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

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University of Thessaly

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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

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Coherent (large scale) flow structures


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Tornado


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A future challenge in fluid flow


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Droplet breakup

Fluid flow is a non-linear phenomenon


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Droplet production from water stream


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“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


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Flow around a cylinderLaminar

Turbulent


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Turbulence is

3D

random isotropic

(?)Experimental evidence caused

revision

of this perception of turbulence


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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


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Wind tunnel of UTH Fluids Lab

Airfoil

NACA 4418


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V Pds + + VdVρt

¶ ¶¶

+ ρdz = 0

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


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Daniel Bernoulli (1700-1782)


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ρ ρ( +U + V )xt

V V V Vα ¶ ¶ ¶ ¶¶ ¶ ¶ ¶

= + Wy z

ρ p ρα g f f vis em = - + + + +...

Equations of fluid motion

(1)


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Equations of fluid motion

(2)

ρ p ρα g = - +

p ρg =

ρ pα = -

Hydrostatic condition

Euler’s equation

Euler

+ gravity


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Leonhard Euler (1707-1783)


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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

time


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(ρ )+ (ρ ) + xt

U U U U¶ ¶ ¶ ¶¶ ¶ ¶ ¶

U V W(ρ ) + (ρ ) =y z

(ρ)+ (ρ ) + Sxt

¶ ¶ ¶ ¶¶ ¶ ¶ ¶

FU V W(ρ ) + (ρ ) =

y z

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


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George Gabriel Stokes (1819-1903)

Claude Louis Marie Henri Navier(1785-1836)


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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¶ ¶ ¶ ¶ ¶- + + +¶ ¶ ¶ ¶ ¶


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Two-dimensional flows

(2)

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

¶ ¶ ¶ ¶+ +

¶ ¶ ¶¶ ¶

+ + +¶ ¶¶


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0

yv

xu

2

2

u u P uρ(u v ) μx y x y

0

yP

Continuity :

x-momentum:

y-momentum:

Energy equation:

2-D

boundary layer

pT T Tρc (u υ ) (k )x y y y

¶ ¶ ¶ ¶+ =

¶ ¶ ¶ ¶2uμ( ) pu

xy+

¶+

¶¶¶


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Ludwig Prandtl (1875-1953)


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Fluid vorticity

x y z

u v w

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


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Vorticity components

xv wz

ω ¶ ¶¶ ¶

= -y

yw ux

ω ¶ ¶¶ ¶

= -z

zu vy

ω ¶ ¶¶ ¶

= -x


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Vorticity Transport

u v w Sφ +t

¶ ¶ ¶ ¶¶ ¶ ¶ ¶

+ + + =Ω Ω Ω Ω

x x x

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


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Two-Phase Flow: Particle DynamicsLagrange ModelLagrange Model

-pD i i

p

d p F pdt

U U U g F

2p

DD Dp24

ReC18F

2

321D Re

aReaaC


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Two-Phase Flow: Particle Heat Transfer

t

CmTh

3pppp

4Rppfg

pp

p

3pppp

4Rppfg

pp

p

pp

3ppp

e

TAAh

Ahdt

dmTAh

tT

TAAh

Ahdt

dmTAh

)tt(T


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Chemical Reaction -

Coal Combustion

p 2 1 2p 0

1 2

dm R R= -πD Pdt R + R

0.75

p o1 1

p

T + T /2R = C

D

2 2p

ER = C exp -R T

KKineticsinetics

rate:rate:

DDiffusioniffusion

raterate::


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Magnetic force on a particle (Lorentz)


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Hendrik Antoon Lorentz (1853-1928)


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ElectrodynamicsElectrodynamics

--

DefinitionsDefinitions


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P = p + |B|2 /2μ

·

V

= 0

·

B = 0

Magneto-hydrodynamics equations

U (ρ )+ (ρ ) + x

+ ( · ) / +

Pt¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶

¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶

F

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


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Magneto-hydrodynamics equations

U + (ρ ) + + + +x x

+ + + S

tΒ U U UΒ Β Β

Β Β Β

¶ ¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶ ¶

¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶

F

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

¶ ¶ ¶¶ ¶ ¶

=y z


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g

U = V = 0

T = 1

U = V = 0

T = 0

U = V = 0

T = 0Y

U = V = 0

T = 0Y

Y, V

X, U

B

Flow configuration

Non-dimensional variables group


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Ha=0 Ha=30 Ha=75

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

= 106


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Turbulence and large structures in air jet


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Boundary layer around a ship’s hull and waves


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Shock wave and turbulence


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Pressure waves and turbulence


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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


