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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Wall-shear stress results from the relative motion of a fluid over a body surface as a consequence of the no-slip condition of the fluid in the vicinity of the wall. To determine the two-dimensional wall-shear stress distribution is of utter importance in theoretical and applied turbulence research. In this article, characteristics of the Micro-Pillar Shear-Stress Sensor MPS^{3}, which has been shown to offer the potential to measure the two-directional dynamic wall-shear stress distribution in turbulent flows, will be summarized. After a brief general description of the sensor concept, material characteristics, possible sensor-structure related error sources, various sensitivity and distinct sensor performance aspects will be addressed. Especially, pressure-sensitivity related aspects will be discussed. This discussion will serve as ‘design rules’ for possible new fields of applications of the sensor technology.

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The interaction of a fluid with a surface creates mechanical stresses, which can be divided into the wall-normal pressure _{wall}_{wall}

To determine the wall-shear stress is of utter importance in theoretical and applied turbulence research. The mean wall-shear stress defines the friction velocity
^{+} or ^{+} and serves as an important reference to judge the quality of turbulence models. The accurate determination of wall friction would allow the more precise identification of scaling parameters and scaling laws, e.g., for the mean velocity field or for turbulent energy spectra.

The fluctuating wall-shear stress distribution represents a footprint of near-wall turbulent structures and its measurement gives insight into the turbulent momentum transfer processes in the proximity of the wall and is as such of fundamental importance for the basic understanding of turbulent processes.

Furthermore, the measurement of the skin friction is essential in many technical applications, e.g., in the field of drag reduction and performance enhancement for transportation vehicles, where the viscous surface drag plays a major role. In flow control applications, the assessment of the local wall-shear stress or of the wall-shear stress distribution is a necessary prerequisite for the formulation of low-dimensional control models.

First preliminary results of a micro-pillar sensor application have been described in [^{3} have been discussed in [

The article is structured as follows. First, a brief general description of the sensor concept will be given in section 2.. The sensor manufacturing will be briefly addressed in section 3. before material characteristics are discussed in section 4.. Sensor-structure related errors and several sensitivity aspects and diverse sensor performances are discussed at length in sections 5., 6. and 7., respectively. This discussion will yields some kind of ‘design rules’ for possible new fields of applications of the sensor technology. Finally, a conclusion will wrap up the article.

The Micro-Pillar Shear-Stress Sensor MPS^{3} is based on thin cylindrical structures, which bend due to the exerted fluid forces, and as such the technique belongs to the indirect group [_{i}

The pillars are manufactured from the elastomer polydimethylsiloxane (PDMS, Dow Corning Sylgard 184) at diameters in the range of microns such that they are flexible and easily deflected by the fluid forces to ensure a high sensitivity of the sensor. Single pillars are shown in

As a consequence of the limited region, in which the linear relation between near-wall velocity gradient and wall-shear stress applies, the sensor length _{p}^{+} = 5÷6, where ^{+} = _{τ} is the non-dimensional wall-distance in viscous units with ν as the kinematic viscosity of the fluid and _{τ} as the friction velocity. The kinematic viscosity of water is approximately 10^{−6} ^{2}/^{−5} ^{2}/^{−2}÷10^{−1} ^{−2} _{b}^{4}÷10^{5}. In boundary layer facilities with air such as that described in [^{0} _{Θ} = 10^{3}÷10^{4} could be performed with the aforementioned pillar length. Note that the size _{p}^{+} should be considered already an upper limit to the possible pillar length. Due to the integration of the flow field along the pillar length it would be desirable to protrude as little as possible into the viscous sublayer. However, it goes without saying, that a shorter sensor structure also influences the sensor sensitivity and its static response.

