Joint EUROMECH / ERCOFTAC Colloquium 549“Immersed Boundary Methods: Current Status and Future Research Directions”Leiden, The Netherlands, 17-19 June 2013

The LS-STAG immersed boundary method for non-Newtonian and viscoelasticflows in irregular geometries. Applications to the flow of shear-thinning liquids

between eccentric rotating cylinders

Olivier Botella1, Yoann Cheny1, Mazigh Ait-Messaoud1, Adrien Pertat1 and Claire Rigal1

1 LEMTA, Universit de Lorraine, CNRS. 2 avenue de la Fort de Haye - TSA 60604, 54518 Vandœuvre-ls-Nancy, Franceolivier.botella@univ-lorraine.fr

ABSTRACT

This communication presents a progress report on an ongoing project aiming at the computation of rheologically complex fluid flowswith a realistic constitutive law, which would take into account the pseudoplastic, viscoelastic and thixotropic behavior of the materials.The flow solver is based on the LS-STAG method, which is an immersed boundary (IB)/cut-cell method that allows the computationof flows in irregular or moving geometries on fixed Cartesian meshes, reducing thus the bookkeeping of body-fitted methods. Thediscretization in the cut-cells (i.e., the computational cells which are cut by the irregular boundary) is achieved by requiring that theglobal conservation properties of the Navier-Stokes equations are satisfied at the discrete level [7], resulting in a stable and accuratemethod and, thanks to the level-set representation of the IB boundary, at low computational costs. The LS-STAG method has beenvalidated in [3, 4] for canonical Newtonian flows in both fixed and moving geometries, and has been presented in the previous IBMcolloquium (Amsterdam, 2009).

Figure 1: Sketch of the 4 generic fluid cells of the LS-STAG mesh, and location of the discrete velocities, normal and shear stresses(pressure is located as the normal stresses).

In a recent work [1, 2] we have applied the LS-STAG method to viscoelastic flows, for which accurate discretization of the viscousstresses up to the cut-cells is of paramount importance for stability and accuracy. For this purpose, the LS-STAG discretization of theNewtonian stresses has been extended to the transport equation of the elastic stresses (Oldroyd-B model), such that the node-to-nodeoscillations of the stress variables are prevented by using a velocity-pressure-stress (v− p− τ) staggered arrangement (see Fig. 1). Thediscretization of the Oldroyd-B constitutive equation is performed by constructing special quadratures for the volumic terms whichyield a globally conservative discretization up to the cut-cells. Results on popular benchmarks for viscoelastic flows shows that our IBmethod demonstrates an accuracy and robustness comparable to body-fitted methods up to large levels of elasticity.

The next step is to incorporate the non-Newtonian (or pseudoplastic) behavior in the LS-STAG code, by considering that the stresstensor takes the form τ = η(γ)D, where the viscosity η is modelled by popular non-Newtonian laws such as the power-law or Crossmodels, and the shear rate γ is related to the second invariant of the rate-of-strain tensor D :

γ =√

12 D : D =

√2[

∂u∂x

]2+

[∂u∂y

+∂v∂x

]2+2[

∂v∂y

]2. (1)

The crucial part for taking into account shear-thinning or shear-thickening effects is the computation of the shear rate in the vicinity ofthe immersed boundary. We have been able to achieve an accurate discretization that fits elegantly in the framework of the v− p− τ

arrangement and the special quadratures developed previously for viscoelastic flows. In this presentation, we intent to give a thoroughexamination on the global accuracy of the LS-STAG method, and especially the computation of the stresses and non-Newtonianviscosity at the IB where the shear is maximal. For this purpose, we will consider the steady 2D Taylor-Couette flow for which ananalytical solution can be obtained for the cases of Newtonian, power-law and Oldroyd-B fluids.

Finally, we will apply the LS-STAG code to the flow of various shear-thinning xanthan solutions in a wide-gap non-coaxial Taylor-Couettereactor (see Fig. 2), for which rheological characterization, experimental flow measurements (PIV) and FLUENT simulations haverecently been performed in our group [5, 6]. Our numerical investigation will give new insight on the flow patterns (onset, size andposition of the recirculation zone), and will firmly correlate them to global flow properties such as shear-thinning index, generalizedReynolds number and torque ratio at the cylinders.

Figure 2: Streamlines for the flow of xanthan 0.10% with PIV visualization of [6, 5] , FLUENT computations [6, 5] and LS-STAGcomputations : (a) xanthan 0.10% at eccentricity δ = 0.60 and inner cylinder rotation Ω = 10rad · s−1 ; (b) xanthan 0.10% at δ = 0.75and Ω = 1rad · s−1 ; (c) xanthan 0.10% at δ = 0.75 and Ω = 10rad · s−1.

REFERENCES

[1] O. Botella and Y. Cheny. The LS-STAG method for viscous incompressible flows in irregular geometries: Basics of thediscretization and application to viscoelastic flows. In American Society of Mechanical Engineers, Fluids Engineering Division(Publication) FEDSM, volume 1, pages 2441–2451, 2010.

[2] Y. Cheny and O. Botella. Application of the LS-STAG Immersed Boundary Method to Viscoelastic Flow Computations. Computers& Fluids, in Review.

[3] Y. Cheny and O. Botella. An immersed boundary/level-set method for incompressible viscous flows in complex geometries withgood conservation properties. Europ. J. Comput. Mech., 18, 561-587, 2009.

[4] Y. Cheny and O. Botella. The LS-STAG method : A new immersed boundary / level-set method for the computation ofincompressible viscous flows in complex moving geometries with good conservation properties. J. Comput. Phys., 229, 1043-1076,2010.

[5] C. Rigal. Comportement de fluides complexes sous coulement : Approche exprimentale par rsonance magntique nuclaire ettechniques optiques et simulations numriques. PhD Thesis, Institut National Polytechnique de Lorraine, 2012.

[6] C. Rigal, C. Baravian, G. Belbelkacem, S. Skali-Lami, and M. Lebouch. Shear thinning influence on the secondary flow in eccentriccylinders: a rheo-PIV study. Rheologica Acta, in revision.

[7] R. W. C. P. Verstappen and A. E. P. Veldman. Symmetry-preserving discretization of turbulent flow. J. Comput. Phys., 187,343-368, 2003.