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Summary of the Group Discussion on Navier-Stokes Solvers Chairman: J. Piquet ENSM Nantes, France Co-cha~rman: Y. Kodama Ship Research Institute Tokyo, Japan First, questions, fields and issues on which this Group Dicussion should be focused are given(Kodama). Then a general table, given hereafter, of the meth- ods presented during the meeting is briefly dis- cussed(Piquet). It appears that, among the fourteen papers presenting numerical solutions of Navier-Stokes equations, only three methodologies are used; an unseg- regated approach (Hoekstra) in which the solenoidal- ity of the flow is enforced at each iteration; other contributions satisfy the incompressibility condition at convergence either by means of the so-called ar- tificial compressibility method(Yang, Kodama) or by means of a pressure correction technique-projection type method(Zhu, Hino, Doi) or simple-like meth- ods(Tzabiras, Larsson, Oh, Masuko, Piquet, Stern). It should be therefore necessary to compare in a more detailed way the methods, in order to isolate their differences and the resulting effects (Stern). As a first important difference, the choice of independent vari- ables is felt significant(Tzabiras) although no clear evi- dence of optimal choice has been provided. Several specific aspects are then addressed in the discussion: geometric singularities created by the curvi- linear structured grid, averaging procedures(Kodama); orthogonality constraints on the grid, needed regular- ity of the control functions in the elliptic grid gen- eration procedure(Ju), convergence problems on fine grids(Piquet). The question of accuracy measures is posed, from a 2D example, for an inner problem where momentum conservation implies strongly different re- sults on the drag forces coefficients CDP and CDM when computed from the integration of forces and from the global momentum balance(Kubota). The problem of conservation of mass, close to the boundaries - e.g. the free surface(Hino) -as well as that of momentum is dis- cussed. Problems connected to the turbulence model are then addressed. Corrrections for the free-surface prom le~ns(Hino) and existence of a model adequate for lift- ing problems(Tzabiras) are questionned. The need to avoid the "highly convenient" wall function approach is 721 emphasized(Hoekstra). In any case, it appears difficult to check the influence of the turbulence model on the numerical results in an unbiased way. The discussion is then displaced towards what should be done now (Tzabiras), given the existence of several Navier-Stokes codes able in principle to deal with complex problems. A few possibilities are putfor- ward and, among the noticed fields of applications, the impact of Navier-Stokes codes on the propeller research is not considered "too optimistically" (Hoekstra). Endly, the boundary conditions are discussed mainly in relation with the numerical scheme consid- ered(Tzabiras, Kodama) although the natural mathe- matical character of the Neumann pressure condition is pointed out (Hoekstra). To try a tentative evaluation, the chairmen of the discussion feel that the brief survey of technical prob- lems that has been attempted gives a good picture of collective weaknesses, given the rather small size of the community working on Navier-Stokes solvers for hydro- dynamic problems. The discussion was felt either too specific - and so could be considered as a disappointing specialist discus- sion(Himeno) - or not detailed enough to allow an ap- preciation of the pros. and cons. of the presented works and methodologies. May be, this can be attributed to the fact that not only our mutual work is not known in enough details, but also that the concerned aspects are so numerous that a complete assessment of each detail of the used methods is difficult. The weaknesses of the discussions appeared also on a conceptual level, for instance in the treatment of boundary conditions and on the views over accuracy. Recognized inadequacies in the treatment of the geom- etry- e.g. the HSVA tanker- did not raised the ques- tion of the use of partially unstructured girds. Recog- nized difficulties in enforcing conservativity (geometri- cal, mass, momentum) did not raised the question of the use of Galerkin - type methods. May be, these problems should call for a better consideration of the literature

