Governing Equations for Turbulent Flow

8/7/2019 Governing Equations for Turbulent Flow

1/15

Governing Equations forTurbulent Flow

Professor Jung-Yang San

Mech. Engrg. Dept.,

National Chung Hsing University


2/15

(i) Boundary Layer on a Flat Plate

NumberReynoldsRe ==

xUx

Rex=5(10)5 Rex=106Rex=0/ 0.99u U =

Thickness of

boundary layer


3/15

The Origin of Turbulence

Turbulence is believed to be induced by small disturbance

in flow. In laminar flow region, the disturbance isrestrained by the flow, thus it does not affect the flowmotion. In turbulent flow region, the disturbance isamplified by the flow motion to form eddies.

For laminar flow, as the derived continuity, momentumand energy equations are used for predicting flow velocityand temperature distributions. The result appears tomatch well with experimental data. Nevertheless, forturbulent flow, this match usually appears to be poor.

Reynolds numbercan be used for indicating theoccurrence of turbulent flow.


4/15

Fluctuation of velocity and temperature

in turbulent flow region

1 1 1

' ' ' '

'

0 0 01 1 1

Define:

u = u + u (t); v = v + v (t); = + (t); P = P + P (t)T = T + T (t)

1 1 1

u u dt , v v dt , dt etc.t t t

t t t

w w w

w w

t1

u

t (time)

at a certained locationu


5/15

(ii) Time-Averaged Continuity Equation

' ' '

u = u + u (t); v = v + v (t); = + (t);

substituting into the above equation,

w w w

Continuity: 0 (incompressible flow)u v w

x y z

+ + =

' '

1

' '

'0

Integrating over period 0 t

'0

0

u u v v w w

x x y y z z

u u v v w w

x x y y z z

u v w x y z

+ + + + + =

+ + + + + =

+ + =

= 0= 0 = 0


6/15


7/15

1

2 2 2 2

2 2 2

2 2 2 2

2 2 2

averaging over period 0 t

( ) ( ) 1( )

( ) ( ) 1 ( )

e

e

u u uv uw P u u u

t x y z x x y z

u uv uw P u u u

x y z x x y z

+ + + = + + +

+ + = + + +

' ' 'u = u + u (t); v = v + v (t); = + (t);

substituting into the above equation,

w w w

2 ' 2 ' ' ' ' 2 2 2

2 2 2

( ) ( ) ( v) ( ) ( ) ( ) 1[ ] [ ] [ ] ( )

eu u u u v u w u w P u u u

x x y y z z x x y z

+ + + + + = + + +

= 0


8/15

2

where

( ) ( v) ( )

= 2 ; = + v ; = +

Substituting back,

u u u v u u w w u

u u u w x x y y y z z z

' 2 ' ' ' '

2 2 2

2 2 2

( ) ( ) ( )[2 ] [ + v ] [ + ]

1( ) (1)

e

u u v u u v w u u wu u u w

x x y y y z z z

P u u u

x x y z

+ + + + +

= + + +

' 2 ' ' ' '

Multiplying the time-averaged continuity equation by ,

( ) 0

Substracting equation (1) with this equation, it yields

( ) ( ) ([ ] [v ] [

u

u v w

u x y z

u u u u v u u wu w

x x y y z

+ + =

+ + + + +

2 2 2

2 2 2

) 1] ( )

eP u u u

z x x y z

= + + +


9/15

' 2 ' ' ' '

2

2 2 22

2 2 2

After arrangement,

1 ( ) ( ) ( )

v + ( )

where

u u u P u u v u w

u w u x y z x x y z

u u uu

x y z

+ + =

+ +

' '2

In tensor notation, i-direction time-averaged momentum equation is

( )1+ ( )

i ji ij

j j j j

u vu P uu x x x x x

=

Additional terms

Similarly, y and z-directional momentum equations can be devired as well!

2 ' '

2

2 2 ' 2 ' '

2 2

For two-dimensional turbulent boundary layer, the x-momentum equation is as follows:

1 ( )v +

( ) ( )where and

(y-direction

u u P u u vu x y x y y

u u u u v

x y x y

+

velocity fluctuation is larger than x-direction velocity fluctuation )


10/15

'

' '

'

For two-dimensional turbulent boundary layer:

1 1v + = ( )

molecular shear stress ; eddy shear stress ( )

momentum

m

m

u u Pu x y x y

u

u u v

uwhere u v

y y

y

+

= =

' '

eddy diffusivity

[Note: velocity fluctuations ( , ) are assumed to be induced by / .]

1 1Hence, v + = ( )

m

u u

y y

u v u y

u u Pu

x y x y

+

+

( ) ( )

( ) appa

1=

rent shear stress

( ) = ( )m m

m

u u

y yy

uwhere

y

y

+ +

+ =

Negative value


11/15

(iv)k Model for Turbulent Flow

m is not a constant. For solving the momentum equation,its value must be determined.

k Model Involves

(i) Turbulence kinetic energy (k) equation(ii) Dissipation energy () equation

The above two equations plus the momentum equationcan be used to solve the m value and velocity distribution.


12/15

1/ 2M

3/ 2

D

Analog to molecular kinetic theory for gas:

where tubulence energy ; = length scale ; empirical coefficient

Dessipation rate = = C ( )

u

u

C k L

k L C

kL

=

= =

D

2

M

where C drag coefficient 1

Sustituting back to eliminate , it yields ( ) uCLk

= =

=

2

22

1 2

1 2

D: ( ) ( )

Dt

D

: ( ) c ( ) ( )Dt

The following values are suggested for :

0.09,

equation

equ

c 1.44, c 1.92, 1,

a

ion

t

M

k

M

M

u

M

k

k

y y y

u

cy y

u

k k

k

y

k

C

= +

= +

=

= =

=1.3

=


13/15

(v) Time-Averaged Energy Equation

2 2 2

2 2 2

Energy equation (neglect dissipation term):

( ) = k( )pT T T T T T T

c u v wt x y z x y z

+ + + + +

' ' '

' '

where u = u + u (t) ; v = v + v (t) ; = + (t)

P = P + P (t) ; T = T + T (t)

w w w

2 2 2

2 2 2

Multiplying continuity equation by T:

T ( ) 0 adding to the above equation

( ) ( ) ( )( )

u v w

x y z

T uT vT wT T T T

t x y z x y z

+ + =

+ + + = + +


14/15

1

2 2 2

2 2 2

2 2 2

2 2 2

Time averaged from 0 t ,

( ) ( ) ( )( )

( ) ( ) ( )After arrangement, ( )

[ ] [ ] [

T uT vT wT T T T

t x y z x y z

uT vT wT T T T

x y z x y z

T u T v T wu T v T w T

x x y y z z

+ + + = + +

+ + = + +

+ + + + +

' ' ' ' ' ' 2 2 2

2 2 2

' ' 2

2

]

( ) ( ) ( )( )

For two-dimensional flow (also using continuity equation), it yields :( )

[Note: In the boundary layer

u T v T wT T T T

x y z x y z

T T v T T u v

x y y y

+ + + = + +

+ + =

2 2 ' ' ' '

2 2

( ) ( ), ; ]

T T u T v T

x y x y


15/15

Documents

Governing Equations for Turbulent Flow