Introduction

Abstract

In this chapter, a brief literature review has been carried out considering those contributions which are aligned with the objectives of the present book. Since, there are thousands of works dealing with internal and external turbulent flows, therefore, we consider a selection of those contributions which are relevant to the understanding of the new hypothesis on the anisotropic Reynolds stress tensor in Chap. 5. For the sake of completeness, the governing equations of incompressible turbulent flows have been derived in conjunction with the generalised Boussinesq hypothesis on the Reynolds stress tensor. Intermediate mathematical steps are included in the derivations to make graduate and postgraduate students familiar with the heart of the closure problem of anisotropic turbulence. The shortcomings of the generalised Boussinesq hypothesis have also been discussed to emphasise the necessity of a new hypothesis on the Reynolds stress tensor.

Many engineers to-day may consider the problem of turbulence merely as an interesting chapter of mathematical physics. They may be right. However, they should remember that if we meet a practical question in aerodynamic design which we are unable to answer, the reason that we are unable to give a definite answer is almost certainly that it involves turbulence

—Theodore von Kármán, 1937

1.1 Historical Background and Literature Review

The origin of the proposal to a new hypothesis on the anisotropic Reynolds stress tensor in this book (see Chap. 5) dates back to the similarity theory of von Kármán [61,62,63, 68, 69] and the vorticity transport theory of Taylor [124]. During the 1930s, von Kármán [64] worked further on the similarity hypothesis and Taylor [131] further developed the vorticity transport theory. The homogeneous isotropic simplification of the mathematical and physical description of turbulent flows was proposed by Taylor [125,126,127,128,129,130], which is a significant simplification compared to real turbulent flows occurring in the nature. The isotropic turbulence approach considers all normal components of the Reynolds stress tensor are equal to each other and all non-diagonal shear stress components are assumed to be equal to zero in the Reynolds stress tensor. Taylor’s isotropic turbulence assumption could be valid far from any solid wall where the effect of shear stresses is negligible. For turbulent flows, e.g. around an aircraft wing or e.g. in a three-dimensional channel near to the wall, shear stresses become dominant in the boundary layer. Therefore, an anisotropic mathematical description of the Reynolds stress tensor is desirable to predict and model the physics of turbulence correctly.

Theoretical and experimental investigations of the boundary layer and shear flows have been in the centre of research interest since the begining of the 20th century [110]. The semi-empirical analysis of turbulent shear flows originates from the eddy viscosity hypothesis of Boussinesq [10] and the mixing-length theory of Taylor [123] and Prandtl [107]. Theoretical analysis on the Reynolds momentum equation for channel and pipe flows were carried out by Prandtl [107] and von Kármán [61, 63, 65, 68]. Simplified analytical solutions of the Reynolds-Averaged Navier-Stokes (RANS) momentum equations (Reynolds equations) relying on the semi-empirical theories of turbulence can be found in the book of Shih-I [111]. Due to the mathematical and physical complexity of the statistical description of anisotropic boundary layer and shear flows, Taylor [125,126,127,128,129,130], von Kármán [66, 67, 70], von Kármán and Howarth [71], Dryden [33], and Heisenberg [47] focused initially on the development of the statistical theory of homogeneous isotropic turbulence. Taylor [132], Kolmogorov [74,75,76] and von Kármán and Lin [72] investigated the spectrum of turbulence. The two-dimensional similarity theory of von Kármán [61,62,63, 68, 69] and its applications to internal flows were further investigated by Goldstein [40] and further extended to compressible boundary layer flows over a flat plate by Lin and Shen [90,91,92]. A comprehensive introduction to the experimental and theoretical developments in conjunction with semi-empirical theories of turbulent flows—including the similarity theory of von Kármán [61,62,63, 68, 69] and the vorticity transport theory of Taylor [124]—was given by Goldstein [41, 42]. The theoretical achievements in the research field of homogeneous isotropic turbulence are discussed in-depth by Batchelor [9], Shih-I [111], Leslie [89], Davidson [32] and McComb [96].

Note that the classical semi-empirical theories [61,62,63, 68, 69, 107, 123, 124] can only be used for two-dimensional boundary layer flows where a simple geometry is considered. Furthermore, the theoretical results achieved in the research field of homogeneous isotropic turbulence can be investigated with a great success in grid generated turbulent flows or in a periodic box mathematical model problem. These physical circumstances do less likely occur near to the wall of an aircraft wing or when three-dimensional shear flows in the boundary layer are concerned. Therefore, the development of three-dimensional advanced turbulence modelling approaches has to take into account the Reynolds stress anisotropy to make an attempt to capture internal and external separating flows in a physically correct way.

Turbulent shear flows were investigated theoretically and experimentally by Townsend [134]. Rotta [109] developed a statistical theory for non-homogenerous turbulence. The results of the early development on the theory of non-isotropic turbulence is discussed by Hinze [49]. Champagne et al. [18] carried out an experimental investigation on nearly homogenerous turbulent shear flows. Oberlack [104] studied anisotropic dissipation in non-homogenerous turbulence. Bradshaw et al. [14] focused on the boundary layer development through the turbulent energy equation. Bradshaw [11,12,13] investigated the structure of boundary layers and shear flows along with their engineering applications. A theoretical analysis of turbulent boundary layer flows including the study on the governing equations and compressibility effects was carried out by Cebeci and Smith [16].

The statistical description of turbulent vortical structures has also been in the centre of research interest since the begining of the the 20th century [94]. Stochastic mathematical tools in the field of turbulence research can be found in the book of Lumley [94]. A comprehensive work on statistical mechanics of turbulent flows was carred out by Monin and Yaglom [101, 102]. The theory of vorticity dynamics was discussed in-depth by Tennekes and Lumley [133]. Since the wall is a vortex generator, Smith and Walker [113] investigated the structure of turbulent wall-layer vorticies. The dynamics and statistics of vortical behaviour of turbulence was studied by Hunt [50]. Novikov focused on turbulent vortical structures and their modelling [103].

We can distinguish four main groups of closure models for predicting the elements of the Reynolds stress tensor. The first group consists of algebraic, one- and two-equation RANS engineering turbulence models relying on the generalization of the Boussinesq-hypothesis. The second one is the group of Reynolds stress models (RSMs) which introduces closure approaches through the solution of Reynolds stress transport equation to make an attempt to completely abandon the Boussinesq-hypothesis. The third group could be considered as the mathematical and physical description of the anisotropic Reynolds stress tensor based on the three-dimensional anisotropic mechanical similarity theory of turbulent oscillatory motions or Galilean invariant velocity fluctuations (see Chap. 4). The fourth one is the group of hybrid hypotheses on the Reynolds stress tensor (see Chap. 5).

Launder and Spalding [86] proposed and developed the standard k-$\epsilon$ two-equation turbulence model to overcome the difficulties with the classical mixing-length approaches and they assumed that the eddy viscosity is isotropic. An earlier work of Jones and Launder [57] focused on the low-Reynolds number phenomena also with a two-equation turbulence model. Daly and Harlow [31] derived transport equations for incompressible turbulent flows in conjunction with the Reynolds stress transport equation. Hanjalić and Launder [45] proposed a Reynolds stress model for computing thin shear flows. Launder et al. [87] developed a Reynolds stress transport closure model to take into account the Reynolds stress anisotropy which is known as LRR model. Pope [105] proposed a more general effective-viscosity hypothesis for two-dimensional flows which related the Reynolds stress tensor to a tensor polynomial through the rate-of-strain (deformation) and vorticity tensors. Hanjalić and Launder [46] studied low-Reynolds number turbulent flows through a Reynolds stress closure model. Gibson and Launder [39] investigated the ground effects on pressure fluctuations in the atmospheric boundary layer through the Reynolds stress transport differential equation. Speziale [115] introduced non-linear K-l and K-$\epsilon$ models to predict the normal Reynolds stresses more accurately than the linear K-l and K-$\epsilon$ models taking into account realisability and invariance requirements. Wilcox [136] carried out a study on the reassessment of the scale-determining equations for advanced turbulence modelling approaches. Speziale [116] reviewed the past and the future of turbulence modelling in the end of the 1980s. Speziale, Sarkar and Gatski [119] proposed a Reynolds stress transport model to take into account the Reynolds stress anisotropy in the near-wall region which is known as SSG model. Analytical models in conjunction with the development of Reynolds stress closure models were reviewed by Speziale [117]. Gatski and Speziale [38] focused on the development of explicit algebraic stress models for three-dimensional turbulent flows with the generalization of the models of Pope [105] and Launder, Reece and Rodi [87]. The theoretical and practical developments on RANS and RSM turbulence models is discussed by Wilcox [137].

The standard k-$\omega$ turbulence model of Wilcox [137] and the k-$\omega$ SST formulation of Menter [97, 98] were successfully used for modelling boundary layer flows for industrial applications. These eddy viscosity RANS turbulence models employ the Boussinesq-hypothesis for the mathematical description of the Reynolds stress tensor. In these turbulence models, the eddy viscosity is a scalar quantity and the Reynolds stress tensor is modelled through the symmetric mean rate-of-strain (deformation) tensor and the isotropic turbulent kinetic energy tensor. It is a well-known fact that the Boussinesq-hypothesis [10] itself does not provide an accurate prediction of Reynolds stress anisotropies from a physical point-of-view. Therefore, researchers carried out work on modelling the Reynolds stress anisotropy in conjunction with the Reynolds stress transport (RST) models because the RST modelling can be considered the most advanced RANS tools nowadays. One can also find relevant contributions to Reynolds stress modelling within the context of two-equation RANS turbulence models. Antonia et al. [3] studied the anisotropy of the dissipation tensor in turbulent boundary layer flows. Craft et al. [20] developed an anisotropic cubic eddy-viscosity model and they proposed a cubic relationship between the stress tensor and the rate-of-strain and vorticity tensors to predict the Reynolds stresses accurately and capture the effect of streamline curvature. An analysis including modelling of anisotropies in the dissipation rate of turbulent flows was carried out by Speziale and Gatski [118]. Craft et al. [21] developed a non-linear eddy viscosity model to predict turbulent flows far from the equilibrium state including the modelling of transition based on an anisotropic cubic relationship between the stress, rate-of-strain (deformation) and vorticity tensors. Jakirlić and Hanjalić [51] derived a new approach for modelling near-wall turbulence energy and stress dissipation. A detailed discussion on second-moment closure turbulence modelling was carried out by Hanjalić and Jakirlić [44]. Abe et al. [2] investigated near wall-anisotropy expressions and turbulent length scale equations in conjunction with non-linear eddy viscosity models. Menter and Egorov [99] introduced a scale-adaptive simulation (SAS) modelling approach based on two-equation turbulence models. Eisfeild and Brodersen [37] proposed first the SSG/LRR-$\omega$ Reynolds stress turbulence model investigating the DLR-F6 configuration. Liu and Pletcher [93] carried out an investigation on the anisotropic behaviour of turbulent boundary layer flows. A hybrid RSM closure model was proposed and developed by Cecora et al. [17] for aerospace applications. Klajbár et al. [73] proposed a modified hybrid SSG/LRR-$\omega$ Reynolds stress model in conjunction with a simplified diffusion model for three-dimensional incompressible turbulent flows around bluff bodies. Vitillo et al. [135] proposed and validated an anisotropic shear stress transport (ASST) formulation related to the two-equation k-$\omega$ SST turbulence model of Menter [97, 98]. One can find more details on the statistical description of turbulent flows including an overview of classical RANS and advanced RSM, LES and DNS computational approaches in the book of Pope [106].

The two-equation k-$\omega$ SST turbulence model of Menter [97, 98] and its further development including transitional flows brought particular attention amongst researchers over the past twenty-five years [43, 48, 84, 85, 95, 100, 112, 114]. Hellsten [48] proposed a further improved version of the two-equation k-$\omega$ SST model to take into account the effects of system rotation and streamline curvature, furthermore a modification was introduced to make the original SST model rotationally invariant. Mani et al. [95] focused on rotation and curvature correction assessment in conjunction with one- and two-equation RANS engineering turbulence models. Spalart and Rumsey [114] proposed an approach for effective inflow conditions for turbulence modelling in aerodynamic computations. Smirnov and Menter [112] carried out investigations on the sensitisation of the SST model to rotation and curvature taking into account the Spalart-Shur correction term in their mathematical formulations. Langtry [84], and Langtry and Menter [85] developed a local correlation-based transition model which is the further extenstion of the k-$\omega$ SST formulation to capture transitional flows. The transitional SST model of Langtry and Menter [85] is a four-equation RANS turbulence model. In addition to the turbulent kinetic energy k and transport of the specific dissipation rate $\omega$ equations, the transitional SST formulation [85] introduces the gamma $\gamma$ and the Reynolds-theta $Re_{\theta }$ transport equations to capture the flow physics of transitional flows in particular to aerospace applications. The physically and numerically correct prediction of transitional flows is amongst the most difficult and challenging scientific and engineering problems, because transition occurs in most cases due to the presence of fluid flow instabilities, e.g. Tollmien-Schlichting waves or cross-flow instabilities. The theoretical establishment of physically correct transitional flow modelling approaches including their accurate mathematical formulations is a state-of-the-art ongoing research field. Due to the fact that semi-empirical correlations have to be taken into account in the development of transitional models, therefore, it is difficult to set up a mathematical formulation for general purposes. Menter et al. [100] carried out a study on transitional flow modelling for general purpose CFD codes including investigations on a three-dimensional transonic wing and a full helicopter configuration. In terms of the development of transitional flow models, another difficulty is to preserve Galilean invariance of the Reynolds stress tensor due to the complexity of the concerned flow physics. Langtry and Menter [85] pointed out that the local correlation-based transition model, the four-equation transitional SST approach, is not Galilean invariant, because of the mathematical formulation of the velocity gradient along the streamline at the boundary layer edge. Grabe et al. [43] proposed a transitional transport modelling approach to an accurate prediction of the cross-flow transition for three-dimensional aerospace applications. One of their modelling approach takes into account the local helicity of the fluid flow, therefore, Grabe et al. [43] emphasized that their local helicity based transitional model is also not Galilean invariant. Thus, the preservation of the Galilean invariant property of RANS transitional flow models in their mathematical formulations could still remain a challenge in their development.

The theorerical development of anisotropic stochastic turbulence modelling including the investigations on fluctuations is in the scope of the current mainstream research interest. Bakosi and Ristorcelli [5, 6] proposed and developed a probability density function (PDF) based method for variable-density turbulent mixing. They highlighted the importance of the presence of small-scale anisotropy which is a non-Kolmogorovian feature of turbulent flows under external acceleration forces. Their approach considers a tensorial diffusion term to capture persistent small-scale anisotropic fluid flow behaviour. The stochastic diffusion process in conjunction with conservation law constraints was also investigated by Bakosi and Ristorcelli [7, 8]. It is important to mention that the investigation of anisotropic properties of astrophysical turbulent flows is also a current mainstream research area. The spectral anisotropy in the solar wind was discussed recently by Bruno and Carbone [15]. Theoretical and practical achievements in the research field of statistical mechanics of turbulent flows and their advanced modelling approaches were discussed in-depth by Heinz [1], and Durbin and Pettersson Reif [35]. Statistical turbulence modelling approaches in conjunction with classical RANS and advanced RSM models—widely used for solving engineering problems—are discussed by Leschziner [88].