The question how far the near-wall velocity field can be considered an adequate representative of the local mean and fluctuating wall-shear stress has been discussed in great detail in [

The sensor structure has a minimum dimension in the wall-parallel plane thereby reducing the spatial averaging. For the range of the above mentioned Reynolds numbers the wall-parallel dimension of the sensor, i.e., its non-dimensionalized diameter _{p}^{+}, in viscous units is _{p}^{+} ≤ 1, where _{p}^{+} = _{τ}_{p}_{p}_{p}

Besides the above mentioned aspects, which determine the maximum allowable sensor length, the pillar length also needs to be chosen reasonably small to consider the sensor structure non-intrusive for the flow field. In turbulent shear flows, it is generally accepted sufficient that if the sensor structures are fully immersed in the viscous sublayer no disturbances are caused outside the viscous sublayer, and hence, global changes of the flow field in the buffer and logarithmic region of the shear layer do not occur. Wall-shear stress statistics, turbulent spectra, and spatial two-point correlations calculated from measurements with the pillars installed in the streamwise direction allowed to confirm the low intrusiveness of the technique and no interaction of the sensor structures. To further corroborate the low intrusive interference of the sensor and to ensure a purely local effect on the flow field near the sensor structure the flow field around the pillars has been examined using _{Dp} ≤ 1. To be more precise, the flow is no longer affected in a region three pillar diameters downstream of the structure. The streaklines possess a symmetric curvature. No separation zone in the wake region of the pillar can be identified. In other words, the flow past the pillar is well in the Stokes-flow or Oseen-flow regime. Consequently, it can be assumed that the flow field is only locally disturbed in a zone of only a few pillar diameters around the structure. That is, in the case of arrays, if a sufficiently large spacing of the pillars has been chosen, no global effect of the presence of the pillars immersed in the viscous sublayer is expected. Generally, pillar spacings of approximately 1÷2 _{p}_{p}^{+} at the range of Reynolds numbers in previous experiments, i.e., compatible to assess characteristic turbulent length scales.

The sensor concept allows the two-directional detection of the fluid forces, since the symmetric geometry has no preferred sensitivity direction and furthermore, the sensor does not suffer from cross-axis sensitivity. Thus, the micro-pillar sensor enables the measurement of the two wall-parallel components of the wall-shear stress.

Most of today’s representatives of wall-shear stress sensors are so called Micro-Electro-Mechanical Systems (MEMS) that transform the mechanical reaction of the sensor to the exerted forces into an electrical signal, i.e., a voltage, by capacitive, inductive, or resistive means. Although these techniques possess a couple of advantages compared to the method described in this work, the mechano-optical principle, on which the Micro-Pillar Shear-Stress Sensor MPS^{3} is based, outperforms in various other regards. Some of the pros and cons will be briefly discussed in the following.

Most other sensor techniques reported in the literature differ from the MPS^{3} sensor design by their need for diverse secondary structures implemented on the ground. This can be either electrical supply wiring or mechanical read-out devices. It has already been mentioned that due to the optical detection principle there is no need for further structure on the wall and as such the assessment of the two-dimensional wall-shear stress distribution at high spatial resolution as small as 5 ^{+} is possible.

The use of more viscous fluids (e.g., the fluid used in the oil-channel facility described by [

Furthermore, the optical detection principle allows the simultaneous determination of all wall-shear stress components without suffering from cross-axis sensitivity, which has been experienced by many other multi-directional devices.

On the other hand, the optical detection requires optical access, which can not in all cases be guaranteed, and limits the sensor at the current state to laboratory applications. Using conventional digital cameras limits the data by the available amount of memory. At full frame size, the number of images recordable at high recording frequency (recent commercially available high-speed cameras offer on-board memory of up to 12

The raw images require a very time-consuming image evaluation before the actual wall-shear stress data is obtained, whereas MEMS devices can output the wall-shear stress or a direct representative, e.g., an electrical voltage, almost immediately. This limits the use of the pillar sensor in flow control application, which would require a real-time evaluation of the data.

Let us summarize the aforementioned sensor properties. Depending on the geometrical properties of the sensor, the detection of characteristic scales of turbulent flow is possible mostly at low to moderate Reynolds numbers. Under certain circumstances, measurements even at high Reynolds numbers can be performed [

To produce filigree structures such as micro-pillars from PDMS elastomer at aspect ratios _{p}_{p}_{p}_{p}

Depending on the aperture used during the laser perforation the hole diameter can be varied from a few ten up to several hundred microns. As mentioned above, since the focus of the laser beam is not moved during the perforation process, holes with a trombone-like shape have been perforated resulting in a ratio between the exit and entry diameter of almost 0.35 ÷ 0.4 (

The entry and exit holes have been inspected manually by microscopy evidencing a high degree of cylindricity. SEM images of final micro-pillar posts confirmed the these findings, however, no statistical evaluation has been done due to the insufficient number of sensors investigated by SEM.

