OVERSET UNSTRUCTURED GRID METHOD FOR FLOW SIMULATION OF COMPLEX AND MULTIPLE BODY PROBLEMS

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1 ICAS 2000 CONGRESS OVERSET UNSTRUCTURED GRID METHOD FOR FLOW SIMULATION OF COMPLEX AND MULTIPLE BODY PROBLEMS Kazuhro Nakahash, Fumya Togash Tohoku Unversty Keywords: CFD, Unstructured grd, overset grd Abstract The use of the overset concept for the unstructured grd method s dscussed for the numercal smulatons of flows around complex and multple bodes n relatve moton. The ntergrd boundares n the overset grds are automatcally localzed usng the wall dstance as a basc parameter. The search for donor cells s effcently performed by the neghbor-toneghbor jump search on a modfed convex doman utlzng a byproduct of the Delaunay trangulaton method. The capablty of the method s demonstrated by smulatons of a rocket-booster separaton from a supersonc expermental arplane. A grd around the rocket booster s overset on the arplane grd that covers the entre flow feld. The complcated confguraton of the rocket booster s fathfully reproduced by a sngle grd. Ths capablty of the unstructured grd sgnfcantly smplfes the overset procedure as compared wth the conventonal structured grds approach. Comparsons of the computed results wth the experment show good agreements n the lft and ptchng moment hstores. 1 Introducton CFD applcatons to real engneerng problems requre the numercal smulaton of flows around complex geometres. The overset grd method [1] provdes a powerful means of handlng complex geometres by structured grd methods. Unstructured grd methods are also well suted to these knds of smulaton due to ther ablty n accurately dscretzng complex computatonal domans and to ther flexblty n refnng the grd n order to match the local flow features. Although these two CFD approaches for complex geometry problems have become wdespread, some dffcultes stll reman n each approach. In the overset structured grd method, the number of subgrds whch overlap ncreases f the geometry of the computatonal models becomes complex. The large number of subgrds complcates the procedure to construct the ntergrd communcatons among oversetstructured grds. Unstructured grd approach has a dffculty n computng the unsteady flow around multple bodes n relatve moton. Although the unstructured grd s capable of treatng movng bodes as well (for example, [2]), a part of the computatonal grd or n some cases the whole grd has to be regenerated at every tme step. Ths procedure may become qute complex and computatonally expensve. In ths paper, we dscuss the applcaton of the overset concept to unstructured grd methods. By use of unstructured meshes, the number of submeshes requred for coverng the flow fled can be sgnfcantly reduced as compared wth that needed n the overset structured grd. It can also extend the applcablty of the unstructured grd method to multple movng-body problems wthout much need for code development. In ths paper, an effcent and relable algorthm to automatcally localze the ntergrd boundares for the overset unstructured grd method s descrbed. The method s tested for a separaton smulaton of a supersonc arplane and a rocket booster

2 K. Nakahash, F. Togash 2 Intergrd Boundary Defnton There are two major steps to establsh ntergrd communcatons n the overset method: Hole-cuttng, whch nvolves dvdng all ponts of each subgrd nto two groups, actve and non-actve ponts. The ntergrdboundary ponts are dentfed as the actve ponts next to non-actve ponts. Identfcaton of nterpolaton stencls, whch nvolves a search of donor cells for all ntergrd-boundary ponts. The second step, dentfcaton of nterpolaton stencls, s straghtforward for unstructured grds. Once a donor cell s dentfed, values on the pont n ths cell are nterpolated from values on the vertex of the cell usng the area co-ordnates for a trangle and the volume co-ordnates for a tetrahedral cell. In the present approach, the donor cell for the nterpolaton at each ntergrd-boundary pont s dentfed durng the process of