REVERSE COMPUTATION OF FORCED CONVECTION HEAT TRANSFER USING ADJOINT FORMULATION

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1 REVERSE COMPUTATION OF FORCED CONVECTION HEAT TRANSFER USING ADJOINT FORMULATION Kazunari Momose and Hideo Kimoo Graduae School of Engineering Science, Osaka Universiy Osaka , Japan ABSTRACT. A reverse analysis based on an adjoin formulaion of unseady forced convecion hea ransfer is proposed. The numerical soluion of an appropriae adjoin problem enables us o predic he hea ransfer characerisic, such as he oal hea ransfer rae or he emperaure a a specific locaion, under arbirary hermal boundary condiions. As a resul, he opimal hermal boundary condiions can be obained in boh ime and space. INTRODUCTION Since forced convecion hea ransfer is one of he fundamenal processes in hea ransfer, a large number of analyical and eperimenal sudies have been made for various configuraions [,2,3]. Mos of he hermal boundary condiions employed in hese sudies, however, have been limied o an isohermal or a uniform hea flu condiion. Indeed he uniform hermal boundary condiion is useful o simplify hea ransfer phenomena and someimes gives a good approimaion for pracical condiions, bu much of hea ransfer problems ha we will encouner in pracical applicaions have nonuniform hermal boundary condiions. In oher words, he hea ransfer characerisics evaluaed under such uniform boundary condiions are no meaningful if he uniform condiion is violaed. On he oher hand, wih recen progress in numerical simulaion echniues [4,5,6], we can numerically evaluae hea ransfer characerisics under arbirary hermal boundary condiions. However, each numerical resul gives only a paricular soluion under a specific boundary condiion; hus, i provides no idea o improve he hea ransfer characerisics. In his paper, we propose a reverse compuaion based on an adjoin formulaion of he forced convecion hea ransfer. In he reverse compuaion, he hea flow is reversed in boh ime and space, jus as smoke reurns o he chimney in a reverse operaing video. As a resul, we can predic he hea ransfer characerisics, such as a oal hea ransfer rae or a emperaure a a specific locaion, under arbirary hermal boundary condiions. GOVERNING EQUATION AND BOUNDARY CONDITIONS Consider a forced convecion field Ω wih boundary Γ. When he fluid is incompressible and has hermally independen properies, he energy euaion for forced convecion problem can be wrien in a nondimensional form as Tb, g + ubg Tb, g = Tb, g 2 () PrRe

2 where T is he emperaure, u is he velociy vecor, and Pr and Re are he Prandl and Reynolds numbers. In his sudy, we assume ha he flow field is seady and is given by a cerain numerical mehod before hea ransfer analyses; hus he energy euaion () is a linear differenial euaion wih he spacedependen coefficien u. Furher, we define he emperaure as he difference from a known iniial condiion, such as, T, T, (2) and suppose ha he boundary is specified as follows: Γ = Γ Γ Γ Γ where Γ and Γ denoe he boundaries specified via variable emperaure and variable hea flu, while Γ and Γ denoe he boundaries via consan ones. Under hese assumpions menioned above, he energy euaion and he boundary condiions adoped in his sudy can be wrien as follows: 2, + u, =, PrRe = Ω iniial condiion (5), b :, g R Γ variable boundary emperaure = S T b, g Γ : consan boundary emperaure (6), : R T, b, g : u n PrRe n = Γ variable boundary hea flu S, Γ : consan boundary hea flu (3) (4) (7) INTEGRAL EQUATION Defining a linear differenial operaor L, for noaional convenience, we wrie E. (4) as L (, ) =, L + u PrRe Then he weak soluion of E. (8) can be epressed as z z L dωd = Ω 2 (8) where is a es funcion or an adjoin emperaure. Applying he divergence heorem o E. (9), we obain he following inegral euaion: z z z z z Ω Γ Ω c h () L dωd = dγd dω where is he adjoin hea flu defined as (9) = PrRe n and L is he adjoin operaor for L, and L possesses he form as ()

