TURBULENT FLOW A BEGINNER S APPROACH Tony Saad March 2004 http://tsaad.uts.edu - tsaad@uts.edu
CONTENTS Introducton Random processes The energy cascade mechansm The Kolmogorov hypotheses The closure problem Some Turbulence Models Concluson
INTRODUCTION Most flows encountered n engneerng practce are Turbulent Turbulent Flows are characterzed by a fluctuatng velocty feld (n both poston and tme). We say that the velocty feld s Random. Turbulence hghly enhances the rates of mng of momentum, heat etc
INTRODUCTION The motvatons to study turbulent flows are summarzed as follows: The vast maorty of flows s turbulent The transport and mng of matter, momentum, and heat n turbulent flows s of great practcal mportance Turbulence enhances the rates of the above processes The basc approaches to nvestgatng turbulent flows are shown n the followng chart
INTRODUCTION Eperment Numercal Methods Tme Consumng Resource Consumng Emprcal Correlatons Hgh Cost $$$$ Averagng of Equatons. Turbulence Modelng Flterng of Equatons. Large Eddy Smulaton Full Resoluton of the Flow Drect Numercal Smulaton M O D E L I N G A P P R O A C H E S
RANDOM PROCESSES Consder a typcal ppe flow where the nstantaneous velocty s to be measured as a functon of tme. (see dye move see velocty profle move)
RANDOM PROCESSES It s clear that the velocty s fluctuatng wth tme We say that the velocty feld s RANDOM What does that mean? Why s t so?
RANDOM PROCESSES Consder the ppe flow eperment that s to be repeated several tmes (say 395804735 tmes) under the same set of condtons: On the same planet In the same country In the same lab Usng the same apparatus Supervsed by the same person
RANDOM PROCESSES We are nterested n measurng the streamwse velocty component at a gven pont A( 1,y 1,z 1 ) & at a gven tme durng the eperment: U(A,t 1 ) Velocty Measurment
RANDOM PROCESSES Consder the event denoted by E such that: { U ( A t < m } 1 E =, ) 10 Three cases tae place: If E nevtably occurs, then E s certan If E cannot occur, then E s mpossble Another possblty s that E may but need not occur, then E s Random s
RANDOM PROCESSES The word Random does not hold any sophstcated sgnfcance as t s usually assgned The event E s random means only that t may or may not occur And ths answers our frst queston;.e. the meanng of a random varable
RANDOM PROCESSES We can now chec the results of our eperment. After 40 repettons of the eperment the measured velocty was recorded as shown below: 20 Magntude of U 15 U (m/s) 10 5 0 0 5 10 15 20 25 30 35 40 Fgure 1.1: Magntude of U as a functon of Eperment Number
RANDOM PROCESSES Oay. Why s the velocty feld random n a turbulent flow?
RANDOM PROCESSES Obvously, f there were no determnsm (at least n the underlyng structure of the unverse), why would we do scence n the frst place? As scence ams @ predctng the behavor of physcal phenomena To answer the queston, we have to resort to the theory of dynamcal systems. After all, the governng equatons of flud mechancs represent a set of dynamcal equatons. Dynamcal systems are systems that are etremely senstve to mnute changes n the surroundng envronment,.e. ntal & boundary condtons
RANDOM PROCESSES An llustratve eample s the Lorenz dynamc system. Consder the followng set of ODEs = σ ( y ) y = ρ y z z = β z y Assume we f σ =10 and β =8/3 and vary ρ
RANDOM PROCESSES We are nterested n solvng ths system for two dfferent acute ntal condtons. [ ( [ ( 1 0 0 ),y( ),y( 0 0 ),z( 0 )] = ),z( 0 )] = [010101].,.,. and [0100000101.,., 01]. Obvously, the two solutons for the dfferent ntal condtons should be very close snce the ntal condtons are appromately the same Ths s what we epect. Let us see the results
RANDOM PROCESSES The soluton s easy to get numercally and s as follows for varous values of ρ 60 ( t) 30 60 1( t) 60 30 0 ρ = 5 30 0 10 20 30 40 50 60 0 t t 60 30 0 ρ = 5 Frst set of ntal condtons Second set of ntal condtons T 60 ( t) 1( t) 30 ρ = 5 60 30 0 ρ = 5 30 0 10 20 30 40 50 60 0 t t Dfference between the two solutons T 30 30 0 10 20 30 40 50 60 0 t T Dfference between the two solutons
RANDOM PROCESSES Nothng much happens for some values of ρ 60 ( t) 30 60 1( t) 60 30 0 ρ = 20 30 0 10 20 30 40 50 60 ( t) 1( t) 0 t t 60 30 0 ρ = 20 Frst set of ntal condtons Second set of ntal condtons T 60 30 60 30 0 ρ = 20 ρ = 20 30 0 10 20 30 40 50 60 0 t t Dfference between the two solutons T 30 30 0 10 20 30 40 50 60 0 t T