The three-dimensional anisotropic mechanical similarity theory of Czibere [22, 23] on oscillatory motions of turbulent flows is a key component in terms of the proposal to a new hypothesis on the anisotropic Reynolds stress tensor in Chap. 5. The new hypothesis proposed in this monograph unifies the generalised Boussinesq hypothesis with the mathematical and physical description of the anisotropic Reynolds stress tensor relying on the three-dimensional similarity theory of the Galilean invariant velocity fluctuations which leads to a new formulation of the Reynolds stress tensor (see Chap. 5). The three-dimensional mechanical similarity theory of turbulent oscillatory motions or velocity fluctuations [22, 23] is the extension of von Kármán’s [61,62,63, 68, 69] two-dimensional similarity theory to three-dimensional incompressible turbulent flows which is a completely different approach compared to other anisotropic eddy viscosity models (see e.g. in [20, 21, 135]). It is important note that Goldstein [41, 42] mentioned in his book—in the footnote on page 348—that the similarity theory of von Kármán [61,62,63, 68, 69] is valid for three-dimensional eddying motion in general and the two-dimensional description of turbulent oscillatory motions is merely a simplification. However, it seems that the three-dimensional mathematical formulation of the similarity theory of oscillatory motions was not derived by other researchers until the begining of the 21st century. This might be explained by the fact that there are controversial statements in the literature beside the difficulties of the three-dimensional mathematical formulation of the similarity theory of turbulent velocity fluctuations. Shih-I [111] stated in his book on page 28 that according to von Kármán, a complete similarity of turbulent flows considering all fluctuating components is not possible. According to Taylor [123,124,125,126,127,128,129,130,131], a general expression for the vorticity fluctuations is intractable which could also imply that the mathematical formulation of turbulent velocity fluctuations is intractable. Shih-I [111] and Goldstein [41, 42] also refer to the intractability of the vorticity fluctuation related to the vorticity transport theory of Taylor [123, 124, 131]. In the first half of the 20th century, these controversial statements on the difficulties of the three-dimensional mathematical formulation of the similarity theory of turbulent velocity fluctuations might be discouraging to carry out further research on the similarity theory of von Kármán [61,62,63, 68, 69]. Moreover, there could be another reason why researchers put aside the three-dimensional further investigation of von Kármán’s similitude of oscillatory motions for a long time, because the two-dimensional mathematical formulation has a direct connection with the semi-empirical mixing-length theory. However, the relevance of the von Kármán similarity theory of oscillatory motions should be more than the deduction of the von Kármán’s length scale [61, 63, 68]. In fact, the three-dimensional similarity theory of turbulent oscillatory motions could give a deeper insight into the internal stochastic mechanism of the mechanically similar local velocity fluctuations, because a symmetrical anisotropic similarity tensor can be deduced in conjunction with the anisotropic Reynolds stress tensor as derived by Czibere [22, 23].

The author of this book was working in the research group of Czibere focusing on stochastic turbulence modelling of internal flows on curvilinear domains from 1999 to 2006 at the University of Miskolc, in Hungary. The Hungarian research group validated the anisotropic similarity theory for wide range of internal flow applications [24,25,26,27,28,29,30, 52,53,54,55,56, 58, 59, 77,78,83, 120,121,122], but not for three-dimensional external flows. For modelling turbulent shear flows in straight and curved channels, Janiga [52, 53] developed a computational method for solving the Reynolds momentum equation in conjunction with the stochastic turbulence model (STM) of Czibere [22, 23]. In addition to an in-house code implementation, Janiga [54] implemented the algebraic version of the STM in the ANSYS-FLUENT environment. Könözsy [77, 82, 83] developed a high-order curvilinear mesh generation method to support the numerical solution of the Reynolds-averaged mean vorticity transport equation for rotationally-symmetric turbulent shear flows on curvilinear domains [83]. Kalmár et al. [59] investigated the model parameters of the two-equation version of the STM. The Hungarian research project on turbulent internal flows was sponsored by the DAAD-MÖB German-Hungarian fund in co-operation with the University of Siegen and the Otto-von-Guericke-University Magdeburg, Institutes of Fluid- and Thermodynamics, in Germany. Further practical applications of the three-dimensional similarity theory [22, 23] related to the numerical simulation of incompressible external turbulent flows were carried out within postgraduate thesis projects [19, 34] from 2012 to 2014 at Cranfield University, in the United Kingdom.

1.2 Governing Equations of Incompressible Turbulent Flows

1.2.1 Mass Conservation (Continuity) Equations in the Instantaneous, Mean and Fluctuating Velocity Fields

To derive the mass conservation (continuity) equation for incompressible turbulent flows, the differential form of the instantaneous general mass conservation equation of turbulent flows has to be considered by

$$\begin{aligned} \frac{\partial \rho _{T}}{\partial t}+\nabla \cdot \left( \rho _{T}\mathbf {u}_{T}\right) =0 , \end{aligned}$$

(1.1)

where the subscript ‘T’ denotes an instantaneous value—which notation is consistent with the notation used in [22, 23]—thus $\rho _{T}$ is the instantaneous density of the fluid, $\mathbf {u}_{T}$ is the instantaneous velocity field, and $\nabla$ is the Hamilton (nabla) vector-type differential operator. For incompressible turbulent flows, the density fluctuation of the fluid flow $\rho ^{\prime }\left( \mathbf {x},t\right)$ is neglected, therefore, the instantaneous density $\rho _{T}$ is assumed to be equal to the density of the fluid $\rho$ which is a constant value. Therefore, the general mass conservation equation of turbulent flows (1.1) can be written as

$$\begin{aligned} \frac{\partial \rho }{\partial t}+&\nabla \cdot \left( \rho \mathbf {u}_{T}\right) =\underset{=0}{\underbrace{\frac{\partial \rho }{\partial t}}}+\rho \cdot \left( \nabla \cdot \mathbf {u}_{T}\right) +\mathbf {u}_{T}\cdot \underset{=0}{\underbrace{\nabla \rho }}=\nonumber \\&\quad =\underset{=0}{\underbrace{\frac{\partial \rho }{\partial t}}}+\rho \cdot \mathrm {div}\,\mathbf {u}_{T}+\mathbf {u}_{T}\cdot \underset{=0}{\underbrace{\mathrm {grad}\,\rho }}=0 , \end{aligned}$$

(1.2)

thus, we can write as follows

$$\begin{aligned} \rho \cdot \left( \nabla \cdot \mathbf {u}_{T}\right) =\rho \cdot \mathrm {div}\,\mathbf {u}_{T}=0 , \end{aligned}$$

(1.3)

which can be divided by the constant density of the fluid $\rho$. Consequently, for incompressible turbulent flows, the mass conservation (continuity) equation holds the incompressibility (divergence-free) constraint which is assumed to be valid in the instantaneous (turbulent) velocity field $\mathbf {u}_{T}$ as

$$\begin{aligned} \nabla \cdot \mathbf {u}_{T}=\mathrm {div}\,\mathbf {u}_{T}=0 . \end{aligned}$$

(1.4)

In other words, Eq. (1.4) is the mass conservation (continuity) equation in the instantaneous velocity field of incompressible turbulent flows.

Fig. 1.1

Reynolds decomposition [108] of the instantaneous (turbulent) velocity vector $\left( \mathbf {u}_{T}=\mathbf {u}+\mathbf {u}^{\prime }\right)$ into the sum of the mean velocity vector $\mathbf {u}$ and the fluctuating velocity vector $\mathbf {u}^{\prime }$ at an arbitrarily chosen point ‘P’ in the space of the turbulent flow field near to or far from a solid boundary: $x_{1}$, $x_{2}$, $x_{3}$ are axes of the physical coordinate system

According to the Reynolds decomposition [108], each physical quantity in the instantaneous flow field can be decomposed into the sum of a mean and a fluctuating component, thus the instantaneous velocity field is

$$\begin{aligned} \mathbf {u}_{T}=\mathbf {u}+\mathbf {u}^{\prime } , \end{aligned}$$

(1.5)

where $\mathbf {u}$ is the mean velocity field and $\mathbf {u}^{\prime }$ is the fluctuating velocity field. The Reynolds decomposition of the instantaneous velocity field (1.5) at an arbitrarily chosen point ‘P’ of the turbulent flow field is shown in Fig. 1.1. Note that Eq. (1.5) follows the triangle law of vector addition in conjunction with the mean $\mathbf {u}$ and the fluctuating $\mathbf {u}^{\prime }$ velocity vectors. Therefore, the velocity triangle of the Reynolds decomposition (1.5) as shown in Fig. 1.1 may also be called as the Reynolds triangle of the turbulent velocity field. In other words, the Reynolds triangle is a geometrical representation of the Reynolds decomposition of the instantaneous (turbulent) velocity field (1.5).

The substitution of the Reynolds decomposition of the instantaneous velocity field (1.5) into the mass conservation equation (1.4) leads to

$$\begin{aligned} \nabla \cdot \mathbf {u}_{T}=\nabla \cdot \left( \mathbf {u}+\mathbf {u}^{\prime }\right) =\nabla \cdot \mathbf {u}+\nabla \cdot \mathbf {u}^{\prime }=0 , \end{aligned}$$

(1.6)

which is the Reynolds decomposition of the continuity equation (1.4) in the instantaneous (turbulent) velocity field (1.5). For incompressible turbulent flows, in order to satisfy the continuity equation (1.6) in the instantaneous velocity field, the mass conservation equation must hold the incompressibility (divergence-free) constraint in the mean velocity field $\mathbf {u}$ as

$$\begin{aligned} \nabla \cdot \mathbf {u}=\frac{\partial u_{1}}{\partial x_{1}}+\frac{\partial u_{2}}{\partial x_{2}}+\frac{\partial u_{3}}{\partial x_{3}}=\sum _{i=1}^{3}\frac{\partial u_{i}}{\partial x_{i}}\equiv \frac{\partial u_{i}}{\partial x_{i}}=0 , \end{aligned}$$

(1.7)

where the equivalent symbol ‘$\equiv$’ denotes the use of Einstein’s summation convention [36] where one can omit the summation symbol using the rules of Cartesian index notation [4, 60]. The incompressible continuity equation must also be satisfied in the fluctuating velocity field $\mathbf {u}^{\prime }$ as well as

$$\begin{aligned} \nabla \cdot \mathbf {u}^{\prime }=\frac{\partial u_{1}^{\prime }}{\partial x_{1}}+\frac{\partial u_{2}^{\prime }}{\partial x_{2}}+\frac{\partial u_{3}^{\prime }}{\partial x_{3}}=\sum _{i=1}^{3}\frac{\partial u_{i}^{\prime }}{\partial x_{i}}\equiv \frac{\partial u_{i}^{\prime }}{\partial x_{i}}=0 . \end{aligned}$$

(1.8)

The instantaneous vorticity vector $\varvec{\Omega }_{T}$ can easily be defined and introduced by taking the rotation (curl) of the instantaneous velocity field (1.5) as

$$\begin{aligned} \varvec{\Omega }_{T}=\mathrm {rot}\,\mathbf {u}_{T}=\mathrm {curl}\,\mathbf {u}_{T}=\nabla \times \mathbf {u}_{T} , \end{aligned}$$

(1.9)

which can also be written as

$$\begin{aligned} \varvec{\Omega }_{T}=\nabla \times \mathbf {u}_{T}=\nabla \times \left( \mathbf {u}+\mathbf {u}^{\prime }\right) =\nabla \times \mathbf {u}+\nabla \times \mathbf {u}^{\prime }=\varvec{\Omega }+\varvec{\Omega }^{\prime } , \end{aligned}$$

(1.10)

thus the Reynolds decomposition [108] is also valid for the instantaneous vorticity field $\varvec{\Omega }_{T}$, which can also be decomposed into the sum of the mean vorticity field $\varvec{\Omega }$ and the fluctuating vorticity field $\varvec{\Omega }^{\prime }$. The mean vorticity vector $\varvec{\Omega }$ is the rotation (curl) of the mean velocity field $\mathbf {u}$ as

$$\begin{aligned}&\varvec{\Omega }=\mathrm {rot}\,\mathbf {u}=\mathrm {curl}\,\mathbf {u}=\nabla \times \mathbf {u}=\left( \frac{\partial u_{3}}{\partial x_{2}}-\frac{\partial u_{2}}{\partial x_{3}}\right) \mathbf {e}_{1}+ \nonumber \\&+\left( \frac{\partial u_{1}}{\partial x_{3}}-\frac{\partial u_{3}}{\partial x_{1}}\right) \mathbf {e}_{2}+\left( \frac{\partial u_{2}}{\partial x_{1}}-\frac{\partial u_{1}}{\partial x_{2}}\right) \mathbf {e}_{3}\equiv \epsilon _{ijk}\frac{\partial u_{k}}{\partial x_{j}} , \end{aligned}$$

(1.11)

and the fluctuating vorticity vector $\varvec{\Omega }^{\prime }$ can be expressed by

$$\begin{aligned}&\varvec{\Omega }^{\prime }=\mathrm {rot}\,\mathbf {u}^{\prime }=\mathrm {curl}\,\mathbf {u}^{\prime }=\nabla \times \mathbf {u}^{\prime }=\left( \frac{\partial u_{3}^{\prime }}{\partial x_{2}}-\frac{\partial u_{2}^{\prime }}{\partial x_{3}}\right) \mathbf {e}_{1}+ \nonumber \\&+\left( \frac{\partial u_{1}^{\prime }}{\partial x_{3}}-\frac{\partial u_{3}^{\prime }}{\partial x_{1}}\right) \mathbf {e}_{2}+\left( \frac{\partial u_{2}^{\prime }}{\partial x_{1}}-\frac{\partial u_{1}^{\prime }}{\partial x_{2}}\right) \mathbf {e}_{3}\equiv \epsilon _{ijk}\frac{\partial u_{k}^{\prime }}{\partial x_{j}} , \end{aligned}$$

(1.12)

where $\epsilon _{ijk}$ is the Levi-Civita permutation symbol which is also known as the alternating symbol [4, 60]. According to the vector analysis and tensor calculus [4, 60], the divergence of the rotation (curl) of an arbitrarily chosen vector field $\mathbf {a}$ is equal to zero as

$$\begin{aligned} \mathrm {div}\,\left( \mathrm {rot}\,\mathbf {a}\right) =\nabla \cdot \left( \nabla \times \mathbf {a}\right) =0 , \end{aligned}$$

(1.13)

which implies that the divergence of the instantaneous vorticity field (1.10) is always equal to zero, therefore we can write

$$\begin{aligned}&\nabla \cdot \varvec{\Omega }_{T}=\nabla \cdot \left( \nabla \times \mathbf {u}_{T}\right) =\nabla \cdot \left[ \nabla \times \left( \mathbf {u}+\mathbf {u}^{\prime }\right) \right] = \nonumber \\&=\nabla \cdot \left( \nabla \times \mathbf {u}\right) +\nabla \cdot \left( \nabla \times \mathbf {u}^{\prime }\right) =\nabla \cdot \varvec{\Omega }+\nabla \cdot \varvec{\Omega }^{\prime }=0 . \end{aligned}$$

(1.14)

Relying on Eqs. (1.13) and (1.14), the mean vorticity field (1.11) and the fluctuating vorticity field (1.12) are always divergence-free (solenoidal) vector fields for both incompressible and compressible fluid flows as

$$\begin{aligned} \nabla \cdot \varvec{\Omega }=\mathrm {div}\,\varvec{\Omega }=0,\,\,\,\,\,\mathrm {and}\,\,\,\,\,\nabla \cdot \varvec{\Omega }^{\prime }=\mathrm {div}\,\varvec{\Omega }^{\prime }=0 . \end{aligned}$$

(1.15)

The incompressibility of turbulent flows means that the instantaneous, mean and fluctuating velocity fields relying on Eqs. (1.4), (1.7) and (1.8) are required to be divergence-free, the fluid density $\rho$ is assumed to be constant, therefore, the density fluctuation function $\rho ^{\prime }\left( \mathbf {x},t\right)$ is equal to zero. Note that the mean velocity vector $\mathbf {u}$ itself is not Galilean invariant, however, the velocity fluctuation vector $\mathbf {u}^{\prime }$ and the instantaneous, mean and fluctuating vorticity fields $\varvec{\Omega }_{T}$, $\varvec{\Omega }$ and $\varvec{\Omega }^{\prime }$ are Galilean invariants [35, 96] (see Sect. 2.2).

