The Phenomena of Fluid Flow

The Phenomena of Fluid Flow Nicholas S. Vlachos Lab. Fluid Mechanics & Turbomachines Department of Mechanical Engineering University of Thessaly Program of Graduate Studies Academic Year 2009-2010 2010 1

Coherent (large scale) flow structures 2

Tornado 3

Coherent (large scale) flow structures 4

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A future challenge in fluid flow 6

Droplet breakup Fluid flow is a non-linear phenomenon 7

Droplet production from water stream 8

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 9

Flow around a cylinder Laminar Turbulent 10

Turbulence is 3D random isotropic (?) Experimental evidence caused revision of this perception of turbulence 11

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 12

Wind tunnel of UTH Fluids Lab Airfoil NACA 4418 13

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Bernoulli equation (valid along fluid path): 0Bernoulli V P t ds + + VdV ρ +ρdz=16

Daniel Bernoulli (1700-1782) 17

+++ρα p ρg f visf z=-= V V V +W V ρα ρ( +U + V y) t x +.Equations of fluid motion (1) em18

Equations of fluid motion (2) =α p ρg =ρ p -Hydrostatic condition Euler s equation =-+ρα p ρg Euler + gravity 19

Leonhard Euler (1707-1783) 20

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 methods Require knowledge of distributions Computational Fluid Dynamics (CFD) Requires large grids and CPU time 21

Navier-Stokes flow equations Mass conservation equation (continuity): (ρv)+ (ρw)=fu(ρ)+ (ρ ) + ys t x (ρ U)+ (ρ U) + (ρv)+u (ρw)=yzu t x U (μ U )+ (μ U )+ (μ U )xfxyefyzef zlinear momentum equation (Navier-Stokes): of Thessaly - Lab. Fluid Mechanics & Turbomachines zuniversity Fe22 + S

Claude Louis Marie Henri Navier (1785-1836) George Gabriel Stokes (1819-1903) 23

Continuity equation: Momentum equation: ρu u u + ρ(u + v ) = t x y p u u - + [ μ( ) + μ( )] + ρgxβδτ x x x y y ρv v v + ρ(u + v ) = t x y Two-dimensional flows ρ ρu ρv + + = 0 t x y p v v - + [ μ( ) + μ( )] + y x x y y ρgyβδτ 24

Two-dimensional flows (2) Energy equation: T T + = x y ρc p(u ρυ ) T T (k ) + (k ) x x y y u u u u x x y y p p (u v ) x y 2 2 + μ[( ) + 2 + ( ) ] + + 25

Continuity : x-momentum: y-momentum: Energy equation: 2-D boundary layer u v 0 x y ρ(u v ) μ x y x y P 0 y T T T ρc p (u + υ ) = (k ) x y y y 2 u u P u u y 2 + μ( ) + 2 p u x 26

Ludwig Prandtl (1875-1953) 27

zuniversity of Thessaly - Lab. Fluid Mechanics & Turbomachines Fluid vorticity Ω V V ì u v w ï iyi i = rot = x =í,, ý x ïî x y z ïþ ü ï 28

v ω x Vorticity components w -yw u = ω y = -zz z x u v ω y = -xuniversity of Thessaly - Lab. Fluid Mechanics & Turbomachines 29

Vorticity Transport Ω Ω Ω + u + v x+ w Ωx= Sφ + t ì ï Ω x1 Ω ü í ( meff x) + y( meff y) + z( meff Ωx) ï ý r ïî ïþ zkolmogorov Kolmogorov scales λ = (ν( 3 /ε) 1/4, υ = (νε( νε) 1/4 30

Two-Phase Flow: Particle Dynamics Lagrange Model dup dt - p F U Up g F D i i p F D 18C 24 p D Dp Re 2 C D a 1 a 2 Re a Re 3 2 31

32 Two-Phase Flow: Particle Heat Transfer t C m T h 3 p p p p 4 R p p fg p p p 3 p p p p 4 R p p fg p p p p p 3 p p p e T A A h A h dt dm T A h t T T A A h A h dt dm T A h ) t t ( T University of Thessaly - Lab. Fluid Mechanics & Turbomachines

Chemical Reaction - Coal Combustion dm dt R R p =-πd 2 P 1 2 p 0 R + R 1 2 Diffusion rate: R =C 1 1 T p +T o /2 D p 0.75 Kinetics rate: R E =C exp - RT 2 2 p 33

Magnetic force on a particle (Lorentz) 34

Hendrik Antoon Lorentz (1853-1928) 35

Electrodynamics - Definitions 36

Magneto-hydrodynamics equations (ρ )+ (ρ U ) + (ρv)+ (ρw)=- P +U U yu zu x t x (μ U )+ (μ U )+ (μ U )μxfxyefyzef+ ( B ) B/ + Se zuniversity of Thessaly - Lab. Fluid Mechanics & Turbomachines F P = p + B 2 /2μ V = 0 B= 0 37

Magneto-hydrodynamics equations Β (ρv) (ρw)=b + (ρ UΒ) + Β + Β U B + U Byz yzy+ U z t x x 1/μσ ( Β ) ( Β ) ( Β )]xx+ yy+ z+ S [of Thessaly - Lab. Fluid Mechanics & Turbomachines FxB By Bz =zx zuniversity + y+ 0 x 38