RANDOM PROCESSES What s so specal about ρ = 24.2 60 ( t) 30 60 1( t) 60 30 0 ρ = 24.2 30 0 10 20 30 40 50 60 0 t ( t) 1( t) T 60 30 0 ρ = 24.2 Frst set of ntal condtons t Second set of ntal condtons 60 30 60 30 0 ρ= 24.2 ρ = 24.2 30 0 10 20 30 40 50 60 0 t t Dfference between the two solutons T 30 30 0 10 20 30 40 50 60 0 t T
RANDOM PROCESSES ρ = 24.2 denotes the onset of senstvty!!! 60 ( t) 30 60 1( t) 30 60 30 0 ρ = 24.3 30 0 10 20 30 40 50 60 ( t) 1( t) 0 t t 60 30 0 ρ = 24.3 30 0 10 20 30 40 50 60 0 t Frst set of ntal condtons Second set of ntal condtons T T 60 30 60 30 0 ρ = 24.3 ρ = 24.3 30 0 10 20 30 40 50 60 0 t See anmaton t Dfference between the two solutons T
RANDOM PROCESSES As we can see, the fgures show the senstvty of the system. In fact, for a crtcal value of the coeffcents (f σ, β) ρ =24.3, the system becomes hghly senstve to small perturbatons. Ths coeffcent corresponds to the Reynolds number n flud flow. Beyond a crtcal value, the flow becomes etremely senstve to any mnute dsturbance whether from the boundary or from the nternal structure of the flow (an eddy)
RANDOM PROCESSES Concluson: A theory that predcts eact values of the velocty feld s defntely prone to be wrong. Such a theory should am at predctng the probablty of occurrence of certan events
THE ENERGY CASCADE One of the frst attacs to theoretcally tacle the problem of turbulence was done by Rchardson n 1922 Hs hypothess s called the Energy Cascade Mechansm & s now one of the fundamental physcal models underlyng any approach to solvng turbulent flow problems Turbulence can be consdered to be composed of eddes of dfferent szes
THE ENERGY CASCADE An eddy s consdered to be a turbulent moton localzed wthn a regon of sze l These szes range from the Flow length scale L to the smallest eddes. Each eddy of length sze l has a characterstc velocty u(l) and tmescale τ(l) = l/u(l) The largest eddes have length scales comparable to L
THE ENERGY CASCADE Move
THE ENERGY CASCADE Each eddy has a Reynolds number For large eddes, Re s large,.e. vscous effects are neglgble. The dea s that the large eddes are unstable and brea up transferrng energy to the smaller eddes. The smaller eddes undergo the same process and so on
THE ENERGY CASCADE
THE ENERGY CASCADE Ths energy cascade contnues untl the Reynolds number s suffcently small that energy s dsspated by vscous effects: the eddy moton s stable, and molecular vscosty s responsble for dsspaton. Bg whorls have lttle whorls, whch feed on ther velocty; and lttle whorls have lesser whorls, and so on to vscosty
THE KOLMOGOROV HYPOTHESES What s the sze of the smallest eddes??? The KOLMOGOROV Hypotheses address ths fundamental queston along wth many others There are three propostons that Kolmogorov made: The hypothess of local sotropy Frst smlarty Second smlarty
THE KOLMOGOROV HYPOTHESES We frst ntroduce some useful scales Denote by L 0 the length scale of an eddy comparable n sze to the flow geometry L EI 1 L Denote by 6 0 the demarcaton scale between the largest (L > L EI ) eddes and the smallest eddes (L < L EI )
THE KOLMOGOROV HYPOTHESES Kolmogorov s hypothess of local sotropy: At suffcently hgh Reynolds number, the smallscale motons (L < L 0 ) are statstcally sotropc. As the energy passes down the cascade, all nformaton about the drectonal propertes of the large eddes (determned by the flow geometry) s also lost. As a consequence, the small enough eddes have a somehow unversal character, ndependent of the flow geometry
THE KOLMOGOROV HYPOTHESES Kolmogorov s frst smlarty hypothess: In every turbulent flow at suffcently hgh Reynolds number, the statstcs of the smallscale motons (L < L EI ) have a unversal form that s unquely determned by ν and ε. Just as the drectonal propertes are lost n the energy cascade, all nformaton about the boundares of the flow s also lost, therefore, the mportant processes responsble for dsspaton are determned unquely by ν and ε
THE KOLMOGOROV HYPOTHESES Kolmogorov s second smlarty hypothess: In every turbulent flow at suffcently hgh Reynolds number, there ests a range of scales whose propertes are unquely determned by ε The three hypothess are shown n the setch to follow
THE KOLMOGOROV HYPOTHESES Unversal Equlbrum Range Energy Contanng Range Dsspaton Range Governed by ν & ε Inertal Subrange Governed by ε η DI EI L 0 The smallest scales
THE CLOSURE PROBLEM Mass Conservaton: ρ τ ρu = 0 Momentum Conservaton: ρu τ Accum. ρu u Convecton = u µ Dffuson p ρg su Source
