Statistics of the Energy Dissipation Rate and Local Enstrophy in Turbulent Channel Flow
Peter E. Hamlington, Dmitry Krasnov, Thomas Boeck, Jörg Schumacher
aa r X i v : . [ phy s i c s . f l u - dyn ] J un Statistics of the Energy Dissipation Rate andLocal Enstrophy in Turbulent Channel Flow
Peter E. Hamlington
Laboratory for Computational Physics and Fluid Dynamics, Naval ResearchLaboratory, Washington D.C. 20375, USA
Dmitry Krasnov, Thomas Boeck, J¨org Schumacher
Institute of Thermodynamics and Fluid Dynamics, Ilmenau University ofTechnology, P.O. Box 100565 D-98684 Ilmenau, Germany
Abstract
Using high-resolution direct numerical simulations, the height and Reynolds num-ber dependence of higher-order statistics of the energy dissipation rate and localenstrophy are examined in incompressible, fully-developed turbulent channel flow.The statistics are studied over a range of wall distances, spanning the viscous sub-layer to the channel flow centerline, for friction Reynolds numbers Re τ = 180 and Re τ = 381. The high resolution of the simulations allows dissipation and enstrophymoments up to fourth order to be calculated. These moments show a dependenceon wall distance, and Reynolds number effects are observed at the edge of the log-arithmic layer. Conditional analyses based on locations of intense rotation are alsocarried out in order to determine the contribution of vortical structures to the dis-sipation and enstrophy moments. Our analysis shows that, for the simulation at thelarger Reynolds number, small-scale fluctuations of both dissipation and enstrophybecome relatively constant for z + & Key words:
Turbulent channel flows, turbulent shear flows, energy dissipationrate, enstrophy
PACS:
Determining the detailed properties of wall-bounded turbulent shear flowshas been the focus of considerable experimental and computational research(see, e.g., [1,2,3] for reviews). Although prior studies have provided insights
Preprint submitted to Elsevier 13 November 2018 nto low-order statistics and coherent structures in these flows, higher-ordervelocity gradient statistics remain relatively unexplored. Velocity gradients re-flect the structure and properties of the turbulent small scales, [4] and theirhigher-order statistics are particularly sensitive to the large-amplitude, inter-mittent fluctuations characteristic of high-Reynolds number flows. While suchhigher-order statistics have been studied in homogeneous isotropic turbulenceusing direct numerical simulations (DNS) for several decades, [5,6,7,8,9] it hasonly recently become feasible to carry out similar analyses in wall-boundedflows. This is due, in large part, to the substantial computational resourcesrequired to resolve the smallest scales of the turbulence. Progress in simulat-ing wall bounded flows has recently been made, however, in Ref. [10], wherevelocity gradient moments up to fourth order are examined using DNS offully-developed turbulent channel flow at Re τ = 180, where Re τ = u τ L/ν , u τ isthe friction velocity, L is the half-width of the channel, and ν is the kinematicviscosity.The present paper refines and significantly extends the prior study in Ref.[10] by using highly resolved turbulent channel flow DNS at Re τ = 180 and Re τ = 381 to examine both the height and Reynolds number dependence ofhigher-order velocity gradient statistics. Particular emphasis is placed on themoments of the energy dissipation rate ε = 2 νS ′ ij S ′ ij , (1)and the local enstrophy (or enstrophy density)Ω = 12 ω ′ i ω ′ i . (2)Taken together, these quantities characterize the straining and rotation asso-ciated with small-scale turbulent fluctuations. [8] The fluctuating strain ratetensor, S ′ ij , in Eq. (1) is given in terms of the fluctuating velocity, u ′ i = u i − h u i i (where h·i is a z -dependent average over time and x - y planes parallel to thechannel walls), as S ′ ij = 12 ∂u ′ i ∂x j + ∂u ′ j ∂x i ! , (3)and the fluctuating vorticity, ω ′ i , in Eq. (2) is given by ω ′ i = ǫ ijk ∂u ′ k ∂x j , (4)where ǫ ijk is the cyclic permutation tensor. The moments of ε and Ω, denoted2 ε n i and h Ω n i , reflect properties of small-scale, high-amplitude fluctuations inthe velocity gradient. For example, h ε i / h ε i and h Ω i / h Ω i can be used toquantitatively assess the degree of small-scale intermittency in the flow. [4]In order to measure moments of ε and Ω up to fourth order, the resolutionsused in the present simulations are substantially higher than in prior sim-ulations at similar values of Re τ . Recent studies of homogeneous isotropicturbulence [8,15] and turbulent channel flow [10] have shown that