AIAA JOURNAL Vol. 40, No. 10, October 2002 Modeling Rotating and Swirling Turbulent Flows: A Perpetual Challenge S. Jakirlic´¤ Darmstadt University of Technology,D-64287 Darmstadt, Germany K. Hanjalic´â Delft University of Technology, 2628 CJ Delft, The Netherlands and C. Tropeaâ¡ Darmstadt University of Technology,D-64287 Darmstadt, Germany Several types of rotatingand swirling ows for a rangeofReynoldsnumbersandrotationrates or swirl intensities have been studied computationally, aimed at identifying speciŽ c features that require special consideration in turbulence modeling. The ows considered include turbulent channel ows subjected to streamwise and spanwise rotation,with stationaryandmovingboundaries;developingand fully developed ows in axiallyrotatingpipes; and swirling ows in combustor geometries and long pipes. Computationsperformed with three versions of the second- moment closure and two eddy-viscosity models show that the second-moment models are superior, especially when the equations are integrated up to the wall. Such models reproduced the main ow parameters for all ows considered in acceptable agreement with the available experimental data and direct numerical simulations. However, challenges still remain inpredictingaccurately somespeciŽ c ow features, suchas capturing the transition from a free vortex to solid-body rotation in a long straight pipewith a weak swirl, or reproducing the normal stress components in the core region. Also, the so-called uw anomaly in fully developed ows with streamwise rotation remains questionable. For rotating ows, the low-Reynolds-number models yield a somewhat premature ow relaminarization at high rotation speeds. I. Introduction T OGETHER with ow separation,swirl and rotationare proba-bly the most frequently encountered ow phenomena in tech- nical applications: aircraft gas combustors, pulverized coal burn- ers, turbomachinery,cyclone separators, centrifugalgas separators, large-scale pipeline systems, etc. Because of their practical and theoreticalimportance,numerousexperimentalinvestigations,com- putational modeling and, more recently, direct numerical simula- tions (DNS) and large-eddy simulations (LES) of these ows have been reported in the literature. We mention here only some rele- vant publications on several ow classes: pipe ows rotating about their own axis,1¡12 rotating channel ows with stationary9;13¡20 and moving boundaries,20;21 and swirling ows in geometries relevant to combustor chambers10;22¡27 and long pipes.27¡29 Although ex- perimental and DNS/LES studies provide useful information and insight into the ow physics, the computationalmethods involving turbulencemodeling offers the broadestprospects for industrial ap- plications. However, the complex ow structure invalidates some of the assumptions on which simple turbulence models are based. Particularly challenging for modeling are ow features that are ab- sent in simple ows in which the models are usually tuned and validated.Examples of such features are the secondaryshear strain, for example, the shear component @W=@r in addition to the com- mon mean shear @U=@r , streamline curvature (with respect to the additionalstrain¡W=r), strongdeparturefrom localenergyequilib- rium, and the effects of turbulenceanisotropy. In ows with system Received 30 January 2001; revision received 10 January 2002; accepted for publication23May 2002.Copyright c° 2002by theAmerican Instituteof Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose- wood Drive, Danvers, MA 01923; include the code 0001-1452/02 $10.00 in correspondence with the CCC. ¤Research Scientist, Chair of Fluid Mechanics and Aerodynamics, Petersenstr. 30. â Professor, Faculty of Applied Physics, Lorentzweg 1. â¡Professor, Chair of FluidMechanics and Aerodynamics, Petersenstr. 30. Member AIAA. rotation, the ow structure is additionallymodiŽ ed due to the action of Coriolis and centrifugal forces. We report here on the investigation of several ow classes that exhibit the speciŽ c ow features mentioned. The choice of the ows was determined by their industrial relevance, but also by the availability of experimental and/or DNS data for model vali- dation. The following ow categories were considered: develop- ing (experimental1) and fully developed ows in an axially rotat- ing pipe (DNS5¡7), swirling ows in short (combustor) geometries (experimental22;23) and in long pipes (experimental28;29), channel ows with streamwise (DNS15) and spanwise rotationwith station- ary (rotatingPoiseuille ow, DNS,14 LES,16 and experimental13 and moving walls (rotating Couette ow, DNS21). The rotating channel ows and the Roback and Johnsonâs,22 swirling ow served as the test cases in the seventhâninth European Research Community on Flow, Turbulenceand Combustion (ERCOFTAC) workshopson re- Ž ned turbulence modeling. Schematic representation of all cases considered is shown in Fig. 1. It is well known that the standard kâ" and other linear two- equation eddy-viscosity models have a number of weaknesses, which are especially pronounced in ows affected by rotation and swirl. In contrast,second-momentclosures,which provide informa- tion about all stress componentsand contain exact terms for rotation effects in the stress equation, are inherently capable of capturing most of the phenomenamentioned earlier. Indeed, in contrast to the kâ" models, in all ow cases reported here the second-momentclo- sures produced acceptable results. However, a systematic scrutiny of a large range of ows revealed several speciŽ c shortcomings: the wrong (negative)sign of the uw shear stress component, the tangen- tial velocity proŽ le retains its initial concentrated vortex shape too long in a long straightpipewith a weak swirl, and thepremature ow laminarizationin axially rotatingpipes and cylinders.Althoughusu- ally not detrimental, these shortcomingspose intriguing challenges to turbulence modeling and may also be of importance for accu- rate prediction of engineering ow parameters, especially at high rotation rates and swirl intensities. The main goal of this work is to provide a systematic survey of the performance of selected second-moment closures in a range of rotating and swirling ows and to establish more clearly their 1984 D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 JAKIRLIC´, HANJALIC´, AND TROPEA 1985 Fig. 1 Schematic of the ow conŽ gurations considered: a) developing rotating pipe ow, b) fully developed rotating pipe ow, c) channel ow with streamwise rotation, d) channel ow with spanwise rotation, e) rotating Couette ow, f ) swirling ow in a combustor chamber, g) strong swirl in a long straight pipe, and h) weak swirl in a long straight pipe. potential and limitations, particularly in comparison with linear eddy-viscositymodels. The study also provides further insight into the physics of system rotation, swirl, transverse shear, and their effect on turbulence, within the framework of the Reynolds aver- agedNavierâStokes (RANS) approach,as well as ideas for possible model improvement. The suitability of the wall function approach for treating the near-wall region is also discussed. II. Numerical Method and Turbulence Models The ows consideredare essentiallyall threedimensional(except the two-dimensional channel ows rotating in orthogonal mode), for which all three velocity components and all six Reynolds stress componentsmustbe accountedfor.However, theReynoldsaveraged continuity and