Non-dimensional variables group Flow configuration U=V=0 T=0 Y g U=V=0 T=0 U=V=0 T=1 Y, V B X, U U=V=0 T=0 Y 39

Ha=0 Ha=30 Ha=75 Stream lines (upper) and isotherms (lower) for Gr = 10 6 40

Turbulence and large structures in air jet 41

Boundary layer around a ship s hull and waves 42

Shock wave and turbulence 43

Pressure waves and turbulence 44

Modeling turbulent flow 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 45

Spectrum of turbulent kinetic energy Kolmogorov scales length - λ = (ν( 3 /ε) 1/4 velocity υ = (νε( νε) 1/4 Reλ = λυ/ν = 1 46

k=k=g-g-university of Thessaly - Lab. Fluid Mechanics & Turbomachines Balance of turbulent kinetic energy Increasing turbulence Equilibrium Decaying turbulence k = ½ (u' 2 +v' 2 + w' 2) ρε>0 t ρε=0 t k 0 t -ρε<=g47

Reynolds decomposition: zreynolds φ = Φ + φ Continuity equation: zcontinuity Time-mean mean φ = Φ (ρv)+ (ρw)=(ρ ) + y0 x U (ρ v (ρw(ρu) + y0 x )+)=48

Osborne Reynolds (1842-1912) 49

50 Reynolds Averaged Equations Reynolds Averaged Equations y z z Φ(ρΦ)+(ρUΦ)+(ρVΦ)+(ρW)=txy y ΦΦΦΦ(Γ-ρu'φ')+(Γ-ρv'φ')xxz z ΦΦΦ+(Γ-ρw'φ')+SUniversity of Thessaly - Lab. Fluid Mechanics & Turbomachines

2D boundary layer Continuity: U x V y 0 x momentum: y - momentum: Energy equation:: 2 U U P U U V u' v' 2 x y x y P 0 y T T T rcp( u + ru ) = ( k )-rcpu' T-rcpv' T x y y y 51

Turbulence models - Standard k-εk Turbulence kinetic energy k(ρ ) + (ρvk)+ (yx Ukk(Γ)+(Γ)xxyΦyk + (Γ )zφ ΦρWk)=zTurbulence kinetic energy zuniversity of Thessaly - Lab. Fluid Mechanics & Turbomachines +G-ρε k = ½ (u' 2 +v' 2 + w' 2 ) 52

Turbulence model (2) Turbulence dissipation rate (ρ ) + x (Γ ε )+ (Γ ε ε Φ)+(ΓΦ)+ ε/k(c 1 G - C 2 ρε) xxyyzzturbulence ε (ρvε)+ (ρwε)=y U ΦzUniversity of Thessaly - Lab. Fluid Mechanics & Turbomachines 53

G2}Uμe{2[( ) x 2V )y 2W Turbulence Model - constants )2(VWUVW[ ]2[ z ]2[ ] y x x z zu (]yuniversity of Thessaly - Lab. Fluid Mechanics & Turbomachines μ e = μ + μ t = C ρ D k2 ε C D = 0,09 C 1 = 1,44 C 2 = 1,92 σ k = 1,0 σ ε = 1,314 μ t 54

Magneto-hydrodynamic turbulence: (ρv) (ρw)=(ρuβ-rub) + yβ-rvb + zβ-rwb x B U B U Bxyyz U z- b u + y- b u + z-xb u x 1/μσ[ ( Β ) ( Β ) ( Β )]F xx+ yy+ z+ S y zx B B B + + =y0 x b b b + + =0 x z x y z zyzuniversity of Thessaly - Lab. Fluid Mechanics & Turbomachines 55

Coherent (large scale ) flow structures (1) 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) 56

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 57

Coherent (large scale) flow structures 58

Modeling turbulent flow 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 59

Magneto-hydrodynamic turbulence: (ρv) (ρw(ρuβ-ru b) + yβ-rv b + zβ-rwb x B U By U ybz U z- b u + y- b u + z-b u x 1/μσ[ ( xβ ) ( Β ) ( Βx)]B xx+ yy+ z+ S y zx B B B + y+ x of Thessaly - Lab. Fluid Mechanics & Turbomachines z =x y z z b b b =zuniversity 0 + y+ x )=0 60

Turbulence model (k-equation) Turbulence kinetic energy k = ½ (u' 2 +v' 2 + w' 2 ) k(ρ ) + (ρvk)+y x U (Γ )+ (ΓΦ )+ (ΓΦ xxyyz Φ(ρWk)=zTurbulence kinetic energy kkk)zb +G-ρε MHD term??----> 4σC ( BB ) k 3 61

Turbulence model (ε-equation)( Turbulence dissipation rate ε(ρ ) + x Udissipation rate (ρvε)+(ρwε)=yzε +(ΓΦε )+(ΓΦε )zturbulence + ε/k(c 1 G - C 2 ρε) (Γ ) xxyyz ΦMHD term??----> 4σC B( 3 BB ) ε 62

GUniversity of Thessaly - Lab. Fluid Mechanics & Turbomachines UVWμ222e{2[( )( )( )] x y VWUVW[ ]2[ z ]2[ ] y x x z zu 2}Turbulence Model - constants μ e = μ + μ t μ = C ρ t D ε ycb k2 C D = 0,09 C 1 = 1,44 C 2 = 1,92 σ k = 1,0 σ ε = 1,314 = 0 or 1 63

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?) 64

MHD - FUSION RESEARCH 65