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I.K. Marchevsky, V.V. Puzikova

26

ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. Естественные науки. 2017. № 5

Then

,

xy

i j

M

can be expressed through the area of base mesh cells:

 

 

 



 

, 1 , 1

1,

1,

,

,

, 1 , 1

,

=

.

xy

i j

i j

i

j i

j

i j i j

i j

i j

i j

M V

V

V

V

Here

,

i j

V

is the area of the cell

,

.

i j

Since

t

and shear Reynolds or subgrid stresses (5) are sampled at the same

points, it follows that

 

,

,

,

,

| =

,

t

t

xy i j

i j

i j

i j

u v

y

x

(8)

whereas averaged values of turbulent viscosity

,

t

i j

and the turbulent kinetic energy

,

i j

k

should be used for the computation of the normal Reynolds or subgrid stresses

(4):

,

,

,

,

,

,

,

,

2

2

| = 2

;

| = 2

;

3

3

t

t

t

t

i j

xx i j

yy i j

i j

i j

i j

i j

i j

u

v

k

k

x

y

 

 

        

   

,

,

,

,

, 1

1,

1, 1

,

,

, 1

1,

1, 1

= (

);

= (

).

t

t

t

t

t

i j

i j

i j

i j

i j

i

j

i

j

i j

i j

i j

i

j

i

j

k

k k k

k

(9)

Formulae for normal stresses

,

i j

u

x

and

,

i j

v

y

computation are the following:

 

  

   

,

1,

,

1,

1,

,

,

,

,

(

)

;

/

u

u

u

u ib

i j

i

j

i j

i

j

i

j

i j

i j

i j

j

i j

u

u

u

u

x

V y

 

 

  

,

, 1

,

, 1

, 1 ,

,

,

,

(

)

.

/

v

v

v

v ib

i j

i j

i j

i j

i j

i j i j

i j

i

i j

v

v

v

v

y

V x

formulae for shear stresses

,

i j

u

y

and

,

i j

v

x

computation depend on the type of

,

xy

i j

cell:

 

 

   

    



 

 

  

 

, 1 ,

,

, 1

1

, 1

,

,

, 1

, 1

,

, 1

1

, 1

,

,

,

,

,

,

, 1

,

,

,

if

0,

0;

(

) / 2

( ,

)

=

, if

= 0,

0;

/ 2

( ,

)

=

,

if

0,

= 0;

/ 2

i j

i j

u

u

i j

i j

u

u

j

j

i j

i j

ib s

ib

i j

i

i j

u

u

i j

i j

u

j

i j

i j

i j

ib n

ib

i

i j

i j

u

u

i j

i j

u

j

i j

i j

u u

y

y

u u x y

u

u

y

y

y

u x y u

u

y

y