Part 22

Immersed Boundary Method

Recent advances in the development of animmersed boundary method for industrialapplications

M.D. de Tullio∗, P. De Palma∗, M. Napolitano∗ and G. Pascazio∗

∗ CEMeC & DIMeG, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy

E-mail: m.detullio@poliba.it, depalma@poliba.it, napolita@poliba.it, pascazio@poliba.it

Abstract This paper provides some recent developments of an immersed bound-ary method for solving flows of industrial interest at arbitrary Mach numbers. Themethod is based on the solution of the preconditioned compressible Favre-averagedNavier–Stokes equations closed by the k-ω low Reynolds number turbulence model.A flexible local grid refinement technique is implemented on parallel machines us-ing a domain-decomposition approach and an edge-based data structure. Thanks tothe efficient grid generation process, based on the ray-tracing technique, and the useof the METIS software, it is possible to obtain the partitioned grids to be assignedto each processor with a minimal effort by the user. This allows one to by-pass thevery time consuming generation process of a body-fitted grid.

1 Introduction

The immersed boundary (IB) method is emerging as a very appealing approach forsolving flows past very complex geometries, like those occurring in most industrialapplications. Its main, very significant, feature is the use of a Cartesian grid em-bodying the complex boundaries of the flow domain, which allows one to generatethe computational mesh within a few minutes, whereas a very complicated bodyfitted grid may require several hours or even days of manpower. The IB techniquewas originally developed for incompressible flows, see, e. g., [1] and the referencestherein, using non-uniform Cartesian grids to take advantage of simple numericalalgorithms. More recently, the authors have contributed to the extension of the IBmethod to the preconditioned compressible Navier–Stokes (NS) equations in orderto solve complex flows at any value of the Mach number [2], and equipped it with alocal mesh refinement procedure to resolve boundary layers and regions with highflow gradients (e.g., shocks) [3]. In this work, some recent improvements and exten-sions of the method of [3] are presented, together with some new interesting results.

1

2 M.D. de Tullio et al.

2 Numerical method

The basic numerical method uses a dual-time-stepping approach to solve unsteadyflows. At every physical time level, the unsteady residual is reduced to any desiredlevel by iterating in pseudo-time. When a steady solution is sought, the physicaltime derivative is removed and the steady residual is reduced by iterating in pseudo-time [3]. Here only the new features are briefly described.

Firstly, the use of an edge-based data structure allows one to refine the grid lo-cally in any desired direction; this is a great advantage with respect to the previousapproach [3], in which the grid had to be refined direction-by-direction throughoutthe entire flowfield, so that a successive coarsening step had to be applied far awayform the high-gradient regions. Moreover, an anisotropic local grid refinement isperformed, namely, each cell can be refined independently in each Cartesian di-rection. This feature complicates the grid topology but renders the approach moreflexible to handle complex geometries with a remarkable reduction of the memoryrequirement with respect to a standard OCTREE data structure.

In more details, starting from a grid with uniform mesh size, a locally refined gridis generated by recursively halving the mesh size at the immersed boundary region,until an assigned target value is reached [4]. This automatic refinement is based onthe following strategy. A tag function, generated using the ray tracing technique, isused to mark the cells inside and outside the immersed body: an integer value ±1is assigned to “fluid” and “solid” cells, respectively. The gradient of this functionis different from zero only at the immersed boundary and depends on the local gridsize. The components of this gradient in the x and y directions are used to select thecells to be refined. The grid is refined until a user specified resolution is achieved atthe boundary. In addition, one can refine other regions of the computational domainaway from the immersed boundary, choosing the local resolution of the mesh, suchas wake or bow-shock regions.

Secondly, the method has been equipped with Sutherland’s law and temperature-variable gas properties to handle hypersonic flows.

Thirdly, a heat conduction solver has been combined with the flow one so as tosolve conjugate-heat-transfer problems, see [6], for details.

Finally, the code is parallelized implementing the communication exchangeamong the processors based on the MPI protocol. Employing the software METIS [5],the mesh is divided into a number of blocks defined by the user, balancing thenumber of computational cells among them. Following a domain decompositionapproach, each block is assigned to a CPU which performs the integration of theNS equations in parallel, exchanging the needed information with its neighboringprocessors. All of the grid properties (coordinates, metrics, pointers for the com-munications among cells and among processors, etc.) are allocated according to theedge-based data structure and the data are provided in output files to be read byeach processor. All operations are performed automatically and the user needs onlyto establish the number of processors employed for the computation.

Recent advances in the development of an immersed boundary method 3

3 Results

All of the results presented in this paper aim at validating the new features of themethod, with particular attention to the domain decomposition and the code paral-lelization.

The transonic flow past the unmanned space vehicle FTB-1 tested by the ItalianCenter for Aerospace Research (CIRA) has been computed at first [7]. The flightconfiguration corresponds to the launch experiment in which the vehicle falls froman altitude of 20 km; 30.8 seconds after the launch, the Mach number is equal to0.94 and the incidence angle is 7.24 (zero elevon and rudder deflections have beenassumed). The computational grid containing about 3.5 million cells is locally re-fined in the leading edge region of the wings and around the nose of the vehicleas shown by figure 1 which provides the local views of the grid at the midplane(y = 0) and at the plane y = 1 m. The grid has been generated automatically and par-titioned into sixteen balanced blocks in about fifteen minutes on a single core of anIntel Xeon X5560 @ 2.80GHz processor The steady version of the code provided aresidual drop of two orders of magnitude within about six CPU hours. Obviously, theresidual cannot be reduced further, insofar as the flow has several unsteady featureswhich need to be time resolved. Nevertheless, the computed complex shock systemnear the wings and in the rear part of the fuselage has been captured well with re-spect to the experimental data provided by CIRA [7]. Figure 2 provides the speedcontours at same two longitudinal planes. The position and strength of the shocksare in good agreement with the data visualized in a wind tunnel at CIRA [7]. Thistest case demonstrates the flexibility of the proposed parallel IB numerical strategyand its applicability to very complex flow fields.

Fig. 1 USV flow: grid at y = 0 (left) and y = 1 m (right).

