THE NUMERICAL SIMULATION OF A 3D FLOW IN THE VKI-GENOA TURBINE CASCADE TAKING INTO ACCOUNT THE LAMINAR-TURBULENT TRANSITION

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1 THE NUMERICAL SIMULATION OF A 3D FLOW IN THE VKI-GENOA TURBINE CASCADE TAKING INTO ACCOUNT THE LAMINAR-TURBULENT TRANSITION SERGIY YERSHOV ANTON DEREVYANKO VIKTOR YAKOVLEV AND MARIA GRYZUN Insttute for Mechancal Engneerng Problems of Natonal Academy of Scences /0 Pozharsky St Kharkv Ukrane Natonal Techncal Unversty Kharkv Polytechnc Insttute Frunze St. 600 Kharkv Ukrane Abstract: Ths study presents a numercal smulaton of a 3D vscous flow n the VKI-Genoa cascade wth takng nto account the lamnar-turbulent transton. The numercal smulaton s performed usng the Reynolds-averaged Naver-Stokes equatons and the two-equaton k-ω SST turbulence model. The algebrac Producton Term Modfcaton model s used for modelng the lamnar-turbulent transton. Computatons of both fully turbulent and transtonal flows are carred out. The contours of the Mach number the turbulence knetc energy the entropy functon as well as lmtng streamlnes are presented. Our numercal results demonstrate the nfluence of the lamnar-turbulent transton on the secondary flow pattern. The comparson between the present computatonal results and the exstng expermental and numercal data shows that the proposed approach reflects suffcently the physcs of the lamnar-turbulent transton n turbne cascades. Keywords: numercal smulaton 3D flow turbne cascade lamnar-turbulent transton turbulence knetc energy secondary flows losses.. Introducton Despte the fact that turbomachnes have been known for a long tme and a large number of scentfc papers have been devoted to ther research and mprovement not all possbltes of such mprovement have been exhausted yet. The applcaton of modern methods for flow computatons usng 3D Reynolds-averaged Naver-Stokes (RANS) solvers [] permts the numercal flow feld smulaton n a turbomachnery and the 3D desgn of ther flowpathes. However f such desgn does not take nto account all phenomena of a 3D turbulent vscous compressble flow then an ncorrect estmaton of the turbomachnery effcency s practcally nevtable and therefore there remans room for ts mprovement. The lamnar-turbulent transton of a turbomachnery flow s one of such nsuffcently studed phenomena and therefore t s usually unaccounted n most studes. The effect of transton on both the knetc energy losses and the turbomachnery effcency s ambguous. As t s known the knetc energy losses are smaller n the lamnar boundary layer when compared wth that n the turbulent boundary layer of the same thckness but the former s more susceptble to separaton n whch the losses can ncrease []. On the other hand the flow acceleraton can cause a relamnarzaton and thnnng of the boundary layer. Thnner boundary layers are more resstant to separaton and even f a separaton occurs then ts thckness wll be smaller than that of a thcker turbulent boundary layer n slghtly-accelerated flow cascades. In ths case agan the knetc energy losses may be less when compared wth the fully turbulent flow case and ths fact s often used to desgn hghly-loaded cascades. Both the effect of the 3D flow pattern on lamnar-turbulent transton and the reverse effect of lamnar-turbulent transton on 3D secondary flows are practcally unstuded. Such ambguty and uncertanty of the nfluence of lamnar-turbulent transtons on the turbomachnery effcency requres a complete 3D flow nvestgaton for every flowpath desgn to evaluate ts respectve performance.

