Jurnal f Applied Mathematics and Physics, 17, 5, 966-989 http://www.scirp.rg/jurnal/jamp ISSN Online: 37-4379 ISSN Print: 37-435 Exact Slutin fr Thermal Stagnatin-Pint Flw with Surface Curvature and External Vrticity Effects Rnald Ming Ch S 1, Elizabeth Wing Sze Kam * 1 Mechanical Engineering Department, the Hng Kng Plytechnic University, Hng Kng, China Department f Mathematical Sciences, Rensselaer Plytechnic Institute, Try, NY, USA Hw t cite this paper: S, R.M.C. and Kam, E.W.S. (17) Exact Slutin fr Thermal Stagnatin-Pint Flw with Surface Curvature and External Vrticity Effects. Jurnal f Applied Mathematics and Physics, 5, 966-989. https://di.rg/1.436/jamp.17.5485 Received: February 19, 17 Accepted: April 7, 17 Published: April 3, 17 Cpyright 17 by authrs and Scientific Research Publishing Inc. This wrk is licensed under the Creative Cmmns Attributin Internatinal License (CC BY 4.). http://creativecmmns.rg/licenses/by/4./ Open Access Abstract Exact slutin f the steady Navier-Stkes equatins has been btained fr the thermal stagnatin-pint flw at the leading edge f a turbine blade under the assumptins f cnstant nse radius and external vrticity, and fluid prperties independent f temperature. The slutins reveal that curvature affects lcal heat transfer and skin frictin while external vrticity des nt. The effect f external vrticity is t shift the zer skin frictin pint away frm the stagnatin pint. This slutin is valid fr all Reynlds number, external vrticity, and nse radius. In the limit f nse radius ging t infinity and external vrticity, ging t zer, the exact slutin fr tw-dimensinal plane stagnatin-pint flw is recvered identically. In additin, it can be shwn that the velcity field arund the stagnatin pint f a rtating curved surface is the same as that arund the stagnatin pint f a statinary curved surface with an external vrticity which equals t twice f the rtatinal speed. This realizatin renders the present slutin equally valid fr thermal stagnatinpint flw at the leading edge f centrifugal impeller blades. Keywrds Navier-Stkes Equatins, Exact Slutins, Thermal Stagnatin-Pint Flw, Displacement Effect 1. Intrductin Frequently, in the past, tw-dimensinal calculatin methds were used t determine the gas-side heat transfer cefficient n turbine blades [1]. All these methds, whether based n the integral r differential equatins, were derived n the assumptin that the flw is steady and the static pressure variatin acrss DOI: 1.436/jamp.17.5485 April 3, 17
the bundary layer has very little effect n the flw, and hence can be neglected []. Measurements f tw-dimensinal turbulent bundary layers alng plane surfaces lend supprt t this assumptin [3]. Hwever, when these methds were used t calculate flws alng curved surfaces, they were fund t be inadequate cmparing with measured data []. The reasn is that existing tw-dimensinal methd neglects the effect f curvature n the mean flw streamlines. This neglect is justifiable in laminar flws [4] [5] if kδ 1. On the ther hand, the neglect f the effect f streamline curvature is nt justifiable in the case f turbulent flws [6] even when kδ 1. Striking effects f curvature have been bserved in turbulent bundary-layer flw ver cnvex and cncave surfaces [7] [8]. Bundary layer measurements shwed that large cnvex curvature in the mean flw streamlines leads t vanishing shear stress in regins where the mean velcity gradient is still substantial [7], while large cncave curvature prmtes the frmatin f Taylr-Grtler type instabilities [8]. As a result, varius attempts were made by different investigatrs t accunt fr the effect f streamline curvature in tw-dimensinal turbulent shear flws [9]-[14]. Attempt t use these techniques t calculate flw arund cmpressr blades was first made in [13], and gd crrelatins were btained between measured and calculated results. Furthermre, these techniques were extended t mdel heat transfer n curved surfaces alne [14] and curved surfaces with swirl [15]. Again, gd crrelatins were btained between measured and calculated heat transfer results. In view f this, the same technique has als been used t predict the flw and heat transfer arund axial flw turbine blades where Crilis effects are absent [15]. Generally, the slutin f the tw-dimensinal plane stagnatin-pint flw [16] [17] [18] is used t prvide initial cnditins fr heat transfer calculatins alng turbine blades. This assumptin is valid as lng as the surface curvature is cnsidered t have negligible effect n the bundary-layer flw arund the blades. With the intrductin f imprved techniques mentined abve [9]-[15], it is evident that, t be cnsistent with the imprved techniques, the effect f surface curvature n the stagnatin-pint flw cannt be neglected. The bundary layer thickness in a stagnatin-pint flw is given by δ =.4 ν a ; therefre, it can be seen that the neglect f curvature effect is valid nly if ν a is much less than the radius f curvature f the surface at the stagnatin pint [18]. In the leading edge f a turbine blade, the nse radius is very small and the abve cnditin is nt necessary true. As a result, surface curvature may have a significant effect n the flw and heat transfer dwnstream f the frward stagnatin f a turbine blade. In additin t being affected by surface curvature (k), the flw and heat transfer at the leading edge are als influenced by free stream vrticity ( Ω ) in the flw appraching the turbine blades. Therefre, t btain the crrect initial cnditins fr subsequent heat transfer calculatins arund the blades using any ne f the previusly mentined techniques [9]-[15], the effect f k and Ω n the flw and heat transfer 967
