Reynolds stress equation models The most complex classical turbulence model is the Reynolds stress equa-

3.7.3 Reynolds stress equation models The most complex classical turbulence model is the Reynolds stress equa-

tion model (RSM), also called the second-order or second-moment closure model. Several major drawbacks of the k– ε model emerge when it is attempted to predict flows with complex strain fields or significant body forces. Under such conditions the individual Reynolds stresses are poorly represented by formula (3.48) even if the turbulent kinetic energy is computed to reasonable accuracy. The exact Reynolds stress transport equation on the other hand can account for the directional effects of the Reynolds stress field.

The modelling strategy originates from work reported in Launder et al. (1975). We follow established practice in the literature and call R ij =− τ ij / ρ = u ′u i j ′ the Reynolds stress, although the term kinematic Reynolds stress would be more precise. The exact equation for the transport of R ij takes the

following form:

3.7 RANS EQUATIONS AND TURBULENCE MODELS

DR ij ∂R

ij +C

ij =P ij +D ij − ε ij +Π ij +Ω ij (3.55)

Dt

∂t

Rate of Transport

Transport of R ij due Transport change of + of R ij by = production + of R ij by − dissipation + to turbulent pressure + of R ij due to R ij = u i ′u j ′ convection of R ij

Rate of

Transport Rate of

– strain interactions rotation Equation (3.55) describes six partial differential equations: one for the trans-

diffusion

of R ij

port of each of the six independent Reynolds stresses ( u ′ 2 1 , u 2 ′ 2 , u ′ 2 3 , u ′u 1 2 ′ , u 1 ′u 3 ′ and , u 2 ′u 3 ′ since u ′u 2 1 ′ = u 1 ′u 2 ′ , u ′u 3 1 ′ = u ′u 1 3 ′ and u 3 ′u 2 ′ = u 2 ′u 3 ′ ). If it is compared with the exact transport equation for the turbulent kinetic energy (3.42) two new physical processes appear in the Reynolds stress equations: the pressure–strain interaction or correlation term Π ij , whose effect on the

kinetic energy can be shown to be zero, and the rotation term Ω ij . In CFD computations with the Reynolds stress transport equations the convection, production and rotation terms can be retained in their exact form. The convective term is as follows:

∂(ρU k u ′u i j ′)

C ij =

= div( ρ u i ′u j ′ U)

∂x k

the production term is

and, finally, the rotational term is given by

(3.58) Here ω k is the rotation vector and e ijk is the alternating symbol; e ijk = +1 if i,

Ω ij = −2 ω k ( u j ′u′ m e ikm + u i ′u′ m e jkm )

j and k are different and in cyclic order, e ijk = −1 if i, j and k are different and in anti-cyclic order; and e ijk = 0 if any two indices are the same.

To obtain a solvable form of (3.55) we need models for the diffusion, the dissipation rate and the pressure–strain correlation terms on the right hand side. Launder et al. (1975) and Rodi (1980) gave comprehensive details of the most general models. For the sake of simplicity we quote those models derived from this approach that are used in some commercial CFD codes. These models often lack somewhat in detail, but their structure is easier to understand and the main message is intact in all cases.

The diffusion term D ij can be modelled with the assumption that the rate of transport of Reynolds stresses by diffusion is proportional to gradients of Reynolds stresses. This gradient diffusion idea recurs throughout turbulence modelling. Commercial CFD codes often favour the simplest form:

∂ A ν t ∂R ij D A ν t

grad(R ) D D ij = B E = div B ij

∂x m C σ k ∂x F C σ

k 2 with ν t =C µ ,C µ = 0.09 and σ k = 1.0

82 CHAPTER 3 TURBULENCE AND ITS MODELLING

The dissipation rate ε ij is modelled by assuming isotropy of the small dissi- pative eddies. It is set so that it affects the normal Reynolds stresses (i =j)

only and each stress component in equal measure. This can be achieved by

2 ε ij = εδ ij

3 where ε is the dissipation rate of turbulent kinetic energy defined by (3.43).

