NOMENCLATURE. n i. nb variable of the present and

Duan, R., Lu, W., Xu, L., Huang, Y., Shen, X., Ln, C.-H., Lu, J., Chen, Q., and Sasanapur, B. 2015 Mesh type and number for CFD smulatons of ar dstrbuton n an arcraft cabn, Numercal Heat Transfer, Part B: Fundamentals, 67(6), 489-506. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 MESH TYPE AND NUMBER FOR CFD SIMULATIONS OF AIR DISTRIBUTION IN AN AIRCRAFT CABIN Ran Duan 1, We Lu 1, Luy Xu 1, Yan Huang 1, Xong Shen 1, Chao-Hsn Ln 2, Junje Lu 1, Qngyan Chen 3,1 and Balasubramanyam Sasanapur 4 1 Tanjn Key Laboratory of Indoor Ar Envronmental Qualty Control, School of Envronmental Scence and Engneerng, Tanjn Unversty, Tanjn 300072, Chna 2 Envronmental Control Systems, Boeng Commercal Arplanes, Everett, WA 98203, USA 3 School of Mechancal Engneerng, Purdue Unversty, West Lafayette, IN 47907, USA 4 ANSYS Inda Pvt Ltd, Pune, Inda Ths nvestgaton evaluated the mpact of three mesh types (hexahedral, tetrahedral, and hybrd cells) and fve grd numbers (3, 6, 12, 24, and >38 mllon cells) on the accuracy and computng costs of ar dstrbuton smulatons n a frst-class cabn. Ths study performed numercal error analyss and compared the computed dstrbutons of arflow and temperature. The study found that hexahedral meshes were the most accurate, but the computng costs were also the hghest. 12- mllon-cell hexahedral meshes would produce acceptable numercal results for the frst-class cabn. Dfferent mesh types would requre dfferent grd numbers n order to generate accurate results. 1. INTRODUCTION NOMENCLATURE α p,α nb coeffcent of the varable at the x spatal coordnates present cell and neghborng cells, Г ϕ,eff effectve dffuson coeffcent respectvely ρ densty of flud b source term or boundary condtons average general varable E ro maxmal round-off error n, n exact soluton, approxmated Err total numercal error soluton N grd number p, nb varable of the present and r dstance from the center pont of cell neghborng cells, respectvely (center of gravty) to the nterfacal center f round-off error R ϕ normalzed resduals Subscrpts and Superscrpts S ϕ source term c center pont of cell t tme f nterfacal center pont n 1 Address correspondence to Xong Shen, Tanjn Key Laboratory of Indoor Ar Envronmental Qualty Control, School of Envronmental Scence and Engneerng, Tanjn Unversty, Tanjn 300072, Chna. E-mal: shenxong@tju.edu.cn 1

