TURBULENCE-INDUCED RELATIVE VELOCITY OF DUST PARTICLES. IV. THE COLLISION KERNEL

Inertial particles suspended in turbulent flows show an inhomogeneous spatial distribution. A well-known explanation for the inhomogeneous distribution is that inertial particles tend to be expelled from vortical structures in the flow (Maxey 1987; Squires & Eaton 1991). Vortices induce rotation of the particles, leading to a centrifugal force that pushes the particles out. Particles are thus collected in regions in between strong vortices, where the strain of the flow velocity dominates over the vorticity (see Figure 2 of Cencini et al. 2006 and Figure 3 of Pan et al. 2011).

The degree of clustering can be quantified by the RDF, g. In Figure 1, we plot the RDFs at fixed Stokes ratios, f, as a function of the Stokes number, St_h, of the larger particle at three distances, r. The upper X-axis normalizes the friction time, τ_{p, h}, of the larger particle to T_L, i.e., Ω_h = τ_{p, h}/T_L. The red lines correspond to the case of equal-size particles, which has already been shown in Paper I. The monodisperse RDF peaks for St ≃ 1 particles for which the friction time couples to the smallest eddies of the flow. In Pan et al. (2011), we showed that the peak of the RDF at St ≃ 1 can be explained by an analysis of the effective compressibility of the particle "flow", which is the largest at St ≃ 1. The RDF of equal-size particles with St ≲ 6 shows a significant dependence on r, increasing as a power law with decreasing r (see Pan et al. 2011; Paper I). For these particles, the relative velocity decreases with decreasing r (see Figure 2 in Section 3.2). The slower relative motions at smaller scales make the spatial dispersion (or diffusion) of the particle concentration less efficient, leading to an increase of the RDF toward smaller r. The r dependence disappears at St ≳ 6, where the particle relative velocity becomes r-independent (see Figure 2).

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At a given St_h, the RDF decreases with decreasing Stokes ratio f. In general, the RDF between two different particles is smaller than the monodisperse RDF of each particle, i.e., $g(r, \rm {St}_{\ell }, \rm {St}_{h}) \le {g}({r}, \rm {St}_{\ell }, \rm {St}_{\ell })$ and $g(r, \rm {St}_{\ell }, \rm {St}_{h}) \le {g}({r}, \rm {St}_{h}, \rm {St}_{h})$ (Zhou et al. 2001). This is because particles of different sizes tend to cluster at different locations. If one continuously changes the particle size, the positions of maximum clustering intensity would shift spatially (see Pan et al. 2011). As the size difference of the two particles increases, the spatial separation between their local concentration peaks becomes larger, leading to a decrease in their "relative" clustering.

The RDF of different particles also has a much weaker r dependence than for particles of the same size. As shown in Pan et al. (2011), the bidisperse RDF as a function of r becomes flat and approaches a constant at small r (see also Chun et al. 2005). Intuitively, the increases of the RDF toward small r would eventually stop at scales below the typical separation between local concentration peaks of two different particle species.⁵ The flattening of the RDF toward small r makes it easier to achieve convergence for particles of different sizes. At f ⩽ 1/2, the RDF already converges at r ≳ η/4, except for the smallest two particles (St_ℓ = 0.1 and St_h = 0.19) in our simulation.

To our knowledge, the effect of turbulent clustering on the collision kernel has not been included in existing dust coagulation models. As will be discussed in Sections 3.4 and 5, neglecting this effect tends to underestimate the collision rate.

In Figure 2, we show 〈|w_r|〉 at fixed values of f. The red lines, corresponding to the monodisperse case, have been studied in Paper I (see Figure 16 of Paper I), to which we refer the reader for details. The black lines plot the radial relative velocity, 〈|w_{f, r}|〉, between the particle and the flow (or tracer) velocity at given distances, corresponding to St_ℓ = 0 or f = 0. All of the curves for 0 < f < 1 lie in a region between the monodisperse and the particle-flow lines, which serve as useful delimiters.

Most theoretical models for the particle relative velocity predict its variance 〈w²〉 (or $\langle w_{\rm r}^2 \rangle$ for the radial component), which is easier to model than the mean relative velocity, $\langle |\boldsymbol {w}|\rangle$ (or 〈|w_r|〉). The variance (or the rms) does not enter the collision kernel, but it is of theoretical importance because understanding it helps reveal the underlying physics. The qualitative behavior of the mean radial relative velocity in Figure 2 is very similar to the rms relative velocity shown in Figure 7 of Paper II for 〈w²〉^1/2 (see also the right panel of Figure 10 in Paper II for the radial rms $\langle w_{\rm r}^2 \rangle ^{1/2}$). Therefore, the successful physical explanation for the rms relative velocity in Paper II based on the Pan & Padoan model can also be used to understand the behavior of 〈|w_r|〉 as a function of St_h and f.

In the PP10 model, the relative velocity between two particles of any arbitrary size has two contributions, named the generalized shear and acceleration contributions.⁶ The generalized shear contribution represents the effect of the particles' memory of the spatial flow velocity difference "seen" by the two particles at given times in the past. Because the flow velocity difference, Δu(ℓ), scales with the length scale, ℓ, the shear contribution depends on the separation, d(t), of the two particles backward in time (i.e., at t < 0). The memory timescale of an inertial particle is essentially its friction time, and the particles "forget" the flow velocity they saw before a friction time ago, i.e., at t ≲ −τ_p. Thus, considering that the particle separation (and hence Δu(d(t))) increases backward in time, the flow velocity difference, Δu, across the particle separation, d(− τ_p), at a friction time ago plays a crucial role in determining the shear contribution.

The generalized acceleration term reflects the different responses of different particles to the flow velocity and is associated with the temporal flow velocity difference, Δ_Tu, along individual particle trajectories (see Figure 1 of Paper II for an illustration). The generalized acceleration contribution increases with the friction time difference, τ_{p, h} − τ_{p, l}. For particles of equal size, it vanishes, and only the generalized shear term contributes (Paper II). The generalized acceleration term is r-independent because it only depends on the temporal flow velocity statistics along individual particle trajectories (see Paper II for details). In Paper II, we showed that the prediction of the PP10 model for the rms relative velocity is in good agreement with the simulation results, confirming the physical picture of the model.

For particles of equal size, 〈|w_r|〉 shows a strong r dependence at St ≲ 1. Because of the short memory/friction time, the particle distance d(− τ_p) a friction time ago is close to r, and thus Δu(d(− τ_p)) ≃ Δu(r), which is linear with r for a small r. This gives rise to the r dependence of the shear contribution. As St increases, d(− τ_p) increases and becomes less sensitive to r(= d(0)). Thus, with increasing St, the relative velocity increases, and the r dependence is weaker. At St ≳ 6, 〈|w_r|〉 becomes r-independent. Finally, for τ_p ≫ T_L, the flow velocity correlation/memory time is shorter than the particle memory, and the flow structures seen by the two particles within a friction time are incoherent. The action of the flow velocity on the particles is similar to random kicks in Brownian motion. This causes a reduction of the relative velocity by (T_L/τ_p)^1/2, leading to a St^−1/2 decrease in the τ_p ≫ T_L limit. Intuitively, in this limit, the motions of larger particles are slower, and their relative velocity decreases with increasing τ_p.

