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A Numerical Scheme Based on an Immersed Boundary Method for Compressible Turbulent Flows with Shocks: Application to Two-Dimensional Flows around Cylinders

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A Numerical Scheme Based on an Immersed Boundary Method for Compressible Turbulent Flows with Shocks: Application to Two-Dimensional Flows around Cylinders
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  See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/260550589 A Numerical Scheme Based on an ImmersedBoundary Method for Compressible TurbulentFlows with Shocks: Application to Two-Dimensional Flows around Cylinders  ARTICLE   in  JOURNAL OF APPLIED MATHEMATICS · MARCH 2014 Impact Factor: 0.72 · DOI: 10.1155/2014/252478 CITATIONS 4 READS 83 3 AUTHORS: Shun TakahashiTokai University 23   PUBLICATIONS   59   CITATIONS   SEE PROFILE Taku NonomuraJapan Aerospace Exploration Agency 116   PUBLICATIONS   367   CITATIONS   SEE PROFILE Kota FukudaTokai University 15   PUBLICATIONS   23   CITATIONS   SEE PROFILE Available from: Taku NonomuraRetrieved on: 14 January 2016  Research Article  A Numerical Scheme Based on an Immersed Boundary Method for Compressible Turbulent Flows with Shocks: Application to Two-Dimensional Flows around Cylinders Shun Takahashi, 1 Taku Nonomura, 2 and Kota Fukuda  3 󰀱 Department of Prime Mover Engineering, Tokai University, Hiratsuka, Kanagawa 󰀲󰀵󰀹-󰀱󰀲󰀹󰀲, Japan 󰀲 Department of Space Flight Systems, Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency,Sagamihara, Kanagawa 󰀲󰀵󰀲-󰀵󰀲󰀱󰀰, Japan 󰀳 Department of Aeronautics and Astronautics, Tokai University, Hiratsuka, Kanagawa 󰀲󰀵󰀹-󰀱󰀲󰀹󰀲, Japan Correspondence should be addressed to Shun Takahashi; takahasi@tokai-u.jpReceived 󰀱󰀲 October 󰀲󰀰󰀱󰀳; Accepted 󰀳󰀱 December 󰀲󰀰󰀱󰀳; Published 󰀶 March 󰀲󰀰󰀱Academic Editor: Lu´ıs GodinhoCopyright © 󰀲󰀰󰀱 Shun Takahashi et al.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal work is properly cited.Acomputationalcodeadoptingimmersedboundarymethodsforcompressiblegas-particlemultiphaseturbulentflowsisdevelopedand validated through two-dimensional numerical experiments. e turbulent flow region is modeled by a second-order pseudoskew-symmetricformwithminimumdissipation,whilethemonotoneupstream-centeredschemeforconservationlaws(MUSCL)schemeisemployedintheshockregion.epresentschemeisappliedtotheflowaroundatwo-dimensionalcylinderundervariousfreestream Mach numbers. Compared with the srcinal MUSCL scheme, the minimum dissipation enabled by the pseudo skew-symmetricformsignificantlyimprovestheresolutionofthevortexgeneratedinthewakewhileretainingtheshockcapturingability.In addition, the resulting aerodynamic force is significantly improved. Also, the present scheme is successfully applied to movingtwo-cylinder problems. 1. Introduction eacousticwavesfromrocketplumesaresufficientlystrongto damage the satellites inside the fairing of a rocket. esewaves are widely assessed by empirical prediction methods[󰀱], but the low accuracy of these methods renders themunsuitablefornewrocketlaunchsites.Toimprovethepredic-tionaccuracyofacousticwavesfromrocketplumes,asophis-ticated model based on the underlying physics is required.Numerical simulations are an essential component of new model development [󰀲–󰀵]. e behavior of acoustic waves from rocket plumes is difficult to predict, because actualplumesareveryhot,withveryhighspeed,andofthemultiplephase conditions. In real rocket systems, acoustic waves aresuppressed by a water injection system installed at the rocketlaunch site. Fukuda et al. [󰀶] showed that rarefaction orabsorptionofacousticwavesbyparticlesexertsnosignificanteffect and