A Level Set Immersed Boundary Method for Water Entry and Exit - Zhang

of 24
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
  Commun. Comput. Phys.doi: 10.4208/cicp.060709.060110aVol. 8 , No. 2, pp. 265-288August 2010 A Level Set Immersed Boundary Method for WaterEntry andExit YaliZhang 1, ∗ , QingpingZou 1 ,DeborahGreaves 1 ,DominicReeve 1 ,Alison Hunt-Raby 1 , David Graham 2 , Phil James 2 and Xin Lv 1 1 School of Marine Science and Engineering, University of Plymouth, UnitedKingdom, PL4 8AA. 2 School of Mathematics and Statistics, University of Plymouth, United Kingdom,PL4 8AA. Received 6 July 2009; Accepted (in revised version) 6 January 2010Communicated by Boo Cheong KhooAvailable online 12 March 2010 Abstract.  The interaction between free surface flow and structure is investigated us-ing a new level set immersed boundary method. The incorporation of an improvedimmersed boundary method with a free surface capture scheme implemented in aNavier-Stokes solver allows the interaction between fluid flow with free surface andmoving body/bodies of almost arbitrary shape to be modelled. A new algorithm isproposedto locate exactforcing points nearsolid boundaries, which providesanaccu-ratenumericalsolution. ThediscretizedlinearsystemofthePoisson pressureequationis solved using the Generalized Minimum Residual (GMRES) method with incom-plete LU preconditioning. Uniform flow past a cylinder at Reynolds number Re=100is modelled using the present model and results agree well with the experiment andnumerical data in the literature. Water exit and entry of a cylinder at the prescribedvelocity is also investigated. The predicted slamming coefficient is in good agreementwith experimental data and previous numerical simulations using a ComFlow model.The verticalslamming forceand pressuredistribution for the freefalling wedge is alsostudiedbythepresentmodelandcomparisonswithavailabletheoreticalsolutions andexperimental data are made. AMSsubjectclassifications : 6D05, 76T10, 65B99,65E05 Key words : Level set method, immersed boundary method, slamming coefficient, water entryand exit, free surface, fluid-structure interaction. ∗ Corresponding author.  Email addresses:  (Y. Zhang),  (Q. Zou),  (D. Greaves),  (D.Reeve),  (A.Hunt-Raby),  (D.Graham),  (P. James),  (X. Lv) 265 c  2010 Global-Science Press  266 Y. Zhang et al. / Commun. Comput. Phys.,  8  (2010), pp. 265-288 1 Introduction Investigation of fluid structure interaction at the free surface is a classical hydrodynamicproblem and has a wide range of applications particularly in the fields of naval architec-ture, civil and ocean engineering and physical oceanography. A flow singularity occurswhen a body impacts the free surface, which gives rise to a high pressure peak localizedatthesprayrootandmakeswaterentryandexitproblems difficult. Ingridbasednumer-ical methods, there are two main strategies to handle a moving or deforming boundaryproblem with topological change, namely body conforming moving grids (Baum et al.1996; Yan et al. 2007) and embedded fixed grids (Yang et al. 1997; Ye et al. 1999; Tuckeret al. 2000; Fadlun et al. 2000; Tseng et al. 2003; Balaras et al. 2004; Yang et al. 2006; Lvet al., 2006). For the former method, the grid can be efficiently deformed in an arbitraryLagrangean-Eulerian (ALE) frame of reference to minimize distortion if the geometricvariation is quite modest. Boundaryconditions can be applied at the exact location oftherigid boundary. However, if the change of topology is complex, it will be very difficultand time consuming to regenerate the mesh. Also difficulties arise in the form of gridskewness and additional numerical dissipation may be a consequence of the redistribu-tion of the field variables in the vicinity of the boundary.An alternative to body conforming moving grids is embedded fixed grids where thegoverning equations are usually discretized on fixed Cartesian grids. The method canalso be divided into two major classes based on the specific treatment of the boundarycells; (1)Cartesiancutcellmethods(Yangetal. 1997; Yeetal. 1999; Tuckeretal. 2000) and(2) Immersedboundarymethods(Fadlunetal. 2000; Tsengetal. 2003; Balaras et al. 2004;Yang et al. 2006; Lv et al., 2006). Although the Cartesian cut cell method was srcinallydeveloped for potential flow, it has been applied and extended to the Euler equations,shallow water equations, Navier-Stokes equations to simulate low speed incompressibleflows and flows with moving interfaces. It has the potential to significantly simplify andautomate the difficulty of mesh generation. There are also a number of disadvantagesinherent in the use of this method. It cuts the solid body out of a background Cartesianmesh, which can generate sharp corners and a variety of different cut cell types. Thus,extending this method to three-dimensions is not a trivial task. In addition, arbitrarilysmall cells arising near solid boundaries due to the Cartesian mesh intersecting a solid body can restrict the stability of the Cartesian solvers.Intheimmersedboundarymethodthemomentumforcing whichis introducedtoen-forcetheboundaryconditionofthebodyinthefluidcanbeprescribedonafixedmeshsothat the accuracy and efficiency of the