Renormalization Group Approach as a Symmetry Transformation for an Analysis of Non-Newtonian Elastic Turbulence

Abstract

Symmetry transformation methods are widely used in fluid flow problems. One such method is renormalization group analysis. Renormalization group methods are used to develop a macroscopic turbulence model for non-Newtonian fluids (Oldroyd-B type). This model accounts for the large-distance and large-time behavior of velocity correlations generated by the momentum equation for a randomly stirred, incompressible flow and does not account for empirical constants. The aim of this mathematical study was to develop a k-ε RNG turbulence model for non-Newtonian fluids (Oldroyd-B type). For the first time, using the renormalization procedure, the transport equations for the large-scale modes and expressions for effective transport coefficients are obtained. Expressions for the renormalized turbulent viscosity are also derived. This model explains the phenomenon of the abrupt growth of the irregularity of velocity at low values of the Reynolds number.

1. Introduction

A moving fluid whose viscosity does not depend on its nature and temperature but depends on the velocity gradient is usually called a non-Newtonian (elastic) fluid. Flows of this type are widespread in oil, chemical, food, and many other fields. Taking into account various hydrodynamic effects and their prediction during the development of these types of flows is an important factor in the design and subsequent operation of equipment for various applications.

The majority of research works in recent decades were devoted to studies of elastic flow motion modes and factors that influence changes in these modes, which, in general, is of a fundamental nature.

When calculating the dynamics of flows of various types and processes, one mainly encounters the calculation of the turbulent flow field. To calculate such flows, it is mainly necessary to solve the equations of motion (Navier–Stokes) [1]. The solution of these equations is based on the use of numerous numerical methodologies [2,3,4,5].

Different empirical and theoretical prediction models are used to simulate turbulence processes. The order of the models is determined by the number of additional differential equations that close the Navier–Stokes system. In this case, depending on the type of flow, the numerical values of the model constants can vary, and additional terms can be added. The simplest empirical models of turbulence are those based on the ideas of von Kármán, Taylor, and Prandtl, which allow for predicting the shear stresses of the shear layers. These methodologies are characterized by the concept of the turbulent transfer coefficient and the mixing length.

The authors of work [6] investigated the transition from laminar to turbulent flows of a non-Newtonian fluid based on the results of numerical simulations. A comparison of the results of numerical modeling and experimental studies showed that the Slatter model based on the Reynolds number and the Hanks model based on the stability parameter provided data closest to the experimental data for non-Newtonian fluids with a yield point and a low behavior index.

Wang et al. [7] presented the results of modeling the transition from laminar to turbulent flows of non-Newtonian fluid using several computational fluid dynamics (CFD) models. The authors found that the SST k–ω turbulence model (shear stress transport) and the RSM (Reynold stress model) yielded the closest results to the experimental data when predicting the torque dynamics of rotating non-Newtonian fluid during the mixing process. The SST model was more effective at predicting the hydrodynamic parameters of the flow of elastic fluids with higher concentrations, especially in the near-wall region. The RSM performed better when simulating mixing processes in solutions with lower component concentrations and at low vessel rotation speeds.

A mathematical model of bubble growth at low Mach numbers in a generalized Newtonian fluid was proposed for the case of using a magnetic field and shear stress in [8]. Based on the proposed mathematical model, it is possible to track the dynamics of bubble growth at different speeds of motion of the generalized Newtonian fluid.

In study [9], the instability of a concentrated polymer liquid was considered using the Maxwell model. The instability of short waves in a viscoelastic liquid flow was investigated. It was shown that the short waves of instability are of the second order of smallness in extremely viscous liquids. It was proven that this type of instability appears in systems with a low surface tension, such as polymer solutions.

The authors of work [10] studied the surface linear instability between the layers of elastic fluid in a channel at high values of the Weisenberg number (Wi) with insignificant inertia. Oldroyd-B or UCM (Upper Convected Maxwell) types of fluid have high viscosity values. In UCM fluids, elastic instability was exclusively revealed for short waves in the absence of surface tension. Pulsations in such a flow decayed according to the exponential law in the direction away from the surface. It was shown that this kind of instability at large Wi numbers persists throughout the channel.

In works [11,12], the Oldroyd-B instability of the Poiseuille fluid of three-layer [11] and multilayer [12] flows were also considered. In study [13], an analytical description of the elastic flow in channels of rectangular and cylindrical configurations is presented. A modification of the solution of the von Kármán equation for a plate is presented and the derivation of nonlinear equations for a thin cylindrical shell is extended. In work [14], the elastic instability of the Taylor–Couette flow is studied. Experiments on the instability of the Couette flow between two cylinders with a highly viscous polymer solution were conducted in the study of Larson et al. [15]. These experiments show that the instability of such a flow is determined by the Deborah number and the polymer concentration, while the effect of the Reynolds number becomes insignificant. The mechanism for the description of this kind of instability was also proposed in [15] based on the Oldroyd-B rheological model.

The film thickness of coating flows of non-Newtonian fluids at low Reynolds numbers were investigated theoretically and experimentally [16]. The analytical method is based on the Criminale–Ericksen–Filbey (CEF) constitutive equation. The CEF method provides a more accurate description of the physical processes, which allows for determining the film thickness value. As said in [16], experimental verification of the analytical data on the film thickness of a non-Newtonian fluid in a single-roller geometry generally shows good qualitative agreement, with a small deviation at low rotation speeds.

The authors of work [17] considered thermo-elastic dynamic instability (TEDI) in sliding fractions of the flow. First, thermoelastic and dynamic instability were considered separately using one-dimensional models, and afterward, their combined influence was studied based on the so-called TEDI model.

Groisman and Steinberg [18] discovered experimentally that in viscoelastic fluids, such as solutions of flexible long-chain polymer compounds, turbulence arises at small Reynolds numbers. Groisman and Steinberg showed that such turbulence is an elastic turbulence. Obviously, this phenomenon is linked to the non-Newtonian (nonlinear) character of a fluid structure. This kind of fluid can be characterized by the relaxation parameters λ and γ, which describe the relaxation times for the shear stress and rate of distortion, respectively.

Unlike empirical models, the RNG (renormalization group) model obtained on the basis of the renormalization analysis of turbulence enables obtaining the theoretical constants of turbulence models and unifying the models themselves for different flows [19,20]. The RNG model is effective for low and high Reynolds numbers (laminar, transitional, and fully turbulent flows). The RNG theory was first developed for quantum field research as a standard method for removing singularities [21]. Afterward, this theory was applied to the investigation of critical phenomena by Wilson [22].

