Entropy generation in bypass transitional boundary layer flows *

Transcription

1 669 4,6(5): DOI:.6/S-658(4)675-5 Entropy generation in bypass transitional bounary layer lows * EORE Joseph, OWEN Lanon D., XIN Tao Department o Mechanical Engineering, ollege o Engineering, University o Iaho, Moscow, Iaho 83843, USA, geor635@vanals.uiaho.eu MELIOT Donal M. University o Iaho, Iaho Falls, Iaho 834, USA REPEAU John. Department o Mechanical Engineering, ollege o Engineering, University o Iaho, Moscow, Iaho 83843, USA BUDWI Ralph S. University o Iaho, Boise, Iaho 837, USA NOLAN Kevin P. Imperial ollege Lonon, Lonon SW7-8Z, UK (ceive June, 4, vise September 5, 4) Abstract: The primary objective o this stuy is to evaluate the accuracy o using computational lui ynamics (FD) turbulence moels to preict entropy generation rates in bypass transitional bounary layers lows uner zero an averse pressure graients. Entropy generation rates in such lows are evaluate employing the commercial FD sotware, ANSYS FLUENT. Various turbulence an transitional moels are assesse by comparing their results with the irect numerical simulation (DNS) ata an two recent FD stuies. A solution veriication stuy is conucte on three systematically reine meshes. The actor o saety metho is use to estimate the numerical error an gri uncertainties. Monotonic convergence is achieve or all simulations. The ynols number base on momentum thicness,, sin-riction coeicient,, approimate entropy generation rates, S, issipation coeicient,, an the intermittency,, are calculate or bypass transition simulations. All ynols average Navier-Stoes (RANS) turbulence an transitional moels show improvement over previous FD results in preicting onset o transition. The transition SST - 4 equation moel shows closest agreement with DNS ata or all low conitions in this stuy ue to a much iner gri an more accurate inlet bounary conitions. The other RANS moels preict an early onset o transition an higher bounary layer entropy generation rates than the DNS shows. Key wors: entropy generation, bypass transition, ynols average Navier-Stoes (RANS), transitional bounary layer, turbulence moels Introuction Entropy is the property that serves as a measure o isorer within a system. Entropy generation thereore causes irreversible loss o energy in lui lows. Determining an minimizing these losses improves the eiciency o a system []. Systems that beneit rom the minimization o entropy generation inclue: coo- * Biography: EORE Joseph (986-), Male, Master aniate orresponing author: XIN Tao, ing@uiaho.eu ling systems or electronic evices an nuclear reactors, thermal heat echangers, an more. Four ierent mechanisms contribute to entropy generation: Mean an luctuating heat lu an Mean an luctuating viscous eects. Steay, unheate, laminar low has zero luctuations so the entropy generation occurs only rom the viscous losses associate with mean velocity graients. Bypass transition occurs when reestream vortical isturbances inuce transition to turbulence in a bounary layer without the intervention o viscous Tollmien-Schlichting waves []. Viscous losses associate with the mean an luctuating velocity graients cause entropy generation in bypass transitional

2 67 bounary layer lows. Many ierent methos eist to preict entropy generation in lui systems. Direct numerical simulation (DNS) is a proven tool in eluciating low physics. DNS completely resolves all o the laminar an turbulent length scales an thus can be use as a numerical benchmar to evaluate the accuracy o simulations using various turbulence moels. McEligot et al. [3] analyze DNS results rom two ierent stuies conucte by Spalart [4,5] o turbulent bounary layer lows with zero an avorable pressure graients with ranging rom 3 to 4. The stuy oun that approimately two-thirs o the entropy generation occurs in the viscous layer o a turbulent bounary layer (eine as y 3). The stuy emonstrate that entropy issipation is nearly universal within the viscous layer o turbulent bounary layer lows with zero an avorable pressure graients. The stuy showe that the methoology evelope by Rotta [6] or approimating S is inaccurate or the given low characteristics. McEligot et al. [7] similarly analyze results rom a DNS [8] o turbulent channel low with zero an avorable pressure graients. McEligot compare two methos or etermining entropy generation. The irst metho evaluate the luctuating graients orming the issipation term in the turbulent enthalpy equation an the secon metho evaluate an approimate analogy to laminar