7.1 Basic Equations of Turbulence 湍流的基本方程[cfd-7-1]

The first approach for the approximate treatment of turbulent flows was presented by Reynolds in 1895. The methodology is based on the decomposition of the flow variables into a mean and a fluctuating part. The governing equations (7.1) are then solved for the mean values, which are the most interesting for engineering applications. Thus, considering first incompressible flows, the velocity components and the pressure in Eq. (7.1) are substituted by [13]

where the mean value is denoted by an overbar and the turbulent fluctuations by a prime. The mean values are obtained by an averaging procedure. There are three different forms of the Reynolds averaging:

1. Time averaging -- appropriate for stationary turbulence (statistically steady turbulence)

As a consequence, the mean value \(\bar{v}_i\) does not vary in time, but only in space. The situation is sketched in Fig. 7.2. In practice, \(T\to\infty\) means that the time interval \(T\) should be large as compared to the typical time-scale of the turbulent fluctuations.

2. Spatial averaging -- appropriate for homogeneous turbulence

with \(\Omega\) being a control volume. In this case, \(\bar{v}_i\) is uniform in space, but it is allowed to vary in time.

3. Ensemble averaging -- appropriate for general turbulence

Here, the mean value \(\bar{v}_i\) still remains a function of time and of space coordinates.

For all three approaches, the average of the fluctuating part is zero, i.e., \(\overline{v'_i}=0\). However, it can be easily seen that \(\overline{v'_iv'_i}\neq 0\). The same is true for \(\overline{v'_iv'_j}\), if both turbulent velocity components are correlated.

In cases where the turbulent flow is both stationary and homogeneous, all three averaging forms are equivalent. This is called the ergodic hypothesis.

In cases where the density is not constant, it is advisable to apply the density (mass) weighted or Favre decomposition [14], [15] to certain quantities in Eq. (7.1) instead of Reynolds averaging. Otherwise, the averaged governing equations would become considerably more complicated due to additional correlations involving density fluctuations. The most convenient way is to employ Reynolds averaging for density and pressure, and Favre averaging for other variables such as velocity, internal energy, enthalpy and temperature. Favre averaged quantities, for example the velocity components, are obtained from the relation [14], [15]

where \(\bar{\rho}\) denotes the Reynolds-averaged density. Hence, the Favre decomposition reads

where \(\tilde{v}_i\) represents the mean value and \(v''_i\) the fluctuating part of the velocity \(v_i\). Again, the average of the fluctuating part is zero, i.e., \(\widetilde{v''_i}=0\). Furthermore, the average of the product of two fluctuating quantities is not zero, if the quantities are correlated. Hence, for example, \(\widetilde{v''_iv''_i}\neq 0\) and in general \(\widetilde{v''_iv''_j}\neq 0\).

The following relationships can be derived for a mix between Favre and Reynolds averaging

These relations will be utilised in later subsections.

If we apply either the time averaging Eq. (7.8) or the ensemble averaging Eq. (7.10) to the incompressible Navier-Stokes equations (7.6), we obtain the following relations for the mass and momentum conservation

These are known as the Reynolds-Averaged Navier-Stokes equations (RANS). The equations (7.14) are formally identical to the Navier-Stokes equations (7.6) with the exception of the additional term

which constitutes the so-called Reynolds-stress tensor. It represents the transfer of momentum due to turbulent fluctuations. The laminar viscous stresses are evaluated according to Eqs. (7.2) and (7.3) using Reynolds-averaged velocity components, i.e.,

The Reynolds-stress tensor consists in 3D of the nine components

However, since \(v'_i\) and \(v'_j\) in the correlations can be interchanged, the Reynolds-stress tensor contains only six independent components. The sum of the normal stresses divided by density defines the turbulent kinetic energy, i.e.,

As we can see, the fundamental problem of turbulence modelling based on the Reynolds-averaged Navier-Stokes equations is to find six additional relations in order to close the equations (7.14). We shall introduce the basic methodologies in the Subsections 7.1.5--7.1.7.

In turbulence modelling, it is quite common to assume that Morkovin's hypothesis [16] is valid. It states that the turbulent structure of a boundary layer is not notably influenced by density fluctuations if \(\rho'\ll\bar{\rho}\). This is generally true for wall-bounded flows up to a Mach number of about five. However, in the case of hypersonic flows or for compressible free shear layers, density fluctuations have to be taken into account. The same holds also for flows with combustion or with significant heat transfer.

Application of the Reynolds averaging (Eq. (7.8) or (7.10)) to density and pressure, and of the Favre averaging Eq. (7.11) to the remaining flow variables in the compressible Navier-Stokes equations (7.1) yields [17]

These are the Favre- and Reynolds-Averaged Navier-Stokes equations. Similarly to the Reynolds averaging, the viscous stress tensor in the momentum (and energy) equation is extended by the Favre-averaged Reynolds-stress tensor, i.e.,

Its form is similar to Eq. (7.17) with Favre instead of Reynolds averaging. The components of the laminar (molecular) viscous stress tensor \(\bar{\tau}_{ij}\) are evaluated by Eq. (7.2) using Favre-averaged velocity components.

If we employ the definition of the Favre-averaged turbulent kinetic energy, i.e.,

we can express the total energy in Eq. (7.19) as

The total enthalpy is defined as

The individual parts of the Favre- and Reynolds-averaged Navier-Stokes equations (7.19) have the following physical meaning [17]:

• \(\dfrac{\partial}{\partial x_j}\left(k\dfrac{\partial\tilde{T}}{\partial x_j}\right)\) - molecular diffusion of heat
• \(\dfrac{\partial}{\partial x_j}\left(\bar{\rho}\widetilde{v''_jh''}\right)\) - turbulent transport of heat
• \(\dfrac{\partial}{\partial x_j}\left(\widetilde{\tau_{ij}v''_i}\right)\) - molecular diffusion of \(\tilde{K}\)
• \(\dfrac{\partial}{\partial x_j}\left(\bar{\rho}\widetilde{v''_jK}\right)\) - turbulent transport of \(\tilde{K}\)
• \(\dfrac{\partial}{\partial x_j}\left(\tilde{v}_i\bar{\tau}_{ij}\right)\) - work done by the molecular stresses
• \(\dfrac{\partial}{\partial x_j}\left(\tilde{v}_i\tau^F_{ij}\right)\) - work done by the Favre-averaged Reynolds stresses

The molecular diffusion and turbulent transport of \(\tilde{K}\) are very often neglected. This is a valid approximation for transonic and supersonic flows. In order to close the Favre- and Reynolds-averaged equations (7.19), we also have to supply six components of the Favre-averaged Reynolds-stress tensor (Eq. (7.20)) and three components of the turbulent heat-flux vector. We shall discuss the three basic approaches in the next subsections.

One of the most significant contributions to turbulence modelling was presented in 1877 by Boussinesq [11], [12]. His idea is based on the observation that the momentum transfer in a turbulent flow is dominated by the mixing caused by large energetic turbulent eddies. The Boussinesq hypothesis assumes that the turbulent shear stress depends linearly on the mean rate of strain, as in a laminar flow. The proportionality factor is the eddy viscosity. The Boussinesq hypothesis for Reynolds averaged incompressible flow (Eq. (7.14)) can be written as

where \(\overline{S}_{ij}\) denotes the Reynolds-averaged strain-rate tensor (Eq. (7.3), cf. also Eq. (7.16)), \(K\) is the turbulent kinetic energy (\(K=(1/2)\overline{v'_iv'_i}\)), and \(\mu_T\) stands for the eddy viscosity. Unlike the molecular viscosity \(\mu\), the eddy viscosity \(\mu_T\) represents no physical characteristic of the fluid, but it is a function of the local flow conditions. Additionally, \(\mu_T\) is also strongly affected by flow history effects.

In the case of the compressible Favre- and Reynolds-averaged Navier-Stokes equations (7.19), the Boussinesq eddy-viscosity hypothesis reads

where \(\tilde{S}_{ij}\) and \(\tilde{K}\) are the Favre-averaged strain rate and turbulent kinetic energy, respectively. Note the similarity to Eq. (7.2). The term \((2/3)\rho K\delta_{ij}\) in Eqs. (7.24) and (7.25) is required in order to obtain the proper trace of \(\tau^R_{ij}\) or \(\tau^F_{ij}\). This means that we must have

in the case of \(\overline{S}_{ii}=0\) (continuity equation) or \(\tilde{S}_{ii}=0\), in order to fulfil the relations Eq. (7.18) or (7.21) for the turbulent kinetic energy. However, the term \((2/3)\rho K\delta_{ij}\) is often neglected, particularly in connection with simpler turbulence models (like algebraic ones).

The approximation, which is commonly used for the modelling of the turbulent heat-flux vector, is based on the classical Reynolds analogy [18]. Hence, we may write

with the turbulent thermal conductivity coefficient \(k_T\) being defined as

In Equation (7.27), \(c_p\) denotes the specific heat coefficient at constant pressure and \(Pr_T\) is the turbulent Prandtl number. The turbulent Prandtl number is in general assumed to be constant over the flow field (\(Pr_T=0.9\) for air).

