COMPUTATION OF CONJUGATE HEAT TRANSFER OVER A 2D CIRCULAR CYLINDER IN A LOW SPEED CROSSFLOW BY A GHOST CELL IMMERSED BOUNDARY METHOD

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1 COMPUAION OF CONJUGAE HEA RANSFER OVER A 2D CIRCULAR CYLINDER IN A LOW SPEED CROSSFLOW BY A GHOS CELL IMMERSED BOUNDARY MEHOD Dartzi Pan Department o Aeronautic and Atronautic, National Cheng Kung Univerit, ainan Cit, aiwan, ROC dpan@mail.ncku.edu.tw ABSRAC A ghot cell immered boundar method i emploed to compute the conjugate heat traner over a circular clinder in crolow. he clinder i internall heated b a circular urace o contant temperature. he conjugate heat traner boundar treatment propoed at luid-olid interace i hown to ield econd order accurate olution. o demontrate the capabilit o the current method, the location o the internal heating urace i varied to eamine it eect on the heat traner. INRODUCION Conjugate heat traner i common in engineering problem involving internall heated olid component and eternal luid low. he conventional bod-itted grid method o olving conjugate heat traner problem i to generate eparate grid repectivel or luid domain and olid domain, to olve luid low equation and olid equation independentl uing dierent numerical method, and to couple the two olution together through boundar condition treatment at the luid-olid interace. Recentl immered boundar method [-5] ha provided a wa to olve both luid and olid equation on one ingle Carteian grid, greatl reducing the compleit involved when olving conjugate heat traner problem. However, there are variou tpe o immered boundar method, each with it own merit or drawback. Hence, an immered boundar method need to be eamined independentl to identi it uitabilit to conjugate heat traner problem. On the luid-olid interace, the no-lip boundar condition or velocit and the continuit o temperature ditribution acro the interace are Dirichlet tpe boundar condition. he conervation o heat conduction acro the interace i Neumann tpe boundar condition. hu, both kind o boundar condition on the interace need to be treated accuratel imultaneoul when olving conjugate heat traner problem. In thi paper, a ghot cell immered boundar method [6, 7, 8] capable o 2 nd order accurac or both Dirichlet and Neumann boundar condition i emploed to olve conjugate heat traner problem. A model conjugate heat traner problem i ued to veri the accurac o the propoed boundar treatment. o demontrate the capabilit o the developed method, an internall heated circular clinder in a crolow i computed. he location o the internal heating urace i varied to eamine it eect on the heat traner over the clinder. GOVERNING EQUAIONS he integral incompreible Navier-Stoke equation are v ds = G G GG G G G G Gr G vdv + vv ds v ds + dv + PdS = t Re θ 2 () CV CV Re dv + v ds ds = t θ θ θ Pr Re CV where v i velocit, P i preure, θ i luid temperature with ubcript, Re i the ree tream Renold number, Pr i the ree tream Prandtl number, G r i the vector orm Graho number or gravitational orce, indicate Control Surace, CV indicate Control Volume. For the olid domain, the governing equation or heat conduction i α dv ( ) ds = t θ θ (2) α Pr Re CV where θ i the olid temperature with ubcript, α i the olid thermal diuivit, α i the luid thermal diuivit. In tead tate, Eq. (2) i a Laplace equation. All variable in Eq. () and (2) are properl non-dimenionalized b repective reerence value. In particular, temperature i normalized a θ =, where i the ree tream temperature, rrr i a re given reerence temperature, here it i the temperature o the heating urace. On the luid-olid interace, the no-lip boundar condition or velocit i applied. For temperature, the boundar condition at the interace are θ = θ k = k (3) where k i the luid thermal conductivit, and n i the ditance along interace normal. he irt equation o Eq. (3) enorce the continuit o temperature ditribution and the econd equation indicate the conervation o conduction heat acro the interace. 87

2 FINIE VOLUME MEHOD One ingle Carteian grid i generated to cover both luid and olid domain. he ame cell-centered inite volume dicretization method [6,7,8] i applied to both luid and olid equation. For low computation, Eq. () i olved b an implicit preure correction method. Keeping the preure ied at the current time and uing implicit time integration, the ull dicretized inite volume equation i * n n cq c2q + c3q * * * n ( + Rconv Rvi H + RP + Fdir ) V = (4) t where the upercript * repreent the intermediate tate, n indicate the current time level; V i the volume o the conidered cell; t i the time increment; Q = [ u v θ ] i the vector o conerved variable per unit volume; R conv, R P and R vi are the vector o urace integral o convective, preure and vicou lue repectivel over the control urace divided b V ; H i vector o the ource term per unit volume; F i a direct orcing per unit volume added to orcing point dir in order to model the preence o immered bodie. he contant are c =.5, c 2 =2 and c 3 =.5 or the econd-order backward dierencing cheme, and c =, c 2 = and c 3 = or the irt-order Euler implicit cheme. he intermediate velocit v generall doe not ati the divergence-ree condition. he velocit and the preure are corrected a n+ n v = v t φ (5) n+ n n P = P + φ n B requiring v + be divergence-ree, we obtain the Poion equation: 2 n v * φ = (6) t which i to be olved or the preure correction φ n. For tead-tate computation, the time integration i continued b n+ taking θ = θ * ater Eq. (4) to Eq. (6). A or the olid temperature computation on the olid domain, the ame dicretization and time marching method a thoe or luid temperature are ued. During each time marching tep, Eq. () and Eq. (2) are olved in equence and multigrid ub-iteration are ued to drive the time accurate luid and olid equation to their convergence independentl. A GHOS CELL IMMERSED BOUNDARY MEHOD he ghot cell tem i deined or luid domain and olid domain eparatel and independentl. For low computation, the ghot cell (in olid domain), their projection point (on interace) and their image point (in luid domain) are depicted in Fig.. Here the empt quare are the luid cell center whoe value are to be olved, the triangle are the ghot cell center whoe value are to be aigned in order to implicitl enorce appropriate boundar condition at their projection point repectivel, and the illed quare are the olid cell which pla no role in low computation. he ghot cell or low computation are the irt laer o cell on the olid ide adjacent to the luid-bod interace with at leat one neighboring luid cell. Along their projection line onto the interace, the image point are a ditance δ into the luid domain, a hown in Fig.. Here δ = 2 i taken in 2D to enure that the image point are urrounded b luid cell center not involving an ghot center. For olid computation, a imilar et o ghot cell (in luid domain), their projection point (on interace) and their image point (in olid domain) will be deined eparatel and independentl. In thi cae the ghot cell are the irt laer o cell on the luid ide adjacent to the interace with at leat one neighboring olid cell. he image point are located a ditance δ rom the interace into the olid domain. he value at ghot cell center are obtained b etrapolation rom the olution domain. heir value are deigned to implicitl enorce the required boundar condition at their projection point on the interace. A general repreentation o the boundar condition at a projection point can be epreed b Q ( ) proj + tq proj = q (7) w where n w indicate the outward urace normal pointing into Q the olution domain (luid domain in thi eample), ( ) proj nw i the normal variable gradient at the projection point, Q proj i the variable value at the projection point, q i the boundar value given at the projection point; the coeicient and t determine the tpe o the boundar condition. It i clear that the contant et (=, t=) indicate Dirichlet condition, (=, t=) indicate Neumann condition, and (=, t=) indicate Robin condition. he ghot-cell direct orcing approach i impl an appropriate olution recontruction procedure to et the ghot cell value uch that Eq. (7) i atiied implicitl on the immered boundar. he variable value Q image can be obtained eail b a bilinear interpolation utilizing the variable value o the our luid center encloing the image point. A bi-linear Q approimation or ( ) can alo be done eail with image nw known direction vector nˆ w. A 2-point econd-order polnomial recontruction along n w can be obtained b olving Q 2δ a ( ) image 2 w 2 δ δ a = Q (8) image t a q or the coeicient a 2, a and a. For a target point on the urace normal paing through the projection point, it variable value Q can be interpolated or etrapolated b t arg et 2 2rn Q t arg et = a + arn + a (9) where r n i the outward normal ditance between the target point and the projection point. he ghot value Q ghot can be 88

3 etrapolated b taking rn r ghot and r proj = r r in Eq. (9), where ghot proj are the poition vector o the ghot center and the projection point, repectivel. Note that here r n i negative and Q ghot i obtained b an etrapolation. It ha been hown [6, 7, 8] that Eq. (8) and Eq. (9) ield 2 nd -order accurate olution in L2 ene or both Dirichlet and Neumann boundar condition. FLUID-SOLID CONJUGAE HEA RANSFER In theor, the two boundar condition in Eq. (3) need be atiied imultaneoul. In practice, ince the low domain and the olid domain are olved in equence during one time tep, the two equation in Eq. (3) are alo enorced in equence. When k > k, it i choen to enorce or olid computation that k = ( ) () k k hat i, ( ) at the interace i etimated b the low k olution θ, then it i aigned to gradient