7. Turbulence Flow Models 7.1 Introduction Turbulent flows are characterized by fluctuating velocity fields. These fluctuations mix transported quantities such as momentum, energy, and species concentration, and cause the transported quantities to fluctuate as well. Since these fluctuations can be of a small scale and high frequency, they are too computationally expensive to simulate directly in practical engineering calculations. Instead, the instantaneous (exact) governing equations can be time-averaged, ensemble-averaged, or otherwise manipulated to remove the small scales, resulting in a modified set of equations that are computationally less expensive to solve. However, the modified equations contain additional unknown variables, and turbulence models are needed to determine these variables in terms of known quantities. FLUENT provides the following choices of turbulence models: Spalart-Allmaras model k-ε models: Standard k-ε model, RNG k-ε model, Realizable k-ε model k-ω models: Standard k-ω model, Shear-stress transport (SST) k-ω model Reynolds stress model (RSM) Large eddy simulation (LES) model 1 7.2 Choosing a Turbulence Model It is an unfortunate fact that no single turbulence model is universally accepted as being superior for all classes of problems. The choice of turbulence model will depend on considerations such as the physics encompassed in the flow, the established practice for a specific class of problem, the level of accuracy required, the available computational resources, and the amount of time available for the simulation. To make the most appropriate choice of model for your application, you need to understand the capabilities and limitations of the various options. The purpose of this section is to give an overview of issues related to the turbulence models provided in FLUENT. The computational effort and cost in terms of CPU time and memory of the individual models are discussed. While it is impossible to state categorically which model is best for a specific application, general guidelines are presented to help you choose the appropriate turbulence model for the flow you want to model. 7.2.1 Reynolds-Averaged Approach vs. LES A complete time-dependent solution of the exact Navier-Stokes equations for high-Reynolds-number turbulent flows in complex geometries is unlikely to be attainable for some time to come. Two alternative methods can be employed to transform the Navier-Stokes equations in such a way that the small-scale turbulent fluctuations do not have to be directly simulated: Reynolds averaging and filtering. Both methods introduce additional terms in the governing equations that need to be modeled in order to achieve "closure". (Closure implies that there are a sufficient number of equations for all the unknowns.) The Reynolds-averaged Navier-Stokes (RANS) equations represent transport equations for the mean flow quantities only, with all the scales of the turbulence being modeled. The approach of permitting a solution for 2 the mean flow variables greatly reduces the computational effort. If the mean flow is steady, the governing equations will not contain time derivatives and a steady-state solution can be obtained economically. A computational advantage is seen even in transient situations since the time step will be determined by the global unsteadiness in the mean flow rather than by the turbulence. The Reynolds-averaged approach is generally adopted for practical engineering calculations and uses models such as Spalart-Allmaras, k-ε and its variants, k-ω and its variants, and the RSM. LES provides an alternative approach in which the large eddies are computed in a time-dependent simulation that uses a set of "filtered" equations. Filtering is essentially a manipulation of the exact NavierStokes equations to remove only the eddies that are smaller than the size of the filter, which is usually taken as the mesh size. Like Reynolds averaging, the filtering process creates additional unknown terms that must be modeled in order to achieve closure. Statistics of the mean flow quantities, which are generally of most engineering interest, are gathered during the time-dependent simulation. The attraction of LES is that, by modeling less of the turbulence (and solving more), the error induced by the turbulence model will be reduced. One might also argue that it ought to be easier to find a "universal" model for the small scales, which