Spectral analysis of turbulent viscoelastic and Newtonian channel flows

Journal of Non-Newtonian Fluid Mechanics 200 (2013) 165–176

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Spectral analysis of turbulent viscoelastic and Newtonian channel flows L. Thais a,b, G. Mompean a,b,⇑, T.B. Gatski c,d a

Université de Lille Nord de France, USTL, F59000 Lille, France Laboratoire de Mécanique de Lille, CNRS, UMR 8107, F59655 Villeneuve d’Ascq, France c Institut Pprime, CNRS–Université de Poitiers–ISAE–ENSMA, F86962 Futuroscope Chasseneuil Cedex, France d Center for Coastal Physical Oceanography and Ocean, Earth and Atmospheric Sciences, Old Dominion University, Norfolk, VA 23529, USA b

a r t i c l e

i n f o

Article history: Available online 16 May 2013 Keywords: Turbulent channel flow Spectra Viscoelastic fluid Drag reduction FENE-P model

a b s t r a c t The one-dimensional spectra in the streamwise direction of the velocity and vorticity fields in turbulent channel flows of Newtonian and non-Newtonian viscoelastic fluids are presented for friction Reynolds numbers up to Res0 = 1000. The most striking feature induced by viscoelasticity is a marked drop, as rapid as k5, in the energy level of the streamwise velocity spectra at high wave-numbers, and in agreement with experimental data by Warholic et al. (1999) [15]. The scaling of the streamwise velocity spectra for viscoelastic flow share some characteristics with the Newtonian spectra, but also exhibit unique properties. In particular, the logarithmic correction to the usual k1 law at the intermediate scales (eddies with size one to ten times the distance from the wall), found by del Álamo et al. (2004) [7] in the case of Newtonian turbulence, still holds in viscoelastic flows; although, with different scaling coefficients. In contrast, the longest modes of the spectra of the streamwise velocity component are found to behave differently. These modes are longer in viscoelastic flows and their scaling with the channel centerline velocity, here confirmed for Newtonian flow, fails for high drag reduction viscoelastic turbulent flows. As for vorticity, it is found that the spectra of its cross-flow component in viscoelastic flows exhibit a significantly higher energy level at large scales, with a tendency towards a k1 law for high drag reduction and high Reynolds number. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Reynolds number similarity is essential in describing the nature of turbulent wall-bounded flows. It is also vital to the engineer who is sometimes prone to extrapolate information from low Reynolds number numerical simulations, or from laboratory experiments. This holds for Newtonian turbulent flows, and there is no reason it should not for viscoelastic turbulent flows, although the Reynolds number range of interest is certainly more reduced for the latter. Within the framework of wall-bounded turbulence in a Newtonian fluid, it is now widely admitted that there are serious similarity failures in the behavior of the velocity fluctuations. In particular, the intensity of the streamwise velocity fluctuation scales poorly with the friction velocity as the Reynolds number is increased. Townsend’s [1] concept of ‘active’ motion, contributing to turbulent energy production, as opposed to ‘inactive’ motion responsible for local stirring of streamwise vortices, is now retrospectively believed to be a plausible explanation of this failure. ⇑ Corresponding author at: Université de Lille Nord de France, USTL, F59000 Lille, France. Tel.: +33 3 28 76 74 64. E-mail addresses: [email protected] (L. Thais), [email protected] (G. Mompean), [email protected] (T.B. Gatski). 0377-0257/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnnfm.2013.04.006

Since the inactive motions are quasi-inviscid and contribute significantly to the streamwise velocity fluctuation, there is no reason that they should scale with the friction velocity. Another aspect concerns the properties of the power spectra of the velocity fluctuations in turbulent shear flows, which will be the main focus of the present contribution. At distances sufficiently far away from boundaries, Kolmogorov’s similarity argument applies, see e.g. [2] where the theory is well outlined. However, for the wall-region of wall-bounded flows, Laufer [3] was the first to provide experimental evidence of a k1 decay for the ‘large’ scales in the one-dimensional power spectra of the streamwise velocity component. The k1 law is further supported by theoretical arguments based on dimensional analysis [4] (see also [5]), and intimately related to Townsend’s attached-eddy hypothesis. Nevertheless, it has also received a variety of criticisms. Incomplete similarity was found in [6], in which high Reynolds number data from turbulent pipe flows are shown to collapse with classical inner and outer scaling, but only over limited ranges of non-overlapping wave numbers. Similarly del Álamo et al. [7], using channel direct numerical simulation (DNS) data, proposed a logarithmic correction to the usual k1 behavior. This scaling of the spectra is consistent with a scaling of the largest modes of velocity with the channel centerline velocity, rather than with the friction velocity.