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 TE truncaton error ndex of coordnate u average velocty nb neghborng cells V flow doman sze p present cell In the past decade, the number of ar travelers worldwde ncreased to 11.3 bllon [1]. Ar dstrbuton n arlner cabns s mportant for the thermal comfort and well-beng of travelers and crew members [2]. However, many recent studes [3, 4] found that thermal comfort n arlner cabns was not satsfactory. The spatal ar temperature dstrbutons n arlner cabns were not unform, and many passengers found that ther upper bodes were too warm and lower bodes too cold. Measurements n a large number of commercal arlner cabns by Guan et al. [5] dentfed many pollutants that are potentally harmful to passengers and crew members and should therefore be removed effectvely from the cabns by ventlaton. Adjustment of ar dstrbuton n cabns n order to mprove thermal comfort and reduce pollutant levels s an mportant subject for arplane cabn desgners and researchers. Expermental measurements and computer smulatons are two of the prmary methods of nvestgatng ar dstrbuton n an arlner cabn [6]. For example, Zhang et al. [7] used a CFD program to study the ar dstrbuton n an arlner cabn mock-up. L et al. [8] measured contamnant dstrbuton expermentally n an arplane cabn. Lu et al. used both expermental measurements [9] and computer smulatons [10] to obtan the ar dstrbuton n a frst-class cabn. These studes showed that, whle expermental measurements n an arlner cabn were relable, t was dffcult to conduct the measurements on board wth suffcent fne spatal resoluton because of regulatons mposed by avaton authortes and the hgh costs assocated wth the experments. Most of the measurements were conducted on the ground n arplanes or cabn mock-ups [8, 11, 12]. CFD smulaton, on the other hand, s less expensve and more effcent [6]. Thus, recent studes of thermal comfort and ar qualty n arlner cabns have been conducted prmarly by CFD [13-15]. Because the geometry of an arlner cabn s very complex and the arflow appears unstable [10], the experence obtaned n smulatng arflow n other enclosed spaces, such as buldngs, cannot be appled to arlner cabns. Therefore, t s mportant to nvestgate the use of CFD for ths applcaton. Sgnfcant effort has been made n recent years n studyng ar dstrbutons n arlne cabns by CFD. For example, Lu et al. [10] evaluated dfferent turbulence models for predctng ar dstrbutons, and Zhang and Chen [16] assessed varous partcle models for predctng contamnant dspersons. However, few studes have evaluated the mesh type and number used n CFD. Because CFD solves dscretze transport equatons for flow (Naver-Stokes equatons), the flow doman n an arlner cabn should be dvded nto a large number of cells. The mesh type and sze can be very mportant factors n the cost of computaton and the accuracy of the numercal results. Snce an arlner cabn s three dmensonal, the commonly appled mesh types are hexahedral [17], tetrahedral [18], and hybrd meshes [19]. The hexahedron, a structured mesh, was frst developed [20] n the 1970s. Compared wth tetrahedral and hybrd meshes, hexahedral meshes can be algned wth the predomnant drecton of a flow, thereby decreasng numercal dffuson [21]. However, t s dffcult to generate hexahedral elements for arlner cabns wth complcated boundares [6], although there are examples of ths applcaton [17]. Developments n meshng technques n the 1980s made the tetrahedron a popular alternatve [22]. Tetrahedral cells are more adaptve to a flow doman wth a complcated boundary [23]. Today, because commercal CFD software can generate tetrahedral 2

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 meshes automatcally, such meshes are favored by nexperenced users. Many researchers [24, 25] have appled these meshes to ar cabns. However, a tetrahedron s not as accurate as a hexahedron wth the same grd number [26, 27]. The grd number of a tetrahedral mesh s larger than that of a hexahedral mesh wth the same cell dmensons. Therefore, hybrd meshes [28, 29] have been developed that use tetrahedral meshes n the flow feld wth a complcated boundary and hexahedral meshes n the other flud doman. Several studes [10, 15] have appled hybrd meshes to the nvestgaton of ar dstrbutons n cabns. Unfortunately, hybrd meshes cannot be automatcally generated, and ntensve labor s requred to buld such a mesh manually. The above revew llustrates the pros and cons of dfferent mesh types. It s mportant to dentfy the type mesh that s most sutable for use n arlner cabns. Another mportant factor n the computng cost and accuracy of CFD smulatons s the number of cells. Many CFD studes have performed grd ndependence tests. For example, Lu et al. [10] compared three grd quanttes for a frst-class cabn, but the maxmum grd number was only 13 mllon, whch was not suffcently fne to obtan grd ndependency. A coarse mesh could lead to a larger spatal dscretzaton error, and refnng the mesh could reduce the numercal dsspaton. However, f the grd number were very large, round-off error could ncrease rapdly and would exceed truncaton errors, and thus the accuracy could also become poor [30]. Therefore, a cell number that s ether too small or too large could lead to poor results. It s necessary to determne the most sutable grd number. On the bass of state-of-the-art CFD smulatons of ar dstrbuton n arlner cabns, ths nvestgaton conducted a systematc evaluaton of mesh type and number. The goal was to dentfy a sutable mesh type and number for studyng ar dstrbuton n an arlner cabn n order to mprove the thermal comfort and well-beng of passengers and crew members. 2. RESEARCH METHOD 86 87 88 89 90 91 92 93 94 95 96 97 98 2.1. Selecton of Grd Type and Number Our study used the frst-class cabn of a sngle-asle arcraft (an MD-82 arplane) to study the mpact of grd type and number on the computng costs and accuracy of numercal smulatons of ar dstrbutons n the cabn. Fgure 1(a) s a schematc of the fully-occuped, frst-class cabn. The role of grd type was nvestgated by usng hexahedral, tetrahedral, and hybrd meshes, as shown n Fgure 1. For evaluatng grd number, a mesh of at least 3 mllon cells s necessary n order to descrbe cabn detals that are crucal for smulatng ar dstrbuton, such as dffusers. We progressvely doubled the grd number n order to study ts mpact on accuracy. Because of lmtatons on our computng resources, the maxmum grd number used was about 48 mllon. Snce t took a long tme to generate the fner grds and t was not easy to control the grd number, the largest grd numbers for the hexahedral, tetrahedral, and hybrd meshes were 59, 50, and 38 mllon, respectvely. Table 1 shows the grd numbers used n ths nvestgaton. 3

