SIMILARITY SOLUTION ON UNSTEADY AXI-SYMMETRIC VISCOUS BOUNDARY LAYER FLOW

Transcription

1 SIMILARITY SOLUTION ON UNSTEADY AXI-SYMMETRIC VISCOUS BOUNDARY LAYER FLOW *P. Sulochana Depatment of Mathematics, Intell Engineeing College, Ananthapuamu *Autho fo Coespondence ABSTRACT In this pape, we have consideed an incompessible viscous fluid in a thee-dimensional axi-symmetic coodinate system. The govening equations ae tansfomed to odinay diffeential equations by using popely intoduced similaity vaiable. Futhemoe, pessue vaiations along the bounday laye thickness ae taken into account. The enegy equation is solved by Numeical technique. The fluid tempeatue, the velocity components and the solidification ate ae pesented fo diffeent values of nondimensional govening paametes. Keywods: Axisymmetic Solidification, Viscous Flow, Exact Solution, Stagnation Point, Unsteady Flow INTRODUCTION The manufactue of almost evey man-made mateial involves the solidification pocess at some stage of the manufactuing pocess. Some of the impotant pocesses that involve solidification ae foundy, welding, casting of ingots and continuous casting. Continuous casting is of such impotance because it is a vey economic method of foming a metallic component. The composition vaiation within a solidified poduct is known as segegation. Segegation in an alloy is a esult of solute ejection at the solidification font followed by its edistibution by diffusion and mass flow. Depending on the extension of composition vaiation, this defect is classified as mico segegation, elative to gain scale, o maco segegation, elative to the poduct scale. Maco segegation affects the mechanical (ductility, stength, etc) and chemical popeties (coosion esistance) of mateials in the poduct extension. It also affects pecipitation of weak second phases, that was not expected if the initial composition was achieved, and poosity may be geneated. Hence, Solidification is a two-phase phenomenon that is used in diffeent natual pocesses and industial applications. Glass, metal, plastic and oil industies, poviding food and othe coesponding industies needs a good insight of solidification behavio as the natue of solid gowth. Studies of phase change in stagnant media fo bette undestanding of convection effect upon the inteface behavio and solidification popeties ae needed by industial demand such as the desie fo moe homogenous semi-conducto cystals, in nuclea industy, as well as the bette undestanding of natual ice fomation. The classic poblem stagnant fluid solidifying on the cold plate is solved by Stefan (Stefan, 89). One dimensional heat fluxes method fo phase change poblem is pesented (Goodich, 978). These methods ae accompanied simplified assumptions such as one dimensionality solid-liquid inteface. An expeimental study fo natual convection in inteface within heat flux contolling due to solidification is povided (Spaow et al., 983). Also, a numeical method fo solidifying in natual convection is used (Lacoix, 989) and thee dimensional poblems fo natual convection accompaniment phase change in ectangula channel is solved by Hadji and Schell (99) in fluid vaiable popeties state with tempeatue. Solidification of a fluid laye confined between two isolated plates is investigated by Hanumanth (99). Anothe way fo calculating of heat flux depended to on time in natual convection is pesented by Oldenbug and Spea (99). A combined model fo phase change upon vaious states of pue substances, melting fluid poblem due to speading and solidifying on the flat plate and numeical modeling of foming and solidifying of a doplet on a cold plate is investigated by Tapaga et al., (99); Watanabe et al., (99); Machi et al., (993). Evolution due to impact on substate plate and solidifying of a doplet (Battkus and Davis, 988) is pesented. But in concentating upon stagnation flow, solidification of an inviscid fluid at inteface and effect of its phenomena on mophological instability is investigated by Rangel and Bian (994). Stefan poblem fo inviscid Cente fo Info Bio Technology (CIBTech) 63