At a given St_h, 〈|w_r|〉 increases with decreasing f. This is due to the generalized acceleration contribution, which increases with increasing Stokes number difference. The PP10 model shows that the acceleration contribution dominates over the shear effect if the Stokes numbers differ by a factor of ≳ 4 (Paper II). In the case of f = 0, w_{f, r} can be estimated by the temporal flow velocity difference, Δu_T(Δτ), at a time lag of Δτ ≃ τ_p along the particle trajectory (see Paper II). Assuming that Δu_T can be approximated by the Lagrangian temporal velocity difference Δu_L and that Δu_L∝Δτ^1/2 for inertial-range time lags, this explains why 〈|w_{f, r}|〉 increases approximately as St^1/2 (dotted line segment) for τ_{p, h} in the inertial range. As Δu_T saturates at large time lags, w_{f, r} becomes constant for τ_p ≫ T_L. The r dependence for particles of different sizes is much weaker than the case of equal-size particles because the acceleration contribution is r-independent. At f ⩽ 1/2, 〈|w_r|〉 already converges at r = (1/4)η. Together with the convergence of the RDF, this makes it easier to estimate the bidisperse collision kernel at r → 0 than in the case of equal-size particles (see Section 3.3).

In Figure 3, we plot the spherical collision kernel per unit cross section, Γ^sph/(πd²) = 2g〈|w_r|〉, normalized to u_η and u' on the left and right Y-axes, respectively. The left panel shows the kernel for equal-size particles with f = 1. The red lines in this panel are essentially the product of the red lines in Figure 1 for g and those in Figure 2 for 〈|w_r|〉. They correspond to the solid lines in Figure 17 of Paper I. For convenience, we will also refer to the kernel per unit cross section as the normalized kernel.

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For particles of equal size with St ≳ 1, the normalized kernel already converges at r ≳ (1/4)η as the r dependences of g and 〈|w_r|〉 cancel out (see Paper I). However, for small particles with St ≲ 1, the kernel shows a significant r dependence at r ⩾ (1/4)η. The decrease of 〈|w_r|〉 with decreasing r is faster than the increase of g, and the normalized kernel keeps decreasing in the r range shown here. Because the dust particle size in protoplanetary disks is much smaller than the Kolmogorov scale, the measured kernel for St ≲ 1 at r ⩾ (1/4)η cannot be directly used in applications. One solution to this is to seek convergence of the normalized kernel by conducting larger simulations with a larger number of particles that allow accurate measurements at smaller scales. Alternatively, one may try to extrapolate the measured kernel to the r → 0 limit using the physical insights and theoretical models for the problem. Because the former is computationally expensive, we made an attempt in Paper I to pursue the latter approach.

Theoretical models have shown that, at a given small distance r, there are two types of particle pairs, whose contributions to the collision kernel have different scaling behaviors with r (see Falkovich et al. 2002; Wilkinson et al. 2006). Following the terminology of Wilkinson et al. (2006), we named the two types of particle pairs "continuous" and "caustic" in Paper I. As explained in detail in Paper I, the two types correspond to the inner and outer parts of the probability distribution of w_r at small and large relative velocities, respectively.

Physically, the continuous particle pairs are located in flow regions with small velocity gradients, where the particles can efficiently respond to the flow. Their relative velocity for St ≲ 1 particles thus roughly follows the flow velocity difference across the particle distance. On the other hand, the caustic pairs correspond to flow regions with velocity gradients larger than ≃ 1/τ_p. In these regions, the particle motions cannot catch up with the rapid flow velocity change, and their trajectories would deviate considerably from the flow elements. For example, particles may be shot out of the flow streamlines of high curvature (e.g., around a vortex), and these particles may cross the trajectories of nearby particles (Falkovich & Pumir 2007). At the trajectory crossing point, the particle velocity becomes multivalued, giving rise to the formation of folds, named caustics, in the momentum–position phase space (see, e.g., Figure 1 of Gustavsson & Mehlig 2011). In contrast, the continuous pairs appear to be smooth in phase space. This difference in the phase space motivated the terminology for the two types of particle pairs (Wilkinson et al. 2006).

The relative velocity of continuous particle pairs decreases linearly with decreasing r, faster than the increase of the RDF. As a result, their contribution to the normalized kernel vanishes as r → 0 (Gustavsson & Mehlig 2011; Hubbard 2012; see Figure 19 in Paper I). On the other hand, the caustic contribution, corresponding to trajectory crossing events, is predicted to be r-independent, and it becomes dominant at r → 0 (Gustavsson & Mehlig 2011). We thus attempted to isolate the r-independent caustic contribution.

Motivated by previous theoretical studies and our simulation data, we use a critical radial relative velocity, w_c(<0), to split approaching particle pairs into continuous (w_c ⩽ w_r ⩽ 0) and caustic (w_r ⩽ w_c) pairs. The contribution of caustic pairs to the kernel is then given by $\Gamma ^{\rm cau} = -4\pi d^2 g \int _{-\infty }^{w_{\rm c}} w_{\rm r} P(w_{\rm r}) dw_{\rm r}$. Theoretical models suggest w_c∝ − r/τ_p (e.g., Falkovich et al. 2002; see PP10) but leave the coefficient in this estimate unfixed. There is thus freedom in the choice of w_c. Because the caustic contribution is expected to be r-independent, we aim to select a value of w_c that minimizes the r dependence of Γ^cau. The best choice is found to be w_c = −0.2 r/τ_p. The blue lines in the left panel of Figure 3 plot Γ^cau with this w_c, and we see that the three lines for different r almost all collapse onto a single curve. In Paper I, we set w_c = −r/τ_p, with which Γ^cau still had a (weak) r dependence (see Figure 19 in Paper I). Setting w_c = −0.2 r/τ_p appears to be a better choice for the purpose of obtaining an r-independent kernel contribution.

In the right panel of Figure 3, the red lines are the caustic contribution for particles of equal size, corresponding to the blue lines in the left panel. All the other lines in the right panel show the normalized kernel for particles of different sizes. Unlike the case of equal-size particles, the kernels for particles of different sizes with f ⩽ 1/2 already converge at r = (1/4)η and can thus be directly used in practical applications. As seen earlier, for f ⩽ 1/2, both the RDF and 〈|w_r|〉 converge at r = (1/4)η.