99 100 (a) (b) 101 102 103 104 105 106 107 108 109 110 (c) (d) Fgure 1. (a) Schematc of the fully-occuped, frst-class cabn; and mesh dstrbuton at the back secton for dfferent mesh grd types: (b) hexahedral mesh, (c) tetrahedral mesh, and (d) hybrd mesh wth 24 mllon cells. Table 1. Grd numbers, dmensons, and Y+ values used n ths study Mesh type Abbrevaton Cell number Global mesh Surface-average (mllons) sze (mm) Y+ HY3 3 64 5.02 HY6 6 48 3.84 Hybrd HY12 12 32 3.32 HY24 24 24 2.86 HY38 38 24 2.21 T3 3 80 4.38 T6 6 64 3.33 Tetrahedral T12 12 48 2.45 T24 24 32 2.04 T50 50 24 1.67 H12 12 24 2.11 Hexahedral H24 24 24 1.89 H59 59 24 1.54 4

111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 Fgure 1 shows the grd dstrbutons of the three mesh types. Dfferent mesh types had smlar mesh dstrbutons. For example, the mesh was very fne n the regons close to the walls, mankns, and ar dffusers because of large gradents n the varables, whle coarse meshes were used n the man flow regon. Ths nvestgaton defned the large mesh dmenson used n the man flow regon as the global mesh dmenson. For the hexahedral mesh, as depcted n Fgure 1(b), the dstrbuton of the meshes was unform n most of the man flow regon. Because the dffuser sze was only 3 mm and the global mesh dmenson was much larger than that, we gradually ncreased the mesh dmenson for the dffusers to the man flow to ensure grd qualty. Fgure 1(c) shows the tetrahedral mesh dstrbuton and Fgure 1(d) the hybrd mesh dstrbuton, under the same strategy as that used for the hexahedral meshes. The hybrd mesh was dvded nto three flow regons: the regon wth regular geometry close to the asle and floor, where hexahedral meshes were used; the regon wth rregular geometry close to the dffusers, walls, celng, seats, and mankns, where tetrahedral meshes were used; and the transton regons, where pyramdal meshes were used. 2.2. Turbulence models and numercal scheme CFD smulatons of ar dstrbutons n arlner cabns would need to use turbulence models, as current computer capacty and speed are nsuffcent to smulate the detals of turbulence flow n arlner cabn. Among varous turbulence models, Lu et al. [10] recommended large-eddy-smulatons (LES) and detached-eddy-smulatons (DES) for arflow smulatons n arlner cabns. However, these models requre long a computng tme and hgh mesh densty. Zhang et al. [31] concluded that the LES model provded the most detaled flow features, whle the v2f and re-normalzaton (RNG) k-ε models could produce acceptable results wth greatly reduced computng tme. Snce the RNG k-ε model s one of the most popular turbulence models used n desgn practce, the current study used ths model to smulate cabn flows. Because the arflow n an arlner cabn can be transtonal, ths study also smulated the flow as transent or unsteady. The governng equatons for the RNG k-ε model for both steady and transent flows can be wrtten n a general form: 140 u [, eff ] S t t x x (1) 141 142 143 144 145 146 147 148 149 150 where ϕ represents the flow varables (ar velocty, energy, and turbulence parameters), Г ϕ,eff s the effectve dffuson coeffcent, and S ϕ s the source term. When ϕ = 1, then equaton (1) becomes the contnuty equaton. Ths study used commercal CFD software FLUENT [32] for all numercal smulatons. The Naver-Stokes equaton was dscretzed by the fnte-volume method [33, 34, 35]. We employed the SIMPLE algorthm to couple the pressure and velocty calculatons. The PRESTO! scheme was adopted for pressure dscretzaton, and the frst-order upwnd scheme was used for all the other varables. We tested the second-order scheme, but the calculaton dd not lead to a converged soluton [10]; ths result was unfortunate, and the scheme should be further nvestgated n the future. Ths 5