1.2.2 The Navier–Stokes Momentum Equation in the Instantaneous Velocity Field

The vectorial form of the Navier–Stokes momentum equation in the instantaneous (turbulent) velocity field $\mathbf {u}_{T}$ can be derived from the instantaneous general Cauchy momentum equation which can be written as

$$\begin{aligned} \rho \frac{\partial \mathbf {u}_{T}}{\partial t}+\rho \nabla \cdot \left( \mathbf {u}_{T}\otimes \mathbf {u}_{T}\right) =\rho \mathbf {g}-\nabla p_{T}+\nabla \cdot \underline{\underline{\tau _{T}}} , \end{aligned}$$

(1.16)

where the symbol ‘$\otimes$’ denotes the dyad (tensor) product and the twice underline ‘$=$’ refers to a second-rank tensor, $\rho$ is the density of the fluid, $\mathbf {u}_{T}$ is the instantaneous velocity vector, $\mathbf {g}$ is the gravitational body force vector, $p_{T}$ represents the instantaneous pressure field. According to the Navier–Stokes hypothesis on the viscous stress tensor—which was proposed for compressible flows—the instantaneous viscous stress tensor can be defined by

$$\begin{aligned} \underline{\underline{\tau _{T}}}=2\mu \underline{\underline{S_{T}}}-\frac{2}{3}\mu \left( \nabla \cdot \mathbf {u}_{T}\right) \cdot \underline{\underline{\mathrm {I}}}\, , \end{aligned}$$

(1.17)

where the dynamic viscosity of the fluid $\mu$ is defined by the product of the density $\rho$ and the kinematic viscosity $\nu$ of the fluid as

$$\begin{aligned} \mu =\rho \nu , \end{aligned}$$

(1.18)

and the instantaneous rate-of-strain (deformation) tensor is given by

$$\begin{aligned} \underline{\underline{S_{T}}}=\frac{1}{2}\left[ \left( \nabla \otimes \mathbf {u}_{T}\right) +\left( \nabla \otimes \mathbf {u}_{T}\right) ^{T}\right] , \end{aligned}$$

(1.19)

which is a symmetrical tensor defined by the half of the sum of the instantaneous velocity gradient tensor ($\nabla \otimes \mathbf {u}_{T}$) and its transpose $\left( \nabla \otimes \mathbf {u}_{T}\right) ^{T}$. The unit tensor $\underline{\underline{\mathrm {I}}}$ can be defined by the sum of the dyad product of each unit vector, and its vectorial, matrix and Cartesian index notation forms are given by

$$\begin{aligned} \underline{\underline{\mathrm {I}}}=\left( \mathbf {e}_{1}\otimes \mathbf {e}_{1}\right) +\left( \mathbf {e}_{2}\otimes \mathbf {e}_{2}\right) +\left( \mathbf {e}_{3}\otimes \mathbf {e}_{3}\right) =\left[ \begin{array}{ccc} 1 &{} 0 &{} 0\\ 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 1 \end{array}\right] \equiv \delta _{ij} , \end{aligned}$$

(1.20)

where $\delta _{ij}$ is the Kronecker delta [4, 60]. By taking into account the mass conservation (continuity) equation (1.4) in the instantaneous velocity field $\mathbf {u}_{T}$ for incompressible turbulent flows, the Navier–Stokes hypothesis on the instantaneous viscous stress tensor (1.17) will be simplified to

$$\begin{aligned} \underline{\underline{\tau _{T}}}=2\mu \underline{\underline{S_{T}}} , \end{aligned}$$

(1.21)

which means that the instantaneous viscous stress tensor is proportional to the instantaneous rate-of-strain (deformation) tensor (1.19) for incompressible turbulent flows. To obtain the vectorial form of the Navier–Stokes equations in the instantaneous (turbulent) velocity field—through the instantaneous general Cauchy momentum equation (1.16) and the viscous stress tensor (1.21)—the tensor divergence of the instantaneous viscous stress tensor (1.21) has to be derived. In order to take the tensor divergence of the viscous stress tensor (1.21), the following vector identity has to be considered by

$$\begin{aligned} \nabla \cdot \left[ \left( \nabla \otimes \mathbf {a}\right) +\left( \nabla \otimes \mathbf {a}\right) ^{T}\right] =\nabla ^{2}\mathbf {a}+\nabla \left( \nabla \cdot \mathbf {a}\right) , \end{aligned}$$

(1.22)

which is valid for any arbitrarily chosen $\mathbf {a}$ vector field, and where $\nabla ^{2}$ denotes the scalar-type second-order Laplace differential operator. By setting $\mathbf {a}=\mathbf {u}_{T}$ and taking under consideration that the dynamic viscosity of the fluid (1.18) is constant for incompressible turbulent flows, the tensor divergence of the instantaneous viscous stress tensor (1.21) can be derived by

$$\begin{aligned}&\mathrm {Div}\,\underline{\underline{\tau _{T}}}=\nabla \cdot \underline{\underline{\tau _{T}}}=\nabla \cdot \left( 2\mu \underline{\underline{S_{T}}}\right) =\nabla \cdot \left\{ 2\mu \frac{1}{2}\left[ \left( \nabla \otimes \mathbf {u}_{T}\right) +\left( \nabla \otimes \mathbf {u}_{T}\right) ^{T}\right] \right\} = \nonumber \\&\qquad =\mu \nabla \cdot \left[ \left( \nabla \otimes \mathbf {u}_{T}\right) +\left( \nabla \otimes \mathbf {u}_{T}\right) ^{T}\right] =\mu \nabla ^{2}\mathbf {u}_{T}+\mu \nabla \underset{=0}{\underbrace{\left( \nabla \cdot \mathbf {u}_{T}\right) }} , \end{aligned}$$

(1.23)

where the second vector divergence term on the right hand side vanishes due to the mass conservation (continuity) equation (1.4) or incompressibility (divergence-free) constraint (1.4) in the instantaneous velocity field $\mathbf {u}_{T}$. Consequently, the tensor divergence of the instantaneous viscous stress tensor (1.23) for incompressible turbulent flows can finally be written as

$$\begin{aligned} \mathrm {Div}\,\underline{\underline{\tau _{T}}}=\nabla \cdot \underline{\underline{\tau _{T}}}=\mu \nabla ^{2}\mathbf {u}_{T} . \end{aligned}$$

(1.24)

The substitution of the tensor divergence (1.24) into the instantaneous general Cauchy momentum equation (1.16) leads to the vectorial form of the Navier–Stokes equations in the instantaneous velocity field $\mathbf {u}_{T}$ as

$$\begin{aligned} \rho \frac{\partial \mathbf {u}_{T}}{\partial t}+\rho \nabla \cdot \left( \mathbf {u}_{T}\otimes \mathbf {u}_{T}\right) =\rho \mathbf {g}-\nabla p_{T}+\mu \nabla ^{2}\mathbf {u}_{T} . \end{aligned}$$

(1.25)

Note that the convective/advective term of the instantaneous Navier–Stokes equation (1.25) can be written in different mathematical forms. By considering the vector identity for the conservative mathematical form of the convective term on the left hand side of the Navier–Stokes equation (1.25) as

$$\begin{aligned} \nabla \cdot \left( \mathbf {a}\otimes \mathbf {b}\right) =\left( \nabla \cdot \mathbf {a}\right) \mathbf {b}+\left( \mathbf {a}\cdot \nabla \right) \mathbf {b}=\left( \nabla \cdot \mathbf {a}\right) \mathbf {b}+\mathbf {a}\cdot \left( \nabla \otimes \mathbf {b}\right) , \end{aligned}$$

(1.26)

and setting $\mathbf {a}=\mathbf {b}=\mathbf {u}_{T}$, furthermore, taking into account the mass conservation (continuity) equation (1.4) of incompressible flows, we can write

$$\begin{aligned}&\nabla \cdot \left( \mathbf {u}_{T}\otimes \mathbf {u}_{T}\right) =\underset{=0}{\underbrace{\left( \nabla \cdot \mathbf {u}_{T}\right) }}\,\,\mathbf {u}_{T}+\left( \mathbf {u}_{T}\cdot \nabla \right) \mathbf {u}_{T}=\nonumber \\&\qquad \;\;\;=\left( \mathbf {u}_{T}\cdot \nabla \right) \mathbf {u}_{T}=\mathbf {u}_{T}\cdot \left( \nabla \otimes \mathbf {u}_{T}\right) , \end{aligned}$$

(1.27)

thus the vectorial form the instantaneous Navier–Stokes equation (1.25) through the convective/advective term (1.27) can be expressed by

$$\begin{aligned} \rho \frac{\partial \mathbf {u}_{T}}{\partial t}+\rho \left( \mathbf {u}_{T}\cdot \nabla \right) \mathbf {u}_{T}=\rho \mathbf {g}-\nabla p_{T}+\mu \nabla ^{2}\mathbf {u}_{T} , \end{aligned}$$

(1.28)

which can also be written as

$$\begin{aligned} \rho \frac{\partial \mathbf {u}_{T}}{\partial t}+\rho \mathbf {u}_{T}\cdot \left( \nabla \otimes \mathbf {u}_{T}\right) =\rho \mathbf {g}-\nabla p_{T}+\mu \nabla ^{2}\mathbf {u}_{T} . \end{aligned}$$

(1.29)

Note that the vector identity (1.26) holds for the mean velocity field $\mathbf {u}$ and the fluctuating velocity field $\mathbf {u}^{\prime }$ by taking into account the mass conservation (continuity) equations (1.7) and (1.8), respectively. Therefore, we can write

$$\begin{aligned} \nabla \cdot \left( \mathbf {u}\otimes \mathbf {u}\right) =\underset{=0}{\underbrace{\left( \nabla \cdot \mathbf {u}\right) }}\,\,\mathbf {u}+\left( \mathbf {u}\cdot \nabla \right) \mathbf {u}=\left( \mathbf {u}\cdot \nabla \right) \mathbf {u}=\mathbf {u}\cdot \left( \nabla \otimes \mathbf {u}\right) , \end{aligned}$$

(1.30)

and the following identity is also valid as

$$\begin{aligned} \nabla \cdot \left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) =\underset{=0}{\underbrace{\left( \nabla \cdot \mathbf {u}^{\prime }\right) }}\,\,\mathbf {u}^{\prime }+\left( \mathbf {u}^{\prime }\cdot \nabla \right) \mathbf {u}^{\prime }=\left( \mathbf {u}^{\prime }\cdot \nabla \right) \mathbf {u}^{\prime }=\mathbf {u}^{\prime }\cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) . \end{aligned}$$

(1.31)

The equalities (1.30) and (1.31) will be employed subsequently when the mathematical description of the convective term will be considered.

1.2.3 The Reynolds Momentum Equation

For incompressible turbulent flows, Reynolds [108] assumed that the Navier–Stokes equations are valid in the instantaneous velocity field (1.5). Therefore, taking into account that the density of the fluid $\rho$ is constant for incompressible flows, the instantaneous Navier–Stokes momentum equation (1.25) can be written with invariant (Gibbs) notation in the form as

$$\begin{aligned} \frac{\partial \left( \rho \mathbf {u}_{T}\right) }{\partial t}+\nabla \cdot \left( \rho \mathbf {u}_{T}\otimes \mathbf {u}_{T}\right) =\rho \mathbf {g}-\nabla p_{T}+\mu \nabla ^{2}\mathbf {u}_{T} . \end{aligned}$$

(1.32)

The Reynolds decomposition [108] is assumed to be valid for the instantaneous velocity and pressure fields as well as

$$\begin{aligned} \mathbf {u}_{T}=\mathbf {u}+\mathbf {u}^{\prime },\,\,\,\,\,\mathrm {and}\,\,\,\,\,p_{T}=p+p^{\prime } . \end{aligned}$$

(1.33)

The substitution of the Reynolds decomposition of the instantaneous velocity and pressure fields (1.33) into the vectorial form of the instantaneous Navier–Stokes momentum equations (1.32), we can write

$$\begin{aligned}&\frac{\partial \left[ \rho \left( \mathbf {u}+\mathbf {u}^{\prime }\right) \right] }{\partial t}+\nabla \cdot \left[ \rho \left( \mathbf {u}+\mathbf {u}^{\prime }\right) \otimes \left( \mathbf {u}+\mathbf {u}^{\prime }\right) \right] = \nonumber \\&\qquad \quad =\rho \mathbf {g}-\nabla \left( p+p^{\prime }\right) +\mu \nabla ^{2}\left( \mathbf {u}+\mathbf {u}^{\prime }\right) , \end{aligned}$$

(1.34)

which can also be expressed by

$$\begin{aligned} \frac{\partial \left[ \rho \left( \mathbf {u}+\mathbf {u}^{\prime }\right) \right] }{\partial t}+\nabla&\cdot \left[ \rho \left( \mathbf {u}\otimes \mathbf {u}\right) +\rho \left( \mathbf {u}\otimes \mathbf {u}^{\prime }\right) +\rho \left( \mathbf {u}^{\prime }\otimes \mathbf {u}\right) +\rho \left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) \right] =\nonumber \\&=\rho \mathbf {g}-\nabla \left( p+p^{\prime }\right) +\mu \nabla ^{2}\left( \mathbf {u}+\mathbf {u}^{\prime }\right) . \end{aligned}$$

(1.35)

At this point, the time-averaging procedure of Reynolds [108, 111] has to be used, which can be denoted with overbars, and the time-averaged form of the momentum equation (1.35) can be written formally as

$$\begin{aligned} \frac{\partial \left[ \rho \overline{\left( \mathbf {u}+\mathbf {u}^{\prime }\right) }\right] }{\partial t}+\nabla&\cdot \left[ \rho \overline{\left( \mathbf {u}\otimes \mathbf {u}\right) }+\rho \overline{\left( \mathbf {u}\otimes \mathbf {u}^{\prime }\right) }+\rho \overline{\left( \mathbf {u}^{\prime }\otimes \mathbf {u}\right) }+\rho \overline{\left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) }\right] =\nonumber \\&=\rho \mathbf {g}-\nabla \overline{\left( p+p^{\prime }\right) }+\mu \nabla ^{2}\overline{\left( \mathbf {u}+\mathbf {u}^{\prime }\right) } , \end{aligned}$$

(1.36)

where according to the rules of the Reynolds time-averaging procedure [108], a time-averaged mean value is equal to the mean value itself ($\overline{\mathbf {u}}=\mathbf {u}$), and a time-averaged fluctuating value is equal to a zero vector ($\overline{\mathbf {u}^{\prime }}=\mathbf {0}$). Thus, the time-averaged instantaneous velocity field $\overline{\mathbf {u}_{T}}$ can be written as

$$\begin{aligned} \overline{\mathbf {u}_{T}}=\overline{\mathbf {u}+\mathbf {u}^{\prime }}=\overline{\mathbf {u}}+\underset{=\mathbf {0}}{\underbrace{\overline{\mathbf {u}^{\prime }}}}=\mathbf {u} . \end{aligned}$$

(1.37)

The time-averaged dyad (tensor) product of the mean velocity field $\mathbf {u}$ in the momentum equation (1.36) can be expressed by

$$\begin{aligned} \overline{\mathbf {u}\otimes \mathbf {u}}=\overline{\mathbf {u}}\otimes \overline{\mathbf {u}}=\mathbf {u}\otimes \mathbf {u} , \end{aligned}$$

(1.38)

and the second and third tensors on the left hand side of the formally written time-averaged momentum equation (1.36) are

$$\begin{aligned} \overline{\mathbf {u}\otimes \mathbf {u}^{\prime }}=\overline{\mathbf {u}}\otimes \underset{=\mathbf {0}}{\underbrace{\overline{\mathbf {u}^{\prime }}}}=\underline{\underline{\mathrm {O}}},\,\,\,\,\,\mathrm {and}\,\,\,\,\,\overline{\mathbf {u}^{\prime }\otimes \mathbf {u}}=\underset{=\mathbf {0}}{\underbrace{\overline{\mathbf {u}^{\prime }}}}\otimes \overline{\mathbf {u}}=\underline{\underline{\mathrm {O}}} , \end{aligned}$$

(1.39)

where $\underline{\underline{\mathrm {O}}}$ is the second-rank zero tensor. The physical meaning of Eq. (1.39) is that there is no statistical correlation between the time-averaged tensor product of the mean velocity field $\mathbf {u}$ and the fluctuating velocity field $\mathbf {u}^{\prime }$. However, Reynolds [108] assumed that there is a statistical correlation between the time-averaged tensor product of the fluctuating velocity field $\mathbf {u}^{\prime }$, which leads to a second-rank non-zero tensor by

$$\begin{aligned} \overline{\mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }}=\overline{\mathbf {u}^{\prime }}\otimes \overline{\mathbf {u}^{\prime }}\ne \underline{\underline{\mathrm {O}}} . \end{aligned}$$

(1.40)

Furthermore, the time-averaged instantaneous pressure field $\overline{p_{T}}$ in the momentum equation (1.36) can also be written as

$$\begin{aligned} \overline{p_{T}}=\overline{p+p^{\prime }}=\overline{p}+\underset{=0}{\underbrace{\overline{p^{\prime }}}}=p . \end{aligned}$$

(1.41)

Therefore, relying on the time-averaged instantaneous vector and tensor fields defined by Eqs. (1.37)–(1.41), the vectorial form of the time-averaged or Reynolds-averaged momentum equation (1.36) is equal to

$$\begin{aligned} \frac{\partial \left( \rho \mathbf {u}\right) }{\partial t}+\nabla \cdot \left[ \rho \left( \mathbf {u}\otimes \mathbf {u}\right) +\rho \overline{\left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) }\right] =\rho \mathbf {g}-\nabla p+\mu \nabla ^{2}\mathbf {u} , \end{aligned}$$

(1.42)

where the fluid density $\rho$ is a constant value for incompressible turbulent flows, therefore, the Reynolds-Averaged Navier–Stokes (RANS) or Reynolds momentum equation (1.42) can also be written as

$$\begin{aligned} \rho \frac{\partial \mathbf {u}}{\partial t}+\rho \nabla \cdot \left( \mathbf {u}\otimes \mathbf {u}\right) =\rho \mathbf {g}-\nabla p+\mu \nabla ^{2}\mathbf {u}+\nabla \cdot \left( -\rho \overline{\mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }}\right) , \end{aligned}$$

