Virtual-mass force

Note that for liquid solid systems, Eq. (20) should also include the short-range lubrication forces and the effects of other forces such as the virtual mass force. But this is beyond the scope of this chapter. 

In the preceding equations, Fp can be expressed as a combination of local averaged drag force and virtual mass force [Anderson and Jackson, 1967]. 

The second important term is the virtual mass coefficient (Cv). When the dispersed phase accelerates (or decelerates) with respect to the continuous phase, the surrounding continuous phase has to be accelerated (or decelerated). For such a motion, additional force is needed, which is called added or virtual mass force. This force was given by the second term in Eq. (8). The constant Cy is called the virtual or added mass coefficient. It is difficult to estimate the value of Cv with the present status of knowledge. Therefore, many recommendations are available in the published literature. In an extreme case of potential flow, the value of Cy is 0.5. 

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)... 

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... 

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... 

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 ... 

The Added mass or virtual mass force on a single rigid sphere in potential flow... 

To define the virtual mass force we postulate a Lagrangian force balance for the sphere ... 

It follows that the virtual mass force is given by [97] ... 

It is emphasized that the virtual mass force accounts for the form drag (shape effects) due to the relative acceleration between the particle and the surrounding fluid. 

In the Eulerian framework the virtual mass force valid for potential flows is normally expressed by [7, 152, 64, 33, 8] ... 

While the virtual mass force accounts for the form drag on the particle due to relative acceleration between the particle and the surrounding fluid, the history term accounts for the corresponding viscous effects. Moreover, the history force originates from the unsteady diffusion of the vorticity around the particle so there is a delay in the boundary layer development as the relative velocity changes with time [96, 97, 22]. This means that when the relative velocity between the particle and the fluid varies, the vorticity present at the particle surface changes and the surrounding flow needs a flnite time to readapt to the new conditions. 

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 virtual mass force term in the gas momentum equation is discretized in the same way as described when considering the liquid phase momentum equation in sect C.4.5. 

Ek,fluid kinetic energy (KE) of fluid surounding a particle in virtual mass force analysis (J)... 

The virtual-mass force is the exception to this rale because it involves the time derivative C7. For this case, must be defined in a Lagrangian sense by taking into account the virtual mass in Eq. (5.9). 

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. 

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 ... 

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. 

In order to complete our discussion on momentum transfer, we must consider the final forms of the mesoscale acceleration models in the presence of all the fluid-particle forces. When the virtual-mass force is included, the mesoscale acceleration models must be derived starting from the force balance on a single particle ... 

On the right-hand sides of these force balances, we include the body and fluid-particle forces discussed in the previous sections. By moving the time derivatives from the virtual-mass force to the right-hand side, we can combine the balance of forces in matrix form ... 

From Eqs. (5.93) and (5.94), it is very interesting to note that while the fluid-particle forces are the same as in the absence of the virtual-mass force, the body forces for the particle and the fluid are now coupled whenever Cym + 0. However, body forces that are linearly proportional to mass such as gravity (i.e. Fp = PpVpg and Ff = pfVfg) will remain uncoupled. In the very dilute limit, we have Vp Vf and thus Eq. (5.93) reduces to the classical particle model in which the virtual-mass force affects only the particle mass. The same is true for the fluid seen by the particle in the limit where pf pp. 

It is very interesting to note that the virtual-mass force mainly acts through the effective volume coefficient y, and tends to reduce the momenffim-exchange terms. The final step is to use the definitions of the mass-average moments introduced in Chapter 4 to derive the force terms in the macroscale momenffim balance. For the disperse-phase momentum balance (see Eq. (4.85) on page 123), this procedure leads to... 

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. 

Sankaranarayanan, K., Shan, X., Kevrekidis, I. G. Sundaresan, S. 2002 Analysis of drag and virtual mass forces in bubbly suspensions using an implicit formulation of the lattice Boltzmann method. Journal ofEluid Mechanics 452, 61-96. 

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