Eulerian mesh

The DNS methods are the most detailed they typically advance the gas-liquid interface through the flow field in an Eulerian mesh and do not require empirical constimtive equations. However, the DNS methods are limited to a very small number of particles/bubbles (e.g., typically 10s of bubbles in the flow field) due to computational limitations. Most industrial applications require high superficial gas velocities, and therefore the Eulerian-Eulerian method is preferred (Dudukovic, 2002 Pan et al., 2000) and will be summarized later. 

FLUENT provides the volume of fluid model (VOF) for the description of separated multiphase flows. The VOF model is based on the resolution of the phase interface in a fixed Eulerian mesh. The conservation equations here are not solved separately for the individual phases but rather for the entire calculation domain with material properties averaged across the phases. For this purpose, an additional conservation equation is introduced for the volume fraction f in the continuous phase. A cell contains either the dispersed phase only f = 0), the continuous phase only f = 1), or the phase interface (0 < / < 1). In order to avoid blurring... 

The equations, written in finite-difference form appropriate to a fixed (Eulerian) mesh, are used to determine the dynamics of the fluid. The fluid is moved by a continuous mass transport method. 

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... 

Governing flow equations, originally written in an Eulerian framework, should hence be modified to take into account the movement of the mesh. The time derivative of a variable / in a moving framework is found as... 

Figure 5.4 The finite element mesh configurations in the Arbitrary Lagrangian-Eulerian scheme...

Arbitrary-Lagrangian-Eulerian (ALE) codes dynamically position the mesh to optimize some feature of the solution. An ALE code has tremendous flexibility. It can treat part of the mesh in a Lagrangian fashion (mesh velocity equation to particle velocity), part of the mesh in an Eulerian fashion (mesh velocity equal to zero), and part in an intermediate fashion (arbitrary mesh velocity). All these techniques can be applied to different parts of the mesh at the same time as shown in Fig. 9.18. In particular, an element can be Lagrangian until the element distortion exceeds some criteria when the nodes are repositioned to minimize the distortion. 

The computational code used in solving the hydrodynamic equation is developed based on the CFDLIB, a finite-volume hydro-code using a common data structure and a common numerical method (Kashiwa et al., 1994). An explicit time-marching, cell-centered Implicit Continuous-fluid Eulerian (ICE) numerical technique is employed to solve the governing equations (Amsden and Harlow, 1968). The computation cycle is split to two distinct phases a Lagrangian phase and a remapping phase, in which the Arbitrary Lagrangian Eulerian (ALE) technique is applied to support the arbitrary mesh motion with fluid flow. 

In the recent years different numerical models for the conversion of wood in a packed bed have been presented, e. g, [3-6], Existing models mostly describe the as a porous media by an Eulerian approach, with the cons vation equations for the solid and the gas phase solved with the same mesh. This approach implies that heat and mass transfer can only be taken into account according to the dimensions of the bed but not within the particles itself. Temperature and species distributions are assumed to be homogenous over the fuel particles. Thus, the influence of the particle dimensions on the conversion process can only be captured by simplified assumptions or macrokinetic data. 

The model formulation used with moving meshes is of the arbitrary-Lagrangian-Eulerian (ALE) type. The integral form of the equations governing the incompressible Newtonian fluid in a time-varying control-volume V(t) is written as ... 

The MAC method, which allows arbitrary free surface flows to be simulated, is widely used and can be readily extended to three dimensions. Its drawback lies in the fact that it is computationally demanding to trace a large number of particles, especially in 3D simulation. In addition, it may result in some regions void of particles because the density of particles is finite. The impact of the MAC method is much beyond its interface capmring scheme. The staggered mesh layout and other features of MAC have become a standard model for many other Eulerian codes (even numerical techniques involving mono-phase flows). 

The IB technique [22] is a combined Eulerian-Lagrangian method, in the sense that the flow is solved using a Eulerian approach on a fixed Cartesian mesh, and the interface is represented by a discrete set of points and advected in a Lagrangian... 

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