Nomenclature
Roman symbols
-
$c$
-
chord length
-
${c_{p,f}}$
-
carrier fluid heat capacity
-
${c_{p,p}}$
-
droplet heat capacity
-
${C_D}$
-
drag coefficient
-
${C_m}$
-
mass transfer coefficient
-
${d_0}$
-
initial droplet diameter
-
${d_p}$
-
droplet diameter
-
$D$
-
drag
-
${\rm{E}}{{\rm{u}}_v}$
-
vortex Euler number
-
${k_f}$
-
thermal conductivity of fluid
-
$L$
-
latent heat of vapourisation
-
${p_0}$
-
far-field pressure
-
${p_f}$
-
local fluid pressure
-
${p_{sat}}$
-
vapour saturation pressure
-
${p_{v,p}}$
-
partial vapour pressure at the particle interface
-
${p_{v,\infty }}$
-
ambient partial vapour pressure
-
${\rm{Pr}}$
-
Prandtl number
-
$r$
-
radial distance from the wingtip vortex axis
-
${R_{cond}}$
-
condensation core radius
-
${\rm{R}}{{\rm{e}}_c}$
-
chord-based Reynolds number
-
${\rm{R}}{{\rm{e}}_p}$
-
particle-based Reynolds number
-
$S$
-
vortex swirl number
-
${\rm{Sc}}$
-
Schmidt number
-
${\rm{Ste}}_0^{\rm{*}}$
-
modified Stefan number
-
${\rm{St}}{{\rm{k}}_0}$
-
droplet initial Stokes number
-
$t$
-
time
-
${T_0}$
-
far-field temperature
-
${T_f}$
-
local fluid temperature
-
${T_p}$
-
droplet temperature
-
${T_{sat}}$
-
vapour saturation temperature
-
$U$
-
flight velocity
-
${u_f}$
-
local flow velocity
-
${u_p}$
-
droplet velocity
-
${x_p}$
-
droplet spatial location
-
$z$
-
downstream distance from the wing
Greek symbols
-
${\rm{\Gamma }}$
-
initial vortex intensity
-
$\eta $
-
normalised radial position (
${r^2}/4z$
) -
$\theta $
-
normalised temperature
-
${\nu _f}$
-
fluid kinematic viscosity
-
${\rho _f}$
-
fluid density
-
${\rho _p}$
-
droplet density
-
$\rho _{v,\infty }^{\rm{*}}$
-
vapour ideal gas density in ambient conditions
-
${\tau _f}$
-
free-stream flow time scale
-
${\tau _{p,0}}$
-
droplet’s initial relaxation time
-
$\phi $
-
relative humidity
1.0 Introduction
While the interactions between droplets and their environment are a fundamental problem in fluid mechanics [Reference Pullin and Saffman1, Reference Dávila and Hunt2], our understanding of such phenomena is incomplete. The complexity of the general problem arises from the strong coupling between the carrier flow and the transport of mass, momentum and heat to the droplet, thus limiting the scope of any general analysis. Past studies have investigated the two-way coupling between simple vortices and droplets with a comparable length scale and found that these vortical structures could significantly alter the droplet evaporation rate in spray combustion systems [Reference Chakroun, Shanbhogue, Dagan and Ghoniem3–Reference Dagan, Arad and Tambour8]. Vortical structures might also completely change the dynamics and thermal behaviour of the single droplet [Reference Dagan9–Reference Avni and Dagan13].
A clear manifestation of such complex interactions is wingtip trailing vortices, which enhance the condensation and freezing of the air’s vapour content and contribute to the formation of condensation trails. The resultant multiphase flow structures might alter the local radiative forcing and the photochemistry of the atmosphere [Reference Jugier, Fontane, Joly and Brancher14]. The interaction between trailing vortices and dispersed particles may also influence the efficiency of aerial agricultural spraying, as the aircraft wake can alter the motion of the spray drops [Reference Atias and Weihs15]. The analysis of multiphase trailing vortices using high-fidelity simulations may require the implementation of Lagrangian particle tracking. However, incorporating a detailed model might have high computational costs, even for relatively simple setups. Notably, Sölch and Kärcher [Reference Sölch and Kärcher16, Reference Unterstrasser and Sölch17] used a Lagrangian particle tracking model in conjunction with high-fidelity LES simulations to investigate the dynamics of ice particles in vortex flows, with a focus on aerosol and ice microphysics in contrail formation. While their study primarily addresses the nucleation and growth of ice particles in moist air within trailing vortices, our work centres on the evaporation and condensation of liquid droplets in these vortices, which occur at lower altitudes, typically during takeoff, landing, or sharp manoeuvers [Reference Holzäpfel, Stephan, Heel and Körner18, Reference Campbell and Chambers19]. Furthermore, even the wake of high-performance racecars can induce sufficient pressure gradients to cause droplet condensation within the rear wingtip vortices.
