Wavelet analysis of coherent structures and intermittency in forced homogeneous isotropic turbulence with polymer additives

Abstract

Large eddy simulation was performed for forced homogeneous isotropic turbulence with/without polymer additives. Wavelet transform in one dimension and two dimensions were performed to investigate the multi-resolution features of coherent structures and intermittency in forced homogeneous isotropic turbulence based on large eddy simulation database. Using wavelet decomposition in one dimension and two dimension, it is found that polymer additives behave inhibitive effect on the intermittent pulse and the amount of coherent structures in forced homogeneous isotropic turbulence. The reconstructions of velocity waveform for coherent structures with strongest intermittence were surveyed at the scale $a=2^6$ obtained by maximum energy criterion, showing that the quasi-periodicity and intermittence for coherent structures in polymer solutions are not as distinct as that in the Newtonian fluid. To detect intermittency, the flatness factor and local intermittency measure for velocity fluctuation signals were calculated by wavelet coefficients. The results for flatness factor in one-dimensional wavelet transform and LIM in both one-dimensional and two-dimensional wavelet transform intuitively show that the local contribution to intermittency in polymer solutions flow is relatively smaller, leading to the suppression of intermittency in forced homogeneous isotropic turbulence with polymer additives. The robustness of wavelet transform method has been demonstrated in exploring the characteristics of turbulent structures and intermittency, which are of key importance in understanding the mechanism of turbulent drag reduction.

Introduction

In 1948, Toms discovered that adding a minute amount of flexible long-chain polymers to water flow can induce significant turbulent drag reduction (DR; named Toms’ effect thereafter). ¹ It is repeatedly obtained that there exists dramatic frictional DR in pipe or channel turbulent flows of polymer or surfactant solution.^2–7 The DR rate can reach to even more than 80% at some situations. ⁸ The dramatic turbulent DR has attracted many researchers to investigate the characteristics of turbulent drag-reducing flow and mechanism of DR since this technique is of great significance in decreasing energy consumption, protecting environment, and so on. However, in order to study the mechanisms of DR, there are yet robust theories or numerical simulation tools (at high Reynolds number) currently.

So far, there have been many numerical studies on the characteristics and mechanism of turbulent DR, most of which are direct numerical simulation (DNS), some of which are Reynolds-averaged Navier–Stokes (RANS) model, and very few of which are large eddy simulation (LES). Sureshkumar et al. ⁹ first used DNS to simulate turbulent drag-reducing flow with the finitely extensible nonlinear elastic in the Peterlin approximation (FENE-P) model based on pseudospectral method. Following that, more and more DNS works have been carried out to resolve the detailed characteristics of turbulent drag-reducing flows and investigate the mechanism of DR.^10–19 Due to the constraint of computer hardware, the performed DNSs had to be limited to most flows with moderate Reynolds number. Regarding RANS method, several researchers paid attention to the proposition of RANS models for viscoelastic fluid flow and the inquiry on the overall characteristics of polymer solution flows.^20–22 Cruz et al. ²² proposed a two-equation $k-\epsilon$ turbulence model for turbulent pipe flow of viscoelastic fluid, which can adequately predict turbulent kinetic energy. About LES, up to now, there have been many subgrid-scale (SGS) models for LES to simulate turbulent flows, such as the Smagorinsky model, ²³ dynamic Smagorinsky model, ²⁴ and mixed model. ²⁵ However, for viscoelastic fluid flow, reported LES studies are very few. The attempt by Thais et al. ²⁶ seems to be the only one. Thais et al. ²⁶ first verified the correctness of temporal approximate deconvolution model (TADM) and then presented the relationships between the Reynolds stress and temporal filter width, the ensemble-averaged mean values, and root-mean-square fluctuations of conformation tensor and investigated the influence of the filter width. The above researches on the characteristics of turbulent drag-reducing flows by DNS, RANS, and LES all proved that in turbulent drag-reducing flows of polymers or surfactant additives, the interaction between viscoelasticity and turbulence results in a modified turbulent kinetic energy cascading process, modified turbulence coherent structures (CSs), and turbulent DR in consequence. The CSs become comparatively regularized and the population density of CSs is reduced; the intermittency caused by the formation of CSs is inhibited in drag-reduced turbulent flows. Moreover, it has been generally accepted that CSs play an important role in momentum, energy, and mass transfers, as well as the maintaining and development of turbulence. ²⁷ Therefore, study on the characteristics of CSs and the intermittency of turbulence with polymer additives are of great importance in understanding turbulent DR mechanism.

