average of Equation 6.50 over a large number of cycles and assuming that the turbulent fluctuations are small, a linearized expression for the periodic wall-shear stress is obtained. To summarize, the classical expression, Equation 6.43, gives a relation between the wall-shear stress and the heat transfer from the wall. This expression assumes steady, laminar, zero-pressure-gradient flow, and is not valid in a turbulent environment. Menendez and Ramaprian (1985) have derived an extended version of Equation 6.43 valid for a periodically fluctuating freestream velocity, Equation 6.50. However, the lat- ter relation contains some assumptions that are questionable for turbulent flows. For instance, in the thermal boundary layer it is assumed that the temperature distribution is self-similar and that the local thickness varies linearly. These assumptions are relevant for a streamwise velocity oscillation and a weak fluctuation, but certainly not in a turbulent flow, which is strongly unstable in all directions. 6.4.3.3 Calibration Several methods and formulas are in use for calibrating hot-film shear probes operated in the constant- temperature mode and the choice of method depends on flow conditions and sensor used. In this subsub- section, two static calibration methods are discussed. Both are based on the theoretical analysis leading to the relation between rate of heat transfer and wall-shear stress, as discussed in the last subsubsection. The challenge is of course to be able to use the shear-stress sensor in a turbulent environment. If a laminar flow facility is used to calibrate the wall-shear-stress sensor, then Equation 6.43 can be re- written more conveniently in time-averaged form: τ ෆ w ෆ 1/3 ϭ Ae 2 – ϩ B (6.54) where τ ෆ w ෆ is the desired mean wall-shear stress, e 2 – is the square of the mean output voltage, and A and B are calibration constants. The term B represents the heat loss to the substrate in a quiescent surrounding, and this procedure is similar to a conventional calibration of a hot-wire [King, 1916]. However, a laminar flow is often difficult to realize in the desired range of turbulence wall-shear stress: it is more practical to calibrate without moving the sensor between calibration site and measurement site. In that case, the calibration is made in a high-turbulence environment and the high-order moments of the voltage must also be considered. Ramaprian and Tu (1983) proposed an improved calibration method, and the instantaneous version of Equation 6.54, can be re-written and time averaged to give: τ ෆ w ෆ ϭ —— (Ae 2 —— ϩB) 3 ϭ A 3 e 6 – ϩ 3A 2 Be 4 – ϩ 3AB 2 e 2 – ϩ B 3 (6.55) This can be rewritten as: τ ෆ w ෆ ϭ C 6 e 6 – ϩ C 4 e 4 – ϩ C 2 e 2 – ϩ C 0 (6.56) where the C’s are the new calibration constants. The high-order moments of the voltage can easily be calcu- lated with a computer, but care must be taken that they are fully converged. The relation between e and τ w can also be represented by a full polynomial function, so if the frequency response of the sensor is suffi- ciently flat, it is possible to write: τ ෆ w ෆ ϭ C 0 ϩ C 1 e ϩ C 2 e 2 – ϩ … ϩ C M e N — (6.57) where M is the number of calibration points and N is the order of the polynomial above. A system of linear equations is obtained where the calibration coefficients can be computed by a numerical least-square method. For example, the mean wall-shear stress on the left-hand side of Equation 6.57 can be measured with a Preston tube using the method of Patel (1965). The second calibration technique described here is called “stochastic” calibration by Breuer (1995), who also demonstrated that the validity of the calibration poly- nomial may extend well beyond the original calibration range, although this requires careful determination of the higher-order statistics as well as a thorough understanding of the sensor response function. The experiments of Bremhorst and Gilmore (1976) showed that the static and dynamic calibration coef- ficients for hot-wires agree to within a standard error of 3% for the velocity range of 3–32 m/s. Thus, they Sensors and Actuators for Turbulent Flows 6-31 © 2006 by Taylor & Francis Group, LLC recommend the continued use of static calibration for dynamic measurements. As pointed out in previous section, this is not true for a wall-mounted hot-film. One solution to this problem may be to calibrate the hot-film in pulsatile laminar flow with a periodic freestream velocity U a (t), and make use of the Menendez and Ramaprian’s formula, Equation 6.50: τ ෆ w ෆ ϭ (Ae 2 – ϩ B) 3 ϩ ϩ c 2 A to obtain the additional calibration constants c 1 and c 2 . This step is necessary in order to characterize the gauge dynamic response at relatively high frequencies. The constants A and B are obtained from a steady- state calibration. A difficulty in performing a dynamic calibration is to generate a known sinusoidal wall-shear stress input. Bellhouse and Rasmussen (1968) and Bellhouse and Schultz (1966) achieved this in two different ways. One method is to mount the hot-film on a plate which can be oscillated at various known frequencies and amplitudes. The main drawback to this arrangement is the limited amplitudes and frequencies that can be achieved when attempting to vibrate a relatively heavy structure. An alternative strategy is to gen- erate the shear stress variations by superimposing a monochromatic sound field of different frequencies on a steady, laminar flow field. A hot-wire close to the wall can be used as a reference. 6.4.3.4 Spatial Resolution Equation 6.50 is an extended version of Equation 6.43, and by assuming a steady flow and zero-pressure- gradient in the streamwise direction it reduces to the classical formula. The latter expression can be used to estimate the effective streamwise length of a hot-film: τ w ϭ ϪA 1 q 3 W (6.58) By assuming a stepwise temperature variation and introducing the average heat flux q ෆ W ෆ over the heated area which is assumed to have a streamwise length L, the desired relation reads: q ෆ W ෆ ϭ 0.807 κ f ∆T w Pr 1/3 ΂ ΃ (6.59) It is convenient to re-write the above expression into a non-dimensional form, and in doing so we keep L explicit: Nu — ϵ ϭ 0.807(PrL ϩ2 ) 1/3 (6.60) where Nu — is the Nusselt number averaged over the heated area, L ϩ is the streamwise length of the heated area normalized with the viscous length-scale v/u τ , and h c is the convective heat transfer coefficient. This equa- tion has been derived for flows with pressure gradient by Brown (1967). The dimensionless sensor length L ϩ is a crucial parameter when examining the assumptions made. The lower limit on L ϩ is imposed by the boundary-layer approximation since there is an abrupt change of temperature close to the leading and trailing edge zones of the heated strip where the neglected diffusive terms in Equation 6.45 become significant. Tardu et al. (1991) have conducted a numerical simulation of the heat transfer from a hot-film, and found a peak of the local heat transfer at the leading and trailing edges. They conclude that if the hot-film is too narrow, the heat transfer would be completely dominated by these edge effects. Ling (1963) has studied the same problem in a numerical investigation, and concludes that the diffusion in the streamwise direction can be neglected if the Péclet number is larger than 5000. (The Péclet number here is defined as the ratio of heat transported by convection and by molecular diffusion, Pe ϭ PrL ϩ2 , [Brodkey, 1967]). Pedley (1972) concludes that, provided 0.5 Pe Ϫ0.5 Ͻ x/L Ͻ (1 Ϫ 0.7Pe Ϫ0.5 ), there exists a central part of the hot-film where the boundary-layer solution predicts the heat transfer within 5%. Figure 6.20 shows the relation proposed by Pedley (1972); it can be seen that the heat transfer is cor- rectly described over a large part of the hot-film area, but as the Péclet number decreases, the influence h c L ᎏ κ f τ w ᎏ Lv µ de 2 – ᎏ dt dU a ᎏ dt c 1 ᎏ Ae 2 – ϩ B 6-32 MEMS: Applications © 2006 by Taylor & Francis Group, LLC from the diffusive terms must be considered. For Péclet numbers larger than 40, the heat transfer is correctly described by more than 80% of the heated area. The upper limit of L ϩ is crucial since there the hot-film thermal boundary layer may not be entirely sub- merged in the viscous sublayer. Equation 6.58 has been included in Figure 6.20, and it can be seen that Péclet numbers larger than 40 correspond approximately to τ ෆ w ෆ 1/3 Ͼ 0.95. A relatively simple calculation of the upper limit of L ϩ can then be made by assuming that the viscous sublayer is about five viscous units, which yields an upper limit of L ϩ which for air can be estimated to be approximately 47. It can be concluded that the streamwise extent of the hot-film cannot be too small, otherwise the boundary-layer approximations made are not applicable. On the other hand, a sensor that is too large will cause the thermal boundary layer to grow beyond the viscous region. Additionally, the spatial resolution will be adversely affected if the sensor is too large, since the smallest eddies imposed by the flow struc- tures above the wall will then be integrated along the sensor length. 