that acoustic waves might be primarily attenuatedby interactions between particles and turbulence. However,this scenario is not well modeled in their study. To moreaccurately evaluate acoustic wave suppression by particle-turbulence interactions, further fundamental analyses arenecessary.erefore,weconsideramultiscaleanalysisofgas-particle multiphase high-speed compressible flows. Becausethe target is a rocket plume, we propose a simultaneoustreatment of the turbulence and the shock waves. Figure 󰀱shows an overview of the proposed numerical approach.e simultaneous high-resolution simulation of the particlesand turbulence is conducted on the microscale, the largeeddy simulation (LES) modeling only the particle behavioris conducted on the intermediate scale, and the complex flow fields are modeled by the Reynolds-averaged Navier-Stokes(RANS) simulation.Speeds of these scattering particles cover a wide range of Mach numbers from subsonic to supersonic, while Reynoldsnumbersarequitelowsincethesizesoftheparticlesaresmall. Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 252478, 21 pageshttp://dx.doi.org/10.1155/2014/252478  󰀲 Journal of Applied Mathematics DNS of gas-particle simulationImmersed boundary-based Euler-EulerLES of gas-particle simulationEuler-LagrangeEuler-EulerRANS of gas-particle simulationEuler-LagrangeEuler-EulerAccurate, expensive(leading to analysis for the small region)Modeled, cheap(leading to analysis for the wider region)Target of this study: F􀁩󰁧󰁵󰁲󰁥 󰀱: Overview of the proposed gas-particle simulation. erefore, flowfields around the small particles can be solvedby flow simulation without any turbulence models thoughthe flowfield is macroscopically turbulent. ere are severalkinds of numerical methods to solve the problem treating anumber of moving objects, such as dynamic mesh method[󰀷] or overset method [󰀸]. However, simple implementation and rapid computation are difficult to achieve in usingthese methods because additional numerical processes areincluded in the flow simulation like mesh regeneration orinterpolation between computational grids. We select levelset method [󰀹] to track a number of moving particles in thisstudy. ere are some other options to trace many movingobjects with high accuracy like phase field method or fronttrackingmethod.However,ourmainfocusisinvestigationof the acoustic wave characteristics under interference betweenturbulence and particles. erefore, level set method isselected based on computational efficiency. e representedboundaries by level set method are imposed by immersedboundary method [󰀱󰀰] in equally spaced Cartesian meshin this study. Immersed boundary method (IBM) is widely used in the community of Cartesian mesh method fromthe simplicity and applicability. e methodology of IBMcan be classified into several categories such as continuousforcing method [󰀱󰀱], direct forcing method, and ghost-cellmethod [󰀱󰀰]. Although IBM was srcinally proposed andused for incompressible flow simulations, it is also appliedto compressible flow simulations [󰀱󰀲–󰀱󰀵], recently. Takahashi et al. have been developing several Cartesian flow solvers[󰀱󰀵–󰀲󰀱] and investigating the performance of this kind of  numerical method. Based on the background, we adoptedghost-cell method [󰀱󰀰, 󰀱󰀵] with equally spaced Cartesian meshbythepointsofsimpleimplementation,robustness,andextensibility.Here, we should recall our flow features that consist of both turbulence and shocks. In general, an upwind schemeis o󰀀en employed to evaluate inviscid fluxes in IBM flow solver to stabilize a flow with numerical dissipation. edissipation is not preferable to solve our flows of turbulencepartclearly,whileitshouldbeaddedappropriatelytocapturethe