Dimensional geometry of the release conditions for scalars C 1 and C 2 upstream of the cylinder wake.
Geometry of the domain for the numerical simulations. Flow is from left to right, and the cylinder is shown as a shaded circle. The scalars were released at , but the filaments included an initial lateral diffusion that gave them an effective release location of . The shaded rectangular region received significant mesh refinement for the scalar transport simulations.
Numerical simulation results for the flowfield at Re ϕ = 100. (a) Streamlines of (u, v), which are dominated by the mean streamwise flow. (b) Streamlines of (u − 0.95 u 0, v), which correspond to a perspective moving with the mean wake, exposing the underlying vortex structure.
Concentration profiles for C 1 and C 2 in the no-cylinder case for small (solid lines) and large (dashed lines). The initial condition for is shown in the left profile. Subsequent diffusive behavior given by Eqs. (15) and (17) are shown in the middle and right profiles.
Analytical solution for the laterally integrated reaction rate (Eq. (23) ), which is independent of filament separation (subject to Da ≪ 1 and ). The reaction rate is plotted as a function of nondimensional streamwise distance λ (Eqs. (16) and (24) ). The plots show the solution on (a) linear axes and (b) log-log axes. Reactions begin to appear near λ ≈ 1. The asymptotic behavior for as λ → ∞ (Eq. (25) ) is indicated with slope triangles.
Integrated reaction rate for the no-cylinder case, using the same axes as Fig. 5 , showing (a) effect of nondimensional filament width for small Damköhler number (Da = 0.01 ≪ 1), and (b) effect of Da for large filament width ( for the range of the plot). Symbols are numerical simulations, solid line is Eq. (23) , and dashed lines in part (a) are numerical integrations of Eq. (13) .
Simulation results showing instantaneous distributions of concentrations C 1 and C 2 (left column) with associated reaction rate R = C 1 C 2 (right column), for five representative cases. The final row shows the color schemes for C 1, C 2, and R. For all five cases shown, Da ϕ = 0.01.
Integrated reaction rate (Eq. (29) ) as a function of streamwise location , showing (a) effect of filament spacing (for ), and (b) effect of filament width (for ). The streamwise location of the cylinder is indicated by the gray circle centered at . For all cases shown, Sc = 10 and Da ϕ = 0.01.
Integrated reaction rate (Eq. (29) ) as a function of streamwise location , showing (a) effect of Schmidt number (for Da ϕ = 0.01), and (b) effect of Damköhler number (for Sc = 10). For all cases shown, and .
Schematic of the initial conditions for the stretched interface model. The interface is stretched longitudinally ( -direction) and compressed in the normal direction ( ). Filaments are shown for simplicity with finite width ; the stretched interface model assumes that is large.
Image-based quantification of interface length using results from the numerical simulations. The lengths L 1, L 2, and L 3 were measured over full filament periods, whereas L 0 was measured over a half-period to avoid the chaotic zone immediately downstream of the cylinder.
Comparison of the simple stretched interface model prediction (solid line) to the full numerical simulation (dashed line) for (a) and (b) , showing qualitative agreement. All cases shown correspond to , Sc = 10, Da = 0.01.
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