6.4.3.5 Temporal Resolution The temporal resolution of the thermal probe is affected by the different time-constants of the hot-film and the substrate. The hot-film usually has a much shorter time-constant than the substrate. The higher the per- centage of the total heat that leaks into the substrate, the lower is the sensitivity of the device to shear-stress fluctuations; this changes the sensor characteristics sufficiently to invalidate the static calibration. An exam- ple of this phenomenon is given by Haritonidis (1989), who showed that a hot-film sensor in a fluctuating wall-shear stress environment will respond quickly to the instantaneous shear stress, while the substrate will react slowly due to its much larger thermal inertia. Haritonidis (1989) also showed that the ratio of the fluctuating sensitivity, S f , to the average sensitivity, S a , can be related to the ratio of the effective lengths under dynamic and static conditions: ϭ ΂ ΃ 2/3 Ͻ 1 (6.61) where L a is the average effective length during static calibration and L f is the effective length during dynamic calibration. Due to this, the hot-film becomes less sensitive to shear-stress fluctuations at higher frequencies and the static calibration in a laminar flow will not give a correct result. These length-scales can be considerably larger than the probe true extent. For example, Brown (1967) reported that the effec- tive length-scale from a static calibration was about twice the physical length. L f ᎏ L a S f ᎏ S a Sensors and Actuators for Turbulent Flows 6-33 0 0.2 0.4 0.6 0.8 1 0.6 0.8 1 1.2 1.4 20 40 60 80 100 x/L Pe 1-0.7Pe −0.5 ␶ w −1/3 ␶ w −1/3 0.5Pe −0.5 FIGURE 6.20 Pedley’s (1972) relation showing the region of validity of the boundary-layer solution as a function of Péclet number. In the area in between the two continuous curves, the boundary-layer approximation predicts the heat transfer accurately to within 5% of the correct value. © 2006 by Taylor & Francis Group, LLC Both the substrate material and the amount of heat that is lost to the substrate are crucial when deter- mining the temporal or frequency response of the hot-film sensor. At low frequencies, the thermal waves through the substrate and into the fluid are quasi-static, which means that the fluctuating sensitivity of the hot-film is determined by the first derivative of the static calibration curve. In this range, on the order of a couple of cycles per second, the heat transfer through the substrate responds without time lag to wall-shear stress fluctuations. Basically, this frequency range does not cause any major problems. For the high-frequency range at the other end of the spectrum, it is possible to estimate the substrate role by considering the propagation of heat waves through a semi-infinite solid slab subjected to periodic temperature fluctuations at one end. This has been reported by Blackwelder (1981), who compared the wave- length of the heat wave to the hot-film length. He showed that the amplitude of the thermal wave would attenuate to a fraction of a percent over a distance equal to its wavelength. A relevant quantity to consider in this context turned out to be the ratio of the wavelength to the length of the substrate since this is an indication of the extent to which the substrate will partly absorb heat from the heated surface and partly return it to the flow. Haritonidis (1989) computed this ratio for a number of fluids and films and con- cluded that at high frequencies the substrate would not participate in the heat transfer process. However, this conclusion should be viewed with some caution because the frequencies studied were the highest that could be expected in a wall flow. The most difficult problem occurs for frequencies in the intermediate range, resulting in a clear sub- strate influence and an associated deviation from the static calibration. Hanratty and Campbell (1996) showed that damping by the thermal boundary layer for pipe flow turbulence is important when: [L ϩ2 (5.65 ϫ 10 Ϫ2 ) 3 Pr] 1/2 р 1 (6.62) For Pr ϭ 0.72, this requires the dimensionless length in the streamwise direction, L ϩ , to be less than 90. For turbulence applications, this is most disturbing since it is in this frequency range where the most ener- getic eddies are situated. The primary conclusion from this discussion is that all wall-shear stress mea- surements in turbulent flows require dynamical calibration of the hot-film sensor. 