151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 nvestgaton started an unsteady-state smulaton that was based on the results of a steady-state smulaton. We estmated that the unsteady-state smulaton took one tme constant of 50 s to reach a stable flow feld. The computaton then contnued for another 100 s, after whch tme-averaged smulaton results could be obtaned. For the regons near the walls, our study used the enhanced wall functon [32], whch requred that the Y+ value be less than 30. Table 1 shows the surface-averaged wall Y+ values, whch were all smaller than 5; thus, the wall functon could be used. The study consdered the solutons to be converged when the sum of the normalzed resduals for all the cells satsfed the condtons shown n Table 2. The normalzed resduals were defned as: cellsp anbnb bapp a cellsp where P and nb are the gven varable at the present and neghborng cells, respectvely; a P s the coeffcent of the varable at the present cell; a nb are the correlaton coeffcents of the varable at the neghborng cells; and b s the source term or boundary condtons. R nb Table 2. Resdual values below whch solutons are consdered to be converged, for the three dfferent mesh grd types Resduals Hexahedral Tetrahedral Hybrd contnuty 10-4 10-4 10-4 velocty 10-3 10-3 10-3 energy 10-6 10-6 10-6 k 10-3 10-3 10-3 ε 10-4 10-4 10-4 P P (2) 6

172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 2.3. Numercal errors Dscretzng the partal dfferental governng equaton (Eq. 1) gves rse to three types of errors: truncaton errors, errors ntroduced by the numercal defntons of boundary condtons, and round-off errors [36]. The followng two sub-sectons present the method we used to estmate the truncaton errors and round-off errors because they are related to grd type and number. 2.3.1. Truncaton errors Fgure 2 shows parameters of two adjacent cells and the truncaton error between typcal neghborng cells for dfferent mesh types. The varable ϕ was chosen as a general varable to account for the truncaton error. The varables f and c are the ndces of the nterfacal and cell center ponts, respectvely. CFD smulatons are used to obtan ϕ f, the ϕ value at f n the nterface of two adjacent cells, through nterpolaton by usng the ϕ values at the two cell centers: r r f, r r 1 c, c, 1 where r s the dstance from the center of cell (center of gravty) to the nterfacal center pont f, and r +1 s the dstance from the center of cell 2 (center of gravty) to the nterfacal center pont f. 1 (3) 192 193 194 195 196 197 198 (a) (b) (c) Fgure 2. Typcal neghborng cells and parameters of two adjacent cells for dfferent mesh grd types: (a) hexahedral mesh (b) tetrahedral mesh and (c) hybrd mesh. By usng a Talor seres, we can express the term on the rght-hand sde of Eq. (3) as: 7

199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 1 2 1 2 r 1 c, rc, 1 r 1 f, rf, r 1 ( r) f, r( r1 ) f, r 1 ( r) f, r ( r+1) f, +... 2! 2! (4) Therefore, the truncaton error of Eq. (3) s: r r, r 1 r1 1 1 8 (5) 2 2 r 1 f, r r f, r1 1 r 1 ( r) f, r( r1) f, TE... r r 2 r r Let us now study four dfferent grd-type scenaros: Scenaro 1: Neghborng cells have the same geometrcal shape and edge length (such as the cubcal and equlateral-trangular shaped cells shown n Fgures 2(a) and (b), respectvely). The drectons of r and r +1 are opposte one another, and r +1 s thus: r 1 r The frst term of the truncaton error n Eq. (6) becomes zero, so the truncaton error s secondorder as follows: 1 r ( r ) r( r ) TE 2 r r 2 2 1 f, 1 f, Scenaro 2: Neghborng cells have dfferent geometrcal shapes, but each cell has equal edge lengths, such as those shown n Fgure 2(c). When hexahedral and pyramdal cells are adjacent to each other, then r +1 can be wrtten as r 1 r r The frst term of the truncaton error n Eq. (6) agan becomes zero, and the truncaton error s also of second order. When tetrahedral and pyramdal cells are adjacent to each other, r +2 and r +3 are not parallel. The frst term of the truncaton error cannot cancel out, and the truncaton error wll be of frst order. In hybrd meshes wth transtons between tetrahedral and pyramdal cells, the truncaton error s of frst order, whle n meshes of a sngle type such as hexahedral and tetrahedral meshes, the leadng term s of second order. Therefore, the truncaton error for hybrd meshes wll be hgher than that for the other two grd types. 1 1 r... (6) (7) (8) (9)