Therefore, this work offers a new Lagrangian analysis of the dynamics of droplets swirling within a simple, analytically described viscous vortical flow structure, the well-known Batchelor vortex. Batchelor [Reference Batchelor20] suggested such a similarity solution for modelling a trailing vortex far downstream from the wing itself owing to the complexity of such a three-dimensional unsteady problem [Reference Saffman21].
Our analysis aims to reveal the complex interaction between Batchelor’s Lamb-Oseen-based vortex and a Lagrangian droplet as it may undergo both evaporation and condensation due to sharp changes in the environmental conditions generated by the vortex. The nonlinear relations between the carrier flow fields, droplet relaxation time, drag forces, mass convection and heat exchange is derived and analysed. The isolation of the interaction between the modelled vortical flow and the droplets allows for a rudimentary, parametric investigation of the transport processes. Thus, the proposed model offers a new analytic tool that might aid in uncovering the underlying physical mechanisms governing the fundamental problem of droplet evaporation and condensation within wingtip trailing vortical structures.
2.0 Governing equations
The transport of discrete micron-sized droplets within a steady, analytically described Batchelor vortex is analysed here using a Lagrangian approach. The single droplet’s spatial location
${\boldsymbol{x}_p}$
, velocity
${\boldsymbol{u}_p}$
, diameter
${d_p}$
and temperature
${T_p}$
are traced and coupled to the local flow
${\boldsymbol{u}_f}$
, pressure
${p_f}$
and temperature
${T_f}$
fields. We assume the droplets are dispersed and dilute enough so that their motion does not affect the flow field; furthermore, any potential interactions between droplets are discounted. The equations for the carrier flow and the Lagrangian droplet are presented as follows.
Figure 1. Carrier flow setup.
2.1 Carrier flow
The investigated carrier flow, a three-dimensional Batchelor vortex, describes an axisymmetric vortical structure decaying due to viscous effects as illustrated by Fig. 1 flow. The initial vortex intensity
${\rm{\Gamma }}$
is determined by the wing’s geometry and the flight conditions. The customary time variable
$t$
is replaced here by
$U$
and
$z$
, the flight velocity and downstream distance from the wing, correspondingly. Assuming the trailing vortex system is generated by a wing on which the total drag force is
$D$
, the velocity field of such flow may be expressed in cylindrical coordinates as
\begin{align}{\tilde{\mathbf{u}}_f} = \left[ {\frac{{\rm{\Gamma }}}{{2\pi \tilde r}}\left( {1 - {e^{ - \frac{{U{{\tilde r}^2}}}{{4{\nu _f}\tilde z}}}}} \right)} \right]{\bar e_\theta } + \left[ {U - \frac{D}{{8\pi {\rho _f}{\nu _f}z}}{e^{ - \frac{{U{{\tilde r}^2}}}{{4{\nu _f}\tilde z}}}}} \right]{\bar e_z}.\end{align}
when considering that the axial and azimuthal velocities decay independently [Reference Batchelor20]. Since the Batchelor vortex is an exact solution of the Navier-Stokes equation, we may extract an analytic expression for the pressure field induced by the vortical flow. As we investigate the droplets’ dynamics near a vortical structure, this model assumes that the pressure drop is induced only by the circulation, i.e. the pressure drop matches the pressure changes within a Lamb-Oseen vortex. The pressure term yields [Reference Saffman21]
\begin{align}{\tilde p_f} = {p_0} - \frac{{{\rho _f}{{\rm{\Gamma }}^2}}}{{8{\pi ^2}{{\tilde r}^2}}}\left[ {{{\left( {1 - {e^{ - \frac{{U{{\tilde r}^2}}}{{4{\nu _f}\tilde z}}}}} \right)}^2} + \frac{{U{{\tilde r}^2}}}{{2{\nu _f}\tilde z}}\left( {{\rm{Ei}}\left( {\frac{{U{{\tilde r}^2}}}{{4{\nu _f}\tilde z}}} \right) - {\rm{Ei}}\left( {\frac{{U{{\tilde r}^2}}}{{2{\nu _f}\tilde z}}} \right)} \right)} \right],\end{align}
given that
${p_0}$
is the far-field pressure and
${\rm{Ei}}(x) = \mathop \smallint \nolimits_{ - x}^\infty {e^{ - \xi }}/\xi d\xi $
is the tabulated exponential integral notation.
