Computational Aeroacoustics

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1 Turbulence and Aeroacoustcs Research team of the Centre Acoustque École Centrale de Lyon & LMFA UMR CNRS Computatonal Aeroacoustcs... Lecture notes - Work n Progress - verson 1.1! Chrstophe Bally Arbus Toulouse, Aprl 2012 Laboratore de Mécanque des Fludes et d Acoustque École Centrale de Lyon & UMR CNRS

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3 3 Contents 1 Statstcal descrpton of turbulent flows Method of takng averages Reynolds averaged Naver-Stokes equatons The flud dynamcs equatons Averaged equatons Knetc energy budget of the mean flow Knetc energy budget of the fluctuatng feld Transport equaton of Reynolds stresses Budget of the turbulent knetc energy Turbulent vscosty: the Boussnesq model An example: the turbulent channel flow Turbulence models Mxng length models The k t ǫ turbulence model Transport equaton of the turbulent knetc energy Transport equaton of the dsspaton Transport equaton of energy Hgh-Reynolds-number form of the model Determnaton of the constant C ǫ Low-Reynolds-number form of the k t ǫ model Realsablty and unsteady smulatons The k t ǫ model for compressble flows Favre-averaged Naver-Stokes equaton Compressble form of the dsspaton rate Compressble form of the k t ǫ model The k t ω t turbulence model The Spalart and Almaras turbulence model Concludng remarks Appendces Transport equaton for the dsspaton Favre-averaged turbulent knetc energy equaton

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5 5 Chapter 1 Statstcal descrpton of turbulent flows Ths second chapter focuses on the statstcal approach of turbulence. On the one hand, t seeks to descrbe the evoluton of mean and turbulent felds, and on the other, to hghlght the transfer terms n between these two felds. Ths splttng, ntroduced n 1883 by Reynolds, s not unque, and nether s t the most satsfyng. Other flow decompostons are exposed n chapters?? and 2. Nevertheless, the Reynolds approach s stll to ths date the only one enablng smple statstcal assessments of flud dynamc equatons. 1.1 Method of takng averages Any varable occurng n a turbulent feld, such as velocty, pressure or temperature, s a random functon of poston x and tme t. The frst method to defne an average s thus based on a probablst approach. Ths means that the same experment s repeated a larger number of tmes provdng ndependent realzatons of the feld. The statstcal mean F(x, t) of a varable f(x, t) s then defned as N 1 F(x, t) = lm N N f () (x, t) =1 where f () s the -th realzaton. Ths average wll be the one employed throughout ths chapter because convenent when manpulatng equatons. It s however dffcult to be mplemented n concrete experments. Two other methods are therefore nvolved n specal cases. When the turbulent feld s statonary,.e. when tme t does not enter nto F, a temporal average s possble, leadng to 1 t0 +T F T (x) = lm f ( x, t ) dt T T t 0 where T s the observaton tme over only one realzaton. As an example, a sensor s placed at locaton x n a et operated by a constant power supply. The duraton T has to

6 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 6 be large to permt F T to approach F. The exact demonstraton reles on the hypothess of ergodcty, whch s developed at large by Lumley n hs book on stochastc tools n turbulence. 7 Practcally, a tangble requrement s that ponts at suffcently large separaton be uncorrelated. As a result, tme T has to be much greater than the largest turbulent tme scale encountered n the turbulent feld. Moreover ths tme T depends on the nature of the consdered varable f. Even ust for velocty, T dffers for measurements dealng wth u(x, t) and measurements nvolvng u 2 (x, t) or u 4 (x, t). When the turbulent feld s homogeneous,.e. when poston x does not enter nto F, a spatal average s possble leadng to 1 F V (t) = lm V V V f ( x, t ) dx As above, the volume V has to be large relatve to the turbulent spatal scales nvolved n the varable f whch s consdered. The spatal average s rather employed when runnng computatons whch generally provde a knowledge of the turbulent feld f at all ponts x, and moreover at all tmes t n the volume V. An example s the turbulence decay n a large box after the ntal exctaton s turned off. In the Reynolds decomposton, any physcal varable f can be splt nto ts mean part F and ts fluctuaton part f,.e. f = F+ f wth f = 0. As a conventon n what follows, captal letters are employed as much as possble when desgnatng mean physcal quanttes, n addton to the bar related to the mean operator. It s mportant to notce that the mean part represents what s reasonably calculable, or at least the determnstc part, as opposed to the random or ncoherent fluctuatons whch wll be ether modeled or measured. Some mportant propertes of the averagng operator are now lsted. For two random varables f = f (x, t) and g = g(x, t) and a constant α, one easly establshes, () () f + g = F+G α f = αf () Fg = F G (v) (v) f = F f dt = f = F x x Fdt f dx = Fdx Moreover, an mportant practcal rule whch wll often be employed later on, concerns the product of two varables f and g, f g = F Ḡ+ f g (1.1)

7 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 7 The alert reader may have notced that the presence of a nonlnear term n the equatons does not allow to express the product as a functon of F and Ḡ only, but ntroduces a new second-moment term f g. 1.2 Reynolds averaged Naver-Stokes equatons Pror to consderng the mplcatons for turbulence of the Reynolds decomposton n a mean part and a fluctuatng part, the flud dynamcs equatons must be clearly stated. The reader can refer to classcal textbooks n order to revew the dfferent local forms of these equatons The flud dynamcs equatons The flud dynamcs equatons take the followng forms whether they concern mass, momentum or energy conservaton, ρ (ρu) + (ρu) = 0 (1.2) + (ρu u) = p+ τ (1.3) (ρh) + (ρhu) = q+ p + u p+ u : τ (1.4) where ρ, u, p and h desgnate densty, velocty, pressure and enthalpy, respectvely. No volume force such as gravty s here consdered. The vscous stress tensor τ s expressed for a Newtonan flud as, τ = µ [ u+( u) t] + λ 2 u = µ [ u+( u) t 2 3 ( u) I] + µ b ( u) I (1.5) where µ, λ 2 and µ b = λ 2 + 2µ/3 desgnate the dynamc vscosty, the second vscosty and the bulk vscosty, respectvely. To make clear the compact tensoral notatons, let us explct the convectve term n (1.3) and the dsspaton term n (1.4), [ (ρu u)] = (ρu u ) u : τ = u τ By reportng expresson (1.5) nto the Naver-Stokes equaton (1.3), the thermodynamc pressure s now dstnct from the effectve or mechancal pressure gven by p µ b u. The classcal approach conssts n takng µ b = 0, accordng to the Stokes hypothess, and the vscous stress tensor s then smply expressed as, τ = µ [ u+( u) t] 2 µ( u) I (1.6) 3

8 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 8 However, except for a monoatomc gas, 3 ths assumpton s expermentally not verfed 4, 5 when the determnaton of µ b s carred out by measurng sound absorpton. It should be noted that ths dscusson only occurs for compressble flows. Expresson (1.5) drectly provdes the expresson of τ for an ncompressble flow, thus satsfyng u = 0. In addton, f the flud s not Newtonan, n such case as water flows contanng polymers or bubbles for drag reducton, other consttutve equatons have to be used. The flud s presumed to act as a perfect gas,.e. p = ρrt, and the conductve heat flux s supposedly descrbed by the Fourer law q = λ T. In these relatons, r s the perfect gas constant of the studed flud and λ ts thermal conductvty. The use of the enthalpy varable h to wrte the conservaton of energy enables to obtan the temperature T drectly by usng the relaton dh = c p dt, where c p s the specfc heat for a constant pressure. The energy conservaton equaton can be wrtten n many other forms such as, (ρe) (ρe t ) + (ρeu) = q p u+ u : τ (1.7) + (ρe t u) = [ q p u+τ u ] (1.8) where e = c v T and e t = e+u 2 /2 are the nternal energy and the total energy respectvely, wth ρe t = p/(γ 1)+ρu 2 /2 for a perfect gas. In addton, the followng rule obtaned by usng equaton (1.2) for any varable f, wll also be frequently employed, (ρ f) ( ) f + (ρ f u) = ρ + u f (1.9) In order to smplfy algebra n ths statstcal study, the flow s assumed ncompressble,.e. u = 0. Compressble flows, whch are more complex, wll be brefly descrbed n chapter 2 as appled to statstcal modellng, see secton Therefore, the followng wll now focus essentally on the Naver-Stokes equaton. The case of a dlatable flud ncludng boyancy effects, that s to say for whch ρ = ρ(t), can be treated as an exercse n order to practce the course. Some results of contnuum mechancs concernng the vscous stress tensor are brefly remnded so as to conclude ths part. The velocty gradent tensor can be decomposed n the sum of a symmetrcal part, the deformaton or rate-of-stran tensor e, and an antsymmetrcal part, the vortcty tensor ω, u = u ) + 1 x 2 ( u }{{} e ( u u ) x }{{} ω