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For viscoelastic drag reducing turbulent flows, the available information is even more scarce than for ’classical’ Newtonian turbulence. In spite of a growing amount of DNS and experimental data (see [8] and the references therein for a recent review), there appears to be no systematic study of Reynolds number similarity. From numerical simulations, most of the existing channel flow DNS are limited to very low Reynolds numbers, usually Res0 = O(300), where Res0 = us0h/m0 is the friction Reynolds number based on the zero-shear friction velocity at the wall us0, the channel halfwidth h, and the total (solvent + polymer) viscosity m0. Noticeable exceptions exist, e.g. Res0 = 530 in Min et al. [9] where drag reduction inception was explored, Housiadas and Beris [10] (see also Housiadas et al. [11]) where Res0 = 590 was reached and in which a Reynolds number similarity study was undertaken. However, increasing the Reynolds number in viscoelastic DNS has always been made at the cost of a reduced channel size that imposes limitations on the ability of the simulation to capture the elongated large scale structures present in drag reducing flows (see also the discussion about the so-called ‘Very Large Scale Motions’ [12] in the context of Newtonian turbulence). The very low-Reynolds number DNS data available makes it difficult to develop generally applicable Reynolds averaged closure models as well. Such closures will inherently need closure coefficients dependent on Reynolds number and not directly applicable to equilibrium turbulent flows independent of Reynolds number effects. The focus of this study is to use recently computed turbulent Newtonian and viscoelastic channel data obtained in a channel of length Lx = 6 p and up to a friction Reynolds number Res0 = 1000. The data set was obtained using a hybrid spatial scheme with Fourier spectral discretization in the two homogeneous directions and high-order finite differences in the wall-normal direction [13]. Part of this data set was recently used [14] to explore the Reynolds number similarity of the flow statistics at high drag reduction of the order of 60%. In [14], it was shown that the friction velocity was a poor choice for scaling the velocity fluctuations in viscoelastic turbulent channel flows. In particular here, the properties of the power spectral density in viscoelastic drag reducing turbulent flows will be analyzed. To date there does not appear to be any such systematic study using numerical simulation data; although, wavenumber and frequency velocity power spectra at the same Reynolds number (Res0 = 1000) from experimental data can be found in Warholic et al. [15]. This, then provides an excellent opportunity to compare DNS and experimental data. Although the emphasis will be put on velocity spectra, the power spectral density of vorticity will also be considered. While it is well-known that the vorticity is correlated only over the dissipation scales in homogeneous, isotropic turbulence [16], it has been suggested that vorticity could be statistically correlated over the same integral length scale as the velocity in inhomogeneous, anisotropic turbulence. This results in large values of the spectral density of vorticity at the low and inertial ranges of wavenumbers in experimental data [17]. Viscoelastic turbulence being more anisotropic than Newtonian turbulence, the question arises whether an increased anisotropy will also affect the vorticity spectra in viscoelastic wall-bounded turbulent flows. The organization of the paper is as follows. Section 2 briefly describes the model equations being solved as well as the main statistics of the flows at Res0 = 1000. Here, three flows at Res0 = 1000 will be studied: the reference Newtonian flow, and two low (30%) and high (60%) drag reduction cases. With this choice, it will be possible to highlight (see Section 3) the effects of viscoelasticity on the spectra at a high Reynolds number. In Section 4, Reynolds number similarity of the spectra will be considered using the high drag reduction data with the friction Reynolds number varying from Res0 = 180 up to 1000.

2. The channel data Fully developed turbulent channel flows of Newtonian and viscoelastic fluids modeled for the latter as a polymer dilution in a Newtonian solvent are considered. The usual notations for this geometry are used, i.e. the channel streamwise direction is x1 = x, the wall-normal direction is x2 = y, and the spanwise direction x3 = z, with the velocity field in the respective directions (u, v, w) = (u1, u2, u3). The channel half-gap is denoted h, while the two other directions are considered of infinite extent and discretized with Fourier expansions. With the length and time respectively scaled by m0/us0 and m0 =u2s0 , where m0 is the total (solvent + polymer) viscosity and us0 the zero-shear friction velocity, the dimensionless conservation equations are:

@uþj @xþj

¼ 0;

ð1aÞ þ

þ @uþi @Pþ @ Nij þ @ui þ þ: þ þ uj þ ¼  @t @xj @xþi @xj

ð1bÞ

The pressure is P+ and the stress tensor Nþ ij is composed of (Newtonian) solvent and (polymeric) viscoelastic contributions,

Nþij ¼ 2b0 sþij þ Npþ ij ;

ð2Þ

  þ þ þ =2; b0 the ratio ¼ @uþ i =@xj þ @uj =@xi of the Newtonian viscosity mN to the total zero-shear viscosity m0 = mN + mp0, and Res0 = us0h/m0 the zero-shear friction Reynolds number. The polymer is based upon the nonlinear kinetic based FENE-P dumbbell model, from which the polymeric stress is given by with the strain rate tensor sþ ij

Npþ ij ¼

1  b0 ½f ðfcgÞcij  dij ; Wes0

ð3Þ

with Wes0 ¼ ku2s0 =m0 the friction Weissenberg number representing the ratio of the elastic relaxation time k relative to the viscous timescale. The polymeric stress of the FENE-P model includes the Peterlin function,

f ðfcgÞ ¼

L2  3 L2  fcg

;

ð4Þ

ensuring shear-thinning behavior and a finite elongational viscosity for a finite value of the extension rate, with L the fully-stretched polymer length and fcg designating the trace of the conformation tensor c. The equation system is closed with a transport equation for c,

    @cij þ @c ij þ þ þ þ þ þ uk þ ¼ c ik skj þ sik ckj  c ik wkj  wik ckj @t @xk 

f ðfcgÞcij  dij ; Wes0

ð5Þ

þ þ þ þ where wþ ij ¼ ð@ui =@xj  @uj =@xi Þ=2 is the rotation rate tensor. A detailed description of the direct numerical simulations and the associated numerical procedure used to solve Eqs. (1)–(5) can be found in [13]. The hybrid spatial scheme has Fourier spectral accuracy in the two homogeneous directions and 6th-order compact finite differences for first and second-order wall-normal derivatives, while time marching is 4-th order accurate. The resulting parallel algorithm is highly scalable, utilizing up to 16384 cores of an IBM Blue Gene/P computer, which allows DNS at (relatively) high Reynolds numbers in large computational boxes for both Newtonian and viscoelastic flows. For the present study, the parameter set b0 = 0.9 at a fixed Reynolds number Res0 = 1000 with two drag reduction cases are first considered: L = 30, Wes0 = 50, corresponding to a medium percentage drag reduction (DR) of 30%, and L = 100, Wes0 = 115, corresponding to a high percentage drag reduction of 58%. The channel extent for these flow cases is

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Fig. 1. (a) Mean velocity profiles across channel half-width for Newtonian (full thick lines) and viscoelastic (dotted lines) flows at Res0 = 1000. (b) Scaled mean velocity gradient over channel half-width. The full thin lines mark the maximum drag reduction (MDR) asymptote on the left panel, and the inverse of the Kármàn constant, 1/0.41, on the right panel.

