10th International Symposium on Ultrasonic Doppler Methods for Fluid Mechanics and Fluid Engineering

Tokyo Japan (28-30. Sep., 2016)

Ultrasonic velocity profiling as a wall shear stress sensor for turbulent boundary layers

Yuichi Murai, Yuji Tasaka, and Hyun Jin Park

Faculty of Engineering, Hokkaido University, N13W8, Kita-ku, Sapporo, 060-8628, Japan

There are a number of technical problems pointed out in wall shear stress measurement as it is directly sensed with a shear transducer. This is because most of the sensors adapt mechanical displacement principle intrusively to boundary layer before converted to electric signals. Here we propose an alternative method that excludes mechanical parts, based on ultrasound velocity profiling (UVP) technique. By introducing universal turbulent log-law theory to UVP data, accurate wall shear stress measurement has been realized, which was successfully demonstrated by application to turbulent channel flows at 104 < Re < 105 in this report. Our demonstrative experiments have confirmed that there is bias error less than 1 % in the wall shear stress while random error takes within 3 % as compared with Blasius’ formula. In addition, we present its extended application to bubbly turbulent channel flows in which bubbles reduced wall shear stress, i.e., drag reduction, via modification of the turbulent velocity profiles in the log-law regions.

Keywords: Wall shear stress, turbulent flow, boundary layer, ultrasound velocity profiling, drag reduction

1. Introduction

   Drag reduction is a major issue in fluid engineering, that contributes to energy saving. In turbulent boundary layers along a wall, frictional drag is originated by active fluid mixing in the buffer layer that lies between viscous sublayer and outer flow regions. For single-phase turbulent flows, plenty number of studies have been reported historically to elucidate the inner layer structure of wall turbulence and its correlation to wall shear stress.As its scientific understanding progresses, there comes up

2. Measurement Principle

   1. Log-law theory of turbulent boundary layer

      In turbulent boundary layer, non-dimensionalized fluid velocity u+ has the following profile within the logarithmic layer,u+ = 1 log y+ + B , (1)whereu l udemands to measure the local wall shear stress accurately.u+ = u( y) ,y+ = y ,and l= v . (2)For example, drag reduction by injection of additives such as polymer and bubbles induces significant fluctuation of wall shear stress both in time and space [1- 3]. Artificial modification of wall surface such as by riblet and wettability also requires assessment with resolving their space-time effects.There are some commercially available produces for local wall shear stress measurement, e.g. shear transducer so-called. Most cases are fluid–contact types which require a small tolerance allowing shear-sensing displacement as flash-mounted on the target wall [4]. If the sensing area is reduced to improve the spatial resolution, the contact problem leads to more serious error especially for high Re-number flows. This does because the tolerance alters original boundary layer structures as it faces with viscous sublayer on the order of a few micrometer. The authors have such rich experiences how natural contamination and artificial mixing of dilute particles/microbubbles lose the accuracy of shear transducers.To exclude such a contact issue, we have heredeveloped a new method of local wall shear stress measurement. It is based fully on ultrasound velocityff f fHere y is the spatial coordinate from the wall surface. uf and v are friction velocity and kinematic viscosity, respectively. The length scale lf is called wall unit. Substituting all the definitions in Eq. (2) into Eq. (1) givesu   v u( y) = 1 log( u f y  + B . (3)f  Two parameters,  and B, are known to be constants since the equation stands universally, and these are approximately given by  = 0.4 (called von Kármán’s universal constant), and B = 0.41. Nishioka [5] suggested the best accurate values on these parameters to be  = 0.379, and B = 0.406, which the present authors employ in this study.Friction velocity uf is defined by the wall shear stressτw and fluid density p asτw / pu f = . (3)Here the wall shear stress is generally described byprofiling (UVP), and therefore the measurementτ = C1 pU 2 , (4)procedure and the applicable targets are the same as UVP.w f 2where Cf and U are friction coefficient and outer flowvelocity, respectively. Substituting Eq. (4) into Eq. (3) gives the following relationship;u f =Cf U . (5)22CfCf2Further substituting Eq. (5) to Eq. (3) obtainsu( y)= 1 log( Uy + B . (6)U  v  

