Flow, Turbulence and Combustion 2005 74: 311–329 DOI: 10.1007/s10494-005-9002-6
C
Springer 2005
New Answers on the Interaction Between Polymers and Vortices in Turbulent Flows YVES DUBIEF1 , VINCENT E. TERRAPON2 , CHRISTOPHER M. WHITE2 , ERIC S.G. SHAQFEH2,3 , PARVIZ MOIN1,2 and SANJIVA K. LELE2 1
Center for Turbulence Research, Bldg 500, Stanford, CA 94305-3035, USA; Mechanical Engineering Department, University of Vermont, 201A Votey Bldg, 33 Colchester Ave, Burlington, VT 05405, E-mail:
[email protected] 2 Department of Mechanical Engineering Stanford 3 Department of Chemical Engineering, Stanford Accepted 21 September 2005 Abstract. Numerical data of polymer drag reduced flows is interpreted in terms of modification of near-wall coherent structures. The originality of the method is based on numerical experiments in which boundary conditions or the governing equations are modified in a controlled manner to isolate certain features of the interaction between polymers and turbulence. As a result, polymers are shown to reduce drag by damping near-wall vortices and sustain turbulence by injecting energy onto the streamwise velocity component in the very near-wall region. Key words: drag reduction, turbulence, polymer additives
1. Introduction Among the many strategies employed for drag reduction in wall-bounded flows [9, 18], polymer additives are the most efficient, yet the most intricate. The complexity of the phenomenon stems from the small amount of microscopic polymer molecules needed to achieve significant drag reduction: drag reduction (DR) up to 80% have been achieved with only a few ppm concentrations. The exact details as to how minute concentrations of polymer molecules can create large reductions in turbulent drag are still a matter of debate. The present paper investigates the manipulation of turbulence by polymers using direct numerical simulations of a channel flow containing homogeneous concentration of polymers. Physical understanding is obtained using the concept of numerical experiment, where boundary conditions or governing equations are modified in a controlled manner to elucidate physical patterns [15, 16]. Over the past 50 years, much speculation has been made concerning the mechanism of polymer drag reduction. Interestingly, the most quoted theoretical approaches have been published based on physical considerations and interpretations of low-order statistical velocity moments [20, 26, 29]. Each work refines or corrects the earlier one(s), yet the picture of the mechanism is still incomplete,
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mainly because the data required to test these theories is insufficient. As a consequence, no predictive model of polymer drag reduction is available. The response of turbulent statistics to polymer addition is well-documented such that drag reduction can be divided into two regimes [34]. The Low Drag Reduction regime (LDR) describes flows experiencing DR 40%. Velocity statistics normal√ ized by the kinematic viscosity ν and skin-friction velocity (u τ ≡ ν(dU/dy)w , where U is the mean streamwise velocity and w refers to wall values), and denoted by the superscript + , exhibit the following behavior: the log-law in the mean velocity profile is translated upward; the maximum of the root-mean squared (RMS) of the streamwise velocity fluctuations, u + , is increased while the fluctuations in the wall-normal v + and spanwise w + and the Reynolds shear stress uv + are reduced. For DR 40%, the High Drag Reduction (HDR) regime, the slope of the log-law increases with increasing DR. Transverse velocity fluctuations and Reynolds shear stress are significantly reduced while the maximum of u + is found to remain close to the Newtonian value and perhaps slightly decreases. Drag reduction is eventually bounded by the so-called Maximum Drag Reduction (MDR) asymptote which was empirically defined by [32] as a function of Reynolds number alone. MDR cannot be surpassed by polymer addition alone, making this asymptotic state one of the most interesting properties of polymer drag reduction. Recent experiments in a channel and a pipe [24, 34] demonstrated low Reynolds shear stress, uv + , which could not be balanced by the viscous stress (dU + /dy + ) in the stress balance equation. This considerable stress deficit, larger in magnitude than uv + was attributed to polymer stress, leading Warholic et al. [34] and Ptasinski et al. [24] to conclude that polymer stress is critical to sustain MDR. Where and how polymers affect turbulence is the focus of the present paper. The starting point in assessing the polymer effect on turbulence must be the self-sustained mechanism of wall-bounded turbulence. Wall turbulence is driven by an autonomous regeneration cycle confined in the inner region of the flow, 0 < y + 100 [16]. The cycle revolves around the mean wall shear, non-linear interactions, streaks and quasi-streamwise vortices (see also [13]). Jim´enez and Pinelli [16] established that the cycle is self-sustained and its interruption, via numerical experiment, leads to relaminarization, regardless of the initial intensity of turbulence in the outer region of the flow. Here, data from numerical simulations is explored to assess how this cycle is affected by polymer addition. Polymer effects are first investigated in terms of the breakdown of the law of the wall scaling for second-order moments of the velocity fluctuations and changes in the anisotropy of the flow. Lastly, the interaction between polymers and turbulence is studied in each of the three spatial directions of space to isolate drag reducing and enhancing phenomena. The motivation of this work and the methodology employed is largely inspired by Prof. R. A. Antonia’s work on the response of turbulent boundary layers to small perturbations of the boundary conditions, including roughness and suction, [1, 3, 4, 10]. This work was presented at the conference organized in honor of Prof. Antonia 60th Birthday.
