Advanced turbulence models Two-equation turbulence models, such as the k– ε model introduced earlier,

3.7.4 Advanced turbulence models Two-equation turbulence models, such as the k– ε model introduced earlier,

give good results for simple flows and some recirculating flows, but research over a period of three decades has highlighted a number of shortcomings. Leschziner (in Peyret and Krause, 2000) and Hanjaliz (2004) summarised the nature and causes of these performance problems:

• Low Reynolds number flows: in these flows wall functions based on the log-law are inaccurate and it is necessary to integrate the k- and ε-equations to the wall. Very rapid changes occur in the distributions

of k and ε as we reach the buffer layer between the fully turbulent region and the viscous sublayer. This requires large numbers of grid points to resolve the changes, and we also need non-linear wall-damping functions to force upon k and ε the correct behaviour as the character

of the near-wall flow changes from turbulence dominated to viscous dominated. As a consequence the system of equations that needs to be solved is numerically stiff, which means that it may be difficult to get converged solutions. Furthermore, the results can be grid dependent.

• Rapidly changing flows: the Reynolds stress − ρ u i ′u j ′ is proportional to the mean rate of strain S ij in two-equation models. This only holds when

86 CHAPTER 3 TURBULENCE AND ITS MODELLING

the rates of production and dissipation of turbulence kinetic energy are roughly in balance. In rapidly changing flows this is not the case.

• Stress anisotropy: the normal Reynolds stresses − ρ u i ′ 2 will all be approximately equal to − – 2 3 ρk if a thin shear layer flow is evaluated

using a two-equation model. Experimental data presented in section 3.4 showed that this is not correct, but in spite of this the k– ε model

performs well in such flows because the gradients of normal turbulent

stresses − ρ u ′ 2 i are small compared with the gradient of the dominant turbulent shear stress − ρ u ′v′ . Consequently, the normal stresses may be large, but they are not dynamically active in thin shear layer flows, i.e. they are not responsible for driving any flows. In more complex flows the gradients of normal turbulent stresses are not negligible and can drive significant flows. These effects cannot be predicted by the standard two-equation models.

• Strong adverse pressure gradients and recirculation regions: this problem

particularly affects the k– ε model and is also attributable to the isotropy of its predicted normal Reynolds stresses and the resultant failure to represent correctly the subtle interactions between normal Reynolds stresses and mean flow that determine turbulent energy production. The k– ε model overpredicts the shear stress and suppresses separation in flows over curved walls. This is a significant problem in flows over aerofoils, e.g. in aerospace applications.

• Extra strains: streamline curvature, rotation and extra body forces all

give rise to additional interactions between the mean strain rate and the Reynolds stresses. These physical effects are not captured by standard two-equation models.

As we have seen, the RSM incorporates an exact representation of the Reynolds stress production process and, hence, addresses most of these problems adequately, but at the cost of a significant increase in computer storage and run time. Below we consider some of the more recent advances in turbulence modelling that seek to address some or all of the above problems.

Advanced treatment of the near-wall region: two-layer k--- εε model The two-layer model represents an improved treatment of the near-wall

region for turbulent flows at low Reynolds number. The intention is, as in the low Reynolds number k– ε model discussed earlier, to integrate to

the wall by placing the near-wall grid point in the viscous sublayer ( y + < 1). The numerical stability problems (Chen and Patel, 1988) associated with the non-linear wall-damping functions, necessary in the low Reynolds number k – ε model to integrate both k- and ε-equations to the wall, are avoided by

sub-dividing the boundary layer into two regions (Rodi, 1991): • Fully turbulent region, Re y =yk/ ν ≥ 200: the standard k–ε model is

used and the eddy viscosity is computed with the usual relationship

(3.44), µ t ,t =C µ ρk 2 / ε • Viscous region, Re y < 200: only the k-equation is solved in this

region and a length scale is specified using ᐉ = κy[1 − exp(−Re y /A)] for the evaluation of the rate of dissipation with ε=C 3/4 µ k 3/2 /ᐉ using

A =2 κC −3/4 µ 1/4 and the eddy viscosity in this region with µ t ,v =C µ ρkᐉ and A = 70

3.7 RANS EQUATIONS AND TURBULENCE MODELS 87

The mixing length formulae are similar in form to the expression in Table 3.2 for the length scale in the viscous sub-layer of a wall boundary layer. In order to avoid instabilities associated with differences between µ t ,t and µ t ,v at the join between the fully turbulent and viscous regions, a blending formula is

used to evaluate the eddy viscosity in τ ij =− ρ u i ′u j ′ =2 µ t S ij − – 2 3 ρkδ ij : µ t =F µ µ t ,t + (1 − F µ ) µ t ,v

(3.64) The blending function F µ =F µ (Re y ) is zero at the wall and tends to 1 in the

fully turbulent region when Re y Ⰷ 200. The functional form of F µ is designed to ensure a smooth transition around Re y = 200.