The above-mentioned works on CSs and intermittency in polymer solution flows are all focused on mean features estimated with the partial derivatives of velocity fluctuation. However, the detailed features of CSs and intermittency should be explored at different scales, since turbulence can be seen as the superposition of vortex structures at different scales and the strongest vortex structure is the so-called CS which exists not only at large scale but also at small scale. To this end, the wavelet transform (WT) method has been utilized because WT can translate turbulent velocity fluctuation signals into local vortex structures and thus make the vortex structures at different regions and scales be easily and clearly observed. Farge and Rabreau ²⁸ first used WT to demonstrate the intermittency of small-scale vorticity filed and the relations between CSs and intermittency in homogeneous turbulent flows. After that, WT has been widely used in examining the characteristics of turbulence, such as turbulent fractal features, CSs, and the intermittency. Bacry et al. ²⁹ obtained negative local scaling index of turbulence by WT, reflecting the singularity of turbulence. For uniform grid turbulence and turbulent jet flow, Camussi and Guj ³⁰ found that there exists connection between small-scale signals with strong intermittency and CSs based on orthonormal WT. Farge et al. ³¹ proposed a numerical method for extracting CSs with wavelet decomposition of vorticity field. By inquiring fully developed turbulent channel flow data, Onorato et al. ³² put forward a turbulent signal decomposition technique by WT. All the above discussions about the applications of WT were for the characteristics of Newtonian fluid turbulent flows. For viscoelastic fluid turbulent flows, WT method is yet rarely utilized in analyzing the characteristics of modified CSs and the intermittency in drag-reduced turbulence. In Wang et al. ³³ and Wu et al., ³⁴ which are almost the only two reported examples, WT method was used to analyze the flow structures in turbulent drag–reducing channel flow based on DNS data.

In this article, wavelet analysis was performed for a database of velocity filed in forced homogeneous isotropic turbulence (FHIT) with drag-reducing polymer additives. The velocity field database of this drag-reduced FHIT was obtained by LES using our newly proposed Mixed SGS model based on Coherent structures and Temporal approximate deconvolution (MCT). ³⁵ The purpose of this study was to investigate the variation in CSs and intermittency influenced by polymer additives in turbulent drag-reducing flows at different scales. In this study, Daubechies wavelet ³⁶ with three-order vanishing moments (db3) was used for one-dimensional (1D) and two-dimensional (2D) WT since db3 has suitable scaling and wavelet function with a certain smoothness and relatively small computation cost. Then, the wavelet decomposition, reconstruction of velocity waveform, flatness factors, and local intermittence measure (LIM) are deeply analyzed for the drag-reduced FHIT with polymer additives in detail.

Numerical models and mathematical method

As mentioned previously, the reported numerical simulation studies on turbulent drag-reducing flows of viscoelastic fluid using LES were very few. We have then proposed a new SGS model particularly for LES of viscoelastic fluid flows named MCT, ³⁵ which has been successfully used to simulate FHIT with polymer additives and turbulent drag-reducing channel flow of surfactant solution. In this section, the LES procedures based on MCT are briefed at first. The WT procedures are then introduced in detail.

LES of FHIT with polymer additives based on MCT SGS model

The MCT SGS model is established through the following procedures. The continuity and modified momentum equations for LES based on spatial filtering in turbulent drag-reducing polymer solution flows are expressed as follows

$$\frac{\partial {\overline{u}}_i}{\partial x_i}=0$$

(1)

$$\begin{array}{l}\frac{\partial {\overline{u}}_i}{\partial t}+\frac{\partial {\overline{u}}_j{\overline{u}}_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial \overline{p}}{\partial x_i}+{\upsilon }^{[s]}\frac{{\partial }^2{\overline{u}}_i}{\partial x_j{x}_j}\\ +{\upsilon }^{[s]}\frac{1-\beta }{\beta }\frac{\partial {\overline{T}}_{ij}^{[p]}}{\partial x_j}-\frac{\partial {\tau }_{ij}^a}{\partial x_j}+{\upsilon }^{[s]}\frac{1-\beta }{\beta }\frac{\partial R_{ij}}{\partial x_j}+f_i\end{array}$$

(2)

where ${\overline{u}}_i(x_i,t)$ is the filtered velocity vector, $\overline{p}(x_i,t)$ is the filtered local pressure, $\rho$ is the solution density, $\beta ={\upsilon }^{[s]}/({\upsilon }^{[s]}+{\upsilon }^{[p]})$ is a dimensionless measure of dilute polymer solution concentration, and smaller $\beta$ corresponds to denser polymer solution; here, ${\upsilon }^{[s]}$ is the solvent viscosity and ${\upsilon }^{[p]}$ is the polymeric zero-shear-rate viscosity; ${\overline{T}}_{\mathrm{ij}}^{[p]}$ is the additional elastic stress tensor caused by polymer additives; ${\tau }_{\mathrm{ij}}^a$ is the traceless SGS stress tensor; $R_{\mathrm{ij}}$ is a subfilter term related to the nonlinear restoring force in the filtered conformation tensor transport equation based on TADM^26,35,37 (see the temporal filtering for the conformation equation as follows); and $f_i$ is the external force (here, Chen et al.’s method ³⁸ is adopted). The Dumbbell model ³⁹ is used to calculate ${\overline{T}}_{\mathrm{ij}}^{[p]}=(f(\overline{r}){\overline{C}}_{\mathrm{ij}}-1)/{\tau }_p$ , where ${\overline{C}}_{\mathrm{ij}}$ is the filtered polymer conformation tensor and $f(\overline{r})=({\ell }^2-3)/({\ell }^2-{\overline{r}}^2)$ is the FENE-P model; here, $\ell$ is the maximum chain extensibility, ${\overline{r}}^2=\mathrm{trace}({\overline{C}}_{\mathrm{ij}})$ is the trace operator of conformation tensor, and ${\tau }_p$ is the polymer relaxation time.