9 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 9 A physcal nterpretaton of the velocty gradent tensor s gven n secton??. The vortcty tensor s lnked to the vortcty vector ω = u through the two relatons below, ω = 1 2 ǫ kω k ω k = 1 2 ǫ kω Moreover, t s convenent to splt the deformaton tensor e nto ts sotropc and devatorc parts by wrttng, e = e + ed = 1 3 e kkδ + (e 13 ) e kkδ (1.10) The sotropc or sphercal part e expresses the volume change snce tr(e ) = e δ = 1 3 e kk(1+1+1) = u k x k = u whereas the devatorc part e d s such as tr(ed ) = 0, and s also symmetrc. From now on, the devatorc part of the velocty gradent tensor wll be denoted s, s e d 1 2 ( u + u ) 1 x 3 ( u)δ (1.11) As clear from the consttutve relaton (1.6), the vscous stress tensor s lnearly lnked to ths tensor s by, τ = 2µs (1.12) for a Newtonan flud satsfyng Stokes s hypothess. At last, the dsspaton term u : τ appearng n the energy conservaton equaton (1.4) s always postve, and ths result s straghtforward by notng that the vscous tensor s symmetrc, u : τ = u τ = u x τ = e τ = s τ and thus u : τ = 2µs 2 0. Ths dsspaton term represent the quantty of mechancal energy transformed nto thermal energy by vscous effects, and t s necessarly postve because of the second thermodynamc prncple. By developng ths expresson, one has ( u : τ = 2µ e 2 1 ) 3 e2 kk = 2µ [e 2 13 ] ( u)2 A fnal remark to observe that there s no dstncton between e and e d s for an ncompressble flow of a Newtonan flud.

10 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally Averaged equatons To derve averaged equatons, the Reynolds decomposton s appled to velocty u = Ū + u and pressure p = P+ p, as well as to the vscous tensor τ = τ + τ. The flow s consdered ncompressble and densty s supposed to be constant. However, we wll try to keep general expressons as far as possble n the developments. Frst, from the ncompressblty condton, one obtans by averagng ths equaton, (Ū + u ) = 0 x Ū x = 0 and then, u x = 0 by substracton of the two frst equatons. Therefore, the nstantaneous fluctuatng velocty feld s ncompressble. Ths property wll be employed very often later on. In the same way, from the mass conservaton equaton (1.2) [ ρ(ū + u )] = 0 and from the momentum conservaton equaton (1.3) [ ρ(ū + u )] + [ ρ(ū + u x )(Ū + u )] = ( P+ p ) + ( τ + τ ) (1.13) x the averagng operaton leads to the Reynolds averaged equatons, (ρū ) ( ρū ) = 0 (1.14) + ( ρū Ū ) = P + [ τ ρu ] x x u (1.15) The new unknown ρu u ssued from the convectve nonlnear term n (1.3), s called the Reynolds stress tensor. Generally ths term s larger than the mean vscous stress tensor except for wall bounded flows, where vscosty effects become preponderant close to the wall. The no-slp boundary condton ndeed requres that u u 0 at the wall. Due to the appearance of the new unknown ρu u, the set of equatons gvng the mean velocty feld Ū cannot be solved, and there are more unknowns than equatons. In short, ths s called a closure problem. Two strateges can then be pursued to get over the dffculty. The frst one s due to Boussnesq, and conssts n modellng the Reynolds

11 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 11 tensor ρu u. Detals are gven n paragraph 1.5. The second one conssts n wrtng an equaton for ths unknow tensor,.e. a transport equaton for the Reynolds stress tensor ρu u. However, as one can guess, ths new equaton wll ntroduce thrd-moment terms n velocty fluctuatons, as shown later wth the trple correlaton term n equaton (1.19) for nstance, whch stll requres a closure. The mean knetc energy of the fluctuatng feld k t, also called the turbulent knetc energy, k t u u 2 = u2 1 + u 2 2+ u2 3 2 can be ntroduced through the contracton of the Reynolds stress tensor (1.16) ρu u δ = 2ρk t It appears therefore nstructve to carry on the presentaton by consderng the knetc energy budget of the mean flow as well as the knetc energy budget of the fluctuatng feld. 1.3 Knetc energy budget of the mean flow The equaton whch descrbes the knetc energy ρū 2 /2 of the mean flow can be obtaned by multplyng the averaged Naver-Stokes equaton (1.15) n the -th drecton by the mean velocty Ū component, Ū { (ρū ) + ( ) ρū Ū = P + [ ] } τ ρu x x u From the mass conservaton equaton (1.14), a rule smlar to relaton (1.9) can easly be derved for an arbtrarly mean flow quantty F, (ρ F) + (ρū x F) = ρ F + ρū F d x dt (ρ F) (1.17) and s here used to rearrange the left-hand sde of the prevous equaton n a conservatve form, { (ρū ) Ū + (ρū } Ū ) Ū = ρū + ρū Ū Ū ) = ρ (Ū2 2 = ( ρū ρū ) + (Ū2 2 ( Ū ρū 2 2 ) )

12 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 12 The other terms can be rewrtten thanks to the ncompressblty condton for the mean flow, and as a result, the knetc energy budget can be recasted n the followng form, ( ρū 2 ) + ( ρū 2 ) Ū 2 2 Ū }{{} (a) = ρu u (Ū P) x }{{ } (c) τ Ū }{{} (b) + ) (Ū τ ) (Ū ρu x u }{{}}{{} (d) (e) (1.18) Intentonally, let us frst consder the last three terms (c), (d) and (e). They respectvely represent the power of pressure forces, vscous forces and Reynolds stress forces. It s mportant to observe that these terms are zero for an homogeneous mean flow. Indeed, they represent pure dffuson transfers and are actually wrtten as a flux dvergence form. As a result, the most mportant terms are term (a) whch represent a transfer between the mean flow and the fluctuatng flow, and the term (b) whch represents the vscous dsspaton of the mean flow. In order to correctly understand the transfer term between the mean feld and the turbulent feld, t s useful to wrte down the equaton governng the turbulent knetc energy of the fluctuatng feld. 1.4 Knetc energy budget of the fluctuatng feld The turbulent knetc energy k t s defned by relaton (1.16) and ts governng equaton can be obtaned by wrtng the general equaton for the Reynolds stress tensor, and then contractng ndces, the mnus sgn of ρu u beng left off Transport equaton of Reynolds stresses The Naver-Stokes equaton governng the component u s obtaned by subtractng to the ntal equaton (1.13) ts averaged equaton (1.15), whch leads to, ( ρu ) + [ ( ρ u x Ū k + Ū u k + )] p u u k = + ( ρu k x x u k + ) τ k k and ths equaton s denoted(σ ). The dummy summaton ndex s now noted k to avod any confuson wth the ndex consdered n the Reynolds stress component ρu u. Smlarly, we can obtan the equaton governng the component u, denoted (Σ ). The transport equaton s then obtaned by formng the u Σ + u Σ and by applyng the average

13 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 13 operator. More specfcally, the three followng groups are frst rearranged, u (ρu x Ūk)+u (ρu k Ūk) = (ρu u k Ūk) k u (ρū x u k )+u (ρū k u k ) = Ū ρu u k + ρu Ū k x u k k x k u (ρu x u k )+u (ρu k x u k ) = (ρu k x u u k ) k then applyng the averagng operator and usng the ncompressblty condton, one gets (ρu u ) + x k ( ρu u Ū k ) = P + T + Π + D ρǫ (1.19) where the dfferent terms n the rght-hand sde are defned as, ( ) P = ρu Ū u k + ρu Ū x u k k x k T = (ρu x u u k ) k ( Π = u p + u p ) x D = ( u τ k + ) u τ k k ρǫ = τ k u + τ u x k k x k An nterpretaton of ths budget s presented n secton 1.6 for the plane channel flow. Ths equaton can be numercally solved, as an alternatve to the modelng of Reynolds stress tensor ρu u. However, and as already mentoned n secton 1.2.2, ths equaton contans a trple correlaton term T, whch requres a closure Budget of the turbulent knetc energy The transport equaton for k t s drectly deduced from (1.19) by contractng the ndces and keepng the half-sum, namely d(ρk t )/ dt = (P +T + Π +D ρǫ )/2. It yelds, d(ρk t ) dt = ρu u k Ū x k } {{ } (a) τ k u 1 x k 2 }{{} (b) x k ρu u u k } {{ } (c) u p x }{{} (d) Moreover, the pressure term (d) can be rearranged as follows, + u x τ k } k {{} (e) (1.20) Π = u p x = x u p p u x = x u p (1.21)