The Kronecker delta δ ij is given by δ ij = 1 if i = j and δ ij = 0 if i ≠ j. The pressure–strain interactions constitute one of the most important terms in (3.55), but the most difficult one to model accurately. Their effect on the Reynolds stresses is caused by two distinct physical processes: (i) a ‘slow’ process that reduces anisotropy of the turbulent eddies due to their mutual interactions; and (ii) a ‘rapid’ process due to interactions between turbulent fluctuations and the mean flow strain that produce the eddies such that the anisotropic production of turbulent eddies is opposed. The overall effect of both processes is to redistribute energy amongst the normal Reynolds stresses (i = j ) so as to make them more isotropic and to reduce the Reynolds shear stresses (i ≠ j ). The simplest account of the slow process takes the rate of return to isotropic conditions to be proportional to the

degree of anisotropy a ij of the Reynolds stresses (a ij =R ij − – 2 3 k δ ij ) divided by

a characteristic time scale of the turbulence k/ ε. The rate of the rapid pro- cess is taken to be proportional to the production processes that generate the anisotropy. The simplest representation of the pressure–strain term in the Reynolds stress transport equation is therefore given by

2 2 Π ij = −C 1 (R ij −k δ ij ) −C 2 (P ij −P δ ij )

(3.61) k

with C 1 = 1.8 and C 2 = 0.6

More advanced accounts include corrections in the second set of brackets in equation (3.61) to ensure that the model is frame invariant (i.e. the effect is the same irrespective of the co-ordinate system).

The effect of the pressure–strain term (3.61) is to decrease anisotropy of Reynolds stresses (i.e. to equalise the normal stresses u 1 ′ 2 , u 2 ′ 2 and u ′ 3 2 ), but we have seen in section 3.4 that measurements indicate an increase of the anisotropy of normal Reynolds stresses in the vicinity of a solid wall due to damping of fluctuations in the directions normal to the wall. Hence, addi- tional corrections are needed to account for the influence of wall proximity on the pressure–strain terms. These corrections are different in nature from the wall-damping functions encountered in the k– ε model and need to be

applied irrespective of the value of the mean flow Reynolds number. It is beyond the scope of this introduction to give all this detail. The reader is directed to a comprehensive model that accounts for all these effects in Launder et al. (1975).

Turbulent kinetic energy k is needed in the above formulae and can be found by simple addition of the three normal stresses:

k = – 1 (R +R +R ) = – 1 ( u ′ 2 + u ′ 2 + u ′ 2 2 11 22 33 2 1 2 3 ) The six equations for Reynolds stress transport are solved along with a

model equation for the scalar dissipation rate ε. Again a more exact form is

3.7 RANS EQUATIONS AND TURBULENCE MODELS 83

found in Launder et al. (1975), but the equation from the standard k– ε model is used in commercial CFD for the sake of simplicity:

ε 2 = div

grad ε+C 1 ε 2 ν t S ij .S ij −C 2 ε (3.62) Dt

where C 1 ε = 1.44 and C 2 ε = 1.92

Rate of Transport Transport Rate of Rate of change + of ε by

+ production − destruction of ε

= of ε by

of ε The usual boundary conditions for elliptic flows are required for the solution

convection diffusion

of ε

of the Reynolds stress transport equations: • inlet:

specified distributions of R ij and ε • outlet and symmetry: ∂R ij / ∂n = 0 and ∂ε/∂n = 0 • free stream:

R ij = 0 and ε = 0 are given or ∂R ij / ∂n = 0 and ∂ε/∂n = 0

• solid wall: use wall functions relating R

ij to either k or u τ ,

e.g. u ′ 2 = 1.1k, u 2 ′ 1 2 = 0.25k, u 3 ′ 2 = 0.66k,

− u 1 ′u 2 ′ = 0.26k

In the absence of any information, approximate inlet distributions for R ij may be calculated from the turbulence intensity T i and a characteristic length L of the equipment (e.g. equivalent pipe diameter) by means of the follow- ing assumed relationships:

u i ′u j ′ = 0 (i ≠ j ) Expressions such as these should not be used without a subsequent test of

the sensitivity of results to the assumed inlet boundary conditions. For computations at high Reynolds numbers wall-function-type bound- ary conditions can be used, which are very similar to those of the k– ε model and relate the wall shear stress to mean flow quantities. Near-wall Reynolds stress values are computed from formulae such as R ij = u ′u i j ′ =c ij k , where the

c ij are obtained from measurements. Low Reynolds number modifications to the models can be incorporated to add the effects of molecular viscosity to the diffusion terms and to account for anisotropy in the dissipation rate term in the R ij -equations. Wall-damping functions to adjust the constants of the ε-equation and Launder and Sharma’s

modified dissipation rate variable 6 ≡ ε − 2ν(∂k 1/2 / ∂y) 2 (see also section 3.7.2) give more realistic modelling near solid walls (Launder and Sharma, 1974). So et al. (1991) gave a review of the performance of near-wall treatments where details may be found.

Similar models involving three further model PDEs – one for every tur- bulent scalar flux u i ′ ϕ′ of equation (3.32) – are available for scalar transport.

84 CHAPTER 3 TURBULENCE AND ITS MODELLING

The interested reader is referred to Rodi (1980) for further material. Com- mercial CFD codes may use or give as an alternative the simple expedient of solving a single scalar transport equation and using the Reynolds analogy by adding a turbulent diffusion coefficient Γ t = µ t / σ φ to the laminar diffusion

coefficient with a specified value of the Prandtl/Schmidt numbers σ φ around

0.7. Little is known about low Reynolds number modifications to the scalar transport equations in near-wall flows.

Assessment of performance RSMs are clearly quite complex, but it is generally accepted that they are

the ‘simplest’ type of model with the potential to describe all the mean flow properties and Reynolds stresses without case-by-case adjustment. The RSM is by no means as well validated as the k– ε model, and because of the high cost of the computations it is not so widely used in industrial flow calculations (Table 3.5). Moreover, the model can suffer from convergence problems due to numerical issues associated with the coupling of the mean velocity and turbulent stress fields through source terms. The extension and improvement of these models is an area of very active research. Once a con- sensus has been reached about the precise form of the component models and the best numerical solution strategy, it is likely that this form of turbu- lence modelling will begin to be more widely applied by industrial users. Figure 3.16 (Leschziner, in Peyret and Krause, 2000) gives a performance comparison of the RSM and k– ε models against measured distributions of pressure coefficient and suction-side skin friction coefficients for an Aérospatiale aerofoil. Leschziner notes that the aerofoil is close to stall at the chosen angle of attack. The diagrams show that the k– ε model (labelled LL k– ε) fails to reproduce several details of the pressure distribution in the leading and trailing edge regions. The prediction of the onset of separation depends crucially on the details of the boundary layer structure just upstream, which are captured much better by the RSM model (labelled RSTM + 1eq, to highlight the chosen treatment of the viscous sub-layer).

This model also gives excellent agreement with the measured distribution of skin friction on the suction side of the aerofoil.

Table 3.5 RSM assessment Advantages:

• potentially the most general of all classical turbulence models • only initial and/or boundary conditions need to be supplied • very accurate calculation of mean flow properties and all Reynolds stresses

for many simple and more complex flows including wall jets, asymmetric channel and non-circular duct flows and curved flows

Disadvantages: • very large computing costs (seven extra PDEs) • not as widely validated as the mixing length and k– ε models

• performs just as poorly as the k– ε model in some flows due to identical problems with the ε-equation modelling (e.g. axisymmetric jets and unconfined recirculating flows)

3.7 RANS EQUATIONS AND TURBULENCE MODELS

Figure 3.16 Comparison of predictions of RSM and standard k – ε model with measurements on

a high-lift Aérospatiale aerofoil: (a) pressure coefficient; (b) skin friction coefficient Source : Leschziner, in Peyret and Krause (2000)