tend to be more isotropic and less affected by the macroscopic flow features than the large eddies. It should, however, be stressed that the application of LES to industrial fluid simulations is in its infancy. As highlighted in a recent review publication, typical applications to date have been for simple geometries. This is mainly because of the large computer resources required to resolve the energy-containing turbulent eddies. Most successful LES has been done using high-order spatial discretization, with great care being taken to resolve all scales larger than the inertial subrange. The degradation of accuracy in the mean flow quantities with poorly resolved 3 LES is not well documented. In addition, the use of wall functions with LES is an approximation that requires further validation. As a general guideline, therefore, it is recommended that the conventional turbulence models employing the Reynolds-averaged approach be used for practical calculations. The LES approach described further in Section 7, has been made available for you to try if you have the computational resources and are willing to invest the effort. The rest of this section will deal with the choice of models using the Reynolds-averaged approach. 7.2.2 Reynolds (Ensemble) Averaging In Reynolds averaging, the solution variables in the instantaneous (exact) Navier-Stokes equations are decomposed into the mean (ensemble averaged or time-averaged) and fluctuating components. For the velocity components: 𝑢𝑖 = 𝑢̅ 𝑖 + 𝑢́ 𝑖 ⋯ (7.1) where 𝑢̅i and 𝑢́ i are the mean and fluctuating velocity components (i =1; 2; 3). Likewise, for pressure and other scalar quantities: ̅ + 𝜙́ 𝜙 = 𝜙 ⋯ (7.2) where ϕ denotes a scalar such as pressure, energy, or species concentration. Substituting expressions of this form for the flow variables into the instantaneous continuity and momentum equations and taking a time (or ensemble) average (and dropping the overbar on the mean velocity, 𝑢̅) yields the ensemble-averaged momentum equations. They can be written in Cartesian tensor form as: 4 𝜕𝜌 𝜕 (𝜌𝑢𝑖 ) = 0 + 𝜕𝑡 𝜕𝑥𝑖 𝜕 𝜕𝑡 (𝜌𝑢𝑖 ) + 𝜕 𝜕𝑥𝑗 𝜕 𝜕𝑥𝑖 ⋯ (7.3) 𝜕𝑝 𝜕 𝜕𝑢 𝜕𝑢𝑗 2 𝜕𝑢 (𝜌𝑢𝑖 𝑢𝑗 ) = − 𝜕𝑥 + 𝜕𝑥 [𝜇 (𝜕𝑥 𝑖 + 𝜕𝑥 − 3 𝛿𝑖𝑗 𝜕𝑥𝑙 )] + 𝑖 𝑗 𝑗 𝑖 ̅̅̅̅̅ ́𝑖 𝑢𝑗́ ) (−𝜌𝑢 𝑙 ⋯ (7.4) Equations (3 and 4) are called Reynolds-averaged Navier-Stokes (RANS) equations. They have the same general form as the instantaneous NavierStokes equations, with the velocities and other solution variables now representing ensemble-averaged (or time-averaged) values. Additional terms now appear that represent the effects of turbulence. This Reynolds ̅̅̅̅̅ stresses, (−𝜌𝑢 ́𝑖 𝑢𝑗́ ), must be modeled in order to close Equation 4. For variable-density flows, Equations 3 and 4 can be interpreted as Favreaveraged Navier-Stokes equations, with the velocities representing massaveraged values. As such, Equations 3 and 4 can be applied to densityvarying flows. 7.2.3 Boussinesq Approach vs. Reynolds Stress Transport Models The Reynolds-averaged approach to turbulence modeling requires that the Reynolds stresses in Equation 4 be appropriately modeled. A common method employs the Boussinesq hypothesis to relate the Reynolds stresses to the mean velocity gradients: ̅̅̅̅̅ −𝜌𝑢 ́𝑖 𝑢𝑗́ = 𝜇𝑡 ( 𝜕𝑢𝑗 𝜕𝑢𝑖 𝜕𝑢𝑗 2 + ) − (𝜌𝑘 + 𝜇𝑡 ) 𝛿𝑖𝑗 𝜕𝑥𝑗 𝜕𝑥𝑖 3 𝜕𝑥𝑖 5 ⋯ (7.5) The Boussinesq hypothesis is used in the Spalart-Allmaras model, the k-ε models, and the k-ω models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, µt. In the case of the Spalart-Allmaras model, only one additional transport equation (representing turbulent viscosity) is solved. In the case of the k-ε and k-ω models, two additional transport equations (for the turbulence kinetic energy, k, and either the turbulence dissipation rate, ε, or the specific dissipation rate, ω) are solved, and µt is computed as a function of k and ε. The disadvantage of the Boussinesq hypothesis as presented is that it assumes µt is an isotropic scalar quantity, which is not strictly true. The alternative approach, embodied in the RSM, is to solve transport equations for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (normally for ε) is also required. This means that five additional transport equations are required in 2D flows and seven additional transport equations must be solved in 3D. In many cases, models based on the Boussinesq hypothesis perform very well, and the additional computational expense of the Reynolds stress model is not justified. However, the RSM is clearly superior for situations in which the anisotropy of turbulence has a dominant effect on the mean flow. Such cases include highly swirling flows and stress-driven secondary flows. 