To appear in International Journal of Numerical Methods in Fluids in Stability analysis of numerical interface conditions in uid-structure therm

Transcription

1 To appear in International Journal of Numerical Method in Fluid in 997. Stability analyi of numerical interface condition in uid-tructure thermal analyi M. B. Gile Oxford Univerity Computing Laboratory Numerical Analyi Group Thi paper analye the numerical tability of coupling procedure in modelling the thermal diuion in a olid and uid with continuity of temperature and heat ux at the interface. A imple onedimenional model i employed with uniform material propertie and grid denity in each domain. A number of dierent explicit and implicit algorithm are conidered for both the interior equation and the boundary condition. The analyi how that, in general, thee are table provided Dirichlet boundary condition are impoed on the uid and Neumann boundary condition are impoed on the olid; in each cae, the impoed value are obtained from the other domain. Oxford Univerity Computing Laboratory Numerical Analyi Group Wolfon Building Park Road Oxford, England OX 3QD April, 997

2 Introduction Thi analyi i motivated by interet in numerical procedure for coupling eparate computation of thermal diuion in a olid and a uid. A typical example application i the computation of heat tranfer to a blade in a ga turbine. The urrounding air in a high preure turbine i on average at a much higher temperature and therefore there i a ignicant heat ux from the uid into the turbine blade. In teady-tate, thi i matched by a correponding heat tranfer from the blade to relatively cold air owing through internal cooling paage. One approach to the numerical approximation of the above ituation would be the ue of a ingle conitent, fully-coupled dicretiation modelling both the olid and the uid, plu the boundary condition at the interface [9]. However, for the olid it i the calar unteady parabolic p.d.e. which decribe the thermal diuion, while for the uid the appropriate equation are the Navier- Stoke equation, with uitable turbulence modelling. Therefore, the production of a ingle fully-coupled code for the combined diuion application can be a much work a writing the individual program for the eparate olid and uid application. Since there are often exiting code which accurately and eciently olve thee individual problem, a more practical approach in many circumtance i to ue thee together to analye coupled problem [, 4, 8, 7, 3, ]. Both CFD code and thermal analyi code uually have the capability to pecify either the temperature or the heat ux at boundarie. A natural choice therefore for coupling thee code i to pecify the urface temperature at the interface in one code, taking the value from the other code, and pecify the boundary heat ux in the econd code, taking it value from the rt code [, 4]. A concern i whether there i any poibility that thi coupling procedure will introduce a numerical intability which doe not exit for the uncoupled problem. Thi i the iue that i addreed in thi tudy. The general theory for the analyi of numerical interface or boundary condition intabilitie i well-etablihed [6, ] but can be very complicated to apply in practice. In D and 3D computation of engineering interet, one cla of error mode which might be untable are thoe whoe variation i purely in the direction normal to the interface between the olid and uid. The nite difference or nite volume equation for thi cla of error mode reduce to being one-dimenional, and therefore in thi paper we implify the analyi for thi diuion problem by retricting attention to a imple D model problem with a uniform grid on either ide of the interface. Since there i no velocity component normal to the olid boundary in D and 3D ow, it i appropriate in the D model problem to omit any convection term. Stability for thi D model problem i a neceary condition for the tability of the real D and 3D computation. It may, or may not, be a ucient condition for tability but undertanding the nature of poible D intabilitie clearly give inight into the potential intabilitie