3 2 L u PrRe (2) In he adjoin operaor derived above, he unseady erm and he convecion erm become negaive, while he diffusion erm is unchanged. This means ha any adjoin problems derived laer should be solved oward he negaive ime direcion, and he soluions will be reversed in boh ime and space. To eliminae he las erm in he righ-hand side of E. (), we se he adjoin emperaure a = as b, g = Ω (3) which can be regarded as an iniial condiion for he adjoin problem. Moreover, for convenience, defining an inegral operaor as z z f g fb, ggb, gdγd (4) Γ Γ we arrive a he following inegral euaion: z z L dωd = (5) Ω Γ Γ BOUNDARY INTEGRAL EQUATIONS According o E. (5), if we can eliminae he lef-hand side inegral, which is a ime-domain inegral, we can obain several ime-boundary inegral relaionships. Firs we adop an adjoin problem, such ha L (, ) = (6) and se he boundary condiions for he adjoin problem as where δ = = is Dirac s dela funcion. Then we obain he following boundary inegral relaionship: Qbg b gdγ = +, = δ Γ,, Γ,, Γ Γ (7) z, (8) Γ Γ Γ Γ Γ Euaion (8) indicaes ha if we numerically solve he adjoin euaion (6) under he iniial and boundary condiions in Es. (3) and (7), we can predic he oal hea ransfer rae on Γ a = under arbirary hermal boundary condiions. In a similar fashion, if we choose he adjoin problem as under L (, ) = δ δ (9) b, g = Γ Γ, b, g = Γ Γ (2) we ge he following relaionship: = + (2) Γ Γ Γ Γ,

4 where is a specific locaion a which we wan o predic he emperaure. Euaion (2) implies ha he soluion of E. (9) enables us o evaluae arbirary hermal boundary condiion effecs on he emperaure a. Moreover, if we replace he poin impulse in he righ-hand side of E.(9) wih ha in a finie area, we can predic he mean emperaure wihin he area. For seady sae problems, he ime-dependen noaions can be simplified as follows:, s,, s, 2 L L s u (22) PrRe b, g sbg, b, g s, 2 L L s u (23) PrRe z Γ Γ Γ δb g, f g f, s gs f s gs dγ (24) Then he relaionships obained in Es. (8) and (2) become z Qs sbgdγ = ds, si c s, sh + cs, sh c s, sh Γ Γ Γ Γ Γ (25) c h d i c h c h =,, +,, for seady sae problems. s s s Γ s s s s s s Γ Γ Γ (26) COMPUTATIONAL IMPLEMENTATION Le us consider an euaion (9) as a general eample for compuaional implemenaion of he adjoin problem. The implemenaion is surprisingly easy. Reversing he velociy vecor and shifing ime, such ha we obain u = u, = + = 2 u + δ δ PrRe (27) (28) This is he same form as an energy euaion having a source erm. Thus, he adjoin problem obained in E. (28) can easily be solved by a sandard hea ransfer code based on a finie difference mehod or a finie elemen mehod. I should be noed ha he soluion of he adjoin problem described in E. (28) converges rapidly in ime, because i is an impulse response from a uni impulse a. In oher wards, only he early sage of he soluion is significan and he laer is negligible. Thus, we can predic (,) in E. (2) a arbirary ime by recycling a runcaed soluion in he convoluion inegral wih respec o he ime, which is generally defined in E. (4). In addiion, using a relaion =, d s z he seady-sae soluion of he adjoin problem can also be obained by he runcaed unseady soluion. (29)

