Drag, virtual mass

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 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 ... 

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

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity /z, which is released and begins to fall (in the x = — z direction) under the influence of gravity. A momentum balance on the particle is simply T,FX = max, where the forces include gravity acting on the solid (T g), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (FD). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the virtual mass (m() of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = jms(p/ps). Thus the momentum balance becomes... 

The first term of Eq. (11-11) is the Stokes drag for steady motion at the instantaneous velocity. The second term is the added mass or virtual mass contribution which arises because acceleration of the particle requires acceleration of the fluid. The volume of the added mass of fluid is 0.5 F, the same as obtained from potential flow theory. In general, the instantaneous drag depends not only on the instantaneous velocities and accelerations, but also on conditions which prevailed during development of the flow. The final term in Eq. (11-11) includes the Basset history integral, in which past acceleration is included, weighted as t — 5) , where (t — s) is the time elapsed since the past acceleration. The form of the history integral results from diffusion of vorticity from the particle. 

The acting forces per unit volume between the phases are due to drag and virtual-mass. 

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

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... 

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... 

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 virtual mass effect relates to the force required for a particle to accelerate the surrounding fluid [65, 170, 26]. When a particle is accelerated through a fluid, the surrounding fluid in the immediate vicinity of the particle will also be accelerated at the expense of work done by the particle. The particle apparently behaves as if it has a larger mass than the actual mass, thus the net force acting on the particle due to this effect has been called virtual mass or added mass force. The steady drag force model does not include these transient effects. 

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. 

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 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). 

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