Introducton to Turbulence by Håkan Gustavsson Dvson of Flud Mechancs Luleå Unversty of Technology
Foreword The ntenton wth these pages s to present the student wth the basc theoretcal concepts of turbulence and derve exact relatons from the governng equatons. The dea s to show that despte the complexty of turbulent flows, some general propertes can be educed from the equatons. Hopefully, ths wll remove some of the mystque that has surrounded turbulence as a topc n undergraduate courses. The materal s used as lecture notes n the course MTM 6 (Advanced Flud Mechancs) gven n the last year of undergraduate studes at Ltu. Luleå, October 006 Håkan Gustavsson Dvson of Flud Mechancs
Contents Page. Introducton. Reynolds decomposton 3 3. Equatons for the mean flow 4 3. Contnuty 4 3. Momentum 4 3.3 Knetc energy 6 4. Equatons for the turbulent fluctuatons 7 4. Momentum 7 4. Knetc energy 8 5. Turbulent channel flow 9 5. Momentum equaton 9 5. Turbulence producton 3 5.3 Mean velocty profle 4 6. Kolmogorov mcroscales 6 7. Turbulence structure 7 References 8
. Introducton Turbulence s generally consdered one of the unresolved phenomena of physcs. Ths means that there s not one model that descrbes the appearance and mantenance of turbulence n all stuatons where t appears. Because of the techncal mportance of turbulence, models based on correlatons of partcular expermental data have been developed to a large extent. The task to develop a general turbulence model s challengng snce turbulence appears almost everywhere: Flows n rvers, oceans and the atmosphere are large scale examples. Flows n ppes, pumps, turbnes, combuston processes, n the wake of cars, arplanes and trans are some techncal examples. Even the blood flow n the aorta s occasonally turbulent. In fact, one can say that turbulence s the general flow type on medum and large scales whereas lamnar flows appear on small scales, and where the vscosty s hgh. For example, the flow of lubrcatng ols n bearngs s lamnar. Before we dscuss the techncal aspects of turbulence t s necessary to state ts man knematc characterstcs. In the lst below, some flows may have one or two features but turbulence has all three. Irregularty. Observng structures n the flow from a smoke stack, or measurng the velocty n a ppe flow, show that any partcular pattern never repeats tself. Ths randomness suggests that a statstcal treatment of turbulence s worthwhle. In fact, statstcal quanttes such as mean values, correlatons etc. are generally repeatable and make statstcal theores attractve. Mxng. A case of randomness s the deflecton of a water surface due to wnd. However, n ths flow flud partcles stay largely n one place whch they do not n a strred cup of coffee. Thus, mxng s a promnent feature of turbulence and nvolves mxng of partcles and all physcal quanttes related to partcles.e. heat, momentum etc. Three-dmensonal vortcty fluctuatons. Turbulent flows always exhbt hgh levels of vortcty. The mantenance of turbulence requres the process of vortex stretchng whch occurs only n 3D. A word of cauton: D turbulence may occur f magnetc felds control the flow. In addton to these knematc characterstcs, we may also add some features of turbulence that help to decde when to expect ts appearance and udge ts physcal sgnfcance.