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2. Overview of the Numerical Methods 2.1. E QUATIONS The bulk of this work consists of viscoelastic simulations of channel flow using the FENE-P (Finite Elastic Non-linear Extensibility-Peterlin) model for polymers. This viscoelastic model assumes that the polymer concentration is uniform so that polymer dynamics can be represented by an evolution equation of the phase-averaged configuration tensor defined as ci j = qi q j , where qi are the components of the end-to-end vector for a polymer molecule. The polymer field is therefore considered as a continuum. The FENE-P model is a single dumbbell model which has been previously used by [28] in numerical simulation of turbulent flows. The evolution of polymers is governed by the balance of stretching and restoring forces in an Eulerian framework, such that, ∂t ci j + u k ∂k ci j = ck j ∂k u i + cik ∂k u j − τi j ,
(1)
ci j 1 τi j = − δi j . W e 1 − cLkk2
(2)
In this equation, u i are the components of the velocity vector, the first term on the rhs. of Equation (1) represents the stretching forces (or polymer stretching due to the straining action of the flow) and δi j is the unit tensor. Polymers extract or release energy from and into the flow through the polymer stress tensor τi j (or restoring force). Polymer stress results from the action of polymer molecules to keep their configuration close to the highest entropic state, i.e. the coiled configuration. In its most stretched configuration, ckk approaches L 2 , L is the polymer length. The Weissenberg number W e is the ratio of the polymer relaxation time λ to the flow time scale, λUc λu 2 ; W eτ = τ . (3) h ν Here the Weissenberg number is given as a function of the outer flow time scale h/Uc and inner region time scale , where h is the half width of the channel and Uc the centerline velocity. In previous numerical simulations, the elasticity is usually characterized with W eτ , based on the wall shear of the Newtonian flow. The momentum equation incorporates polymer effects via an additional term, the polymer body force f i , which is the divergence of the polymer stress tensor, We =
∂t u i + u j ∂ j u i = −∂i p +
β 1−β ∂ j ∂ j ui + ∂ j τi j , Re Re
(4)
fi
where β is the ratio of the solvent viscosity ν to the total viscosity and consequently depends on the concentration of polymers. The Reynolds number is defined as Rec = Uc h/ν.
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2.2. TIME
ADVANCEMENT
The numerical schemes for solving both the flow and polymer configuration tensors are based on [23] with some modifications to accommodate large Weissenberg numbers and very long polymer chains. The flow solver uses classical second-order central finite differences on a staggered grid. The choice of finite differences over pseudo-spectral methods was motivated by the perspective of simulations in complex geometries where periodic conditions, required for spectral methods, are no longer valid. The time advancement scheme relies on a semi-implicit third-order Runge Kutta/second-order Crank Nicolson, necessary to relax the time-step constraint on dissipative terms in the wall-normal direction for Equation (4). An important modification was made to the original [23]’s method in the form of a novel implicit scheme for the conformation equation tensor (Equation (1)) in order to ensure that the solution of the trace ckk remains upper bounded by L 2 . With an explicit temporal scheme for Equation (1), ckk can occasionally exceed L 2 in regions where polymer stretching and numerical errors due to advection (as explained in the following) may be large. In our algorithm, the conformation tensor equations are solved first, as written here for each sub-time steps, (l) (l−1) ci(l)j − ci(l−1) c c α j i j i j l = γl Ri(l−1) + ζl Ri(l−2) − + (l−1) − 2δi j , j j ckk t W e 1 − ckk(l) 1 − 2 L L2 (5) (l) (l) (l) (l) (l) where Ri(l)j = −u (l) k ∂k ci j + (cik ∂k u j + ck j ∂k u i ). The index l is the substep of 8 the RK3 and γl , ζl and αl the corresponding coefficients: γ1 = 15 ; ζ1 = 0; α1 = 4 5 17 1 3 5 1 ; γ2 = 12 ; ζ2 = − 60 ; α2 = 15 ; γ3 = 4 ; ζ3 = − 12 ; α3 = 6 . By summing the 15 (l) /L 2 , equations for the diagonal components and using the variable ψ (l) = 1 − ckk Equation (5) can be simply recast into a second order polynomial,
ψ (l)
2
+ Bψ (l) − C = 0,
(6)
with the following coefficients, t 1 αl t 6 (l−1) (l−2)