at the leading edge have t be cnsidered and analyzed. Effect f surface curvature n stagnatin-pint flw was first examined in [4] [5]. Assuming that Re 1 and kδ 1, the analysis in [4] [5] prceeded t expand the stream functin in terms f 1/Re. The prblem was then slved using the technique f matched asympttic expansins and the results gave the secnd-rder effect n heat transfer cefficient and skin frictin due t surface curvature. Further, secnd-rder effect due t flw displacement and free- stream Ω was als evaluated [4] [5]. Subsequently, ther investigatrs [19] [] [1] have als carried ut analysis n the same prblem invking the same assumptins, but using different methds t slve the prblem. The results btained by these investigatrs were n different than thse presented earlier [4] [5]. In sme cases [19] [], the results were shwn t be less accurate because f the assumptins the investigatrs had t invke t simplify the prblem. The effect f cnstant external Ω n tw-dimensinal plane stagnatin- pint flw was als attempted and an exact slutin t the gverning equatins was btained []. Unlike the perturbatin slutin mentined previusly [4] [5], the exact slutin, withut accunting fr the effect f flw displacement, is valid fr all Re []. Further, it was fund that the effect f external Ω n the flw was t shift the zer skin frictin pint away frm the stagnatin pint; hwever, n attempt had been made t analyze the heat transfer prblem []. On the ther hand, a cmplete secnd-rder analysis f the prblem was carried ut and it shwed that if displacement effect were als included [3], the shift f the zer skin frictin pint frm the stagnatin pint is greater than that predicted in []. A thrugh discussin f mst secnd-rder effects has been presented in [3] where the flws cnsidered are nt limited t stagnatin-pint flws. In spite f the fairly cmplete secnd-rder treatment f stagnatin-pint flw n curved surfaces [4] [5] [3], the analysis is rather limiting because f the assumptins f Re 1 and kδ 1. These assumptins culd lead t tw surces f errr in the predictin f gas-side heat transfer cefficients n turbine blades. The bvius surce is the inaccurate estimate f the heat transfer cefficient at the leading edge, which may r may nt be t critical depending n the value f Re and kδ at the nse. A secnd surce f errr is assciated with the incrrect estimate f initial cnditins fr subsequent cnvective heat transfer calculatins n the rest f the blade. The effect f this errr n the calculated heat transfer cefficient is difficult t estimate, and it culd be mre severe as Re decreases and kδ increases at the leading edge. Cnsequently, there is a need fr a mre exact thery that can crrectly accunt fr the effect f surface curvature and external Ω n thermal stagnatin-pint flws at the leading edge f turbine blades. The bjective f the present paper, therefre, is t investigate the effect f surface curvature and cnstant external Ω n heat transfer at the leading edge f axial flw turbine blades. T simplify the prblem, the flw near the leading edge f the blade is apprximated by the flw arund a tw-dimensinal cylinder f 968
the same radius as the nse. Als, the flw is assumed t be steady and laminar, and the fluid prperties are assumed cnstant ver the temperature range f interest. As the first attempt, the assumptin f laminar flw at the stagnatin pint and its immediate vicinity is reasnable. It is shwn that exact slutins t the gverning equatins culd be btained and that these slutins wuld reduce t the exact slutin with cnstant Ω [] as k, and t the classical slutins [15] [16] [17] [18] in the limit f zer surface curvature and zer external Ω. Thus frmulated, the slutins are valid fr all values f Re, kδ and Ω. Once this slutin is btained, it is shwn that the frmulatin can be extended t analyze the thermal stagnatin-pint flw at the leading edge f centrifugal impeller blades.. The Gverning Equatins The gverning equatins describing the steady incmpressible flw and temperature fields near the frward stagnatin pint f a tw-dimensinal cylinder f cnstant radius R can be written with respect t a c-rdinate system attached t the cylinder as shwn in Figure 1. In cmpnent frm, the full set f steady Navier-Stkes equatins [4] is reduced t: Figure 1. Curvilinear c-rdinate system fr a statinary surface. 969
u x ( hv), + = (1) y 1 1 1 1 ( ) ρ uux + v hu = px + ν h uxx + huyy + ku y k h u + kh v x, y () 1 1 1 1 uvx + hvvy ku = ρ hpy + ν h vxx + hvyy + k vy k h v kh u x, (3) 1 utx + hvty = α h Txx + htyy + kt y, (4) where subscripts dente partial differentiatin, k is psitive fr cnvex and negative fr cncave curvature, and viscus dissipatin f heat has been assumed negligible. The bundary cnditins are n slip at the wall, unifrm flw and temperature far upstream f the stagnatin pint, and inviscid flw with a cnstant Ω utside a shear layer clse t the wall. Assuming cnstant fluid prperties allw the velcity field equatins t be decupled frm the temperature field equatin. A cmplete slutin t the aerdynamic prblem, i.e. Equatins (1)-(3), can be btained by first finding a slutin t the uter inviscid flw and then prceeding t determine the flw in the viscus regin near the wall. Once the velcity field near the stagnatin pint is knwn heat transfer t r frm the cylinder surface can be btained by slving Equatin (4) with the apprpriate temperature bundary cnditins. 3. The Inviscid Slutin Since the flw in the uter regin is inviscid, the stream functin, Ψ ( xy, ), satisfies the Pissn equatin, which can be written as ( x) ( y) h h Ψ + h hψ = Ω, (5) 1 1 1 x where the stream functin is defined by y u = Ψ, (6) y hv = Ψ. x (7) The bundary cnditins are Ψ= n the bdy, (8) Ψ( xy, ) ~ Ψ far upstream. (9) This prblem can be cnsidered prperly set fr an elliptic differential equatin, and a slutin fr Ψ can be sught in the frm Ψ ( xy, ) = PxQ ( ) ( y) +Ω S( y). (1) Substituting (1) int (5) gives P h Q khq = + = n, (11) P Q Q h S + khs = h, (1) where the primes dente rdinary differentiatin with respect t x fr P, and t y fr Q and S. If the slutin fr Ψ ( xy, ) is t apprach the classical tw-dimensinal plane stagnatin-pint flw slutin when k and Ω, then 97