6.4.3.6 MEMS Thermal Sensors Kälvesten (1996) and Kälvesten et al. (1996b) have developed a MEMS-based, flush-mounted wall-shear stress sensor that relies on the same principle of operation as the micro-velocity sensor presented by Löfdahl et al. (1992). The shear sensor is based on the cooling of a thermally insulated, electrically heated part of a chip. As depicted in Figure 6.21, the heated portion of the chip is relatively small, 300 ϫ 60 ϫ 30 µm 3 , and is thermally insulated by polyimide-filled, KOH-etched trenches. The rectangular top area, with a side- length to side-length ratio of 5:1, yields a directional sensitivity for the measurements of the two perpen- dicular in-plane components of the fluctuating wall-shear stress. Due to the etch properties of KOH, the 30 µm-deep, thermally-insulating trenches have sloped walls with a bottom and top width of about 30 µm. The sensitive part of the chip is electrically heated by a polysilicon piezoresistor and its temperature is measured by an integrated diode. For the ambient temperature, a reference diode is integrated on the sub- strate chip, far away from the heated portion of the chip. (Note that for backup, two hot diodes and two cold diodes are fabricated on the same chip.) Kälvesten (1996) performed a static wall-shear stress calibration in the boundary layer of a flat plate. A Pitot tube and a Clauser plot were used to determine the time-averaged wall-shear stress. The power con- sumed to maintain the hot part of the sensor at a constant temperature was measured and Figure 6.22 shows the data for two different probe orientations. For a step-wise increase of electrical power, the response time was about 6 ms, which is double the calculated value. This response was considerably shortened to 25 µs when the sensor was operated in a constant-temperature mode using feedback electronics. Table 6.2 lists some calculated and measured characteristics of the Kälvesten MEMS-based wall-shear stress sensor. Jiang et al. (1994; 1996) have developed an array of wall-shear stress sensors based on the thermal princi- ple.The primary objective of their experiment was to map and control the low-speed streaks in the wall region of a turbulent channel flow. To properly capture the streaks,each sensor was made smaller than a typical streak width. For a Reynolds number based on the channel half-width and centerline velocity of 10 4 , the streaks are estimated to be about 1 mm in width, so each sensor was designed to have a length less than 300µm. 6-34 MEMS: Applications © 2006 by Taylor & Francis Group, LLC Figure 6.23 shows a schematic of one of Jiang et al.’s sensors. It consists of a diaphragm with a thickness of 1.2 µm and a side-length of 200 µm. The polysilicon resistor wire is located on the diaphragm and is 3 µm wide and 150 µm long. Below the diaphragm there is a 2 µm-deep vacuum cavity so that the device will have a minimal heat conduction loss to the substrate. When the wire is heated electrically, heat is trans- ferred to the flow by heat convection resulting in an electrically measurable power change which is a function of the wall-shear stress. Figure 6.24 shows a photograph of a portion of the 2.85 ϫ 1.00 cm 2 streak-imaging chip containing just one probe. The sensors were calibrated in a fully-developed channel flow with known average wall-shear stress values. Figure 6.25 depicts the calibration results for 10 sensors in a row. The out- put of these sensors is sensitive to the fluid temperature, and the measured data must be compensated for this effect. Measurements of the fluctuating wall-shear stress using the sensor of Jiang et al. (1997) have been reported by Österlund (1999) and Lindgren et al. (2000). 6.4.3.7 Floating-Element Sensors The floating-element technique is a direct method for sensing skin friction, which means a direct mea- surement of the tangential force exerted by the fluid on a specific portion of the wall. The advantage of this method is that the wall-shear stress is determined without having to make any assumptions about either the flow field above the device or the transfer function between the wall-shear stress and the measured quantity. The sensing wall-element is connected to a balance which determines the magnitude of the applied force. Basically two arrangements are distinguished to accomplish this: displacement balance, which is Sensors and Actuators for Turbulent Flows 6-35 Temperature- sensitive diode Electrically heated chip Aluminum Buried silicon dioxide