all the input quantities can be found in the spleen open-source database [22].

A set of *N*_{i} Nu_{corr} − Re_{w} and Nu − Re_{w} − Kn calibration laws was generated and used to solve the sensitivity system. Due to the scarcity of the calibration points, the error imputable to the two fittings was propagated in the sensitivity system estimating the 95% confidence interval of the calibration laws around each measuring point (in terms of average Mach, Reynolds, and Knudsen numbers) and treating it as a random contribution, as suggested by Boufidi et al. [7]. The convergence on the turbulence statistics was reached approximately after *N*_{i} = 5000 iterations. The resulting distributions of turbulence intensity were found to be mildly positively skewed, following a log-log-normal distribution. The 95% confidence intervals around the Tu mean value were found to be [$\u22120.35%$ ; $+0.48%$] for the tests with the TG, and [$\u22120.09%$ ; $+0.14%$] in the absence of the TG. Similarly, the non-symmetric 95% confidence intervals for the integral length scales were determined to be [−0.6 mm; +2.3 mm] and [−1.3 mm; +0.3 mm] for the tests with and without the TG, respectively.

### 5.3 Unsteady Measurements.

The results presented in this section refer to the unsteady tests, carried out with the WG, a set of 96 rotating bars shedding wakes at engine-representative reduced frequencies (*f*^{+} = 0.95), corresponding to a bar-passing frequency around 5.3 kHz. All tests were performed with the SW probe; thus, Eq. (26) was exploited to retrieve the turbulent velocity fluctuations. As highlighted in the previous section, neglecting the density fluctuations is an oversimplification that led to an underestimation of the Tu of ∼ 1% for the tests performed with the TG. As a consequence, the unsteady measurements presented in this section are expected to be less accurate, compared to the steady ones, and can provide qualitative trends, but not high-fidelity quantitative results.

*N*

_{per}. The period of each event was discretized in several classes,

*N*

_{class}, and the points belonging to the same class were averaged to obtain the final ensemble-averaged signal. The PLA method, for a velocity signal

*u*(

*t*), can be defined mathematically in Eq. (28):

*u*

_{PLA}was then removed from the original velocity time series, eliminating the deterministic contributions while preserving the turbulent fluctuations

*u*

_{turb}(

*t*):

*u*

_{turb}(

*t*).

In this work, the velocity time series were ensemble-averaged using two different approaches, considering as repeating periods both the full rotor revolution (disk passing frequency method (DPF)) and the single bar passage (bar passing frequency method (BPF)). Each signal was acquired for 3 s, corresponding to approximately 165 complete rotor revolutions and 15800 bar passages. The PSD of the velocity time series *u*_{turb}(*t*) are shown for the two methods in Fig. 13. The peaks associated to the periodic blade passages were completely eliminated by both approaches. However, when the BPF method was employed, the PLA failed to identify the smaller peaks, suggesting that these frequency components were associated either to unsteady vibrations of the WG disk or to small differences in the wakes generated by the different bars.