The concept of coherent-structure Smagorinsky model (CSM)^40–42 is adopted to calculate the SGS stress tensor

$${\tau }_{\mathrm{ij}}^a=-2C{\overline{\mathrm{\Delta }}}^2\left|\overline{S}\right|{\overline{S}}_{\mathrm{ij}}$$

(3)

where C is the model parameter, $\overline{\mathrm{\Delta }}$ is the filter width, $\overline{\mathrm{\Delta }}=({\overline{\mathrm{\Delta }}}_i{\overline{\mathrm{\Delta }}}_j{\overline{\mathrm{\Delta }}}_k{)}^{1/3}$ ; ${\overline{S}}_{\mathrm{ij}}=(\partial {\overline{u}}_j/\partial x_i+\partial {\overline{u}}_i/\partial x_j)/2$ is the filtered rate-of-strain tensor; $\left|\overline{S}\right|$ is the magnitude of ${\overline{S}}_{\mathrm{ij}}$ and $\left|\overline{S}\right|=\sqrt{2{\overline{S}}_{\mathrm{ij}}{\overline{S}}_{\mathrm{ij}}}$ . The model parameter C can be locally determined and is always positive which is defined as follows^35,40

$$C=C_{\mathrm{CSM}}{\left|F_{\mathrm{CS}}\right|}^{3/2}{F}_{\mathrm{\Omega }}$$

(4)

with

$$\begin{array}{c}{F}_{\mathrm{CS}}=\frac{Q}{E}\hfill \\ F_{\mathrm{\Omega }}=1-F_{\mathrm{CS}}\hfill \end{array}$$

(5)

$$\begin{array}{l}Q=\frac{1}{2}\left({\overline{W}}_{ij}{\overline{W}}_{ij}-{\overline{S}}_{ij}{\overline{S}}_{ij}\right)=-\frac{1}{2}\frac{\partial {\overline{u}}_j}{\partial x_i}\frac{\partial {\overline{u}}_i}{\partial x_j},\\ E=\frac{1}{2}\left({\overline{W}}_{ij}{\overline{W}}_{ij}+{\overline{S}}_{ij}{\overline{S}}_{ij}\right)=\frac{1}{2}{\left(\frac{\partial {\overline{u}}_j}{\partial x_i}\right)}^2\end{array}$$

(6)

where $C_{\mathrm{CSM}}(=1/22)$ is a fixed model constant and ${\overline{W}}_{\mathrm{ij}}=(\partial {\overline{u}}_j/\partial x_i-\partial {\overline{u}}_i/\partial x_j)/2$ is the filtered vorticity tensor. The upper and lower limits of $F_{\mathrm{CS}}$ and $F_{\mathrm{\Omega }}$ are specified in order to obtain permanently positive model parameter C and ensure steadiness in numerical simulation process ³⁹

$$-1\le F_{\mathrm{CS}}\le 1,0\le F_{\mathrm{\Omega }}\le 2$$

(7)

However, the spatial filtering method described above is not suitable for the conformation tensor transport equation. Therefore, the concept of temporal filtering used in TADM is adopted to filter the conformation tensor of polymers^35,37

$$\begin{array}{c}\frac{\partial {\overline{C}}_{\mathrm{ij}}}{\partial t}+{\overline{u}}_k\frac{\partial {\overline{C}}_{\mathrm{ij}}}{\partial x_k}={\overline{C}}_{\mathrm{ik}}\frac{\partial {\overline{u}}_j}{\partial x_k}+{\overline{C}}_{\mathrm{kj}}\frac{\partial {\overline{u}}_i}{\partial x_k}\hfill \\ +P_{\mathrm{ij}}+Q_{\mathrm{ij}}-R_{\mathrm{ij}}+{\chi }_c({\overline{\gamma }}_{\mathrm{ij}}-{\overline{C}}_{\mathrm{ij}})\hfill \\ -\frac{1}{{\tau }_p}[\alpha f\left(\overline{r}\right){\overline{C}}_{\mathrm{ik}}{\overline{C}}_{\mathrm{kj}}+(1-\alpha )\hfill \\ f\left(\overline{r}\right){\delta }_{\mathrm{ik}}{\overline{C}}_{\mathrm{kj}}-\alpha {\overline{C}}_{\mathrm{ik}}{\delta }_{\mathrm{kj}}-(1-\alpha ){\delta }_{\mathrm{ij}}]\hfill \end{array}$$

(8)

where $P_{\mathrm{ij}}$ and $Q_{\mathrm{ij}}$ are subfilter terms caused by stretching; $R_{\mathrm{ij}}$ is a subfilter term related to the nonlinear restoring force, which is the same as that in the filtered momentum equation; ${\chi }_c({\overline{\gamma }}_{\mathrm{ij}}-{\overline{C}}_{\mathrm{ij}})$ is a second-order regularization term considering the kinetic energy transfer between the scales that cannot be recovered by the deconvolution procedure; ${\chi }_c$ is the dissipative coefficient for conformation tensor transport equation ( ${\chi }_c=1$ in this study); $\alpha$ is the mobility factor ( $\alpha =0$ in FENE-P model); and ${\delta }_{\mathrm{ij}}$ is the Kronecker symbol.