3 3 in D and 3D computation. Analytic problem The parabolic p.d.e. decribing unteady thermal diuion @x q = (.) Here T (x; t) i the temperature, q(x; t) i the heat ux, c(x) i the heat capacity and k(x) i the conductivity. Thee equation are valid for arbitrary, piecewie continuou poitive function c(x); k(x). The nite volume algorithm to be analyed are all baed on the integral verion of thi equation, d dt Z x x c T dx = [q] x x : (.) At any interface at which c and/or k are dicontinuou, the equation are augmented by the requirement that T and q mut be continuou. The boundary condition a x! are that the temperature aymptote to a contant value, T, and o the heat ux tend to zero. Dening T max (t) = max x T (x; t); T min (t) = min x T (x; t); (.3) an important property of olution of the unteady diuion equation i that for any non-uniform initial condition T (x; 0) and for all t > 0, dt max dt 0; dt min dt 0: (.4) Furthermore, if T = T + then T min (t)! T max (t) a t!. The behaviour of the maximum and minimum temperature will be important in dening the numerical tability of the coupled ytem. Although the above theory i given for general c(x); k(x), in thi paper we will now retrict attention to a ingle interface at x = 0 with c and k having uniform value c ; k for x <0, and c + ; k + for x >0. 3 Fully-coupled dicretiation In thi ection we examine the tability of fully-coupled dicretiation of the model problem. The theory for thi i well etablihed ince it i imply a pecial cae of the more general problem of the dicretiation of a parabolic p.d.e. with patially varying diuivity [5, 0]. There are everal reaon for doing thi analyi even though it i believed that the fully-coupled approach i not the mot

4 4 practical approach to real coupled application. The rt i to how that in a good fully-coupled dicretiation there are no intabilitie aociated with the interface treatment, that tability of the dicretiation on the uniform meh on either ide of the interface i a neceary and ucient for tability of the fullycoupled dicretiation. The econd i to have a benchmark againt which to compare the `weakly-coupled' dicretiation in the next ection. Thee will be hown to have interface intabilitie under certain condition, and it i informative to ee how thee are related to dierence in the interface treatment relative to the fully-coupled dicretiation. Uing a computational grid with uniform pacing x for x <0 and uniform pacing x + for x >0, the location of grid node i given by x = ( x ; 0 x + ; 0 (3.) Aociated with each grid node i the dicrete temperature variable T n to approximate the analytic olution T (x; t) at x=x ; t=nt. which i 3. An explicit algorithm Uing forward Euler time dierencing and conervative patial dierencing baed on the integral form of the unteady diuion equation on the interval x xx + give the following explicit algorithm, C (T n+ T n ) = (q n + q n ); q n + = K + (T+ T n n ) (3.) where 8 >< C = c x t c x t ; < 0 + c +x + ; = 0 t (3.3) and >: K + c + x + ; > 0 t = 8 >< >: k x ; + < 0 k + x + ; + > 0 (3.4) Note that the equation for = 0 involve the conductivity and heat capacity on both ide of the interface. In particular, t C 0 i the heat capacity of the whole nite volume computational cell extending from x to x +.

5 5 For 6=0, the dierence equation reduce to T n+ = T n + d T n + T n + T n (3.5) where d = k t cx : (3.6) Standard Fourier tability analyi on either ide of the interface how that a dicrete Fourier mode i table provided d. We will now prove that if the requirement of Fourier tability are atied on each ide of the interface, then the fully-coupled dicretiation i table in the ene that T n+ max T n max ; T n+ min T n min ; (3.7) where T n max max T n ; We begin by noting that if d T n min min T n : (3.8) then for any poitive value r d + rd + ( + r) =) d + rd + + r : (3.9) We will ue thi reult with r dened a the ratio of the heat capacitie of the computational cell on either ide of the interface, where r = c +x + c x : (3.0) The next tep i to re-write the full dierence equation a T n+ = ( a b ) T n + a T n + + b T n (3.) a = b = d ; < 0 a 0 = d + r ; b 0 = rd + + r ; (3.) a = b = d + ; > 0 0 < d o for all, a ; b and a b are poitive quantitie and thu T n+ i a poitive weighted average of T n + ; T n ; T n. Hence, T n min min(t n + ; T n ; T n ) T n+ max(t n + ; T n ; T n ) T n max (3.3) Thi i true for all, and o taking the maximum over all, and the minimum over all, give the deired reult, Equation (3.7).