very fineresolutions are required to fully resolve higher-order statistics of ε and Ω. Inparticular, using simulations of a Re τ = 180 channel flow at three differentresolutions, it was shown in Ref. [10] that the lowest resolution was sufficientto measure low-order statistics such as mean velocities and Reynolds stresses, u ′ i u ′ j . Finer resolutions, however, were required to accurately measure higher-order velocity gradient statistics such as the mean value of ε , particularlynear the channel walls. The resolutions used in the present simulations areeven higher than those in Ref. [10], and allow calculation of up to fourth ordermoments of ε and Ω.A wide range of wall distances are examined in the present study in order todetermine the height dependence of the statistics. Turbulent channel flows aretypically divided into four regions; the viscous sublayer for z + <
5, the bufferregion for 5 < z + <
30, the logarithmic layer for z + >
30 and z . . L , andthe outer flow for z & . L , [11] where z + ≡ zu τ /ν and z is the coordinate inthe wall-normal direction. For both values of Re τ considered here, statisticsare examined from z + = 2, well within the viscous sublayer, to the channelcenterline. The inhomogeneity in the z -direction requires that the statistics of ε and Ω be calculated in planes parallel to the channel walls, in order to allowaveraging over homogeneous flow directions. Compared to studies of three-dimensional homogeneous isotropic turbulence, where full volume averagingis possible, this places significant restrictions on the statistical convergence ofthe results. Consequently, particular attention is paid in the following to theconvergence of the moments of ε and Ω as a function of z + and Re τ .While variations in the moments of ε and Ω with Re τ and z + are importantfor understanding the small-scale structure of the channel flow, the presenceof coherent vortical structures is also expected to play a role in determin-ing these moments. Theodorsen [17] first proposed a hairpin shape for thesestructures, and subsequent experimental (e.g., Refs. [18,19,20]) and numericalstudies (e.g., Refs. [12,21,22]) have characterized hairpin vortices throughoutwall bounded flows (see Ref. [2] for a review). Although these vortices havebeen identified as important in fluid transport and the generation of Reynoldsstresses, [22] relatively little is known about their contribution to the higher-order, small-scale statistics of ε and Ω. In the following, we examine this issueusing conditional analyses of the moments of ε and Ω based on locations ofintense rotation in the flow. We also consider the moments at locations away3rom regions of intense rotation in order to determine the extent to whichdifferences in the moments as a function of Re τ and z + can be attributed tovortical structures.The manuscript is organized as follows. Details of the numerical simulationsare presented in the next section. The moments of ε and Ω are then presentedup to fourth order for both values of Re τ , accompanied by an analysis of thestatistical convergence of these moments. Conditional analyses based on loca-tions of intense rotation are outlined in Section 4. The method by which theselocations are identified is briefly discussed, and results from conditional analy-ses of the ε and Ω moments are presented. Finally, a summary and conclusionsare provided at the end. The numerical simulations used in the present study solve the incompressibleNavier Stokes equations for a fully developed turbulent channel flow. Theseequations are written in non-dimensional form as ∂u i ∂x i = 0 , (5) ∂u i ∂t + u j ∂u i ∂x j = − ∂p∂x i + 1 Re ∂ u i ∂x j ∂x j , (6)where u i is the velocity field and p is the kinematic pressure. The Reynoldsnumber, Re , is given as Re = U L/ν , where U is the total mean velocity in thechannel.As described in Ref. [10], the Navier-Stokes equations in Eqs. (5) and (6)are decomposed in poloidal-toroidal form and then solved using a pseudo-spectral method. This method uses Fourier expansions in the x and y di-rections, which are parallel to the channel walls, and Chebyshev polynomialexpansions in the z direction, normal to the walls. [23,24] De-aliasing with the2 / f n +1 − f n + f n − t = L f n +1 + 2 N ( f n ) − N ( f n − ) , (7)where L is a linear operator, N represents nonlinear terms, and ∆ t is the timestep. The approximation for the time derivative, ∂f /∂t , on the left side of Eq.4 e τ Re N x × N y × N z N t ∆ x + ∆ y + ∆ z + c