14 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 14 and the three terms (c), (d) and (e) can then be rewrtten as the dvergence of a flux vector J = T + Π+D to show that there are transport terms. The trple velocty correlaton T represents the dffuson of the turbulent knetc energy by the fluctuatng velocty, and the terms Π and D can be assocated wth dffuson effects through the pressure and the vscous stresses. For an homogeneous turbulence, J 0, the convecton of the turbulent knetc energy s then only balanced by, d(ρk t ) dt = ρu u Ū τ u (1.22) namely the transfer term P between the mean feld and the turbulent feld on the one hand, see comments of equaton (1.18), and the vscous dsspaton per unt of mass of the turbulent knetc energy on the other hand, denoted ρǫ. Ths quantty s always postve, ρǫ = τ u = 2µs 2 x 0 (1.23) A smplfed expresson of the dsspaton ǫ, whch s often used n practce, s brefly ntroduced n paragraph 1.26, and dscussed later n secton??. By comparng equatons (1.18) and (1.20), the energy exchanged between the mean feld and the turbulent feld can only be carred on through the term P. Ths term s generally postve, correspondng to an energy supply from the mean feld to the turbulent one, and s called the producton term by abuse of language. A smple heurstc argument shows that ths s true for a sheared flow. Let us take for example the case of a boundary layer where dū 1 /dx 2 > 0, and magne that a flud partcle gets through the grey lne from bottom to top. x 2 Ū1 dū 1 dx 2 > 0 { u 2 > 0 u 1 < 0 u 1 u 2 < 0 x 1 P = ρu 1 dū 1 u 2 > 0 dx 2 Thus u 2 > 0, and ths partcle fnds tself n the mdst of a materal anmated by a greater mean velocty, thus havng a defct n longtudnal velocty, leadng to u 1 < 0 and < 0. The same reasonng can be appled to a flud partcle gong through to the grey u 1 u 2 lne from top to bottom, or to other flows wth a negatve mean velocty gradent such as ets. Fnally, the P producton term appears to be rather postve. The prncpal terms that have been hghlghted n the analyss of the knetc energy budgets for the mean and the turbulent felds are resumed n fgure 1.1. The rule (1.1) s

15 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 15 mean feld P = ρu u Ū fluctuatng feld Ū τ ρǫ = τ x u heat (nternal energy) Fgure 1.1 : Smplfed dagram of energy transfers between the mean and turbulent felds, when neglectng all dffuson terms. The term P s generally a producton term for the fluctuantng feld. agan llustrated here wth the dsspaton nduced by vscous effects, τ u = τ Ū + τ u = τ Ū + ρǫ whch s splt nto two contrbutons n the budget of the knetc energy of the mean flow (1.18) and of the turbulent flow (1.20). These two terms are producton terms n the transport equaton for the mean temperature, as shown later n paragraph Turbulent vscosty: the Boussnesq model The most famous and wdely used closure for the Reynolds stress tensor s based on the concept of a turbulent vscosty, ntroduced by Boussnesq. Ths hypothess conssts n expressng the Reynolds stress tensor by analogy wth the vscous stress τ. It s therefore assumed that, ρu u = 2µ t S 2 3 ρk tδ }{{} (a) (1.24) where µ t s called the dynamc turbulent vscosty or the eddy vscosty. The sotropc term (a) s necessary to satsfy the condton ρu u = 2ρk t by contracton of the ndces, that s for = and takng the summaton over. The turbulent vscosty seems on frst approach as a functon of the flow µ t = µ t (x, t), as opposed to the molecular vscosty whch s an ntrnsc physcal property of the flud. A great challenge n turbulence s to derve a comprehensve formulaton for µ t, and examples wll be gven n the next chapters. Closure based on a turbulent vscosty model s presented here because t s the bass of most turbulent models used n numercal smulatons of the Reynolds averaged Naver-Stokes equatons. Chapter 2 treats n detal ths

16 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 16 ssue, but t seems mportant to already notce some consequences of such modelng for the Reynolds stress tensor snce the reference to a turbulent vscosty s made throughout the textbook. Note also that approaches followng n large eddy smulaton are defntvely dfferent, and wll be dscussed n chapter??. Replacng the sx unknowns of the Reynolds stress tensor, three normal stresses ρu 2 and three shear stresses ρu u for =, by a unque unknown functon µ t nduces a one-way energy transfer from the mean flow feld towards the fluctuatng feld. Indeed, the term P takes the form, P = ρu Ū u = ( 2µ t S 2 3 ρk tδ ) Ū = 2µ t S 2 0 as µ t > 0, whch s generally assumed n turbulent models. Moreoever, n the case of a mean shear flow, wth Ū 1 = Ū 1 (x 2 ) and Ū 2 = Ū 3 = 0, the model mposes the Reynolds stress to be null for an extremum of the mean velocty, as the total stran τ t seen by the flud s gven by, τ t = τ 12 ρu 1 u 2 = µdū 1 dx 2 + µ t dū 1 dx 2 = (µ+µ t ) dū 1 dx 2 (1.25) There s no dffculty for symmetrcal mean flow felds n 2-D and for axsymmetrcal mean felds n 3-D. Fgure 1.2 reproduces for example the Ū z et u r u z profles measured n a round et by Hussen, Capp & George.? Notce that u ru z s null on the axs and that the producton term P s postve. However, at least two famous counterexamples exst regardng asymmetrcal mean flows. The frst s the wall et, the expermental profles of whch appear n fgure 1.3. Clearly ρu 1 u 2 and dū 1 /dx 2 are null for dfferent x 2 postons. The second example concerns a channel flow wth a smooth wall on one sde and a rough wall on the other, studed by Hanalć & Launder.?. In both these cases, the locaton of the pont where ρu 1 u 2 = 0 s dsplaced further than the pont where dū 1 /dx 2 = 0 by the turbulent regon whch possesses the most ntense velocty fluctuatons, that s the et compared to the wall n the frst example, and the rough sde compared to the smooth one n the second example. 1.6 An example: the turbulent channel flow In order to conclude ths chapter and llustrate the turbulent knetc energy assessment that has been establshed n a general case, the followng wll focus on a fully developed statonary turbulent flow between two flat parallel walls, see fgure 1.4. Ths turbulent channel flow s often used as a reference for drect numercal smulaton, such for example the calculatons of Kmet al.,? Mansour et al.,? Moser et al.? or Hoyas & Jménez,?

17 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 17 Ūz/Ūaxs r/(z z 0 ) u ru z/ū2 axs r/(z z 0 ) Fgure 1.2 : Subsonc round et at Mach M = 0.16 and Reynolds number Re D = Radal profles of the mean velocty Ū z /Ū axs and of the u r u z /Ū2 axs term aganst the radal dstance r/(z z 0 ) n the fully developed regon of the et, at a dstance larger than 25D from the nozzle ext. Data from Hussen, Capp & George,? see also chapter??. wall et x 2 /η 1/ τ t /ρū 2 1max 1.0 Ū 1 /Ū 1max 0.5 Ū 1max τ w Fgure 1.3 : Wall et, Reynolds number at the nozzle ext Re Mean velocty Ū 1 and total shear stress τ t = µdū 1 /dx 2 ρu 1 u 2 profles as a functon of the dstance to the wall, x 2 /η 1/2, where η 1/2 s the dstance for whch Ū 1 = 0.5 Ū 1max, and τ w s the wall shear stress. By movng away from the wall, one frst fnds the pont where τ t = 0, then the one where dū 1 /dx 2 = 0. Data from Talland & Matheu.?