Polyimide-filled, KOH-etched trench Flat plate Temperature reference diode Heating resistor Temperature- sensitive diode Aluminum (a) (b) FIGURE 6.21 Flush-mounted wall-shear stress sensor. (a) Top-view; (b) schematic cross-section. (Reprinted with permission from Kälvesten, E. [1996] “Pressure and Wall Shear Stress Sensors for Turbulence Measurements,” Royal Institute of Technology, TRITA-ILA-9601, Stockholm Sweden.) © 2006 by Taylor & Francis Group, LLC 6-36 MEMS: Applications Wall shear stress, ␶ w (Pa) 42 43 44 45 46 0 0.5 1 1.5 2 2.5 3 3.5 4 300 µm 60 µm Power dissipation, P (mW) Parallel flow P = 42.6 + 1.09␶ w (mW) Perpendicular flow ∆T = 100 K 0.50 P = 42.4 + 1.09␶ w (mW) 0.52 FIGURE 6.22 Total steady-state power dissipation calibration as a function of the wall-shear stress for two different probe orientations. (Reprinted with permission from Kälvesten, E. [1996] “Pressure and Wall Shear Stress Sensors for Turbulence Measurements,” Royal Institute of Technology, TRITA-ILA-9601, Stockholm Sweden.) TABLE 6.2 Some Calculated and Measured Characteristics of the MEMS-based Wall-Shear Stress Sensors Fabricated by Kälvesten et al. (1994) Theory Measurements Heated chip top-area, A ϭ w ϫ ᐉ (µm 2 ) 300 ϫ 60 1200 ϫ 600 300 ϫ 60 1200 ϫ 600 Heated chip thickness, h (µm) 30 30 30 30 Thermal conduction conductance, G c (µW/K) 372 510 426 532 Thermal convection conductance, G f (µW/K) Perpendicular Configuration at 50 m/s 6,7 207 21.0 231 Parallel Configuration at 50 m/s 5.6 192 17.3 160 Thermal capacity, (µJ/K) 1.2 37 — — Thermal time-constant at zero flow, (ms) 3 72 6 55 Time response (electronic feedback), (µs) — — 25 — Vacuum cavity Silicon substrate Aluminum Polysilicon wire Nitrid e FIGURE 6.23 Cross-sectional structure of a single shear-stress sensor. (Reprinted with permission from Jiang, F., Tai, Y-C, Walsh, K., Tsao, T., Lee, G.B., and Ho, C H. [1997] “A Flexible MEMS technology and Its First Application to Shear Stress Skin,” in Proc. IEEE MEMS Workshop (MEMS ’97), pp. 465–470.) © 2006 by Taylor & Francis Group, LLC a direct measurement of the distance the wall-element is moved by the wall-shear stress; or null balance which is the measurement of the force required to maintain the wall-element at its original position when actuated on by the wall-shear stress. The principle of a floating-element balance is shown in Figure 6.26. In spite of the fact that the force measurement is simple, the floating-element principle is afflicted with some severe drawbacks which strongly limit its use as has been summarized by Winter (1977). It is difficult to choose the relevant size of the wall-element in particular when measuring small forces and in turbulence applications. Misalignments and the gaps around the element, especially when measuring small forces, are constant sources of uncertainty and error. Effects of pressure gradients, heat transfer, and suction or blowing cause large uncertainties in the measurements as well. If the measurements are conducted in a moving frame of reference, effects of Sensors and Actuators for Turbulent Flows 6-37 Diaphragm Polysilicon thermistor wire FIGURE 6.24 SEM photo of a single wall-shear stress sensor. (Reprinted with permission from Jiang, F., Tai, Y-C, Walsh, K., Tsao, T., Lee, G.B., and Ho, C H. [1997]“A Flexible MEMS technology and Its First Application to Shear Stress Skin,” in Proc. IEEE MEMS Workshop (MEMS ’97), pp. 465–470.) 2.2 2 1.8 1.6 1.4 Mean output voltage (V) 1.2 1 0.2 0.4 0.6 Shear-stress, ␶ (Pa) 0.8 1 FIGURE 6.25 Calibration curves of 10 different sensors in an array. (Reprinted with permission from Jiang, F., Tai, Y-C, Walsh, K., Tsao, T., Lee, G.B., and Ho, C H. [1997] “A Flexible MEMS technology and Its First Application to Shear Stress Skin,” in Proc. IEEE MEMS Workshop (MEMS ’97), pp. 465–470.) © 2006 by Taylor & Francis Group, LLC gravity, acceleration, and large transients can also severely influence the results. Haritonidis (1989) dis- cussed the mounting of floating-element balances and errors associated with the gaps and misalignments. In addition, floating-element balances fabricated with conventional techniques have in general poor fre- quency response and are not suited for measurements of fluctuating wall-shear stress. To summarize, the idea of direct force measurements by a floating-element balance is excellent in principle, but all the draw- backs taken together make them difficult and cumbersome to work with in practice. It was not until the introduction of microfabrication in the late 1980s that floating-element force sensors achieved a revital- ized interest in particular for turbulence studies and reactive flow control. 