holecuttng. Therefore, the frst step s descrbed here. 2.1 Automatc Hole-Cutng The dentfcaton of the ntergrd boundary must be performed completely automatcally f unstructured grds are used for the overset approach. Manual creatons or correctons of the hole-cuttng for the overset unstructured grds are almost mpossble because of the unstructured numberng of the node ponts. Here, the wall dstance s used as a parameter to construct the ntergrd boundary. In the mesh overlappng regon, a node pont havng a shorter dstance to the sold boundary whch belongs to the same grd of the node s selected. The procedure of the ntergrd boundary defnton s schematcally shown n Fg. 1. Suppose that the dotted lnes show a grd (Grd- A) generated around Body-A, and the sold lnes show a grd (Grd-B) for Body-B. Before the hole-cuttng, the mnmum dstance of each node pont to ts body surface s computed. Then, the hole-cuttng procedure s dvded nto two steps. The frst step s to desgnate all nodes ponts as actve or non-actve ones. The second (c) Fg. 1 Determnaton of ntergrd-boundary: Grd A (dotted lnes) and Grd B (sold lnes) are overlapped. After the node dentfcaton. Crcles are the actve nodes n Grd A and squares are the ones n Grd B, (c) After removal of non-actve cells

3 OVERSET UNSTRUCTURED GRID METHOD FOR FLOW SIMULATIONS OF COMPLEX AND MULTIPLE BODY PROBLEMS step s to classfy all cells nto three groups: actve cells, non-actve cells and ntergrdboundary cells. Let s consder node pont n Grd-A n Fg. 1. Suppose we know the donor cell n Grd-B for ths node. In Fg. 1, the donor cell s ndcated by abc. The dstance to wall-b from poston n cell abc s then evaluated by a lnear nterpolaton from ts vertex values. Ths dstance to wall-b of the donor cell s compared wth the dstance to Body-A of the node. Snce the dstance of ths node to wall-a s shorter than that of the donor cell to wall-b, we select ths node as an actve node. In contrast, node j n Fg. 1 wll be selected as a non-actve node. Ths assgnment procedure s repeated for all node ponts n both grds. In Fg. 1, nodes shown by crcles are actve nodes n Grd-A, and those shown by squares are actve nodes n Grd-B. The remanng node ponts are nonactve. By desgnatng all node ponts n the grds as actve or non-actve, the next step s to classfy all cells nto three groups: actve cells, non-actve cells and ntergrd-boundary cells. An actve cell s a cell whose vertex nodes are all actve, whle a non-actve cell s the one whose vertexes are all non-actve nodes. The remanng cells are the ntergrd-boundary cells whch construct the overlappng layers among subgrds for ntergrd communcatons. Fgure 1(c) shows the grds after removal of the nonactve cells. The above-mentoned procedure s very smple, yet t automatcally defnes the ntergrd boundary and overlappng layer between grds. The overlappng layer has a wdth of mostly one or two cells as shown n Fg. 1(c). 2.2 Neghbor-to-neghbor search The use of the wall dstance for the automatc defnton of the ntergrd boundary s smple and very relable. However, all node ponts must fnd ther donor cells n the overset meshes. The number of searches easly surpasses one mllon for three-dmensonal problems. Therefore, an effcent and relable search algorthm must be developed. Fg. 2 Neghbor-to-neghbor search. The searches from B and C wll fal. Fg. 3 Neghbor-to-neghbor search n a convex doman. The neghbor-to-neghbor jump search algorthm [3] s effcently utlzed n the present method. The procedure s schematcally shown n Fg. 2 for a trangular grd. Startng from an ntal guess, cell-a n Fg. 2 for example, the method s to repeat a jump to the neghborng cell that s located on the target sde of the current cell. Ths search s very effcent because the search path s one dmensonal even n a threedmensonal feld. However, t easly fals dependng on the startng pont. As shown n Fg. 2, a search from pont A succeeds n reachng the target. Searches startng from B and C, however, get stuck at the body boundary or the outer boundary. For these cases, the search has to be restarted by changng the cell from whch the search s commenced. To avod such uncertanty of the search, the search doman s modfed to be a convex hexahedron for any computatonal geometry. Ths s done so as to add subsdary grds nto the bodes and outsde of the computatonal regon as shown n Fg. 3. If we use the 263.3

