1. Instructor: André Bakker
Lecture 18 - Eulerian Flow Modeling Applied Computational Fluid Dynamics
Instructor: André Bakker
© André Bakker (2002)
© Fluent Inc. (2002)

2. Contents Overview of Eulerian-Eulerian multiphase model.
Spatial averaging.
Conservation equations and constitutive laws.
Interphase forces.
Heat and mass transfer.
Discretization.
Solver basics.
The lecture will cover the following:
Overview of Eulerian Eulerian Multiphase Model
Spatial Averaging methodology
Application of the spatial and temporal averaging to the Conservation Equations and derivation of the Constitutive Laws
Discussion on the applicable Interphase Forces
Some Examples
Heat and Mass Transfer
Discretization
Solver Basics

3. Eulerian-Eulerian multiphase - overview
Used to model droplets or bubbles of secondary phase(s) dispersed in continuous fluid phase (primary phase).
Allows for mixing and separation of phases.
Solves momentum, enthalpy, and continuity equations for each phase and tracks volume fractions.
Uses a single pressure field for all phases.
Uses interphase drag coefficient.
Allows for virtual mass effect and lift forces.
Multiple species and homogeneous reactions in each phase.
Allows for heat and mass transfer between phases.
Can solve turbulence equations for each phase.
The Eulerian Eulerian model is appropriate for modeling multiphase flows involving gas-liquid or liquid-liquid flows. For example, droplets or bubbles of the secondary phase dispersed in the primary or continuous phase. The phases mix or separate and the secondary phase volume fraction can be vary anywhere between zero to 100%. Practical application of the Eulerian Eulerian model would be to evaporation, boiling, separators and aeration. The Eulerian multiphase model is inappropriate for modeling situations where the accurate description of the interface boundary is important, in the case of stratified of free-surface flows.
The Eulerian multiphase model takes the continuity equation and momentum equation, as I had just posed, and species equation for each phase, solves them and tracks the volume fraction, k . It uses a single pressure field for all phases. So the pressure term or the pressure field is the same regardless if you're in the continuous phase or in the dispersed phase. The interaction between the mean flow of both phases is modeled through the interaction term which includes a drag force, and virtual mass effect when there exists acceleration of secondary phase relative to the primary and additional lift forces. It provides several formulations for drag and these will be described later. Alternative drag laws can be introduced to Fluent 4.5 through user defined subroutines. You're not restricted by any means, to the drag formulations that we have in Fluent and it's quite simple or straightforward to implement. As well as you can introduce additional body forces to the momentum equations through a user defined subroutines.

4. Methodology
A general multiphase system consists of interacting phases dispersed randomly in space and time. Interpenetrating continua.
Methods:
Use of continuum theory and thermodynamical principles to formulate the constitutive equations (ASMM).
Use of microstructural model in which macroscopic behavior is inferred from particle interaction: Eulerian-Granular.
Use of averaging techniques and closure assumptions to model the unknown quantities:
Space averaging with no time averaging.
Time averaging with no space averaging.
Ensemble averaging with no space averaging.
space/time or ensemble/space averaging: Eulerian-Eulerian.
The modeling of each phase in your system as separate continuum requires that you treat them as interpenetrating because they are mixing and interpenetrating into one another. And we use statistical mechanics. We treat each phase based on a microstructural model. We extrapolate from that microspace a macroscopic behavior of the fluids. So we're not dealing on the molecular level, but we are extrapolating into an interaction between the granular particles and basing that on kinetic theory. We also require the use of averaging techniques and some closure assumptions, because we introduce some unknowns into the governing equations for each phase. And so…the averaging techniques we would use could be time averaging, volume and spatial averaging or ensemble averaging.
There are two methods of approaching this. One is to use a deductive approach the other an inductive approach.

5. Two-fluid model (interpenetrating continua)
Deductive approach:
Assume equations for each pure phase.
Average (homogenize) these equations.
Model the closure terms.
Inductive approach:
Assume equations for interacting phases.
So the deductive approach assumes an equation for each pure phase. So you treat each phase as separate. You then average these equations if you're trying to model turbulent flows. So essentially you're homogenizing them. Then you need to model the closure equations. The inductive approach goes by first assuming equations for interacting phases and then modeling the closure terms.
Next we derive the governing equations.