   2. Estimation of friction coefficient

      As velocity profile u(y) is measured, all the values in Eq.(6) is fully given except the friction coefficient Cf. Therefore, Cf can be determined and Eq. (6) is satisfied. Unfortunately Eq. (6) cannot be converted to explicit equation regarding Cf, it needs graphical work or numerical approach to solve. Clauser [6] proposed graphical way, which is today known as Clauser’s method. In principle, a single velocity data u at an arbitrary position of y within the logarithmic layer is enough for Eq. (6) to estimate Cf value. However, before knowing the velocity profile, y-coordinate range of the buffer layer is not judged as in practical applications. Thus, advantage of UVP takes place here. UVP obtains velocity profile u(y) that constitutes the left-hand side of Eq. (6), and the logarithmic range can be identified.Not only for the profile judgment, but also for accurate estimation of Cf, UVP has another advantage. That is, many equations can stand for Eq. (6) onto all the points of the measurable coordinate y. Hence, least square approach is introducible. We define local residual of Eq.2CfCf2(6) as two functions of Cf and y asFig. 1 Overview of horizontal channel flow facility(a) Spanwise view (b) Streamwise view Fig. 2 Measurement line of UVPFig. 1 shows overview of the experimental facility. The main channel flow section is L = 6 m in total length, H = 40 mm in height, and W = 160 mm in span width. Water flow rate is varied with a pump at less than Q =0.01 m3/s (600 l/min.). In case of bubbly flow experiments, air bubbles are injected from a hole- arranged plate mounted on the top wall of the channel.Fig. 2 shows how the UVP measurement line was set( ) u( y)1 ( Uy . (7)at the rectangular channel section. The head of theg Cf , y =U–  log v + Btransducer is submerged in a small water jacket to allow To have the minimum residual along all the range of logarithmic layer, we further define a squared cumulative function to best estimate Cf value assufficient quality of ultrasound pulse. Setting parameters of UVP operation are summarized in Table 1. The beam angle uncertainty is estimated around 0.5 degree, but which does not affect the wall shear stress estimation significantly because of logarithmic impact as aforeG (Cf ) =g (Cf , y ) dy2y2y1→ min., (8)mentioned . We employ 4 MHz in basic frequency so that UVP covers all log-law region considering futurewhere y1 and y2 are the lower and the upper borders of the logarithmic layer. Consequently, the friction coefficient Cf is determined to minimize the cumulative residual. Partial derivative of Eq. (8) respect to Cf only produces an implicit equation which needs numerical search to find the best estimate of Cf. After the search, the wall shear stress is immediately obtained by Eq. (4). Some other approaches are examined using DNS database [7], but which assumes zero-noise in measurement, being inapplicable to experimental measurement.

3. Channel Flow Measurement

   The proposed method has been validated by application to a water channel flow measurement at turbulent flow states. In this section, applications to single-phase and bubbly two-phase turbulent flows are presented.application to ship boundary layers.         Table 1 Setting parameters of UVP          Base frequency 4.0 MHzTemporal resolution 17 msSpatial resolution 0.78 mm