4 K. Nakahash, F. Togash Delaunay trangulaton for the grd generaton, the subsdary grds can be obtaned automatcally as a byproduct of the grd generaton procedure. By utlzng the subsdary grds, the neghbor-to-neghbor search becomes relable and more effcent. Snce the effcency of the neghbor-toneghbor search depends on the ntal guess, the computatonal load for the second search after the ntal hole-cuttng becomes sgnfcantly small. Once the ntal hole-cuttng has been done, the search of the donor cells after a small relatve movement of the subgrd can be lmted to nodes around the current ntergrd boundares. 3 Soluton Algorthm 3.1 Flow solver The Euler equatons for compressble nvscd flows are wrtten n an ntegral form as follows: t where ƒ Q dv + F( Q) nds = 0, (1) ƒ T Q s the vector of = [ ρ, ρu, ρv, ρw, e] conservatve varables, ρ the densty, u, v, w the velocty components n the xyz,, drectons, and e the total energy. The vector F (Q) represents the nvscd flux vector and n s the outward normal of Ω whch s the boundary of the control volume Ω. Ths system of equatons s closed by the perfect gas equaton of state. The equatons are solved by a fnte volume cell-vertex scheme. The control volume s a non-overlappng dual cell. For the control volume, Eq. (1) can be wrtten n an algebrac form as follows: Q t 1 = V j( ) S h Q j + (, Q, n ) j j j, (2) where S s the segment area of the control j volume boundary assocated wth the edge connectng ponts and j. Ths segment area, S j, as well as ts unt normal, n j, can be computed by summng up the contrbuton from each tetrahedron sharng the edge. The term h s an nvscd numercal flux vector normal to ± the control volume boundary, and Q are values j on both sdes of the control volume boundary. The subscrpt of summaton, j (), means all node ponts connected to node. The numercal flux, h, s computed usng an approxmate Remann solver of Harten-Lax-van Leer-Enfeldt-Wada[4]. The second order spatal accuracy s realzed by a lnear reconstructon of the prmtve gas dynamc varables wth Venkatakrshnan s lmter [5]. The LU-SGS mplct method [6] s appled to ntegrate Eq. (2) n tme. Wth n+1 n Q = Q Q and a lnealzaton of the umercal n+1 n + flux term as hj = hj + A Q + A j Q, the j fnal form of the LU-SGS method on an unstructured grd becomes, Forward sweep: * * ( h ) * 1 = D R 0.5 Sj j A Q j L( ) Q ρ, j (3a) Backward sweep: ( h ) Q ρ, where * 1 = Q 0.5D S j j A Q j j U ( ) j( ) (3b) n R = Sh, h = h( Q + Q) h( Q) and j j D s a dagonal matrx derved by Jameson- Turkel approxmaton of Jacoban [7] as ± A = 0.5( A ± ρ AI), where ρ s a spectral radus A of Jacoban A. And D s gven as follows: V j D = Sj ρ I. A (4) t j( ) The lower/upper splttng of Eq. (3), namely j L() and j U (), for the unstructured grd s realzed by usng a grd reorderng technque [6] to mprove the convergence and the vectorzaton. 3.2 Overset mplementaton The above flow solver must be modfed to account for the use of multple meshes. In 263.4

5 OVERSET UNSTRUCTURED GRID METHOD FOR FLOW SIMULATIONS OF COMPLEX AND MULTIPLE BODY PROBLEMS addton to the boundares of the computatonal doman, subgrds may have holes and ntergrd boundares wth the neghborng donorsubgrds. The non-actve cells must be excluded or blanked from the flow feld soluton. All node ponts have nformaton as to whether they belong to the actve or non-actve cells. Namely, 1, f a pont sn t blanked; IBLANK= (5) 0, f a pont s blanked. Ths value s 1 or 0 dependng on the area nsde or outsde of the computatonal subregon. In the flow solver, the rght-hand sde vector R n Eq. (3) s multpled by the value IBLANK (). Namely the ƒ Q n the outsde regon (hole regon) s set to be zero. 4 Computatonal Results The Natonal Aerospace Laboratory (NAL) of Japan s