2 There exst several models that descrbe lamnar-turbulent transtons and could be used wth the RANS equatons. Suffcently detaled revews concernng transton models are presented n [3-5]. Such models are manly sem-emprcal and are not unversal so each model has a range of applcablty. Therefore the development of new transton models and ther verfcaton wth respect to a varety of turbulence models s an mportant problem. Ths study consders a numercal smulaton of a 3D transtonal flow n a turbne cascade. A smple algebrac model s chosen as a lamnar-to-turbulent transton model. It requres less computatonal resources n comparson wth approaches that ntroduce addtonal dfferental equatons [6]. The numercal solutons for two flow cases namely a fully turbulent flow and a transtonal flow are compared wth each other as well as wth the exstng expermental and numercal data.. Mathematcal model of the flow A 3D vscous compressble flow through a turbne cascade s descrbed by the RANS equatons wrtten n a local curvlnear coordnate system that rotates at a constant speed Ω: QJ F + = H t ξ where Q s the vector of conservatve varables n the Cartesan coordnates; flux vector n the curvlnear coordnates; F = Fξ J s the F s the flux vector n the Cartesan coordnates; t s tme; ξ are the curvlnear coordnates; ξ are the metrc coeffcents; H s the source term vector; J s the Jacoban of the coordnate transformaton between the Cartesan and curvlnear coordnate systems. If the rotaton occurs around the x3 -axs then the vector of conservatve varables Q the flux vector F and the source term vector H n the Cartesan coordnate system x are of the form: ρ ρu 0 ρu ρuu + δ p τ ρu Ω + ρω r Q = ρu F = + ρuu δ p τ H = J ρu Ω + ρω r ρu3 ρh ( ) ρuu3 + δ3 p τ3 0 ρh + p u τkuk + q 0 where ρ u and p are the densty the components of the velocty vector and the pressure respectvely; τ k s the effectve stress tensor; q s the effectve heat flux; γ s the specfc heat rato; r r and r are the dstance from the rotaton axs and ts proectons on the coordnate axes p u u Ω r x and x respectvely; h = + + k ; k s the turbulence knetc energy (TKE); γ ρ δ s the Kronecker delta. The metrc coeffcents and the Jacoban of the coordnate transformaton are wrtten as 3 ξ ξ ξ ξ ξ 3 k l ξ = = lm J = ξ = ξ ξ ξ = ε klξξ ξ 3 x x 0 x 3 ξ ξ ξ 3 3 where ε kl s the Lev-Cvta symbol. The effectve stress tensor represents a sum of the vscous and Reynolds (turbulent) stress tensors: ) τ = τ + τ k k 3 k

3 where τ k = µ Sk Sllδk s the vscous stress tensor; u u S k = + s the stran rate 3 x x tensor; µ = ρν s the dynamc vscosty; ν s the knematc vscosty. Smlarly the effectve heat flux s a sum of the molecular and turbulent heat fluxes: ) q = q + q T where q = λ s the molecular heat flux; q ) s the turbulent heat flux; λ s the molecular x thermal conductvty; T s the temperature. Here and further the dervatves n the Cartesan coordnates are determned through the curvlnear coordnate dervatves and vce versa accordng to the followng coordnate transformatons: ξ x = = ξ = = x. x ξ x ξ ξ x ξ x 3. Turbulence modelng For modelng turbulence n the present study we use the two-equaton k-ω SST model developed by Menter [7]. The model s wrtten n the low-reynolds form [8] that takes nto account the Producton Term Modfcaton () [9-]. The turbulence and flow equatons n the resultng system are solved separately whch enables more effcent computatonal algorthms. The dfferental equatons of the model are wrtten n the form: VJ W + = P D L t ξ where V s the vector of the conservatve turbulent varables n the Cartesan coordnates; W = Wξ J s the turbulent flux vector n the curvlnear coordnates; W s the turbulent flux vector n the Cartesan coordnates; P D and L are the source term vectors. These vectors can be wrtten as ) k ρk V = W ρω ρku = ρωu ( µ + σ µ ) k ) ( µ + σ µ ) where ω s the specfc dsspaton rate; P k and ω dsspaton respectvely; D k and dsspaton respectvely; L ω s cross-dffuson. The turbulence producton s wrtten as follows P = α ) τ S ω x P k ω Dk P = P Dk D = ω D 0 D = ω D L = ω Lω x P are producton of turbulence and producton of D ω are the dsspaton of turbulence and the dsspaton of k where ) τ = µ ) S Snnδ ρkδ 3 3 s the Reynolds stress tensor; α s the producton term modfer [0] whch equals n the case of the standard hgh- and low-reynolds models. The dsspaton producton could be wrtten n the followng form: P = αρ ) τ) S ω µ. The dsspaton of turbulence and the dsspaton of dsspaton could be defned usng the turbulence varables k and ω: D k β * = ρωk D ω = βρω.