2 stagnation flow by two methods and solidifying of supe-cooled liquid stagnation inviscid flow is consideed by Lambet and Rangel (3); Yoo (), espectively. Recently, Shokgoza and Rahimi () studied the two-dimensional solidification of a viscous stagnation flow. In this pape, the exact solution of the momentum equations (Shokgoza and Rahimi, 3) is used fo numeical solution of the enegy equation. Imagine the fluid fom fa field moves pependiculaly appoaches to a cold plate and afte impinging on the plate the solid phase will fomed on it gadually (Figue ). In this study, the solidification pocess of a viscous stagnation flow is investigated in a thee-dimensional axi-symmetic coodinate whee a new method is implemented fo validation of the numeical esults. In this method, the exact solution of a heat pofile is used as a quasi-steady solution fo the poblem. A paametic study is pefomed to examine influences of govening dimensionless paametes on the esults as well. An exact solution is pefomed fo solving momentum equations (Shokgoza et al., 6) while the enegy equation in liquid phase, solid-liquid inteface and solid phase is solved by using finite diffeence method. The exact solution of the enegy equation is used fo validation of the numeical solution of enegy equation, too. Foth ode Runge-Kutta algoithm is used fo solving momentum and enegy equations. In addition, Numeical solution is needed to finding unsteady tempeatue pofiles at each time step. Fomulation and Solution of the Poblem We conside unsteady viscous incompessible lamina stagnation flow with stain a( t ) pependiculaly appoaches to a plate, along z-diection, initially positioned at z when t. Fo all times of consideation, the fluid is solidified with vaiable solidification velocity and acceleation, S( t ) and S( t ), espectively, that of an imaginay plate at solid-liquid inteface is moved towads fluid whee S (t) is the plate distance, at time t, fom the plate oigin at z. Figue : Axi-symmetic Stagnation Flow (Coodinate System) Figue epesents thee-dimensional axi-symmetic coodinates with coesponding ( u,w ) velocities elated to (,z ). The imaginay plate is consideed as a flat one because the only mechanism of heat tansfe in the inteface is conduction with the same tempeatue diffeence so the substate emains flat. The inviscid flow can be assumed as potential flow within displacement thickness in bounday laye egion. Whee, a is the stain ate at fa field. Fo a Newtonian fluid with constant density and viscosity, unsteady thee-dimensional axi-symmetic Navie-Stokes equations govening the flow and heat tansfe ae given as: Cente fo Info Bio Technology (CIBTech) 64

3 Mass, Momentum: w u z u w t z z u u u p u u u u w w w p w w w u w (3) t z z z In liquid phase: Enegy (dissipation and adiation heat tansfe ae neglected without intenal souce): T T T T T T u w (4) t z z In solid phase T T T T (5) t z At the inteface d S(t ) Ts Tl hls ks kl (6) d t z z The conductivity and heat capacity coefficients ae constant (k and c espectively) also du c dt is assumed whee p,,, and ae the fluid pessue, density, kinematic viscosity, and themal diffusivity, espectively. The dissipation tems ae neglected in the enegy equation because of the flow velocities being too small. Also, subscipts s and l denote solid and liquid, espectively. Accoding to Shokgoza et al., (6), viscous pats of the velocity components ae as: u a( t ) f ( ) (7) w a a( t ) f ( ) (8) a, and z S( t ) (9) In which the tems involving f ( ) in (7), (8) compise the axi-symmetic similaity fom fo unsteady stagnation flow, and pime denotes diffeentiation with espect to. Tansfomations (7)-(9) satisfy () automatically and thei insetion into ()-(3) yields an odinay diffeential equation in tems of f ( ) along with an expession fo the pessue, as follow: d a P f f S a f a f f ' a d a a a p p a af f Sf a f d () In which, P a a a a () () () () a S S WS (3) Cente fo Info Bio Technology (CIBTech) 65