In a second step, the PDMS elastomer is cast into the mold and cured. Both processes are performed under vacuum. Subsequently, the master mold is removed from the cured elastomer structure. Partly the removal by peel-off is possible, however, only at aspect ratios in the order of 10÷15, thereby allowing to keep the mold intact. At larger aspect ratios, the master mold was removed by wash-off.

To allow for a better optical detection of the micro-pillar, in a final step highly reflective hollow spheres are attached to the pillar tip. Further details of the manufacturing are described in great detail in [

The micro-pillar sensors are manufactured from Dow Corning’s two-component silicone elastomer Sylgard^{®}184, which belongs to the group of polydimethylsiloxanes (PDMS). Due to its mechanical, chemical, and optical properties, PDMS has become widely spread as material for nano- and microfluidic devices and for micro-structural mechanical sensor devices such as tactile [

PDMS possesses a specific gravity of 1050 ^{3}, a tensile strength of 6.2

Its water absorption is less than 0.1% after seven days of immersion such that mechanical properties can be expected to not be influenced by a sensor being positioned in water flow facilities. The brittle point of the material is low with −65°

Young’s modulus (_{0} of the test specimen decreases by approximately 11% at elongations of ɛ = 0.20÷0.25. This has been accounted for in the calculation of Young’s modulus. The results for different curing cycles at a constant silicone to curing-agent ratio of 10÷1 revealed Young’s modulus to vary between 0.5÷2.0

This sensitivity of Young’s modulus on the curing cycle temperature strongly affects the correct determination of the parameter and tests revealed Young’s modulus to differ by up to 30% between probes made of the same charge of silicone and under assumedly identical curing conditions. As such, it is critical to transfer Young’s modulus from a specimen to the actual pillar structure, if the curing procedure has not meticulously been identical. In consequence, this makes a static calibration of the sensor structures necessary, since Young’s modulus can not be determined from a specimen of the same material at a sufficient degree of accuracy. Furthermore, the determination of Young’s modulus directly from tests of the micro-pillar by means of static deflection to a known force or dynamic excitation, assuming it as a clamped cantilever, is not possible, because of a large uncertainty in the exact determination of the pillar’s geometry, and because of the intricacy and reliability of micro-structure mechanical tests.

To test the influence of different ambient fluids Young’s modulus of a probe has been evaluated after the pillar was immersed in water and water/glycerine mixtures for four hours up to seven days. The results indicated Young’s modulus not to be affected by this treatment, that is, Young’s modulus can be assumed constant under the impact of these fluids.

Hysteresis describes the continuation of an effect after omission of its cause, e.g., the path-dependence of the reaction of a mechanical system to an oscillating force. To test the hysteresis of PDMS, load-unload cycles at strain rates in the range of 4÷30%/

Furthermore, the loss tangent ^{−3} [

The results reported by [

The linear and volume coefficient of thermal expansion are 3.0·10^{−4} %/°^{−4} %/°

This section discusses possible error sources related to the sensor structure itself. Measurement errors and the achievable accuracy of the optical acquisition principle have in detail been discussed in [

To discuss the different aspects of possible sensor misalignments, first the procedure of the sensor positioning in turbulent flow fields will be shortly elucidated. The actual micro-pillar sensor posts and the surface, on which they are mounted, are manufactured in one single step. This sensor ‘chip’ can be either directly flush-mounted in suitable grooves of a flow facility wall or in wall adapters, which can be further placed in the wall of the flow facility wall.

In the studies reported in [_{p}

The positioning of the sensor mount can be performed using micro-manipulating devices and visual inspection at microscopic magnification allows the detection of maximum vertical offsets of the sensor mount of less than 5 ^{+} at typical Reynolds numbers in the experiments performed up to now such that no global flow field disturbance is expected and furthermore, the flow on the sensor ‘chip’ can be considered to be non-affected by the existence of vertical offsets of this size. As such, errors due to a misalignment of the structure can be considered negligible.