low employing assume bounary layer (an other) approimations [9]. Both methos preict similar S values. The secon metho uner-preicte entropy generation in the linear layer an over-preicte entropy generation in the rest o the viscous layer. Another stuy by McEligot et al. [] compare the entropy generation preicte rom a DNS o turbulent bounary layer low to the entropy generation preicte rom a DNS o channel low [8,]. The stuy emonstrate that the pointwise entropy generation at the bounary o the viscous layer is relatively insensitive or both bounary layer an channel lows with large avorable pressure graients. The integral over the area o the viscous layer ecrease moerately only or bounary layer lows. Walsh an McEligot [] improve an eisting correlation or the issipation coeicient,, using ata rom multiple DNS stuies o low turbulent bounary layer an channel lows with zero an avorable pressure graients [4,8,3,4]. Walsh et al. [] analyze a DNS o bypass transitional bounary layer lows or ranging rom 5 to 5 [5,6]. The stuy emonstrate that the term or turbulent convection in the turbulent inetic energy (TKE) balance is signiicant within the transition region. This is as a consequence o more turbulent energy being prouce than issipate. The stuy showe that a popular approimation metho over-estimates the issipation coeicient by as much as 7%. The stuy emonstrate that the approach evelope by Rotta [6] is more accurate or transitional bounary layers. A DNS o bypass transition was perorme by Zai an Durbin []. This simulation showe that highrequency, reestream luctuations are ept rom entering the bounary layer ue to shear sheltering. The stuy evaluate the coupling coeicient between continuous spectrum Orr-Someriel an squire moes. The stuy emonstrate that a strongly an wealy couple high-requency moe is require to simulate the transition process completely. The bypass transition simulations here are compare to a DNS by Nolan an Zai [7]. The DNS stuy use a computational omain size o ( L, Ly, Lz)/ (9,4,3) with a gri resolution o ( n, ny, n z) (37,9,9). The spatial resolution was (, y, z ) (.7,.4,6). The inlet mean velocity Blasius proile was create base on 8 with a turbulent intensity o 3%. The stuy trace own turbulent spots resulting rom high-amplitue streas upstream. The stuy oun that the volumetric growth rate o turbulent spots is insensitive to the pressure graient. Two recent FD stuies by hasemi et al. [8,9] evaluate the accuracy o ierent turbulence an transitional moels or preicting bounary layer behavior an entropy generation in bypass transition, incluing the - moel, the SST - moel, the - 4 equation moel, the -l- 3 equation moel, an the ynols stress moel (RSM). The mesh use in the stuy ha gri points on a two-imensional, zero pressure graient omain equal to L / 9 in the streamwise irection an an averse pressure graient omain equal to L / 6 in the streamwise irection. The inlet bounary conition speciie a turbulent intensity o 3% with a turbulent length scale equal to the bounary layer thicness at the inlet. These stuies showe that the RANS moels preict the onset o transition much earlier than the corresponing DNS [7]. The RANS moels over-preicte the integral entropy generation rate an the sin riction coeicient in the transition region. The objective o the current stuy is to evaluate the accuracy o various turbulence moels to preict entropy generation an location o transition within a bypass transitional bounary layer. The commercial FD sotware ANSYS FLUENT is employe or simulations. The low moele with RANS turbulence moel is steay, incompressible, two-imensional bypass transitional bounary layer low. The RANS moels employe in the stuy are the - moel, - SST moel, RSM moel an transitional 4 equation SST - moel. Quantitative solution veriication is conucte using three systematically reine structu-

3 67 re gris, with the inest gri containing about 6 gri points. The low characteristics are compare to the DNS results rom Nolan an Zai [7] an two recent FD stuies by hasemi et al. [8,9].. omputational methos. Turbulence moels an numerical methos The non-linear ynols stress term is close in the RANS moels with the Boussinesq ey viscosity hypothesis. The isplacement thicness, momentum thicness, corresponing ynols number an the ynols stresses are calculate as, * u y U, s U s{} u u i j uu i j ij t 3 j i u u y U s U, s () () where the variable u i is the velocity along the, y or z ais an u is the velocity along an ais ie- j rent rom the irection o u i. This similarly applies to i as