On the applicability of LDV profile-sensors for periodic open wet clutch flow scenarios

The main difference between a profile sensor compared to a conventional LDV set-up is the use of two overlapping fringe systems with different spatial gradients, wherewith one velocity component and the position of particles passing through the measurement volume can be detected. The beam waist positions of the two laser systems are, therefore, shifted longitudinally away from the intersection point in opposite directions. Thus, the fringes are diametrically opposed and the fringe spacings \(d_{\mathrm {f,}i}\) for \(i=1,2\) are a function of the \(x_3\)-position as illustrated in Fig. 1, where the complete used LDV-system is decomposed for the different laser beams. The \(x_3\)-position of the particles can be determined by the quotient q of the received Doppler frequencies \(f_\mathrm {i}\) which is given by \(q(x_3)=f_2/f_1=d_{\mathrm {f},1}(x_3)/d_{\mathrm {f},2}(x_3)\), if the velocity \(u_1\) is equal to \(u_2\). \(x_3\) can then be found by means of a calibration function. Once the \(x_3\)-position of a particle is found the local fringe spacings \(d_{\mathrm {f},i}\) and the resulting velocity u can be calculated retroactively. Therefore, the calibration procedure is of major importance for the measurement process, which is normally done already by the manufacturer. According to Czarske et al. (2002) the uncertainty of the position can be stated asA possible location error depends on the slope of the calibration function q and the ratio of frequency uncertainty \(f^{\sigma }\) and the frequency f. Possible measurement errors are a low Signal-to-Noise ratio (SNR), a undersampled burst signal or—in case of a industry-near application—vibrations of the test rig. With the newly developed Bragg-shifted version of the LDV-PS it becomes furthermore possible to detect quasi-stationary particles, as occur e.g. in recirculation bubbles (cp. Burgmann et al. 2021).

For the conducted experiments an ILA R &D 1D2C-LDV Profile Sensor with two Nd:YAG lasers (\(\lambda =532\,\) nm and \(\lambda =553\,\) nm) and an additional laser in a general LDV set-up (\(\lambda =561\,\) nm) to determine a second velocity component is employed as shown in Fig. 1. The two lasers featuring the profile sensor functionality detect the circumferential velocity and the general LDV set-up detects the radial and axial velocity. All components are Bragg-shifted with a frequency of 5 MHz (40 MHz native modulation with downsampling) and the focal length of the applied front lens of the sensor is 160 mm. Additional profile-sensor parameters are summarized in Table 1. The calibration quotient q together with the fringe distances for the used LDV sensor is depicted in Fig. 2. For the velocity measurements, the coincidence mode of the LDV-sensor was enabled, resulting in the measurement of only those particles, that were registered with all three laser components. Each channel’s validation option must be met in order to lead to a valid coincidence signal. Moreover, the position limit of the profile sensor (600 \(\upmu\)m) must be satisfied.

The set-up for all conducted experiments is an open wet clutch facility, which comprises a single-stage rotor-stator configuration as shown in Fig. 3. The stator plate serves as optical access with anti-reflection coated float glass window to guarantee a low distortion of the inbound and scattered light. As rotating counterpart, an interchangeable disk construction serves the purpose to operate different groove patterns or non-grooved clutch plates. For a repeatable and precise movement of the rotating disk unit a nanotec ST11018L8004 stepping motor was used. Angular-resolved measurements are enabled due to a WL100L-F2231 light barrier sensor.

For the uncertainty quantification of the LDV-PS system the test rig is equipped with variant A, according to Fig. 3. A wire of diameter \(d_{\mathrm {w}}=10\) \(\upmu\)m is mounted on the rotating plate at different rotational velocities. The sensor is tilted to face the wire orthogonal and the stationary housing of the test rig with the glass plate is removed. The set-up for these studies is shown in Fig. 3(a) and aims to mimic a particle at exactly the same angular and lateral position, as well as a known and constant velocity. Similar test set-ups were conducted by Czarske et al. (2002) and are carried out during the calibration of the LDV-PS at the manufacturer. This set-up is referred to as wire set-up or variant A below.