currently workng on a project to develop expermental supersonc arplanes [8] as a basc study for the next generaton supersonc transport. The frst model of the expermental arplanes s unpowered and a sold rocket booster wll be used to launch t to a hgh alttude at a speed of about Mach 2.5. Fgure 4 shows the confguraton for launch. The fuselage length of the arplane s 11.5 m, and the wngspan s m. The present method was appled to the numercal smulaton of ths expermental supersonc arplane s separaton from a rocket booster. The geometry shown n Fg. 4 was produced by NAL usng the CATIA CAD software. It ncludes detaled components, such as the attachments of the arplane/booster, fttngs for the launcher, and frnges on the rocket booster. To compute the flow around ths complex geometry by the conventonal overset structured grd method wll need a tme consumng work for generatng the grd due to the small components attached on the model. Thus ths s a good test case for the present method n order to evaluate ts capablty for a multple movng-body problem wth complex geometry. At the begnnng, the unstructured surface grds on the arplane and the rocket booster were generated separately by the drect advancng front method combned wth the geometrcal feature reconstructon [9] on the STL format fle produced by the CATIA. Ths surface grd generaton approach sgnfcantly reduces the surface meshng burden for complex geometry. Fgure 4 s an enlarged vew of the surface grd at the wng tralng edge regon of the arplane. The unstructured surface grd approprately covers the detaled components of the arplane-booster connecton parts as well as the frnge and the wre cover on the rocket booster. After the surface meshng, two unstructured volume grds, each of whch covers the arplane and the rocket booster, respectvely trangulatons as shown n Fg. 5, were generated usng the Delaunay method [10]. The Fg. 4 NAL s expermental supersonc arplane wth rocket booster for launch. Entre vew, Enlarged vew showng the surface grd

6 K. Nakahash, F. Togash Fg. 5 Overset grds for the supersonc arplane (outer cylndrcal regon) and rocket booster (nner cylndrcal regon). (c) outer cylndrcal grd was generated for the arplane and the nner cylndrcal grd for the rocket booster. The number of node ponts and cells of these grds are shown n Table 1. For a smulaton of the arplane/rocket booster separaton, the nner grd moves wth the rocket booster n the statonary outer cylndrcal grd. Table 1 Grd data and average numbers of cell jumps for search Arplane grd 599,203 nodes ponts, 3,268,529 cells Booster grd 262,513 node ponts, 1,436,487 cells Donor cell search 100 cells/node (ntal) from cells/node (the arplane to booster followng steps) Donor cell search 192 cells/node (ntal) from 0.13 cells/node (the booster to arplane followng steps) Fgure 6 shows the ntergrd boundares of the booster grd for several relatve locatons between the arplane and the booster. The relatve postons and angles of attack were Fg. 6 Intergrd boundares of the booster subgrd: angles of attack of the arplane at 2 deg and the booster at 0 deg and Z=0.4m, Z=1.0m, (c) Z=3.0m, where Z s the relatve dstance between the arplane and the booster. prescrbed. In the fgure, the dark rough surfaces are the cut surfaces (ntergrd boundares) of the booster grd due to the exstence of the arplane. As shown n the fgure, the ntergrd boundary becomes complex because of the large fns attached to the end of the rocket booster. Fgure 7 shows the grds on symmetrcal plane and a cut plane perpendcular to the axs of the 263.6