6.4.3.8 MEMS Floating-Element Sensors Schmidt et al. (1988) were the first to present a MEMS-based floating-element balance for operation in low-speed turbulent boundary layers. A schematic of their sensor is shown in Figure 6.27. A differential capacitive sensing scheme was used to detect the floating element movements. The area of the floating element used was 500 ϫ 500 µm 2 , and it was suspended by four tethers, which acted both as supports and restoring springs. The floating element had a thickness of 30 µm, and was suspended 3 µm above the sil- icon substrate on which it was fabricated. The gap on either side of the tethers and between the element and surrounding surface was 10 µm, while the element top was flush with the surrounding surface within 1 µm. The element and its tethers were made of polyimide, and the sensor was designed to have a band- width of 20 kHz. A static calibration of this force gauge indicated linear characteristics, and the sensor was able to measure a shear stress as low as 1 Pa. However, the sensor showed sensitivity to electromagnetic interference due to the high-impedance capacitance used, and drift problems attributed to water-vapor absorption by the polyimide were observed. No measurements of fluctuating wall-shear stress were made because the signal amplitude available from the device itself was too low in spite of the fact that the first- stage amplification was fabricated directly on the chip. Since the introduction of Schmidt et al.’s sensor, other floating-element sensors based on transduction, capacitive, and piezoresistive principles have been developed [Ng et al., 1991; Goldberg et al., 1994; Pan et al., 1995]. Ng et al.’s sensor was small and had a floating element with a size of 120 ϫ 40 µm 2 . It operated on a transduction scheme and was basically designed for polymer-extrusion applications so it operated in the shear stress range of 1–100 kPa. Goldberg et al.’s sensor had a larger floating element size, 500 ϫ500 µm 2 . It had the same application and the same principle of operation as the Ng et al.’s balance. Neither of these two sensors is of interest in turbulence and flow control applications since their sensitivity is far too low. The capacitive floating-element sensor of Pan et al. (1995) is a force-rebalance device designed for wind- tunnel measurements, and is fabricated using a surface micromachining process. Unfortunately, this particu- lar fabrication technique can lead to non-planar floating-element structures. The sensor has only been tested in laminar flow and no dynamic response of this device has been reported. Recently, Padmanabhan (1997) has presented a floating-element wall-shear stress sensor based on optical detection of instantaneous element displacement. The probe is designed specifically for turbulent boundary 6-38 MEMS: Applications ␶ x (a) ␶ x (b) FIGURE 6.26 Schematic drawing of a floating-element device, with (a) pivoted support; and (b) parallel linkage support. (Reprinted with permission from Haritonidis, J.H. [1989] “The Measurements of Wall-Shear Stress,” in Advances in Fluid Mechanics Measurements, M. Gad-el-Hak, ed., pp. 229–261, Springer-Verlag, Berlin.) © 2006 by Taylor & Francis Group, LLC layer research and has a measured resolution of 0.003 Pa and a dynamic response of 10 kHz. A schematic illustrating the sensing principle is shown in Figure 6.28. The sensor is comprised of a floating element which is suspended by four support tethers. The element moves in the plane of the chip under the action of wall-shear stress. Two photodiodes are placed symmetrically underneath the floating element at the lead- ing and trailing edges, and a displacement of the element causes a “shuttering” of the photodiodes. Under uniform illumination from above, the differential current from the photodiodes is directly proportional to the magnitude and sign of shear stress. Analytical expressions were used to predict the static and dynamic response of the sensor; based on the analysis, two different floating-element sizes were fabricated, 120 ϫ 120 ϫ 7µm 3 and 500 ϫ 500 ϫ 7 µm 3 . The device has been calibrated statically in a laminar flow over a stress range of four orders of magnitude, 0.003–10 Pa. The gauge response was linear over the entire range of wall-shear stress. The sensor also showed good repeatability and minimal drift. A unique feature of the shear sensor just described is