18 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 18 x 2 2h Ū 1 (x 2 ) Fgure 1.4 : Sketch of the geometry of a turbulent channel flow. x 1 because t s expermentally well documented, see Laufer,? Comte-Bellot? or Johansson & Alfredsson.? A channel flow s found fully developed n experments when the consdered secton s far enough from the entrance, x 1 120h, where h s the half-wdth of the channel. All mean quanttes are then statonary and do not depend on x 1 except for the mean pressure. The mean velocty has the form Ū 1 (x 2 ) and Ū 2 = Ū 3 0. Moreover, the flow s statstcally ndependent from x 3, meanng that x 3 s an homogenous drecton for the turbulent feld. The orentaton of the x 3 axs has no nfluence, whch leads to u 1 u 3 = u 2 u 3 = 0 and p u 3 = 0. Therefore, x 3 s a prncpal drecton for the tensors, and the Reynolds tensor smplfes as, ρu u = ρ u 1 2 u 1 u 2 0 u 1 u 2 u u 3 2 Equaton (1.19) whch governs the velocty correlatons u u, s wrtten for the = case, where the ndce s here replaced by the greek letter α so as to ndcate that two repeated ndces are not summed, d ( ) dt ρu 2 α = Pαα + T αα + Π αα +D αα ρǫ αα The last two terms are often rearranged for ncompressble flows. By notng that, D αα = µ 2 u 2 α + 2µ 2 u k u α x k x k x k x α ρǫ αα = 2µ u α x k u α x k + 2µ 2 u k u α x k x α the vscous dffuson and dsspaton terms are thus combned as follows, D αα ρǫ αα = µ 2 u 2 α x k x k 2µ u α x k u α x k D h αα ρǫ h αα (1.26)

19 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 19 The new dsspaton term ǫαα h s not the correct thermodynamcally expresson of the dsspaton as remnded by Corrsn 2. Ths term s sometmes called the sotropc dsspaton, even though only homogenety s requred to have the equalty between ǫ αα and ǫ h αα. Ths approxmaton of the exact dsspaton s classcaly used n turbulence models snce the transport equaton of ǫ h αα s much smpler, see paragraph??. At least for the case of the channel flow, the dfference between the two expressons remans reasonably small.? The transport equaton for the u α 2 component can then be recasted as, d dt (ρu2 α ) = 2ρu αu Ū α k ρu x k x αu αu k p 2u α + µ 2 u 2 α 2µ u α u α k x α x k x k x k x k = P αα +T αα + Π αα +D h αα ρǫ h αα (1.27) In ths equaton,p αα s the producton, T αα s the turbulent dffuson, Π αα s the velocty pressure-gradent correlaton, D h αα s the vscous dffuson and ǫ h αα s the dsspaton. The term Π αα can also be decomposed as the sum of a pressure dffuson term and a pressure velocty-gradent correlaton term, n a smlar way as n expresson (1.21), ( Π αα = 2 u x α p + p u α α x α to hghlght the specfc role of the fluctuatng pressure. ) = Π d αα + Π s αα (1.28) Equaton (1.27) s now partcularzed for the case of a fully developed turbulent channel flow. For each normal stress composent u α 2, t thus yelds 0 = 2u 1 dū 1 u 2 + d ( ) u 2 dx 2 dx 1 u 2 + νdu2 1 2 dx 2 0 = 0 + d dx 2 0 = 0 + d dx 2 ( u 2 ( u 2 2 u 2 + νdu2 2 dx 2 ρ u 2 p u 2 + νdu2 dx 2 ) + 2 ρ p u 1 2ν u 1 u 1 x 1 x k x k ) + 2 ρ p u 2 x 2 2ν u 2 x k u 2 x k + 2 ρ p u 3 x 3 2ν u 3 x k u 3 x k (1.29) In these equatons, the terms are organzed n order to have successvely the producton term, the dffuson and pressure - velocty correlaton terms, the pressure - velocty gradent term and fnally the vscous dsspaton term. It s essental to notce that only the longtudnal component u 1 receves energy from the mean flow through the producton term P 11. The transverse velocty components u 2 and u 3 can therefore only receve energy from u 1 through pressure fluctuatons. The latter beng scalar, there s no prvleged drecton, whch explans why both u 2 and u 3 can receve energy. Pressure therefore has a redstrbuton role relatvely to the turbulent knetc energy between the three components of velocty.

20 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 20 Fnally, the half-sum of all three equatons (1.29) provdes the turbulent knetc energy budget for the channel flow, 0 = ρu 1 dū 1 u 2 1 d ρu dx 2 2 dx u u 2 + k t µd2 2 dx2 2 d u dx 2 p µ u u 2 x k x k (1.30) 0 = P + T + D h + Π d ρǫ h Notce that pressure has an nfluence only through the transport term Π d = Π d /2 = Πd 22 /2 n (1.30). One has ndeed, Π s = Π s /2 = 0, and so t s expected that Πs 11 = p x1 u 1 s negatve whereas Π22 s and Πs 33 are expected to be rather postve terms. These conectures are confrmed by experment, and results wll be presented n detal n chapter??. It s however nterestng to notce here that u 1 2 s greater than u2 2 and u2 3 n rms snce u 1 2.5u τ, u 2 u τ and u 3 1.3u τ, where u τ s the frcton velocty defned by u τ = τ w /ρ and τ w s the shear stress at the wall, namely τ w = τ 12 (x 2 = 0). Drect numercal smulaton also complete these analyses, specfcally for correlaton terms nvolvng pressure, whch n general cannot be measured. The budgets of u 1 2, u2 2, u 3 2 and u 1 u 2 computed by Hoyas & Jménez?,? are shown n fgure 1.5. For u 1 2, the producton term s found postve P 11 > 0. At the wall, the dsspaton term s counterbalanced by the vscous dffuson. Moreover, the pressure - velocty correlaton gradent term s negatve as expected, Π11 s < 0. The orders of magntude of the terms nvolved n the budget of u 2 2, u2 3 and u 1 u 2 are smaller. Concernng the analyss of the role of pressure, Π22 s s found postve as soon as x+ 2 > 10, Πs 33 > 0 and Πd 22 + Πs 22 > 0. One also knows that Π11 d = Πd 33 = 0. The producton term P 12 n the budget of u 1 u 2 s found negatve, and accordng to (1.19), t can be wrtten as, P 12 = u 2 2 dū 1 < 0 dx 2 but ths matches a postve producton on the Reynolds stress u 1 u 2. Fnally, the turbulent knetc energy budget s represented n fgure 1.6, stll accordng to Hoyas & Jménez et al s calculatons.?,? The role of pressure through the dffuson term s found very weak, even f pressure fluctuatons are at the orgn of the velocty components u 2 and u 3. It can be also observed that for x+ 2 30, there s a quas-equlbrum between dsspaton and producton. However ths balance cannot extend to the center of the channel, where clearly the dsspaton stll exsts whereas the turbulent knetc energy producton tends to zero. In the budget of k t shown n fgure 1.7 as functon of the dstance x 2 /h to the wall, the equlbrum subssts up to x 2 /h 0.5. Ths nterval over whch producton and dsspaton are balanced, s characterstc of a wall flow, and ths mportant result s developed n chapter??.

21 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally T + 11 D h+ 11 ǫ h+ 11 P + 11 u 2 1 Π s D h+ 22 Π d+ 22 T + 22 Π s+ 22 ǫ h+ 22 u x x D h+ 33 T + 33 Π s+ 33 ǫ h+ 33 u T + 12 D h+ 12 Π + 12 ǫ h+ 12 P + 12 u 1 u x x + 2 Fgure 1.5 : Budgets of u 1 2, u2 2, u2 3 and u 1 u 2 for the channel flow at Reynolds number Re + = hu τ /ν = 2003, computed by Hoyas & Jmenez.?,? See equatons (1.27) and (1.29) for the notatons. Profles are plotted as a functon of the wall varable x + 2 = x 2 u τ /ν, where u τ s the frcton velocty, and all terms are made dmensonless wth u τ, ν and ρ.

22 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally P Π d+ 0.1 D + h T + ǫ + h x + 2 Fgure 1.6 : Turbulent knetc energy budget (1.30) for a channel flow at Reynolds number Re + = hu τ /ν = 2003, computed by Hoyas & Jmenez.?,? plotted as a functon of the wall varable x + 2 = x 2u τ /ν. Profles are P ǫ + h x 2 /h Fgure 1.7 : Turbulent knetc energy budget (1.30) for a channel flow at Reynolds number Re + = 2003, computed by Hoyas & Jmenez.?,? Profles are plotted as a functon of x 2 /h, and all terms are made dmensonless usng u τ, ν and ρ.