233 234 235 236 237 238 239 240 241 242 243 244 245 Scenaro 3: Neghborng cells have the same geometrcal shape, but each cell has dfferent edge lengths (such as a rectangular parallelepped and scalene-trangular shaped cell). The frst term of the errors arsng on opposte hexahedral cell faces cancels out completely on the bass of Eq. (9), snce the cell faces are parallel. However, because the cell faces are not parallel for tetrahedral meshes, the truncaton error s stll of frst order. Hence, hexahedral meshes are superor to tetrahedral meshes wth a smlar resoluton [21]. Scenaro 4: Neghborng cells have dfferent geometrcal shapes, and each cell has dfferent edge lengths. The truncaton error s always of frst order. Refnng the meshes would reduce the truncaton error. When the mesh s suffcently fne, mesh type has lttle nfluence on the accuracy of smulaton results because lm ( TE) O( r ) 0 r 0 (10) 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 2.3.2. Round-off errors Round-off error, n, s the dfference between the exact soluton n and the approxmated n soluton of the governng equaton, as shown n Eq. (11). Lmted computer word length would lead to the round-off error. As the tme step sze and cell dmenson decrease, the round-off error ncreases whle the truncaton error decreases. Decreasng the cell dmenson and tme step sze does ensure more accurate results. When the tme step sze and cell dmenson are very small, the accuracy s compromsed because the round-off error may overtake the truncaton error. Therefore, the grd number should be small enough to prevent round-off error. n n n (11) It s necessary to dentfy the relatonshp between numercal errors (ncludng round-off and truncaton errors) and grd number. Snce the cell dmenson may not be constant over an entre computatonal doman because of the uneven mesh dstrbuton, let us use an average cell dmenson to estmate the average truncaton error. In the case n whch r and r +1 have the same drecton and the second and hgher order terms n Eq. (6) can be neglected, the averaged truncaton error n the computatonal doman wll be maxmal: 266 267 TE V f N 3 (12) 9

268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 where N s the grd number and V s the flow doman sze (15.5 m 3 for the frst-class cabn). In the case n whch r and r +1 have the opposte drecton and the second and hgher order terms n Eq. (6) can be neglected, the averaged truncaton error wll be zero. For a CFD program wth double precson parameters, the storage accuracy of a computer can be as hgh as 10-15. If we terate 20,000 tme steps for a 150 s unsteady-state smulaton n the frst-class cabn, the maxmal round-off error s: 15 Ero N 20,000 /10 (13) The total numercal error s then: 11 V Err N 210 3 f (14) N A sutable grd number for achevng the mnmal numercal error can be determned by equatng the dervatve of the rght-hand term of Eq. (14) to zero. By usng V = 15.5 m 3, we obtan N=9.2 10 f 7 4 3 (15) Eq. (15) shows that a sutable grd number s a functon of ϕ for the ar cabn. 3. RESULTS Ths secton frst compares the smulated results from the steady- and unsteady-state RNG k-ε models wth the measured data, and then dscusses the mpact of mesh type and number on the smulated results. 3.1. Steady- and unsteady-state turbulent flow modelng The steady-state RNG k-ε model (RANS) and unsteady-state RNG k-ε model (URANS) should yeld the same results for stable flow. However, as shown n Fgure 3, dfferent ar velocty profles were obtaned n the three selected vertcal postons n the cabn wth hexahedral meshes of 24 mllon cells. Because the URANS results were obtaned by averagng them over 100 s (two tme constants), the dfferences n the two smulated results suggest that the flow n the cabn was unstable. Kumar and Dewan [37] found that thermal plumes can be ntermttent and gve rse to tme-dependent fluctuated flow felds around human bodes. Fgure 3 also shows that the predcton by URANS s better than that by RANS when the smulated results are compared wth the expermental data. However, t s mportant to note that the expermental data contaned some uncertantes resultng from the complex boundary condtons, as reported by Lu et al. [6]. The expermental data should be 10