The black lines in the right panel plot the kernel between inertial particles and tracers, corresponding to f = 0. In this case, the RDF is unity, and the normalized kernel is simply calculated as 2〈|w_{f, r}|〉 with 〈|w_{f, r}|〉 the average of the radial component of the particle-flow relative velocity. The black lines and the red ones for f = 1 provide useful limits, between which the curves for 0 < f < 1 reside.

All curves of different f appear to cross each other in a Stokes number range of 7 ≲ St_h ≲ 14. In particular, at St_h = 12.4, the variation of the normalized kernel is very small within ≲ 10% as f changes from 0 to 1. We can thus assume that, for all values of f, the normalized kernels approximately cross at a common point around St_h = 12.4 with Γ^sph/(πd²) ≃ u' = 6.8u_η. This means that, if the larger particle has a friction time τ_{p, h} ≃ T_L = 14.4τ_η, the kernel per unit cross section may be estimated by the 1D rms flow velocity, u', independent of the size of the smaller particle.⁷ The f independence at τ_{p, h} ≃ T_L can be viewed as being due to the cancelation of the dependences of the RDF and 〈|w_r|〉 on f. At St_h = 12.4, 〈|w_r|〉 increases by a factor of 1.7 as f goes from one to zero, while g decreases by the same amount (Figures 1 and 2).

Considering that the motions of particles with τ_p ≃ T_L are insensitive to turbulent eddies at small scales, the constancy of the kernel with f at τ_{p, h} ≃ T_L is likely independent of the Reynolds number, Re, of the flow. However, as pointed out in Section 2, these particles couple to the driving scales of the turbulent flow, and their dynamics may be affected by how the flow is driven. Therefore, it remains to be confirmed whether the f independence of the kernel at τ_{p, h} ≃ T_L is universal with respect to the flow-driving mechanism. For such a confirmation, one needs to conduct a new set of simulations with different driving schemes, which are computationally expensive and beyond the scope of the current work.

For τ_{p, h} ≳ T_L, the kernel is larger at smaller f because of the increase of 〈|w_r|〉 with decreasing f. Interestingly, the opposite is seen for 1 ≲ St_h ≲ 12.4, where the kernel is the smallest for f = 0. As f increases toward 1, the clustering effect is stronger, and the increase of the RDF more than compensates for the decrease of 〈|w_r|〉, lifting all of the kernels for f > 0 above the black lines. The particle-tracer kernel (black lines) simply follows the scaling of 〈|w_{f, r}|〉 (because g = 1) and goes like $\rm {St}_{h}^{1/2}$ (dotted black line segment) in the inertial range. At f > 0, the RDF increases as St_h decreases toward one, and the increase is more rapid at larger f (see Figure 1). Thus the slope of the normalized kernel with St_h in the inertial range becomes continuously shallower as f increases. For particles of equal size in the inertial range, the opposite trends of g and 〈|w_r|〉 with St (red lines in Figures 1 and 2) lead to a very flat slope: the normalized kernel scales with St_h roughly as $\rm {St}_{h}^{0.15}$ (the red dotted-line segment). For 0 < f < 1, the slope of the kernel is in between 0.5 and 0.15. For example, the measured slope is approximately 0.22, 0.3, 0.35, 0.4, and 0.43 for f = 1/2, (1/4), (1/8), 1/16, and (1/32), respectively. These features are not captured by the kernel formulation commonly used in dust coagulation models because the clustering effect is neglected.

In the right panel, we see that, for small particles with St ≲ 1, the kernels at different f almost cross at another common point with St_h = 0.6 and Γ_sph/(πd²) ≃ u_η. Below St_h = 0.6, the normalized kernel decreases as f increases from 0 to 1, suggesting that, for small particles in the dissipation range, collisions of different particles are more frequent than between similar ones.

The collision kernel commonly used in dust coagulation models ignores the effect of clustering and adopts the model of Völk et al. (1980) and its successors for the particle relative velocity. At a given St_h, the Völk-type models predict a larger relative velocity between particles of different sizes than between particles of equal size (see, e.g., Ormel & Cuzzi 2007). Therefore, the commonly used kernel is smallest at f = 1 and the largest at f = 0 for any St_h. This trend is consistent with our results for St_h ≲ 1 and τ_{p, h} ≳ T_L, but it is incorrect for τ_{p, h} in the inertial range, where the clustering effect makes the kernel largest at f = 1. Equation (28) of Ormel & Cuzzi (2007) for the rms relative velocity of inertial-range particles suggests that, in the commonly used prescription, the normalized collision kernel for f = 0 is larger by a factor of 1.5 than for f = 1. On the other hand, our simulation data indicate that, in the inertial range, the kernel for f = 1 is larger than the f = 0 case by a factor of a few. As discussed in Section 3.5, in a realistic disk turbulence with a much larger Reynolds number than our simulated flow, the collision kernel for inertial-range particles of equal size (f = 1) may exceed the kernel at f → 0 by a significantly larger amount if the RDF, g, has a significant Reynolds number dependence.

We also computed the cylindrical kernel Γ^cyl and found that it coincides with Γ^sph, except for small particles of similar sizes with St_h ≪ 1 (see Section 3.4). As mentioned earlier, the two formulations give equal estimates for the kernel if the direction of ${\boldsymbol {w}}$ is isotropic. In the cylindrical formulation, we can also split particle pairs of equal size into two types using a critical value, $|{\boldsymbol {w}}|_{\rm c}$, for the 3D relative velocity amplitude, $|{\boldsymbol {w}}|$. Particle pairs with $|{\boldsymbol {w}}| <|{\boldsymbol {w}}|_{\rm c}$ and $|{\boldsymbol {w}}| \ge |{\boldsymbol {w}}|_{\rm c}$ are counted as continuous and caustic pairs, respectively. With the choice of $|{\boldsymbol {w}}|_{\rm c} = 0.27 r/\tau _{\rm p}$, the caustic contribution to the cylindrical kernel per unit cross section, $g \int _{|{\boldsymbol {w}}|_{\rm c}}^{\infty } |{\boldsymbol {w}}| P(|{\boldsymbol {w}}|) d |{\boldsymbol {w}}|$, is r-independent and is almost equal to the caustic contribution from the spherical formulation. Note that the critical value $|{\boldsymbol {w}}|_{\rm c} = 0.27 r/\tau _{\rm p}$ is different from the one used in the spherical formulation, and it is also different from the choice of $|{\boldsymbol {w}}|_{\rm c} = r/\tau _{\rm p}$ by Hubbard (2013).