Foster et al. [20] applied the RNG technique for power-law forcing in cases where the small scales act at large scales, like eddy viscosity. Utilizing the RNG theory, Yakhot and Orszag [23] developed a closed model of turbulence. Following Foster et al. [20], they developed a theory for large scales in which the effect of small scales was represented by a modified transport coefficient. The equations for the large scales were obtained by using averaging over an infinitesimal band of small scales. After this, the small scales were removed from consideration. This removal process was iterated and it yielded a differential equation for effective transport coefficients. The large-scale equation after the removal of small scales retained its form. Using the RNG theory, Yakhot and Orszag [23] obtained expressions for the turbulent viscosity, the turbulent Prandtl number, the Batchelor constant, the equations for a passive scalar, the turbulence kinetic energy, and the dissipation rate. The RNG technique is used for the investigation of turbulent flow for special cases. Avramenko and Kuznetsov [24,25] created a renormalization group model of macroscopic turbulence in porous media. The RNG model was also developed for flows with nanoparticles [26] and microflows [27].

The objective of this study was to utilize the RNG theory that was earlier successfully applied to obtain turbulence models for Newtonian fluids to further develop a macroscopic turbulence model for non-Newtonian (elastic) fluids. A renormalization group is defined as a set of symmetry transformations operating on a space of parameters.

Based on the above, this article includes a detailed mathematical description of the renormalization group transformation of momentum and heat transfer equations in order to obtain renormalized coefficients of kinematic viscosity and thermal conductivity. The renormalization procedure consists of several mathematical stages, namely, in the first stage, the velocity, temperature, and force fields are divided into fast and slow modes. Then, fast modes are excluded from the expressions for slow modes. As a result, we obtain differential expressions for the kinematic viscosity and for the thermal conductivity coefficient.

2. Basic Equations in Fourier Space

RNG analysis is used for a renormalization of the Navier–Stokes equations with the purpose of “pumping-over” short-wavelength modes with an efficient transport coefficient, which is turbulent viscosity. The Navier–Stokes equation in divergent form looks like [23]

$${\displaystyle \frac{\partial u_n}{\partial t}}+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial p}{\partial x_n}}+{\displaystyle \frac{\partial u_n{u}_m}{\partial x_m}}-f={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\tau }_{nm}}{\partial x_m}},$$

(1)

where f is an external random force, which is assumed to be Gaussian, isotropic in space, and homogeneous white noise in time; p is the pressure; t is the time; u_n are the projections of the velocity vector on the orthogonal coordinate axes x_n; ρ is the fluid density; and ${\tau }_{nm}$ is the shear stress.

The analysis is carried out in the Fourier space. The Fourier decomposition of the velocity, pressure, random force, and shear stress is performed as follows [23]:

$$\begin{gathered} u_n={\displaystyle \frac{1}{{\left(2\pi \right)}^{d+1}}}{\displaystyle {\int }_{\kappa \le {\kappa }_c}{d}^d\kappa {\displaystyle \int d}\omega {\mathrm{U}}_n\left(\overrightarrow{\mathsf{\kappa }},\omega \right)\exp \left(i\overrightarrow{\mathsf{\kappa }}\cdot \overrightarrow{\mathrm{x}}-i\omega t\right)}, \\ p={\displaystyle \frac{1}{{\left(2\pi \right)}^{d+1}}}{\displaystyle {\int }_{\kappa \le {\kappa }_c}{d}^d\kappa {\displaystyle \int d}\omega \mathrm{P}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)\exp \left(i\overrightarrow{\mathsf{\kappa }}\cdot \overrightarrow{\mathrm{x}}-i\omega t\right)}, \\ {u}_n{u}_m={\displaystyle \frac{1}{{\left(2\pi \right)}^{d+1}}}{\displaystyle {\int }_{\kappa \le {\kappa }_c}{d}^d\kappa {\displaystyle \int d}\omega {\mathrm{W}}_{nm}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)\exp \left(i\overrightarrow{\mathsf{\kappa }}\cdot \overrightarrow{\mathrm{x}}-i\omega t\right)}, \\ f={\displaystyle \frac{1}{{\left(2\pi \right)}^{d+1}}}{\displaystyle {\int }_{\kappa \le {\kappa }_c}{d}^d\kappa {\displaystyle \int d}\omega \mathrm{F}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)\exp \left(i\overrightarrow{\mathsf{\kappa }}\cdot \overrightarrow{\mathrm{x}}-i\omega t\right)}, \\ {\tau }_{nm}={\displaystyle \frac{1}{{\left(2\pi \right)}^{d+1}}}{\displaystyle {\int }_{\kappa \le {\kappa }_c}{d}^d\kappa {\displaystyle \int d}\omega {\mathsf{{\rm T}}}_{nm}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)\exp \left(i\overrightarrow{\mathsf{\kappa }}\cdot \overrightarrow{\mathrm{x}}-i\omega t\right)}, \end{gathered}$$

(2)

where ${\mathrm{U}}_n$, P, W_nm, F, and T_nm are the Fourier transforms of $u_n$, p, $u_n{u}_m$, f, and τ_nm, respectively; κ is a wave number (a component of the d-dimensional wave vector κ); $\omega$ is the frequency; x the d-dimensional position coordinate vector; i is the imaginary unit; and κ_c is the ultraviolet cutoff (it is assumed that velocity modes vanish when κ > κ_c [23]).

Substituting Equation (2) into Equation (1) gives

$$-i\omega {\mathrm{U}}_n=\mathrm{F}-i{\kappa }_n{\displaystyle \frac{\mathrm{P}}{\rho }}-i{\kappa }_m{\mathrm{W}}_{nm}+i{\kappa }_m{\displaystyle \frac{{\mathsf{{\rm T}}}_{nm}}{\rho }}.$$

(3)

The non-Newtonian model of the fluid has the following form [28]:

$${\displaystyle \frac{\lambda }{\rho }}{\displaystyle \frac{\partial {\tau }_{nm}}{\partial t}}+{\displaystyle \frac{1}{\rho }}{\tau }_{nm}={\nu }_0\left({\displaystyle \frac{\partial u_n}{\partial x_m}}+\gamma {\displaystyle \frac{\partial }{\partial t}}{\displaystyle \frac{\partial u_n}{\partial x_m}}\right),$$

(4)

where λ and γ are the relaxation times for the shear stress and rate of distortion, respectively; ν₀ is the fluid kinematic viscosity (the subscript “0” means that the renormalization process begins from this value).

