The temporal evolution of the energy flux across scales in homogeneous turbulence
José I. Cardesa, Alberto Vela-Martín, Siwei Dong, Javier Jiménez
aa r X i v : . [ phy s i c s . f l u - dyn ] N ov The temporal evolution of the energy flux across scales inhomogeneous turbulence
J.I. Cardesa, a) A. Vela-Mart´ın, S. Dong, and J. Jim´enez
School of Aeronautics, Universidad Polit´ecnica de Madrid, 28040 Madrid,Spain (Dated: 9 May 2018)
A temporal study of energy transfer across length scales is performed in 3D numer-ical simulations of homogeneous shear flow and isotropic turbulence. The averagetime taken by perturbations in the energy flux to travel between scales is measuredand shown to be additive. Our data suggests that the propagation of disturbancesin the energy flux is independent of the forcing and that it defines a ‘velocity’ thatdetermines the energy flux itself. These results support that the cascade is, onaverage, a scale-local process where energy is continuously transmitted from onescale to the next in order of decreasing size.The difficulty in understanding a multiscale problem such as turbulence has been fre-quently tackled by assuming a priori some type of simplified phenomenology. Take, forinstance, Richardson’s cartoon based on concepts such as eddies whose energy cascades bythe successive breakup of larger eddies into smaller ones, until it is dissipated by viscosity. It was later used as the basis for the more quantitative work of Kolmogorov. We now knowthat in 3D turbulence the energy does cascade towards the smallest scales, at least onaverage. But several models have been discussed in the literature which are consistent withthis average trend yet differ in their detailed mechanism. For example, the energy couldjump directly from large eddies to much smaller ones, contradicting the scale localityassumed by Richardson, or include frequent excursions from smaller to larger scales – aprocess coined backscatter – questioning a unique directionality. Such alternative roadsleading to the same average behavior urge an improved understanding of the cascade dy-namics. In this letter, we report on the time taken by disturbances in the the energy fluxto travel in scale space – an essential ingredient in unsteady phenomenological models. To overcome the difficulty in generating a wide dynamic range in space but especially intime, our direct numerical simulations are purposely run for very long times to perform atemporal cross-correlation analysis between energy fluxes at various length scales.Our approach is based on the large- and small-scale decomposition of the instantaneousvelocity field according to u i ( x i , t ) = e u i ( x i , t ) + u ′ i ( x i , t ), where e u i is the i-th component ofthe spatially low-pass filtered velocity. In an incompressible flow with kinematic viscosity ν , the kinetic energy of the large-scale field evolves as (cid:18) ∂∂t + e u j ∂∂x j (cid:19) e u i e u i = − ∂∂x j (cid:16)e u j e p + e u i τ ij − ν e u i e S ij (cid:17) − ν e S ij e S ij − Σ + e u i e f i , (1)where e S ij = ( ∂ e u i /∂x j + ∂ e u j /∂x i ) / − τ ij e S ij , τ ij = g u i u j − e u i e u j is the subgrid-scale stress tensor and f i is the forcing term.Eq. (1) has been studied extensively in the context of large-eddy simulations (LES), wherecapturing the behavior of Σ is at the cornerstone of most modeling strategies. In the mean,Σ is positive and acts as a net energy removal from the large scales by the small ones.
