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Abstract

In order to study the polymer effect on the behavior of nonlinearities in decaying homogeneous isotropic turbulence (DHIT), direct numerical simulations were carried out for DHIT with and without polymers. We investigate the nonlinear processes, such as enstrophy production, strain production, polymer effect, the curvature of vortex line, and many others. The analysis results show that the nonlinear processes like enstrophy production (and many others) are strongly depressed in regions dominated by enstrophy as compared to those dominated by strain either in the Newtonian fluid case or in polymer solution case. Polymers only decrease the values of these parameters in the strongest enstrophy and strain regions. In addition, polymer additive has a negative effect on enstrophy and strain production, that is, depression of nonlinearity in DHIT with polymers.

1. Introduction

Kraichnan and Panda [1] suggested comparing the key nonlinear terms in real turbulent flows with their Gaussian counterparts, which are involved in the description of nonlinear dynamics in physical space, and firstly introduced the notion of depression of nonlinearity. In decaying homogeneous isotropic turbulence (DHIT), they found that $J \equiv 〈 {| u \times ω |}^{2} 〉 / N_{4}$ (where u and ω are velocity and vorticity vectors, resp.; ω = ∇ × u) drops to 87% of the value for a Gaussianly distributed velocity field with the same instantaneous velocity spectrum and $H \equiv 〈 {| u \cdot ω |}^{2} 〉 / N_{4}$ rises to 120% of the Gaussian value. Here $N_{4} = 〈 {| u |}^{2} 〉〈 {| ω |}^{2} 〉$ is a normalization factor. And also they observed a depression of normalized mean-square total nonlinear term $Q = 〈 {| u \times ω - \nabla (p + {(1 / 2) u}^{2}) |}^{2} 〉 / N_{4}$ (where p and u are pressure and velocity magnitude, resp.) to 57% of the Gaussian value [1]. Some other investigations on the depression of nonlinearity from the nonlinear term $〈 {| u \times ω - \nabla (p + (1 / 2) u^{2}) |}^{2} 〉$ [2] and the Lamb vector ω × u [3, 4] have been carried out. It is thus known that a study on the depression of nonlinearity is of profound importance for fluid dynamics in general and for the dynamics of turbulence in particular. Besides, the numerical results of Kraichnan and Panda [1] showed that the alignment of velocity and vorticity in physical space is not the sole source of the depression of nonlinearity. Shtilman and Polifke [5] argued that another source of the depression of nonlinearity seems to be a tendency of the Lamb vector ω × u in Fourier space to align with the wave vector k. These aforementioned investigations mainly focused on the relationship between velocity and vorticity.

However, as compared with velocity, the field of velocity gradient is much more sensitive to the non-Gaussian nature of turbulence or more generally to its structure and hence reflects more of its physics [6]. It is also a Galilean invariance containing significant fluid mechanics information independent of the reference of a moving observer. Moreover, its dynamical behavior governs the mechanism of vortex stretching which in turn contributes to the energy cascade process in turbulent flows (as pointed in [6] “The turbulent kinetic cascade in Fourier space must be replaced with the generation of velocity gradients (i.e., both vorticity and strain) in physical space.”). Therefore, its evolution is of primary importance in the understanding of the kinematics and dynamics of turbulence. The velocity gradient includes two parts: vorticity and strain. Therefore, it is very important to investigate the nonlinearity process from vorticity and strain transport equations. Tsinober [7] defined all four regions in real turbulent flow: (i) regions of concentrated vorticity; (ii) regions of “structureless” background; (iii) regions of strong vorticity/strain (self) interaction and strong enstrophy generation; (iv) regions with negative enstrophy production. The results of turbulent grid experiment [8] and DHIT [7] demonstrated that all four regions are strongly non-Gaussian, dynamically significant, and possess structures, and it is also argued that due to the strong nonlocality of turbulence in physical space all the four regions are in continuous interaction and are strongly correlated. Tsinober [7] also highly stressed the role of regions of strong vorticity/strain (self) interaction and argued that important regions of concentrated vorticity are not as important as is commonly believed. Then he [9, 10] investigated the behavior of key nonlinearities related to velocity gradients in flow regions dominated by enstrophy and strain, such as the magnitude of the vortex stretching vector $W = | ω_{j} S_{i j} |$ (where S_ij = (∂ u_i/ ∂ x_j + ∂ u_j/ ∂ x_i)/2 is the rate-of-strain tensor), the enstrophy generation ω_iω_jS_ij, and the curvature of vortex lines. It is shown that the nonlinear processes like enstrophy production (and many others) are strongly depressed in regions dominated by enstrophy as compared to those dominated by strain.