However, if measurements at higher Reynolds numbers, i.e., smaller absolute geometric dimensions of ^{+} are performed, even higher accuracy needs to be achieved. This could for example be managed by directly manufacturing the complete sensor on the flow facility wall without having the need to manually position sensor ‘chips’ on the wall or in wall adapters.

It goes without saying that any change in structure-mechanically relevant sensor parameters and particularly Young’s modulus will modify the sensors sensitivity. The long-term (30 days-180 days) and short-term (30 min-7 days) repeatability tests at fluctuating and constant load evidenced an excellent agreement of the mean detected pillar deflections within ±2÷3% in all flow media used in the present studies, hence glycerine, water and air, i.e., no material aging was observed. Note, the sensor was stored in air between the tests. However, sensor calibrations have always been performed prior to measurements to ensure that any kind of sensor degradation or aging is accounted for.

Furthermore, changes of the sensor material due to long-term exposure to different environmental influences could lead to errors. Effects due to water absorption are negligible following the manufacturer’s information [

It is well known that elastic materials tend to yield under constant stress causing the sensitivity of the sensor to change with time. It can be expected that altering mechanical properties and yielding would influence the mean detected wall-shear stress. To check for a possible drift of the sensor due to mechanical yielding, long-term (up to 30 _{p}_{p}_{p}

It would be desirable to calibrate the micro-pillar structure in-situ, that is, to position the sensor for static calibration directly in the turbulent flow field. This would allow to avoid measurement errors arising from variances in the flow conditions during the rheometer calibration and the actual flow case, e.g., different temperatures, strongly differing Reynolds numbers _{Dp} determining the local flow field around the sensor structure, or from a possible sensor misalignments.

To obtain the relation between pillar deflection and wall-shear stress in the flow facility, it is necessary to quantitatively know the mean wall-shear stress at a high enough accuracy in the turbulent flow field, in which the calibration is performed. This requires on the one hand, the simultaneous application of an already calibrated device to assess the wall-shear stress or on the other hand, the wall-shear stress to be determined from an analytical relation assuming it to be valid to within the desired accuracy.

A problem of the in-situ calibration is the non-linearity of the static sensor response at low or high deflections. The velocity field in turbulent shear flows in the vicinity of the wall and hence, the wall-shear stress, are known to fluctuate strongly around their mean values. Simply assuming the arithmetic mean of the measured sensor deflections to be a linear-proportional representative of the mean wall-shear stress would yield an error especially at low deflections due to the non-linearity of the static response. Hence, an approach similar to that used for calibrations of hot-films or hot-wires - in case they are calibrated in highly fluctuating flow fields - is necessary. In [

Commonly, sensors are not only sensitive to one form of excitation. That is, a sensor response, e.g., in the case of the micro-pillar its deflection, not solely originates from the wall-shear stress, i.e., from the drag forces of the local small-scale flow field around the structure, but it is rather the consequence of several contributing effects. To judge the possibility of such a multi-sensitivity, the influence of secondary contributions will be discussed in the following.

Two different kinds of sensitivity-related aspects are possible. On the one hand, the direct impact of forces other than the drag force resulting from the fluid field surrounding the sensor structure, which we relate to the wall-shear stress, needs to be accounted for. As external forces, accelerations (e.g. due to accelerated flow facilities or test structures, in/on which the sensor is installed) and pressure forces need to be addressed. On the other hand, changes in the sensor sensitivity itself might have an deteriorating effect on the sensor function. Only temperature-related effects will be discussed in this section, since chemical and load-related changes of the sensor material and its mechanical properties have already been discussed in the preceding section and showed to have negligible effects on the sensor sensitivity.

Most recent floating-element wall-shear stress sensors suffer from a certain degree of sensitivity to pressure forces. On the one hand, slightly differing pressure forces in pressure-driven flows act on the trailing and leading edges of floating-element sensors, and on the other hand, a pressure gradient between the sensor surface and the gap between the sensor and the substrate might be present, resulting in a wall-normal force, which contributes to the total load acting on the tethering springs. Similarly, it is possible that pressure forces act on the pillar structure (

If the Reynolds number _{Dp} reaches a certain level, Stokes or Oseen flow around the structure can no longer be assumed and a detachment of the flow field at the lee-site of the pillar (_{Dp} ≤ 1, the Stokes condition can be assumed valid such that the flow field should symmetrically follow the pillar contour allowing to determine the total drag forces exerted by the local flow field around the sensor structure by analytical estimates given e.g. in [

In pressure-gradient driven flows, the mean pressure gradient Δ

It has been shown in [

The assumption of Oseen flow around the sensor structure is valid if the Reynolds number _{Dp}= _{p}_{p}_{p}_{p}_{p}_{Dp} ≤ 1.