By applying the eddy-viscosity approach to the Reynolds- (and Favre-) averaged form of the governing equations (2.19) or Eq. (7.1), the dynamic viscosity coefficient \(\mu\) in the viscous stress tensor Eq. (2.15) or Eq. (7.2) is simply replaced by the sum of a laminar and a turbulent component, i.e.,

The laminar viscosity \(\mu_L\) is computed, for example, with the aid of the Sutherland formula (2.30). Furthermore, according to the Reynolds analogy given by Eq. (7.26), the thermal conductivity coefficient \(k\) in Eq. (2.24) or Eq. (7.2) is evaluated as

The eddy-viscosity concept of Boussinesq is, at least from the engineering point of view, very attractive since it requires ``only'' the determination of \(\mu_T\) (the turbulent kinetic energy \(K\) needed for the term \((2/3)\rho K\delta_{ij}\) in Eq. (7.24) or (7.25) is either obtained as a by-product of the turbulence model or is simply omitted). Once we know the eddy viscosity \(\mu_T\), we can easily extend the Navier-Stokes equations (2.19) or (7.1) to the simulation of turbulent flows by introducing averaged flow variables and by adding \(\mu_T\) to the laminar viscosity. Therefore, Boussinesq's approach became the basis for a large variety of first-order turbulence closures. However, there are applications for which the Boussinesq hypothesis is no longer valid (see, e.g., [17] p. 214 or [19] p. 111):

• flows with sudden change of mean strain rate,
• flows with significant streamline curvature,
• flows with rotation and stratification,
• secondary flows in ducts and in turbomachinery,
• flows with boundary layer separation and reattachment.

The limitations of the eddy-viscosity approach are caused by the assumption of equilibrium between the turbulence and the mean strain field, as well as by the independence on system rotation. The results can be notably improved by using appropriate correction terms in the turbulence models [20], [21]. Further increased accuracy of predictions can be achieved through the application of non-linear eddy-viscosity models which are described next.

In order to remove the restrictions imposed by the assumption of equilibrium between the turbulence and the mean strain rate, Lumley [22], [23] proposed to extend the linear Boussinesq approach by higher-order products of strain and rotation tensors. This can viewed as a Taylor series expansion. Following the idea of Lumley, numerous non-linear eddy-viscosity models were proposed, see, for example, Refs. [24]--[29].

In the following, we shall present one recent approach proposed by Shih et al. [25]. It includes up to third-order terms in the general eddy-viscosity formulation and is particularly suited to swirling flows. As already pointed out in [19], p. 194, cubic terms are essential for high accuracy. The Reynolds stresses \(\tau^R_{ij}\) can be expressed as [25], [30] (cf. Eq. (7.24))

with \(\overline{S}_{ij}\), \(\overline{\Omega}_{ij}\) according to Eqs. (7.3) and (7.4). Furthermore,

The values of the turbulent kinetic energy \(K\) and the dissipation rate \(\varepsilon\) are obtained from low-Reynolds \(K\)-\(\varepsilon\) turbulence model (cf. Subsection 7.2.2). The factors \(C_1\) to \(C_5\) in Eq. (7.30) are provided in Refs. [25], [30].

In comparison to the linear eddy-viscosity approach, the non-linear models are computationally only slightly more expensive, but they offer a substantially improved prediction capabilities for complex turbulent flows.

It is possible to derive exact equations for the Reynolds stresses by taking the time average (second-order moment)

where \(\mathcal{N}(v_i)\) denotes the Navier-Stokes operator, i.e.,

Using the average Eq. (7.32) together with Eq. (7.33), we obtain the following Reynolds-stress transport equation [31]

for incompressible flow. The formulation for compressible flows can be found in Ref. [17], p. 179, or in [32], [33]. The production of the turbulent kinetic energy \(P_{ij}\), the pressure-strain term \(\Pi_{ij}\), the dissipation-rate term \(\varepsilon_{ij}\), and the third-order diffusion term \(C_{ijk}\) in Eq. (7.34) are defined as

In Eq. (7.35), \(S'_{ij}\) denotes the fluctuating part of the strain-rate tensor. The first part of \(C_{ijk}\), the triple velocity term, represents transport driven by fluctuating convection, the two other parts are the pressure transport terms (pressure-velocity correlations), respectively.

As we can see, the exact Reynolds-stress equation contains new unknown higher-order correlations (e.g., \(\overline{v'_iv'_jv'_k}\)). Therefore, Equation (7.34) can be closed only by using empirical models. This is caused by the non-linear nature of the Navier-Stokes equations. The second-order closures -- the Reynolds-stress models -- provide the necessary framework for solving Eq. (7.34). Examples of implementations can be found, e.g., in Refs. [34]--[36].