The Fourier decomposition of Equation (4) is

$${\displaystyle \frac{-i\omega \lambda }{\rho }}{\mathsf{{\rm T}}}_{nm}+{\mathsf{{\rm T}}}_{nm}={\nu }_0\left(i{\kappa }_m{\mathrm{U}}_n-ii{\kappa }_m\omega \gamma {\mathrm{U}}_n\right).$$

(5)

From here, one can obtain

$${\displaystyle \frac{1}{\rho }}{\mathsf{{\rm T}}}_{nm}={\nu }_{0}i{\kappa }_m{\mathrm{U}}_n{\displaystyle \frac{1-i\omega \gamma }{1-i\omega \lambda }}$$

(6)

The substitution of Equation (6) into Equation (3) gives

$$G_0^{-1}{\mathrm{U}}_n=\mathrm{F}-i{\kappa }_n{\displaystyle \frac{\mathrm{P}}{\rho }}-i{\kappa }_m{\mathrm{W}}_{nm},$$

(7)

where

$$G_0={\left(-i\omega +{\nu }_0{\kappa }^2{\displaystyle \frac{1-i\omega \gamma }{1-i\omega \lambda }}\right)}^{-1}$$

(8)

is the propagator of order zero and

$${\kappa }^2={\displaystyle \sum _{j=1}^d{\kappa }_j^2=}{\kappa }_1^2+{\kappa }_2^2+\mathrm{\dots }+{\kappa }_d^2.$$

(9)

After eliminating the Fourier transforms P and W_nm from Equation (7), we have

$$G_0^{-1}\left(\kappa \right){\mathrm{U}}_n\left(\overrightarrow{\mathsf{\kappa }},\omega \right)=\mathrm{F}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)+{\lambda }_0{M}_{nml}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle {\int }_{\kappa \le {\kappa }_c}\frac{d^d\sigma }{{\left(2\pi \right)}^d}{\displaystyle \int \frac{d\varpi }{2\pi }{\mathrm{U}}_m\left(\overrightarrow{\mathsf{\sigma }},\omega \right){\mathrm{U}}_l\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }},\omega -\varpi \right)}},$$

(10)

where λ₀ = 1 is introduced for the perturbation analysis:

$$M_{nml}={\displaystyle \frac{1}{2i}}\left({\kappa }_m{M}_{nl}+{\kappa }_l{M}_{nm}\right)\mathrm{and}{M}_{nl}={\delta }_{nl}-{\displaystyle \frac{{\kappa }_n{\kappa }_l}{{\kappa }^2}}.$$

(11)

The random force f is assumed to be Gaussian and white noise in time. It is specified by the following Fourier transform of its two-point correlation function [18]:

$$〈{\mathrm{F}}_n\left(\kappa ,\omega \right){\mathrm{F}}_m\left({\kappa }^{\prime },{\omega }^{\prime }\right)〉={\displaystyle \frac{2{\left(2\pi \right)}^{d+1}}{{\kappa }^{d-4+\mathsf{\epsilon }*}}}{D}_0{M}_{nm}\left(\kappa \right)\mathsf{\delta }\left(\kappa +{\kappa }^{\prime }\right)\mathsf{\delta }\left(\omega +{\omega }^{\prime }\right).$$

(12)

In the case of ε* = 4, the constant D₀ has the dimension of the energy dissipation rate, which is the only significant parameter that determines the universal turbulence regime. We can say that ε* = 4 corresponds to the “real theory”.

3. Renormalization Procedure and Calculation of the Turbulent Viscosity

To proceed with the renormalization procedure [20], the velocity and force fields are separated into long-wavelength and short-wavelength modes so that

$$\mathrm{U}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)=\left\{\begin{array}{l}{\mathrm{U}}^{<}\left(\kappa ,\omega \right)0<\kappa <{\kappa }_c\exp \left(-\tau \right)\\ {\mathrm{U}}^{>}\left(\kappa ,\omega \right){\kappa }_c\exp \left(-\tau \right)<\kappa <{\kappa }_c\end{array}\right.$$

(13)

$$\mathrm{F}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)=\left\{\begin{array}{l}{\mathrm{F}}^{<}\left(\kappa ,\omega \right)0<\kappa <{\kappa }_c\exp \left(-\tau \right)\\ {\mathrm{F}}^{>}\left(\kappa ,\omega \right){\kappa }_c\exp \left(-\tau \right)<\kappa <{\kappa }_c\end{array}\right.$$

(14)

$$\mathrm{T}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)=\left\{\begin{array}{l}{\mathrm{T}}^{<}\left(\kappa ,\omega \right)0<\kappa <{\kappa }_c\exp \left(-\tau \right)\\ {\mathrm{T}}^{>}\left(\kappa ,\omega \right){\kappa }_c\exp \left(-\tau \right)<\kappa <{\kappa }_c\end{array}\right.$$

(15)

where τ is a positive parameter; ${\mathrm{U}}^{<}$, ${\mathrm{T}}^{<}$, and ${\mathrm{F}}^{<}$ denote the long-wavelength modes; and ${\mathrm{U}}^{>}$, ${\mathrm{T}}^{>}$, and ${\mathrm{F}}^{>}$ denote the short-wavelength modes.

In accordance with the above, we substitute (13) and (14) into (10). As a result, we obtain

$$\begin{array}{l}{G}_0^{-1}\left(\kappa \right){\mathrm{U}}_n^{<}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)={\mathrm{F}}^{<}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)+\\ +{\lambda }_0{M}_{nml}^{<}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle {\int }_{\kappa \le {\kappa }_c}\frac{d^d\sigma }{{\left(2\pi \right)}^d}{\displaystyle \int \frac{d\varpi }{2\pi }\left[{\mathrm{U}}_m^{<}\left(\overrightarrow{\mathsf{\sigma }},\omega \right){\mathrm{U}}_l^{<}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }},\omega -\varpi \right)+\right.}}\\ \left.+{2\mathrm{U}}_m^{<}\left(\overrightarrow{\mathsf{\sigma }},\omega \right){\mathrm{U}}_l^{>}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }},\omega -\varpi \right)+{\mathrm{U}}_m^{>}\left(\overrightarrow{\mathsf{\sigma }},\omega \right){\mathrm{U}}_l^{>}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }},\omega -\varpi \right)\right],\end{array}$$