Lx = 6ph, Ly = 2h, Lz = 1.5ph, with grid resolutions 8.8 6 dx+ 6 12.3, 0.2 6 dy+ 6 10.4, 5.4 6 dz+ 6 7.3. Fig. 1 displays the mean velocity profiles (left panel) as well as the rescaled wall-normal velocity gradient (right panel). Here, the overbar denotes averaging in time, in wall-parallel planes. A broad loglaw region appears for the 3 flows considered at Res0 = 1000, which has not been observed in previous viscoelastic DNS at lower Reynolds numbers, see e.g. [10,19,20]. Also shown in Fig. 1a is the asymptotic limit of maximum drag reduction (MDR) theoretically predicted by Benzi et al. [18], and closely matching the experimental observations obtained by Virk [21]. One notices a steepening of the viscoelastic velocity profiles in the near-wall layer, the high drag reduction flow case getting close to the MDR asymptote in this region. Such behavior suggests a significant extension of the sublayer/buffer layer regions into the channel as viscoelasticity increases. Fig. 2 shows the corresponding root-mean-squared (r.m.s.) velocities across the channel half-width. The well-known anisotropic effect of the polymer which contributes to the increase in the turbulent fluctuations in the streamwise direction, while suppressing fluctuations in the other two directions is shown. These effects are also shown to be more pronounced with increasing viscoelasticity. In addition, there is a shift in the peak amplitude location for the streamwise r.m.s. velocity relative to the Newtonian case. A local peak in the spanwise direction for Newtonian flow also appears, and is not observed at lower Reynolds numbers, confirming earlier findings in Newtonian turbulence [22]. Viscoelasticity tends to suppress the existence of this local peak in the spanwise direction. Since the primary focus in this analysis is the power spectra of the velocity field, it is advantageous to exploit the Fourier-Galerkin spatial scheme used in the two homogeneous directions x and z, and straightforwardly evaluate in Fourier space the discrete premultiplied two-dimensional power spectral density of the ui velocity component,





Uii ðkx ; kz ; yÞ ¼ kx kz u^ i u^i ;

ð6Þ

with kx, kz the discrete wavenumbers in each respective direction.   ^i u ^ i is the time-averaged two-dimensional power The quantity u spectral density1 of the ui velocity component, which is here taken  ^i u ^ i is time-averaged power per unit spectral ray, or time-averaged power u spectral density, the overhat standing for the unscaled 2D-discrete Fourier transform coefficients. The star is complex conjugation, and the angle brackets are for timeaveraging. 1



in wall-parallel slabs over 500 flow snapshots spanning 5–6 eddy turnover times. From the 2D-power spectral density, it is also straightforward to evaluate the discrete 1D-power spectral densities in each direction x and z by integration in the respective orthogonal direction. Here, we shall restrict ourselves to the 1D-x power spectral density in the streamwise direction, which can be evaluated through integration in the cross-channel direction p N   ð2LX Þ gz z   2p ^i u ^ i ðkx ; kz ; yÞ; Eii ðkx ; yÞ ¼ u Lz kz ¼ð2LpÞ:N gz z

ð7Þ

where Lz is the channel width, and Ngz the number of Fourier Galerkin modes in the same direction. Similarly, one can define the 1D-streamwise power spectral density of the ith component of vorticity p N   ð2LX Þ gz z   2p ^ ix ^ i ðkx ; kz ; yÞ; /ii ðkx ; yÞ ¼ x Lz kz ¼ð2LpÞ:N gz z

ð8Þ

  ^ ix ^ i is the 2D-power spectral density of the ith compowhere x nent of the vorticity vector x = r  u. 3. Power spectral densities at Res0 = 1000 Both the velocity and vorticity power spectral densities are examined at two different locations within the channel. One location is at y+ = 60 where significant effects of drag reduction on the flow structure are expected to be found and the other is at the channel centerline where both geometric symmetry and turbulence stress isotropy can influence the flow structure. Alternate scalings, using either wall-normal distance or channel half-width, of streamwise velocity spectra are also investigated in order to identify and quantify the power law behavior. 3.1. Newtonian vs. viscoelastic power spectral density Before examining the scaling of the velocity spectral density for each flow case, it is interesting to compare velocity spectral density at fixed positions from the wall for the 3 flows considered at Res0 = 1000. Fig. 3 shows such a comparison at the wall-normal position y+ = 60, which is in the region where a strong influence of the drag reduction mechanisms are assumed to take place. Note that the spectral densities are here normalized using wall units, which

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Fig. 2. Distributions of r.m.s. velocity fluctuations for Newtonian (full lines) and viscoelastic (dotted lines) flows at Res0 = 1000.

allows a direct comparison of the relative weight of the spectral rays for each flow. The most striking feature of Fig. 3 is the pronounced drop of energy in all three components in the high wavenumber range as viscoelasticity increases. In the high drag reduction case, there is a 5 clear tendency towards a kx power law over a full decade in the 2 wavenumber range 2  10 [ kxg [ 2  101 for the streamwise and spanwise velocity spectral densities (g is the Kolmogorov viscous scale). No such behavior is observed in the Newtonian velocity spectra. This observation is in line, at least qualitatively, with the experimental data of Warholic et al. [15]. In particular, their Fig. 15 shows a similar dramatic drop in the energy density function of the streamwise velocity component. Although their plot is a frequency spectrum, one can extrapolate, using the Taylor frozen-turbulence hypothesis, a power law region approximately fol4 5 lowing kx to kx for their 3 high drag reduction flow cases (55%, 64% and 69% percent drag reduction). Another characteristic of the viscoelastic spectra is that the low wavenumber energy content in the streamwise direction is markedly higher for the high drag reduction flow case with respect to Newtonian flow, which globally contributes to the enhancement of the streamwise velocity fluctuation as was shown in Fig. 2a. This feature, however, is only partially in line with the experimental data shown in Fig. 15 of Ref. [15], where the low frequency spectral content is seen to increase with respect to the Newtonian flow only for their 55% drag reduction case, and not for the other two drag reduction cases at 64% and 69%. While the small-scale structures