The principle of TADM is to take advantage of values at current and earlier time steps to calculate the deconvolved values which approximate the unfiltered values. The deconvolved velocity $v_i$ and the deconvolution conformation tensors ${\varphi }_{\mathrm{ij}}$ and ${\gamma }_{\mathrm{ij}}$ are built as follows, respectively^26,35,37

$$\begin{array}{l}{v}_i={\displaystyle \sum _{m=0}^p{C}_m{\overline{u}}_i^{(m+1)}},{\varphi }_{ij}={\displaystyle \sum _{m=0}^p{C}_m{\overline{C}}_{ij}^{(m+1)}},\\ {\gamma }_{ij}={\displaystyle \sum _{m=0}^q{D}_m{\overline{C}}_{ij}^{(m+1)}}\end{array}$$

(9)

In this study, $p=3$ and $q=2$ are used. According to the binomial theorem, the optimal deconvolution coefficients for $C_m$ and $D_m$ are $[C_0,C_1,C_2,C_3]=[0,\sqrt{6},\sqrt{4+2\sqrt{6}}-2\sqrt{6},1-\sqrt{4+2\sqrt{6}}+\sqrt{6}]$ and $[D_0,D_1,D_2]=[15/8,-9/8,1/4]$ .

The subfilter terms $P_{\mathrm{ij}}$ , $Q_{\mathrm{ij}}$ , and $R_{\mathrm{ij}}$ are given in the literature^26,35,37

$$P_{\mathrm{ij}}=\overline{\frac{\partial v_i}{\partial x_k}{\varphi }_{\mathrm{kj}}}-\frac{\partial {\overline{v}}_i}{\partial x_k}{\overline{\varphi }}_{\mathrm{kj}},Q_{\mathrm{ij}}=\overline{\frac{\partial v_j}{\partial x_k}{\varphi }_{\mathrm{ki}}}-\frac{\partial {\overline{v}}_j}{\partial x_k}{\overline{\varphi }}_{\mathrm{ki}}$$

(10)

$$T_{\mathrm{ij}}=\frac{f(\varphi ){\varphi }_{\mathrm{ij}}-1}{{\tau }_p},R_{\mathrm{ij}}={\overline{T}}_{\mathrm{ij}}-\frac{f(\overline{\varphi }){\overline{\varphi }}_{\mathrm{ij}}-1}{{\tau }_p}$$

(11)

Then, filterings of the above variables are based on time-domain^26,35,37

$$\frac{\partial {\overline{u}}_i^{(m+1)}}{\partial t}=\frac{{\overline{u}}_i^{(m)}-{\overline{u}}_i^{(m+1)}}{{\mathrm{\Delta }}_u},1\le m\le max(p,q)$$

(12a)

$$\frac{\partial {\overline{C}}_{\mathrm{ij}}^{(m+1)}}{\partial t}=\frac{{\overline{C}}_{\mathrm{ij}}^{(m)}-{\overline{C}}_{\mathrm{ij}}^{(m+1)}}{{\mathrm{\Delta }}_c},1\le m\le max(p,q)$$

(12b)

$$\begin{array}{l}\frac{\partial {\overline{v}}_i}{\partial t}=\frac{v_i-{\overline{v}}_i}{{\mathrm{\Delta }}_u},\frac{\partial {\overline{\varphi }}_{ij}}{\partial t}=\frac{{\varphi }_{ij}-{\overline{\varphi }}_{ij}}{{\mathrm{\Delta }}_c},\\ \frac{\partial {\overline{\gamma }}_{ij}}{\partial t}=\frac{{\gamma }_{ij}-{\overline{\gamma }}_{ij}}{{\mathrm{\Delta }}_c},\frac{\partial {\overline{T}}_{ij}}{\partial t}=\frac{T_{ij}-{\overline{T}}_{ij}}{{\mathrm{\Delta }}_c}\end{array}$$