The two-layer model is less grid dependent and more numerically stable than the earlier low Reynolds number k– ε models and has become quite popular in more complex flow simulations where integration to the wall of the flow equations is necessary.

Strain sensitivity: RNG k--- εε model The statistical mechanics approach has led to new mathematical formalisms,

which, in conjunction with a limited number of assumptions regarding the statistics of small-scale turbulence, provide a rigorous basis for the extension of eddy viscosity models. The renormalization group (RNG) devised by Yakhot and Orszag of Princeton University has attracted most interest. They represented the effects of the small-scale turbulence by means of a random forcing function in the Navier–Stokes equation. The RNG procedure sys- tematically removes the small scales of motion from the governing equations by expressing their effects in terms of larger scale motions and a modified viscosity. The mathematics is highly abstruse; we only quote the RNG k– ε model equations for high Reynolds number flows derived by Yakhot et al. (1992):

∂(ρk) + div( ρkU) = div[α k µ eff grad k] + τ ij .S ij − ρε

(3.65) ∂t

ε 2 + div( ρεU) = div[α ε µ eff grad ε] + C* 1 ε τ ij .S ij −C 2 ε ρ

(3.66) ∂t

k with

C µ = 0.0845 α k = α ε = 1.39 C 1 ε = 1.42 C 2 ε = 1.68 (3.67)

98 CHAPTER 3 TURBULENCE AND ITS MODELLING

The literature is too extensive even to begin to review here. The main sources of useful, applications-oriented information are: Transactions of the American Society of Mechanical Engineers – in particular the Journal of Fluids Engineering, Journal of Heat Transfer and Journal of Engineering for Gas Turbines and Power – as well as the AIAA Journal, the International Journal of Heat

and Mass Transfer and the International Journal of Heat and Fluid Flow.

3.8 Large eddy simulation

In spite of century-long efforts to develop RANS turbulence models, a general-purpose model suitable for a wide range of practical applications has so far proved to be elusive. This is to a large extent attributable to differences in the behaviour of large and small eddies. The smaller eddies are nearly isotropic and have a universal behaviour (for turbulent flows at sufficiently high Reynolds numbers at least). On the other hand, the larger eddies, which interact with and extract energy from the mean flow, are more anisotropic and their behaviour is dictated by the geometry of the problem domain, the boundary conditions and body forces. When Reynolds-averaged equations are used the collective behaviour of all eddies must be described by a

single turbulence model, but the problem dependence of the largest eddies complicates the search for widely applicable models. A different approach to the computation of turbulent flows accepts that the larger eddies need to be computed for each problem with a time-dependent simulation. The universal behaviour of the smaller eddies, on the other hand, should hope- fully be easier to capture with a compact model. This is the essence of the large eddy simulation (LES) approach to the numerical treatment of turbulence.

Instead of time-averaging, LES uses a spatial filtering operation to separ- ate the larger and smaller eddies. The method starts with the selection of a filtering function and a certain cutoff width with the aim of resolving in an unsteady flow computation all those eddies with a length scale greater than the cutoff width. In the next step the spatial filtering operation is performed on the time-dependent flow equations. During spatial filtering information relating to the smaller, filtered-out turbulent eddies is destroyed. This, and interaction effects between the larger, resolved eddies and the smaller unre- solved ones, gives rise to sub-grid-scale stresses or SGS stresses. Their effect on the resolved flow must be described by means of an SGS model. If the finite volume method is used the time-dependent, space-filtered flow equa- tions are solved on a grid of control volumes along with the SGS model of the unresolved stresses. This yields the mean flow and all turbulent eddies at scales larger than the cutoff width. In this section we review the methodo- logy of LES computation of turbulent flows and summarise recent achieve- ments in the calculation of industrially relevant flows.