The effect of collision-coagulation on the mean relative velocity of particles in turbulent flow: systematic results and validation of model.

The collision and coagulation of inertial particles in turbulent flow is a common and important physical process in both scientific research and industrial applications. For example, the collision rate between small droplets in atmospheric clouds affects precipitation (Shaw, 2003; Grabowski & Wang, 2013). In astrophysical environments such as protoplanetary disks, the collision of dust grains is a step in the formation of planetesimal (Johansen et al., 2009; Pan et al., 2011). In turbulent spray combustion, the distribution and the size of fuel droplets affect the efficiency of energy conversion (Smith et al., 2002). (Saffman & Turner, 1956) studied the collision rate of no-inertia particles in turbulent flow and obtained the collision rate: $N_{c}=n_{1}n_{2}(r_{1}+r_{2})^{3}(\frac{8\pi\varepsilon}{15\nu})^{1/2}$, in which $n_{1}$ and $n_{2}$ are the mean concentrations of two kinds of particles, respectively, $r_{1}$ and $r_{2}$ are their radii. The mean radial relative velocity of zero-inertia particles is $(r_{1}+r_{2})^{3}(\frac{8\pi\varepsilon}{15\nu})^{1/2}$. Unlike fluid particles, inertial particles are preferentially concentrated in the high strain and low vorticity region of turbulent flow, resulting in an inhomogeneous distribution, which is also called particle clustering (Squires & Eaton, 1991; Saw et al., 2008). In addition, the relative velocity of inertial particles increases at contact in turbulent flow (Bragg & Collins, 2014). These two effects make the collision rate of inertial particles in turbulent flow is larger than that of fluid particles and difficult to calculate. In (Sundaram & Collins, 1997), the relationship among the collision rate, particle clustering and particle relative radial velocity was presented: $N_{c}/(n_{1}n_{2}V)=4\pi d^{2}g(d)\langle w_{r}(d)\rangle$, in which $d$ is the particle diameter, $g(d)$ is the radial distribution function (RDF) when the particle separation distance $r$ is equal to $d$, and $\langle w_{r}(d)\rangle$ is the mean radial relative velocity (MRV) of particles. $V$ is the spatial volume of the domain. This finding makes it possible to study the particle RDF and MRV independently and to predict the collision rate of inertial particles in different situations. This paper focuses on the MRV of inertial particles, and the following also focuses on the current state of research related to the relative velocity of particles.

The particle Stokes number is a dimensionless quality characterizing the inertia of particles, is equal to the ratio of the particle relaxation time $\tau_{p}$ to the Kolmogorov time scale $\tau_{\eta}$ of the flow. The theoretical studies show that for particles with medium Stokes number, they have a path-history effect in turbulent flow. When particles are at the same position, their velocities have low correlation, resulting in high relative velocity. This phenomenon is called sling effect or caustics (Wilkinson & Mehlig, 2005; Wilkinson et al., 2006; Falkovich et al., 2002). The mean radial relative velocity is the relative velocity of particles projected in the radial direction: $\langle w_{r}\rangle=\langle(\vec{v}_{2}-\vec{v}_{1})\cdot\vec{r}\rangle$, which is also the first-order structure function of particle relative velocity. If the collision-coagulation of particles is not considered, since the turbulent velocity is isotropic at small scale, $\langle w_{r}\rangle=0$. In most numerical simulation studies, collision and coagulation of particles are not considered, therefore these works often investigated higher order particle relative velocity structure function, the second-order structure function $S_{\parallel}^{2}$ for example, or particles mean negative radial relative velocity $\langle w_{r}|_{w_{r}<0}\rangle$, to indirectly obtain the trend of the MRV. The results show that the value of $S_{\parallel}^{2}$ and $\langle w_{r}|_{w_{r}<0}\rangle$ of the particles increase gradually with the increase of the particle Stokes number, and the growth is significant when $St>0.2$, which is consistent with the finding of the sling effect/caustics (Bewley et al., 2013; Ireland et al., 2016). In (Falkovich et al., 2002) and (Wilkinson & Mehlig, 2005), they think that the sling effect or caustics becomes more significant with the increase of the turbulent Reynolds number. However, the numerical results show that for small $St$ particles, $S_{\parallel}^{2}$ and $\langle w_{r}|_{w_{r}<0}\rangle$ are only weakly dependent on turbulent Reynolds number (Ireland et al., 2016). For particles with $St>3.0$, they decrease with increasing Reynolds number (Bec et al., 2010; Rosa et al., 2013).

In (Saw & Meng, 2022), they find that when particles collide and coagulate, the magnitude of MRV increases significantly in the range of the separation distance $r$ is close to the particle diameter $d$. They also propose a MRV phenomenological model to illustrate the relationship between MRV and the separation distance $r$. Based on their results, we first make an improvement for the MRV phenomenological model. Then, the collision-coagulation is considered in our numerical simulation, the relationship between the MRV with particles and turbulent parameters is studied directly. The contents of this paper are organized as follows. In Section 2, we describe the numerical method that are used to simulate homogeneous isotropic turbulence and particle motion with the consideration of particle collision-coagulation. The improvement of the MRV phenomenological model is shown in section 3. The results from DNS are discussed in Section 4. A summary and main conclusion are given in Section 5.