We point out that there are uncertainties in how to split the continuous and caustic pairs for small particles of equal size (St ≲ 1, f = 1). Theoretical models suggest that the critical relative velocity w_c (or $|{\boldsymbol {w}}|_{\rm c}$) scales as ∝r/τ_p, but they do not predict its exact value. We selected a value w_c to obtain an r-independent contribution to the kernel, but there is no theoretical guarantee of the accuracy of our choice for w_c. It is possible that the r-independent caustic contribution we obtained at r ≳ (1/4)η may not precisely represent the realistic kernel as r → 0. In other words, as r decreases below (1/4)η, the critical relative velocity w_c that minimizes the r dependence could be different from the value we selected. Another uncertainty is that it is not clear whether, at the distance range we measured, the two types of pairs can be precisely divided simply by a single critical value for the relative velocity. The transition from continuous to caustic types may not occur sharply at a single relative velocity. Instead, the transition could be gradual, occurring over a relative velocity range, in which particle pairs may not be definitely counted as continuous or caustic. Because of these uncertainties, the caustic kernel we obtained should be tested by future simulations that can measure the kernel at smaller scales, where the caustic contribution dominates.

Despite the uncertainties, we think that the method of splitting two types of pairs is useful to provide physical insights for understanding the r dependence of the collision kernel at St ≲ 1 and f = 1 in the r range explored. We also found that it gives a reasonable estimate for small equal-size particles at r → 0. Wilkinson et al. (2006) predict that, for these particles, the normalized kernel at r → 0 is given by ∝exp (− A/St), similar to an activation process. Our measured caustic kernel is in good agreement with this prediction with an activation threshold of A ≃ 1.0.

Hubbard (2012) argued that, in the monodisperse case, inertial particles that show significant clustering do not contribute to the collision rate, i.e., the particles that cluster are noncollisional. In our terminology, this argument essentially assumes that clustering is mainly caused by the continuous particle pairs, which do not contribute to the collision rate at r → 0, while the caustic pairs that contribute to the collision rate do not significantly contribute to clustering. This is likely true for particles of equal size with St ≲ 1. For these small particles, caustic pairs are very rare and are only a tiny fraction of the total number of particle pairs at a given small distance. Thus, the main contribution to clustering must be due to the rest of the particles, i.e., particle pairs of the continuous type. As discussed earlier, the continuous pairs do not contribute to the collision kernel as the particle distance (or size), r, approaches zero (Wilkinson et al. 2006; Hubbard 2012).

However, our data indicate that the argument of Hubbard (2012) is invalid for inertial-range particles with τ_η ≲ τ_p ≲ T_L. In Paper I, we showed that caustic pairs start to dominate at St ≳ 1, and, at St ≳ 3.11, most particle pairs belong to the caustic type. Therefore, if significant clustering occurs for inertial-range particles, it must be from caustic pairs. Figure 1 shows that the RDF, g, is significantly above unity for inertial-range particles of equal size, suggesting that caustic particles do contribute to clustering. In fact, it is the clustering of the caustic pairs that makes the normalized kernel for particles of equal size in the inertial range larger than that between particles of different sizes. Thus, for inertial-range particle of equal size, it is not true that the clustering particles do not collide.

For particle of different sizes, the RDF, the relative velocity, and hence the kernel all converge at r ≳ (1/4)η, and it is thus not necessary to split the pairs into the two types. In Figure 1, we see that, for a Stokes ratio of f ≳ (1/4), there is a moderate degree of clustering at St_h ≃ 1. Because the kernel for f ≳ (1/4) and St_h ≃ 1 already converges at r = (1/4)η, all particle pairs at that distance contribute to the collision rate. Therefore, clustering at f ≳ (1/4) and St_h ≃ 1 would contribute to increasing the collision rate, again in contrast to the argument of Hubbard (2012). In summary, the claim of Hubbard (2012, 2013) that the clustering particles do not contribute to the collision rate is not valid in general and applies only for the special case of small equal-size particles with St ≲ 1.

The kernel formula commonly used in dust coagulation models is Γ^com = πd²〈w²〉^1/2, where the rms relative velocity, 〈w²〉^1/2, is usually taken from the model of Völk et al. (1980) or its successors. The variable Γ^com is in a simpler form than Γ^{sph, cyl}, and in this section we use our simulation data to assess the accuracy of the simplified formula for the collision kernel. We defer a systematic test of Völk-type models for 〈w²〉^1/2 to a later work.

In comparison to the spherical and cylindrical kernels Γ^{sph, cyl}, Γ^com neglects the clustering effect and uses the 3D rms of the relative velocity rather than the mean relative velocity, 〈|w_r|〉, or $\langle |{\boldsymbol {w}}| \rangle$. The ratios of Γ^{sph, cyl} to Γ^com can be written as $g C_{12}^{\rm sph, cyl}$, where $C_{12}^{\rm sph} \equiv 2\langle |w_{\rm r}|\rangle / \langle w^2\rangle ^{1/2}$ and $C_{12}^{\rm cyl} \equiv \langle |{\boldsymbol {w}}| \rangle /\langle w^2\rangle ^{1/2}$. The ratios $C_{12}^{\rm sph, cyl}$ compare the mean and rms relative speeds, corresponding to the first- and second-order moments of ${\boldsymbol {w}}$, respectively. In the left panel of Figure 4, we show $C_{12}^{\rm sph}$ (solid) and $C_{12}^{\rm cyl}$ (dashed) computed from our simulation data at r = η/4. For all Stokes pairs, the ratios are smaller than unity, meaning that using the rms relative velocity tends to overestimate the kernel. Except for a small difference of 5% at St_h ≪ 1 and f = 1, the solid lines almost coincide with the dashed ones, suggesting that $2 \langle | w_{\rm r}\rangle = \langle |{\boldsymbol {w}}| \rangle$, and hence Γ^sph = Γ^cyl within an uncertainty of only 5%. The near equality Γ^sph ≃ Γ^cyl was expected under the assumption of isotropy for the direction of ${\boldsymbol {w}}$.

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The ratios $C_{12}^{\rm sph, cyl}$ depend on the shape of the PDF of ${\boldsymbol {w}}$, particularly on the central part of the PDF, because both the mean and rms are low-order moments. If ${\boldsymbol {w}}$ is Gaussian, the PDF of the 3D amplitude $|{\boldsymbol {w}}|$ is Maxwellian, and it is easy to show $C_{12}^{\rm cyl} = ({8/3\pi })^{1/2} \simeq 0.92$. The same value is expected for $C_{12}^{\rm sph}$. Because the radial rms relative speed, $\langle w_{\rm r}^2\rangle ^{1/2}$, is typically smaller than the 3D rms, 〈w²〉^1/2, by a factor of $\sqrt{3}$ (see Paper II), we have $C_{12}^{\rm sph} \simeq (4/3)^{1/2} \langle |w_{\rm r}|\rangle /\langle w_{\rm r}^2\rangle ^{1/2}$. With a Gaussian distribution for w_r, $\langle |w_{\rm r}|\rangle /\langle w_{\rm r}^2\rangle ^{1/2} = ({2/\pi })^{1/2}$, and thus $C_{12}^{\rm sph}$ is also equal to (8/3π)^1/2. A value of ≃ 0.92 for $C_{12}^{\rm sph, cyl}$ is observed in the left panel of Figure 4 at τ_{p, h} ≫ T_L, where the PDF of ${\boldsymbol {w}}$ is approximately Gaussian (see Paper III).