(l−1) 2 − (l−1) + 2 + 2 γl Rii + ζl Rii + B = −ψ We ψ L L αl t C= . We The roots of Equation (6) are real and of opposite sign, therefore the positive root ensures that the trace ckk remains upper-bounded as the root approaches zero as 2C 2αl t . (7) ψ (l) ∼ 2 = 2 B B We
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The velocity field is subsequently solved using a classical fractional step method [17],
u i(∗) − u i(l−1) = −γl Ni(l−1) − ζl Ni(l−2) + αl L i(l) + L i(l−1) + Ti(l) + Ti(l−1) t (8) ∂ k ∂k φ =
1 ∂k u (∗) k αl t
u i(l) = u i(∗) − αl t∂i φ. 2.3. T HE
(9) (10)
ADVECTION PROBLEM
While the time-advancement scheme described above proves to be robust in regions of highly stretched polymers, numerical oscillations can be observed in the same regions for large Weissenberg number. The occurrence of these oscillations in the ci j fields can easily be explained by a physical analysis of each term in the conformation tensor equations (Equation 1). On the rhs. of Equation (1), the stretching (cik ∂k u j + ck j ∂k u j ) and the relaxation (−τi j ) terms are local quantities; no spatial diffusion occurs via these terms. On the lhs., the advection term contains spatial derivatives of ci j ; Equation (1) has therefore the characteristics of an advection equation with a source term consisting of the rhs. of Equation (1). For the source term, the stretching term is governed by the velocity gradient tensor, thus polymers are stretched by all the scales of turbulence from the largest to the smallest, the Kolmogorov scale defined as η K ≡ (ν 3 /¯ε)1/4 , where ε¯ is the average dissipation of turbulent energy (¯ε = 2ν∂ j u i ∂i u j ). For polymers to be stretched, the polymer relaxation time needs to be larger that the Kolmogorov time scale. To this extent, [20] argued that W eτ should be larger or equal to unity for drag reduction to occur. In a Newtonian channel flow, [2] established that the Kolmogorov scale is of the order of ν/u τ in the viscous sublayer and the buffer region, the Kolmogorov time scale is of the order of ν/u 2τ which corresponds to W eτ = 1. In simulations presented here, Weissenberg numbers are much larger than one. In this context, the source term (rhs. of Equation 1) is expected to produce scales as small as the Kolmogorov but not smaller. Since the FENE-P model does not have a diffusion term, the gradients are likely to become sharper, including shock-like structures, under the influence of advection, as it would happen for a passive scalar with low diffusivity. In the case of a passive scalar with a very low√diffusivity κ, [5] determined that the smallest scale (Batchelor scale) is θ = η K / Sc, where the Schmidt number is the ratio of the kinematic viscosity to the scalar diffusion coefficient. Batchelor also showed that the spectrum for a passive scalar decays as k −1 after the inertial
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range for the kinetic energy (k denotes the wavenumber). For a polymer solution, the Schmidt number is of the order of 106 , hence the absence of diffusion in the FENE-P model. Numerically, accurate resolution of sub-Kolmogorov scales is out the question, especially when Sc 1. To solve the advection term, we use [23]’s technique which is inspired from compressible flow calculations. This technique is presented and discussed at the end of this section. To support the claim that small scales are governed by advection, we first compare our viscoelastic DNS to our Brownian dynamics code [30, 31] The Brownian dynamics (BD) code simulates the Lagrangian advection of large number of polymers, represented as particles. In spite of the use of 100,000 particles, the concentration is too low to derive the polymer force term in Equation (4). The current BD simulations cannot therefore predict drag reduction in a fully turbulent flow. The particles are advected on the velocity fields obtained with the viscoelastic DNS. Two cases are considered: an uncoupled simulation (β = 1) where turbulence intensities are large and an HDR simulation, where turbulence has been significantly damped. For each particle in BD simulations, the Lagrangian version of the FENEP model Equation (1) is solved, and statistics are collected in 100 bins throughout the half-width of the channel. As shown in Figure 1a, the polymer stress (normal stress here) predicted by DNS is dramatically overestimated for the uncoupled simulation when compared to BD data, while at HDR the two methods nearly collapse. In the uncoupled case, the coarsest resolution 64 × 129 × 32 is obviously not grid dependent but the finer grids 96 × 151 × 48 (not shown here) 128 × 151 × 64 and 256 × 161 × 128 