the location along a given ais, y or z. The variable ij in the equation is the Kronecer elta an not the bounary layer thicness. The turbulent inetic energy,, is eine as, u v w (3) The transport equations or the ierent moels can be summarize as, t ( ) ( ui) Y t i j j (4) ( [ DV ]) ( [ DV ] ui ) t i t [ DV ] Y D j D j D D D (5) where [ DV ] is the corresponing turbulence issipation variable or the moel. The ormulations or the - moel are escribe in the ANSYS FLUENT Theory uie []. The transport equations or the - moel require the ollowing, u [ DV ], u iuj D.44, Y D i j, Y,.9, D D,.,.3 (6) D where the turbulent viscosity is calculate as,.9 (7) t are, The transport equations or the SST - moel u j [ DV ], min u iuj,y, i Y.9, D,, D F F.76. t YD, F F..68 The turbulent viscosity is calculate as, t SF T ma,.3 (8) (9) In these equations, not all the variables are constants as is the case or the - moel. The transition SST moel couples two aitional transport equations with the SST - transport equations. The irst aitional transport equation is or the intermittency,, eine as, t ( ) ( uj ) P t j j j E P E ()

4 67 The onset o transition is controlle by, ys T V where, R T S T is the strain rate magnitue given as, ().3 Iterative an statistical convergence Statistical convergence o the running mean on the time history o the resistance establishes statistically stationary unsteay solutions []. Statistical convergence or the unsteay simulation is etermine using the rag coeicient,, eine as, D ST SijS ij, S ij u u i j j i () The secon aitional transport equation or the transition momentum thicness ynols number, is, t t ( ) ( u ) ( ) P t j j j (3) where t j t t t p is a proprietary empirical correlation or the transition onset an F t is a unction base on the bounary layer correlations. The transport equation or the RSM is, ( uu i j) ( uuu i j) [ uuu i j t p( ju i iuj )] ( u i uj ) uj u i u u i j uu i uu j p j i u u i j ( uu j m im uu i m jm) (4) where im an jm are permutation symbols. More inormation on the moels is available within the ANSYS FLUENT Theory uie [].. High perormance computing A thir-orer MUSL scheme is applie or the momentum an turbulence solvers with the pressurevelocity couple scheme. A convergence tolerance o is set or all simulations to ensure the iterative errors are much smaller than the gri errors such that the ormer can be neglecte. Simulations are conucte using several local worstations an on University o Iaho s HP computing resource, Big-STEM, using 8- core PU s. sults are post-processe using Scilab-5.4. an Tecplot D (5) The rag coeicient is monitore uring the simulation. Data are collecte once the rag coeicient oscillations aroun the mean value vary by only % o the mean value..4 Solution veriication metho Solution veriication is important to estimate the numerical errors an gri uncertainties o a FD simulation. Numerical errors are ue to the numerical solution o the mathematical equations. Aspects o the simulation that cause numerical errors inclue: iscretization, artiicial issipation, incomplete iterative an gri convergence, an computer roun-o. To etermine numerical errors generally involves perorming a sensitivity stuy by varying the mesh spacing an/or time step size to a smaller value an evaluating the solution ierences. Here S, S, S 3 represent the ine, meium, an coarse gri solutions o any variable in the simulations, respectively. The relative percentage ierence ( %) between FD results an correlation values, represente below as A, is calculate as, A S A % % (6) The solution veriication metho in place is the actor o saety metho [,3] which requires the use o the ollowing equations with the use o L norm or proiles [4], S S (7) 3 S3 S (8) R p where 3 3 ln ln( r ) r is the gri reinement ratio, (9) () / an

5 673 /, ## is the hange between Ŝ # or ie- 3 rent gris, S # is the value o a given variable with the gri reinement speciie in the subscript. Monotonic convergence is achieve when ( R ). The ratio o the estimate orer o accuracy to the theoretical orer o accuracy o the numerical scheme is eine as, P where p p th () p is the proile average orer o accuracy, p th is the theoretical orer o accuracy, R is the proile average convergence ratio. The closer P is to the closer the FD simulation to the asymptotic range. The estimate error ( RE ) an gri uncertainty ( U ) are eine as, r RE p () U [.6P.45( P)], P (3a) U [.6P4.8( P )], P (3b) RE RE where the gri uncertainty, U, is presente as a percentage o the correlation value or value rom the ine gri solution