In the present analysis, effort was put toward determining the phase-resolved turbulence intensity $Tu~$. When the ensemble-averaging was performed, the standard deviation of the velocity signal was computed for each PLA class, i.e., for every phase of the repeating event (the disk revolution for the DPF method and the bar passage for the BPF method). The phase-resolved $Tu~$ was then obtained applying Eq. (24) to the phase-resolved standard deviation vector. An example of phase-resolved turbulence intensity $Tu~$ is shown in Fig. 14. In Fig. 14(a), the ensemble-average was performed using the DPF method, so the 96 peaks and valleys representative of the 96 bar passages can be distinguished. The peak-to-peak and valley-to-valley variability of $Tu~$ highlighted how the 165 rotor revolution events of the DPF method were not enough to achieve a statistical convergence on the velocity standard deviation. This was confirmed by the results obtained with the BPF method, as shown in Fig. 14(b). The statistical convergence was indeed achieved after 85 rotor revolutions, corresponding to 8200 bar passage events. This result suggested that approximately 8200 full rotor revolutions would be necessary to accurately phase-resolve the turbulence intensity exploiting the DPF method. For completeness, the $Tu~$ obtained ensemble-averaging the DPF phase-resolved $Tu~$ over the 96 blade passages is reported in Fig. 14, and no significant differences were detected in the maximum and minimum values when compared to the BPF method. The peaks and valleys in Fig. 14 are associated to regions of wake-perturbed and clean flow, respectively. If the phase-locking was performed over the full rotor disk, the maximum level of $Tu~$ associated to the wake was approximately in the order of 6.5%, while the lower values in the clean flow region were found to be around 2%. On the other hand, using the BPF method, the maximum and minimum levels of $Tu~$ were 5.5% and 3%, respectively. In Fig. 15, the results of the time-averaged $Tu~$ for the full experimental dataset are shown by the blue points for the DPF method and by the green points for the BPF method. The time-average turbulence intensity was found to be in the order of 4.5% for all the experiments and for the two averaging approaches. The slightly higher values obtained using the BPF method were due to the deterministic fluctuations associated to the disk, which were not eliminated by this approach as previously shown in Fig. 13. The bar plot represents the minimum and maximum values of the phase-resolved $Tu~$, showing the range of the turbulence intensity oscillations for each test. For the DPF method, the peak-to-valley variation of $Tu~$ amounted approximately to 4.5%, again with maximum values of ∼ 6.5% and minimum values around ∼ 2%. The test-to-test variability was imputed to the statistical convergence not achieved by the DPF method. If the BPF method was used, maxima and minima were found to be almost constant for all the experiments, again in the order of 5.5% and 3%, respectively. For the sake of completeness, the turbulence intensity computed without ensemble-averaging the signal is presented in Fig. 15 in red. As expected, this value was significantly higher compared to the one obtained removing the PLA of the signal. Indeed, if Eq. (29) was not applied, the deterministic fluctuations caused by the WG were treated as stochastic and thus used to compute $Tu~$, leading to an overestimation of the turbulence intensity in the order of 3%.

To conclude, the DPF method allowed the removal of not only the deterministic fluctuations associated to the bar-passing frequency but also the ones due to the WG disk rotation and bar-to-bar variations. It was thus considered more reliable to compute the time-average turbulence intensity. On the other hand, the BPF method provided a significantly better statistical convergence, i.e., a better phase-resolved turbulence intensity, since the number of blade passing events was 96 times higher, when compared to the 165 disk revolutions.

## 6 Conclusions

In this work, the application of hot-wire anemometry in compressible rarefied flows is discussed in detail. The experiments were performed in the von Karman Institute S-1/C high-speed low-Reynolds number facility, at engine-representative flow conditions of a modern low-pressure turbine, with an inlet Mach number around 0.46 and a Knudsen number, computed with respect to the diameter of the hot wire, around 0.1. In compressible and rarefied flow regimes, a hot wire is strongly sensitive to the density fluctuations occurring in the flow, while the sensitivity to velocity is typically smaller. So, the traditional calibration and data reduction approaches suffer from inaccuracies and lead to erroneous results if the density fluctuations effect is neglected. In this article, expanding the work reported in Refs. [3,6] to slip flow applications, a novel nondimensional calibration procedure, based on the dimensionless groups Nu − Re_{w} − Kn, and a sensitivity-based postprocessing technique were derived and applied to the experimental dataset. In the absence of a turbulence grid and a WG, the turbulence intensity computed with the proposed methodology amounted to ∼0.55%. This value was underestimated by ∼0.25% due to the neglect of the density fluctuations. In the presence of the turbulence grid, the resulting turbulence intensity was ∼2.5%, in fair agreement with the value estimated by a correlation proposed by Roach [17]. In the presence of the grid, if the density fluctuations were neglected, a severe underestimation of the turbulence intensity of ∼1% occurred. According to these results, neglecting the contribution of the density fluctuations provided nonnegligible differences. Consequently, the use of a cross-wire probe, coupled with the proposed calibration and postprocessing methodology, was necessary in order to decouple the instantaneous fluctuations of density and velocity and to compute turbulence parameters. The integral length scales were found to be of 12 mm and 17 mm for the tests without and with the turbulence grid, respectively. In the presence of the turbulence grid, the results suggested that the most energetic structures were associated to a length that was in the order of the grid mesh size, amounting to 12 mm.