show little difference. Under the assumption that the smallest scale of ci j is of the order of the Batchelor scale, the resolution would need to increase by a few orders of magnitude to notice a difference in statistics of ci j or τi j . The spectrum of τ yy , Figure 1b, shows almost no decay in the uncoupled case while the velocity spectrum drops 5 decades for high wavenumbers. The lack of resolution makes uncoupled simulations inaccurate quantitatively. In drag reduced flows, the τ yy spectrum decays much more strongly at high wavenumbers. Note that the Kolmogorov scale is also bigger in drag reduced flow with the v-spectrum decaying at a much faster rate for large k than in the uncoupled case. Although the simulation should be under-resolved according to the present reasoning, the spectrum of τ yy for the uncoupled simulation shows a decay close to k −1 in the high-wavenumbers. Only the wall-normal direction is shown here since the energy of τ yy was found to decay in the high wavenumbers at a slower rate than any other components of the tensor. 2.4. N UMERICAL
TREATMENT OF POLYMER ADVECTION
The advection issue deserves particular attention in the context of numerical simulation of turbulence. Stiff equations, such as pure advection, requires special treatment of the advection term to take into account the shock-like gradients which may occur. The most common approach is the upwinding approach, which reduces the risk
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Figure 1. (a) Normal stress profiles in a minimal channel flow (π h × 2h × h): comparison between Brownian dynamics (BD) and viscoelastic simulations (DNS). Uncoupled simulation (β = 0, W eτ = 35, L = 60): DNS, (arrow indicates increasing resolution 64×129×32, 128 × 151 × 64, 256 × 161 × 128); BD, (upper curve). High drag reduction (HDR) simulation (β = 0.9, L = 100, W eτ = 120): DNS, ; BD, (lower curve). (b) Symbols and lines show spectra of wall-normal velocity and polymer stress τ yy , respectively. DNS: ◦, (DR = 0%, β = 1, W eτ = 35, L = 60) ; , : DR = 65% (β = 0.9, W eτ = 84, L = 60). The spectra are shown for y + = 15 and are normalized by their respective variance. Rec = 7500, L = 60.
of oscillations but adds artificial dissipation. On a fixed grid based on flow scales and using Eulerian methods, upwinding is the only option available to us which is computationally viable. Min et al. [23] tested different schemes and favored a third-order compact upwind scheme for its low dissipation and its spectral-like nature [19]. We used a slightly modified version of Min et al.’s scheme which ensures third-order and fourth-order accuracy for the upwind and centered schemes, respectively. The upwinding coefficient,
1 = (s − + s + ), 2
(11)
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is a function of s − and s + which denotes the sign of velocity at the interface of the cell. The compact scheme is written as (2 − 3)φi+1 + 8φi + (2 + 3)φi−1 1 [(1 − )φi+1 + 2φi − (1 + )φi−1 ], = 6
(12)
To illustrate the benefit of using compact schemes, Figure 2 shows the modified wavenumber of the second-order finite difference scheme and of Equation (12) when discretized on pressure nodes. The modified wavenumber k is obtained by transforming the numerical scheme in Fourier space and writing the resultˆ Figure 2 indicates which wavenumbers are accurately ing equation as φˆ = k φ. discretized. Compact schemes increase by almost a factor two the range of wellresolved wavenumbers compared to the 2nd-order scheme. Min et al. stressed the need for an additional dissipation to stabilize ci j fields. The dissipation consists of the addition of a tensor Di j to Equation (1). They compared a global artificial dissipation (GAD), Di j = κ∂k ∂k ci j ,
(13)
to a second-order local artificial dissipation (LAD), Di j = κ2k ∂k2 ci j ,
(14)
which is applied only to locations where the tensor ci j experiences a loss of positiveness. The criterion used by Min et al. is det(ci j ) < 0. These authors used visualizations to show that GAD smears ci j gradients more significantly than LAD. Note that GAD is commonly used with pseudo-spectral methods, i.e. non-dissipative and non-dispersive, with Schmidt numbers (Sc = ν/κ) below unity in order to ensure stability [28]. Based on the argument that
Figure 2. Modified wavenumbers of the second-order finite difference scheme ( ), the fourth-order compact scheme ( , Equation (12), = 0) and the third-order compact upwind scheme ( , Equation (12), = 0). The straight line denotes spectral differentiation.
