Two-dimensional, homogeneous, isotropic fluid turbulence with polymer additives
We carry out the most extensive and high-resolution direct numerical simulation, attempted so far, of homogeneous, isotropic turbulence in two-dimensional fluid films with air-drag-induced friction and with polymer additives. Our study reveals that the polymers (a) reduce the total fluid energy, enstrophy, and palinstrophy, (b) modify the fluid energy spectrum both in inverse- and forward-cascade régimes, (c) reduce small-scale intermittency, (d) suppress regions of large vorticity and strain rate, and (e) stretch in strain-dominated regions. We compare our results with earlier experimental studies and propose new experiments.
Polymer additives have remarkable effects on turbulent flows: in wall-bounded flows they lead to drag reduction (1); (2); in homogeneous, isotropic turbulence they give rise to dissipation reduction, a modification of the energy spectrum, and a suppression of small-scale structures (3); (4); (5); (7); (8); (9); (10); (11); (6); (12); (13). These effects have been studied principally in three-dimensional (3D) flows; their two-dimensional (2D) analogs have been studied only over the past decade in experiments (14); (15); (16) on and direct numerical simulations (DNSs) (17); (18); (19); (20) of fluid films with polymer additives. It is important to investigate the differences between 2D and 3D fluid turbulence with polymers because the statistical properties of fluid turbulence in 2D and 3D are qualitatively different (21): the inviscid, unforced 2D Navier-Stokes (NS) equation admits more conserved quantities than its 3D counterpart; one consequence of this is that, from the forcing scales, there is a flow of energy towards large length scales (an inverse cascade) and that of enstrophy towards small scales (a forward cascade). We have, therefore, carried out the most extensive and high-resolution DNS study of homogeneous, isotropic turbulence in the incompressible, 2D NS equation with air-drag-induced friction and polymer additives, described by the finitely-extensible-nonlinear-elastic-Peterlin (FENE-P) model for the polymer-conformation tensor. We find that the inverse-cascade part of the energy spectrum in 2D fluid turbulence is suppressed by the addition of polymers. We show, for the first time, that the effect of polymers on the forward-cascade part of the fluid energy spectrum in 2D is (a) a slight reduction at intermediate wave numbers and (b) a significant enhancement in the large-wave-number range, as in 3D; the high resolution of our simulation is essential for resolving these spectral features unambiguously. In addition, we find dissipation-reduction-type phenomena (7); (8): polymers reduce the total fluid energy and energy- and mean-square-vorticity- or enstrophy-dissipation rates, suppress small-scale intermittency, and decrease high-intensity vortical and strain-dominated régimes. Our probability distribution functions (PDFs) for and , the squares of the strain rate and the vorticity, respectively, agree qualitatively with those in experiments (16). We also present PDFs of the Okubo-Weiss parameter , whose sign determines whether the flow in a given region is vortical or strain-dominated (22); (23), and PDFs of the polymer extension; and we show explicitly that polymers stretch preferentially in strain-dominated regions.
The remaining part of this paper is organized as follows. In Sec. II we define the equations we use for polymer additives in a fluid and we describe the numerical methods we use to study these equations. Section III is devoted to the results of our study and Sec. IV contains a discussion of our principal results.
Ii Model and Numerical Methods
The 2D incompressible NS and FENE-P equations can be written in terms of the stream-function and the vorticity , where is the fluid velocity at the point and time , as follows:
Here , the uniform solvent density , is the coefficient of friction, the kinematic viscosity of the fluid, the viscosity parameter for the solute (FENE-P), and the polymer relaxation time; to mimic experiments (16), we use a Kolmogorov-type forcing , with amplitude ; the energy-injection wave vector is (the length scale ); the superscript denotes a transpose, are the elements of the polymer-conformation tensor (angular brackets indicate an average over polymer configurations), is the identity tensor, is the FENE-P potential, and and are, respectively, the length and the maximal possible extension of the polymers; and is a dimensionless measure of the polymer concentration (24) .
