In this paper, we study an exponential time differencing method for solving the gauge system of incompressible viscous flows governed by Stokes or Navier-Stokes equations. The momentum equation is decoupled from the kinematic equation at a discrete level and is then solved by exponential time stepping multistep schemes in our approach. We analyze the stability of the proposed method and rigorously prove that the first order exponential time differencing scheme is unconditionally stable for the Stokes problem. We also present a compact representation of the algorithm for problems on rectangular domains, which makes FFT-based solvers available for the resulting fully discretized system. Various numerical experiments in two and three dimensional spaces are carried out to demonstrate the accuracy and stability of the proposed method.

In this paper, an alternative lattice Boltzmann (LB)model for incompressible flows is proposed. By modifying directly the moments of the equilibrium distribution function (EDF), the continuous expression of the EDF in tensor Hermite polynomials is derived using the moment expansion and then discretizedwith the discrete velocity vectors of the D2Q9 lattice. The present model as well as its counterpart, the incompressible LB model proposed by Guo, reproduces the incompressible Navier-Stokes (N-S) equations for both steady and unsteady flows. Besides, an alternative pressure formula, which represents the pressure as the diagonal part of the stress tensor, is adopted in the present model. Furthermore, in order to enhance the stability of the present LB model, an additional relaxation time pertaining to the non-hydrodynamic mode is added to the BGK collision operator. The present LB model is validated by two benchmark tests: the cavity flow with different Reynolds number (Re) and the flow past an impulsively started cylinder at Re=40 and 550.

This paper presents the extension of a well-established Immersed Boundary (IB)/cut-cell method, the LS-STAG method (Y. Cheny & O. Botella, J. Comput. Phys. Vol. 229, 1043-1076, 2010), to viscoelastic flow computations in complex geometries. We recall that for Newtonian flows, the LS-STAG method is based on the finite-volume method on staggered grids, where the IB boundary is represented by its level-set function. The discretization in the cut-cells is achieved by requiring that global conservation properties equations be satisfied at the discrete level, resulting in a stable and accurate method and, thanks to the level-set representation of the IB boundary, at low computational costs.

In the present work, we consider a general viscoelastic tensorial equation whose particular cases recover well-known constitutive laws such as the Oldroyd-B, White-Metzner and Giesekus models. Based on the LS-STAG discretization of the Newtonian stresses in the cut-cells, we have achieved a compatible velocity-pressure-stress discretization that prevents spurious oscillations of the stress tensor. Applications to popular benchmarks for viscoelastic fluids are presented: the four-to-one abrupt planar contraction flows with sharp and rounded re-entrant corners, for which experimental and numerical results are available. The results show that the LS-STAG method demonstrates an accuracy and robustness comparable to body-fitted methods.

High-order discretization techniques offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lower-order methods. However, efficient and universally-applicable high-order discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes and for incompressible/low-speed flows. A novel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for the solution of the conservation equations of viscous, incompressible flows on three-dimensional unstructured meshes. Similar to finite element methods, coordinate transformations are used to maintain the scheme’s order of accuracy even when dealing with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes equations and the resulting discretized equations are solved with a parallel implicit Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach is adopted and the resulting temporal derivatives are discretized using the family of high-order backward difference formulas (BDF). The proposed finite-volume scheme for fully unstructured mesh is demonstrated to provide both fast and accurate solutions for steady and unsteady viscous flows.

We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows.Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes.Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem.An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh.Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms.Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem.It has been developed following two main ideas.On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes.On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitkäranta technique.We prove that, under usual regularity assumptions for the solution (each component of the velocity in H2(∞) and pressure in H1(∞)), the scheme is first order convergent in the usual finite volume discrete H1 norm and the L2 norm for respectively the velocity and the pressure, provided, in particular, that the approximation of the mass balance flux is of second order.With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles triangulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions.Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete L2 norm.