Then, the classical hypersonic flow over a 2D compression ramp [8] has beencomputed. The ramp geometry is employed to study the effects of flap deflectionon the flow past a space vehicle. This configuration shows the typical featuresof shock wave-boundary layer interaction with flow separation and re-attachment.Slip velocity conditions are computed using the first-order relation provided in [9].

4 M.D. de Tullio et al.

Fig. 2 USV flow: velocity magnitude contours at y = 0 (left) and y = 1 m (right).

The distance xc, between the leading edge of the flat plate and the ramp corner isequal to 71.4 mm and the ramp angle and length are 35-deg and 71.4 mm, respec-tively [8]. The flow of nitrogen gas is considered with the following free-streamconditions [8]: ρ∞ = 1.401×10−4 kg/m3, V∞ = 1521 m/s, T∞ = 9.06 K. The corre-sponding free-stream Mach number and Reynolds number are 24.8 and 12,020, re-spectively. The wall is perfectly diffusing with the wall temperature set at 403.2 K.Two-dimensional computations have been performed using a locally refined gridwith about 80000 cells clustered at the leading edge of the plate and in the recircu-lation region. The height of the first cell along the wall is 0.02 mm. About 20 wall-clock CPU minutes are needed to generate the grid and to drop the steady residualby two orders of magnitude using two 8-core Intel Xeon @2.80Ghz processors. Fig-ure 3 reports the streamlines (superposed to the density contours) and the pressurecoefficient distribution which is in good agreement with the Direct Monte-CarloSimulation of Moss [8].

Fig. 3 Hypersonic ramp flow: streamlines (left); pressure coefficient (right).

Finally, the code has been applied to the the simulation of a highly-loaded cooledtwo-dimensional turbine cascade. The geometry of the blade, known as the T106

Recent advances in the development of an immersed boundary method 5

turbine cascade [10], has been modified by adding three cooling channels. The flowis subsonic, with isentropic exit Mach number equal to 0.3, inlet flow angle equalto 37.7, and Reynolds number, based on the chord length and on the exit condi-tions, equal to 3× 105. Air and stainless steel are considered for the fluid and forthe solid, respectively. At the inlet boundary points, the total pressure and tempera-ture are assigned, together with the flow direction, whereas only the static pressureis prescribed at the outlet points. The two side holes have assigned wall tempera-ture, equal to Tc = 200 K, whereas, cooling air flows from the central hole into asecondary channel so as to form a cooling film on the suction side of the blade.The inlet flow condition for the secondary air are set at midspan, where total tem-perature and pressure conditions are imposed corresponding to an inlet temperatureTc = 200 K and an inlet velocity normal to the endwall, vc = 5 m/s.

Thanks to the versatility of the present IB approach, the complete geometry of theblade can be discretized easily and efficiently. The computational grid, using about66000 cells (33700 in the solid region), shown in figure 4(a), is refined at the leadingedge of the blade, at the region of maximum curvature, and near the cooling holes,see figure 4(b). Figures 5(a) and (b) provide the computed temperature countours inthe solid and in the fluid, and the velocity-vector field in and around the main centralcooling channel. This test case demonstrates the capability of the present method tosolve conjugate-heat-transfer problems of industrial interest.

Fig. 4 Cooled T106 cascade: locally refined grid (a); temperature contours (b).

4 Conclusions and future work

This work has presented some recent improvements and applications of an immersed-boundary method suitable for solving industrial problems from incompressible tohypersonic flow conditions. The method has been tested versus three applications ofaerospace and industrial interest. Current work aims at coupling the unsteady ver-

6 M.D. de Tullio et al.

Fig. 5 Cooled T106 cascade: local view of the grid (a); velocity vectors (b).

sion of the proposed methodology with an integral method for solving the FfowcsWilliams and Hawkings equations, so as to predict aerodynamic noise.

Acknowledgements This work has been supported by the MIUR and the Politecnico di Bari,Cofinlab 2000 and by PRIN-2007 grants.

References

1. R. Mittal, G. Iaccarino, Annu. Rev. Fluid Mech., 37, 239 (2005).2. P. De Palma, M. D. de Tullio, G. Pascazio, M. Napolitano, Comput. Fluids, 35, 693, (2006).3. M. de Tullio, P. De Palma, G. Iaccarino, G. Pascazio, M. Napolitano, J. Comput. Phys., 225,

2098-2117, (2007).4. S. Kang, G. Iaccarino, F. Ham, J. Comput. Phys., 228, 3189 (2009).5. G. Karypis, V. Kumar, SIAM J. Sci. Comput., 20, 359 (1998).6. M. D. de Tullio, S. S. Latorre, P. De Palma, M. Napolitano, G. Pascazio, An Immersed Bound-

ary Method for Conjugate Heat Transfer Problems, ASME 2010 FEDSM2010-ICNMM2010,August 2-4, 2010, Montreal, Canada.

7. G. C. Rufolo, M. Marini, P. Roncioni, S. Borrelli, In flight aerodynamic experiment for theunmanned space vehicle FTB-1, First CEAS European Air and Space Conference, Berlin,September 10-13 2007.

8. J. N. Moss, D. F. G. Rault, J. M. Price, in Rarefied Gas Dynamics: Space Science and En-gineering, edited by B. D. Shizgal and D. P. Weaver, 160 of Progress in Aeronautics andAeronautics, AIAA, Washington, 1993, pp. 209-220.

9. N. G. Hadjiconstantinou, Physics of Fluids, 18, 111301, (2006).10. H. Hoheisel, R. Kiock, H. J. Lichtfuss, L. Fottner, ASME J. Turbomach., 109, 210219 (1987).

