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Characteristics of temperature dissipation rate in a turbulent near wake
Experimental Thermal and Fluid Science 114 (2020) 110050
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Characteristics of temperature dissipation rate in a turbulent near wake J.G. Chen
a,b
c
, R.A. Antonia , Y. Zhou
a,b,⁎
, T.M. Zhou
T
d
a
Institute for Turbulence-Noise-Vibration Interactions and Control, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China Digital Engineering Laboratory of Offshore Equipment, Shenzhen 518055, China School of Engineering, University of Newcastle, NSW 2308, Australia d Department of Civil, Environmental and Mining Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia b c
A R T I C LE I N FO
A B S T R A C T
Keywords: Temperature dissipation rate Turbulent wake Mixing
This work aims to provide in-depth understanding of the effects the quasi-periodical Kármán vortices produce on the temperature dissipation rate in a turbulent cylinder near wake. Measurements are made at x/d = 10, 20 and 40, where x is the streamwise distance from the cylinder axis and d is the cylinder diameter, with a Reynolds number of 2.5 × 103 based on d and the free-stream velocity. A multi-wire probe is deployed to measure simultaneously the fluctuating temperature and its gradient vector, at nominally the same spatial point in the plane of the mean shear. It is found that the coherent streamwise and spanwise temperature derivatives are similarly distributed with respect to the spanwise vortex, exhibiting twin peaks at the temperature fronts, while the coherent lateral component is linked to the rib-like structures. The temperature variance dissipation rate is found to be statistically independent of the temperature fluctuation when the Kármán vortex is so weak (say at x/d = 40) that the large-scale temperature front resulting from the vortex entrainment ceases to be present and the coherent strain rate at the saddle region is relatively small. In addition, the most effective turbulent mixing is found to take place around the temperature front near the wake centerline, which is in contrast to the conjecture by Hussain and Hayakawa (1987).
1. Introduction The mean dissipation rate of a passive scalar variance θ¯2 is defined as
χ = κ (θ¯,12 + θ¯,22 + θ¯,32 )
(1.1)
where an overbar denotes time-averaging and θ ,i ≡ ∂θ ∂x i , i = 1, 2, 3 represents the streamwise, lateral and spanwise directions, respectively, which are used interchangeably with x, y and z in the present study. In (1.1), κ is the scalar diffusivity or the thermal diffusivity if the passive scalar is temperature. χ has been extensively studied in the past few decades, as it plays a central role in the turbulent combustion modeling, both in the Reynolds-averaged Navier-Stockes (RANS) and Large-eddy simulation (LES) approaches (see the review of [17]). In his review, Pitsch [17] pointed out that production/dissipation balance assumption which seems to be inherently used in all models for the scalar dissipation rate in LES of turbulent combustion is not always applicable due to the effect of the large-scale structures in the flow. For instance, the ‘ramp-cliff’ structures of the scalar resulting from the large-scale turbulent motions (e.g. [9,10]), instead of the energy cascade, cannot be described by the production/dissipation balance assumption. It is
⁎
thus naturally of great interest to examine how the behavior of χ is influenced by the large-scale structures in the turbulent flow, which is the primary motivation of the present study. There is evidence that the scalar dissipation is largely anisotropic, particularly in shear flows (e.g. [21,22]). Even in the self-preserving far wake, Antonia and Browne [1] observed that the mean-square value of the transverse derivative of temperature is larger than that of the longitudinal derivative. One may expect that the three components of the scalar dissipation rate are affected differently by the large-scale motions. Jayesh and Warhaft [12] found that for grid turbulence without a mean temperature gradient, χ is statistically independent of θ , leading to a Gaussian distribution of θ . As is known, the expectation of χ conditioned on θ plays an important role in closing the transport equation of the probability density function (pdf) of θ [13,2,20]. It is expected that in the turbulent near wake which is characterized by an appreciable streamwise decay of anisotropic large-scale organized motions, such as the spanwise Kármán vortices and the quasi-streamwise ribs, the situation could be very different from the self-preserving far wake or grid turbulence. This raises several questions regarding the effect of these organized motions on the temperature dissipation. How are the three components of χ spatially distributed with respect to the
Corresponding author at: Institute for Turbulence-Noise-Vibration Interactions and Control, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China. E-mail address: [email protected] (Y. Zhou).
https://doi.org/10.1016/j.expthermflusci.2020.110050 Received 12 September 2019; Received in revised form 4 January 2020; Accepted 15 January 2020 Available online 29 January 2020 0894-1777/ © 2020 Published by Elsevier Inc.
Experimental Thermal and Fluid Science 114 (2020) 110050
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organized structures in the near wake? How is the dependence between χ and θ affected by these coherent vortices? How does the dependence between χ and θ evolve with x as the structures weaken? These issues have yet to be clarified. A resolution of these issues would provide us with valuable insight not only into the physics of the small-scale temperature dissipation rate itself, but also its interaction with the temperature fluctuation, especially when the transporting flow is perturbed by organized motions. This work also attempts to shed light on the influence of the coherent motions on the turbulent mixing of the passive scalar. Hussain & Hayakawa [11] showed the topological scheme for the coherent structures of a turbulent plane wake (see also [24]). It presents a clear picture of the spatial distribution of the turbulence production and mixing, in the context of the organized vortex structures, showing that turbulence production took place in the saddle region due to the vortex stretching, and the turbulent kinetic energy was transported along the separatrix and accumulated within the vortices. They also conjectured that the turbulent mixing occurred mostly in the region where the streamwise and spanwise vortices are in contact with each other. Experimental evidence is actually still lacking on this point. In the present experiment in the near wake, relative to the more homogeneous and self-preserving far wake, of a circular cylinder the ambient cool fluid is engulfed by the Kármán vortex into the heated wake, so the turbulent mixing region can be identified by examining the mixing performance between the warm wake and the engulfed cool fluid in different local regions with respect to the large-scale vortices. This could also provide a new understanding of the physics of turbulent mixing in the present flow.
Fig. 1. (a) Experimental arrangement and the coordinate system; (b) side and (c) front views of the multi-wire probe.
‘wake’ of the cold wires on the X-hotwires is negligible. With this arrangement, the velocity gradients involved in the three vorticity components ωi (i = 1, 2 and 3) and the temperature dissipation rate χ can be measured simultaneously. Please refer to Chen et al. [6] for more details about the probe and its performance of measuring ui, ωi and θ . A detailed assessment of the spatial resolution of the probe for both velocity and temperature derivatives is available in Chen et al. [5] and will not be repeated here. Note that only the results for θ and θ,i are discussed in this study. A phase-averaging technique is used to extract the coherent structures from the flow field. The technique is the same as that used in Chen et al [6]. Briefly, the instantaneous quantity Γ may be viewed as the sum of the time-averaged component Γ , and the fluctuating component ∼ β , which can be further decomposed into a coherent fluctuation β and a remainder βr , namely:
2. Experimental details and phase-averaging technique The detailed information on the experimental arrangement was given in Chen et al. [6]. Some experimental details that are relevant to the present study are briefly recalled below. Experiments were conducted in an open-loop wind tunnel with a working section of 1.2 m (width) × 0.8 m (height) and 2.0 m long. The outer diameter of the circular cylinder used to generate the wake is 12.7 mm. A nickel wire with a diameter of about 0.5 mm was coiled along the thread of a ceramic tube of about 11 mm in diameter, which was then placed inside the cylinder and acted as a heating element. The cylinder was heated in such a way that the temperature on the surface of cylinder was constant. The free-stream velocity U1 was 3.0 m/s, corresponding to a Reynolds number Re (≡ U1d ν ) of 2.5 × 103, where ν is the kinematic viscosity. At this Reynolds number, the transition takes place at the intermediate part of the free shear layer, and the wake has become fully turbulent [24]. The coordinate system and some symbols are defined in Fig. 1a. Measurements were conducted within the cylinder mid-span plane at x/d = 10, 20 and 40 where x is the streamwise distance from the cylinder axis. The maximum mean temperature excess Θ0, relative to the ambient fluid, was approximately 1.6 °C, 1.5 °C and 1.3 °C on the centerline for x* = 10, 20 and 40, respectively. This excess is small enough to avoid any buoyancy effects and hence allow temperature to be treated as a passive scalar. Hereafter, an asterisk ‘*’ denotes normalization by d, U1 and/or Θ0 . A probe consisting of four X-hotwires (X1-X4 in Fig. 1b and c) and four cold wires (C1-C4 in Fig. 1b and c) was used to measure simultaneously the fluctuating velocity ui and temperature θ at nominally the same spatial point. Two of the X-wires, X2 and X4, were aligned in the (x, y) plane and separated by Δz = 2.7mm in the z direction; the other two, X1 and X3, were in the (x, z) plane and separated by Δy = 2.0mm in the y direction. The separation between the two wires of each Xprobe was approximately 0.6 mm. Four cold wires with a diameter of 1.27 μm were orthogonally arranged in the (y, z) plane and perpendicular to the planes of the four X-hotwires and placed at approximately 1 mm (approximately 800 diameters of the cold wire) upstream of the latter, which is sufficiently far to ensure that the influence from the
∼ β = β + βr
(2.1)
The phase average of an instantaneous quantity Γ at a particular N phase ϕ is given by Γ (ϕ) = (1 N ) ∑i = 1 Γi (ϕ) , where N is the total number of detections, which is approximately 1980, 1700 and 1050 for x* = 10, 20 and 40 respectively. It has been checked (not shown) that the number of detections has been large enough for Γ (ϕ) to be converged across the flow. The vortex shedding period is divided into 20 phases. The reference phase is the phase of the local velocity fluctuation ∼ v at each y* position. The coherent component β ≡ β reflects the effect from the large-scale coherent motions, while βr is associated with the remaining smaller-scale motions. 3. The components of the temperature dissipation rate In the turbulent near wake, the flow is dominated by the large-scale spanwise Kármán vortices and the streamwise rib-like structures. Our interest in this section is primarily on how the behavior of the three components of the temperature dissipation rate differ from each other under the effect of these coherent motions. In addition, the effect of these coherent events on the connection between the small-scale temperature dissipation rate and temperature fluctuation is also examined. ∼ We start by presenting the isocontours of θ∗ (Fig. 2a) at x* = 10 for a visual perspective of the large-scale temperature concentration. It was first shown in Chen et al. [6]. In Fig. 2a and the following iso-contours figures, the positions of the centers and saddle points, identified from 2