momentum equations are solvedwith assumption of axisymmetry (@=@âD 0): 1 r @.rU j / @x j D 0 (1) 1 r @.rU jUi / @x j D 1 r @ @x j µ r ³ º @Ui @x j ¡ u iu j ´¶ ¡ 1 ½ @P @xi C SUi (2) Table 1 Sources in mean momentum equations SUi Convection Viscous transport Turbulent transport SU ââ ââ ââ SV CW 2=r ¡ºV=r2 Cw2=r SW ¡VW=r ¡ºW=r2 ¡wv=r where the vector xi .r; z/ denotes the coordinate directions and Ui .Ur ;Uâ;Uz/ [´.V;W;U /] is the mean velocity vector. The source terms SUi , arising from the coordinate transformation, are given in Table 1. All computations were performed by a computer code based on a Ž nite volume numerical method for solving the RANS equa- tions in orthogonal coordinate systems with a collocated variable arrangement. (See Jakirlic´ et al.30 for more details.) For discretiza- tion of the convective uxes, a blended upwind-centraldifferencing scheme, implemented in the so-called deferred-correctionmanner was used. For most ows considered, we used a high portion of central differencing (CD): The blending factor was typically be- tween 0.7 and 1.0 for all variables. Because of a weak ow-to- grid skewness, and because of the very Ž ne grid used, in most D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 1986 JAKIRLIC´, HANJALIC´, AND TROPEA Table 2 Grids and dimensions of the solution domains for the ows computed Grid (control volumes) Solution domain High-Reynolds- Low-Reynolds- Radius R, mm Flow type number model number model Length L , mm Width 2h, mm Flows in axially rotating pipes Ref. 4 (experiment), Refs. 5â7 (DNS) Rem D 4:9£ 103â2£ 104; N D 0:0â10.0 2£ 20â40 2£ 60â100 2 100 Ref. 1 (experiment) 6000 Rem D 6£ 104 ; N D 0:0, 0.5, 0.83 100£ 70 100£ 130 (L of rotating partD 4120) 50 Flows in rotating channels Pouseuille ow with spanwise rotation: Ref. 14 (DNS) RoD 0:0â0.5 10£ 40 10£ 120 10 200 Couette ow with spanwise rotation: Ref. 21 (DNS) RoD 0:0â0.5 10£ 40 10£ 120 10 200 Channel with streamwise rotation: Ref. 15 (DNS) Ro¿ D 0:0â10.0 2£ 40 2£ 120 2 200 Swirling ows in combustor chambers Ref. 23 (experiment) SD 2:25 100£ 70 100£ 130 348 62.5 Ref. 22 (experiment) SD 0:45 200£ 70 220£ 120 750 61 Swirling ows in long straight pipes Ref. 28 (experiment) Rem D 5£ 104 , SD 1:0 200£ 60 200£ 140 6150 75 Ref. 29 (experiment) Rem D 5£ 104 , SD 0:1 120£ 60 120£ 100 5300 35 Table 3 Summary of the model coefŽ cients Parameter GL SSG HJ C1 1.8 1.7 2.5AF 1 4 f C A 12 E2 C 01 0.0 1.05 0.0 C 02 0.0 0.9 0.0 C3 0.8 0.8â0.65 A 1 2 2 1.067 A 1 2 C4 0.6 0.625 0.8 A 1 2 C5 0.6 0.2 0.8 A 1 2 Cw1 0.5 0.0 max.1¡ 1:75AF 1 4 f I 0:3/ Cw2 0.3 0.0 min.AI 0:3/ ® 1.0 1.0 1.4 Cs 0.22 0.22 0.22 fs 0.0 0.0 1¡ A 1 2 E 2 C"1 1.44 1.44 1.44 C"2 1.92 1.83 1.92 C" 0.18 0.18 0.18 Q" " " Q" f" 1.0 1.0 1¡ C"2 ¡ 1:4 C"2 exp ( ¡ ³ Ret 6 ´2) S" 0 0 0:25º k " u juk @2Ui @x j@xl @2Ui @xk@xl cases the solutions are virtually indistinguishable from those ob- tained by pure upwind differencing. [The only exception was the swirling ow discharging into a sudden expansion (experiment dis- cribed by Roback and Johnson22).] The solution domain used for the computation of the swirling ows and of the ows in an axi- ally rotating pipe/cylinder typically has the shape of the axisym- metric pipe geometry, with a length L and a height RD D=2, D being a pipe, that is, combustor diameter. (See Table 2 for dimensions of the solution domain and sizes of the numerical grids used.) For rotating channel ows, it was necessary to account for both walls. In the case of fully developed ows, periodic inlet/ outlet boundary conditionswere applied. Computationswere performedwith several two-equationmodels: the standardkâ" high-Reynolds-numbermodelandwith its two low- Reynolds-number extensions, due to Launder and Sharma31 (LS) and Chien,32 as well as with three types of Reynolds stress models: 1) the basic high-Reynolds-number version [see Gibson and Launder33(GL)], 2) the model of Speziale et al.34 (SSG), and 3) the low-Reynolds-number version of the second-moment (Reynolds stress) closure model (RSM) denoted as the Hanjalic´âJakirlic´ (HJ) low-Reynolds-numberRSM. The latter model contains the modiŽ - cations for viscosity and wall blockage effects. (See Hanjalic´ and Jakirlic´35 for more details.) A summary of the model coefŽ cients is given in Table 3. The second-moment closure models are deŽ ned by the transport equationsfor the turbulent stress tensor and turbulenceenergydissi- pation (for equations in cylindricalcoordinates,see Jakirlic´ et al.30): @ @xk .Ukuiu j / D @ @xk µ³ º±kl C Cs k " ukul ´ @u iu j @xl ¶ ¡ u iuk @U j @xk ¡ u juk @Ui @xk C 8i j ¡ "i j (3) @ @xk .Uk"/ D @ @ xk µ³ º±kl C C" k " ukul ´ @" @xl ¶ ¡C"1 " k ³ ukul @Uk @xl ´ ¡ C"2 f" " Q" k C S" (4) Amajor differencebetweenvarioussecond-momentclosuremodels is in the treatment of the pressure strain term 8i j and the stress dissipation rate "i j : 8i j D ¡C1"ai j C C 01" £ ai kak j ¡ 13 A2±i j ¤ ¡C 02Pkai j C C3kSi j C C4k £ ai kS jk C a jk Sik ¡ 23 ±i jakl Skl ¤ CC5k.aikW jk C a jkWik / CCw1 fw ."=k/ £ ukumnknm±i j ¡ 32u iuknkn j ¡ 32 uku jnkni ¤ CCw2 fw £ 8km ;2nknm±i j ¡ 328ik ;2nkn j ¡ 328k j;2nkn i ¤ (5) "i j D fs"¤i j C .1¡ fs/ 23 ±i j " (6) where "¤i j D ."=k/[u iu j C .uiukn jnk C u juknink C ukulnknlnin j / fd ] 1C 32 .u puq=k/n pnq fd Of course, in the high-Reynolds-number models, fs D 0 so that Eq. (6) reduces to "i j D 2"±i j=3. The model coefŽ cients are shown in Table 3, and the functions are as follows: fw D min ( k 3 2 2:5"xn I ® ) ; fd D .1C 0:1Ret / F D minf0:6I A2g; f D min »³ Ret 150 ´ 3 2 I 1 ¼ A D 1¡ 9 8 .A2 ¡ A3/; A2 D ai ja ji D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 JAKIRLIC´, HANJALIC´, AND TROPEA 1987 A3 D ai ja jkaki ; ai j D uiu jk ¡ 2 3 ±i j ; Ret D k 2 º" Si j D 1 2 ³ @Ui @x j C @U j @xi ´ ; E D 1¡ 9 8 .E2 ¡ E3/ E2 D ei je ji ; E3 D ei je jkeki ; ei j D "i j " ¡ 2 3 ±i j Q" D " ¡ 2º @ p k @ x j @ p k @x j ; Wi j D 1 2 ³ @Ui @x j ¡ @U j @xi ´ In the remainder of the paper we present a summary of selected results, focusing on the speciŽ c effects discussed earlier and the performance of different models in reproducing those effects. III. Discussion Logarithmic Law It is well known that swirl and rotationcause a strongdeviationof the velocity proŽ le from the logarithmic law, invalidating the stan- dardwall functionsfor the treatmentof thewall boundarycondition. In such cases it seems unavoidable to use a turbulence model that accounts for low-Reynolds-numberand wall-proximity effects, al- lowing integrationof the governingequationsup to the wall. This is especially the case in ows in axially rotating pipes and cylinders, which are prone to laminarization. This is shown in Figs. 2, 3a, and 4, where the computed velocity proŽ les are compared with the semilogarithmic line and with the available DNS and experimental data for the ow in an axially rotating pipe, for the swirling ow in a long straight pipe and the rotating Poiseuille and Couette ows. For axially rotating pipe ows, we show results for a range of rotation rates, deŽ ned by the ratio of the circumferential wall velocity Wwall to the axial bulk velocity Ub , N DWwall=Ub . The DNS shows that the axial velocity UCDU=U¿ follows the log- a) b) Fig. 2 Departure of the mean axial U++ , circumferential W++ , and the resultant R++ velocity from the logarithmic law for the ow in an axially rotating pipe. a) b) Fig. 3 a) Departure of themean axialU++ , circumferentialW++ , and the resultant R++ velocity from the logarithmic law and b) pressure proŽ les at selected positions in the swirling ow in a long straight pipe. arithmic slope for the lower rotation rates but departs at higher rotation rates (Fig. 2a). However, the circumferential velocity (WC D jWwall ¡W j=W¿ ) shows no similarity with the logarithmic law. The computationsusing the HJ model reproduce this behavior well (Fig. 2a). Figure 2b shows the mean resultant velocity RCD .UC2 CWC2/1=2 computed by the basic high-Reynolds-number, second-momentclosuremodel33 using wall functions.Whereas the computations of the cases with lower rotation rates (N < 1:0; here N D 0:32 shown)result in goodagreementwithDNSdata, the inade- quacyof thewall functionconceptbecomesvisibleat higherrotation rates (N ¸ 1:0; here N D 2:0 shown).Note that the velocitygradient in the Ž rst near-wall grid point is prescribed to dU=dyDU¿ =.