The FCS (Figure 1b) is bounded by two concentric cylinders; the outer cylinder radius reduces from r_E = r_D = 3.81 cm to r_F = 1.11 cm over the axial length HG, causing a conical constriction of the flow, before the combined flow enters the flow classification section of the instrument. The nozzle is radially asymmetrical in shape, flaring towards the outer cylinder, with the outer edge terminating at BC and the longer inner edge extending to IJ. The radial location of this annular nozzle is closer to the outer cylinder than to the inner cylinder. Dimensionless radii, r*, and axial lengths, x*, are scaled to the initial outer cylinder radius r_E = r_D = 3.81 cm (Table S1). Dimensionless velocities, v*, are scaled to the initial outer sheath velocity, v_{out sh} = 3.89 cm/s. The pressure and axial shear stress, σxr=−μ(∂vr∂x+∂vx∂r), are made dimensionless by reference to ½ ρ v_{out sh}².

Grid generation is an important part of any CFD simulation. The grid has a significant impact on the rate of convergence, on solution accuracy and on the CPU time required for the solution. An appropriate grid needs to resolve the inlet region of the entering annular flows and the regions near the wall. Pointwise 16.02 was used to generate a high-quality structured multi-block mesh for the complex FCS geometry, which has several high-gradient regions. Details are discussed in the Supplemental Material (Tables S2, S3).

The governing equations for the present flow are the continuity and the Navier-Stokes equations. The flow is treated as incompressible and is laminar (Reynolds number Re < 2000): for Case 1, at the aerosol nozzle entrance, Re_AK ≈ 14; at the outer sheath inlet, Re_CD ≈ 25; at the inner sheath inlet, Re_HI ≈ 50; at the FCS exit, Re_FG ≈ 400.

For an incompressible fluid, the continuity equation reduces to (1)∇․v→=0 and the momentum equation becomes (2)∂(ρv→)∂t+∇․(ρv→v→)=−∇p+∇․(τ¯¯) where the stress tensor is (3)τ¯¯=μ[(∇v→+∇v→T)−23∇․v→I¯¯], and v⃗ is the velocity field, p is the static pressure, ρ and μ are, respectively, the density and the dynamic viscosity of air, and I̿ is the identity tensor. Gravity has negligible impact on the air flow, so there is no body-force term in Eq. (2).

The equations were solved numerically using a finite-volume solver (Fluent 6.3, based on an algorithm of Patankar, 1980). A pressure-based solver (incompressible flow) is used, with SIMPLE (semi-implicit method for pressure-linked equations) as the pressure-velocity coupling method, with the default values for the relaxation parameters (0.3 for continuity and 0.7 for momentum)—we note that convergence was fastest when these relaxation parameters were used. A second-order up-winding scheme for convective terms, and a second-order central-differenced scheme for diffusion terms, were used to integrate the equations in space (our results do not appear to change when a third-order upstream centered scheme, MUSCL, is used instead for the convection terms). Temporal discretization was performed using a second-order implicit scheme.

In our calculations, all three fluids (aerosol, inner and outer sheaths) are taken to be dry air at T = 20°C (The fiber concentration in the NIOSH instrument is n_aero ~ 6 * 10⁴ fibers/cm³, which corresponds to a solids volume fraction of ϕ ~ 6 * 10⁻⁷; this low solids volume fraction motivates a purely aerodynamic treatment of the aerosol flow.). In the operation of the Baron classifier, both the aerosol and sheath flow air need to be humidified to RH = 50%, so as to facilitate fiber polarization (Wang et al. 2005), as well as to minimize Coulombic fiber-fiber interactions (Turkevich and Deye, unpublished). However, for our simulations, we have neglected humidity; while humidity influences the fiber polarization, it is not expected to alter the aerodynamics of the aerosol.

The aerosol is introduced at the nozzle inlet (AK) with a uniform inlet velocity. The nozzle axial length AB = 3.17 cm is 25 times longer than its hydraulic diameter at the inlet: the entrance length (calculated for a straight annulus, using the hydraulic diameter), for the Case 1 flow rates, is 0.085 cm. The uniform inlet velocity profile (at AK) will thus become fully developed flow by the nozzle exit (BJ). The sheath flows are introduced via foam diffusers (at CD and HI) so that these flows enter the FCS with uniform inlet velocity. The nozzle and FCS surfaces are non-porous solid walls, with zero tangential (no-slip) and normal components of the velocities at the walls. The outlet pressure (at FG) is set at gauge pressure (p = 0).

The solution of the equations was considered to be converged when the scaled residuals for the continuity and momentum equations decreased to 10⁻¹² (from initial values ~ 10⁻¹). All simulations were performed under this convergence criterion.

In cylindrical coordinates, the axial velocity, v_x, is related to the stream function, ψ, by (4)vx=1r∂ψ∂r which is easily integrated at the inlet, as the inlet axial velocity is uniform. If ψ₁ and ψ₂ represent values of the stream function at radial locations r₁ and r₂ at the inlet, then (5)Δψ=ψ2−ψ1=vxr22−r122 This permits an identification of streamline contour values (Figures S3 and 6, where we have used uniform contour spacing, given in Table S7).