are already significantly affected at low drag reduction, the effect is further enhanced at high drag reduction. In contrast, the large turbulent structures are relatively unaltered by viscoelasticity in the low drag reduction flow, with only a marginal increase in the streamwise direction in the wavenumber range 103 [ kxg [ 102. There is also an increase in the low wavenumber energy level in the spanwise velocity component spectrum at high drag reduction, but the increase is clearly less pronounced than in the streamwise component. The spectra of the streamwise and spanwise velocity components in the viscoelastic cases cross the Newtonian spectra at approximately kxg  102; the crossing wavenumber being slightly smaller at high drag reduction. The spanwise velocity energy drop at high wavenumber is such that it is not compensated by the modest energy increase at low wavenumber, resulting in a net attenuation of the spanwise velocity fluctuations as seen in Fig. 2b. In contrast with the power spectral density of the velocity components in the two homogeneous directions, the viscoelastic power spectral densities of the wall-normal velocity (Fig. 3c) are attenuated at all wavenumbers, and the (relative) attenuation is more pronounced at low wavenumber. Second, there is a high wave4 5 number kx power law, which is less steep than the kx law observed in the streamwise and spanwise components. The confining effect of the channel walls is a plausible explanation to the energy attenuation of the largest turbulent structures in the wall-normal velocity, while a similar polymer-turbulence interaction mechanism must be at play at small scales in the 3 spatial

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169

+ Fig. 3. 1D-power spectral density of the 3 velocity components Eþ ii ¼ Eii =ðm0  us0 Þ at wall-normal position y = 60 for Newtonian and viscoelastic flows at Res0 = 1000. g is the Kolmogorov viscous scale based on the zero-shear total viscosity: (a) streamwise velocity-u; (b) spanwise velocity-w; and (c) wall-normal velocity-v.

directions. The overall consequence of the energetics in the wallnormal direction is of course a marked attenuation of the wall-normal velocity fluctuations as seen in Fig. 2c. Vorticity spectra, Eq. (8), can provide further information in the analysis of viscoelastic turbulent flows. In Fig. 4, the 1D-power spectral densities of the 3 components of vorticity at the same wall-normal position y+ = 60 are plotted for the Newtonian and viscoelastic flows at Res0 = 1000. For the spectra of the streamwise component of vorticity, one finds a similar behavior to the spectra of the wall-normal velocity component (Fig. 3c). The large scales are little affected by viscoelasticity; whereas, viscoelasticity in3 duces a drop in the energy content following a kx power law from the middle of the inertial subrange, kxg J 102, down to the smallest resolved scales. The spectra of the other two components of vorticity have a distinct behavior. The spectra of the spanwise and wall-normal vorticity components in the viscoelastic cases cross the Newtonian spectra around kxg  102; the crossing wavenumber being slightly smaller at high drag reduction, which is similar to the results obtained for the velocity spectra. One will also notice the 4 existence of a steeper drop following a power law kx in the inertial subrange. Viscoelasticity effects are most noticeable in the crossstream component of vorticity (Fig. 4b). For this component, the large scale energy content is increased by more than an order of magnitude in the high drag reduction flow with respect to the 1 Newtonian flow. Also, there is a tendency towards a kx behavior

for the high drag reduction flow in the low wavenumber range 2  103 [ kxg [ 8  103. This result is in line with the findings of Morris and Foss [17] who reported large values of the spectral density of the cross-flow vorticity at the low and inertial ranges of wavenumbers in inhomogeneous high Reynolds number Newto1 nian turbulent flow. It is surprising there is no evidence of a kx decay in the Newtonian case here, which is probably due to the modest value of the Reynolds number. This suggests that such a decay law in the viscoelastic case is a result of the increased anisotropy levels. Another feature of the heightened anisotropy of these flows is the existence of power laws in the vorticity spectra at high wavenumbers. In homogeneous, isotropic turbulence no such power law appears in the inertial subrange [2]. Finally, Fig. 5 shows the velocity power spectral densities as above, but at the channel mid-plane. Here, the Newtonian and low drag reduction spectra are virtually the same, indicating that the polymer has little, if any, influence on the turbulent structures outside the log-layer in the low drag reduction flow. In contrast, the high drag reduction spectra again exhibit a high wavenumber energy drop, and a low wavenumber energy increase; although, both effects are less pronounced than observed in the inner layer. Here, the polymer effect on turbulent structures is isotropic, in contrast with the observation in the inner layer. There is no tendency towards a power law drop of the energy for the spanwise and wall-normal velocity components. However, for Newtonian flow and for low drag reduction flow, there is a discernible inertial