An Overview of the LS-STAG ImmersedBoundary Method for Viscous IncompressibleFlows

Olivier Botella and Yoann Cheny

Abstract The LS-STAG method is an immersed boundary method for viscous in-compressible flows based on the staggered MAC arrangement for Cartesian grids,where the irregular boundary is sharply represented by its level-set function. Thelevel-set function enables us to compute efficiently all relevant geometry parame-ters of the so-called “cut-cells”,i.e. the cells that are cut by the immersed boundary,reducing thus the bookkeeping associated to the handling ofcomplex geometries.One of the main features of the LS-STAG method is the use of a consistent and uni-fied discretization of the flow equations in both Cartesian and cut-cells, which hasbeen obtained by enforcing the strict conservation of global invariants of the flowsuch as total mass, momentum and kinetic energy in the whole fluid domain. Aftera short discussion on the salient features of the LS-STAG method, we will presenta recent application : the computation of viscoelastic flowsin planar contractiongeometries.

1 INTRODUCTION

This paper presents an overview of the LS-STAG method [3, 2],which is a novelimmersed boundary (IB) method for flows in moving irregular geometries on fixedCartesian grids. In IB methods (see [5] for a recent review),the irregular bound-ary is not aligned with the computational grid, and the treatment of thecut-cells,cells of irregular shape which are formed by the intersection of the Cartesian cellsby the immersed boundary, remains an important issue. Indeed, the discretizationin these cut-cells should be designed such that :(a) the global stability and accu-

Olivier BotellaLEMTA, Nancy-University, CNRS, 2 avenue de la Foret de Haye, 54504 Vandœuvre, France. e-mail: Olivier.Botella@ensem.inpl-nancy.fr

Yoann ChenyCERFACS, 42 rue Gaspard Coriolis, 31057 Toulouse Cedex 01, France. e-mail: cheny@cerfacs.fr

1

2 O. Botella and Y. Cheny

racy of the original Cartesian method are not severely diminished and(b) the highcomputational efficiency of the structured solver is preserved.

Two major classes of IB methods can be distinguished on the basis of their treat-ment of cut-cells. Classical IB methods such as themomentum forcing methods[5],use a finite volume/difference structured solver in Cartesian cells away from the ir-regular boundary, and discard the discretization of flow equations in the cut-cells. In-stead, special interpolations are used for setting the value of the dependent variablesin the latter cells. Thus, strict conservation of quantities such as mass, momentumor kinetic energy is not observed near the irregular boundary. The most severe man-ifestations of these shortcomings is the occurrence of non-divergence free velocitiesor unphysical oscillations of the pressure in the vicinity of the immersed bound-ary. Numerous revisions of these interpolations are still proposed for improving theaccuracy and consistency of this class of IB methods.

A second class of IB methods (also calledcut-cell methodsor simplyCartesiangrid methods[5], aims for actually discretizing the flow equations in cut-cells. Thediscretization in the cut-cells is usually performed byad hoctreatments which havemore in common with the techniques used on curvilinear or unstructured body-conformal grids than Cartesian techniques. Such treatments of the cut-cells gener-ate a non-negligible bookkeeping to discretize the flow equations and actually solvethem, and it is difficult to evaluate the impact of these treatments on the computa-tional cost of the flow simulations.

The purpose of this communication is to present an overview of the LS-STAGmethod [3, 2] for incompressible viscous flows which takes the best aspects of bothclasses of IB methods. This IB method, which is based on thesymmetry preserv-ing finite-volume method of Verstappen & Veldman [8] for non-uniform Cartesiangrids, has the ability to preserve up to the cut-cells the conservation properties (fortotal mass, momentum and kinetic energy) of the original MACmethod. After ashort discussion on the salient features of the method, we will present its most recentapplication : the computation of viscoelastic flows in planar contraction geometriesgoverned by the Oldroyd-B constitutive equation [7].

2 Basics of the LS-STAG Method for Newtonian and viscoelasticflows in irregular geometries

The LS-STAG method presented in [3, 2] is a finite volume method for computingfluid flows in the irregular fluid domainΩ f = Ω \Ω ib, whereΩ ib is a solid do-main immersed in the rectangular computational domainΩ (see Fig. 1). As shownin this figure, the irregular boundaryΓ ib = ∂Ω ib is implicitly represented by itssigned distance functionφ(x,y) (i.e. the level-set function [6]), which is discretizedat the vertices of the rectangular cells. The level-set function enables us to computeefficiently all relevant geometry parameters of the cut-cells (such as their volume,projected areas, boundary conditions, . . . ), reducing thusthe bookkeeping associ-ated to the handling of complex geometries.

Overview of the LS-STAG Immersed Boundary Method 3

Fig. 1 Staggered arrangement of the variables near the trapezoidalcut-cellΩi, j on the LS-STAGmesh. The control volume forui, j is shown with dashed lines, and non-homogeneous velocityboundary conditions are discretized at the vertices () of the cut-cells.

Fig. 2 Sketch of the 3 generic cut-cells and location of the normal and shear stresses.

Fig. 2 shows the 3 generic types of cut-cells which are present in the LS-STAG mesh. For building our discretization in each type of cut-cells, we haverequired that our scheme strictly conserves global quantities of the flow such astotal mass

∫

Ω f ∇ · vdV, total momentumP(t) = ρ∫

Ω f vdV and total kinetic energyEc(t) = 1

2

∫

Ω f |v|2 dV (when viscosityη becomes negligible), which are crucial prop-erties for obtaining physically realistic numerical solutions, see [8] and referencestherein. It is well known that the original MAC method on uniform Cartesian mesheswith central differencing conserve these global invariants. On the other hand, formore general grid systems or higher-order methods the construction of “globally

4 O. Botella and Y. Cheny

conservative” methods is not a trivial task, and one needs toenforce the conser-vation properties to the discretization scheme [8]. For example, let us consider thefollowing semi-discretization of the incompressible Navier-Stokes equations :

ρddt

(MU)+C [U ]U +G P−ηK U = 0, (1a)

DU = 0, (1b)

where the diagonal mass matrixM is built from the volume of the fluid cells, matrixC [U ] represents the discretization of the convective fluxes,G is the discrete pressuregradient,K represents the Newtonian stresses, andD is the discrete divergence. Forsimplicity, we have discarded here the influence of the boundary conditions, but thecomplete discussion can be found in Ref. [3]. The budget for the discrete kineticenergy Ehc(t) is obtained from the discrete momentum equation as :

dEhc

dt= −UT C [U ]T +C [U ]