at the same streamwise location. A lower magnitue o U usually inicates a better quality o FD results..5 Analysis metho The viscous issipation or the mean velocity proile is the only contributor to entropy generation in laminar low []. Thereore, pointwise entropy generation rate equation applie or steay, two-imensional, laminar bounary layer lows without signiicant luctuations is, U TS{} y y (4) However, the low consiere herein is unheate. Hence, the entropy generation occurs only ue to the square o the graients o the mean streamwise velocity. The integral over the bounary layer o the pointwise entropy generation rate provies the entropy generation rate per unit area, (5) TS S y The issipation coeicient,, is a imensionless variable that represents the entropy generation rate per unit area. The correlation by McEligot an Walsh estimate the issipation coeicient multiplie by as, (6) Both the isplacement thicness an momentum thicness are integrate to in place o the upper ineinite boun. Fluctuations in bypass transitional lows necessitate aitional terms to the entropy generation equations use or laminar low. These equations are outline urther by Walsh et al. [].The imensionless entropy generation rate per unit area or a transitional low is calculate as, U U ( S { }) y ( uv ) y y y [( ) ( ) ] U u v y U ( q ) y (7) The variables in Eq.(7), are eine as,, u ( S ) TS 3, ( S ) T S 4 u U U w, u u, u, y yu, q u v w (8) where u, v, w are the velocity luctuations in the, y an z irections, respectively. The imensionless orm o Eq.(7) is the issipation coeicient, 3/ ( S { }) where as, (9), the sin-riction coeicient, is calculate U {} w s (3) Intermittency is a measure or etermining the laminar, transition, an turbulent regions o the low an is calculate as,

6 674,turb,lam,lam (3) where the sin riction coeicient variables or the laminar an turbulent regions are calculate respectively as,.664,lam,.5,turb gth,.455 ln (.6 ) (3) The intermittency is compare to transition len- e s s (33) where the value or the beginning o transition, s, is when.5 an the value or the en o transition, g, is when.95. graient (ZP) an averse pressure graient (AP) cases, respectively. A curve top wall is use to generate the esire pressure graients within the low iel. The curvature o the top wall is etracte rom the DNS gri points an the top wall shape matches the geometry o the DNS [7]. An outlow bounary conition is applie to the omain outlet. A slip wall bounary conition is applie at the top wall whereas a no-slip wall bounary is applie at the plate surace. A velocity inlet bounary conitions is applie to the omain inlet by speciying a mean inlet Blasius velocity proile at U s / v 8. The mean proiles o velocities ( uv, ) an turbulent structure properties ( an or ) are speciie as bounary conition at the inlet to the omain an the inlet turbulence is base on the mean ynols stresses rom the DNS mean statistics. The DNS employs an unsteay inlet using Orr-Somerile an squire moes whereas the current stuy is at steay state. The imensionless inlet proiles are shown in Fig.-Fig.3. Fig. Mean velocity proile at inlet Fig.3 ynols shear stress proile at inlet Fig.4 eometry an mesh representation Fig. ynols normal stress proiles at inlet. Simulation esign an veriication. eometry an low conitions The length o the plate, L, is the same as the DNS, i.e., L / 9 an 6 or the zero pressure The an values at the inlet an or all moels, are estimate using the equations rom the ANSYS FLUENT User s uie [5] as,.9.4 3/4 3/ /4.9,.4 (34) While the ynols shear stresses are speciie irectly at the inlet or the RSM. The use o the inlet mean

7 675 Table Simulation esign table Flow type Viscous moels L Ly Lz (m) n ny nz y ZP ( AP wea :.8, AP strong :.4) K -, SST K -, RSM, 4 equation K - SST K -, SST K -, RSM, 4 equation K - SST proiles rom DNS in the current stuy is more accurate bounary conitions compare to those applie by hasemi et al. [8] wherein the inlet bounary conition is speciie with a constant turbulent intensity o 3% an a turbulent length scale equal to the bounary layer thicness.. Mesh an simulation table The mesh is create in Pointwise v7.r. The gri points in the streamwise irection are uniorm an the gri points in the plate-normal irection are clustere near the plate surace, as shown in Fig.4. To assign more gri points towar the wall ensures that enough gri points eist within the bounary layer to capture the high velocity graients in the bounary layer. A meium mesh an a coarse mesh were create or the solution veriication stuy using a constant gri reinement ratio. Figure 4 shows a schematic representation o the omain with an eaggerate curvature o top wall. The igure is a representation o an averse pressure graient geometry an mesh. The coorinate ais an bounaries are labele. A general overview o the ierent simulations perorme in this stuy is containe in Table. 