Analysis of the local truncation error in the pressure‐free projection method for incompressible flows: a new accurate expression of the intermediate boundary conditions

Abstract

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.

Time‐accurate intermediate boundary conditions for large eddy simulations based on projection methods

“…After having solved (9), the second step consists in determining an auxiliary gradient ÿeld such that v * = v n+1 + t∇ n+1 is the decomposition prescribed by the HHD theorem. However, as addressed in References [10,11], the orthogonality of this decomposition, that is v n+1 · ∇ n+1 dV = 0, is not always guaranteed but depends on the prescribed boundary condition n · v n+1 @ , where n is the normal to the boundary oriented in outward direction. Thus, by solving the Poisson problem…”

“…As already addressed in the Introduction, in order for problem (10) to be well posed it is su cient to prescribe that the normal component n · v * @ , equals the normal component of the gradient ÿeld, i.e. t(n · ∇ n+1 @ ), added with that of the exact velocity, i.e.…”

“…The present paper focuses only on the so-called PFP method [7,8,10,11], which is adopted for solving the LES equations for incompressible ows. For unconÿned ows, this method is a second-order time-accurate version of the original Chorin's method [4,14].…”

Spurious transients of projection methods in microflow simulations

“…Over the years, many authors [5,[13][14][15][16][17] studied the effects of different update schemes for the pressure to obtain higher-order (in time) methods. The treatment of boundary conditions also focused research efforts, as they impact on the time accuracy [15,18,19,7,[20][21][22].…”

Análise e implementação de métodos implícitos e de projeção para escoamentos com superfície livre.

“…Em particular, em [13,19,7,6,76,35,94,65,61] exemplos desse tipo de exercício podem ser vericados. Inicialmente, para construir um método semi-implícito estável, consistente e agora de ordem dois, considere a discretização temporal para as equações de Navier-Stokes, baseada em uma discretização de ordem dois, aplicando os parâmetros da tabela 2.1, apresentada no capítulo 2,…”