Turbulence Model

The prediction of turbulent flows in complex flow passages such as turbo- machinery is a great challenge due to the influence of the complex geometry

of the passage. To achieve the objective of prediction, the accuracy of the turbulence model employed is a key issue in the numerical simulation of the flow field [42, 43]. This section is not aiming to cover all turbulence models, more attentions is given on the turbulence model that has been utilized in our numerical studies. The materials cited here are mainly from Launder & Spalding (1972) [44] and Wilcox (1998) [45].

Turbulent flows is strictly governed by the Navier-Stokes equations. In practice, it is usually solved by the Reynolds-averaged Navier-Stokes equations together with a proper turbulence models. A turbulence model is defined as a series of equations (algebraic or differential) which determine the turbu- lent transport terms in the mean flow equations and thus close the system of equations. Complexity of different turbulence models strongly depends on the information what one wants to achieve and also on the nature of Navier- Stokes equation (i.e. theN−S equation) which is inherently nonlinear, time- dependent, three-dimensional PDE. Turbulence models are all based on hy- potheses about the turbulent processes and to some extent rely on empirical formula; they do not resolve the details of the turbulent motion, but only the effect of turbulence on the mean flow behavior. Therefore, the concept of Reynolds averaging is the basis of turbulence modeling. For modeling very complex phenomena, one of the most important issue is how to ob-

tain the useful information by using a model as simple as possible.

Generally, the simulation methods can be classified by Figure 2.411_.

Most extensive work has been done by Daly and Harlow (1970) [46] and Launder and Spalding (1972) [44] on two-equation turbulence model. k −

model is the most common used two-equation turbulence model, although it

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Figure 2.4: Classification of simulation methods of turbulent flow

does not perform very well in the flow with large adverse pressure gradients. It has two extra transport equations to represent the turbulent properties of the flow, allowing a two equation model to account for history effects like convection and diffusion of turbulent energy.

The first transported variable is turbulent kinetic energy, k. The sec- ond transported variable in this case is the turbulent dissipation, , which determines the scale of the turbulence.

The basis for all two equation models is the Boussinesq eddy viscosity assumption, which postulates that the Reynolds stress tensor, τij, is propor-

tional to the mean strain rate tensor,Sij, and can be written in the following

way:

τij = 2µtSij +

2

3ρkδij (2.5) Where µt is a scalar property called the eddy viscosity which is normally

computed from the two transported variables. The last term is included for modeling incompressible flow to ensure that the definition of turbulence kinetic energy is obeyed: k = 1 2u 0 iu 0 i. And

Sij = ∂ui ∂xj + ∂uj ∂xi (2.6)

Whereδij is the symbol of ‘Kronecker delta’ (wheni=j, δij = 1; when i6=j,

δij = 0.

There are varieties of two-equation turbulence models, such as Standard

k− model, Realisable k− model, RNGk−model, k−ω model, Wilcox’s

k −ω model, Wilcox’s modified k −ω model, and SST k − ω model. For unsteady turbulent flow calculations, RNGk−model and SST k−ω model are widely used [11].

RNG k− model

The RNG model was developed using a rigorous statistical technique (called Renormalization Group (RNG) theory) by Yakhot et al. [47] to renormalise the Navier-Stokes equations to account for low-Reynolds-number effects. In the standardk−model the eddy viscosity is determined from a single turbulence length scale, so that the calculated turbulent diffusion is only for the specified scale, whereas in reality all scales of motion will contribute to the turbulent diffusion. The RNG model has an additional term in itsequation, accounting for the different scales of motion, which significantly improves the accuracy for rapidly strained flows. Besides, the effect of swirl on turbulence is included in the RNG model, enhancing accuracy for swirling flows. An analytical formula for turbulent Prandtl numbers was applied, while only user-specified, constant values was used in Standard k− model [48]. These features make the RNG

k−model more accurate and reliable for a wider class of flows12_{than Standard}

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For example, high streamline curvature and strain rate; transitional flows; wall heat and mass transfer.

k− model. But it still lacks the accuracy for predicting the spreading of a round jet.

The momentum equation is

ρ∂ui ∂t +ρuj ∂ui ∂xj =ρFi− ∂p¯ ∂xi +µ ∂ 2_u i ∂xj∂xj −ρ ∂ ∂xj u0 iu 0 j . (2.7) Here,−ρu0 iu 0

j is the Reynolds stress of turbulent flow as shown in Eqn. (??); ¯p

is the averaged pressure; ρis the fluid density; and F is the body force acting on the unit volume fluid.

The transport equations for turbulent kinetic energy k are

ρDk Dt = ∂ ∂xj αkµef f ∂k ∂xj + 2µtSij ∂ui ∂xj −ρε (2.8)

and for dissipation rate are

ρDε Dt = ∂ ∂xj αεµef f ∂ε ∂xj + 2C1ε ε kµtSij ∂ui ∂xj −C2ερ ε2 k −R. (2.9)

Here, the strain tensor components: Sij =

∂ui ∂xj + ∂uj ∂xi

; the effective viscosity

µef f =µt+µ, where the eddy viscosity is µt =ρCµk

2

ε and µis the molecular

viscosity of fluid; and the additional term R = Cµη3(1−η/η0)

1+βη3 ε

2

k with η = S k ε.

The coefficients above are evaluated as η0 = 4.38, Cµ = 0.0845, β = 0.012,

C1ε= 1.42, C2ε= 1.68, αk = 1.0 and αε= 0.769 [47, 49].

SST k−ω model

The k−ω model is one of the most common turbulence models, which is a two-equation eddy-viscosity model. The first transported variable is turbulent kinetic energy, k. The second transported variable in this case is the specific

dissipation frequency,ω. It is the variable that determines the scale of the tur- bulence, whereas the first variable,k, determines the energy in the turbulence. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy. The k −ω based Shear-Stress-Transport (SST) model was originally used for aeronautic applications, providing highly accurate predictions of the onset and the amount of flow separation under adverse pressure gradients by the inclusion of transport effects into the formu- lation of the eddy-viscosity. This results in a major improvement in terms of flow separation predictions [50]. It becomes an industrial, commercial and re- search codes which has been widely applied to accurate computations of flows with pressure induced separation far beyond aerodynamics.

The use of a k − ω formulation in the inner parts of the boundary layer makes the model directly usable all the way down to the wall through the viscous sub-layer, hence the SST k−ω model can be used as a Low-Re turbulence model without any extra damping functions. The SST formulation also switches to a k−ω behaviour in the free-stream and thereby avoids the commonk−ω problem that the model is too sensitive to the inlet free-stream turbulence properties. The SST k −ω model does produce a bit too large turbulence levels in regions with large normal strain, like stagnation regions and regions with strong acceleration. This tendency is much less pronounced than with a normal k−ω model though.