The LS-STAG Method : A new Immersed Boundary / Level-Set Method for the Computation of Incompressible Viscous Flows in Complex Geometries

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1 The LS-STAG Method : A new Immersed Bondary / Level-Set Method for the Comptation of Incompressible Viscos Flows in Complex Geometries Yoann Cheny & Olivier Botella Nancy Universités LEMTA - UMR 7563 (CNRS-INPL-UHP) 2, avene de la Forêt de Haye, B.P. 160, Vandœvre-lès-Nancy (France) Yoann.Cheny@ensem.inpl-nancy.fr ; Olivier.Botella@ensem.inpl-nancy.fr Abstract : This paper concerns the development of a new Cartesian grid / immersed bondary (IB) method for the comptation of incompressible viscos flows in irreglar geometries. In IB methods, the irreglar bondary is not aligned with the comptational grid, and of pmost importance for accracy and stability is the discretization in cells which are ct by the bondary, the so-called ct-cells. In this paper, we present a progress report on a new finite volme discretization where the irreglar bondary is implicitly represented by its signed distance fnction (the level-set fnction). With the help of level-set calcls tools, we are able to preserve the Cartesian strctre of the discretized Navier-Stokes eqations in the ct-cells. The reslting discrete systems are efficiently solved with a black box mltigrid solver for strctred grids. The method is validated for the circlar cylinder flow at low Reynolds nmber. Key-words : Incompressible flows ; Comptational flid dynamics ; Immersed bondary methods 1 Introdction Mch attention has recently been devoted to the extension of Cartesian grid flow solvers to complex geometries by immersed bondary (IB) methods (see [7] for a recent review). In these methods, the irreglar bondary is not aligned with the comptational grid, and the treatment of the cells which are ct by the bondary remains an important isse. Indeed, the discretization in these ct-cells shold be designed sch that : (a) the overall accracy of the method is not severely diminished and (b) the high comptational efficiency of the strctred solver is preserved. Two major classes of IB methods can be distingished on the basis of their treatment of ct-cells. Classical IB methods se a finite volme/difference strctred solver in Cartesian cells away from the irreglar bondary, and discard the discretization of flow eqations in the ct-cells [7]. Instead, special interpolations are sed for setting the vale of the dependent variables in the latter cells. Ths, strict conservation of qantities sch as mass and momentm is not observed near the irreglar bondary. Nmeros revisions of these interpolations are still proposed for improving the accracy and consistency of this class of IB methods [5]. A second class of IB methods (also called ct-cell methods, see e.g. [3]) aims for actally discretizing the flow eqations in ct-cells. However, the calclation of flxes in ct-cells relies sally on nstrctred techniqes, and their negative impact on the comptational efficiency of the code is difficlt to evalate. The prpose of this commnication is to present a new IB method for incompressible viscos flows which takes the best aspect of both approaches. As the ct-cell method of Ref. [3], or method is based on the symmetry preserving finite-volme discretization on Cartesian grids by Verstappen & Veldman [10]. However, the major difference is that we have ndertaken 1

2 representing the irreglar bondary by its level-set fnction [8]. With the help of level-set calcls tools, the flxes in Cartesian and ct-cells are discretized in a nified fashion, ensring that the Cartesian strctre of the stencils is preserved. Or method has the following distinctive featres : In contrast to standard IB methods, flow variables are actally compted near the irreglar bondary, and not interpolated [5]. This featre is of pmost importance for compting the bondary layer of viscos flows. It is even more important for the viscoelastic flow comptations we intend to perform in the ftre, for which the basic solver we present in this paper is the bilding block. The Cartesian strctre of the discrete systems enables the se of efficient black box solvers for strctred grids, reslting in a highly comptationally efficient method. In the final part of the paper, or method is validated for the circlar cylinder flow in the laminar regime, which has become the standard benchmark test for IB methods. 2 Nmerical method Let Ω be a rectanglar comptational domain and its srface. The governing eqations are the incompressible Navier-Stokes eqations in integral form. In the following, we will consider the finite-volme discretization of the continity eqation : v n ds = 0, (1) where v = (, v) is the velocity, and of the -momentm eqation : d dv + F c + p e x n ds 1 dt Ω Re F d = 0, (2) where p is the pressre, Re is the Reynolds nmber, F c (v n) ds and F d n ds are the convective and diffsive flx respectively. Basic discretization on the MAC mesh The Cartesian method on which or IB method is based is the second-order finite volme discretization of Verstappen & Veldman [10], which has the ability to preserve on non-niform cell distribtions the conservation properties (for total mass, momentm and kinetic energy) of the original MAC method [4]. The staggered arrangement of the discrete velocities on the MAC mesh is illstrated in Fig. 1. The comptational domain is divided in non-niform Cartesian cells Ω i,j = ] [ ] [ x i 1, x i yj 1, y j, of size Vij = x i y j, and center x c ij = (x c i, yj). c The srface of cell Ω i,j is divided in 4 elementary faces as i,j = e w n s sing the sal compass notations. Cell Ω i,j is sed as the control volme for discretizing the continity eqation (1), while the staggered control volme Ω i,j = ] [ ] [ x c i, x c i+1 yj 1, y j, of size Vij = x i y j with x i = 1 x 2 i + 1 x 2 i+1, is sed for the -momentm eqation (2). Compass notations are sed for intermediate steps of the discretization of (2). For example, we denote Fe c the convective flx throgh the east face e of the staggered control volme, i.e. : F c e yj y j 1 (v n e ) (x c i+1, y) dy. (3) 2