2U −PT

GTU −UT η(K T +K )

2U, (2)

and if we require that this discrete budget mimics the dissipation of energy by theviscous forces in the whole fluid domain, then we have to require that the discretiza-tion of the convective terms leads to a skew-symmetric matrix : C [U ] = −C [U ]T,the pressure gradient has to be dual to the divergence operator : G = −DT, andthe viscous matrixK T +K should be positive definite. In Ref. [3], we have devel-oped the LS-STAG immersed boundary method such as the convective, pressure andviscous fluxes in the cut cells of Fig. 2 are unambiguously determined by these re-quirements, and such that non-homogeneous boundary conditions at the immersedboundary are naturally embedded. On Cartesian grids, the LS-STAG method re-duces to the Cartesian method of Verstappen & Veldman [8]. From the algorithmicpoint of view, one of the main consequences is that the LS-STAG discretizationpreserves the 5-point structure (in 2D) of Cartesian methods, and allows the use ofefficient black-box solvers for structured grids, where noad hocmodifications hadto be undertaken for taking account of the immersed boundary.

Recently, the LS-STAG method has been extended in [2] to the computation ofviscoelastic flows [7]. The shear stress tensor is decomposed in a Newtonian (sol-

vent) and elastic part asτ = ηs∇v+τe, where the extra-stress tensorτe =

(

τxxe τxy

e

τxye τxy

e

)

is governed by the Oldroyd-B transport equation, whose generic expression may bewritten as :

λ( d

dt

∫

Ωτ dV +

∫

Γ(v·n)τ dS

)

=

∫

ΩSτ dV, (3)

whereλ is the elastic characteristic time and∫

Ω Sτ dV is a volumic terms comingfrom the definition of the upper-convected derivative. The nondimensional numberthat measures the level of elasticity of the flow is the Weissenberg number We=λU/L, whereL andU are characteristic length and velocity.

The solution of the viscoelastic flow system (1) and (3) presents several numer-ical challenges [7], one of the most severe being related to the velocity-pressure-

Overview of the LS-STAG Immersed Boundary Method 5

stress coupling. Indeed, as for the velocity and pressure coupling for Newtonianflows, the discretization of the velocity and stress has to becompatible for prevent-ing nonphysical node-to-node oscillations of the extra-stress variables. For a finite-volume method on a Cartesian grid, a compatible discretization is usually achievedby an adequate staggering of the normal and shear stress unknowns at the locationof their Newtonian counterparts. For enforcing this property in the cut-cells of theLS-STAG mesh, we have to position the viscoelastic shear-stress unknownsτxy

i, j atthe vertices of the cut-cells, and the normal stresses at their centroid, see Fig. 2.

All details of the discretization of the Oldroyd-B system (3) are given in [2].Firstly, staggered control volumes are defined for the normal and shear stressesin the cut-cells and, for the volumic integrals in (3) which were absent from theNavier-Stokes equations, novel quadratures with good conservation properties areconstructed for the 3 generic cut-cells of Fig. 2.

3 Numerical results for viscoelastic Oldroyd-B flows

Fig. 3 Sketch of the 4:1 planar contraction flow and close up of the mesh near the re-entrant corner.

In Ref. [3], the accuracy and the robustness of the LS-STAG method has been as-sessed on benchmark Newtonian flows at low to moderate Reynolds number (Cou-ette Taylor flow, flows around cylinders, . . . ), including thecase where the complexgeometry is moving. This last case is one of the most appealing features of IB meth-ods, since the computations are performed on fixed grids without domain remeshingat each time-step. In the following, we present some unpublished results from [2]that concern a popular benchmark for viscoelastic fluids : the creeping flow of anOldroyd-B fluid in a 4:1 planar contraction with rounded re-entrant corner [4, 1],see the sketch in Fig. 3 (left). Numerical results up to We= 8 have been obtainedon the 150×71 mesh shown in Fig. 3 (right), which has 64% of fluid cells. Fig. 4shows that the LS-STAG method predicts accurately the reduction of the salientvortex when the level of elasticity increases, and that the contours are free from any

6 O. Botella and Y. Cheny

spurious oscillations thanks to the fully staggered arrangement of the flow variables.Table 1 reports quantitative results of the flow for increasing values of the elasticity

0.20.2 0.4

0.4

0.6

0.6

0.8

0.8

1

1

1

1.00039

We= 20.2

0.2 0.4

0.4

0.6

0.6

0.8

0.8

1

1

1

We= 40.2

0.20.4

0.4

0.6

0.6

0.6

0.8

0.8

1

1

1

We= 8

Fig. 4 Streamlines contours for increasing values of the Weissenberg number We.

level. When available, these results are compared to those obtained with conformalgrid methods [4, 1], and where they are available, a good agreement is met with theliterature.

XR ∆Ψ ×10−3 CWe=2 LS-STAG 1.095 0.368 −1.469

Ref. [1] − 0.342 −1.72Ref. [4] − 0.48 −0.69

We=4 LS-STAG 0.923 0.249 −2.843Ref. [1] − 0.225 −3.54Ref. [4] − 0.33 −1.44

We=6 LS-STAG 0.764 0.207 −3.154Ref. [4] − 0.26 −2.27

We=8 LS-STAG 0.602 0.236 −2.834

Table 1 Properties of the corner vortex and Couette coefficientC for increasing values of We.

References

1. M. Aboubacar, H. Matallah, and M.F. Webster. Highly elasticsolutions for Oldroyd-B andPhan-Thien/Tanner fluids with a finite volume/element method: Planar contraction flows.J.Non-Newtonian Fluid Mech., 103, 65-103, 2002.

2. Y. Cheny.La MethodeLS-STAG : une nouvelle Approche de type Frontiere Immergee/Level-Set pour la Simulation d’Ecoulements Visqueux Incompressibles en Geometries Complexes.Application aux Fluides Newtoniens et Viscoelastiques. PhD Thesis, Nancy-University, 2009.

3. Y. Cheny and O. Botella. The LS-STAG method : A new immersed boundary / level-set methodfor the computation of incompressible viscous flows in complex moving geometries with goodconservation properties.J. Comput. Phys., 229, 1043-1076, 2010.

4. H. Matallah, P. Townsend, and M. F. Webster. Recovery and stress-splitting schemes for vis-coelastic flows.J. Non-Newtonian Fluid Mech., 75, 139-166, 1998.

5. R. Mittal and G. Iaccarino. Immersed boundary methods.Annu. Rev. Fluid Mech., 37, 239-261,2005.

6. S. Osher and R. P. Fedkiw.Level Set Methods and Dynamic Implicit Surfaces. Springer, 2003.7. R. G. Owens and T. N. Phillips.Computational Rheology. Imperial College Press, 2002.8. R. W. C. P. Verstappen and A. E. P. Veldman. Symmetry-preservingdiscretization of turbulent

flow. J. Comput. Phys., 187, 343-368, 2003.

A two-dimensional embedded-boundary methodfor convection problems with moving boundaries

Yunus Hassen and Barry Koren

Abstract A 2D embedded-boundary algorithm for convection problems is presented.A moving body of arbitrary boundary shape is immersed in a Cartesian finite-volumegrid, which is fixed in space. The boundary surface is reconstructed in such a way thatonly certain fluxes in the immediate neighbourhood indirectly accommodate effects ofthe boundary conditions valid on the moving body. Over the majority of the domain,where these boundary conditions have ‘no’ effect, the fluxesare computed using standardschemes. Examples are given to validate the method.

1 Introduction

Recently, immersed-boundary methods have been favourablypopularised by their rela-tively simple ideas and ease of implementation. The immersed-boundary method, alsosynonymously known as embedded-boundary method, in general, is a method in whichboundary conditions are indirectly incorporated into the governing equations. It is verysuitable for simulating flows around flexible, moving and/orcomplex bodies (see [5] fora comprehensive review).

In this work, we present a new embedded-boundary approach for advection prob-lems. As is standard in the immersed-boundary methods, moving bodies are embeddedin a fixed, Cartesian grid. We employ the method of lines: a higher-order, cell-averaged,fixed-grid, finite-volume method for the spatial discretization, and the explicit Eulerscheme for the time integration. The essence of the present method is that body geome-tries are, without loss of generality, effectively simplified and their presence is restrictedto a minimal zone in the computational region so that standard discretization schemescan be readily applied elsewhere. The boundary conditions valid on a possibly movingbody are indirectly accommodated by specific fluxes in the vicinity of the boundary.

Yunus Hassen, Barry KorenCentrum Wiskunde & Informatica, Amsterdam, The Netherlandse-mail: yunus.hassen@cwi.nl, barry.koren@cwi.nl

1

2 Y. Hassen, B. Koren

2 The embedded boundary method

As in the previous one-dimensional work [1], our approach uses a finite-volume dis-cretization that embeds the boundary of a body in a regular, fixed grid. Dividing thecurrent 2D computational domainD , of dimensionℓx ×ℓy, into Nx ×Ny uniform, rectan-gular finite volumes that are fixed in space, we have:

Di, j :=

[xi−1

2,x

i+ 12]× [y

j−12,y

j+ 12]∣

∣

∣x

i−12

=(

i− 12

)

hx + x0,

yj−1

2=

(

j− 12

)

hy + y0; i ∈ [1,Nx], j ∈ [1,Ny]

, (1)

wherehx = ℓx/Nx andhy = ℓy/Ny are cell sizes, andx0 andy0 some constants. The totalsimulation timeT > 0 is equally divided intoNt time steps of sizeτ = T /Nt .

Having generated the (Cartesian) grid, Eq. (1), the body is immersed inside the grid.To obtain discrete embedded boundaries (EBs), at a given timetn, n = 0,1, · · · ,Nt , firstly,the finite volumes that contain (a part of) the boundary of theimmersed body are iden-tified, and then the points of intersection of the boundary ofthe immersed body withthe faces of these computational cells,xn

B, are detected. That is, the coordinates of the

boundary pointsxk,nB := (xk,n

B ,yk,nB ), k = 1,2, · · · ,NB, whereNB is the total number of

boundary-face intersections, are computed.Care is required in making the underlying uniform fixed grid detect boundary points

that lie exactly at or very close to grid vertices. Grid vertices are shared by more than onecontrol volume, and a boundary point lying exactly at a grid vertex, or in the immediateneighbourhood, is, due to the round-off error, arbitrarilyassigned to any of the cells thatshare the vertex. The prevailing arbitrariness can lead to erroneous absence of an EB ina cell, due to discount of a second boundary point (or two boundary points altogether)within the cell. This undesirable scenario can be remedied by taking the precision of themachine into account; see [2] for details.

2.1 Determination of EBs

Once the boundary points have been detected, the actual boundary of the body is read-ily degenerated as a piece-wise continuous, closed or open,poly-line, with NB andNB−1 segments, respectively. This representation facilitates explicit association of eachside/segment of the polygon/poly-line with individual control volumes, resulting in onediscrete EB, at most, in a cell.

The manner in which a generic 2D EB segment, situated in a cell(Fig. 1a), is projectedonto the grid (coordinate) directions is crucial. To take advantage of the 1D methodproposed in [1], we resort to dimensional splitting. The procedure is illustrated in Fig. 1and described in detail in [2].

To project the discrete EB, shown in Fig. 1a, into the relevant grid direction and to geta single orthogonal EB in a cell, two geometrical properties– orientation andlocation –are used. The oblique discrete EB (Fig. 1a) is next rotated parallel to the grid directionto which it is closest (Fig. 1b or Fig. 1c). The actual location of the orthogonalized EB,

An embedded-boundary method for convection 3

δx

δy

(a) Continuous / discrete EBδx

(b) δx ≤ δy

δy

(c) δx > δy

Fig. 1: Alignment of a two-dimensional EB, situated in a cell, with the relevant grid (coordinate) direc-tion.

at tn, inside a cell is represented byβ n = (β nx ,β n

y ), a normalised variable that discernsthe orthogonalized EB’s position relative to the left- or bottom-face of the cell, and it isdetermined by the area, subset in the cell by the non-orthogonalized EB.