(12c)

$$\begin{array}{l}\frac{\partial }{\partial t}\left(\overline{\frac{\partial v_i}{\partial x_k}{\varphi }_{kj}}\right)=\frac{1}{{\mathrm{\Delta }}_c}\left(\frac{\partial v_i}{\partial x_k}{\varphi }_{kj}-\overline{\frac{\partial v_i}{\partial x_k}{\varphi }_{kj}}\right),\frac{\partial }{\partial t}\left(\overline{\frac{\partial v_j}{\partial x_k}{\varphi }_{ki}}\right)\\ =\frac{1}{{\mathrm{\Delta }}_c}\left(\frac{\partial v_j}{\partial x_k}{\varphi }_{ki}-\overline{\frac{\partial v_j}{\partial x_k}{\varphi }_{ki}}\right)\end{array}$$

(12d)

where ${\mathrm{\Delta }}_u$ and ${\mathrm{\Delta }}_c$ are the filter widths for velocity and conformation tensor transport equation, respectively. For more details about MCT method, one refers to Li et al. ³⁵

In this study, a periodic cubic ${\mathbb{R}}^3=(2\pi {)}^3$ with ${64}^3$ collocation points is used as simulation domain. Spectral method is adopted to solve the filtered continuity and momentum equations for LES. Finite difference method is used to solve the conformation tensor transport equation due to its hyperbolic characteristics. A second-order Kurganov–Tadmor scheme ⁴³ is adopted for discretization of the convective term. Meanwhile, for both the filtered momentum and conformation tensor transport equations, a second-order Adams–Bashforth scheme is adopted for time evolution, and a second-order Runge–Kutta scheme is used for filtering on time-domain. The Taylor micro-scale Reynolds number ${\mathrm{Re}}_{\lambda }$ and the Weissenberg number $\mathrm{Wi}$ based on the Newtonian fluid as a benchmark are defined as follows

$${\mathrm{Re}}_{\lambda }=\frac{\sqrt{20}{\xi }^{[N]}}{\sqrt{3{\upsilon }^{[s]}{\epsilon }^{[N]}}},\mathrm{Wi}={\tau }_p\sqrt{\frac{{\epsilon }^{[N]}}{{\upsilon }^{[s]}}}$$

(13)

where ${\xi }^{[N]}=〈{\overline{u}}^2〉/2=({\int }_0^{\infty }E(k)\mathrm{dk})/3$ and ${\epsilon }^{[N]}={\upsilon }^{[s]}{\int }_0^{\infty }{k}^{2}E(k)\mathrm{dk}$ ( $E(k)$ is the energy spectrum) are the turbulent kinetic energy and the turbulent energy dissipation rate in the Newtonian fluid flow, respectively. In addition, the initial turbulent kinetic energy spectrum is chosen as $E_0(k)=0.01k^4\exp (-0.14k^2)$ . In this study, the Newtonian fluid at ${\mathrm{Re}}_{\lambda }=202$ and polymer solution flows at $\mathrm{Wi}=0.950,\beta =0.7$ and ${\mathrm{Re}}_{\lambda }=202$ are discussed.

WT

WT is a multi-scale analysis method for fluctuating functions or signals by scale and translation. Compared with Fourier transform, WT is a kind of local transformation on both time domain and frequency domain, which can obtain more effective information. WT can be classified into continuous WT (CWT) and discrete WT (DWT). For CWT, the wavelet function ${\psi }_{a,\tau }(x)$ and wavelet coefficient are as follows

$${\psi }_{a,\tau }(x)=\frac{1}{\sqrt{\left|a\right|}}\psi \left(\frac{x-\tau }{a}\right),a,\tau \in R,s\ne 0$$

(14)

$$W_f(a,\tau )\le f,{\psi }_{a,\tau }\ge \underset{R}{\int }f(x){\psi }_{a,\tau }^{*}(x)\mathrm{dx}$$

(15)

where $f(x)$ is a signal, ${\psi }_{a,\tau }^{*}(x)$ is the conjugate function of ${\psi }_{a,\tau }(x)$ , and scale a and translation $\tau$ are both continuous variation in CWT. CWT has great relevance, causing that wavelet coefficient $W_f(a,\tau )$ of signal $f(x)$ is redundant. And in fact, sampling signal points are all discrete, especially for computer and experimental sampling. Therefore, it is necessary to disperse CWT into DWT for the scale and translation are discrete. When $a=a_0^j$ and $\tau =k{\tau }_0{a}_0^j(a>1,{\tau }_0\in R,j,k\in Z)$ for equation (14), the wavelet function for DWT can be built as

$${\psi }_{j,k}(x)=\frac{1}{\sqrt{a_0^j}}\psi \left(\frac{x-k{\tau }_0{a}_0^j}{a_0^j}\right),j,k\in Z$$

(16)

In order to make WT adapt the non-stationary of signal by changing the time and frequency resolutions, the scale and translation must be adjusted for DWT. If the scale is binary discrete and the translation is continuous, the binary wavelet as a kind of DWT is formed. In binary wavelet, $a_0=2$ and ${\tau }_0=1$ correspond to the scale $a=2^j$ and the translation $\tau =2^{j}k$ ( $k=0,1,\dots ,((N/2^j)-1)$ , where N is the length of the input signal) at each grid node, respectively. So, the expressions of wavelet function and wavelet coefficient for binary wavelet are as follows

$${\psi }_{j,k}(x)=\frac{1}{\sqrt{2^j}}\psi \left(\frac{x-k{2}^j}{2^j}\right)$$

(17)

$$W_f(a,\tau )\le f(x),{\psi }_{2^j}(x)\ge \frac{1}{2^j}\underset{-\infty }{\overset{\infty }{\int }}f(x){\psi }^{*}(2^{-j}x-k)\mathrm{dx}$$