(16)

$$\begin{array}{l}{G}_0^{-1}\left(\kappa \right){\mathrm{U}}_n^{>}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)={\mathrm{F}}^{>}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)+\\ +{\lambda }_0{M}_{nml}^{>}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle {\int }_{\kappa \le {\kappa }_c}\frac{d^d\sigma }{{\left(2\pi \right)}^d}{\displaystyle \int \frac{d\varpi }{2\pi }\left[{\mathrm{U}}_m^{<}\left(\overrightarrow{\mathsf{\sigma }},\omega \right){\mathrm{U}}_l^{<}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }},\omega -\varpi \right)+\right.}}\\ \left.+{2\mathrm{U}}_m^{<}\left(\overrightarrow{\mathsf{\sigma }},\omega \right){\mathrm{U}}_l^{>}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }},\omega -\varpi \right)+{\mathrm{U}}_m^{>}\left(\overrightarrow{\mathsf{\sigma }},\omega \right){\mathrm{U}}_l^{>}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }},\omega -\varpi \right)\right]\end{array}$$

(17)

Furthermore, it is necessary to eliminate the short-wavelength modes, which for this purpose are expanded as a perturbation series in the ordering parameter ${\lambda }_0$:

$${\mathrm{U}}^{>}\left(\tilde{\kappa }\right)={\displaystyle \sum _{s=0}^{\infty }{\lambda }_0^s}{\mathrm{U}}_s^{>}\left(\tilde{\kappa }\right)$$

(18)

where $\tilde{\kappa }=\left(\kappa ,\omega \right)$.

We substitute (18) into both sides of (16) and equate the terms at the same powers. This gives

–

s = 0:

$${\mathrm{U}}_0^{>}\left(\tilde{\kappa }\right)=G_0\left(\tilde{\kappa }\right){\mathrm{F}}^{>}\left(\tilde{\kappa }\right),$$

(19)

–

s = 1:

$$\begin{array}{l}{\mathrm{U}}_1^{>}\left(\tilde{\kappa }\right)=G_0\left(\tilde{\kappa }\right)M^{>}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle \int \frac{d^{d+1}\tilde{\sigma }}{{\left(2\pi \right)}^{d+1}}\left[{\mathrm{U}}^{<}\left(\tilde{\sigma }\right){\mathrm{U}}^{<}\left(\tilde{\kappa }-\tilde{\sigma }\right)+\right.}\\ \left.+{2\mathrm{U}}^{<}\left(\tilde{\sigma }\right){\mathrm{U}}_0^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)+{\mathrm{U}}_0^{>}\left(\tilde{\sigma }\right){\mathrm{U}}_0^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)\right],\end{array}$$

(20)

–

s = 2:

$$\begin{array}{l}{\mathrm{U}}_2^{>}\left(\tilde{\kappa }\right)=G_0\left(\tilde{\kappa }\right)M^{>}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle \int \frac{d^{d+1}\tilde{\sigma }}{{\left(2\pi \right)}^{d+1}}\left[{2\mathrm{U}}^{<}\left(\tilde{\sigma }\right){\mathrm{U}}_1^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)+\right.}\\ \left.+2{\mathrm{U}}_0^{>}\left(\tilde{\sigma }\right){\mathrm{U}}_1^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)\right]\end{array}$$

(21)

etc.

The next step is to eliminate the fast modes from Equation (16) for the slow modes by substituting the series (18) into (16), which gives

$$\begin{array}{l}{G}_0^{-1}\left(\tilde{\kappa }\right){\mathrm{U}}^{<}\left(\tilde{\kappa }\right)={\mathrm{F}}^{<}\left(\tilde{\kappa }\right)+\\ +{\lambda }_0{M}^{<}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle \int \frac{d^{d+1}\tilde{\sigma }}{{\left(2\pi \right)}^{d+1}}\left[{\mathrm{U}}^{<}\left(\tilde{\sigma }\right){\mathrm{U}}^{<}\left(\tilde{\kappa }-\tilde{\sigma }\right)+{2\mathrm{U}}^{<}\left(\tilde{\sigma }\right){\mathrm{U}}_0^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)+\right.}\\ \left.+{\mathrm{U}}_0^{>}\left(\tilde{\sigma }\right){\mathrm{U}}_0^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)\right]+2{\lambda }_0^2{M}^{<}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle \int \frac{d^{d+1}\tilde{\sigma }}{{\left(2\pi \right)}^{d+1}}\left[{\mathrm{U}}^{<}\left(\tilde{\sigma }\right){\mathrm{U}}_1^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)+\right.}\\ \left.+{\mathrm{U}}_0^{>}\left(\tilde{\sigma }\right){\mathrm{U}}_1^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)\right]+O\left({\lambda }_0^3\right).\end{array}$$

(22)

Now, by means of averaging, it is possible to exclude the effects of fast modes. In this case, the following rules must be taken into account in the averaging process:

(1)

Slow modes are statistically independent with respect to fast modes and are also invariant with respect to the averaging operation:

$$\overline{{\mathrm{F}}^{<}}={\mathrm{F}}^{<},\overline{{\mathrm{U}}^{<}}={\mathrm{U}}^{<}.$$

(2)

Averaging can be replaced by averaging, taking into account the linearity of relationship (19).

(3)

The exciting force is statistically homogeneous, and therefore,

$$M^{<}\left(\overrightarrow{\mathsf{\kappa }}\right)\overline{{\mathrm{U}}_0^{>}\left(\tilde{\sigma }\right){\mathrm{U}}_0^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)}=0,$$

due to the presence of the Dirac delta function in (12) since

$$M^{<}\left(0\right)=0.$$

(4)

For fast modes, the following averaging rules apply:

$$\overline{{\mathrm{F}}^{>}}=0,\overline{{\mathrm{U}}^{>}}=0.$$

(5)

The disturbing force obeys a Gaussian probability distribution. Therefore,

$$\overline{{\mathrm{F}}^{>}{\mathrm{F}}^{>}{\mathrm{F}}^{>}}=0,$$

hence,

$$\overline{{\mathrm{U}}^{>}{\mathrm{U}}^{>}{\mathrm{U}}^{>}}=0.$$

Due to the noted rules, when averaging (22), the term on the right side at the zero power of λ₀ does not change, and in the term at the first power of λ₀, only the first term remains. Let us consider in more detail what happens to the term at the square of λ₀. In the process of averaging, the fast mode (20) is replaced in accordance with (22), as follows:

$$\begin{array}{l}{\mathrm{U}}_1^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)=G_0\left(\tilde{\kappa }-\tilde{\sigma }\right)M^{>}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }}\right){\displaystyle \int \frac{d^{d+1}\tilde{\upsilon }}{{\left(2\pi \right)}^{d+1}}\left[{\mathrm{U}}^{<}\left(\tilde{\upsilon }\right){\mathrm{U}}^{<}\left(\tilde{\kappa }-\tilde{\sigma }-\tilde{\upsilon }\right)+\right.}\\ \left.+{2\mathrm{U}}^{<}\left(\tilde{\upsilon }\right){\mathrm{U}}_0^{>}\left(\tilde{\kappa }-\tilde{\sigma }-\tilde{\upsilon }\right)+{\mathrm{U}}_0^{>}\left(\tilde{\upsilon }\right){\mathrm{U}}_0^{>}\left(\tilde{\kappa }-\tilde{\sigma }-\tilde{\upsilon }\right)\right].\end{array}$$