3 Figre 1: Location of the variables near the cell Ω i,j on the MAC mesh. The LS-MAC mesh for discretization in irreglar domains We consider now an irreglar flid domain Ω f which is embedded in the comptational domain Ω. To keep track of the irreglar bondary f, we employ a signed distance fnction φ(x) (i.e., the level set fnction [8]) sch that φ is negative in the flid region Ω f, φ is positive in the solid region Ω s = Ω \ Ω f, and sch that the bondary f corresponds to the zero level-set of this fnction, i.e. :, x Ω f, φ(x) 0, x f, (4) +, x Ω s, where is the distance from x to the closest point on the irreglar bondary. This leads to the modification of the MAC mesh that is described in Fig. 2, and that will be sbseqently referred to as the LS-MAC mesh. In this figre, the cell Ω i,j which is part flid / part solid, is commonly called a ct cell. The discretization of the flxes in ct cells is performed with the help of an additional variable, denoted φ i,j, that stores the vales of the level-set fnction φ at the pper right corner of the cells. The level-set vales are sed to efficiently compte qantities relevant to the irreglar bondary and the ct-cells, sch as otward normal vectors or srface integrals over f, areas and volmes of the ct-cells, etc... In this respect, the most important qantity for the LS-MAC discretization is the flid portion of the cell faces. For example in Fig. 2, by sing one-dimensional linear interpolation of φ(x i, y) in [y j 1, y j ], the length of the face e which belongs to the flid domain is : y n y j 1 = θ i,j y j, with θ i,j = φ i,j 1 φ i,j 1 φ i,j since φ(x i, y n ) = 0. The qantity θi,j [ 0, 1 ], which is called the cell-face ratio, shall be sed to : Detect whether the discrete velocities in the LS-MAC mesh are in the flid region. For example in Fig. 2, velocity i,j has to be compted since θi,j > 0, bt v i,j shold not since θi,j v = 0. Discretize the srface and volme integrals of the Navier-Stokes eqations (1)-(2) in the ct-cells. For example, the volme of the trapezoidal ct-cell in Fig. 2 is : V i,j = 1 ( ) θ 2 i 1,j + θi,j xi y j. (5) 3

Figre 2: Location of the variables near the ct cell Ω i,j on the LS-MAC Mesh. The intersection of the east face with the irreglar bondary is (x i, y n ). See Fig. 1 for complementary notations.

2 is compted as : Fn d y dx = x (x i, y n ) i,j i 1, (6) n θ 2 i,j y j with (x i, y n ) given by the Dirichlet conditions at the irreglar bondary.

4 Figre 2: Location of the variables near the ct cell Ω i,j on the LS-MAC Mesh. The intersection of the east face with the irreglar bondary is (x i, y n ). See Fig. 1 for complementary notations. Implement bondary conditions in the fashion of the ghost flid method [8]. For example, the diffsive flx throgh the solid north bondary n in Fig. 2 is compted as : Fn d y dx = x (x i, y n ) i,j i 1, (6) n θ 2 i,j y j with (x i, y n ) given by the Dirichlet conditions at the irreglar bondary. These points will be discssed in greater lengths in a forthcoming paper (see also [1]). In the following, we will mainly detail the discretization of the continity eqation (1). Discretization of the continity eqation As in the Cartesian method of Verstappen & Veldman [10], the starting point of the LS-MAC discretization concerns the continity eqation (1), that reads in cell Ω i,j : i,j i 1,j + v i,j v i,j 1 = 0, (7) where mass flxes across cell faces are denoted with a bar ; for example the mass flx throgh the east face e in Fig. 2 is : i,j yn y j 1 (x i, y) dy. (8) In order to easily discretize this integral, we first locate the discrete nknown i,j in the middle of the flid part of the face as i,j (x i, y j θ 2 i,j y j ). Then, by sing midpoint qadratre, we finally discretize (8) as : i,j = θ i,j y j i,j. (9) After similar calclations for the other cell faces, the discretization of (7) is : y j ( θ i,j i,j θ i 1,j i 1,j ) + xi ( θ v i,j v i,j θ v i,j 1 v i,j 1 ) = 0, (10) and, in the case of a reglar Cartesian cell (θ = 0 or 1 only), the discrete continity eqation redces to the one of the standard MAC discretization. 4

Discretization of the convective flx follows the lines of the symmetry preserving discretization of [10]. For example, Eq.