305 306 307 308 used only as a reference, rather than a crteron. Because of the unstable flow features, ths nvestgaton used URANS to study the mpact of grd type and number on the predcton of ar dstrbutons n the arlner cabn. P1 P2 P5 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 Fgure 3. Comparson of the smulated ar velocty profles obtaned by the RNG k-ε and unsteadystate RNG k-ε models wth those measured n the occuped frst-class cabn. 3.2. Impact of grd number on the smulated results Fgure 4 compares the smulated velocty profles at fve vertcal samplng lnes wth tetrahedral meshes of dfferent grd number. As the grd number ncreased, the truncaton error decreased. However, mesh densty was hgh n the regons close to the walls and ar dffusers, such as at P1 and P5, and local truncaton errors n these regons were smaller. Hence, a further ncrease n grd number had lttle nfluence on the smulated velocty profles at P1 and P5. However, n the regons wth large cell dmensons, such as at P2, P3, and P4, the correspondng truncaton errors were large. The dfferent grd numbers led to dfferent smulated results. The results were very dfferent from those wth fner grds, especally when the grd numbers were low (3 and 6 mllon cells). The smulated ar velocty profles were smlar wth meshes of 12, 24, and 50 mllon cells, whch meant that the truncaton errors were smlar. 11

327 328 329 330 Fgure 4 also shows the measured ar velocty profles at the fve vertcal postons. The smulated and measured results show smlar trends. The dfferences between the two results can be attrbuted to numercal errors and expermental uncertantes. 1.6 2.0 2.0 1.2 1.6 1.6 Z (m) 0.8 P1 P2 P3 1.2 1.2 Z (m) Z (m) 0.8 0.8 0.4 0.4 0.4 331 0.0 0 0.1 0.2 0.3 0.4 0.5 Velocty (m/s) 0.0 0 0.1 0.2 0.3 Velocty (m/s) 0.0 0 0.1 0.2 0.3 Velocty(m/s) 2.0 1.6 1.6 P4 1.2 P5 Z (m) 1.2 0.8 0.4 Z (m) 0.8 0.4 332 333 334 335 336 337 338 339 340 341 0.0 0.0 0 0.1 0.2 0.3 0 0.1 0.2 0.3 Velocty (m/s) Velocty (m/s) Fgure 4. Comparson of the vertcal ar velocty profles computed usng dfferent tetrahedral meshes wth the expermental data at the fve locatons n the occuped cabn. Fgure 5 compares the smulated temperature profles at the fve vertcal lnes wth tetrahedral meshes of dfferent grd number. The smulated temperature was less affected by truncaton errors than was velocty, as a result of the small temperature gradent n the cabn. The temperature profles for grd numbers of 3 and 6 mllon cells dffered from those for fner grds, although the dfference was not as evdent for temperature as for velocty. 12

1.6 2.0 2.0 1.2 1.6 P1 P2 P3 1.6 Z (m) 0.8 Z (m) 1.2 Z (m) 1.2 0.8 0.8 0.4 0.4 0.4 342 0.0 19 21 23 25 27 T( ) 0.0 19 20 21 22 23 24 T ( ) 0.0 20 22 24 T ( ) 2.0 1.6 1.6 P4 1.2 P5 Z (m) 1.2 0.8 0.4 Z (m) 0.8 0.4 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 0.0 0.0 20 22 24 20 22 24 26 T ( ) T ( ) Fgure 5. Comparson of the vertcal ar temperature profles computed usng dfferent tetrahedral meshes wth the expermental data at the fve locatons n the occuped cabn. Although not dscussed n ths paper, the results for the hexahedral and hybrd meshes exhbted patterns that were smlar to those of the tetrahedral meshes. Because of the complcated geometrcal boundary, we found that hexahedral meshes wth three mllon cells were nsuffcent for generatng a reasonable mesh dstrbuton. The correspondng smulatons dd not lead to converged solutons. A mesh sze of at least 12 mllon cells was requred for a hexahedral mesh n the frst-class cabn. 3.3. Impact of mesh type on the smulated results Fgure 6 compares the smulated and measured arflow dstrbutons wth coarse (3 mllon cells), medum (12 mllon cells) and fne (more than 38 mllon cells) meshes at a cross secton n the frstclass cabn. (The locaton of the cross secton s shown n Fgure 1(a).) Fgure 6(a) presents only the results for 3-mllon-cell tetrahedral and hybrd meshes, because the smulaton wth 3 mllon hexahedral cells dd not lead to a converged result. Snce hybrd meshes have many transtonal 13

