The LS-STAG method is an immersed boundary/cut-cell method for viscous incompressible flows based on the staggered MAC arrangement for Cartesian grids, where the irregular boundary is sharply represented by its level-set function. The 2D version of the LS-STAG method is now well-established, and this talk presents its extension to 3D geometries with translational symmetry in the z direction (subsequently called quasi-3D configurations), which can be viewed as an intermediate step before fully 3D geometries, where both numerical and HPC implementation issues are tackled at this stage of development. The discretization in the 3 basic types of quasi-3D cut-cells is performed by following the principles of the energy-conserving discretization of Cheny & Botella (J. Comput. Phys. Vol. 229, 1043-1076, 2010). We will present applications on the flow of non-Newtonian fluids in a circular duct with sudden expansion, for which experimental results performed in our team are available. Finally, the performance of the LS-STAG code on high-performance parallel computers will be discussed

Botella, Olivier

Cheny, Yoann

Nikfarjam, Farhad

I-Revues

22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015
The LS-STAG immersed boundary/cut-cell method
for non-Newtonian flows in quasi-3D geometries
F. NIKFARJAMa, Y. CHENY & O. BOTELLA
LEMTA, Université de Lorraine, CNRS
2 avenue de la Forêt de Haye - TSA 60604, 54518 Vandoeuvre Cedex, France.
a. farhad.nikfarjam@univ-lorraine.fr
Résumé :
La méthode LS-STAG est une méthode de type frontières immergées/ cut-cell pour les écoulements vis-
queux incompressibles, basée sur laméthodeMAC sur grilles cartésiennes décalées, telle que la frontière
irrégulière du domaine est représente par sa fonction level-set. La version 2D de la méthode LS-STAG
est maintenant bien éprouvée, et cette présentation portera sur son extension aux géométries 3D avec
symétrie de translation dans la troisième direction (appelées par la suite géométries quasi-3D). La dis-
crétisation dans les 3 types de cut-cells d’un maillage quasi-3D sont obtenues en suivant les principes de
la discrétisation energy-conserving de Cheny & Botella (J. Comput. Phys. Vol. 229, 1043-1076, 2010).
Nous présenterons des applications de la méthode à l’écoulement de fluides non-Newtoniens dans une
conduite circulaire à expansion brusque, pour laquelle des expériences ont été réalisées dans notre
équipe. Finalement, les performances du code LS-STAG dans un environnement de calcul intensif se-
ront discutées.
Abstract :
The LS-STAG method is an immersed boundary/cut-cell method for viscous incompressible flows based
on the staggered MAC arrangement for Cartesian grids, where the irregular boundary is sharply repre-
sented by its level-set function. The 2D version of the LS-STAG method is now well-established, and this
talk presents its extension to 3D geometries with translational symmetry in the z direction (subsequently
called quasi-3D configurations), which can be viewed as an intermediate step before fully 3D geome-
tries, where both numerical and HPC implementation issues are tackled at this stage of development.
The discretization in the 3 basic types of quasi-3D cut-cells is performed by following the principles of
the energy-conserving discretization of Cheny & Botella (J. Comput. Phys. Vol. 229, 1043-1076, 2010).
We will present applications on the flow of non-Newtonian fluids in a circular duct with sudden expan-
sion, for which experimental results performed in our team are available. Finally, the performance of
the LS-STAG code on high-performance parallel computers will be discussed.
Keywords :Computational fluid dynamics, Immersed boundarymethod, Cut-
cellmethod, Incompressible flows, non-Newtonian fluids, 3D geometries, High
performance computing
22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015
Advanced computational fluid dynamics (CFD) applications concern flows in highly irregular three-
dimensional geometries, whose process of grid generation, from the early stage of CAD design to fi-
nal volume meshing, is very time-consuming and requires highly trained users. Typical unstructured
body-conformal meshes are made of elements (cells of heaxaedral, tetraedral shape, . . .) with irregular
connectivity, which makes the solution algorithms extremely memory and CPU intensive. Moreover, it
is difficult to assess the quality of the grid (smooth, undistorted cells properly clustered at the wall, . . .),
which may negatively impact the convergence and accuracy of the numerical solution.
In contrast, cut-cell/immersed boundary CFD codes (see [1] for a review) are based onCartesianmethods
which are extremely robust and optimized, and thus requires only a fraction of the computer resources
of an unstructured code. Cartesian grid generation is greatly simplified compared to the body-conformal
approach, and the grid regularity and quality is not significantly impacted by the complexity of the
geometry. Moreover, the computation of flows with moving boundaries can be performed on fixed grids,