We choose it as our real-space marker of cross-scale energy transfer, and study it in twodifferent flows: homogeneous shear turbulence (HST) and homogeneous isotropic turbulence(HIT). The details of the simulations are given in Table I. The low-pass filtered velocityin HIT was obtained by multiplying the Fourier modes ˆ u i ( k , t ) by an isotropic Gaussiankernel ˆ G ( k ) = exp[ − ( r k ) / where k is the wavevector and r the filter width. In a) [email protected] TABLE I. Parameters of the simulations. Re λ is the Reynolds number based on the Taylor-microscale. N i and L i are the number of real Fourier modes and the domain size in directions i = x, y, z . Length scales are η = (cid:0) ν /ε (cid:1) / and L o = K / /ε . Times are normalized by T o = K/ε . T simu is the simulation time, and ∆ t tot is the average delay between h K i and h ε i . T KK is anautocorrelation time for h K i , defined in the text. K = u i u i / Re λ N x × N y × N z ( L x × L y × L z ) /η L o /η T simu /T o ∆ t tot /T o T KK / ∆ t tot HST 107 768 × ×
255 1117 × ×
372 267 213 0.49 9.04HIT1 146 256
425 165 0.53 2.32HIT2 236 512
876 16 0.40 2.36HIT3 384 1024 HST this method could only be applied in the streamwise and spanwise directions, alongwhich the flow was Fourier-discretized. Since seventh-order compact finite differences wereused in the vertical (sheared) direction, filtering along it was implemented by convolvingthe velocity with the real-space transform of ˆ G ( k ): G ( x ) = P exp[ − x /r ], where P isa constant chosen to meet the normalization condition. For comparison, a sharp spectralfilter was used on some HIT fields with ˆ G ( k ) = 0 when | k | ≥ π/r , ˆ G ( k ) = 1 otherwise.Other energy transfer markers exist which could have been suited to the HIT simulation.An obvious candidate comes from the spectral energy equation in isotropic turbulence ∂E ( k, t ) ∂t = F ( k, t ) − νk E ( k, t ) + Ξ( k, t ) , (2)where E ( k, t ) is the 3D instantaneous energy spectrum, k = | k | and Ξ is the forcing. Thespectral energy flux Π( k ) = R k F ( k ) d k is often invoked in energy cascade studies. For asingle flow field of isotropic turbulence, Π( k ) and h Σ( r ) i are equal when a sharp spectralfilter is used to compute Σ with a cut-off wavenumber k = π/r . (Hereafter, h θ i is thetime-dependent spatial average of θ over the computational domain, while θ is the temporalmean of h θ i .) The difference between the two energy transfer markers thus amounts toa choice of filter type. Since Π is of limited use in flows other than HIT and given therelevance of Σ in LES, we favored Σ for comparison between the two flows.The probability density function ρ (Σ) can be seen on Fig. 1(a). Its positive skewnessindicates that strong events are more likely when Σ acts as a sink in Eq. (1) than as asource. The tails of ρ become narrower with increasing filter width, in agreement withRef. 11. Whereas the volume ratio of forward cascade to backscatter is known to favorthe former over the latter for Gaussian filters, the ratio of the total forward energy fluxΦ F = R ∞ Σ ρ (Σ) dΣ to that of backscatter Φ B = R −∞ Σ ρ (Σ) dΣ is not documented. Theirratio is given on Table II, which reveals an even stronger preponderance of the forwardcascade than what could be inferred from the volume ratios alone. To emphasize this point,Fig. 1(c) displays the value of the integrand defining Φ F and Φ B . The sharp spectral filterleads to a much more symmetric picture of the cascade, as inferred by comparing Fig. 1(c)and (d). Fig. 1(b), with a sharp filter, shows that ρ (Σ) is less skewed than on Fig. 1(a),with a Gaussian filter, yet the wider tails for decreasing r occur with both filters. TABLE II. Ratio Φ F / Φ B of forward to reverse energy flux. Numbers in parentheses are thecorresponding volume ratios, R ∞ ρ (Σ) dΣ / R −∞ ρ (Σ) dΣ . r/η N/A