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Figure 3. Spectra of polymer stress τ yy at y + = 15 in an uncoupled (β = 0) simulation using LAD ( ) and GAD ( ). Spectra are.
Equation (1) behaves like a passive scalar, low-Schmidt numbers force the smallest scale produced in ci j fields to be larger than the Kolmogorov scale [6], which is at odds with the physical analysis of Equation (1) developed earlier. [23] showed that polymer stretching occurs at the same location for GAD and LAD, only the extent of regions of large stretch are smeared by GAD yielding a lower magnitude of the polymer body force f i in Equation (1). To illustrate the impact of the dissipation used, Figure 3 shows two spectra of τ yy for the uncoupled simulations, one obtained with LAD while the other results from a simulation performed using GAD for Sc = 0.5. The spectra are taken in the buffer layer (y + = 15) and are normalized by the variance predicted by the LAD simulation. As shown in Figure 1b, an uncoupled simulation is the most stringent test case, for which the polymer stress’ spectra hardly decay. For the same polymer parameters (W eτ = 35, L = 60), the spectra in Figure 3 have very different shape. When GAD is used, the spectrum of τ yy decays rapidly at high wavenumbers, supporting the earlier argument that small scales are governed by the advection term. GAD results in lower variance of polymer stress and polymer force (Equation 4) and leads to lower DR (not shown here). The local artificial dissipation is therefore adopted in this work. The combination of the compact upwind scheme and LAD can be interpreted as a MILES technique (Monotone Integrated Large Eddy Simulation), which allows for the simulation of Schmidt numbers larger than unity. This method however does not permit an exact measure of the actual Schmidt number. At best, spectra of ci j can only distinguish between the two classical regimes: Sc < 1 and Sc > 1. The decay at high wavenumbers of the polymer stress spectra is probably the best test for choosing κ: τi j should decay at a slower rate than the kinetic energy, according to the analogy between low κ passive scalar and the FENE-P model. The constant κ in Equation (14) was found to be a function of the grid. The choice is determined by the average number of computational nodes where LAD is required. If about 10% of the nodes are affected by LAD at the onset of polymer
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interaction with the flow, the fraction of nodes requiring LAD drops below 5% at LDR and much lower than 1% for HDR. Depending on the grid resolution, κ is chosen in the range [10, 0.01]; most of the simulations here were performed with κ = 1. To be consistent with the need for improved resolution at the small scales, the divergence term in the polymer body force f i is discretized with a fourth-order compact scheme whose equation is given by Equation (12) with = 0. 3. Statistical Description of HDR Flows The simulations were performed in channel flows for Rec = 7500 using a constant mass flow. The friction Reynolds number of the reference case, or Newtonian flow noted DR = 0%, is Reτ = hu τ /ν = h + = 300. The resolution for all simulations is x + = 9, y + = 0.1 − 5 and z + = 6, when normalized by the skin friction at DR = 0%. The dimensions of the computational domain are 4π h × 2h × 4h. Larger domains were simulated, yielding minor changes in drag reduction. Two different sets of polymer parameters are simulated: (L = 60, W eτ = 84) and (L = 100, W eτ = 120), representing realistic polymers. The first set produces DR = 47% while the second achieves DR = 60% for β = 0.9. Increasing the Weissenberg number to 150 for L = 100 fails to surpass DR = 60%. From Figure 4 which shows the asymptotic profile of [32], it is obvious that for the Reynolds number of interest the maximum drag reduction should be higher than 60%. The inability of DNS to reach MDR could be caused by the low Reynolds number. Another probable cause is the absence of internal modes in the FENE-P model. At HDR, average extensions ckk do not exceed 35% of L 2 , resulting in weak polymer stress (see Figure 1). In this regime of small extensions, the FENE-P model departs from more accurate models such as FENE multi-chain. As MDR is approached, turbulence is anticipated to be weaker and weaker yielding ¨ smaller polymer extensions. Herrchen and Ottinger [14] showed that the FENE-P model, based on a single dumbbell, fails for weak transient strains compared to FENE multichain for which a polymer is represented as a connected series of
Figure 4. Mean velocity profiles in wall variables. , DR = 60%; , DR = 72%.