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∼∗ that the maximum concentration of negative θ,1 takes place at a loca∼∗ tion where θ changes sign (positive to negative, at ϕ = −0.25π , y* = 0.5 in Fig. 2a) in the x direction, while the maximum con∼∗ ∼∗ centration of positive θ,1 occurs when θ switches sign (at ϕ = 0.25π , ∼∗ y* = 0.5) in the x direction. The iso-contours of θ,2 (Fig. 3b) are quite ∼∗ ∼∗ different from those of θ,1 and θ,3 , exhibiting concentrations along both sides of the diverging separatrix. It is noted that the alignment between the quasi-streamwise vortices or the ribs and the diverging separatrix has long been confirmed (e.g. [4,11,6]). Therefore, the footprint of the quasi-streamwise vortex can be identified with the position of the diverging separatrix. The difference in the spatial distribution between ∼∗ ∼∗ ∼∗ ∼∗ ∼∗ the θ,1 - (or θ,3 -) and θ,2 -contours suggests that both θ,1 and θ,3 are ∼∗ mainly associated with the large-scale temperature front, whereas θ,2 is more likely linked to the ribs along the diverging separatrix. The remainders θ,ir2 ∗ (i = 1, 2, 3) of the mean-square velocity derivatives at x* = 10 (Fig. 3d–f) display similar iso-contours with comparable maximum concentrations near the vortex center, in distinct ∼∗ contrast to the different spatial distributions between the θ,2 (Fig. 3b) ∼∗ ∼∗ and θ,1 or θ,3 (Fig. 3a, c). This similarity indicates that the remaining smaller-scale random fluctuations θ,ir2 ∗ (i = 1, 2, 3) are mostly concentrated in the Kármán vortices and, as expected, behave more isotropically than their corresponding coherent part. Also, the maximum concentrations of the remainders θ,ir2 ∗ are much larger than their ∼∗2 coherent counterparts of θ,i (Fig. 3a–c). This is consistent with the fact that the temperature dissipation rate mostly resides at small scales. ∼∗ ∼∗ While an obvious correlation exists between θ,1 and θ,3 , θ,ir2 ∗ (i = 1, 2, 3) show little interconnection. This is confirmed by the cospectra and the phase lag of (θ,1, θ,2 ) and (θ,1, θ,3 ) at the lateral position of the vortex center (y* = 0.39) at x* = 10 (Fig. 4). The co-spectrum or the coincident spectral density function of fluctuations (α, β ) , Coαβ (f ) , is the real part of their cross-spectral density function which is the Fourier transform of the cross-correlation function of (α, β ) [3]. In the present ¯ = ∫∞ Coαβ (f ) df . The study, the co-spectrum of (α, β ) is such that αβ 0 phase lag between (α, β ) , defined as the phase angle of their crossspectral density function, reflects their phase difference at the corresponding frequency [3]. The co-spectrum of (θ ,1 , θ ,2 ) shows pronounced negative peaks at frequencies associated with the vortex shedding frequency and its second harmonic, i.e. St (≡ fsheddingd/U1, fshedding being the vortex shedding frequency) = 0.21 and 2St, indicating that the correlation θ ,1 ¯θ ,2 is largely contributed by the scales com∼∗ parable to the Kármán vortices. Note that the iso-contours of both θ,1 ∼∗ and θ,3 are characterized by two concentrations for y* ≲1, within one vortex shedding period (Fig. 3a, c). Naturally, one pronounced peak occurs at 2St. The negative peaks are consistent with the observation ∼∗ that the isocontours of θ,1 correspond approximately to the opposite∼∗ signed θ,2 -contours (Fig. 3b). A similar explanation applies for the positive peaks at frequencies corresponding to St and 2St in the co-spectrum of (θ ,1 , θ ,3 ) . The phase lag between θ,1 and θ,3 is much smaller in magnitude than that between θ,1 and θ,2 over the frequency range of 0.1 < f* < 1.0 (Fig. 4b) where θ,1 and θ,2 or θ,1 and θ,3 show appreciable correlations (Fig. 4a). The small phase lag between θ,1 and θ,3 ∼∗ is internally reconcilable with the similarity between the θ,1 - and ∼∗ θ,3 -contours (Fig. 3a, c).
Fig. 2. (a) Color-filled isocontours of the phase-averaged temperature fluctua∼∗ tion θ at x* = 10. Here and in subsequent figures, ‘+’ and ‘×’ represent the vortex center and saddle points respectively. The thicker dashed lines denote the periphery of the coherent spanwise vorticity ω3∗. The dash-dotted line through the saddle point represents the diverging separatrix. The flow is from left to right. (b) Phase-averaged sectional streamline at x* = 10.
the phase-averaged sectional streamlines (Fig. 2b), are marked by ‘+’ and ‘×’, respectively. The thick dashed lines give an approximate idea of the boundary of the spanwise vortex, which is about 25% of the ∼∗| (not shown). This contour provides a reference maximum value of |ω 3 for the periphery of the Kármán vortex. The inclined dash dotted line passing through the saddle point represents the diverging separatrix which is identified via the sectional streamline (Fig. 2b). The phase ϕ can be interpreted in terms of a longitudinal distance based on Taylor’s hypothesis; ϕ = 0 − 2π corresponding to the average vortex wavelength. The flow is left to right. The warm fluid (positive contours in Fig. 2a) is contained within the Kármán vortex, while the cool fluid (negative contours) is entrained from the freestream on either side of the wake. Large-scale temperature fronts, which can be identified with the contours with a value of 0 in Fig. 2a, form between the warm fluid in the vortex and the entrained cool fluid. These fronts are associated closely with the formation of the largescale structures of the temperature dissipation rate. Fig. 3 shows the ∼∗ three phase-averaged coherent temperature derivatives θ,i and the remainders of the mean-square temperature derivatives θ,ir2 ∗ at ∼∗ ∼∗2 x* = 10. Note that θ,i , instead of θ,i , is shown to reveal the large-scale features of the three components of the temperature dissipation rate, ∼∗ mainly because the signs of the θ,i -contours facilitate the identification ∼∗ ∼∗ ∼∗ of the relationship between θ,i and θ . The contours of θ,1 (Fig. 3a) and ∼∗ θ,3 (Fig. 3c) are qualitatively similar to each other, except that the ex∼∗ ∼∗ treme concentrations of θ,1 are about twice those of θ,3 . The similarity ∼∗ ∼∗ between θ,1 and θ,3 suggests that the large-scale temperature front in the present flow is distorted in the spanwise direction, presumably due to the spanwise distortion of the vortex rolls (e.g. [11]). Their extreme concentrations with opposite signs, occurring on either side of the vortex center, are located approximately in the regions where the warm fluid in the vortex and the entrained cool fluid meet, i.e. at the temperature fronts (the red dashed contours in Fig. 3a–c which correspond ∼∗ to the zero contours in Fig. 2a). The connection between θ,1 and the large-scale temperature front can also be identified by the observation
4. The dependence of χ on θ Sinai and Yakhot [19] derived an exact solution of the pdf equation of a passive scalar with no mean scalar gradient, in the limit of time t → ∞, viz. −1
p (θ θrms ) = C ⎡ ⎢ ⎣
E (χ θ) ⎤ ·exp ⎛⎜ χ ⎥ ⎦ ⎝
∫0
θ θrms
−
βdβ ⎞ ⎟ E (χ β ) χ ⎠
(4.1)
where C is a constant. It essentially describes the distribution of the passive scalar in a self-similar state, and is well satisfied in the grid 3
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∼ ∼ ∼ Fig. 3. Iso-contours of (a–c) the phase-averaged coherent temperature derivatives θ ,1∗ , θ ,∗2 , θ ,∗3 and (d–f) the remainders 〈θ,12r 〉∗, 〈θ,22r 〉∗, 〈θ,32r 〉∗ at x* = 10. Contour intervals: (a–c) 0.15, 0.15, 0.084; (d–f) 0.22, 0.28, 0.30. The red dashed contours (contours with a value of 0 in Fig. 2a) are the positions of the large-scale temperature fronts. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
turbulence at θ θrms > 2. For a clearer picture of the effect of coherent events on the correlation between χ and θ as the flow develops downstream, we employed the conditional correlation coefficient, viz.
turbulence without a mean scalar gradient (e.g. [12]). It indicates that the pdf of the passive scalar, p (θ θrms ) , is uniquely determined by the expectation of its conditional dissipation rate E (χ θ) for a given value of θ. When χ and θ are independent of each other, E (χ θ) χ is constant and equal to 1, leading to a Gaussian distribution of p (θ θrms ) . The large ∼∗ extreme concentration of θ,1 at x* = 10 (Fig. 3a) indicates that the small-scale temperature derivative feels the influence of the coherent events in the flow, implying that the coupling between χ and θ may be enhanced by the coherent motions. The distribution of E (χ θ) χ on the centerline from x* = 10 to x* = 40 confirms this point (Fig. 5). The data in a grid flow [12] without a mean temperature gradient are also included for comparison. At x* = 10 and 20, where the coherent motions are strong, the conditioned scalar dissipation rate displays a marked dependence on θ θrms ; the expectation of the dissipation rate at x* = 10 varies almost proportionally to θ θrms . The temperature dissipation rates where θ θrms (negative) is smallest approach zero for x* = 10 and 20. This is reasonable. The cool fluid entrained from the freestream is present on the centerline (Fig. 2a). The unmixed cool fluid comes from the potential freestream and the temperature dissipation rate in that fluid is expected to be zero [13]. In contrast, at x* = 40, where the coherent motions are rather weak, the distribution of E (χ θ) χ appears to be independent of θ θrms , with a value of approximately unity; in fact, this distribution is very similar to that in the grid flow, except at θ θrms > 2. At large θ θrms , the intermittency of the dissipation rate increases and E (χ θ) χ rises well above unity, which accounts for the large E (χ θ) χ at all three x* stations. The intermittency caused by the extreme events of the temperature field is expected to be flow-dependent and thus it is reasonable to see the departure of the distributions of E (χ θ) χ at x* = 40 and in the grid
¯ χ − E (χ )) ⎤ (θ − E (θ))( ρ (θ , χ ϕ) ≡ ⎡ ϕ ⎢ ⎥ σθ σχ ⎣ ⎦
(4.2)
Here, σθ and σχ are the standard deviations of θ and χ at phase ϕ , respectively. The ρ (θ , χ θ) quantifies the correlation between θ and χ at the same phase. It should be noted that, at a given phase ϕ , E (θ) and E (χ ) are equal to their corresponding phase-averaged values at this ∼ χ (ϕ) . Fig. 6 shows the distribution of ρ (θ , χ θ) at phase, i.e. θ (ϕ) and ∼ ∼∗ all three x* positions. The isocontours of θ at each x* position are also superposed onto the color-filled isocontours of ρ (θ , χ θ) for reference. At x* = 10 (Fig. 6a), the large correlation coefficient between θ and χ , of the order of 0.7, is found in the saddle region along the diverging separatrix where the temperature front forms, while a much smaller value, close to 0, is associated with the downstream half of the vortex. The distribution of ρ (θ , χ θ) suggests that the evident dependence between θ and χ at x* = 10 in Fig. 5 is due to the combined effect of the coherent strain rate and the temperature front at the saddle region. Note that only when the temperature front is present, e.g. immediately downstream of the vortex, ρ (θ , χ θ) is quite small. A similar distribution of ρ (θ , χ θ) is found for x* = 20 (Fig. 6b), although the magnitude is reduced, which is consistent with the dependence between θ and χ at x* = 20 in Fig. 5. However, for x* = 40, the situation changes: ρ (θ , χ θ) is generally below 0.1, except in the region above the saddle point near the boundary between the wake and the freestream. This is mainly because at x* = 40, the coherent strain rate at the saddle region 4
Experimental Thermal and Fluid Science 114 (2020) 110050
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Fig. 4. (a) Co-spectra and (b) the phase-lag between θ,1 and θ,2 and between θ,1 and θ,3 at the lateral position corresponding to the spanwise vortex center (y* = 0.39) at x* = 10.