As discussed in Paper I, the mean-to-rms ratios are smaller if the PDF of ${\boldsymbol {w}}$ is non-Gaussian with a sharper inner part and fatter tails (see Figure 16 of Paper I). For particles of equal size with τ_p ≲ T_L, the PDF is highly non-Gaussian with very fat tails, and the ratios $C_{12}^{\rm sph, cyl}$ are significantly smaller than 0.92. In particular, the degree of non-Gaussianity peaks at St ≃ 1, leading to a dip in the ratio at St ≃ 1, where $C_{12}^{\rm sph, cyl} \simeq 0.3\hbox{--}0.4$. In Paper III, we showed that the non-Gaussianity of the relative velocity PDF decreases with decreasing Stokes ratio f, and this is responsible for the increase of $C_{12}^{\rm sph, cyl}$ as f decreases toward zero. In the case of f = 0, the inner part of the PDF for the particle-flow relative velocity is nearly Gaussian for all particles (see Figure 1 of Paper III), which explains why the black lines for f = 0 are close to 0.9 at all St. The left panel of Figure 4 suggests that using the rms instead of the mean relative velocity in the collision kernel tends to overestimate the collision rate, especially for particles of similar sizes.

We also computed $C_{12}^{\rm sph, cyl}$ at other distances, r. For f ≲ 1/2, the ratio converges at r = (1/4)η, corresponding to the convergence of the PDF shape found in Section 4.2 of Paper III. However, for particles of equal size with St ≲ 6, $C_{12}^{\rm sph, cyl}$ have not converged and decrease with decreasing r because of the fatter PDF shape at smaller r (Paper I). For these particles, the ratios at r ≲ η/4 could be significantly smaller than the values shown in the left panel of Figure 4.

In the right panel of Figure 4, we plot the ratio Γ^{sph, cyl}/Γ^com in our simulation. In the calculation of Γ^com, we used the 3D rms relative velocity measured from our data. Again, the near coincidence of the solid and dashed lines for Γ^sph/Γ^com and Γ^cyl/Γ^com indicates that the two formulations give similar estimates for the collision rate. In the limit f → 0 (black lines), the ratio of the commonly used formula to Γ^{sph, cyl} is about 0.9 at all St_h, meaning that it overestimates the kernel slightly by ≲ 10%. For this particle-tracer case, the RDF g is unity, and the overestimate is because 〈w²〉^1/2 is larger than 2〈|w_r〉 by 10% (see the left panel). For f ≲ 1/4, Γ^com is a good approximation to the collision kernel, within an uncertainty of ≲ 40%.

For particles of similar sizes (f = 1 and 1/2), the ratios of Γ^{sph, cyl} to Γ^com are close to unity for τ_p ≳ T_L, but they become significantly larger as St_h decreases in the inertial range toward ≃ 1. This means that Γ^com underestimates the kernel for particles of similar sizes in the inertial range and in particular at St_h ≃ 1. The ratios Γ^{sph, cyl}/Γ^com converge at r = (1/4)η except for f = 1. For equal-size particles, Γ^{sph, cyl}/Γ^com keeps increasing as r decreases to (1/4)η, and the underestimate of Γ^com would be more severe at r < (1/4)η. In general, how Γ^com compares to Γ^{sph, cyl} depends on two opposite effects: using the rms relative velocity instead of the mean tends to overestimate the collision rate, and neglecting the clustering effect causes an underestimate. Both effects are strongest for particles of similar sizes at St_h ≃ 1 (see the left panel of Figure 4 and Figure 1), but it turns out that the second effect dominates for St_h in the inertial range, where g is significantly larger than $1/C_{12}^{\rm sph,cyl}$. Another interesting consequence of neglecting clustering is that, as f decreases from one to zero, Γ^com increases because 〈w²〉 is larger at smaller f (see Figure 7 of Paper II). This is in contrast to the result shown in the right panel of Figure 3, where the kernel decreases with decreasing f for St_h in the inertial range.

In summary, the commonly used kernel formula provides a good approximation for very different particles with the Stokes ratio f ≲ 1/4 or if the larger particle has τ_{p, h} ≳ T_L. However, it significantly underestimates the collision rate for particles of similar sizes (with f ≳ 1/2) with τ_{p, h} ≲ T_L, especially for St_h ≃ 1.

Because of the limited numerical resolution, the Reynolds number, Re, of our simulated flow is only ≃ 10³, much smaller than the typical value of Re ≃ 10⁶–10⁸ in protoplanetary disks. To study dust particles in protoplanetary turbulence, the results of our simulation should be appropriately extrapolated to a realistic Re. An accurate extrapolation requires the Re dependence of the collision kernel, which is currently unknown. Here we offer some speculations on how to extrapolate the measured results to higher Re. Studies of the Re dependence using higher-resolution simulations will be conducted in future work.

In our simulation, the inertial range is short, and the timescale separation between the driving scale and the Kolmogorov scale is only one order of magnitude. More precisely, we have T_L/τ_η ≃ 14.4 in our flow. Although we count all particles with τ_η ≲ τ_p ≲ T_L inertial-range particles, strictly speaking, there is only one species of particle in our simulation that is definitely not affected by either the dissipation or the driving scales of the flow (Paper III). This species is particles with τ_p = 6.21τ_η, or equivalently τ_p = 0.43T_L. Particles with τ_p ≳ 0.43T_L are coupled to eddies significantly above the dissipation range, and their dynamics are insensitive to the smallest eddies in the flow. Thus, when normalized to the flow properties at driving scales, the collision statistics of these particles are expected to be independent of Re. Our measured kernel for these particles can be directly applied to corresponding particles with Ω ≡ τ_p/T_L ≳ 0.43 in protoplanetary disks. Therefore, we only need extrapolations for Ω ≲ 0.43 particles in the inertial and dissipation ranges of the real flow.

As Re increases, the separation between τ_η and T_L increases as Re^1/2. Considering that Re ≃ 10³ and T_L/τ_η ≃ 14.4 in our flow, we have Ω ≃ 2.2 Re^−1/2St for larger Re. This suggests that, if the driving of the flow is fixed, then, with increasing Re, St = 1 particles correspond to smaller size with Ω ≃ 2.2 Re^−1/2. Particles with 2.2 Re^−1/2 ≲ Ω ≲ 1 lie in the inertial range, and we need to extrapolate the kernel down to Ω ≃ 2.2 Re^−1/2.

