1.3 Fundamentals of Computational Fluid Dynamics
1.3.2 Discrete liquid phase
stopped. A target for the global balances, called conservation target, can additionally be set. In a transient calculation the solver then proceeds to the next timestep.
• The virtual or added mass force FV M,i due to the no-slip condition for the sur-rounding fluid on the particle surface: the particle has to move and accelerate not only its own mass but also the fluid mass. This leads to an additional flow resistance, i.e. drag. FV M,i is significant especially for bubbles.
• The time-dependent Basset-history termFB,i which accounts for the past particle motion and flow patterns. It is also particularly important for bubbles.
• The Saffman force FS,i for a particle in a shear flow. The non-uniform pressure distribution across the particle surface leads to a lateral force in the reverse direc-tion of the pressure gradient. It is important particularly near walls or in flows with high velocity gradients.
• The Magnus force FM,i for rotating particles in a parallel flow. The pressure difference between the particle side which moves in the same direction as the flow due to its rotation and the opposite side provokes a force perpendicular to the flow direction.
• External forcesFEXT ,i, such as gravity, buoyancy or electromagnetic forces. In a rotating frame of reference, centripetal and Coriolis forces also occur.
In the considered two-phase flows with small liquid drops the drag force is usually dominant. Besides gravity it is the only force taken into account in this work.
The influence of particle deformations on the different force terms is hardly known and the particles are hence assumed spherical. They are considered non-rotating and drop/drop interactions, i.e. collisions, are not accounted for, see Section 1.3.2.4. Drop breakup is not modelled either.
In the discretised problem of numerical simulations the particle displacement is calcu-lated by simple forward Euler integration of the particle velocity components in CFX:
xp,i(t+δt) =xp,i(t) +vp,i(t)·δt . (1.27) δtdenotes the particle timestep which is defined locally: a characteristic length scale of the element where the particle momentarily resides is divided by the particle velocity and a user-defined factor, the so-called “number of integration time steps per element”, which is set to 10 by default. When the particle crosses an element border or if an Eulerian timestep ends, the value of δt is shortened to synchronise both phases and a new value is calculated.
To determine the new particle velocity vp,i at time t+δt, the equation of motion 1.24 is numerically integrated. All fluid properties, like velocity or density, are taken from the start of the Eulerian timestep. In space, they are interpolated from the vertices to the particle position. To perform the integration, CFX simplifies the appearing force terms in linearising them with respect to the particle velocity components vp,i.
1.3.2.2 Influence of turbulence
Solving the equation of motion for the dispersed particle phase it would be rather re-stricted to consider only averaged properties of the continuous phase, e.g. v¯f,i (i = 1,2,3). Yet, using Reynolds averaging, the equations of the Eulerian phase are only solved for these averaged quantities while the turbulent fluctuationsvf,i0 are not known.
τe = le· 2·k
3
−1/2
. (1.29)
le denotes the eddy length and τe the eddy lifetime. During the particle/eddy interac-tion, the turbulent fluctuations are determined as
v0f,i=prand,i· 2·k
3 1/2
(i= 1,2,3) (1.30)
in CFX. Due to the introduced random numbersprand,i, which are normally distributed, particle trajectories are not deterministic. If the particle/eddy interaction time reaches τeor if the displacement of the particle relative to the eddy gets as large asle, the particle leaves the considered virtual eddy and enters a new one with other characteristics and a new value of v0f,i is calculated.
In reality, particles may also dampen or excite the turbulence field of the continuous phase. Yet, there are no appropriate models describing these effects so far and therefore these are usually neglected. No source terms due to phase coupling appear in the turbulent transport equations 1.15 and 1.19 respectively. Particles affect the turbulence field of the Eulerian phase only indirectly by influencing the velocity field.
1.3.2.3 Coupling of the continuous and the dispersed phase
Besides the turbulent dispersion, i.e. the virtual eddies, which are related to the tur-bulence of the continuous gas phase and which affect the discrete liquid phase, the following interactions between Eulerian and Lagrangian phase occur:
• momentum transfer due to viscous drag.
• Heat transfer which comprises three terms:
– The convective heat transfer (forced convection, unit[J/s]) due to a temper-ature difference is given as
Q˙C =hc·π·Dp2·(Tf −Tp), (1.31) where the sign has to be chosen such that the heat flow is directed from the warmer to the colder phase. hcdenotes the heat transfer coefficient and can be expressed by the Nusselt number
Nu= hc·Dp
λf (1.32)
which defines the ratio between convective and conductive heat transfer. λf is the thermal conductivity of the gas phase.