7.3 The Spalart-Allmaras Model The Spalart-Allmaras model is a relatively simple one-equation model that solves a modeled transport equation for the kinematic eddy (turbulent) 6 viscosity. This embodies a relatively new class of one-equation models in which it is not necessary to calculate a length scale related to the local shear layer thickness. The Spalart-Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. It is also gaining popularity for turbomachinery applications. In its original form, the Spalart-Allmaras model is effectively a lowReynolds number model, requiring the viscous- affected region of the boundary layer to be properly resolved. In FLUENT, however, the SpalartAllmaras model has been implemented to use wall functions when the mesh resolution is not sufficiently fine. This might make it the best choice for relatively crude simulations on coarse meshes where accurate turbulent flow computations are not critical. Furthermore, the near-wall gradients of the transported variable in the model are much smaller than the gradients of the transported variables in the k-ε or k-ω models. This might make the model less sensitive to numerical error when non-layered meshes are used near walls. On a cautionary note, however, the Spalart-Allmaras model is still relatively new, and no claim is made regarding its suitability to all types of complex engineering flows. For instance, it cannot be relied on to predict the decay of homogeneous, isotropic turbulence. Furthermore, oneequation models are often criticized for their inability to rapidly accommodate changes in length scale, such as might be necessary when the flow changes abruptly from a wall-bounded to free shear flow. Turbulence quantities that can be reported for the Spalart-Allmaras model are as follows: Modified Turbulent Viscosity, Turbulent Viscosity, Effective Viscosity, Turbulent Viscosity Ratio, Effective Thermal Conductivity, Effective Prandtl Number, Wall Yplus 7 7.4 k-ε Model 7.4.1 The Standard k-ε Model The simplest "complete models" of turbulence are two-equation models in which the solution of two separate transport equations allows the turbulent velocity and length scales to be independently determined. The standard kε model in FLUENT falls within this class of turbulence model and has become the workhorse of practical engineering flow calculations. Robustness, economy, and reasonable accuracy for a wide range of turbulent flows explain its popularity in industrial flow and heat transfer simulations. It is a semi-empirical model, and the derivation of the model equations relies on phenomenological considerations and empiricism. As the strengths and weaknesses of the standard k-ε model have become known, improvements have been made to the model to improve its performance. Two of these variants are available in FLUENT: the RNG k-ε model and the realizable k-ε model. 7.4.2 The RNG k-ε Model The RNG k-ε model was derived using a rigorous statistical technique (called renormalization group theory). It is similar in form to the standard k-ε model, but includes the following refinements: The RNG model has an additional term in its ε equation that significantly improves the accuracy for rapidly strained flows. The effect of swirl on turbulence is included in the RNG model, enhancing accuracy for swirling flows. 8 The RNG theory provides an analytical formula for turbulent Prandtl numbers, while the standard k-ε model uses user-specified, constant values. While the standard k-ε model is a high-Reynolds-number model, the RNG theory provides an analytically-derived differential formula for effective viscosity that accounts for low-Reynolds-number effects. Effective use of this feature does, however, depend on appropriate treatment of the near-wall region. These features make the RNG k-ε model more accurate and reliable for a wider class of flows than the standard k-ε model. 