Large Reynolds number. Typcal for all turbulent flows s that a relevant Reynolds number (UL/ν) for the flow s large. It s the obectve for stablty theory to determne the crtcal Reynolds number over whch turbulence may appear. Ths s a part of the turbulence engma. Dsspaton. Turbulent flows loose mechancal energy due to the acton of shear stresses (dsspaton) at a much larger rate than lamnar flows. The flow losses are much larger. The energy s converted nto nternal energy and thus shows up as an ncrease of temperature. Wthout a contnuous supply of energy, turbulence decays (cf. the strred coffee n a cup when the spoon s removed). Contnuum. The smallest scales of turbulence are generally far larger than molecular length scales. Turbulence s therefore a contnuum phenomenon, and should be possble to descrbe by the equatons of moton of flud mechancs. Turbulent flows are flows. Turbulence s a feature of the flow and not of the flud. Thus, dfferent fluds show the same propertes gven the (non-dmensonal) flow parameters are the same. For the analyss of turbulence phenomena we use the equatons of moton (contnuty, momentum and energy) but the added complexty of randomness makes statstcal tools necessary. Mathematcally, we wll make frequent use of tensor notaton n wrtng the equatons of moton. Ths smplfes the dervatons and reduces the wrtng consderably. A powerful tool s also dmensonal analyss whch relates the ngredent varables through ther dmensons. Wth smple assumptons about the flow very far-reachng conclusons can be drawn wth dmensonal arguments. We wll use ths technque manly when dervng the smallest dsspatve scales n turbulence, the Kolmogorov mcroscales. In choosng specfc flows to analyse, we pck wall bounded turbulent flows (channel flow) for whch analytcal results may be derved and some crucal concepts appear. The next step would be to treat free turbulence (wakes, ets and free shear layers) where ust a modest level of turbulence modellng (ed vscosty) leads to surprsngly useful results. The further steps of turbulence modellng and the modern approach of drect numercal smulaton of turbulence (DNS) are treated at the graduate level.
3. Reynolds decomposton Typcal sgnals from measurements of velocty components near a sold wall are shown n fgure.. The data llustrate the randomness of the sgnal but one can generally produce a Fgure.: Near-wall data of u- v- and uv sgnals and the correspondng short-tme varances of u and v for T=0t*.(From Alfredsson & Johansson 984, JFM 39) tme-average defned as T f = F = lm f (t)dt, (.) T T 0 where f can be any of the flud mechancal varables of nterest (velocty component, pressure etc.). We use the over-bar notaton, or captal letter, for the tme-average. Usng ths defnton, we splt each nstantaneous component nto ts tme-average and a tme-dependent fluctuaton. For the velocty and the pressure ths gves u = U u (Note that u = 0, by defnton) p = P p (.a,b) Ths splttng of a turbulent sgnal s denoted Reynolds decomposton of a turbulent flow.
4 3. Equatons for the mean flow To descrbe the average and namc propertes of turbulent flows, use s made of the basc conservaton laws of flud mechancs,.e. the equatons for mass, momentum and energy. These relatons have n general to be complemented wth a state condton for the flud. In ths text we wll not treat energy through the thermonamc energy equaton but restrct the stu to the knetc energy. Ths can be done wthout nvokng thermonamcs but rather by multplyng the momentum equaton wth a sutable velocty component. 3. Contnuty In ths presentaton we wll assume the flow to be ncompressble.e. dv u = u, = 0 (3.) Usng the decomposton n (.a), t s deduced that the followng relatons apply for the mean flow and the fluctuatons: U, = 0 (3.) and, = 0 (3.3) Thus, both the mean flow and the fluctuatons satsfy the ncompressble condton. 3. Momentum We frst formulate the averagng process for the general momentum equaton and then specalze to the Naver-Stokes equatons. Thus, we have Du Dt u t u σ ρ = ρ u = ρf (3.4) where F s a mass force (typcally g) and σ s the stress tensor. Usng the decomposton n (.a) for u and applyng the averagng process (.) the left hand sde of (3.4) becomes u (U ) U ρ u = ρ (U ) = ρ U u (3.5) In the averagng process, the two terms nvolvng only one prmed quantty become zero so only the mean term and the doubly prmed term reman. Usng contnuty, the last term can also be wrtten as ρu (ρ ) = (3.6)
