Title: Two-dimensional off-lattice Boltzmann model for van der Waals fluids with variable temperature,
Abstract: We develop a two-dimensional Lattice Boltzmann model for liquid-vapour
systems with variable temperature. Our model is based on a single particle
distribution function expanded with respect to the full-range Hermite
polynomials. In order to ensure the recovery of the hydrodynamic equations for
thermal flows, we use a fourth order expansion together with a set of momentum
vectors with 25 elements whose Cartesian projections are the roots of the
Hermite polynomial of order Q = 5. Since these vectors are off-lattice, a
fifth-order projection scheme is used to evolve the corresponding set of
distribution functions. A fourth order scheme employing a 49 point stencil is
used to compute the gradient operators in the force term that ensures the
liquid-vapour phase separation and diffuse reflection boundary conditions are
used on the walls. We demonstrate at least fourth order convergence with
respect to the lattice spacing in the contexts of shear and longitudinal wave
propagation through the van der Waals fluid. For the planar interface, fourth
order convergence can be seen at small enough lattice spacings, while the
effect of the spurious velocity on the temperature profile is found to be
smaller than 1.0%, even when T w ' 0.7 T c . We further validate our scheme by
considering the Laplace pressure test. Galilean invariance is shown to be
preserved up to second order with respect to the background velocity. We
further investigate the liquid-vapour phase separation between two parallel
walls kept at a constant temperature T w smaller than the critical temperature
T c and discuss the main features of this process.