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Spectrum of turbulent kinetic energySpectrum of turbulent kinetic energy

KolmogorovKolmogorov

scalesscales

lengthlength

--

λλ

= (= (νν33//εε))1/41/4

velocityvelocity

––

υυ

= (= (νενε))1/41/4

ReReλλ

= = λυλυ//νν

= 1= 1


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Balance of turbulent kinetic energyBalance of turbulent kinetic energy

t

0k = G - ρε > Increasing turbulenceIncreasing turbulence

EquilibriumEquilibriumt

0k = G - ρε =

Decaying turbulenceDecaying turbulence

t

0k = G - ρε <

k = ½

(u' 2

+v' 2

+ w' 2)


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(ρ ) + 0x¶ ¶ ¶¶ ¶ ¶

U V W(ρ ) + (ρ ) =y z

Reynolds decompositionReynolds decomposition: :

φφ

= = ΦΦ

+ + φφ’’

TimeTime--mean mean φφ

= = ΦΦ

(ρu) + 0x¶ ¶ ¶¶ ¶ ¶

v w(ρ ) + (ρ ) =y z

Continuity equationContinuity equation::


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Osborne Reynolds (1842-1912)


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Reynolds Averaged EquationsReynolds Averaged Equations

y z z¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶

Φ(ρΦ) + (ρUΦ) + (ρVΦ) + (ρW ) =

t x

y y¶ ¶ ¶ ¶¶ ¶ ¶ ¶Φ Φ

Φ Φ(Γ -ρu'φ') + (Γ - ρv'φ')

x x

z z¶ ¶¶ ¶

ΦΦ

Φ+ (Γ -ρw'φ') + S


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0U Vx y

2

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

0yP

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

r ru r r¶ ¶ ¶ ¶

+ = - -¶ ¶ ¶ ¶

Continuity:

x –

momentum:

y -

momentum:

Energy equation::

2D

boundary layer


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Turbulence modelsTurbulence models

--

Standard kStandard k--εε

(ρ ) + x

Uk (ρVk) + (ρWk) =y z

Turbulence kinetic energyTurbulence kinetic energy

k = ½

(u' 2

+v' 2

+ w' 2)

+G-ρε Φ Φ Φ

k k k(Γ ) + (Γ ) + (Γ )

x x y y z z


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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


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Turbulence ModelTurbulence Model

--

constantsconstants

CD = 0,09

C1 = 1,44 C2 = 1,92

σk

= 1,0 σε

= 1,314

ε2kρDC =tμ

μe

= μ

+ μt

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


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Magneto-hydrodynamic

turbulence:

(ρU ub) + vb + wbx

- b u + - b u + b ux

+ + + S

Β Β Β

U U U

Β Β Β

¶ ¶ ¶r r r

¶ ¶ ¶¶ ¶ ¶¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶

F

- - -

-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

¶ ¶ ¶¶ ¶ ¶

=y z

x y zb b b + + 0x

¶ ¶ ¶¶ ¶ ¶

=y z


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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)


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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


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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

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solution leads to higher-order correlations

Modeling turbulent flow


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Magneto-hydrodynamic

turbulence:

(ρU u ) + v + wx

- b u + - b u + b ux

+ + + S

¶ ¶ ¶r r r

¶ ¶ ¶¶ ¶ ¶¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶

- - -

-

B

Β b Β b Β b

U U U

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¶ ¶ ¶¶ ¶ ¶

=y z

x y zb b b+ + x

0¶ ¶ ¶¶ ¶ ¶

=y z


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Turbulence modelTurbulence model

(k(k--equation)equation)

(ρ ) + x

Uk (ρVk) + (ρWk) =y z

Turbulence kinetic energyTurbulence kinetic energy k = ½

(u' 2

+v' 2

+ w' 2)

+G-ρε Φ Φ Φ

k k k(Γ ) + (Γ ) + (Γ )

x x y y z z

B4σC ( ) k3

B B

MHD term ??---->


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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

B B

MHD term ??---->


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Turbulence ModelTurbulence Model

--

constantsconstants

CD = 0,09

C1 = 1,44 C2 = 1,92

σk

= 1,0 σε

= 1,314

ε2kρDC =tμμe

= μ

+ μt

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

or 1


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Modelling

MHD Flows & Transport for ITER

Need to understand physics of:-

Hydrodynamic turbulence

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Electromagnetic field effects-

Electromagnetic turbulence

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Coherent structure dynamics

Need to improve modelling

techniques:-

Incorporate better multi-physics

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Develop better numerical schemes-

Perform parallel computing (clusters?)


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MHD -

FUSION RESEARCH


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