that its dynamic response has been experimen- tally determined to 10 kHz. Padmanabhan (1997) described how oscillating wall-shear stress of a known magnitude and frequency can be generated using an acoustic plane-wave tube. A schematic of the cali- bration experiment is shown in Figure 6.29. The set-up is comprised of an acrylic tube with a speaker- compression driver at one end and a wedge-shaped termination at the other end. The latter is designed to minimize reflections of sound waves from the tube end and thereby set up a purely travelling wave in the tube. A signal generator and an amplifier drive the speaker to radiate sound at different intensities and frequencies. At some distance downstream, the waves become plane; at this location a condenser micro- phone (which measures the fluctuating pressure) and the shear-stress sensor are mounted. The flow field inside the plane-wave tube is very similar to a classical fluid dynamics problem — the Stokes second problem. The only difference is that instead of an oscillating wall with a semi-infinite stationary fluid, the plane-wave tube has a stationary wall and oscillating fluid particles far away from the wall. Solutions to the Stokes problem can be found in many textbooks [Brodkey, 1967; Sherman, 1990; and White, 1991]. Padmanabhan (1997) converted the boundary conditions and derived corresponding analytical expres- sion for the plane-wave tube. The analytical solution of the fluctuating wall-shear stress was compared to the measured output of the shear-stress sensor as a function of frequency and a transfer function of Sensors and Actuators for Turbulent Flows 6-39 Embedded conductor Floating-element Silicon Cps1 Cdp Cps2 Passive electrodes Csb1 Csb2 V d On chip Off chip V DS + − FIGURE 6.27 Schematic of the MEMS floating-element balance. of Schmidt et al. (1988).(Reprinted with permis- sion from Schmidt, M., Howe, R., Senturia, S., and Haritonidis, J. [1988] “Design and Calibration of a Microfabricated Floating-Element Shear-Stress Sensor,” IEEE Trans. on Electron Devices 35, pp. 750–757.) © 2006 by Taylor & Francis Group, LLC the sensor was determined. As expected, the measured shear stress showed a square-root dependence on frequency. 6.4.3.9 Outlook for Shear-Stress Sensors Since coherent structures play a significant role in the dynamics of turbulent shear flows, the ability to con- trol these structures will have important technological benefits such as drag reduction, transition control, mixing enhancement, and separation delay. In particular, the instantaneous wall-shear stress is of interest for reactive control of wall-bounded flows to accomplish any of those goals. An anticipated scenario to realize this vision would be to cover a fairly large portion of a surface, for instance parts of an aircraft wing or fuse- lage, with sensors and actuators. Spanwise arrays of actuators would be coupled with arrays of wall-shear stress sensors to provide a locally controlled region. The basic idea is that sensors upstream of the actuators detect the passing coherent structures, and sensors downstream of the actuators provide a performance mea- sure of the control. Fast, small, and inexpensive wall-shear stress sensors like the microfabricated thermal or floating-element sensors discussed in this section would be a necessity in accomplishing this kind of futuris- tic control system. Control theory, control algorithms, and the use of microsensors and microactuators for reactive flow control are among the topics discussed in several chapters within this handbook. 6-40 MEMS: Applications Tether W e Photodiode X X Flow W diode Floating-element Free-standing structure Anchored to substrate Photodiode L diode W t L t L e L t Incident light from a laser source Flow t n-type Si W diode Floating-element shutter 2.1 µm Oxide passivation g = 1.2 µm Photodiodes n + p-type Si substrate n + FIGURE 6.28 Schematic of the floating-element balance of Padmanabhan (1997). (Reprinted with permission from Padmanabhan, A. [1997] Silicon Micromachined Sensors and Sensor Arrays for Shear-Stress Measurements in Aero- dynamic Flows, Ph.D. Thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts.) © 2006 by Taylor & Francis Group, LLC [...]... 0.5 40 Ͼ20 7 .8 45 10 10 2 15 15 4 10 Ͼ20 16 14 2.5 18 Ͼ25 14 17 10 20 25 0.25 4.3 0.05 0.025 7.5 25 1 19 4 8 13 0.025 at 6 V 5 0.4 10 2 3.5 0.92 0.0009 at 10 V 5 11 0 .1 0.065 0. 01 66 54 72 68 25 — — 60 30 31. 