N/A
HIT2(Gauss) 15(3) 15(4) 21(5) 39(8) 59(10) 37(9)HIT2(sharp) 1.4(1.2) 1.7(1.3) 2.2(1.6) 2.9(1.9) 3.7(2.3) 5.0(2.7) −10 −5 0 5 10 1510 −4 −3 −2 −1 Σ / Σ ′ Σ ′ ρ ( Σ ) (a) r = 16 η r = 31 η r = 62 η r = 125 η −10 −5 0 5 10 1510 −4 −3 −2 −1 Σ / Σ ′ Σ ′ ρ ( Σ ) (b) r = 16 η r = 31 η r = 62 η r = 125 η −4 −2 0 2 4 6 8−0.100.10.20.3 Σ / Σ ′ Σ ρ ( Σ ) / Σ ′ (c) r = 16 η r = 31 η r = 62 η r = 125 η −4 −2 0 2 4 6 8−0.100.10.20.3 Σ / Σ ′ Σ ρ ( Σ ) / Σ ′ (d) r = 16 η r = 31 η r = 62 η r = 125 η FIG. 1. Top: probability density functions (PDFs) of Σ in HIT2, normalised by the standarddeviation Σ ′ . (a) Gaussian filter. (b) Sharp filter. Bottom: weighted PDF, whose integral definesΦ F and Φ B . (c) Gaussian filter. (d) Sharp filter. We now move towards the dynamics of h Σ i , based on its time series which we computedfor all flows with a Gaussian filter - the sharp filter was used only on a few HIT fieldswidely spaced in time. They are shown on Fig. 2(a) at two filter widths, together withthe kinetic energy h K i = h u i u i i / h ε i . It is clear that thesignals are correlated, but that there is a delay separating them. h K i and h ε i behaveas the earliest and latest signals, while the delay of h Σ i with respect to h K i increaseswith decreasing r . To visualize this effect more clearly, Fig. 2(c) displays the temporalevolution of h Σ i as a color map where the abscissae are time. Color bands correspondingto h Σ i have been ordered vertically with r decreasing logarithmically downwards, and h ε i added at the bottom. The propagation across r and t of disturbances in h Σ i is evident.To quantify this process, we compute the temporal cross-correlation of all these signalswith each other, as well as their temporal autocorrelation. A few such correlations areillustrated on Fig. 2(b), where the peak appears at the average delay between the twochosen signals. We start by looking at ∆ t tot , the average time taken by a change in h K i to propagate and appear as a change in h ε i . It is compiled in Table I for all our flows,which shows that ∆ t tot is approximately half the integral dissipation time T o ≡ K/ε .Similar data are scarce in the literature, so that a comparison is not straightforward. In acomputational study of homogeneous shear flow, the delay between the time histories of h K i and h ε i was estimated to be of the order of ∆ t tot /T o ≈ . Re λ ≈
50. Since thisdisagrees with our ∆ t tot /T o = 0 .
49 at Re λ ≈ t tot /T o = 0 .
52. The discrepancy can probably be attributed to the differentestimation methods. Cross-correlations were not computed in Ref. 14, where ∆ t tot wasnot the main focus of the study. A value of ∆ t tot /T o = 0 .
44 can be extracted from thedata in Ref. 15 where a DNS of HIT was ran at Re λ = 122, which agrees well with ourfindings. In Ref. 16, a ∆ t tot was defined as the delay between h K i / /L int and h ε i , where L int = (3 π/ h K i ) R k − E ( k ) d k . Using temporal cross-correlations in HIT at Re λ = 219– comparable to our HIT2, they found ∆ t tot /T o ≈ .
