PSI - Issue 3

Alberto Lorenzon et al. / Procedia Structural Integrity 3 (2017) 370–379 A. Lorenzon et al. / Structural Integrity Procedia 00 (2017) 000–000

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2.1.2. LES models In LES, the large eddies are fully resolved and the smallest, subgrid-scale eddies are modeled. Instead of time averaging, the resolved (large-eddy) field is separated from the small-eddy (sub-grid) field by using a spatial filtering and providing the following governing equations:   1 i i i j ij j i j j u P u u u t x x x x                           (9)

where

1 3

 

  

(10)

Q Q

   





ij

ij

kk ij

1 3

(11)

kk ij P p Q    

(12)

ij ij ij Q R C  

where C ij is the cross term stress tensor, R ij and Q ij are Sub-Grid Scales (SGS) Reynolds stress tensors. Smagorinsky developed a model for the SGS stresses assuming that they follow a gradient-diffusion process similar to molecular motion, whose governing equations are:

1 2 x x               j i i u u j

(13)

2 ij T ij S    ,

S

ij

where S ij is the resolved strain rate, ν T is the Smagorinsky eddy viscosity

2

(14)

(   

) s ij ij C S S

T

and C s is the Smagorinsky coefficient. The accuracy of LES is much dependent on the use of an accurate SGS stress model and in the correct representation of boundary conditions. In alternative to Smagorinsky’s model, Wall Adapting Eddy Viscosity model has been proposed in Ducros et al. (1998), which improves the prediction of the wall stress rate, as well as turbulent intensities. 2.1.3. PANS models Partially Averaged Navier-Stokes model allows for a smooth variation from RANS equations to Navier-Stokes equations, depending on the values of the filter-width control parameters defined in Girimaji (2006):

K



u

u

(15)

,

f

f





k



K



where f k is the ratio of the unresolved ( K u ) to total ( K ) kinetic energy, f ε is the ratio of resolved ( ε u ) to total ( ε ) dissipation. These models have shown good preliminary results, with results comparable to even the most computationally expensive LES models but have not yet been used extensively in the field of civil engineering.