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issued from applied mathematics. Because a lot of time has been spent on the tech- nical aspects of the work, and probably also because of the way the discussion has been conducted, the practi- cal importance of the whole set of aforementioned prom lems was not evaluated. A fortiori, the ability to use Navier-Stokes solvers to understand flow situations was not considered, although these solvers offer a unique opportunity to get information on the flow at a level of details not possible with experiments. Table of the 14 papers related to Nav~er-Stokes solvers presented during INC-5 (Part 1~. AUTHORS VARIABLES GRID INCOMPRESS. PRESSURE MOMENTUM & LAYO UT GENERATION CONSTRAINT SOLVER SO EVER HOEKSTRA covariant VW Schwarz multiple relax. contravariant U Christoffel Unsegregated sweep CSIP collocated transv. ortho. YANG UVWP Algebraic artif. comp. relaxation relaxation et A1. collocated + IAF ~ IAF KODAMA UVWP Geometrical artif. comp. IAF(Implicit IAF collocated [interp. btw.] Approximate surface grids Factorization) ZH U contravariant Elliptic MAC relaxation explicit HINO UVWP collocated Algebraic MAC relaxation explicit node-centered KINOSHITA UV\VP Algebraic MAC relaxation explicit et A1. collocated Ix U BO TA U V VV P Geometrical compressible rel axation explicit et A1. collocated (cavitation) DOI No Projection (channel flow) TZABIRAS UVWP Elliptic SIMPLE relaxation relaxation LOUKAKIS staggered LARSSON contravariant Elliptic SIMPLER relaxation relaxation et A1. staggered OH phys. polar Elliptic SINIPLE relaxation relaxation et A1. staggered MASU KO UVWP Elliptic SIMPLE relaxation relaxation OGI\VARA staggered PIQU ET UV\NJP Transfinite PISO ILU-PBCG relaxation VISONNEAU collocated ~ Elliptic STERN UV\VP Elliptic SILIPLER plane ADI plane ADI IxIM staggered 722

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Table of the 14 papers related to Navier-Stokes solvers presented during INC-5 (Part 2,). AUTHORS TURBU LENCE WALL SPACE TIME FREE SURFACE HOEhSTRA Mixing length damping pressure implicit no (CS) factor downstream 2nd-order V upstream YANG Baldwin-Lomax damping TVD(Roe) implicit no et A1. factor if\ form) KODAMA Baldwin-Lomax damping centered 2nd-order implicit no factor artif.dissipation (~ form) ZHU SGS damping centered 4th-order explicit no et A1. factor artif.dissipation HINO Baldwin-Lomax wall 2nd-order P explicit yes function 3rd-order cony. Fn=0.25 KINOSHITA none no slip 2nd-order P explicit no et A1. 3rd-order cony. KU BOTA none no slip 2nd-order P explicit cavitation et A1. . 3rd-order cony. DOI SGS damping 3rd-order cony. Adams-Bashforth no factor artif.dissipation (explicit) TZABIRAS k-e wall ftn. Hybrid implicit no LOUKAKIS LARSSON k-e wall fin. Finite-Analytic implicit no et A1. OH k-e wall fin. Finite-Analytic implicit no et A1. MASUKO k-e wall ftn. Hybrid implicit no OGIWARA PIQUET k-e wall ftn. Finite-Analytic implicit no VISONNEAU STERN laminar wall ftn. Finite-Analytic implicit no KIM k-e 723

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Table of the 14 papers related to Navier-Stokes solvers presented during INC-5 (Part 3~. AUTHORS START INLET FAR GRID NO. of CPU TIME TEST CASES FIELD (, 0, ~ ITER. HOEKSTRA potential thin B.L. potential 45x49x29 24 .5-lie Cray2 HSVA tanker P77 = 0 midship 10-3 resid YANG 91x25x29 220 17' Afterbodies 1,2,5 et A1. Cray YMP Bodies at incidence 40"/iter. Flat plate,Wigley, IxODAMA far uniform 5000 Stellar Series 60 l | upstream l l l | 4~S1000 | (cb=o.6~17'o8) ZH U rest far uniform 170x30x50 20h Wigley hull et A1. upstream (255000 to Hitac 340000 pts) S820/80 HINO rest far uniform 100x20x38 11000 2h/ 1000stps Wigley hull upstream ACOS 910 Series 60 (Cb=0.6) KIN OSHITA rest rest rest 140x60 40000 70' Oscillating et A1. VP- 100 circular cylinder I&U BOTA uniform far uniform 101x31x3 Hitac NACA0015 et A1. upstream M-680H wing section DOI rest periodic no Channel flow LES TZABIRAS rest thin B.L. potential 44x32x30 35 (20% 60hrs. SSPA ship liner LOUKAKIS midship ,u Vax / iter) SSPA ship liner LARSSON rest thin B.L. 60x21x15 et A1. midship OH rest 1/7 + eq uniform 54~32x25 190 1600" SSPA ship liner et A1. midship VP-100 MASUKO rest far uniform 94x25x21 Series 60 (Cb=0.6) OGIWARA | | ups ream | l l l | IHI-BO anker PIQUET rest thin B.L. uniform 80x40x31 ;2hrs HSVA tanker VISONNEAU midship Cray 2 STERN rest thin B.L. uniform 80000 Cray XMP Propeller shaft KIM +rot. (relative case frame ~ 724