6 6 3. An implicit algorithm Replacing the forward Euler time dierencing with backward Euler time dierencing give the following implicit algorithm. C (T n+ T n ) = (q n+ + q n+ ) q n+ + = K + (T n+ + T n+ ); (3.4) with C ; K + a dened before. Fourier tability analyi of the dicretiation on either ide of the interface how it to be unconditionally table. The fully-coupled dicretiation i alo unconditionally table in the ame ene a before. To prove thi, the dierence equation i re-written a where T n+ = ( a b ) T n + a T n+ + + b T n+ ; (3.5) a = b = K + K + + K K + C ; ; (3.6) K + + K + C a b = C K + + K + C : It i clear that a ; b and a b are poitive quantitie, for all. We now chooe J uch that T n+ J = Tmax. n+ Subtracting Tmax n+ from both ide of the dierence equation give ( a J b J )(T n J T n+ max) + a J (T n+ J+ T n+ max) + b J (T n+ J T n+ max) = 0: (3.7) Becaue T n+ J ; T n+ J+ Tmax n+ and a J ; b J ; a J b J are all poitive, either TJ n > T max n+ or TJ n = T n+ J+ = T n+ J+ = Tmax. n+ In the rt cae, we immediately get the reult that Tmax n > T n+ max. In the econd cae, we can repeat the argument with = J. By further repetition if neceary, we conclude that either Tmax n > T max n+ or T n = T n+ = Tmax, n+ for all, in which cae Tmax n = Tmax. n+ Exactly the ame argument can be ued to prove that Tmin n T n+ min equality occurring only in the trivial cae in which T n i contant. with 4 Looely-coupled dicretiation In the looely-coupled dicretiation, each half of the domain i olved eparately with boundary condition containing information from the other. The natural boundary condition for a diuion problem are either Dirichlet (the pecication

7 7 of the boundary temperature) or Neumann (the pecication of the boundary heat ux). Therefore we will conider a looely-coupled procedure in which the calculation for x 0 ue Dirichlet data obtained from the olution for x 0, while the calculation for x 0 ue Neumann data obtained from the olution for x An explicit algorithm Given exiting olution at time level n in both halve of the domain, the implet and mot natural explicit numerical algorithm for determining T n+ for 0 i c x t c x t (T n+ T n ) = k x (T n + T n +T n ); < 0 (T n+ 0 T n 0 ) = q w k x (T n 0 T n ); (4.) where q w i the heat ux pecied a the interface boundary condition. Uing a nite volume derivation, the equation for < 0 correpond to the control volume [x ; x + ] of width x, wherea the equation for =0 correpond to the control volume [x ; 0] of width x. The implet conitent equation for determining the heat ux at the interface from the data in 0 i q w = k + x + (T n T n 0 ): (4.) Thi one-ided approximation to the temperature gradient at the urface i only rt order accurate during unteady tranient. However, it i typical of the numerical method ued for practical computation [8, 7]. The correponding explicit numerical algorithm for imultaneouly determining T n+ for >0 i c + x + t (T n+ T n ) = k + x + (T n + T n +T n ): (4.3) The equation for = require the variable T n 0 and thi i et by the Dirichlet boundary condition T n 0 = T w ; (4.4) where T w i the interface temperature. The obviou value for thi i imply T n 0 from the computation for 0. To ummarie the communication between the two calculation for 0 and 0, at each timetep there i an exchange of data, with the program or ubroutine performing the calculation for 0 upplying the value of T w to the other program or ubroutine performing the calculation for 0, while the latter