Having all the discrete EBs in the domain aligned with the relevant grid directionand appropriately positioned inside a cell, we achieve the desired sub-cell resolution ofthe immersed body boundary. In the latter respect, the present method essentially differsfrom the stair-case approach wherein the boundary is projected on cell faces [5].

2.2 Merging of EBs

Occasionally, it might arise a scenario with two successivecells, along a column or rowof cells, each having orthogonalized EBs in the same direction. Technically, with the out-lined procedure, this is a natural outcome and it can be accommodated. However, this sit-uation may be non-physical. It might cause‘numerical spray’ and, as such, it perturbs anevolving solution, not to mention the algorithmic (flux-computation and time-stepping)complications it creates during implementation.

In the event of two such distinct, orthogonalized EBs, we caneffectively get rid ofthe ‘numerical spray’ by properly merging these EBs, obtaining a single equivalent EBin one of the cells. The‘numerical spray’ is superseded by reuniting it with its‘parentmaterial’ that originally gave it off. Therefore, the merging is done in the direction ofthe ‘parent material’ and such that the conservation law is satisfied. Further details aregiven in [2].

3 Finite-volume discretization

A multi-dimensional convection problem, in conservation form, with the associated ini-tial condition, can be written as:

4 Y. Hassen, B. Koren

∂c∂ t

+∇ ·F = 0, in D ∈ R2× (0,T ], (2a)

c(x, t =0) = c0(x), in D ∈ R2, (2b)

wherec(x, t) is the scalar field that is convected by the flow fieldu =(

u(x) ,v(x))T

. The

components of the flux vectorF =(

f (u,c) ,g(v,c))T

are defined, attn, as: f n(u,c) :=

u(x)c(x, tn) andgn(v,c) := v(x)c(x, tn). For a cell-averaged discrete solution inDi, j, attn, i.e.,cn

i, j, an integral form of (2a) can be formulated. Assuming fluxes to be constantalong cell faces, we have the semi-discrete equation:

hxhydci, j

dt+

(

fi+1/2, j(t)− fi−1/2, j(t))

+(

gi, j+1/2(t)−gi, j−1/2(t))

= 0. (2c)

Certain fluxes in the immediate neighborhood of an orthogonalized EB are modifiedto accommodate the corresponding embedded-boundary conditions, see [1] for details.Elsewhere, where the EBs have ’no’ effect, the standard MUSCL scheme [4, 3] is used.With a very small Courant numberν , Eq. (2c) yields a temporally accurate solution,using the forward Euler scheme:

cn+1i, j = cn

i, j −τ

hxhy

(

f n

i+ 12 , j

− f n

i−12 , j

)

−τ

hxhy

(

gn

i, j+ 12−gn

i, j−12

)

. (2d)

4 Numerical examples

To validate the algorithm presented in this article, we consider two test cases, a translat-ing rectilinear discontinuity and a revolving cylindricaldiscontinuity, with prescribed 2Dflow fields inside a rectangular domain. Settings of the problems are depicted in Fig. 2.Here, we takeℓx = ℓy = 2, with D = [−1,1]× [−1,1].

EB

x

y

ϑ

u

v

ϑ

0 < ϑ < π2

x

y

ω

u

v

Fig. 2: Domain, flow fields and problem settings. A rectilinear (left) and cylindrical (right) discontinuties.

An embedded-boundary method for convection 5

4.1 A translating rectilinear discontinuity

Consider a rectilinear discontinuity of arbitrary orientation ϑ ∈ [0,π/2], initially situatedat the bottom-left corner of the domain, and moving in a uniform 2D flow field withvelocity u = (cosϑ , sinϑ)T. The flow field, shown in Fig. 2 (left), is normal to thediscontinuity.

The discontinuity, which goes with the flow, is assumed to model a rigid, infinitelythin plate that separates two quantities of different values, i.e.,c = 1 (at the upstreamside) andc = 0 (at the downstream side). These solution values are taken as embeddedboundary conditions, i.e., cEBl = 1 andcEBr = 0, to be used in the relevant fixed-gridfluxes in the immediate neighbourhood of the embedded boundary.

Fig. 3 shows results for the translating rectilinear discontinuity, of arbitrary orientationϑ = π/6, on a grid(Nx,Ny) = (20,20) and at final timeT =

√2. The results obtained

with the current method appear to be significantly more accurate than those obtainedwith the standard method. Results for other orientations inthe rangeϑ ∈ [0,π/2] aregiven in [2].

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

(a) Standard solution

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

(b) EB solution

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

(c) Exact discrete solution

Fig. 3: Results for a translating rectilinear discontinuity, of orientationϑ = π/6, obtained with limitedfluxes andν ≪ 1.

4.2 A revolving cylindrical discontinuity

Consider a cylindrical discontinuity of radiusR = 0.2, and unit height, initially located at(x,y) = (1/2,1/2), shown in Fig. 2 (right), which revolves with a circular flow-field withvelocity u = (−ωy,ωx)T, whereω = 2π is the angular velocity (a solid-body rotation).

Similarly, as in§ 4.1, the discontinuity, which goes with the flow, is assumed to modela rigid, infinitely thin-walled cylinder that separates twoquantities of different values,i.e., c = 1 andc = 0, inside and outside the cylinder, respectively. The solution valuesc = 1 andc = 0 are appended to the boundary of the immersed cylinder;cEBl = 1 at theinner, andcEBr = 0 at the outer side of the cylinder’s wall. Fig. 4 shows the results forthe revolving cylindrical discontinuity obtained on a gridof (Nx,Ny) = (40,40), at a finaltimeT = 1 (i.e., after one full revolution). Clearly, the results ofthe current EB method,

6 Y. Hassen, B. Koren

on this grid, have higher resolution than those of the standard method, but they are notyet monotone.

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y

(a) Standard solution

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

x

y

(b) EB solution

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

(c) Exact discrete solution

Fig. 4: Results for a revolving cylindrical discontinuity after one full revolution on a 40×40-grid, ob-tained with limited fluxes andν ≪ 1.