(1.43)

where the symmetrical Reynolds stress tensor [108] is defined by

$$\begin{aligned} \underline{\underline{\tau }}^{R}=-\rho \overline{\mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }} . \end{aligned}$$

(1.44)

Using Eq. (1.30) for the convective/advective term of Eq. (1.43), thus the vectorial form of the Reynolds (RANS) momentum equation (1.43) can also be written with invariant (Gibbs) notation as

$$\begin{aligned} \rho \frac{D\mathbf {u}}{Dt}=\rho \frac{\partial \mathbf {u}}{\partial t}+\rho \mathbf {u}\cdot \left( \nabla \otimes \mathbf {u}\right) =\rho \mathbf {g}-\nabla p+\mu \nabla ^{2}\mathbf {u}+\nabla \cdot \underline{\underline{\tau }}^{R} , \end{aligned}$$

(1.45)

where the unsteady term on the left hand side of Eq. (1.45) is

$$\begin{aligned} \rho \frac{\partial \mathbf {u}}{\partial t}=\rho \frac{\partial }{\partial t}\left( u_{1}\mathbf {e}_{1}+u_{2}\mathbf {e}_{2}+u_{3}\mathbf {e}_{3}\right) =\rho \frac{\partial }{\partial t}\sum _{i=1}^{3}\left( u_{i}\cdot \mathbf {e}_{i}\right) \equiv \rho \frac{\partial u_{i}}{\partial t} , \end{aligned}$$

(1.46)

and the non-linear convective/advective term can be expressed by

$$\begin{aligned}&\qquad \;\;\rho \mathbf {u}\cdot \left( \nabla \otimes \mathbf {u}\right) =\rho \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}=\rho \left( u_{1}\frac{\partial u_{1}}{\partial x_{1}}+u_{2}\frac{\partial u_{1}}{\partial x_{2}}+u_{3}\frac{\partial u_{1}}{\partial x_{3}}\right) \mathbf {e}_{1}+ \nonumber \\&+\rho \left( u_{1}\frac{\partial u_{2}}{\partial x_{1}}+u_{2}\frac{\partial u_{2}}{\partial x_{2}}+u_{3}\frac{\partial u_{2}}{\partial x_{3}}\right) \mathbf {e}_{2}+\rho \left( u_{1}\frac{\partial u_{3}}{\partial x_{1}}+u_{2}\frac{\partial u_{3}}{\partial x_{2}}+u_{3}\frac{\partial u_{3}}{\partial x_{3}}\right) \mathbf {e}_{3}= \nonumber \\&\qquad \qquad \qquad \qquad \qquad =\rho \sum _{j=1}^{3}\left[ \sum _{i=1}^{3}\left( u_{i}\frac{\partial u_{j}}{\partial x_{i}}\cdot \mathbf {e}_{j}\right) \right] \equiv \rho u_{i}\frac{\partial u_{j}}{\partial x_{i}} . \end{aligned}$$

(1.47)

The first term on the right hand side of the Reynolds momentum equation (1.45) represents the gravitational body force which is an external conservative force field and its vectorial form can be given by

$$\begin{aligned} \rho \left( g_{1}\mathbf {e}_{1}+g_{2}\mathbf {e}_{2}+g_{3}\mathbf {e}_{3}\right) =\rho \sum _{i=1}^{3}\left( g_{i}\cdot \mathbf {e}_{i}\right) \equiv \rho g_{i} , \end{aligned}$$

(1.48)

and the second term is the mean pressure gradient of the turbulent flow as

$$\begin{aligned} \mathrm {grad}\,p=\nabla p=\frac{\partial p}{\partial x_{1}}\mathbf {e}_{1}+\frac{\partial p}{\partial x_{2}}\mathbf {e}_{2}+\frac{\partial p}{\partial x_{3}}\mathbf {e}_{3}=\sum _{i=1}^{3}\left( \frac{\partial p}{\partial x_{i}}\cdot \mathbf {e}_{i}\right) \equiv \frac{\partial p}{\partial x_{i}} , \end{aligned}$$

(1.49)

and the third second-order Laplacian term represents the molecular diffusion due to viscous effects which can be expressed by

$$\begin{aligned}&\mu \nabla ^{2}\mathbf {u}=\mu \left( \frac{\partial ^{2}u_{1}}{\partial x_{1}^{2}}+\frac{\partial ^{2}u_{1}}{\partial x_{2}^{2}}+\frac{\partial ^{2}u_{1}}{\partial x_{3}^{2}}\right) \mathbf {e}_{1}+\mu \left( \frac{\partial ^{2}u_{2}}{\partial x_{1}^{2}}+\frac{\partial ^{2}u_{2}}{\partial x_{2}^{2}}+\frac{\partial ^{2}u_{2}}{\partial x_{3}^{2}}\right) \mathbf {e}_{2}+ \nonumber \\&+\mu \left( \frac{\partial ^{2}u_{3}}{\partial x_{1}^{2}}+\frac{\partial ^{2}u_{3}}{\partial x_{2}^{2}}+\frac{\partial ^{2}u_{3}}{\partial x_{3}^{2}}\right) \mathbf {e}_{3}=\mu \sum _{j=1}^{3}\left[ \sum _{i=1}^{3}\left( \frac{\partial ^{2}u_{j}}{\partial x_{i}^{2}}\cdot \mathbf {e}_{j}\right) \right] \equiv \mu \frac{\partial ^{2}u_{j}}{\partial x_{i}\partial x_{i}} . \end{aligned}$$

(1.50)

Using Eqs. (1.46)–(1.50), the Reynolds (RANS) momentum equation (1.45) can also be written by Cartesian index notation as

$$\begin{aligned} \rho \frac{Du_{i}}{Dt}=\rho \frac{\partial u_{i}}{\partial t}+\rho u_{i}\frac{\partial u_{j}}{\partial x_{i}}=\rho g_{i}-\frac{\partial p}{\partial x_{i}}+\mu \frac{\partial ^{2}u_{j}}{\partial x_{i}\partial x_{i}}+\frac{\partial \tau _{ij}^{R}}{\partial x_{i}} . \end{aligned}$$

(1.51)

where the Reynolds stress tensor (1.44) can also be given by

$$\begin{aligned} \tau _{ij}^{R}=-\rho \overline{u_{i}^{\prime }u_{j}^{\prime }} . \end{aligned}$$

(1.52)

Note that the Reynolds stress tensor expressed by Eqs. (1.44) and (1.52) is a symmetrical tensor which means that it is equal to its transpose $\tau _{ij}^{R}=\tau _{ji}^{R}$. Therefore, the tensor divergence in the last term on the right hand side of the Reynolds momentum equation (1.51) with respect to the indicies i and j remains unchanged, respectively. Therefore, we can also write

$$\begin{aligned} \frac{\partial \tau _{ij}^{R}}{\partial x_{i}}=\frac{\partial \tau _{ij}^{R}}{\partial x_{j}} . \end{aligned}$$

(1.53)

The relationship between the Reynolds decomposition [108] of the instantaneous velocity field (1.5) at an arbitrarily chosen point $'P'$ in the space and simulation techniques/modelling approaches is shown in Fig. 1.2.

Fig. 1.2

Reynolds decomposition [108] of the instantaneous (turbulent) velocity field $\left( \mathbf {u}_{T}=\mathbf {u}+\mathbf {u}^{\prime }\right)$ at an arbitrarily chosen point $'P'$ in the turbulent flow field related to simulation techniques (DNS, ILES) and modelling approaches (URANS, RANS, STM)

The instantaneous velocity $\mathbf {u}_{T}$ and pressure $p_{T}$ fields can be computed with Direct Numerical Simulation (DNS) and Implicit Large-Eddy Simulation (ILES) techniques (see Fig. 1.2). The mean $\mathbf {u}$ and fluctuating velocity $\mathbf {u}^{\prime }$ fields can be predicted through URANS, RANS and Stochastic Turbulence Modelling (STM) approaches, respectively (see Chaps. 3, 4, and 5).

1.2.4 The Reynolds Stress Tensor and Its Relation to Isotropic and Anisotropic Turbulent Flows

For incompressible turbulent flows, the symmetrical Reynolds stress tensor expressed by Eqs. (1.44) and (1.52) can also be written in matrix form by

$$\begin{aligned} \underline{\underline{\tau }}^{R}=-\rho \overline{\mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }}=-\rho \left[ \begin{array}{ccc} \overline{u_{1}^{\prime }u_{1}^{\prime }}\,\,\,\, &{} \overline{u_{1}^{\prime }u_{2}^{\prime }}\,\,\,\, &{} \overline{u_{1}^{\prime }u_{3}^{\prime }}\\ \overline{u_{2}^{\prime }u_{1}^{\prime }}\,\,\,\, &{} \overline{u_{2}^{\prime }u_{2}^{\prime }}\,\,\,\, &{} \overline{u_{2}^{\prime }u_{3}^{\prime }}\\ \overline{u_{3}^{\prime }u_{1}^{\prime }}\,\,\,\, &{} \overline{u_{3}^{\prime }u_{2}^{\prime }}\,\,\,\, &{} \overline{u_{3}^{\prime }u_{3}^{\prime }} \end{array}\right] , \end{aligned}$$

(1.54)

where the homogeneous isotropic simplification of its mathematical and physical description was proposed by Taylor [125,126,127,128,129,130]. It means that all normal components of the Reynolds stress tensor (1.54) are equal to each other, therefore, we can write

$$\begin{aligned} -\rho \overline{u_{1}^{\prime }u_{1}^{\prime }}=-\rho \overline{u_{2}^{\prime }u_{2}^{\prime }}=-\rho \overline{u_{3}^{\prime }u_{3}^{\prime }} , \end{aligned}$$

(1.55)

and due to the isotropic simplification of the physical description of turbulent flows, all non-diagonal shear stress components in the Reynolds stress tensor (1.54) are assumed to be equal to zero as

$$\begin{aligned} -\rho \overline{u_{1}^{\prime }u_{2}^{\prime }}=-\rho \overline{u_{2}^{\prime }u_{1}^{\prime }}=0 , \end{aligned}$$

(1.56)

$$\begin{aligned} -\rho \overline{u_{1}^{\prime }u_{3}^{\prime }}=-\rho \overline{u_{3}^{\prime }u_{1}^{\prime }}=0 , \end{aligned}$$

(1.57)

$$\begin{aligned} -\rho \overline{u_{2}^{\prime }u_{3}^{\prime }}=-\rho \overline{u_{3}^{\prime }u_{2}^{\prime }}=0 . \end{aligned}$$

(1.58)

Taylor’s isotropic simplification in the mathematical and physical description of the Reynolds stress tensor (1.54) suggests that Eqs. (1.55)–(1.58) may approximately be satisfied by considering only special physical circumstances, e.g. turbulence far from any solid boundary. Even if we consider the differences between all normal Reynolds stresses (1.55) to be very small, it may also be difficult to believe by physical intuition that all isotropic turbulent flow conditions, i.e. Eqs. (1.55)–(1.58), can be exactly satisfied. Taylor’s isotropic turbulence assumption is a significant simplification compared to real turbulent flows, because all isotropic turbulent flow conditions according to Eqs. (1.55)–(1.58) could easily break down near to any solid boundary when the effect of shear stresses becomes dominant. Therefore, for those turbulent flows, when the effect of any solid boundary is present, an anisotropic mathematical and physical description of the Reynolds stress tensor (1.54) is required. It means that all normal stress components of the Reynolds stress tensor (1.54) are not equal to each other, therefore, we can write

$$\begin{aligned} -\rho \overline{u_{1}^{\prime }u_{1}^{\prime }}\ne -\rho \overline{u_{2}^{\prime }u_{2}^{\prime }}\ne -\rho \overline{u_{3}^{\prime }u_{3}^{\prime }} , \end{aligned}$$

(1.59)

and because of the anisotropic mathematical and physical description of turbulent flows, all non-diagonal shear stress components in the Reynolds stress tensor (1.54) are assumed to be not equal to zero as

$$\begin{aligned} -\rho \overline{u_{1}^{\prime }u_{2}^{\prime }}=-\rho \overline{u_{2}^{\prime }u_{1}^{\prime }}\ne 0 , \end{aligned}$$

(1.60)

$$\begin{aligned} -\rho \overline{u_{1}^{\prime }u_{3}^{\prime }}=-\rho \overline{u_{3}^{\prime }u_{1}^{\prime }}\ne 0 , \end{aligned}$$

(1.61)

$$\begin{aligned} -\rho \overline{u_{2}^{\prime }u_{3}^{\prime }}=-\rho \overline{u_{3}^{\prime }u_{2}^{\prime }}\ne 0 . \end{aligned}$$

(1.62)

The heart of the closure problem of anisotropic turbulence is to find a plausible mathematical and physical description of the anisotropic Reynolds stresses expressed by Eqs. (1.59)–(1.62). In other words, the closure problem of anisotropic turbulence is about the physically correct mathematical description of all fluctuating components. When a hypothesis is proposed to describe the anisotropic physical behaviour of the Reynolds stress tensor (1.54), the tensor divergence of the Reynolds stress tensor will be changed in the last term of the Reynolds momentum equation, see Eqs. (1.45) and (1.51). Since, there is no unified physical theory to describe anisotropic turbulent flows in conjunction with three-dimensional complex problems in general, therefore, the scientific research on this subject is in the centre of interest nowadays. The anisotropic conditions for the Reynolds stresses according to Eqs. (1.59)–(1.62) could represent more realistic physical circumstances of turbulent flows in general compared to Taylor’s isotropic turbulence assumptions relying on Eqs. (1.55)–(1.58). Therefore, a new anisotropic hypothesis on the Reynolds stress tensor with the addition of a new tensorial term have been proposed in the present book (see Chap. 5). The new hypothesis is relying on the unification of the generalised Boussinesq [10, 32] and the fully Galilean invariant revised three-dimensional similarity hypothesis of oscillatory motions [22, 23]. In other words, the new hypothesis is an anisotropic modification of the generalised Boussinesq hypothesis on the Reynolds stress tensor [10, 32] based on the three-dimensional similarity theory of turbulent velocity fluctuations [22, 23]. The anisotropic hypothesis proposed in Chap. 5 is a different approach to the physics of anisotropic turbulence compared to other recent works in [3, 5, 6, 20, 21, 135].

1.2.5 Mathematical Derivation of the Turbulent Kinetic Energy Transport Equation

The derivation of the turbulent kinetic energy transport equation k includes the consideration of more intermediate mathematical steps compared to the derivation of the mass conservation (continuity) equation (1.7) and the Reynolds momentum equation (1.45) in Sects. 1.2.1 and 1.2.3, respectively. In most textbooks, the full derivation of the turbulent kinetic energy transport equation is left with the reader as an exercise due to its lengthy character. However, the precise understanding of each intermediate derivation step is crucial for undergraduate and postgraduate students in order to develop new one- or two-equation closure models for turbulent flows. Therefore, the full mathematical derivation of the turbulent kinetic energy transport equation and the physical explanation of each term have been provided here.

The turbulent kinetic energy k is the kinetic energy of turbulent fluctuations per unit mass which is defined by the half of the sum of the diagonal elements of the Reynolds stress tensor (1.54) as

$$\begin{aligned} k=\overline{\frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}}=\frac{1}{2}\left( \overline{u_{1}^{\prime }u_{1}^{\prime }}+\overline{u_{2}^{\prime }u_{2}^{\prime }}+\overline{u_{3}^{\prime }u_{3}^{\prime }}\right) =\frac{1}{2}\sum _{i=1}^{3}\overline{u_{i}^{\prime }\cdot u_{i}^{\prime }}\equiv \frac{1}{2}\overline{u_{i}^{\prime }u_{i}^{\prime }} , \end{aligned}$$

(1.63)

where the equivalent symbol ‘$\equiv$’ denotes again the use of Einstein’s summation convention [36] where one can omit the summation symbol using the Cartesian index notation [4, 60]. The transport equation of the turbulent kinetic energy k can be derived from the Reynolds stress transport equation [16, 32, 106, 137]. However, the multiplication of the instantaneous Navier–Stokes equations (1.28) by the fluctuating velocity vector $\mathbf {u}^{\prime }$ and using the Reynolds time-averaging procedure [108] will lead to the same result.