4 The cross-dffuson term whch s usually not taken nto account n most of the k ε -type turbulence models s represented as follows k ω Lω = ( F ) ρσ ω ω x x where F s the frst Menter s blendng functon whch depends on the dstance from the wall y and s calculated usng the followng formulas: 4 k 500ν 4ρσ F = max[ tanh( arg ) F4 ] ωk arg = mn max * βωy ωy CDk ω y 8 k ω 0 R CD = k ω max ρσ ω ;0 0 ω x x y y k F 4 = exp R y =. 0 ν The correcton functon F 4 whch depends on the local turbulent Reynolds number R y s ntroduced as a necessary modfcaton for modelng the lamnar-turbulent transton and should be equal to zero when the standard hgh- and low Reynolds models are used. The dynamc turbulent vscosty s calculated usng the turbulence parameters k and ω takng nto account the Bradshaw s hypothess [7] and the realzablty constrants []: ) ρk where µ =α * * α F S max ω; ; a 3 S = S S s the magntude of the mean stran rate tensor. The second Menter s blendng S 3 S nn functon F s computed as follows k 500ν F = tanh( arg ) arg = mn *. βωy ωy * * Values α αβ βσ k σ are found usng a lnear nterpolaton: φ = ω F φ ( F ) φ * * + φ = α αβ βσ σ where the parameters φ and φ correspond to the k ω and k ω T k ε models respectvely; Menter s blendng functon F s the nterpolaton weght coeffcent here. In the standard hgh-reynolds Menter s k ω SST model the values φ and φ are constants. In the low-reynolds model some of these quanttes that are related to the k ω model and determne the behavour of the boundary layer turbulence are defned usng the local turbulent R t = k νω : Reynolds number ( ) = α * 40 + Rt /6 + R /6 t = α Rt /7 * 5/8 + ( Rt /8) β = 0.09 β 4 = R /7 + ( R /8) t σ k = 0.85 σ ω = 0. 5 α * = α = β * = β = σ k =. 0 σ ω = a = The lamnar-turbulent transton model The basc dea behnd the transton model s n the reducton of the turbulence producton n the lamnar and transtonal boundary layers by a factor of α whch can be consdered as an analogue of the turbulence ntermttency coeffcent. It s assumed that the lamnar-turbulent transton s affected by the turbulence ntensty of an external flow and the local pressure gradent. Thus the boundary layer turbulzaton s reproduced due to the mpact of a hgh external flow turbulence ntensty and of separaton or t

5 preseparaton flow condtons wth an adverse pressure gradent. The functon P tm accounts for the nfluence of the external flow turbulence ntensty and t s calculated as follows [( ) Rν ( ) Rν + (.43 0 ) Rν ] Rν < 000 P tm = c 5 [ 0. + ( 0 0 ) R ] R 000 ν y S where R ν = s the Reynolds number determned by the dstance from the nearest wall and the ν magntude of the mean stran rate tensor; c s a constant. The functon P tm estmates the nfluence of the pressure gradent: 0.4 Rν K K < 0 µ dp P tm = 80 K = ( M ) 3 0 K 0 ρ U ds where U s the flow velocty; M s the Mach number; s s the streamwse coordnate. The combned effect of the two factors s taken nto account usng the followng relaton: + Ptm = 0.94( Ptm + Ptm ) F3 th[ ( y /7) ] b where F = exp[ ( R / a ) ][ P( R )] /P( R ) s a functon that trggers the TKE producton 3 t t t when the local Reynolds number reaches ts crtcal value; + y = yu τ ν ν s the dmensonless dstance from the nearest wall; u = τ τ w ρ s the frcton velocty; 5 ( R t 3) P ( R exp w t ) = π s a functon of the turbulent Reynolds number R t ; a and b are constants. Usually the values of the constants are the followng: a = 3.45 ; b =. 0 ; c =. 0. Accordng to the physcs of transtonal flows the producton term modferα smlar to the turbulence ntermttency should be lmted between zero and one. Therefore n ths study the followng restrcton s mposed: α = mn. ( ) 5. Numercal approach The governng equatons are ntegrated numercally usng an mplct second-order accurate ENO-scheme [3] that apples an exact Remann solver to calculate fluxes at dscretzaton cell boundares. A smplfed multgrd method local tme steppng and Newton teratons [4] are used to accelerate the convergence of the proposed scheme. When performng calculatons usng a large Courant number the tme step should be adusted for excessvely elongated cells. The consdered approach has been mplemented n the CFD solver F [5 6]. 6. Methodology of carryng out the calculatons In ths study we found that the accuracy and the relablty of numercal results depend on several factors. Frst n order to model adequately the lamnar-turbulent transton t s necessary to use physcally grounded turbulence models. In partcular an mportant pont s the use of the realzablty constrants for the Reynolds stress tensor wthout whch the soluton can be nfeasble. Second n order to successfully model transtonal behavour the number of cells and the dscretzaton mesh qualty must be severely restrcted. In the transton zone of the boundary layer flow characterstcs are subected to rapd changes: the boundary layer thckness and ts profle vary sgnfcantly wthn very short dstances. Therefore t s necessary to ensure that the mesh resoluton s hgh along the streamwse and wall-normal drectons. Accordng to author s experence n computng transtonal flows an acceptable descrpton of the lamnar-turbulent P tm