it can be shwn that the nly meaningful slutin fr (11) is given by n =. With n =, slutins fr P( x ) and Q( y ) are P ( x ) = ax, (13) 1 a ( ) ln h Q y =, (14) k where a 1 and a are arbitrary cnstants. Since the hmgeneus slutin f (1) is identical t that btained fr Q and Ω is cnstant, it can be easily shwn that nly the particular integral f Equatin (1) is f interest. A particular slutin f (1) is given by ( ) S y ( + ) y 1 h ln h = +. (15) 4k k As a result, the slutin fr Ψ with bundary cnditins (8) and (9) can be written as: ( 1 h) ax Ω y + Ω Ψ ( xy, ) = ln h + ln h, (16) k 4k k where a is an arbitrary cnstant. It shuld be pinted ut that in the limit f k, Equatin (16) reduces t the slutin given in [], r p 1 Ψ= Ω (17) lim axy y. k Therefre, when Ω=, Equatin (17) reduces t the classical ptential slutin fr tw-dimensinal plane stagnatin-pint flw. The velcity field is given by ( 1 h) ax Ω y + u = h h, (18) aln h v =, kh (19) and the pressure field is btained by integrating the Euler equatins which are the inviscid cunterparts f () and (3). With the help f (18) and (19), the pressure field can be written as: a ln h ρ Ω + Ω h k h 4k h kh ( ) 4 1 a x a x h ln h h 1 h 4h ln h 1 p = + + +, () where p is the pressure and p is the pressure at the stagnatin pint. It shuld be nted that even when Ω the velcity u dwnstream f the stagnatin pint is nt unifrm as culd be seen frm Equatin (18). This is necessary because curvature gives rise t a centrifugal frce that is balanced by a nrmal pressure gradient. 4. Viscus Flw and Heat Transfer Equatins (18)-() represent a cmplete slutin t the inviscid prblem. Once the inviscid slutin is knwn, the next task is t fcus n the viscus flw and 971
heat transfer near the surface. Since the inviscid slutin des nt satisfy the n-slip cnditin at the wall, a viscus slutin that is valid at and near the surface will be sught. This slutin has t apprach (18) and (19) fr large y, and in this limit, the vrticity als appraches Ω in the external flw. In rder nt t impse any cnditins n Re, kδ, and Ω, an exact slutin t the gverning Navier-Stkes Equatins (1)-(3) is sught in the viscus layer. The pressure terms in the mmentum equatins are eliminated by crss-differentiating () and (3), and the resulting vrticity equatin is given by h uu + h vu + kh uv + kh vu huv h vv + hkvv + khuu 3 xy yy y y xx xy x x 3 3 ν huxxy kuxx h uyyy kh uyy k huy k u khvxy k vx vxxx h vxyy = + + + + +, where use has been made f Equatin (1). Stagnatin-pint flw near the surface is, therefre, given by the slutin f (1) subject t the bundary cnditins uxy (, ) = vx (, ) =, (a) ( 1 h) ax y uxy (, ) h Ω + h 1 1 h vx h ( hu) Ω y (1) fr large y. (b, c) Once the velcity field is knwn, the stagnatin-pint heat transfer prblem can be analyzed by slving Equatin (4) subject t the cnditins f an isthermal bdy and a unifrm temperature in the external flw. As in the case f plane flws with r withut the effect f Ω [15] [16] [17] [18] [], a similar slutin t Equatin (1) is sught such that the velcity and temperature fields can be written as: where ( a ν ) y K ( ν a) k G ( η ), ν u = axf ( η ) Ω (3) a h F ( η ) v = aν, (4) h T = T + ( T T ) Θ ( η ), (5) w w η = is the similarity variable, h= 1+ ky= 1+ Kη and =. Substituting Equatins (3)-(5) int (1) and (4), the fllwing equatins fr F ( η ), G ( η ), and Θ ( η ) are btained, F + K K ( F K ) hf K ( F K ) + KhF F + F + F F =, (6) 3 h h h ( K) F K K F hf + KF G + G + G G =, h h h (7) F Pr + K Θ+ Θ=, h (8) where the primes dente rdinary differentiatin with respect t η. The cnditins that T( x,) = T, w T( x, large y) = T and () give the necessary bundary cnditins fr (6)-(8). It is nt cnvenient t chse G ( η ) at large η as ne f the uter bun- 97
dary cnditins fr G because its value at large η is dependent n displacement. As a result, the uter bundary cnditins fr G are defined by the functins G ( η ) and G ( η ) at large η. In terms f η, the bundary cnditins fr F, G, and Θ can then be written as at η = : F ( ) =, (9a) F ( ) =, (9b) G ( ) =, (9c) Θ ( ) =, (9d) 1 ( η) h at large η: F =, (3a) ( η ) Kh, F = (3b) G G ( η ) = h, (3c) ( η ) = K, (3d) Θ ( η ) = 1. (3e) The viscus flw and heat transfer prblem at the stagnatin pint is reduced t slving three rdinary differential equatins gverning F ( η ), G ( η ) and Θ ( η ), and subject t bundary cnditins (9) and (3). Thus frmulated, slutins f these equatins will represent exact slutins t the thermal stagnatin-pint prblem at the leading edge f axial flw turbine blades. 5. Numerical Integratin f the Gverning Equatins The rdinary differential equatins gverning F, G, and Θ are highly nnlinear and analytic slutins fr (6)-(8) are nt easily btainable and are nt presently available. Hwever, numerical technique culd be used t simulate the slutins fr (6)-(8). Since numerical slutins f rdinary differential equatins is far better develped than thse f highly nnlinear partial differential equatins, such as the Navier-Stkes equatins, and a lt mre accurate, therefre, numerical slutins f (6)-(8) are bth accurate and reliable. T further simplify the gverning equatins fr numerical analysis, a transfrmatin first put frward in [] is adpted. If the new similarity variable is dented by ln ( 1+ Kη ) ζ =, (31) K and the new functins fr F, G, and Θ are written as f ( ζ ) = F( η), (3) g ( ζ ) = G( η), (33) ( ) = Θ ( η), (34) θ ζ then it can be easily shwn that in terms f f, g, and θ, (6)-(8) becme ( ) ( ) f + f 4K f + 4K Kf f f =, (35) 973