at the interace via the Neumann boundar condition treatment or the olid k temperature. Note that the condition < indicate k <, which give a better tabilit when enorcing Neumann boundar condition in the olid computation. he other boundar condition i enorced in luid computation, that i, θ = θ () he interace temperature i etimated uing the olid temperature θ at the interace, and then it i aigned to θ at the interace via the Dirichlet boundar condition treatment or the luid temperature. k In the cae when <, the Neumann boundar k condition on the interace hould be enorced in the low computation, leaving the Dirichlet boundar condition to the olid computation. VALIDAEION he heat traner problem o among three coaial circular clinder i depicted in Fig. 2, which i a cop rom Re. []. he inner clinder ha a radiu R i =. 45, the middle clinder ha R m =. 9, and the outer clinder ha R o =. 8. he inner clinder urace i ied at a urace temperature θ i =. he annular pace between R i and Rm i illed with a olid, while the annular pace between R m and R o i illed with a luid. he outer clinder urace at R i rotating at a circumerential o peed U =. It drive a rotational low in the annular pace between R m and R o. he urace temperature at R o i ied at θ. here i conjugate heat traner acro the middle o = clinder urace. he analtical olution o thi model problem can be ound in Re. []. Note that the tead tate temperature ditribution i olel determined b the heat conduction proce. he rotational peed U ha no eect on the tead tate temperature ditribution. Hence thi problem i computed here on Carteian grid o variou ize with U =. he computational domain i a quare o length 4 covering the three clinder urace. Figure 3 how the temperature ditribution along the horizontal line paing through clinder center on a grid or k = k, and Fig. 4 i or k =.k. he mbol are computed reult, which are in ecellent agreement with the eact olution. Uing data computed along the horizontal grid line paing through clinder center, Fig. 5 how the convergence tet uing 64 64, and grid. he etimated order o accurac i.94 in L2 norm or k =.k and.7 or k = k. hi validate that the current ghot cell method i econd order accurate when applied to conjugate heat traner problem. Although not hown here, but i onl a irt-order etimation o temperature gradient at the projection point on the interace, then onl irt order accurac i oberved. INERNALLY HEAED CIRCULAR CYLINDER he conjugate heat traner characteritic o a clinder with one or multiple laer o olid material between the internal heating element and the eternal crolow have important implication to crolow heat echanger [9]. In thi paper, a computational model o internall heated clinder i made o a clinder o unit diameter with an internal circular urace o diameter.2 at ied temperature θ=. Heat i tranported rom the internal heating urace to the clinder urace b pure conduction, and eventuall i tranported awa rom the clinder urace b convection and conduction. he conductivit ratio i taken to be k / k = 9. he Ranold number baed on clinder diameter, the ree tream velocit and the luid dnamic vicoit i Re=4. he Prandtl number i et to.7. he Graho number i zero. We are intereted in tead tate temperature ditribution, where the olid equation i a Laplace equation. he Carteian grid ha a total o 572 cell covering a computational domain o ize 6 6. he grid i locall reined uch that there are about 68.3 cell along the clinder diameter and about 3.6 cell along the internal urace diameter. he ree tream condition i et to the inlow boundar and the two ide boundarie. he ree tream temperature i θ =. he downtream boundar ollow the upwind dierenced equation o ( Q / t ) + U n ( Q / ) =, where U n i the computed normal outlow velocit at the boundar. Since onl the tead tate olution i o interet, the implicit Euler method i ued in time integration or luid and 89

4 α olid. he diuivit ratio α can be arbitraril choen to accelerate the convergence. Figure 5 to Fig. how the temperature contour inide and outide the clinder with the heating urace center (c, c) located in dierent poition. With zero Graho number, the temperature equation in Eq. () i decoupled rom the momentum equation. hu the velocit ield computed or all cae how here are identical to the computer machine accurac. he identical treamline pattern in Fig. 5 to Fig. relect thi act. Note that the temperature ield i mmetric with repect to ai in Fig. 5 to Fig. 7, and ammetric in Fig. 8 to Fig. due to the poition o the heating urace. he dicontinuou lope in temperature contour acro the clinder urace due to the jump in conductivit can be clearl een in thee igure. o compute the heat traner H Bod over the bod urace, the ollowing urace integral are ued: H Bod = θ S (2) urace Pr Re where the ummation i perormed on all boundar egment o the bod urace. he temperature