All of the flows studied are steady and laminar; the aerosol streamlines are preserved, and the aerosol never mixes with the sheath flows. For a typical fiber (d ~ 1 µm, L ~ 10 µm, which yields an aerodynamic diameter d_aero ~ 1 µm, we can calculate a Stokes number, Stk = (ρ_part/ρ_air)*(d_part² U)/(18 ν_air D), where we consider glass fibers (ρ_part ~ 2.58 g/cm³), and where we take the nozzle width as the sharpest length scale over which the flow can be diverted, D ~ r_A – r_K = 0.64 cm. For Case 1 (the usual operation of the Baron instrument), the aerosol velocity at the nozzle is U ~ 15.4 cm/sec, whence Stk ~ 2 * 10⁻⁴; for Case 4 (the most extreme case considered), Stk ~ 4 * 10⁻³. These estimates hardly change when asbestos is used instead of glass. Figures 6 show the streamline contours (uniform contour spacing, given in Table S7) for the different cases studied.

The streamlines for Case 1 (Q_aero/Q_sheath = 0.22, v_aero/v_{in sh} = 3.96, v_aero/v_{out sh} = 3.12) are shown in Figure 6a. The flow is free from recirculation. The fibers are confined near the middle region at the exit, away from the walls. The presence of the sheath flows (Figure 6a) inhibits the formation of recirculation regions (torroidal vortices) which develop in the entrance region near the outer and inner cylinders when these sheath flows are absent (Figure S3). Also, in the absence of sheath flows, the aerosol, emerging from the nozzle, tends to spread out along with the edges (at the exit, the aerosol extends over the entire annular region 0.210 < r* < 0.291). The sheath layer is effective in confining the aerosol, preventing deposition on the walls (at the exit, the aerosol is confined to the central annular region 0.249 < r* < 0.255). The bending of the streamlines near the nozzle edges is strongly affected by the sheath flows: without the protection of the outer sheath flow, the aerosol would deposit on the outer wall (as suggested by Figure S3) as it emerges from the upper end of the nozzle; the sheath flows (Figure 6a) refocus the aerosol annulus and inhibit wall deposition.

In Case 2 (no illustration), the aerosol flow rate has been increased 10-fold (Q_aero/Q_sheath = 2.22, v_aero/v_{in sh} = 39.57, v_aero/v_{out sh} = 31.17) without vortex formation; at this higher aerosol flow, at the exit, the aerosol now extends over a slightly expanded annular region 0.233 < r* < 0.271.

Case 3, which represents a further increase (15-fold) in the aerosol flow rate (Q_aero/Q_sheath = 3.33, v_aero/v_{in sh} = 59.35, v_aero/v_{out sh} = 46.76), suggests a possible flow rate limitation of the FCS. Figure 6b shows the streamlines at t = 5 seconds following the flow initiation (steady state has been achieved at this time). A weak recirculation region has developed near the outer wall. The origin of this recirculation may be tracked with earlier time solutions (t = 0.1, 0.2, 0.3 seconds, computed with convergence at each time step). At the earlier times, the velocity mismatch, between the aerosol and the outer sheath flow, initiates the formation of a vortex near the entrance region, which is then pushed downstream, close to the outer cylinder.

Figure 6c shows the streamlines for Case 4. At this further (20-fold) increase in the aerosol flow rate (Q_aero/Q_sheath = 4.44, v_aero/v_{in sh} = 79.14, v_aero/v_{out sh} = 62.35), a second weak recirculation region develops, initiated by the large velocity difference between the aerosol and the inner sheath flow; this is accompanied by a strengthening of the outer vortex. This structure, of a weak inner vortex accompanying a stronger outer vortex, mimics the same structure that was found (Figure S3) for the case in the absence of sheath flows. The relative strengths of the two vortices is given by the maximum values of the stream function (Table S7).

The critical aerosol flows for the onset of these recirculations were determined to be: i) outer vortex formation at Q_aero ~ 13.2 L/min (Q_aero/Q_sh ~ 2.93, v_aero/v_sh ~ 52); ii) inner vortex formation at Q_aero ~ 15.9 L/min (Q_aero/Q_sh ~ 3.54, v_aero/v_sh ~ 50)—these critical aerosol flows are confirmed by the full 3-dimensional simulation (Table S8 of the Supplemental Material). Additional simulations (not reported here) confirm that these vortices may be suppressed by further increasing the sheath flows.

Figure 7 displays the pressure along the length of the FCS (at a distance Δr* = 0.042 from the outer cylinder) for the four simulations studied (at the exit, this corresponds to a radius midway between the inner and outer cylinders). The inviscid contribution to the pressure for Case 1 is estimated (applying Bernoulli’s equation along a streamline close to the converging cylinder) to be ~ 62 % of the total pressure. For all of the simulated cases, the pressure (along the line plotted) is effectively constant along the first 80% of the FCS length, with almost 90% of the pressure drop occurring over the last 20% of the FCS length. The pressure is insensitive to the detailed structure of the upstream combination of the aerosol and sheath flows. The pressure shows no obvious signature of the recirculation regions (Cases 3 and 4). The pressure is radially quite constant (not shown here), with maximal variation of ~ 8 % (5 %) at x* ~ 4.2 for Case 1 (Case 4); near the inlets, the radial variation of the pressure is also a small (~ 0.2 %).