   Beam angle	7	degree
   Number of cycles	4	-
   Number of repetitions	32	-

   1. Single-Phase Flow Conditions

      For a single-phase flow, channel flow structures keep dynamic similarity characterized by Reynolds number. We here define it using the channel central fluid velocity U and the channel half height H/2 asRe = UH / 2 . (9)νFig. 3 depicts water velocity distribution measured by UVP, which is expanded in space–time domain. At Re = 28000, we can confirm significant velocity fluctuation activated by wall turbulence in the channel flow. Fig. 4 represents time-averaged velocity profiles as water flow rate Q increases, i.e., Re number increases. The data points at y/H < 0.25 include structured noises due to near- field beam characteristics of the ultrasound transducer which is set outside the channel wall with 10 mm in thickness. To the contrary, the data at y/H > 0.25 is obtained without noise, and we target this zone for the wall shear stress analysis.Fig. 3 Velocity distribution at Re=28000.Fig. 4 Mean velocity profiles as water volume flow rate changes. Ultrasound transducer is outside the left edge of the graph. The wall surface coordinate was judged by an echo profile.Fig. 5 shows a velocity profile obtained by UVP at Re= 54000, represented in semi-log graph. Many inclined lines are theoretical velocity profiles of Eq. (6) as various Cf -values are assumed. We made a numerical software which automatically finds the best Cf value. The matching accuracy has five significant numbers in digits.Fig. 6 shows the friction coefficients Cf measured by the present method at eight different Re numbers. A curve in the graph is Blasius formula of the friction coefficient for a turbulent pipe flow in the same range of pipe- equivalent Re number. It is confirmed that the present method and Blasius theory agree to each other very well. There is no significant bias error while a small random error less than 3% comes up but which seems to be negligible in the authors’ point of view as compared withunstable performance of existing shear transducers. Fig. 7 shows the wall shear stress, which is our final goal of the measurement. On the graph, error bars mean ±5% in relative error.Fig. 5 Semi-log representation of measured velocity profile at Re = 54000 compared with theoretical log-law profiles with different friction coefficient Cf assumed in the process of numerical search for Eq. (8).Fig. 6 Friction coefficients measured from UVPFig. 7 Friction coefficients measured from UVP

   2. Bubbly Two-Phase Flow Conditions

      We have applied the present method to bubbly two-phase flow using the same channel flow facility. Bubble size ranges from 1 mm to 20 mm, subject to a broad deviation. We understand that Clauser’s method is valid only for single-phase flow, but here we discuss its extensibility tomultiphase flows as engineering purpose, expecting practical applications.Fig. 8 UVP data analyzer for multiphase flowFig. 9 Drag reduction performance at low speed flowFig. 10 Drag reduction performance at high speed flow Fig. 8 shows a program window to process UVP dataobtained for bubbly two-phase turbulent flow conditions. The process starts with interface detection based on Sobel filtering [8], and ends with wall shear stress estimation via log-law fitting. Details are explained in the presentation in ISUD.Fig. 9 shows measured wall shear stresses as bulk void fraction of the channel increases. UC is time-average flow speed of liquid phase at the center of the channel. Solid circles indicate the data of liquid in-phase value,and open circles are entire averages of the wall shear stress where the local wall shear stress is assumed to be approximately zero inside bubble passing periods (i.e. free-slip wall, evidenced by Murai et al [4]). The dotted line in the graph means linear fitting of the drag reduction, which has 5.2 factor to bulk void fraction. As entire drag is reduced, we can see that the liquid in-phase drag also decreases at around 10–30 %. To the contrary, a high- speed flow condition (see Fig. 10) has smaller impact to the in-phase wall shear stress but higher factor to bulk void fraction at around 8.2. These results infer that bubbles in high-speed flow, i.e. high Weber number bubbles (We > 200), can reduce entire drag effectively.

4. Conclusions

We proposed in this paper a method of wall shear stress measurement from UVP data as applied for velocity profiling of turbulent boundary layers. The measurement principle of Clauser’s graphical approach has been converted to data processing software which numerically finds log-law region automatically and extracts corresponding friction coefficient. The method is applicable to any liquid which UVP can measure. By application to turbulent water channel flows, the present measurement principle has been validated successfully. The method was extendedly applied to bubbly two-phase channel flow, and drag reduction performance due to injection of bubbles has been obtained only by UVP information.

Acknowledgment

The present study is supported by Japan Soc. Promotion of Science: JSPS-KAKENHI (Grant No. 24246033). The authors also thank to Mr. J. Nagao for his supporting work for UVP measurement of two-phase channel flows.

References

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