( ) ( ) g + f 4K g K f K g f g =, (36) θ + f Prθ =. (37) The bundary cnditins can be btained frm (9) and (3) with the help f (31)-(34). The transfrmed bundary cnditins are given by ζ = f ( ) = (38a) at :, f ( ) =, (38b) g ( ) =, (38c) θ ( ) =, (38d) at large ζ : f 1, g f ( ζ ) = (39a) ( ζ ) =, (39b) ( ζ ) = exp ( Kζ ), (39c) ( ζ ) = exp ( ζ ), (39d) g K K θ( ζ ) = 1, (39e) where the primes nw dente differentiatin with respect t ζ. Equatins (35)-(37) are much simplified cmpared t (6)-(8). Hwever, they are still highly nnlinear and tw additinal initial cnditins each, namely, f ( ), f ( ) and g ( ), g ( ) are required fr the numerical integratin f (35) and (36). Frtunately, a first integral f (35) and (36) can be btained, whereas such is nt pssible fr (6) and (7). These integrals can be used t estimate ne f the required initial cnditins fr the integratins f (35) and (36). Integrating (35) and (36) nce frm t ζ and making use f (38a-38d) gives 4 4 d ( ) 4 ( ), (4) ς f + ff f K f Kf + ff ζ = f Kf = C ς g + fg f g K 4g ( f K ) g gf dζ g ( ) 4Kg + ( ) C = = 1. (41) The cnstants C and C 1 can be determined by evaluating (4) and (41) at large ζ and making use f (39a-39e) and the fllwing cnditins. Since the effect f displacement have nt been included in the present frmulatin fr f ( ζ ), it fllws frm (39a) that f ζ = ζ δ at large ζ, (4a) ( ) * * where δ is the displacement thickness fr the K but Ω= case. Als, the cnditin f cnstant in the external flw gives f ( ζ ) = at large ζ. (4b) On the ther hand, the displacement effect n g ( ζ ) can be easily analyzed and it will be carried ut in the fllwing analysis. Frm (39c) it can be shwn that 1 * g( ζ ) = exp ( Kζ ) exp ( Kδ ) at large ζ, K (43a) 974
if displacement effect n g (ζ) are neglected, and 1 g( ζ ) = exp( Kζ ) 1 at large ζ, K (43b) if displacement effect n g ( ζ ) are included. In the absence f curvature, (43a) reduces t the result given in [] where displacement effect is nt included, and (43b) reduces t the result presented in [4] [5] where displacement effect is accunted fr. Therefre, tw sets f slutins fr g ( ζ ) will be btained depending n whether g ( ζ ) is taken t satisfy (43a) r (43b) at large ζ in the evaluatin f C 1. With the cnditins fr f and g defined at large ζ, it can be easily shwn that C and C 1 are given by ( ) ( ) ζ ζ C = f 4Kf = 1 K ff d + 4 K, (44) ( ) 4 ( ) C1 = g Kg 1 * * * = exp ( Kδ ) 1 ( δ K) exp ( Kδ ) K ζ 1 * + K gf { exp ( Kζ ) exp ( Kδ )} d ζ, K if displacement effect n g is neglected, r { ( ) ( )} (45a) * ζ 1 * C1 = δ K 1 gf exp Kζ exp Kδ d ζ, K (45b) if displacement effect n g is included. It shuld be pinted ut that in the limit * f K, as expected, (45a,b) reduce t C 1 = and C1 = δ, respectively. In the limit f K, (4) and (41) tgether with (44) and (45a) reduce t the equatins analyzed in []. Accrding t [4], the slutin given in [] can nly accunt fr the kinematic effect f external Ω. If the dynamic effect were t be analyzed, then the displacement effect n g cannt be neglected. In ther wrds, C 1 has t be evaluated frm (45b). The secnd-rder equatin fr vrticity effect analyzed in [4] [5] is inhmgeneus. Hwever, it was shwn in [4] [5] * that the inhmgeneus part f the secnd-rder equatin is given by δ. This is different frm the present analysis. Accrding t (45b), the result * C1 = δ is true nly when K = ; in ther wrds, nly fr tw-dimensinal plane stagnatin-pint flw. Therefre, the apprximate treatment [4] [5] can nly be interpreted as accunting fr the displacement effect resulting frm a tw-dimensinal plane stagnatin-pint flw, and nt that at the leading edge f turbine blades. Frm this discussin, it can be seen that the present frmulatin fr the stagnatin-pint flw prblem including surface curvature and external Ω effects indeed appraches the tw limiting cases f K =, Ω and K =, Ω= crrectly. Therefre, the slutins btained are exact and are valid fr all values f Re, kδ, and Ω. In additin, the slutins crrectly accunt fr the bundary-layer displacement effect at the stagnatin pint. 975
Althugh ne additinal initial cnditin each is required fr the integratin f (4) and (41), these tw equatins are nt cnvenient t slve numerically because they invlve the integrals f f and g. As a result, (35) and (36) are slved, and (44) and (45) are used t estimate the initial values fr f ( ) and θ ( ) nce the initial guesses n f ( ) and g ( ) are knwn. The integrals in (44) and (45) are evaluated frm the previus iteratins fr f and g. A furth-rder Runge-Kutta technique is used t start the integratin f (35) - (37) using the apprximate results given in [4] [5] as initial guesses fr f ( ), g ( ), and θ ( ). Subsequent integratin f the equatins is perfrmed by a predictr-crrectr methd that has accuracy in bth predictr and crrectr f 5 O ( ζ ). Details f this methd are utlined in [5]. The cmplete integratin f equatins (35)-(37) is iterated with different initial cnditins until the uter bundary cnditins (39a)-(39e) are satisfied. The uter bundary cnditins are cnsidered satisfied when numerical values at the last tw cnsecutive uter steps agree t within 1 5. When this cnditin is reached, cnvergent slutins fr f, g, and θ are btained. 