The flow structure investigation, as actual test scenario, is progressed with a radially-grooved rotor disk and oil in between the gap of the rotor and stator, as sketched and referred to as variant B in Fig. 3. The clutch is operated at an angular velocity of \(\varOmega =12.6\) 1/s, which equals a low-to-moderate circumferential disk velocity (according to Iqbal et al. 2013, for instance) and corresponds to an expected disk velocity of \(u_{\varphi }=1.1\,\mathrm {m/s}\) at the mean radius \(R_{\mathrm {m}}=(R_1+R_2)/2\). The inner and outer radii of the disk are \(R_1=82.5\) and \(R_2=93.75\) mm. The disk features 32 evenly distributed radial grooves with a bottom groove width of \(W=1.35\,\)mm and height of \(H=0.97\) mm. The gap between rotor and stator is adjusted to \(h=320\) \(\upmu\)m. The measurement ellipsoid length is adjusted to 600 \(\upmu\)m and traversed in 200 \(\upmu\)m steps through the gap along the sensor axis. Despite full coverage of the entire gap with the ellipsiod, the traversing appears necessary, since the measuring region in z-direction spans a length of \(h+H=1.29\,\)mm at the groove locations. The adjustments of the ellipsoid length resulted from the findings of the conducted wire experiment as will be elaborated further below. The radial position of the measurement ellipsoid for both inclination angles was located at the mean radius \(R_{\mathrm {m}}\) and validated with the expected disk velocity.

The applied white mineral oil \((\rho _{\mathrm {o}}=850\,\)kg/m\(^3\), dynamic viscosity \(\mu =0.0136 \, \mathrm {kg/ms}\), refractive index \(n=1.4\)) has been seeded with silver-coated hollow glass spheres \((d= 14\,\) \(\upmu\)m; \(\rho _{\mathrm {p}}=1700 \, \mathrm {kg/m^3})\), which corresponds to a characteristic time scale of \(\tau _{\mathrm {p}}=1.36\times 10^{-6} \, \mathrm {s}\) and a Stokes number of \(S_k=0.0047\), considering the gap height as characteristic length scale. The gap and the inner part of the clutch were filled with oil to ensure a single-phase flow through the gap, but no additional external volumetric flow rate was applied. The corresponding gap Reynolds number as already discussed by Daily and Nece (1960) results in \(Re_{\mathrm {h}}=\varOmega h^2/\nu =0.08\). The corresponding lubrication Reynolds number is \(Re_{\mathrm {l}}=\varOmega R_{\mathrm {m}} h/\nu =22.0\), which is considered for the present flow scenario (cp. also Leister et al. 2020).

To extent the velocity-component measurement and also to avoid influential light reflections, the sensor is tilted forwards and backwards to an angle of \(\alpha =30^{\circ }\) compared to the z-axis. The \(x_3\)-axis as indicated in Fig. 1 is rotated with \(\pm 30^{\circ }\) compared to the z-axis of the test rig as indicated in Fig. 4. This way of measurement enables particle detection in proximity to the disk where normally the reflected light limits the detection of the particle’s Mie scattering. Additionally, this enables the retroactive determination of the third flow component, which would otherwise, be impossible to capture with an LDV sensor and this complex geometric set-up. The two considered sensor positions as used for the flow measurements are shown in Fig. 4.

While the circumferential velocity component \(u_{\varphi }\) is measured with the profile sensor lasers and directly with either sensor positions I and II, both components \(u_r\) and \(u_z\) are combined in the second conventional recorded velocity component perpendicular to the sensor axis in the \(r-z\)-plane. According to the sensor orientations, this second component is referred to as \(u_{\text {2,p\,I}}\) and \(u_{\text {2,p\,II}}\), for sensor orientation I and II, respectively; see Fig. 4. The difference of \(2 \alpha \ge 60^{\circ }\) has been recommended for the extraction of two velocity components by Morrison et al. (1991) and Orloff and Snyder (1982). A summary of the different sources contributing to error propagation is summarized by Tropea (1995). Several additional influences are caused by the test rig and the inclination of the sensor. These circumstances lead to a change in the interference pattern for all components. The calibration curve as shown in Fig. 2 could not be directly applied to the output z-position but some systematic deviation must be corrected to reach meaningful results. The main sources of deviation are summarized in the following. First, the glass plate leads to a shift of the actual sensor position in comparison to a purely geometric consideration as e.g. outlined by Shirai et al. (2008). Second, the refractive index of the oil (and the glass plate) leads to an angular deviation, which must be addressed accordingly via a calculation with the law of refraction. The z-position of each detected particle is thus corrected. Additionally, the velocity information is transformed and corrected.

Table 2 shows the measured and calculated velocity components for clarity. The detected velocities of the second component can be divided into the radial and axial velocity according to As such, the measured velocities of each particle can be split into the radial and axial velocity component, which are then expressed as:

Note that the refractive-index matched angle \(\beta =\arcsin ( \sin \alpha /n)\) is applied in (2) and (3) to account for the fluid properties of the chosen oil.

Equation set (3) directly emphasizes the interplay between the consecutively recorded data sets, where any possible error comprised in the results from either sensor orientation, can immediately affect both determined velocity components \(u_r\) and \(u_z\). Consequently, the quality of both calculated velocity components relies on both measured orientations.

The conversion from (2) to (3) is illustrated in the insert of Fig. 4, which displays the smooth part of the gap without grooves for clarity.

For the spatial uncertainty study the mentioned \(d_{\mathrm {w}}=10\) \(\upmu\)m wire and variant A of the set-up was used; see Fig. 3(a). The uncertainty of z-direction is directly linked to the profile sensor function, while the \(\varphi\)-coordinate is allocated to the recorded LDV signals from the timing unit of the encoder.

For this experiment, the sample rate was fixed at 50 MHz and the signal window was set to 8K. The higher velocities could have been sampled with a smaller signal window, however, all velocities are present in the gap flow, so fixed values for these quantities fulfill the applicational requirements. A more precise detection for the low velocities would be easily manageable with further extension of the sample rate and the signal window for each specific velocity range.

Figure 5 shows all detections of the wire for five different velocities and their calculated spatial locations. A clear trend to increasingly scattered location estimation is observed for lower velocities. Table 3 shows this variation of the results in terms of the standard deviation \(\sigma\). Approximately 1500 wire detections have been recorded for each velocity to ensure statistical significance of the data evaluation process. Possible sources for the occurring scatter are a low signal-to-noise ratio of the light signal caused by strong reflections within the sub-millimeter gap; an unadjusted signal window for low velocities, which leads to an undersampled frequency signal and vibrations caused by the test rig in this slightly rough industry-related scenario. The uncertainty can be systematically diminished with a high number of overall detections in the low velocity regime, which will be considered for the flow measurements. As a result of the above insight—the measurements of the fluid flow have been partitioned into measurement ellipsoid lengths of \(600\,\) \(\upmu\)m, as already indicated in the previous section, thus matching every velocity range with suitable parameters. This partially compensates for the high uncertainty for low velocities.

An additional determination strategy arises from the scattered data of Fig. 5 in terms of possible averaging methods. In principle averaging in space or velocity is possible, where either the determination of the velocity statistics can be performed for a given spatial binning of the data, or—vice versa—the location statistics can be determined for preset velocity bins. For low velocities definitively an average over certain velocity bins make sense since this quantity varies much less compared to the spatial distribution. Additionally, a velocity averaging doesn’t coop with this strong change in uncertainty, which improves the averaging results in general. Averaging in space appears suitable when the velocity is high enough and the possible arising error can be considered as small enough for the examined flow. For more complex flows and flows with a non-injective functional space-velocity relationship (e.g. local maxima/minima) more elaborated evaluation strategies appear suitable. Particularly, the application of fitting functions and/or advanced averaging methods, which e.g. combine velocity and spatial averaging relative to the occurring gradients, can be envisioned. In the present context, the flow in the groove reveals a local velocity maximum slightly higher than the rotational velocity of the disk and thus only a weakly-pronounced gradient in z-direction, as further elaborated below. Consequently, a \(4^{\text {th}}\)-order polynomial has been chosen to approximate the velocity distribution.

Since the spatial location is the most crucial quantity is discussed sufficiently, the following part focuses on all velocity and spatial components that were conducted with variant A of the set-up and are essential for the considered flow scenario. Figure 6 shows multiple histograms of all extracted results for a nominal angular velocity of \(\varOmega =12.57\) 1/s in detail. To compare the possible errors and their propagation, the standard deviation of the stepping motor was specified by the variation of the light barrier signal, which was additionally fed into the LDV software. The mean frequency and corresponding standard deviation results in \(f_{\mathrm {m}}=1.9951 \pm 0.00015\) Hz, which corresponds to a standard-deviation contribution of \(u_{{\varphi },\;SM}^{\mathrm {\sigma }}=0.000083\) m/s of the stepper-motor variation within the \(u_{\varphi }\)-histogram.

As shown in Fig. 6(a) and (b) the circumferential and radial velocities were detected with low uncertainties, as expected for LDV systems. Although the \(u_{\varphi }\)-component features the profile sensor function with four partial laser beams, the velocity uncertainty is within the same range as for the \(u_r\) velocity, which is provided by a classical LDV system (cp. Fig.  1). An expected higher velocity in the groove of \(5\%\) compared to the disk velocity, as uncovered by Leister et al. (2021), equals a \(\varDelta u_{\varphi ,\mathrm {OS}}\) of 0.055 m/s in this flow scenario, which is around 10 times higher then the standard deviation. In conclusion this quantity of clutch flow pivotal element is expected to be resolved accordingly even for fairly low disk velocities which were chosen in this study.

Figure 6(c) presents the uncertainty distribution of the phase angle \(\varphi\). Here only six different values with a difference of \(\varDelta \varphi _{\mathrm {theo}}=0.029^{\circ }\) of the 10 \(\upmu\)m wire have been distinguished, which renders a possible theoretical uncertainty of \(\varDelta \varphi _{\mathrm {theo}} R_{\mathrm {m}}=44.6\) \(\upmu\)m. Compared to the groove width W as characteristic length scale in circumferential direction the relative uncertainty is \(\varDelta \varphi _{\mathrm {theo}} R_{\mathrm {m}}/W=0.033\). Hence, approximately 30 different positions can be distinguished within the groove, which leads to a sufficient resolution for the given flow scenario.

The constant z-position of the 10 \(\upmu\)m diameter wire has been detected with a standard deviation of \(z^{\sigma }=80.46\)  \(\upmu\)m at a wire velocity of \(1.11\,\)m/s; see Fig. 6(d). This measure indicates only limited uncertainty, when compared to the gap height of \(h=320\) \(\upmu\)m as characteristic length scale in z-direction. The relative uncertainty of \(z^{\sigma }/h = 80.46\) \(\upmu\)m/320 \(\upmu\)m \(=25\%\) for the 1300 registered burst from the wire, emphasizes the necessity to further evaluate the required sample number. For the given problem, a relative uncertainty of \(5\%\) is considered sufficiently accurate, which corresponds to an absolute error of 16.0 \(\upmu\)m. Consequently, more than 97 individual measurements are required assuming a Gaussian error distribution.

Figure 7 shows the detected particles (blue dots) in a \(z-u_{\varphi }\)-diagram for \(\varphi \in [7^{\circ },11.25^{\circ }]\), corresponding to a smooth region of the disk sufficiently displaced from the groove. The dashed black line shows a linear profile for comparison. The revealed linear Couette-like velocity profile is characteristic of the smooth regions of the clutch gap at low rotational velocities as elaborated by Lance and Rogers (1962), for instance. Moreover, the superimposed red bullets show the respective standard deviation \(z^{\mathrm {\sigma }}\) of the location estimations from the wire experiment for the tested velocities, as summarized in Table 3.