·y/ by applying the wall functions (Fig. 2b). The ratio of the turbulentkinetic energyproduction to its dissipa- tion rate is presented in Fig. 5 for both rotation rates computed. In both cases, the equilibrium value Pk="¼ 1 was taken at the node closest to the wall, as follows from the wall function con- cept. Whereas for N D 0:32 this value of unity holds almost up to yC ¼ 100, in accordancetoDNSdata, this is not thecase for N D 2:0. The ratio drops suddenly to a very low value, leading Ž nally to zero turbulent kinetic energy production in the remainder of the cross section, in spite of using the Ž xed turbulent boundary condition. The shape of the resultant velocity proŽ le in a strongly swirling ow in a long pipe (Fig. 3a) shows a behavior that suggests oppo- site signs of the streamwise pressure gradient in the annular (wall) and core ow regions (Fig. 3b). In the annular region, the pressure gradient is favorable, in uencing the velocity proŽ le in such a way that it overshoots the log-line at the edge of the buffer zone, similar to the behavior encountered in accelerated ows. Contrary to this, the ow in the core region is decelerated,indicatingthe existenceof an adverse pressure gradient. [The disagreementwith experimental results correspondsclosely to the disagreementof both the normal- to-the-wall and spanwise stress components in this ow region (for example, see Fig. 6a), @P=@r D ½[@.¡rv2/=@r CW 2 Cw2]=r .] Figure 4 shows the semilog proŽ les of the mean axial velocity in the channel ows rotatingabout the orthogonalaxis, with stationary D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 1988 JAKIRLIC´, HANJALIC´, AND TROPEA a) b) Fig. 4 Departure of the mean axial U++ velocity from the logarithmic law for the rotating a) Poiseuille and b) Couette ows. Fig. 5 Ratio of the kinetic energy production to its dissipation rate for the ow in an axially rotating pipe. (Fig. 4a) and moving walls (Fig. 4b). On the suction side of the rotatingPoiseuille ow (Fig. 4a), the rotationhas a stabilizingeffect on turbulence, as indicated by a positive SD¡2Ã=.@U=@y/> 0, where Ãi .0; 0;Ã/ is the system vorticity and @U=@y represents the local mean shear vorticity. Here the velocity proŽ le exhibits a laminarlike shape, approaching the linear law (UC D yC) with increasing bulk rotation number RoD 2Ãh=Ub , where h is a half channel width. The velocity proŽ le in the lower half of the channel (pressure side, with a destabilizingeffect on turbulence, S< 0) lies completely below the log-law proŽ le. For rotating Couette ow (Fig. 4b, RoD 2Ãh=Uwall), the devia- tion of the velocity proŽ le from the log-law is also obvious.Unlike the rotating channelwith stationarywalls, the ratio S is always neg- ative, implying a continuousdestabilizingeffect on turbulence.The a) b) Fig. 6 ProŽ les of the Reynolds stress components at selected positions in the swirling ow in a long, straight pipe (experiment of Steenbergen29). rotation rates presented do not exhibit a laminarlike proŽ le as in the earlier case despite the decay of turbulent stress components observed (Fig. 7b). (See the works of Pettersson and Andersson20 and Bech and Andersson21 for further details.)Because of this tran- sitional nature, the wall functions are here totally unsuitable. Fully Developed Channel Flows Rotating in Orthogonal Mode Turbulent ow in a channel, rotating around a spanwise axis z, represents an idealization of the ow in blade passages of ra- dial ow pumps and compressor impellers. These ows are com- monly considered in the rotating frame of reference. In that case, one has to account for the exact production of turbulent stresses Ri j D¡2Ãk .u jum²ikm C u ium² jkm /. Furthermore, the model of the redistribution among the Reynolds stress components needs to be modiŽ ed to account for body forces originating from system rotation.17 For the case considered here with constant angular ve- locity X .0; 0;Ã/), these are as follows. Coriolis force: ¡2½ X £U.2½ÃV ;¡2½ÃU; 0/ (7a) Centrifugal force: ¡½ X £ . X £ x/.¡½Ã2x;¡½Ã2 y; 0/ (7b) where the vector x.x; y; z/ denotes the coordinate directions and U.U; V ; 0/ is the mean velocity vector in the rotating frame of reference. Both forces act on the mean ow inducing a secondary motion and on turbulence, modifying its structure. This recircu- lating motion (roll cells) is particularly intensive if the channel walls are moving (rotating Couette ows). Both ow cases, with stagnant and moving walls, were recently investigated by DNS by Kristoffersen and Andersson14 and Bech and Andersson.21 Sev- eral rotational intensities, expressed in terms of rotation number Ro ranging between 0.0 and 0.5, were considered, all at relatively low bulk Reynolds numbers, Rem D Reb DUb2h=º¼ 6000, that is, D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 JAKIRLIC´, HANJALIC´, AND TROPEA 1989 a) b) Fig. 7 Turbulence stress components in the channel ows with spanwise rotation with a) stationary and b) moving walls. Rew DUwallh=ºD 2600. The RANS computations of the rotating channel ows at such a low Reynolds number show premature laminarization18;19 as compared with DNS, which for the rotating Poiseuille ow shows that the onset of laminarization on the suc- tion channel side occurs at RoD 0:5. The introduction of a new, system-rotation related term in the dissipation equation cured this deŽ ciency. (See section about model modiŽ cations.) At the higher Reynolds numbers investigated experimentally by Johnston et al.13 and by Kim16 using LES, which are much closer to the conditions encountered in turbomachinery [where Reynolds number based on the blade cord length c is about Rec D 105 ¡ 106 (e.g., Mayle36)], there was no necessity to introduce such a term, at least for the rotation rates considered. Figure 8 shows axial mean velocity proŽ les for both channel ows, with and without wall movement for different rotation rates, but also for different bulk Reynolds numbers. The in uence of ro- tation on the proŽ le shape is obvious,making it increasinglyasym- metric with an increasing in rotational intensity. Over a portion of the channel cross section, the proŽ les are linear, with a (normal- ized) slope corresponding to the actual rotation number Ro. This is in compliancewith the theory for symmetries in inhomogeneous turbulent shear ows, which yields U DC5ÃyCC6 , with C5 and C6 denoting the integrationconstants.37 The width of this region in- creases at higher values of Ro. It is clear from the results presented in Figs. 8 that the model computationsresult in an overpredictionof the mean shear rate in