5 Frther comptational details Discretization of the momentm eqation (2) is performed in staggered control volmes as shown in Fig. 2. Volme integrals are discretized as : d dv dt = V d i,j i,j Ω dt, (11) and the volme of the ct-cells is efficiently compted with the cell face ratios (see Eq. (5)). Discretization of the convective flx follows the lines of the symmetry preserving discretization of [10]. For example, Eq. (3) is discretized as Fe c ) = 2( 1 i,j + i+1,j e, and central interpolation with constant weights for e is sed. The discretization of the diffsive flxes is more involved and is not detailed in this paper. The pressre gradient in Eq. (2) is simply compted as : 1 V i,j [ e p dx w ] p dx p i+1,j p i,j = x p, (12) i with x p i = 1 2 x i x i+1 for the ct-cell of Fig. 2. The time stepping method we se is the semi-implicit AB / BDF 2 projection scheme (see e.g. [2]). With or LS-MAC discretization, the pressre Poisson eqation of the projection step is a symmetric linear system, as in the Cartesian case. This linear system is efficiently solved with a black-box mltigrid solver for Cartesian grids. 3 Nmerical reslts Figre 3: At left : geometry and bondary conditions of the flow over a circlar cylinder. At right : meshes from or simlations. One mesh line ot of three is shown in both direction. We have validated or method on both steady and nsteady flows arond a circlar cylinder in a free-stream (see Fig. 3 (left)). The Reynolds nmber is based on the constant inlet velocity and the cylinder diameter D. For all comptations, the inlet bondary is located at X = 8 length nits pstream of the obstacle ; the vale of the time step is t = We se non-niform Cartesian meshes refined in the wake of the cylinder as shown in Fig. 3 (right), with N x N y cells. In the vicinity of the cylinder, the cell distribtion is niform of size h = For a shape as simple as a cylinder profile with nit diameter, the level set fnction takes the simple analytic expression : φ(x, y) = 1 2 (x x C ) 2 + (y y C ) 2, 5

6 N x N y A X d L w min C D S S Experiments [11] Linnick & Fasel [6] Table 1: Smmary of the steady comptations at Re = 20 and comparison with reslts pblished in the literatre. L w : wake length, min : velocity extremm in the near wake. C D : drag coefficient. N x N y A X d St C D C Lmax S S S Experiments [11] Persillon & Braza [9] Linnick & Fasel [6] Table 2: Smmary of the nsteady comptations at Re = 100 and comparison with reslts pblished in the literatre. St : Strohal nmber, C D : time average of drag coefficient. C Lmax : peak vale of lift coefficient. All qantities have been compted from a time signal eqal to 280 nits, which gives a freqency resoltion eqal to ± where (x C, y C ) is the cylinder center. This expression is discretized at cell corners prior to the time integration. For both steady flow at Re = 20 and time-periodic flow at Re = 100, we have stdied the inflence of the locations of the otflow bondary X d and solid blockage A (see Tables 1 and 2). For Re = 100 in particlar, it can be observed that grid independent reslts are reached for A = 16 and X d = 15. These tables also compares or reslts with selected experimental and nmerical stdies. Or comptations show an overall good agreement with these reslts. References [1] O. Botella and Y. Cheny. On the treatment of complex geometries in a Cartesian grid flow solver with the level set method. in Eropean Conference on Comptational Flid Dynamics ECCOMAS CFD 06, P. Wesseling, E. Oñate & J. Périax eds. (CD-ROM & WEB) [2] O. Botella, M. Y. Forestier, R. Pasqetti, R. Peyret, and C. Sabbah. Chebyshev methods for the Navier- Stokes eqations : Algorithms and applications. Nonlinear Analysis, 47, , [3] M. Dröge and R. Verstappen. A new symmetry-preserving Cartesian-grid method for compting flow past arbitrarily shaped objects. Int. J. Nmer. Meth. Flids, 47, , [4] F. H. Harlow and J. E. Welch. Nmerical calclation of time-dependent viscos incompressible flow of flid with free srfaces. Phys. Flids, 8, , [5] S. Kang, G. Iaccarino, and P. Moin. Accrate and efficient immersed-bondary interpolations for viscos flows. In Center for Trblence Research Briefs, pp , [6] M. N. Linnick and H. F. Fasel. A high order immersed interface method for simlating nsteady incompressible flows on irreglar domains. J. Compt. Phys., 204, , [7] R. Mittal and G. Iaccarino. Immersed bondary methods. Ann. Rev. Flid Mech., 37, , [8] S. Osher and R. Fedkiw. Level Set Methods and Dynamic Implicit Srfaces. Springer, New-York, [9] H. Persillon and M. Braza. Physical analysis of the transition to trblence in the wake of a circlar cylinder by three dimensional Navier-Stokes simlation. J. Flid Mech., 365, 23-88, [10] R. W. C. P. Verstappen and A. E. P. Veldman. Symmetry-preserving discretization of trblent flow. J. Compt. Phys., 187, , [11] M. M. Zdravkovich. Flow Arond Circlar Cylinders. Volme 1 : Fndamentals. Oxford University Press, Oxford,

Investigation of cross flow over a circular cylinder at low Re using the Immersed Boundary Method (IBM)

Computational Methods and Experimental Measurements XVII 235 Investigation of cross flow over a circular cylinder at low Re using the Immersed Boundary Method (IBM) K. Rehman Department of Mechanical Engineering,