Reynolds equations for turbulent flow

Reynolds equations for

turbulent flow

Turbulence

Jet in cross-flow

Instantaneous Vizualization

Turbulence

• Statistics – time mean quantities

– Mean value – Variance

Reynolds decomposition

• Dec.

– Mean value – Fluctuation

• Properties

• NS eq.

 

a t

a

 

a t

   

a t  a a t  

  ⁰

a t a a t



 

    ²

D a  a t 

 ^,t ^   ^{ } ^,t

u x u x u x

 ^,    ^,

p x t  p x  p x t

Ensemble average

• Quantity

• Ensemble average

• Ergodic process

• Estimator

Choice of T, n

 

a t

1

1 ⁿ

i i

a a

n _





T  

a t a a

   

lim _T lim _T

T n

a a t a t

 

 

  ₀    

1

1 1

n .

T T

i

a t a t d a t n dt

T   n









EA – choice of n and T

• Statistically independent n > 200

• Pseudo-periodical sig. T >> T_smax

T_smax T_smax

spectrum

Properties of averaging

• Linearity

• Commutativity

– with der.

– with int.

• Double averaging

• Averaging of product

• ATTENTION!!!

F of time const

Any nonlinear function!!!

a b a b a a

a dx  a dx

 

a a

x x

 

  

a b a  b ^{a t} 

 

b t



a b a b

Continuity equation

Both mean and fluktuation velocity fields are solenoidal Averaging

instantaneous velocity field Is solenoidal

Reynolds:

0

k k

u div x

  

 u

k k k

u  u  u

k 0

k

u x

 



 

k k 0

k k k

k k k k

u u

u u u

x x x x

   

  

   

   

k 0

k

u x

 



0

k k

u x

 

 

Substantial derivative

Averaging

?

NONLINEAR !

Integration per-partés:

= 0 (continuity)

i i i

k

k

Du u u

Dt t u x

 

 

 

i k

k i i k

k k k

u u

u u u u

x x x

    

   ^k _kⁱ _k ^{i k}

u u u u

x x

 

  

i i i

k

k

Du u u

Dt t u x

 

 

 

i 0 u 

Substantial derivative

Extra term !

 

     

 

i i i i i k

k

k k

i i k k

i i

k

i k i k i k i k

i

k

i i k i k

k k

i i k

k

Du u u u u u

Dt t u x t x

u u u u

u u

t x

u u u u u u u u u

t x

u u u u u

t x x

Du u u

Dt x

   

    

   

   

  

    

  

 

   

   

   

 

    

  

  

  

 



i i k

k

k k

u u u

u x x

  

 

Reynolds equations

Reynolds 1894

Extra term ! N-S eq.

Averaging

Reynolds eq.

0

2 2

i i i 1 i i k

k

k i k k

Du u u p u u u

Dt t u x x  x x



      

     

    

2 2

i 1 i

i k

Du p u

Dt x  x



 

  

 

2 2

i 1 i

i k

Du p u

Dt x  x



 

  

 

Solvability of RANS

• Equations: 3 + 1

• Unknowns:

3 1 6 (symm)

Total 10 PROBLEM!!!

Extra eqs are needed!

2 2

1 ;

i i i i i k

k

k i k k

Du u u p u u u

Dt t u x x  x x



      

     

     ^k 0

k

u x

 



, 1, 2,3

ui i  p u u_{i k} , i k, 1, 2,3

Extra eqs for correlations

Eq for fluctuations (subtract RANS from NS):

Multiply by than averaginguj

Eqs for Reynolds stress

• I the time change of the local correlation tensor,

• II advection of Reynolds stress through a mean flow (not total!),

• III interaction between the mean and fluctuation components of the flow and is related to the production of Reynolds stress,

• IV advection in relation to the fluctuation part of the flow,

• V effect of pressure,

• VI diffusion and dissipation resulting from viscosity, which is expressed mainly in small scale perturbations

Eqs for Reynolds stress

• 4 + 6 = 10 eqs

• Reynolds stress

• Unknown tensors of 2nd order

• Unknown tensor of 3rd order

Overall

75 unknowns !!!

Averaged Poisson eq.

• Poisson eq.