180 2800 512 × × . . . × × . . . Re τ , global Reynolds number Re = U L/ν , grid dimen-sions N x × N y × N z , number of temporal snapshots, N t , horizontal resolutions,∆ x + ≡ ∆ xu τ /ν and ∆ y + ≡ ∆ yu τ /ν , and vertical resolution at the centerline,∆ z + c ≡ ∆ z ( z/L = 1) u τ /ν , for the two simulations analyzed herein. (7) gives f at time level n + 1 using values at the two previous levels, denoted f n and f n − . The simulations are parallelized using MPI.The channel flow simulation domain consists of a rectangular box of non-dimensional size ( L x × L y × L z ) /L = 4 π × π × Re τ = 180 and 2 π × π × Re τ = 381. In Ref. [10], the Re τ = 180 case was examined at a maximumresolution of N x × N y × N z = 512 × ×
513 collocation points in physical space.Due to de-aliasing, the number of Fourier or Chebyshev modes is 2 / N x,y,z . In the present study, this resolution has been increasedto 512 × × Re τ = 381case, which was not examined in Ref. [10], is 1024 × × Re τ .The Re τ = 180 channel flow simulated by Kim et al. [12], for example, used192 × ×
129 grid points in a domain of size 4 π × π × z is used as the wall-normal coordinate). Similarly, the Re τ = 395 channel simulation by Moser et al. [13]) was performed on a gridconsisting of 256 × ×
193 points in a domain of size 2 π × π × . L/U , for the Re τ = 180 simulation, with a total of N t = 403 snapshots. Thetime interval between snapshots for the Re τ = 381 simulation is 0 .
19 convec-tive time units, with N t = 134 total snapshots. The statistical analysis for asingle horizontal plane (and its symmetry counterpart in the upper half of thechannel) can thus be made over a set of up to 2 . × data points. Due tothe larger N t for the Re τ = 180 case and the greater N x for Re τ = 381 (seeTable 1), we use a similar amount of data in the analysis of each value of Re τ .Figure 1 shows the mean velocity in the x direction and the Reynolds shearstress τ +13 = u ′ x u ′ z + = u ′ x u ′ z /u τ as a function of z + for Re τ = 180 at a resolutionof 128 × × ε and Ω to be calculated.5 ig. 1. Mean streamwise velocity u + (a) and Reynolds shear stress τ +13 = u ′ x u ′ z + (b)from Re τ = 180 numerical simulation at resolution 128 × × Re τ = 180. The statistics of ε and Ω are examined in the following at various wall dis-tances, z + , by carrying out the analysis in x - y planes parallel to the channelwalls. Within each of these planes, the flow is essentially homogeneous, thusallowing an examination of the flow statistics similar to that employed forhomogeneous isotropic turbulence. In the following, the average h·i denotesan x - y average at a particular value of z + . It is always combined with anarithmetic average over the full sequence of temporal snapshots. Statisticalconvergence is further improved by using symmetric planes from both the topand bottom halves of the channel, where the velocity and velocity gradientfields in the top half are reflected about the centerline. Figure 2 shows probability density functions (pdfs) of ε/ h ε i and Ω / h Ω i forboth values of Re τ at wall distances spanning the viscous sublayer ( z + = 2) tothe channel centerline, where the averages h ε i and h Ω i are functions of z + . Thepdfs of both ε and Ω vary as the wall is approached from the centerline, andFigure 2 shows that the degree of intermittency, as indicated by the width ofthe pdf tails and the corresponding probability of obtaining large amplitudesof ε and Ω, generally decreases as the wall is approached, with a minimum inthe buffer layer at approximately z + = 10. Immediately at the wall, for z + = 2,however, the pdfs of ε and Ω for both values of Re τ are wider than those for z + = 10. For Re τ = 381, the tails of the ε and Ω pdfs are most prominent inthe logarithmic layer for z + ≈ −
90, while for Re τ = 180 the most prominenttails occur further from the wall at z + ≈ −
20 40 60 80 10010 −8 −6 −4 −2 P ( ε / h ε i ) ( a ) R e τ = 1 8 0 −8 −6 −4 −2 P ( Ω / h Ω i ) ( b) R e τ = 1 8 0 −8 −6 −4 −2 P ( ε / h ε i ) ε / h ε i ( c) R e τ = 3 8 1 −8 −6 −4 −2 P ( Ω / h Ω i ) Ω / h Ω i ( d) R e τ = 3 8 1 Fig. 2. Probability distribution functions of dissipation rate ε/ h ε i and local enstro-phy Ω / h Ω i for Re τ = 180 ( ε : (a) and Ω: (b)) and Re τ = 381 ( ε : (c) and Ω: (d)). of increased intermittency at these values of z + , and may be connected to thebursting of coherent vortical structures away from the wall. [2]With respect to the differences between the pdfs of ε and Ω, Figure 2 showsthat for both values of Re τ and wall distances down to z + ≈
40, the tails of theΩ pdfs are more pronounced than those for the ε pdfs. This is consistent withprior results in homogeneous isotropic turbulence [8] and turbulent channelflow [10]. For z + .