5 For the stress tensor we haveσ = Σ σ, where Σ s the tme-average. The average of the fluctuatng part depends on the partcular (rheologcal) model that s used to descrbe the relaton between the stress and the velocty feld. As we wll consder only Newtonan fluds, the stress s lnearly related to the velocty gradent. Thus, averagng gves σ = 0. (3.7) Combnng (3.4)-(3.7) then result n the followng equaton for the average quanttes U ρu = ρf ( Σ ρ ) (3.8) where the term (3.6) has been moved to the rght hand sde so that a physcal nterpretaton of ts sgnfcance can be gven. (3.8) s denoted the Reynolds equaton for a turbulent (mean) flow and may be nterpreted as f the stress s gven an extra contrbuton due to the turbulent fluctuatons, ρu. Ths extra stress s denoted the Reynolds stress n honour of Osborne Reynolds (84-9) who was the frst to dentfy ths contrbuton of the turbulent fluctuatons. It should be noted that the Reynolds stress has ts orgn n the non-lnear (advectve) term n the momentum equaton and s thus a property of the flow and not of the flud. It s also noted that the Reynolds stress s a second rank tensor and properly should be denoted the Reynolds stress tensor. It s the obectve of turbulence modellng to connect the Reynolds stress to other flow quanttes. For the partcular case of an ncompressble Newtonan flud, the stress depends on the pressure and the velocty gradent through ( u u ) σ = p δ μ (3.9), where μ s the namc vscosty. For the mean flow ths reduces to and (3.8) becomes, ( U U ) Σ = P δ μ (3.0) U,, ( ρ ) ρ U = P ρf μu (3.8),, Dvdng through by ρ, we obtan the momentum equaton for a Newtonan flud that wll be used frequently n the sequel. U ( ) U = P F νu (3.),, Often, the mass force s neglected. ρ
6 3.3 Knetc energy If we multply (3.) by U, neglectng F and usng contnuty, the followng terms are obtaned U U U ( U U / ) = U (3.) U P = ( U P) (3.3), ρ ρ U ( U U ), ν U = ν νu U (3.4),,, ( ) ( U ) U = U, (3.5) Collectng the terms (3.)-(3.5) we obtan U ( UU /) = ( U P/ρ νu,u U ) νu,u, U, (3.6) The terms n (3.6) have been arranged so that a physcal nterpretaton s possble. The left hand sde s the advected varaton of the knetc energy. On the rght hand sde, the frst term s denoted a transport term snce an applcaton of Gauss theorem shows that t s only the changes at the surfaces that contrbute to ths term. The second term contans only squared terms and because of the mnus sgn, t represents a loss of knetc energy due to vscous dsspaton. The last term represents ether a gan or a loss of knetc energy and shows how the Reynolds stresses, together wth the mean shear, act to change the knetc energy of the mean flow.
7 4. Equatons for the turbulent fluctuatons Snce the Reynolds stresses contan the turbulent fluctuatons, t s necessary to derve equatons also for the development of these quanttes. We wll do ths n a general form and n the next chapter specalze to plane channel flow. 4. Momentum The nstantaneous velocty and pressure felds (.) must satsfy the momentum equaton so (3.4) becomes (U ) (U ) = t ρ ( Σ σ ) (4.) Here, the average (captal letters) satsfy (3.8) so a subtracton of ths equaton gves the equaton for the turbulent quanttes: U t U = ρ ( σ ρ ρ ) (4.) For a Newtonan medum and (4.) reduces to ( u ) σ = p δ μ U t, U, p = ν ρ ( ) (4.3) Ths s the momentum equaton for the turbulent fluctuatons. The last term on the rght hand sde s non-lnear and represents the devaton from the mean of the Reynolds stress and may therefore be denoted the fluctuatng Reynolds stress. (4.3) llustrates the problem wth the analyss of turbulent flows: n order to determne the turbulent fluctuatons we need to know ther average expressed as the Reynolds stress. It s therefore necessary to go to hgher order whereby new unknown quanttes appear. Endng ths sequence of equatons s the closure problem of turbulence and s so far unresolved. Because of ths dffculty, t has become necessary to fnd smpler models for turbulence. Despte the ntroducton of large scale smulatons for turbulence flows, turbulence modellng s stll a very actve research area.
8 4. Knetc energy If we multply eq. (4.3) by, we can derve an equaton for the knetc energy of the fluctuaton feld. Averagng over tme (carry out the detals!), result n the followng expresson: q u ν = (p /ρ q) ν U (4.4), U where q = u /. The frst term on the rght hand sde s nterpreted as due to turbulent transport; ts contrbuton s zero f ntegrated between sold walls. The second term s turbulent producton and should be compared wth the smlar term n (3.6). Just note the dfferent sgns! What appears as a gan n energy for the fluctuatons s seen as a loss of energy for the mean flow. Ths gves a physcal sgnfcance to the Reynolds stresses: they act together wth the mean shear (U, ) to transfer knetc energy between the mean flow and the turbulent fluctuatons. The last term n (4.4) s always negatve and represents vscous dsspaton, generally denoted ε.