6 transton n a three-dmensonal cascade requres ensurng the followng condtons: y + value for the cell nearest to the wall must be of the order of ; there must be 30 or more cells across the boundary layer n the transton zone; there must be at least 50 cells along each blade surface n the stream-wse drecton; cell szes must change smoothly. These requrements are usually realzable for computatonal meshes of sze greater than between a few mllon and a few tens of mllon cells nsde one 3D blade-to-blade passage. Thrd the convergence rate n the case of the transtonal flows s qute slow and t s recommended to use a fully turbulent flow feld as an ntal approxmaton. Such computatons requre a sgnfcant of runnng tme and t s hghly desrable to apply the numercal methods that are able to operate on very fne meshes usng the Courant number that s much greater than. 7. Computatonal results We consder the three-dmensonal flow n the cascade VKI-Genoa whch has been prevously studed expermentally [7] and computatonally [6 8]. We use a computatonal mesh of H-type wth an approxmate orthogonalzaton of mesh lnes near sold walls. The consdered mesh contans on the order of 4. mllon cells (8x8x56). The value of y + of the cell nearest to the wall s approxmately ; more than 30 cells are located n the transton regon across the boundary layer and there are 68 cells along the blade surface n the streamwse drecton wth refnement at the leadng and tralng edges. The flow through the consdered cascade s subsonc wth the ext Mach number 6 M s = 0.4 and the Reynolds number Re =.6 0. The turbulence ntensty at the nlet equals percent. The experment dd not determne the endwall boundary layer thckness n front of the cascade. Snce an assgnment of ths value s mandatory for a three-dmensonal cascade flow calculaton n the present study we arbtrarly assume that the boundary layer orgnates at the nlet boundary of the computatonal doman whch was axal chord upstream of the leadng edges. Shown n Fgure s the dstrbuton of the dmensonless adabatc speed U at the blade surface n the md-span secton. The nondmensonalzaton s performed by dvdng by the nlet velocty U 0. The surface coordnate S s measured along the blade contour from the leadng edge. The value S max corresponds to the surface length from the leadng edge to the tralng edge. The dots dsplay the expermental data [7] the black lnes present the numercal results of Langtry [6] the red and blue lnes demonstrate the numercal results for the fully turbulent and transtonal flow cases obtaned n the present study respectvely. There s a good agreement between these results for the most part of the blade surface except for the tralng edge regon. Dsplayed n Fgure are the Mach number contours n the md-span secton of the cascade whch are calculated n fully turbulent and transtonal flow cases. The behavour of man flow s smlar enough n both cases and flow structures appear to be qualtatvely and quanttatvely very close. Fgure 3 presents the dstrbuton of the dmensonless frcton velocty u = τ ρ along the blade surface n the md-span secton. Here τ w s the wall shear stress. The nondmensonalzaton s performed by dvdng by the local adabatc speed. The dots show the expermental data [7] the black lne dsplays the numercal results of Langtry [6] the red and blue lnes demonstrate the results for the fully turbulent and transtonal flow cases obtaned n the present study respectvely. We took the results of other authors shown n the Fgure from [6]. Fgure 4 gves the analogous graph but the data of other authors are taken from [8]. As can be seen the nterpretatons of the expermental results n [6] and [8] are somewhat dfferent. Nevertheless the followng ponts can be concluded: the transton occurs n the non-separated boundary layer at the blade sucton sde of the VKI- Genoa cascade; the model of the lamnar-turbulent transton that s used n the present study reflects qualtatvely correctly the transton phenomenon; τ w