Fig. 6. Color-filled isocontours of the conditional correlation coefficient between θ and χ at the same phase. (a) x* = 10, (b) 20, (c) 40. The scales of the ∼∗ color bar are the same for (a)-(c). The iso-contours of θ at each x* position are ∼∗ superposed for reference. Solid contours correspond to θ > 0 and dashed ∼∗ contours are for θ < 0.
weakens due to the degraded rotational motion of the vortex; the intensity of the vortex entrainment, which gives rise to the temperature front, reduces and now occurs away from the wake centerline. The small magnitude of ρ (θ , χ θ) at x* = 40 explains why θ and χ are independent of each other at x* = 40 in Fig. 5. In view of the independence between χ and θ shown in Figs. 5 and 6c at x* = 40 as well as Eq. (4.2), it is inferred that the distribution of θ is closer to be Gaussian at x* = 40 than x* = 10 or 20. This expectation is confirmed by the relationship between the departure of the pdf of θ from the Gaussian distribution and the level of correlation between χ and θ 2 (Fig. 7). The parameter δ ≡ Sθ + Fθ −3 is employed to characterize the departure of the pdf of θ from the Gaussian distribution (Sθ =0 and Fθ = 3), where Sθ and Fθ are the skewness and kurtosis of θ , respectively (e.g. [15]). The non-centered correlation coefficient between θ 2 and χ , given by
Fig. 5. Expectation of the temperature dissipation rate conditioned on the temperature fluctuation at the wake centerline from x* = 10 to 40. The conditional temperature dissipation rate in grid turbulence [12] is also included. The horizontal dashed line corresponds to the value of 1.
R≡
θ 2¯χ −1 θ¯2χ
(4.3)
quantifies the statistical dependence between θ 2 and χ (e.g. [8]). R is somewhat similar to the centered correlation coefficient 2 ρ (=(θ 2 − θ¯2¯)(χ − χ ) (θrms χrms )) but is easier to evaluate accurately [12] 5
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between θ 2 and χ corresponds to a nearly Gaussian distribution of θ at x* = 40. Note that Mi [15] did not distinguish between different distributions of (R, δ) at different x* positions as he focused mainly on differences in (R, δ) among different flows. In contrast, the concentration of (R, δ) at x* = 10 and 20 lies generally away from the origin. Consistent with the distributions of (R, δ), the pdfs of θ along the centerline at x* = 10 (and 20) are evidently skewed, while those at x* = 40 are basically Gaussian (shown later in Fig. 8 of Section 5). The linear relationship between R and δ reflects the direct connection between the dependence of the scalar dissipation rate χ on the scalar θ and the pdf of θ . This is consistent with the analytical finding of O’Brien and Jiang [16] that the pdf of θ being Gaussian is a sufficient and necessary condition of χ to be independent of θ in homogenous turbulence, although the present flow is far from being homogenous, especially at x/d = 10 and 20. The dynamical reason behind the ‘linear’ behavior of the connection of R and δ is still unclear. Mi [15] suggests that the linear relationship between R and δ represent a general feature of the scalar property in turbulence. It would be interesting in the future to derive the analytical connection between R and δ based on the governing equations of the passive scalar in turbulence and compare with the experimental results observed in different flows.
Fig. 7. The relationship between δ and R across the flow at x* = 10 to 40. The turbulent plane wake data of Mi [15] are also included.
since it is usually much larger than ρ . The points (R, δ) in the present flow exhibit roughly a linear distribution, i.e. the departure of the pdf of θ from Gaussian distribution is proportional to the extent of the correlation between θ 2 and χ . This is quite similar to what Mi [15] has observed in a plane wake (Re = 3200). The concentration of (R, δ) for x* = 40 is generally around the origin, i.e. the weak correlation
5. Topological features of the passive scalar mixing In this section, we attempt to identify the most efficient turbulent mixing region in the wake by examining the mixing process between the engulfed cool fluid and the warm wake, which provides an
∼∗ Fig. 8. Iso-contours of θ (the left column) and the conditional probability density function of the temperature, P (θ ϕ) , with respect to the Kármán vortex (the right column). (a, d) x* = 10; (b, e) 20, (c, f) 40. 6
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departure from the Gaussian distribution of θ at this x* in Fig. 7. In the topological picture of the coherent structures proposed by HH in the near wake of a cylinder, they presumed that the direct interaction between the rib fluid, mostly with longitudinal vorticity, and the fluid in the spanwise vortex rolls produces strong three-dimensional vorticity fluctuations in region A, and enhances the mixing in this region. However, Fig. 8d–f suggests that the mixing between cool and warm fluids in regions C & D is more effective than that in region A. This feature implies that the region of most effective mixing in the plane wake is different from that surmised by HH, which suggests that the mechanism for the turbulent mixing could be different from that conjectured by HH. Turbulent mixing can be viewed as a three-stage process of entrainment, dispersion and diffusion [7]. In the present flow, the cool fluid is entrained into the wake by the large-scale vortices, while the consequent mixing between the cool and warm fluids, i.e. the thermal dispersion and diffusion, at the interface, is due to the intermediateand small-scale motions. Zhou et al. [25] studied the two rows of the vortex streets of the wake of two side-by-side cylinders. They observed that the inner vortices from either cylinder wake decays more rapidly than the outer vortices because of the vorticity cancellation caused by the vorticity flux between the opposite-signed inner vortices. Similarly, there is also intense vorticity fluxes (from the positive vortex to the negative vortex) in both regions C and D, especially region C at x* = 10 and 20 [6]. This vorticity transport should accelerate the thermal dispersion and diffusion more effectively than the interaction between the streamwise ribs and the spanwise rolls, which can be confirmed by the much larger incoherent heat flux away from the vortex in region C than in region A [14]. This is very similar to the reason why the turbulent mixing of the passive scalar in regions C (and D) is more effective than in region A. The entrained fluid at x* = 10 is mixed with the wake fluid continuously as the flow develops downstream. At x* = 40 (Fig. 8c), the sharp termperature front near the wake centerline has virtually gone, indicating a well mixed stage of the warm and cool fluids, as confirmed by the reasonable agreenment between the pdfs of θ and the Gaussian pdf (Fig. 8f). At last, it is worth commenting on the evolution of the characteristics of the turbulent mixing downstream. The present observation that the turbulent mixing of the scalar with freestream fluid is faster near the centerline depends largely on the entrainment of the coherent vortices and the vorticity fluxes between the opposite-signed vortices in this flow range (x/d = 10–40). In further downstream where the freestream fluid cannot be entrained into the wake centerline due to the degraded coherent vortices, the scalar mixing would mainly happen in the turbulent/non-turbulent interface away from the centerline (e.g. Westerweel et al., 2009). Also, the readers should bear in mind that the scalarθ is trapped inside the vortex is largely due to the effect of the initial condition that the scalarθ is injected into the flow via heating the cylinder. Warm fluid is thus kept inside when the thermal shear layer warps into Kármán vortices. Since the random turbulent kinetic energy has its maximum concentration within the vortex [11], the flow region inside the vortex could also be associated with effective turbulent mixing when the initial condition of θ changes, e.g. the fluctuation of temperature being introduced by a toaster or a mandoline (e.g. [12]), such that both cool and warm fluids could be present within the vortex.
experimental evaluation of the turbulent mixing scenario in the present flow proposed by Hussain and Hayakawa [11], (hereinafter referred to as HH). The physics behind the scalar mixing process is also discussed. Before the mixing process is investigated, it is necessary to find a reliable means of assessing how the turbulent mixing takes place. Roshko [18] examined the mixing process in a mixing layer between two different passive scalars (nitrogen and helium-argon), and found that in the region between the two passive scalars where the mixing is most sufficient, the pdf of either scalar concentration is symmetric with respect to its mean value but the pdf of the scalar in the unmixed region away from the middle region is highly skewed. This feature provides us with a simple qualitative way of assessing the mixedness of the passive scalar in the present flow, i.e. the degree of mixing between warm and cool fluids in a particular flow region can be inferred from the departure from symmetry of the temperature pdf. In order to identify the region where turbulent mixing is most efficient, we divide the flow region with respect to the Kármán vortex into five representative areas, denoted by A-E in Fig. 8. Area A is the region where the longitudinal rib structures and the spanwise vortex rolls interact with each other and where HH speculated that the most effective mixing takes place; B is the region downstream of the vortex at the same y* position as A; C is the region downstream of the vortex near the temperature front close to the centerline; D is a region upstream of the vortex close to the temperature front at the same y* position as C; E is at the vortex center. The pdfs of θ conditioned on the phases ϕ of A–E, denoted by P (θ ϕ) , are calculated using the temporal traces of θ measured at these five locations (ϕ , y*). We have checked that P (θ ϕ) in the vicinity of A–E are qualitatively similar to the pdfs at A–E, so P (θ ϕ) at A–E are reasonable representatives of the temperature distribution in the corresponding region. The number of data points used for calculating the conditional pdf of θ is the same as that for estimating the phase average of θ , i.e. approximately 1980, 1700 and 1050 for x* = 10, 20 and 40 respectively. The convergence of the conditional pdf of θ has also been confirmed by the collapse of the pdfs with the number of samples beyond 750 (not shown). Note that P (θ ϕ) is transformed to P (θ ϕ) with the corre∼ sponding mean θ (ϕ) and standard deviation σθ (ϕ) , i.e. ∼ θ ϕ=(θ (ϕ) − θ (ϕ)) σθ (ϕ) , so that the distribution P (θ ϕ) can be compared with a standard Gaussian distribution in Fig. 8d–f. At x* = 10 (Fig. 8d), the pdfs at four locations on the periphery of the vortex (A–D) are highly skewed. The sharp cut-off on the negative side of the pdfs is due to the presence of ambient cool fluid. The skewness of the pdfs indicates that the warm and cool fluids are not well mixed at these locations, except at the vortex center (E) where the pdf displays good symmetry and follows the Gaussian distribution reasonably well. Note that the warm fluid is predominant around the vortex center (Fig. 8a) due to the initial condition, i.e. the thermal shear layer shedding from the cylinder and wrapping into vortices. The high turbulence around the vortex center (cf. Fig. 8 of HH) leads to the passive scalar in this region being adequately mixed. The excellent agreement between the pdf at the vortex center and the Gaussian distribution also confirms the effectiveness of the pdf method in assessing mixing. At x* = 20 (Fig. 8e), the pdfs of A and B remain similar to their counterparts at x* = 10; however, the pdfs at C and D are closer to the pdf at E or the Gaussian distribution and their sharp cutoff at x* = 10 has virtually disappeared. The different behavior of P (θ ϕ) at A (or B) and D (or C) from x* = 10 to 20 implies that the mixing between the entrained cool fluid and the warm wake is more rapid at C and D than at A and B. At x* = 40 (Fig. 8f), all distributions of P (θ ϕ) collapse reasonably well onto the Gaussian distribution, although the cut-off is still discernible on the negative side of P (θ ϕ) at B. It is worth noting that region B is where the cool fluid has just been entrained into the wake (Fig. 8c), although the entrainment is rather weak at x* = 40. Therefore, the persistence of the cut-off is not surprising. The Gaussian distribution of P (θ ϕ) at x* = 40 is also consistent with the small
6. Conclusions The characteristics of the passive scalar (temperature) variance dissipation rate are studied in a slightly heated turbulent wake based on data obtained with a multi-wire probe. We have examined the effects of the coherent vortices on the components of χ and on the relationship between χ and θ . Besides, the turbulent mixing process with respect to the Kármán vortex is also investigated. Three major conclusions are drawn from this work. 7
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Declaration of Competing Interest
(1) The three components of the temperature gradient feel the influ∼ ence of the coherent vortices in different ways. The component θ ,1∗ is maximum at or near the large-scale temperature front which forms due to the entrainment of cool fluid from the freestream by ∼ the spanwise vortices, whereas θ ,∗2 is associated with the quasistreamwise vortex and is concentrated along the diverging separa∼ ∼ trix. The similarity between the contours of θ ,∗3 and θ ,1∗ is most likely due to the spanwise kink of the temperature front, which is caused by the distortion of the vortex roll [11]. (2) The χ and θ exhibit a correlation at x* = 10 and 20, but tend to be statistically independent of each other at x* = 40. The difference in the inter-connection between χ and θ from x* = 10 or 20 to 40 indicates that the large and small scales in the temperature field are strongly coupled in the presence of the organized Kármán vortices. The phase-averaged correlation coefficient between χ and θ , ρ (θ , χ ϕ) , shows that this is mainly due to the strain rate and the temperature front at the saddle region at x* = 10 and 20. The roughly linear distribution of (δ, R) at x* = 10 and 20 indicates that the coupling between χ and θ is proportional to the extent to which the distribution of θ departs from Gaussianity; the data for the combination (δ, R) at x* = 40 is clustered mostly around the origin, suggesting that the distribution of θ is basically Gaussian and independent of χ when the distance is large enough (say x* ⩾ 40) from the cylinder. It is thus inferred that the assumption of independence of χ and θ usually made in turbulent combustion modeling (e.g. [17]) is valid only when, at least for the present flow, the effects of the rate of strain and temperature front caused by the large-scale vortex can be ignored. (3) The region where the most effective mixing between the entrained cool fluid and the warm fluid takes place is around the temperature front in the inner half of the vortices near the centerline, not in the region where the ribs and rolls contact with each other, which is conjectured by Hussain and Hayakawa [11]. Between x* = 10 and x* = 40, the pdfs of θ in regions C and D (Fig. 8) evolve towards a Gaussian distribution more rapidly than those in regions A and B, indicating that the turbulent mixing is more efficient in regions C and D. This result also suggests that the mechanism responsible for the turbulent mixing between the wake and the entrained ambient fluid is different from that surmised by Hussain and Hayakawa [11]. The vorticity transport between opposite-signed vortices is likely to be the dominant physics which is responsible for the efficient turbulent mixing in regions C and D, particularly region C. Although the present experiment is conducted at a moderate Reynolds number, the topological features of the flow are essentially unchanged as long as Re is smaller than its critical value, approximately 2.5 × 105 [24]. As a matter of fact, Yiu et al. [23] studied the Re effect on the three dimensional vorticity in a turbulent cylinder wake with Re = 2500, 5000 and 10,000 and showed that the topology of the wake (x/d = 10–40) is essentially the same for this range of Reynolds numbers although the three-dimensionality of the flow is enhanced at higher Re.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements YZ wishes to acknowledge support given to him from NSFC through grants 11632006, 91752109 and U1613226 and from Research Grants Council of Shenzhen Government through grants JCYJ20160531193220561 and JCY20160531192108351. References [1] R.A. Antonia, L. Browne, Anisotropy of the temperature dissipation in a turbulent wake, J. Fluid Mech. 163 (1986) 393–403. [2] J. Bakosi, P. Franzese, Z. Boybeyi, Probability density function modeling of scalar mixing from concentrated sources in turbulent channel flow, Phys. Fluids 19 (2007) 115106. [3] J.S. Bendat, A.G. Piersol, Random Data: Analysis and Measurement Procedures, fourth ed., Wiley, Hoboken, N.J., 2010. [4] B.J. Cantwell, Organized motion in turbulent flow, Annu. Rev. Fluid Mech. 13 (1981) 457–515. [5] J.G. Chen, T.M. Zhou, R.A. Antonia, Y. Zhou, Comparison between passive scalar and velocity fields in a turbulent cylinder wake, J. Fluid Mech. 813 (2017) 667–694. [6] J.G. Chen, Y. Zhou, T.M. Zhou, R.A. Antonia, Three-dimensional vorticity, momentum and heat transport in a turbulent cylinder wake, J. Fluid Mech. 809 (2016) 135–167. [7] P.E. Dimotakis, Turbulent mixing, Annu. Rev. Fluid Mech. 37 (2005) 329–356. [8] V. Eswaran, S.B. Pope, Direct numerical simulations of the turbulent mixing of a passive scalar, Phys. Fluids 31 (1988) 506–520. [9] C.H. Gibson, C.A. Friehe, S.O. McConnell, Structure of sheared turbulent fields, Phys. Fluids 20 (1977) S156. [10] M. Holzer, E.D. Siggia, Turbulent mixing of a passive scalar, Phys. Fluids 6 (5) (1994) 1820–1837. [11] A.K.M.F. Hussain, M. Hayakawa, Eduction of large-scale organized structures in a turbulent plane wake, J. Fluid Mech. 180 (1987) 193–229. [12] Warhaft, Z. Jayesh, Probability distribution, conditional dissipation, and transport of passive temperature fluctuations in grid-generated turbulence, Phys. Fluids Fluid Dyn. 4 (1992) 2292–2307. [13] P. Kailasnath, K.R. Sreenivasan, J.R. Saylor, Conditional scalar dissipation rates in turbulent wakes, jets, and boundary layers, Phys. Fluids Fluid Dyn. 5 (1993) 3207–3215. [14] M. Matsumura, R.A. Antonia, Momentum and heat transport in the turbulent intermediate wake of a circular cylinder, J. Fluid Mech. 250 (1993) 651–668. [15] J. Mi, Correlation between non-Gaussian statistics of a scalar and its dissipation rate in turbulent flows, Phys. Rev. E 74 (2006) 016301. [16] E.E. O’Brien, T. Jiang, The conditional dissipation rate of an initially binary scalar in homogeneous turbulence, Phys. Fluids A: Fluid Dynam. 3 (1991) 3121–3123. [17] H. Pitsch, Large-Eddy simulation of turbulent combustion, Annu. Rev. Fluid Mech. 38 (2006) 453–482. [18] A. Roshko, Structure of turbulent shear flows: a new look, AIAA J. 14 (1976) 1349–1357. [19] Ya.G. Sinai, V. Yakhot, Limiting probability distributions of a passive scalar in a random velocity field, Phys. Rev. Lett. 63 (1989) 1962–1964. [20] N. Soulopoulos, Y. Hardalupas, A.M.K.P. Taylor, Mixing and scalar dissipation rate statistics in a starting gas jet, Phys. Fluids 27 (2015) 125103. [21] K.R. Sreenivasan, On local isotropy of passive scalars in turbulent shear flows, Proc. R. Soc. Lond. Ser. A 434 (1991) 165–182. [22] Z. Warhaft, Passive scalars in turbulent flows, Annu. Rev. Fluid Mech. 32 (2000) 203–240. [23] M.W. Yiu, Y. Zhou, T. Zhou, L. Cheng, Reynolds number effects on three-dimensiional vorticity in a turbulent wake, AIAA J. 42 (2004) 1009–1016. [24] M.M. Zdravkovich, Flow Around Circular Cylinders, Oxford science publications, Oxford University Press, Oxford; New York, 1997. [25] Y. Zhou, H.J. Zhang, M.W. Yiu, The turbulent wake of two side-by-side circular cylinders, J. Fluid Mech. 458 (2002) 303–332.
CRediT authorship contribution statement J.G. Chen: Formal analysis, Methodology, Visualization, Writing original draft. R.A. Antonia: Conceptualization, Writing - review & editing. Y. Zhou: Writing - review & editing, Supervision. T.M. Zhou: Investigation, Data curation, Validation.