6. Discussin f Results Equatins (35)-(37) with bundary cnditins (38) and (39) are slved fr thirteen different values f K; namely, K =, ±.15, ±.3, ±.5, ±.1, ±., ±.3, at a Pr =.7. The integratin f f, g, and θ are carried ut t ζ = 1 fr all values f K except K = -.1, -. and -.3, and the uter bundary cnditins are applied there instead f at ζ. Fr large negative values f K, integratins are carried ut t ζ = 1; this chice f ζ is satisfactry because f, g, and θ apprach their uter bundary cnditins very rapidly even fr large values f K. Fr all the cases cnsidered, it is fund that f, g, and θ apprach their free stream value arund ζ = 8. Hence, the chice f ζ = 1 r 1 fr all integratins is mre than adequate. Lcal heat transfer and skin frictin can be evaluated frm T q = κ, (46) y y= T τ w = µ y y=. (47) With the help f (3), (5) and equatins (3)-(34), (46) and (47) can be reduced t a q = κ( T Tw ) θ (, ) (48) ν ( ) ( ) τw = ρ a aνxf µ Ω g. (49) It can be seen frm (49) that the effect f external Ω is t shift the zer shear pint away frm the stagnatin pint, which is lcated at x =. The shear τ and is given by, stress that is respnsible fr this shift is designated ( ) w v 976
( τ ) µ ( ) = Ω. (5) w v g The calculated value is ( τ w ) = 1.465µ Ω with displacement effect n g v included, and is ( τ w ) =.679µ Ω when displacement effect is neglected. v These results shw that ( τ w ) is nt affected by K but depends n Ω and the v displacement. They validate the apprximate analysis results f [4] [5], and shw that, indeed, the effect f curvature n g ( ) is negligible. If the rigin f the crdinate axes is lcated at the zer-shear pint and the new x crdinate is dented by x, then (49) can be rewritten as ( ) τ = ρ a aνxf. (51) Denting the lcal heat transfer and skin frictin f the K = case by a subscript, the fllwing relatins fr the rati f lcal heat transfer and skin frictin with and withut curvature effect included are btained q q θ = θ ( ) ( ) ( ) ( ), (5) τ f =. (53) τ f These results are shwn in Figure and Figure 3, respectively. In the apprximate analysis f [4] [5], the inviscid surface velcity in the immediate vicinity f the stagnatin pint was expanded in the frm 3 ( ) 11 1 ( ) u x, = u x+ u x + O x, (54) where u 11 has the dimensin f (time) 1 and u 1 has the dimensin f (length) 1 (time) 1. Using this expansin, the apprximate analysis gives rise t the fllwing expressins fr the lcal heat transfer and skin frictin ratis; namely, q ν Ωu = 1. 58356K + 1. 81687, q 1 1 5 u11 1 1 5 u11 (55) τ ν Ωu = 1 1. 556K + 4. 743353. (56) τ On first examinatin, it seems that the apprximate results f [4] [5] are quite reasnable because ne wuld indeed expect the effect f external Ω t be felt by the lcal heat flux and the wall shear. The apprximate analysis [4] [5] shwed that, in additin t shifting the zer-shear pint away frm the stagnatin pint, external Ω als influences lcal heat transfer and skin frictin. Hwever, n clser examinatin, it culd be seen that the inviscid slutin (18)-() deduced frm the present analysis gives an inviscid surface velcity f u ( x,) = ax nly. This implies that the last term in (55) and (56) shuld be identically zer; i.e., u1. In ther wrds, assuming (54) fr u( x,), as suggested in [4] [5], is incrrect; at least in the immediate vicinity f the stagnatin pint. Setting u 1 = reduce the equatins fr lcal heat flux (55) and lcal shear stress (56) t: 977
q 1.58356 K, q = (57) τ 1 1.556 K. τ = (58) These results are shwn in Figure and Figure 3 fr cmparisn. It can be seen in Figure that the exact wall heat flux decrease with K fr K > is faster than that predicted by the apprximate analysis [4] [5], and the ppsite is true fr K <. But as shwn in Figure 3, there is n discernible difference between the exact result fr wall shear stress and that btained frm the apprximate analysis [4] [5]. Hwever, n clse examinatin, a cnsistent difference des exist, especially fr large K. This is evident frm the results tabulated in Table 1 where the values f f ( ) and θ ( ) are reprted t the furth decimal pint. The apprximate results [4] [5], which are dented by a subscript VD, are als listed in Table 1 fr cmparisn. It can be seen that the apprximate results are crrect fr values f K up t K =.15. Thereafter, they deviate frm the exact slutins. The variatins are small fr small values f K, Figure. Wall heat flux, qq, at different values f K (red slid line apprximate analysis [4] [5]; blue dashed line exact slutins). Figure 3. Wall shear stress, τ τ, at different values f K (red slid line apprximate analysis [4] [5]; blue dashed line exact slutins). 978
Table 1. A cmparisn f f ( ) and ( ) [4] [5] fr different values f K. θ with apprximate results deduced frm K.3..1.5.3.15.15.3.5.1..3 f () 1.89 1.6188 1.45 1.386 1.91 1.613 1.36 1.39 1.1753 1.137 1.46.8553.6717 f ( VD ) 1.866 1.615 1.439 1.383 1.9 1.613 1.36 1.39 1.175 1.1369 1.413.8499.6586 θ ( ).568.5781.578.51.4996.4978.4959.4959.49.489.48.4656.4457 θ ( ).5343.515.587.53.4997.4978.4959.4959.49.4895.4895.47.4574 VD but increase as K increases. These small differences are significant because in the curse f numerically integrating (35) and (37), it is fund that, if the apprximate values are used fr f ( ) and θ ( ), the numerical results fail t apprach the bundary cnditins at ζ = 1 r 1 crrectly and still satisfy the accuracy criteria impsed n f ( ζ ), f ( ζ ), and θ( ζ ). The situatin gets wrse when K increases and at K.1 the numerical cmputatins fail t cnverge t a meaningful slutin. Cnsequently, it can be cncluded that the