8 8 end q w to the former. It i then poible that the computation for the two halve could proceed in parallel (perhap uing eparate procee on eparate worktation) until they again exchange data before the next timetep. By comparing Equation (4.,4.3) with Equation (3.), it can be een that the only dierence i the omiion of the term c + x + t in the equation for = 0. If c + x + c x, then thi omitted term i negligible compared to the retained term c x t and o it eem likely that no intability will be introduced by it omiion. On the other hand, if c + x + c x, then the omitted term may be very ignicant. Thi indicate very imply that a key parameter in the following analyi will be the variable r, dened earlier in Equation (3.0) a the ratio of thee two quantitie. For the purpoe of analyi it i more convenient to conolidate and implify the equation into the following form, T n+ = T n + d T n + T n + T n ; < 0 T n+ 0 = T n 0 d T n 0 T n + rd+ (T n T n 0 ) ; (4.5) T n+ = T n + d + T n + T n + T n ; > 0 where d and r are a dened previouly. In applying the tability theory of Godunov and Ryabenkii [6, ], the tak i to invetigate the exitence of eparable normal mode of the form T n = z n f : (4.6) The dicretiation i untable if the dierence equation admit uch olution which atify the far-eld boundary condition, f! 0 a!, and have z >, giving exponential growth in time. The form of the olution i very imilar to the aumed Fourier mode, except that the amplitude of the patial ocillation decay exponentially with away from the interface. For thi application the normal mode mut be of the form 8 < T n = : z n ; 0 z n +; 0 : (4.7)

9 9 The dierence equation, Equation (4.5) are atied provided the three variable z; ; + atify the following equation. z = + d ( + ) z = + d ( ) + rd + ( + ) (4.8) z = + d + ( ) Solving the rt of thee equation to obtain give 0 = 4d d z A : (4.9) To atify the far-eld boundary condition a! it i neceary to chooe the negative quare root when the argument i real and poitive. When it i complex, the choice of root i dened by the requirement that <. Similarly, olving the third of the equation give + = z d + 4d + A : (4.0) z Subtituting thee into the econd equation give the following nonlinear equation for z. 0 4d 4d + A = 0 (4.) z z There i no imple cloed form olution to thi, giving z a an explicit function of the parameter d ; d + ; r. Intead, we conider aymptotic olution under dierent aumption. When d ; d +, the quare root term can be expanded to give the following approximate equation and olution. d z rd + z 0; =) z d rd + : (4.) The requirement for tability i z <. The olution z(r) lie inide z = for uciently mall value of r, but then croe it at z = when r = d +. Thu for d ; d + the tability requirement i r < d +. Expanding the analyi to conider arbitrary value for d ; d +, we begin by conidering the aymptotic behaviour when r and r. When r, the econd term in Equation (4.) i relatively mall, and the approximate olution i 4d z 0; =) z 4d : (4.3)

10 0 Since d mut atify 0 < d for the dicretiation to be table according to Fourier tability analyi, it follow that z. Thu, there i no coupled intability when r. When r, the rt term in Equation (4.) i relatively mall, and o to a rt approximation the olution i 4d + z 0; =) 4d + z 0 =) z : (4.4) To get a more accurate approximate olution, the rt term i approximated uing z to obtain 4d + z r =) 4d + z r =) z rd + : (4.5) Thu for xed d + and uciently large r, there i an intability with z being large, real and negative. The correponding value of and + will be mall, real and negative, o the intability will appear a a `awtooth' ocillation mode, both patially and in time, with an amplitude which decay exponentially away from the interface, but grow exponentially in time. Since the looely coupled ytem i table for r and untable for r the remaining quetion i the value of r at which the intability begin. Thi correpond to the lowet poitive real value of r for which z =. Becaue of the requirement that r i real, it can be hown from Equation (4.) that thi again require z =, in which cae p d r = p (4.6) d + Thu the condition for tability i r < p d p d + (4.7) A typical calculation with a timetep cloe to the Fourier tability limit might have d = d + = 3, for which the coupled tability limit i r <. The key to 8 obtaining tability in practical computation i the correct choice of which half of the domain ue the Dirichlet boundary condition and which half ue the Neumann boundary condition. The uual practice for the coupled blade/air computation dicued in the Introduction i to ue Neumann boundary condition for the olid computation, and Dirichlet boundary condition for the uid computation. For thi choice, the correponding value of r i given by r = c uid x uid c olid x : (4.8) olid

