Lattice Boltzmann Methods for Multiphase Flow Simulations across Scales
Falcucci Giacomo
;
Ubertini Stefano
;
Biscarini Chiara
;
Di Francesco Silvia
;
Chiappini Daniele
;
Palpacelli Silvia
;
De Maio Alessandro
;
Succi Sauro
The simulation of multiphase flows is an outstanding challenge, due to the inherent complexity of the underlying physical phenomena and to the fact that multiphase flows are very diverse in nature, and so are the laws governing their dynamics. In the last two decades, a new class of mesoscopic methods, based on minimal lattice formulation of Boltzmann kinetic equation, has gained significant interest as an efficient alternative to continuum methods based on the discretisation of the NS equations for non ideal fluids. In this paper, three different multiphase models based on the lattice Boltzmann method (LBM) are discussed, in order to assess the capability of the method to deal with multiphase flows on a wide spectrum of operating conditions and multiphase phenomena. In particular, the range of application of each method is highlighted and its effectiveness is qualitatively assessed through comparison with numerical and experimental literature data.
Bartlett's decomposition provides the distributional properties of the elements of the Cholesky factor of $A=G^TG$ where the elements of $G$ are i.i.d. standard Gaussian random variables.
In this paper the most general case where the elements of $G$ have a joint multivariate Gaussian density is considered.
For G open bounded subset of R^2 with C^1 boundary, we study the regularity of the variational solution u in H^1_0(G) to the quasilinear elliptic equation of Leray-Lions type: -div A(x,Du)=f , when f belongs to the Zygmund space L(log L)^{\delta}, \delta>0. As an interpolation between known results for \delta=1/2 and \delta=1 of [Stampacchia] and [Alberico-Ferone], we prove that |Du| belongs to the Lorentz space L^{2, 1/\delta}(G) for \delta in [1/2, 1].
Elliptic equations; Gradient regularity; Grand lebesgue spaces; Zygmund spaces