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Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs
Characteristics of temperature dissipation rate in a turbulent near wake J.G. Chen
a,b
c
, R.A. Antonia , Y. Zhou
a,b,⁎
, T.M. Zhou
T
d
a
Institute for Turbulence-Noise-Vibration Interactions and Control, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China Digital Engineering Laboratory of Offshore Equipment, Shenzhen 518055, China School of Engineering, University of Newcastle, NSW 2308, Australia d Department of Civil, Environmental and Mining Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia b c
A R T I C LE I N FO
A B S T R A C T
Keywords: Temperature dissipation rate Turbulent wake Mixing
This work aims to provide in-depth understanding of the effects the quasi-periodical Kármán vortices produce on the temperature dissipation rate in a turbulent cylinder near wake. Measurements are made at x/d = 10, 20 and 40, where x is the streamwise distance from the cylinder axis and d is the cylinder diameter, with a Reynolds number of 2.5 × 103 based on d and the free-stream velocity. A multi-wire probe is deployed to measure simultaneously the fluctuating temperature and its gradient vector, at nominally the same spatial point in the plane of the mean shear. It is found that the coherent streamwise and spanwise temperature derivatives are similarly distributed with respect to the spanwise vortex, exhibiting twin peaks at the temperature fronts, while the coherent lateral component is linked to the rib-like structures. The temperature variance dissipation rate is found to be statistically independent of the temperature fluctuation when the Kármán vortex is so weak (say at x/d = 40) that the large-scale temperature front resulting from the vortex entrainment ceases to be present and the coherent strain rate at the saddle region is relatively small. In addition, the most effective turbulent mixing is found to take place around the temperature front near the wake centerline, which is in contrast to the conjecture by Hussain and Hayakawa (1987).
1. Introduction The mean dissipation rate of a passive scalar variance θ¯2 is defined as
χ = κ (θ¯,12 + θ¯,22 + θ¯,32 )
(1.1)
where an overbar denotes time-averaging and θ ,i ≡ ∂θ ∂x i , i = 1, 2, 3 represents the streamwise, lateral and spanwise directions, respectively, which are used interchangeably with x, y and z in the present study. In (1.1), κ is the scalar diffusivity or the thermal diffusivity if the passive scalar is temperature. χ has been extensively studied in the past few decades, as it plays a central role in the turbulent combustion modeling, both in the Reynolds-averaged Navier-Stockes (RANS) and Large-eddy simulation (LES) approaches (see the review of [17]). In his review, Pitsch [17] pointed out that production/dissipation balance assumption which seems to be inherently used in all models for the scalar dissipation rate in LES of turbulent combustion is not always applicable due to the effect of the large-scale structures in the flow. For instance, the ‘ramp-cliff’ structures of the scalar resulting from the large-scale turbulent motions (e.g. [9,10]), instead of the energy cascade, cannot be described by the production/dissipation balance assumption. It is
⁎
thus naturally of great interest to examine how the behavior of χ is influenced by the large-scale structures in the turbulent flow, which is the primary motivation of the present study. There is evidence that the scalar dissipation is largely anisotropic, particularly in shear flows (e.g. [21,22]). Even in the self-preserving far wake, Antonia and Browne [1] observed that the mean-square value of the transverse derivative of temperature is larger than that of the longitudinal derivative. One may expect that the three components of the scalar dissipation rate are affected differently by the large-scale motions. Jayesh and Warhaft [12] found that for grid turbulence without a mean temperature gradient, χ is statistically independent of θ , leading to a Gaussian distribution of θ . As is known, the expectation of χ conditioned on θ plays an important role in closing the transport equation of the probability density function (pdf) of θ [13,2,20]. It is expected that in the turbulent near wake which is characterized by an appreciable streamwise decay of anisotropic large-scale organized motions, such as the spanwise Kármán vortices and the quasi-streamwise ribs, the situation could be very different from the self-preserving far wake or grid turbulence. This raises several questions regarding the effect of these organized motions on the temperature dissipation. How are the three components of χ spatially distributed with respect to the
Corresponding author at: Institute for Turbulence-Noise-Vibration Interactions and Control, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China. E-mail address: [email protected] (Y. Zhou).
https://doi.org/10.1016/j.expthermflusci.2020.110050 Received 12 September 2019; Received in revised form 4 January 2020; Accepted 15 January 2020 Available online 29 January 2020 0894-1777/ © 2020 Published by Elsevier Inc.
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organized structures in the near wake? How is the dependence between χ and θ affected by these coherent vortices? How does the dependence between χ and θ evolve with x as the structures weaken? These issues have yet to be clarified. A resolution of these issues would provide us with valuable insight not only into the physics of the small-scale temperature dissipation rate itself, but also its interaction with the temperature fluctuation, especially when the transporting flow is perturbed by organized motions. This work also attempts to shed light on the influence of the coherent motions on the turbulent mixing of the passive scalar. Hussain & Hayakawa [11] showed the topological scheme for the coherent structures of a turbulent plane wake (see also [24]). It presents a clear picture of the spatial distribution of the turbulence production and mixing, in the context of the organized vortex structures, showing that turbulence production took place in the saddle region due to the vortex stretching, and the turbulent kinetic energy was transported along the separatrix and accumulated within the vortices. They also conjectured that the turbulent mixing occurred mostly in the region where the streamwise and spanwise vortices are in contact with each other. Experimental evidence is actually still lacking on this point. In the present experiment in the near wake, relative to the more homogeneous and self-preserving far wake, of a circular cylinder the ambient cool fluid is engulfed by the Kármán vortex into the heated wake, so the turbulent mixing region can be identified by examining the mixing performance between the warm wake and the engulfed cool fluid in different local regions with respect to the large-scale vortices. This could also provide a new understanding of the physics of turbulent mixing in the present flow.
Fig. 1. (a) Experimental arrangement and the coordinate system; (b) side and (c) front views of the multi-wire probe.
‘wake’ of the cold wires on the X-hotwires is negligible. With this arrangement, the velocity gradients involved in the three vorticity components ωi (i = 1, 2 and 3) and the temperature dissipation rate χ can be measured simultaneously. Please refer to Chen et al. [6] for more details about the probe and its performance of measuring ui, ωi and θ . A detailed assessment of the spatial resolution of the probe for both velocity and temperature derivatives is available in Chen et al. [5] and will not be repeated here. Note that only the results for θ and θ,i are discussed in this study. A phase-averaging technique is used to extract the coherent structures from the flow field. The technique is the same as that used in Chen et al [6]. Briefly, the instantaneous quantity Γ may be viewed as the sum of the time-averaged component Γ , and the fluctuating component ∼ β , which can be further decomposed into a coherent fluctuation β and a remainder βr , namely:
2. Experimental details and phase-averaging technique The detailed information on the experimental arrangement was given in Chen et al. [6]. Some experimental details that are relevant to the present study are briefly recalled below. Experiments were conducted in an open-loop wind tunnel with a working section of 1.2 m (width) × 0.8 m (height) and 2.0 m long. The outer diameter of the circular cylinder used to generate the wake is 12.7 mm. A nickel wire with a diameter of about 0.5 mm was coiled along the thread of a ceramic tube of about 11 mm in diameter, which was then placed inside the cylinder and acted as a heating element. The cylinder was heated in such a way that the temperature on the surface of cylinder was constant. The free-stream velocity U1 was 3.0 m/s, corresponding to a Reynolds number Re (≡ U1d ν ) of 2.5 × 103, where ν is the kinematic viscosity. At this Reynolds number, the transition takes place at the intermediate part of the free shear layer, and the wake has become fully turbulent [24]. The coordinate system and some symbols are defined in Fig. 1a. Measurements were conducted within the cylinder mid-span plane at x/d = 10, 20 and 40 where x is the streamwise distance from the cylinder axis. The maximum mean temperature excess Θ0, relative to the ambient fluid, was approximately 1.6 °C, 1.5 °C and 1.3 °C on the centerline for x* = 10, 20 and 40, respectively. This excess is small enough to avoid any buoyancy effects and hence allow temperature to be treated as a passive scalar. Hereafter, an asterisk ‘*’ denotes normalization by d, U1 and/or Θ0 . A probe consisting of four X-hotwires (X1-X4 in Fig. 1b and c) and four cold wires (C1-C4 in Fig. 1b and c) was used to measure simultaneously the fluctuating velocity ui and temperature θ at nominally the same spatial point. Two of the X-wires, X2 and X4, were aligned in the (x, y) plane and separated by Δz = 2.7mm in the z direction; the other two, X1 and X3, were in the (x, z) plane and separated by Δy = 2.0mm in the y direction. The separation between the two wires of each Xprobe was approximately 0.6 mm. Four cold wires with a diameter of 1.27 μm were orthogonally arranged in the (y, z) plane and perpendicular to the planes of the four X-hotwires and placed at approximately 1 mm (approximately 800 diameters of the cold wire) upstream of the latter, which is sufficiently far to ensure that the influence from the
∼ β = β + βr
(2.1)
The phase average of an instantaneous quantity Γ at a particular N phase ϕ is given by Γ (ϕ) = (1 N ) ∑i = 1 Γi (ϕ) , where N is the total number of detections, which is approximately 1980, 1700 and 1050 for x* = 10, 20 and 40 respectively. It has been checked (not shown) that the number of detections has been large enough for Γ (ϕ) to be converged across the flow. The vortex shedding period is divided into 20 phases. The reference phase is the phase of the local velocity fluctuation ∼ v at each y* position. The coherent component β ≡ β reflects the effect from the large-scale coherent motions, while βr is associated with the remaining smaller-scale motions. 3. The components of the temperature dissipation rate In the turbulent near wake, the flow is dominated by the large-scale spanwise Kármán vortices and the streamwise rib-like structures. Our interest in this section is primarily on how the behavior of the three components of the temperature dissipation rate differ from each other under the effect of these coherent motions. In addition, the effect of these coherent events on the connection between the small-scale temperature dissipation rate and temperature fluctuation is also examined. ∼ We start by presenting the isocontours of θ∗ (Fig. 2a) at x* = 10 for a visual perspective of the large-scale temperature concentration. It was first shown in Chen et al. [6]. In Fig. 2a and the following iso-contours figures, the positions of the centers and saddle points, identified from 2