(18)

From the above description, it can be known that the translation invariant for a signal on the time domain is not damaged in binary wavelet. And $2^{-j}$ can reflect the magnification for signal. In order to further inspect detailed information of the signal, j should be decreased; on the contrary, if it just needs to understand the variation trend of the signal with coarse resolution, bigger j can be adopted. This incarnates the microscope functionality for binary wavelet. ⁴⁴

For the 2D signal $f(x,y)$ , the wavelet function and wavelet coefficient for binary wavelet are given by

$${\psi }_{j,k}(x,y)=\frac{1}{\sqrt{2^j}}\psi (\frac{x-k{2}^j}{2^j},\frac{y-k{2}^j}{2^j})$$

(19)

$$\begin{array}{l}{W}_f(a,\tau )\le f(x,y),{\psi }_{2^j}(x,y)\ge \\ \frac{1}{2^j}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f(x,y){\psi }^{\ast }((2^{-j}x-k),(2^{-j}y-k))}dxdy\end{array}$$

(20)

In this article, db3 as a kind of binary wavelet in 1D and 2D WT is applied. For 1D WT, the time-varying velocity fields obtained by LES between $t=20\mathrm{s}$ (from which the simulated FHIT reaches its statistically steady state) and $t=120\mathrm{s}$ ( $\mathrm{\Delta }t=2.5\times {10}^{-3}$ , and thus, the number of sampling points is $4\times {10}^4$ ) are inspected. For 2D WT, the velocity fluctuation fields for statistically steady state in the $x-z$ plane at $y=\pi$ are used. It is noteworthy that although LES filters out a portion of the vortex structures under the filter width, it retains most vortex structures above the filter width, which are analyzed by 1D and 2D WT in this work.

Results and discussions

In WT, the most distinctive characteristic is the wavelet decompositions: the original 1D signal $f(t)$ and 2D signal $f(x,y)$ can be divided into average component and detailed components as follows

$$\begin{array}{cc}\multicolumn{1}{c}{f(t)\mathrm{or}f(x,y)}& =A_1+D_1=A_2+D_2+D_1\hfill \\ \multicolumn{1}{c}{}& =A_3+D_3+D_2+D_1\hfill \\ \multicolumn{1}{c}{}& =A_n+D_n+D_{n-1}+\cdots +D_2+D_1\hfill \\ \multicolumn{1}{c}{}& =A_n+\sum _{i=1}^n{D}_i\hfill \end{array}$$

(21)

where $A_n$ is the average component, representing low-frequency portion of the original signal and $D_n$ is the detailed component representing high-frequency portion of the original signal. The wavelet decompositions by equation (21) of the time-varying streamwise velocity fluctuation for the Newtonian fluid and polymer solution flows are given in Figures 1 and 2 (the signal was decomposed into 15 scales and only 12 scales are shown here), respectively. It shows that the larger the scale, the lower the pulsation frequency for both the Newtonian fluid and polymer solution flows. WT of a signal is equivalent to local cross-correlation analysis between the signal and wavelet function. From Figures 1 and 2, it can be clearly seen that the intermittent pulse phenomena exist for both the Newtonian fluid and polymer solution flows, and the interval of intermittent pulse is slightly longer in the latter. In order to clearly explain the above-mentioned phenomenon, the fast Fourier transforms (FFTs) for D₁–D₄ in the Newtonian fluid and polymer solution flows are given in Figure 3. It can be found that the amplitudes for D₁ and D₂ in polymer solution are obviously smaller and smoother than those in the Newtonian fluid, indirectly demonstrating that CSs in polymer solutions are more regular and polymers behave an inhibitive effect on the intermittency in viscoelastic fluid turbulent flow.

Figure 1.

Wavelet decompositions of the streamwise velocity fluctuation time series for the Newtonian fluid case.

Figure 2.

Wavelet decompositions of the streamwise velocity fluctuation time series for the polymer solution case.

Figure 3.

FFTs of the detailed components of the signal for (a) the Newtonian fluid case and (b) polymer solution case.

As we know, the isoline or isosurface of velocity fluctuations can extract vortex structure. For 2D WT, the detailed components in equation (21) can image the relatively small-scale CSs. ⁴⁵ The wavelet decompositions of velocity fluctuations in the $x-z$ plane for the Newtonian fluid and polymer solution flows are shown in Figure 4 (the signal is decomposed into eight scales and only four scales are exhibited here). It can be discovered that at the same scale, the quantity of relatively small-scale CSs is fewer in polymer solution flow, which is in accordance with the ever reported analysis on the characteristics of CSs in turbulent drag–reducing flows.^10–19,37 This also explains that the amount of CSs is inhibited by polymer additives, and the intermittency in viscoelastic fluid flow is likewise suppressed since vortex structures induce the intermittency in turbulence.