The flow rates for which Deye et al. (1999) performed their calculation, correspond to our Case 1. The pattern of the streamlines for Case 1 (Figure 6a) is similar to that of Deye et al. (1999); in particular, Deye et al. (1999) did not detect any vortex formation. The focusing, by the sheath flows, of the aerosol streamlines, as they exit from the nozzle, is common to both works. However, our axial velocity contours (Figure 8a) differ from those of Deye et al. (1999) near the inlet. Their contours indicate a higher velocity near the outer cylinder upstream. However, since the outer sheath inlet area is larger than the inner sheath inlet area, the outer sheath inlet velocity must be smaller than the inner sheath velocity. We believe the contours of Deye et al. (1999) are in error near the inlet. The general structure of the axial velocity contours (Case 1) is a ramp, increasing from the inlet nozzle to the flow combination exit, and falling off as either cylindrical wall is approached. With the increase in aerosol velocity (Figure 8b for Case 4), the recirculation regions introduce axial velocity contour wells, radially on either side of the overall ramp, and these wells tend to pinch the ramp upstream.

It is instructive to consider the aerodynamics of the FCS confined laminar annular jet in two ways: i) with a time-dependent description; ii) within a steady-state description.

When a circular jet discharges into an unconfined quiescent fluid, the boundary layer on the jet nozzle forms a free shear layer, originating at the nozzle lip and extending downstream (Reynolds et al., 2003). This shear layer is subject to a Kelvin-Helmholtz instability (Schlichting, 1933; Ho & Huerre, 1984). Axisymmetric ring vortices form in the jet shear layers. These vortices are advected downstream, interact, detach, become unstable, and break down, giving rise to downstream turbulence in the far field.

When the jet is confined (as by the outer and inner cylinders of the FCS), the time-dependent behavior may be stabilized due to competition between vorticity generation at, and its diffusion away from, the jet boundary layers. If the out-diffusion dominates (Fig. S2a: Re_aerosol ~ 13, Re_out ~ 25, Re_in ~ 50), vortex rings do not form. If the out-diffusion balances the vorticity generation (Fig. S2b: Re_aerosol ~ 117, Re_out ~ 25, Re_in ~ 50), a steady state recirculation develops downstream of the nozzle; in the FCS, the radial asymmetry of the nozzle introduces asymmetry in the generation of the inner and outer vortices. Presumably, at still higher Re (which we did not investigate), the flow will become time-dependent, unstable and, ultimately, turbulent.

The effects of confinement may also be understood within a steady-state description (Landfried et al., 2012). When a steady-state jet discharges into an unconfined quiescent fluid, the external fluid is entrained, and the jet spreads (Pai, 1954). Confinement of the external fluid restricts this entrainment. If the jet is confined purely by a lateral cylindrical wall (a ‘ventilated jet’), fluid upstream from the jet nozzle may still be entrained, and the jet spreads to the confining wall. If, however, there is also a back wall, obstructing the axial flow exterior to the jet nozzle, recirculation of the ‘dead fluid’ is induced as the jet spreads. If the back wall is sufficiently far upstream from the nozzle, the recirculation occurs exterior to and upstream of the nozzle, and the jet spreads, essentially as it would were there no back wall present (Landfried et al., 2012). If, however, the back wall is flush with the nozzle, then the recirculation is forced downstream and interferes with the spreading of the jet—this is precisely the geometry of the Baron FCS. The recirculation transition may be understood as a cross-over from the low aerosol flow regime (Fig. S2a), where there is sufficient exterior sheath flow to permit entrainment, to the high aerosol flow regime (Fig. S2b), where there is insufficient exterior sheath flow for the entrainment, and recirculation occurs downstream of the nozzle.

Of importance for a particular aerosol instrument is the prevention/control of this recirculation through various instrumental innovations. The tapered FCS design delays the appearance of both the outer recirculation to Re_aerosol ~ 171 (ΔRe = Re_aerosol − Re_out ~ 146) and the inner recirculation to Re_aerosol ~ 204 (ΔRe = Re_aerosol − Re_in ~ 154). The CFD simulation demonstrates how this conical convergence in a real instrument stabilizes the flow against formation of outer and inner vortices (Q_aero/Q_sheath ~ 8.1, 9 goes to 13.2, 15.8).

The usefulness of the tapered geometry to stabilize flows, where aerosol and sheaths are introduced, has long been appreciated by aerosol instrument designers; however, to our knowledge, elucidation of the alternative flow patterns that actually develop within the mixing region, and their quantitative suppression by the tapered geometry, has not been explicitly pointed out to the aerosol community.