It is well known the Oseen iteration for the stationary Navier–Stokes equations is unconditionally stable. However, it is a coupled type scheme where the velocity and pressure are coupled together at each iteration. By treating pressure explicitly would lead to a decoupled iteration, but this treatment is unstable. In this article, we construct a decoupled and unconditionally stable iteration method to solve the stationary Navier–Stokes equations by adopting the pressure projection method to the temporal disturbed Navier–Stokes system whose solution approximates the steady state solution over time (). We also rigorously prove its unconditional stability. Numerical simulations demonstrate that our iterative method is more efficient and stable than the extensively used T-S and Oseen iterations, and could solve the fluid flow with high Reynolds number.
We present a fully-explicit, iteration-free, weakly-compressible method to simulate immiscible incompressible two-phase flows. To update pressure, we circumvent the computationally expensive Poisson equation and use the general pressure equation which is solved explicitly. In addition, a less diffusive algebraic volume-of-fluid approach is used as the interface capturing technique and in order to facilitate improved parallel computing scalability, the technique is discretised temporally using the operator-split methodology. Our method is fully-explicit and stable with simple local spatial discretization, and hence, it is easy to implement. Several two- and three-dimensional canonical two-phase flows are simulated. The qualitative and quantitative results prove that our method is capable of accurately handling problems involving a range of density and viscosity ratios and surface tension effects.
A modified fifth-order WENO-Z scheme is proposed by analogy with the non-normalized weights of the reformulated fifth-order adaptive order (AO) WENO scheme. We show that if the original fifth-order WENO-AO scheme is rewritten as the form of the conventional WENO combination, the resulting non-normalized weights can be divided into three parts: a constant one term, a local stencil smoothness measure term and a global stencil smoothness measure term. In order to make use of the latter two terms for constructing a modified WENO-Z scheme with enhanced performance, we change the form of the third term and introduce an adaptive scaling factor to adjust the contributions from the second and third terms. Numerical examples show that the modified fifth-order WENO-Z scheme has the advantage of high resolution in smooth regions and sharp capturing of discontinuities, and it can obtain evidently better results for shocked flows with small-scale structures compared with the recently developed WENO-Z+ and WENO-Z+M schemes.
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