5 — 62 Ͻ25 35 35 57 90 30 30 60 58 69 6-4 6 MEMS: Applications piezoresistive gauges on a diaphragm is the high sensitivity that is accomplished with this arrangement The drawback is the high temperature... et al (19 96) Bull (19 67) Schewe (19 83 ) Blake (19 86 ) Lauchle (19 87 ) Spalart (19 88 ) Choi & Moin (19 90) Farabee (19 91) 7 6 prms /␶w 5 4 7.2 21. 6 3 33 66 19 2 1 1 40 256 18 0 333 0 10 2 10 3 10 4 10 5 Reu FIGURE 6.36 Dependence of normalized rms pressure fluctuations on Reynolds number Uncertainty less than 10 % The numbers next to selected data points indicate the diaphragm side-length in wall units (Reprinted... (19 88 ) Sprenkels et al (19 89 ) Hohm and Hess (19 89 ) Voorthuyzen et al (19 89 ) Murphy et al (19 89 ) Bergqvist and Rudolf (19 91) Schellin and Hess (19 92) Kuhnel (19 91) Kuhnel and Hess (19 92) Scheeper et al (19 92) Bourouina et al (19 92) Ried et al (19 93) Kälvesten (19 94) Scheeper et al (19 94) Bergqvist (19 94) Schellin et al (19 95) Kovacs and Stoffel (19 95) Kronast et al (19 95) © 2006 by Taylor & Francis Group,... Thomas, 19 76; Schewe, 19 83 ; Blake, 19 86 ; Lauchle and Daniels, 19 84 ; Farabee and Casarella, 19 91] A clear shortcoming in many of the experiments designed to measure pressure fluctuations is the quality of the data In the low-frequency range, the data may be contaminated by facility-related noise, while in the high-frequency range the spatial resolution of the transducers limits the accuracy The former... Level [dB(A)] Piezoelectric Capacitive Piezoelectric Piezoelectric Capacitive Capacitive Capacitive Capacitive Capacitive Capacitive Piezoresistive FET Capacitive Capacitive Capacitive Piezoelectric Piezoresistive Capacitive Capacitive Piezoelectric Capacitive FET 3ϫ3 0 .8 ϫ 0 .8 3ϫ3 0 .8 ϫ 0.9 3ϫ3 3ϫ3 0 .8 ϫ 0 .8 2.45 ϫ 2.45 3ϫ3 2ϫ2 1 1 0 .85 ϫ 1. 3 0 .8 ϫ 0 .8 1. 5 ϫ 1. 5 1 1 2.5 ϫ 2.5 0 .1 ϫ 0 .1 2ϫ2 2ϫ2 1 1. .. noise levels Although simple in design, they have fairly low sensitivity and relatively large spatial extension of their diaphragms The piezoresistive sensors seem to be most flexible since the major advantage of locating Table 6.3 Summary of Silicon Micromachined Microphones in Chronological Order Author Royer et al (19 83 ) Hohm (19 85 ) Muller (19 87 ) Franz (19 88 ) Sprenkels (19 88 ) Sprenkels et al (19 89 ) Hohm... at 10 V (µV/Pa) Static sensitivity at 10 V (µV/Pa) Noise (A-weighted), [dB(A)] 10 0 0.4 2 ϫ 10 Ϫ3 13 10 1. 0 1. 7 — Measurements 300 0.4 5 ϫ 10 Ϫ0 89 4 0.2 2.2 — 10 0 0.4 10 a 25a 0.9 1. 2 Ϸ90 300 0.4 10 a 25a 0.3 1. 4 Ϸ90 a Limits set by the calibration measurement setup The expected theoretical acoustical pressure sensitivity, S, for small pressure amplitudes has been derived by Kälvesten (19 94) and reads: 1. .. applications A sensitivity of 50–250 µV/Pa, and a frequency response in the range of 10 Hz 10 kHz (flat within 5 dB) were recorded Other researchers have presented similar piezoelectric silicon microphones [Kim et al., 19 91; Kuhnel, 19 91; Schellin and Hess, 19 92; Schellin et al., 19 95], and the sensitivities of these microphones were in the range of 0.025 1 mV/Pa However, applications of the piezoelectric... of the correlation as shown in Figure 6. 38 © 2006 by Taylor & Francis Group, LLC Sensors and Actuators for Turbulent Flows 6-5 1 1.2 1. 0 13 Hz 13 kHz 30 Hz 10 00 Hz 0 .8 300 Hz 13 kHz Rpp 0.6 0.4 0.2 0.0 −0.2 0 0.5 1. 0 1. 5 2.0 2.5 3.0 x1/␦* FIGURE 6. 38 Longitudinal spatial correlation of the pressure fluctuations in three frequency ranges: 30 Hz Ͻ f Ͻ 10 00 Hz; 300 Hz Ͻ f Ͻ 13 kHz; and 13 Hz Ͻ f Ͻ 13 ... Space-Time Correlation Measurements in a Flat Plate Boundary Layer,” J Fluids Eng 11 8, pp 457–463.) of about 0.9 and 0.3 µV/Pa for diaphragm side-lengths of, respectively, 10 0 and 300 µm The deviations from the theoretical estimates for the acoustical sensitivities can be explained by the approximations made in the sensitivity calculation and the uncertainty of the level of the built-in stresses in the . (19 96) Lofdahl et al. (19 96) Bull (19 67) Schewe (19 83 ) Blake (19 86 ) Lauchle (19 87 ) Spalart (19 88 ) Choi & Moin (19 90) Farabee (19 91) p rms /␶ w Re u 8 7 6 5 4 3 2 1 0 10 2 10 3 10 4 10 5 1 333 19 33 66 7.2 21. 6 40 18 0 256 . (19 83 ) Piezoelectric 3 ϫ 3 40 0.25 66 Hohm (19 85 ) Capacitive 0 .8 ϫ 0 .8 Ͼ20 4.3 54 Muller (19 87 ) Piezoelectric 3 ϫ 3 7 .8 0.05 72 Franz (19 88 ) Piezoelectric 0 .8 ϫ 0.9 45 0.025 68 Sprenkels (19 88 ) Capacitive. interest in particular for turbulence studies and reactive flow control. 6.4.3 .8 MEMS Floating-Element Sensors Schmidt et al. (19 88 ) were the first to present a MEMS- based floating-element balance