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m0 =u3s0 at the wall-normal position y+ = 60 for Newtonian and viscoelastic flows at Res0 = 1000. pffiffiffiffiffiffiffiffiffiffi wavenumber range where the k5/3 decay law applies on the limit kz =y ¼ a1 kx =y, with a1  2.5. In the mean time, the lower streamwise spectra. In contrast, no such k5/3 law applies to the part of the 2D-spectral density is bounded atphigher ffiffiffiffiffiffiffiffiffiffi wavelengths high drag reduction flow over a significant wavenumber range. by a parallel square-root frontier kz =y ¼ a0 kx =y, with a0  0.5 This means that for this particular flow the turbulence has not another constant. In [7], this somewhat surprising nonlinear reached an homogeneous-isotropic state on the channel midsquare-root behavior was explained as the result of random stirplane. Similar conclusions can be drawn concerning the spectra ring of the mean velocity gradient by eddies with size O(y). of vorticity at the channel mid-plane, but these are not shown. The concurrent 2D-power spectral densities Uþ 11 for viscoelastic turbulent flows are plotted in Fig. 6b and c. There seems to subsist a similar global qualitative behavior for viscoelastic 2D-power 3.2. Scaling of streamwise velocity spectra with distance off the wall spectral densities. However, there is a clear shift of the spectra towards the right, i.e. towards larger scales. In particular, the transiA logarithmic correction to the usual k1 behavior of E11(kx) tion between wall-attached and wall-detached eddies seems to was suggested in [7] at high wave-numbers. The correction was occur at scales 30–50% larger than in Newtonian flow. One also nobased on the hypothesis that the pre-multiplied 2D-power spectices that the linear upper bound kz = kx is considerably narrower as tral density U11 was approximately constant over a significant drag reduction increases. spanwise wavelength range [k0, k1], with the lower and upper Another striking feature induced by polymer is an overall widbounds of this range scaling, respectively, as k0 ’ (kxy)1/2 and ening of the space between the nonlinear upper and lower bounds, k1 ’ kx. Fig. 6 displays equal isolines of Uþ 11 (in wall units) as a which suggests that the energy containing eddies also increase in function of the streamwise and spanwise wavelengths kx/y and size in the spanwise direction. A closer inspection of the nonlinear kz/y. The chosen iso-value corresponds to approximately 25% of lower and upper bounds of the spectra reveals another interesting the peak spectral density for the 3 flows considered at property. While these nonlinear bounds have a square root behavRes0 = 1000, and the 3 plots are at the same wall-normal position ior in Newtonian flow, their shape in viscoelastic turbulent flow y+ = 159. follows a nonlinear law of the form kz/y = (ai)i=0,1(kx/y)n. The nonFig. 6a confirms the findings in [7] for Newtonian flow, i. e. the Newtonian exponent n is smaller than 0.5 (n = 0.4 for 30% drag 2D-power spectral density Uþ is bounded by k = k at short z x 11 reduction, n = 0.35 for 58% drag reduction, see caption of Fig. 6 wavelengths in the range y [ kx [ 10y. Beyond kx  10y, which for more details), which means that the slope of the bounds is corresponds to the transition between wall-attached and wall-demilder than in Newtonian flow. tached eddies, the ridge of the spectrum approaches a square-root Fig. 4. 1D-power spectral density of the 3 components of vorticity /þ ii ¼ /ii 

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Fig. 5. 1D-power spectral density of the 3 velocity components Eþ ii ¼ Eii =ðm0  us0 Þ at the mid-channel center plane for Newtonian and viscoelastic flows at Res0 = 1000. On the zero-shear total viscosity: (a) streamwise velocity-u; (b) spanwise velocity-w; and (c) wall-normal velocity-v.

2 With the assumptions that Uþ 11 ¼ U11 =us0  c, with c some n 1n other constant, k0 ’ a0 kx y and k1 ’ kx, a logarithmic correction for the pre-multiplied streamwise 1D-power spectral density can be derived as

þ

kx Eþ11 ðkx ; yÞ ’

Z

k1

k0

Uþ11 ðkx ; kz ; yÞ

  1 k dkz x ; ’ ð1  nÞc log a0n1 kz y

ð9Þ

Eq. (9) generalizes Eq. 3.2 in [7] for non-Newtonian turbulent flow. The exponent value n = 0.5 reverts back to the Newtonian behavior with a square-root dependence of the lower wavelength bound k0. Note that when n < 0.5 (non-Newtonian flow), the nature of the correction is unchanged, it is still a logarithmic correction in kx/y, but with a different slope. In order to probe the above mentioned logarithmic correction to the k1 energy spectrum hypothesis, Fig. 7a shows the pre-multiplied 1D-power spectral density for Newtonian flow using inner variable scaling, i.e. the similarity variables us0 and y for velocity and length, respectively. The spectra are presented in pre-multiplied form on linear-log axes since a linear ordinate enables a closer scrutiny of scalings than that afforded by a log–log representation, any k1 dependence now showing as a horizontal line. The logarithmic horizontal axis is the non-dimensional streamwise wavelength kx = 2p/kx, such that small to large turbulent scales are read from left to right. The spectra are given for 5 wall-normal positions extending from the upper bound of the buffer layer to the outer part of the loglayer.

As can be seen, there is a significant wavelength range over which the pre-multiplied power spectral densities are collapsed along a straight line with positive slope. The logarithmic correction (9), with a0  0.7 and c  0.6, fits the Newtonian data in Fig. 7a for over a decade in the wavelength range 0.6y [ kx [ 6y. It should be pointed out here that in [7], values a0  0.5 and c  0.4 were suggested (rescaling the coefficients to conform with Eq. (9)). Although the data used in [7] for Reynolds number Res0 = 950 was in a wider channel than considered here, Lz = 3ph vs. Lz = 1.5ph, this raises some doubt as to the ‘‘universality‘‘ of these constants. As suggested in [23], in the other spectral limit the large turbulent eddies should scale with outer variables, namely the channel centerline mean velocity U c and the channel half-width h. This is confirmed for the Newtonian flow in Fig. 7b where the spectra collapse reasonably well in the high wavelength range kx J 5h and for y+ in the range [100, 200]. The spectrum at position y+ = 402 does not appear to scale, but this corresponds to y/h ’ 0.4h and presumably too deep in the outer layer. Although there exist significant fluctuations in the large scale part of the spectra, probably due to intermittency and a too short averaging time, all the pre-multiplied spectra reach a plateau above kx  h. This implies that the k1 energy spectrum hypothesis seems to hold for Newtonian flow for eddies with size above the channel half-width. Fig. 8 portrays the pre-multiplied 1D-power spectral densities for the low drag reduction flow using the same inner and outer

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+ Fig. 6. Isoline of the two-dimensional pre-multiplied power spectral densities Uþ 11 ¼ 0:16 at wall-normal position y = 159 for flows at Reynolds number Res0 = 1000 as a function of kx/y and kz/y. (a) Newtonian flow (b) viscoelastic flow (L = 30, Wes0 = 50, 30% drag reduction) (c) viscoelastic flow (L = 100, Wes0 = 115, 60% drag reduction). The full straight line is the linear bound kz = kx for the 3 flows. The dashed lines represent the nonlinear bounds following the general expression kz/y = (ai)i=0,1(kx/y)n, with a0 the coefficient of the lower bound, a1 the coefficient of the upper bound: a0 = 0.50, a1 = 2.5, n = 0.5 for Newtonian flow ; a0 = 0.55, a1 = 4.5, n = 0.4 for 30% drag reduction ; a0 = 0.50, a1 = 6.0, n = 0.35 for 58% drag reduction.