9 5. Turbulent channel flow In ths secton we wll wrte the momentum equaton n the smple (but mportant!) geometry of a plane channel flow. The mean flow s drven by a pressure gradent n the x-drecton (cf fgure 5.) and after a certan dstance from the channel entrance we assume the flow characterstcs to be ndependent of x ( fully developed flow ) and z (D). y=h U(y) y x y Fgure 5.: Geometry for turbulent channel y=-h flow, drven by a pressure gradent. Mathematcally, ths s wrtten as U = U(y), V = 0 (5.) = 0 (except for P) and = 0 (5.) z The condton V = 0 follows from contnuty assumng U ndependent of x. 5. Momentum equaton The two components of Reynolds equaton for the mean flow (3.) become, x-dr: P d U d = ν ( ) (5.3) ρ 0 P d y-dr: 0 = ( ) (5.4) ρ y Here, (5.4) can be ntegrated n y leadng to P 0 P (x) = (5.5) ρ ρ where P 0 may be seen as an ntegraton constant but t also shows that the pressure across a boundary layer may change due to turbulent fluctuatons. Elmnatng P between (5.3) and (5.5) gves dp0 d U d = ν ( ) (5.6) ρ dx 0
0 (5.6) can be ntegrated n y and choosng the ntegraton nterval... we obtan dp du du 0 = (y h) ν ρ dx 0 y= h y h ( 0), (5.7) where we have used the fact that the Reynolds stress vanshes on the wall. The evaluaton of the mean shear on the wall may be expressed n terms of the wall shear stress snce we have du ν y= h du = μ = τw/ρ u (5.8) ρ y= h Here, we have defned a characterstc velocty,, whch s denoted the frcton velocty (or wall velocty). It has a fundamental role when scalng the velocty close to the wall. u The relaton of u to other veloctes n a flow can be derved usng the local frcton τ coeffcent C f,x w ρu /, where U m s a typcal (mean) velocty. Substtutng for u and m knowng that C f,x s dependent on the Reynolds number, one obtans u /U m = /. For a turbulent boundary layer C f,x = 0.059Re x -/5 and n a turbulent ppe flow C f,x = 0.079Re -/4, respectvely. Thus the rato /U m wll vary weakly wth the Reynolds number. u C f, x To see the use of u n the scalng of the momentum equaton we frst elmnate P 0 by puttng y = 0 n (5.7), usng that U s symmetrc and u = 0 there. Ths yelds dp0 0 = h u* (5.9) ρ dx Combnng wth (5.7) gves y du 0 = u* ν (5.0) h (5.0) s vald n the whole nterval y h but the dfferent terms balance each other dependent on where n the nterval we are. As the wall proxmty s of most nterest, t s useful to ntroduce the dstance from the wall as a new varable, y = y h. In terms of y, (5.0) thus becomes y du 0 = u* ν (5.) h
(5.) s of consderable nterest snce t couples the Reynolds stress to the mean flow. However, the balance between the dfferent terms depend on the y poston and to elucdate ths t s necessary to scale the equaton. Ths can be done n (at least) two ways, one usng h as the characterstc length, the other u /ν, denoted the wall length. The velocty s scaled by u n both cases. Outer scalng: Scalng wth h (and u ) (5.) reduces to 0 y d(u/u ) /h) u = h uh d(y Introducng the Reynolds no., Re 0 y ν = = h Re d(y * uh, ths expresson may also be wrtten as ν d(u/u ) /h) u * (5.) Snce Re n general s a large number, (5.) s reduced n the center porton of the channel y to 0 = (5.3) h u * whch shows that the Reynolds stress vares lnearly wth y n ths regon. Ths result has been verfed expermentally (cf. fg 5.). Fgure 5.: Varaton of Reynolds stress across a channel (from Rechardt) It s observed that the lnear relatonshp (5.3) holds n almost 70% of the channel and s thus a surprsngly good approxmaton to (5.). Closer to the wall, however, another balance
exsts between the terms n (5.). In partcular, the vscous term must be mportant and t can be ncorporated by another scalng of (5.). Inner scalng: Scalng wth ν / u (and u ). The second scalng that may be done s such that the vscous term s weghted by unty. The proper length scale s then ν/ u and (5.) then becomes y u d(u/u ) 0 = (5.4) ν Re d(y u / ν) u It s customary to ntroduce the new wall dstance and the new velocty y u y = (5.5) ν = U U (5.6) u (5.4) can then be wrtten as u * * y du 0 = (5.4) Re Wth a large value of the Reynolds no., (5.4) reduces to du 0 = (5.7) u * The dstrbuton of uv close to the wall can been measured n detal wth LDV and the results are shown n fgure 5.3. It s observed that the uv-value s almost constant n a large porton of the wall layer. Fgure 5.3: Dstrbuton of Reynolds stress close to a wall. (from Karlsson et al.)