23 23 Chapter 2 Turbulence models Ths chapter focuses on turbulent models, whch are wdely used n a large verety of engneer studes, ncludng atmosphere dynamcs and weather forecastng. The obectve s to determne the turbulent mean flow from the averaged equatons (1.14) - (1.15) establshed n chapter 1. A turbulence model s however requred to express the unknown Reynolds stress tensor. The most popular approaches are based on Boussnesq s hypothess (1.24) wth the ntroducton of a turbulent vscosty. A survey of these so-called eddyvscosty models s presented n ths chapter. For clarty, the Reynolds averaged Naver-Stokes equaton s frst recalled, (ρū ) + ( ) ρū Ū = P + ( τ x ρu u ) (2.1) where Ū s the mean velocty, τ = 2µ S s the vscous stress and ρu u the Reynolds stress. The mean flow s assumed to be ncompressble. Secton 2.3 s devoted to the specfc treatment of the compressble case. Followng Boussnesq s hypothess, see secton 1.5, the unknown Reynolds stress tensor s expressed as, ρu u = 2µ t S 2 3 ρk tδ (2.2) where µ t = ρν t s the dynamc turbulent vscosty and k t the turbulent knetc energy. By notng the smlarty played by µ and µ t, equaton (2.1) can be rearranged as follows, (ρū ) + ( ) ρū Ū = ( 2 P+ x 3 ρk ) [ ] t + 2(µ+µt ) S (2.3) The only remanng unknown s now the turbulent vscosty. Turbulence models dffer n the way to determne ts expresson. A classcal way conssts n wrtng the turbulent vscosty as the product of a velocty scale by a length scale, and to solve two addtonal transport equatons to determne these scales. The standard k t ǫ model or k t ω t models fall nto ths famly for nstance. An alternatve approach s to drectly wrte a transport equaton on the turbulent vscosty, as n Spalart-Allmaras model. These models and some extensons are ntroduced n what follows.

24 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally Mxng length models In mxng length models ntroduced by Prandtl, the turbulent vscosty s wrtten as the product of a velocty scale u by a length scale l m by analogy wth the molecular vscosty. It s also assumed that the tme scale assocated wth the mean shear flow Ū 1 / x 2 1 s proportonal to the tme scale bult from u and l m, namely u l m dū 1 dx 2 From the relaton ν t u l m, the expresson of the turbulent vscosty can thus be rearranged as follows, ν t = lm 2 Ū 1 x (2.4) 2 where the absolute value of the mean flow gradent s taken to ensure a postve value of ν t. The Reynolds shear stress ρu 1 u 2 s then determned from Boussnesq s hypothess (2.2), u 1 u 2 = l2 Ū 1 Ū 1 m x (2.5) 2 x 2 Ths model has already been appled n secton?? to calculate the mean velocty profle of a turbulent boundary layer. It has been shown that the mxng length must be equal to l m = κx 2 n the logarthmc regon, and that t must be reduced close to the wall by usng a dampng functon as proposed by Van Drest, 67 see expresson??. In a plane mxng layer, the mxng length s l m 0.07 δ(x 1 ) where δ s the local thckness of the layer. For other free shear flows, as n ets or wakes, the length scale l m s also lnked to the shear layer thckness. For nstance, one approxmately has l m 0.09 δ(z) n a round et, where δ s the wdth of the annular mxng layer. Further examples can be found n the book by Wlcox.? Expresson (2.4) can be generalzed to an arbtrarly mean shear flow by wrttng ν t = l 2 m S, where S s defned by S 2 S S The magntude of the mean vortcty Ω can also be used to defne the eddy-vscosty, that s ν t = l 2 m Ω wth Ω 2 2 Ω Ω, as n the Baldwn-Lomax model 9 used n aerodynamcs. 34, 70 Mxng length models are easy to mplement snce the turbulent vscosty s smply determned by an algebrac relaton. They are however ncomplete snce the user must prescrbe the sutable evoluton of l m (x, t). Ths drawback can be removed by solvng addtonal transport equatons n order to predct the two scales u and l m for all ponts of the computatonal doman.

25 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally The k t ǫ turbulence model One of the most popular two-equaton models s certanly the k t ǫ turbulence model, ntroduced by Jones & Launder 31 and formulated by Launder & Spaldng 38 n ts standard form. In ths model, the expresson of the turbulent vscosty s bult by takng for the velocty scale the square root of the turbulent knetc energy, u k t, whle the length scale s calculated from the dsspaton rate of the turbulent knetc energy. By notng that for a homogeneous turbulence, ǫ h u 3 /L f where u 2 = (2/3)k t, an expresson of the turbulent vscosty can then be derved by chosng l m proportonal to the ntegral length scale L f. It yelds, ν t u l m k 1/2 t k 3/2 t ǫ 1 k 2 t h and hence, ν t C µ (2.6) ǫ h where C µ s a constant of the model, whch needs to be determned. The estmaton of ν t also requres the knowledge of k t and ǫ h. The next sectons are devoted to derve transport equatons for these two quanttes Transport equaton of the turbulent knetc energy The budget of the turbulent knetc energy k t has already been establshed n chapter 1, see equaton (1.20). For the sake of clarty, ths equaton s rewrtten below, d(ρk t ) dt = ρu u k u k Ū τ 1 ρu x k x }{{ k 2 x u u k u } k x p + u x τ k } k {{} (a) (b) (2.7) The dsspaton term (a) and the dffusve transport term (b) can be rearranged as follows, u x τ k τ k k u = x k x k ( µ k ) t + µ u u = ( µ k ) t ρǫ x k x k x k x k x h (2.8) k snce τ k = 2µs k, and also by usng the ncompressblty condton x k u k = 0. The homogeneous part of the dsspaton ǫ h has already been ntroduced when studyng the turbulent channel flow, see expresson 1.26 and assocated comments. It represents a good approxmaton of the total dsspaton rate ǫ for an nhomogeneous turbulent flow at hgh Reynolds number, 58 refer also to secton (??). The transport equaton for the turbulent knetc energy (2.7) thus becomes, d(ρk t ) dt = ρu u k Ū ρǫ x h + ( µ k ) t 1 k x k x k 2 x k ρu u u k x u p (2.9)

26 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 26 Only the two last terms need to be modelled n ths expresson. Turbulent transport terms such as the trple-correlaton term n equaton (2.9) are usually closed by a gradentdffuson model. For an arbtrary quantty ϕ, t s assumed that ϕu k = a ϕ t x k where a t s a dffuson coeffcent assocated wth ϕ. Wth ϕ = u u /2 n the present case, and by smply ncludng the pressure-velocty correlaton n ths gradent model, t yelds ρ 2 u u u k p u k = µ t σ kt k t x k where σ kt s a turbulent Prandtl number, by analogy wth the molecular transport, and ν t /σ t s a turbulent dffusvty. The pressure-velocty correlaton cannot be drectly measured but can be computed by numercal smulatons. Its contrbuton s found small n channel flows or n et flows, see llustratons n paragraph 1.6 and **TKEet** respectvely, whch can ustfy ths approxmaton. Fnally the followng transport equaton s retaned for the turbulent knetc energy k t d(ρk t ) dt = [( µ+ µ ) ] t kt +P ρǫ x k σ kt x h (2.10) k where P = ρu u k Ū / x k s the producton term Transport equaton of the dsspaton Dervaton of the transport equaton of the dsspaton ǫ h for an ncompressble homogeneous flow requres much more efforts than for the turbulent knetc energy. Startng from the Naver-Stokes equaton on the fluctuatng velocty establshed n chapter 1 (see paragraph 1.4.1), ( ρu ) + [ ( ρ u x Ū k + Ū u k + )] p u u k = + ( ρu k x x u k + ) τ k k (2.11) the followng equaton s then bult ν u (2.11)

27 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 27 and averaged to form an equaton on ρǫ h = µ x u x u, see equaton (2.8). Detals are reported n appendx 2.7. It s shown that the transport equaton can be wrtten as, d(ρǫ h ) dt = 2µ Ū ( k u u + u u ) k x k x x }{{} () 2µu u k 2 Ū x k }{{} () 2µ u u k u µ u u x k x k k }{{}}{{} () (v) u 2ν u p x }{{} (v) +µ 2 ǫ h x k x k }{{} (v) 2ρν 2 2 u 2 u x k x k }{{} (v) (2.12) It s not easy to nterpret the dfferent terms and therefore, to propose a ratonal modellng of ths equaton. Moreover, numercal smulatons do not often provde dsspatonrate budgets.? Only a subtle dmensonal analyss can be performed to model the dsspaton transport equaton (2.12), whch would be too long to reproduce n what follows. An argumentaton can be found n Hanalć & Launder 24, 25 and n Launder. 39 Fnally, the modelled transport equaton for the dsspaton s taken as follows d(ρǫ h ) dt = [( µ+ µ ) ] t ǫh x k σ ǫ x }{{ k } (a) + ǫ h k t (C ǫ1 P C ǫ2 ρǫ h ) }{{} (b) (c) (2.13) n the k t ǫ model, where σ ǫ, C ǫ1 et C ǫ2 are three constants of the model. Ths sememprcal equaton has the same structure that equaton (2.10) for the turbulent knetc energy k t. Transport of the dsspaton rate along the mean flow s balanced by three terms. The turbulent dffuson term (a) represents terms (v), (v) and (v) n equaton (2.12), where the vscous dffuson term (v) can be neglected at hgh Reynolds numbers. The producton term (b) s assocated wth term (), one can ndeed refer to secton?? where t s shown that ths term s assocated wth the vortex stretchng. The destructon term (c) s assocated wth terms (v). These two terms (b) and (c) domnate all the other terms n the transport equaton but, n vew of ther apparent dfference, they should be both kept. At last, t can be shown 25, 39 that the terms () and () vary as Re 1/2 t and as Re 1 t respectvely, and are thus neglected relatve to other terms. The turbulent Reynolds number s defned by Re t = k 2 t /(νǫ h).