7 OVERSET UNSTRUCTURED GRID METHOD FOR FLOW SIMULATIONS OF COMPLEX AND MULTIPLE BODY PROBLEMS Fg. 7 Overset grds on: symmetrc plane, a cross-sectonal plane. arplane fuselage. Two subgrds, one for the arplane and one for the booster, overlap each other. The overlappng layer between two grds has a wdth of mostly one or two cells. Table 1 shows the average numbers of cell jumps requred for one search path measured for the confguraton of zero relatve angle of attack. The search from the booster grd to the arplane grd s to fnd a donor cell n about 3.3 mllon cells. Even wth ths large number of cells, one search path requred 192 cell jumps n average. The ntal guess for each search uses the search result of the prevous node. Therefore, ths search path length may be reduced some more f the node numberng s well ordered. The searches after the ntal ntergrd setup become sgnfcantly small as (c) Fg. 8 Computed pressure contours of supersonc arplane/rocket booster separaton at M = 2.5, angles of attack of the arplane at 2 deg and the booster at 0 deg, and Z =0.4m, Z =2.4m, (c) Z =5.0m shown n the Table 1. Computatons were performed at a Mach number of 2.5 wth an assumpton of the quas263.7

8 K. Nakahash, F. Togash steady flow. The angle of attack of the arplane was fxed at 2.0 degrees and the angle of attack of the rocket booster relatve to the arplane was at 2.0 degrees (0 degree to freestream). The relatve horzontal locaton of the arplane and rocket was fxed to 0, the relatve vertcal dstances between the arplane and the booster ( Z ) were ncreased n a prescrbed moton. Fgure 8 shows the computed pressure contours around the arplane and booster. Shock waves generated at the noses of the arplane and the booster create a complex reflecton pattern n the narrow regon between the bodes. At the begnnng of the booster separaton, the shock wave from the booster nose hts the forward part of the lower surface of the arplane wng. Ths ntally causes an ncrease n the ptchng moment of the arplane. Ths ptchng moment then decreases to a negatve value as the mpngng pont of the booster-nose shock on the arplane wng moves downward. Fgure 9 s an enlarged pcture of the computed pressure contours at Z =0.1m. A strong shock wave generated at the booster nose hts the lower surface of the arplane fuselage and the reflectng shock return to the booster surface. The arplane/booster attachment part also generates a strong shock wave whch nteracts wth the reflectng nose shock. Fgure 10 shows comparsons of the lft and ptchng moment coeffcents between computatonal and wnd tunnel results. In these fgures, CL denotes the dfference between the lft coeffcent of the multple body case and the one of the solated body case. The value CM denotes the same dfference of the ptchng moment coeffcents. At the begnnng of the booster separaton from the arplane, the shock wave from the booster nose hts the forward part of the lower surface of the arplane wng. Ths causes an ncrease n the ptchng moment and lft of the arplane. Then, the ptchng moment of the arplane moves to be negatve as the mpngng pont of the booster-nose shock on C L C M arplane (exp.) booster (exp.) arplane (no frnge) booster (no frnge) arplane (wth frnge) booster (wth frnge) Z m arplane (exp.) booster (exp.) arplane (no frnge) booster (no frnge) arplane (wth frnge) booster (wth frnge) Z m Fg. 9 Computed pressure contours of supersonc arplane/rocket booster separaton at M = 2.5, angles of attack of the arplane at 2 deg and the booster at 0 deg, and =0.1m Z Fgure 10 Comparsons of lft and ptchng moment coeffcents between the expermental and computatonal results at M = 2. 5, angles of attack of the arplane at 2deg and the booster at 0deg

9 OVERSET UNSTRUCTURED GRID METHOD FOR FLOW SIMULATIONS OF COMPLEX AND MULTIPLE BODY PROBLEMS the lower surface of the arplane moves downward. At Z =2m, the ptch-down moment of the arplane becomes the maxmum value, and then t decreases temporarly when the mpngng pont of the booster shock moves downstream of the arplane wng. However the ptch-down tendency of the arplane occurred agan when the booster shock hts the tal wng of the arplane. The ptchng moment and lft of the booster are decreased due to the effect of not only the shock wave from the arplane but also the reflectng shock wave of tself as shown n Fg. 8 and Fg. 9. Then, the ptchng moment of the booster moves to be postve as the mpngng pont of the shock wave from the arplane moves downward. The computatons were executed n two