The pressure forces per unit length _{P}_{p}_{p}^{−1} _{Dp} is of order 10^{−1}, and thereby the shear load per unit length _{P}^{−4}÷10^{−3} ^{−3} _{p}^{−5}÷10^{−4} _{p}^{−4}÷10^{−3} ^{2}, the resulting pressure force per unit length ^{−7} _{P}_{P}^{−4}÷10^{−3}, and hence, can be considered negligible. _{P}_{P}^{−2}.

In the following the influence of mean wall-normal pressure gradients ∂_{2} is parameter defining the curvature of the wall-normal velocity profile, which will be discussed in further detail later in this section. The second part in _{p}_{p}_{P}_{P}

At the present experimental conditions _{P}_{P}_{p}_{p}

In turbulent flows, fluctuating pressure forces resulting from turbulent fluid motion, will exert on the sensor structure. Such pressure fluctuations ^{′} can, for example, arise from acoustic pressure waves in gases or from turbulent momentum transfer, i.e., result from velocity fluctuations. Second, turbulent fluctuations cause local pressure fluctuations, which could exert on the sensor structure. Both possible contributions will be discussed in the following.

Similar to the considerations in [^{′} smaller than the pillar dimension will effectively impose pressure forces on the structure. It is reasonable to assume that the pillar diameter _{p}_{a}_{a}

At the Reynolds numbers, at which the sensor is applied, the pillar diameter _{p}^{+} and as such, pressure fluctuations induced by turbulent structures can be expected to not significantly contribute to the wall-parallel pillar load. However, to study the impact of turbulence induced pressure fluctuations ^{′} ≡ ∂_{i}

First, the magnitude of pressure fluctuations need to be assessed. From the Navier-Stokes equations the rms of local pressure gradients can be expressed by (see e.g. [

The factor of 2 in this equation is representative for the level of fluctuations at ^{+} = 10 and rather overestimates the pressure fluctuations in the viscous sublayer. The second part of _{1} being essentially 0. From

From DNS results of turbulent channel flow at Reynolds numbers similar to those in the present experiments [_{2} in _{2} ≈ 0.06. Note, this value represents the curvature of the velocity profile through the entire viscous sublayer, or, to be more precise its rms value (around zero). It has already been discussed that the mean velocity profile can be assumed linear, but instantaneous velocity profiles apparently possess a slight curvature, hence a value of α_{2} different from zero. The value of β_{2} is approximately 0.04 (rms around zero). _{2} and β_{2} as a function of τ^{′} /τ̄, respectively. While α_{2} appears to slightly depend on the value of τ^{′} /τ̄,it is evident that the magnitude of β_{2} does not increase. This suggests that the curvature in the streamwise velocity profile becomes a problem before the wall-normal pressure gradient does.

Let us first discuss the effect of streamwise pressure gradients ∂^{′}(_{P}_{P}_{P}_{P}_{P}_{P}

It is evident from _{τ}, i.e., with Reynolds number. Generally valid limits of the Reynolds number, at which the pressure contribution remains negligible, can hardly be given here due to the interdependence of sensor and flow characteristics, but the equations allow to estimate the influence of pressure forces on the total sensor load. At the experimental conditions in [_{P}_{P}

Note again, that the way, in which wall-normal pressure forces contribute is different from that of wall-parallel fluctuations. The ratio between load contributions resulting from the mean wall-normal pressure gradient and the effect of turbulent wall-normal pressure fluctuations is approximately β_{2}^{2}^{+3} : β_{2} and hence 10 : 1. That is, wall-normal pressure fluctuations contribute an order of magnitude less to the total pillar bending.

Let us conclude the above findings. The pressure contribution to the pillar load resulting from wall-parallel and wall-normal pressure gradients have been discussed. These pressure gradients can represent mean pressure gradients in the flow or they might be a consequence/cause of turbulent motion. At the present experimental configurations pressure forces can reliably be neglected. However, with increase in Reynolds number, pressure contributions might represent a non-negligible contribution to the pillar bending. Formula to estimate the influence of pressure gradients have been extensively discussed in this section.

It is well known that a sphere [_{p}^{+} < 1, lateral velocity gradients across the sensor diameter, which would cause lift-induced deflections of the sensor structure, can be considered negligible. Note, at scales smaller than the Kolmogorov length scale or viscous length scale, _{k}^{+}, respectively, the fluid motion can be considered uniform, and lateral velocity gradients, and hence, lift-force inducing shear-flow conditions, will only arise at larger dimension.

Due to its own mass the sensor is sensitive to inertial effects arising from exterior accelerations. Similar to the considerations in the previous section, one possible approach to estimate the sensor sensitivity to acceleration is to relate a deflection caused by inertial forces to a corresponding wall-shear stress, which would have caused the same deflection.

The force _{inert}_{inert}_{p}_{p}_{p}_{p}_{Lp} along the sensor geometry, which is good enough as a first rough estimate, and further applying the relation τ_{wall}_{Lp}/_{p}

For Reynolds numbers _{Dp} of 10^{−3}÷10^{0}

As such, the equivalent wall-shear stress to a one-g acceleration of the sensor can be expressed by

At typical pillar dimensions _{p}^{−3} _{p}^{−5}÷10^{−4} _{p}^{3} ^{3} the equivalent shear stress becomes approximately 0.01

The sensor material is known to react with an increased shear modulus

Eperiments of micro-pillars in laminar shear flow in a plate-cone rheometer and experiments in turbulent boundary layer flow showed temperature-related problems. In both studies a 100

Exemplary results evidencing the temperature sensitivity of the sensor are given in 8. _{p}

Furthermore, a strong influence of temperature on the elastic behavior of the pillars was observed in turbulent boundary layer air flow measurements (_{∞} < 9 _{∞} reduces the problem. At 10

It goes without saying that in water due to the strongly increased thermal convection, the problem of structural heating is less dominant and could indeed not be observed. To completely eliminate thermal effects, it is advisable to use cold-light illumination systems.

In conclusion, it needs to be stated that the sensor evidences to be sensitive to temperature. Therefore, special care has to be taken to minimize systematic errors resulting from thermal effects. To be more precise, the temperature during measurements needs to be kept constant. This, however, is a typical requirement in fluid flow experiments to ensure constant measurement conditions, e.g., a constant Reynolds number. Furthermore, the temperature difference between static and dynamic calibration and measurements should be kept identical.

Assuming a temperature sensitivity (Δ

Thermal expansion effects are, as long as the temperature is kept constant to within ±1°

In the following the sensor performance and design rules for an optimum sensor layout will be discussed. It has become evident that similar to almost all fluid measurement techniques diverse restrictions need to be accomplished for an optimum wall-shear stress measurement with micro-pillars. Since many of these aspects are closely related to the geometry of the flow facilities, e.g., bulk scales, fluid viscosities, etc., no general Reynolds number range can be given here, at which the sensor can be used. Up to now, sensor applications at moderate Reynolds numbers have been successfully performed [

The sensitivity describes the minimum magnitude of an input signal required to produce a specified output signal. The dynamic range describes the ratio between the smallest and largest possible detectable wall-shear stress. From _{wall}_{wall}

In many applications the value of the wall-shear stress is very low and to increase the sensor’s sensitivity most flow cantilevers and floating-element based sensors cope with the issue of ^{+} has been generally accepted sufficient in the literature [

To explicitly specify the dynamic range of the micro-pillar concept is not an easy task since many aspects such as the sensor sensitivity, the optical resolution, and the quality of the recorded images contribute to the effective dynamic range. Under optimum conditions, the sensor concept has been shown to detect a range of 10^{2}÷10^{3} of magnitude of wall-shear stress at a signal-to-noise (SNR) of approximately

This implies that the sensor and the optical setup need to be specified in compliance with the shear-stresses present in the flow field. Generally, at each configuration, the maximum detectable shear stress is limited by the endurable mechanical load of the structures. On the other hand, a lower limit is given by the noise of the chosen optical resolution and the image detection and evaluation processes.