5 Conclusion

In this work, a new immersed-boundary method, which effectively embeds boundaryconditions, valid on a moving body, only in certain fluxes in the immediate neighbour-hood, has been introduced. The algorithm has been tested with two problems and theresults obtained are promising. They have higher resolution compared to those obtainedby standard methods, but are not yet entirely monotone. It isa simple and elegant al-gorithm, and we anticipate to use it for 2D Euler flows, which we foresee to considernext.

Acknowledgements The first author’s research is funded by theDelft Centre for Computational Science& Engineering (DCSE), TU Delft.

References

1. Hassen, Y., Koren, B.: Finite-volume discretizations and immersed boundaries. In: B. Koren, C. Vuik(eds.) Advanced Computational Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng.,vol. 71, pp. 229–268. Springer, Heidelberg (2010)

2. Hassen, Y., Koren, B.: A two-dimensional embedded-boundary method for convection problems withmoving boundaries. Report MAC-1003, CWI, Amsterdam (2010)

3. Koren, B.: A robust upwind finite-volume method for advection, diffusion and source terms. In: C.B.Vreugdenhil, B. Koren (eds.) Numerical Methods for Advection-Diffusion Problems,Notes on Num.Fl. Mech., vol. 45, pp. 117–138. Vieweg, Braunschweig (1993)

4. van Leer, B.: Upwind-difference methods for aerodynamic problems governed by the Euler equa-tions. In: B.E. Engquist, S. Osher, R.C.J. Somerville (eds.) Large-Scale Computations in Fluid Me-chanics,Lect. Appl. Math., vol. 22.2, pp. 327–336. Am. Math. Soc., Providence, RI (1985)

5. Mittal, R., Iaccarino, G.: Immersed boundary methods. Ann. Rev. Fl. Mech.37, 239–261 (2005)

A second-order immersed boundarymethod for the numerical simulation oftwo-dimensional incompressible viscousflows past obstacles

Francois Bouchon, Thierry Dubois and Nicolas James

Abstract We present a new cut-cell method, based on the MAC scheme onCartesian grids, for the numerical simulation of two-dimensional incompress-ible flows past obstacles. The discretization of the nonlinear terms, writtenin conservative form, is formulated in the context of finite volume methods.While first order approximations are used in cut-cells the scheme is globallysecond-order accurate. The linear systems are solved by a direct method basedon the capacitance matrix method. Accuracy and efficiency of the method aresupported by numerical simulations of 2D flows past a cylinder at Reynoldsnumbers up to 9 500.

1 Introduction

Avoiding the use of curvilinear or unstructured body-conformal grids, im-mersed boundary (IB) methods provide efficient solvers, in terms of compu-tational costs, on Cartesian grids for flows in complex geometries. IB methodscan be classified in two groups. Classical IB methods add in the momentumequation a forcing term accounting for the presence of an obstacle in thecomputational domain. Cut-cell methods discretize the momemtum and con-tinuity equations in mesh cells cut by the solid. The scheme proposed in thispaper lies in this class of IB methods and differs from other cut-cell methodsin the treatment of the diffusive and convective terms in cut-cells. The schemeis globally second-order accurate for the velocity and pressure variables.

This paper is organized as follows. The first section is devoted to thedescription of the problem. Then the IB/cut-cell method is detailed and finalysome numerical simulations are given.

Laboratoire de Mathematiques, Universite Blaise Pascal and CNRS (UMR6620), Campus Universitaire des Cezeaux, 63177 AUBIERE, France, e-mail:name.surname@math.univ-bpclermont.fr

1

2 Francois Bouchon, Thierry Dubois and Nicolas James

2 The settings of the problem

Let Ω be a rectangular domain. We consider an irregular fluid domain ΩF

which is embedded in the computational domain Ω and its complement ΩS ,the solid domain. The interface between solid and fluid is denoted Γ . Thedecoupling between velocity and pressure variables is achieved by applying asecond-order (BDF) projection scheme to the incompressible Navier-Stokesequations. First, the prediction step consists in computing the velocity fielduk+1 which is solution of the following equation :

3uk+1 − 4uk + uk−1

2δt−4uk+1/Re = −∇pk + fk+1 − 2 div

(uk ⊗ uk

)+ div

(uk−1 ⊗ uk−1

)(1)

with appropriate boundary conditions on ∂ΩF . Then we solve the projectionstep :

uk+1 = uk+1 − 2δt

3∇(pk+1 − pk

), (2)

div uk+1 = 0, (3)(uk+1 − uk+1

)|∂Ω .n = 0. (4)

3 The IB/cut-cell method

3.1 Staggered arrangement of the unknowns

As in Cheny and Botella [1], a signed distance to the obstacle d is usedto represent solid boundaries in the computational domain. The pressure isplaced at the center of every Cartesian cell either filled by the fluid or cutby the solid boundaries (see Figure 1). The velocity components are placedat the middle of the part of the edges located in the fluid. Unlike for cellslocated in the fluid part of the computational domain, velocity and pressureare not aligned in cut-cells. Cell-face ratios rui, j , r

vi, j are calculated from the

distance of the mesh point to the obstacle, denoted di, j . The interface Γh,linear in each cell, approaches the regular solid boundary Γ (see Figure 1).

3.2 Discretization of the prediction step

In fluid cells, the classical five-point stencil approximation of the viscousterms is used. In cells sharing a face with a cut-cell, we propose a first-

A second-order immersed boundary method... 3

Γh

pi j

ui j

vi j

xixi−1

yj−1

yj

di j−1

di j

rui j hy

ΩF

ΩS

Fig. 1 Staggered arrangement of the unknowns in cells cut by an obstacle

order Finite Difference approximation. More precisely, we consider V =O,N, S,E,W,P with O the position of ui j , N,S,E,W are the locationof unknowns close to O or on the boundary, and P is arbitrarily chosen (seeFigure 2). Then, we search coefficients αM such that

∑M∈V αMu(M) is a first

Fig. 2 Six points areused for the discretizationof the diffusive term inmesh cells cut by the solidboundary

O

N

S

E

W

P

order approximation of 4u(O). This leads to a linear system of six equationswith six unknowns.