gradient are obtained b the bi-linear approimation decribed previoul uing the urrounding cell verte value. able lit the computed heat traner time the contant RePr with varing heating urace location. Note that heat traner increae when heating urace move toward the uptream tagnation (6th row to 2nd row). hi i reaonable, becaue the heat convection tart rom the uptream tagnation, and more heat will be convected downtream when the heat ource i cloer to the uptream tagnation. CONCLUSIONS hi paper demontrate the application o a ghot cell immered boundar method in olving conjugate heat traner problem. he ame inite volume dicretization method i ued on one ingle Carteian grid to olve the governing equation or both luid and olid phae. hi approach i advantageou when compared with the traditional approach that ue dierent dicretization method and dierent bod-itted grid or dierent phae. It i hown that the preent treatment acro the olid-luid interace ield econd-order accurate olution in L2 norm. he eample o a circular clinder in a low peed crolow with an internal heating urace located in variou location demontrate the capabilit o the preent method to treat comple geometr. Solving Conjugate Heat raner Problem in urbomachiner, Proceeding o ECCOMAS CFD 2, Libon, Portugal,4-7 June 2. [2] Xie, W. and DeJardin, P. E., A Level Set Embedded Interace Method or Conjugate Heat raner Simulation o Low Speed 2D low, Computer & Fluid, vol. 37, 28, pp [3] Nagendra, K., ati, D. K. and Viwanath, K., A New Approach or Conjugate Heat raner problem Uing Immered Boundar Method or Curvilinear Grid-baed Solver, Journal o Computational Phic, vol. 267, 24, pp [4] Hu,Y., Li, D., Shu, S., Niu, X., Simulation o Stead Fluid olid Conjugate Heat raner Problem via Immered Boundar-Lattice Boltzmann method, Computer and Mathematic with Application, vol. 7, 25, pp [5] Jeon, B. J., Kim, Y.S., and Choi, H.G., Eect o the Renold Number on the Conjugate Heat raner Around a Circular Clinder with Heat Source, Journal o Mechanical Science and echnolog, vol. 26, No. 2, 22, pp [6] Pan, D., A Comparion o Fluid-cell and Ghot-cell Direct Forcing Immered Boundar Method or Incompreible Flow with Heat raner, Numerical Heat raner, Part B, Fundamental, Vol. 68, 25, pp [7] Pan, D., A General Boundar Condition reatment in Immered Boundar Method or Incompreible Navier- Stoke Equation with Heat raner, Numerical Heat raner, Part B, Fundamental, Vol. 6, 22, pp [8] Pan, D., A Simple and Accurate Ghot Cell Method or the Computation o Incompreible Flow over Immered Bodie with Heat raner, Numerical Heat raner, Part B, Fundamental, Vol. 58, 2, pp [9] Kaptan, K., Buruk, E., and Ecder, A., Numerical Invetigation o Fouling on Cro-low Heat Echanger ube with Conjugated Heat raner Approach, International Communication in Heat and Ma raner, vol. 35, 28, pp ACKNOWLEDGEMENS hi work i unded b the Minitr o Science and echnolog o aiwan, ROC under project MOS4-222-E-6-8. he upport i highl appreciated. REFERENCES [] De ullio, M. D., Latorre, S. S., De Palma, P., Napolitano, M., Pacazio, G., An Immered Boundar Method or 8

5 able Heat traner over clinder urace with internal heating urace at variou poition, Re=4, K/K=9 emperature Along Horizontal Line d=/6, K/K= Center (c, c) H Bod Re Pr (-.25, ) 8.3 (-.77,.77) (,.25) 7.82 (.77,.77) (.25, ) (, ) , _eact _eact Figure 3 emperature ditribution along central horizontal grid line, center at =2, K/K=, mbol: computed, line: eact. emperature Along Horizontal Line d=/6, K/K=..8 _eact, _eact Figure Depict o luid low domain and it ghot cell tem Figure 3 emperature ditribution along central horizontal grid line, center at =2, K/K=., mbol: computed, line: eact. 3-Clinder Conjugate Heat raner -4 Figure 2 Depict o heat traner problem or the olid and the rotational low in annular pace. (Cop rom Re. []) L2 Error Norm t-order 2nd-order K/K= K/k= d Figure 4 Order anali uing temperature along central horizontal grid line, Square: K/K=., triangle: K/K=. 8

6 emperature Contour and Streamline k/k=9, Re=4, Center (-.25, ) emperature Contour and Streamline k/k=9, Re=4, Center (-.77,.77) Figure 5 emperature contour and treamline, heating urace at (-.25, ), K/K=9, Re=4 emperature Contour and Streamline k/k=9, Re=4, Center (, ) Figure 8 emperature contour and treamline, heating urace at (-.77,.77), K/K=9, Re=4 emperature Contour and Streamline k/k=9, Re=4, Center (,.25) Figure 6 emperature contour and treamline, heating urace at (, ), K/K=9, Re=4. emperature Contour and Streamline k/k=9, Re=4, Center (.25, ) Figure 9 emperature contour and treamline, heating urace at (,.25), K/K=9, Re=4 emperature Contour and Streamline k/k=9, Re=4, Center (.77,.77) Figure 7 emperature contour and treamline, heating urace at (.25, ), K/K=9, Re=4-2 Figure emperature contour and treamline, heating urace at (.77,.77), K/K=9, Re=4 82