In the present paper, the purpose is to investigate the polymer effect on the behavior of nonlinearity in DHIT and study which region (as Tsinober defined in [7]) is influenced by polymers. We use the dataset from direct numerical simulation in the periodic cubic domain of size ℝ = 2π computed with a pseudo-spectral code for Navier-Stokes equations and finite difference code for FENE-P (finitely extensible nonlinear elastic Peterlin) constitutive model with resolution 96³ using the Adams-Bashforth scheme [11]. The Taylor-scale Reynolds number Re_λ and the Weissenberg number Wi are defined as $R e_{λ} = \sqrt{20} ξ_{m}^{[N]} / \sqrt{3 υ^{[s]} ε_{m}^{[N]}}$ and $Wi = τ_{p} \sqrt{ε_{m}^{[N]} / υ^{[s]}}$ . Here ξ_m^[N] and ε_m^[N] are the turbulent kinetic energy and energy-dissipation rate for the Newtonian fluid flow at t = t_m, where t_m corresponds to the moment at which ε^[N] reaches to its maximum amplitude; υ^[s] is the solvent kinetic viscosity; τ_p is the polymer relaxation time. In this paper, $β = υ^{[s]} / (υ^{[s]} + υ^{[p]})$ is a dimensionless measure of dilute polymer solution concentration, the smaller β is, the denser polymer solution is; here υ^[p] is the polymer viscosity. The results presented in this paper are at a time right after the total enstrophy has reached its maximum and at which time Re_λ = 26 and for polymer solution flow, Wi = 0.62, β = 0.6. For more information, the reader can be referred to [11].

2. Enstrophy and Strain Transport Equations

In one of our previous studies [12], we have paid our attention to velocity gradient, especially for vorticity and strain field. These two quantities must be considered in parallel, as they are weakly correlated in isotropic turbulence and they are tied by a strongly nonlocal relation. So we firstly give the fundamental evolution equations for the enstrophy $ω^{2} / 2$ and the total strain $S_{i j} S_{i j} / 2$ .

The enstrophy transport equation of DHIT with polymers is as follows:

\begin{matrix} \frac{\partial Ω}{\partial t} + u_{j} \frac{\partial Ω}{\partial x_{j}} = ω_{i} ω_{j} S_{i j} + υ^{[s]} ω_{i} \frac{\partial^{2} ω_{i}}{\partial x_{j} \partial x_{j}} + ω_{i} \frac{\partial^{2} T_{m j}^{[p]}}{\partial x_{m} \partial x_{n}} ς_{n j i}, \end{matrix}

(1)

where ς_nji is the permutation symbol; Ω = ω_iω_i/2 the enstrophy, here $ω_{i} = (\partial u_{k} / \partial x_{j}) ς_{i j k}$ ; S_ens = ω_iω_jS_ij the enstrophy production due to the interaction between vorticity and strain; $V_{ens} = υ^{[s]} ω_{i} (\partial^{2} ω_{i} / \partial x_{j} \partial x_{j})$ the enstrophy viscous dissipation; $W_{i}^{[p]} = ω_{i} (\partial^{2} T_{m j}^{[p]} / \partial x_{m} \partial x_{n}) ε_{n j i}$ the polymer effect vector; P_ens = ω_iW_i^[p] the polymer effect due to the interaction between vorticity and polymer conformation and does not appear in the Newtonian fluid case.

The strain transport equation of DHIT with polymers is as follows:

\begin{matrix} 2 \frac{\partial S}{\partial t} + 2 u_{j} \frac{\partial S}{\partial x_{j}} = 2 (- S_{i k} S_{k j} S_{i j} - \frac{1}{4} ω_{i} ω_{j} S_{i j} - S_{i j} \frac{\partial p}{\partial x_{i} \partial x_{j}} + υ^{[s]} S_{i j} \frac{\partial^{2} S_{i j}}{\partial x_{j} \partial x_{j}} + \frac{\partial^{2} T_{i k}^{[p]}}{\partial x_{k} \partial x_{j}} S_{i j}), \end{matrix}

