OTRAS INSTANCIAS ORGANIZACIONALES 1. COORDINADOR (A)

Determine the application of the turbulence transport schemes:

Ž model thermally buoyant flow in a cavity using a two-dimensional plane mesh with a height to width ratio of 50:1

Ž employ various turbulence models for thermal convection, i.e. k-ε turbulence, Reynolds stresses etc

Ž model buoyant gas phase motion in a manner that is similar to that of the thermal convection simulation except that a two-dimensional plane mesh with a height to diameter ratio of 5:1 is used

Ž use both the mixture model available in Fluent9 _{and a modified scalar equation form of} the model

The purpose of this section is to provide an assessment of the use of turbulence models for buoyancy driven flow through the analogy between thermal convection and multiphase flow in bubble columns41. To develop the use of a scalar equation for the transport of the gas phase with the need for further application to modelling the motion of more than one discrete phasei.

4.2 Mathematical models

Energy transport

Heat transport has long been established in the modelling of turbulent flows particularly when simulating natural convection in cavities with large height to diameter ratios39-40. The simulation of lateral convection of heat the across the width of the mesh provides an excellent example of how to simulate the turbulence in buoyancy driven flows. As there is an analogy between both multiphase flow in bubble columns and natural convection it would seem prudent to attempt to model the transport of the discrete phase in a similar manner41_{. To use} this technique effectively it would be wise to test the capacity of Fluent9 _{to simulate natural}

convection, utilising the work of Chait and Korpela as an example of how to simulate such flows for reasons of geometric consistency40.

The thermal transport equation (Equation 41) has the general form of a transport equation with time dependent and the non-linear velocity terms on the left hand side. On the right hand side is the temperature gradient and the velocity coupling term. These last two terms are intrinsic to the accurate prediction of the structure of both the velocity and temperature fields and how each of the variables influences the other parameter. Along with the energy transport equation, the conservation of mass and momentum (Equation 42 and Equation 43) are used to predict and check the pressure and velocity fields. The fluid density varies with temperature and this variation is calculated through the use of the Boussinesq approximation (Equation 45), which is then applied to the transport equations (Equation 41 to Equation 43).

Transport of turbulence

The k-ε turbulence equations are used to depict the transport of energy between large and small vortices in the velocity field. The energy transport is modelled through the use of a pair of coupled transport equations (Equation 25 and Equation 26) that describes the generation of energy due to motion (conversion from one source i.e. density difference to another) in the one equation (Equation 25) and dissipation of that energy in the other (Equation 26). This effectively characterises the growth and evolution of vortices caused by some form energy input be that agitation from heat, another fluid or the motion of a solid object.

Previous investigations into gas-liquid motioni suggest the use of a more complex model for the closure of the transport equations, by the use of Reynolds stresses; Equation 31 shows the exact Reynolds stresses turbulence transport equation. The equation utilised in Fluent9,11 is simplified into several different equations to reduce the complexity of the formulation and to enable closure of the exact equation in the mathematical models. This includes the use of both the k-ε equations (Equation 25 and Equation 26) and the inclusion of the effects of buoyancy, pressure, pressure-strain and any rotation (Equation 32 to Equation 40). Note that the number equations used depends on whether the domain used is two or three-dimensional.

In the prediction of natural convection it is beneficial to the numerical solution of the transport equations to split the flow into basic and secondary flow (Equation 50) quantities39- 40_{. The basic flow quantity is a known profile that was determined by the laminar flow} conditions. The secondary flow is determined by taking an x-y average of the Navier-Stokes equation and this then represents the effect of the Reynolds stresses. Using such a profile is relevant to multiphase flow based on the density difference driving force by analogy with natural convection41. The profiles used for both types of flow can be found for temperature, volume fraction and the velocity parallel to the direction of the flow (Equation 51, Equation 52 and Equation 60). A series of functions in Fluent9 are employed to update parameters such as temperature, velocity and volume fraction of the gas phase. The functions used are defined as the DEFINE_ADJUST, DEFINE_INIT and DEFINE_ON_DEMAND functions, where the DEFINE_ADJUST functions updates the variable for every iteration and the DEFINE_ON_DEMAND function is utilised when an execute command is exercised in a defined sequence enabling the quantity to be updated when the user specifies. The DEFINE_INIT function is used to initialise the flow-field with the basic profilesi.

Algebraic slip mixture model9,26

The investigations of Zuber, Findlay, Ishii and Mishima provide the physical and theoretical basis for the application of a continuum mixture approach to the simulation of multiphase flow24,27-28. Allowing the discrete and continuous phases to be considered as a pseudo- continuous mixture, a single continuity equation (Equation 1) and a single momentum equation (Equation 2) were employed to assess mixture phase transport. To predict the mixture phase composition a volume fraction equation for the discrete phase (Equation 3), a mixture density (Equation 4) and a mixture viscosity (Equation 5) must be employed to characterise gas-liquid or solid-liquid interactions. However, the phases have distinct interactions that influence the transport of each phase and this is characterised by the use of mass-averaged, drift and slip velocities (Equation 6, Equation 7 and Equation 8) for the mixture. The slip or relative velocity was obtained by averaging a combined momentum equation for the discrete phase and the mixture according to the principles of local equilibrium and Favre averaging. Assumptions made as part of the averaging procedure are that the pressure is the same for all phases and that only viscous drag influences particle

motion leading to fluctuating form of slip velocity. A constitutive equation is then employed to account for the fluctuating terms to further simplify the relative velocity25.

The formulation of the mixture phase for one flow regime is not the same for another and this has a critical influence on how the driving forces and frictional effects (employing Equation 9 where the Reynolds number, Equation 10, is based on the particle diameter as the characteristic dimension) are considered when modelling such flows24-28. The regimes vary according to discrete particle size, the volume fraction and the distribution of particle sizes. Predicting flow phenomena in the heterogeneous regime increases the complexity of the description of the mixture phase, as different bubble sizes are more prevalent. Therefore, to simplify the models employed we concentrate on flow predictions in the homogeneous regime, where a single bubble size is assumed.

Modified scalar equation mixture model

The scalar transport model can be used to model the transport of any parameter such as volume fraction of a phase, heat, the influence of electromagnetism etc9 and is used in conjunction with basic continuity (Equation 42) and momentum (Equation 53) equations. The scalar equation employed here models the transport of a pseudo-continuous discrete phase that has the form of Equation 54 where the diffusion coefficient used, Γ, was defined as 0.1 for the discrete phase volume fraction equation. To depict the transport of the discrete phase, Equation 55 and Equation 56 are source terms that are applied to Equation 53 and Equation 54 respectively. The first term in both source terms is the deviatoric stress tensor and the last term is the inter-phase interaction term as calculated by the drift velocity. The second term in Equation 56 is the convective flux of the discrete phase noting that there are two convective fluxes with one for the mixture phase. The scalar volume fraction equation is different to the volume fraction formulation used in algebraic slip mixture model26 where the diffusive flux term and deviatoric stress tensor are not included. The drift velocity is calculated through the use of the mixture density (Equation 4), mass-averaged, drift and slip velocity formulations (Equation 6 to Equation 10). The mixture viscosity is calculated using

Equation 57 to Equation 59, this differs from the algebraic slip mixture model9,26 which only employs Equation 5i.