Assuming homogeneous, steady gas conditions around a spherical drop and a spatially constant temperature inside the drop, the Nusselt number is often expressed as
Nu= 2 + 0.6·Re0.5·Pr0.33, (1.33) where the influence of the drop motion is modelled empirically, see [54], [42].
The Prandtl number is given as Pr= µpλ·cP
p calculated with drop properties.
– Latent heat transfer due to evaporation of a drop is expressed by the heat flow
Q˙m = dmp
dt ·∆hvap. (1.34)
The value of the latent heat per mass unit, ∆hvap, depends on the ambient temperature. It is withdrawn from the gas phase in case of evaporation and inserted in case of condensation assuming a drop of homogeneous tempera-ture, i.e. omitting cooling/heating of the drop liquid.
– Radiative heat transfer can be described as
Q˙R = (A,p·Tf4−E,p·Tp4)·σS ·π·D2p, (1.35) where σS = 5.67·10−8W/(m2·K4) is the Stefan-Boltzmann constant. E,p denotes the emissivity of the liquid (which equals 1 for a black body and is smaller else) and A,p its absorptivity. Radiation is neglected in this work.
The sum of all contributions constitutes the collective heat transfer rate for the liquid mp·cP ·dTdtp which is equal to the enthalpy change of the drop.
Further terms, e.g. due to a time-variable or inhomogeneous temperature field or due to drop acceleration (i.e. rotation) are neglected, see [42].
• Mass transfer, where the vapour pressure is calculated using the empirical Antoine equation:
pvap =pref·exp
A− B
Tp+C
. (1.36)
A,B andC are material dependent constants. Ifpvap is larger than the surround-ing gaseous pressure, the particle boils and the mass transfer rate is determined by the convective heat transfer
dmp
dt = 1
∆hvap ·Q˙C. (1.37)
Else it can be described analogous to the heat transfer rate in Equation 1.31 as dmp
dt =βc·π·D2p·(ρf −ρp), (1.38) where βc is the mass transfer coefficient. It can be replaced by the Sherwood number
Sh= βc·Dp
D , (1.39)
The equations in this subsection have been presented for one liquid material component only, because CFX is currently not able to describe multi-component evaporation which is generally difficult, cf. [72].
Apart from turbulence, two-way coupling is always considered throughout this work:
besides the influence of the continuous fluid on particles, these in turn affect the Eulerian phase. Therefore, the discussed source terms already appeared in the equations of the continuous phase, see above. They are non-zero for any control volume where a particle is located during the timestep. The general definition of a particle source term Sp is given as
dSp
dt =CS ·Φp+RS. (1.41)
CS ·Φp marks all contributions linear in the solution variable Φp and the term RS summarises the rest. The frequency with which particle source terms are introduced to the continuous phase can be controlled by the user and for large source terms only every second iteration might be advisable. Moreover, underrelaxation factors can be used to gradually increase the introduced source terms with the number of iterations per Eulerian timestep from a reduced to their full value. It is also possible to smear them over several volume elements, which might help to avoid divergence in case of strong source accumulations, e.g. on wall impact. Yet, it strongly reduces the accuracy and is generally not applied in this work.
1.3.2.4 Discrete Droplet Model
The calculation of all single spray drops is usually much too costly with respect to com-putational time and storage. Therefore, the ensemble is represented by a comparatively small number of Lagrangian parcels. In this Discrete Droplet Model (DDM), see [18]
and [3], every parcel stands for a number n of real drops with identical properties. In CFX, n is called “number rate”, because the quantity is scaled with the timestep value of the continuous phase.
The considerations of the previous subsection remain valid with parcels considered in-stead of individual drops.
A significant simplification in the considered Lagrangian approach is the neglect of the liquid volume fractions in the calculation of the Eulerian phase. It is only valid in dilute flows where the liquid volume fractions are small. This restriction imposes in principle also a limit to grid refinement in an area of liquid accumulations, e.g. near injection regions.
Additionally, inter-particle collisions, e.g. statistical models of [51] or [74], become im-portant in dense flows. They are usually computationally expensive and not addressed in this work.