3. sults an iscussion The bypass transition simulation results are compare with the DNS results rom Nolan an Zai [7]. Aitionaly, the ZP results are compare to the FD results by hasemi et al. [8] an AP results with hasemi et al. [9]. The current simulations employ a more accurate inlet conitions an much iner mesh than the simulations by hasemi et al. The,, z, proiles an ynols stress values are prescribe at the inlet, epening on the moel in use, to match the conitions o the DNS simulation. hasemi et al. applie a velocity inlet bounary with a speciie turbulent intensity o 3% an a turbulent length scale, whereas the mean velocity an turbulent structure proiles obtaine rom DNS [7] ata are speciie at the inlet in the current stuy. Aitionally, this stuy also eamines both FD preictions or entropy generation rates compare to that post-processe rom DNS results. Table Solution veriication or bypass transitional bounary layer low R P P U (% ) S Solution veriication The results rom the solution veriication stuy or the - moel are shown in Table. The istance to the asymptotic range ( P ) is shorter or than. Monotonic convergence is achieve. The gri uncertainty is below.6%s or both variables. The solution veriication stuy shows that the bypass transition results are inepenent o the gri resolution an thus all results are presente on the ine gri. Fig.5 versus 3. Zero pressure graient (ZP) Figure 5 shows how varies with. Figure 6 shows a more etaile view near the inlet region. The DNS ata is linear in the log-log scale, the slope remains small up to 45 an then the slope becomes much steeper inicating the onset o transition rom laminar to turbulent bounary layer proile. The - moel ollows the DNS results near the inlet until 8 where it transitions to tur-

8 676 bulence. However, in that range results rom ashemi et al. [8] show a steep slope an transition right at the inlet an much urther away rom DNS results. The RSM moel shows similar tren lie the - moel but the results rom ashemi et al. are urther away rom the DNS. throughout the low iel. Fig.7 versus Fig.6 versus (etaile view near inlet) The - moel shows very close agreement to the DNS until 5 an then eviates ownstream whereas ashemi et al. shows transition much earlier at 85 an has a much larger ierence in magnitue o. The - 4 equation moel results closely resemble the DNS results compare to all other simulations but the magnitue remains slightly lower than DNS ata throughout the omain an the change in slope occurs at 575 later than 475 rom DNS. The - 4 equation moel or ashemi et al. shows transition much earlier than DNS at 35. The results in this stuy agree much better with the DNS results than the results rom hasemi et al. [8] in terms o overall magnitue o preicte an the location o transition or all corresponing moels. Figure 7 an Fig.8 show how an vary with, respectively. The issipation coeicient,, provies a measure o the pointwise entropy generation rate, S (in non-imensional orm), within the bounary layer or ZP case as escribe earlier. The DNS ata has a linear slope in the turbulent regime. The laminar region is the initial ownwar slope, the rise inicates the transition region, an the small oscillations ownstream are within the ully turbulent region. Figure 7 also shows the analytical laminar an turbulent lines. Similar to the trens seen in Fig.5, the - moel an RSM transition to turbulent proile very close to the inlet an remain turbulent Fig.8 versus In Fig.7, the - moel shows an initial laminar proile near the inlet, similar to DNS, beore transition occurs ownstream. The - moel shows a laminar region until 5, where the onset o transition is preicte by the moel. The - moel shows close agreement o preicte to the DNS ata rom the inlet until the onset o transition an also in the turbulent region but transition occurs upstream compare to the - 4 equation moel an the DNS ata. The - 4 equation moel shows better agreement with the DNS ata or both an. The an preicte by the - 4 equation moel is very accurate compare to the DNS ata until the transitional point in DNS at 45. The moel, however, over preicts the location o the onset o transition which occurs much later than DNS at 575.

