Turbulence models  I . Introduction

In Fluent, turbulence is modeled using one of two "two-equation" turbulence models, or the Reynolds stress model. In two-equation models, effects of turbulence are represented by an isotropic turbulent viscocity which is evaluated using two quantities, k and e, are obtained from the solutions of "modeled" transport equations. In the Reynolds stress model, transport equations ( 6 in 3D, 4 in 2D) are explicitly solved for the Reynolds stresses.Fluent provides the following choices of turbulences models :

• Standard k- e model
• Renormalization-Group(RNG)-based k-e model
• Reynolds stress model (RSM)

Before starting any simulation, we have to determine a critic temperature difference DTc to have a turbulent flow. The following picture gives the rate of flow for differents values of e :  According to this scale, the flow is turbulent for eturbulent >2000, i.e., DT turbulent > 2001.DTc .But with a critical temperature difference about 0,2 degree, we should have DT turbulent = 500 degrees. Consequently, if we use liquid water, this conditions will be unachievable because the computation time will be very high and the fluid might boil ! So we found a case non-laminar by starting with low DT and increasing it untill the results were not physical because the flow was no longer laminar.We can see it with the temperature field with a laminar study :

 D T = 0,1 e ~-0,5 no rolls DT = 0,2 e ~0 2 rolls DT = 5 e ~24 2 rolls DT = 10 e ~49 4 rolls DT = 20 e ~99 2 rolls In this last caseI, the computations are converged with 39 iterations !
All this part , we will work with these conditions :  II . k- e model

I-1 Description

• Closing law • Equations on k and e In these previous equations, five constants appears. The k-e standard model is based on the following values :

• Cm=0,09
• sk=1
• se =3
• Ce1=1,41
• Ce2=1,92

II-2 Results

The simulation are done with 20 degrees between the two walls.

 Turbulent cinetic energy Velocity field Temperature We can see that a kind of roll lices on. This phenomena is due to the oversight the term pu' in the k-equation.

In order to avoid these problems, we added the following option : The results are :

 Turbulent cinetic energy Velocity field Temperature The convection rolls don't appear. Moreover if we look at the turbulent cinetic energy, we find that the highest is localised inthe center of the box although it is near the wall that there are gradient of temperature.
Actually, this model gives good results for Reynolds numbers of the turbulence ReT >>1 and far from any partitions. In fact, the velocity can be descirbed by a logarithmic law : The validity field is y+>30. But an estimation of y+ can be done with the following values :
y =10-3 u* =10-2 v =10-6 so y+=10 < 30.
Moreover if we try to optimise the mesh by increasing the number of nodes, y+ will decreased . III . RNG k- e

III-1 Description

Renormalization group (RNG) methods are a general framework for "model building" in which the complex dynamics of physical problems is described in terms of ''coarse-grained" equations of motion governing a large scale, long-time behavior of the physical system. The mathematics of RNG theory allows similar coarse-graining of physical phenomena and has been applied to a range of physical processes including critical phenomena, high-energy particle physics, and, in the context of fluid dynamics, turbulence, combustion, and heat transfert. The key idea is that the RNG method is applicable to scale-invariant phenomena lacking externally-imposed characteristic length and time scales. For turbulence, this means that the method can describe the small scales that should be statistically independent of the external initial conditions and dynamical forces that create them through various kinds of instability phenomena.

III-2 Results

 Turbulent cinetic energy Velocity field Temperature We can see that the differents field have the same than previously. But if we look at the maximum of turbulent cinetic energy and velocity , we find:

 Model Ec (m2/s2 ) V (m/s) k-e standard 10-5 2,24.10-3 k-e standard + boyancy effects 6,79.10-3 3,30.10-8 RNG 3,5.10-4 3,5.10-4

The problems encounted with this last model can be partly explained with stricted mesh conditions .