40, however, the pdfs of ε and Ω are more similar, indicatinga closer correspondence between the statistics of ε and Ω near the wall.The connection between ε and Ω at each value of z + can be further examinedusing joint probability distributions of ε and Ω, shown in Figure 3 for Re τ =381. Similar joint pdfs are shown for Re τ = 180 in Ref. [10]. The joint pdfs,denoted P ( ε, Ω), are normalized by P ( ε ) and P (Ω) as [10]Π( ε, Ω) = P ( ε, Ω) P ( ε ) P (Ω) , (8)where values of Π( ε, Ω) greater than 1 indicate a higher correlation between ε and Ω than if the two quantities were statistically independent. [25] Consistent7 a ) z + = 3 8 0 l og ( ε / h ε i ) −4 −2 0 2−4−202 ( b) z + = 2 0 0 −4 −2 0 2−4−202 ( c) z + = 1 2 0 −4 −2 0 2−4−202 ( d) z + = 6 0 l og ( ε / h ε i ) l o g ( Ω / h Ω i ) −4 −2 0 2−4−202 ( e) z + = 3 0l o g ( Ω / h Ω i ) −4 −2 0 2−4−202 ( f ) z + = 1 0l o g ( Ω / h Ω i ) −4 −2 0 2−4−202 Fig. 3. Joint pdfs of ε and Ω at six values of z + for Re τ = 381. The joint pdfs arenormalized by the one-dimensional pdfs of ε and Ω, as in Eq. (8), and log [Π( ε, Ω)]is shown. The color contours are the same in all panels, and range from 10 − (blue)to 10 (red). The black diagonal line in panel (f) corresponds to the relation ε = 2 ν Ωobtained from Eq. (9). with prior results for Re τ = 180 [10] and with simulations of homogeneousisotropic turbulence, [25] Figure 3 shows that intense ε and Ω are correlatedat all values of z + . As the wall is approached, however, Figure 3 shows thatthe support of the joint pdfs decreases. As noted in Ref. [10], the averages of ε and Ω are connected in a wall-bounded shear flow by h ε i = 2 ν h Ω i + 2 ν ∂ h u ′ z i ∂z . (9)Since the second term on the right-side of (9) is small compared to 2 ν h Ω i inthe channel flow, [10] we obtain h ε i ≈ ν h Ω i . Figure 3 shows that the values of ε and Ω fall increasingly close to this relation as the wall is approached. Theseresults at Re τ = 381 are qualitatively consistent with the findings in Ref. [10]for Re τ = 180. The statistical convergence of a moment h ϕ n i can be assessed from plots of ϕ n P ( ϕ ) versus ϕ , where ϕ is an arbitrary velocity gradient quantity such as ε or Ω. [8] These distributions are shown for n = 3 and n = 4 for the Re τ = 180and Re τ = 381 simulations in Figures 4 and 5, respectively. Figure 4 shows8
20 40 60 80 1000123 ( ε / h ε i ) P ( ε / h ε i ) ( a ) n = 3 ( Ω / h Ω i ) P ( Ω / h Ω i ) ( b) n = 3 ( ε / h ε i ) P ( ε / h ε i ) ε / h ε i ( c) n = 4 ( Ω / h Ω i ) P ( Ω / h Ω i ) Ω / h Ω i ( d) n = 4 Fig. 4. Moment convergence results for ε and Ω for Re τ = 180. The n = 3 and n = 4moments are shown for ε (a) and (c), and Ω (b) and (d), respectively. that, for Re τ = 180, the n = 3 moments are converged for both ε and Ω atall z + , as indicated by the decrease in the curves to zero for large ε/ h ε i andΩ / h Ω i . While the n = 4 moments of ε are also converged at all z + , the n = 4moments of Ω show a lack of convergence for z + ≈ − ε moments, and suggests thatthe non-converged fourth moments of Ω in Figure 4(d) may have a physicalorigin.For Re τ = 381, Figures 5(a) and (c) show that the moments of ε are adequatelyresolved up to n = 4 at all values of z + . Figures 5(b) and (d) further show thatthe n = 3 and n = 4 moments of Ω are generally better converged than thosefor Re τ = 180 in Figure 4. Given the fact that approximately the same amountof data is used in the analysis of both runs, this suggests that the cross-channel bursting of vortex structures may be less prevalent for higher Re τ . Inparticular, there is an extended bulk region for the higher Reynolds numbercase that separates the two logarithmic layers in the upper and lower halvesof the channel. 9
