Virtual-mass and lift forces

Invoking conservation of momentum at the mesoscale then leads to the following expression for the fluid velocity seen by the particle  

Collecting the terms in Eq. (5.82) involving the time derivative of U leads to the mesoscale model for the virtual-mass force (in the absence of other forces ) in the monokinetic-fluid limit  

Particles moving in a fluid with mean shear experience a lift force perpendicular to the direction of fluid flow. The shear lift originates from inertia effects in the viscous flow around the particle and depends on the mean vorticity of the fluid phase evaluated at the particle location x = X (r). For a spherical particle, the particle acceleration due to the lift force (also known as the Saffman lift force) is equal to (Auton, 1987 Drew Lahey, 1993 Drew Passman, 1999 Saffman, 1965) 

Summarizing the forces introduced above, tests carried out in different multiphase systems have shown that the order of importance of the different forces involved typically ranks buoyancy and drag in the first positions and then lift and virtual-mass forces for fluid-solid systems and virtual-mass and lift forces for fluid-fluid systems (see, for example, the studies on non-drag forces by Diaz et al (2008) and Barton (1995)), whereas the most common values for the corresponding constants are Cl = 0.25 and Cv = 0.5 both for fluid-fluid and for fluid-solid systems. Naturally, since it is straightforward to implement all the forces in a computational code (Vikas et al, 201 lb), it is best to include them all for the sake of generality. 

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Drew DA, Lahey RT Jr (1987) The virtual mass and lift force on a sphere in rotating and straining inviscid flow. Int J Multiphase Flow 13 (1) 113-121... 

Drew DA, Lahey RT Jr (1979) Application of general constitutive principles to the derivation of multidimensional two-phase flow equations. Int J Multiph How 5 243-264 Drew DA (1983) Mathematical modeling of two-phase flow. Ann Rev Huid Mech 15 261-291 Drew DA, Lahey RT Jr (1987) The virtual mass and lift force on a sphere in rotating and straining inviscid flow. Int J Multiph Flow 13(1) 113-121... 

Drew DA, Lahey RT Jr (1990) Some supplemental analysis concerning the virtual mass and lift force on a sphere in a rotating and straining flow. Int J Multiph How 16 1127-1130 Drew DA (1992) Analytical modeling of multiphase flows modern developments and advances. In Lahey RT jr (ed) Boiling heat transfer. Elsevier Science Publishers BV, Amsterdam, pp 31-83... 

In particular, referring to the introduction of the external forces as presented in sect 4.1.3 there are still no complete consensus in the literature regarding the treatment of the interfacial coupling terms like the steady drag-, added mass- and lift forces. In one view it is considered convenient to split the net force exerted by the interstitial fluid on the particle into two different contributions One virtual force applied by an undisturbed flow on a imaginary fluid particle which coincides with the solid particle in volume and shape, and a second contribution that represents the forces due to the perturbations in the flow. These flow disturbances are created by the presence of the particles. The phrase undisturbed flow thus refers to the flow that would be observed if the particle was not present. Neglecting the effects of the perturbations in the flow, the net force exerted on a particle (4.57) might be approximated by ... 

The interphase forces considered were steady drag, added (virtual) mass and lift. The steady drag force on a collection of dispersed bubbles with a given average diameter was described by (5.48) and (5.34). The transversal lift force was determined by the conventional model (5.65), whereas the added mass force was approximated by (5.112). 

One therefore has to decide here which components of the phase interaction force (drag, virtual mass, Saffman lift, Magnus, history, stress gradients) are relevant and should be incorporated in the two sets of NS equations. The reader is referred to more specific literature, such as Oey et al. (2003), for reports on the effects of ignoring certain components of the interaction force in the two-fluid approach. The question how to model in the two-fluid formulation (lateral) dispersion of bubbles, drops, and particles in swarms is relevant... 

The terms on the right-hand side of Eq. (11.4) correspond to interphase drag force, virtual mass force. Basset force and lift force, respectively, /l is a transversal lift... 

It must be noted here that even for Eulerian-Lagrangian simulations, although there is no complexity of averaging over trajectories, the accuracy of simulations of individual bubble trajectories depends on lumped interphase interaction parameters such as drag force, virtual mass force and lift force coefficients. All of these interphase interaction parameters will be functions of bubble size and shape, presence of other bubbles or walls, surrounding pressure field and so on. Unfortunately, adequate information is not available on these aspects. To enhance our understanding of basic... 

As can be seen from Eq. (5.100), the virtual-mass force reduces the drag and lift forces by a factor of 1 /y. The buoyancy force is not modified because we have chosen to define it in terms of the effective volume Vpy. We remind the reader that the mesoscale acceleration model for the fluid seen by the particle A j must be consistent with the mesoscale model for the particle phase A p in order to ensure that the overall system conserves momentum at the mesoscale. (See Section 4.3.8 for more details.) As discussed near Eq. (5.14) on page 144, this is accomplished in the single-particle model by constraining the model for Apf given the model for Afp (which is derived from the force terms introduced in this section). Thus, as in Eqs. (5.98) and (5.99), it is not necessary to derive separate models for the momentum-transfer terms appearing in Apf. 