Taking into account this expression and the averaging rules given above, we transform the expression at the square power of λ₀. The result is as follows:

$$\begin{array}{l}2M^{<}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle \int \frac{d^{d+1}\tilde{\sigma }}{{\left(2\pi \right)}^{d+1}}\left[{\mathrm{U}}^{<}\left(\tilde{\sigma }\right){\mathrm{U}}_1^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)+{\mathrm{U}}_0^{>}\left(\tilde{\sigma }\right){\mathrm{U}}_1^{>}\left(\tilde{\kappa }-\tilde{\sigma }\right)\right]}=\\ =M^{<}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle \int \frac{G_0\left(\tilde{\kappa }-\tilde{\sigma }\right)M^{>}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }}\right)d^{d+1}\tilde{\sigma }}{{\left(2\pi \right)}^{d+1}}{\displaystyle \int \frac{d^{d+1}\tilde{\upsilon }}{{\left(2\pi \right)}^{d+1}}\times }}\\ \times \left[2{\mathrm{U}}^{<}\left(\tilde{\sigma }\right){\mathrm{U}}^{<}\left(\tilde{\kappa }-\tilde{\sigma }-\tilde{\upsilon }\right){\mathrm{U}}^{<}\left(\tilde{\upsilon }\right)+4{\mathrm{U}}_0^{>}\left(\tilde{\sigma }\right){\mathrm{U}}_0^{>}\left(\tilde{\kappa }-\tilde{\sigma }-\tilde{\upsilon }\right){\mathrm{U}}^{<}\left(\tilde{\upsilon }\right)\right].\end{array}$$

The last term on the right-hand side of this equality can be transformed by taking into account (12) and (19) when the fast modes of velocity are replaced by fast modes of the random force.

The renormalization procedure gives [23]

$$G^{-1}\left(\kappa \right){\mathrm{U}}_n^{<}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)={\mathrm{F}}^{<}\left(\overrightarrow{\mathsf{\kappa }},\omega \right)+{\lambda }_0{M}_{nml}^{<}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle {\int }_{\kappa \le {\kappa }_c}\frac{d^d\sigma }{{\left(2\pi \right)}^d}{\displaystyle \int \frac{d\varpi }{2\pi }\left[{\mathrm{U}}_m^{<}\left(\overrightarrow{\mathsf{\sigma }},\omega \right){\mathrm{U}}_l^{<}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }},\omega -\varpi \right)\right]}},$$

(23)

where

$$G\left(\kappa \right)={\left[-i\omega +{\kappa }^2\left({\nu }_0{\displaystyle \frac{1-i\omega \gamma }{1-i\omega \lambda }}+\mathsf{\Delta }\nu \right)\right]}^{-1},$$

(24)

$$\mathsf{\Delta }\nu =8{\lambda }_0^2{\kappa }^{-2}{D}_0{M}_{hml}^{<}\left(\overrightarrow{\mathsf{\kappa }}\right){\displaystyle \int \frac{d^d\overrightarrow{\mathsf{\sigma }}}{{\left(2\pi \right)}^d}{\displaystyle \int \frac{d\varpi }{2\pi }{G}_0\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }},\omega -\varpi \right)\frac{{\left|G_0\left(\overrightarrow{\mathsf{\sigma }},\varpi \right)\right|}^2{M}_{lth}^{>}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }}\right)M_{mt}\left(\overrightarrow{\mathsf{\sigma }}\right)}{{\sigma }^{d-4+\mathsf{\epsilon }*}}}}$$

(25)

where $\mathsf{\Delta }\nu$ is the correction-renormalized effective viscosity.

As the next step, a differential equation for the effective viscosity (which equals the sum of molecular and turbulent viscosities) needs to be obtained. For this purpose, integral (25) must be calculated with respect to the frequency and wave number. This gives

$$\mathsf{\Delta }\nu =A_d{\displaystyle \frac{{\lambda }_0^2{D}_0}{{\nu }^2{\kappa }_c^{\mathsf{\epsilon }*}}}\left[{\displaystyle \frac{\exp \left(\mathsf{\epsilon }*\tau \right)-1}{\mathsf{\epsilon }*}}+2{\displaystyle \frac{\exp \left(\left(\mathsf{\epsilon }*-2\right)\tau \right)-1}{\mathsf{\epsilon }*-2}}\left(\gamma -\lambda \right)\nu {\kappa }_c^2\right],$$

(26)

where

$$A_d={\tilde{A}}_d{\displaystyle \frac{S_d}{{\left(2\pi \right)}^d}},{\tilde{A}}_d={\displaystyle \frac{d-1}{2\left(d+2\right)}},S_d={\displaystyle \frac{2{\pi }^{d/2}}{\mathsf{\Gamma }\left(d/2\right)}},$$

(27)

Γ is the gamma function. τ is the renormalization step, which is linked to the local ultraviolet cutoff wave numbers by the following relation:

$${\kappa }_c^{\prime }={\kappa }_c\exp \left(-\tau \right)<\kappa <{\kappa }_c.$$

(28)

Here, ${\kappa }_c$ and ${\kappa }_c^{\prime }$ are the previous and subsequent cutoff wave numbers.

To obtain a differential equation for the effective viscosity, Equation (26) is differentiated with respect to τ in the limit τ → 0 [23]. In this case, the difference between the local ultraviolet cutoff wave numbers, ${\kappa }_c-{\kappa }_c^{\prime }$, approaches zero. Then, from Equation (26), it follows that

$${\displaystyle \frac{d\nu }{d{\kappa }_c}}=-A_d{\displaystyle \frac{{\lambda }_0^2{D}_0}{{\nu }^2{\kappa }_c^{1+\mathsf{\epsilon }*}}}\left[1+2\left(\gamma -\lambda \right)\nu {\kappa }_c^2\right]$$

(29)

For a Newtonian fluid, when the second term in square brackets is absent, the solution for the turbulent viscosity is obtained by Yakhot and Orszag [23]. Generally speaking, Equation (29) is the Riccati equation, which has an exact solution only at particular values of numerical coefficients.

