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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2.1.1. RANS models RANS models have been for many years the only tool available to calculate the features of a turbulent flow in relevant, complex applications (Hanjalic (2005)) The fundamental concept of RANS methods lies in Reynolds decomposition of Navier-Stokes equations: turbulent flow is described as a random variation around a mean value. The Reynolds-averaged Navier-Stokes equations can be written as Wilcox (2006):

U

U P





x x    

' ' j i S u u    ji

i

i

(3)

(2

)

U







  

j

t

x





j

i

j

while time-averaged mass-conservation is identical to the instantaneous:

i U x



i

(4)

0





where U i is the time-averaged velocity, u i ’ is the fluctuating velocity, µ is molecular viscosity, S ij is the deformation tensor. In RANS equation the quantity ' ' j i u u  is known as the Reynolds Stress tensor, ' ' j i ij u u    . The unknowns are 10: one pressure, three velocity components, six Reynolds stress tensor components, while the equations are four. In order to solve the prolem it is necessary to introduce more equations. This is called the “closure problem” and many models have been proposed, most of which are part of the two-equation models family. The most popular two-equation model is the Standard k   model and is based to the physical hypothesis that the production of dissipation should be proportional to the production of turbulent kinetic energy. The following equations define a standard k   model:

j i i u u x x               t j

2 3

(5)

,

K





ij  

ij

2

(6)

,

t C K    

 

 

 

K K 

u

K



 





i

2 , K

T

(7)

u



  

  



 

i

ij

t

x

x

i     K i x x





i

j

2

 

 



u





















i

2   ,

T

(8)

u

C

C





 





 

i

1

2

ij





t

x

K x 

K x

i     i x  





i

j

Where K is the turbulent kinetic energy, ε is the turbulent dissipation rate, T  is the eddy viscosity, i u is the mean velocity vector,  is the kinematic viscosity of the fluid and the constants assume the following approximate values of 0.09 C   , 1.0 K   , 1.3    , 1 1.44 C   , 1 1.92 C   . Many improvements have been done over the years to the standard k- ε model. Other closure models differ essentially by the choice of the modeled equations along with the equation of turbulent kinetic energy. At the moment, one of the RANS model that has proven to be more efficient is v 2 -f proposed in Durbin (1991), which is based on the elliptic relaxation concept and uses two additional equations, one for the velocity scale and one for the elliptic relaxation function. This model improved the modeling in proximity of the wall.