Fig. 7. Pre-multiplied 1D-Power spectral density of the streamwise velocity component for Newtonian flow at Res0 = 1000 at various wall-normal positions as a function of the streamwise wavelength kx = 2p/kx using: (a) inner scaling (us0 as velocity scale, y as length scale) and (b) outer scaling (channel centerline mean velocity U c as velocity scale, and channel half-gap h as length scale). The black solid line on the left panel is the logarithmic fit 0.3log(2.0kx/y), see Eq. (9).

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Fig. 8. Pre-multiplied 1D-Power spectral density of the streamwise velocity component for viscoelastic flow (L = 30, Wes0 = 50, 30% drag reduction) at Res0 = 1000 at various wall-normal positions as a function of the streamwise wave-length kx = 2p/kx using: (a) inner scaling (us0 as velocity scale, y as length scale) and (b) outer scaling (channel centerline mean velocity U c as velocity scale, and channel half-gap h as length scale). The black solid line on the left panel is the logarithmic fit 0.45log(1.0kx/y), see Eq. (9).

Fig. 9. Pre-multiplied 1D-Power spectral density of the streamwise velocity component for viscoelastic flow (L = 100, Wes0 = 115, 58% drag reduction) at Res0 = 1000 at various wall-normal positions as a function of the streamwise wave-length kx = 2p/kx using: (a) inner scaling (us0 as velocity scale, y as length scale) and (b) outer scaling (channel centerline mean velocity U c as velocity scale, and channel half-gap h as length scale). The black solid line on the left panel is the logarithmic fit 0.99log(0.45kx/y), see Eq. (9)

Fig. 10. Pre-multiplied 1D-Power spectral density of the streamwise velocity component for viscoelastic flow (L = 100, Wes0 = 115,  60% drag reduction) at various wallnormal positions as a function of the streamwise wave-length kx = 2p/kx using outer scaling (channel centerline mean velocity U c as velocity scale, and channel half-gap h as length scale) at Reynolds numbers (a) Res0 = 180 (b) Res0 = 395.

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Fig. 11. Power spectral densities (left panels) and pre-multiplied power spectral densities (right panels) of the streamwise velocity component at the wall-normal position y+ = 99 for Newtonian flow (upper panels) and viscoelastic high drag reduction flow of order 60% (bottom panels) at various friction Reynolds numbers up to Res0 = 1000.

Fig. 12. 1D-Power spectral density of the cross-stream vorticity component at the wall-normal position y+ = 99 for Newtonian flow (left panel) and viscoelastic high drag reduction flow of order 60% (right panel) at various friction Reynolds numbers up to Res0 = 1000.

scalings. Here, similar conclusions to the Newtonian flow can be drawn regarding the small scale structures. The difference lies in the logarithmic correction coefficients, which are now a0  1.0 and c  0.75, thus confirming the generalization of Eq. (9) to

non-Newtonian flow. In particular, the slope of the logarithmic correction is decreasing with increasing drag reduction. It is worth noting that the wavelength range over which this fitting applies is shifted to the right (towards larger scales), y [ kx [ 8y,

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indicating again an overall increase in the size of these turbulent scales in the viscoelastic flow. As for large scale structures, the outer scaling is not as good as for the Newtonian case. The scaling remains reasonable over a considerably narrower wavelength range (beyond kx  8h), but the two spectra at positions y+ = 99 and y+ = 402 are not collapsed by this scaling. There is still the occurrence of a plateau in the pre-multiplied spectra, indicating the existence of a k1 power law in the range 2h [ kx [ 10h. The spectra for high drag reduction flow shown in Fig. 9 exemplify the low drag reduction results. The inner scaling for the premultiplied spectra still holds, but with the logarithmic correction coefficients now a0  1.6 and c  1.5. The wavelength range over which the logarithmic correction applies is shifted even further towards the large scales. As for large scale structures, the traditional outer scaling here clearly fails; there is no noticeable wavelength range over which the spectra collapse. However, there is again an occurrence of a plateau in each pre-multiplied spectrum in the wavelength range 3h [ kx [ 10h, although this remains subject to interpretation owing to the presence of even larger oscillations in the spectra, as a probable consequence of increased intermittency in the viscoelastic high drag reduction flow.

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For the viscoelastic spectra at Reynolds number Res0 = 1000, a k1 region can still be identified, but over a narrower wavenumber range stopping at kxh  2. The difference between the Res0 = 590 and Res0 = 1000 spectral density (Newtonian and viscoelastic) is quite small in the k1 region, proving that Reynolds number similarity has almost been reached for the large anisotropic flow scales. 4.2. Spectral density of the cross-stream vorticity component If Reynolds number similarity has been reached for the large anisotropic scales, it is also reached for the small turbulent scales if one uses inner scaling for plotting the spectra rather than outer scaling as in Fig. 11. This is best illustrated in Fig. 12 where the spectra of the cross-stream vorticity component have been plotted at the wall-normal position y+ = 99 for increasing Reynolds numbers up to Res0 = 1000. Note that the same y-axis range was intentionally used on both panels (Newtonian flow on the left, viscoelastic flow on the right), showing the impressive drop in the high wavenumber spectral level of vorticity in viscoelastic turbulence. In this figure, one can see that self-similarity is fully obtained for Newtonian and viscoelastic flows for almost the whole wavenumber range resolved.