Figure 4.

Wavelet decompositions of the velocity fluctuations in $x-z$ plane for (a) the Newtonian fluid case and (b) polymer solution case.

As is known, WT is an effective tool to detect CSs that have strongest energy locally and play essential role in turbulence. One feature of WT is the energy conservation, as expressed in the following

$$\underset{-\infty }{\overset{+\infty }{\int }}{|f(x)|}^2\mathrm{dx}=\frac{1}{C_{\psi }}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{|W_f(a,\tau )|}^2\frac{\mathrm{dad}\tau }{a^2}$$

(22)

According to the content of energy density, the following equation satisfies

$$\underset{-\infty }{\overset{+\infty }{\int }}{|f(x)|}^2\mathrm{dx}=\underset{-\infty }{\overset{+\infty }{\int }}E(a)\frac{\mathrm{da}}{a}$$

(23)

Coupling equations (22) and (23), the wavelet energy for WT at each scale is obtained as

$$E(a)=\frac{1}{C_{\psi }}\underset{-\infty }{\overset{+\infty }{\int }}{|W_f(a,\tau )|}^2\frac{d\tau }{a}$$

(24)

In order to extract CSs, the maximum wavelet energy must be determined. In this study, the maximum energy criterion proposed by Liandrat and Moret-Bailly ⁴⁶ was adopted and its expression is as follows

$$\frac{E(a_m)}{a_m^2}=\underset{a>0}{max}\frac{E(a)}{a^2}$$

(25)

where $a_m$ is the individual scale corresponding to each of CSs in turbulence. The distribution of $E(a)/a^2$ along with scale a is plotted for the Newtonian fluid and polymer solution flows in Figure 5. It shows that the maximum $E(a)/a^2$ for the Newtonian fluid is about 16% larger than that for polymer solution flow, stating that CSs in viscoelastic fluid flow have been inhibited (drag-reducing effect) on account of the fact that the biggest $E(a)/a^2$ represents CSs in turbulence. Figure 5 also displays that the energy fluctuations first increase and then decrease from $a=2^6$ for both the Newtonian fluid and polymer solution cases, meaning that there exist the largest energy fluctuations at scale $a=2^6$ . So, velocity waveform reconstruction for CSs can be realized at scale $a=2^6$ , as shown in Figure 6. It can be observed that CSs in the Newtonian fluid and polymer solution flows both possess quasi-periodicity and intermittent pulse. However, quasi-periodicity and intermittent pulse in viscoelastic fluid flow is less evident as compared to that in the Newtonian fluid flow. To clarify the aforementioned phenomenon, the FFTs for velocity wavelet reconstruction in the Newtonian fluid and polymer solution flows are given in Figure 7. It can be seen that the number of obvious amplitudes in polymer solution is less than that in the Newtonian fluid, indicating that the intermittency in FHIT with polymer additives has been inhibited.

Figure 5.

The distribution of the energy fluctuations with scale: (a) the Newtonian fluid case and (b) the polymer solution case.

Figure 6.

Velocity waveform reconstruction for CS: (a) the Newtonian fluid case and (b) the polymer solution case.

Figure 7.

FFTs of velocity wavelet reconstruction for (a) the Newtonian fluid case and (b) polymer solution case.

Turbulence is a nonlinear phenomenon and possesses CSs leading to intermittency in the space and time domains. The detections of vortex structures by wavelet decomposition can indirectly represent the intermittency characteristics. A typical quantity, the flatness factor, characterizing intermittency in both the Newtonian fluid and viscoelastic fluid flows is quantitatively analyzed with WT. The definition of the flatness factor in DWT is as follows ⁴⁷

$$\mathrm{FF}(a)=\frac{<{|W_f(a,\tau )|}^4{>}_t}{<{|W_f(a,\tau )|}^2{>}_t^2}$$

(26)

where $<\cdots {>}_t$ is the time average between $t=20s$ and $t=120s$ . The flatness factors of the streamwise velocity fluctuation for the Newtonian fluid and polymer solution cases are given in Figure 8. The results demonstrate that the intermittency exists at all scales, and the flatness factor gradually decreases with the scale a. As is known, the flatness factor is equal to 3 (the dashed line plotted in Figure 8) for a Gaussian distribution at each scale. Figure 8 shows that flatness factors deviate from 3 for all the scales, and the deviation increases with the decrease in scale for $a<2^{10}$ , implying that there exists stronger intermittency at smaller scales or small scales contribute much more to the intermittency in velocity fluctuation field. We can also discover that the flatness factor for the Newtonian fluid flow is nearly double that for polymer solution flow, and the maximum deviation is 846.50 for the Newtonian fluid and 449.30 for polymer solution, indicating that the extent of intermittency is decreased by 46.92% by polymer additives and an inhibition of the intermittency in turbulence by polymer additives has occurred.