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∼∗ that the maximum concentration of negative θ,1 takes place at a loca∼∗ tion where θ changes sign (positive to negative, at ϕ = −0.25π , y* = 0.5 in Fig. 2a) in the x direction, while the maximum con∼∗ ∼∗ centration of positive θ,1 occurs when θ switches sign (at ϕ = 0.25π , ∼∗ y* = 0.5) in the x direction. The iso-contours of θ,2 (Fig. 3b) are quite ∼∗ ∼∗ different from those of θ,1 and θ,3 , exhibiting concentrations along both sides of the diverging separatrix. It is noted that the alignment between the quasi-streamwise vortices or the ribs and the diverging separatrix has long been confirmed (e.g. [4,11,6]). Therefore, the footprint of the quasi-streamwise vortex can be identified with the position of the diverging separatrix. The difference in the spatial distribution between ∼∗ ∼∗ ∼∗ ∼∗ ∼∗ the θ,1 - (or θ,3 -) and θ,2 -contours suggests that both θ,1 and θ,3 are ∼∗ mainly associated with the large-scale temperature front, whereas θ,2 is more likely linked to the ribs along the diverging separatrix. The remainders θ,ir2 ∗ (i = 1, 2, 3) of the mean-square velocity derivatives at x* = 10 (Fig. 3d–f) display similar iso-contours with comparable maximum concentrations near the vortex center, in distinct ∼∗ contrast to the different spatial distributions between the θ,2 (Fig. 3b) ∼∗ ∼∗ and θ,1 or θ,3 (Fig. 3a, c). This similarity indicates that the remaining smaller-scale random fluctuations θ,ir2 ∗ (i = 1, 2, 3) are mostly concentrated in the Kármán vortices and, as expected, behave more isotropically than their corresponding coherent part. Also, the maximum concentrations of the remainders θ,ir2 ∗ are much larger than their ∼∗2 coherent counterparts of θ,i (Fig. 3a–c). This is consistent with the fact that the temperature dissipation rate mostly resides at small scales. ∼∗ ∼∗ While an obvious correlation exists between θ,1 and θ,3 , θ,ir2 ∗ (i = 1, 2, 3) show little interconnection. This is confirmed by the cospectra and the phase lag of (θ,1, θ,2 ) and (θ,1, θ,3 ) at the lateral position of the vortex center (y* = 0.39) at x* = 10 (Fig. 4). The co-spectrum or the coincident spectral density function of fluctuations (α, β ) , Coαβ (f ) , is the real part of their cross-spectral density function which is the Fourier transform of the cross-correlation function of (α, β ) [3]. In the present ¯ = ∫∞ Coαβ (f ) df . The study, the co-spectrum of (α, β ) is such that αβ 0 phase lag between (α, β ) , defined as the phase angle of their crossspectral density function, reflects their phase difference at the corresponding frequency [3]. The co-spectrum of (θ ,1 , θ ,2 ) shows pronounced negative peaks at frequencies associated with the vortex shedding frequency and its second harmonic, i.e. St (≡ fsheddingd/U1, fshedding being the vortex shedding frequency) = 0.21 and 2St, indicating that the correlation θ ,1 ¯θ ,2 is largely contributed by the scales com∼∗ parable to the Kármán vortices. Note that the iso-contours of both θ,1 ∼∗ and θ,3 are characterized by two concentrations for y* ≲1, within one vortex shedding period (Fig. 3a, c). Naturally, one pronounced peak occurs at 2St. The negative peaks are consistent with the observation ∼∗ that the isocontours of θ,1 correspond approximately to the opposite∼∗ signed θ,2 -contours (Fig. 3b). A similar explanation applies for the positive peaks at frequencies corresponding to St and 2St in the co-spectrum of (θ ,1 , θ ,3 ) . The phase lag between θ,1 and θ,3 is much smaller in magnitude than that between θ,1 and θ,2 over the frequency range of 0.1 < f* < 1.0 (Fig. 4b) where θ,1 and θ,2 or θ,1 and θ,3 show appreciable correlations (Fig. 4a). The small phase lag between θ,1 and θ,3 ∼∗ is internally reconcilable with the similarity between the θ,1 - and ∼∗ θ,3 -contours (Fig. 3a, c).
Fig. 2. (a) Color-filled isocontours of the phase-averaged temperature fluctua∼∗ tion θ at x* = 10. Here and in subsequent figures, ‘+’ and ‘×’ represent the vortex center and saddle points respectively. The thicker dashed lines denote the periphery of the coherent spanwise vorticity ω3∗. The dash-dotted line through the saddle point represents the diverging separatrix. The flow is from left to right. (b) Phase-averaged sectional streamline at x* = 10.
the phase-averaged sectional streamlines (Fig. 2b), are marked by ‘+’ and ‘×’, respectively. The thick dashed lines give an approximate idea of the boundary of the spanwise vortex, which is about 25% of the ∼∗| (not shown). This contour provides a reference maximum value of |ω 3 for the periphery of the Kármán vortex. The inclined dash dotted line passing through the saddle point represents the diverging separatrix which is identified via the sectional streamline (Fig. 2b). The phase ϕ can be interpreted in terms of a longitudinal distance based on Taylor’s hypothesis; ϕ = 0 − 2π corresponding to the average vortex wavelength. The flow is left to right. The warm fluid (positive contours in Fig. 2a) is contained within the Kármán vortex, while the cool fluid (negative contours) is entrained from the freestream on either side of the wake. Large-scale temperature fronts, which can be identified with the contours with a value of 0 in Fig. 2a, form between the warm fluid in the vortex and the entrained cool fluid. These fronts are associated closely with the formation of the largescale structures of the temperature dissipation rate. Fig. 3 shows the ∼∗ three phase-averaged coherent temperature derivatives θ,i and the remainders of the mean-square temperature derivatives θ,ir2 ∗ at ∼∗ ∼∗2 x* = 10. Note that θ,i , instead of θ,i , is shown to reveal the large-scale features of the three components of the temperature dissipation rate, ∼∗ mainly because the signs of the θ,i -contours facilitate the identification ∼∗ ∼∗ ∼∗ of the relationship between θ,i and θ . The contours of θ,1 (Fig. 3a) and ∼∗ θ,3 (Fig. 3c) are qualitatively similar to each other, except that the ex∼∗ ∼∗ treme concentrations of θ,1 are about twice those of θ,3 . The similarity ∼∗ ∼∗ between θ,1 and θ,3 suggests that the large-scale temperature front in the present flow is distorted in the spanwise direction, presumably due to the spanwise distortion of the vortex rolls (e.g. [11]). Their extreme concentrations with opposite signs, occurring on either side of the vortex center, are located approximately in the regions where the warm fluid in the vortex and the entrained cool fluid meet, i.e. at the temperature fronts (the red dashed contours in Fig. 3a–c which correspond ∼∗ to the zero contours in Fig. 2a). The connection between θ,1 and the large-scale temperature front can also be identified by the observation
4. The dependence of χ on θ Sinai and Yakhot [19] derived an exact solution of the pdf equation of a passive scalar with no mean scalar gradient, in the limit of time t → ∞, viz. −1
p (θ θrms ) = C ⎡ ⎢ ⎣
E (χ θ) ⎤ ·exp ⎛⎜ χ ⎥ ⎦ ⎝
∫0
θ θrms
−
βdβ ⎞ ⎟ E (χ β ) χ ⎠
(4.1)
where C is a constant. It essentially describes the distribution of the passive scalar in a self-similar state, and is well satisfied in the grid 3
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∼ ∼ ∼ Fig. 3. Iso-contours of (a–c) the phase-averaged coherent temperature derivatives θ ,1∗ , θ ,∗2 , θ ,∗3 and (d–f) the remainders 〈θ,12r 〉∗, 〈θ,22r 〉∗, 〈θ,32r 〉∗ at x* = 10. Contour intervals: (a–c) 0.15, 0.15, 0.084; (d–f) 0.22, 0.28, 0.30. The red dashed contours (contours with a value of 0 in Fig. 2a) are the positions of the large-scale temperature fronts. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
turbulence at θ θrms > 2. For a clearer picture of the effect of coherent events on the correlation between χ and θ as the flow develops downstream, we employed the conditional correlation coefficient, viz.
turbulence without a mean scalar gradient (e.g. [12]). It indicates that the pdf of the passive scalar, p (θ θrms ) , is uniquely determined by the expectation of its conditional dissipation rate E (χ θ) for a given value of θ. When χ and θ are independent of each other, E (χ θ) χ is constant and equal to 1, leading to a Gaussian distribution of p (θ θrms ) . The large ∼∗ extreme concentration of θ,1 at x* = 10 (Fig. 3a) indicates that the small-scale temperature derivative feels the influence of the coherent events in the flow, implying that the coupling between χ and θ may be enhanced by the coherent motions. The distribution of E (χ θ) χ on the centerline from x* = 10 to x* = 40 confirms this point (Fig. 5). The data in a grid flow [12] without a mean temperature gradient are also included for comparison. At x* = 10 and 20, where the coherent motions are strong, the conditioned scalar dissipation rate displays a marked dependence on θ θrms ; the expectation of the dissipation rate at x* = 10 varies almost proportionally to θ θrms . The temperature dissipation rates where θ θrms (negative) is smallest approach zero for x* = 10 and 20. This is reasonable. The cool fluid entrained from the freestream is present on the centerline (Fig. 2a). The unmixed cool fluid comes from the potential freestream and the temperature dissipation rate in that fluid is expected to be zero [13]. In contrast, at x* = 40, where the coherent motions are rather weak, the distribution of E (χ θ) χ appears to be independent of θ θrms , with a value of approximately unity; in fact, this distribution is very similar to that in the grid flow, except at θ θrms > 2. At large θ θrms , the intermittency of the dissipation rate increases and E (χ θ) χ rises well above unity, which accounts for the large E (χ θ) χ at all three x* stations. The intermittency caused by the extreme events of the temperature field is expected to be flow-dependent and thus it is reasonable to see the departure of the distributions of E (χ θ) χ at x* = 40 and in the grid
¯ χ − E (χ )) ⎤ (θ − E (θ))( ρ (θ , χ ϕ) ≡ ⎡ ϕ ⎢ ⎥ σθ σχ ⎣ ⎦
(4.2)
Here, σθ and σχ are the standard deviations of θ and χ at phase ϕ , respectively. The ρ (θ , χ θ) quantifies the correlation between θ and χ at the same phase. It should be noted that, at a given phase ϕ , E (θ) and E (χ ) are equal to their corresponding phase-averaged values at this ∼ χ (ϕ) . Fig. 6 shows the distribution of ρ (θ , χ θ) at phase, i.e. θ (ϕ) and ∼ ∼∗ all three x* positions. The isocontours of θ at each x* position are also superposed onto the color-filled isocontours of ρ (θ , χ θ) for reference. At x* = 10 (Fig. 6a), the large correlation coefficient between θ and χ , of the order of 0.7, is found in the saddle region along the diverging separatrix where the temperature front forms, while a much smaller value, close to 0, is associated with the downstream half of the vortex. The distribution of ρ (θ , χ θ) suggests that the evident dependence between θ and χ at x* = 10 in Fig. 5 is due to the combined effect of the coherent strain rate and the temperature front at the saddle region. Note that only when the temperature front is present, e.g. immediately downstream of the vortex, ρ (θ , χ θ) is quite small. A similar distribution of ρ (θ , χ θ) is found for x* = 20 (Fig. 6b), although the magnitude is reduced, which is consistent with the dependence between θ and χ at x* = 20 in Fig. 5. However, for x* = 40, the situation changes: ρ (θ , χ θ) is generally below 0.1, except in the region above the saddle point near the boundary between the wake and the freestream. This is mainly because at x* = 40, the coherent strain rate at the saddle region 4
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Fig. 4. (a) Co-spectra and (b) the phase-lag between θ,1 and θ,2 and between θ,1 and θ,3 at the lateral position corresponding to the spanwise vortex center (y* = 0.39) at x* = 10.