(2)

where S = S_ijS_ij/2 is the total strain; S_str = – S_ikS_kjS_ij the strain production from strain self-amplification; W_str = ω_iω_jS_ij/4 the enstrophy production effect on the total strain; R_str = S_ij (∂ ²p/ ∂ x_i ∂ x_j) the interaction of strain with pressure Hessian; V_str = υ^[s]S_ij ∇ ²S_ij the strain viscous dissipation; $B_{i j}^{[p]} = \partial^{2} T_{i k}^{[p]} / \partial x_{k} \partial x_{j}$ the polymer effect tensor influencing on the generation of strain; P_str = B_ij^[p]S_ij the polymer effect due to the interaction between strain and polymer elastic stress (or polymer conformation) and does not appear in the Newtonian fluid case.

It is found that ω_iω_jS_ij term appears in (1) and (2). When ω_iω_jS_ij is positive, the production of total strain is decreased and the production of total enstrophy is increased; when ω_iω_jS_ij is negative, the results are contrary. However, $〈 ω_{i} ω_{j} S_{i j} 〉$ is always positive [11]. So enstrophy production results from the interaction of vorticity with the strain field, whereas the production of total strain mainly comes from self-amplification of the strain field. The relationship among velocity, velocity gradient tensor, rate-of-strain tensor, vorticity, enstrophy production, and strain production is schematically shown in Figure 1 [6]. And also there exists the relationship as follows: $〈 S_{i j} S_{j k} S_{k i} 〉 = (- 3 / 4) 〈 ω_{i} ω_{j} S_{i j} 〉$ and 〈S_ij (∂ p/ ∂ x_i ∂ x_j)〉 = 0 due to homogeneity and incompressibility; – S_ijS_jkS_ki = – (α₁³ + α₂³ + α₃³) = – 3α₁α₂α₃, where α_i (i = 1, 2, 3) are the corresponding eigenvalues reordered of rate-of-strain tensor S. Moreover, the eigenvalues of velocity gradient tensor, λ₁ > 0, and λ₂ is positively skewed [10]. Due to the nonlocal relation between the rate-of-strain tensor and vorticity, it is useful to investigate the above quantities in parallel, such as the third moments ω_iω_jS_ij, S_ijS_jkS_ki, which are of key importance for turbulence dynamics. However, for DHIT with polymers, we must highly investigate the polymer effect on the important terms in these transport equations, such as $ω_{i} (\partial^{2} T_{m j}^{[p]} / \partial x_{m} \partial x_{n}) ε_{n j i}$ and $(\partial^{2} T_{i k}^{[p]} / \partial x_{k} \partial x_{j}) S_{i j}$ .

Figure 1:

Schematic view of the velocity-gradient self-amplification process in isotropic turbulence.

3. Results and Discussion

We take dissipation rate, enstrophy production, strain production, and vortex stretching herein as the measurement of nonlinearity in DHIT. Conditional averages of all quantities here are conditioned on $ω / 〈 ω 〉$ and $s / 〈 s 〉$ , here $ω = | ω |$ and $s = {(S_{i j} S_{i j})}^{1 / 2}$ . Firstly, we define the regions of the strongest interaction of vorticity and strain where $3 \leq ω / 〈 ω 〉 \leq 4$ and $3 \leq s / 〈 s 〉 \leq 4$ . We give the conditional average of dissipation rate, as shown in Figure 2 (a). It is clearly suggested that the addition of polymers into DHIT can only change dissipation rate in the regions of the strongest enstrophy and strain. And also, dissipation rate appears smaller in the regions dominated by enstrophy than that in the regions dominated by strain (as pointed out by ellipse identifier in Figure 2 (a)). Besides, the result of conditional average of polymer elastic energy is shown in Figure 2 (b). It is also found that e^[p] is much smaller in the enstrophy dominated regions than that in regions dominated by strain (ellipse identifier in Figure 2 (b)).

Figure 2:

Conditional averages of (a) dissipation rate and (b) polymer elastic energy in slots of $ω / 〈 ω 〉$ and $s / 〈 s 〉$ .