360 361 362 363 regons between mesh types, where the truncaton errors are large accordng to Eq. (5), the two arflow patterns obtaned wth the tetrahedral and hybrd meshes are very dfferent, and one of the hybrd meshes appears to be wrong when compared wth the expermental data shown n Fgure 6(d). 364 365 (a) 3 mllon cells 366 367 (b) 12 mllon cells 368 369 (c) More than 38 mllon cells 14

370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 (d) Expermental data Fgure 6. Comparson of the arflow patterns smulated usng dfferent grd types and numbers wth the expermental data at a cross secton n the cabn. Because of the large cell dmenson for T3 (Please see abbrevatons for T3, H12, T12, HY3, HY12, etc. n Table 1.), the truncaton error was large. The grd resoluton and numercal dffuson were nsuffcent for correctly descrbng the crculaton flow drven by the thermal plumes from the human bodes and the jets from the dffusers on the rght sde of the cabn. Fgure 6(b) shows that H12 and T12 led to reasonable solutons, but HY12 could not predct the crculaton on the rght sde of the cabn. Only when the grd number was suffcently hgh dd the three mesh types lead to smlar results, as shown n Fgure 6(c). Fgure 7 compares the smulated and measured temperature felds wth dfferent grd types and numbers. Because HY3 dd not accurately smulate the jet flow from the dffusers on the rght sde of the cabn, the ar temperature n the regon was hgh. When the grd number was ncreased, the predcted ar temperature dstrbutons agreed well wth the measured dstrbuton, as shown n Fgure 7(). Predcton accuracy wth the hybrd meshes was poorer than wth the other two mesh types, but the dfferences between smulated and expermental results were not as evdent as those for ar velocty. 15

(a) Hybrd - 3 mllon (b) Hybrd - 12 mllon (c) Hybrd - 38 mllon (d) Tetrahedral - 3 mllon (e) Tetrahedral - 12 mllon (f) Tetrahedral - 50 mllon (g) Hexahedral -12mllon (h) Hexahedral -59 mllon () Expermental data 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 Fgure 7. Comparson of the ar temperature dstrbutons smulated usng dfferent grd types and numbers wth the expermental data at a cross secton n the cabn. A comparson of ar velocty and temperature dstrbutons showed that, because of truncaton errors, dfferent mesh types produced smulaton results of varyng accuracy. Among the three mesh types, the hexahedral grd had the hghest accuracy, whle the hybrd grd had the lowest. As the grd number ncreased, the truncaton error decreased. At suffcently hgh grd numbers, the effect of mesh type on the smulaton results was small. 3.4. Impact of mesh type and grd number on the numercal errors and computng costs Table 3 shows the numercal errors wth the fnest hexahedral meshes (H59), whch were calculated by Eq. (12, 13). The maxmal round-off error was determned by assumng a double precson smulaton wth a storage accuracy of 10-15 and a grd number of 59 mllon. The truncaton error was determned from the gradent dstrbuton of ϕ and cell dmenson. When the largest gradent of parameter ϕ n the asle regon wth the largest cell dmenson was used n ths calculaton, the maxmal truncaton error was found to be 0.012 f. Table 3 provdes the truncaton errors for dfferent ϕ. The round-off errors were comparable to the truncaton errors wth the fnest grd. 16