apprximate analysis f [4] [5] gives the crrect slpe fr τ τ and qq at K = nly. This was pinted ut in [1] where the authrs cncluded that there is actually n justificatin fr attaching any significance t the curvature f the curves fr τ τ and qq when they are deduced frm any ne f the knwn apprximate methds [19] [] [1]. The higher wall shear given by the exact analysis is a direct cnsequence f the mre favrable pressure gradient seen by the flw fr a given K. Frm () it can be deduced that the pressure gradient at the wall alng the flw directin is given by ρ 1 p u u = ν νk + x y y y = y= y = With the help f (3) and equatins (31) - (33), it can be shwn that (59) reduces t 1 p = a x f ( ) Kf ( ) νω g ( ) Kg ( ). x (6) ρ y= Again, the effect f external Ω is t shift the zer pressure gradient pint away frm the stagnatin pint. Cnsequently, (6) can be written as ( p x) ( ) Kf ( ). (59) y= = f, (61) ρa x while the crrespnding result deduced frm the apprximate analysis [4] [5] is ( p x) y= 1 1. 88488 K. = + ρa x VD (6) These results are pltted in Figure 4. They shw that the decrease in favrable pressure gradient with K, fr K >, is faster fr the exact slutin than fr the 979
Figure 4. Wall pressure gradient rati, ( p x ) ρ a x, at different values f K (red slid line apprximate analysis [4] [5]; blue dashed line exact slutins). y= apprximate analysis. The ppsite is true fr K <. Sme sample plts f f, f, f, g, g, and q fr three different values f K are shwn in Figures 5-1, respectively. They clearly demnstrate the effect f curvature n these prfiles. Therefre, if the plane stagnatin-pint-flw slutin is used as initial cnditins fr subsequent heat transfer calculatins arund turbine blades, the errr incurred culd be substantial, depending n the nse curvature and the external Ω. The latter effect wuld shift the zer shear pint and the zer pressure gradient pint away frm the stagnatin pint, and this culd have an impact n transitin t turbulence and lcal separatin. Hwever, curvature wuld affect the wall shear, the wall heat flux, the pressure distributin arund the leading edge and, mre imprtantly, the bundary layer thickness in the vicinity f the stagnatin pint (Figure 6). This means that if the K = slutin is used as initial cnditins, a wrng estimate f the velcity prfile and mmentum and displacement thicknesses wuld have been used fr subsequent heat transfer calculatin arund turbine blades. The seriusness f this errr is best illustrated by examining the variatin f the displacement thickness rati, * * δ δ, with K fr the case Ω= (Figure 11), which clearly underlines the imprtance f including surface curvature effect in the calculatin f leading edge heat transfer. Finally, cnsider the determinatin f a which is as yet undefined. T accmplish this, cnsider the flw alng the stagnatin streamline tward a circular cylinder in the absence f external vrticity. In the vicinity f the stagnatin pint, the velcity alng the stagnatin streamline as given by (i) the plane stagnatin-pint flw inviscid slutin, (ii) the present inviscid slutin, and (iii) the ptential slutin arund a cylinder, can be written as p V ay (i) = (plane wall) V V c V ln ( 1 ) (ii) ar + y R = (curved wall) V V R+ y ( ) 98
Figure 5. Effects f surface curvature n f (red slid line K =.3; blue dashed line K = ; black slid line K =.3). Figure 6. Effects f surface curvature n f (red slid line K =.3; blue dashed line K = ; black slid line K =.3). Figure 7. Effects f surface curvature n f (red slid line K =.3; blue dashed line K = ; black slid line K =.3). 981
(a) (b) Figure 8. (a) Effects f surface curvature n g with displacement effect (red slid line K =.3; blue dashed line K = ; black slid line K =.3); Effects f surface curvature n g withut displacement effect (red slid line K =.3; blue dashed line K = ; black slid line K =.3). ( + ) ( + ) V (iii) y y R = (cylinder) V R y p c where V is the velcity far upstream and V, V, V are the velcities alng the stagnatin streamlines fr the three cases (i), (ii), (iii) cnsidered, respectively. Fr very small y, the abve relatins reduce t p c V V V y ay = = = =, (63) V V V R V thus giving, V a =. (64) R Therefre, a can be determined frm knwledge f the apprach flw and the nse radius. 98
(a) (b) Figure 9. (a) Effects f surface curvature n g with displacement effect (red slid line K =.3; blue dashed line K = ; black slid line K =.3); (b) Effects f surface curvature n g withut displacement effect (red slid line K =.3; blue dashed line K = ; black slid line K =.3). At this pint, it can be cncluded that the thermal stagnatin-pint flw prblem at the leading edge f axial flw turbine blades has been crrectly and cmpletely slved. This leads t exact slutins fr the gverning Navier-Stkes equatins under the assumptin f steady flw, small nse radius, cnstant external Ω, and any Re. 7. Extentin t Centrifugal Impeller Blades The present analysis is frmulated fr a statinary curved surface with a unifrm flw having a cnstant Ω appraching the surface. Therefre, it is directly applicable t leading edge prblems in axial flw turbines where Crilis frce effect are absent in the viscus flw and heat transfer arund turbine blades. Hwever, this is nt true fr a radial flw machine, and it wuld seem that the present analysis culd nt be applied t study the leading edge prblem f cen- 983