7.4.3 The Realizable k-ε Model The realizable k-ε model is a relatively recent development and differs from the standard k-ε model in two important ways: The realizable k-ε model contains a new formulation for the turbulent viscosity. A new transport equation for the dissipation rate, ε, has been derived from an exact equation for the transport of the mean-square vorticity fluctuation. The term "realizable" means that the model satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of turbulent flows. Neither the standard k-ε model nor the RNG k-ε model is realizable. An immediate benefit of the realizable k-ε model is that it more accurately predicts the spreading rate of both planar and round jets. It is also, likely to provide superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation. 9 Both the realizable and RNG k-ε models have shown substantial improvements over the standard k-ε model where the flow features include strong streamline curvature, vortices, and rotation. Since the model is still relatively new, it is not clear in exactly which instances the realizable k-ε model consistently outperforms the RNG model. However, initial studies have shown that the realizable model provides the best performance of all the k-ε model versions for several validations of separated flows and flows with complex secondary flow features. One limitation of the realizable k-ε model is that it produces non-physical turbulent viscosities in situations when the computational domain contains both rotating and stationary fluid zones (e.g., multiple reference frames, rotating sliding meshes). This is due to the fact that the realizable k-ε model includes the effects of mean rotation in the definition of the turbulent viscosity. This extra rotation effect has been tested on single rotating reference frame systems and showed superior behavior over the standard k-ε model. However, due to the nature of this modification, its application to multiple reference frame systems should be taken with some caution. Turbulence quantities that can be reported for the k-ε models are as follows: Turbulent Kinetic Energy (k), Turbulence Intensity, Turbulent Dissipation Rate (Epsilon), Production of k, Turbulent Viscosity, Effective Viscosity, Turbulent Viscosity Ratio, Effective Thermal Conductivity, Effective Prandtl Number, Wall Yplus, Wall Ystar 7.5 k-ω Model 7.5.1 The Standard k-ω Model The standard k-ω model in FLUENT is based on the Wilcox k-ω model, which incorporates modifications for low-Reynolds-number effects, 11 compressibility, and shear flow spreading. The Wilcox model predicts free shear flow spreading rates that are in close agreement with measurements for far wakes, mixing layers, and plane, round, and radial jets, and is thus applicable to wall-bounded flows and free shear flows. A variation of the standard k-ω model called the SST k-ω model is also available in FLUENT, and is described in Section 5.2. 7.5.2 The Shear-Stress Transport (SST) k-ω Model The shear-stress transport (SST) k-ω model was developed to effectively blend the robust and accurate formulation of the k-ω model in the nearwall region with the free-stream independence of the k-ε model in the far field. To achieve this, the k-ε model is converted into a k-ω formulation. The SST k-ω model is similar to the standard k-ω model, but includes the following refinements: The standard k-ω model and the transformed k-ε model are both multiplied by a blending function and both models are added together. The blending function is designed to be one in the near wall region, which activates the standard k-ω model and zero away from the surface, which activates the transformed k-ε model. The SST model incorporates a damped cross-diffusion derivative the term in the ω equation. The definition of the turbulent viscosity is modified to account for the transport of the turbulent shear stress. The modeling constants are different. These features make the SST k-ω model more accurate and reliable for a wider class of flows (e.g., adverse pressure gradient flows, airfoils, transonic shock waves) than the standard k-ω model. 