28 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally Transport equaton of energy Several forms of the energy equaton have been ntroduced n chapter 1. The enthalpy equaton (1.4) s frst consdered, (ρh) + u (ρh) = ( ) λ h + p c p + u p u + τ (2.14) wrtten for a perfect gas, that s dh = c p dt. As usual, the Reynolds decomposton of the enthalpy s ntroduced h = H+h and the averagng operator s appled to derve a transport equaton of the mean enthalpy, d(ρ H) dt = ( ) λ H ρu c p x h + P + (Ū P) + u p Ū + τ + ρǫ (2.15) In numercal smulatons, the turbulent dffuson term through fluctuatng pressure s often neglected n the energy equaton. Moreover, the dsspaton s approxmated by ǫ ǫ h accordng to equaton (2.13). The velocty-enthalpy correlaton term s expressed from a gradent model, ρu h µ t σ t h where σ t s a turbulent Prandtl number defned by σ t ν t /a t, and a t s a turbulent thermal dffusvty. Ths number can be estmated expermentally n sheared mean flows for nstance.? From the followng relaton σ t = ν t a t = u 1 u 2 Ū 1 / x 2 ( h u 2 h/ x 2 ) 1 (2.16) values of the turbulent Prandtl number are found between 0.6 σ t 1, and the recommended value?, 76 n the k t ǫ model s σ t 0.85 for boundary layer flows and σ t 0.5 for free shear flows. It s also convenent to ntroduce a turbulent thermal conductvty λ t, wth a t = λ t /(ρc p ), to model the correlaton between fluctuatng velocty component and temperature. From the Reynolds decomposton of the temperature T = T+T, t s assumed that u T = a t T x The partcular case σ t = 1 corresponds to the Reynolds analogy,? for whch turbulent momentum and thermal transfers lead to smlar turbulent boundary layer profles for the mean velocty and temperature. Fnally, the law for a perfect gas provdes a relaton between the mean temperature and pressure, that s P = ρr T. As a further llustraton, the transport equaton of the specfc nternal energy e s also brefly examned for a perfect gas, that s, wth p = (γ 1) ρe. Ths equaton can be

29 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 29 wrtten as, ( ) ρe+ρ u2 + [( ] ρe+ρ u2 )u p = ( λ T ) u + τ The Reynolds decomposton s then ntroduced, wth e = Ē+e, and the total mean energy s defned by Ē t Ē+Ū Ū /2+k t. A mean equaton s obtaned by applyng the averagng operator. More detals can be found n revews by Vandromme 68 or by Knght 34 for nstance. Only the smplfed usual form of ths equaton s gven below, (ρē t ) + [( ρē t + P+ 2 ) ] 3 ρ k Ū = [ (λ+λ t ) ] T + 2(µ+µ t ) x S Ū where the gradent-dffuson assumpton has agan been used for second-order correlaton terms. As mentoned n ntroducton, when the densty s assumed constant, the mean temperature s a passve scalar. The mean velocty feld s then not coupled to the transport equaton for energy. Ths s no more the case for an ncompressble flow wth varable densty, that s ρ = ρ( T), and when buoyancy effects are taken nto account. The gravty force ρgδ 3 s then ncluded n the Naver-Stokes equaton, and the Boussnesq approxmaton?, 42 s usually appled to derved averaged equatons. Densty varatons are thus neglected n the flud dynamcs equatons, except for terms nvolvng buoyant effects. For nstance, ths leads to an addtonal term n the turbulent knetc energy equaton (2.7). Usually denoted G, ths term s expressed as G = ρgβθu 3, where the coeffcent of thermal expanson s β = 1/ T for a perfect gas. A transport equaton for the temperature varance T 2 can also be establshed and modelled, a revew s provded by Hanalć Hgh-Reynolds-number form of the model Equatons of the k t ǫ turbulence model for an ncompressble flow at hgh-reynolds number are summarzed n what follows. The mean turbulent flow feld s gouverned by, Ū = 0 x d(ρū ) dt d(ρ H) dt = x ( P+ 2 = [( µ σ + µ t σ t 3 ρk ) t + ) ] H [ (µ+µt ) S ] + P + (Ū P) Ū + τ + ρǫ x h where the turbulent vscosty s expressed from the turbulent knetc energy k t and the dsspaton for a homogeneous flow ǫ h as µ t = C µ ρk 2 t /ǫ h. The heart of the model s

30 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 30 consttuted by the two transport equatons for k t and ǫ h, whch take the followng form, d(ρk t ) = [( µ+ µ ) ] t kt +P ρǫ dt x k σ kt x h k d(ρǫ h ) dt = [( µ+ µ ) ] t ǫh + ǫ (2.17) h (C ǫ1 P C ǫ2 ρǫ x k σ ǫ x k k h ) t where the producton term s gven by P = 2µ t S 2. The model contans fve constants whch have been determned by comparson wth expermental data of smple flows. 38 As llustraton, the case of the constant C ǫ2 s examned n the next secton. Fnally, the followng values are retaned n the standard form of the model, C µ = 0.09 C ǫ1 = 1.44 C ǫ2 = 1.92 σ kt = 1.0 σ ǫ = 1.3 (2.18) The knowledge of k t and ǫ h provdes an estmaton of the turbulent velocty scale u = 2k t /3 and of an ntegral length scale from L f u 3 /ǫ h. Moreover, from the expresson of the turbulent vscosty ν t = u l m = C µ k 2 t /ǫ h, the mxng length s gven by l m 3/2C µ k 3/2 t /ǫ h. The k t ǫ turbulence model has been used wth success n varous confguratons. The mean flow feld computed behnd a ducted daphragram s shown n fgure 2.1 for nstance. More precsely, streamlnes are plotted n dfferent vertcal planes to take nto account the three-dmensonalty of the mean velocty feld. The et-lke flow comng from the daphragm aperture s attached to the top wall through the Coanda effect, and there s a reattachment to the bottom wall n the second part of the outlet duct. Unfortunately, as most of the turbulence models, adaptatons are requred for specfc flows as for axsymmetrc ets 49, 63, 64 or for radal ets 43, 51 to menton a few of them. Boundary layer flows are dscussed n secton Determnaton of the constant C ǫ2 The k t ǫ model can be wrtten for specfc flows n order to determne the emprcal constants. Consder for nstance the case of decayng turbulence behnd a grd, for whch expermental results have been presented n secton??. The two transport equatons (2.17) take the smplfed form, U 0 k t x 1 ǫ ǫ U h ǫ 2 h 0 C ǫ2 x 1 k t where U 0 s the unform mean flow velocty. These two equatons can the be combned as follows, 1 ǫ h U 0 ǫ h x 1 = C ǫ2 ǫ h k t = C ǫ2 1 k t U 0 k t x 1

31 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 31 Fgure 2.1 : Computed mean flow feld behnd a daphragm? by usng the standard k t ǫ model. The duct heght s 2.29h and the duct wdth s w = 2.86h where h = 35 mm s the aperture sze. The Reynolds number based on h and on the maxmum mean velocty Ū m = 20 m.s 1 s Re h = Streamlnes n xy-planes computed wth the Ū 1 and Ū 2 components of the 3-D mean velocty feld at three spanwse locatons z/w = 0.1, 0.5 and 0.9. and can be solved to obtan, ( ) ǫ Cǫ2 h kt = ǫ h0 k t0 where k t0 and ǫ h0 are two constants. Therefore, the soluton for the turbulent knetc energy and the dsspaton rate are, [ k t = 1+(C ǫ2 1) ǫ ] 0 x 1 ( C 1 ǫ2 1 = 1+ x ) n 1 k t0 k t0 U 0 Θ c0 U 0 and ( ǫ h = 1+ x ) (n+1) 1 ǫ h0 Θ c0 U 0 where C ǫ2 = n+1 Θ c0 = (C ǫ2 1) k t0 = n k t0 n ǫ h0 ǫ h0 The constants k t0 and ǫ h0 correspond to the ntal values at tme t = x 1 /U 0 = 0, and an ntegral tme scale of turbulence s defned by Θ c = k t /ǫ h. The value of n found n measurements?,? s n 1.3, whch yelds C ǫ Ths s not exactly the recommended standard value n (2.18), whch results from other addtonal factors. Fnally, the evoluton of the turbulent vscosty ν t = C µ k 2 t /ǫ h s gven by ( ν t = 1+ x ) 1 n 1 ν t0 Θ c0 U 0 whch ndcates that decayng turbulence s not a self-smlar flow snce n = 1, as already noted n secton??.

