Journal of Fluid Science and Technology

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1 Bulletin of the JSME Vol.9, No.3, 24 Journal of Fluid Science and Technology Re-evaluating wake width in turbulent shear flow behind an axisymmetric cylinder by means of higher order turbulence statistics Kuan-Huang LEE* and Keh-Chin CHANG* * Department of Aeronautics and Astronautics, National Cheng Kung University No., University Road, Tainan, Taiwan kcchang@mail.ncku.edu.tw Received 6 October 23 Abstract The free turbulent shear flow behind a long axisymmetric cylinder, i.e., a wake, is two-dimensional in nature and is composed of inner shear turbulence and outer potential, irrotational flow (free stream) regions. The conventional definitions of wake width, l u, are based on the mean velocity field, namely, the first order statistics of turbulence. Two alternative definitions of wake width, l S = 2y S and l F = 2y F, are proposed in terms of skewness and flatness factors, respectively, which are considered as third- and fourth-order statistics of turbulence in this study. Between these two new definitions of wake width, l F can represent a more proper sectional range of shear turbulence in a wake. Key words : Wake width, Skewness factor, Flatness factor, Experimental fluid mechanics, Statistics of turbulence. Introduction Flow around cylinders is encountered in many engineering problems. Circular cylinder flow seems to be one of the simplest geometrics, however, it generates highly complex flow structures and has attracted a great deal of attention (Zdravkovich, 997). When the fluid flows pass over a circular cylinder, a boundary layer forms and is attached to the forward body surface. Experiencing the adverse pressure gradient, the boundary layers separate (in the range of 4 to < Re D < 3 to 48) and continue to develop downstream along with the free shear layers. A wake forms behind the cylinder and develops stream-wisely through near-wake and far-wake regions. The most dominant feature of a wake is the vortex shedding phenomenon when the Reynolds number becomes larger than 3 to 48. As reported by Roshko and Fiszdon (969), the development of transition occurs in different disturbed flow regions with Re D being increased, by the wake, free shear layer and boundary layer. The characteristic length scale of wake is generally represented by the wake width, l u, which is defined by l u = 2y Δ with the centerline position of the wake being set at y = in each section. Here the half-width y Δ is defined as the transverse position of (U - U c ) / ΔU being equal to the value of Δ (ΔU = U - U c, U and U c are the mean stream-wise velocities in the free stream and at the centerline, respectively). For example, Townsend (98) set the value of Δ as e -/2 while Kline (96) set it as.99. However, these conventional definitions of wake width are based on the mean velocity field, namely, the first order statistics of turbulence. It was reported (Zdravkovich, 997; Townsend, 98) that the phenomenon of intermittency in the outer regions of the wake width, defined by l u, can still be observed. Free turbulent shear flow such as wakes, jets, mixing layers and so on, are bounded by ambient fluid which is not turbulent and is usually in irrotational flow. The intermittency surface (Townsend, 98) divides the fluid of turbulent, vortical motion from the fluid of potential, irrotational motion. Within the intermittency surface, the flow is fully turbulent. Outside the surface, the velocity fluctuations arise from irrotational flow fields induced by the turbulent eddies, which exhibit motion that is not turbulence in the usual sense. Since this bounding surface may change in shape and position, an anemometer at a fixed position may experience intermittent signals from either fully turbulent flow or fluctuations in the potential flow. Thus, we can conclude that a wake width defined in terms of l u is incapable of Paper No.3-26 [DOI:.299/jfst.24jfst3] 24 The Japan Society of Mechanical Engineers

Lee and Chang, Journal of Fluid Science and Technology, Vol.9, No.3 (24) covering all the flow regions which could be influenced by the shear turbulence within the wake.