11 Turbulence quantities that can be reported for the k-ω models are as follows: Turbulent Kinetic Energy (k), Turbulence Intensity, Specific Dissipation Rate (Omega), Production of k, Turbulent Viscosity, Effective Viscosity, Turbulent Viscosity Ratio, Effective Thermal Conductivity, Effective Prandtl Number, Wall Ystar, Wall Yplus 7.6 The Reynolds Stress Model (RSM) The Reynolds stress model (RSM) is the most elaborate turbulence model that FLUENT provides. Abandoning the isotropic eddy-viscosity hypothesis, the RSM closes the Reynolds-averaged Navier-Stokes equations by solving transport equations for the Reynolds stresses, together with an equation for the dissipation rate. This means that four additional transport equations are required in 2D flows and seven additional transport equations must be solved in 3D. Since the RSM accounts for the effects of streamline curvature, swirl, rotation, and rapid changes in strain rate in a more rigorous manner than one-equation and two-equation models, it has greater potential to give accurate predictions for complex flows. However, the fidelity of RSM predictions is still limited by the closure assumptions employed to model various terms in the exact transport equations for the Reynolds stresses. The modeling of the pressure-strain and dissipation-rate terms is particularly challenging, and often considered to be responsible for compromising the accuracy of RSM predictions. The RSM might not always yield results that are clearly superior to the simpler models in all classes of flows to warrant the additional computational expense. However, use of the RSM is a must when the flow 12 features of interest are the result of anisotropy in the Reynolds stresses. Among the examples is cyclone flows, highly swirling flows in combustors, rotating flow passages, and the stress-induced secondary flows in ducts. Turbulence quantities that can be reported for the RSM are as follows: Turbulent Kinetic Energy (k), Turbulence Intensity, UU Reynolds Stress, VV Reynolds Stress, WW Reynolds Stress, UV Reynolds Stress, VW Reynolds Stress, UW Reynolds Stress, Turbulent Dissipation Rate (Epsilon), Production of k, Turbulent Viscosity, Effective Viscosity, Turbulent Viscosity Ratio, Effective Thermal Conductivity, Effective Prandtl Number Wall Yplus, Wall Ystar, Turbulent Reynolds Number (Ret) 7.7 Large eddy simulation (LES) model Turbulent flows are characterized by eddies with a wide range of length and time scales. The largest eddies are typically comparable in size to the characteristic length of the mean flow. The smallest scales are responsible for the dissipation of turbulence kinetic energy. It is theoretically possible to directly resolve the whole spectrum of turbulent scales using an approach known as direct numerical simulation (DNS). DNS is not, however, feasible for practical engineering problems. To understand the large computational cost of DNS, consider that the ratio of the large (energy-containing) scales to the small (energy dissipating) scales are ⁄ proportional to Re3𝑡 4 , where Ret is the turbulent Reynolds number. Therefore, to resolve all the scales, the mesh size in three dimensions will ⁄ be proportional to Re9𝑡 4 . Simple arithmetic shows that, for high Reynolds numbers, the mesh sizes required for DNS are prohibitive. Adding to the 13 computational cost is the fact that the simulation will be a transient one with very small-time steps since the temporal resolution requirements are governed by the dissipating scales, rather than the mean flow or the energycontaining eddies. As explained in Section 7.2.1, the conventional approach to flow simulations employs the solution of the Reynolds-averaged Navier-Stokes (RANS) equations. In the RANS approach, all the turbulent motions are modeled, resulting in significant savings in computational effort. Conceptually, large eddy simulation (LES) is situated somewhere between DNS and the RANS approach. Basically, large eddies are resolved directly in LES, while small eddies are modeled. The rationale behind LES can be summarized as follows: Momentum, mass, energy, and other passive scalars are transported mostly by large eddies. Large eddies are more problem-dependent. They are dictated by the geometries and boundary conditions of the flow involved. Small eddies are less dependent on the geometry, tend to be more isotropic, and are consequently more universal. The chance of finding a universal model is much higher when only small eddies are modeled. Solving only for the large eddies and modeling the smaller scales results in mesh resolution requirements that are much less restrictive than with DNS. Typically, mesh sizes can be at least one order of magnitude smaller than with DNS. Furthermore, the time step sizes will be proportional to the eddy-turnover time, which is much less restrictive than with DNS. In practical terms, however, extremely fine meshes are still required. It is only due to the explosive increases in computer hardware performance coupled with the availability of parallel processing that LES can even be considered as a possibility for engineering calculations. 