20 40 60 80 1000123 ( ε / h ε i ) P ( ε / h ε i ) ( a ) n = 3 ( Ω / h Ω i ) P ( Ω / h Ω i ) ( b) n = 3 ( ε / h ε i ) P ( ε / h ε i ) ε / h ε i ( c) n = 4 ( Ω / h Ω i ) P ( Ω / h Ω i ) Ω / h Ω i ( d) n = 4 Fig. 5. Moment convergence results for ε and Ω for Re τ = 381. The n = 3 and n = 4moments are shown for ε (a) and (c), and Ω (b) and (d), respectively. Figure 6 shows the resulting moments for n = 2 − Re τ = 180 and Re τ =381 simulations throughout the entire channel. Estimates of 90% confidenceintervals for the calculated moments have been included in Figure 6 using amoving-block bootstrap method [26] on time-series of h ε n i xy,s and h Ω n i xy,s ,where h·i xy,s denotes a time- and z -dependent average over x - y planes andsymmetric halves of the channel ( s ). The block sizes used in the bootstrapanalyses are approximately equal to one convective time unit ( L/U ) for bothvalues of Re τ , giving ten snapshots per block for Re τ = 180 and five snapshotsper block for Re τ = 381.For both values of Re τ , Figure 6 shows that the ε and Ω moments are large veryclose to the wall at z + = 2, but reach their minimum values at z + = 10 withinthe buffer layer. The large amplitudes of the moments at z + = 2 arise from acombination of effects, including normalization of the moments by the meanvalues of ε and Ω, and the fact that z + = 2 corresponds to locations within theviscous sublayer where the flow is not fully turbulent. Moreover, while velocityfluctuations become small near the wall, fluctuating velocity gradients becomevery steep, resulting in very large values of ε and Ω at z + = 2. For both valuesof Re τ , the moments of ε and Ω are similar up to z + ≈ −
40, but for largervalues of z + , the moments of Ω are substantially greater than those for ε . This10
50 100 150 20010 Ω ε h ε n i / h ε i n , h Ω n i / h Ω i n ( a ) R e τ = 1 8 0 , n = 2 Ω ε ( b) R e τ = 1 8 0 , n = 3 Ω ε ( c) R e τ = 1 8 0 , n = 4 Ω ε h ε n i / h ε i n , h Ω n i / h Ω i n z + ( d) R e τ = 3 8 1 , n = 2 Ω εz + ( e) R e τ = 3 8 1 , n = 3 Ω εz + ( f ) R e τ = 3 8 1 , n = 4 Fig. 6. Moments h ε n i / h ε i n (black lines) and h Ω n i / h Ω i n (red lines) as a functionof z + for n = 2 − Re τ = 180 (a)-(c) and Re τ = 381 (d)-(f) simulations.Legend is shown in panel (c). Error bars show estimates of 90% confidence intervalsobtained using a moving-block bootstrap method, [26] with block sizes equal to oneconvective time unit ( L/U ) for both values of Re τ . last result is consistent with the wider pdfs of Ω compared to those for ε for z + &
40 in Figure 2, and has also been observed in studies of homogeneousisotropic turbulence. [8]While Figure 6 shows that the dependence of the moments on z + is similarin certain respects for Re τ = 180 and Re τ = 381, there are slight differencesbetween the two values of Re τ . For Re τ = 381, the moments of ε reach a localmaximum near z + ≈
50 before decreasing until z + ≈ z + , the ε moments remain relatively constant to the channel flow centerline.For Re τ = 180, by contrast, the moments of ε do not show a pronouncedpeak anywhere in the channel and remain relatively constant outside of thebuffer layer. The local maxima in the ε moments at z + ≈
50 for Re τ = 381could indicate a qualitative change in the small-scale turbulence, in accordancewith Yakhot et al. , [27] where it was suggested that small-scale fluctuationsoutside the buffer layer should be similar to those in isotropic turbulence. Theinvariance of the ε and Ω moments for z + &
100 in the Re τ = 381 simulationprovides support for this idea.Figures 6(a)-(c) further show that the moments of Ω for Re τ = 180 reach a11aximum near z + ≈ −
150 before decreasing again at the channel centerline.This pronounced maximum is likely due to the bursting of coherent vortices,but is not observed for Re τ = 381 in Figures 6(d)-(f), where the moments of Ωremain relatively constant for z + & Re τ = 180 are, however, largest in this range of z + , indicating that, particularlyfor the fourth moments, additional data is required to fully capture the strongtemporal variability of the vorticity field created by these bursting structures. The statistics of ε and Ω in the previous section are obtained using all pointsin each plane of the channel. Certain features of these statistics, such as thelocal maxima in the moments of Ω for Re τ = 180 at z + ≈ −