4 p(,z,t ) a( t ) P(,z,t ), S( t ) a S( t ), a( t ), a a S( t ) S( t ) a, a (4) Whee dot denotes diffeentiation with espect to t, Also, P,S,S and ae dimensionless foms of P,S,S and, espectively. The bounday conditions fo the diffeential equation () ae: : f, f (5) : f (6) It is woth mentioning that elation () which epesents pessue is obtained by integating Equation (3) in z-diection and by use of the potential flow solution as bounday conditions. To tansfom the enegy equation into a non-dimensional fom fo the case of defined wall tempeatue, we intoduce: T( ) T (7) Tw T Making use of tansfomations (7) - (9), this equation may be witten as: " S( ) f P. ' (8) With bounday conditions as: at (9), at () Whee, is dimensionless tempeatue, the subscipt w and efe to the conditions at the wall and in the fee steam, espectively, and pime indicates diffeentiation with espect to. Using the non-dimensional quantities fo tempeatue as, time as, distance fom axis as ~, and distance fom z axis as ~ z, equations (4)-(6) become: Fo liquid phase: l l l l l l u w l () z z Fo solid phase: s s s s s () z And fo thei intesection: P d S s l (3) St d k z z At fist, the momentum equation of liquid () is solved numeically using a shooting method based on Runge-Kutta algoithm. Resulted velocities ae used in enegy equation () in liquid egion in ode to convet this nonlinea equation to a linea one. Then, this linea equation is dischetized by using Powe Law scheme so fo small Pe ( Pe ) and Lage Pe ( Pe ), scheme is cental and upwind, espectively. Fo Pe, scheme is composite of these two. Fo solving the algebaic system of equations, TDMA i within ADI ii method is used. In addition, the esulted velocities of momentum solution ae used to obtain the exact solution of enegy equation (Shokgoza et al., 6). Howeve, this exact solution is not used to captue the fluid tempeatue in the computational domain at each time step. In the next sections, the eason of this phenomenon will be discussed. Cente fo Info Bio Technology (CIBTech) 66

5 RESULTS AND DISCUSSION In this section, in ode to validate the enegy equation numeical esults of ou study, the obtained esults ae compaed with exact solution and pevious studies. Compaison of esults between this study and exact solution is new and ceative method. Fo compaisons, Shokgoza et al., (3) study esults ae the most complete study and best selection. This efeence was selected to validate the achieved esults. Fo simplification and moe excellent compaisons, paamete intoducing in this study is the same as (Shokgoza et al., 3) study. The esults of these two studies ae pesented togethe in figue 3 fo P, St,,, k. i Liquid Liquid Bounday Final Solid Thickness Figue a: Final Solid Thickness, Low P Numbe Bounday Final Solid Thickness Figue b: Final Solid Thickness, High P Numbe Accoding to this figue, thee is a diffeence in ultimate solid thickness between two-dimensional and thee-dimensional cases, as expected; howeve, the tend of evolution is the same in both gaphs. In addition, the exact solution of enegy equation can be used fo validation of the numeical solution of the same equation. Figue 8 shows the compaison between exact and numeical solutions of tempeatues pofiles. Indeed, the exact solution is a quasi steady solution of the enegy equation. This means the numeical solution is the same as the exact solution if thee is enough time fo evolution while othe conditions ae maintained constant. In this figue, two pofiles ae matched completely, at fist and last times. At the beginning, the time of evolutions is vey small and diffeences between these two pofiles ae negligible (two pofiles ae matched completely). At the end, thee is enough time to complete the evolution of tempeatue pofile while vaiations of othe paametes ae not consideable as they ae vey close to each thei steady conditions. In the middle times, thee is a noticeable diffeence between two pofiles expectedly; howeve, the tend of the pofiles evolution at both figues is the same. Figue 4 epesents the tempeatue pofiles of liquid and solid phases fo diffeent times due to advancement of solidifying font. The slope of each chat is tg( ) in the fist node of liquid o solid phase. The figue eveals that solidification is stopped just as values of these two slopes become equal (note that ). Figue 5 pesents the exact solution of themal pofile fo diffeent times and diffeent solidification font velocity. Next, Figues 6 and 7 povide the velocity pofiles in and z diections fo diffeent times and Cente fo Info Bio Technology (CIBTech) 67