6. Spatial averaging: basic equations
Application of the general transport theorem to a property k gives the general balance law and its jump condition:
Continuity equation:
Momentum equation:
The basic equations for multiphase modeling, the continuity and momentum equations, are derived from the general transport theorem. By making the appropriate substitutions you get the continuity and momentum equations. The subscript k refers to the phase.
As discussed, modeling the interpenetrating continua through a deductive approach defines governing equations for each phase which are then averaged in time and/or space. The methods for averaging will now be discussed.

7. Spatial averaging
Consider an elementary control volume d bounded by the surface dS.
Length scales:
Volumes:
Averaging volume and coordinate system:
To define the spatial averaging of the continuum equations it is necessary to consider an elementary control volume dW bounded by surface i where the length scale of this control volume is much less than the computational domain and much larger than the length scales of the phenomena occurring in the control volume. Consider the pink phase control volume dW which contains multiple occurrences of intrinsic phase Wk. The sum of the intrinsic phase volumes at an instant in time is equal to the control volume dW

8. Space averaging: basic equations
Definitions:
Volume average Intrinsic phasic average.
Volume phase fraction:
There are three methods of space averaging: volume average, intrinsic phasic averaging and volume phase fraction. The volume fraction is defined as the volume integral of the variable divided by the control volume.
The product of the intrinsic phasic average and the volume phase fraction gives the volume average of the variable.

9. Space averaging: averaging theorems
For all k-volumes that are differentiables, (Gray and Lee (1977), Howes and Whitaker (1985)).
Temporal derivative:
Spatial derivative:
Next, apply averaging to the conservation equations.
For all k-volumes that are differentiable, the definitions of the temporal and spatial averages are defined here. The spatial averaging of the temporal derivative of function f is not equal to the temporal derivative of the spatially average of function f. The difference being the volume-averaged surface integrated flux .Similarly, the grad of the spatial average of function f is not equal to the spatial average of grad f.
These averaging methods can now be applied directly to the temporal and spatial averaging of the continuity and momentum equations.

10. Space averaging: conservation equations
Continuity equation:
Momentum equation:
mass transfer term
The spatially averaged conservation equations are presented here. The continuity equation shows the appropriate spatial averaging where the term on the RHS accounts for the mass transfer between the phases.
The momentum equation shows the spatially averaged time-derivative, convective, pressure, diffusive and body force terms in addition to the interaction term between the phases. The drag is described through the interaction term.
interaction term

11. Interphase forces Drag is caused by relative motion between phases.
Commonly used drag models (fluid-fluid multiphase).
Schuh et al. (1989).
Schiller and Naumann (1935).
Morsi and Alexander (1972).
Schwarz and Turner (1988; for bubble columns).
Symmetric law.
Many researchers devise and implement their own drag models for their specific systems.
The interaction term in the momentum equation describes the interphase forces or drag caused by the relative motion between the phases. The sum of all relative motion between all present phases is zero. The coefficient K account for the drag function that depends on the shape, size and density of the particle and the viscosity of the continuous phase. Empirical relationships are typically used to describe the fdrag term in the relation.
The drag can be applied through a number of models in FLUENT. Custom drag laws can be implemented through user-defined subroutines.

12. Interphase forces: drag force models
Schiller and Naumann
Schuh et al.
The drag laws mentioned in the previous slide are plotted here for illustration. The drag function is plotted against the particle Reynolds number. The Schiller and Naumann correlation for the drag covers a range of Reynolds number below and above The main differences between the drag laws show up for particle Reynolds numbers greater than These drag laws were derived for smooth, spherical, rigid particles.
Morsi and Alexander

13. Interphase forces: virtual mass and lift
Virtual mass effect: caused by relative acceleration between phases Drew and Lahey (1990).
Virtual mass effect is significant when the second phase density is much smaller than the primary phase density (i.e., bubble column).
Lift force: caused by the shearing effect of the fluid onto the particle Drew and Lahey (1990).
Lift force usually insignificant compared to drag force except when the phases separate quickly and near boundaries.
Two additional forces can be modeled in addition to the drag. These are the virtual mass force of mass effect and the lift force. If there exists relative acceleration rather then just relative motion between the phases then the virtual mass effect is significant. And this would be cases like bubble columns where your secondary phase density is much smaller then the primary phase density. And the lift force is an additional force that's introduced due to the shearing effect of your continuous on the secondary phase. And again shows like this. And the lift force is usually insignificant compared to drag force near boundaries or where the phases separate quickly.