150 and thedifferences between the moments for ε and Ω outside of the buffer layer, maybe due, in part, to the presence of intense vortical structures in the channel.This issue can be examined by calculating the statistical moments at pointsboth within and outside regions of strong rotation in the flow, which are takenhere to correspond to vortical structure locations. There has been considerable research over the last several decades on the mostappropriate methods by which to identify intense vortical structures in turbu-lent flows. Chakraborty et al. [28] showed, however, that most vortical struc-ture identification procedures give similar results for homogeneous isotropicturbulence. Since our primary interest here is not in characterizing the prop-erties of the structures or assessing the merits of various identification pro-cedures, we simply classify vortical structures as intense, rotation-dominatedregions of the flow. The analysis of the ε and Ω moments is then carried outonly over points that fall within these regions, or points completely outsidethese regions, which we term the background flow.As first proposed by Hunt et al. , [29] rotation-dominated regions can be iden-tified as locations where Q >
0, where Q is the second invariant of the velocitygradient tensor, A ij = ∂u i /∂x j , and is written for an incompressible flow as Q = 12 ( − S ij S ij + R ij R ij ) . (10)12he strain rate tensor, S ij , is given by S ij = 12 ∂u i ∂x j + ∂u j ∂x i ! , (11)and R ij is the anti-symmetric part of A ij given by R ij = 12 ∂u i ∂x j − ∂u j ∂x i ! . (12)Here we use a closely related identification method, originally proposed byZhou et al. , [30] that requires A ij to have two complex conjugate eigenvalues, λ r ± iλ ci , and that the complex part, λ ci >
0, be larger than a prescribed cutoff.The restriction that A ij have two complex conjugate eigenvalues is equivalentto requiring that ∆ >
0, where ∆ is given by [22]∆ = Q + 274 R , (13)and R = −
13 ( S ij S jk S ki + 3 R ij R jk S ki ) , (14)is the third invariant of A ij for an incompressible flow. Regions of intenserotation – and, by extension, vortical structure locations – are then identifiedby requiring that λ ci > α ( λ ci ) max , where ( λ ci ) max is the maximum value of λ ci at each value of z + . The pre-factor α determines, in large part, the numberand magnitude of vortical structures identified in the flow, and we use α =0 .
05 herein. The resulting locations of intense rotation, identified as pointswhere λ ci is large, are generally similar to those obtained using other criteria,including the Q criterion. [28]Due to the presence of a mean shear in the channel, the entire vortical structureidentification procedure is carried out using the fluctuating velocity gradient, A ′ ij ≡ ∂u ′ i /∂x j . This approach has been used previously by Robinson [31] andPirozzoli et al. , [22] and is particularly important near the channel walls wherethe mean shear can generate large vorticity not associated with coherent vor-tical structures. The fluctuating tensors S ′ ij , given in Eq. (3), and R ′ ij , givenby R ′ ij = 12 ∂u ′ i ∂x j − ∂u ′ j ∂x i ! , (15)13 x y z y x ω θ xy θ e z y x z x y ω θ zx θ zy Fig. 7. Schematic showing vorticity orientation angles defined in Eq. (16). are thus used in the expressions for Q , R , and ∆ in Eqs. (10), (14), and (13).The difference between using the full velocity gradient tensor and the fluctu-ating tensor is small for much of the channel, and only becomes significant inthe near-wall region where the mean shear is large. In the present analysis, the orientation of intense vortical structures is inferredfrom the orientation of the fluctuating vorticity field, ω ′ i , at rotation-dominatedlocations in the flow. This is in contrast to prior approaches (e.g., Ref. [20])which have attempted to treat coherent vortices as connected regions with asingle orientation associated with the structure as a whole. Following Pirozzoli et al. , [22] the orientation of the vorticity is characterized using the angles θ ij = tan − ω ′ i ω ′ j ! , θ e = sin − ω ′ z ω ′ ! , (16)where ω ′ ≡ ( ω ′ i ω ′ i ) / , which are shown schematically in Figure 7. The orien-tation of intense vortical structures can then be determined from the jointpdfs of θ xy and θ e , denoted P ( θ xy , θ e ). Since an isotropic vorticity field gives P ( θ xy , θ e ) ∼ cos( θ e ), we consider normalized joint pdfs of P ( θ xy , θ e ) / cos( θ e ),following the approach used in Ref. [22].Joint pdfs of P ( θ xy , θ e ) / cos( θ e ) for Re τ = 381 in Figure 8 show that the dis-tribution of vorticity within rotation-dominated regions varies substantiallyas the wall is approached from the channel centerline. There is essentially nopreferred orientation of the vorticity near the centerline, as shown by the lackof any clear maxima or minima in Figure 8(a). At z + ≈