the channel core (correspondingto the slope of velocity proŽ le), indicating a larger in uence of rotation, which can consequently lead to premature ow laminarization. The excessive in uence of rotation on the model results is par- ticularly visible in turbulence intensities, and especially for the ro- tating Couette ow (Fig. 7). The model results show much lower turbulence levels then computed by DNS (Fig. 7b). Similar results a) b) Fig. 8 Mean axial velocity proŽ les in the channel ows with spanwise rotation with a) stationary and b) moving walls. were obtained by application of the Ristorcelli, Lumley, and Abid second-momentclosuremodel in the frameworkofDurbinâs elliptic relaxationmethod (see Pettersson and Andersson20). In addition to possible model deŽ ciencies, this discrepancywith DNS results can be partlyascribedto the computationalprocedureused.Whereas the DNS accounted for a very strong, rotationally induced secondary motion14;21 (cross ow and streamwise secondary ow), whose tur- bulence intensity is much higher for the Couette ow than in the Poiseuille ow, the steady RANS method assumed that the mean motion is unidirectional. Laminarization Phenomena in an Axially Rotating Pipe Flow The most interesting issues in ows in rotating pipes/cylinders are the change in stress anisotropyand ow laminarization.A num- ber of experimental investigationsof fully developed and develop- ing ows in an axially rotating pipe can be found in the literature. From themeasurementsof mean velocity,Nishiboriet al.2 andHirai et al.3 observed changes in mean ow properties, which indicated a tendency toward ow laminarization.The stabilizingeffect of the positive gradient of the tangential velocity in the radial direction, corresponding to the positive Richardson number Ri D 2.W=r 2/@.rW /=@r .@U=@r /2 C [
[email protected]=r/=@r]2 reduces the turbulence level, producing, in turn, a decrease in the gradientof the axial velocity componentclose to the wall and a con- sequent reduction in the friction coefŽ cient. The centerlinevelocity UCL increases with an increase in the rotation intensity and the ve- locity proŽ le approaches the laminar form: UCL¼ 2Ubulk (Fig. 9a). AccordingtoNishiboriet al.,2 thisprocessis completedwhen the ro- tationrate reachesthevalueN ¼ 3:5.The computationof these ows D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 1990 JAKIRLIC´, HANJALIC´, AND TROPEA a) b) Fig. 9 Evolution of a) the axial velocity and b) kinetic energy proŽ les in an axially rotating pipe. usingthe low-Reynolds-numberReynoldsstressmodel supportsthis tendency(Fig. 9a, lines),but the laminarsolutionis obtainedatmuch lower rotation rates, already at N ¼ 1:0. An illustrative example is given in Fig. 9b, where the kinetic energy proŽ les for two rotation rates N D 0:5 and 1.0 are shown.Whereas the former agrees reason- ablywell with the DNS data of Orlandi and Fatica,6 the latter shows a much lower turbulence level. For any higher rotation rate, the model computations show complete laminarization.The same out- come was observed by other authors using different low-Reynolds- number eddy viscosity model (EVM) and RSM, as reported, for example, by Shih et al.10 (nonlinear kâ" model), Pettersson et al.11 (second-momentclosurewith Durbinâs elliptic relaxationmethod), andPoroseva et al.12 (structure-basedmodel).Recently,Orlandi and Ebstein7 performedDNS for rotation rates up to N D 10. In spite of a close resemblanceof themean axial velocityproŽ le to the laminar shape, theprocessof laminarizationis, contrarytowork ofNishibori et al.,2 still not complete.The turbulencestresses, althoughstrongly modiŽ ed, show considerable levels. The failure of all models to reproduce the laminarization at the appropriate rotation intensity indicates a deŽ ciency in the current turbulence models. The premature ow laminarization is closely connected to the negative (false) sign of the uw stress compo- nent, causing also a negative production rate of the uv shear stress: P12 ¡C12D 2uwW=r . (See section about uw anomaly.)DNS com- putations of fully developed ow in an axially rotating pipe by Orlandi and Fatica6 and Orlandi and Ebstein7 for rotation rates up to N D 10, provide more details about the turbulent stress Ž eld and shed more light on the actual process of turbulence Ž eld modiŽ ca- tion due to rotation. Stress Anisotropy The DNS results also show a weakening of the stress anisotropy with an increase in the rotation speed, in particular in the near-wall region outside the viscous zone. This is clearly indicated by an in- a) b) c) Fig. 10 Two-componentality factor, invariant map, and evolution of the coefŽ cient W = f"C"2 in an axially rotating pipe. crease in Lumleyâs two-componentalityfactor AD1¡ 98 .A2¡ A3/ (Fig. 10a). The weakening of the stress anisotropy was also con- Ž rmed by Ferziger and Shaanan.38 The peak stress component nor- mal to the rotationplane (streamwise direction)shows a continuous decrease, whereas the components in the plane of rotation (espe- cially the spanwise one) are enhanced, implying in fact a trend toward equalization of stress components. Very close to the wall, where the wall-normal stress diminishes faster and the turbulence approachesthe two-component (2-C) limit for all Ž ve rotation rates considered,the turbulenceŽ eld showsa clear tendencytoward the2- C isotropicstate,as shownbysymbolsin theLumleyâs invariantmap (IID¡A2=8, IIID A3=24) (Fig. 10b). Along the straight line (AD 0) A2D A3 C 89 ) the turbulence also has two components but is generallyanisotropic.This feature is beyond the reach of any eddy- viscosity models, but surprisingly, nor can it be fully reproduced by the second-moment closures, despite satisfactory prediction of the stress anisotropy in nonrotatingwall ows at different pressure gradients (e.g., Hanjalic´ et al.39). This indicates a deŽ ciency in the pressure-redistribution model to account for the rotation effects. Another interesting Ž nding in the axially rotating pipe ows is the modiŽ cation of the destruction term in the dissipation equation D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 JAKIRLIC´, HANJALIC´, AND TROPEA 1991 ¡9"2=k. Note that 9 DC"2 f" in conventional notation. It has al- ready been shown for homogeneous rotating turbulence,where no productionof dissipationexists,40 that9 is affected by the rotation. On the other hand, the DNS of nonrotating channel ow41 shows also a strong modiŽ cation of 9 in the near-wall region, where the turbulence approaches the 2-C state (Fig. 10c; symbols). Note that in all cases9D 0 at the wall: 9 D ¡£¡P4" ¡ Y ¢¯"2=k¤ / y4 (8) where .P4" ¡ Y // y2 represents the differencebetween the produc- tion of dissipation by self-stretchingof vortex Ž laments P4" and the destruction of dissipation due to viscosity Y . Recently Jovanovic´ et al.42 proposed to express9 as a function of the stress anisotropy invariantsin the frameworkof thehomogeneousdissipationconcept, satisfying the 2-C turbulence limit. For a fully developed nonrotat- ing channel ow, this model function, evaluated fromDNS data for anisotropy invariants,41 shows excellent agreement with the DNS data for9 (Fig. 10c; symbols and dotted line, respectively).More- over, Jovanovic´ et al.42 argue that this function should be univer- sal, irrespective of the reasons for the stress anisotropy evolution. Figure 10c shows also the variationof9 for the axially rotatingpipe ow, computedby the model of Jovanovic´ et al. using the DNS data for stress anisotropy of Orlandi and Fatica.6 Although the function does not explicitly contain any rotation parameter, Fig. 10c shows a clear increase in9 with the rotation rate. Unfortunately, the DNS data for .P4" ¡Y /, that is,9 , are not available for rotating ows, so that a direct comparisonis not possible.However, an a priori evalua- tion fromDNS data indicatesthat the9 functionof Jovanovic´ et al.42 reproduces well the effect of rotation through the modiŽ cation of the stress anisotropy. The models for 9, that is, C"2 proposed by Bardina et al.40 [Eq. (10)] andHallba¨ck and Johansson43 [Eq. (11)], resulted in a similar increasing tendency (not shown here). Swirling Flows in Long, Straight Pipes We considered two swirling ows entering long straight pipes, which were investigated experimentally by Kitoh28 and Steenbergen.29 In both cases, the proŽ le of the tangential velocity componentimposedat thepipeentrancecorrespondsto the so-called free vortex type. The only, but important, difference is in the swirl intensity, S D 2¼ Z R 0 UWr 2 dr ¼ R3U 2m For Kitohâs case SD 1:0, and for Steenbergenâs case SD 0:1. Figures 6 and 11 show the Reynolds stress and mean velocity pro- Fig. 11 ProŽ les of the tangential velocity components at selected positions in the swirling ow in a long, straight pipe, experiment of Steenbergen,29 - - - -, low-Reynolds-numberChien kâ", key as in Fig. 6. Ž les at several selected cross sections downstream of the pipe in- let for the Steenbergenâs case. Generally, the computational results obtained (except those with the Chien32 low-Reynolds-number kâ" model) are in reasonable agreement with the experiments. In the annular region (0:3< r=R< 1:0), there is a strong contributionof the negativegradientof the tangential velocity to the productionof both normal-to-the-wall and spanwise stress components,so that these two stressesaremuch higher than in the nonswirlingpipe ow. HJ low-Reynolds-number and SSG high-Reynolds-numbersecond-momentclosure computa- tions reproducedthisbehaviorverywell (Fig. 6a). In the core region, however, up to the length of about x=DD 35, all three normal stress componentsshow very high values, especially the wall-normal one. This ow region is characterized by a very high production rate (production plus convection): P22 ¡C22 D 4vwW=r . In addition to a very strong radial gradient of the tangential velocity, the contri- bution of the axial swirl decay (@U=@x > 0; @W=@x < 0) is also signiŽ cant. This region extends up to x=D¼ 35. Compared to the wall region, the agreement between computational and experimen- tal results for turbulent stresses is poor, although both the axial (not shown here) and the tangentialmean velocity components are well predicted.Approximatelyat this position, the ow evolves into a solid-body motion (Fig. 11), and the agreement is much better (Fig. 6b). This kind of transition is characteristicfor the ows with aweakerswirl intensity,as in thecaseofSteenbergenâs experiment29 and because of this, these ows are very difŽ cult to model. For the higher swirl intensities (Kitohâs experiment28), the ow remains of the free vortex type. The reason for the poor performance of the model in this region is probably because the mean shear @U=@r and, consequently, the production rate P11 at the axis of symme- try become zero, whereas most of the models were calibrated in ows with a strong mean shear. The computations yield an almost isotropic stress Ž eld: u2 ¼ v2Dw2 . This behaviorwas also found in the computationof ow in a combustor chamber (experiment by So et al.23) by Hogg and Leschziner.24 According to their conclusion, the simple linear formulation of the rapid part of the redistribution term is responsible for the underpredictionof the normal Reynolds stress components.A similar conclusionwas made by Kitoh,28 who attributes this deŽ ciency to a low value of the coefŽ cient C2 in the LaunderâReeceâRodi (LRR)44 model. However, the application of theSSGmodelwith a quadraticformulationof the rapidpartbrought no improvement, and, clearly,more detailed analysis is needed. The transition of the free vortex ow type to solid-body rotation due to swirl decayhas alreadybeenmentioned.The second-moment closures somewhat overpredicted the mean velocity proŽ les in the core region of the pipe at the position at which transition should occur x=D¼ 35. The velocity proŽ les retain their free vortex form, indicating a retarded decay of the swirl. The shear stress uv (Fig. 6) shows a direct proportionality to the gradient of the axial veloc- ity (not shown here), with a change of the sign corresponding to the shape of the velocity proŽ le. The wall-normal and spanwise stress components change the sign of the gradient also in accord with the observedvelocityproŽ les. Similar, unrealisticvelocitypro- Ž les, were obtained by Steenbergen29 when applying an algebraic stress model. It is again believed that the linear model of the rapid part might be responsible for such a behavior. The performance of some cubic models for the rapid part (e.g., Fu and Wang9), having a stronger response to the streamline curvature (expressed in terms of additional straining¡W=r , which has a very strong in uence in the pipe core), are to be tested. uw Anomaly One of the perpetuating failures of all second-moment closures when applied to swirling and rotating ows is predicting the proper sign of the shear stress in the tangential plane, uw. In the 1970s, Launder and Morse45 in their computation of swirling free jets us- ing an LRR quasi-isotropic(QI)44 second-momentclosure obtained the negative sign of uw in contrast to experimental results. This anomaly has been later conŽ rmed by several subsequent computa- tions in differentswirlingand rotating ows. For example,Oberlack et al.15 recentlyreportedthe sameproblemin a fully developedchan- nel ow with streamwise rotationwhen applying the SSG model in D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 1992 JAKIRLIC´, HANJALIC´, AND TROPEA a) b) c) Fig. 12 Fully developed ow in an axially rotating pipe: comparison between theDNS dataandresults obtainedbyHJ low-Reynolds-number RSM for a) the nw shear stress; b) the pressure redistribution, produc- tion, and convection forN = 0.61; and c) DNS budget of the uw equation for N =10. conjunctionwith an elliptic relaxationmethod. [This was also con- Ž rmed by our computations using the Reynolds stress (RS) models considered here.] We also obtained the negative uw in the axially rotating fully developedpipe ow, irrespectiveof the model type used (Fig. 12a), whereas the analysis of the exact equation for the stress uw (Eggels et al.5) suggests a positive sign. However, the opposite signs in pre- dictions and experiments of uw seem to appear only in fully devel- oped owsor not far fromthatcondition.In thedevelopingpartof the ow in an axially rotatingpipe studied experimentallybyKikuyama et al.,1 the HJ second-moment closure yielded both the (negative) sign and amplitude in accordwith experiments (Fig. 13a), but at the end of the rotatingpipe (28.5D from inlet),where the conditionsare very close to fully developed, the experiments show a tendency to change the sign (Fig. 13b). The negative sign was obtained also by Imao et al.4 in their laser Doppler velocimetrymeasurementsof the meanvelocitiesand turbulent uctuationsin an axially rotatingpipe. They concluded that a decrease in the U