9 677 by as much as % in the ully turbulent region. All moels ten to preict a much steeper slope in the transition region compare to the much smoother slope in the DNS ata. Fig.9 versus Fig. versus AP wea or various RANS moels or Figure 9 shows that all turbulence moels eamine in this stuy preict transition onset (.5) earlier than the DNS ata. The -, -, an RSM s emonstrate very similar trens with steeper slopes than the - 4 equation moels. The - 4 equation moel is the closest to the DNS ata in preicting the transition onset location but over-preicts 3. Averse pressure graient (AP) The bypass transition simulation results or AP cases are also compare to the DNS results rom Nolan an Zai [7] an the FD results by hasemi et al. [9]. The AP results are evaluate using the sinriction coeicient,, an the approimate pointwise entropy generation rate, S. 3.. AP wea :.8 Figure shows comparison o preicte rom various moels along the length o the lat plate versus on a log-log scale or the wea AP case. The rom DNS eviates rom the analytical Blasius laminar approimation at about an is lower in the laminar region than the analytical approimation. The DNS preicts the onset o transition at about 4 an ully evelope turbulent low beyon 55. Figure (a) an Figure (b) show the -, RSM, - SST an transitional - 4 equation moels compare with results rom DNS an hasemi et al. [9] or the same moels. The -, RSM, - SST an - 4 equation moels preict onset o transition at 7,, an 35, respectively. All the above moels compare better with DNS in preicting transition occurrence urther ownstream than by the corresponing moels rom hasemi et al.. The transition - 4 equation moel preicts the low very accurately an ollows the DNS preiction very closely throughout the omain especially in the laminar an transition regimes. All our moels in the present stuy uner preict the sin-riction coeicient magnitue in the ully turbulent regime. Overall the results rom this stuy are comparatively more accurate an in line with the DNS results than the results rom hasemi et al. [9] in terms o trens an magnitue or all corresponing moels especially, the transitional - 4 equation moel. The uner preiction o in the ully turbulent regime compare to hasemi et al. [9] is ue to the ierences in the inlet bounary conitions speciie. Speciying a constant turbulent intensity o 3% an a speciic length scale rather a mean proile or turbulent structures as in the simulations by hasemi et al. coul result in over preiction o ully turbulent regime compare to the actual capabilities o each RANS moel.

10 678 the wall an eeping y at the irst gri point away rom the plate. The preictions rom the -, RSM an - SST are consierably closer to DNS values than hasemi et al.. The transitional - 4 equation moel is the most accurate among all moels although it slightly over-preicts the magnitue o S. Fig. S versus y or various RANS moels or near location o transition shown by values o AP wea Figure shows the comparison o approimate point-wise entropy generation rates, S, as preicte by each moel within the bounary layer plotte nor- mal to the wall in terms o y. Since ierent moels preict varying locations o transition, the entropy generation rate ( S ) comparison is mae at ierent locations along the lat plate or each moel. These lo- cations (inicate by values) are selecte at a point near the onset o transition as preicte by each moel. Figure (a) shows the - an RSM moel an Figure (b) shows the - SST an transitional - 4 equation moels compare with results rom DNS an hasemi. As seen rom the igures the preictions rom the current stuy are more accurate in terms o trens, magnitue an location than rom hasemi or all corresponing moels compare to DNS values. This is a irect result o better resolution within the bounary layer using more gri points near Fig. versus AP Strong or various RANS moels 3.. AP strong :.4 Figure shows the comparison o prei- cte rom various moels along the length o the lat plate versus on a log-log scale. The DNS [7] results show that eviates rom the Blasius laminar approimation at approimately 8 an preicts a lower value in the laminar region as seen in the previous AP wea case. The stronger averse pressure graient causes an earlier shit on preicte rom the analytical app-