We set
$c$
, the wing’s chord length as the characteristic radial length,
$U$
as the characteristic velocity, and introduce the dimensionless variables
\begin{align}r = \tilde r/c;{\rm{\;\;\;\;\;\;}}z = \tilde z/c{\rm{R}}{{\rm{e}}_c};{\rm{\;\;\;\;\;\;}}\eta = {r^2}/4z;{\rm{\;\;\;\;\;\;}}u = \tilde u/U;{\rm{\;\;\;\;\;\;}}p = \tilde p/{p_0},\end{align}
where
${\rm{R}}{{\rm{e}}_c} = cU/{\nu _f}$
. Hence, Equation (1) and Equation (2) may be written in a dimensionless form as
\begin{align}{\boldsymbol{u}_f} = \frac{S}{r}\left( {1 - {e^{ - \eta }}} \right){\bar e_\theta } + \left( {1 - \frac{{{C_D}{e^{ - \eta }}}}{{16\pi z}}} \right){\bar e_z}.\end{align}
\begin{align}{p_f} = 1 - \frac{{{\rm{E}}{{\rm{u}}_v}}}{{{r^2}}}\left[ {{{\left( {1 - {e^{ - \eta }}} \right)}^2} + 2\eta ( {{\rm{Ei}}( \eta ) - {\rm{Ei}}( {2\eta } )} )} \right],\end{align}
using three dimensionless parameters: the wing’s drag coefficient
\begin{align}{C_D} = \frac{D}{{\frac{1}{2}{\rho _f}{U^2}{c^2}}},\end{align}
the vortex swirl number
\begin{align}S = \frac{{\rm{\Gamma }}}{{2\pi cU}},\end{align}
and the vortex Euler number
\begin{align}{\rm{E}}{{\rm{u}}_v} = \frac{{{\rho _f}{{\rm{\Gamma }}^2}}}{{8{\pi ^2}{c^2}{p_0}}} = \frac{{\frac{1}{2}{\rho _f}{U^2}{S^2}}}{{{p_0}}},\end{align}
which signifies the ratio between the dynamic pressure drop and the total pressure.
Figure 2 pressure_drop demonstrates the relation between
${\rm{E}}{{\rm{u}}_v}$
and the pressure field around the idealised trailing vortex; an increase of
${\rm{E}}{{\rm{u}}_v}$
manifests in lower absolute pressures at the origin of the vortex. The derived pressure field is used hereafter to couple the Lagrangian droplet transport with the thermodynamic gradients induced by the vortical flow. We also consider the vortex to be adiabatic; hence, one may couple the temperature field within the vortical viscous core to the pressure field
\begin{align}{\tilde T_f} = {T_0}{\left( {\frac{{{{\tilde p}_f}}}{{{p_0}}}} \right)^{2/7}}.\end{align}
Figure 2. Normalised pressure distribution
${p_f}( {r,z} )$
for various values of the non-dimensional Euler number Eu, as predicted by Equation (5). (a) Radial pressure distribution at axial location
$z = 1$
. (b) Pressure distribution along the vortex axis at radial location
$r = 1$
.
The temperature field outside the droplet is crucial for finding the droplet temperature, as it is dictated by, among other mechanisms, heat diffusion to the droplet.