For incompressible turbulent flows, the starting point of the derivation of the turbulent kinetic energy transport equation k is the vectorial form of the instantaneous Navier–Stokes momentum equation (1.28) given by

$$\begin{aligned} \rho \frac{\partial \mathbf {u}_{T}}{\partial t}+\rho \left( \mathbf {u}_{T}\cdot \nabla \right) \mathbf {u}_{T}=\rho \mathbf {g}-\nabla p_{T}+\mu \nabla ^{2}\mathbf {u}_{T} , \end{aligned}$$

(1.64)

where the Reynolds decomposition [108] of the instantaneous velocity and pressure fields is assumed again to be valid as

$$\begin{aligned} \mathbf {u}_{T}=\mathbf {u}+\mathbf {u}^{\prime },\,\,\,\,\,\mathrm {and}\,\,\,\,\,p_{T}=p+p^{\prime } . \end{aligned}$$

(1.65)

Using the Reynolds decomposition (1.65), the Navier–Stokes equations in the instantaneous velocity field (1.64) can be expressed by

$$\begin{aligned}&\rho \frac{\partial \left( \mathbf {u}+\mathbf {u}^{\prime }\right) }{\partial t}+\rho \left[ \left( \mathbf {u}+\mathbf {u}^{\prime }\right) \cdot \nabla \right] \left( \mathbf {u}+\mathbf {u}^{\prime }\right) = \nonumber \\&\quad \;\;\;=\rho \mathbf {g}-\nabla \left( p+p^{\prime }\right) +\mu \nabla ^{2}\left( \mathbf {u}+\mathbf {u}^{\prime }\right) , \end{aligned}$$

(1.66)

which can also be written as

$$\begin{aligned}&\qquad \qquad \qquad \qquad \rho \frac{\partial \mathbf {u}}{\partial t}+\rho \frac{\partial \mathbf {u}^{\prime }}{\partial t}+\rho \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}+\rho \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}^{\prime }+ \nonumber \\&+\rho \left( \mathbf {u}^{\prime }\cdot \nabla \right) \mathbf {u}+\rho \left( \mathbf {u}^{\prime }\cdot \nabla \right) \mathbf {u}^{\prime }=\rho \mathbf {g}-\nabla p-\nabla p^{\prime }+\mu \nabla ^{2}\mathbf {u}+\mu \nabla ^{2}\mathbf {u}^{\prime } . \end{aligned}$$

(1.67)

To derive the transport equation of the turbulent kinetic energy k, the first step of the derivation is to multiply the instantaneous Navier–Stokes equations (1.67) by the fluctuating velocity vector $\mathbf {u}^{\prime }$, thus we can write

$$\begin{aligned}&\rho \mathbf {u}^{\prime }\cdot \frac{\partial \mathbf {u}}{\partial t}+\rho \mathbf {u}^{\prime }\cdot \frac{\partial \mathbf {u}^{\prime }}{\partial t}+\rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}+\rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}^{\prime }+ \nonumber \\&\qquad \qquad \;+\rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \nabla \right) \mathbf {u}+\rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \nabla \right) \mathbf {u}^{\prime }= \nonumber \\ =\,&\rho \mathbf {u}^{\prime }\cdot \mathbf {g}-\mathbf {u}^{\prime }\cdot \nabla p-\mathbf {u}^{\prime }\cdot \nabla p^{\prime }+\mu \mathbf {u}^{\prime }\cdot \nabla ^{2}\mathbf {u}+\mu \mathbf {u}^{\prime }\cdot \nabla ^{2}\mathbf {u}^{\prime } . \end{aligned}$$

(1.68)

In order to obtain an appropriate mathematical form for the physical interpretation of each term of the turbulent kinetic energy transport equation k, further transformations are required with the use of vector identities. Therefore, the second fluctuating temporal derivative term on the left hand side of the transport equation (1.68) can be expressed with the identity by

$$\begin{aligned} \rho \mathbf {u}^{\prime }\cdot \frac{\partial \mathbf {u}^{\prime }}{\partial t}=\rho \frac{\partial }{\partial t}\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) =\frac{1}{2}\rho \mathbf {u}^{\prime }\cdot \frac{\partial \mathbf {u}^{\prime }}{\partial t}+\frac{1}{2}\rho \mathbf {u}^{\prime }\cdot \frac{\partial \mathbf {u}^{\prime }}{\partial t}=\rho \mathbf {u}^{\prime }\cdot \frac{\partial \mathbf {u}^{\prime }}{\partial t}, \end{aligned}$$

(1.69)

and the fourth non-linear convective/advective term on the left hand side of the transport equation (1.68) can be derived by

$$\begin{aligned}&\qquad \qquad \qquad \rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}^{\prime }=\rho \left( \mathbf {u}\cdot \nabla \right) \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) = \nonumber \\&=\frac{1}{2}\rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}^{\prime }+\frac{1}{2}\rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}^{\prime }=\rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}^{\prime } . \end{aligned}$$

(1.70)

To express the fifth term on the left hand side of the transport equation (1.68) in another mathematical form, the following vector identity—which is valid for any arbitrarily chosen $\mathbf {a}$ and $\mathbf {b}$ vector field–has to be considered by

$$\begin{aligned} \rho \mathbf {a}\cdot \left( \mathbf {a}\cdot \nabla \right) \mathbf {b}=\rho \left( \mathbf {a}\otimes \mathbf {a}\right) \cdot \cdot \left( \nabla \otimes \mathbf {b}\right) , \end{aligned}$$

(1.71)

where the symbol of two consecutive dots ‘$\cdot \cdot$’ denotes the double inner dot (scalar) product of two second-rank tensors, which is often denoted by the symbol ‘ : ’ in the literature [4, 60]. By setting $\mathbf {a}=\mathbf {u}^{\prime }$ and $\mathbf {b}=\mathbf {u}$ in the vector identity (1.71), the fifth non-linear convective/advective term on the left hand side of the transport equation (1.68) can be written as

$$\begin{aligned} \rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \nabla \right) \mathbf {u}=\rho \left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}\right) . \end{aligned}$$

(1.72)

To transform the sixth term on the left hand side of the transport equation (1.68) into a different form, another vector identity—which is also valid for any arbitrarily chosen $\mathbf {a}$ vector field–has to be taken into account by

$$\begin{aligned} \left( \mathbf {a}\cdot \nabla \right) \mathbf {a}=\nabla \left( \frac{\mathbf {a}\cdot \mathbf {a}}{2}\right) -\mathbf {a}\times \left( \nabla \times \mathbf {a}\right) , \end{aligned}$$

(1.73)

which can be multiplied by $\rho \mathbf {a}$, thus we can write

$$\begin{aligned} \rho \mathbf {a}\cdot \left( \mathbf {a}\cdot \nabla \right) \mathbf {a}=\rho \mathbf {a}\cdot \nabla \left( \frac{\mathbf {a}\cdot \mathbf {a}}{2}\right) -\rho \mathbf {a}\cdot \left[ \mathbf {a}\times \left( \nabla \times \mathbf {a}\right) \right] , \end{aligned}$$

(1.74)

where the second term on the right hand side can be expressed by another mathematical form using the vector identity of the scalar triple product as

$$\begin{aligned} \mathbf {a}\cdot \left( \mathbf {b}\times \mathbf {c}\right) =\mathbf {b}\cdot \left( \mathbf {c}\times \mathbf {a}\right) =\mathbf {c}\cdot \left( \mathbf {a}\times \mathbf {b}\right) , \end{aligned}$$

(1.75)

therefore, the vector identity (1.74) can also be written as

$$\begin{aligned}&\qquad \rho \mathbf {a}\cdot \left( \mathbf {a}\cdot \nabla \right) \mathbf {a}=\rho \mathbf {a}\cdot \nabla \left( \frac{\mathbf {a}\cdot \mathbf {a}}{2}\right) -\rho \mathbf {a}\cdot \left[ \mathbf {a}\times \left( \nabla \times \mathbf {a}\right) \right] = \nonumber \\&=\rho \mathbf {a}\cdot \nabla \left( \frac{\mathbf {a}\cdot \mathbf {a}}{2}\right) -\rho \left( \nabla \times \mathbf {a}\right) \cdot \underset{=\mathbf {0}}{\underbrace{\left( \mathbf {a}\times \mathbf {a}\right) }}=\rho \mathbf {a}\cdot \nabla \left( \frac{\mathbf {a}\cdot \mathbf {a}}{2}\right) . \end{aligned}$$

(1.76)

By setting $\mathbf {a}=\mathbf {u}^{\prime }$ in Eq. (1.76), the sixth non-linear convective/advective term on the left hand side of the transport equation (1.68) can be derived as

$$\begin{aligned}&\qquad \rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \nabla \right) \mathbf {u}^{\prime }=\rho \mathbf {u}^{\prime }\cdot \nabla \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) -\rho \mathbf {u}^{\prime }\cdot \left[ \mathbf {u}^{\prime }\times \left( \nabla \times \mathbf {u}^{\prime }\right) \right] = \nonumber \\&=\rho \mathbf {u}^{\prime }\cdot \nabla \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) -\rho \left( \nabla \times \mathbf {u}^{\prime }\right) \cdot \underset{=\mathbf {0}}{\underbrace{\left( \mathbf {u}^{\prime }\times \mathbf {u}^{\prime }\right) }}=\rho \mathbf {u}^{\prime }\cdot \nabla \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) . \end{aligned}$$

(1.77)

Using Eqs. (1.69), (1.70), (1.72) and (1.77), transport equation (1.68) can be transformed into another mathematical form which can be written as

$$\begin{aligned}&\rho \mathbf {u}^{\prime }\cdot \frac{\partial \mathbf {u}}{\partial t}+\rho \frac{\partial }{\partial t}\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) +\rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}+\rho \left( \mathbf {u}\cdot \nabla \right) \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) + \nonumber \\&\qquad \qquad \quad +\rho \left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}\right) +\rho \mathbf {u}^{\prime }\cdot \nabla \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) = \nonumber \\&\quad =\rho \mathbf {u}^{\prime }\cdot \mathbf {g}-\mathbf {u}^{\prime }\cdot \nabla p-\mathbf {u}^{\prime }\cdot \nabla p^{\prime }+\mu \mathbf {u}^{\prime }\cdot \nabla ^{2}\mathbf {u}+\mu \mathbf {u}^{\prime }\cdot \nabla ^{2}\mathbf {u}^{\prime } , \end{aligned}$$

(1.78)

which mathematical form can be re-arranged as

$$\begin{aligned}&\rho \mathbf {u}^{\prime }\cdot \frac{\partial \mathbf {u}}{\partial t}+\rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}+\rho \frac{\partial }{\partial t}\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) +\rho \left( \mathbf {u}\cdot \nabla \right) \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) = \nonumber \\&\qquad \;\;=\rho \mathbf {u}^{\prime }\cdot \mathbf {g}-\rho \left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}\right) -\rho \mathbf {u}^{\prime }\cdot \nabla \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) \nonumber \\&\qquad \qquad \quad -\mathbf {u}^{\prime }\cdot \nabla p-\mathbf {u}^{\prime }\cdot \nabla p^{\prime }+\mu \mathbf {u}^{\prime }\cdot \nabla ^{2}\mathbf {u}+\mu \mathbf {u}^{\prime }\cdot \nabla ^{2}\mathbf {u}^{\prime } . \end{aligned}$$

(1.79)

Before using the time-averaging procedure of Reynolds [108], the last term on the right hand side of the transport equation (1.79)—which is a second-order Laplacian term of the fluctuating velocity field—has to be transformed into another mathematical form for its adequate physical interpretation. For the transformation of the Laplacian term in Eq. (1.79), the vector identity for any arbitrarily chosen $\mathbf {a}$ vector field has to be considered [60] by

$$\begin{aligned} \nabla ^{2}\mathbf {a}+\nabla \left( \nabla \cdot \mathbf {a}\right) =\nabla \cdot \left[ \left( \nabla \otimes \mathbf {a}\right) +\left( \nabla \otimes \mathbf {a}\right) ^{T}\right] , \end{aligned}$$

(1.80)

which can be multiplied by $\mu \mathbf {a}$, thus we can write

$$\begin{aligned}&\qquad \qquad \mu \mathbf {a}\cdot \nabla ^{2}\mathbf {a}+\mu \left( \mathbf {a}\cdot \nabla \right) \left( \nabla \cdot \mathbf {a}\right) =\mu \mathbf {a}\cdot \nabla \cdot \left[ \left( \nabla \otimes \mathbf {a}\right) +\left( \nabla \otimes \mathbf {a}\right) ^{T}\right] = \nonumber \\&=\mu \nabla \cdot \left\{ \mathbf {a}\cdot \left[ \left( \nabla \otimes \mathbf {a}\right) +\left( \nabla \otimes \mathbf {a}\right) ^{T}\right] \right\} -\mu \left[ \left( \nabla \otimes \mathbf {a}\right) +\left( \nabla \otimes \mathbf {a}\right) ^{T}\right] \cdot \cdot \left( \nabla \otimes \mathbf {a}\right) = \nonumber \\&\qquad \qquad \qquad \qquad =\mu \nabla \cdot \left[ \mathbf {a}\cdot \left( \nabla \otimes \mathbf {a}\right) \right] +\mu \nabla \cdot \left[ \mathbf {a}\cdot \left( \nabla \otimes \mathbf {a}\right) ^{T}\right] - \nonumber \\&\qquad \qquad \qquad \qquad \qquad -\mu \left[ \left( \nabla \otimes \mathbf {a}\right) +\left( \nabla \otimes \mathbf {a}\right) ^{T}\right] \cdot \cdot \left( \nabla \otimes \mathbf {a}\right) , \end{aligned}$$

(1.81)

where two vector identities are valid [4, 60] for

$$\begin{aligned} \mathbf {a}\cdot \left( \nabla \otimes \mathbf {a}\right) =\left( \mathbf {a}\cdot \nabla \right) \mathbf {a} , \end{aligned}$$

(1.82)

$$\begin{aligned} \mathbf {a}\cdot \left( \nabla \otimes \mathbf {a}\right) ^{T}=\left( \nabla \otimes \mathbf {a}\right) \cdot \mathbf {a}=\nabla \left( \frac{\mathbf {a}\cdot \mathbf {a}}{2}\right) , \end{aligned}$$

(1.83)

therefore, the first two terms on the right hand side of the vector identity (1.81) can also be expressed with vectorial equalities by

$$\begin{aligned} \mu \nabla \cdot \left[ \mathbf {a}\cdot \left( \nabla \otimes \mathbf {a}\right) \right] =\mu \left( \nabla \otimes \mathbf {a}\right) \cdot \cdot \left( \nabla \otimes \mathbf {a}\right) +\mu \left( \mathbf {a}\cdot \nabla \right) \left( \nabla \cdot \mathbf {a}\right) , \end{aligned}$$

(1.84)

$$\begin{aligned} \mu \nabla \cdot \left[ \mathbf {a}\cdot \left( \nabla \otimes \mathbf {a}\right) ^{T}\right] =\mu \nabla \cdot \left[ \left( \nabla \otimes \mathbf {a}\right) \cdot \mathbf {a}\right] =\mu \nabla ^{2}\left( \frac{\mathbf {a}\cdot \mathbf {a}}{2}\right) . \end{aligned}$$

(1.85)

By using Eqs. (1.84), (1.85), and setting $\mathbf {a}=\mathbf {u}^{\prime }$, Eq. (1.81) can be written as

$$\begin{aligned}&\qquad \qquad \qquad \qquad \mu \mathbf {u}^{\prime }\cdot \nabla ^{2}\mathbf {u}^{\prime }+\mu \left( \mathbf {u}^{\prime }\cdot \nabla \right) \left( \nabla \cdot \mathbf {u}^{\prime }\right) = \nonumber \\&=\mu \nabla ^{2}\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) +\mu \left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) +\mu \left( \mathbf {u}^{\prime }\cdot \nabla \right) \left( \nabla \cdot \mathbf {u}^{\prime }\right) \nonumber \\&\qquad \qquad \quad -\mu \left[ \left( \nabla \otimes \mathbf {u}^{\prime }\right) +\left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}\right] \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) , \end{aligned}$$

(1.86)

where the mathematical term $\mu \left( \mathbf {u}^{\prime }\cdot \nabla \right) \left( \nabla \cdot \mathbf {u}^{\prime }\right)$ will cancel out on the left and right hand sides of Eq. (1.86), thus the Laplacian term on the right hand side of the transport equation (1.79) can be expressed by

$$\begin{aligned}&\mu \mathbf {u}^{\prime }\cdot \nabla ^{2}\mathbf {u}^{\prime }=\mu \nabla ^{2}\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) +\mu \left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) \nonumber \\&\qquad \qquad -\mu \left[ \left( \nabla \otimes \mathbf {u}^{\prime }\right) +\left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}\right] \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) . \end{aligned}$$