7 the transton model predcts the transton pont somewhat upstream compared wth the model [6] and somewhat downstream compared wth the model [8]; t should be noted that the dfferental models of [6] and [8] dffer manly n the emprcal calbraton of the correlaton functons; Fg.. The dmensonless adabatc velocty at the blade surface n the md-span secton a b Fg.. The Mach number contours n the md-span secton a fully turbulent flow; b transtonal flow

8 Fg. 3. The dmensonless frcton velocty at the blade sucton surface n the md-span secton. Based on data from [6] the transton modelng accuracy can be consdered to be qute acceptable snce n the experment [7] the transton pont locates at S/S max =0.48 whch corresponds approxmately to the mddle of the transton regon predcted n ths study; the peak poston msalgnment near the tralng edge as well as that observed n Fgure s apparently due to the dfferent defnton of the surface length S max but n any case the error does not exceed n order the tralng edge thckness. It should be noted that the prelmnary D flow calculatons have shown a later transton along wth slghtly larger both the ext flow velocty and the Reynolds number. Dsplayed n Fgure 5 are the TKE contours for the cascade md-span secton whch are obtaned n smulatng the fully turbulent and transtonal flow cases. It s seen that n the case of the fully turbulent flow (Fgure 5 a) the TKE growth at the blade sucton surface begns almost mmedately downstream of the leadng edge but at the pressure sde t starts to grow somewhat later. When takng nto account the transton (Fgure 5 b) the TKE growth occurs approxmately at the cascade throat: at the tralng edge of the pressure sde and near the mddle of the chord at the sucton sde. Thus the flow upstream of the cascade throat remans rather lamnar. It should be noted that n the case of the transtonal flow the maxmum TKE values downstream of the transton pont are hgher and the turbulent boundary layer thckness s somewhat less than those of the fully turbulent flow case. Demonstrated n Fgure 6 s the lmtng streamlnes on the endwall surface. Here and below streamlne tracng s performed usng the freeware Paravew [9]. The flow patterns of the fully turbulent and transtonal flow cases seem to be qualtatvely smlar. The man dfference s that the pressure-drven cross-flow n the endwall boundary layer from the blade pressure sde to the sucton sde s more ntensve n the case of the transtonal flow especally n the area between the branches of the horseshoe vortces. In ths connecton the horseshoe vortex branch that runs along the sucton sde flows on the blade sucton sde somewhat upstream n the case of the transtonal

9 flow when compared wth the fully turbulent flow case. The postons of the second branch of the horseshoe vortex are about the same n both flow cases. We may assume that the endwall boundary layer n the regon of the horseshoe vortex s rather lamnar. The profle of such the boundary layer s less flled than that of the turbulent boundary layer and therefore t s more susceptble to both separaton and cross-flow. The turbulent flow develops downstream of the horseshoe vortex. The profle of the boundary layer n ths area becomes more flled. Therefore n the consdered fully turbulent and transtonal flow cases the dfference n the ntensty of the endwall cross-flow n the regon between the blade pressure sde and the nearest horseshoe vortex branch s less pronounced. Fg. 4. The dmensonless frcton velocty at the blade sucton surface n the md-span secton Based on data from [8] Some confrmaton of ths assumpton can be found n Fgure 7 whch gves the TKE contours n the endwall boundary layer at the dstance of percent of the blade span from the endwall. Snce the endwall boundary layer starts to emerge from the nlet boundary of the computatonal doman t remans suffcently thn and quas-lamnar n the blade-to-blade passage even when modelng a fully turbulent flow. It s seen that the flow turbulzaton (TKE ncreasng) begns n the horseshoe vortex upstream of the leadng edge. The values of TKE n ths area are relatvely small and are about the same for both the turbulent and transtonal flow cases. Downstream n these flow cases there s a sgnfcant ncrease of TKE n the regon of the boundary layer cross-flow. Ths process n the case of the transtonal flow starts upstream when compared wth the fully turbulent flow case and approxmately n the same zone where the boundary layer s turbulzed at the blade sucton surface.