One of the aspects of the problem in question concerns the enstrophy production S_ens = ω_iω_jS_ij, enstrophy viscous dissipation $V_{ens} = υ^{[s]} ω_{i} (\partial^{2} ω_{i} / \partial x_{j} \partial x_{j})$ , and the polymer effect on enstrophy P_ens = ω_iW_i^[p] and vortex stretching W². These parameters are very important for investigating the nonlinear processes and the polymer effect in DHIT, and the conditional averages of which are shown in Figure 3. It can be seen that the addition of polymers only depresses the values of these parameters in the strongest regions of enstrophy and strain. And also, it is shown that the values of these parameters are smaller in regions dominated by enstrophy than those in regions dominated by strain (as pointed out by ellipse identifiers in Figure 3). This adequately suggests that the depression of the nonlinearity is strengthened due to the existence of polymers. In addition, the main contribution to vortex stretching and enstrophy production comes from the regions dominated by strain. So it is out of question to say that the regions dominated by strain (the regions of the strongest interaction of vorticity and strain) are more important than the enstrophy dominated regions. And the polymer additive has the same influence on strong vorticity/strain (self) interaction and has a negative effect on enstrophy production, suggesting that the decrease on enstrophy production is the depression of nonlinearity in DHIT with polymers. That is to say, the negative effect of polymers on enstrophy production (causing the decrease of vortex stretching) in DHIT is similar to a negative torque on vortices as investigated in the polymeric channel flow [13, 14].

Figure 3:

Conditional averages of (a) enstrophy production; (b) enstrophy viscous dissipation; (c) the polymer effect; (d) vortex stretching in slots of $ω / 〈 ω 〉$ and $s / 〈 s 〉$ .

It is equally important to investigate the nonlinear behavior of strain production (S_str = – S_ikS_kjS_ij and 2 (S_str – W_str) = 2(-S_ikS_kjS_ij – ω_iω_jS_ij/4)), strain viscous dissipation (V_str = υ^[s]S_ij ∇ ²S_ij), and the polymer effect on strain (P_str = B_ij^[p]S_ij), and the conditional averages of which are shown in Figure 4. It follows that the mean rate of strain production $〈 2 (- S_{i k} S_{k j} S_{i j} - ω_{i} ω_{j} S_{i j} / 4 - S_{i j} \partial p / \partial x_{i} \partial x_{j}) 〉 = 〈 ω_{i} ω_{j} S_{i j} 〉$ is equal to that of enstrophy, hence (and also due to $〈 ω^{2} 〉 = 2 〈 S_{i j} S_{i j} 〉$ ) a coefficient 2 is used in (2) on both sides. From Figure 4, it can be seen that the addition of polymers also only depresses the values of these parameters in the strongest regions of enstrophy and strain, which is the same tendency as the results of conditional averages of those parameters in enstrophy transport equation. And also, these parameters are smaller in the enstrophy dominated regions than those in regions dominated by strain (ellipse identifier in Figure 4). And the polymer additive has the same influence in strong vorticity/strain (self) interaction and almost has a negative effect on strain production, suggesting the decrease on the strain production, that is, the polymer effect depressing the nonlinear processes in DHIT.

Figure 4:

Conditional averages of (a) strain production; (b) strain production and enstrophy production effect on the total strain; (c) strain viscous dissipation; (d) the polymer effect in slots of $ω / 〈 ω 〉$ and $s / 〈 s 〉$ .

Another manifestation of the depression of nonlinearity is the decrease of curvature of vortex lines in the enstrophy dominated regions. As is known, it is easy to obtain three-dimensional vorticity ω, its magnitude $ω = | ω |$ , its direction vector $\hat{ω} = ω / ω$ , and the curvature of vortex line $R = | \hat{ω} \cdot \nabla \hat{ω} |$ . So we get the conditional average of curvature R in slots of $ω / 〈 ω 〉$ and $S / 〈 S 〉$ , as shown in Figure 5. It is seen that the curvature is decreasing with $ω / 〈 ω 〉$ , and it is firstly increasing and then decreasing with $S / 〈 S 〉$ . And also, the curvature of vortex line in the enstrophy dominated regions is smaller than that in the strain dominated regions (ellipse identifier in Figure 5). This suggests that vortex structures are distorted tempestuously in the strong vorticity/strain regions.

Figure 5:

Conditional average of curvature in slots of $ω / 〈 ω 〉$ and $s / 〈 s 〉$ .

Also we consider that the conditional averages of enstrophy, strain, and dissipation rate conditioned on $e^{[p]} / 〈 e^{[p]} 〉$ , as shown in Figure 6. It is clearly shown that these parameters firstly are increasing and then decreasing with $e^{[p]} / 〈 e^{[p]} 〉$ , but mainly existing in the elastic regions where $e^{[p]} \approx 10 〈 e^{[p]} 〉$ . In addition, strain is much larger than enstrophy in the polymer elastic energy regions, suggesting much well correlation between strain and polymer elastic energy.