without the domain remeshing which is required by body-conformal methods.
The major difficulty of traditional IB methods is the treatment of the cut-cells (cells of irregular shape
which are formed by the intersection of the Cartesian mesh with the immersed boundary). Classical
IB methods such as the momentum forcing method [1] discard the discretization of flow equations in
the cut-cells, and use special interpolations instead. Thus, strict conservation of mass is not observed,
which leads to non-divergence free velocities or non-physical presssure oscillations in the vicinity of the
immersed boundary [2, 3]. On the other hand, cut-cell methods aim for discretizing the flow equations
in cut-cells by ad hoc treatments, whose major drawback is the loss of the 5-points structure (in 2D) of
Cartesian methods. Moreover, these methods rely on cell-merging techniques for which 3D extension is
known to be nontrivial [1]. Therefore, the development of an accurate and cost-effective 3D solver based
upon the IB/cut-cell approach is still a challenging task.
Figure 1 – Cartesian cell and 3 basic types of cut-cells present in the LS-STAG mesh in 2D (a) and
quasi-3D (b). For the quasi-3D mesh, the velocities are discretized at cell faces, pressure and normal
stresses at cell center (●), shear stresses at edge centers : ∂u∂y , ∂v∂x (∎), ∂u∂z , ∂w∂x (∎) and ∂v∂z , ∂w∂y (∎).
In Refs. [4, 5] we have developed an innovative cut-cell discretization for 2D incompressible flows called
“LS-STAG” (Level-Set, STAGgered), which is based on the classical MACmethod for Cartesian stagge-
red grids, where the irregular boundary is sharply represented by its level-set function. In the three basic
types of 2D cut-cells (pentagonal, trapezoidal and triangular, see Fig. 1(a)), the discretization is designed
22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015
such as the global invariants of the flow (total mass, momentum and kinetic energy) are preserved at the
discrete level. The LS-STAG discretization in the cut-cells is consistent with the MAC discretization
used in Cartesian fluid cells, and has the ability to preserve the 5-point Cartesian structure of the sten-
cil, resulting in a highly computationally efficient method. We have successfully applied the LS-STAG
method to Newtonian flows at moderate Reynolds number in fixed and moving geometries [5], pseudo-
plastic flows [6], and viscoelastic flows [7, 8]. In addition, we mention that the LS-STAG discretization
has also been implemented in the ship hydrodynamics software COMFLOW [9] for the handling of ir-
regular geometries, and has been extended to the computation of coupled aeroelastic problems [10] and
turbulent flows with RANS modelling [11].
The purpose of this communication is to present a progress report towards a fully 3D LS-STAG code,
totally optimized for high-performance computing. One of the major difficulty in cut-cell discretization
for 3D geometries is the large number of cases to consider : we have enumerated 16 different cut-cells in
3D whereas there are only 3 in 2D, see Fig. 1(a)). In order to mitigate this issue, we first consider buil-
ding the LS-STAG discretization for 3D configurations with translational symmetry in the z direction
(subsequently called quasi-3D configurations), where are only present the 4 types of cells depicted in
Fig. 1(b). This intermediate step towards the fully 3D implementation can be applied to a wide variety
of canonical flows (Taylor-Couette flow, flow past cylinder of arbitrary cross section shape, abrupt pla-
nar/axisymmetric contraction flow, . . .) and is regarded as the keystone for the full 3D solver, since both
numerical and HPC implementation issues are tackled at this stage of development. In quasi-3D geo-
metries, only extruded 2D cut-cells are present, which enables us to easily extend the principles of the
energy-conserving discretization of Ref. [5] to the cut-cells of Fig. 1(b) : the discretization of the conti-
nuity equation, pressure gradient, convective fluxes, normal stresses are straightforward extensions of
the 2D LS-STAG formulas. The discretization of shear stresses terms such as ∂w/∂x and ∂w/∂y, which
were absent from the 2D case, needs further attention. These terms are tentatively discretized with the
formula proposed by van der Heiden et al. [9]. Preliminary computations of the quasi-3D LS-STAG
method for benchmark confined cylinder flows were presented in Ref. [12].
−1 −0.5 0 0.5 10
0.5
1
1.5
2
r/R
w
Experiments
Present
Figure 2 – At left : flow in a cylinder duct with sudden expansion at Re = 273. At right : axial velocity
profile at z/D1 = −0.9 obtainedwith the LS-STAGmethod and comparisonwith the experimental results
of [13].
In this talk, we will present the extension of the quasi-3D LS-STAG method to the computation of
pseudoplastic flows, where the stress tensor takes the form τ = η(γ˙)D and the viscosity η is modelled
22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015
by popular non-Newtonian laws such as the power-law or Cross models. The crucial part for taking