Under optimum mechanical conditions, i.e., the sensor bandwidth is not capped by a mechanical overload of the structure, the dynamic range is mostly limited by the image-evaluation routines and is as such comparable to the bandwidth of standard Particle-Image Velocimetry (PIV) [^{3} and more with a remaining SNR of 10 even at the smallest fluctuations.

From _{p}_{p}_{p}

Characteristic dimensions and characteristics of sensors used in recent studies are _{p}_{p}^{6} ^{2}. With values of the wall-shear stress in the range of 10÷1000 ^{2} at reasonable pillar deflections in the order of 10^{0}÷10^{1}

Note again, that _{p}_{p}

To detect the complete frequency spectrum of the fluctuating wall-shear stress, a high enough dynamic bandwidth of the sensor structure is required. Depending on the flow characteristics, it can be necessary that the sensor possesses a bandwidth that allows to detect frequencies of a few _{k}

It has been shown in [_{0} of the structure to be a sufficient parameter to determine the frequency range, at which the sensor possesses a reasonably constant gain. To be more precise, the gain up to approximately 0.3 _{0} was nearly constant.

The transfer function of the structure in water resembles a low-pass filter, i.e., the gain drops at frequencies higher than the damped eigenfrequency _{D}

The quantity λ_{1} = 1.875 is the first eigenvalue for a clamped beam. Note,

That is, again, an exact determination of the sensor properties is impeded by the remaining uncertainty in the determination of characteristic geometric and mechanic parameters of the pillar sensor as has been discussed in [

It has already been mentioned that the dynamic response function of the wall-shear stress sensor needs to be chosen in compliance with the Reynolds number and the highest expected frequencies of the investigated flow field. As such, it is necessary to make a rough estimate of the frequency spectrum of turbulent fluctuations existent in the flow at the chosen Reynolds number. The highest characteristic frequencies are related to the smallest-scale structures in turbulent flows. These smallest scales are defined by the Kolmogorov length scale _{k}_{k}_{t}_{t}_{t}_{t}_{∞} is a characteristics bulk-scale velocity, e.g., the freestream velocity in boundary layers or the bulk velocity in pipe or channel flow. The ratio of the convective time scale (_{∞}/^{−1} and the Kolmogorov time scale _{k}

At Reynolds numbers of _{b}_{k}_{k}

Measurements in a turbulent boundary layer flow in air at Reynolds numbers _{Θ} = 7800÷21000 reveal the general applicability of the sensor technique to such flows [

Cross-axis sensitivity describes the mechanical coupling of perpendicular axis sensitivities. A one-directional sensor device can be sensitive to forces exerted along the axis perpendicular to the axis along which the force is applied. To minimize this kind of cross-axis sensitivity the stiffness of the sensor structure along the perpendicular direction can be chosen much higher than that of the primary axis causing parasitic off-axis contributions to be negligible.

A second kind of cross-axis sensitivity may arise in the case of multi-directional devices where mechanical receptors, e.g., strain gages or piezo-resistive devices, detecting the deflection of sensor elements are sensitive to deflections along perpendicular directions making an identification of the originating force direction difficult or impossible.

Due to its symmetric shape the pillar sensor possesses an identical stiffness along the two perpendicular in-plane directions and is as such a multi-directional sensor at constant sensitivity along all radial directions. Consequently the pillar deflection can be considered a direct representative of the exerted forces, in magnitude and angular orientation. Furthermore, the optical detection principle allows a distinct identification of the two perpendicular wall-shear stress components. That is, a cross-axis sensitivity of the sensor of the second type is not expected.

In consequence of these theoretical considerations, the cross-axis sensitivity has not extensively been studied. However, tests of pillar deflections under varying angular orientations of the sensor in a magnetic field performed in the context of the dynamic calibration described in [

According to the ‘Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results’ the repeatability of measurement results is defined as the closeness of the agreement between the results of successive measurements of the same measurand carried out under the same conditions of measurement. Repeatability can be expressed as

In this article the Micro-Pillar Shear-Stress Sensor MPS^{3}, which offers the potential to measure the two-directional dynamic wall-shear stress distribution in turbulent flows, has been discussed in detail.