Fig. 6. Color-filled isocontours of the conditional correlation coefficient between θ and χ at the same phase. (a) x* = 10, (b) 20, (c) 40. The scales of the ∼∗ color bar are the same for (a)-(c). The iso-contours of θ at each x* position are ∼∗ superposed for reference. Solid contours correspond to θ > 0 and dashed ∼∗ contours are for θ < 0.
weakens due to the degraded rotational motion of the vortex; the intensity of the vortex entrainment, which gives rise to the temperature front, reduces and now occurs away from the wake centerline. The small magnitude of ρ (θ , χ θ) at x* = 40 explains why θ and χ are independent of each other at x* = 40 in Fig. 5. In view of the independence between χ and θ shown in Figs. 5 and 6c at x* = 40 as well as Eq. (4.2), it is inferred that the distribution of θ is closer to be Gaussian at x* = 40 than x* = 10 or 20. This expectation is confirmed by the relationship between the departure of the pdf of θ from the Gaussian distribution and the level of correlation between χ and θ 2 (Fig. 7). The parameter δ ≡ Sθ + Fθ −3 is employed to characterize the departure of the pdf of θ from the Gaussian distribution (Sθ =0 and Fθ = 3), where Sθ and Fθ are the skewness and kurtosis of θ , respectively (e.g. [15]). The non-centered correlation coefficient between θ 2 and χ , given by
Fig. 5. Expectation of the temperature dissipation rate conditioned on the temperature fluctuation at the wake centerline from x* = 10 to 40. The conditional temperature dissipation rate in grid turbulence [12] is also included. The horizontal dashed line corresponds to the value of 1.
R≡
θ 2¯χ −1 θ¯2χ
(4.3)
quantifies the statistical dependence between θ 2 and χ (e.g. [8]). R is somewhat similar to the centered correlation coefficient 2 ρ (=(θ 2 − θ¯2¯)(χ − χ ) (θrms χrms )) but is easier to evaluate accurately [12] 5
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between θ 2 and χ corresponds to a nearly Gaussian distribution of θ at x* = 40. Note that Mi [15] did not distinguish between different distributions of (R, δ) at different x* positions as he focused mainly on differences in (R, δ) among different flows. In contrast, the concentration of (R, δ) at x* = 10 and 20 lies generally away from the origin. Consistent with the distributions of (R, δ), the pdfs of θ along the centerline at x* = 10 (and 20) are evidently skewed, while those at x* = 40 are basically Gaussian (shown later in Fig. 8 of Section 5). The linear relationship between R and δ reflects the direct connection between the dependence of the scalar dissipation rate χ on the scalar θ and the pdf of θ . This is consistent with the analytical finding of O’Brien and Jiang [16] that the pdf of θ being Gaussian is a sufficient and necessary condition of χ to be independent of θ in homogenous turbulence, although the present flow is far from being homogenous, especially at x/d = 10 and 20. The dynamical reason behind the ‘linear’ behavior of the connection of R and δ is still unclear. Mi [15] suggests that the linear relationship between R and δ represent a general feature of the scalar property in turbulence. It would be interesting in the future to derive the analytical connection between R and δ based on the governing equations of the passive scalar in turbulence and compare with the experimental results observed in different flows.
Fig. 7. The relationship between δ and R across the flow at x* = 10 to 40. The turbulent plane wake data of Mi [15] are also included.
since it is usually much larger than ρ . The points (R, δ) in the present flow exhibit roughly a linear distribution, i.e. the departure of the pdf of θ from Gaussian distribution is proportional to the extent of the correlation between θ 2 and χ . This is quite similar to what Mi [15] has observed in a plane wake (Re = 3200). The concentration of (R, δ) for x* = 40 is generally around the origin, i.e. the weak correlation
5. Topological features of the passive scalar mixing In this section, we attempt to identify the most efficient turbulent mixing region in the wake by examining the mixing process between the engulfed cool fluid and the warm wake, which provides an
∼∗ Fig. 8. Iso-contours of θ (the left column) and the conditional probability density function of the temperature, P (θ ϕ) , with respect to the Kármán vortex (the right column). (a, d) x* = 10; (b, e) 20, (c, f) 40. 6
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departure from the Gaussian distribution of θ at this x* in Fig. 7. In the topological picture of the coherent structures proposed by HH in the near wake of a cylinder, they presumed that the direct interaction between the rib fluid, mostly with longitudinal vorticity, and the fluid in the spanwise vortex rolls produces strong three-dimensional vorticity fluctuations in region A, and enhances the mixing in this region. However, Fig. 8d–f suggests that the mixing between cool and warm fluids in regions C & D is more effective than that in region A. This feature implies that the region of most effective mixing in the plane wake is different from that surmised by HH, which suggests that the mechanism for the turbulent mixing could be different from that conjectured by HH. Turbulent mixing can be viewed as a three-stage process of entrainment, dispersion and diffusion [7]. In the present flow, the cool fluid is entrained into the wake by the large-scale vortices, while the consequent mixing between the cool and warm fluids, i.e. the thermal dispersion and diffusion, at the interface, is due to the intermediateand small-scale motions. Zhou et al. [25] studied the two rows of the vortex streets of the wake of two side-by-side cylinders. They observed that the inner vortices from either cylinder wake decays more rapidly than the outer vortices because of the vorticity cancellation caused by the vorticity flux between the opposite-signed inner vortices. Similarly, there is also intense vorticity fluxes (from the positive vortex to the negative vortex) in both regions C and D, especially region C at x* = 10 and 20 [6]. This vorticity transport should accelerate the thermal dispersion and diffusion more effectively than the interaction between the streamwise ribs and the spanwise rolls, which can be confirmed by the much larger incoherent heat flux away from the vortex in region C than in region A [14]. This is very similar to the reason why the turbulent mixing of the passive scalar in regions C (and D) is more effective than in region A. The entrained fluid at x* = 10 is mixed with the wake fluid continuously as the flow develops downstream. At x* = 40 (Fig. 8c), the sharp termperature front near the wake centerline has virtually gone, indicating a well mixed stage of the warm and cool fluids, as confirmed by the reasonable agreenment between the pdfs of θ and the Gaussian pdf (Fig. 8f). At last, it is worth commenting on the evolution of the characteristics of the turbulent mixing downstream. The present observation that the turbulent mixing of the scalar with freestream fluid is faster near the centerline depends largely on the entrainment of the coherent vortices and the vorticity fluxes between the opposite-signed vortices in this flow range (x/d = 10–40). In further downstream where the freestream fluid cannot be entrained into the wake centerline due to the degraded coherent vortices, the scalar mixing would mainly happen in the turbulent/non-turbulent interface away from the centerline (e.g. Westerweel et al., 2009). Also, the readers should bear in mind that the scalarθ is trapped inside the vortex is largely due to the effect of the initial condition that the scalarθ is injected into the flow via heating the cylinder. Warm fluid is thus kept inside when the thermal shear layer warps into Kármán vortices. Since the random turbulent kinetic energy has its maximum concentration within the vortex [11], the flow region inside the vortex could also be associated with effective turbulent mixing when the initial condition of θ changes, e.g. the fluctuation of temperature being introduced by a toaster or a mandoline (e.g. [12]), such that both cool and warm fluids could be present within the vortex.
