L

ehrstuhl für

Modellierung und Simulation

Physics of turbulence

Lecture 2 UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

Experiments of Osborne Reynolds

Reynolds number Re 

UD

 Re_transition=2400 in the pipe flow UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION 2

Definition

• • • Turbulent motion is the three dimensional unsteady flow motion with chaotical trajectories of fluid particles, fluctuations of the velocity and strong mixing arisen at large Re numbers due to unstable vortex dynamics. Difference between vorticity and concentrated vortices  0 Vortex motion Vortex structures

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Turbulence in free flows

Confined jets Diffuser

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Vortex structures in a free jet close to the nozzle UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Helmholtz Kelvin Instability UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Vortex structures in a free jet in a far field UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Vortex Structures in a free jet with acoustic impact With acoustic waves Free jets UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Vortices in a jet mixer Resolution 300 µ LES: Knots number versus resolution Resolution 31 µm 300 µm Grid points 23 000 000 000 23 000 000 2D

2.72 mm

Resolution 31 µm 90000 points in measur. window

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x/D=1

Fascinating world of vortices Spatial Resolution 31µ.

x/D=2

Smallest vortices of the flow kolmogorov vortices

x/D=3 x/D=5

PLIF Measurement of the LTT Rostock UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION 10

What vortices are present in the flow?

Resolution 300 µ

From scales compared to macrosize of the flow (pipe diameter) …..

2D

x/D=3

to Kolmogorov scales (microns) 250 200 150 Geschwindigkeit 0.9 m/s 1.2 m/s 1.75 m/s 100 50 0 0 2 4 x/D 6 8

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Near Wall Turbulence Vortex cascade. Laminar-turbulent transition.

One of the possible schemes of the laminar turbulent transition in the boundary layer.

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Structure of the turbulent boundary layer.

τ T

Far from the wall where the viscous stresses are much less compared with turbulent ones

 ρ x / y  ρRl l ux uy   du dy x   2 ρl 2   du dy x   2

l

 du dy x  

y u

*  y

u *



τ 0 ρ

τ



τ 0  0

u

x 

u

*  

u

*  ln

u

*  0 

u

*  ln

u

*   lnC 0 

u

* C

u

x

u

*

y

 

u

* y   1  lny +  C   0.41;

C

 5.0

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Structure of the turbulent boundary layer.

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Coherent structures

Smoke visualization of streaks in transition under the high level free-stream turbulence in a boundary layer (from Matsubara & Alfredsson, 2001).

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Coherent structures

Streaks

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From Smith&Walker (1995)Fluid vortices

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Coherent structures

From Smith&Walker (1995)Fluid vortices

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Coherent structures. Hairpin vortices

From Vincent and Meneguzzi (2001)

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  

u

 Vorticity is solenoidal 

u )



0

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Vector algebra. Operator Nabla of Hamilton

 Exercise. Prove the following expressions

A )



0, 0,



A )

   

2

      

NS equations:



u



t f



p



( )u

a mistake is here !!!! Pls correct !

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Vorticity transport equation NS equations:

  

t )

 

(

 

( )



Vortex convection Vortex diffusion Vortex stretching

D



Dt



(



Vortex convection



( )



D



Dt



0

Rotation and amplification Diffusion (thickening)

D

 

Dt D

 

Dt (

 

)u )



Source of the turbulence !!!

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Vortex induced velocities UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Vortex cascado. Vortex reconnection UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Sample of the Vortex reconnection. Crow Instability (1971) UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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