# CFD Simulations AC3-12

## Overview of CFD Simulations

Detailed numerical calculations were also performed by Sommerfeld et al. (1992) and Sommerfeld and Qiu (1993) using the two-dimensional axially-symmetric Euler/Lagrange approach without two-way coupling. The fluid flow calculation is based on the time-averaged Navier-Stokes equations in connection with a closure assumption for the turbulence modelling. The solution of the above equations is obtained by using the so-called FASTEST-code (Dimirdzic and Peric, 1990) which incorporates the well-known k-ε two-equation turbulence model and uses a finite-volume approach to descretize the equations. In order to minimize the effects of numerical diffusion in the present calculations, the quadratic, upwind-weighted differencing scheme (QUICK) was used for differencing the convection terms. Furthermore, flux-blending techniques, where the convective flux can be calculated as a weighted sum of the flux expressions from the "upwind" and QUICK differencing schemes (Peric et al., 1988), was used to avoid instabilities and convergence problems that sometimes appear when using higher order schemes. The choice of the solution procedure described above was based on the recommendations of Durst and Wennerberg (1991) who also concluded that for moderate swirl intensities the k-ε turbulence model performs reasonably well.

## Computational Domain and Boundary Conditions Fluid Flow

The present calculations have been performed on a mesh of 80 by 78 grid points in the stream-wise and radial directions, respectively. For two-dimensional axis-symmetric calculations this grid resolution was found to be sufficient as demonstrated by Durst and Wennerberg (1991). The computational domain corresponds exactly to the experimental configuration given in Figure 1. However, in the stream-wise direction it was only extended up to 1.0 m downstream from the inlet. The applied inlet conditions correspond to the measured mean velocity components (i.e. available for all three components) and the measured turbulent kinetic energy. At the walls no-slip conditions were applied in connection with the standard wall function. At the outflow boundary zero-gradients have been assumed.

## Modelling of Particle Phase

Details on the treatment of the dispersed phase can be found in Sommerfeld and Qiu (1993). Here only a brief summary of the main issues is given. The converged solution of the gas flow field was used for the simulations of the particle phase based on a Lagrangian formulation of the basic equations, and a stochastic model was used for simulation the interaction of the particles with the fluid turbulence. For the calculation of the particle phase mean properties, a large number of particles were traced through the flow field, typically around 100,000.

In order to take into account the effect of the wide size spectrum of the glass beads used in the experiments on the particle mean velocities, the velocity fluctuations, and the dispersion process, the numerical calculations were performed considering the particle size distribution.

The effect of the particle phase on the fluid flow was neglected in the present calculations since only very small particle loadings were considered (see Table 1). Furthermore, some simplifications in the equation of motion for the particles have been made, since a gas-solid two-phase flow with a density ratio of ${\displaystyle {\displaystyle \left.\rho _{p}/\rho \sim 2000\right.}}$ was considered. This implies that the added mass effect and the Basset history force have been neglected in the present calculations. As a consequence only the drag force, considering a non-linear term for higher particle Reynolds numbers, and the gravity force were taken into account.

The equations of motion were solved by an explicit Euler method, where the maximum allowable time step was set to be 10 percent of the following characteristic time scales:

• the Stokesian response time of the particle,
• the time required for a particle to cross the mesh and
• the local eddy life-time

The instantaneous fluid velocity components in the above equations are obtained from the local mean fluid velocities and the velocity fluctuations which are randomly sampled from a Gaussian distribution function characterized by and the fluid rms value, σ. The latter is evaluated from the turbulent kinetic energy by assuming isotropic turbulence. The instantaneous fluid velocities seen by the particles are randomly generated by the "discrete eddy concept" (see Sommerfeld et al. 1993; Sommerfeld 2008) and are assumed to influence the particle movement during a certain time period, the interaction time, before new instantaneous fluid velocities are sampled from the Gaussian distribution function. In the present model, the successively sampled fluid velocity fluctuations and the individual components are assumed to be uncorrelated.

The boundary conditions for the particle tracking procedure are specified as follows. At the inlet, the particle velocities and the mass flux are specified according to the experimental conditions. This implies that the actual injected particle size is sampled from the measured local size distributions and the particle velocities are sampled from a normal velocity distribution considering the measured local size-velocity correlations for all three components. A particle crossing the centreline is replaced by a particle entering at this location with opposite radial velocity. For the particle interaction with the solid wall, elastic reflection is assumed (i.e., ${\displaystyle {\displaystyle \nu _{p2}=-\nu _{p1}}}$).

Contributed by: Martin Sommerfeld — Martin-Luther-Universität Halle-Wittenberg