Figure 6:

Conditional averages of (a) enstrophy and strain; (b) dissipation rate in slot of $e^{[p]} / 〈 e^{[p]} 〉$ .

In order to expatiate the distribution of enstrophy production, enstrophy viscous dissipation, and the polymer effect in the polymer elastic energy region, the conditional averages of these parameters are carried out, as shown in Figure 7. It is clearly shown that the enstrophy production and enstrophy viscous dissipation are firstly increasing with $e^{[p]} / 〈 e^{[p]} 〉$ and then decreasing with $e^{[p]} / 〈 e^{[p]} 〉$ . But for the polymer additive, it always has a negative effect on enstrophy production in regions where $e^{[p]} \leq 10 〈 e^{[p]} 〉$ and sometimes a positive effect on enstrophy production in regions where $e^{[p]} > 10 〈 e^{[p]} 〉$ . So it is sufficient to say that the strongest contribution of the polymer effect exists in regions where $e^{[p]} \approx 10 〈 e^{[p]} 〉$ .

Figure 7:

Conditional averages of (a) enstrophy production and enstrophy viscous dissipation; (b) the polymer effect in slot of $e^{[p]} / 〈 e^{[p]} 〉$ .

Also, we show the conditional averages of strain production, strain viscous dissipation, and the polymer effect in the polymer elastic energy regions, as shown in Figure 8. Compared with the results in Figure 7 (a), it can be seen that the values of 2 (S_str – W_str) and V_str are much larger than the values of S_ens and V_ens. This indicates again the well correlation between strain and polymer elastic energy, which is consistent with the conclusion of polymers mainly stretching in strain dominated regions in our previous study [11]. The polymer additive always has a negative effect on strain, suggesting the inhibition of strain production by polymer additives. The value of conditional averages of P_str is much larger than that of P_ens, indicating that the polymer effect on strain field is stronger than that on enstrophy field. Overall, polymers have an important and negative effect on enstrophy and strain fields. From this viewpoint, the nonlinear processes in DHIT of polymer solution are depressed due to the existence of polymer additives.

Figure 8:

Conditional averages of (a) strain production and enstrophy production effect on the total strain and strain viscous dissipation; (b) the polymer effect in slot of $e^{[p]} / 〈 e^{[p]} 〉$ .

4. Conclusions

DNS of DHIT with and without polymers have been carried out based on Navier-Stokes equation coupled with FENE-P constitutive model. We investigated the polymer effect on the nonlinear processes based on the important parameters in the enstrophy/strain transport equations and the curvature of vortex line. Some important conclusions have been drawn as follows.

The existence of polymers in DHIT only decreases the values of the important parameters in the strongest regions of enstrophy and strain. And also, polymers have a negative effect on enstrophy production and strain production, suggesting the depression of nonlinearity in DHIT with polymers.

The curvature of vortex line in the enstrophy dominated regions is smaller than that in the strain dominated regions. This suggests that vortex structures are distorted tempestuously in the strong vorticity/strain regions.

The strongest contribution of the polymer effect exists in regions where $e^{[p]} \approx 10 〈 e^{[p]} 〉$ . And also, strain is much larger than enstrophy in the polymer elastic energy regions, suggesting much well correlation between strain and polymer elastic energy.

Footnotes

Acknowledgments

The authors thank Professor B. Yu of China University of Petroleum (Beijing) and Dr. Y. Yamamoto of Kyoto University, for their discussion on DNS. This study was supported by National Natural Science Foundation of China (Grant no. 51206033), the Fundamental Research Funds for the Central Universities (Grant no. HIT.NSRIF.2012070), the China Postdoctoral Science Foundation (Grant no. 2011M500652), and the Heilongjiang Postdoctoral Science Foundation (Grant no. 2011LBH-Z11139). The authors are also very grateful for the enthusiastic help of all members of Complex Flow and Heat Transfer Laboratory of Harbin Institute of Technology.

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The Polymer Effect on Nonlinear Processes in Decaying Homogeneous Isotropic Turbulence

Abstract

1. Introduction

2. Enstrophy and Strain Transport Equations

3. Results and Discussion

4. Conclusions

Footnotes

Acknowledgments

References