velocity and an increase in the W velocity in the radial direction (opposite signs of the ve- a) b) Fig. 13 ProŽ les of the tangentialvelocity and the shear stress uw in the developing ow in an axially rotating pipe. locity gradients @U=@r and @W=@r ) lead to opposite signs of their uctuations u and w and, hence, the likely negative sign of their correlation uw. The origin of this anomaly is attributed to the inadequacy of the model of the pressure-strainterm813, as seen in Fig. 12b. The (neg- ative) productionP13 (D¡vw @U=@r ¡ uv @W=@r) and convection ¡C13 (D¡uvW=r), which balance the positive813 in the equation for uw (see also Fig. 12c for very high rotation rate), are reproduced well in accord with the DNS results. Note that neither P13 nor C13 are explicitly in uenced by the uw. The model yields positive, but very much underpredicted,813. (Note that DNS data are shown for the velocity pressure gradient correlation513, which differs from 813 only for the pressurediffusion term, here considered to be neg- ligible in the bulk of the ow cross section.) The solid line shows the predictionsusing the full HJ model, that is, with computed neg- ative uw, whereas the broken line represents a priori evaluation of 813 using DNS data for all variables (including positive uw) in the expression for the model of 813 . Therefore, irrespective of the sign of uw, the model of813 is unsatisfactory,and it is even worse when uw> 0 from DNS is used. Because of negative production and convection, the underpredictionof813 results in negative uw. For the axially rotating pipe ow considered here, only data for the complete 8i j are available, and so it is not possible to diag- nose precisely whether the main deŽ ciency lies in its slow, rapid, or wall-re ection part. The simple linear model of the slow part 813;1 D¡C1"uw=k used here would imply that 813;1 should be negative, hence of an opposite sign from the total813. Application of the nonlinear formulation [Eq. (5)] in the framework of the SSG model and also in combination with the HJ low-Reynolds-number RSM did not result in any improvement. There is not much one can do to change the sign of the slow part because all model forms pro- posed in the literature (linear, quadratic) are based on a relatively sound hypothesis of return to isotropy of anisotropic turbulence in the absence of external forcing.This means that813;1 is most prob- ably negative, and the rapid part813;2 should be positive and much larger than predicted by the GL and SSG models. D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 JAKIRLIC´, HANJALIC´, AND TROPEA 1993 Fig.14 ProŽ les ofmeanvelocities andall three shearstress components on a selected position in the swirling ow in a long straight pipe with strong swirl. LaunderandMorse45 associatedtheproblemwith themean-strain contribution to 813 that is, with its rapid part 813;2 . They used the QI model, so that the transport equation for uw (C13 D P13C 813;1 C813;2) reduces to [practicallyidenticalfor the isotropization of production (IP) formulation used in the GL model] C1 " k uw D .® ¡ 1/ ³ uv @W @r C vw@U @r ´ ¡ .1C ¯/uvW r (9) and succeeded in obtainingboth the proper sign of the stress and its intensity in a free swirl ow, in reasonablygood agreementwith the experiments by reducing ® by 40%. In the case of fully developed axially rotating pipe, such a remedy would have an opposite effect, that is, only an increase in the mean-strain contribution to the 813 would make uw positive. A numerical experiment, performed by the authors, conŽ rmed this statement. However, because of a very weak in uence of the uw stress on the mean ow and turbulence, the rate of ampliŽ cation of the mean-straineffect needed to produce the desirableuw would be much higher compared to the experience of Launder and Morse.45 Figure 13 shows that the negative sign of the uw-stress is closely connected to the positive gradient of the W velocity in the radial direction. For the ows in rotating pipes (Fig. 13), as well as in the ow regions with solid-body rotation (not shown here), this gradi- ent is always positive (with relatively weak @U=@r in this region), contributing to the negative production rate P13 . In the portions of cross section (annular regions) with the negative values of @W=@r , as in the Kitohâs28 strongly swirling ow, the production rate P13 takes a positivevalue, leadingto the positiveuw stress (Fig. 14). In a part of this annular regionwith relativelyweak strain rate contribu- tion (r=R¼ 0:85), the uw stress takes a small negative value, being in uenced mostly by additional production¡uvW=r [see Eq. (9)]. Apart from larger scatter in the experimental data (in particular in the core region), all three shear stress components show reason- able agreement with measurements related to both their signs and amplitudes. In the strongly swirling ow in a long straight pipe, Kitoh28 observed a weak in uence of this shear stress on the stress uv and consequentlyon the mean velocity Ž eld. In the fully developed ow in an axially rotating pipe, its in uence is not so strong for lower rotationintensities(N · 1:0).The computationsof these ows, with the stress uw taken fromDNS, did not result in signiŽ cant changes. However, the negative sign of this stress component causing nega- tive production rate P12 ¡C12 could signiŽ cantly contribute to the premature laminarization at moderate and higher rotation rates. From the precedinganalysis it seems obvious that the inadequate modelingof the pressure scramblingprocess,and in particularof its rapid part 8i j;2 , is the main cause for wrong prediction of the uw stress. However, the application of more advancedmodels, such as SSG quadratic model, or cubic formulations by Fu, Launder, and Tselepidakis (see Ref. 9) and Ristorcelli, Lumley, and Abid, either in conjunction with the Durbinâs elliptic relaxation method (see Ref. 11) or a structure-based model,12 did not cure this anomaly, and the problem still awaits further clariŽ cation. Fig. 15a Axial velocity proŽ les in the swirling ow in a long, straight pipe. Fig. 15b Evolution of the centerline velocity along a combustor chamber with sudden expansion. Comments on the kâ"Model In the standard kâ" model, the k and " equations are not directly sensitiveto rotationand swirl becausethere are no speciŽ c terms ac- counting for these effects.Even in ows with system rotation (when considered in rotating frame of reference), the exact source term in the Reynolds stress equations Ri j contracts to zero and disappears from the kinetic energy equation.The same is true with the " equa- tion. Hence, both the k and " equations retain the same form as in nonrotating and nonswirling ows. Of course, there are indirect effects due to the modiŽ cation of the mean velocity and pressure Ž eld, but they are usually too weak to account fully for the effect of rotation and swirl on turbulence. This can be illustrated by the computational proŽ les of the axial velocity in an axially rotating pipe ow (not shown here) or in swirling ows in combustor cham- bers and long, straight pipes (Fig. 15a), which differ marginally from the nonrotating cases, irrespective of the rotation rate or the imposed initial conditions. One of the reasons for such a behavior is the simple linear relationship between Reynolds stress and the mean rate of strain tensors, implying essentially that the eddy vis- cosity is isotropic. However, it is well