3 5. Turbulence producton Accordng to (4.4), the producton of turbulent knetc energy s gven by u U,. For the channel flow here, ths reduces to du P = (5.8) The mnus-sgn n (5.8) needs a comment. If we consder the lower half of fgure 5. and let a flud partcle move upwards, ts component s > 0. But, snce the partcle starts from a regon wth lower mean velocty t wll cause a negatve where t ends up. Thus, the product s negatve and, snce du > 0, there wll be a producton of turbulent energy due to ths moton. A smlar argument for a partcle movng downward also gves a postve producton. We can now use the results of the nner scalng to estmate where the maxmum turbulent producton s to be expected. Close to the wall, (5.) reduces to du 0 = u νu U = (5.9) * Multplyng (5.9) by U we get P = U = (u νu ) U (5.0) Seen as a functon of U, (5.0) can be dfferentated to gve optmum (maxmum) for P. We have Thus, at dp = u νu νu = u νu du U = u / ν (5.) there s maxmum n P whch becomes Snce u / ν P max = ( u / ν ) (5.) s the wall-value of U, (5.) shows that P max occurs where ths value s halved. The only thng that must be asserted wth ths analyss s that the maxmum pont actually les wthn the wall regon so that (5.9) s vald. Before ths can be asserted, some knowledge of the velocty profle s necessary.
4 5.3 Mean velocty profle The nformaton ganed from the two types of scalng leads to a way to estmate the turbuelent velocty profle. Frst, (5.7) ndcates that close to the wall both the mean velocty and the Reynolds stress are functons of y, only. Thus, U = f(y ) and / u = g(y ) (5.3) Ths s denoted a wall-law. In the outer regon, where (5.3) apples, there s no nformaton on U snce t s absent n the equaton. However, some nformaton can be obtaned from the energy equaton for the fluctuatons (4.4). In the outer regon, t reduces to du = d p q ε ρ (5.4) (5.3) shows that u s of the order whch we can expect also q and p /ρ to be. Thus we can wrte du u df = h dη Integratng ths relaton from the center of the channel gves U U 0 = F( η) u u (5.5) (5.6) Ths type of relaton s denoted a velocty defect law and s vald n the center porton of the channel. Closer to the wall, the wall law (5.3) apples and t s natural to ask what happens n the ntermedate regon. Ths can be gven some nput by consderng the velocty dervatve du/ derved from the two relatons (5.3) and (5.6). Law of the wall (5.3) du = u df u = ν df (5.7) Law of the wake (5.6) du df dη u df u = dη h dη = (5.8) In the overlap regon both relatons must hold. Multplyng both sdes wth df df y = η (5.9) dη y / u gves then Here, the left hand sde s a functon of y and the left hand sde a functon of η. Thus they must be a constant. Ths constant s denoted von Karmáns constant (κ) and s roughly 0.4. Integratng the two relatons then gve
5 f (y ) = ln y constant (5.30) κ and smlarly for F. The presence of a logarthmc velocty profle s a promnent character of turbulent boundary layers. In fgure 5.4 we show an example; note that the y -coordnate s logarthmc. Ths makes the wall regon exaggerated. Fgure 5.4: Turbulent velocty profle close to a wall. Scalng n wall varables; note the logarthmc dstance. Ths exaggerates the near wall regon. (From Karlsson et al.) In secton 5. we derved the condton for maxmum turbulence producton assumng ths to be n the near-wall regon and showed ths to be where U = u / ν. Expressed n wall varables ths may also be wrtten as du / = /. For the lnear profle closest to the wall, du / = and n the logarthmc regon du / =.44/y. Thus we conclude that the value / s obtaned n a pont between these two regons confrmng the assumpton that the maxmum turbulence producton s obtaned n the near wall regon. It turns out that maxmum turbulence producton occurs at y -5.