4. Reynolds number similarity 5. Summary and conclusions The above results overall suggest that the large scale motions in the high-drag reduced flow would be different from those in the Newtonian flow. This finding was illustrated at Reynolds number Res0 = 1000. In Fig. 10, the pre-multiplied power spectral density of the streamwise velocity component are plotted for high drag reduction flows at smaller Reynolds numbers Res0 = 180 and Res0 = 395. The large scale motion segregation of energy for highdrag reduced flow is also observed at smaller Reynolds numbers. Note however, that at such small Reynolds numbers, there does not seem to be any discernible wave-length range where the usual k1 law would be valid. As pointed out in the introduction, Reynolds number similarity is essential in describing the nature of turbulent wall-bounded flows because turbulence is mainly driven by the Reynolds number [24]. Now, the DNS data for the Newtonian and high drag reduction (of order 60%) viscoelastic flows obtained at 4 different friction Reynolds numbers Res0 = 180, 395, 590, and 1000 will be systematically examined. The Reynolds number similarity of the statistics for this high drag reduction flow case was extensively discussed in [14] and will not be repeated here. As for power spectra which is the focus here, the k1 dependence of the streamwise velocity spectra will be probed in the constant stress layer at the wall-normal position y+ = 99.

4.1. Spectral density of the streamwise velocity component Figs. 11a, c show power spectral densities and Figs. 11b, d premultiplied power spectral densities of the streamwise velocity component for Newtonian and high drag reduction viscoelastic flows at the wall-normal position y+ = 99. Here, outer scaling has been used together with the conventional log–log axes, with the x-axis representing the non-dimensional wavenumber, such that small to large turbulent scales read from the right to the left. The use of log–log axes significantly flattens the low wavenumber fluctuations in the spectra in such a way that it is now possible to identify a significant range of wavenumber (kxh [ 10) where the k1 slope holds for the Res0 = 1000 Newtonian spectra. As expected, it is difficult to identify such a wide k1 region in the Newtonian spectra at lower Reynolds numbers, although such a region does start to build up at Res0 = 395 and 590.

A spectral analysis of both the velocity and vorticity fields was performed using direct numerical simulation data of fully developed turbulent channel flows of Newtonian and viscoelastic fluids. The viscoelastic fluid was modeled using the FENE-P constitutive equation, with a ratio of the Newtonian viscosity to the total zero-shear viscosity of b0 = 0.9. Two different regimes of drag reduction were first considered at the highest friction Reynolds number available to date, Res0 = 1000: a flow with medium percentage drag reduction (30%) and one with a high percentage of drag reduction (58%), corresponding respectively to fully-stretched polymer lengths L = 30 and L = 100, and friction Weissenberg numbers Wes0 = 50 and Wes0 = 115. The DNS results showed a marked drop in the energy level of the velocity field at high wavenumbers induced by the presence of the polymer in the constant stress layer, in qualitative agreement with the experimental results of Warholic et al. [15]. For the high drag reduction case, within the buffer layer region (y+ = 60) where the turbulent kinetic energy reaches a maximum (see 14, Fig. 4b), a clear tendency towards a k5 power law over a full wavenumber decade is noted. This behavior is valid for the streamwise and spanwise velocity components, but is anisotropic since a smaller slope of k4 is observed in the wall-normal velocity spectra. Another feature is the markedly higher level of energy content in the low wavenumbers in the streamwise velocity spectra compared with the Newtonian case. The spanwise vorticity spectra of the viscoelastic flow were also found to have some interesting properties. The large-scale energy content was increased by more than one order of magnitude in the high drag reduction regime when compared with the Newtonian flow. It was also found that viscoelasticity induces a drop of the energy for all vorticity components at high wavenumbers, starting in the inertial subrange. These observations are the signature of the long streamwise coherent structures, accompanied by a dampening of the small-scale turbulent structures. This is not surprising given that previous calculations of the Karhunen–Loeve (K– L) eigenmodes have demonstrated a dramatic decrease in the K–L dimensionality accompanying the presence of viscoelasticity in turbulent channel flows 25,26 (see also [27] where experimental evidence based on the K–L analysis of particle image velocimetry data is given).

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Effects of scaling, using inner and outer variables, on the streamwise velocity spectra at different distances from the wall in the constant stress layer was also investigated. The friction velocity us0 and the distance from the wall y were used as inner scales. For the outer scaling, the channel centerline mean velocity and the channel half-gap h were used. Using inner scaling, the logarithmic correction to the usual k1 behavior in the streamwise velocity spectra, first suggested in [7] for Newtonian turbulence, was confirmed for both Newtonian and viscoelastic flows; although with different scaling coefficients. An overall shift towards larger scales of the wavenumber range for which the log correction applied was also observed in viscoelastic flows. At larger scales, the k1 energy spectrum hypothesis was confirmed for the Newtonian case for eddies with wavelength kx larger than the channel half-width. For the viscoelastic case in the low drag reduction regime, the outer scaling was not as good as for the Newtonian case, but there was still an occurrence of the k1 power law in the range 2h < kx < 10h. In the high drag reduction regime, the outer scaling failed and there was no wavelength range where the k1 law held. The Reynolds number similarity of the power spectra was explored for both the Newtonian flow and for the high percentage drag reduction flow computed at 4 different friction Reynolds numbers, Res0 = 180, 395, 590, and 1000. Results were analyzed at the wall-normal position y+ = 99. As expected, it was difficult to identify a wide k1 region in the Newtonian spectra at the lowest Reynolds numbers; although, such a region did start to build up at Res0 = 395. For the Newtonian fluid at Res0 = 1000, a significant range of wavenumbers (kxh 6 10) showed the k1 slope. In contrast, for the viscoelastic fluid, a narrower wavenumber region with a k1 slope was identified, stopping at kxh  2. Given the empirical observations provided here, there are a number of remaining open questions. All in all, viscoelasticity induces a strengthening of the large-scale energy containing eddies, and a dampening of the small-scale turbulent structures characterized by a rapid energy drop (k5) in the inertial subrange of the velocity spectra. This suggests an alteration of the energy cascade as put forward in the elastic theory of drag reduction by Tabor and De Gennes [28] in the framework of shear-free homogeneous turbulence, and later extended to wall-bounded turbulence in [29]. Further investigations are necessary to identify whether this theory is able to predict (quantitatively) the original spectral properties observed here.