Figure 8 provides an overview of the detected particles and determined quantities within one of the 32 groove-to-groove sections. Particularly, the spatial distribution across the section is shown in Fig. 8(a) in a normalized \(\varphi -z\) coordinate system for all particles. The physical dimensions of the gap and the groove are illustrated for clarity. The \(\varphi\)-coordinate can be resolved sufficiently precise so that the start and end of the groove, namely \(\varphi R_{\mathrm {m}}/W=0\) and \(\varphi R_{\mathrm {m}}/W=1\), can be located. The origin of the \(\varphi\)-coordinate is set to the left side of the groove and normalized with \(R_{\mathrm {m}}/W\), for further clarity. In addition, the detected particles in the groove region appear color-coded along the \(\varphi\)-coordinate, to distinguish different slices of the flow for further (color-coded) evaluation of the averaged velocity profiles.

Particle velocities and locations across the gap and in the groove are shown in Fig. 8(b) in a normalized \(z-u_{\varphi }\) coordinate system, which comprises particles from both the smooth and the grooved parts. As before, the rotor velocity of \(\varOmega R_{\mathrm {m}}\) can be identified in the diagram such that particles within this small velocity band were excluded for subsequent average-velocity calculations. The particle distributions demonstrate that the velocity in the groove region is well below \(\varOmega R_{\mathrm {m}}\) at the gap-groove interface and then advances beyond the rotor velocity well inside the groove cavity. This intra-groove vortex system has already been identified by Leister et al. (2021) and was referred to as cavity roller. The corresponding velocity field becomes particularly obvious from the color-coded angular-averaged velocity profiles inside the groove.

The mean velocities arise from a fitting function of higher order, which originates at (0, 0). A simple piece-wise averaging would lead to a inaccurate representation of the flow, so a more complex combination seems suitable for this flow scenario.

Figure 9 provides a comprehensive flow-field overview of all three velocity components for the selected groove-to-groove section in a normalized \(\varphi -z\) coordinate system, showing each velocity component separately as contours. For visual purposes only one periodic section of the \(360^{\circ }\)-distribution with 32 grooves is shown. The circumferential velocity component is shown in Fig. 9(a) and reveals a Couette-like flow in nearly the complete gap. This insight is valuable information for clutch manufacturers, which up-to-now was missing in the literature. The findings now substantiate existing clutch flow models with experimental data (cp. e.g. Iqbal et al. 2013b). Interestingly, the Couette-like distribution is only disturbed in vicinity of the groove. A reasonable approach for further in-detail studies might therefore focus on the region in the range of one groove width W up- and downstream of the groove (i.e. \(\varphi R_{\mathrm {m}} /W \in [-1,2]\)) for the influence of radial grooves.

An additional aspect of the flow-structure investigation is the slightly higher circumferential velocity in the groove compared to the disk velocity. This region belongs to the above-mentioned occurrence of the cavity roller, which results from the fact that the radial groove of the present investigation is oriented perpendicular to the shear-flow direction in the gap. The cavity vortex is not limited to the groove but expands into the gap. In accordance with earlier observations by Leister et al. (2021) the highest circumferential velocity can be found in the groove’s rear part and has a value of \(6 \%\) above \(\varOmega R_{\mathrm {m}}\). As a counterpart of a higher velocity at the groove’s bottom, fluid with lower velocity than \(\varOmega R_{\mathrm {m}}\) expands into the anterior part of the groove (cp. color-coded profiles in Fig. 8(b)).

The conversion of the second recorded component \(u_2\) from two consecutive measurements into radial and axial velocity components \(u_r\) and \(u_z\) via equation set (3) has been successfully applied, such that either component can be evaluated separately. The angular-resolved radial velocity component is shown in Fig. 9(b). The values are located in the range of 0 and \(0.04\; \varOmega R_{\mathrm {m}}\), with values of the smooth gap region in the lower range. The velocity at the rotor and stator tends towards zero, while the values in the middle of the gap show slightly higher velocities. This finding supports the fact that a radial outflow is present due to moderate centrifugal forces of the rotor, which for the given parameters counteracts the flow induced by gravitational forces. Additional valuable information is the higher radial outflow velocity in the groove, which is about twice as high than the smooth-gap counterpart. The highest values of around 3–4% of the disk velocity can be found in the posterior part of the groove. This is in consistency with the centrifugal forces since also the highest values of circumferential velocity are detected in this part. The findings support the fact that the radial flux is mainly dominated by the flow in vicinity of the groove and should be focused on predominantly for aeration predictions of open wet clutches.