For example, by analyzing the intermittency factor (Townsend, 949), which is estimated with the proportion of intermittent time by placing the sensor in the turbulent flow field, one can extract the

2 Lee and Chang, Journal of Fluid Science and Technology, Vol.9, No.3 (24) covering all the flow regions which could be influenced by the shear turbulence within the wake. Different ways to determine the interface between the inner shear turbulence and the outer free stream have been reported in the literature. For example, by analyzing the intermittency factor (Townsend, 949), which is estimated with the proportion of intermittent time by placing the sensor in the turbulent flow field, one can extract the interface of the activity of entrainment (Bevilaqua, 97). Li, et al. (2) investigated the transition from the shear layer to the two outer free stream regions in the planar mixing layer by monitoring the variation of the roundness of the distribution of the velocity joint probability density function. In addition, our previous work (Chang, et al., 22) determined the mixing length of the planar mixing layer by monitoring the sectional distributions of high-order turbulence statistics such as the skewness factor (S u ) or the flatness (Kurtosis) factor (F u ) of the turbulent velocity distribution for the stream-wise component. These factors are defined as u u 3 S u () u 3 RMS u 4 F u (2) 4 RMS where the lower and upper cases denote the instantaneous and mean quantities, respectively, while the superscript, the subscript RMS and the symbol < > denote the fluctuating quantity, root-mean-square value and ensemble averaging operator, respectively. This study follows our previous work (Chang, et al., 22) and re-evaluates the wake width in the turbulent shear flow behind an axisymmetric cylinder by means of skewness and flatness factors. 2. Experimental Aspects 2. Experimental facilities The experimental facility used in this study is a vertically downward, rectangular, suction-type wind tunnel, schematically shown in Fig., which is composed of a settling chamber, contraction, test section, and noise reduction chamber. This wind tunnel has a test section with a cross-sectional area of mm and length of 6 mm, where the cylinder is mounted at the middle of the cross section, 7 mm down from the top of the test section. A Cartesian coordinate is selected such that the transverse coordinate y is positive toward the right and the stream-wise coordinate x is positive downward with the origin at the center of the cylinder. The diameter of the cylinder, D, is mm such that the slenderness ratio is equal to, which was reported by (Lin, et al., 22) to be capable of yielding wake flow with two-dimensional characteristics in the central span-wise regions of the domain of interest. Fig. Schematics of the experimental set-up 2.2 Measurement instrumentation Two-component velocity measurements are made with X-array hot-wire anemometry (HWA, the StreamLine Pro System of Dantec Dynamics Inc.) which is composed of a mainframe (9N) with a temperature probe included and the [DOI:.299/jfst.24jfst3] 24 The Japan Society of Mechanical Engineers 2

3 Lee and Chang, Journal of Fluid Science and Technology, Vol.9, No.3 (24) two modules of a constant temperature anemometer (CTA, 9C). The X-array hot-wire probe has two μm tungsten sensing elements, which are about.4 mm long and are positioned about mm apart. The sampling rate used in the measurement is khz. It was reported (Chang, et al., 22) that a large number of raw data are required to achieve the stationary statistics of S u and F u around the edge of the shear turbulence in a planar mixing layer. In order to significantly reduce the sample number of raw data when evaluating S u and F u, the following procedure, which is un-correlative with other procedures, is used to keep each piece of raw data in the ensemble operation of either Eq. () or Eq. (2) as follows. The time interval between the consecutive inputs of raw data must be longer than the integral time scale at that position. Here the integral time scale is estimated by integrating the auto-correlation function constructed by the measured raw data. Figure 2 shows the variations of the standard deviations for the skewness factor (σ S ) and flatness factor (σ F ) versus the number of raw data at four purposely selected positions which are located adjacent to the edge of the wake region (cf. Fig. ). It is observed that both σ S and σ F asymptote almost to their own constant values; in other words, they are statistically stationary, with 3 raw data. All the turbulence statistics reported here are thus made with 3 raw data in this way. Uncertainty of HWA measurements is estimated within 2. % for both the mean and root-mean-square fluctuating velocity x/d =, y/d = 3 x/d =, y/d = 3.3 x/d = 2, y/d = 3.6 x/d = 2, y/d = x/d =, y/d = 3 x/d =, y/d = 3.3 x/d = 2, y/d = 3.6 x/d = 2, y/d = S. F Sample numbers Sample numbers (a) Skewness (b) Flatness Fig. 2 Dependence of sample number on the standard deviations of both skewness and flatness factors at four position around the edge of the wake 2.3 Experimental conditions The experiment is conducted with the free stream (stream-wise) velocity of m/s, which causes the Reynolds number Re D to be equal to. 4. The free stream turbulent intensity is about %. According to the summary provided by Zdravkovich (997), the transition occurs along the free shear layer while the boundary layers on the cylinder remain fully laminar for the flow conditions of Re D ranging from ~2 3 < Re D < 2~4 4 in which the present test case falls. The flow becomes fully turbulent in the downstream station of about. to.9 D. The measured stream-wise stations in this study are x/d =,, 2, and 2, all of which are definitely located in the turbulent flow region. It was reported (Townsend, 98) that a wake requires to D to reach a self-preserving condition for the mean flow quantities. Obviously, the stations measured here are all located in the near (or developing) flow field of the wake. 3. Results and Discussion Figures 3 and 4 present the sectional distributions of mean and root-mean-square fluctuating velocity for both stream-wise and transverse components, respectively, which are all normalized by ΔU at the four measured stream-wise stations of x/d =,, 2, and 2. Note that the investigated wake is made behind an axisymmetric cylinder. Only the half sectional results are reported here. As observed from the results in Figs. 3 and 4, it is clear that the wake has not reached the self-preserving state in the measured flow regions. [DOI:.299/jfst.24jfst3] 24 The Japan Society of Mechanical Engineers 3