into account shear-thinning or shear-thickening effects is the accurate computation of the shear rate
γ˙ =√12D ∶ D in the vicinity of the immersed boundary, since the velocity gradients appear at different
locations inside a cut-cell (see Fig. 1(b)). To perform this, we had to adapt to the 3D staggering the
special interpolations developed in [6] for 2D pseudoplatic flows. The accuracy of the discretization
will be evaluated on Poiseuille pipe flow, and we will present preliminary results on the flow of xanthan
fluids in a circular duct with sudden expansion (see Fig. 2), for which MRI 1 velocity measurements
performed in our group [13] are available.
Finally, we will discuss the HPC optimization of the LS-STAG code, which is made necessary by the the
large computational resources required by 3D computations. The principal issue in the optimization of a
Cartesian code for incompressible flows is the parallelisation of the iterative solution of linear systems,
most notably the Poisson pressure equation. For that purpose, the LS-STAG quasi-3D code has been
developed in an MPI parallel environment on the HPC cluster of the Université de Lorraine, and the
linear systems are solved with the multigrid and Krylov solvers of the Hypre library [14]. The parallel
implementation will be discussed and results on its scalability properties will be presented.
Références
[1] R. Mittal and G. Iaccarino. Immersed boundary methods. Annu. Rev. Fluid Mech., 37, 239-261,
2005.
[2] F. Muldoon and S. Acharya. A divergence-free interpolation scheme for the immersed boundary
method. Int. J. Numer. Meth. Fluids., 56, 1845-1884, 2008.
[3] S. Kang, G. Iaccarino, and P. Moin. Accurate and efficient immersed-boundary interpolations for
viscous flows. In Center for Turbulence Research Briefs, NASA Ames/Stanford University, pp.
31-43, 2004.
[4] Y. Cheny and O. Botella. An immersed boundary/level-set method for incompressible viscous
flows in complex geometries with good conservation properties. Europ. J. Comput. Mech., 18,
561-587, 2009.
[5] Y. Cheny and O. Botella. The LS-STAG method : A new immersed boundary / level-set method
for the computation of incompressible viscous flows in complex moving geometries with good
conservation properties. J. Comput. Phys., 229, 1043-1076, 2010.
[6] O. Botella,M. Ait-Messaoud, A. Pertat, C. Rigal, andY. Cheny. The LS-STAG immersed boundary
method for non-Newtonian flows in irregular geometries : Flow of shear-thinning liquids between
eccentric rotating cylinders. Theoretical and Computational Fluid Dynamics, 29, 93-110, 2015.
[7] O. Botella and Y. Cheny. The LS-STAG method for viscous incompressible flows in irregular
geometries : Basics of the discretization and application to viscoelastic flows. In American Society
of Mechanical Engineers, Fluids Engineering Division (Publication) FEDSM, volume 1, pages
2441–2451, 2010.
[8] O. Botella, F. Nikfarjam, M. Stoica, and Y. Cheny. Entry flow computations of shear-thinning and
viscoelastic liquids with the LS-STAG immersed boundary method. Chengdu, China, 14-18 July
2014. Eighth International Conference on Computational Fluid Dynamics (ICCFD8).
[9] P. van der Plas, H.J.L. van der Heiden, A.E.P. Veldman, R. Luppes, and R.W.C.P. Verstappen. Ef-
ficiently simulating viscous flow effects by means of regularization turbulence modeling and local
1. MRI : Magnetic resonance imaging.
22ème Congrès Français de Mécanique Lyon, 24 au 28 Août 2015
grid refinement. Seventh International Conference on Computational Fluid Dynamics (ICCFD7),
Hawaii, paper ICCFD-2503, 2012.
[10] V. Puzikova and I. Marchevsky. Application of the LS-STAG immersed boundary method for
numerical simulation in coupled aeroelastic problems. Proceedings of the 11th World Congress
on Computational Mechanics (WCCM XI), E. Oñte, J. Oliver and A. Huerta (Eds), 20-25 July
2014, Barcelona, Spain., 2014.
[11] V. Puzikova and I. Marchevsky. Extension of the LS-STAG cut-cell immersed boundary method
for RANS-based turbulence models. Proceedings of the International Summer School-Conference
"Advanced Problems in Mechanics", June 30-July 5, St. Petersburg, Russia., 2014.
[12] Y. Cheny, F. Nikfarjam, and O. Botella. Towards a fully 3D version of the LS-STAG immersed
boundary/cut-cell method. Chengdu, China, 14-18 July 2014. Eighth International Conference on
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[13] C. Rigal. Comportement de fluides complexes sous écoulement : Approche expérimentale par
résonance magnétique nucléaire et techniques optiques et simulations numériques. PhD Thesis,
Institut National Polytechnique de Lorraine, 2012.
[14] R. Falgout, A. Cleary, J. Jones, E. Chow, V. Henson, C. Baldwin, P. Brown, P. Vassilevski, and
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The LS-STAG immersed boundary/cut-cell method for non-Newtonian flows in quasi-3D geometries

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The LS-STAG immersed boundary/cut-cell method for non-Newtonian flows in quasi-3D geometries

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