120 in Figure 8(b),however, the vorticity shows a weak preference for θ e ≈ ± ◦ and θ xy ≈ ± ◦ ,14 a ) z + = 3 8 0 θ e ( ◦ ) −180 −90 0 90 180−90−4504590 ( b) z + = 1 2 0 −180 −90 0 90 180−90−4504590 ( c) z + = 6 0 −180 −90 0 90 180−90−4504590 ( d) z + = 3 0 θ e ( ◦ ) θ xy ( ◦ ) −180 −90 0 90 180−90−4504590 ( e) z + = 1 0 θ xy ( ◦ ) −180 −90 0 90 180−90−4504590 ( f ) z + = 2 θ xy ( ◦ ) −180 −90 0 90 180−90−4504590 Fig. 8. Joint pdfs of the elevation angle θ e and the orientation angle θ xy , defined inEq. (16), at six values of z + for Re τ = 381. Only vorticity identified to be withinvortical structures is included in the pdfs (i.e. points for which λ ci > . λ ci ) max ,where ( λ ci ) max is determined at each z + ). Contours of log [ P ( θ xy , θ e ) / cos θ e ] areplotted, with levels from 10 − (blue) to 10 (red). approximately corresponding to the “necks” of hairpin vortices discussed inprevious studies. [20,22] As the wall is approached, Figure 8(c) shows thatthe peaks in the distributions begin to shift towards θ e = 0 ◦ , corresponding toquasi-streamwise vortices [11,22] which are oriented in the x -direction parallelto the channel walls. At the same time, the probability for θ xy > | ◦ | also be-gins to increase, and by z + = 10 in Figure 8(e) there is a strong probability ofobtaining vorticity with θ xy close to ± ◦ . At z + = 2 in Figure 8(f), much ofthe vorticity is oriented in the spanwise direction parallel to the channel walls(giving θ e ≈ θ xy ≈ ◦ ). There are also less pronounced peaks at | θ xy | = 180 ◦ ,which are due to weaker vorticity not associated with intense structures (asindicated by the disappearance of these secondary peaks when using largervalues of α ). There is thus a rapid change in the orientation of intense vorticesbetween the buffer layer and viscous sublayer, a result that is mirrored in thelarge changes in the moments of ε and Ω between z + = 10 and z + = 2 in Figure6. Figure 9 shows the moments of ε and Ω conditioned on vortical structurelocations, which are identified as rotation-dominated points in the flow using15
50 100 150 20010 h ε n i / h ε i n , h Ω n i / h Ω i n ( a ) R e τ = 1 8 0 , n = 2 ( b) R e τ = 1 8 0 , n = 3 ( c) R e τ = 1 8 0 , n = 4 h ε n i / h ε i n , h Ω n i / h Ω i n z + ( d) R e τ = 3 8 1 , n = 2 z + ( e) R e τ = 3 8 1 , n = 3 z + ( f ) R e τ = 3 8 1 , n = 4 F u l l ε Ω Vor t e x ε Ω NV ε Ω Fig. 9. Moments h ε n i / h ε i n and h Ω n i / h Ω i n for n = 2 − Re τ = 180 (a)-(c) and Re τ = 381 (d)-(f) simulations. The curves correspond to moments obtained fromthe full fields (black lines, denoted “Full”), points at rotation-dominated locations(red lines, denoted “Vortex”), and points where the rotation is weak (blue lines,denoted “NV”). the procedure outlined in Section 4.1. Moments are also shown for points inthe background field, where the rotation is small (see the λ ci criterion definedimmediately below Eq. (14)). The moments for both subsets are normalizedwith respect to the means of ε and Ω obtained from the full field.For both ε and Ω, Figure 9 shows that the moments obtained at locations ofintense rotation exceed those for both the full and background fields. Althoughthe increase in the Ω moments at these locations is to be expected (since theidentification procedure essentially selects points with large vorticity magni-tude), the accompanying increase in ε can be understood from the joint pdfsin Figure 3. These pdfs show that intense ε and Ω are statistically correlated,and thus the moments of ε at locations of intense rotation tend to be largerthan the corresponding full field values. From studies of isotropic turbulence(see, e.g., [32]), it is also well-known that large amplitudes of both ε and Ωappear in close spatial proximity. At the same time, however, it should benoted that the moments of ε remain smaller than those of Ω at all z + , for allorders, and at both values of Re τ .Substantial differences between the moments of Ω and ε