experimental evaluation of the turbulent mixing scenario in the present flow proposed by Hussain and Hayakawa [11], (hereinafter referred to as HH). The physics behind the scalar mixing process is also discussed. Before the mixing process is investigated, it is necessary to find a reliable means of assessing how the turbulent mixing takes place. Roshko [18] examined the mixing process in a mixing layer between two different passive scalars (nitrogen and helium-argon), and found that in the region between the two passive scalars where the mixing is most sufficient, the pdf of either scalar concentration is symmetric with respect to its mean value but the pdf of the scalar in the unmixed region away from the middle region is highly skewed. This feature provides us with a simple qualitative way of assessing the mixedness of the passive scalar in the present flow, i.e. the degree of mixing between warm and cool fluids in a particular flow region can be inferred from the departure from symmetry of the temperature pdf. In order to identify the region where turbulent mixing is most efficient, we divide the flow region with respect to the Kármán vortex into five representative areas, denoted by A-E in Fig. 8. Area A is the region where the longitudinal rib structures and the spanwise vortex rolls interact with each other and where HH speculated that the most effective mixing takes place; B is the region downstream of the vortex at the same y* position as A; C is the region downstream of the vortex near the temperature front close to the centerline; D is a region upstream of the vortex close to the temperature front at the same y* position as C; E is at the vortex center. The pdfs of θ conditioned on the phases ϕ of A–E, denoted by P (θ ϕ) , are calculated using the temporal traces of θ measured at these five locations (ϕ , y*). We have checked that P (θ ϕ) in the vicinity of A–E are qualitatively similar to the pdfs at A–E, so P (θ ϕ) at A–E are reasonable representatives of the temperature distribution in the corresponding region. The number of data points used for calculating the conditional pdf of θ is the same as that for estimating the phase average of θ , i.e. approximately 1980, 1700 and 1050 for x* = 10, 20 and 40 respectively. The convergence of the conditional pdf of θ has also been confirmed by the collapse of the pdfs with the number of samples beyond 750 (not shown). Note that P (θ ϕ) is transformed to P (θ ϕ) with the corre∼ sponding mean θ (ϕ) and standard deviation σθ (ϕ) , i.e. ∼ θ ϕ=(θ (ϕ) − θ (ϕ)) σθ (ϕ) , so that the distribution P (θ ϕ) can be compared with a standard Gaussian distribution in Fig. 8d–f. At x* = 10 (Fig. 8d), the pdfs at four locations on the periphery of the vortex (A–D) are highly skewed. The sharp cut-off on the negative side of the pdfs is due to the presence of ambient cool fluid. The skewness of the pdfs indicates that the warm and cool fluids are not well mixed at these locations, except at the vortex center (E) where the pdf displays good symmetry and follows the Gaussian distribution reasonably well. Note that the warm fluid is predominant around the vortex center (Fig. 8a) due to the initial condition, i.e. the thermal shear layer shedding from the cylinder and wrapping into vortices. The high turbulence around the vortex center (cf. Fig. 8 of HH) leads to the passive scalar in this region being adequately mixed. The excellent agreement between the pdf at the vortex center and the Gaussian distribution also confirms the effectiveness of the pdf method in assessing mixing. At x* = 20 (Fig. 8e), the pdfs of A and B remain similar to their counterparts at x* = 10; however, the pdfs at C and D are closer to the pdf at E or the Gaussian distribution and their sharp cutoff at x* = 10 has virtually disappeared. The different behavior of P (θ ϕ) at A (or B) and D (or C) from x* = 10 to 20 implies that the mixing between the entrained cool fluid and the warm wake is more rapid at C and D than at A and B. At x* = 40 (Fig. 8f), all distributions of P (θ ϕ) collapse reasonably well onto the Gaussian distribution, although the cut-off is still discernible on the negative side of P (θ ϕ) at B. It is worth noting that region B is where the cool fluid has just been entrained into the wake (Fig. 8c), although the entrainment is rather weak at x* = 40. Therefore, the persistence of the cut-off is not surprising. The Gaussian distribution of P (θ ϕ) at x* = 40 is also consistent with the small
6. Conclusions The characteristics of the passive scalar (temperature) variance dissipation rate are studied in a slightly heated turbulent wake based on data obtained with a multi-wire probe. We have examined the effects of the coherent vortices on the components of χ and on the relationship between χ and θ . Besides, the turbulent mixing process with respect to the Kármán vortex is also investigated. Three major conclusions are drawn from this work. 7
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Declaration of Competing Interest
(1) The three components of the temperature gradient feel the influ∼ ence of the coherent vortices in different ways. The component θ ,1∗ is maximum at or near the large-scale temperature front which forms due to the entrainment of cool fluid from the freestream by ∼ the spanwise vortices, whereas θ ,∗2 is associated with the quasistreamwise vortex and is concentrated along the diverging separa∼ ∼ trix. The similarity between the contours of θ ,∗3 and θ ,1∗ is most likely due to the spanwise kink of the temperature front, which is caused by the distortion of the vortex roll [11]. (2) The χ and θ exhibit a correlation at x* = 10 and 20, but tend to be statistically independent of each other at x* = 40. The difference in the inter-connection between χ and θ from x* = 10 or 20 to 40 indicates that the large and small scales in the temperature field are strongly coupled in the presence of the organized Kármán vortices. The phase-averaged correlation coefficient between χ and θ , ρ (θ , χ ϕ) , shows that this is mainly due to the strain rate and the temperature front at the saddle region at x* = 10 and 20. The roughly linear distribution of (δ, R) at x* = 10 and 20 indicates that the coupling between χ and θ is proportional to the extent to which the distribution of θ departs from Gaussianity; the data for the combination (δ, R) at x* = 40 is clustered mostly around the origin, suggesting that the distribution of θ is basically Gaussian and independent of χ when the distance is large enough (say x* ⩾ 40) from the cylinder. It is thus inferred that the assumption of independence of χ and θ usually made in turbulent combustion modeling (e.g. [17]) is valid only when, at least for the present flow, the effects of the rate of strain and temperature front caused by the large-scale vortex can be ignored. (3) The region where the most effective mixing between the entrained cool fluid and the warm fluid takes place is around the temperature front in the inner half of the vortices near the centerline, not in the region where the ribs and rolls contact with each other, which is conjectured by Hussain and Hayakawa [11]. Between x* = 10 and x* = 40, the pdfs of θ in regions C and D (Fig. 8) evolve towards a Gaussian distribution more rapidly than those in regions A and B, indicating that the turbulent mixing is more efficient in regions C and D. This result also suggests that the mechanism responsible for the turbulent mixing between the wake and the entrained ambient fluid is different from that surmised by Hussain and Hayakawa [11]. The vorticity transport between opposite-signed vortices is likely to be the dominant physics which is responsible for the efficient turbulent mixing in regions C and D, particularly region C. Although the present experiment is conducted at a moderate Reynolds number, the topological features of the flow are essentially unchanged as long as Re is smaller than its critical value, approximately 2.5 × 105 [24]. As a matter of fact, Yiu et al. [23] studied the Re effect on the three dimensional vorticity in a turbulent cylinder wake with Re = 2500, 5000 and 10,000 and showed that the topology of the wake (x/d = 10–40) is essentially the same for this range of Reynolds numbers although the three-dimensionality of the flow is enhanced at higher Re.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements YZ wishes to acknowledge support given to him from NSFC through grants 11632006, 91752109 and U1613226 and from Research Grants Council of Shenzhen Government through grants JCYJ20160531193220561 and JCY20160531192108351. References [1] R.A. Antonia, L. Browne, Anisotropy of the temperature dissipation in a turbulent wake, J. Fluid Mech. 163 (1986) 393–403. [2] J. Bakosi, P. Franzese, Z. Boybeyi, Probability density function modeling of scalar mixing from concentrated sources in turbulent channel flow, Phys. Fluids 19 (2007) 115106. [3] J.S. Bendat, A.G. Piersol, Random Data: Analysis and Measurement Procedures, fourth ed., Wiley, Hoboken, N.J., 2010. [4] B.J. Cantwell, Organized motion in turbulent flow, Annu. Rev. Fluid Mech. 13 (1981) 457–515. [5] J.G. Chen, T.M. Zhou, R.A. Antonia, Y. Zhou, Comparison between passive scalar and velocity fields in a turbulent cylinder wake, J. Fluid Mech. 813 (2017) 667–694. [6] J.G. Chen, Y. Zhou, T.M. Zhou, R.A. Antonia, Three-dimensional vorticity, momentum and heat transport in a turbulent cylinder wake, J. Fluid Mech. 809 (2016) 135–167. [7] P.E. Dimotakis, Turbulent mixing, Annu. Rev. Fluid Mech. 37 (2005) 329–356. [8] V. Eswaran, S.B. Pope, Direct numerical simulations of the turbulent mixing of a passive scalar, Phys. Fluids 31 (1988) 506–520. [9] C.H. Gibson, C.A. Friehe, S.O. McConnell, Structure of sheared turbulent fields, Phys. Fluids 20 (1977) S156. [10] M. Holzer, E.D. Siggia, Turbulent mixing of a passive scalar, Phys. Fluids 6 (5) (1994) 1820–1837. [11] A.K.M.F. Hussain, M. Hayakawa, Eduction of large-scale organized structures in a turbulent plane wake, J. Fluid Mech. 180 (1987) 193–229. [12] Warhaft, Z. Jayesh, Probability distribution, conditional dissipation, and transport of passive temperature fluctuations in grid-generated turbulence, Phys. Fluids Fluid Dyn. 4 (1992) 2292–2307. [13] P. Kailasnath, K.R. Sreenivasan, J.R. Saylor, Conditional scalar dissipation rates in turbulent wakes, jets, and boundary layers, Phys. Fluids Fluid Dyn. 5 (1993) 3207–3215. [14] M. Matsumura, R.A. Antonia, Momentum and heat transport in the turbulent intermediate wake of a circular cylinder, J. Fluid Mech. 250 (1993) 651–668. [15] J. Mi, Correlation between non-Gaussian statistics of a scalar and its dissipation rate in turbulent flows, Phys. Rev. E 74 (2006) 016301. [16] E.E. O’Brien, T. Jiang, The conditional dissipation rate of an initially binary scalar in homogeneous turbulence, Phys. Fluids A: Fluid Dynam. 3 (1991) 3121–3123. [17] H. Pitsch, Large-Eddy simulation of turbulent combustion, Annu. Rev. Fluid Mech. 38 (2006) 453–482. [18] A. Roshko, Structure of turbulent shear flows: a new look, AIAA J. 14 (1976) 1349–1357. [19] Ya.G. Sinai, V. Yakhot, Limiting probability distributions of a passive scalar in a random velocity field, Phys. Rev. Lett. 63 (1989) 1962–1964. [20] N. Soulopoulos, Y. Hardalupas, A.M.K.P. Taylor, Mixing and scalar dissipation rate statistics in a starting gas jet, Phys. Fluids 27 (2015) 125103. [21] K.R. Sreenivasan, On local isotropy of passive scalars in turbulent shear flows, Proc. R. Soc. Lond. Ser. A 434 (1991) 165–182. [22] Z. Warhaft, Passive scalars in turbulent flows, Annu. Rev. Fluid Mech. 32 (2000) 203–240. [23] M.W. Yiu, Y. Zhou, T. Zhou, L. Cheng, Reynolds number effects on three-dimensiional vorticity in a turbulent wake, AIAA J. 42 (2004) 1009–1016. [24] M.M. Zdravkovich, Flow Around Circular Cylinders, Oxford science publications, Oxford University Press, Oxford; New York, 1997. [25] Y. Zhou, H.J. Zhang, M.W. Yiu, The turbulent wake of two side-by-side circular cylinders, J. Fluid Mech. 458 (2002) 303–332.
CRediT authorship contribution statement J.G. Chen: Formal analysis, Methodology, Visualization, Writing original draft. R.A. Antonia: Conceptualization, Writing - review & editing. Y. Zhou: Writing - review & editing, Supervision. T.M. Zhou: Investigation, Data curation, Validation.
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