Basset terms

Basset term in Tchen s original equation. Lee (1981) derived the values of the rms particle fluctuating velocity, Mp, and fluid-particle relative turbulent velocity, Mr, based on the assumption that the particle experiences Lagrangian fluid energy spectrum for particle size up to 1000 pm. The experimental data of Snyder and Lumley... 

Extensive reviews of turbulent diffusion were provided by Levich andHinze. Tchen ° was the first investigator who modified the Basset-Boussinesq-Oseen (BBO) equation and applied it to study motions of small particles in a turbulent flow. Corrsin and Lumley pointed out some inconsistencies of Tchen s modifications. Csanady showed that the inertia effect on particle dispersion in the atmosphere is negligible, but the crossing trajectory effect is appreciable. Ahmadi and Ahmadi and Goldschmidt smdied the effect of the Basset term on the particle diffusivity. Maxey and Riley obtained a corrected version of the BBO equation, which includes the Faxen correction for unsteady spatially varying Stokes flows. 

This chapter deals with the movement of a small solid particle in a fluid flow. We start by presenting the equations governing particle movements, which we refer to as the Basset, Boussinesq, Oseen, and Tchen (BBOT) equations, to name a few key contributors to this modeling. Rather than deriving the equations, we endeavor to identify and discuss the physical meaning of the different terms acceleration, added mass, Basset term, etc. 

Terms VI are called "Basset terms . Indeed, they render an effect of the history of the particle s movement with respect to the fluid. The Basset terms that appear in... 

The transient stage during which the particle accelerates is described by the BBOT equations. Neglecting the Basset term in a first step, diffeiential equation [16.8] simplifies into the form ... 

The Basset term is a friction term which is added in a transient regime. In the present case, dJr) /dr >o, the Basset term is positive and the friction associated with... 

This formulation shows that the relative significance of the Basset term depends only on the density ratio between the particle and the fluid (more specifically, it does not depend on the size of the particle or on the viscosity of the fluid). Figure 16.1 compares the time evolution of the velocity of a particle dropped with zero velocity in the gravity field, obtained by taking into account the Basset term and by omitting it. The two cases with a Basset term correspond to a particle in air or water. It is verified that the Basset term has almost no effect for the particle in air Pp Pf = 3,000). For tlie particle in water // / = 3), the Basset term slows down the particle s acceleration, but the characteristic time of the acceleration remains short (for the values indicated in Table 16.1). The characteristic time (defined by E(f )/tF stokes =(i- xp(-i)) )> obtained by taking into account the Basset... 

The Basset term incorporates, via the time integral, the recent history of the particle s movement. The quantity H sjn t-T) inside the integral allows for a... 

In order to simplify the calculations, we have omitted the Basset terms, which have been found not to alter the nature of the physics. These essentially amount to an additional friction term, which slows down the adaptation of the particle s movement to the surrounding flow, but this adaptation remains very rapid anyway. On the left-hand side, the second term appears when the acceleration is expressed while changing the reference frame. Transferring this term to the right, we infer ... 

The left-hand-side terms in equations [16.43], [16.47], and [16.49] are in identical form to those derived when we stndied the fall of a small particle within the gravity field (section 16.3) or its displacement in a nnidirectional fluid flow (section 16.4). The time periods characterizing the acceleration phase of the particle are identical (equations [16.11] and [16.18]). We recall that these were short. Our discussion regarding the Basset terms (section 16.3) also transposes to the case of centrifiigatioa... 

The Basset terms (terms VI in eqnation [16.5]) are additional drag terms which intervene during a transient regime. Their significance depends on the value of the density ratio pp pf. They are negligible for particles in a gas. They are not... 

The influence of the Basset term on dispersion coefficients and phenomena (Picart et al 1982 Picart, 1984). 

If the motion is steady, the left-hand side is zero. The steady motion of the particle is therefore determined by Stk, and pp does not need to be included explicitly in the analysis, as mentioned in the discussion following Eq. (8.1.4). Also the density ratio Ap/p does not occur in the equations when the added mass and Basset terms are neglected. 

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 existence of crystal lamellae in melt-crystallised polyethylene was independently shown by Fischer [28] and Kobayashi [39]. They observed stacks of almost parallel crystal lamellae with amorphous material sandwiched between adjacent crystals. At the time, another structure was well known, the spherulite (from Greek meaning small sphere ). Spherulites are readily observed by polarised light microscopy and they were first recognised for polymers in the study of Bunn and Alcock [40] on branched polyethylene. They found that the polyethylene spherulites had a lower refractive index along the spherulite radius than along the tangential direction. Polyethylene also shows other superstructures, e.g. structures which lack the full spherical symmetry referred to as axialites, a term coined by Basset et al. [41]. 

Picart. A., Berlemont, A. and Gouesbet, G. (1982). De I infulence du terme de Basset sur la diepersion de particules discretes dans le cadre de la theorie de Tchen, C.R. Acad. Sci., Paris, Series //, T295, 305. 

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... 

In Eq. (29), Vd represents the dispersed phase velocity, Fq is the drag force, Fg denotes the force of gravity, Fl is the lift force, Fs represents effects of the fluid stress gradients, Fh is the Basset history term, and F-w represents interactions with the wall. The review paper by Loth (42) presents and discusses all the forces present in Eq. (29). Flere we limit ourselves to the most important effect of drag forces. In the case of spherical solid particles of diameter d, Fd can be expressed as... 

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. 

In summary, the Boussinesq-Basset, Brownian, and thermophoretic forces are rarely used in disperse multiphase flow simulations for different reasons. The Boussinesq-Basset force is neglected because it is needed only for rapidly accelerating particles and because its form makes its simulation difficult to implement. The Brownian and thermophoretic forces are important for very small particles, which usually implies that the particle Stokes number is near zero. For such particles, it is not necessary to solve transport equations for the disperse-phase momentum density. Instead, the Brownian and thermophoretic forces generate real-space diffusion terms in the particle-concentration transport equation (which is coupled to the fluid-phase momentum equation). 

Here, the first term is the added mass contribution, the second is Stokes law, and the third is known as the Basset memory integral contribution. Evaluate this expression for... 

---