are observed in the16ackground field, where the rotation is small. In particular, there is near agree-ment between the ε moments in the full and background fields. This indicatesthat locations of intense rotation, and hence vortical structures, do not con-tribute substantially to the full statistics of ε . By contrast, Figure 9 showsthat the background moments of Ω are significantly smaller than the corre-sponding full field values, indicating that locations within vortical structuresmake a substantial contribution to the full field moments of Ω. Contrary tothe moments in the full field and at rotation dominated locations, the localmaxima in the moments of Ω at z + ≈
150 for Re τ = 180 are absent in thebackground field. This reinforces the connection between these maxima andintense vortices in the flow. Such pronounced maxima are not observed in anyof the fields shown in Figure 9 for Re τ = 381, and it remains to be seen infuture studies whether these results undergo additional changes for even larger Re τ . The present high-resolution DNS study of fully-developed turbulent channelflow has examined velocity gradient statistics as a function of wall distance, z + , at friction Reynolds numbers Re τ = 180 and Re τ = 381. An emphasis hasbeen placed on the statistics, and in particular the higher-order moments, ofthe energy dissipation rate, ε , and the local enstrophy, Ω. The probabilitydensity functions (pdfs) of ε and Ω for the Re τ = 381 case are qualitativelysimilar to previous results obtained for Re τ = 180. [10] The pdfs of both ε and Ω generally become less intermittent as the wall is approached from thechannel centerline, as indicated by the less broad tails of the pdfs. For z + & ε , although the pdfs for bothquantities are similar near the channel walls. Joint pdfs show that both largeand small values of ε and Ω are correlated, and that the support of the pdfsis reduced as the wall is approached.The high resolution of the present simulations has allowed moments of ε andΩ up to fourth order to be calculated. The moments of ε and Ω are similar inthe viscous sublayer and buffer layer, but, for locations further from the wall,the moments of Ω are substantially larger than those of ε . Reynolds numbereffects are observed in the moments of both ε and Ω. In particular, there isa local maximum in the moments of ε at the beginning of the logarithmiclayer for Re τ = 381, which is not present in the moments of ε for Re τ = 180.For both values of Re τ , the moments of ε remain relatively constant from z + ≈
100 to the channel centerline. The Ω moments also remain relativelyconstant over this range for Re τ = 381, but for Re τ = 180 the moments of Ωincrease and reach a maximum at z + ≈ − Re τ are17ost likely due to bursting of vortical structures across the channel, which isan effect due to the low Reynolds number. In particular, it is possible that bothwall regions may not be fully decoupled. This bursting causes large temporalvariations of the local enstrophy, which complicates the statistical convergenceof higher order moments. This lack of convergence may thus have a physicalfingerprint, and substantially more temporal snapshots (requiring significantadditional computational effort) are necessary to resolve this issue in futureinvestigations.Using conditional analyses based on regions of intense rotation, which aretaken here to correspond to vortical structure locations, the moments of both ε and Ω are shown to be larger in rotation-dominated regions than in thefull field. The increase in ε , in particular, is due to the correlation betweenevents of intense ε and Ω shown in Figure 3. At the same time, momentscalculated in the background field, where the rotation is not intense, indicatethat vortical structures make only a small contribution to the moments of ε inthe full field. Differences between the moments for the two values of Re τ in theconditional intense rotation fields are similar to the differences in the full field.In particular, the conditional moments of ε and Ω are relatively constant for z + &
100 in the Re τ = 381 case, but local maxima, particularly in the momentsof Ω, are still observed at z + ≈
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