11 679 roimation. The DNS preicts the onset o transition at about 33 an ully evelope turbulent low beyon 45. The Stronger AP also shows an increase the maimum magnitue o S rom. in the AP wea case to a value o.5. Figure (a) show the - an RSM moel an Fig.(b) shows the - SST an transitional - 4 eqation moels. The -, RSM, - SST an - 4 equation moels preict onset o transition at 7, 95, an 9, respectively. The - moel transitions to ully turbulent low near the inlet or the current stuy as in the stuy by hasemi et al.. This may be a result o the stronger pressure graient being impose on the low an thereby inicating the moels incapability in hanling strong averse pressure graients eectively. The RSM, - SST an - 4 equation moels ollow similar trens as seen in the AP wea case with better comparison to DNS values than that by hasemi et al. [9]. Fig.3 S versus y or various RANS moels AP Strong Following the tren seen in previous results the moels in current stuy uner preict magnitue o in the ully turbulent regime. Possible causes or such uner preiction maybe as note earlier in AP Wea results. Figure 3 shows the comparisons o preicte approimate point-wise entropy generation rate S within the bounary layer normal to the wall near the location o transition point in terms o y. It is noteworthy that, since the - moel transitions near the inlet uner the strong averse pressure graient, Fig.3(a) shows a turbulent proile o preicte entropy generation rate at 75 or this moel. The RSM, - SST an - 4 equation moels preict more accurate comparable proiles or S near their transition location than the moels by hasemi et al.. 4. onclusions an uture wor This stuy evaluates the capability o various RANS moels to preict entropy generation rates in bypass transitional bounary layer lows with an without pressure graients. The results show signiicant improvements over the RANS results by hasemi et al. [8,9] or all comparable moels ue to the employment o a much iner gri an more accurate inlet bounary conitions or velocity an turbulent structures. Overall, the results rom this stuy are more accurate an comparable to the DNS results than the results rom hasemi or both AP wea an AP strong cases with respect to trens an magnitue or all corresponing RANS moels ecept or slight uner preiction o magnitue within the ully turbulent regime. Better gri resolution within the bounary layer also helps preict the approimate pointwise entropy generation rate proiles or averse pressure graient cases more closely to DNS than hasemi et al. [9]. AP has a higher maimum value o S in the strong bounary layer than AP wea case inicating a irect relationship between the pressure graient an entropy generation rates. The results suggest that the - 4 equation moel accurately preicts the bounary layer behavior an entropy generation or bypass transitional lows. All other moels preict transition onset upstream o the location shown by the DNS ata. However, the - transition 4 equation moel slightly over preicts the onset o transition rom the DNS ata base on the sin riction coeicient or the ZP case an uner preicts in the AP cases. In the uture, the capability o using ierent

12 68 LES moels to preict entropy generation rates or bypass transitional lows with an without streamwise pressure graients will be evaluate. Sensitivity o LES moels to gri resolution an time step size will be eamine ollowing the recent general ramewor or LES veriication an valiation [6,7]. The use o unsteay hyroynamic instabilities in velocity an turbulent structure proiles at the inlet may lea to more accurate FD preictions or bypass transitional bounary layer lows or both RANS an LES moels. Acnowlegements This wor was supporte by the U.S. Department o Energy, Oice o Science, Basic Energy Sciences, uner Awar # DE-S475. The authors woul also lie to than Dr. Tamer Zai, Dr. Kevin Nolan, an Dr. Emon Walsh or meaningul contributions. erences [] WALSH E. J., MELIOT D. M. an BRANDT L. et al. Entropy generation in a bounary layer transitioning uner the inluence o reestream turbulence[j]. Journal o Fluis Engineering,, 33(6): 63. [] ZAKI T. A., DURBIN P. A. Moe interaction an the bypass route to transition[j]. Journal o Flui Mechanics, 5, 53(): 85-. [3] MELIOT D. M., WALSH E. J. an LAURIEN E. et al. Entropy generation in the viscous parts o turbulent bounary layers[j]. Journal o Fluis Engineering, 8, 3(6): 65. [4] SPALART P. R. Direct simulation o a turbulent bounary layer up to 4 [J]. Journal o Flui Mechanics, 988, 87(): [5] SPALART P. R. Numerical stuy o sin-low bounary layers[j]. Journal o Flui Mechanics, 986, 7(): [6] ROTTA J. Turbulent bounary layers in incompressible low[j]. Progress in Aerospace Sciences, 96, (): -95. [7] MELIOT D. M., WALSH E. J. an LAURIEN E. et al. Entropy generation in the viscous layer o a turbulent channel low[r]. Iaho National Laboratory (INL), 6. [8] ABE H., KAWAMURA H. an MATSUO Y. Direct numerical simulation o a ully evelope turbulent channel low with respect to the reynols number epenence[j]. Journal o Fluis Engineering,, 3(): [9] KRAUSE E., OERTEL H. J. an SHLIHTIN H. Bounary-layer theory[m]. New Yor, USA: Springer, 4. [] MELIOT D. M., NOLAN K. P. an WALSH E. J. Eects o pressure graients on entropy generation in the viscous layers o turbulent wall lows[j]. International Journal o Heat an Mass Transer, 8, 5(5-6): 4-4. [] TSUKAHARA T., SEKI Y. an KAWAMURA H. et al. DNS o turbulent channel low at very low ynols numbers[]. Proceeings o the 4th International Symposium on Turbulence an Shear Flow Phenomena. Williamsburg, USA, 5, [] WALSH E. J., MELIOT D. M. A New correlation or entropy generation in low ynols number turbulent shear layers[j]. International Journal o Flui Mechanics search, 9, 36(6): [3] ABE H., KAWAMURA H. an MATSUO Y. Surace heat-lu luctuations in a turbulent channel low up to with Pr.5 an.7[j]. International Journal o Heat an Flui Flow, 4, 5(3): [4] HOYAS S., JIMÉNEZ J. Scaling o the velocity luctuations in turbulent channels up to 3 [J]. Physics o luis, 6, 8(): 7. [5] SHLATTER P., BRANDT L. an De LANE H. et al. On strea breaown in bypass transition[j]. Physics o luis, 8, (): 55. [6] BRANDT L., SHLATTER P. an HENNINSON D. S. Transition in bounary layers subject to ree-stream turbulence[j]. Journal o Flui Mechanics, 4, 57: [7] NOLAN K., ZAKI T. A. onitional sampling o transitional bounary layers in pressure graients[j]. Journal o Flui Mechanics, 3, 78: [8] HASEMI E., MELIOT D. an NOLAN K. et al. Entropy generation in a transitional bounary layer region uner the inluence o reestream turbulence using transitional RANS moels an DNS[J]. International ommunications in Heat an Mass Transer,, 4(): -6. [9] HASEMI E., MELIOT D. M. an NOLAN K P. et al. Eects o averse an avorable pressure graients on entropy generation in a transitional bounary layer region uner the inluence o reestream turbulence[j]. International Journal o Heat an Mass Transer, 4, 77(): [] ANSYS. FLUENT theory guie v4... [R].. [] XIN T., BHUSHAN S. an STERN F. Vortical an turbulent structures or KVL at rit angle,, an 3 egrees[j]. Ocean Engineering,, 55(3): [] XIN T., STERN F. losure to Discussion o Factors o saety or Richarson etrapolation (, Journal o Fluis Engineering, 33, 55)[J]. Journal o Fluis Engineering,, 33(): 55. [3] XIN T., STERN F. Factors o saety or richarson etrapolation[j]. Journal o Fluis Engineering,, 3(6): 643. [4] WILSON R. V., STERN F. an OLEMAN H. W. et al. omprehensive approach to veriication an valiation o FD simulationspart : Application or rans simulation o a cargo/container ship[j]. Journal o Fluis Engineering,, 3(4): [5] ANSYS. FLUENT user guie v4... [R].. [6] XIN T., EORE J. Quantitative veriication an valiation o large ey simulations[]. ASME 4 Veriication an Valiation Symposium. Las Vegas, Nevaa, USA, 4. [7] XIN Tao. A general ramewor or veriication an valiation o large ey simulations (eynote speaer)[]. Proceeings o the 3th National ongress on Hyroynamics an 6th onerence on Hyroynamics. Qingao, hina, 4, 4-58.