Lattice Boltzmann methods for complex fluids and multiphase flow
Background
Simple fluids are characterised by a linear, isotropic relation between the
stress and the instantaneous strain rate. Complex or non-Newtonian fluids are characterised
by a more complicated relationship. These include fluids with an instantaneous but
anisotropic stress-strain relation, such as strongly magnetised plasmas and liquid
crystals, and fluids whose stress depends on their deformation history, such as
viscoelastic materials. Reliable numerical predictions of flows of complex fluids
require intensive computations and are notoriously difficult to obtain. This is
particularly challenging for fluids with memory, such as polymer melts and solutions,
where the deviatoric stress evolves according to an additional, generally highly
nonlinear, partial differential equation (PDE).
Techniques and Challenges
The lattice Boltzmann equation (LBE) is a promising alternative to the traditional methods of computational fluid dynamics (CFD). Rather than discretising the system of PDEs of classical continuum mechanics directly, the LBE is derived from a velocity-space truncation of the Boltzmann equation of classical kinetic theory. The kinetic formulation yields a linear, constant coefficient hyperbolic system where all nonlinearities are confined to algebraic source terms. The linear differential operators may be discretised exactly by integrating along their characteristics, while the hydrodynamic equations with their nonlinear convection terms are recovered by seeking slowly varying solutions to the kinetic equations.
The locality of the resulting algorithm
allows the LBE to exploit massively parallel modern computer architectures, including
graphics processing units (GPUs), leading to very fast computations. This feature
is highly desirable in fields such as complex fluids where intensive, time-consuming,
computations are commonplace. The major challenge is to embed the complicated constitutive
equation consistently into the discrete kinetic equation without sacrificing the
inherent advantages of the lattice Boltzmann model.
Results
Solutions to practical flow problems require boundary conditions. Numerical
methods for complex flows must accurately satisfy these conditions. In a departure
from the more traditional technique used in lattice Boltzmann methods, we have considered
imposing boundary conditions directly upon the hydrodynamic moments of the discrete
kinetic equation, and then translating these into the particle basis. This technique
has been successfully applied to a variety of flows, including rarefied flows
in microchannels.
By extending this methodology
further, we have derived lattice Boltzmann stress conditions which include Knudsen
and non-Newtonian effects. We also have an ongoing interest in multiphase LBEs, particularly
when applied to flows in porous media (see REE1).
The Future
The main objectives of this project are summarised as follows:
- To study plasmoid reconnection and magneto-hydrodynamic turbulence
- To develop lattice Boltzmann models for fluids with instantaneous but anisotropic stress-strain relations and in particular, to extend to liquid crystals the recently developed approach for Braginskii magneto-hydrodynamics
- To extend the analogy between model viscoelastic equations such as Oldroyd-B and the evolution equation for the deviatoric stress in kinetic theory, aiming to construct an objective, frame invariant LBE for memory fluids
- To further study lattice Boltzmann methodologies for multiphase flow in porous media
To achieve these goals collaborations are active with
- Dr Paul Dellar, Oxford
- Prof. Ravi Samtaney, KAUST
- Prof. Shuyu Sun and Ms Rebecca Allen, KAUST
- Dr Apala Majumdar, Bath
- Prof. Tim Phillips, Cardiff
To help strengthen the UK lattice Boltzmann community and encourage further collaborations we have organised informal workshops and discussion groups. Our 2010 meeting was held in New College and in Wadham college in 2011.
References
[12/25] Reis T., Wilson H.J.: Rolie-Poly fluid flowing through constrictions: Two distinct instabilities, J. Non-Newt. Fluid Mech., 195 77-87 2013
[12/23] Reis T., Dellar P.J.: Lattice Boltzmann simulations of pressure-driven flows in microchannels using Navier-Maxwell slip boundary conditions, Phys. Fluids 24 112001, 2012
[10/08] Reis T., Dellar P.J.: A volume-preserving sharpening approach for the propagation of sharp phase boundaries in multiphase lattice Boltzmann simulations, Computers & Fluids 46 417-421, 2011
[1] Dellar P.J.: Lattice kinetic schemes for magnetohydrodynamics, J. Comput. Phys. 179 95-126, 2002
[2] Samtaney R., Loureiro N.F., Schekochihin A.A., Uzdensky D., Cowley S.C.: Formation of plasmoid chains in magnetic reconnection, Phy Rev Lett, 103, 105004
[3] García-Colín L.S., Velasco R.M., Uribe F.J.: Beyond the Navier–Stokes equations: Burnett hydrodynamics, Physi Rep 465 149–189
[4] Bird R., Hassager O.: Dynamics of Polymeric Liquids, Wiley 1987