THE CLOSURE PROBLEM The nfntude of scales n a turbulent flow renders the flow equatons mpossble to tacle mathematcally n that the velocty feld represents a random varable A clever proposton, addressed by Reynolds reduces the numbers of scales from nfnty to 1 or 2 After all, for all practcal purposes, we are only nterested n the mean flow and n the mean flud propertes
THE CLOSURE PROBLEM Problems: Ellptc & Non-Lnear Equatons; Pressure-Velocty Couplng & Temperature-Velocty Couplng; Turbulence: Unsteady, 3-D, Chaotc, Dffusve, Dsspatve, Intermttent,...; Infnte Number of Scales (Grd Re 9/4 ), Etc. Soluton: Reduce the number of scales Reynolds Decomposton. ϕ = Φ ϕ'
Turbulence: RANS Result: Reynolds Averaged Naver-Stoes (RANS) Equatons. Closure Problem: Turbulent Stresses ( ) are unnowns Turbulence Models. 0 U = ρ τ ρ u S P u u U U U U = ' ' ρ µ ρ τ ρ u u ' ' ρ
THE CLOSURE PROBLEM One can suggest to wrte transport equatons for the reynolds stresses, however, one would end up wth trple correlatons and so forth The dea s that we have to stop somewhere and model the correlaton usng physcal arguments
THE CLOSURE PROBLEM ( ) ( ) ( ) ( ) ( ) ( ) ( ) ε u' u' u' u' μ u' u' μ Φ u' p u' p D u' μu u' μu u' u' μ u p δ u p δ u u ρu G t u ρβg t u ρβg P U u ρu U u ρu u' u' ρu τ u' ρu' = 2 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' Eample Of transport Equaton For the Turbulent Stress
THE CLOSURE PROBLEM Physcal mechansms governng the change of turbulent stresses & flues. Convecton Vscous Generaton Net Balance Dffuson Turbulent Pressure Dstrbuton Dsspaton
THE CLOSURE PROBLEM Unnown Correlatons: -) Turbulent & Pressure Dffuson (D, D t ) -) Pressure-Stran Correlatons (Φ, Φ t ) -) Dsspaton Rates (ε, ε t ) -) Possble Sources (S t ) Modelng s requred n order to close the governng equatons. Turbulence Models: -) Algebrac (zero-equaton) models: Mng length, etc. -) One-equaton models: -model, µ t -model, etc. -) Two-equaton models: -ε, -l, -ω 2, low-reynolds -ε, etc. -) Algebrac stress models: ASM, etc. -) Reynolds stress models: RSM, etc.
THE CLOSURE PROBLEM Frst Order Models Second Order Models Reynolds Stress Models Algebrac Stress Models Two-Equaton Models One-Equaton Models Zero-Equaton Models u' u' / = Constant ε P = ε = 3/ 2 / L = ( L du dy) 2
Frst-Order Models Governng equatons of turbulent stresses and flues are very comple Analogy between lamnar and turbulent flows. Lamnar Flow: Turbulent Flow: Generalzed Boussnesq Hypothess u 3 2 u u µδ = µ τ 3 2 U U u u t t ρ δ = µ = ρ τ
Frst-Order Models Zero Equaton Models As the name mples, these models do not mae use of any addtonal equaton. In such models we contend wth a smple algebrac relaton for the turbulent vscosty Zero equaton models are based on the mng length theory whch s the length over whch there s hgh nteracton between vortces Note that the turbulent vscosty s not a property of the flud! It s a flow property
Zero-Equaton Models Mng Length Theory: Dmensonal analyss ν t = µ t ρ lu = l m l m s determned epermentally. For boundary layers : l m du dy l m = κy for y<δ l m = δ for y δ δ y l m δ
Frst-Order Models One Equaton Models As the name mples, we develop one equaton for a gven varable. The varable of choce s the turbulent netc energy We wrte a transport equaton for turbulent netc energy and use algebrac modelng for the new varables ntroduced
One-Equaton Models Turbulent Knetc Energy: Dmensonal analyss 1 ( 2 2 2 = u v w 2 ) µ t ul u : velocty scale - proportonal to 1/2 l : length scale mng length l m µ t = C µ l m 1 2 Other choces: (Bradshaw et al.), etc. u uv Drect calculaton of µ t usng a transport equaton for the eddy vscosty (Nee & Kovasznay).
One-Equaton Models Turbulent Knetc Energy: Unnown Correlatons that requre modelng: -) Turbulent & Pressure Dffuson; -) Dsspaton Rates. ( ) ( ) ( ) 2 u u D pu u u 2 1 G P t u g U u u U ε µ µ ρ ρ β ρ = ρ τ ρ
One-Equaton Models Turbulent Knetc Energy Equaton: Dmensonal analyss (Harlow & Naayama) ρε σ µ µ = ρ τ ρ t G P U p t t T c g G ; U U U P = β = µ m 2 3 d 2 3 C l l = ε
Concluson Turbulence s a very mportant phenomenon Turbulence modelng s a hghly actve area of research The future of CFD has a hgh dependence on turbulence models LES seems to be the prevalng method for the net two decades DNS, wth the ever ncreasng computatonal power and parallel computng technologes, remans a dream for all researchers. Wll t ever be feasble on a personal computer?