6 Solid Thickness solid thicknesses. As it is shown, when the solidification velocity is vey high, that is just fo initial moments, the slope of velocity pofile in bounday laye is vey steep and the velocity appoaches towad potential flow vey fast and so the thickness of viscous bounday laye is vey thin. By deceasing solidification velocity, Hiemenz flow is appeaed moe and moe. Also, a compaison is made in Figue 8 between exact and numeical solutions of heat tansfe pofiles at diffeent times. S (Shokgoza et al, 6) P = (D) Pesent study P = (3D) Solid Thickness =. Solid Thickness =.3 Solid Thickness =.5 Solid Thickness =.45 Solid Thickness = log ( ) Figue 3: Compaison of Two-Dimensional (Shokgoza et al., 6) and Thee- Dimensional (Pesent Study) Results fo P,St,K,, Figue 4: Numeical Solution, Themal Liquid and Solid Pofiles fo P,St,K,, P= Solid Thickness =. Solid Thickness =.3 Solid Thickness =.8 Solid Thickness =.7 Solid Thickness = Solid Thickness =. Solid Thickness =.5 Solid Thickness =.3 Solid Thickness =.7 Solid Thickness =.8 Solid Thickness =.8,Stain(t) =,Stain(t) =.,Stain(t) =.,Stain(t) =.,Stain(t) =.,Stain(t) =.5 f ' Figue 5: Exact Solution, Themal Pofile fo P,St,K,, Figue 6: Velocity Pofile in x Diection fo P,St,K,, Cente fo Info Bio Technology (CIBTech) 68

7 - Solid Thickness =. Solid Thickness =.3 Solid Thickness =.7 Solid Thickness =.8 Solid Thickness = Exact Solution, Last Times Numeical Solution, Last Times Exact Solution, Middle Times Numeical Solution, Middle Times Exact Solution, Initial Times Numeical Solution, Initial Times f Figue 7: Velocity Pofile in z Diection fo P,St,K,, Figue 8: Compaison between Tempeatues Pofile of Exact Solution and Numeical Solution P,St,K,, The paametic studies ae applied fo diffeent values of P,St, I,K, while study is concentated on the advancement of solidifying font vesus time that is the most impotant phenomenon in solidification. In Figue 9, a compaison is made between the solidification pocess of this study and Ref. (Shokgoza et al., 3) fo P, P and P., espectively. The tend is the same in the two gaphs so the solid uppe limit deceases as P numbe inceases elatively to basic P gaph. Mathematical analysis can confim the validity of the numeical solution. Solidification will stop when the steady conduction heat tansfe establishes in intesection. Onedimensional steady conduction heat tansfe at intesection eads: P. z St liquid solid that the just uppe and lowe nodes tempeatue in fluid and solid egions ae intoduced by liquid and solid, espectively. In this equation, dimensionless time ( ) tends to infinity as and solidification is stopped consequently while is assumed fo simplification. It can be efeed to (Shokgoza et al., 3) fo moe details. Figues A and B detemine the ultimate solid thickness that is equal to tg ( ) (whee tg ( ) is slope of tempeatue pofile at fist node). It is evident that when the thickness of the tempeatue bounday laye inceases, the solid uppe limit inceases and this inceasing is due to P numbe decease and vice vesa. Moeove, it was captue that by taking into account solely the effect of St numbe does not change the solid uppe limit as St numbe does not appea in (4) but St numbe vaiations changes the solidification time. Howeve, a complete match of the heat tansfe pofiles obtained by two methods of exact solution and numeical solution at the beginning and ending times can be consideed as the best eason fo validation of the numeical esults. Cente fo Info Bio Technology (CIBTech) 69 liquid solid

8 7 S D, P=. 3D, P=. D, P=. 3D, P=. D, P=. 3D, P=. S.4.6. K o =.5 K o = K o = log ( ) Figue 9: Compaison between Two- Dimensional and Thee-Dimensional; Effect of P Numbe upon Solidification Font fo St,K,, log ( ) Figue : Effect of k o Vaiations upon Solidification Font fo P,St, I 3.5 i =. i =. i =.5.5 P =. P = P =. S.5 S log ( ) Figue : Effect of I Vaiations upon Solidification Font fo P,St,K, log ( ) Figue : Effect of P Numbe upon Solidification Font fo St,K,, Equation (4) shows that deceasing of k o by half, inceases solid uppe limit two times exactly of that tg ( ) inceases two times (figue ). Also, Figue shows the effect of k and vaiations upon solidification font thickness. In this case, deceasing of k and by half, inceases thickness of solid two times exactly. Figue epesents esults fo I. 5 and I ( P, St, k, ). The vaiations of I have inteesting esults. When I tends to zeo, ultimate fozen thickness tends to infinity and this case equies anothe study sepaately. Figue epesents effect of P numbe vaiations in font solidification moe clealy. As peviously mentioned, this figue shows inceasing P numbe deceases solid uppe limit and vice vesa. Figue 3 epesents St numbe has no effect on the ultimate thickness in font solidification as fome discussion. Howeve, the St numbe has only effect upon the solidifying time. Cente fo Info Bio Technology (CIBTech) 7