11 Given typical value for the parameter involved, r i uually very mall and o thi i table. If, on the other hand, one were to ue Dirichlet boundary condition for the olid computation and Neumann boundary condition for the uid computation, then the appropriate value for r would be the invere of the above quantity, which would be very large. In thi cae the coupled calculation would be untable unle one ued an extremely mall timetep. Uing the approximate olution for r in Equation (4.5), the timetep tability limit i given by d + r ; (4.9) o tability of the coupled ytem would require the ue of a timetep very much maller than that needed for Fourier tability. Thi analyi i upported by the numerical reult preented in Figure and. The computation ue the nite domain , initial condition T 0 = for < 0 and T 0 = for 0 and boundary condition T n 000 = ; T n 000 =. In addition, all of the computation ue d = d + = 3 for 8 which the analyi above predict the coupled ytem to be table only for r <. Figure how two et of reult with T n plotted for the rt 0 iteration in each cae. In a), r = and the olution i clearly table, with an initial tranient at the interface decaying very quickly, while in b), r = and the olution i very untable. Figure how another two et of reult with T n plotted every 5 iteration. In a), r = 0:99 and the olution appear to be table, although with the interface tranient decaying more lowly in thi cae, while in b), r = :0 and the olution i clearly untable. 4. A hybrid algorithm The next algorithm to conider i a hybrid one, in which the computation i unaltered for >0, but the algorithm for 0 i replaced by the correponding implicit method baed on a backward Euler time dicretiation. c x t c x t (T n+ T n ) = k n+ (T+ T x n+ +T n+ ); < 0 (T n+ 0 T n 0 ) = q w k n+ (T0 T x n+ ): (4.0) The boundary heat ux q w i again dened explicitly by q w = k + x + (T n T n 0 ): (4.) The dierence equation for >0 are unchanged, a i the communication of data between the calculation for 0 and >0.

12 The conolidated, implied form of the equation i T n+ = T n + d T n+ + T n+ T n+ 0 = T n 0 d T n+ 0 T n+ T n+ = T n + d + T n + T n + T n and the normal mode i again of the form 8 < T n = : + T n+ ; < 0 z n ; 0 + rd+ (T n T n 0 ) ; (4.) ; > 0 z n +; 0 : (4.3) The dierence equation, Equation (4.) are atied provided the three variable z; ; + atify the following equation. = z + d ( + ) = z + d ( ) + rd + z ( + ) (4.4) z = + d + ( ) The third of thee equation require that + depend on z in exactly the ame way a for the purely explicit algorithm. Solving the rt of thee equation ubect to the far-eld boundary condition give = + z d Subtituting thee into the econd equation give + 4d z r When r, the aymptotic olution i and o the dicretiation i table for all value of d. When r, the aymptotic olution i + 4d A : (4.5) z 4d + A = 0 (4.6) z z = ( + 4d ) + O(r); (4.7) z rd + p + 4d + O(r ); (4.8) and o the coupled dicretiation i till untable for uciently large value of r.

13 3 The cro-over from tability to intability again occur when z =, giving p + d r = p (4.9) d + Thu the condition for tability i r < p + d p d + (4.30) Comparing thi reult with the correponding reult for the purely explicit algorithm, it can be een that the new tability region i greater except when d. Thi ha a phyical interpretation; when d i not mall, the trong implicit coupling of the computational cell for 0 increae the eective thermal capacity of the cell aected in one timetep by the interface heat ux. Numerical experiment were performed on the ame domain and with the ame initial and boundary condition a before, and with d = 4; d + = 3 8. The analyi above predict tability provided r < 6, and thi i upported by Figure 3 which how two et of reult with T n plotted every 5 iteration. In a), r = 5:95 and the olution i table with a lowly decaying interface tranient, while in b), r =6:05 and the olution i clearly untable. 4.3 An implicit algorithm We now conider an algorithm which i implicit on each ide of the interface, but with explicit updating of the data ued for the interface boundary condition. The implicit numerical algorithm for 0 i again c x t c x t with q w dened explicitly by (T n+ T n ) = k n+ (T+ T x n+ +T n+ ); < 0 (T0 n+ T0 ) n = q w k n+ (T x 0 T n+ ): (4.3) q w = k + x + (T n T n 0+): (4.3) An important point in the above equation i the ditinction between T0, n the value of T n at =0 a calculated for the domain 0, and T0+, n the value of T n at = 0 for the domain 0. In the previou dicretiation thee two value have been identical but thi will not be true in thi cae. The correponding implicit numerical algorithm for imultaneouly determining T n+ for >0 i c + x + t (T n+ T n ) = k + x + (T n+ + T n+ +T n+ ): (4.33)