10 Fgure 8 shows the TKE contours at the dstance of 0. percent of the cascade ptch from the blade sucton surface n both flow cases. It s clearly seen that as mentoned earler the growth of TKE n the case of the fully turbulent flow begns mmedately downstream of the leadng edges whereas n the transtonal flow case t takes place near the cascade throat. In both flow cases the turbulzaton of the blade sucton sde boundary layer occurs almost smultaneously over a whole blade span. The observed zones of the turbulzaton delay and advance n the secondary flows regon correlate well wth the poston of the boundary layer cross-flow at the blade sucton sde and wth the related areas of hgh and low flow velocty. a Fg. 5. The TKE contours n the md-span secton a fully turbulent flow; b transtonal flow b a b Fg. 6. The lmtng streamlnes at the endwall surface a fully turbulent flow; b transtonal flow

11 Gven n Fgure 9 are lmtng streamlnes at the tralng edge and n the downstream endwall regon n the cases of the fully turbulent and transtonal flow. The vortex flow pattern s complex enough n ths area we may observe focuses correspondng to the two counter-rotatng vortces n the base regon of the tralng edge (a D separaton) and abound 7 pars of saddle and spreadng ponts whch ndcate the poston of the vortex zones of a 3D separaton. We wll not descrbe such a flow n excessve detal; however we note the most mportant of ts features. Frst despte the fact that n the transtonal flow case the cross-flow nsde the endwall boundary layer starts upstream when compared wth the fully turbulent flow case t penetrates n spanwse drecton approxmately to the same dstance n both flow cases. Perhaps ths s due to the fact that n the transtonal flow case the boundary layer at the blade sucton sde s thnner than that of the fully turbulent flow case. At the same tme ts profle downstream of the cascade throat becomes turbulent and more flled. The mentoned phenomena prevent the further advance of the cross-flow wthn the boundary layer along the blade sucton sde n the spanwse drecton. a b Fg. 7. The TKE contours n the boundary layer at the dstance of percent of the blade heght from the endwall; a fully turbulent flow; b transtonal flow Second on the tralng edge the spanwse cross-flows of alternatng drectons are observed whch are separated by sngular ponts (two spreadng ponts and one saddle pont). Ths flow dscontnuty results n a dscreteness of the vortex wake downstream of the cascade. The cross-flow that s drected along the tralng edge to the md-span sectons of the blade channel forms an extensve dscrete wake vortex whch we shall call the man dscrete wake vortex. Ths cross-flow and consequently the man vortex are more ntensve n the fully turbulent flow case. Closer to the endwall surface the near-endwall vortex of the opposte rotaton drecton s formed. As t s seen from the locatons of the sngular ponts at the tralng edge and endwall surfaces the near-endwall vortex n the fully turbulent flow case when compared wth the transtonal flow case has a slghtly larger dmenson n the span drecton but t s smaller n the ptch drecton. In the corner whch s formed by the tralng edge and endwall surfaces the smaller corner wake vortex s nduced. Thrd the typcal dmensons of the corner vortex whch s nduced by the passage vortex n the corner area between the sucton sde and the endwall surface are larger n the case of the fully turbulent flow that s apparently also caused by a greater thckness of the boundary layer n ths flow case.

12 a b Fg. 8. The TKE contours at the dstance of 0. percent of the cascade ptch from the blade sucton surface a fully turbulent flow; b transtonal flow γ Presented n Fgure 0 are the contours of the entropy functon p ρ n a cross-secton downstream of the tralng edge n fully turbulent and transtonal flow cases. The flow s suffcently symmetrc relatve to the mddle of the cascade channel. The numbers n the Fgure refer to the areas of ncreased entropy the physcal meanng of whch wll be explaned below. Demonstrated n Fgure are the secondary flow patterns n the cascade. The lmtng streamlnes at the endwall surface are shown to be smlar to those n Fgure 6. The entropy functon contours at the perpendcular transversal surface are presented as t was done n Fgure 0. Also n the endwall boundary layer regon the volume streamlnes wthn the horseshoe passage and near-endwall vortces are dsplayed. Analyss of these plots and of the entropy functon dstrbutons at several cross-sectons n the proxmty of the tralng edges and downstream of them allows us to note the followng ponts. The wake makes the man contrbuton to the growth of entropy and consequently to the entropy losses n the consdered cascade. The local maxma of the entropy functon n Fgures 0 and correspond to the dscrete wake vortces descrbed above. The effect of the secondary flows as well as the dmensons of the regons occuped by them are much smaller. Ths s explaned by both a large thckness of the tralng edge and the low cascade loadng. The streamlnes of the passage vortex shown n Fgure n lght color come manly n the hgh-entropy zone whch corresponds to the man dscrete wake vortex. Both of these vortces have the same rotaton drecton and lkely they are merged and mxed downstream of the tralng edge. It should be noted that n the transtonal flow case whch s characterzed by a larger

