Olivier BOTELLA(olivier.botella@univ-lorraine.fr)

Yoann CHENY

Mazigh AIT-MESSAOUD

Adrien PERTAT

Claire RIGAL

LEMTA – Université de Lorraine, CNRS

NANCY (France)

EUROMECH Colloquium 549

Immersed Boundary Methods: Current Status and Future Research Directions

17-19 June 2013, Leiden, The Netherlands

The LS-STAG Cut-Cell / Immersed Boundary Method. Basics of the discretization and application to

non-Newtonian and viscoelastic flows

Outline of the talk :

• Presentation of the LS-STAG immersed boundary method• Basics of the method for Newtonian fluids • Principle of energy-conserving discretization in the cut-cells

• Application to rheologically complex fluids (shear-thinning, viscoelastic)• Overview of the method for viscoelastic fluids • Extension to non-Newtonian fluids

• Accuracy tests for Taylor-Couette analytical solution• Newtonian fluid• non-Newtonian (power-law) fluid• Viscoelastic (Oldroyd-B) fluid

• Unified discretization of in Cartesian & cut-cellsTotally staggered mesh (no spurious oscillations)

Cartesian cell Cut-cell Stress tensor in cut-cell

Overview of the LS-STAG immersed boundary (IB) / cut-cell method

• Sharp representation of IB boundary by level-set (LS) – Efficient computation of geometry parameters of

cut-cells (volume, face areas, boundary conditions, …)– Flow equations actually solved in cut-cells (not

interpolated) : velocity gradients at IB boundary are accurately computed.

– No domain remeshing for flows in moving gemetries

• LS-STAG means Level Set-STAGgered : Extension of the MAC method to irregular geometries• Finite-volume skew-symmetric discretization in Cartesian & cut-cells (Vertappen & Veldman,

JCP 2003)

Principles of the LS-STAG discretization for incompressible flows

• Discretization of the Navier-Stokes equations in the 4 types of fluid cells …

… such as the global conservation properties (conservation of total mass, momentum, kinetic energy) are preserved at the discrete level :

Skew-symmetric discretization of convective flux

• Usual central scheme in half CV (i,j), e.g. :

• LS-STAG discretization in half CV (i+1,j)

• Coeffs. α,β, γ determined by skew-symmetry condition . For line (i,j), it yields:

• The methodology can be applied for each half CV independently.

• The central discretization gives :

by local conservation of the fluxes, and :

• Finally, LS-STAG flux is completely defined as :

• Special quadrature/ weighted interpolation in each control volumes, i.e. :

where the weights are defined for each type of cut-cells such as the method is conservative

Extension of LS-STAG method to viscoelastic flows:

• Fully staggered arrangement for the elastic stresses to prevent spurious oscillations :

• Oldroyd-B transport Eq. for elastic stress tensor

• The method has been validated for benchmark viscoelastic flows in contraction geometries (ASME 2010, ICCFD6, 2012)

• As in Newtonian method, midpoint quadrature on each face gives :

• Shear rate at faces center is interpolated w/ quadratures :

• Stress tensor in Navier-Stokes eqns: with shear-rate • Direct application of the numerical tools developed forNewtonian and viscoelastic flows :

• Newtonian difference quotients for • Interpolations/quadratures developed for

viscoelastic constitutive equation

• Discretization of viscous flux for u :

Extension of LS-STAG method to non-Newtonian flows (1/2) :

• Newtonian difference quotient for normal and shear stresses, except for at IB boundary where there is no straightforward formula !

Extension of LS-STAG method to non-Newtonian flows (2/2) :

• Discretization of depends on the type of 2 adjacent cut-cells. If is trapezoidal and can be :

• Case 1 : is a pentagonal cut-cell:

(one sided-quotient)

• Case 2 : is a trapezoidal cut-cell:

(as if the IB of both cut-cells was alignated)

• Case 3 : is a triangular cut-cell:

• Same methodology is used for the computation of forces (skin-friction), moments, etc …, acting at IB boundary.

0,001

0,01

0,1

1

10

0,01 0,1 1 10 100

Visc

osity

(Pa.

s)

Shear stress (Pa)

experimental data

Cross model

Constant viscosity (1.62 Pa.s)

0.40%

0.10%

T= 20°C0.30%

0.20%

• Simulation with LS-STAG code(non-Newtonian Cross model)

PIV velocimetry

Flow of Xanthan 10%,

FLUENT (220,000 cells)

LS-STAG (88,000 cells)

• Experimental database by C. Rigal (PhD, 2012) for Xanthan flow (shear thinning, elastic)

Application : Shear-thinning fluids between eccentric cylinders (ICCFD6 2012, AERC 2013)

• Comparison with experiments and study of the recirculation zone presented at ICCFD6 2012, AERC 2013

• Concentric Taylor-Couette flow:• Power-Law viscosity :

( n : power-law index)

• Analytical solution available for all models :

• Newtonian fluid :• Non-Newtonian (shear-thinning) fluid :

• Viscoelastic (Oldroyd-B) fluid : . Analytical solution for elastic stress is :

Validation of LS-STAG method : spatial accuracy (1/4)

• Numerical parameters :• 4 uniform grids :

• Taylor number is 1000.

• Symmetry of the numerical domain is broken to avoid superconvergence effects . Center of concentric cylinders offset : . Play it fair !

Spatial accuracy (2/4) : Newtonian Fluid• Comparison of the LS-STAG method with the Staircase method :

LS-STAG method is near second-order accurate, Staircase is only first order

Pressureu-velocity

Whole fluid domain(i.e. error in the cut-cells)

90 % of fluid domain 90 % of fluid domain

u-velocity

Whole fluid domain

• Comparison of the LS-STAG method with the Staircase method :

90 % of fluid domain90 % of fluid domain

u-velocity u-velocity Torque at inner cylinder

(i.e. error in the cut-cells)

Shear-rate and torque accurately computed at IB boundary by the LS-STAG method

Spatial accuracy (3/4) : Non-Newtonian Fluid

Whole fluid domain

• Comparison of the LS-STAG method with the Staircase method :

90 % of fluid domain90 % of fluid domain

u-velocity u-velocity Shear stress

(i.e. error in the cut-cells)

Spatial accuracy (4/4) : Viscoelastic (Oldroyd-B) Fluid

Only a slight improvement regarding the absolute error. Error dominated by Upwind scheme (1st order) !

Concluding Remarks

• LS-STAG method has been successfully applied to :

• Newtonian flows in fixed and moving geometries (JCP, 2010)

• Viscoelastic flows (ASME 2010, paper in review)

• non-Newtonian flows(ICCFD6 2012, AERC 2013)

• Further topics to be addressed :• Towards higher order of accuracy

→ talk of N. James et al.• Extension to 3D

→ talk of H.J.L. van der Heiden et al.• Coupled fluid-solid computations

→ talk of I. Marchevsky & V. Puzikova

Outline of the talk :Principles of the LS-STAG discretization for incompressible flows Extension of LS-STAG method to viscoelastic flows:Extension of LS-STAG method to non-Newtonian flows (1/2) :Extension of LS-STAG method to non-Newtonian flows (2/2) :Validation of LS-STAG method : spatial accuracy (1/4)Spatial accuracy (2/4) : Newtonian FluidSpatial accuracy (3/4) : Non-Newtonian FluidSpatial accuracy (4/4) : Viscoelastic (Oldroyd-B) FluidConcluding Remarks