shocks in a part of our flows. In other words, we shouldminimize numerical dissipation in the turbulent region,while the dissipation must be added to prevent spuriousoscillations around shocks. In resolving both the turbulenceand shock waves, dissipation switching scheme can play amajorrole.Inthisstudy,aswitchingprocedurefornumericaldissipation is introduced and examined through the two-dimensional test cases. is study overviews the computa-tional code and demonstrates its efficacy in simulations of two-dimensional static cylinder flows under various Machnumber conditions. e present numerical method is devel-oped to three-dimensional problem of interference amongturbulence, shocks, and particles with high-performanceparallel computing based on the previous studies [󰀱󰀶, 󰀱󰀷, 󰀲󰀰, 󰀲󰀱]. 2. Computational Methods 󰀲.󰀱. Governing Equations and Numerical Method.  In thepresent study, flows are governed by two-dimensional com-pressible Navier-Stokes equations. No averaging or filteringprocess is involved and the flows are solved without any turbulence models:  +  +  =  V   +  V   = 󐁛󰁛󰁛􀁛 V  󐁝󰁝󰁝􀁝,  = 󐁛󰁛󰁛􀁛 2 + V  (+)󐁝󰁝󰁝􀁝,  =󐁛󰁛󰁛􀁛 V   V   V  2 +(+) V  󐁝󰁝󰁝􀁝,  Journal of Applied Mathematics 󰀳  V   = 󐁛󰁛󰁛􀁛0      +  V   +  󐁝󰁝󰁝􀁝,  V   =󐁛󰁛󰁛􀁛0      +  V   +  󐁝󰁝󰁝􀁝, (󰀱)where    and    are the inviscid fluxes in the   - and   -directions, respectively,   V   and   V   are the corresponding viscous fluxes, and    contains the conservative variables.Here the stress tensor components are given as    = 23󰀨2   − V   󰀩,    =    = 󰀨   + V   󰀩,   = 23󰀨2 V    −  󰀩.  (󰀲)e pressure    is related to the total energy     per unit massby the equation of state:  = −1 + 12󰀨 2 + V  2 󰀩.  (󰀳)eheatfluxtermintheequationoftotalenergyiscomputedby     =  Pr −1􀀨􀀩  ,    =  Pr −1􀀨􀀩  ,  ()where the equation is transformed by the ideal gas equationand Prandtl number as follows:    = +  − 12 󰀨 2 + V  2 󰀩,  Pr  =    .  (󰀵)All variables are nondimensionalized by the freestreamconditions of density, sound speed, and unit length. eabove equations are discretized on an equally spaced Carte-sian mesh with a cell-centered arrangement. To eliminateadditional numerical dissipation everywhere, except in the vicinities of shock waves and potential flows, the inviscidterms are computed by a hybrid scheme that combines thepseudo skew-symmetric central difference scheme [󰀲󰀲] withthe monotone upstream-centered scheme for conservationlaws(MUSCL)-Roescheme[󰀲󰀳,󰀲].efluxintheturbulent regionismodeledbyapseudoskew-symmetriccentraldiffer-ence scheme with a minimum dissipation term as follows:  +1/2, turbelent  =  +1/2, cent  + +1/2, cent , +1/2, cent  =󐁛󰁛󰁛󰁛󰁛󰁛󰁛󰁛󰁛􀁛14 (   + +1 )(   + +1 )18 (   + +1 )(   + +1 ) 2 + 12 (   + +1 )18 (   + +1 )(   + +1 )( V    + V  +1 )14 (     + +1  +1  +   + +1 )(   + +1 )󐁝󰁝󰁝󰁝󰁝󰁝󰁝󰁝󰁝􀁝, +1/2, cent  = 14(   + +1 )(   + +1 )󐁛󰁛󰁛􀁛0 +1/2,  − +1/2, V  +1/2,  − V  +1/2,  +1/2,  − +1/2, 󐁝󰁝󰁝􀁝. (󰀶)Herethesubscript  denotesthequantityonthe  thgridpointand subscripts    and    denote the le󰀀- and right-side states,respectively, interpolated by the third-order MUSCL scheme[󰀲] with van Albada’s limiter [󰀲󰀵]. Ontheotherhand,thefluxintheshockandpotentialflow region, computed by the second-order MUSCL-Roe scheme,is written as follows:  +1/2, shock   = 12 (   + +1  + +1/2, Roe ( +1/2,  − +1/2, )), (󰀷)where    is the flux Jacobian and the subscript Roe denotesthe Roe-averaged quantity of the le󰀀 and right states. Here || = |Λ|,  (󰀸)where    and    are the right and le󰀀 eigenmatrices of    ,respectively, and  Λ =   is a diagonal matrix.e symmetric central difference part of (󰀸) can bereplacedbythatofthepseudoskew-symmetricformwithoutlosing the formal order of accuracy of the equation. eproposed scheme adopts the following new form of (󰀷):  +1/2, shock  = +1/2, cent  + 12 ( +1/2, Roe ( +1/2,  − +1/2, )). (󰀹)Combiningthiswiththedigitalswitchingfunction,weobtainthe following hybrid scheme:  +1/2  = (1− +1/2 ) +1/2, turbulent  + +1/2  +1/2, shock  =  +1/2, cent  +(1− +1/2 ) +1/2 + +1/2 12 ( +1/2, Roe ( +1/2,  − +1/2, )). (󰀱󰀰)Excessive dissipation is added to the shock or potential flow region when beta is unity, whereas dissipation is minimizedwhen beta is zero.   +1/2  is defined in terms of the Ducros-type sensor [󰀲󰀶] as follows:  +1/2  =  min (1,   + +1 ), =󰁻󰁻󰁻󰁻􀁻󰁻󰁻󰁻󰁻󐁻0 |∇⋅| 2 |∇⋅| 2 +|∇×| 2 + < 1 |∇⋅| 2 |∇⋅| 2 +|∇×| 2 + ≥ . (󰀱󰀱)Here   = 10 −15 is a small value that prevents division by zero and   = 0.6  is the switching threshold. e divergenceand rotation in (󰀱󰀱) are computed by a second-order finitedifference scheme. In the present study, the Ducros-typesensor[󰀲󰀶]aloneisusedinboththeshockandpotentialflow regions,althoughpreviousstudieshavecombinedthissensorwith the Jameson sensor [󰀲󰀷] in the shock region [󰀲󰀶, 󰀲󰀸]. Furthermore, in one of our proposed schemes,   +1/2  is setto unity in cells close to the body. Finally, the flux derivativeis approximated as follows:  =  +1/2  − −1/2 Δ .  (󰀱󰀲)   Journal of Applied Mathematics Ghost cellObject cellFluid cellBoundary  ΔxΔxd OC d FC d GC yx F􀁩󰁧󰁵󰁲󰁥󰀲:Cellconstructionandclassificationinthepresentlevelsetmethod. e derivative /  is obtained similarly.e diffusive terms are treated by a second-order, cen-tral difference scheme using the mid-point flux. e timemarching is conducted by the three-stage total variationdiminishing Runge-Kutta scheme [󰀲󰀹]. In this study, timeincrement is determined as follows: Δ =  Δ ∞  + ∞ + V  ∞ + obj + V  obj .  (󰀱󰀳) 󰀲.󰀲. Boundary Representation.  e boundary is defined by the level set method [󰀹, 󰀱]. e level set function is deter- mined in whole cells as a signed distance from the objectboundary. A schematic of the cells around a boundary isshown in Figure 󰀲. e level set method effectively computesthe normal vector from the object surface on the basis of a gradient operation. In the present study, flows aroundmultiple moving objects are solved by extending the levelset method to multiple level set functions based on simpleminimum distance approach [󰀸].On the basis of the level set function (󰀱), all cells areclassifiedintothreecategories:fluidcell,ghostcell,andobjectcell,asshowninFigure 󰀲.eghostcellsbehaveasguardcellsbetween the fluid and object regions and are assigned in twolayers under the present definition as follows:  FC  > 0,  GC  ≤ 0,  GC  ≥ −2√ 2Δ, OC  < −2√ 2Δ.  (󰀱)Ghost cells are used for imposing boundary conditionin the present method [󰀱󰀰, 󰀱󰀵]. An image point set in the region of fluid cells is used to collect flow information fora ghost cell. A primary advantage of the present ghost-cell Image point xy Ghost cellObject cellFluid cell d GC d IP   Boundary  ΔxΔxV IP V IB V GC F􀁩󰁧󰁵󰁲󰁥 󰀳: Schematic of the present ghost cell approach with imagepoint. Image point’ xy Ghost cellObject cellFluid cell ΔxΔSΔx Boundary  d IP’ F􀁩󰁧󰁵󰁲󰁥 : Image point projected from a fluid cell to compute thefluid force on the surface. method adopting the image point approach is its robustness.A schematic of the present immersed boundary method isshown in Figure 󰀳. e image point is located at the edgeof a probe that extends from a ghost cell through the objectboundary in the direction normal to the surface. e lengthof the probe, denoted as   IP  in Figure 󰀳, is an importantparameter that eliminates recursive interpolation. Here wefix the length of    IP  as 󰀱.󰀷󰀵 times the mesh size, considering
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