4 Lee and Chang, Journal of Fluid Science and Technology, Vol.9, No.3 (24).2.2 (U - U c ) / U x / D = x / D = x / D = 2 V / U x / D = x / D = x / D = (a) stream-wise component (b) transverse component Fig. 3 Sectional distributions of normalized mean velocity.4.4 u' RMS / U x / D = x / D = x / D = 2 v' RMS / U x / D = x / D = x / D = (a) stream-wise component (b) transverse component Fig. 4 Sectional distributions of normalized root-mean-square fluctuating velocity Figure. presents the sectional distributions of skewness and flatness factors for the stream-wise component, calculated with Eqs. () and (2), respectively, at the four measured stream-wise stations. It is known that S u = and F u = 3 are at the axisymmetric axis (y = ) which matches approximately with the experimental results reported in Fig.. Furthermore, in the outer potential flow regions, the probability density function of u, P(u ), can be represented in Gaussian distributions, i.e., S u = and F u = 3. Both distributions of S u and F u show these trends when moving further outward after passing their peak positions at each investigated station. The skewness factor (S u ) is known as a measure of lopsidedness (or asymmetry) of P(u ). The positive/negative sign of S u indicates whether u is more likely to cling to u = (in other words, at the mean velocity U) for negative/positive excursions (Davidson, 24). Since it is an axisymmetric wake, the sectional distributions of S u must be symmetric with respect to the wake axis (at the centerline, y = ). Thus there exists another peak of S u in the left-half plane of each section, which was not shown in Fig. a. One can imagine that a pair of vortices, each rotating in the opposite direction, is generated in the axisymmetric wake. Each pairing vortex is located on a different (positive/negative) side of sectional plane. The peak position of S u can be taken as the boundary of the dominant vortical motion and move more outward as the flow is more downstream (see Fig. a). The flatness factor F u indicates how far, and for how much time, u departs from u =. A so-called intermittent signal is a signal which is quiescent (or dormant) for much of the time, and occasionally burst into life. A signal in which F u is large implies extreme intermittence, exhibiting periods of quiescence interspersed by large transient excursions away from u = (Davidson, 24). Thus, the positions of peaks of F u, as shown in Fig. b, can be taken as the farthest positions where the turbulent fluctuations stemmed from the wake (shear turbulence) region can reach. Similar to that observed from the sectional distributions of S u (Fig. a), the peak positions of F u moves more outwards [DOI:.299/jfst.24jfst3] 24 The Japan Society of Mechanical Engineers 4