A simple extrapolation is to assume that the measured slope for the normalized kernel in the inertial range (see Figure 3) is independent of the Reynolds number. With this assumption, the normalized kernel for equal-size particles (f = 1) in the inertial range is approximately given by ≃ u'Ω^0.15, while in the f → 0 limit it scales as ≃ u'Ω^1/2 (see Figure 3 and Section 3.3). In this case, the effect of clustering significantly enhances the kernel for all inertial-range particles of similar sizes. Interestingly, with this extrapolation, we find that, for a realistic value of Re ≳ 10⁶–10⁸, the normalized kernel for equal-size particles at St_h = 1 (corresponding to Ω_h = 2.2 × 10⁻³–2.2 × 10⁻⁴) is larger by a factor of 10–20 than the particle-tracer limit with f → 0. This suggests a significantly higher collision frequency between similar particles than between particles of different sizes, as the particles just grow into the inertial range. Note that, in our flow with low Re, the variation of the kernel at St_h = 1 is quite small, increasing only by a factor of ⩽2 as f increases from zero to one (see Figure 3).

In the extrapolation above for equal-size particles, an implicit assumption was made for the scaling of the RDF with Ω in the inertial range. Because the relative velocity is expected to scale as ∝Ω^1/2 (or St^1/2) in the inertial range,⁸ the assumed Ω^0.15 scaling for the normalized kernel, Γ^sph/πd² (≡ 2g〈|w_r|〉), holds only if the RDF increases with decreasing Ω as ∝Ω^−0.35 in the inertial range. The RDF for f = 1 in our flow does not have a clear power-law range with Ω (see Figure 1). As mentioned earlier, strictly only one species of particles (St = 6.21) lies in the inertial range in our flow, and it is possible that g may show an extended power-law scaling with Ω in a high-Re flow with a wider range of scales unaffected by the dissipation or the driving scales. The possibility, however, needs to be verified by simulations at higher resolutions.

If the scaling g∝Ω^−0.35 is valid for particles in the inertial range, it would imply an Re dependence of the RDF for particles at fixed Stokes numbers, St. Inserting Ω∝Re^−1/2St to this scaling gives g(St ≃ 1)∝Re^0.175 for St ≃ 1 particles in turbulent flows of different Re. The Re dependence of the RDF at fixed Stokes numbers around or below unity have been studied in the literature, but, to our knowledge, a conclusive answer is still lacking. For example, the simulation of Collins & Keswani (2004) indicated that the RDF of St ≃ 1 particles converges already at Re ≃ 600 (see also Hogan & Cuzzi 2001), while Falkovich & Pumir (2004) found that the clustering intensity shows a significant increase with increasing Re.

Clearly, if the Re dependence of the RDF at St ≃ 1 is weaker than Re^0.175, the increase of g with decreasing Ω would be slower than Ω^−0.35, and thus the scaling of the normalized kernel for f = 1 would be steeper than Ω^0.15. An extreme case is that the RDF for St fixed around ≃ 1 is independent of Re. In that case, the RDF for inertial-range particles with τ_p significantly away from the dissipation scales can only vary slowly with Ω, and, for these particles, the normalized kernel would scale as Ω^1/2, as it just follows scaling of the relative velocity. This scaling is similar to the f → 0 limit. Therefore, we would expect the kernels for f = 1 and f → 0 to remain close to each other as Ω_h decreases in the inertial range. In the inertial range, clustering does not considerably increase the collision rate of similar-size particles. Only as St_h approaches one would the monodisperse RDF start to increase significantly, which would tend to increase the kernel for f = 1 above the f → 0 case by a factor of two or so. This behavior is in contrast to the extrapolation based on the slope of the normalized kernel measured in our simulation.

In summary, the extrapolation of the measured kernel from our simulation to a realistic flow with much larger Re is uncertain. If we assume that the measured slope of the normalized kernel at fixed f can be applied to larger Re, then the decrease of the kernel for f = 1 with decreasing Ω in the inertial range is slower than the limit f → 0, and, at τ_{p, h} ≃ τ_η, the collision rate between similar particles is much larger than between different particles. On the other hand, if the RDF at a fixed Stokes number St around unity is Re-independent, then the normalized kernels for f = 1 and f = 0 have a scaling similar to Ω_h and likely stay close to each other for any Ω_h in the inertial range. This question will be settled with future higher-resolution (⩾1024³) simulations, where the scalings of g and of the normalized kernel with Ω (in the inertial range) and the Re dependence of g at fixed Stokes numbers can be checked.

We start by considering the collision-rate weighted rms relative velocity, $w^{\prime }_{\rm sph}$ and $w^{\prime }_{\rm cyl}$, in the two formulations. In the cylindrical formulation, $w^{\prime }_{\rm cyl} \equiv (\int _0^{\infty } |{\boldsymbol {w}}|^2 P_{\rm cyl} (|{\boldsymbol {w}}|) d |{\boldsymbol {w}}|)^{1/2} = (\langle |{\boldsymbol {w}}|^3 \rangle / \langle |{\boldsymbol {w}}| \rangle)^{1/2}$ (see Hubbard 2012). Similarly, we have $w^{\prime }_{\rm sph} \equiv (\int _0^{\infty } \boldsymbol {w}^2P_{\rm sph} (|\boldsymbol {w}|) d|\boldsymbol {w}|)^{1/2} = (\langle w_{\rm r}{\boldsymbol {w}}^2 \rangle _-/ \langle w_{\rm r}\rangle _-)^{1/2}$ for the spherical formulation. The subscript "-" in the ensemble averages indicates that only approaching particle pairs with w_r < 0 are counted, e.g., $\langle w_{\rm r}\rangle _- \equiv \int _{-\infty }^{0} w_{\rm r} P(w_{\rm r}) dw_{\rm r}/\int _{-\infty }^{0} P(w_{\rm r}) dw_{\rm r}$. With collision-rate weighting, the rms relative velocity, $w^{\prime }_{\rm sph}$ and $w^{\prime }_{\rm cyl}$, provides a measure for the average collisional energy per collision. Although the rms itself is not sufficient for modeling collisional growth of dust particles, a calculation of $w^{\prime }_{\rm sph}$ and $w^{\prime }_{\rm cyl}$ helps illustrate the effect of the collision-rate weighting. Also, the collision-rate weighted rms is likely more appropriate than the unweighted one to use in coagulation models that ignore the distribution of the collision velocity.

Figure 5 shows results for $w^{\prime }_{\rm sph}$ (solid) and $w^{\prime }_{\rm cyl}$ (dashed) measured at r = 1/4η. The solid and dashed lines in Figure 5 almost coincide. The equality of the weighted rms relative velocity in the two formulations was expected from the isotropy of ${\boldsymbol {w}}$. Slight differences (≲ 10%) are found at small St_h and f ≳ 1/2, because for these Stokes pairs, the assumption of perfect isotropy of ${\boldsymbol {w}}$ is not satisfied.