We use periodic boundary conditions, a square simulation domain with side and collocation points, a fourth-order, Runge-Kutta scheme, with time step , for time marching, an explicit, fourth-order, central-finite-difference scheme in space, and the Kurganov-Tadmor (KT) shock-capturing scheme (25) for the advection term in Eq. (3); the KT scheme (Eq. (7) of Ref. (8)) resolves sharp gradients in and thus minimizes dispersion errors, which increase with and . We solve Eq. (2) in Fourier space by using the FFTW library (26). We choose so that does not become larger than (Table 1). We preserve the symmetric-positive-definite (SPD) nature of by adapting to 2D the Cholesky-decomposition scheme of Refs. (24); (7); (8): We define , so Eq. (3) becomes
where , , , and Given that and hence are SPD matrices, we can write , where is a lower-triangular matrix with elements , such that for ; Eq.(4) now yields and
We have tested explicitly that the statistical properties we measure do not depend on the resolutions we use for our DNS. We check this both by increasing and decreasing this resolution. Indeed, our DNS uses the highest resolution that has been attempted so far for this problem (it uses 256 times as many collocation points as those in Ref. (18)). Furthermore, the Kurganov-Tadmor shock-capturing scheme that we use controls any dispersive errors, because of sharp gradients in the polymer-conformation tensor, as in similar three-dimensional studies (7); (24).
We maintain a constant energy-injection rate with ; the system attains a nonequilibrium, statistically steady state after , where the box-size eddy-turnover time and is the root-mean-square velocity.
In addition to , , and we obtain , the fluid-energy spectrum , where indicates a time average over the statistically steady state, the total kinetic energy , enstrophy , and palinstrophy , where denotes a spatial average, the PDF of scaled polymer extensions , the PDFs of , , and , where , and , the PDF of the Cartesian components of , and the joint PDF of and . We obtain the isotropic part of order-, structure function from longitudinal velocity increments as described in Ref (22). We concentrate on and the hyperflatness ; the latter is a measure of the intermittency at the scale .
In Fig. (1a) we show how (top panel), (middle panel), and (bottom panel) fluctuate about their mean values , , and for (pure fluid) and . Clearly, , , and decrease as increases. Thus, polymers increase the effective viscosity of the solution; but this naïve conclusion has to be refined, as will be shown later, because the effective viscosity depends on the length scale (6); (7); (8).
In Fig. (2a), we plot versus for (blue circles and run ) and (green asterisks and run ); the dashed line, with slope 2, is shown to guide the eye; this slope agrees with the form that we expect, at small , by Taylor expansion. At large values of , deviates from this behavior, more so for than for , in accord with experiments (16). Plots of versus (Fig. (2b)), for (blue circles) and (green asterisks and run ), show that, on the addition of polymers, small-scale intermittency decreases as increases.
In Fig. (3a), we show how changes, as we increase : at low and intermediate values of (e.g., and , respectively), decreases as increases; but, for large values of (e.g., ), it increases with . Figure (3b) shows how changes, as we increase with held fixed at . At low values of (e.g., ), decreases as increases; but for large values of (e.g., ) it increases with .
In Fig. (4a) we give plots, for , of the spectra for (red triangles and run ) and (green asterisks and run ); for comparison we also plot for ; as increases, the difference between and increases at large values of . We see that the larger the value of the more pronounced is the rise of the large- tail of (cf. the plots in Fig. (4a) with red triangles and green asterisks for and , respectively).
We can understand these trends qualitatively by noting that, even at maximal extension, the size of a polymer is (the dissipation scale). Thus, the polymers stretch at the expense of the fluid energy, which cascades from the intermediate length scales to dissipative scales; this leads to a reduction of at the values of that correspond to these intermediate scales. As the polymers relax, they feed energy to the fluid at the deep-dissipation, i.e., large-, scales; this leads to an enhancement in the tail of at large values of . The reduction of energy in the inverse-cascade, low- regime can be understood by noting that polymers enhance the overall, effective viscosity of the fluid. Indeed, in the limit , (27).
To understand quantitatively the effect of polymers on , in different regimes of , we must compare the fluid-energy spectra, with and without polymers (Fig. (1b)). This leads us naturally to define (6); (7); (8) the effective, scale-dependent viscosity , with
and the Fourier transform of . Figure (1c) shows that for , where , whereas, for large values of , , where ; the superscripts and stand, respectively, for the fluid without and with polymers. To understand this dependence on we plot, in Fig. (4b), the scale-dependent viscosity for these two representative values, namely, (red triangles and run ) and (green asterisks and run ). We find that is positive and higher for , at small values of , than its counterpart for ; this explains why is smaller for than for at small . For large values of , is more negative for than for , so is larger for than for . Note that changes its sign, from positive to negative, at a smaller value of for than for ; therefore, the large- tail of rises above that of at a smaller value of for than for . By using , which we obtain from our NS+FENE-P run , we carry out a DNS of the NS equation with replaced by ; in Fig. 5 we present plots of the energy (left panel), energy spectra (middle panel), PDFs of (right panel and Fig. 8), to compare the results of this DNS with those of run (NS+FENE-P); the good agreement of these results shows that the NS equation with the scale-dependent viscosity captures the essential effects of polymer additives on fluid turbulence in run (NS+FENE-P). The form of our effective viscosity indicates that, at large length scales, in addition to the friction, polymers also provide a dissipative mechanism. By contrast, at small length scales, polymers inject energy back into the fluid.