Let us consider another limiting case: when the reason for the generation of turbulence is the non-Newtonian nature of a fluid [18]. In this case, Equation (29) is transformed as follows:

$${\displaystyle \frac{d\nu }{d{\kappa }_c}}=-2A_d{\displaystyle \frac{{\lambda }_0^2{D}_0}{{\nu }^2{\kappa }_c^{\mathsf{\epsilon }*-1}}}\left(\gamma -\lambda \right)\nu .$$

(30)

It should be noted that elastic turbulence, according to [18], behaves differently from non-Newtonian fluid, which is determined by Equation (30). Therefore, such results are obtained for turbulent viscosity.

The solution of Equation (30), assuming that that ε* > 2 and with the boundary condition $\nu ={\nu }_0$ at ${\kappa }_c\to \infty$, looks like

$$\nu =\sqrt{{\displaystyle \frac{4A_d}{\mathsf{\epsilon }*-2}}{\displaystyle \frac{{\lambda }_0^2{D}_0}{{\kappa }_c^{\mathsf{\epsilon }*-2}}}\left|\gamma -\lambda \right|+{\nu }_0^2}.$$

(31)

As a reminder, ε* = 4 for a Newtonian fluid. In this case (and under the assumption that ν₀ → 0), the energy spectrum of turbulence is [23]

$$E\left(\kappa \right)={\displaystyle \frac{d-1}{2}}{\displaystyle \frac{S_d}{{\left(2\pi \right)}^d}}{D}_0{\displaystyle \frac{{\kappa }^{1-\mathsf{\epsilon }*}}{\nu \left(\kappa \right)}}={\displaystyle \frac{d-1}{4}}{\displaystyle \frac{S_d}{{\left(2\pi \right)}^d}}\sqrt{{\displaystyle \frac{D_0\left(\mathsf{\epsilon }*-2\right)}{A_d\left|\gamma -\lambda \right|}}}{\kappa }^{-\mathsf{\epsilon }*/2}={\displaystyle \frac{d-1}{4}}{\displaystyle \frac{S_d}{{\left(2\pi \right)}^d}}\sqrt{{\displaystyle \frac{2D_0}{A_d\left|\gamma -\lambda \right|}}}{\kappa }^{-2}.$$

(32)

Now, it is necessary to establish a correlation between D₀ and the dissipation rate ε. In order to do this, the following equation for the dissipation rate is used:

$$\epsilon =2\nu \left({\kappa }_c\right){\displaystyle \underset{0}{\overset{{\kappa }_c}{\int }}{\kappa }^{2}E\left(\kappa \right)d\kappa }.$$

(33)

The substitution of relations (31) for viscosity and (32) for the spectrum in Equation (33) yields

$$2D_0{\displaystyle \frac{S_d}{{\left(2\pi \right)}^d}}={\displaystyle \frac{2}{d-1}}\epsilon .$$

(34)

Thus, the energy spectrum of the turbulence is determined by the expression

$$\begin{array}{l}E\left(\kappa \right)={\displaystyle \frac{d-1}{2}}{\displaystyle \frac{S_d}{{\left(2\pi \right)}^d}}{D}_0{\displaystyle \frac{{\kappa }^{1-\mathsf{\epsilon }*}}{\nu \left(\kappa \right)}}={\displaystyle \frac{d-1}{4}}{\displaystyle \frac{S_d}{{\left(2\pi \right)}^d}}\sqrt{{\displaystyle \frac{D_0\left(\mathsf{\epsilon }*-2\right)}{A_d\left|\gamma -\lambda \right|}}}{\kappa }^{-\mathsf{\epsilon }*/2}=\\ =\sqrt{{\displaystyle \frac{d-1}{8}}{\displaystyle \frac{S_d}{{\left(2\pi \right)}^d}}{\displaystyle \frac{\epsilon }{A_d\left|\gamma -\lambda \right|}}}{\kappa }^{-2}=\sqrt{{\displaystyle \frac{d-1}{8}}{\displaystyle \frac{\epsilon }{{\tilde{A}}_d\left|\gamma -\lambda \right|}}}{\kappa }^{-2}.\end{array}$$

(35)

One can further obtain the following for the turbulent viscosity using Equations (31) and (34):

$$\nu =\sqrt{{\displaystyle \frac{2}{d-1}}{\tilde{A}}_d{\displaystyle \frac{{\lambda }_0^2\epsilon \left|\gamma -\lambda \right|}{{\kappa }_c^2}}+{\nu }_0^2}$$

(36)

Let us eliminate a wave number from Equation (36). For this purpose, we use the expression for the energy of turbulence:

$$k={\displaystyle \underset{{\kappa }_c}{\overset{\infty }{\int }}E\left(\kappa \right)d\kappa =}{\displaystyle \underset{{\kappa }_c}{\overset{\infty }{\int }}\sqrt{\frac{d-1}{8}\frac{\epsilon }{{\tilde{A}}_d\left|\gamma -\lambda \right|}}{\kappa }^{-2}d\kappa =}\sqrt{{\displaystyle \frac{d-1}{8}}{\displaystyle \frac{\epsilon }{{\tilde{A}}_d\left|\gamma -\lambda \right|}}}{\kappa }_c^{-1}.$$

(37)

Based on Equation (37), one can eliminate the wave number from Equation (29), which results in

$$\nu =\sqrt{{\left({\displaystyle \frac{4}{d-1}}{\tilde{A}}_d\left|\gamma -\lambda \right|k\right)}^2+{\nu }_0^2}.$$

(38)

Thus, the renormalized equation of motion is

$${\displaystyle \frac{\partial u_n}{\partial t}}+{\displaystyle \frac{\partial u_n{u}_m}{\partial x_n}}={\displaystyle \frac{\partial }{\partial x_n}}\left[\left({\nu }_0+{\nu }_t\right){\displaystyle \frac{\partial u_n}{\partial x_n}}\right],$$

(39)

where ${\nu }_t$ is the turbulent kinematic viscosity.

4. Renormalization of the Energy Equation

The next step is the renormalization of the energy equation

$$\left({\displaystyle \frac{\partial }{\partial t}}-a_0{\nabla }^2\right)T+{\lambda }_0{\displaystyle \frac{\partial \left(T{u}_n\right)}{\partial x_n}}=0,$$

(40)

where a₀ is the thermal diffusivity (subscript “0” is used again because this value is a starting point for the renormalization procedure).

In the Fourier space, Equation (40) has the following form [23]:

$$G_{T0}^{-1}\left(\overrightarrow{\mathsf{\sigma }},\omega \right)T\left(\overrightarrow{\mathsf{\sigma }},\omega \right)=-i{\kappa }_l{\lambda }_0{\displaystyle {\int }_{\kappa \le {\kappa }_c}\frac{d^d\sigma }{{\left(2\pi \right)}^d}{\displaystyle \int \frac{d\varpi }{2\pi }{\mathrm{U}}_l\left(\overrightarrow{\mathsf{\sigma }},\omega \right)\mathrm{T}\left(\overrightarrow{\mathsf{\kappa }}-\overrightarrow{\mathsf{\sigma }},\omega -\varpi \right)}},$$

(41)

where

$$G_{T0}={\left(-i\omega +a_0{\kappa }^2\right)}^{-1},$$

(42)

and T is the Fourier transform of the temperature.

We carry out the procedure of temperature renormalization similar to that for velocity and force, namely, we divide the temperature field into slow and fast modes. Then, we carry out the renormalization operation, which consists of excluding fast temperature modes by introducing a formal expansion in a series in terms of the parameters λ′₀ and λ₀ similar to series (18). As a result, we obtain an equation for the slow temperature modes

$$G_T^{-1}\left(\tilde{\kappa }\right){\mathrm{T}}^{<}\left(\tilde{\kappa }\right)=-i\kappa {\lambda }_0{\displaystyle \int \frac{d\tilde{\sigma }}{{\left(2\pi \right)}^{d+1}}{\mathrm{U}}_l^{<}\left(\tilde{\sigma }\right){\mathrm{T}}^{<}\left(\tilde{\kappa }-\tilde{\sigma }\right)},$$

(43)

where

$$G_T\left(\kappa \right)={\left[-i\omega +{\kappa }^2\left(a_0+\mathsf{\Delta }a\right)\right]}^{-1},$$

(44)

$$\mathsf{\Delta }a=2D_0{\displaystyle \frac{{\kappa }_l{\kappa }_n}{{\kappa }^2}}{\lambda }_0^2{\displaystyle \int \frac{d\tilde{\sigma }}{{\left(2\pi \right)}^{d+1}}\frac{G_{T0}\left(\tilde{\kappa }-\tilde{\sigma }\right)}{{\sigma }^y}{\left|G_0\left(\tilde{\sigma }\right)\right|}^2{M}_{nl}\left(\overrightarrow{\mathsf{\sigma }}\right)},$$

(45)

and y = d − 4 + ε*. After integration with respect to the frequencies and decomposition of the obtained expression in McLaurin series on κ and ω, one can obtain

$$\mathsf{\Delta }a\approx {\displaystyle \frac{{\kappa }_n{\kappa }_l}{{\kappa }^2}}{\displaystyle \frac{{\lambda }_0^2{D}_0}{{\left(2\pi \right)}^d\nu }}{\displaystyle \int d^d\overrightarrow{\mathsf{\sigma }}\left(1+2a\left[1+\nu {\sigma }^2\left(\gamma -\lambda \right)\right]\frac{\overrightarrow{\mathsf{\kappa }}\cdot \overrightarrow{\mathsf{\sigma }}}{{\sigma }^2}\right)\frac{M_{nl}\left(\overrightarrow{\mathsf{\sigma }}\right){\sigma }^{-y-4}}{a+\nu +a\nu {\sigma }^2\left(\gamma -\lambda \right)}}.$$

(46)

The integration of Equation (46) gives

$$\begin{array}{l}\mathsf{\Delta }a\approx {\displaystyle \frac{d-1}{d}}{\displaystyle \frac{S_d{\lambda }_0^2{D}_0}{{\left(2\pi \right)}^d\mathsf{\epsilon }*\nu \left(a+\nu \right){\kappa }_c^{\mathsf{\epsilon }*}}}\left[\exp \left(\mathsf{\epsilon }*\tau \right){}_2\mathrm{F}_1\left(-{\displaystyle \frac{\mathsf{\epsilon }*}{2}},1,1-{\displaystyle \frac{\mathsf{\epsilon }*}{2}},-{\displaystyle \frac{a\nu \left(\gamma -\lambda \right)}{\left(a+\nu \right){\kappa }_c^{-2}}}\exp \left(-2\tau \right)\right)-\right.\\ \left.-{}_2\mathrm{F}_1\left(-{\displaystyle \frac{\mathsf{\epsilon }*}{2}},1,1-{\displaystyle \frac{\mathsf{\epsilon }*}{2}},-{\displaystyle \frac{a\nu \left(\gamma -\lambda \right)}{\left(a+\nu \right){\kappa }_c^{-2}}}\right)\right]\end{array}$$

(47)

This expression represents the renormalized effective thermal diffusivity. In Equation (47), ₂F₁ is the hypergeometric function.

Differentiation of Equation (47) with respect to τ in the limit τ → 0 yields the following result:

$${\displaystyle \frac{da}{d\tau }}={\displaystyle \frac{d-1}{d}}{\displaystyle \frac{S_d}{{\left(2\pi \right)}^d}}{\displaystyle \frac{{\lambda }_0^2{D}_0}{{\nu }^2{\kappa }_c^{\mathsf{\epsilon }*}}}{\displaystyle \frac{\nu }{\nu +a\left(1+\nu \left(\gamma -\lambda \right){\kappa }_c^2\right)}}.$$

(48)

Equation (48) is further recast by introducing the turbulent Prandtl number Pt_t = ν/a:

$$\begin{array}{l}{\displaystyle \frac{da}{d{\kappa }_c}}={\displaystyle \frac{d\left({\mathrm{Pr}}_t^{-1}\nu \right)}{d{\kappa }_c}}=\nu {\displaystyle \frac{d{\mathrm{Pr}}_t^{-1}}{d{\kappa }_c}}+{\mathrm{Pr}}_t^{-1}{\displaystyle \frac{d\nu }{d{\kappa }_c}}=\\ -{\left(2A_d\right)}^{-1}{\displaystyle \frac{d-1}{d}}{\displaystyle \frac{S_d}{{\left(2\pi \right)}^d}}2A_d{\displaystyle \frac{{\lambda }_0^2{D}_0}{{\nu }^2{\kappa }_c^{\mathsf{\epsilon }*-1}}}{\displaystyle \frac{\nu \left(\gamma -\lambda \right)}{\left(\gamma -\lambda \right){\kappa }_c^2}}{\displaystyle \frac{1}{\nu +a\left(1+\nu \left(\gamma -\lambda \right){\kappa }_c^2\right)}}=\\ {\displaystyle \frac{d-1}{2d}}{\tilde{A}}_d^{-1}{\displaystyle \frac{d\nu }{d{\kappa }_c}}{\displaystyle \frac{1}{\nu \left(\gamma -\lambda \right){\kappa }_c^2}}{\displaystyle \frac{1}{1+{\mathrm{Pr}}_t^{-1}\left(1+\nu \left(\gamma -\lambda \right){\kappa }_c^2\right)}}\end{array}$$

(49)

It follows from here that

$${\displaystyle \frac{d{\mathrm{Pr}}_t^{-1}}{d{\kappa }_c}}={\displaystyle \frac{1}{\nu }}{\displaystyle \frac{d\nu }{d{\kappa }_c}}\left({\displaystyle \frac{d-1}{2d}}{\tilde{A}}_d^{-1}{\displaystyle \frac{1}{\nu \left(\gamma -\lambda \right){\kappa }_c^2}}{\displaystyle \frac{1}{1+{\mathrm{Pr}}_t^{-1}\left(1+\nu \left(\gamma -\lambda \right){\kappa }_c^2\right)}}-{\mathrm{Pr}}_t^{-1}\right).$$

(50)

It is necessary to solve this differential equation at the boundary condition ${\mathrm{Pr}}_t=\mathrm{Pr}$ at $\nu ={\nu }_0$. Equation (50) has no exact analytical solution. However, it is quite easy to obtain a numerical solution that depends on the parameter (γ − λ).

Thus, the renormalized energy equation looks like

$${\displaystyle \frac{\partial T}{\partial t}}+{\displaystyle \frac{\partial u_{n}T}{\partial x_n}}={\displaystyle \frac{\partial }{\partial x_n}}\left[\left(a_0+{\displaystyle \frac{{\nu }_t}{{\mathrm{Pr}}_t}}\right){\displaystyle \frac{\partial T}{\partial x_n}}\right].$$

(51)

5. Turbulence Kinetic Energy and Dissipation Rate Equations

For the closure of the turbulence model, equations for the kinetic energy of turbulence k and the dissipation rate ε must be obtained. The renormalization procedure of equations for the kinetic energy and the dissipation rate is described in detail in Yakhot and Orszag [23]. The final renormalized forms of the equations for the kinetic energy of turbulence and dissipation rate are

$${\displaystyle \frac{\partial k}{\partial t}}+{\displaystyle \frac{\partial u_{n}k}{\partial x_n}}=2{\nu }_t{S}_{nm}^2-\epsilon +{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\nu }_0+{\nu }_t}{{\mathrm{Pr}}_K}}{\displaystyle \frac{\partial k}{\partial x_n}}\right),$$

(52)

$${\displaystyle \frac{\partial \epsilon }{\partial t}}+{\displaystyle \frac{\partial u_n\epsilon }{\partial x_n}}=2C_{1\epsilon }{\nu }_t{\displaystyle \frac{\epsilon }{k}}{S}_{nm}^2-C_{2\epsilon }{\displaystyle \frac{{\epsilon }^2}{k}}+{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\nu }_0+{\nu }_t}{{\mathrm{Pr}}_{\epsilon }}}{\displaystyle \frac{\partial \epsilon }{\partial x_n}}\right),$$

(53)

where

$$S_{nm}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial {\overline{u}}_n}{\partial x_m}}+{\displaystyle \frac{\partial {\overline{u}}_m}{\partial x_n}}\right),$$

(54)

Pr_K and Pr_ε are determined by equations similar to Equation (42), C_1ε = 1.42, and C_2ε = 1.68 [23,29,30].

6. Turbulence Model for Large Eddy Simulation

Equation (29) can be used for the large eddy simulation (LES) of turbulence. For this purpose, according to Yakhot and Orszag [23], let us neglect molecular viscosity in Equation (29) and replace the dissipation rate using the relation

$$\epsilon ={\nu }_t{\left({\displaystyle \frac{\partial u_n}{\partial x_m}}\right)}^2.$$

(55)

Furthermore, it is necessary to eliminate all high-frequency scales with wave numbers larger than κ_c. For this purpose, one can use a Gauss filter with the width

$$L={\displaystyle \frac{2\pi }{{\kappa }_c}},$$

(56)

where L is the size of a mesh for the numerical calculations of turbulent flows. Taking into account Equations (55) and (56), one can recast Equation (36) to produce

$${\nu }_t=c_S\left|\gamma -\lambda \right|L^2{\left({\displaystyle \frac{\partial u_n}{\partial x_l}}\right)}^2,$$

(57)

where

$$c_S={\displaystyle \frac{2}{{\left(2\pi \right)}^2\left(d-1\right)}}{\tilde{A}}_d.$$

(58)

7. Results

Thus, using the mathematical technology of renormalization group analysis, we obtained a system of renormalized equations, which is a k-ε turbulence model. It consists of the equations of momentum (39), energy (51), kinetic energy of turbulence (52), and dissipation rate (53). An expression for the thermal conductivity coefficient and the coefficient of turbulent kinematic viscosity for a large eddy simulation (LES) was also obtained.

8. Conclusions

In the present work, symmetry transformation properties for fluid flow equations are used for non-Newtonian elastic turbulence.

A closed model of turbulence for non-Newtonian fluids is obtained by using the renormalization group approach. The model includes the momentum equation and energy equation, as well as equations for the turbulence kinetic energy and dissipation rate. This system is closed by Equation (38) for turbulent viscosity or alternatively by Equation (49).

According to Equation (38), turbulent viscosity depends only on the kinetic energy of turbulence and properties of the non-Newtonian fluid, i.e., turbulent viscosity does not depend on the dissipation rate. Hence, the entire kinetic energy is used for the generation of turbulent pulsations. Obviously, this explains the abrupt decrease in shear stress at low values of Reynolds numbers in polymeric solutions, which was observed in experiments [31].

A similar conclusion follows from Equation (57) for a large eddy simulation (LES). In the case of a Newtonian fluid in a large eddy simulation, the turbulent viscosity is proportional to the tensor of the distortion rate. In this case, for non-Newtonian fluids, the dependence between the viscosity and the tensor of the distortion rate has a quadratic character. Consequently, a very rapid increase in stresses are observed with the appearance of heterogeneities in the velocity fields.

Another conclusion is that the form of the functional dependence for the energy spectrum of turbulence does not affect the resulting relationship for turbulent viscosity.