Closing remarks --- RANS turbulence models The field of turbulence modelling provides an area of intense research

3.7.5 Closing remarks --- RANS turbulence models The field of turbulence modelling provides an area of intense research

activity for the CFD and fluid engineering communities. In the previous sections we have outlined the modelling strategy of the most prominent RANS turbulence models that are applied in or under development for commercially available general-purpose codes. Behind much of the research effort in advanced turbulence modelling lies the belief that, irrespective of boundary conditions and geometry, there exists a (limited) number of universal features of turbulence, which, when identified correctly, can form the basis of a complete description of flow variables of interest to an engineer. The emphasis must be on the word ‘belief ’, because the very existence of

a classical model – based on time-averaged equations – of this kind is con- tested by a number of renowned experts in the field. Encouraged by, for example, the early successes of the mixing length model in the external aerodynamics field, they favour the development of dedicated models for limited classes of flow. These two viewpoints naturally lead to two distinct lines of research work:

1 Development and optimisation of turbulence models for limited categories of flows

2 The search for a comprehensive and completely general-purpose turbulence model

Industry has many pressing flow problems to solve that will not wait for the conception of a universal turbulence model. The k– ε model is still widely

used in industrial applications and produces useful results in spite of earlier observations relating to its limited validity. Fortunately many sectors of industry are specifically interested in a limited class of flows only, e.g. pipe flows for the oil transportation sector, turbines and combustors for power engineering. The large majority of turbulence research consists of case- by-case examination and validation of existing turbulence models for such specific problems.

78 CHAPTER 3 TURBULENCE AND ITS MODELLING

stresses take over from turbulent Reynolds stresses at low Reynolds numbers and in the viscous sub-layer adjacent to solid walls. The equations of the low Reynolds number k– ε model, which replace (3.44)–(3.46), are given below:

G A µ t D J + div( ρkU) = div H B µ+ E grad k

µ ij ∂t

K +2 t S .S ij − ρε (3.52)

+ div( ρεU) = div ∂t

H B µ+ E grad ε K

ε 2 +C 1 ε f 1 2 µ t S ij .S ij −C 2 ε f 2 ρ

k The most obvious modification, which is universally made, is to include the

molecular viscosity µ in the diffusion terms in (3.52)–(3.53). The constants

C µ ,C 1 ε and C 2 ε in the standard k– ε model are multiplied by wall-damping functions f µ , f 1 and f 2 , respectively, which are themselves functions of the turbulence Reynolds number (Re = ϑᐉ/ ν=k 2 /( εν)), Re =k t 1/2 y y / ν and/or similar parameters. As an example we quote the Lam and Bremhorst (1981) wall-damping functions:

2 A )] 20.5 1 f D

µ = [1 − exp(− 0.0165 Re y B +

Re C E t F

A 3 0.05 D

f 1 2 =1+ B E f 2 = 1 − exp(−Re t )

C f µ F Equations (3.51) – (3.53) and the RANS equations need to be integrated to

the wall, but the boundary condition for ε gives rise to problems. The best available measurements suggest that the rate of dissipation of turbulent energy rises steeply as the wall is approached and tends to an (unknown) con- stant value. Lam and Bremhorst use ∂ε/∂y = 0 as the boundary condition.

Other low Reynolds number k– ε models are based on a modified dissipation rate variable defined as 6 = ε − 2ν(∂ k/∂n) 2 , introduced by Launder and Sharma (1974), which allows us to use the more straightforward boundary condition 6 = 0. It should be noted that the resulting equation set is numer- ically stiff and the further appearance of non-linear wall-damping functions regularly gives rise to severe challenges to achieve convergence.

Assessment of performance The k– ε model is the most widely used and validated turbulence model.

It has achieved notable successes in calculating a wide variety of thin shear layer and recirculating flows without the need for case-by-case adjustment of

3.7 RANS EQUATIONS AND TURBULENCE MODELS

the model constants. The model performs particularly well in confined flows where the Reynolds shear stresses are most important. This includes a wide range of flows with industrial engineering applications, which explains its popularity. Versions of the model are available which incorporate effects of buoyancy (Rodi, 1980). Such models are used to study environmental flows such as pollutant dispersion in the atmosphere and in lakes and the modelling of fires. Figure 3.15 ( Jones and Whitelaw, 1982) shows the results of early calculations with the k– ε model of turbulent combusting flows for an axisymmetric combustor. Computed contours of axial velocity and temperature are compared with experimental values showing good general agreement but differences in detail. The flow pattern in the combustor is dominated by turbulent transport and hence its correct prediction is vitally important for the development of the flow field and the combustion process. We come back to this issue in Chapter 12 where we examine different models of turbulent combustion.

Figure 3.15 Comparison of predictions of k– ε model with measurements in an axisymmetric combustor: (a) axial velocity contours; (b) temperature contours Source : Jones and Whitelaw (1982)

In spite of the numerous successes, the standard k– ε model shows only moderate agreement in unconfined flows. The model is reported not to perform well in weak shear layers (far wakes and mixing layers), and the spreading rate of axisymmetric jets in stagnant surroundings is severely overpredicted. In large parts of these flows the rate of production of turbulent kinetic energy is much less than the rate of dissipation, and the difficulties can only be over- come by making ad hoc adjustment to model constants C.

Bradshaw et al. (1981) stated that the practice of incorporating the pressure transport term of the exact k-equation in the gradient diffusion expression of

80 CHAPTER 3 TURBULENCE AND ITS MODELLING

the model equation is deemed to be acceptable on the grounds that the pressure term is sometimes so small that measured turbulent kinetic energy budgets balance without it. They noted, however, that many of these meas- urements contain substantial errors, and it is certainly not generally true that pressure diffusion effects are negligible.

We can expect that the k– ε model, and all other models that are based on Boussinesq’s isotropic eddy viscosity assumption, will have problems in swirling flows and flows with large rapid extra strains (e.g. highly curved boundary layers and diverging passages) that affect the structure of turbu- lence in a subtle manner. Secondary flows in long non-circular ducts, which are driven by anisotropic normal Reynolds stresses, can also not be predicted due to the same deficiencies of the treatment of normal stresses within the k – ε model. Finally, the model is oblivious to body forces due to rotation of

the frame of reference.

A summary of the performance assessment for the standard k– ε model is given in Table 3.4.

Table 3.4 Standard k– ε model assessment Advantages:

• simplest turbulence model for which only initial and/or boundary conditions need to be supplied • excellent performance for many industrially relevant flows • well established, the most widely validated turbulence model

Disadvantages: • more expensive to implement than mixing length model (two extra PDEs) • poor performance in a variety of important cases such as:

(i) some unconfined flows (ii) flows with large extra strains (e.g. curved boundary layers, swirling

flows) (iii) rotating flows (iv) flows driven by anisotropy of normal Reynolds stresses (e.g. fully

developed flows in non-circular ducts)