21. We found ∆ t tot /T o ≈ .
28 usingour HIT2 and the same definition of ∆ t tot as in Ref. 16, confirming that the lag between t/T o (a) h K ih Σ i , r = 125 η h Σ i , r = 62 η h ε i t/T o (b) η , η η , η η , εK, ε l o g r η t/T o h ε i (c) FIG. 2. (a) Temporal evolution of spatially-averaged quantities, centered and normalized by theirstandard deviation; HIT2. (b) Cross-correlation curves between time series of h Σ i at various filterwidths and between h K i and h ε i ; HIT2. (c) Time-scale diagram of h Σ i , with r decreasing from topto bottom and h ε i added as the bottom band; HIT3 with r/η values from Table II. The dash-dottedline corresponds to ε / ∆ t = (250 η ) / − r / - see Eq. (6). h K i / /L int and h ε i is shorter than that between h K i and h ε i .A comment on the origin of the time dependence of h K i , h ε i and h Σ i is in order. Thetemporal oscillations of h K i and h ε i in HST are known to be physically caused and relatedto bursting . In the HIT simulations, the turbulence is sustained by a deterministic forceˆ f i ( k , t ) = ( ε ˆ u i ( k , t ) / [2 E f ( t )] , if 0 < k < k f , , otherwise , (3)where ε is the target mean dissipation, E f ( t ) = R k f E ( k, t ) d k and k f = 4 π/L x . Thiscommonly used scheme is mildly unstable, because of the delay between ˆ f i and h ε i . Itgenerates time oscillations of the energy while maintaining a constant resolution of k max η =1 . t tot onthe large-scale forcing. We define the characteristic time scale T KK of the kinetic energyas the width of the temporal autocorrelation of h K i at half its peak height. The ratio T KK / ∆ t tot is between 2 and 2.5 for all our HIT simulations, but about 9 for the HST –see Table I. Yet changes in h K i appear as changes in h ε i within half a large-eddy turnovertime in the two differently forced flows, suggesting that ∆ t tot /T o is a common feature of theenergy cascade when the large scales fluctuate with periods in our range of T KK /T o . Thedependence of our measured ∆ t tot /T o on the large-scale period could be studied further byextending this range of T KK /T o with the addition of a modulating frequency in the forcing,as done Ref. 18, or with a stochastic forcing. TABLE III. Symbol legend for Figs. 3 and 5. r a /η = 10 a √ /π so that r ≈ η , r ≈ η , etc. r → h ε i a = 1 a = 2 a = 4 a = 8 a = 16HST + N/A × ∗ N/A
HIT1 (cid:3) △ ▽ N/A N/A
HIT2 (cid:4) N
H ◮
N/A
HIT3 (cid:4) N
H ◮ ◭ r/ η ∆ t / t k o l (a) ∝ r ∝ r .940.960.981 r/ η ∆ t s t e p / ∆ t s t e p (b) FIG. 3. (a) Delay between h Σ i and h ε i , measured either as the one-step delay, or as the sum oftwo intermediate steps involving r a . (b) Ratio between two- and one-step delays. See Table III forsymbols of r a and flow. We next test the additivity of the delay times. We want to see if the time taken by adisturbance in the energy flux in going from scale size r to r is equal to the sum of thedelays in going from r to r and from r to r , where r > r > r . Fig. 3(a) displays theaverage time needed for disturbances in the energy fluxes at a given scale r to travel downto h ε i . This time is computed for all available combinations of two intermediate jumpsstarting at r and ending in h ε i . The agreement between the different jump combinationswithin the same flow and across different flows is satisfactory. In order to highlight anyresidual discrepancy, we display the value of the ratios between one- and two-step cascadingtimes on Fig. 3(b). A ratio close to unity implies additivity of the delays, which is confirmedfor all the starting scales r and flows examined. Note the narrow range of the vertical axis.Note also that Fig. 3(a) hints at two different regimes for r above and below approximately30 η . This is consistent with the results of Ref. 20, who showed that viscous eddies below r/η ≈
30 are enslaved to larger ones above that scale. In essence, r/η = 30 is the lowerlimit of the inertial cascade.In an influential paper, Lumley discusses two cascade models, and proposes a forcingexperiment similar to the present one to distinguish between them. He starts by consideringa hierarchy of discrete eddies of decreasing size. In the first model, each eddy transfers itsenergy to the one immediately below. The transfer occurs at a rate determined by thecorresponding scale-dependent eddy turnover time, [ k E ( k )] − / ∼ k − / ∼ r / , so thatthe propagation of energy from the large towards the small scales develops into a front-likediffusion through scale-space with a finite scale-dependent velocity. In the