13 ntensty of the cross-flow n the endwall and sucton sde boundary layer there s a greater dsperson of the passage vortex streamlnes. As a result n the fully turbulent flow case the entropy values and the knetc energy losses of the zone whch contans two vortces are hgher when compared wth the transtonal flow case despte the fact that a cross-flow n the boundary layer and thereby the passage vortex are more ntensve n the latter case. a b Fg. 9. The lmtng streamlnes n the tralng edge/endwall corner regon a fully turbulent flow; b transtonal flow The streamlnes that are panted n blue n Fgure extend from the near-endwall vortex downstream of the tralng edge nto the hgh-entropy zone of Fgure 0. In ths vortex on the contrary the streamlnes are more dspersed n the case of the fully turbulent flow whereas the ntensty of the vortex s larger n the transtonal flow case. The hgh-entropy zone 3 n Fgure 0 corresponds to the passage corner and wake corner vortces whch are not marked n Fgure because of ther small sze. The branch of the horseshoe vortex (ts core s ndcated n red n Fgure ) that s formed near the blade pressure sde extends towards to the lower boundary of the hgh-entropy zone. In the transtonal flow case t s closer to the endwall surface when compared wth the fully turbulent flow case. The branch of the horseshoe vortex that s formed near the blade sucton sde extends towards to the hgh-entropy zone 4 and n the fully turbulent flow case t s closer to the endwall surface. In the case of the fully turbulent flow the entropy values and the flow devaton angle n the md-span sectons of the wake are larger than those of the transtonal flow case. Ths s due to the fact that n the fully turbulent flow case the boundary layers on both blade surfaces are thcker. Fgure shows the dstrbuton of the knetc energy losses along the blade span. The red and blue lnes ndcate the present numercal results n fully turbulent and transtonal flow cases respectvely. The postons of local maxma of the losses correspond to those of the entropy functon dstrbuton n the wake shown n Fgure 0. The values of the total and md-span knetc energy losses are gven n the Table. The losses were determned at 40 percent and 00 percent of the axal chord downstream of the tralng edge. The latter corresponds to the ext boundary of the computatonal doman. It s seen that by takng nto account the lamnar-turbulent transton we have mproved the estmate of the total knetc energy losses by more than percentage pont and of the md-span knetc energy losses by more than 0.5 percentage ponts.

14 a b Fg. 0. The entropy functon contours at the cross-flow secton at the dstance of 0 percent of the axal chord downstream of the tralng edges; a fully turbulent flow; b transtonal flow Table The knetc energy losses Md-span Total The dstance downstream the tralng edges the axal chord 40 % 00 % 40 % 00 % Fully turbulent flow Transtonal flow

15 a b Fg.. The secondary flows structures a fully turbulent flow; b transtonal flow

16 Fg.. The knetc energy losses dstrbuton along the blade span at the dstance of 40 percent of the axal chord downstream of the tralng edges 8. Concluson An algebrac transton model permts the turbne cascade flow smulaton takng nto account the phenomenon of the lamnar-turbulent transton. The locaton of the transton pont calculated usng the approach suggested n ths study s n a satsfactory agreement wth the wellknown expermental and numercal data. The TKE rse at both the sucton and pressure blade sdes s observed suffcently downstream n the transtonal flow case when compared wth the fully turbulent flow case. In general the physcal representaton of fully turbulent and transtonal flows that s obtaned numercally s consstent wth the known deas about ths knd of flows. Takng nto account the lamnar-turbulent transton we have mproved the estmate of the total knetc energy losses by more than percentage pont and of the md-span knetc energy losses by more than 0.5 percentage ponts. In the transtonal flow case due to the greater flow susceptblty to separaton the boundary layer cross-flow from the endwall surface to the blade sucton sde starts upstream and the ntensty of the near-endwall wake vortex s substantally hgher but n general the growth of the knetc energy losses n the flow s sgnfcantly less. Consequently the secondary flow patterns n the transtonal and fully turbulent flow cases are somewhat dfferent. Therefore an mportant ssue for the further research s the study of mechansm of the transton effect on the secondary flows and dentfyng varous ways to reduce the knetc energy losses n the turbne cascades by controllng the transton n the boundary layer. The study found that transton modelng mposes severe restrctons on the adequacy of the turbulence model the relablty and the operaton speed of the numercal method the resoluton

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