FHP, Poiseuille flow LBM, Karman vortex streetmaq/zajecia/cfd2015/Exam/... · Karman vortex street...

FHP, Poiseuilleflow

LBM, Karmanvortex street

Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results

Lattice BoltzmannmethodThe Boltzmann Equation

BGK

Reflection

Karman vortexstreet

Simulation results

Frequancy ofvortex structures

FHP, Poiseuille flowLBM, Karman vortex street

Beata Kowal

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Outline

FHP

Poiseuille flow

Simulation results

Lattice Boltzmann method

Karman vortex street

Simulation results

Frequency of vortex structures

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


FHP

The FHP model of fluid flow is a cellular automaton inwhich particles move in triangular grid.Two- and threebody collision rules (FHP1):

Reflections: bounce back and specular

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


FHP

FHP 2Stationary particles

FHP 3

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Poiseuille flow

The initial velocity of the fluid is directed along axis X. Aftersome time the velocity of the fluid has parabolic profile.

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Program

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Simulation results

FHP 1

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Simulation results

FHP 2

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Simulation results

FHP 3

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results



Lattice Boltzmann methods is a class of computational fluiddynamics methods for fluid simulation.

Instead of solving the Navier–Stokes equations, we solve thediscrete Boltzmann equation.

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results



f (x,u, t) - distribution function

dN = f (x,u, t)d3xd3u - number of particles in finite elementof momentum/position space

The Boltzmann Equation

(∂t + u∇)f (x,u, t) = (∂t f )coll

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results



Bhatnagar-Gross-Krook (BGK) collision model.

BGK collision operator:

(∂t f )coll =1τ

(f eq − f )

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results



f eqi is the discrete, equilibrium distribution function.

f eq = ρ2πRT exp

(− (e−u)2

2RT

)f eqi = ρwi exp

(3eiuc2 − 3u2

2c2

)≈ ρwi

(1 + 3eiu

c2 − 3u2

2c2 + 9(eiu)2

2c4

)weights wi :w0,w1,w2,w3 = 1/9w4,w5,w6,w7 = 1/36w8 = 4/9

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results



elementary vectors of on a two-dimensional rectangular grid.ei :e0 = (1, 0) e1 = (0, 1) e2 = (−1, 0) e3 = (0,−1)e4 = (1, 1) e5 = (−1, 1) e6 = (−1,−1) e7 = (1,−1)e8 = (0, 0)

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results



The collision and transition step are defined:

collision step

fi (x , t + δt) = fi (x , t) +1τ

(f eqi − fi )

transition step

fi (x + eiδt, t + δt) = fi (x , t + δt)

i - directions of momentum.

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results



ReflectionBorder is located halfway between vertices.

Fluid directed toward the border isn’t translated but it’sreflected in the opposite direction.

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results



Karman vortex street in the Greenland Sea (NASA image by Jeff Schmaltz,

http://earthobservatory.nasa.gov/NaturalHazards/view.php?id=77654)

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results



Karman vortex street is a repeating pattern of vorticescaused by the separation of flow of a fluid by obstacles.

Vortex structures are seen in a case of sufficiently high valueof Reynolds number.

Re =Vdν

ν = 13

(τ − 1

2

)- viscosity

d - diameter of the cylinderV - flow velocity.

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Simulation results

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Simulation results

Grid 150x50, 10’000 steps, velocity visualisation, τ = 0.551000 steps

2000 steps

3000 steps

4000 steps

5000 steps

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Simulation results

Grid 150x50, 10’000 steps, velocity visualisationτ = 0.6 ν = 0.033 Re = 50

τ = 0.57 ν = 0.023 Re = 70

τ = 0.56 ν = 0.02 Re = 80

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Simulation results

Grid 150x50, 10’000 steps, velocity visualisationτ = 0.55 ν = 0.017 Re = 100

τ = 0.53 ν = 0.01 Re = 160

τ = 0.52 ν = 0.0067 Re = 240

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Simulation results

Grid 150x50, 10’000 steps, density visualisationτ = 0.6 ν = 0.033

τ = 0.57 ν = 0.023 Re = 70

τ = 0.56 ν = 0.02 Re = 80

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Simulation results

Grid 150x50, 10’000 steps, density visualisationτ = 0.55 ν = 0.017 Re = 100

τ = 0.53 ν = 0.01 Re = 160

τ = 0.52 ν = 0.0067 Re = 240

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Frequancy of vortex structures

f - vortex frequency.d - diameter of the cylinderV - flow velocity.Strouhal number: St = fd

V

Strouhal number, for a range of Reynolds number between 250 < Re < 105, can be expressed as(calculated by G. I. Taylor (1886-1975)):

St = 0.198

(1−

19.7

Re

)”Simple Karman Street model”, Cecilia Tapia S. and Ryad Chellali,DOI: 10.1109/OCEANSSYD.2010.5603671

Model of a vortex street, Tubes, crossflow over, Bengt Sunden

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Frequancy of vortex structures

τ Re fs from simulation ff from formula |∆f |/fs0.55 100 0.00185 0.00153 17%0.53 160 0.00181 0.00167 8%0.52 240 0.00174 0.00175 0.6%

0.515 325 0.00178 0.00179 0.6%

FHP, Poiseuilleflow


Beata Kowal

Outline

FHP

Poiseuille flow

Simulation results


BGK

Reflection

Karman vortexstreet

Simulation results


Bibliography

Sebastian Szczecina, Własności hydrodynamiczne modelugazu sieciowego FHP-III

Josue Njock, A Method of Evaluating the Presence ofFan-Blade-Rotation Induced Unsteadiness in Wind TunnelExperiments

Amir Masoud Abdol, Lattice Gas Automata

A Practical Introduction to the Lattice Boltzmann Method ,Alexander J.Wagner

Badanie zjawisk zachodzących w cieczach nieściśliwychmetodą cząstek znaczonych, M.Matyka

Simple Karman Street model, Cecilia Tapia S. and RyadChellali

Tubes, crossflow over, Bengt Sunden

FHP, Poiseuille flow LBM, Karman vortex streetmaq/zajecia/cfd2015/Exam/... · Karman vortex street...

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