14 4 The equation for = require the variable T n+ 0+ and thi i et by the Dirichlet boundary condition T n+ 0+ = T w ; (4.34) where T w i the interface temperature. Uing explicit updating of boundary data, T w = T n 0 ; (4.35) p T 0+ lag T 0 by one iteration. The pattern of communication between the calculation for 0 and 0 i exactly the ame a for the explicit algorithm. They exchange the value of T w and q w at the beginning of the timetep, perform the timetep calculation independently (poibly in parallel on eparate worktation) and then repeat the proce for the next timetep. For the purpoe of analyi it i again more convenient to conolidate and implify the equation into the following form, T n+ = T n + d T n+ + T n+ + T n+ T0 n+ = T0 n d T n+ 0 T n+ T n+ = T n + d + T n+ + T n+ + T n+ T n+ 0+ = T n 0 : ; < 0 + rd+ T n T n 0+ ; > 0 The form of the normal mode olution for thi cae i 8 < T n = : z n ; = 0 ; ; ; 3; : : : z n +; = 0+; ; ; 3; : : : ; (4.36) : (4.37) The fourth equation in Equation (4.36) i automatically atied by the above choice of normal mode. The other three equation require that the variable z; ; + atify the following equation. = z + d ( + ) = z + d ( ) + rd + z ( + ) (4.38) = z + d + ( ) Solution of the rt and third of thee equation, ubect to the far-eld boundary condition, give = + z d + = + z d d z A ; + 4d + A : (4.39) z

15 5 Subtituting thee into the econd equation give + 4d z + rz 0 When r, the aymptotic olution i and o the dicretiation i table for all value of d. When r, the aymptotic olution + 4d + A = 0: (4.40) z z = ( + 4d ) + O(r); (4.4) z i p r p! + 4d+ p + O(): (4.4) + 4d Thu for xed d ; d + and uciently large r, the coupled ytem i untable. It i not poible for general value of d ; d + to determine explicitly the value of r above which the olution procedure i untable. It i poible however to obtain an aymptotic olution under the aumption d ; d +. Thi i a reaonable aumption ince the motivation in uing implicit method i to ue much larger timetep than would be table uing explicit method. Under the aumption d ; d +, Equation (4.40) reduce to qd + rz q d + 0; =) z i p r d + d! 4 : (4.43) Hence, under thee condition the approximate tability limit i which can alo be re-expreed a r < c 3 +x 4 + k + d d + ; (4.44) < c 3 x 4 k : (4.45) Provided, a before, that the correct choice i made a to which domain ue the Neumann b.c.' and which ue the Dirichlet b.c.', then r hould be uciently mall that practical computation will be table. Numerical experiment were performed on the ame domain and with the ame initial and boundary condition a before, and with d = d + = 50. The approximate tability limit for thee value i r <, and thi i upported by Figure 4 which how two et of reult with T n plotted every 5 iteration. In a), r = : and the olution i table with a lowly decaying interface tranient, while in b), r =: and the olution i clearly untable.