known that swirl causes a strong anisotropyof both the stress and dissipation tensors, as well as a highly anisotropic eddy viscosity (e.g., Kitoh28). This leads to the failure of all models based on the conventional linear eddy- viscosity concept.Another consequenceof isotropic eddy viscosity is the âsolid-body rotationâ form of the tangential velocity proŽ le (e.g., Fig. 11), which is always obtained, independent of the ini- tial velocity proŽ le. Associated with these features is the failure of the standard kâ" model to reproduce accurately the free recircula- tion zone encountered in ows in combustor chambers (in fact a swirling ow discharginginto a sudden expansion;for example, see Lai26). However, note that in the case of higher expansion ratios (Lai investigated a case with ERD 1:5), for example, in the Roback and Johnson22 experiment (ER¼ 2:1), good agreement relating to the length of this free vortex was obtained, also by the standard kâ" model (Fig. 15b). One possible explanation for this feature is that the superposition of the two ow phenomena, ow separation D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 1994 JAKIRLIC´, HANJALIC´, AND TROPEA and swirl, which are both incorrectly predicted by the kâ" models, results in a compensation. Some improvements have been achieved with nonlinear eddy- viscosity formulations.10 With these models, the tangential veloc- ity does not follow the solid-body rotation pattern any more, but the models still do not properly account for the Reynolds stress anisotropy. Model ModiŽ cations for Rotational Effects VariousmodiŽ cations for rotationaleffects, proposed in the liter- ature, aremostly related to the eddy-viscositymodels to compensate for their insensitivity to system rotation.They are modeled in terms of rotational Richardson number (accounting for streamline curva- ture) and are usually introduced via additional term(s) in the scale supplyingequation,that is, bymodifyingitsmodelcoefŽ cients(e.g., Launder et al.46 and Howard et al.47) or in the model for turbulent viscosity (Kim and Chung48). Recently, Shur et al.49 introduced an empirical function multiplying the production rate in the eddy- viscositytransportequationof the SpalartâAllmarasmodel unifying in such a way the system-rotationand streamline-curvatureeffects. Although there is no term in the exact dissipation rate equation that accounts directly for system rotation, " is indirectly in uenced through changes in anisotropies of the dissipation correlation "i j . The dynamic equation for the tensor "i j comprises a term governed by the angular velocity vector, similar to the transport equation for turbulent stresses. This conŽ rms the necessity for the modiŽ cation of the dissipation rate equation also in the framework of second- moment closure modeling, though here the problem is less acute: The exact rotation term Ri j in the stress equation accounts a great deal for the rotational effect on the stress anisotropy and indirectly to the dynamics of ". However, for higher rotation rate this is not sufŽ cient. Here we discuss some common modiŽ cations of the " equation and their effects on the turbulence scale. The major effect of system rotation is in reduction of the energy dissipation rate ", due to the retardationof the energy transfer from large to small scales.50 The change in " naturallyaffects the turbulent stresses and the kinetic energy. Based on DNS and LES of rotating homogeneous turbulence, Bardina et al.40 proposed the following modiŽ cation of the dissipationdestructionterm, supposed to mimic the retardation of spectral energy transfer: ¡C"2 ¡ 1C C"Ãä ¢ ."2=k/; ä D Ãk="; C"à D 0:15 ¯ C"2 (10) The coefŽ cient C"2 takes its standard value 1.92. Cambon et al. 51 pointed out that this modiŽ cation leads to unrealistic nonzero k for larger decay times, contrary to the pure viscous decay at vanishing rotation numbers. Hallba¨ck and Johansson43 conŽ rmed the signiŽ - cant overpredictionof k using the Bardina et al.40 term, particularly in caseswith high rotation rates. They proposed a newmodiŽ cation of the coefŽ cient C"à in terms of Reynolds number of turbulence Ret D k2=.º"/: C"à D 0:6 p Ret C"2 .25C 2ä/ (11) Shimomura52 arrived at a similar conclusion. He also investigated the performance of the dissipation equation proposed by Hanjalic´ and Launder53 for nonrotating ows, in which the term captur- ing the in uence of irrotational straining on " was introduced: ¡2C"4 kWi jWi j (C"4 D 0:27). By transformingthis term into a rotat- ing frame of reference (Wi j !Wi j ¡ ²i jkÃk ), one obtains ¡2C"4kWi jWi j ¡ 4C"4 kÃ2 C 4C"4k²i j k @Ui @x j Ãk (12) The analytical solution for k for homogeneous turbulent ow with- outmean shear(Dk=Dt D¡") obtainedby introducingthis terminto the dissipation equation (D"=Dt D¡C"2"2=k¡ 4C"4 kÃ2) yields a sinusoidal behavior causing negative values of ". This formulation was made realizable by expressing the coefŽ cientC"4 as a function ofä. This remedy could Ž nally be interpretedas a modiŽ cation of the destruction term in the dissipation equation (4C"4 !C"à ): ¡C"2 ¡ 1CC"Ãä2 ¢ ."2=k/; C"à D 0:1 ¯£ C"2 ¡ 1C 0:1ä2¢¤ (13) This model formulation was shown to be superior to the Bardina et al.40 model in computation of several homogeneous ow cases. Recently,Rubinsteinand Zhou54 proposeda furthermodiŽ cation of the dissipation destruction term: ¡C"2 ¡ 1C C"Ãä2 ¢ 1 2 ."2=k/ (14) This modiŽ cation arose from the interpolation of the series expan- sions of the destruction term under the conditions of weak rotation (expansion in positive powers ofÃ) and strong rotation (expansion in negative powers of Ã). No proposition for the coefŽ cient C"à , nor computational validation, were reported. Our test with model (14) did not produce the desirable improvement. In view of the fact that the rotational effects increase progressively with the rotation intensity, it seems more likely that an exponent in Eq. (14) should be higher than 1, instead of 1 2 , to capture the nonlinear rotational effects. All modiŽ cations discussed were originally developed for the rotating homogeneous ows without mean rate of strain. Hence, they necessarily focus on the destruction term as the only existing term in the source of the " equation.An increase in its magnitude in proportionto the rotationintensitywill reduce" and retard the decay of k as compared with the standard kâ" model, in accord with the DNS and experimental Ž ndings. However, for more general wall- bounded rotating and swirling ows, in which the rotation rate can varywith position, the effect of themodiŽ cations discussed is not so clear cut. Bardina et al.40 argued that in such ows the magnitudeof the system rotation vector X [D .