Acknowledgments This research has granted access to the HPC resources of [CCRT/ CINES/IDRIS] under the allocation i2012022277 made by GENCI (Grand Equipement National de Calcul Intensif). The data used herein was produced on the IBM Blue Gene/P computer Babel at the IDRIS/CNRS computing center, Orsay, France. Cross-channel statistics from the DNS database can be publicly accessed on-line at http://lml.univ-lille1.fr/channeldata/.

References [1] A.A. Townsend, The structure of Turbulent Shear Flow, second ed., Cambridge University Press, Cambridge, UK, 1976. [2] S.B. Pope, Turbulent Flows, Cambridge University Press, Cambridge, UK, 2000. [3] J. Laufer, The Structure of Turbulence in Fully Developed Pipe Flow, Technical Report 1174, Nat. Adv. Ctee Aero., Washington, USA, 1955. [4] A.E. Perry, C.J. Abell, Asymptotic similarity of turbulence structures in smoothand rough-walled pipes, J. Fluid Mech. 79 (1977) 785–799. [5] A.E. Perry, C.J. Abell, Scaling laws for pipe-flow turbulence, J. Fluid Mech. 67 (1975) 257–271. [6] J.F. Morrison, W. Jiang, B.J. McKeon, A.J. Smits, Reynolds number dependence of streamwise velocity spectra in turbulent pipe flow, Phys. Rev. Lett. 88 (2002) 214501/1–214501/4. [7] J.C. del Álamo, J. Jiménez, P. Zandonade, R.D. Moser, Scaling of the energy spectra in turbulent channels, J. Fluid Mech. 500 (2004) 135–144. [8] C.M. White, M.G. Mungal, Mechanics and prediction of turbulent drag reduction by polymer additives, Annu. Rev. Fluid Mech. 40 (2008) 235–256. [9] T. Min, J.Y. Yoo, H. Choi, Maximum drag reduction in a turbulent channel flow by polymer additives, J. Fluid Mech. 492 (2003) 91–100. [10] K.D. Housiadas, A.N. Beris, Polymer-induced drag reduction: effects of the variations in elasticity and inertia in turbulent viscoelastic channel flow, Phys. Fluids 15 (2003) 2369–2384. [11] K.D. Housiadas, A.N. Beris, R.A. Handler, Viscoelastic effects on higher order statistics and on coherent structures in turbulent channel flow, Phys. Fluids 17 (2005) 035106–035125. [12] B.J. McKeon, K.R. Sreenivasan, Introduction: scaling and structure in high Reynolds number wall-bounded flows, Phil. Trans. Roy. Soc. 365 (2007) 635– 646. [13] L. Thais, A. Tejada-Martinez, T.B. Gatski, G. Mompean, A massively parallel hybrid scheme for direct numerical simulation of turbulent viscoelastic channel flow, Comput. Fluids 43 (2011) 134–142. [14] L. Thais, T.B. Gatski, G. Mompean, Some dynamical features of the turbulent flow of a viscoelastic fluid for reduced drag, J. Turbul. 13 (2012) 1–26. [15] M.D. Warholic, J. Massah, T.J. Hanratty, Influence of drag reducing polymers on turbulence: effects of Reynolds number, concentration and mixing, Exps. Fluids 27 (1999) 461–472. [16] G.K. Batchelor, Homogeneous Turbulence, Cambridge University Press, Cambridge, UK, 1954. [17] S.C. Morris, J.F. Foss, Vorticity spectra in high reynolds number anisotropic turbulence, Phys. Fluids 17 (2005) 088102/1–088102/4. [18] R. Benzi, E.D. Angelis, V.S. L’vov, I. Procaccia, Identification and calculation of the universal asymptote for drag reduction by polymers in wall-bounded turbulence, Phys. Rev. Lett. 95 (2005) 194502-1–194502-4. [19] K.D. Housiadas, A.N. Beris, Extensional behavior influence on viscoelastic turbulent channel flow, J. Non-Newton. Fluid Mech. 140 (2006) 41–56. [20] Y. Dubief, C.M. White, V.E. Terrapon, E.S.G. Shakfeh, P. Moin, S.K. Lele, On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows, J. Fluid Mech. 514 (2004) 271–280. [21] P.S. Virk, Drag reduction fundamentals, AIChE 21 (1975) 625–656. [22] S. Hoyas, J. Jiménez, Scaling of velocity fluctuations in turbulent channels up to Res = 2000, Phys. Fluids 18 (2006) 011702-1–011702-4. [23] J.C. del Álamo, J. Jiménez, Spectra of the very large anisotropic scales in turbulent channels, Phys. Fluids 15 (2003) L41–L44. [24] A.S. Monin, A.M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, The MIT Press, Cambridge, MA, 1975. [25] G. Samanta, G.M. Oxberry, A.N. Beris, R.A. Handler, K.D. Housiadas, Timeevolution K–L analysis of coherent structures based on DNS of turbulent newtonian and viscoelastic flows, J. Turbul. 9 (41) (2008) 1–25. [26] G. Samanta, A.N. Beris, R.A. Handler, K.D. Housiadas, Velocity and conformation tensor statistics based on reduced Karhunen–Loeve projection data from DNS of viscoelastic turbulent channel flow, J. Non-Newton. Fluid Mech. 160 (1) (2009) 55–63. [27] W.H. Cai, F.C. Li, H.N. Zhang, X.B. Li, B. Yu, J.J. Wei, Y. Kawaguchi, K. Hishida, Study on the characteristics of turbulent drag-reducing channel flow by particle image velocimetry combining with proper orthogonal decomposition analysis, Phys. Fluids 21 (11) (2009) 115103-1–115103-12. [28] M. Tabor, P.G. De Gennes, A cascade theory of drag reduction, Europhys. Lett. 2 (1986) 519–522. [29] P.G. De Gennes, Towards a scaling theory of drag reduction, Physica D 140 (1986) 9–25.