5 ILLUSTRATIVE EXAMPLE As an illusraive eample of he adjoin analysis, namely he reverse analysis, we compued he adjoin euaion defined in E. (28) in a suare caviy, he flow of which is well known as a lid-driven caviy flow as shown in Fig.. The purpose of his eample is o predic he emperaure a or 2 when he caviy walls ecep he op wall are nonuniformly heaed. To solve E. (28) under he boundary condiions (2), which become Γ u,, = l b r, = Γ Γ Γ (3) in his eample, we assumed ha he Prandl and Reynolds numbers are.7 and 2, and employed a sandard finie difference mehod [4], in which he compuaional domain was uniformly divided ino 29 grids in boh horizonal and verical direcions and he compuaional ime sep was.. The compued resuls for = and for = 2 were saved in files a inervals of. unil = 5, a which all he adjoin variables were converged o almos zero, and were enough o calculae he convoluion inegrals in E. (2). We noe ha all resuls presened below, ecep some direc simulaion resuls for confirmaion, can be obained from only hese wo numerical compuaions for he adjoin problem defined in E. (28). From he resul of firs adjoin compuaion for =, we obained a seady-sae adjoin emperaure s bg using he relaion in E. (29). The adjoin emperaure field in he caviy and he adjoin emperaure disribuions on he side and boom walls are shown in Fig. 2. As shown in Fig. 2(a), he adjoin hea flow generaed a he cener of caviy goes oward he upper par of he righ wall. From he consideraion of reverse compuaion, he mos effecive hea flow oward he cener of caviy will come from he upper par of he righ wall. This can also be confirmed uaniaively in Fig. 2(b). Thus, if we insall a heaer on a wall in a seady-sae siuaion, he upper par of he righ wall is bes o raise he emperaure a he cener of caviy. As he ne problem, le us consider unseady wall-heaing effecs on he emperaure a 2. Alhough he emperaure a 2 can be prediced a any and for arbirary wall heaing from E. (2), we assume here ha he side and boom walls are divided ino four segmens, which are numbered from o 2 as (,) u =, = (,) Seg.4 Seg.3 Seg.2 Seg. Γ u 2 3 Γ l Γ r y Γ b (,) Seg.5 Seg.6 Seg.7 Seg.8 (,) Ω Seg.2 Seg. Seg. Seg.9 Figure. Configuraion of model forced convecion field (Pr =.7, Re = 2)

6 u = -, = 4 = = Adjoin emperaure righ wall boom wall lef wall = , y (a) Adjoin-emperaure field in a caviy (b) Adjoin-emperaure disribuions on caviy walls Figure 2. Seady adjoin-emperaure disribuion generaed by a poin hea source a shown in Fig.. If he hea generaion in each segmen is uniform, E. (2) can be simplified o where z 2 b 2 g seg=, = seg seg d (3) seg z = b, g d Γ seg =, 2, L, 2 (32) Γseg The seg bg obained for all segmens are indicaed in Fig. 3. Since Fig. 3 shows an impulse response from each segmen, a sep response can be obained by inegraing he impulse response wih respec o he ime, such as segb 2, g z = segbgd seg =, 2, L, 2 (33) The sep response seg b 2, g, which is a response when each segmen is independenly heaed wih uni hea flu, is indicaed in Fig. 4. seg Seg.2 Seg.3 Seg. Seg.4 Seg.6 Seg.5 Seg.7 Seg.8 Seg.2 Seg. Seg. Seg (a) lef-wall segmens (b) boom-wall segmens (c) righ-wall segmens Figure 3. Impulse response o emperaure a 2 from each segmen

7 seg b 2, g Seg.3 Seg.2 Seg.4 Seg Seg.5 Seg.6 Seg.7 Seg Seg. Seg.2 Seg. Seg (a) lef-wall segmens (b) boom-wall segmens (c) righ-wall segmens Figure 4. Sep response o emperaure a 2 from each segmen Fig. 4 suggess ha if a single segmen is used and is fied during he operaion, we can deermine he opimal segmen o maimize he emperaure a by selecing he segmen having maimum value of seg b 2, g a. For eample, he segmen 3 is bes o maimize he emperaure a =, while he segmen is bes a = 2. Moreover, using he impulse response in Fig. 3, we can deermine he opimal ime seuence of heaing segmens o maimize he emperaure a ; he opimal emperaure variaion is obained as b 2 g =z seg ma, ma d seg (34) where he hea flu a he seleced segmen is assumed o be uniy. The opimal ime seuence of heaing segmen for = and he resuling ime change of emperaure a 2 are shown in Fig. 5. In Fig. 5(b), he emperaure variaions obained by he opimal heaing a fied segmen menioned above and ha by uniform heaing on all segmens are also indicaed; in boh cases, he oal heaing rae a any ime is uniy. As shown in Fig. 5(b), he emperaure by he opimal heaing is 2.2 imes higher han ha by uniform heaing a =. The resuls obained by he adjoin compuaion are compared wih hose by direc numerical simulaions, which are indicaed by dashed lines in Fig. 5(b); he agreemen is uie good. The ime change of emperaure field in he opimizaion process is shown in Fig. 6, which is obained by a direc numerical simulaion using he ime char of heaing segmen shown in Fig. 5(a). Segmen [[[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[ [[[[[[ Fied segmen [[[[[[[[[[[[[[[ [[[[[[[[[[ Opimized [[[[[[[[[[[[[[ [[[[[[[[[[[[[[[[[[ [ b 2, g Fied segmen (Seg.3) Presen mehod Direc simlaion Opimized Uniform heaing (a) Time char of acive segmen (b) Time change of emperaure a 2 Figure 5. Opimizaion of acive segmen o maimize emperaure a 2 and a =