ÃkÃk /1=2] should be replaced by .Wi jWi j=2/ 1=2 . However, such a term remains active also in ows without rotation, and it would deteriorate the predictions of ows that arewell reproducedeven by the standardkâ" or Reynoldsstress (RST) models. In fact, introducingsuch a term resembles the earlier discussed proposal of Hanjalic and Launder,53 who showed that inclusion of an additional term containing mean vorticity may be beneŽ cial in nonequilibrium ows without rotation, but requires readjustment of other coefŽ cient(s) to reproduce the canonic test cases in which the models are usually calibrated. This was also demonstrated by Jones and Pascau,25 who showed that the Bardina et al.40 term deteriorates signiŽ cantly the prediction of a conŽ ned swirl ow as comparedwith the standardRST model. Hence, it can be concludedthat thesemodiŽ cationsare not very suitable for ows in the blade passages of turbomachinery, which are featured by a rotation-induced asymmetric in uence on the turbulence. In spite of this, these terms have been used by some researcherssupposedly to capture rotational effects (e.g., Jones et al.55). Another model dissipation equation proposed by Shimomura56 contains an additional source termCÃk²i jk@Ui=@x jÃk with the co- efŽ cient CÃD 0:074 being theoretically determined. Note that this term actually represents a part of the term proposed by Hanjalic´ and Launder53 [Eq. (12)]. It is recalled here that some of the low- Reynolds number second-momentclosures(e.g., Craft andHanjalic´ and Jakirlic´35) when applied to the computation of the rotating Poiseuille ow at relatively low bulk Reynolds numbers, for exam- ple, Rem ¼ 6000 by DNS of Kristoffersenand Andersson,14 lead to premature ow laminarization19 (privatecommunication,T. J.Craft, 1998) as a consequenceof the combined in uence of viscosity and system rotation.This does not happenat higher bulkReynolds num- bers. See earlier section on rotating channel ows. Introduction of such a term preventspremature laminarization,but at the same time deteriorates the results for lower rotation numbers. As an illustra- tion of its effect on the turbulent stresses, the computational results of the rotating Poiseuille ow by HJ low Reynolds number RSM, with addition of the Shimomuraâs56 term with model coefŽ cient en- larged to 0.12 are shown in Fig. 16. The value 0.12 is still too low to prevent the laminarization at the highest rotation number available, RoD 0:5. As in homogeneous ows, the addition of this term causes an increase in the destruction term of " at the suction side. However, because of zero convection and negligible diffusion of k in fully developedchannel ow, " is completely in balancewith the energy D ow nl oa de d by C O LU M BI A U N IV ER SI TY o n A pr il 9, 2 01 3 | ht tp: //a rc. aia a.o rg | D OI : 1 0.2 514 /2. 156 0 JAKIRLIC´, HANJALIC´, AND TROPEA 1995 Fig.16 ProŽ les ofReynoldsstress componentsin therotatingPoiseuille ow obtained using Shimomuraâs56 term (CX = 0.12). Fig.17 ProŽ les ofReynoldsstress componentsin therotatingPoiseuille ow; effect of Shimomuraâs56 term (CX = ¡ ¡ 0.12). production Pk . (Note that the molecular diffusion is in uential only in the immediatewall vicinityfor yC· 2.)Hence,unlike in homoge- neous ows without production,here an increase in the destruction term of " will lead also to an increase in " and, consequently, to an increasein productionrateof k in the largestportionof the ow cross section. This leads to an enhancement in the turbulence intensity, which prevents or delays the laminarization. It is noted that, unlike other terms already discussed the term ²i j k@Ui=@x j differentiates the orientationof the rotation rate vector, modifying the turbulence energy selectively depending on the sign of the velocitygradient.For this reason, the termhas oftenbeenused in the framework of different kâ" models and second-momentclo- suresfor rotating ows, for example,Shimomura,56 Wizman et al.,18 Pettersson and Andersson,20 and Nagano and Hattori.57 However, they all used a negative value of the coefŽ cient Cà . This substan- tially increases the sensitivity of some models to the rotational ef- fects. This is particularly the case when the kâ" modeling concept is employed. Without this term, completely symmetric proŽ les of mean and turbulence ow properties were obtained, (e.g., Nagano and Hattori57). The effect on turbulence is now opposite to the sit- uation explained earlier, meaning that the turbulence level on the suction side will be reduced (Fig. 17). A slight enhancement of the wall-normal stress component is also captured, pointing to the ef- fect of the isotropizationof the near-wall turbulencedue to rotation. (See Figs. 10a and 10b and correspondingexplanation.) We may conclude here, that despite the theoretical derivation of Shimomura,56 a negative Cà yields the desired improvements in capturing asymmetric effects of rotation in rotating channel ows at higher Reynolds numbers. At low Reynolds numbers, a positive Cà would be more desirable to avoid premature laminarization,but this is probably the consequence of inappropriatemodeling of the viscous effects. IV. Conclusions The potential of single-point turbulence closures for predicting the ow and turbulence in swirling and rotating conŽ ned ows was investigatedby consideringseveral ows with different rotationand swirl intensities and rotation vector orientation. The ows consid- ered includedshort geometries (combustorchamber type) and long, straight pipes, fully developed and developing ows in axially ro- tating pipes, and ows in channels with streamwise and spanwise rotation. The following general and speciŽ c conclusionshave emerged: 1) The low-Reynolds-number version of the Reynolds stress model is superior to other models tested, particularly for ows in rotating pipes and for higher swirl intensities,where transitionphe- nomenaareobserved.Themajor advantageof thismodel is its ability to capture the stress anisotropy in the near-wall region, which ap- pears to be a necessary prerequisite for reproducing these types of ow. 2) The application of the standard kâ" high-Reynolds-number model and its low-Reynolds-numberextensions in swirling and ro- tating pipe ows results in a solid-body rotation ow, thus failing to re ect important features of the ows considered. 3) Good agreementwith experimentwas found for the combustor geometry,whereas in long pipes, the swirl decay was somewhat too slow. For weak swirling ows, the circumferential velocity proŽ les remain too long of the free vortex type. 4) The negative sign of the shear stress uw, obtained in (nearly) fully developed ows, is in contradiction with the DNS results. This has a very weak in uence on other ow quantities for lower rotation rates. However, at higher rotation intensity, the negativeuw causes negative production of the uv stress, leading eventually to the premature ow relaminarization. 5) A tendencytoward laminarlikesolutionsfor the ows in axially rotating pipes and channels with spanwise rotation was observed, particularly at lower bulk Reynolds numbers (based on axial ve- locity) and at higher rotation rates, indicating a need for further reŽ nement of the dissipation equation. 6) The deŽ ciency of wall functions becomes more pronounced when the swirl numbers that is, rotation rates increase. In general, if the ow is driven by the near wall phenomena, for example, fric- tion, as in the rotating pipes and channels with both stationary and moving walls, and, in particular, if transition phenomena occurs, the wall function concept was proven to be totally unsuitable. If the extra strain rates causing nonequilibrium effects arise from the inner part of the ow as in swirling ows entering pipes or cylin- ders, and in particular, if the bulk Reynolds number is high, the wall functionconceptrepresentsa reasonablealternativefor treatingwall boundaries. Acknowledgments We thankP. Orlandi and J. Eggels formakingavaliabletheir DNS data for the fully developed ow in an axially rotating pipe.We are grateful for the numerous speciŽ c comments and suggestions from the reviewers. 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