Elsevier

Computers & Fluids

Volume 198, 15 February 2020, 104396
Computers & Fluids

A SIMPLE-based algorithm with enhanced velocity corrections: The COMPLEX method

https://doi.org/10.1016/j.compfluid.2019.104396Get rights and content

Highlights

  • A SIMPLE-based algorithm, named COMPLEX, is developed using a Taylor expansion.

  • A Fourier analysis shows that COMPLEX has better performance than SIMPLE/SIMPLEC.

  • The stability results are verified solving incompressible flow simulations.

  • A better convergence rate is found using COMPLEX with high relaxation factors.

Abstract

This paper introduces a new pressure-velocity coupling algorithm based on the SIMPLEC method. The new approach considers the neighbour velocity corrections of SIMPLEC as a Taylor series expansion, introducing a first-order term to increase the accuracy of the approximation. The new term includes a velocity correction gradient which is assumed to be a scalar matrix constrained by means of a mass conservation equation. The stability of the method is analyzed via a Fourier decomposition of the error showing a better convergence rate than SIMPLE and SIMPLEC for high relaxation factors. The new method is tested in two incompressible laminar flow problems. Then, the analysis is extended to a turbulent flow case. In all cases, the conclusions of the stability analysis are verified. The current proposal sets a theoretical baseline for further improvements of SIMPLE-based algorithms.

Introduction

A major numerical difficulty for solving the incompressible Navier-Stokes equations arises from the coupling between the pressure and velocity fields. Several strategies have been proposed to tackle this problem which can be classified as coupled and segregated methods. The first group solves all the governing equations at once for all the unknowns [1], [2], [3], [4], [5], [6]. Hence, pressure and velocity are assembled together and the non-linearity of the convective term is treated iteratively. Segregated algorithms, in contrast, solve the equations in a sequential fashion for a single variable at a time. The procedure is iterative and ends when the unknowns satisfy a certain convergence criterion [7], [8], [9]. In general, segregated strategies are superior in terms of memory saving and computational performance but their numerical stability may not be guaranteed when solving strongly-coupled physical problems [6], [10]. Within these group of methods, the penalty method [11], the artificial compression method [12] and projection methods [11], [13], [14] are among the most adopted in CFD codes nowadays [10]. In the projection methods, an initial estimation of the velocity is corrected by means of a pressure equation where the velocity field is forced to satisfy the divergence-free condition. In particular, the Fractional Step Methods [15] rely on a temporal splitting on the velocity where a first estimate is obtained with the momentum equation using an initial guess of the pressure. Then, the remaining velocity contribution is computed by solving a pressure equation based on the mass balance.

Under the scope of the projection methods, a new line of strategies was introduced by Patankar and Spalding with the development of the SIMPLE algorithm [16]. In this method, a pressure equation is derived after introducing the momentum equation into the mass balance. Then, the velocity field is updated based on the momentum equation including the new values of the pressure field. This procedure is iterative and employs relaxation factors to improve the stability and convergence rate of the method. Focusing on these last aspects, further improvements have been developed taking over the years the SIMPLE algorithm as a starting point. Among them, the SIMPLER algorithm [17] enhances the pressure-velocity coupling by solving an additional pressure equation. Another relevant improvement on SIMPLE was proposed by Van Doormal and Raithby [18], who included an approximation of the neighbour velocity corrections to build the pressure equation and the velocity update. This algorithm, called SIMPLEC, suppresses the need for relaxing the pressure field to guarantee stability. In order to solve transient problems, Issa proposed the PISO method [19], [20] which consists on including corrector sub-steps to enforce mass conservation without relaxation. Many variants of the SIMPLE method were developed seeking to enhance the robustness and convergence rate of the standard SIMPLE algorithm [21], [22], [23], [24]. Their particular advantages and drawbacks can be found in the literature [9], [25], [26]. Despite the aforementioned efforts to improve SIMPLE-like algorithms, the errors introduced by the decoupling of the variables cannot be completely eliminated which makes the topic still relevant nowadays [27], [28], [29], [30], [31], [32].

The error introduced by the SIMPLE algorithm comes from neglecting the neighbour velocity to decouple pressure and velocity. In practice, this approximation requires relaxing the pressure to achieve a stable behaviour towards convergence of the iterative process. Regarding this, the SIMPLEC algorithm considers the neighbour velocity correction equal to the one of the current cell. In other words, it considers the velocity correction as a constant field in the proximity of each cell. This process, in most cases, enhances the ranges of stability avoiding the need to relax the pressure field. The SIMPLEC approximation may be seen as an estimation of the neighbour velocities via a zero-order truncation of a Taylor series expansion. In this context, this work proposes to employ this numerical insight by taking into account a first-order term of the same Taylor series expansion. The approach brings up the need for determining the derivative of the velocity correction, which is a topic of discussion of the present work.