2.2 Lagrangian equations
Maxey and Riley [Reference Maxey and Riley22] formulated the generalised equations of motion for small particles in nonuniform, unsteady flows; they considered both gravity, drag, virtual mass and the Basset ‘history’ force. This study concerns the motion of a small liquid droplet in a gaseous medium; the particle-medium density ratio is large, and the droplet’s characteristic length is much smaller relative to the vortex viscous core size. Hence, we neglect the forces due to undisturbed flow, virtual mass, Faxen’s drag correction, rotational inertia and particle history terms. Additionally, we do not account for gravity and assume a linear drag term; the influence of both on the droplet dynamics will be relaxed and examined in future studies. We may now reduce the general form of the equations to the following:
\begin{align}\frac{{d{\mathbf{x}_{\mathbf{p}}}}}{{dt}} = {\mathbf{u}_{\mathbf{p}}},\end{align}
\begin{align}\frac{{d{\mathbf{u}_{\mathbf{p}}}}}{{dt}} = \frac{{{\mathbf{u}_{\mathbf{f}}} - {\mathbf{u}_{\mathbf{p}}}}}{{{\rm{St}}{{\rm{k}}_0}d_p^2}},\end{align}
where
${\mathbf{x}_{\mathbf{p}}}$
and
${\mathbf{u}_{\mathbf{p}}}$
are the droplet location and velocity vectors in the vortex frame of reference, respectively.
${d_p}$
is the particle diameter normalised by its initial diameter
${d_0}$
, and
\begin{align}{\rm{St}}{{\rm{k}}_0} = \frac{{{\tau _{p,0}}}}{{{\tau _f}}} = \frac{{U{\rho _p}d_0^2}}{{18{\mu _f}c}}\end{align}
is the droplet initial Stokes number: the ratio between the particle’s initial relaxation time and free-stream flow time scale. While valid for low particle Reynolds numbers
${\rm{R}}{{\rm{e}}_p} = \left| {{\mathbf{u}_{\mathbf{f}}} - {\mathbf{u}_{\mathbf{p}}}} \right|{d_p}/{\nu _f}$
, the linear drag model offers a good approximation for the drag forces acting on small droplets as their velocity relative to the local flowfield is typically low [Reference Crowe, Schwarzkopf, Sommerfeld and Tsuji23].
The size of the Lagrangian particle is governed by heat and mass transfer processes. Kulmala [Reference Kulmala, Majerowicz and Wagner24, Reference Kulmala and Vesala25] formulated the diffusive mass transfer at the droplet-gas interface for a quasi-stationary case while assuming that the medium is an ideal gas, the droplet-gas interface is saturated and a zeroth-order mass fraction profiles around the droplet. Although droplet ventilation may enhance the mass transfer out of the droplet, we aim to isolate the role of mass diffusivity and consider it the primary transfer mechanism. In terms of droplet diameter, the dimensionless mass equation is
\begin{align}\frac{{d\!\left( {d_p^2} \right)}}{{dt}} = \frac{{4\rho _{v,\infty }^{\rm{*}}}}{{9{\rm{ScSt}}{{\rm{k}}_0}{\rho _f}}}{\rm{ln}}\!\left( {\frac{{{p_0}{p_f} - {p_{v,p}}}}{{{p_0}{p_f} - {p_{v,\infty }}}}} \right);\end{align}
$Sc$
is the non-dimensional Schmidt number,
$\rho _{v,\infty }^{\rm{*}}$
is the vapour ideal gas density in ambient conditions,
${p_{v,p}}$
is partial vapour pressure at the particle interface, and
${p_{v,\infty }}$
is the ambient partial vapour pressure. The difference between the vapour partial pressure at the interface and the vapour partial pressure at the far field dictates the mass flux to and from the droplet. Thus, we shall estimate both by assuming the Lagrangian particle consists of pure liquid water and the gaseous media is an air-vapour mixture with a relative humidity of
$\phi $
. The diffusive driving force reduces to a single dimensionless number, the normalised mass transfer coefficient
${C_m}$
[Reference Avni and Dagan12]; its generalised term is
\begin{align}\frac{{{p_0}{p_f} - {p_{v,p}}}}{{{p_0}{p_f} - {p_{v,\infty }}}} = 1 + \frac{{\phi {p_{sat}}\left( {{T_0}} \right) - {p_{sat}}\left( {{{\tilde T}_p}} \right)}}{{{p_0}{p_f} - \phi {p_{sat}}\left( {{T_0}} \right)}} = 1 + {C_m}.\end{align}
${T_0}$
is the far-field carrier fluid temperature,
${\tilde T_p}$
is the (dimensional) droplet temperature, and
${p_{sat}}(T)$
is the vapour saturation pressure at a given temperature. The mass transfer coefficient sign and value indicate the nature of the mass transfer: evaporation occurs when
${C_m} \lt 0$
and the droplet interface vapour pressure is higher than far-field pressure, condensation occurs for
${C_m} \gt 0$
and the far-field vapour pressure is higher than the droplet interface pressure. Generally, the mass flux is proportional to the transfer coefficient value
$\dot m \propto \left| {{C_m}} \right|$
.