• Averaging

2

1 2 _{k l k l k l}

l k l k k l

u u u u u u

p x x x x x x



      

    

     

1 2 _{k l}

l k

u u

p x x



    

 

Reynolds stress

Stress

Stress 1

Mean pressure

Stress 2

Mean viscous

Stress 3

Velocity fluctuations

Reynolds stress

Rearrangement

2 2

i 1 i i k

i k k

Du p u u u

Dt x  x x



   

    

  

i 1 i k

ik i k

k k i

Du u u

p u u

Dt x   x x 



     

  

       

      

Reynolds stress

• Reynolds stress tensor

• 2nd order tensor

– Symmetric – Semidefinite

 

2

1 1 2 1 3

2

2 1 2 2 3

2

3 1 3 2 3

ij ij i j

u u u u u

R u u u u u u u

u u u u u

  

      

 

 

      

    

 

    

 

 

x

Reynolds stress

• Physical meaning

• Mean momentum flux in i direction induced by fluctuations in j direction

• Mean momentum flux in j direction induced by fluctuations in i direction

• Always 3D!

ij i j

R  u u^{ }

Reynolds stress

• Covariance tensor properties

• Symmetric

• Decomp. (isotropy):

– Isotropic part – Anisotropic part

Kinetic energy

IRROTATIONAL

ROTATIONAL

i j

u u 

i j ij ij

u u   i a 2

ij 3 ij

i  k

2

ij i j 3 ij

a  u u^{ }  k

1 1

2 2 ^{k k}

k  u u   u u 

Reynolds stress

• A:

• B:

For any transversal movement within BL

the POSITIVE Reynolds stress is generated

x₂

x₁ A B

2 0, 1 0 12 1 2 0

u^  u^   R   u u^{ } 

2 0, 1 0 12 1 2 0

u^  u^   R   u u^{ } 

2 0

u 

1  2

u x

1 0

u 

2 0

u 

 

1 2

u x

1 0

u 

Reynolds stress

x₂

x₁ x₂

x₁

Isotropic fluctuations R₁₂ = 0

Anisotropic fluctuations R₁₂ > 0

BOUNDARY LAYER ISOTROPIC TURBULENCE

Quantities

• Kinetic energy

• Intensity of fluctuations

Isotropic turbulence

Intensity of turbulence

1 2 _{k k} k  u u

2 2 2

1 2 3

3 2

u u u

Iu Tu

u

    

 

2 2 2 2

1 2 3

u  u  u  u Tu  u² u

Dissipation rate

• rate of dissipation

Specific dissipation rate

Izotropic turbulence

2 s s_{kl kl}

   1

2

i j ij

j i

u u

s x x

     

     

  k 1

2

k l k l

l k l k

u u u u

x x x x

   ^ ^ ^  ^{  } ^ ^  ^{  }

2 1 1

15 u

   ^ ^x^{ }

Reynolds equations

Anisotropic part R_ij

Joseph Valentin Boussinesq 1877 Hypotheses about RS:

1. >> turbulent viscosity

2. proportional to mean strain rate

i 1 i k

ik

k k

i k i

Du u u

Dt x p x u

x u

 

 ^{     }  ^

          



 

2 3

i j

ij T

j i

i j

u u u u

k x x

  

   ^ ^  ^ ^





 

i 1 i k

eff

i k k i

Du P u u

Dt x x  x x



    

 

      

      

 ^,  ^,

eff t T t

 x    x

 ^{2 3}

P p  k

2 2

i 1 i k

i k

i k

Du u u u

D x

p

t x  x



  

 

    

 

Turbulent viscosity

• Molecular viscosity – material „constant“, property of fluid

• Turbulent viscosity – property of flow

– Function of space

– Function of time (for nonstationary case) – Is solution

– Is input

 ^,  ^,

eff t T t

 x    x

Prandtl mixing length

• Estimate

x₁ x₂

u₁

Boundary layer

Border of the BL

x₁

_tot

_t x₂

2 1

2

t mix

l u

  ^x



t

Transport equation

• Advection

• Production

• Transport

• Dissipation

0 A P T D   

1 2 2 i

i

u q

x

 



i i j

j

u u u

x

  

 

1 2 2 j

j

u q p

x 

   

      

   

1 2

k l k l

l k l k

u u u u

x x x x

   ^ ^ ^  ^ ^{ } ^ ^  ^{  }