This paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.

In this article we detail the methodology developed to construct an efficient interface description technique — the robust conservative level set (RCLS) — to simulate multiphase flows on mixed-element unstructured meshes while conserving mass to machine accuracy. The approach is tailored specifically for industry as the three-dimensional unstructured approach allows for the treatment of very complex geometries. In addition, special care has been taken to optimise the trade-off between accuracy and computational cost while maintaining the robustness of the numerical method. This was achieved by solving the transport equations for the liquid volume fraction using a WENO scheme for polyhedral meshes and by adding a flux-limiter algorithm. The performance of the resulting method has been compared against established multiphase numerical methods and its ability to capture the physics of multiphase flows is demonstrated on a range of relevant test cases. Finally, the RCLS method has been applied to the simulation of the primary breakup of a flat liquid sheet of kerosene in co-flowing high-pressure gas. This quasi-DNS/LES computation was performed at relevant aero-engine conditions on a three-dimensional mixed-element unstructured mesh. The numerical results have been validated qualitatively against theoretical predictions and experimental data. In particular, the expected breakup regime was observed in the simulation results. Finally, the computation reproduced faithfully the breakup length predicted by a correlation based on experimental data. This constitutes a first step towards a quantitative validation.