, DR = 0%;
, DR = 47%;
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dumbbells. The development of a constitutive model which incorporates internal modes will be necessary to understand which of the two assumptions discussed above, low-Reynolds number or including internal modes, is the most important in controlling the HDR/MDR transition. In the course of the validation of our numerical methods, we found that the + + minimal channel flow [15], whose dimensions, L + x × L y × L z = 1000 × 300 × 300 are based on the Newtonian friction velocity, is large enough to contain a weak selfsustained turbulence whose statistics approach that of MDR. The resulting drag reduction is much more significant (72%) than in the large computational domain (60%). Large scale structures at the center of the channel are larger than the domain with its periodic conditions in homogeneous directions. This simulation can be considered as our first numerical experiment. It shows that very large scale structures + longer than L + x ∼ 500 and wider L z ∼ 160 are irrelevant to the mechanism by which turbulence is sustained at HDR. It is interesting to note that, in the minimal channel flow, the mean velocity profile roughly agrees with [32]’s empirical law (Figure 4). As discussed in the introduction, HDR flows are characterized by a steeper slope in the log region than in the Newtonian case. Mean velocity and RMS profiles are plotted in wall variables (Figures 4 and 5a) that illustrates the breakdown of the
Figure 5. RMS of velocity fluctuations in wall variables (a) and relative to the local mean streamwise velocity (b). Lines are as in Figure 4. ◦, u ; , v ; , w .
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inner scaling in drag reduced flows. The log-law is modified, the maximum of u + experiences an increase, v + and w + are significantly reduced, as DR increases. In outer variables (not shown here), the maximum of u retains a magnitude very close to the Newtonian case, slightly smaller as drag is reduced, while the transverse turbulent intensities decrease, as much as 40, 62 and 83% for the maximum of w computed from DR = 40, 60 and 72%, respectively. Close to the wall, the “law of the wall” is still qualitatively valid for the mean velocity and the turbulent intensities, for which u i /u τ may be expanded in powers of y + : u i + = ai y + + bi y +2 + . . . ,
(15)
where a2 = 0. Figure 5b displays the RMS of velocity fluctuations relative to the local mean streamwise velocity, whose asymptotic behavior can be written as U + = y + (1 − y + /2h + ) + O(y +4 ). Here it is observed that the extent of the viscous sublayer increases with increasing DR. The coefficients a1 , b2 (not shown here) and a3 strongly depend upon DR, showing that quantitatively, the equilibrium of drag reduced flows is significantly different than that in Newtonian flows. In a simplified picture of near-wall turbulence, u is associated with streaks, while the major contribution to v and w comes from quasi-streamwise vortices. According to this picture, Figure 5b indicates that the turbulent intensity of the streaks decreases, yet the reduction is much less than for vortices (transverse velocity components). Reynolds shear stress (Figure 6a) is also considerably affected by polymers. At DR ≥ 60%, −uv + is reduced by a factor of 4 compared to the Newtonian case. [24] measured experimentally the same reduction whereas [34] obtained zero Reynolds shear stress. Though quantitatively different, a reduction in the Reynolds shear stress contribution to the force balance, −uv + + β
dU + y+ + + (1 − β)τ = 1 − xy dy + h+
(16)
(here written from Equation (4)), is offset by an increased role of polymer stress, τx y as shown in Figure 6. For DR = 60%, the near-wall polymer stress is comparable to Reynolds shear stress while, at DR = 72%, polymer stress is larger than Reynolds shear stress everywhere. This confirms the importance of polymer stress in the upper HDR and MDR regimes. The behavior of the fluctuations of the velocity components is evidently reflected in the anisotropy of the Reynolds stress tensor. Figure 6b represents an anisotropy invariant map (AIM) [21] of the Reynolds stress anisotropy tensor, bi j = u i u j /u k u k − δi j . The map consists in cross-plotting the second II and third III invariants of bi j : −bi j b ji /2 and bi j b jk bki /3, respectively. The top boundary (II + 3III = −1/9) characterizes a two-component turbulent state (as seen in Figure 2 1 5b, v u , w in the near-wall region of a channel flow). The top cusp ( 27 , 3) designates the one component state, most anisotropic state, while the bottom cusp, (0, 0), represents the isotropic state. The boundary in between, (III/2)(−II/3)3/2 = 1,
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Figure 6. Behavior of the Reynolds stress tensor in drag reduced flows. (a) Reynolds shear stress, ◦, (off-diagonal component) and comparison to polymer stress, , in wall variables. (b) Anisotropy invariant map for the Reynolds stress tensor. , DR = 0%; ◦, DR = 47%; , DR = 60%; ×, DR = 72%. The arrows point at the measure of anisotropy calculated at the first computational node from the wall. The AIM is bounded by .