8 acive segmen =. = 2. = 3. = 4. = 5. = 6. = 7. = 8. = 9. =. Figure 6. Time change of emperaure field in opimizaion process CONCLUSION For general evaluaion of hermal boundary condiion effecs on forced convecion hea ransfer, we propose a reverse analysis based on an adjoin formulaion of he energy euaion. The reverse compuaion enables us o predic he hea ransfer characerisic, such as he oal hea ransfer rae or he emperaure a a specific locaion, for arbirary ime changes of hermal boundary condiions. Moreover, using he adjoin soluion, we can obain he opimal hermal boundary condiions in boh ime and space o maimize he hea ransfer a arbirary ime. Alhough he opimizaion problem presened in his paper is sill simple, he reverse compuaion will be able o applied o more pracical opimizaion problems when combined wih an opimal conrol heory [7] or inverse-problem approaches [8]. REFERENCES. MacAdams, W. H., Hea Transfer Transmission (3rd Ed.), MacGraw-Hill, Zukauskas, A., Advances in Hea Transfer, Vol. 8, Bejan, A., Convecion Hea Transfer (2nd Ed.), John Wiley & Sons, Paankar, S. V., Numerical Hea Transfer and Fluid Flow, MacGraw-Hill, Flecher, C. A. J., Compuaional Techniues for Fluid Dynamics, Springer, Ferziger, J. H. and Peric, M., Compuaional Mehods for Fluid Dynamics, Springer, Luenberger, D. G., Opimizaion by Vecor Space Mehods, John Wiley & Sons, JSME Ediion, Compuer Analyses of Inverse Problems, Corona-sha, 99. (in Japanese)

9 APPENDIX A. Eension o Nonlinear Seady-Sae Problems For naural or mied convecion hea ransfer problems, is adjoin operaor canno be derived direcly, because he coupling of he flow and emperaure fields causes nonlinear hea ransfer characerisics. Thus, we inroduce perurbaions from he base boundary condiions, and hen derive he adjoin operaor for he perurbaion problem. Le us suppose ha he emperaure, hea flu, velociy and sress on heir given boundaries change from he base disribuions, such ha = + on Γ, = + on Γ, u = u + u on Γ, u s = s + s on Γs (A-) where denoes he perurbaion. Then we assume ha he velociy, pressure and emperaure will also change slighly from heir base disribuions o u = u + u, = + p p p, = + in Ω (A-2) Subsiuing Es. (A-2) ino he coninuiy, Navier-Sokes and energy euaions, we obain he firsorder perurbaion euaions, which can be epressed in a mari form as Af = (A-3) where L M, T f = p u, A = u + u j, T Gr M u Pr NM O QP (A-4) Since he governing euaions of he perurbaions are linear as shown in E. (A-4), we define a es funcion vecor as, T, f p u (A-5) and consider he following weak soluion of E. (A-3): z f T Af dω = (A-6) Ω Applying divergence heorem o E. (A-6), we arrive a he following inegral euaion: zω zγ d i (A-7) T f A f dω = + s u u s dγ where A is he adjoin operaor mari corresponding o A, and A possesses he form as L M NM T A = u + u + M Gr j u + Pr O QP (A-8) Euaion (A-7) derived above correspond o E. (5) in he seady forced convecion problem. Thus, our basic idea presened in Es. (25) and (26) can easily be eended o seady mied convecion problems for arbirary hermal and flow boundary perurbaions.