Figure 8.

Flatness factor for fluctuation velocity in streamwise direction.

Generally speaking, intermittency is caused by CSs in turbulence. In order to quantitatively detect the contribution of local CSs to the intermittency in turbulence, Farge ⁴⁸ defined an LIM which can reflect the ratio of local wavelet energy spectrum to the whole-wavelet energy spectrum as follows

$$\mathrm{LIM}(a,b)=\frac{{|W_f(a,\tau )|}^2}{<{|W_f(a,\tau )|}^2{>}_t}$$

(27)

It is defined that the locations at which $\mathrm{LIM}~O(1)$ have no obvious contribution to the intermittency in turbulence. In contrast, if $\mathrm{LIM}>>1$ , contributions from the corresponding locations to the intermittency are significant. The estimated LIMs of the Newtonian fluid and polymer solution cases at scales $2^7$ and $2^{11}$ for 1D WT are shown in Figures 9 and 10, respectively. Comparing the maximum values of LIM in 1D WT, it can be seen that the maximums of LIMs in the Newtonian fluid case are 55.02 at a = 2⁷ and 14.38 at a = 2¹¹ (the small figure in Figure 9(a) is the original results), while they are 34.60 at a = 2⁷ and 11.59 at a = 2¹¹ for polymer solution, explaining that the contributions of the local CSs to the intermittency are much larger in the Newtonian fluid case for the streamwise velocity fluctuation time series and the extents are decreased by 37.11% at a = 2⁷ and 19.40% at a = 2¹¹. From the information of pulse rate shown in Figures 9 and 10, it can be perceived that the interval of strong pulse period for polymer solution flows is longer. For further explanations, the FFTs for LIM in the Newtonian fluid and polymer solution flows are given in Figure 11. It can be discovered that compared with the Newtonian fluid, the number of large amplitudes is obviously smaller and the whole tendency is quite smoother in polymer solution. The estimated LIMs of the Newtonian fluid and polymer solution cases at scales $2^2$ and $2^5$ for 2D WT are shown in Figures 12 and 13, respectively. It shows that the region with relatively large LIM value meaning the existence of prominent contribution to the intermittency is smaller in polymer solution case as compared with LIM in the Newtonian fluid case at scale $2^2$ . Then, the results of probability density functions (PDF) for the Newtonian fluid and polymer solution flows are shown in Figure 14. It can be clearly seen that the number of LIM with relatively large values in the Newtonian fluid is larger than that in polymer solution. The above consequences also illustrate that the intermittency in turbulent drag-reducing flow is inhibited.

Figure 9.

LIM by 1D WT for the Newtonian fluid at scale (a) $a=2^7$ and (b) $a=2^{11}$ .

Figure 10.

LIM by 1D WT for the polymer solution flow at scale (a) $a=2^7$ and (b) $a=2^{11}$ .

Figure 11.

FFTs of LIM for (a) the Newtonian fluid case and (b) polymer solution case at scale $a=2^7$ .

Figure 12.

LIM by 2D WT for the Newtonian fluid at scale (a) $a=2^2$ and (b) $a=2^5$ .

Figure 13.

LIM by 2D WT for the polymer solution flow at scale (a) $a=2^2$ and (b) $a=2^5$ .

Figure 14.

PDFs of LIM for the Newtonian fluid and polymer solution at scale $a=2^2$ .

Conclusion

The drag-reducing effect on the CSs and intermittency in FHIT with polymer additives has been investigated by means of 1D and 2D WT analyses of LES database. Through discussing the wavelet decompositions for the Newtonian fluid and polymer solution flows in 1D and 2D WT, it is found that the intermittent pulse and relatively small-scale vortex structures exist in both the Newtonian fluid and polymer solution flows. Nevertheless, the intermittency and the quantity of CSs in polymer solution flows are both obviously inhibited. Through the detection of CSs in turbulence by maximum energy criterion, the velocity waveforms for CSs are reconstructed at scale $2^6$ which possesses the largest energy for 1D WT. It is observed that the intermittent pulse and quasi-periodicity are less obvious in viscoelastic fluid flow compared with those in the Newtonian fluid flow. The flatness factors show that there exists stronger intermittency at small scales and small scales contribute much more to the intermittency in the streamwise fluctuating velocity component. The drag-reducing polymer additives show an inhibitive effect on the intermittency in polymer solutions case, and the extent of intermittency is decreased by 46.92% by polymer additives. Finally, it is also known from LIM in 1D and 2D WT that the local contribution to the intermittency and the local region where prominent contribution to the intermittency occurs are relatively smaller for polymer solution case than that for the Newtonian fluid case. All the above analyses have demonstrated that the CSs and the intermittency in FHIT with polymer additives are inhibited. However, this study manifested that WT is a robust method for analyses of the characteristics of turbulence structures and intermittency, which is of key importance in understanding the mechanism of turbulent DR of viscoelastic fluid flows.