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Interestingly, if the larger particle is in the inertial range, i.e., if τ_η ≲ τ_{p, h} ≲ T_L, $w^{\prime }_{\rm cyl}$ and $w^{\prime }_{\rm sph}$ are almost independent of f, or the friction time, τ_{p, l}, of the smaller particle. In Paper II, we showed that the unweighted rms relative velocity, $\langle {\boldsymbol {w}}^2 \rangle ^{1/2}$, increases with decreasing  f  because the generalized acceleration contribution to the relative velocity, corresponding to different responses of different particles to the flow velocity, increases with increasing friction time difference (see Figure 7 of Paper II). To understand the invariance of $w^{\prime }_{\rm sph, cyl}$ with f for St_h in the inertial range, it is helpful to compute the ratio of $w^{\prime }_{\rm sph, cyl}$ to the unweighted rms, $\langle {\boldsymbol {w}}^2 \rangle ^{1/2}$, which is shown in the right panel of Figure 5. The ratio depends on the PDF shape of ${\boldsymbol {w}}$. Because $w^{\prime }_{\rm sph}$ and $w^{\prime }_{\rm cyl}$ correspond to higher (third) order statistics than the unweighted rms, the ratio $C_{32} \equiv w^{\prime }_{\rm sph, cyl}/\langle {\boldsymbol {w}}^2 \rangle ^{1/2}$ is larger if the PDF of ${\boldsymbol {w}}$ has fatter tails. In Paper III we found that at a given St_h, the PDF shape of the relative velocity is the fattest for particles of equal size (f = 1) and becomes continuously thinner as f decreases. This explains why the ratios shown in the right panel increase as f increases from zero to one. For St_h in the inertial range, the larger ratio C₃₂ at larger f happens to approximately cancel out the decrease of the unweighted rms, $\langle {\boldsymbol {w}}^2 \rangle ^{1/2}$, with increasing f (see Figure 7 of Paper II), leading to almost invariance of $w^{\prime }_{\rm sph, cyl}$ with f. This f independence simplifies the estimate of the average collision energy of inertial-range particles. For τ_{p, h} ≃ T_L, the average collision velocity for each collision is found to be ≃ 1.3u', independent of f. The collision velocity roughly follows a $\rm {St}_{h}^{1/2}$ scaling for St_h in the inertial range, which, however, needs to be verified by larger simulations with a broader inertial range.

For St_h ≲ 1 and τ_{p, h} ≳ T_L, the increase of C₃₂ with increasing f is not sufficient to compensate for the decrease of $\langle {\boldsymbol {w}}^2 \rangle ^{1/2}$, and the weighted rms is smaller at larger f. At f = 1, the ratio $w^{\prime }_{\rm sph, cyl}/\langle {\boldsymbol {w}}^2 \rangle ^{1/2}$ strongly peaks at St ≃ 1, corresponding to the result of Paper III that the PDF is the fattest at St ≃ 1. As shown in Paper III, the PDF of ${\boldsymbol {w}}$ becomes nearly Gaussian for sufficiently large τ_{p, h} (≫T_L), and, for a Gaussian distribution, it is easy to see that $w^{\prime }_{\rm cyl}/\langle {\boldsymbol {w}}^2 \rangle ^{1/2} = \sqrt{4/3} \simeq 1.15$. A ratio of 1.15 is actually observed at large St_h in the right panel of Figure 5. The right panel of Figure 5 also suggests that using the unweighted rms $\langle {\boldsymbol {w}}^2 \rangle ^{1/2}$ significantly underestimates the average collision velocity per collision, especially for the collisions between particles of similar size with St ≃ 1.

For f ⩽ 1/2, the measured rms velocity with collision-rate weighting converges at r = (1/4)η. The same is found for equal-size particles with St ≳ 0.8. However, complexities arise for smaller particles of equal size with St ≲ 0.4. For these particles, the weighted rms collision velocity has not reached convergence at r = (1/4)η, and a refinement is needed before it can be used in practical applications. In Section 3.3, in order to estimate the kernel in the r → 0 limit, we attempted to isolate an r-independent kernel contribution by separating out the caustic pairs. However, we find that the method does not apply here to estimate the weighted rms collision velocity at r → 0 or to obtain an r-independent contribution to the rms.

The idea of selecting the caustic pairs for small particles of equal size with St ≪ 1 is to approximate the collision statistics at r → 0 by the measured statistics of the relative velocity of particle pairs at a finite r above a threshold. Although it was useful to obtain an r-independent kernel contribution, this approximation is not justified in general. When a threshold relative velocity is applied to select caustic pairs, the continuous pairs with low relative velocity are excluded, which pushes the rms relative velocity to higher values than the overall rms that includes all pairs at a given r. However, the rms collision velocity at r → 0 is expected to be smaller than the overall rms at finite r because the overall rms decreases with decreasing r. This suggests that the rms collision velocity of the caustic pairs does not correctly represent the rms in the r → 0 limit. Thus, the relative velocity PDF that excludes the continuous pairs at a finite r is not equivalent to the PDF in the r → 0 limit; instead it overestimates the collision velocity at r → 0. We note that, when computing the rms collision velocity in simulations, Hubbard (2012) proposed counting only particle pairs with relative velocity above a threshold. Based on the above discussion, there is uncertainty in the rms collision velocity measured this way as an estimate for r → 0, and some care is needed when using such a method.

It turns out that isolating the caustic pairs from the continuous pairs does not help the convergence of the rms collision velocity. The threshold relative velocity is larger at larger r because it is chosen to be linear with r (see Section 3.3). It turns out that, with a stronger threshold, the weighted rms relative velocity of caustic pairs at a larger r exceeds the overall rms by a larger amount than at smaller r. This means that the convergence of the rms relative velocity of caustic pairs is actually slower than the overall rms in the r range ((1/4)η ⩽ r ⩽ 1η) explored here.⁹ In this range of r, the majority of particle pairs with St ≲ 0.4 belong to the continuous type, and the accuracy of approximating the actual statistics in the r → 0 limit by the collision velocity of caustic pairs is poor. As r decreases to a sufficiently small value so that most particle pairs are of the caustic type, the problem is expected to be less severe.

The above problem concerning the use of caustic pairs to approximate the r → 0 statistics adds further uncertainty to the validity and accuracy of the criterion we adopted to isolate the caustic pairs, in addition to those already discussed in Section 3.3. Nevertheless, the classification of two types of particle pairs and the method used to split them based on theoretical models do offer physical insight to understand the r dependence of the kernel in the r range we explored. It is still possible that, despite this problem with the weighted rms collision velocity, our method to obtain the r-independent caustic kernel provides a reasonable approximation for the kernel at r → 0 (see Section 3.3). Larger simulations that can measure the particle statistics at small scales are needed to test this possibility and to resolve the convergence issue.