The sensor is based on flexible micro-pillars protruding into the near-wall region of turbulent flows and bending in reaction to the exerted drag forces. The deflection of the pillars is detected by optical means and is a representative of the local wall-shear stress. It needs no additional infrastructure on the wall thereby reducing additional flow disturbance such that the pillar technique allows extremely high spatial resolutions of 10^{0} ÷ 10^{1} viscous units and the measurement of the wall-shear stress distribution with up to 1000 sensor posts. It possesses the advantage of very low flow interference. Depending on the geometry and material characteristics of the sensor turbulent scales down to less than 50 _{0} = 400÷2000 _{0}. Sensors with even higher eigenfrequencies of up to 5000

The present article has discussed in detail material characteristics, possible sensor-structure related errors, various sensitivity and distinct sensor performance aspects. Some guideline to apply micro-pillar sensors to new fields of application has also been given.

The development of the Micro-Pillar Shear-Stress Sensor MPS^{3} can not be considered finished and further improvements will be an exciting challenge for future work. The manufacturing of pillar arrays demands for a better automated positioning of reflective hollow spheres on top of the sensor posts. The implementation of sub-pixel window shifting and adaptive cross-correlation routines will allow for higher achievable accuracy in the detection of the pillar-tip deflection in the order of 0.01

_{p}

Pillar diameter

Young’s modulus

Frequency

_{O}

Undamped eigenfrequency

_{D}

Damped eigenfrequency

_{k}

Kolmogorov frequency

Shear modulus

Dynamic fluid viscosity

_{k}

Kolmogorov length scale

_{t}

Integral length scale

^{+}

Viscous length scale (^{+} = _{τ}/ν)

_{p}

Pillar length

Kinematic fluid viscosity

Pressure fluctuations

Fluid pressure

_{P}

Pressure load per unit length [

_{P}

Shear load per unit length [

_{Dp}

Reynolds number based on a local velocity (e.g., _{Lp}) and the pillar diameter

_{b}

Reynolds number based on the bulk velocity _{b}

_{t}

Large-scale turbulent Reynolds number

_{Θ}

Reynolds number based on _{∞} and momentum-loss thickness Θ

Fluid density

_{p}

Pillar material density

Mean wall-shear stress

_{wall}

Wall-shear stress

_{i}

Wall-shear stress along the direction

_{i}

Wall-shear stress fluctuations along the direction _{i}_{i}_{i}

Velocity fluctuations along the streamwise, wall-normal and spanwise direction

_{τ}

Friction velocity

Mean local streamwise velocity

_{b}

Bulk velocity e.g. in channel/duct/pipe flow

_{Lp}

Velocity at the pillar tip (≡ velocity at the edge of the viscous sublayer)

_{∞}

Freestream velocity in boundary layer flow

Lateral pillar deflection

_{Lp}

Lateral pillar-tip deflection

Wall-normal variable/along pillar length

^{+}

Non-dimensional (viscous) distance from the wall (^{+} = _{τ}

The financial support of this project by the DFG Focus Research Program ‘Nano- and Microfluidic’SPP 1164 is greatfully acknowledged. We also thank Dr. rer. nat. Georg J. Schmitz from the ACCESS e.V. (RWTH Aachen University) and Dipl.-Chem. Philipp Jacobs from the Fraunhofer Institute for Laser Technology (ILT) for providing the SEM and microscopic images used in this work. We would also like to thank Prof. Jonathan F. Morrison from Imperial College London for the valuable discussion on pressure sensitivity related aspects, which has helped to greatly improve the quality of this paper.

^{3}

^{3}

^{3}

^{3}

^{3}for Turbulent Flows

(

(

View of perforations into a wax foil at the entry and exit side of the laser beam. The entry diameter of the perforation is

(

Two-dimensional flow field around a circular obstacle at different Reynolds number _{Dp} based on the diameter _{p}_{Dp} ≥4 the flow detaches.

(_{2} of the non-dimensionalized streamwise velocity profile ^{+}(^{+}). (_{2} of the non-dimensionalized profile ^{+}(^{+}) of wall-normal velocities.

(

(_{∞} = 4, 6, 8 and 10