9 .4. Stefan =. Stefan =. Stefan = S Figue 3: Effect of St log ( ) Numbe upon Solidification Font fo P,St,K, Conclusion We have consideed an incompessible viscous fluid in a thee-dimensional axi-symmetic coodinate system. The govening equations ae tansfomed to odinay diffeential equations by using popely intoduced similaity vaiable. The conclusions ae made as the following.. The esults show steady tempeatue bounday laye o, by moe exact wods, stat of steady tempeatue pofile slope detemines the ultimate solidification thickness.. The atio of liquid to solid tempeatue diffusivity and, moe impotantly, P numbe has effect upon this tempeatue bounday laye thickness. 3. Vey small effect of convection tems at nea of inteface leads to flatting solidification font but these tems ae vey impotant as appoaching to the edge of bounday laye. 4. The final solid thickness in a thee-dimensional stagnation flow is about. 75 times of that of a two-dimensional case. REFERENCES Battkus K and Davis SH (988). Flow induced mophological instabilities: stagnation-point flows. Jounal of Cystal Gowth Goodich LE (978). Efficient numeical technique fo one dimensional themal poblems with phase change. Intenational Jounal of Heat Mass Tansfe Hadji L and Schell M (99). Intefacial patten fomation in the pesence of solidification and themal convection. Physical Review A Hanumanth GS (99). Solidification in the pesence of natual convection. Intenational Communications in Heat and Mass Tansfe Lacoix M (989). Computation of heat tansfe duing melting of a pue substance fom an isothemal wall. Nume. Heat Tansfe B Lambet RH and Rangel RH (3). Solidification of a supe cooled liquid in stagnation-point flow. Intenational Jounal of Heat Mass Tansfe Cente fo Info Bio Technology (CIBTech) 7

10 Machi CS, Liu H, Lavenia EJ and Rangel RH (993). Numeical analysis of the defomation and solidification of a sigle doplet impinging on to a flat substate. Jounal of Mateials Science Oldenbug CM and Spea FJ (99). Hybid model fo solidification and convection. Numeical Heat Tansfe B 7-9. Rangel RH and Bian X (994). The inviscid stagnation-flow solidification poblem. Intenational Jounal of Heat Mass Tansfe 39(8) Shokgoza A and Rahimi AB (). Investigation of Two-dimensional unsteady stagnation pointflow and heat tansfe impinging on an acceleated flat plate. Jounal of Heat Tansfe Shokgoza A and Rahimi AB (3). Solidification of Two-Dimensional viscous, incompessible Stagnation flow. Intenational Jounal of Heat Tansfe Shokgoza A, Rahimi AB and Mozayyeni H (6). Investigation of thee-dimensional axisymmetic unsteady stagnation point-flow and heat tansfe impinging on an acceleated flat plate. Jounal of Applied Fluid Mechanics Spaow EM, Ramsey JW and Hais S (983). The tansition fom natual convection contolled feezing to conduction contolled feezing. Jounal of Heat Tansfe Stefan J (89). Ube die theoie de eisbildung, insbesondee ube die eisbildung in polamaee. Annalen de Physik und Chemie Tapaga G, Matthys EF, Valecia JJ and Szekely J (99). Fluid Flow, heat tansfe and solidification of molten metal doplets impinging on substates: compaison of numeical and expeimental esults. Metallugical Tansactions B 3B Watanabe T, Kuibayashi I, Honda T and Kanzawa A (99). Defomation and solidification of a doplet on a cold substate. Chemical Engineeing Science 47() Yoo JS (). Effect of viscous plane stagnation flow on the feezing of fluid. Intenational Jounal of Heat and Fluid Flow Cente fo Info Bio Technology (CIBTech) 7