3.3 Turbulence Modelling 湍流建模[cfd-3-3]

The solution of the governing equations (2.19) does not raise any fundamental difficulties in the case of inviscid or laminar flows. The simulation of turbulent flows, however, presents a significant problem. Despite the performance of modern supercomputers, a direct simulation of turbulence by the time-dependent Navier-Stokes equations (2.19), called the Direct Numerical Simulation (DNS), is still possible only for rather simple flow cases at low Reynolds numbers (\(Re\)). The restrictions of the DNS become quite obvious when recalling that the number of grid points needed for sufficient spatial resolution scales as \(Re^{9/4}\) and the CPU-time as \(Re^{3}\). This does not mean that DNS is completely useless. It is an important tool for understanding the turbulent structures and the laminar-turbulent transition. DNS also plays a vital role in the development and calibration of new or improved turbulence models. However, in engineering applications, the effects of turbulence can be taken into account only approximately, using models of various complexities.

The first level of approximation is reached for the Large-Eddy Simulation (LES) approach. The development of LES is founded on the observation that the small scales of turbulent motion posses a more universal character than the large scales, which transport the turbulent energy. Thus, the idea is to resolve only the large eddies accurately and to approximate the effects of the small scales by relatively simple subgrid-scale models. Since LES requires significantly less grid points than DNS, the investigation of turbulent flows at much higher Reynolds numbers becomes feasible. But because LES is inherently three-dimensional and unsteady, it still remains computationally very demanding. Thus, LES is still far away from becoming an engineering tool. However, LES is well suited for detailed studies of complex flow physics including massively separated unsteady flows, large scale mixing (e.g., fuel and oxidiser), aerodynamic noise, or for the investigation of flow control strategies. LES is also very promising for more accurate computations of flows in combustion chambers or engines, heat transfer and of rotating flows. An overview of research activities in LES was recently published in [210].

The next level of approximation is represented by the so-called Reynolds-Averaged Navier-Stokes equations (RANS). This approach, which was presented by Reynolds in 1895, is based on the decomposition of the flow variables into mean and fluctuating parts, followed by time or ensemble averaging [211] (see also [212], [213]). In cases where the density is not constant, it is advisable to apply the density (mass) weighted or Favre decomposition [214], [215] to the velocity components. Otherwise, the averaged governing equations would become considerably more complicated due to additional correlations involving density fluctuations. It is common to assume that Morkovin's hypothesis [216] is valid, which states that the turbulence structure of boundary layers and wakes is not notably influenced by density fluctuations for Mach numbers below 5.

By inserting the decomposed variables into the Navier-Stokes equations (2.19) and averaging, we obtain formally the same equations for the mean variables with the exception of two additional terms. The tensor of the viscous stresses is extended by one term - the Reynolds-stress tensor [211]

where \(v_i''\), \(v_j''\) denote the density-weighted fluctuating parts of the velocity components \(u,v,w\); \(\overline{\phantom{x}}\) and \(\widetilde{\phantom{x}}\) stand for ensemble and density weighted averaging, respectively. The Reynolds-stress tensor represents the transport of mean momentum due to turbulent fluctuations. Furthermore, the diffusive heat flux \(k\nabla T\) in the energy equation (cf. Eq. (2.8)) is enhanced by the so-called turbulent heat-flux vector [43]

Thus, we can see that the solution of the Reynolds-averaged Navier-Stokes equations requires the modelling of the Reynolds stresses (3.13) and of the turbulent heat flux (3.14). The advantages of this approach are that considerably coarser grids can be used as compared to LES, and that stationary mean solution can be assumed (at least for attached or moderately separated flows). Clearly, both features significantly reduce the computational effort in comparison to LES or even DNS. Therefore, the RANS approach is very popular in engineering applications. Of course, because of the averaging procedure, no detailed information can be obtained about the turbulent structures.

A large variety of turbulence models was devised to close the RANS equations and the research still continues. The models can be divided into first- and second-order closures, respectively.

The most complex, but also the most flexible, are second-order closure models. The Reynolds-Stress Transport (RST) model, which was first proposed by Rotta [217], solves modelled transport equations for the Reynolds-stress tensor. The partial differential equations for the six stress components have to be closed by one additional relation. Usually, an equation for the turbulent dissipation rate is employed. The RST models are able to account for strong nonlocal and history effects. Furthermore, they are able to capture the influence of streamline curvature or system rotation on the turbulent flow.

Closely related to the RST approach are the Algebraic Reynolds-Stress (ARS) models. They can be viewed as a combination of lower level models and the RST approach. The ARS models employ only two transport equations, mostly for the turbulent kinetic energy and the dissipation rate. The components of the Reynolds-stress tensor are related to the transport quantities by non-linear algebraic equations [218]. The ARS approach is capable of predicting rotational turbulent flows and secondary flows in channels with accuracy similar to the RST models. Detailed overviews of the RST and ARS models can be found in [219], [220].

Because of numerical problems with the RST and ARS models, which are primarily caused by the stiffness of the RST and the non-linearity of the ARS equations, first-order closures are more widely used in practice. In these models, the Reynolds stresses are expressed by means of a single scalar value, the so-called turbulent eddy viscosity. This approach is based on the eddy viscosity hypothesis of Boussinesq [221], [222], which assumes a linear relationship between the turbulent shear stress and the mean strain rate, similar to laminar flow. Herewith, the dynamic viscosity \(\mu\) in the viscous stress tensor (2.15) or in the governing equations (2.19) is replaced by the sum of a laminar and a turbulent component

As described earlier, the laminar viscosity is computed, for example, with the aid of the Sutherland formula (2.30). In analogy, the turbulent heat-flux vector (3.14) is modelled as

where \(k_T\) denotes the turbulent thermal conductivity coefficient. Hence, the thermal conductivity coefficient in Eq. (2.24) is evaluated as

The turbulent Prandtl number is in general assumed to be constant in the flow field (\(Pr_T=0.9\) for air). The coefficient of the turbulent eddy viscosity \(\mu_T\) has to be determined with the aid of a turbulence model. The limitations of the eddy viscosity approach are given by the assumption of equilibrium between the turbulence and the mean strain field, and by the independence on system rotation. The accuracy of the eddy-viscosity based models can be significantly improved either by using correction terms [223], [224], or by employing non-linear eddy viscosity approaches [225]-[227].

The first-order closures can be categorised into zero-, one-, and multiple-equation models, corresponding to the number of transport equations they utilise. Within the zero-equation or, as they are also denoted, algebraic models, the turbulent eddy viscosity is calculated from empirical relations, which employ only local mean flow variables. Therefore, no history effects can be simulated, which prevents a reliable prediction of separated flows. The most popular algebraic model, which is still in use for some applications, was developed by Baldwin and Lomax [228].

History effects are taken into account by the one- and two-equation models, where the convection and the diffusion of turbulence is modelled by transport equations. The most widely used one-equation turbulence model is due to Spalart and Allmaras [229], which is based on an eddy-viscosity like variable. The model is numerically very stable and easy to implement on structured as well as on unstructured grids.

In the case of the two-equation models, practically all approaches employ the transport equation for the turbulent kinetic energy. Among a large number of two-equation models, the \(K-\varepsilon\) model of Launder and Spalding [230] and the \(K-\omega\) model of Wilcox [231] are most often used in engineering applications. They offer a reasonable compromise between computational effort and accuracy. An interesting comparison between the Spalart-Allmaras model and various two-equation turbulence models was recently published in [232].