In order to process the unsteady tests, performed with the WG, two different ensemble-averaging techniques, based on the full rotor revolution and on the single bar passage, were exploited to distinguish the deterministic fluctuations from the background turbulence. The first method allowed the elimination from the spectra of the deterministic frequency components associated both to the bar-passing frequency as well as to the rotor itself, providing thus better results in terms of time-averaged turbulence intensity downstream of the rotor. However, due to the limited number of full rotor revolutions during each test, the statistical convergence was not achieved and the phase-resolved turbulence intensity was poorly measured. If the ensemble-average was performed over a single bar, only the blade passing frequency and its harmonics were eliminated in the spectra, but the phase-resolved turbulence intensity reached the statistical convergence. Finally, if the deterministic noise was not decoupled, the turbulence intensity was overestimated by ∼3%.

## Footnote

## Acknowledgment

This research is part of the large EU-funded SPLEEN project (Secondary and Leakage Flow Effects in High-SPeed Low-PrEssurE TurbiNes) in collaboration between the von Karman Institute for Fluid Dynamics and Safran Aircraft Engines. The authors thankfully acknowledge the Clean Sky 2 Joint Undertaking for funding of the SPLEEN project (Grant No. 820883) under the European Union’s Horizon 2020 research and innovation program. The authors also acknowledge Filippo Biondi, Mariana Lousada, Terence Boeyen, and Oswald Divers for their significant contribution to this work.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data and information that support the findings of this article are freely available.^{3}

## Nomenclature

*d*=wire’s diameter (

*μ*m)*g*=cascade pitch (mm)

*k*=thermal conductivity (W K

^{−1}m^{−1})*l*=length (mm)

*p*=pressure (Pa)

*s*=blade span (mm)

*u*=velocity (ms

^{−1})*x*=location along axial chord (mm)

*y*=location along pitch direction (mm)

*z*=location along span direction (mm)

*C*=blade chord (mm)

*N*=number (–)

*R*=resistance (Ω)

*S*=sensitivity coefficient (–)

*T*=temperature (K)

*E*_{b}=bridge voltage (V)

*K*_{1,2,3}=parameters of Eq. (13)

*E*(*f*) =velocity PSD (dB/Hz)

- Tu =
turbulence intensity (%)

*α*=yaw angle (deg)

*β*=pitch angle (deg)

*γ*=heat capacity ratio (–)

*η*=recovery factor (–)

*θ*=overheat ratio (

*T*_{w}/*T*_{0}) (–)- Λ
_{int}= integral length scales (mm)

*μ*=dynamic viscosity (kg m

^{−1}s^{−1})*ρ*=density (kg m

^{−3})*τ*_{w}=overheating parameter (

*T*_{w}−*ηT*_{0})/(*ηT*_{0}) (–)*ϕ*=Dewey’s corrective factor (–)

### Dimensionless Groups

### Abbreviations

### Superscripts and Subscripts

- (·)′ =
fluctuating component

- $(\u22c5)\xaf$ =
time-averaged quantity

- $(\u22c5)~$ =
phase-resolved quantity

- (·)
_{0}= total flow quantity

- (·)
_{1,2}= wire 1, wire 2

- (·)
_{ax}= axial direction

- (·)
_{b}= number of blades

- (·)
_{class}= number of PLA classes

- (·)
_{corr}= corrected with Dewey’s correlation

- (·)
_{eff}= effective quantity

- (·)
_{i}= number of Monte Carlo iterations

- (·)
_{in}= cascade inlet

- (·)
_{out}= cascade outlet

- (·)
_{per}= number of PLA periods

- (·)
_{s}= probe support

- (·)
_{t}= bridge top resistance

- (·)
_{turb}= turbulent fluctuations, PLA removed

- (·)
_{w}= wire’s property

- $(\u22c5)\alpha 0$ =
reference position

## References

*Ph.D. thesis*