5 Lee and Chang, Journal of Fluid Science and Technology, Vol.9, No.3 (24) as the flow is further downstream (Fig. b). We set the peak positions of S u and F u as y S and y F and record the evolution of y S and y F along the stream-wise distance in Fig. 6. The results of y.9 which are the transverse positions at (U U c )/ΔU =.9 are also plotted in Fig. 6 for comparison. The figure shows that y F y S > y.9. In particular y F > y S when moving further downstream. The results reveal that the boundary of the dominant vertical motion generated inside the wake is at most the same as the boundary of the shear turbulence. The results of y F > y.9 are consistent with what were reported by, for example, Zdravkovich (997) and Townsend (98) that the phenomenon of intermittency in the outer regions of the wake boundary defined by means of mean velocity such as y.9 can still be observed. Since y F denotes the farthest position that transient excursions can reach in a wake, l F = 2 y F can, thus, be taken as a more proper scale of a wake, i.e., wake width, which bounds shear turbulence inside than the other two definitions examined in this study. This is also consistent with the idea of intermittency surface (Townsend, 98) which divides the fluid of turbulent vortical motion from the fluid of potential, irrotational motion x / D = x / D = x / D = 2 S u - F u 2-2 x / D = x / D = x / D = (a) Skewness (b) Flatness Fig. Sectional distributions of skewness and flatness factors for the stream-wise component y S y F y x / D 4. Conclusions Fig. 6 Stream-wise evolution of peak positions of skewness and flatness factors as well as that of y.9 Two alternative definitions of the half-width of a wake, y S and y F, are proposed in terms of skewness and flatness factors, respectively, which are the high-order statistics of turbulence in this study. Since y F and -y F are the transverse positions of two interfaces between the inner shear turbulence and outer potential flow (free stream) regions in a stream-wise station, the wake width l F = 2 y F can represent a more proper sectional range of shear turbulence in a wake. [DOI:.299/jfst.24jfst3] 24 The Japan Society of Mechanical Engineers

6 Lee and Chang, Journal of Fluid Science and Technology, Vol.9, No.3 (24) References Bevilaqua, P. M. and Lykoudis, P. S., Mechanism of entrainment in turbulent wakes, AIAA Journal, Vol.9, No.8 (97), pp Chang, K-C., Li, K-H. and Chang, T-C., Re-evaluating mixing length in turbulent mixing layer by means of high-order statistics of velocity field, International Journal of Modern Physics: Conference Series, Vol.9 (22), pp.4 6. Davidson, P. A., Turbulence : an introduction for scientists and engineers (24), Chap., Oxford University Press. Kline, S. J., Some remarks on turbulent shear flows, Proceedings of the Institution of Mechanical Engineers, Conference Proceedings, Vol.8, No. (96), pp Li, C-T., Chang, K-C. and Wang, M-R., Experimental study on evolution of joint velocity PDF in planar mixing layer, Experimental Thermal and Fluid Science, Vol.34, No.8 (2), pp Lin, C., Hsieh, S., Lin, W. and Raikar, R., Characteristics of recirculation zone structure behind an impulsively started circular cylinder, Journal of Engineering Mechanics, Vol.38, No.2 (22), pp Roshko, A. and Fiszdon, W., On the persistence of transition in the near-wake (969), Society for Industrial and Applied Mathematics. Townsend, A. A., The fully developed wake of a circular cylinder, Australian Journal of Chemistry, Vol.2, No.4 (949), pp Townsend, A. A., The structure of turbulent shear flow (98), Chap.6, Cambridge University Press. Zdravkovich, M. M., Flow around circular cylinders : a comprehensive guide through flow phenomena, Experiments, Applications, Mathematical Models, and Computer Simulations (997), Chaps. and, Oxford University Press. [DOI:.299/jfst.24jfst3] 24 The Japan Society of Mechanical Engineers 6

PROPERTIES OF THE FLOW AROUND TWO ROTATING CIRCULAR CYLINDERS IN SIDE-BY-SIDE ARRANGEMENT WITH DIFFERENT ROTATION TYPES

THERMAL SCIENCE, Year, Vol. 8, No. 5, pp. 87-9 87 PROPERTIES OF THE FLOW AROUND TWO ROTATING CIRCULAR CYLINDERS IN SIDE-BY-SIDE ARRANGEMENT WITH DIFFERENT ROTATION TYPES by Cheng-Xu TU, a,b Fu-Bin BAO