(1.87)

Taking into account that the double inner dot (scalar) product of two tensors is commutative, the second and third terms on the right hand side of the inhomogeneous Laplace equation (1.87) can also be written as

$$\begin{aligned}&\mu \left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) -\mu \left[ \left( \nabla \otimes \mathbf {u}^{\prime }\right) +\left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}\right] \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) = \nonumber \\&\qquad \qquad =\underset{=0}{\underbrace{\mu \left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) -\mu \left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) }} \nonumber \\&\quad -\mu \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}\cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) =-\mu \left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T} , \end{aligned}$$

(1.88)

therefore, the mathematical form of the Laplacian term (1.87) on the right hand side of the transport equation (1.79) is finally obtained by

$$\begin{aligned} \mu \mathbf {u}^{\prime }\cdot \nabla ^{2}\mathbf {u}^{\prime }=\mu \nabla ^{2}\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) -\mu \left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T} . \end{aligned}$$

(1.89)

Relying on the obtained second-order Laplacian expression (1.89), the transport equation (1.79)—before time-averaging—can be expressed by

$$\begin{aligned}&\rho \mathbf {u}^{\prime }\cdot \frac{\partial \mathbf {u}}{\partial t}+\rho \mathbf {u}^{\prime }\cdot \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}+\rho \frac{\partial }{\partial t}\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) +\rho \left( \mathbf {u}\cdot \nabla \right) \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) = \nonumber \\&\qquad \;\;=\rho \mathbf {u}^{\prime }\cdot \mathbf {g}-\rho \left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}\right) -\rho \mathbf {u}^{\prime }\cdot \nabla \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) \nonumber \\&\qquad \qquad \qquad \quad \;\;-\mathbf {u}^{\prime }\cdot \nabla p-\mathbf {u}^{\prime }\cdot \nabla p^{\prime }+\mu \mathbf {u}^{\prime }\cdot \nabla ^{2}\mathbf {u}+ \nonumber \\&\qquad \qquad \quad +\mu \nabla ^{2}\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) -\mu \left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T} . \end{aligned}$$

(1.90)

The second step of the derivation of the transport equation of the turbulent kinetic energy k is the use of the time-averaging procedure of Reynolds [108] for the instantaneous Navier–Stokes equations multiplied by the fluctuating velocity vector (1.68). In other words, the Reynolds time-averaging of the transport equation (1.68) is equivalent to the time-averaging of the transport equation (1.90). As a consequence, the time-averaged form of the previously obtained transport equation (1.90) can be written as

$$\begin{aligned}&\rho \underset{=\mathbf {0}}{\underbrace{\overline{\mathbf {u}^{\prime }}}}\cdot \overline{\frac{\partial \mathbf {u}}{\partial t}}+\rho \underset{=\mathbf {0}}{\underbrace{\overline{\mathbf {u}^{\prime }}}}\cdot \left( \overline{\mathbf {u}}\cdot \nabla \right) \overline{\mathbf {u}}+\rho \frac{\partial }{\partial t}\underset{\ne 0}{\underbrace{\overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }}}+\rho \left( \overline{\mathbf {u}}\cdot \nabla \right) \underset{\ne 0}{\underbrace{\overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }}}=\nonumber \\&\qquad =\rho \underset{=\mathbf {0}}{\underbrace{\overline{\mathbf {u}^{\prime }}}}\cdot \mathbf {g}-\rho \underset{\ne 0}{\underbrace{\overline{\left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) }\cdot \cdot \left( \nabla \otimes \overline{\mathbf {u}}\right) }}-\rho \underset{\ne 0}{\underbrace{\overline{\mathbf {u}^{\prime }\cdot \nabla \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }}}\nonumber \\&\qquad \qquad \qquad \quad -\underset{=\mathbf {0}}{\underbrace{\overline{\mathbf {u}^{\prime }}}}\cdot \nabla \overline{p}-\underset{\ne 0}{\underbrace{\overline{\mathbf {u}^{\prime }\cdot \nabla p^{\prime }}}}+\mu \underset{=\mathbf {0}}{\underbrace{\overline{\mathbf {u}^{\prime }}}}\cdot \nabla ^{2}\overline{\mathbf {u}}+\nonumber \\&\qquad \qquad \quad +\mu \nabla ^{2}\underset{\ne 0}{\underbrace{\overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }}}-\mu \underset{\ne 0}{\underbrace{\overline{\left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}}}} , \end{aligned}$$

(1.91)

where it is assumed again that a time-averaged mean value is equal to the mean value itself ($\overline{\mathbf {u}}=\mathbf {u}$), and a time-averaged fluctuating value is equal to a zero vector ($\overline{\mathbf {u}^{\prime }}=\mathbf {0}$). Furthermore, it is assumed that there is a correlation between the average value of any two fluctuating physical quantities. Thus, the Reynolds-averaged transport equation (1.91) can also be written as

$$\begin{aligned}&\qquad \qquad \qquad \rho \frac{\partial }{\partial t}\overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }+\rho \left( \mathbf {u}\cdot \nabla \right) \overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }=\nonumber \\&=-\rho \overline{\left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) }\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) -\rho \overline{\mathbf {u}^{\prime }\cdot \nabla \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }-\overline{\mathbf {u}^{\prime }\cdot \nabla p^{\prime }}+\nonumber \\&\qquad \qquad +\mu \nabla ^{2}\overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }-\mu \overline{\left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}} . \end{aligned}$$

(1.92)

Without loss of generality, the second term on the right hand side of the transport equation of the turbulent kinetic energy (1.92)—by taking into account the incompressible mass conservation (continuity) equation for the fluctuating velocity field (1.8)—can be expressed by

$$\begin{aligned}&\qquad \qquad \qquad \quad -\rho \overline{\mathbf {u}^{\prime }\cdot \nabla \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }=-\nabla \cdot \left[ \frac{1}{2}\rho \overline{\mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }\right) }\right] =\nonumber \\&=-\frac{1}{2}\rho \overline{\left( \mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }\right) \cdot \underset{=0}{\underbrace{\left( \nabla \cdot \mathbf {u}^{\prime }\right) }}}-\frac{1}{2}\rho \overline{\mathbf {u}^{\prime }\cdot \nabla \left( \mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }\right) }=-\rho \overline{\mathbf {u}^{\prime }\cdot \nabla \left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) } , \end{aligned}$$

(1.93)

and the third term on the right hand side of Eq. (1.92) can be expressed by

$$\begin{aligned} -\overline{\mathbf {u}^{\prime }\cdot \nabla p^{\prime }}=-\nabla \cdot \overline{\left( \mathbf {u}^{\prime }p^{\prime }\right) }=-\overline{p^{\prime }\underset{=0}{\underbrace{\left( \nabla \cdot \mathbf {u}^{\prime }\right) }}}-\overline{\mathbf {u}^{\prime }\cdot \nabla p^{\prime }}=-\overline{\mathbf {u}^{\prime }\cdot \nabla p^{\prime }} , \end{aligned}$$

(1.94)

and the Laplacian term on the right hand side of Eq. (1.92) can be written as

$$\begin{aligned} \mu \nabla ^{2}\overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }=\nabla \cdot \left[ \mu \nabla \overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }\right] . \end{aligned}$$

(1.95)

By using Eqs. (1.93), (1.94) and (1.95), the turbulent kinetic energy transport equation (1.92) can be written in another mathematical form as

$$\begin{aligned}&\qquad \qquad \qquad \quad \rho \frac{\partial }{\partial t}\overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }+\rho \left( \mathbf {u}\cdot \nabla \right) \overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }=\nonumber \\&=-\rho \overline{\left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) }\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) -\nabla \cdot \left[ \frac{1}{2}\rho \overline{\mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }\right) }\right] -\nabla \cdot \overline{\left( \mathbf {u}^{\prime }p^{\prime }\right) }+\nonumber \\&\qquad \qquad +\nabla \cdot \left[ \mu \nabla \overline{\left( \frac{\mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }}{2}\right) }\right] -\mu \overline{\left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}} , \end{aligned}$$

(1.96)

which can be re-arranged by using the definitions of the turbulent kinetic energy (1.63) and the dynamic viscosity of the fluid (1.18) as

$$\begin{aligned} \rho \frac{\partial k}{\partial t}+\rho \left( \mathbf {u}\cdot \nabla \right) k&=-\rho \overline{\left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) }\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) -\rho \nu \overline{\left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}}+\nonumber \\&+\nabla \cdot \left[ \mu \nabla k-\frac{1}{2}\rho \overline{\mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }\right) }-\overline{\mathbf {u}^{\prime }p^{\prime }}\right] , \end{aligned}$$

(1.97)

which is the transport equation of the turbulent kinetic energy k given by invariant (Gibbs) notation. The first term on left hand side of the turbulent kinetic energy equation (1.97) is the unsteady term, and the second term is the convective term expressed by Cartesian index notation as

$$\begin{aligned} \rho \left( \mathbf {u}\cdot \nabla \right) k=\rho u_{1}\frac{\partial k}{\partial x_{1}}+\rho u_{2}\frac{\partial k}{\partial x_{2}}+\rho u_{3}\frac{\partial k}{\partial x_{3}}=\sum _{i=1}^{3}\left( \rho u_{i}\frac{\partial k}{\partial x_{i}}\right) \equiv \rho u_{i}\frac{\partial k}{\partial x_{i}} . \end{aligned}$$

(1.98)

The first term on the right hand side of the turbulent kinetic energy transport equation (1.97) represents the kinetic energy production by the anisotropic Reynolds stresses (1.54) which can also be written as

$$\begin{aligned}&P_{k}=-\rho \overline{\left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) }\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) =\underline{\underline{\tau }}^{R}\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) =\nonumber \\ =&-\rho \sum _{i=1}^{3}\left( \sum _{j=1}^{3}\overline{u_{i}^{\prime }u_{j}^{\prime }}\cdot \frac{\partial u_{i}}{\partial x_{j}}\right) \equiv -\rho \overline{u_{i}^{\prime }u_{j}^{\prime }}\frac{\partial u_{i}}{\partial x_{j}}=\tau _{ij}^{R}\frac{\partial u_{i}}{\partial x_{j}} . \end{aligned}$$

(1.99)

The second term on the right hand side of the turbulent kinetic energy transport equation (1.97) represents the kinetic energy dissipation through viscous effects and velocity fluctuations which is defined by

$$\begin{aligned} \varepsilon _{k}=\nu \overline{\left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}}=\nu \sum _{i=1}^{3}\left( \sum _{j=1}^{3}\overline{\frac{\partial u_{j}^{\prime }}{\partial x_{i}}\cdot \frac{\partial u_{j}^{\prime }}{\partial x_{i}}}\right) \equiv \nu \overline{\frac{\partial u_{j}^{\prime }}{\partial x_{i}}\frac{\partial u_{j}^{\prime }}{\partial x_{i}}} , \end{aligned}$$

(1.100)

where $\nu$ is the kinematic viscosity of the fluid. The third term on the right hand side of the turbulent kinetic energy transport equation (1.97) represents the diffusion of kinetic energy, which can also be expressed by

$$\begin{aligned}&\qquad \qquad \qquad \qquad \quad D_{k}=\nabla \cdot \left[ \mu \nabla k-\frac{1}{2}\rho \overline{\mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }\right) }-\overline{\mathbf {u}^{\prime }p^{\prime }}\right] =\nonumber \\&\qquad \qquad \qquad =\nabla \cdot \left( \mu \nabla k\right) -\nabla \cdot \left[ \frac{1}{2}\rho \overline{\mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }\right) }\right] -\nabla \cdot \overline{\left( \mathbf {u}^{\prime }p^{\prime }\right) }=\nonumber \\&=\sum _{i=3}^{3}\frac{\partial }{\partial x_{i}}\left( \mu \frac{\partial k}{\partial x_{i}}\right) -\sum _{j=1}^{3}\left\{ \sum _{i=1}^{3}\frac{\partial }{\partial x_{i}}\left[ \frac{1}{2}\rho \overline{u_{i}^{\prime }\cdot \left( u_{j}^{\prime }\cdot u_{j}^{\prime }\right) }\right] \right\} -\sum _{i=1}^{3}\frac{\partial }{\partial x_{i}}\overline{\left( u_{i}^{\prime }\cdot p^{\prime }\right) }\equiv \nonumber \\&\qquad \qquad \qquad \equiv \frac{\partial }{\partial x_{i}}\left( \mu \frac{\partial k}{\partial x_{i}}\right) -\frac{\partial }{\partial x_{i}}\left[ \frac{1}{2}\rho \overline{u_{i}^{\prime }\cdot \left( u_{j}^{\prime }\cdot u_{j}^{\prime }\right) }\right] -\frac{\partial }{\partial x_{i}}\overline{\left( u_{i}^{\prime }p^{\prime }\right) }=\nonumber \\&\qquad \qquad \qquad \qquad \qquad \;=\frac{\partial }{\partial x_{i}}\left[ \mu \frac{\partial k}{\partial x_{i}}-\frac{1}{2}\rho \overline{u_{i}^{\prime }\cdot \left( u_{j}^{\prime }\cdot u_{j}^{\prime }\right) }-\overline{u_{i}^{\prime }p^{\prime }}\right] , \end{aligned}$$

(1.101)

where $p^{\prime }$ is the pressure fluctuation. By using the definition of the Reynolds stress tensor (1.54) and the turbulent kinetic energy dissipation (1.100), the turbulent kinetic energy transport equation (1.97) can also be written as

$$\begin{aligned}&\rho \frac{\partial k}{\partial t}+\rho \left( \mathbf {u}\cdot \nabla \right) k=\underline{\underline{\tau }}^{R}\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) -\rho \varepsilon _{k}+\nonumber \\&\quad +\nabla \cdot \left[ \mu \nabla k-\frac{1}{2}\rho \overline{\mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }\right) }-\overline{\mathbf {u}^{\prime }p^{\prime }}\right] . \end{aligned}$$

(1.102)

Using Eqs. (1.98)–(1.101), the transport equation of the turbulent kinetic energy (1.97) and (1.102) can be expressed with Cartesian index notation by

$$\begin{aligned}&\rho \frac{\partial k}{\partial t}+\rho u_{i}\frac{\partial k}{\partial x_{i}}=\tau _{ij}^{R}\frac{\partial u_{i}}{\partial x_{j}}-\rho \nu \overline{\frac{\partial u_{j}^{\prime }}{\partial x_{i}}\frac{\partial u_{j}^{\prime }}{\partial x_{i}}}+\nonumber \\&+\frac{\partial }{\partial x_{i}}\left[ \mu \frac{\partial k}{\partial x_{i}}-\frac{1}{2}\rho \overline{u_{i}^{\prime }\cdot \left( u_{j}^{\prime }\cdot u_{j}^{\prime }\right) }-\overline{u_{i}^{\prime }p^{\prime }}\right] , \end{aligned}$$

(1.103)

or we can also simply write

$$\begin{aligned} \rho \frac{Dk}{Dt}=P_{k}-\rho \varepsilon _{k}+D_{k} , \end{aligned}$$

(1.104)

where the velocity fluctuations $\mathbf {u}^{\prime }$ have to be physically described or modelled in the production $P_{k}$, kinetic energy dissipation $\varepsilon _{k}$ and the diffusion $D_{k}$ terms on the right hand side of the turbulent kinetic energy equation (1.104).