cases: one for the full detaled confguraton shown n Fg. 4, and one for a clean confguraton whch does not have any small components such as the attachments of the arplane/booster, fttngs for the launcher, and frnges on the rocket booster. In the conventonal structured grd CFD, these small parts are often neglected because of the dffculty of the grd generaton. It s also thought that the effect of those small components to the aerodynamc coeffcents s neglgble. In the present overset unstructured grd approach, to nclude the small components nto the computaton s straghtforward. The wnd tunnel tests were conducted for the full detaled confguraton. By comparng two computed results, wth and wthout the frnges, the effect of the connectng frnges s relatvely small for the arplane aerodynamc coeffcents. However, as shown n Fg.10, the peak values of CL and C M of the booster are apparently affected by the frnges and the computed results wth the detaled confguraton agree better wth the experment. 5 Conclusons The use of the overset concept for the unstructured grd method was dscussed for the numercal smulatons of a booster separaton from a supersonc expermental arplane. A grd around the rocket booster was overset on the arplane grd that covers the entre flow feld. Owng to the unstructured grd, a sngle grd can fathfully reproduce detaled components of the rocket booster. Ths capablty of the unstructured grd sgnfcantly smplfes the overset procedure as compared wth the conventonal approach to overlap a number of structured grds. An effcent and robust algorthm to localze the ntergrd boundares for the overset unstructured grd method has been developed usng the wall dstance as a basc parameter. The use of subsdary grds, whch are generated as a byproduct of the Delaunay trangulaton method, makes the neghbor-to-neghbor jump search relable and effcent. The computed result of the arplane/booster separaton clearly reproduced the complex reflecton shock wave patterns between two bodes. Comparsons wth the expermental results showed good agreements n the lft and ptchng moment hstores. The use of the overset concept wth the unstructured grd methods holds great promse for extendng the applcablty of the unstructured grd method for real engneerng problems wthout much needed for code development. 6 Acknowledgements The authors wsh to thank Dr. Iwamya of Natonal Aerospace Laboratory and Dr. Shnbo of Mtsubsh Heavy Industry for provdng us wth the arplane and rocket booster geometry data used for the wnd tunnel experment. The authors also wsh to thank to Mr. Fujta and Mr. Ito, graduate students of Tohoku Unversty, of ther helps n the CAD operatons and the grd generatons. References [1] Steger, J. L., Dougherty, F. C., and Benek, J. A. A Chmera grd scheme. ASME Mn-Symposum on Advances n Grd Generaton, [2] Lohner, R. and Baum, J. D. Three-Dmensonal Store Separaton Usng a Fnte Element Solver and Adaptve Remeshng. AIAA Paper ,

10 K. Nakahash, F. Togash [3] Lohner, R. Robust, Vectorzed Search Algorthms for Interpolaton on Unstructured Grds. Journal of Computatonal Physcs, Vol.118, pp , [4] Obayash, S., and Guruswamy, G. P. Convergence acceleraton of an aeroelastc Naver-Stokes solver. AIAA Paper , [5] Venkatakrshnan, V. On the accuracy of lmters and convergence to steady state solutons. AIAA Paper , January [6] Sharov, D. and Nakahash, K. Reorderng of Hybrd Unstructured Grds for Lower-Upper Symmetrc Gauss-Sedel Computatons. AIAA Journal, Vol.36, No.3, pp , [7] Jameson, A., and Turkel, E. Implct schemes and LU decompostons. Mathematcs of Computaton, Vol.37, No.156, pp , [8] Iwamya, T. NAL SST project and aerodynamc desgn of expermental arcraft. Proc. Computatonal Flud Dynamcs 98, Vol. 2, ECCOMAS 98, John Wley & Sons, Ltd., pp , [9] Ito, Y. and Nakahash, K. Drect surface trangulaton usng stereolthography (STL) data. AIAA Paper , [10] Sharov, D. and Nakahash, K. A Boundary Recovery Algorthm for Delaunay Tetrahedral Meshng. 5th Int. Conf. on Numercal Grd Generaton n Computatonal Feld Smulatons, pp ,