The Phenomena of Fluid Flow - mie.uth.gr2009-2010)a.pdf · The Phenomena of Fluid Flow ... that has...

University of Thessaly

Lab. Fluid Mechanics & Turbomachines

The Phenomena of Fluid FlowThe Phenomena of Fluid FlowNicholas S. VlachosNicholas S. Vlachos

LabLab. . Fluid Mechanics & TurbomachinesFluid Mechanics & TurbomachinesDepartment of Mechanical EngineeringDepartment of Mechanical Engineering

University of ThessalyUniversity of Thessaly

Program of Graduate StudiesProgram of Graduate StudiesAcademic Year Academic Year 20020099--20102010

Coherent (large scale) flow structures

Tornado

A future challenge in fluid flow

Droplet breakup

Fluid flow is a non-linear phenomenon

Droplet production from water stream

“Observe the motion of the water surface, which resembles that of hair, that has two motions: one due to the weight of the shaft, the other to the shape of the curls; thus, water has eddying motions, one part of which is due to the principal current, the other to the

random and reverse motion.”- Leonardo da Vinci, ca.1510

Flow around a cylinderLaminar

Turbulent

Turbulence is

random isotropic

(?)Experimental evidence caused

revision

of this perception of turbulence

Methods for the study of fluid flow

Theoretical/Analytical: Requires solution of partial differential equations

Solutions exist only for simple flows

Phenomenological/Experimental: Requires many resources and much time

There are no powerful measuring methods

Computational: Requires knowledge of physical phenomena

Wind tunnel of UTH Fluids Lab

Airfoil

NACA 4418

V Pds + + VdVρt

¶ ¶¶

+ ρdz = 0

Bernoulli equation (valid along fluid path):Bernoulli equation (valid along fluid path):

Daniel Bernoulli (1700-1782)

ρ ρ( +U + V )xt

V V V Vα ¶ ¶ ¶ ¶¶ ¶ ¶ ¶

= + Wy z

ρ p ρα g f f vis em = - + + + +...

Equations of fluid motion

ρ p ρα g = - +

p ρg =

ρ pα = -

Hydrostatic condition

Euler’s equation

+ gravity

Leonhard Euler (1707-1783)

Solution methods for flow equations

Lagrange: Requires solution of motion equations

of many particle

(ordinary differential equations)

Euler: Solution of fluid motion as continuum(partial differential equations)

Lattice Boltzmann

methodsRequire knowledge of distributions

Computational Fluid Dynamics (CFD)Requires large grids and CPU

(ρ )+ (ρ ) + xt

U U U U¶ ¶ ¶ ¶¶ ¶ ¶ ¶

U V W(ρ ) + (ρ ) =y z

(ρ)+ (ρ ) + Sxt

¶ ¶ ¶ ¶¶ ¶ ¶ ¶

FU V W(ρ ) + (ρ ) =

Linear momentum

equation (Navier-Stokes):

Mass conservation equation

(continuity):

Navier-Stokes flow equations

+ U U U S¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶

Feff eff eff(μ ) + (μ ) + (μ )

x x y y z z

George Gabriel Stokes (1819-1903)

Claude Louis Marie Henri Navier(1785-1836)

Momentum equation:

ρu u uρ(u v )t x y

¶ ¶ ¶+ + =

¶ ¶ ¶

ρv v vρ(u v )t x y

¶ ¶ ¶+ + =

¶ ¶ ¶

ρ ρu ρv 0t x y

¶ ¶ ¶+ + =

¶ ¶ ¶

Two-dimensional flows

Continuity equation:

xp u u[ μ( ) μ( )]x x x y y

ρg βΔΤ¶ ¶ ¶ ¶ ¶- + + +¶ ¶ ¶ ¶ ¶

yp v v[ μ( ) μ( )]y x x y y

ρg βΔΤ¶ ¶ ¶ ¶ ¶- + + +¶ ¶ ¶ ¶ ¶

Two-dimensional flows

Energy equation:

pT Tρc (u ρυ )x y

¶ ¶+ =

¶ ¶T T(k ) (k )

x x y y¶ ¶ ¶ ¶

+¶ ¶ ¶ ¶

2 2u u u uμ[( ) 2 ( ) ]x x y

p p(u v )x yy

¶ ¶ ¶ ¶+ +

¶ ¶ ¶¶ ¶

+ + +¶ ¶¶

u u P uρ(u v ) μx y x y

Continuity :

x-momentum:

y-momentum:

Energy equation:

boundary layer

pT T Tρc (u υ ) (k )x y y y

¶ ¶ ¶ ¶+ =

¶ ¶ ¶ ¶2uμ( ) pu

¶¶¶

Ludwig Prandtl (1875-1953)

