Theoretical analysis for achieving high‐order spatial accuracy in Lagrangian/Eulerian source terms

Abstract

SUMMARYIn a fully coupled Lagrangian=Eulerian two-phase calculation, the source terms from computational particles must be agglomerated to nearby gas-phase nodes. Existing methods are capable of accomplishing this particle-to-gas coupling with second-order accuracy. However, higher-order methods would be useful for applications such as two-phase direct numerical simulation and large eddy simulation. A theoretical basis is provided for producing high spatial accuracy in particle-to-gas source terms with low computational cost. The present work derives fourth-and sixth-order accurate methods, and the procedure for even higher accuracy is discussed. The theory is also expanded to include two-and three-dimensional calculations. One-and two-dimensional tests are used to demonstrate the convergence of this method and to highlight problems with statistical noise. Finally, the potential for application in computational uid dynamics codes is discussed. It is concluded that high-order kernels have practical beneÿts only under limited ranges of statistical and spatial resolution. Additionally, convergence demonstrations with full CFD codes will be extremely di cult due to the worsening of statistical errors with increasing mesh resolution.

An analysis of the convergence of stochastic Lagrangian/Eulerian spray simulations

“…Early work by Subramaniam and O'Rourke [4] concluded that the convergence of spray simulations was conditional on employing a sufficient number of parcels. Subsequent analysis by Schmidt [5] saw similar trends. If the number of parcels is held constant while the mesh is refined, the error begins to increase after passing an optimum, as shown in Fig.…”

“…They then combined these elements into a full Lagrangian/Eulerian simulation and demonstrated second order convergence. Later, Schmidt [5] studied how second order and higher accuracy schemes converge considering both spatial and statistical error. However, Schmidt only considered liquid-to-gas coupling and only for fixed numbers of parcels per cell.…”

Comparison between Lagrangian and mesoscopic Eulerian modelling approaches for inertial particles suspended in decaying isotropic turbulence

“…Such projection procedure from Lagrangian to Eulerian quantities are widely used to handle the two-way coupling in DPS approach [13][14][15][16]. The salient feature of the configuration is the discontinuous distribution of the Lagrangian velocities due to the particle RUV.…”

“…Firstly, higher order projector weight function should have negative loops leading to unphysical negative particle number density values when very few particles are present. Secondly, the corresponding convergence rate in terms of the particle number is slower [16]. Integral of the different projector kernels over the control volume is unity to ensure the globally (ie.…”

An Eulerian CFD model and X-ray radiography for coupled nozzle flow and spray in internal combustion engines

“…Note that a 0.5 mm mesh resolution or coarser is widely used for Lagrangian-Eulerian spray and engine simulations. Schmidt (2006) showed that a diminishing grid size and very high number of parcels are required to resolve gas-droplet interaction and reduce the statistical error due to Largrangian parcels to achieve grid convergence. The other issue is the dense spray near nozzle region which may not be accurately described with discrete droplet by the Lagrangian approach due to dominant mixing (Siebers, 1998(Siebers, , 1999.…”