In fluid cells, a second-order centered approximation for the nonlinearterms is used. In cells sharing a face with a cut-cell, we propose a first-order Finite Volume approximation of nonlinear terms. The integral of thefirst component of the nonlinear term over a cut-cell Ku

i, j = Kui, j ∩ ΩFh is

expressed in terms of fluxes through cell edges, namely :∫K

i+12, j

(∂x(u2) + ∂y(uv)

)dx =

∫∂K

i+12, j

(u2nx + (uv)ny

)dS (5)

= FEi+1, j − FEi, j + FNi, j − FNi, j−1 + FBi, j . (6)

Second-order interpolations of velocity components are used to approximatethe fluxes at the center of cut-edges (see Figure 3). This leads to a pointwisefirst-order approximation of the convective terms.

4 Francois Bouchon, Thierry Dubois and Nicolas James

xi−1 xi xi+1

yj−1

yj

ui j FEi j

FEi+1 j

FNi j

FNi j−1

FBi j

Γh

ΩFh ΩS

h

xi−1 xi xi+1

yj−1

yj

ui j FEi j

FEi+1 j

FNi j

FNi j−1

FBi j

Γh

ΩFh ΩS

h

Fig. 3 Discretization of the convective term using fluxes reconstruction

3.3 Discretization of the projection step

The continuity equation is decomposed as the net mass flux through eachface of the computational cells. Due to the locations of velocity components,a second-order approximation follows immediately. The discretization of (3)on a cut-cell is (Dobsu)i, j = (D0

obsu)i, j +Dsuppi, j = 0, where the linear part of

the discrete divergence is

(D0obsu)i, j = hy(rui, jui, j − rui−1, jui−1, j) + hx(rvi, jvi, j − rvi, j−1vi, j−1) (7)

and the contribution to the divergence due to the boundaries is Dsuppi, j =

`i, j g(M) .ni, j . Note that `i, j , M and ni, j are respectively the length, themiddle point and the external normal of the edge of the cut-cell shared withthe boundary (see Figure 1). In the case of a fluid cell, this expression reducesto the standard MAC discretization.

The velocity correction step requires the computation of the pressure gra-dients at the location of the velocity. For faces cut by solid boundaries, asecond-order interpolation Pφ is used. As in the classical MAC scheme, a

discrete Poisson-type equation for the pressure increment δpk+1 = pk+1 − pkis obtained by applying the discrete divergence operator to the velocity cor-rection equations :

D0obs(Pφ(Gδpk+1)) =

3 h2

2 δtDobs(u

k+1), (8)

with the classical discrete gradient

(Gδpk+1)i, j =((δpk+1i+1, j − δpk+1

i, j

)/hx ,

(δpk+1i, j+1 − δpk+1

i, j

)/hy

). (9)

It follows that the velocity correction

uk+1 = uk+1 − 2 δt

3 h2Pφ(Gδpk+1) (10)

A second-order immersed boundary method... 5

ensures that the incompressibility condition Dobs(uk+1) = 0 is satisfied in

the whole domain up to computer accuracy.

3.4 Computational efficiency

The non-symmetric linear systems are efficiently solved by a direct method,based on the capacitance matrix method [2]. First, we solve a preprocess-ing step, requiring O(n3) operations. Assuming that the obstacle does notmove, this step is solved once per simulation. Then, at every time step, thistechnique allows to reduce the overall cost of the resolution to O(n2log(n))operations, which is the number of operations needed to solve the linear sys-tems corresponding to five-point stencil operators on the whole Cartesianmesh without obstacle.

The method is tested on the Taylor-Couette flow between two concentriccylinders : second-order spatial convergence for velocity and pressure is found.In Figure 4, we have reported the L∞ error for the velocity, when the error ismeasured on the whole fluid computational domain. Unlike in [1], the second-order accuracy is also satisfied in cut-cells.

Fig. 4 Error for the ve-locity versus grid sizeh. Present study (cir-cles), Cheny and Botellaresults [1] (squares), first-order slope (dash) andsecond-order slope (dot-ted).

0,01 0,1h

1e-05

0,0001

0,001

0,01

0,1

Err

or

Numerical simulations of 2D flows past a cylinder have been performedat Reynolds numbers up to 9 500. As it is shown on Figure 5, an excellentagreement is found with the experimental results presented in Bouard andCoutenceau [3]. Streamlines of the flow past a cylinder at Re = 9 500 at timet = 0.75, 1.0 and 1.25 are represented on Figure 5. We have also studied theflow past a NACA aerofoil at Re = 1 000. Like in [4], a Karman vortex streetdevelops behind the obstacle (see Figure 6) : the flow is well resolved evennear the sharp ending edge. For these numerical simulations, the mesh sizenear the obstacle is 1.6 10−3. The value of the time step, satisfying a CFLstability condition, is 10−4.

6 Francois Bouchon, Thierry Dubois and Nicolas James

Fig. 5 Evolution of the boundary layer : comparison with experimental results atRe = 9 500.

Fig. 6 Flow behind a NACA aerofoil at Re = 1 000 : comparison with experimentalresults.

References

1. Yoann Cheny and Olivier Botella : The LS-STAG method: A new immersedboundary/level-set method for the computation of incompressible viscous flowsin complex moving geometries with good conservation properties. J. of Comput.Phys. 229–4, 1043–1076 (2010)

2. B.L. Buzbee, F.W. Dorr, J.A. George and G.H. Golub : The direct solution ofthe discrete Poisson equation on irregular regions. J. Num. Anal. 8, 722–736(1971)

3. R. Bouard and M. Coutanceau : The early stages of development of the wakebehind an impulsively started cylinder for 40 < Re < 104. J. Fluid. Mech. 101,583–607 (1980)

4. O. Daube, Ta Phuoc Loc, P. Monnet and M. Coutanceau : Ecoulement insta-tionnaire decolle dun fluide incompressible autour dun profil : une comparaisontheorie - experience, AGARD CP 386, Paper 3, (1985)