Magnus lift force

Fig. 5.3. The Magnus lift force on a rotating particle. The sketch illustrates the flow pattern induced by a rotating particle. The initial particle rotation is not caused by the flow.

The question is what the expression for the Magnus force is when the ambient medium is not stagnant (initially) but also exhibits vorticity of its own. The idea is that in such a case it is the particle rotational velocity relative to the fluid vorticity, or the relative sHp, that drives the Magnus lift force. This idea found its way into the textbook by Crowe et al (1998) and can also be found in, e.g., Yamamoto et al (2001) and Derksen (2003). The most general expression for the Magnus Hft force then runs as follows ... 

In a general hydrodynamic system, the vorticity w is perpendicular to the velocity field v, creating a so-called Magnus pressure force. This force is directed along the axis of a right-hand screw as it would advance if the velocity vector rotated around the axis toward the vorticity vector. The conditions surrounding a wing that produce aerodynamic lift describe this effect precisely (see Fig. 2). 

The first term on the right-hand side of Eq. (3.80) is the total drag force in the opposite direction of Up, including both Stokes drag and Oseen drag. The second term represents a lift force in the direction perpendicular to Up. Thus, the lift force or Magnus force for a spinning sphere in a uniform flow at low Reynolds numbers is obtained as... 

At high Reynolds numbers, the rotation of the sphere yields an asymmetric wake, as shown in Fig. 3.4. In this case, the theoretical analysis of the Magnus force and the drag force becomes rather complex because of the difficulties in obtaining the expressions for the pressure and velocity distributions around the surface of the sphere. Thus, the determination of the lift force as well as the drag force relies mainly on the empirical approach. 

In particular applications alternative relations for the slip velocity (3.428) can be derived introducing suitable simplifying assumptions about the dispersed phase momentum equations comparing the relative importance of the pressure gradient, the drag force, the added mass force, the Basset force, the Magnus force and the Saffman lift force [125, 119, 58]. For gas-liquid flows it is frequently assumed that the last four effects are negligible [201, 19[. 

Rigid spheres sometimes experience a lift force perpendicular to the direction of the flow or motion. For many years it was believed that only two mechanisms could cause such a lift. The first one described is the so-called Magnus force which is caused by forced rotation of a sphere in a uniform flow field. This force may also be caused by forced rotation of a sphere in a quiescent fluid. The second mechanism is the Saffman lift. This causes a particle in a shear flow to move across the flow field. This force is not caused by forced rotation of the particle, as particles that are not forced to rotate also experience this lift (i.e., these particles may also rotate, but then by an angular velocity induced by the flow field itself). 

Rubinow and Keller [123] calculated the flow around a rotating sphere moving in a viscous fluid for small Reynolds numbers. They determined the drag, torque, and lift force (Magnus) on the sphere to O(Rep). The results were ... 

Other workers have shown that initially non-rotating particles also experienced lateral motion [79, 80]. The shear flow thus gives rise to a lift force which is caused by the shear solely or rotation induced by the flow. Hence, these observations implied that a Magnus t3rpe of force was not adequate. Lawler and Lu [85] and Lahey [79, 80] gave an overview over lateral distributions of positive, neutrally, and negative buoyant particles. The result is sketched in Fig 5.4. 

Saffmann [125] (p 394) compared the Magnus force developed by Rubinow Keller [123] with the shear force given above and showed that unless the rotational speed is much larger than the rate of shear k = dv /dy, and for a freely rotating particle 17 = the lift force due to shear is an order... 

Lawler and Lu [85] reviewed the classical experimental observations on transversal migration of spherical particles and concluded that neither the original Magnus nor the Saffman force models are capable of explaining all these observations. They thus propose that the lift forces might be expressed in terms of the relative particle-fluid angular velocity rather than the absolute angular velocity of the particle as used in all the classical models. Crowe et al [26] also made similar extensions of the classical lift force models. 

Maxey and Riley pointed out that the Magnus force is of order and, for that reason, it is less important than the Saffman lift force. 

Figure 16.3. Configurations for studying the lift force exerted on a particle in a fluid flow in an infinite medium, (a) Magnus effect in a uniform flow the particle is rotated about a fixed axis, (b) The particle moves in the direction of a fluidflow with constant shear. Two sub-cases are considered (i) the particle is prevented from rotating w = 0) ...

The lift forces (Magnus forces) Fi, Fi y, and Flz, which represent the forces of generating a sidewise force on the spinning bubble in the liquid phase by the liquid velocity gradient, are given by Auton et al. [32] as... 

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... 

Saltation of solids occurs in the turbulent boundary layer where the wall effects on the particle motion must be accounted for. Such effects include the lift due to the imposed mean shear (Saffman lift, see 3.2.3) and particle rotation (Magnus effect, see 3.2.4), as well as an increase in drag force (Faxen effect). In pneumatic conveying, the motion of a particle in the boundary layer is primarily affected by the shear-induced lift. In addition, the added mass effect and Basset force can be neglected for most cases where the particle... 

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