solution procedure on simple grids is maintained.Bodies of almost arbitrary shape can be dealt with and flows with multiple bodies orislands can be computed at reasonable computational cost (Fadlun et al. 2000). The im-mersed boundary method has the advantage of simplified grid generation and inherentsimplicity which allows the study of moving bodies (Mittal et al. 2005) on fixed Carte-sian grids. Furthermore, the appropriate treatment in the immersed boundary methodleads to a convenient method for computing forces acting on a body, namely lift and  Y. Zhang et al. / Commun. Comput. Phys.,  8  (2010), pp. 265-288 267 drag forces. These advantages suggest that it is well suited to study problems involvinga moving body with the free surface flow.The work here builds on earlier work by Zhang et al. (2009) in which a level setmethod with global mass correction was developed for two fluid flows and applied tosimulate water column collapse and free surface waves over a submerged structure. Amajor contribution of the present work is the incorporation of an improved immersed boundary method with the free surface model. This makes it straightforward to under-take a variety of fluid structure interaction problems. A new algorithm is proposedto lo-cateexactforcingpointsnearthesolidboundaries,whichprovidesanaccuratenumericalsolution. To accelerate the convergence of the solution of the Poisson pressure equationthe Generalized Minimum Residual (GMRES) method with incomplete LU factorizationfor preconditioningis applied. This paperis organized as follows. First, governingequa-tions are discussed in Section 2. Then numerical methodologies for the Navier-Stokes,free surface and immersed boundary method are described in Section 3. The problemsarising from the classification of the grid points, direct forcing for the forcing points, hy-drodynamic forces on the body and interaction between the fluid and structure are alsodiscussed in this section in detail. Results are presented in Section 4; initially uniformflow over a circular cylinder at Re = 100 is simulated to demonstrate the accuracy of thepresent level set immersed boundary method. Next, it is used to calculate water exit andentry of a cylinder. Snapshots of the simulations have been compared with photographsof experiments by Greenhow et al. (1983). The slamming coefficients of water entry of acylinder are compared to the theory, experiment and ComFlow (Kleefsman et al. 2005).Results of a free falling wedge where a full coupling between the fluid and body are pre-sented and compared with experimental results and previous numerical simulations inthe literature. Conclusions and future work are given in Section 5. 2 Governing equations 2.1 Navier-Stokes equations The governing equations for an incompressible fluid flow are the mass conservationequation and the Navier-Stokes momentum conservation equations written as ∂ u i ∂ x i = 0, (2.1)and ∂ u i ∂ t  + ∂  u i u  j  ∂ x  j = − 1  ρ∂  p ∂ x i + 1  ρ∂τ  ij ∂ x  j +  f  i , (2.2)where Cartesian tensor notation is used, ( i = 1,2),  u  j ,  p  and  x  j  are the velocities, pressureand spatial coordinates respectively,  f  i  represents momentum forcing components.  τ  ij  is  268 Y. Zhang et al. / Commun. Comput. Phys.,  8  (2010), pp. 265-288 the viscous term given by τ  ij  = µ  ∂ u i ∂ x  j + ∂ u  j ∂ x i  , (2.3)where  ρ  and  τ   are the density and viscosity respectively appropriate for the phase that isoccupying the particular spatial location at a given instant. 2.2 Free surfaceequations Due to the existence of steep gradients in density and viscosity across the free surface,excessive numerical diffusion is experienced when computing viscous flows (Ferziger,2002; Wang, et al. 2009). Here, the level set method is used to capture the interface between the two phases. The evolution of the level-set function,  φ , is governed by ∂φ∂ t  + u  j ∂φ∂ x  j = 0. (2.4)The solutionof the Navier-Stokesequations will yield unwantedinstabilities at the inter-face if density and viscosity are discontinuous there. To overcome this, a region of finitethickness over which a smooth but rapid change of density and viscosity occurs acrossthe interface is introduced. This is achieved by defining a smoothed Heaviside function.  H  ( φ )=  0,  φ < − ε , φ + ε 2 ε  +  12 π  sin  πφε  ,  − ε  φ  ε ,1,  φ > ε ,(2.5)where  ε  is related to the grid size. Using the smoothed Heaviside function, these proper-ties are calculated using  β =( 1 −  H  )  β 1 +  H   β 2 , (2.6)where  β  can be density, viscosity or another property of interest. Since  φ  is the signednormal distance from the interface, it satisfies ∇| φ | = 0. (2.7)When Eq. (2.4) moves the level set  φ = 0 at the correct velocity,  φ  may become irregularafter some period of time (Sussman et al. 1994) and its properties as a distance functionmay be lost. Thus, to ensure that  φ  remains a distance function that satisfies Eq. (2.4), re-distancing mustbe performed. Thisis achieved by solving fora seconddistance function φ ′  given by Eq. (2.8): ∂φ ′ ∂ ¯ t  + s ( φ )  ∇ φ ′  − 1  = 0. (2.8)

Kenneth Bags

Jul 24, 2017

Broch Engines

Jul 24, 2017
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks