5 GENERAL TURBULANCE MODELS

40 To solve CFD problems it consist of three main components which are geometry and grid generation, setting up a physical model and post processing the compute data. In the turbulence it results in increasing energy dissipation, mixing, heat transfer and the drag. The way geometry and the grid are generated and the set problem is computed are very well known. Precise theories are available. But it is not true for setting up a physical model for turbulence flow. There for it need to create the ideal model with the minimum amount of complexity. The complexity of the model will increase with the amount of information required about the flow field. The key elements of turbulence are time dependent and the three dimensional. 17

41 Turbulence models can be categorized in to several different approaches which are by solving the Reynolds-averaged Navier-Stokes equations with suitable models for turbulent quantities or by computing them directly.

Reynolds-Averaged Navier-Stokes (RANS) Models

Eddy Viscosity Model (EVM)

Non-linear Eddy Viscosity Model (NLEVM)

Differential Stress Model (DSM)

Detached eddy simulation (DES)

Large-eddy simulation (LES)

Direct numerical simulation (DNS)

Reynolds stress transport models

Direct numerical simulations

2.6 REYNOLDS-AVERAGED NAVIER-STOKES MODELS (RANS)

42 This method is the mainly use method in Engineering industry. This can be categorized according to the wall function, number of variables and their types. So we mainly focus on following models in RANS.

Spalart-Allmaras

K-Epsilon(?) Model

K-Omega(?) Model

43 Here this k-Epsilon model further divided in to two types of models, which are standard K-Epsilon model (SK-?) and the Realizable K-Epsilon model (RNGK-?). And also this K-omega model also divided in to two models which are standard K-omega model (SK-?) and the shear stress transport K-Omega model (SSTK-?).

2.6.1 SPALART-ALLMARAS

44 This equation solves a modelled transport equation for kinematic eddy turbulent viscosity. It easy to resolve near the wall. From this model it shows good results for boundary layer subjected to adverse pressure gradient in especially wall bounded flows involve in aerospace applications. This could be used for the supersonic and transonic applications. This model is not calibrated for the general industrial flows. This model is very effective in low Reynolds numbers. Minimum boundary layer resolution of 10-15 cells should be there to resolve the equation. The formulation provide wall shear stress and heat transfer coefficient. This model cannot rely on the turbulence isotropic calculations. 18

2.6.2 K-EPSILON(?) MODEL

45 This model mainly focus on the affect the turbulent kinetic energy. In this model it take the kinetic viscosity is isotropic as an assumption, or the ratio between rater of deformation and the Reynolds’ number is same in all directions. This model used commonly in industrial applications rather than the other two models. This model gives reasonably accurate results. Under different pressure gradients it gives the equilibrium boundary layers and free shear flows. This usually use for free shear layer flow with small pressure gradient. This model poorly perform in strong separations, large pressure gradients, unconfined flows, curved boundary layers, rotating flows and flows in non-circular ducts. Among the two type of this model (RNG) K-? model perform better than the SK-? model.

For k and ? it use two transport equations for turbulent length and the viscosity.

Equation for turbulent length;

l=k^(3/2)/? (6)

Equation for turbulent viscosity;

v_t=c_? k^(1/2) l=c_? k^2/? (7)

Turbulent kinetic energy;

?/?t (?k)+ ?y/(?x_i ) (?ku_i )= ?y/(?x_i )(?+?_t/?_k ) ?k/(?x_i )+P_k+P_d+??+Y_M+S_k (8)

Dissipation ?;

?y/?x (??)+ ?y/(?x_i ) (??u_i )= ?/(?x_j )(?+?_t/?_? ) ??/(?x_j )+C_1? ?/k (P_k+?C_3? P?_b )-C_2? ? ?^2/k+S_k (9)

Where,

C1? = 1.44, C2? = 1.92, C3? = 0.09, ?k = 1.0, ?? = 1.3

2.6.3 K-OMEGA (?) MODEL

46 It is two equation model which means it use two transport equations to represent the turbulent properties of the flow. This also a common equation model. This can be integrated to the wall without using the wall functions. From this equations, it accounts history effects such as diffusion and convection of turbulence energy. Here kinetic energy (k) is one of variable. It determines the energy in turbulence. The other variable is dissipation (?), it determine the scale of turbulence.

For kinematic eddy viscosity;

v_t=(a_1 k)/(max?(a,?,?SF?_2)) (10)

Turbulence kinetic energy;

?y/?x+U_j ?k/(?x_j )=P_k-?^* k?+?/(?x_j )(v+?_k+v_T ) ?k/(?x_j ) (11)

Specific dissipation rate;

??/?t+U_j ??/(?x_j )=?S^2-??^2+?/(?x_j ) (v+?_k v_T ) ??/(?x_j )+2(1-F_1)?_(?^2 ) 1/? ?k/(?x_j ) ??/(?x_j ) (12)

?