32 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally Low-Reynolds-number form of the k t ǫ model Treatment of near wall flows requres modfcatons of the k t ǫ turbulence model n ts standard form (2.17). The frst low-reynolds versons of the model were ntroduced by Jones & Launder 31 and by Launder & Sharma. 37 The model of Chen 16 s one of the several sgnfcant developments, and revews can be found n Patel et al., 47 n Celć & Hrschel 14 and n Wlcox s book.? Two man changes are ntroduced n these models. Frst, the constants are now functons of a turbulent Reynolds number defned as Re t = k 2 t /(νǫ), whch tends to zero as x 2 0, and also of the wall dstance x + 2. For nstance, C µ becomes C µ f µ (Re t, x + 2 ). The Van Drest model (??) ntroduced n chapter?? s an example of such sem-emprcal dampng functons. Second, an addtonal term s ncluded n the dsspaton equaton. From the equaton for k t, the value of the dsspaton at the wall verfes 25 ǫ h (x 2 = 0) = ν 2 k ( ) 2 t kt x2 2 = 2ν x 2 x2 =0 x2 =0 and moreover, t can be shown that ǫ = ǫ h at x 2 = 0. The dsspaton equaton s then often wrtten for an auxlary varable ǫ h ǫ h x2 =0 for numercal purposes, n order to apply a Drchlet boundary condton and to ensure a bound value of the Reynolds number at the wall. An alternatve to low-reynolds-number forms of turbulence models, whch requres a relatve fne mesh near the wall wth x + 2 1, s to use wall functons. Ths alternatve s smple to mplement but s based on a strong hypothess. The frst pont of the mesh n the normal drecton to the wall s ndeed assumed to be n the logarthmc regon of a boundary layer. Values of ν t and k t can thus be mposed at ths pont, as well as the value of ǫ h or ǫ h / x Realsablty and unsteady smulatons Turbulence models should satsfy addtonal constrants, mposed by prncple of physcs such as the nvarance of the formulaton by a Gallean transformaton, 61 or by realsablty condtons for the Reynolds stress tensor for nstance. An ntroducton to ths topc can be found n Schumann, 57 and t follows that, u αu α 0 u αu β2 u α u α u β u β det(u αu β ) 0 where the second relaton s the Schwarz nequalty. Recall that there s no summaton on Greek ndces. The renormalsaton group (RNG) approach provdes a more general framework, and was appled by Yakhot, Orszag, Thangam, Gatsk & Spezale 74 to the k t ǫ turbulence

33 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 33 model. In the resultng RNG k t ǫ model, a new set of constants, see (2.18), s obtaned n the hgh-reynolds number approxmaton, C µ = C ǫ1 = 1.42 C ǫ2 = 1.68 σ kt = σ ǫ = Moreover, an addtonal term must be consdered n the dsspaton equaton, and can be ncluded by replacng the constant C ǫ2 by the functon Cǫ2 = C ǫ2 + C µ η 3 (1 η/η 0 )/(1 + βη 3 ) where η = Sk t /ǫ, η 0 = 4.38 and β In pratce, ths smart approach has a rather weak nfluence on turbulence model closures and results. At ths step, t can be the opportunty to make a dgresson about unsteady smulatons of Reynolds-averaged Naver-Stokes equatons. Unsteady RANS (URANS) are obtaned through the use of a tme-marchng algorthm for nstance, and often by reducng the value of C µ and as a result, ν t.? Consder a mean shear flow Ū 1 = Ū 1 (x 2 ) wth Ū 2 = Ū 3 = 0. Schwarz s nequalty for α = 1 and β = 2 gves (2ν t S 12 ) 2 (2ν t S 12 ) 2 (2k t /3) 2 ( )( ) k t 2ν t S 11 3 k t 2ν t S 22 9C 2 µ S 2 12 ǫ 2 h /k2 t by notng that S 12 = 0 and S 11 = S 22 = 0. There s thus an mplct low-pass flter mposed by the mean shear and controlled by the value of C µ. As a result, no development of energy cascade can occur wth grd rafnement n numercal smulatons. Examples and dscussons can be found n Spalart 60 and Iaccarno et al The k t ǫ model for compressble flows Favre-averaged Naver-Stokes equaton The opportunty s taken n ths chapter to ntroduce the Favre average n order to extend the k t ǫ turbulence model to compressble flows or to varable densty flows. A frst natural dea to generalse Reynolds-averaged Naver-Stokes equatons for compressble flows conssts n applyng the Reynolds decomposton to the densty, by wrttng ρ = ρ+ρ. Consder for nstance the conservaton of mass. One obtans, ρ + ( ρū + ρ x u ) = 0 A new unknown appears wth the correlaton ρ u and needs to be modelled. It s relatvely easy to guess consequences from other governng equatons when the densty

34 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 34 decomposton s ntroduced. To overcome at least n part ths dffculty, Favre 22 proposed to defne a new averagng operator defned by, F ρ f ρ (2.19) that s a densty-weghted statstcal averagng. Ths means that for any varable f, one has the followng Favre decomposton f = F+ f wth ρ f = 0 but wth f = ρ f ρ = 0 Note that the fluctuaton f s non-centered. Returnng to the conservaton equaton of mass, the Favre decomposton s then ntroduced for the velocty, that s u = Ũ + u, whch yelds ( ρ+ρ ) + ( ρũ + ρu ) = 0 x By takng the statstcal average to ths equaton, one then obtans ρ + ( ρũ ) = 0 Ths compressble form of the mass conservaton has the same structure than the ncompressble form obtaned by usng the Reynolds varable decomposton, whch clearly llustrates the nterest of the Favre averagng. The Reynolds and Favre averagng are equvalent for ncompressble flows, and only dffer when sgnfcant densty fluctuatons occur. The Favre decomposton s now consdered for all varables, except for densty and pressure. Moreover, the property (1.1) regardng the product of two varables f and g, takes the form ρ f g = ρ F G+ρ f g (2.20) From these rules, the Favre-averaged Naver-Stokes equaton can be straghtforward derved, ( ρũ ) + ( ρũ Ũ ) = P + ( ) τ ρu x u and agan, the structure of the averaged Naver-Stokes equaton s preserved for compressble flows. The Boussnesq assumpton for the closure of the Reynolds tensor ρu u s generalsed as follows, ρu u = 2µ t S 2 ( 3 ρ kδ Ũ where S = 2 + Ũ ) 1 Ũ k δ x 3 x k