10 A.2 An Eample for Nonlinear Mied Convecion Problem Le us consider a nonlinear mied convecion field in a suare caviy wih several inles and an oule as shown in Fig. A-. The purpose of his eample is o predic he change of emperaure a a specific locaion, when he inle emperaure and inle flow are varied from heir base condiions ( =, u = (,) T ). Afer a compuaion of he base problem, we numerically solved he adjoin problem, which can eplicily be wrien as u = T + = u u u p u e j = u Gru j + δ Pr and he boundary condiions adoped are = on Γ Γ Γ Γ Γ Γ 2 3, u b o = l r on Γ Γ, (A-9) u = on Γ Γ2 Γ3 Γu Γb Γl Γ, r s = on Γ o (A-) Then we obain he following boundary inegral relaionship from E. (A-7): = + s u dγ + + s u dγ + + s u dγ (A-) z z z d i d i d i Γ Γ Γ 2 3 Euaion (A-) implies ha if we ge adjoin hea flues and adjoin sress s a he inles, we can predic he emperaure change a for arbirary emperaure and flow perurbaions a he inles. =, u = y, v =,u = = +, u = u + u = +, u = u + u = +, u = u + u Γ l Γ 3 Γ 2 Γ Γ u Pr =. 7 j Gr = 5 4 Γ b =, u = Γ o Γ r =,u = Fig. A- Configuraion of mied convecion hea ransfer in a suare caviy ( =, u = (, ) T ), u y Γ 3 Γ 2 Γ y Γ 3 Γ 2 Γ y Γ 3 Γ 2 Γ Adjoin hea flu Adjoin sress σ Adjoin sress σ y (a) Influence of emperaure (b) Influence of horizonal (c) Influence of verical velociy componen velociy componen Fig. A-2 Influences of hermal and flow perurbaions a inles on emperaure a he cener of caviy

11 Figure A-2 shows he adjoin hea flu and he adjoin sress disribuions obained on he lefhand side of he caviy. According o Fig. A-2(a), he adjoin hea flu disribuions a lower wo inles are posiive. This indicaes ha if he emperaures a lower wo inles increase, he emperaure a he cener of caviy also increases. On he oher hand, he increase of emperaure a inle 3 causes he emperaure decrease a he cener of caviy, because he adjoin hea flu a his inle is negaive. From Fig. A-2(b), we can raise he emperaure a he cener of caviy by simply increasing he horizonal velociy componens a all inles. Moreover, Fig. A-2(c) suggess ha he verical velociy componens a he upper wo inles should be decreased o increase he emperaure a he cener of caviy, while ha a inle should be increased. In order o confirm he predicions discussed above, we carried ou direc numerical simulaions of mied convecion fields wih small hermal and flow perurbaions a he inles. Figure A-3 shows he resuls obained under ypical inle condiions; hese are chosen from he predicions in Fig. A-2 for he case of emperaure increase (a) and for he case of emperaure decrease (b). As shown in Fig. A-3, he emperaures a he cener of caviy can be well conrolled by slighly changing he inle condiions suggesed by he presen mehod. u = U =.9 =. =. =. =. =. =. =. =.9 (a) Temperaure increase condiions Base condiions (no perurbaion) (b) Temperaure decrease condiions Fig. A-3 Thermal and flow inle condiions o increase and decrease emperaure a I should be noed ha he adjoin variables obained in mied convecion problems are sensiiviies for hermal and flow perurbaions. Alhough he sensiiviy provides no idea for he limiaion of he perurbaion, i will provide useful informaion for hermal design, especially when combined wih gradien-based opimizaion sraegies. References Momose, K., Ueda, M.and Kimoo, H., Trans. Jpn. Soc. Mech. Eng., Vol.66, No. 646, B(2), p Momose, K. and Kimoo, H., Proceedings of he 4 h JSME-KSME Thermal Engineering Conference, Vol.(2), p.55-5.