14. Modeling heat transfer
Conservation equation of phase enthalpy.
Energy sources
e.g., radiation
Originates from
work term for
volume fraction
changes
The topic is heat transfer in the Eulerian multiphase model. The conservation equation for the phase enthalpies is shown in this slide as derived from general transport theorem. On the LHS are the time derivative term and the convective transport term. The first term on the right hand side originates out of a work term for volume fraction changes. The second term here is the viscous dissipation term. The third term introduces the radiation contribution. The next slide will describe the remaining three terms that contribute to the energy balance.
Viscous dissipation
term

15. Modeling heat transfer
Conservation equation of phase enthalpy.
Heat conduction
Interphase heat
transfer term
The 4th and 5th terms on the RHS introduces the heat conduction and interphase heat transfer terms. The semi empirical derivations of these terms will be described in the next two slides. The final term on the RHS accounts for energy transfer due to mass transfer.
The description of the contributing terms in the energy equation illustrate the variety of heat transfer phenomena that can be modeled in the multiphase system.
Energy transfer with
mass transfer

16. Heat conduction
Assume Fourier’s law: For granular flows kk is obtained from packed bed conductivity expression, Kuipers, Prins and Swaaij (1992).
Near wall heat transfer is calculated as in single phase.
All standard boundary conditions for temperature can be implemented for multiphase.
The heat conduction in the multiphase system can be accounted for through basic Fourier law relationship. The application of heat conduction is more significant in granular flows. The thermal conductivity is obtained from the work of Kuipers, Prins and Swaaij. It is based on a packed bed conductivity expression. The near wall heat transfer is calculated as in single phase using the wall treatment boundary conditions. The application of heat conduction is made most significant in packed beds in which the packing of the granular phase provides the

17. Interphase heat transfer
The interphase heat transfer coefficient is given by.
Granular Model (Gunn, 1978).
Fluid-fluid model.
For the interphase heat transfer, the heat transfer coefficient is given by a semi empirical formulation with a Nusselt number dependency. For the granular model, the Nusselt number is function of the volume fraction, Reynolds number and the Prandtl number. For fluid-fluid model Nusselt number is as a function of Reynolds number and Prandtl number. In both cases, the Reynolds number is computed from the relative velocity of the phases and the volume fraction term.

18. Conservation of species
Conservation equation for the mass fraction of the species i in the phase k:
Here is the diffusivity of the species in the mixture of the respective phase, is the rate of production/destruction of the species.
Thermodynamic relations an state equations for the phase k are needed.
When calculating mass transfer the shadow technique from Spalding is used to update diameter of the dispersed phase.
Up to this point we have discussed momentum and heat transfer. The species transport equation for the mass fraction of a species I in the a phase k is shown. Here D is the diffusivity of the species in the mixture of the respective phase. m dot appears as a source term in this conservation equation and it's the rate of production or destruction of a species. Thermodynamic relations and state equations for the k phase are needed. When modeling mass transfer, the size or diameter of the dispersed phase can change and we model that change of diameter through the shadow technique which came from Spalding in 1983.

19. Modeling mass transfer
Evaporation and condensation:
For liquid temperatures  saturation temperature, evaporation rate:
For vapor temperatures  saturation temperature, condensation rate:
User specifies saturation temperature and, if desired, “time relaxation parameters” rl and rv (Wen Ho Lee (1979)).
Unidirectional mass transfer, r is constant:
The mass transfer, that was the final term on the RHS of the species equation, can be applied to evaporation and condensation as well as unidirectional mass transfer. For the evaporation process, when the liquid phase temperatures exceed the user-specified saturation temperature, evaporation occurs. When the vapor temperature is less then the saturation temperature, then you have condensation,
To enhance the mass transfer capabilities for your application, you can use a user-defined subroutine for your own mass transfer mechanism.