second model,most of the energy is still transferred to the immediately smaller eddy below, but a fractionis passed to other eddies further along the cascade. Hence the smallest eddies receive a smallamount of energy almost immediately after it is injected into the system, and increasingamounts as time goes on. The difference between the two models is that all the energyin the first one has to pass through each eddy size, resulting in additive cascade times,while this additivity is not guaranteed in the second model. Theoretical arguments wereput forward both for and against long-range energy transfer, and attempts were madeto carry out the forcing experiment, but they were hindered by the low Reynolds numbersavailable from simulations at the time. The matter has remained controversial until now,and our additive data on Fig. 3 favors the local model.We now focus on the scaling of ∆ t . We start by introducing the strong assumption thatthe values of ∆ t we measure are between scales r within an inertial range, where r and ε arethe only relevant quantities. Within this simplistic framework, then, a velocity in r -spacecan be defined as ˙ r = ε/ρ E , (4)where the energy density ρ E ( r ) is a real-space equivalent of E ( k ). We put forward the −1 r/ η − ( d q / d r ) ε − / η / (a) HST, GaussHIT3, sharpHIT3, Gauss ε / r − / r/ η ( − ∆ q ) / ε / / η / (b) HST, GaussHIT3, sharpHIT3, Gauss ∝ r / FIG. 4. (a) Derivative of q = e u i e u i / r . The dashed line corresponds to ε / r − / .(b) Energy content within a band of scales between r and r + ∆ r , where ∆ r goes from a given r to the next bigger r in the plotted series. ∆ q = e u i e u i ( r + ∆ r ) − e u i e u i ( r ). following candidate for ρ E , based on q = e u i e u i / ρ E = − dqdr . (5)Other densities have been introduced in physical space. Townsend’s r -derivative of thecorrelation function, or the signature function found in Ref. 24 are two examples. Wechose our expression as it is based on the filtering approach we use. The data on Fig. 4(a)shows that the energy density − dq/dr in our two flows is a positive quantity. In Fig. 4(b)we see that the energy − ∆ q contained between r and r + ∆ r is a quantity which growsproportionaly to r / within a reasonable range in HIT3 - not so in HST with much smallerscale separation. Such r / behaviour is consistent with the Kolmogorov-Obukhov theory, yet it is based on a completely different flow decomposition from the structure function. ( r/r a ) / − ε / ∆ t r → r a / r / a FIG. 5. ε / ∆ t r → r a /r / a against ( r/r a ) / −
1, where ∆ t r → r a is the average delay between h Σ( r ) i and h Σ( r a ) i , with r > r a . Symbols as in Table III. The solid line corresponds to Eq. (6) Substituting ρ E in Eq. (4) by ε / r − / , and integrating from r a to r leads to∆ t r → r a = ε − / (cid:16) r / − r / a (cid:17) , (6)which implies that our data should fall on a straight line when plotted logarithmicallyas done in Fig. 5. The agreement is not completely unsatisfactory. Particularly whenconsidering the inertial range assumptions used which have no reason to apply if r or r a arebelow the viscous limit of 30 η , or given the poor compliance of HST to the inertial rangescaling with r - see Fig. 4. A dashed line following Eq. (6) for HIT3 was added on Fig. 2(c).A derivation of Eq. (6) carried out in spectral space can be found in Ref. 12, leading to a k − / dependence of ∆ t . Ref. 25 studied Lagrangian time correlations of Σ( k ) and ε , whichhinted at a k − / dependence of the peaks in their correlations - see inset of their Fig. 2.Earlier, the same group measured the temporal correlation between the energy at a givenscale and the energy at a smaller scale found later by following the flow in both forced anddecaying HIT. Their conclusion that the peak in correlation happens later for increasingscale separation is consistent with what we observe. A difficulty with that work was the useof correlations of energy rather than energy flux. Fluxes are the quantities conserved acrosscascades, and the natural objects for their study. Furthermore, they could only considerone-jump delays, ruling out the additivity test on Fig. 3 which supports the locality of theenergy cascade in an average sense.
ACKNOWLEDGMENTS
This work was supported by the Multiflow grant ERC-2010.AdG-20100224. Computa-tional time was provided on GPU clusters at the BSC (Spain) under projects FI-2014-2-0011,FI-2015-1-0001 and in Tianjin’s NSC (China). L. F. Richardson,
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