1.2.6 Reynolds-Averaged Governing Equations of Incompressible Turbulent Flows

The general set of the Reynolds-averaged governing equations of incompressible turbulent flows has been derived in Sects. 1.2.1–1.2.5. The system of governing equations consists of the mass conservation (continuity) equation (1.7), the Reynolds (RANS) momentum equation (1.45) and the turbulent kinetic energy equation (1.97) which can be summarised by

$$\begin{aligned} \nabla \cdot \mathbf {u}=0 , \end{aligned}$$

(1.105)

and the Reynolds momentum equation with invariant (Gibbs) notation is

$$\begin{aligned} \rho \frac{\partial \mathbf {u}}{\partial t}+\rho \mathbf {u}\cdot \left( \nabla \otimes \mathbf {u}\right) =\rho \mathbf {g}-\nabla p+\mu \nabla ^{2}\mathbf {u}+\nabla \cdot \underline{\underline{\tau }}^{R} , \end{aligned}$$

(1.106)

and the turbulent kinetic energy equation is

$$\begin{aligned} \rho \frac{\partial k}{\partial t}&+\rho \left( \mathbf {u}\cdot \nabla \right) k=\underline{\underline{\tau }}^{R}\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) -\rho \nu \overline{\left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}}+\nonumber \\&\qquad \qquad \;+\nabla \cdot \left[ \mu \nabla k-\frac{1}{2}\rho \overline{\mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }\right) }-\overline{\mathbf {u}^{\prime }p^{\prime }}\right] . \end{aligned}$$

(1.107)

The general set of the Reynolds-averaged governing equations (1.105), (1.106) and (1.107) can also be expressed with Cartesian index notation by

$$\begin{aligned} \frac{\partial u_{i}}{\partial x_{i}}=0 , \end{aligned}$$

(1.108)

and the Reynolds momentum equation with Cartesian index notation is

$$\begin{aligned} \rho \frac{\partial u_{i}}{\partial t}+\rho u_{i}\frac{\partial u_{j}}{\partial x_{i}}=\rho g_{i}-\frac{\partial p}{\partial x_{i}}+\mu \frac{\partial ^{2}u_{j}}{\partial x_{i}\partial x_{i}}+\frac{\partial \tau _{ij}^{R}}{\partial x_{i}} , \end{aligned}$$

(1.109)

and the transport equation of the turbulent kinetic energy is

$$\begin{aligned}&\rho \frac{\partial k}{\partial t}+\rho u_{i}\frac{\partial k}{\partial x_{i}}=\tau _{ij}^{R}\frac{\partial u_{i}}{\partial x_{j}}-\rho \nu \overline{\frac{\partial u_{j}^{\prime }}{\partial x_{i}}\frac{\partial u_{j}^{\prime }}{\partial x_{i}}}+\nonumber \\&+\frac{\partial }{\partial x_{i}}\left[ \mu \frac{\partial k}{\partial x_{i}}-\frac{1}{2}\rho \overline{u_{i}^{\prime }\cdot \left( u_{j}^{\prime }\cdot u_{j}^{\prime }\right) }-\overline{u_{i}^{\prime }p^{\prime }}\right] . \end{aligned}$$

(1.110)

For incompressible turbulent flows, we can see from the general set of the Reynolds-averaged governing equations (1.105)–(1.110) that there are more unknowns than partial differential transport equations. Therefore, a hypothesis has to be considered for the anisotropic Reynolds stress tensor (1.54) and the fluctuating velocity $\mathbf {u}^{\prime }$ and pressure $p^{\prime }$ components appearing in the turbulent kinetic energy equation (1.107) and (1.110). The generalised Boussinesq hypothesis on the Reynolds stress tensor [10, 32] is the most widely considered and employed hypothesis within the context of the solution of the Reynolds-averaged governing equations (1.105)–(1.110). Since, the anisotropic modification of the generalised Boussinesq hypothesis is in the centre of the research interest nowadays, see e.g. in [135], therefore, the generalised Boussinesq hypothesis has been discussed subsequently.

1.2.7 The Generalised Boussinesq Hypothesis on the Physical Description of the Reynolds Stress Tensor

For turbulent flows, the generalisation of the Boussinesq hypothesis [10] on the symmetrical Reynolds stress tensor can be given by

$$\begin{aligned} \underline{\underline{\tau }}^{R}=-\rho \overline{\mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }}=2\mu _{t}\underline{\underline{S}}-\frac{2}{3}\mu _{t}\left( \nabla \cdot \mathbf {u}\right) \cdot \underline{\underline{\mathrm {I}}}-\frac{2}{3}\rho k\underline{\underline{\mathrm {I}}}\, , \end{aligned}$$

(1.111)

where the scalar dynamic eddy viscosity is defined by the product of the fluid density $\rho$ and the scalar kinematic eddy viscosity $\nu _{t}$ as

$$\begin{aligned} \mu _{t} = \rho \nu _{t} , \end{aligned}$$

(1.112)

and $\underline{\underline{S}}$ the the second-rank mean rate-of-strain (deformation) tensor, $\mathbf {u}$ represents the mean velocity field, k is the turbulent kinetic energy defined by Eq. (1.63), and $\underline{\underline{\mathrm {I}}}$ is the unit tensor given by Eq. (1.20).

For incompressible turbulent flows, taking into account the mass conservation (continuity) equation (1.7), the generalised Boussinesq hypothesis on the symmetrical Reynolds stress tensor (1.111) becomes

$$\begin{aligned} \underline{\underline{\tau }}^{R}=-\rho \overline{\mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }}=2\mu _{t}\underline{\underline{S}}-\frac{2}{3}\rho k\underline{\underline{\mathrm {I}}}\, . \end{aligned}$$

(1.113)

In other words, according to the generalised Boussinesq hypothesis for incompressible flows (1.113), the Reynolds stress tensor (1.54) is assumed to be related to the symmetrical second-rank mean rate-of-strain (deformation) tensor $\underline{\underline{S}}$ and the turbulent kinetic energy (1.63). The mean rate-of-strain (deformation) tensor $\underline{\underline{S}}$ is the symmetric part of the mean velocity gradient tensor—which is known from the fluid flow kinematics—is defined by

$$\begin{aligned} \underline{\underline{S}}&=\frac{1}{2}\left[ \left( \nabla \otimes \mathbf {u}\right) +\left( \nabla \otimes \mathbf {u}\right) ^{T}\right] =\frac{1}{2}\sum _{i=1}^{3}\left[ \sum _{j=1}^{3}\frac{\partial u_{j}}{\partial x_{i}}\cdot \left( \mathbf {e}_{i}\otimes \mathbf {e}_{j}\right) \right] +\nonumber \\&\qquad \;+\frac{1}{2}\sum _{i=1}^{3}\left[ \sum _{j=1}^{3}\frac{\partial u_{i}}{\partial x_{j}}\cdot \left( \mathbf {e}_{i}\otimes \mathbf {e}_{j}\right) \right] \equiv \frac{1}{2}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) , \end{aligned}$$

(1.114)

where the velocity gradient tensor can be given by

$$\begin{aligned} \nabla \otimes \mathbf {u}=\mathrm {Grad}\,\mathbf {u}=\sum _{i=1}^{3}\left[ \sum _{j=1}^{3}\frac{\partial u_{j}}{\partial x_{i}}\cdot \left( \mathbf {e}_{i}\otimes \mathbf {e}_{j}\right) \right] \equiv \frac{\partial u_{j}}{\partial x_{i}} , \end{aligned}$$

(1.115)

and the transpose of the velocity gradient tensor is defined by

$$\begin{aligned} \left( \nabla \otimes \mathbf {u}\right) ^{T}=\left( \mathrm {Grad}\,\mathbf {u}\right) ^{T}=\sum _{i=1}^{3}\left[ \sum _{j=1}^{3}\frac{\partial u_{i}}{\partial x_{j}}\cdot \left( \mathbf {e}_{i}\otimes \mathbf {e}_{j}\right) \right] \equiv \frac{\partial u_{i}}{\partial x_{j}} . \end{aligned}$$

(1.116)

Relying on the definition of the mean rate-of-strain tensor (1.114), the generalised Boussinesq hypothesis on the symmetrical Reynolds stress tensor for incompressible turbulent flows (1.113) can also be written as

$$\begin{aligned} \underline{\underline{\tau }}^{R}=-\rho \overline{\mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }}=2\mu _{t}\underline{\underline{S}}-\frac{2}{3}\rho k\underline{\underline{\mathrm {I}}}=\mu _{t}\left[ \left( \nabla \otimes \mathbf {u}\right) +\left( \nabla \otimes \mathbf {u}\right) ^{T}\right] -\frac{2}{3}\rho k\underline{\underline{\mathrm {I}}} , \end{aligned}$$

(1.117)

which can also be given in a matrix form by

$$\begin{aligned} \underline{\underline{\tau }}^{R}=\left[ \begin{array}{ccc} 2\mu _{t}\frac{\partial u_{1}}{\partial x_{1}}-\frac{2}{3}\rho k\,\,\,\, &{} \mu _{t}\left( \frac{\partial u_{2}}{\partial x_{1}}+\frac{\partial u_{1}}{\partial x_{2}}\right) \,\,\,\, &{} \mu _{t}\left( \frac{\partial u_{3}}{\partial x_{1}}+\frac{\partial u_{1}}{\partial x_{3}}\right) \\ \mu _{t}\left( \frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial u_{2}}{\partial x_{1}}\right) \,\,\,\, &{} 2\mu _{t}\frac{\partial u_{2}}{\partial x_{2}}-\frac{2}{3}\rho k\,\,\,\, &{} \mu _{t}\left( \frac{\partial u_{3}}{\partial x_{2}}+\frac{\partial u_{2}}{\partial x_{3}}\right) \\ \mu _{t}\left( \frac{\partial u_{1}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{1}}\right) \,\,\,\, &{} \mu _{t}\left( \frac{\partial u_{2}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{2}}\right) \,\,\,\, &{} 2\mu _{t}\frac{\partial u_{3}}{\partial x_{3}}-\frac{2}{3}\rho k \end{array}\right] . \end{aligned}$$

(1.118)

In order to obtain the Reynolds momentum equation (1.45) in conjunction with the generalised Boussinesq hypothesis (1.113), the tensor divergence of the Reynolds stress tensor (1.117) has to be taken as

$$\begin{aligned} \mathrm {Div}\,\underline{\underline{\tau }}^{R}=\nabla \cdot \underline{\underline{\tau }}^{R}=\nabla \cdot \left\{ \mu _{t}\left[ \left( \nabla \otimes \mathbf {u}\right) +\left( \nabla \otimes \mathbf {u}\right) ^{T}\right] -\frac{2}{3}\rho k\underline{\underline{\mathrm {I}}}\right\} , \end{aligned}$$

(1.119)

which can also be derived by using the matrix form (1.118) as

$$\begin{aligned} \nabla \cdot \underline{\underline{\tau }}^{R}&=\left[ \begin{array}{ccc} \frac{\partial }{\partial x_{1}}&\frac{\partial }{\partial x_{2}}&\frac{\partial }{\partial x_{3}}\end{array}\right] \cdot \left[ \begin{array}{ccc} 2\mu _{t}\frac{\partial u_{1}}{\partial x_{1}}-\frac{2}{3}\rho k\,\,\, &{} \,\,\,\mu _{t}\left( \frac{\partial u_{2}}{\partial x_{1}}+\frac{\partial u_{1}}{\partial x_{2}}\right) &{} \,\,\,\mu _{t}\left( \frac{\partial u_{3}}{\partial x_{1}}+\frac{\partial u_{1}}{\partial x_{3}}\right) \\ \mu _{t}\left( \frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial u_{2}}{\partial x_{1}}\right) \,\,\, &{} \,\,\,2\mu _{t}\frac{\partial u_{2}}{\partial x_{2}}-\frac{2}{3}\rho k &{} \,\,\,\mu _{t}\left( \frac{\partial u_{3}}{\partial x_{2}}+\frac{\partial u_{2}}{\partial x_{3}}\right) \\ \mu _{t}\left( \frac{\partial u_{1}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{1}}\right) \,\,\, &{} \,\,\,\mu _{t}\left( \frac{\partial u_{2}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{2}}\right) &{} \,\,\,2\mu _{t}\frac{\partial u_{3}}{\partial x_{3}}-\frac{2}{3}\rho k \end{array}\right] =\nonumber \\&=\sum _{j=1}^{3}\left\{ \sum _{i=1}^{3}\frac{\partial }{\partial x_{i}}\left[ \mu _{t}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) \right] \cdot \mathbf {e}_{j}\right\} -\frac{2}{3}\rho \sum _{i=1}^{3}\left( \frac{\partial k}{\partial x_{i}}\cdot \mathbf {e}_{i}\right) \equiv \nonumber \\&\qquad \qquad \qquad \qquad \equiv \frac{\partial }{\partial x_{i}}\left[ \mu _{t}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) \right] -\frac{2}{3}\rho \frac{\partial k}{\partial x_{i}} . \end{aligned}$$

(1.120)

In other words, Eqs. (1.119) and (1.120) will appear on the right hand side of the Reynolds momentum equation (1.45) in conjunction with the generalised Boussinesq hypothesis (1.113) for incompressible turbulent flows.

Since, the Reynolds stress tensor (1.117) is present in the production term (1.99) of the turbulent kinetic energy transport equation (1.107), therefore, the production term $P_{k}$ has to be derived and expressed in conjunction with the generalised Boussinesq hypothesis (1.117). Therefore, we can write

$$\begin{aligned}&\qquad \qquad \qquad P_{k}=-\rho \overline{\left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) }\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) =\underline{\underline{\tau }}^{R}\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) =\nonumber \\&\qquad \qquad =\left\{ \mu _{t}\left[ \left( \nabla \otimes \mathbf {u}\right) +\left( \nabla \otimes \mathbf {u}\right) ^{T}\right] -\frac{2}{3}\rho k\underline{\underline{\mathrm {I}}}\right\} \cdot \cdot \left( \nabla \otimes \mathbf {u}\right) =\nonumber \\&=\mu _{t}\left\{ \left[ \left( \nabla \otimes \mathbf {u}\right) +\left( \nabla \otimes \mathbf {u}\right) ^{T}\right] \cdot \cdot \left( \nabla \otimes \mathbf {u}\right) \right\} -\frac{2}{3}\rho k\left[ \underline{\underline{\mathrm {I}}}\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) \right] =\nonumber \\&\qquad =\mu _{t}\sum _{i=1}^{3}\left[ \sum _{j=1}^{3}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) \frac{\partial u_{i}}{\partial x_{j}}\right] -\frac{2}{3}\rho k\sum _{i=1}^{3}\left( \sum _{j=1}^{3}\delta _{ij}\frac{\partial u_{i}}{\partial x_{j}}\right) \equiv \nonumber \\&\qquad \qquad \qquad \qquad \quad \equiv \mu _{t}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) \frac{\partial u_{i}}{\partial x_{j}}-\frac{2}{3}\rho k\delta _{ij}\frac{\partial u_{i}}{\partial x_{j}} , \end{aligned}$$

(1.121)

where $\delta _{ij}$ is the Kronecker delta [4, 60]. For incompressible flows, the double dot scalar product of the unit tensor and the mean velocity gradient tensor vanishes due to the continuity equation (1.7), because

$$\begin{aligned} \underline{\underline{\mathrm {I}}}\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) =\nabla \cdot \mathbf {u}=0 , \end{aligned}$$

(1.122)

which means that the turbulent kinetic energy term vanishes in the turbulent kinetic energy production term (1.121), thus we can write

$$\begin{aligned} -\frac{2}{3}\rho k\left[ \underline{\underline{\mathrm {I}}}\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) \right] =-\frac{2}{3}\rho k\delta _{ij}\frac{\partial u_{i}}{\partial x_{j}}=0 . \end{aligned}$$

(1.123)

Therefore, taking into account Eqs. (1.122) and (1.123), the turbulent kinetic energy production term (1.121) can simply be defined by

$$\begin{aligned}&\qquad \qquad \qquad P_{k}=-\rho \overline{\left( \mathbf {u}^{\prime }\otimes \mathbf {u}^{\prime }\right) }\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) =\underline{\underline{\tau }}^{R}\cdot \cdot \left( \nabla \otimes \mathbf {u}\right) = \nonumber \\&=\mu _{t}\left[ \left( \nabla \otimes \mathbf {u}\right) +\left( \nabla \otimes \mathbf {u}\right) ^{T}\right] \cdot \cdot \left( \nabla \otimes \mathbf {u}\right) \equiv \mu _{t}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) \frac{\partial u_{i}}{\partial x_{j}} . \end{aligned}$$

(1.124)

It is important to note regarding the validity and applicability of the generalised Boussinesq hypothesis on the Reynolds stress tensor defined by Eqs. (1.113) and (1.117)—as mentioned in Sect. 1.1—that it is a well-known fact that the Boussinesq-hypothesis [10] itself does not provide an accurate prediction of Reynolds stress anisotropies from a physical point-of-view. Furthermore, the generalised Boussinesq hypothesis (1.113) is also unlikely to be valid for predicting anisotropic turbulent flows as highlighted by Davidson [32] amongst others. To overcome the isotropic limitation of the Boussinesq hypothesis [10] itself, an anisotropic modification of the generalised Boussinesq hypothesis (1.113) is required. Thus, a new hypothesis on the anisotropic Reynolds stress tensor needs to be proposed (see Chap. 5).