410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 Table 3. Analyss of numercal errors Error Parameters Maxmum Medum Mnmum Truncaton error Velocty 10-3 10-4 0 Temperature 10-2 10-3 0 Round-off error All parameters 10-3 n/a 10-15 n/a = not avalable. The accuracy of the smulaton results for dfferent grds s compared further n Table 5. Ths study used the relatve error between the key predcted and measured results as a crteron and ranked the error n the range of Grade A to Grade D. Grades A, B, C, and D represent relatve errors of 0-10%, 10-20%, 20-30%, and greater than 30%, respectvely. Table 4 shows that smulatons wth the hexahedral meshes most closely match the expermental data for both ar velocty and temperature. A grd number of at least 12 mllon cells were necessary for convergence wth the hexahedral meshes. When the grd number was ncreased to more than 38 mllon, all three grd types had smlar results. Table 4. Accuracy of the smulatons wth dfferent grd types and numbers Mesh type Hybrd Tetrahedral Hexahedral Parameter Grd number (mllons of cells) 3 6 12 24 >38 Temperature ( o C) C B B B B Velocty (m/s) D D D D B Temperature ( o C) C B B B B Velocty (m/s) D D C C B Temperature ( o C) B B B n/c Velocty (m/s) B B B A = good ( 10%), B = acceptable (10%, 20%), C = margnal (20%, 30%), D = poor (>30%), n/c = not converged. Table 5 summarzes the computng tme requred. All the smulatons were performed on a standalone computer wth 32 cores and 128G memory. It s clear that the larger the grd number, the longer the computng tme. The tme was nearly proportonal to the grd number. The computng tme was also related to the node numbers of the cells. The hexahedral meshes had more cells than the tetrahedral meshes, whch led to a longer computng tme. In addton, the hgh aspect rato for the tetrahedral meshes may have nfluenced computng tme. In summary, the hexahedral meshes requred the longest computng tme, and the hybrd meshes the shortest. The computng tme and the accuracy of the smulated results wth 24-mllon-cell hybrd meshes were smlar to the tme and accuracy, respectvely, wth 12-mllon-cell hexahedral meshes. We can apparently conclude that, regardless of the grd type used, smlar computng tmes are requred to acheve a gven level of accuracy. Table 5. Computng tme for dfferent grd types and numbers Grd type Grd number (mllons of cells) 17

439 440 3 6 12 24 >38 Hybrd 10.2h 24.8h 47.0h 92.4h 197.6h Tetrahedral 15.4h 37.6h 72.8h 141.7h 374.7h Hexahedral n/c 81.2h 188.2h 565.5h n/c = not converged 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 4. CONCLUSION Ths study evaluated the performance of three mesh types and fve grd numbers for predctng arflow and temperature dstrbutons n the frst-class cabn of an MD-82 arplane. The nvestgaton led to the followng conclusons: The hexahedral meshes were the most accurate, whle also beng the most tme-consumng. The hybrd meshes were the least accurate but used the least computng tme. By ncreasng the grd number of the hybrd mesh to obtan the same accuracy as that wth the hexahedral meshes, a smlar computng tme s acheved. The results suggest that n smulatons wth 12-mllon-cell hexahedral meshes, 24-mllon-cell hybrd meshes, and tetrahedral meshes of approxmately 15 mllon cells, the accuracy would be the same. Furthermore, the computng tme for each of these smulatons would be about 80-90 hours on the computer cluster used for ths nvestgaton. For the frst-class cabn, ths study found that a grd number of at least 12 mllon cells were needed to produce acceptable results. When the grd was suffcently fne (>38 mllon cells), all the three mesh types produced smlar results. The truncaton errors were typcally larger than the round-off errors. When the grd number was suffcently large (>38 mllon), the round-off errors were comparable to the truncaton errors. ACKNOWLEDGEMENTS Ths study was supported by the Natonal Basc Research Program of Chna (the 973 Program) through Grant No. 2012CB720100 and the Center for Cabn Ar Reformatve Envronment (CARE) at Tanjn Unversty, Chna. 463 464 465 466 467 468 469 470 471 REFERENCES [1] World Bank Group (Ed.), World Development Indcators 2013, World Bank Publcatons, 2013. [2] A. Mangl and M. A. Gendreau, Transmsson of nfectous dseases durng commercal ar travel, The Lancet, vol. 365, pp. 989-996, 2005. [3] S. Park, R. T. Hellwg, G. Grün and A. Holm, Local and overall thermal comfort n an arcraft cabn and ther nterrelatons, Buldng and Envronment, vol. 46, pp. 1056-1064, 2011. [4] W. Cu, Q. Ouyang and Y. Zhu, Feld study of thermal envronment spatal dstrbuton and passenger local thermal comfort n arcraft cabn, Buldng and Envronment, vol. 80, pp. 213-220, 2014. 18

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