Fluid vorticity

rot x , ,x

¶ ¶ ¶¶ ¶ ¶

ì üï ïï ïï ïï ïï ï= = =í ýï ïï ïï ïï ïï ïî þ

Ω V V

i i iy z

Vorticity components

ω ¶ ¶¶ ¶

Vorticity Transport

u v w Sφ +t

¶ ¶ ¶ ¶¶ ¶ ¶ ¶

+ + + =Ω Ω Ω Ω

1 ( ) ( ) ( )eff eff eff¶ ¶ ¶ ¶ ¶ ¶

m m mr ¶ ¶ ¶ ¶ ¶ ¶ì üï ïï ï+ +í ýï ïï ïî þ

Ω Ω Ωx x y y z z

KolmogorovKolmogorov

scales scales λλ

= (= (νν33//εε))1/41/4

, , υυ

= (= (νενε))1/41/4

Two-Phase Flow: Particle DynamicsLagrange ModelLagrange Model

-pD i i

d p F pdt

U U U g F

DD Dp24

ReC18F

321D Re

aReaaC

Two-Phase Flow: Particle Heat Transfer

4Rppfg

Chemical Reaction -

Coal Combustion

p 2 1 2p 0

dm R R= -πD Pdt R + R

p o1 1

T + T /2R = C

ER = C exp -R T

KKineticsinetics

rate:rate:

DDiffusioniffusion

raterate::

Magnetic force on a particle (Lorentz)

Hendrik Antoon Lorentz (1853-1928)

ElectrodynamicsElectrodynamics

DefinitionsDefinitions

P = p + |B|2 /2μ

Magneto-hydrodynamics equations

U (ρ )+ (ρ ) + x

+ ( · ) / +

Pt¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶

¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶

U U U U

U U U B B Seff eff eff

V W(ρ ) + (ρ ) = - +y z x

(μ ) + (μ ) + (μ ) μx x y y z z

Magneto-hydrodynamics equations

U + (ρ ) + + + +x x

+ + + S

tΒ U U UΒ Β Β

Β Β Β

¶ ¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶ ¶

¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶

x y zV W(ρ ) (ρ ) = B B By z y z

1/μσ[ ( ) ( ) ( )] x x y y z z

x y zB B B + + 0x

¶ ¶ ¶¶ ¶ ¶

U = V = 0

T = 0Y

U = V = 0

T = 0Y

Flow configuration

Non-dimensional variables group

Ha=0 Ha=30 Ha=75

Stream lines (upper) and isotherms (lower) for Gr

Turbulence and large structures in air jet

Boundary layer around a ship’s hull and waves

Shock wave and turbulence

Pressure waves and turbulence

Navier-Stokes for instantaneous, laminar/turbulent,

fluid motion

Reynolds decomposition

equations of time-averaged motion

Terms of the Reynolds stress tensor appear

The problem of turbulence modeling does not close (closure)

Solution of Reynolds-stress transport equations, leads to higher-order correlations

Modeling turbulent flow

Spectrum of turbulent kinetic energySpectrum of turbulent kinetic energy

KolmogorovKolmogorov

scalesscales

lengthlength

= (= (νν33//εε))1/41/4

velocityvelocity

––

= (= (νενε))1/41/4

ReReλλ

= = λυλυ//νν

= 1= 1

Balance of turbulent kinetic energyBalance of turbulent kinetic energy

0k = G - ρε > Increasing turbulenceIncreasing turbulence

EquilibriumEquilibriumt

0k = G - ρε =

Decaying turbulenceDecaying turbulence

0k = G - ρε <

k = ½

+ w' 2)

(ρ ) + 0x¶ ¶ ¶¶ ¶ ¶

U V W(ρ ) + (ρ ) =y z

Reynolds decompositionReynolds decomposition: :

= = ΦΦ

+ + φφ’’

TimeTime--mean mean φφ

= = ΦΦ

(ρu) + 0x¶ ¶ ¶¶ ¶ ¶

v w(ρ ) + (ρ ) =y z

Continuity equationContinuity equation::

Osborne Reynolds (1842-1912)

Reynolds Averaged EquationsReynolds Averaged Equations

y z z¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶

Φ(ρΦ) + (ρUΦ) + (ρVΦ) + (ρW ) =

y y¶ ¶ ¶ ¶¶ ¶ ¶ ¶Φ Φ

Φ Φ(Γ -ρu'φ') + (Γ - ρv'φ')

z z¶ ¶¶ ¶

Φ+ (Γ -ρw'φ') + S

0U Vx y

2 ' 'U U P UU V u vx y x y

( ) ( ) ' 'p p pT T Tc u k c u T c v Tx y y y

r ru r r¶ ¶ ¶ ¶

+ = - -¶ ¶ ¶ ¶

Continuity:

momentum:

Energy equation::

boundary layer

Turbulence modelsTurbulence models

Standard kStandard k--εε

(ρ ) + x

Uk (ρVk) + (ρWk) =y z

Turbulence kinetic energyTurbulence kinetic energy

k = ½

+ w' 2)