1.2.8 Reynolds-Averaged Governing Equations Using the Generalised Boussinesq Hypothesis

Using Eqs. (1.119), (1.120) and Eqs. (1.123), (1.124), the Reynolds-averaged governing equations of incompressible turbulent flows in conjunction with the generalised Boussinesq hypothesis on the Reynolds stress tensor (1.117) can be obtained. The system of governing equations consists of the mass conservation (continuity) equation, the Reynolds momentum equation and the turbulent kinetic energy equation which can be summarised by

$$\begin{aligned} \nabla \cdot \mathbf {u}=0 , \end{aligned}$$

(1.125)

and the Reynolds momentum equation with invariant (Gibbs) notation is

$$\begin{aligned}&\rho \frac{\partial \mathbf {u}}{\partial t}+\rho \mathbf {u}\cdot \left( \nabla \otimes \mathbf {u}\right) =\rho \mathbf {g}-\nabla p+\mu \nabla ^{2}\mathbf {u}+ \nonumber \\&+\nabla \cdot \left\{ \mu _{t}\left[ \left( \nabla \otimes \mathbf {u}\right) +\left( \nabla \otimes \mathbf {u}\right) ^{T}\right] \right\} -\frac{2}{3}\rho \nabla k , \end{aligned}$$

(1.126)

and the turbulent kinetic energy equation is

$$\begin{aligned}&\qquad \qquad \qquad \qquad \qquad \quad \rho \frac{\partial k}{\partial t}+\rho \left( \mathbf {u}\cdot \nabla \right) k= \nonumber \\&=\mu _{t}\left[ \left( \nabla \otimes \mathbf {u}\right) +\left( \nabla \otimes \mathbf {u}\right) ^{T}\right] \cdot \cdot \left( \nabla \otimes \mathbf {u}\right) -\rho \nu \overline{\left( \nabla \otimes \mathbf {u}^{\prime }\right) \cdot \cdot \left( \nabla \otimes \mathbf {u}^{\prime }\right) ^{T}}+ \nonumber \\&\qquad \qquad \qquad \qquad +\nabla \cdot \left[ \mu \nabla k-\frac{1}{2}\rho \overline{\mathbf {u}^{\prime }\cdot \left( \mathbf {u}^{\prime }\cdot \mathbf {u}^{\prime }\right) }-\overline{\mathbf {u}^{\prime }p^{\prime }}\right] . \end{aligned}$$

(1.127)

The system of the Reynolds-averaged governing equations (1.125)–(1.127) can also be expressed with Cartesian index notation by

$$\begin{aligned} \frac{\partial u_{i}}{\partial x_{i}}=0 , \end{aligned}$$

(1.128)

and the Reynolds momentum equation with Cartesian index notation is

$$\begin{aligned}&\rho \frac{\partial u_{i}}{\partial t}+\rho u_{i}\frac{\partial u_{j}}{\partial x_{i}}=\rho g_{i}-\frac{\partial p}{\partial x_{i}}+\mu \frac{\partial ^{2}u_{j}}{\partial x_{i}\partial x_{i}}+ \nonumber \\&\qquad +\frac{\partial }{\partial x_{i}}\left[ \mu _{t}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) \right] -\frac{2}{3}\rho \frac{\partial k}{\partial x_{i}} , \end{aligned}$$

(1.129)

and the transport equation of the turbulent kinetic energy is

$$\begin{aligned}&\rho \frac{\partial k}{\partial t}+\rho u_{i}\frac{\partial k}{\partial x_{i}}=\mu _{t}\left( \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}}\right) \frac{\partial u_{i}}{\partial x_{j}}-\rho \nu \overline{\frac{\partial u_{j}^{\prime }}{\partial x_{i}}\frac{\partial u_{j}^{\prime }}{\partial x_{i}}}+ \nonumber \\&\qquad \quad +\frac{\partial }{\partial x_{i}}\left[ \mu \frac{\partial k}{\partial x_{i}}-\frac{1}{2}\rho \overline{u_{i}^{\prime }\cdot \left( u_{j}^{\prime }\cdot u_{j}^{\prime }\right) }-\overline{\left( u_{i}^{\prime }p^{\prime }\right) }\right] . \end{aligned}$$

(1.130)

In order to implement the set of governing equations (1.125)–(1.127) related to the generalised Boussinesq hypothesis on the Reynolds stress tensor (1.117), it is necessary to obtain the scalar form of each conservation equation. In other words, the continuity equation (1.125), the Reynolds momentum equation (1.126) with the eddy viscosity hypothesis on the Reynolds stress tensor (1.117) and the turbulent kinetic energy equation (1.127) have to be written component-wise in order to use an appropriate numerical discretisation method to implement them either in an in-house code or in a commerical software environment. Therefore, the scalar forms of the continuity equation (1.125), the Reynolds momentum equation (1.126) and the turbulent kinetic energy transport equation (1.127) have to be summarised.

For incompressible turbulent flows, the scalar form of the mass conservation (continuity) equation (1.125) can be expressed by

$$\begin{aligned} \frac{\partial u_{1}}{\partial x_{1}}+\frac{\partial u_{2}}{\partial x_{2}}+\frac{\partial u_{3}}{\partial x_{3}}=0 , \end{aligned}$$

(1.131)

and the momentum equation (1.126) of the velocity component $u_{1}$ is

$$\begin{aligned}&\qquad \rho \frac{\partial u_{1}}{\partial t}+\rho \left( u_{1}\frac{\partial u_{1}}{\partial x_{1}}+u_{2}\frac{\partial u_{1}}{\partial x_{2}}+u_{3}\frac{\partial u_{1}}{\partial x_{3}}\right) =\rho g_{1}-\frac{\partial p}{\partial x_{1}}+ \nonumber \\&\qquad \qquad +\mu \left( \frac{\partial ^{2}u_{1}}{\partial x_{1}^{2}}+\frac{\partial ^{2}u_{1}}{\partial x_{2}^{2}}+\frac{\partial ^{2}u_{1}}{\partial x_{3}^{2}}\right) +\frac{\partial }{\partial x_{1}}\left( 2\mu _{t}\frac{\partial u_{1}}{\partial x_{1}}\right) + \nonumber \\&+\frac{\partial }{\partial x_{2}}\left[ \mu _{t}\left( \frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial u_{2}}{\partial x_{1}}\right) \right] +\frac{\partial }{\partial x_{3}}\left[ \mu _{t}\left( \frac{\partial u_{1}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{1}}\right) \right] -\frac{2}{3}\rho \frac{\partial k}{\partial x_{1}} , \end{aligned}$$

(1.132)

and the momentum equation (1.126) of the velocity component $u_{2}$ is

$$\begin{aligned}&\rho \frac{\partial u_{2}}{\partial t}+\rho \left( u_{1}\frac{\partial u_{2}}{\partial x_{1}}+u_{2}\frac{\partial u_{2}}{\partial x_{2}}+u_{3}\frac{\partial u_{2}}{\partial x_{3}}\right) =\rho g_{2}-\frac{\partial p}{\partial x_{2}}+ \nonumber \\&+\mu \left( \frac{\partial ^{2}u_{2}}{\partial x_{1}^{2}}+\frac{\partial ^{2}u_{2}}{\partial x_{2}^{2}}+\frac{\partial ^{2}u_{2}}{\partial x_{3}^{2}}\right) +\frac{\partial }{\partial x_{1}}\left[ \mu _{t}\left( \frac{\partial u_{2}}{\partial x_{1}}+\frac{\partial u_{1}}{\partial x_{2}}\right) \right] + \nonumber \\&+\frac{\partial }{\partial x_{2}}\left( 2\mu _{t}\frac{\partial u_{2}}{\partial x_{2}}\right) +\frac{\partial }{\partial x_{3}}\left[ \mu _{t}\left( \frac{\partial u_{2}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{2}}\right) \right] -\frac{2}{3}\rho \frac{\partial k}{\partial x_{2}} , \end{aligned}$$

(1.133)

and the momentum equation (1.126) of the velocity component $u_{3}$ is

$$\begin{aligned}&\;\;\rho \frac{\partial u_{3}}{\partial t}+\rho \left( u_{1}\frac{\partial u_{3}}{\partial x_{1}}+u_{2}\frac{\partial u_{3}}{\partial x_{2}}+u_{3}\frac{\partial u_{3}}{\partial x_{3}}\right) =\rho g_{3}-\frac{\partial p}{\partial x_{3}}+ \nonumber \\&\qquad +\mu \left( \frac{\partial ^{2}u_{3}}{\partial x_{1}^{2}}+\frac{\partial ^{2}u_{3}}{\partial x_{2}^{2}}+\frac{\partial ^{2}u_{3}}{\partial x_{3}^{2}}\right) +\frac{\partial }{\partial x_{1}}\left[ \mu _{t}\left( \frac{\partial u_{3}}{\partial x_{1}}+\frac{\partial u_{1}}{\partial x_{3}}\right) \right] + \nonumber \\&\;+\frac{\partial }{\partial x_{2}}\left[ \mu _{t}\left( \frac{\partial u_{3}}{\partial x_{2}}+\frac{\partial u_{2}}{\partial x_{3}}\right) \right] +\frac{\partial }{\partial x_{3}}\left( 2\mu _{t}\frac{\partial u_{3}}{\partial x_{3}}\right) -\frac{2}{3}\rho \frac{\partial k}{\partial x_{3}} . \end{aligned}$$

(1.134)

The scalar form of the turbulent kinetic energy equation (1.127) is

$$\begin{aligned}&\qquad \qquad \qquad \qquad \qquad \qquad \rho \frac{\partial k}{\partial t}+\rho u_{1}\frac{\partial k}{\partial x_{1}}+\rho u_{2}\frac{\partial k}{\partial x_{2}}+\rho u_{3}\frac{\partial k}{\partial x_{3}}=\nonumber \\&\qquad =2\mu _{t}\left[ \left( \frac{\partial u_{1}}{\partial x_{1}}\right) ^{2}+\left( \frac{\partial u_{2}}{\partial x_{2}}\right) ^{2}+\left( \frac{\partial u_{3}}{\partial x_{3}}\right) ^{2}+\frac{\partial u_{2}}{\partial x_{1}}\frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial u_{3}}{\partial x_{1}}\frac{\partial u_{1}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{2}}\frac{\partial u_{2}}{\partial x_{3}}\!\right] +\nonumber \\&\qquad +\mu _{t}\left[ \left( \frac{\partial u_{1}}{\partial x_{2}}\right) ^{2}+\left( \frac{\partial u_{1}}{\partial x_{3}}\right) ^{2}+\left( \frac{\partial u_{2}}{\partial x_{1}}\right) ^{2}+\left( \frac{\partial u_{2}}{\partial x_{3}}\right) ^{2}+\left( \frac{\partial u_{3}}{\partial x_{1}}\right) ^{2}+\left( \frac{\partial u_{3}}{\partial x_{2}}\right) ^{2}\right] \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad -\rho \nu \left[ \overline{\left( \frac{\partial u_{1}^{\prime }}{\partial x_{1}}\right) ^{2}}+\overline{\left( \frac{\partial u_{2}^{\prime }}{\partial x_{1}}\right) ^{2}}+\overline{\left( \frac{\partial u_{3}^{\prime }}{\partial x_{1}}\right) ^{2}}+\right. \nonumber \\&\qquad \left. +\overline{\left( \frac{\partial u_{1}^{\prime }}{\partial x_{2}}\right) ^{2}}+\overline{\left( \frac{\partial u_{2}^{\prime }}{\partial x_{2}}\right) ^{2}}+\overline{\left( \frac{\partial u_{3}^{\prime }}{\partial x_{2}}\right) ^{2}}+\overline{\left( \frac{\partial u_{1}^{\prime }}{\partial x_{3}}\right) ^{2}}+\overline{\left( \frac{\partial u_{2}^{\prime }}{\partial x_{3}}\right) ^{2}}+\overline{\left( \frac{\partial u_{3}^{\prime }}{\partial x_{3}}\right) ^{2}}\right] +\nonumber \\&\qquad \qquad \qquad \qquad \; +\frac{\partial }{\partial x_{1}}\left( \mu \frac{\partial k}{\partial x_{1}}\right) +\frac{\partial }{\partial x_{2}}\left( \mu \frac{\partial k}{\partial x_{2}}\right) +\frac{\partial }{\partial x_{3}}\left( \mu \frac{\partial k}{\partial x_{3}}\right) \nonumber \\&\qquad \qquad -\frac{\partial }{\partial x_{1}}\left[ \frac{1}{2}\rho \overline{\left( u_{1}^{\prime }u_{1}^{\prime }u_{1}^{\prime }\right) }\right] -\frac{\partial }{\partial x_{2}}\left[ \frac{1}{2}\rho \overline{\left( u_{2}^{\prime }u_{1}^{\prime }u_{1}^{\prime }\right) }\right] -\frac{\partial }{\partial x_{3}}\left[ \frac{1}{2}\rho \overline{\left( u_{3}^{\prime }u_{1}^{\prime }u_{1}^{\prime }\right) }\right] \nonumber \\&\qquad \qquad -\frac{\partial }{\partial x_{1}}\left[ \frac{1}{2}\rho \overline{\left( u_{1}^{\prime }u_{2}^{\prime }u_{2}^{\prime }\right) }\right] -\frac{\partial }{\partial x_{2}}\left[ \frac{1}{2}\rho \overline{\left( u_{2}^{\prime }u_{2}^{\prime }u_{2}^{\prime }\right) }\right] -\frac{\partial }{\partial x_{3}}\left[ \frac{1}{2}\rho \overline{\left( u_{3}^{\prime }u_{2}^{\prime }u_{2}^{\prime }\right) }\right] \nonumber \\&\qquad \qquad -\frac{\partial }{\partial x_{1}}\left[ \frac{1}{2}\rho \overline{\left( u_{1}^{\prime }u_{3}^{\prime }u_{3}^{\prime }\right) }\right] -\frac{\partial }{\partial x_{2}}\left[ \frac{1}{2}\rho \overline{\left( u_{2}^{\prime }u_{3}^{\prime }u_{3}^{\prime }\right) }\right] -\frac{\partial }{\partial x_{3}}\left[ \frac{1}{2}\rho \overline{\left( u_{3}^{\prime }u_{3}^{\prime }u_{3}^{\prime }\right) }\right] \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad -\frac{\partial }{\partial x_{1}}\overline{\left( u_{1}^{\prime }p^{\prime }\right) }-\frac{\partial }{\partial x_{2}}\overline{\left( u_{2}^{\prime }p^{\prime }\right) }-\frac{\partial }{\partial x_{3}}\overline{\left( u_{3}^{\prime }p^{\prime }\right) } . \end{aligned}$$

(1.135)

We can also see from the set of three-dimensional scalar governing equations (1.131)–(1.135) of eddy viscosity models that the computer code implementation of these equations can be a challenging task. As mentioned above, we have more unknowns than scalar partial differential transport equations, thus physically correct hypotheses have to be imposed on the eddy viscosity $\mu _{t}$, on the fluctuating velocity components $u_{1}^{\prime }$, $u_{2}^{\prime }$, $u_{3}^{\prime }$ and on the fluctuating pressure field $p^{\prime }$ behaviour in the set of scalar equations (1.131)–(1.135).

In this book, three different physically plausible closure models—(a) the k-$\omega$ Shear-Stress Transport (SST) turbulence model of Menter [97, 98], (b) the anisotropic stochastic turbulence model (STM) of Czibere [22, 23] which is relying on the three-dimensional similarity theory of velocity fluctuations, and (c) the anisotropic hybrid k-$\omega$ SST/STM turbulence model based on a new hypothesis on the anisotropic Reynolds stress tensor—have been discussed in Chaps. 3, 4 and 5, respectively.

1.3 Summary

In this chapter, a step-by-step full mathematical derivation of the general set of the Reynolds-averaged governing equations of incompressible turbulent flows has been carried out—see Sects. 1.2.1, 1.2.3 and 1.2.5—because many intermediate derivation steps are omitted in most textbooks. For graduate and postgraduate students, the minimum requirement is to understand the basics of the vector analysis and tensor calculus [4, 60]. The understanding of the mathematical derivation of the governing equations of incompressible turbulent flows—including the way of thinking presented here—is crucial to develop a skill to be able to unify, hybridise and modify different existing theories and models, e.g., to propose a new hypothesis on the Reynolds stress tensor (1.54). The validity and the isotropic limitation of the generalised Boussinesq hypothesis on the Reynolds stress tensor (1.113) were discussed briefly, and the need for a new hypothesis on the anisotropic Reynolds stress tensor (1.54) was highlighted. For incompressible turbulent flows, the general governing equations in conjunction with the generalised Boussinesq hypothesis on the Reynolds stress tensor (1.113) were also derived in this chapter. The reader can see further details on the governing equations of turbulent flows in the books of Goldstein [41, 42], Shih-I [111], Hinze [49], Monin and Yaglom [101, 102], Tennekes and Lumley [133], Wilcox [137], Pope [106], Davidson [32] and Leschziner [88].