SUMMARYThis paper is concerned with the development of a new high-order ÿnite volume method for the numerical simulation of highly convective unsteady incompressible ows on non-uniform grids. Speciÿcally, both a high-order uxes integration and the implicit deconvolution of the volume-averaged ÿeld are considered. This way, the numerical solution e ectively stands for a fourth-order approximation of the point-wise one. Moreover, the procedure is developed in the framework of a projection method for the pressure-velocity decoupling, while originally deriving proper high-order intermediate boundary conditions. The entire numerical procedure is discussed in detail, giving particular attention to the consistent discretization of the deconvolution operation. The present method is also cast in the framework of approximate deconvolution modelling for large-eddy simulation. The overall high accuracy of the method, both in time and space, is demonstrated. Finally, as a model of real ow computation, a two-dimensional time-evolving mixing layer is simulated, with and without sub-grid scales modelling.

SUMMARYThe numerical integration of the Navier-Stokes equations for incompressible ows demands e cient and accurate solution algorithms for pressure-velocity splitting. Such decoupling was traditionally performed by adopting the Fractional Time-Step Method that is based on a formal separation between convective-di usive momentum terms from the pressure gradient term. This idea is strictly related to the fundamental theorem on the Helmholtz-Hodge orthogonal decomposition of a vector ÿeld in a ÿ-nite domain, from which the name projection methods originates. The aim of this paper is to provide an original evaluation of the local truncation error (LTE) for analysing the actual accuracy achieved by solving the de-coupled system. The LTE sources are formally subdivided in two categories: errors intrinsically due to the splitting of the original system and errors due to the assignment of the boundary conditions. The main goal of the present paper consists in both providing the LTE analysis and proposing a remedy for the inaccuracy of some types of intermediate boundary conditions associated with the prediction equation. Such evaluations will be directly performed in the physical space for both the time continuous formulation and the ÿnite volume discretization along with the discrete AdamsBashforth=Crank-Nicolson time integration. A new proposal for a boundary condition expression, congruent with the discrete prediction equation is herein derived, fulÿlling the goal of accomplishing the closure of the problem with fully second order accuracy. In our knowledge, this procedure is new in the literature and can be easily implemented for conÿned ows. The LTE is clearly highlighted and many computations demonstrate that our proposal is e cient and accurate and the goal of adopting the pressure-free method in a ÿnite domain with fully second order accuracy is reached.

SUMMARYA general methodology is presented in order to obtain a hierarchy of high-order filter functions, starting from the standard top-hat filter, naturally linked to control volumes flow simulations. The goal is to have a new filtered variable better represented in its high resolved wavenumber components by using a suitable deconvolution. The proposed formulation is applied to the integral momentum equation, that is the evolution equation for the top-hat filtered variable, by performing a spatial reconstruction based on the approximate inversion of the averaging operator. A theoretical analysis for the Burgers' model equation is presented, demonstrating that the local de-averaging is an effective tool to obtain a higher-order accuracy. It is also shown that the subgrid-scale term, to be modeled in the deconvolved balance equation, has a smaller absolute importance in the resolved wavenumber range for increasing deconvolution order. A numerical analysis of the procedure is presented, based on high-order upwind and central fluxes reconstruction, leading to congruent control volume schemes. Finally, the features of the present high-order conservative formulation are tested in the numerical simulation of a sample turbulent flow: the flow behind a backward-facing step.

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