We have so far described drag and lift forces acting on a suspended particle. There are, however, additional hydrodynamic forces, such as Basset history, Faxen correction, and virtual mass effects that act on the particles. Some of these forces could become important especially for the particles suspended in a liquid. The general equation of motion of a small spherical particle suspended in fluid as obtained by Maxey and Riley is given as... 

Linear stability analyses were also carried out for bubble columns by, e.g., Shnip et al (1992), Minev et al (1999), Leon-Becerril and Line (2001), Sankaranarayanan and Sundaresan (2002), Lucas et al (2005, 2006), and Monahan and Fox (2007). The reader is also referred to the review paper by Joshi (2001). The purpose of these analyses was to learn more about the stabihty of homogeneous (or uniform) bubble flow and the transition to heterogeneous flow regimes. In most cases, the results of the analyses were again strongly dependent on the exact formulation of the fluid—bubble interaction force, particularly of its lateral component. Several of these authors pinpoint the crucial effects of virtual mass and of (the sign of) the lift force. 

Apart from the drag force, there are three other important forces acting on a dispersed phase particle, namely lift force, virtual mass force and Basset history force. When the dispersed phase particle is rising through the non-uniform flow field of the continuous phase, it will experience a lift force due to vorticity or shear in the continuous phase flow field. Auton (1983) showed that the lift force is proportional to the vector product of the slip velocity and the curl of the liquid velocity. This suggests that lift force acts in a direction perpendicular to both, the direction of slip velocity... 

FIGURE 4.4 Lift (a) and virtual mass forces, (b) on dispersed phase particles. 

Scmi the net source due to dispersed phase particles (Eq. (4.11)). Fd, Fi and Fvm are drag, lift and virtual mass forces (Section 4.2.1). It must be noted that Eq. (7.9) assumes that the volume-averaged momentum transfer (from the dispersed phase)... 

There are two main approaches for the numerical simulation of the gas-solid flow 1) Eulerian framework for the gas phase and Lagrangian framework for the dispersed phase (E-L) and 2) Eulerian framework for all phases (E-E). In the E-L approach, trajectories of dispersed phase particles are calculated by solving Newton s second law of motion for each dispersed particle, and the motion of the continuous phase (gas phase) is modeled using an Eulerian framework with the coupling of the particle-gas interaction force. This approach is also referred to as the distinct element method or discrete particle method when applied to a granular system. The fluid forces acting upon particles would include the drag force, lift force, virtual mass force, and Basset history force.Moreover, particle-wall and particle-particle collision models (such as hard sphere model, soft sphere model, or Monte Carlo techniques) are commonly employed for this approach. In the E-E approach, the particle cloud is treated as a continuum. Local mean... 

The net hydrodynamic force, frequently also referred to as the generalized drag force, is usually further divided into numerous contributions like the steady drag, virtual mass, lift, and history forces ... 

Drew and Lahey [33] derived the forces on a sphere in inviscid flow. They employed the objectivity principle which implies that the forces should be independent of coordinate system. They found the same formulation for the lift force as given by (5.65), but they stressed that it was not objective in that form. When written in combination with the virtual mass force as ... 

We reiterate that for a dispersed flow Fp the macroscopic generalized drag force normally contains numerous contributions, as outlined in chap 5. However, for gas-solid flows the lift force the virtual mass force fy, and the Besset history force components are usually neglected [39]. The conventional generalized drag force given by (5.27) thus reduces to ... 

The steady-drag, virtual mass, turbulent dispersion, and wall lift forces were approximated using a semi-implicit time discretization scheme ... 

The change of momentum for a particle in the disperse phase is typically due to body forces and fluid-particle interaction forces. Among body forces, gravity is probably the most important. However, because body forces act on each phase individually, they do not result in momentum transfer between phases. In contrast, fluid-particle forces result in momentum transfer between the continuous phase and the disperse phase. The most important of these are the buoyancy and drag forces, which, for reasons that will become clearer below, must be defined in a consistent manner. However, as detailed in the work of Maxey Riley (1983), additional forces affect the motion of a particle in the disperse phase, such as the added-mass or virtual-mass force (Auton et al., 1988), the Saffman lift force (Saffman, 1965), the Basset history term, and the Brownian and thermophoretic forces. All these forces will be discussed in the following sections, and the equations for their quantification will be presented and discussed. 

Tomiyama, a. 2004 Drag, lift and virtual mass forces acting on a single bubble, in 3rd International Symposium on Two-Phase Flow Modeling and Experiments, Pisa. 

The model closes interaction phase Mkj with effective viscosity //eff, and it is considered that the slurry phase of FTS is in the turbulent zone. The effective viscosity can be considered as turbulence viscosity, which can usually be solved by standard model, where turbulence induced by bubbles is also taken into consideration. Interaction force between the phases consists of drag, lift, and virtual mass force. The main force of interaction between phases often is overlooked. Drag force, caused by gas, drives the serious movements and the type of which is written as Eq. (28) ... 

At low gas volume fraction (<0.01), the forces acting on bubbles in a liquid are similar to those acting on a single bubble. Due to their low inertia as compared with the Hquid phase, bubbles are subject to a large number of forces, i.e., forces due to drag, lift, and virtual mass. In the definition of the interfacial forces, it is customary to characterize the bubble size with the sphere equivalent diameter. That is, ah effects due to nonsphericity are lumped in the closure. The closures that are most commonly applied... 

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