Figure 1. Effects f surface curvature n q (red slid line K =.3; blue dashed line K = ; black slid line K =.3). Figure 11. Effects f surface curvature n the displacement thickness rati, δ δ, fr * * the case Ω = (red slid line withut displacement effect; blue dashed line displacement effect included). trifugal impeller blades. Frtunately, there are similarities between a unifrm shear flw tward a statinary surface and a unifrm flw tward a rtating surface that wuld allw the use f the present results fr the study f flws tward rtating curved surfaces. T appreciate this, cnsider the steady thermal stagnatin-pint flw with a unifrm velcity appraching a tw-dimensinal curved surface rtating at a cnstant speed f Ω. The equatins gverning the tw-dimensinal flw can be written with respect t a crdinate system attached t the rtating surface (Figure 1) and beying the right-hand rule. If u and v are again used t dente the relative velcities alng x and y directin, respectively, then the steady state equatins can be written as x ( hv), u + = (65) y 984
Figure 1. Curvilinear c-rdinate system fr a rtating surface with cnstant speed f Ω. 1 * 1 1 1 ( ) ρ uux + v hu hω v = px + ν h uxx + huyy + ku y k h u + kh v x, y 1 * 1 1 1 uvx + hvvy ku + hω u = ρ hpy + ν h vxx + hvyy + kvy k h v kh u x, (67) (66) 1 utx + hvty = α h Txx + htvyy + kt y, (68) where * 1 Ω p = p ρr, is the reduced pressure and r is the radial distance frm the axis f rtatin. As befre, k is taken t be cnstant. Since the flw is tw-dimensinal, there is nly ne cmpnent f vrticity and it is nrmal t the plane f flw. This cmpnent is given by 1 1 ( ) (69) h v h hu =Ω, (7) x in the inviscid flw regin. With the prblem thus frmulated, it can be easily seen that the inviscid flw is again gverned by the Pissn Equatin (5) with bundary cnditins given by (8) and (9). Therefre, the inviscid velcity field is given by (18) and (19) and the pressure field is btained by integrating (66) and (67) with the kinematic viscsity ν. The result is y ( ln h) * * a p p = ρ + + h k h 4k h kh 4 1 ax aωx4h ln h h + 1 Ω h + 1. (71) 985
* By crss-differentiating Equatins (66) and (67) t eliminate p, it can be shwn that the resulting vrticity equatin is again given by (1) because the Crilis frce terms are zer as a result f the cntinuity equatin (65). The bundary cnditins are the same as (). In view f this, the velcity field arund the stagnatin pint f a rtating curved surface is the same as that arund the stagnatin pint f a statinary curved surface with an apprach flw having a cnstant Ω equal t twice the rtatinal speed. Since the temperature field as defined by (68) nly depends n the velcity field, the resulting slutin f (68) wuld als be the same as that btained befre, prvided the thermal bundary cnditins remain the same. The pressure field will be different and is given by the integral f (66) and (67) nce the velcity field is knwn. Frm the abve discussin, it can be seen that the present exact analysis can als be extended t study the steady heat transfer and viscus flw arund the stagnatin pint at the leading edge f centrifugal impeller blades. 8. Cnclusins The prblem f a steady thermal stagnatin-pint flw at the leading edge f an axial flw turbine under the influence f a cnstant external Ω has been analyzed. It is shwn that exact slutins t the steady gverning Navier- Stkes equatins can be btained if the nse radius is cnstant. Heat transfer and skin frictin results btained fr the curvature parameter K ranging frm.3 K.3 shw that, within this range, the linear relatin between wall shear and K given by the apprximate analysis f [4] [5] is essentially crrect. Hwever, the decrease f wall heat flux with K fr K > is faster than that predicted in [4] [5], and the ppsite is true fr K <. Cnsequently, the exact results shw that the apprximate analysis is crrect nly in the determinatin f the slpes f the variatin f wall shear and lcal heat transfer with K at K =. External Ω gives rise t a wall shear stress that is respnsible fr shifting the zer skin frictin pint away frm the stagnatin pint. Althugh the wall shear is a functin f external Ω and displacement, it is independent f curvature. Besides this effect, external Ω has n ther effect n wall shear and lcal heat transfer. This is cntrary t the results given in [4] [5], which revealed that external Ω als has a secnd-rder effect n wall shear and wall heat flux. This errr in the analysis detailed in [4] [5] culd be traced t its incrrect prpsed expansin fr the inviscid surface velcity in the immediate vicinity f the stagnatin pint. If a crrect expansin is prpsed, the apprximate results are cnsistent with the exact slutins, at least t the lwest rder. Surface curvature als influences the wall static pressure distributin in the vicinity f the stagnatin pint. It is fund that cnvex curvature decreases the favrable pressure gradient, but cncave curvature increases it. This implies that the flw near the leading edge f turbine blades (whse curvature is cnvex) will be mre susceptible t laminar separatin than the crrespnding flw tward a plane surface. In additin, surface curvature affects the velcity and temperature prfiles and the thickness f the viscus layer. All these underline the impr- 986