6 6. Kolmogorov mcroscales Turbulence s generated at farly large scales but due to non-lnear nteractons smaller and smaller scales are nvolved. At the smallest level, the energy s lost to vscous dsspaton and t was orgnally suggested that ths process was sotropc,.e. does not have a preferred orentaton. At ths level, the scales should be governed by vscosty (ν) and the dsspaton rate (ε) only. These quanttes have dmensons m /s and m /s 3, respectvely, so the desgn of length, tme and velocty scales based on these quanttes s straght forward and yeld the followng set: 3 Length scale: ( ) / 4 Tme scale: η = τ = ν / ε ( ν / ε) / Velocty scale: ϑ = ( νε) / 4 (6.a-c) These are called the Kolmogorov mcroscales. The length scale s the smallest scale n a turbulent flow. A couple of ponts related to these scales are of nterest. The Reynolds number based on the mcroscales s equal to unty. For turbulence n balance, the dsspaton rate must be equal to the energy transfer from the largest scales, where energy s put n. By knowng the energy nput n e.g. a strrng process t s thus possble to determne the mcroscales.
7 7. Turbulence structures Startng n the 960s, much research n turbulence has been drected to fnd structures and events that have some determnstc features. One such process s the so-called burstng process n the near-wall regon. Durng ths event most of the turbulence energy s produced. The sequence s depcted n fgure 9.. The startng pont s the appearance of longtudnal (low-speed) streaks on whch develops 3D and hghly unstable structures (harpn vortces). An underlyng dea s that the appearng dstorton of the mean profle produces an nflexon n the profle whch, from lnear stablty theory, s known to be unstable. However, later stablty research has establshed that other mechansms may be actve; one s the process of transent growth whch s able to explan the appearance of the longtudnal streaks. Fgure 9.: The burstng sequence; from Klne et al. (967)
8 References There s a multtude of books on turbulence of whch the lst below s ust a short example. Despte ts age, the book by Tennekes and Lumley s stll a qute useful opener to the subect and may be recommended. A more modern book, where turbulence modellng and smulatons are treated, s the book by Pope. Bradshaw P. 985, An Introducton to Turbulence and ts Measurement, Pergamon Press Durbn P. and Pettersson Ref B.A. 00, Statstcal Theory and modellng for Turbulent Flows, Wley. Landahl M.T. and Mollo-Chrstensen E. 986, Turbulence and random processes n flud mechancs, Cambrdge Unversty Press. Leseur M. 990, Turbulence n Fluds, Kluwer. McComb W.D. 990, The Physcs of Flud Turbulence, Oxford Scence Publcatons. Pope S.B. 000, Turbulent Flows, Cambrdge Unversty Press Rotta J.C. 97, Turbulente Strömungen, B.G. Teubner, Stuttgart Schlchtng H. 979, Boundary layer theory, 7 th edton, McGraw Hll Tennekes H. and Lumley J.L. 97, A frst course n turbulence, MIT Press. Townsend A.A. 980, The structure of turbulent shear flow, nd edton, Cambrdge Unversty Press. Wlcox D.C. 993, Turbulence Modelng for CFD, DCW Industres.