defines a disk-like turbulence (axisymmetric state). Drag reduced flows are dramatically more anisotropic than the Newtonian flow, almost reaching the top vertex. In the outer region, they approach the axisymmetric boundary more closely than the Newtonian flow. Figure 6b is another illustration of how significantly different the equilibrium of polymer drag reduced flows is to that of Newtonian flows. Previous studies on kinetic energy budgets [8] have established that the contribution of pressure strain reduces dramatically as drag decreases. Since the role of pressure strain is to distribute energy among velocity components, the AIM is therefore another measure of this reduction.
4. Polymer Interaction with Coherent Structures 4.1. TURBULENCE
REDUCING / ENHANCING PROPERTIES OF POLYMERS
In the previous section, we have suggested that vortices may be damped by polymers. To examine this claim, we investigate the behavior of the polymer body force,
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Figure 7. Correlation velocity-polymer body force ∗ for different drag reductions. β. , ρx ; , ρy ; , ρz . Polymer parameters: ◦: W eτ = 120, β = 0.9, DR = 72%; : W eτ = 120, β = 0.99, DR = 38%; : W eτ = 12, β = 0.99, DR = 13%.
f i = (1 − β)(∂ j τi j )/Re, in the momentum Equation (4). The correlation, ρi =
u i fi , u i f i
(17)
depicted in Figure 7, for different drag reductions, shows that polymer body forces in both wall-normal and spanwise directions are anti-correlated across the channel This suggests that the drag reducing action of polymers is to produce stresses which act in opposition to transverse fluctuations. In the streamwise direction, the polymer body force is correlated with the corresponding velocity component in the viscous sublayer, thus enhancing fluctuations in this particular region. Away from the wall, the streamwise polymer body force also tends to oppose turbulent fluctuations, except for the highest DR, where the action of f x relative to u is indeterminate for y + > 30. Figure 7 supports the conclusion from Figure 6 in the demonstration that MDR is sustained by polymer stress. A similar study was performed for LDR flows by [7]. Actually there has been recently a fair amount of evidence supporting that polymers affect vortices by controlling transverse velocity fluctuations. [25] demonstrated that large polymer stress is found in upwash and downwash flows. Using “exact coherent states” in plane Couette flow, which captures a periodic turbulent-like state at a subcritical Reynolds number [27] have also reached the conclusion that velocity fluctuations and polymer body force are anticorrelated. 4.2. N UMERICAL
EXPERIMENTS
One of the main efforts carried out by Prof. Antonia in near-wall turbulence has been to investigate the response of turbulence to various manipulations. These manipulations have mainly consisted in modifications of the boundary condition, such as the study of the flow over transverse square cavities [10]. In this study, the
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authors showed that the main characteristics of near-wall turbulence are retained, yet the succession of grooves makes the low-speed streaks more stable. Such studies are extremely useful in understanding how turbulence can be affected by various parameters and to investigate different scalings. Considering the complexity of our problem, we try to narrow the field of investigation to the momentum equation (Equation 1). In this equation, polymers appear via f i , the polymer body force. From the previous section, we know that, in the wall-normal and spanwise directions, f i and u i are anti-correlated (Figure 7) and u and f x are positively correlated very close to the wall. Figures 8b–d are representative of the different structures of f i . It is striking that large fluctuations of polymer body force are clustered in the immediate neighborhood of near-wall
Figure 8. Snapshot of vortices and polymer body force in the near wall regions. Vortices are identified using isosurfaces of the positive second invariant of the velocity gradient tensor (white isosurfaces in (a) and mesh isosurfaces elsewhere). (a) Contours of wall-shear stress fluctuations (−0.5 ≤ ∂u + /∂ y + ≤ +0.5, black to white). (b) Isosurfaces of f x ( f x+ = ±0.084). (c) Isosurfaces of f y ( f y+ = ±0.035). (d) Isosurfaces of f z ( f z+ = ±0.07). (b)–(d) Negative threshold, light isosurfaces; positive threshold, dark isosurfaces. DR = 38%.
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vortices, regardless of the component of f i . The spatial scales are similar to those of vortices for f y and f z . These visualizations combined with correlation data between f i and u i suggest that polymers oppose vertical and spanwise flows generated by near-wall vortices, while the dynamics of f x is mostly confined below the vortices. The polymer body force’s behavior may therefore be decomposed crudely in two separate actions, in the same manner as v and w fluctuations may be attributed to vortices and u to streaks in near-wall turbulence, as mentioned earlier. We assume that f y and f z reduce drag by damping vortices and f x sustains turbulence in the near-wall region by enhancing u-fluctuations. To test this hypothesis, two simple numerical experiments are carried out, similar to the ones performed at LDR by [11]. In the first experiment, f y and f z are set to zero, while in the second, f x is set to zero. The reference case is DR = 72% in a minimal channel flow for which turbulence is found to be sustained even though large scales in the core of the flow vanish due to the small dimensions of the domain. The temporal evolution of the pressure drop is plotted in Figure 9. At the onset of polymer interaction with the flow t = 0, the pressure drop experiences an overshoot then decreases over 200h/Uc and fluctuates around the steady-state drag. On the same figure, the pressure drop for a laminar channel flow is shown. Note that for the first experiment, we also vary β from 0.9 at the wall to 1 in the log-layer to isolate the region shown in Figure 7 where f x and u are positively correlated. When the streamwise component of the polymer body force is switched off, the pressure drop rises rapidly to more than twice its initial value. This simulation is delicate to perform since large fluctuations of u occur very near the wall, requiring a very small time step. When f y = f z = 0 throughout the entire channel flow, eventually the pressure drop decreases and fluctuates at a higher magnitude than the reference case. The amount of drag reduction in that case is around 20%. Due to the strong mean shear close to the wall, the polymers, which are set in coil
Figure 9. Temporal evolution of the pressure gradient for numerical experiments in a minimal channel flow: , reference case; , f x = 0; , f y = f z = 0; (horizontal line), laminar −d P/d x.
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configuration at the beginning of the simulation, stretch much faster in the nearwall region and consequently are more rapidly active in this region. The subsequent decrease of pressure drop can be attributed to the drag-reducing nature of f x away from the wall as shown by the weak anticorrelation with u in Figure 7. Another numerical experiment was performed where f x was removed for y + > 20 which lead to a monotonous increase of −d P/d x until the turbulence became too intense for the polymer solver to handle it with reasonable amount of nodes requiring LAD. When only f x is set to zero, the pressure drop decreases monotonically toward the laminar value which is reached around t = 1000h/Uc . Note the absence of overshoot at the start of the simulation. This indicates that, in the reference case, the overshoot is the consequence of a delay between the drag reducing activity and turbulence enhancing activity caused by polymers. It should be noted that f y and f z were also tested independently by [11] and both components showed a similar turbulence-reducing behavior. The physical picture, anticipated from the correlation between polymer body force and velocity, is therefore confirmed by these two numerical experiments. In [12], statistical evidence of the two opposite behaviors, namely drag reduction and turbulence enhancement, found in polymer dynamics in turbulent flows gives further support to the present work. Although we can only simulate MDR using a minimal channel flow, the fact that (i) turbulence is sustained in the reference case and (ii) f x = 0 leads to relaminarization indicates that the turbulence enhancing activity in the streamwise direction that we have isolated is the key of the selfsustained generation process of turbulence at MDR. This point was also inferred by [22] by using FENE-P bead-spring chain in turbulent flow. 5. Conclusion Direct numerical simulations and numerical experiments have been used to gain further insight in the mechanism of polymer drag reduction. By investigating the polymer body force, it is established that polymers reduce turbulence by opposing the downwash and upwash flows generated by near-wall vortices, while they enhance streamwise velocity fluctuations in the very near-wall region. The numerical experiments allow us to characterize the importance of each component of the polymer body force on the drag-reducing and turbulence-enhancing properties found or suggested in other publications [7, 22, 27]. The exact localization of polymer drag-reducing activity is now crucial for the development of a predictive model for polymer drag reduction. The relaxation time scale of the polymers needs obviously to be related to the scale of the quasi-streamwise vortices. While it seems that a model for the onset of drag reduction may be within reach, the main effort in the near future will be to map out the response of velocity and vorticity fluctuations to various drag reductions and to quantify the response of polymer molecules according to flow and polymer characteristics.
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