20. Solution algorithms for multiphase flows
Coupled solver algorithms (more coupling between phases).
Faster turn around and more stable numerics.
High order discretization schemes for all phases.
More accurate results.
Implicit/Full Elimination
Algorithm v4.5
TDMA Coupled
Multiphase Flow Solution
Algorithms
Only Eulerian/Eulerian
model
Due to the very strong coupling between the governing PDE's for multiple fluid equations special multiphase flow solution algorithms are required. The two solvers implemented in FLUENT are the Implicit/Full Elimination Algorithm and the TDMA Coupled Algorithm. The coupled solver algorithms have faster turnaround and more stable numerics. Using higher order discretization as previously discussed will result in more accurate results.

21. Full elimination algorithm
Momentum equation for primary and secondary phase:
Elimination of secondary phase gives primary phase:
Secondary phase has similar form.
Applicable to N phases.
The full elimination solver calculates the effective central convection-diffusion coefficient from the following matrix. If you consider the momentum equations for the primary and secondary phases, eliminating the secondary phase and rearranging we can pose a discretized equation for a_p, the primary phase, and for the secondary equation. Note that the convection-diffusion coefficients for both phase equations appear in coupled form and are represented by an effective a_p.
This formulation is applicable to N number of phases.

22. Coupled TDMA-algorithm
Discretized equations of primary and secondary phase are in matrix form:
Results in a tri-diagonal matrix consisting of submatrices
Closer coupling in each iteration gives faster convergence
The coupled TDMA solver presents the discretized equations in matrix form. In this case, the primary phase velocity and secondary phase velocity are solved in a coupled fashion at the same iteration level making the coupling stronger. This results in a tridiagonal matrix consisting of where the diagonal entries are themselves submatrices as shown. This closer coupling accelerates convergence.
This solution method is only applicable to 2 phases with the interactions between the 2 phase without mass transfer.
The TDMA solver should also only be used when the primary phase strongly affects the secondary phase and the secondary phase strongly affects the primary phase. This solver should not be used when ther is one-way coupling where the secondray phase has a weak influence on the primary phase.

23. Solution algorithm for multiphase
Typical algorithm:
Get initial and boundary conditions.
Perform time-step iteration.
Calculate primary and secondary phase velocities.
Calculate pressure correction and correct phase velocities, pressure and phase fluxes. Pressure is shared by all phases.
Calculate volume fraction.
Calculate other scalars. If not converged go to step three.
Advance time step and go to step two.
The FLUENT algorithm for multiphase flows is as follows:
Get the initial and boundary conditions and then perform the time-step iterations. Calculate the primary and secondary phase velocities using either the full elimination or the TDMA coupled algorithms. Then calculate a pressure correction and based on this pressure correction, update your phase velocities, update your pressure and update all the phase fluxes. The pressure is shared by all phases, as we had mentioned. Then calculate the volume fraction. Calculate other scalars, like species. And then advance to the next highest level and so on.

24. Solution guidelines All multiphase calculations:
Start with a single-phase calculation to establish broad flow patterns.
Eulerian multiphase calculations:
Copy primary phase velocities to secondary phases.
Patch secondary volume fraction(s) as an initial condition.
Set normalizing density equal to physical density.
Compute a transient solution.
Use multigrid for pressure.
All multiphase calculations:
Start with a single-phase calculation to establish broad flow patterns.
Eulerian multiphase calculations:
Use COPY-PHASE-VELOCITIES to copy primary phase velocities to secondary phases.
Patch secondary volume fraction(s) as an initial condition.
For a single outflow, use OUTLET rather than PRESSURE-INLET; for multiple outflow boundaries, must use PRESSURE-INLET for each.
Set the “false time step for underrelaxation” to 0.001
Set normalizing density equal to physical density
Compute a transient solution
Use multigrid for pressure

25. Summary Eulerian-Eulerian is the most general multiphase flow model.
Separate flow field for each phase.
Applicable to all particle relaxation time scales.
Includes heat/mass exchange between phases.
Available in both structured and unstructured formulations.
The Eulerian-Eulerian multiphase model is the most general multiphase flow model.
Separate flow field for each phase are solved and the interaction between the phases modeled through drag and other terms.
The Eulerian-Eulerian multiphase model is applicable to all particle relaxation time scales and Includes heat/mass exchange between phases.
It is available in structured FLUENT 4.5 and also in FLUENT 6. The implementation in FLUENT 6 will the hydrodynamics of the multiphase flow.