14 The following sections give details of the governing equations for LES, present the two options for modeling the subgrid-scale stresses (necessary to achieve closure of the governing equations), and discuss the relevant boundary conditions. Turbulence quantities that can be reported for the LES model are as follows: Subgrid Turbulent Kinetic Energy, Subgrid Turbulent Viscosity, Subgrid Effective Viscosity, Subgrid Turbulent Viscosity Ratio, Effective Thermal Conductivity, Effective Prandtl Number, Wall Yplus 7.8 Wall Functions vs. Near-Wall Model Turbulent flows are significantly affected by the presence of walls. Obviously, the mean velocity field is affected through the no-slip condition that has to be satisfied at the wall. However, the turbulence is also changed by the presence of the wall in non-trivial ways. Very close to the wall, viscous damping reduces the tangential velocity fluctuations, while kinematic blocking reduces the normal fluctuations. Toward the outer part of the near-wall region, however, the turbulence is rapidly augmented by the production of turbulence kinetic energy due to the large gradients in mean velocity. The near-wall modeling significantly impacts the fidelity of numerical solutions, inasmuch as walls are the main source of mean vorticity and turbulence. After all, it is in the near-wall region that the solution variables have large gradients, and the momentum and other scalar transports occur most vigorously. Therefore, accurate representation of the flow in the nearwall region determines successful predictions of wall-bounded turbulent flows. The k-ε models, the RSM, and the LES model are primarily valid for turbulent core flows (i.e., the flow in the regions somewhat far from walls). 15 Consideration therefore needs to be given as to how to make these models suitable for wall-bounded flows. The Spalart-Allmaras and k-ω models were designed to be applied throughout the boundary layer, provided that the near-wall mesh resolution is sufficient. Numerous experiments have shown that the near-wall region can be largely subdivided into three layers. In the innermost layer, called the "viscous sublayer", the flow is almost laminar, and the (molecular) viscosity plays a dominant role in momentum and heat or mass transfer. In the outer layer, called the fully-turbulent layer, turbulence plays a major role. Finally, there is an interim region between the viscous sublayer and the fully turbulent layer where the effects of molecular viscosity and turbulence are equally important. Figure 7.1 illustrates these subdivisions of the near-wall region, plotted in semi-log coordinates. Figure 7.1 Subdivisions of the Near-Wall Region Traditionally, there are two approaches to modeling the near-wall region. In one approach, the viscosity-affected inner region (viscous sublayer and buffer layer) is not resolved. Instead, semi-empirical formulas called \wall 16 functions" are used to bridge the viscosity-affected region between the wall and the fully-turbulent region. The use of wall functions obviates the need to modify the turbulence models to account for the presence of the wall. In another approach, the turbulence models are modified to enable the viscosity-affected region to be resolved with a mesh all the way to the wall, including the viscous sublayer. For purposes of discussion, this will be termed the "near-wall modeling" approach. These two approaches are depicted in Figure 7.2 Figure 7.2 Near-Wall Treatments in FLUENT In most high-Reynolds-number flows, the wall function approach substantially saves computational resources, because the viscosity-affected 17 near-wall region, in which the solution variables change most rapidly, does not need to be resolved. The wall function approach is popular because it is economical, robust, and reasonably accurate. It is a practical option for the near-wall treatments for industrial flow simulations. The wall function approach, however, is inadequate in situations where the low-Reynoldsnumber effects are pervasive in the flow domain in question, and the hypotheses underlying the wall functions cease to be valid. Such situations require near-wall models that are valid in the viscosity-affected region and accordingly integrable all the way to the wall. FLUENT provides both the wall function approach and the near-wall modeling approach. 18