Figure (1d) shows the suppression, by polymer additives, of , where and . The suppression of the spectrum in the small- régime, which has also been seen in experiments (14) and low-resolution DNS (Fig. (4.12) of Ref. (17)), signifies a reduction of the inverse cascade; the enhancement of the spectrum in the large- régime leads to the reduction in and shown in Fig. (1a); to identify this enhancement unambigouosly requires the run , which is by far the highest-resolution DNS of Eqs. (1)-(3) (with times more collocation points than, say, Ref. (18)).
We now plot the PDF versus in Fig. (6) for and (red triangles and run ), and (black squares and run ), and and (green asterisks and run ). The extension of the polymers is bounded between . The lower bound, , corresponds to polymers in a coiled state; near the upper bound, with , the polymers are in a stretched state. In Fig. (6), we show that shows a distinct, power-law regime, with exponents that depends on and . As increases, this exponent can go from a negative value to a positive value, thus signalling a coil-stretch transition.
In Figs. (7a), (7b), and (7c) we present PDFs of , , and , respectively, for (blue circles and run ) and (red triangles and run ) to show that the addition of polymers suppresses large values of , , and . If we make scaled plots of PDFs such as , then they fall on top of each other for different values of ; this also holds for and . The inset of Fig. (7c) shows that the PDF of any Cartesian component of u is very close to a Gaussian.
The Fig.(8a) shows a conditional PDF of () conditioned on for run ; this illustrates that polymers stretch predominantly in strain-dominated regions; this is evident very strikingly in Fig. (8b), which contains a superimposition of contours of on a pseudocolor plot of (for a video sequence of such plots see (28)).
We have carried out the most extensive and high-resolution DNS of 2D, homogeneous, isotropic fluid turbulence with polymer additives. We have used the incompressible, 2D NS equation with air-drag-induced friction and polymer additives; the latter have been modelled by using the finitely-extensible-nonlinear-elastic-Peterlin (FENE-P) model for the polymer-conformation tensor. We find that the inverse-cascade part of the energy spectrum in 2D fluid turbulence is suppressed by the addition of polymers. We demonstrate, for the first time, that the effect of polymers on the forward-cascade part of the fluid energy spectrum in 2D is (a) a slight reduction at intermediate wave numbers and (b) a significant enhancement in the large-wave-number range, as in 3D; these features are resolved unambiguously by our high-resolution DNS. In addition, we find dissipation-reduction-type phenomena (7); (8): polymers reduce the total fluid energy and energy- and mean-square-vorticity- or enstrophy-dissipation rates. However, as we have emphasized above, dissipation reduction is not the only notable effect of polymer additives; our extensive, high-resolution DNS of 2D fluid turbulence with polymer additives yields good qualitative agreement, in the low- régime, with the fluid-energy spectra of Ref. (14), and the of Ref. (16). In addition, our study obtains new results and insights that will, we hope, stimulate new experiments, which should be able to measure (a) the reduction of , , and (Fig.(1a)), (b) the modification of at large (Fig.(1b)), (c) the , and dependences of (Figs.(3a),(3b) and (4a)), (d) the PDFs of , , , and , (e) the stretching of polymers in strain-dominated regions (Fig. (8b)), and (f) the suppression of at small (Fig. (2)).
Two-dimensional fluid turbulence with polymer additives has been studied in channel flows, both in experiments (15) and via DNS (20); this DNS study uses the Oldroyd-B model, which does not have a maximal-polymer-extension length and is, therefore, less realistic than the FENE-P model we use. These studies obtain energy spectra and second-order structure functions that are qualitatively similar to those we obtain, except at small length scales, which are not resolved in these channel-flow studies. This shows, therefore, that energy spectra and structure functions, obtained far away from walls, are not affected significantly by the walls. Thus, our studies are relevant to the bulk parts of wall-bounded flows too.
We thank D. Mitra for discussions, CSIR, UGC, DST (India), and the COST Action MP006 for support, and SERC (IISc) for computational resources. AG thanks the grant from European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement N. 279004.
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- See Supplemental Material at URL for the Video sequence showing pseudocolor plots of the Okubo-Weiss parameter superimposed on contour plots of , which measures the polymer extension. Note that polymers stretch predominantly in the strain-dominated regions of the flow.