16 6 5 Concluding remark The tability analyi in thi paper ha hown the viability of a looely-coupled approach to computing the temperature and heat ux in coupled uid/tructure interaction. The key point to achieving numerical tability i the ue of Neumann boundary condition for the tructural calculation and Dirichlet boundary condition for the uid calculation. Although the analyi wa performed here for the D model diuion equation, the reult are believed to be applicable to the real ituation in which the 3D diuion equation i ued to model the heat ux in the tructure and the 3D Navier-Stoke equation are ued to model the behaviour of the uid. Thi i upported by the practical experience of 3D computation performed uing thi coupling procedure [, 4]. The analyi alo aumed a time-accurate modelling of the uid/tructure interaction. In practical computation, the point of engineering interet i often the teady-tate temperature and heat ux ditribution. In uch cae, the computation in the tructure and uid can both proceed with dierent timetep given by their repective Fourier tability limit. The coupled normal mode analye remain valid uing the value of d ; d + baed on the timetep t ; t + ued in the two domain. Acknowledgement Thi reearch wa upported by Roll-Royce plc and ha benetted from dicuion with Dr. Peter Stow of Roll-Royce plc and Dr. Mehmet Imregun of Imperial College. Reference [] R.S. Amano, K.D. Wang, and V. Pavelic. A tudy of rotor cavitie and heat tranfer in a cooling proce in a ga turbine. Journal of Turbomachinery, 6:333{338, 994. [] D. Bohn, G. Lang, H. Schonenborn, and B. Bonho. Determination of thermal tre and train baed on a combined aerodynamic and thermal analyi for a turbine nozzle guide vane. ASME Paper 95-CTP-89, 995. [3] D. Bohn, H. Schonenborn, B. Bonho, and H. Wilhelmi. Prediction of the lm-cooling eectivene in ga turbine blade uing a numerical model for the coupled imulation of uid ow and diabatic wall. ISABE Conference Paper , 995.

17 7 [4] J. Chew, I.J. Taylor, and J.J. Bonell. CFD development for turbine blade heat tranfer. In 3rd International Conference on Reciprocating Engine and Ga Turbine, I. Mech E., London, number C , 994. [5] J. Crank. The Mathematic of Diuion. Clarendon Pre, nd edition, 975. [6] S.K. Godunov and V.S. Ryabenkii. The Theory of Dierence Scheme{An Introduction. North Holland, Amterdam, 964. [7] A. Heelhau and D.T. Vogel. Numerical imulation of turbine blade cooling with repect to blade heat condution and inlet temperature prole. AIAA Paper , 995. [8] A. Heelhau, D.T. Vogel, and H. Krain. Coupling of 3D Navier-Stoke external ow calculation and internal 3D heat conduction calculation for cooled turbine blade. In AGARD CP-57, Heat Tranfer and Cooling in Ga Turbine, 99. [9] J. Moore, J. G. Moore, G. S. Henry, and U. Chaudry. Flow and heat tranfer in turbine tip gap. Journal of Turbomachinery, :30{309, July 989. [0] K.W. Morton and D.F. Mayer. Numerical Solution of Partial Dierential Equation { an Introduction. Cambridge Univerity Pre, Cambridge, 994. [] R.D. Richtmyer and K.W. Morton. Dierence Method for Initial-Value Problem. Wiley-Intercience, nd edition, 967. Reprint edn (994) Krieger Publihing Company, Malabar.

18 8 :5 a) r = 0:5 T 0:5 : : b) r = 75: T 0: 75: Figure : Explicit algorithm with reult every iteration

19 9 :5 a) r = 0:99 0:5 T 0:5 : : b) r = :0 4: T 0: 4: Figure : Explicit algorithm with reult every 5 iteration

20 0 :5 a) r = 5:95 0:5 T 0:5 : : b) r = 6:05 : T : 6: Figure 3: Hybrid algorithm with reult every 5 iteration

21 :5 a) r = : 0:5 T 0:5 : : b) r = : 5: T 5: 5: Figure 4: Implicit algorithm with reult every 5 iteration