This article is organized as follows. Section 2 describes the SIMPLE and SIMPLEC algorithms and then, in Section 3, the basis and theoretical background of the new method are explained. Subsequently, in Section 4, a stability analysis based on Fourier decomposition is performed for the three mentioned algorithms. Next, in Section 5, a set of numerical problems are solved to calibrate and investigate the performance of the new proposal. Finally, a discussion of the results and final comments are presented in the last section.

Section snippets

Theoretical basis of the SIMPLE-based algorithms

This section presents a general approach for solving the steady-state, incompressible Navier-Stokes equations via segregated pressure-based methods of the SIMPLE family. This is done under the framework of the Finite Volume Method (FVM) on collocated grids with the Rhie-Chow correction [33]. First, the standard SIMPLE algorithm for the pressure and velocity coupling is explained [16]. Then, the SIMPLEC algorithm is introduced [18] following the same formalism and focusing on the approximation

A COupling Method for Pressure Linked Equations based on series eXpansions: COMPLEX

The approximation of the neighbour velocity corrections of SIMPLEC given by Eq.  (2.22) can be seen as a zero-order truncation of a Taylor series expansion over uN. This work proposes to enhance this approximation by preserving first order terms of the same expansion,uN=uP+xPN·uf,where xPN is a vector that goes from the cell centroid P to the cell centroid N. Thus, HP may be written as,HP=(NaN)uP+(NaN)(xPN·uf),which combined with Eq.  (2.24) leads to,HP=(a˜PaP)a˜P(pPpP0)+aPa˜P

Stability analysis

In order to investigate the numerical stability of the new method, an analysis based on a Fourier decomposition of the error is performed. This technique has proven to be a powerful tool to determine the numerical and physical conditions that guarantee a stable solution for linear equations. Moreover, the methodology is extendable to the analysis of multi-variable discrete systems that are addressed by segregation of the variables. The main feature of this approach is to condense the spatial

Test cases

In this section, the COMPLEX method is analysed in incompressible and stationary Newtonian flow problems. The numerical performance of the method is compared with SIMPLE and SIMPLEC by measuring the total number of iterations required to achieve a convergence criterion.

The details of each problem are explained followed by the definition of the convergence criterion adopted. Two different analysis over the COMPLEX method are performed in laminar flow problems: first, a series of simulations are

Conclusions

This paper presented a new segregated algorithm to solve incompressible and stationary flow problems in the context of the FVM, called COMPLEX. It is based on the SIMPLEC method where the neighbour velocity correction is enhanced using a first-order Taylor series expansion. The current procedure requires computing a velocity derivative which is approximated as a scalar matrix defined by a mass balance equation. The contribution of the first-order term of the Taylor expansion is included in the

Declaration of Competing Interest

None.

Acknowledgments

The authors would like to thank ANPCyT for the grant PICT 2016-2908 and CONICET for the funding through doctoral and postdoctoral scholarships. Also, the authors wish to thank the Universidad Nacional del Litoral, Universidad Tecnológica Nacional and Universidad Nacional de Rosario for their support.

References (43)

  • J Aoussou et al.

    Iterated pressure-correction projection methods for the unsteady incompressible Navier–Stokes equations

    J Comput Phys

    (2018)
  • HJ Aguerre et al.

    An oscillation-free flow solver based on flux reconstruction

    J Comput Phys

    (2018)
  • T.F. Miller et al.

    A Fourier analysis of the IPSA/PEA algorithms applied to multiphase flows with mass transfer

    Comput fluids

    (2003)
  • C.M. Venier et al.

    On the stability analysis of the PISO algorithm on collocated grids

    Comput Fluids

    (2017)
  • Z Mazhar

    A procedure for the treatment of the velocity-pressure coupling problem in incompressible fluid flow

    Numerical Heat Transfer Part B

    (2001)
  • Z Mazhar

    An enhancement to the block implicit procedure for the treatment of the velocity-pressure coupling problem in incompressible fluid flow

    Numer Heat Transfer Part B

    (2002)
  • M Darwish et al.

    A fully coupled Navier–Stokes solver for fluid flow at all speeds

    Numer Heat Transfer Part B

    (2014)
  • J.H. Ferziger et al.

    Computational methods for fluid dynamics

    (2012)
  • HK Versteeg et al.

    An introduction to computational fluid dynamics: the finite volume method

    (2007)
  • F Moukalled et al.

    The finite volume method in computational fluid dynamics: an advanced introduction with openfoam(r) and matlab

    (2015)
  • H Wang et al.

    Literature review on pressure–velocity decoupling algorithms applied to built-environment CFD simulation

    Build Environ

    (2018)
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