+G-ρε Φ Φ Φ

k k k(Γ ) + (Γ ) + (Γ )

x x y y z z

Turbulence model (2)Turbulence model (2)

1 2ε ε ε + ε/k(CG - Cρε)¶ ¶ ¶ ¶ ¶ ¶

¶ ¶ ¶ ¶ ¶ ¶Φ Φ Φ(Γ ) + (Γ ) + (Γ )

x x y y z z

Turbulence dissipation rateTurbulence dissipation rate

(ρ ) + x¶ ¶ ¶¶ ¶ ¶

Uε (ρVε) + (ρWε) =y z

Turbulence ModelTurbulence Model

constantsconstants

CD = 0,09

C1 = 1,44 C2 = 1,92

= 1,0 σε

= 1,314

ε2kρDC =tμ

U V W2 2 2G μ 2[( ) ( ) ( ) ]e x y z

U V W U V W2 2 2[ ] [ ] [ ] y x x z z y

Magneto-hydrodynamic

turbulence:

(ρU ub) + vb + wbx

- b u + - b u + b ux

+ + + S

Β Β Β

¶ ¶ ¶r r r

¶ ¶ ¶¶ ¶ ¶¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶

-x x y y z z

V W(ρ ) (ρ ) = y z

B B B y z

1/μσ[ ( ) ( ) ( )] x x y y z z

x y zB B B + + 0x

¶ ¶ ¶¶ ¶ ¶

x y zb b b + + 0x

¶ ¶ ¶¶ ¶ ¶

It has been observed experimentally that in every flow under critical conditions

(i.e critical Re, Ra, etc) especially

in shear flows (velocity gradients), there appear large scale structures

(coherent structures)

These structures modify the flow field and all its parameters of technological importance

(eg. skin friction

coefficient, coefficient of heat/mass transfer etc)

Coherent (large scale ) flow structures (1)

Coherent (large scale ) flow structures (2)

They have been observed in turbulent flows (free jets, mixing layers, boundary layers)

Need to understand their dynamics and effect on the flow

Development of modelling

methods in the CFD environment

Navier-Stokes

for instantaneous, laminar/turbulent,

fluid motion

Reynolds decomposition

equations of time-averaged motion

(Reynolds stress tensor appears)

Closure problem

(turbulence modeling cannot close)

Reynolds-stress transport equations

solution leads to higher-order correlations

Modeling turbulent flow

Magneto-hydrodynamic

turbulence:

(ρU u ) + v + wx

- b u + - b u + b ux

+ + + S

¶ ¶ ¶r r r

¶ ¶ ¶¶ ¶ ¶¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶

Β b Β b Β b

Β Β Β

x x y y z z

V W(ρ ) (ρ ) = y z

B B B y z

1/μσ[ ( ) ( ) ( )] x x y y z z

x y zB B B + + x

0¶ ¶ ¶¶ ¶ ¶

x y zb b b+ + x

0¶ ¶ ¶¶ ¶ ¶

Turbulence modelTurbulence model

(k(k--equation)equation)

(ρ ) + x

Uk (ρVk) + (ρWk) =y z

Turbulence kinetic energyTurbulence kinetic energy k = ½

+ w' 2)

+G-ρε Φ Φ Φ

k k k(Γ ) + (Γ ) + (Γ )

x x y y z z

B4σC ( ) k3

MHD term ??---->

Turbulence model (Turbulence model (εε--equation)equation)

1 2ε ε ε + ε/k(CG - Cρε)¶ ¶ ¶ ¶ ¶ ¶

¶ ¶ ¶ ¶ ¶ ¶Φ Φ Φ(Γ ) + (Γ ) + (Γ ) x x y y z z

Turbulence dissipation rateTurbulence dissipation rate

(ρ ) + x¶ ¶ ¶¶ ¶ ¶

Uε (ρVε) + (ρWε) =y z

B4σC ( ) ε3

MHD term ??---->

Turbulence ModelTurbulence Model

constantsconstants

CD = 0,09

C1 = 1,44 C2 = 1,92

= 1,0 σε

= 1,314

ε2kρDC =tμμe

U V W2 2 2G μ 2[( ) ( ) ( ) ]e x y z

U V W U V W2 2 2[ ] [ ] [ ] y x x z z y

CB = 0

Modelling

MHD Flows & Transport for ITER

Need to understand physics of:-

Hydrodynamic turbulence

Electromagnetic field effects-

Electromagnetic turbulence

Coherent structure dynamics

Need to improve modelling

techniques:-

Incorporate better multi-physics

Develop better numerical schemes-

Perform parallel computing (clusters?)

FUSION RESEARCH

The Phenomena of Fluid Flow - mie.uth.gr2009-2010)a.pdf · The Phenomena of Fluid Flow ... that has...

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