Olivier BOTELLA([email protected])
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
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