We also note that the red lines in the left panel of Figure 5 rise slightly as St decreases toward St = 0.1. This behavior is unexpected and is likely a numerical artifact because the trajectory integration of St = 0.1 particles, the smallest in our simulation, is the least accurate (see Papers I and II). The integration accuracy depends on the ratio of the simulation time step to the particle friction time, which is the largest for the smallest particles. This issue can be solved by future simulations with a better temporal resolution (see Paper II).

In Paper III, we computed the unweighted PDF of turbulence-induced relative velocity for all Stokes number pairs available in our simulation and showed that the PDF of ${\boldsymbol {w}}$ is generically non-Gaussian, exhibiting fat tails. In that paper, we discussed in detail the trend of non-Gaussianity with the particle friction times and interpreted the results using the physical picture of the PP10 model. Here we examine the collision-rate weighted PDF of $|{\boldsymbol {w}}|$.

In Figure 6, we illustrate the effect of collision-rate weighting using a Stokes number pair, St_ℓ = 0.78 and St_h = 1.55, as an example. The dotted line shows the unweighted PDF, and the solid and dashed lines are the weighted PDFs, P_sph and P_cyl, in the spherical and cylindrical formulations, respectively. Clearly, with the weighting factor, the PDF shows lower and higher probabilities at small and large collision speeds, respectively. Because it favors large-velocity collisions, the collision-rate weighting increases the fraction of collisions leading to fragmentation and reduces the probability of sticking. Consequently, the growth of dust particles would be more difficult than predicted by dust coagulation models that ignore the collision-rate weighting. The solid and dashed lines almost coincide with each other. For this Stokes pair, the direction of ${\boldsymbol {w}}$ is nearly isotropic, and the weighted PDFs in the spherical and cylindrical formulations are expected to be equal.

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Figure 7 plots P_sph (solid lines) and P_cyl (dashed lines) at more Stokes number pairs. The left and right panels show the measured PDFs for f = 1 and 1/4, respectively. The weighed PDFs in the two formulations are in good agreement for all Stokes pairs. In cases where the two PDFs show differences, P_cyl has slightly higher and lower probabilities than P_sph at small and large $|{\boldsymbol {w}}|$, respectively. For f = 1, the difference between P_sph and P_cyl is largest at St = 0.79 and then decreases with increasing St. A comparison of the two panels shows that P_sph and P_cyl are in better agreement at smaller f. As shown in Paper II, the isotropy of ${\boldsymbol {w}}$ with respect to ${\boldsymbol {r}}$ improves with increasing St or decreasing f.

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The finding that P_sph ≃ P_cyl at all Stokes number pairs suggests that either formulation can be used to determine the collision outcome in coagulation models for dust particle growth. The geometry assumed in the spherical formulation is physically more reasonable, but the weighted PDF in the cylindrical formulation appears to be more convenient because it is directly related to the unweighted PDF as $|{\boldsymbol {w}}| P(|{\boldsymbol {w}}|)/\langle |{\boldsymbol {w}}| \rangle$. With this relation, one can easily determine the shape of $P_{\rm cyl}(|{\boldsymbol {w}}|)$ from the unweighted PDF presented in Paper III. It is expected that the trend of the shape of $P_{\rm cyl}(|{\boldsymbol {w}}|)$ (and hence P_sph) with f and St_h would be similar to the unweighted PDFs discussed in Paper III. For example, if the shape of the unweighted PDF of $|{\boldsymbol {w}}|$ has a higher degree of non-Gaussianity, the weighted PDF would also have more probabilities at both left and right tails, corresponding to small and large collision velocities. We refer the interested reader to Paper III for a detailed discussion on the shape and non-Gaussianity of the unweighted PDF as a function of f and St_h.

An important application of the collision velocity PDF is to determine the outcome of dust particle collisions induced by turbulence in protoplanetary disks. In Paper III, we computed the fractions of collisions leading to sticking (F_s), bouncing (F_b), and fragmentation (F_f) using typical disk and turbulence parameters at 1 AU, and particular attention was given to the effect of the non-Gaussianity of the relative velocity PDF on the fractions. These fractions need to be incorporated in coagulation models for a realistic prediction of the dust size evolution. The calculation of the fractions was based on the PDF of $|{\boldsymbol {w}}|$ measured in our simulation using a weighting factor ($\propto |{\boldsymbol {w}}|$) from the cylindrical formulation. Several simplifying assumptions were used to extrapolate the measured PDFs in the simulated flow to the disk turbulence with much larger Reynolds number (see Paper III). In Appendix A, we compute F_s, F_b, and F_f from the weighted PDF, $P_{\rm sph} (|{\boldsymbol {w}}|)$, in the spherical formulation and compare the result from the cylindrical formulation. The calculation in Appendix A adopts exactly same parameters and assumptions as in Paper III. In Figure 8 of Appendix A, we see that F_s, F_b, and F_f computed from the two formulations are consistent with one another, as expected from the agreement between P_sph and P_cyl.

The spherical and cylindrical formulations yield very similar results for all of the statistical quantities discussed so far, including the collision kernel; the collision-rate weighted rms and PDF of the 3D amplitude, $|\boldsymbol {w}|$, of the relative velocity; and the fractions of collisions leading to sticking, bouncing, and fragmentation (Appendix A). All these quantities computed from the two formulations coincide with each other, except for small differences for Stokes pairs with f close to one and St_h ≲ 1.

In Appendix B, we find an interesting difference between the two formulations concerning the collision-rate weighted PDFs of the radial and tangential components of ${\boldsymbol {w}}$. The statistics of the radial and tangential components is of interest because the collision outcome may depend on the angle at which the two particles collide. Appendix B shows that in the spherical formulation, the weighted PDF, P_sph(|w_r|), of the radial velocity amplitude of approaching particle pairs is very close to the PDF, P_sph(w_tt), of the total amplitude, w_tt, of the two tangential components. This suggests that, on average, the collision energy in the radial direction is the same as the total tangential collision energy. On the other hand, the weighted radial PDF, P_cyl(|w_r|), in the cylindrical formulation is found to almost coincide with the PDF, P_cyl(|w_t|), of the amplitude, |w_t|, of each single tangential component.¹⁰ Thus, each of the three directions provides an equal amount of collision energy, which is in sharp contrast to the spherical formulation. As the weighting factor ∝w_r, the spherical formulation gives more weight to collisions with larger radial relative speed. Consequently, it favors head-on collisions and predicts more head-on collisions than the cylindrical formulation.

Different predictions for the weighted radial and tangential PDFs by the two formulations can be used to test which one is physically more realistic. The test requires directly measuring the PDFs of the radial and tangential relative speeds for particle pairs that are actually colliding. The direct measurement can be conducted using the method of Wang et al. (2000), which counts how many pairs of particles of given finite sizes come into contact during each small time step. Analyzing the statistics of the radial and tangential collision velocities of particle pairs in contact and comparing the results with the predictions by the two formulations would tell which formulation provides a better prescription for turbulence-induced collisions.