35 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 35 accordng to relaton (1.11) for the devatorc part of the mean velocty gradent tensor. The turbulent knetc energy s now defned by k t = 1 2ũ u = 1 2 ρu u ρ To keep notatons as smple as possble, we wll stll contnue to denote k t the turbulent knetc energy when Favre averagng s used. There s generally no ambguty n nterpretaton. The transport equaton for the conservaton of enthalpy (2.14) s establshed n the same way. Ths yelds for a perfect gas, ( ρ H) + ( ρ HŨ ) = ( ) λ H ρu c p x h + P + Ũ P + u P + u p Ũ x + τ + ρǫ (2.21) The Reynolds decomposton s appled to the heat flux q = (λ/c p ) h/, but the mean component s approxmated by Q (λ/c p ) H/. Ths s a qute reasonable approxmaton for mean molecular transport terms, whch s also used for the dsspaton term τ τ = 2µ S. Fnally, the law of perfect gas s consstently gven by P = rρt = ρr T. Usng the densty-weghted averagng clearly allows to reduce efforts to derve the mean flow equatons wth respect to the usual Reynolds decomposton for the densty. Some dffcultes nevertheless appear as the new term G f u x P snce fluctuatons are non-centered, or the correlaton term between the pressure gradent and the fluctuatng velocty feld, u x p. By observng that u = u u, the latter can be rearranged as follows, u p = u x p = u p }{{} (a) p u }{{} (b) The frst term (a) s a turbulent transport term whle the second term (b) s the so-called pressure-dlataton correlaton, where the dlataton s the dvergence of the fluctuatng velocty feld, denoted d u. The transport equaton for the turbulent knetc energy s more tedous to establsh, and detals are provded n appendx 2.8. Ths equaton can take the followng form, ( ρ k) + ( ρ kũ ) = ρu Ũ u x u τ 1 ρu 2 x u u p x u + u x τ u P p x u (2.22) x

36 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 36 and can be compared to ts ncompressble form (2.9). The convecton of k t by the mean flow s now balanced by a producton term related to the mean shear, a destructon term related to vscous effects, three dffuson transport terms, and two new contrbutons arsng from the non-centered fluctuaton of the velocty, that s u = 0, and from the compressble feature of ths fluctuatng feld, that s d = u = 0. The term G f n equatons (2.21) and (2.22), whch s related to the turbulent mass flux, can be modelled through a dffuson-gradent assumpton, G f = u P = ρ u x ρ P = 1 ρ µ t ρ P (2.23) x σ t x x as proposed by Jones 30 or by Sarkar & Balakrshnan 53 for nstance, where σ t s the turbulent Prandtl number prevously ntroduced, see expresson (2.16). Other strateges are possble, and the reader may refer to Krshnamurty & Shyy 35 or to the book by Wlcox.? Modellng of the pressure-dlataton correlaton p d s dscussed n the next secton, snce a smlar dlatatonal term also appears n the equaton for the dsspaton rate Compressble form of the dsspaton rate The transport equaton for the dsspaton rate n the ncompressble form of the k t ǫ model s sem-emprcal, as dscussed prevously n ths chapter. It s therefore llusve to derve an exact equaton n the compressble case, and to model ths equaton n a second step. The approxmaton of the exact dsspaton rate ǫ by ts expresson for an homogeneous flow ǫ h s stll retaned, but must be now reformulated for compressble turbulence. As a startng pont, one has u ρǫ = τ = 2µs e x = 2µe e 2 3 µdd 2νρe e 2 3 νρd2 by assumng a constant molecular vscosty ν. The dsspaton rate s then spltted n two contrbutons assocated wth the ncompressble part of the velocty n the one hand, and wth dlataton n the other hand. The followng relatonshp can be obtaned, ρǫ = 2µ e e 2 µ ρd 2 3 = µ ω ω µ d 2 + 2µ 2ũ u 4µ ũ x x d }{{} when the underlned term s usually neglected. Ths assumpton holds for hgh-reynolds number flows or for an homogeneous turbulence. The dsspaton rate s then gven by ǫ h = ǫ s h + ǫd h wth ρǫ s h µ ω ω and ρǫ d h 4 3 µ d 2 (2.24)

37 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 37 The dsspaton rate related to the ncompressble part of the velocty, also called the solenodal dsspaton rate ǫh s, s computed from a transport equaton smlar to those of the k t ǫ model n ts ncompressble form. Therefore, only the dlatatonal dsspaton ǫ d h ncludng compressblty effects, needs to be modelled. Varous modellngs have been proposed n lterature for ǫ d h, as well as for the pressure-dlataton correlaton p d. Many of them are bult as follows, ǫh d = f(m t) ǫh s where f s a functon of the turbulent Mach number M t defned by 2kt M t = c The reader may refer to the ponerng work by Zeman and by Sarkar et al Varants and applcatons to shear flows can be found n Lakshmanan & Abdol-Hamdn 36 n Dash et al. 19 or n Krshnamurty & Shyy 35 among others. A more general overvew can be found n Chassang, Anselmet, Joly & Sarkar 15 or n Gatsk & Bonnet? for turbulence modellng aspects Compressble form of the k t ǫ model Equatons of the k t ǫ turbulence model n ts compressble form are fnally collected below. Equatons for the mean flow are smlar to the ncompressble case provded that the Favre averagng (2.19) s used, and that the complete expresson of the devatorc part of the mean velocty gradent tensor (1.11) s consdered. Thus, the conservaton of mass and of momentum can be wrtten as follows, ( ρũ ) ρ + ( ρũ ) = 0 + ( ρũ Ũ ) = ( 2 P+ x 3 ρk ) [ ] t + (µ+µt ) S These equatons must be assocated wth a conservaton equaton for energy, as equaton (2.21) for the mean enthalpy. The turbulent vscosty µ t s now defned from the turbulent knetc energy and from the total dsspaton ǫ h = ǫ s h + ǫd h, µ t = C µ ρ k2 t ǫ h as explaned n the prevous paragraph. The two transport equatons for k t and the ncompressble part of the dsspaton rate ǫh s take the followng form, ( ρk t ) + ( ) ρk t Ũ = [( µ+ µ ) ] t kt +P +G σ kt x f ρǫ h ( ρǫ h s) + ( ρǫs hũ) = [( µ+ µ ) t ǫ s ] h + ǫs [ h Cǫ1 (P +G σ ǫ k f ) C ǫ2 ρǫ s] t

38 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally 38 The compressble part of the dsspaton rate s computed from a specfc model, whch usually takes the followng general form ǫh d = f(m t) ǫh s. The producton termp s gven by P = ρu u Ũ ( ) 2 2 ( = 2µ t S µt.ũ+ ρ k )(.Ũ ) 3 by generalsng Boussnesq s assumpton. Fnally, the values (2.18) of the constants establshed for the ncompressble form of the model are stll used. Applcatons to free shear flows of these compressble and hgh-reynolds number forms of the k t ǫ model can be found n Dash et al. 17, 18 or n Thes & Tam. 64 Regardng perfectly expanded supersonc ets, numercal predctons obtaned by dfferent approaches are dscussed n Barber & Chappetta. 10 As llustraton, the case of a supersonc round et at Mach number M = 2 s consdered n fgure 2.2. The turbulent mean flow s calculated from a compressble form of the k t ǫ model ncludng the model of Zaman 77 for the compressble part of the dsspaton and the term (2.23). The turbulent knetc energy feld s represented through the quantty k 1/2 t /U, whch ndcates a normalsed level of velocty fluctuatons. In the present case for the hot et at T /T 2.8, k 1/2 t /U reaches around 15% n the annular shear-layer. The et ext temperature and the ambant temperature are respectvely denoted T and T, U s the et ext velocty. The faster decrease of the velocty wth ncreasng et temperature s reasonably well retreved by the model. Note that the small wggles for the hot-et mean velocty profle are caused by a slght msmatch between the et ext pressure and the ambant pressure n the calculaton. 2.4 The k t ω t turbulence model Turbulence models based on k t and ω t are currently among the most wdely used twoequaton eddy-vscosty models. The turbulent vscosty s stll bult through the knowledge of two turbulent scales, namely the turbulent knetc energy k t and the specfc dsspaton rate ω t ǫ h /k t. Ths varable ω t has the dmenson of frequency, and can be lnked to the square of enstrophy, where enstrophy s the root mean square fluctuatng vortcty. Ths quantty has been ntroduced n chapter??. The frst two-equaton turbulence model was hstorcally proposed by Kolmogorov,?,? and t s precsely based on k t and ω t. Varous formulatons have been developed and mproved n lterature, but the transport equaton for the turbulent knetc energy often keeps the followng usual form, d(ρk t ) dt = [( µ+ µ t σ kt ) ] kt +P β ρω t k t (2.25) where β s a constant of the model. Its structure s qute smlar to equaton (2.10), and

39 Arbus Toulouse, Lecture notes cb1, Aprl 2012 Ch. Bally Ūz/U z/d Fgure 2.2 : Turbulent mean flow of a round supersonc et at M = 2 computed from a compressble form of the k t ǫ model.?,? At the top, maps of the Mach number and of k t /U for a hot et, T /T 2.8. At the bottom, computed mean axal velocty profles for a cold et T /T 1 and a hot et T /T 2.8, and measurements of Sener et al.?. All axs are normalsed by the nozzle dameter D, and the Reynolds number of the cold et s Re D