tance f including surface curvature and external Ω effect in the heat transfer calculatin arund turbine blades. Finally, it is shwn that the present analysis can als be applied t study the steady leading edge stagnatin-pint flw prblem in centrifugal impeller blades. With the exceptin f the pressure field, the slutin t the impeller blade prblem is identical t that f the axial flw turbine blade. Acknwledgements The first authr wuld like t acknwledge the supprt given him by the Mechanical Engineering Department, the Hng Kng Plytechnic University, Hng Kng, during his stay as Visiting Chair Prfessr frm September t Octber f 16. References [1] van Driest, E.R. (1959) Cnvective Heat Transfer in Gases. In: Lin, C.C., Ed., High Speed Aerdynamics and Jet Prpulsin, Vl. 5, Princetn University Press, Princetn, NJ, 339. [] Kline, S.J., Mrkvin, M.V., Svran, G. and Cckrell, D.J. (1968) v.1. Methds, Predictins, Evaluatin, and Flw Structure. Prceedings f the Stanfrd Cnference n Cmputatin f Turbulent Bundary Layers, AFOSR-IFP, 1, 18-19. [3] Cles, D.E. and Hirst, E.A. (1968) v..cmpiled Data. Prceedings f the Stanfrd Cnference n Cmputatin f Turbulent Bundary Layers, AFOSR-IFP,. [4] Van Dyke, M. (196) Higher Apprximatins in Bundary Layer Thery, Part I: General Analysis. Jurnal f Fluid Mechanics, 14, 161-177. https://di.rg/1.117/s1161147 [5] Van Dyke, M. (196) Higher Apprximatins in Bundary Layer Thery, Part II: Applicatin t Leading Edges. Jurnal f Fluid Mechanics, 14, 481-495. https://di.rg/1.117/s1161391 [6] Bradshaw, P. (1973) Effect f Streamline Curvature n Turbulent Flw. AGARDgraph 169. [7] S, R.M.C. and Mellr, G.L. (1973) Experiment n Cnvex Curvature Effect in Turbulent Bundary Layers. Jurnal f Fluid Mechanics, 6, 43-6. https://di.rg/1.117/s11733 [8] S, R.M.C. and Mellr, G.L. (1975) Experiment n Turbulent Bundary Layers n a Cncave Wall. Aernautical Quarterly, 6, 5-4. https://di.rg/1.117/s19597174 [9] Bradshaw, P. (1969) The Analgy between Streamline Curvature and Buyancy in Turbulent Shear Flw. Jurnal f Fluid Mechanics, 36, 177-191. https://di.rg/1.117/s11691583 [1] Rastgi, A.K. and Whitelaw, J.H. (1971) Prcedure fr Predicting the Influence f Lngitudinal Curvature n Bundary Layer Flws. ASME Paper 71-WA/FE-37. [11] Irwin, H.P.A.H. and Arnt Smith, P. (1975) Predictin f the Effect f Streamline Curvature n Turbulence. Physics f Fluids, 18, 64-63. https://di.rg/1.163/1.861 [1] S, R.M.C. and Mellr, G.L. (1978) Turbulent Bundary Layers with Large Streamline Curvature Effect. Zeitschrift für angewandte Mathematik und Physik ZAMP, 987
9, 54-74. https://di.rg/1.17/bf179733 [13] Jhnstn, J.P. and Eide, S.A. (1976) Turbulent Bundary Layers n Centrifugal Cmpressr Blades, Predictin f the Effect f Surface Curvature and Rtatin. Jurnal f Fluids Engineering, 98, 374-381. [14] S, R.M.C. (1981) Heat Transfer Mdeling fr Turbulent Shear Flws n Curved Surfaces. Zeitschrift für angewandte Mathematik und Physik ZAMP, 3, 514-53. https://di.rg/1.17/bf94718 [15] Ksinlin, M.L., Launder, B.E. and Sharma, B.I. (1974) Predictin f Mmentum, Heat and Mass Transfer in Swirling, Turbulent Bundary Layers. Jurnal f Heat Transfer, 96, 4-9. https://di.rg/1.1115/1.345165 [16] Hiemenz, K. (1911) Die Grenzschicht an einem in den gleichfrmigen Flussigkeitsstrm eingetauchten geraden Kreiszylinder. Dingler's Plytechnical Jurnal, 36, 31. [17] Hwarth, L. (1938) On the Calculatin f the Steady Flw in the Bundary Layer near the Surface f a Cylinder in a Stream. Aernautical Research Cuncil, R. & M. 163. [18] Schlichting, H. (196) Bundary Layer Thery. 4th Editin, McGraw-Hill Bk Cmpany, New Yrk, Lndn, 78. [19] Massey, B.S. and Claytn, B.R. (1965) Laminar Bundary Layers and Their Separatin frm Curved Surfaces. Jurnal f Basic Engineering, 87, 483-493. https://di.rg/1.1115/1.36558 [] Schultz-Grunw, F. and Breuer, W. (1965) Laminar Bundary Layers n Cambered Walls. In: Hlt, M., Ed., Basic Develpments in Fluid Dynamics, Vl. 1, 377-436. [1] Narasimha, R. and Ojha, S.K. (1967) Effect f Lngitudinal Surface Curvature n Bundary Layers. Jurnal f Fluid Mechanics, 9, 187-199. https://di.rg/1.117/s116771 [] Stuart, J.T. (1959) The Viscus Flw Near a Stagnatin Pint When the External Flw Has Unifrm Vrticity. Jurnal f the Aerspace Sciences, 6, 14-15. https://di.rg/1.514/8.7963 [3] van Dyke, M. (1969) Higher-Order Bundary Layer Thery. Annual Review f Fluid Mechanics, 1, 65-9. https://di.rg/1.1146/annurev.fl.1.1169.145 [4] Gldstein, S. (1965) Mdern Develpments in Fluid Dynamics. V. 1, Dver Publicatins, Inc., New Yrk, 119. [5] Ralstn, A. and Wilf, H.S. (196) Mathematical Methds fr Digital Cmputers. Jhn Wiley Sns, Inc., New Yrk, Lndn, 93. 988
a 1 Arbitrary cnstant defined in Equatin (13) a Arbitrary cnstant defined in Equatin (14) a Arbitrary cnstant defined in Equatin (16) C, C 1 Integratin cnstants defined in Equatin (4) and Equatin (41) C p Specific heat f fluid at cnstant pressure f(ζ) Transfrmed similar velcity functin defined in Equatin (3) F(η) Similar velcity functin defined in Equatins (3) and (4) g(ζ) Transfrmed similar velcity functin defined in Equatin (33) G(η) Similar velcity functin defined in Equatin (3) h = 1 + ky Metric cefficient k Surface curvature K = k ν a Nrmalized surface curvature L Characteristic length p Static pressure p Static pressure at the stagnatin pint Pr = να Prandtl number q Wall heat flux q Wall heat flux fr the k = case Re = U r L ν Reynlds number T Fluid temperature T w Wall temperature T Fluid temperature far away frm the wall u Velcity alng x-directin U r Characteristic velcity v Velcity alng y-directin x, y Crdinates measured alng the wall and nrmal t the wall, respectively α Thermal diffusivity f fluid δ Thickness f viscus layer δ Displacement thickness fr the Ω= case η = y a ν Similarity variable Θ ( η ) Nndimensinal temperature functin defined in Equatin (5) θ( ζ ) Transfrmed nndimensinal temperature functin defined in Equatin (34) κ Thermal cnductivity f fluid μ Viscsity f fluid ν Fluid kinematic viscsity ρ Fluid density τ w Wall shear stress τ Mdified wall shear stress τ Mdified wall shear stress fr the k = case Ψ Stream functin fr inviscid flw Ω External vrticity ζ = ln ( 1+ Kη) K Transfrmed similarity variable 989