Direct numerical simulation (DNS), which fully solves the Navier-Stokes equation in both spatial and temporal space, is used to study the particle laden flow in this paper. The incompressible Navier-Stokes equations are solved in a cube which has periodic boundary condition in three directions and the length of which is $2\pi$:

Where $\vec{u}$, $p$, $\rho$ and $\nu$ are velocity, pressure, density and the kinetic viscosity of turbulent flow. $\vec{f}$ is the forcing scheme, added in the large scale region of turbulence to maintain the statistics stationary (Eswaran & Pope, 1988). The N-S equations are transformed from physical space to wavenumber space, which is called the pseudo-spectral method. The 2/3-method is used to deal with aliasing error raised by the convection term in 1 (Rogallo, 1981).

The influence of the turbulent Reynolds number on the particle MRV is studied in this paper, the range of Reynolds number we choose is $Re_{\lambda}=84\sim 189$, where $Re_{\lambda}$ is the Taylor microscale Reynolds number. In different cases, $k_{max}\eta=1.59$, 1.21, 1.38, where $k_{max}$ is the maximum wavenumber, and Courant number $C=0.0248$, 0.0401, and 0.0865 respectively. The detailed parameters are shown in table 1. The simulation method and the DNS parameters are the same with (Meng & Saw, 2023). In (Meng & Saw, 2023), a higher resolution simulation is conducted to study the possible effects of the sub-Kolmogorov intermittency on the RDF of particle, using $N=1024$ at $Re_{\lambda}=124$. The result indicates that the effect of unresolved sub-Kolmogorov intermittency may modify slightly the inertial clustering exponent but would not cause any significant changes to the qualitative trends of particle RDF. Therefore, it can be considered that the sub-Kolmogorov intermittency will not have effect on the main results in this paper.

Particles in this paper are small and heavy, the particle diameter $d$ is much smaller than the Kolmogorov length scale $\eta$ and the particle density $\rho_{p}$ is much larger than the density of flow $\rho$. The basic physical process of particle collision-coagulation is considered in this paper, therefore the gravitational effect and the inter-particle hydrodynamic interaction are not included in simulation system. Particles and turbulent flow are only one-way coupling. Under these hypotheses, particles are only conducted by the Stokes drag force, the motion equation of particles is shown below (Maxey & Riley, 1983):

where $\vec{v}$ is particle velocity, $\vec{u}(\vec{x})$ is the flow velocity at particle position, and $\tau_{p}$ is particle relaxation time.

Particles will collide and create a new particle with the conservation of momentum and mass when the particle separation distance $r$ is equal to or less than the sum of the particle radii. To avoid successive collisions with larger particles created by collision-coagulation, larger particles are removed from the system as soon as they are created. In this situation, in order to maintain the equilibrium of the particles in the system, a fixed number of particles are added to the system at certain time steps, and these new particles have the same parameters as the initial particles. After 10 turbulent characteristic integration times, the particles in the system reach stability. It is important to note that, only monodisperse particles are considered in the calculation of the MRV, which means the relative velocity between particles with different $St$ is not taken into account. The influences of particle Stokes number and the particle size on MRV are studied in this paper, the range of $St$ we choose is $St=0.01~{}2.0$ and the particle diameters are $d=1/3d_{*}$, $d_{*}$, and $3d_{*}$, where $d_{*}=9.49\times 10^{-4}$. It should be noted that the diameter size of the particles is only used to determine whether collisions occur between the particles, the inter-particle hydrodynamic is not considered.

Here, we present a correction (improvement) of the MRV phenomenological model proposed by Saw & Meng (2022) and show that this substantially improve the model’s accuracy. We begin with a brief introduction of the model. The model describes the relationship between MRV and the particle separation distance $r$ in the $St\ll 1$ limit, such that:

where $p_{+}$ ($p_{-}$) is the probability for a sample of $w_{r}$ to turn out positive (negative) and the constants $\xi_{+}$, $\xi_{-}$ depends on the flow energy dissipation rate (Saw & Meng, 2022). As a first-order account, the skewness in the distribution of particle relative velocities were neglected which leads to $p_{+}\equiv\int_{0}^{\frac{\pi}{2}}P_{\theta}d\theta=0.5$, and $p_{-}\equiv 1-p_{+}=0.5$. $P_{\theta}^{+}$ is a conditional probability density function (PDF) of $\theta$ defined as $P_{\theta}^{+}\equiv P(\theta\,\,|\,w_{r}\leq 0)\equiv P(\theta\,\,|\,\theta\in\left[0,\pi/2\right])$. $\theta$ is the angle between the relative velocity $\vec{w}$ and the relative displacement $\vec{r}$ of a particle-pair. Saw & Meng (2022) argue that PDF of $\theta$ may be constructed using the (statistical) central-limit theorem, such that:

where $Nexp[…]$ is the circular normal distribution, and $K$, a variance parameter, may be inferred by matching the model produced transverse-to-longitudinal ratio of structure functions (TLR) of the particle relative velocities to the one measured directly in experiment or DNS (Saw & Meng, 2022). In (Saw & Meng, 2022), the model predicts MRV that agrees best with DNS outcome when $K$ is calibrated using the fourth-order structure functions instead of the second-order which one would normally expect since MRV is a low order statistics. This suggests that the model may be incomplete(Saw & Meng, 2022).

According to the idealized picture of particle collision-coagulation process described in (Saw & Meng, 2022), as shown in Figure 1, the small scale trajectory of particle can be regarded as rectilinear at separation $r$ close to the particle diameter $d$. Since particles collide and coagulate when $r=d$, according to the geometrical analysis, the angle between $\vec{w}$ and $\vec{r}$ cannot be smaller than $\theta_{m}$, which is a value related to the ratio of $d$ to $r$: $\theta_{m}=sin^{-1}(d/r)$. This lower bound of $\theta$ affects the symmetry of the PDF of particle relative velocity, therefore $p_{+}$ (also $p_{-}$) should not be fixed at 0.5 but should equals a value that varies with $\theta_{m}$ (which in turns varies with $r$). Thus, we change the probability of a realization of $w_{r}$ being positive and negative to:

which also satisfy $p_{+}+p_{-}=1$.

Bringing the revised $(p_{+},p_{-})$ into Eq.4 and using Eq.5 where $K$ is calibrated by the measured (via DNS) second-, fourth-, and sixth-order structure functions of particle relative velocity respectively, the new MRV model predictions is compared with the DNS results, as shown in Figure 1. We see that the second-order calibration produces result with the best agreement with DNS, and which is significantly better than that in (Saw & Meng, 2022). This provide strong support for the validity of the MRV model since second-order statistics are the most appropriate (orthodox) reference for the determination of $K$, which is itself a variance parameter analogous to $1/\sigma^{2}$ in the Gaussian distribution.

In this paper, the mean radial relative velocity (MRV) of inertial particles is investigated when the particle collision-coagulation is considered. First, the direct numerical simulation is used to investigate the relationship between the MRV with the particle Stokes number, the turbulent Reynolds number, and the particle diameter. The results show that the magnitude of MRV increases with increasing $St$, and the growth is pronounced when $St>0.2$. We attribute this increase to the scale at which particles with different inertia are affected by non-local turbulent flow in their historical trajectories. Based on this analysis, the relationship function of the value of MRV at $r=d$ with $St$ is obtained: $\langle w_{r}\rangle_{r=d}=\frac{u_{\eta}^{s}}{\eta}(d+f(St))$, where $f(St)$ represents the distance between the scale at which particles are affected by non-local turbulent velocity and their collision radius (e.g., particle diameter $d$). Next, we study the relationship between the value of MRV and the turbulent Reynolds number, and also analyze it from the perspective of the scale change of the non-local turbulent flow that affects inertial particles. We find that for particles whose Stokes number is much less than 1, they are mainly affected by the local turbulent velocity, and $f(St)$ is independent of $Re_{\lambda}$. While for particles with larger $St$, the coefficients $\alpha$ and $\beta$ in $f(St)$ decrease as $Re_{\lambda}$ increases. However, since this paper only selects three cases with different $Re_{\lambda}$, it is impossible to obtain an accurate mathematical relationship between $f(St)$ and $Re_{\lambda}$, which needs further research in the future. In addition, this paper analyzes the MRV for particles with different diameters $d$. The result shows that for particles with the same $St$, although the diameters are different, the MRV under the same $d/r$ has the same value, and $f(St)$ is independent of the particle diameter.

Finally, we make a correction (improvement) on the MRV phenomenological model proposed by Saw & Meng (2022) to captures the effect of collision-coagulation on MRV in the limit of $St\ll 1$. As a result, the corrected model, coupled with proper inputs from the turbulent flow’s velocity statistics, makes prediction that is in close agreement with the DNS results, an outcome that is substantially better than that in (Saw & Meng, 2022). Furthermore, based on the relationship between the value of MRV at $r=d$ and $St$ mentioned above, a coefficient $\gamma$ is defined. We found that by incorporating $\gamma$ into the MRV model, the MRV for cases with any $St$ can be accurately predicted (up to $St=2$) . Our results also suggest that the model could also accurately account for any possible Reynolds number effect through a simple generalization.

[Acknowledgements]This work was mainly supported by the National Natural Science Foundation of China (grand 11872382) and by the Thoudand Young Program of China.