Finally, Fig. 9(c) reveals the axial component, as likewise calculated from the equation set (2). The determined velocities largely range around zero in the smooth part of the gap. As for the groove region, however, clear patches of non-zero velocity can be identified, which are comparable in magnitude to the observed circumferential velocity deviations from \(\varOmega R_{\mathrm {m}}\) inside the groove. These maxima are indeed mandatory to fulfill continuity of the cavity vortex. The highest velocity is located on the left side of the groove, which indicates a clockwise rotation of the cavity roller in the diagram.

The potential benefit lies in the angular resolved manner of the LDV-PS, so that the influence of the respective groove patterns and correspondingly occurring flow patterns can be quantified along the entire circumference. To evaluate this footprint on the resulting drag torque, only the circumferential component is to be considered for the wall shear stress (WSS, \(\tau _{\text {w}}\)) calculations, which leads tofor the given rotor-stator shear flow. The idealized linear Couette-flow assumption in fact is an oversimplification that can be applied for very low rotational velocity components and small gap heights, as already elaborated by Daily and Nece (1960) and recently confirmed experimentally by Leister et al. (2021) for the given clutch flow scenario. Even though an ideal rotational Couette flow, consequently, is not present in this gap flow scenario—neither in the smooth part, nor the groove—the linearized profile nonetheless serves as reference, which is defined asTo account for the mildly non-linear character of the velocity profile while maintaining a robust estimate for the WSS, a linear fit of the circumferential velocity \(u_{\varphi }\) in proximity of the stator in the range \(z \in [70; 100]\) \(\upmu\)m has been chosen for the determination of \(\tau _{\text {w}}\). Since the velocity is low in this region and the scatter around the mean velocity is accordingly strong compared to higher velocities, a direct extraction of the WSS might be heavily biased by location-estimation outliers; see Fig. 7. To overcome this issue, the standard deviation, as occurs for higher velocities, has been chosen as robust filter margin around the linear Couette approximation, to eliminate these outliers from velocity fluctuations. Consequently, the data were filtered with a validation margin of \(\pm 19\) \(\upmu\)m in z-direction compared to \(\tau _{\mathrm {w,\, lin}}\), to avoid a biased result. Figure 7 shows this region visualised with green data points for an easier comprehension. The extracted data points are binned to \(\varphi R_{\text {m}}/W=0.5\) sections. Despite good overall angular resolution of \(\varDelta \varphi =0.029^{\circ }\) and preserved full resolution over the disk revolution, this binning appears rather coarse in comparison with the groove width W. Consequently, the formerly hidden complete groove-to-groove picture of WSS comes at the price of (circumferentially) low-pass filtered extraction of intra-groove structures and resulting variations of WSS in terms of accordingly diminished \({\partial \tau _{\mathrm {w}}}/{\partial \varphi }\) resolutions.

Figure 10 shows the extracted normalised and angular-resolved WSS \(\tau _{\mathrm {w}}\). In accordance with earlier findings by Leister et al. (2021) the WSS values are found to be around \(3\%\) on average below the linearized \(\tau _{\mathrm {w,\, lin}}\) in the smooth region of the gap, far enough from the groove. Assuming a constant value in this part the uncertainty of the WSS yields \(\sigma =1.29 \%\), which—due to the lack of an additional reference experiment—can only serve as an estimate. Despite the relatively high uncertainty the higher value in the region of the groove can be resolved accordingly. The values are found to be \(2 \%\) above the reference value of \(\tau _{\text {w}}/\tau _{\text {w,\,lin}}=1\), which—even though circumferentially low-pass filtered—preserves the observed trends from Leister et al. (2021). The highest value appears on the left side of the groove, which confirms the above-stated recommendation to consider the up- and downstream proximity of the groove in the range \(\varphi R_{\text {m}}/W \in [-1,2]\) to study the influence of radial grooves on the WSS and, thus, the impact on overall drag torque.