The droplet temperature is regulated by heat conduction and convection to the carrier medium at the droplet surface, the sensible heat stored within it, and heat advection due to mass transport. One may derive [Reference Avni and Dagan13] the normalised energy conservation equation for a single droplet, assuming a zeroth-order temperature profile around the droplet. First, we introduce the non-dimensional temperature
\begin{align}\theta = \frac{{\tilde T - {T_0}}}{{{T_{sat}}\left( {{p_0}} \right) - {T_0}}} = \frac{{\tilde T - {T_0}}}{{{\rm{\Delta }}{T_0}}},\end{align}
where
${T_{sat}}\left( {{p_0}} \right)$
is the vapour saturation temperature at the ambient pressure. Now, the Lagrangian energy equation yields
\begin{align}\frac{{d{\theta _p}}}{{dt}} = \frac{2}{{3d_p^2}}\left[ {{\rm{Ste}}_0^{\rm{*}}\frac{{d\!\left( {d_p^2} \right)}}{{dt}} + \frac{{4{c_{p,f}}}}{{9{\rm{PrSt}}{{\rm{k}}_0}{c_{p,p}}}}\left( {{\theta _f}\left( {{\mathbf{x}_{\mathbf{p}}}} \right) - {\theta _p}} \right)} \right],\end{align}
where
${\theta _f}\left( {{{\bar x}_p}} \right)$
is the normalised carrier temperature at the droplet location,
${c_{p,f}}$
is the carrier fluid heat capacity,
${c_{p,p}}$
is the droplet heat capacity and
${\rm{Pr}} = {c_{p,f}}{\nu _f}/{\rho _f}{k_f}$
is the dimensionless Prandtl number. We also introduce a modified Stefan number, defined here as the ratio between the droplet substance’s latent heat of evaporation and sensible heat maxima
\begin{align}Ste_0^{\rm{*}} = \frac{L}{{{c_{p,p}}{\rm{\Delta }}{T_0}}},\end{align}
denoting the latent heat of vapourisation as
$L$
.
Both the thermodynamic conditions and the vortex properties may affect the mass transfer coefficient
${C_m}$
field within it. The pressure drop due to the rotating flow may give rise to distinct condensation zones within the vortex core; the following section analyses the influence of the dynamics of Lagrangian water droplets in and around vortices sustaining such conditions.
3.0 Results
3.1 Condensation core
The interaction between ambient thermodynamic conditions and localised variations induced by the vortex dictates the behaviour of the mass transfer coefficient, potentially leading to a change in its sign near the vortex centre. Namely, given the right conditions, condensation initiates in the vortex viscous core – as exhibited in many natural phenomena and industrial applications. Figure 3 illuminates the relation between ambient properties and the evolving condensation core of the vortex. Near the wing, where
$z$
tends to
$0$
, the condensation core reaches the maximal radial value. However, as the circulation decays along the vortex axis, a decrease in pressure drop occurs. Thus, the condensation core radius progressively shrinks, and eventually, far from the wing, the core disappears as expected.
Figure 3. Condensation core radius
${{\rm{R}}_{{\rm{cond}}}}$
along the vortex axis for different vortex Euler numbers, relative humidities and ambient temperatures. (a) Constant Euler number
${\rm{E}}{{\rm{u}}_{\rm{v}}} = 0.15$
and varying relative humidities
$\phi $
. (b) Constant air relative humidity
$\phi = 0.8$
and varying vortex Euler numbers
${\rm{E}}{{\rm{u}}_{\rm{v}}}$
. Dashed lines denote results obtained for ambient temperature of
${{\rm{T}}_0} = 290{\rm{K}}$
, while solid and dotted lines denote results obtained for
${{\rm{T}}_0} = 280{\rm{K}}$
and
${{\rm{T}}_0} = 274{\rm{K}}$
, correspondingly.
The influence of air relative humidity is depicted in Fig. 3(a). Lower relative humidity leads to smaller condensation cores that decay rapidly along the vortex axis. Moreover, a nonlinear relation between the changes in the relative humidity and the core size emerges; the core size significantly increases and subsequently diverges as
$\phi \to 1$
. Figure 3(b) illustrates how the vortex Euler number, representing the ratio between ambient and dynamic pressures within the vortex, affects the behaviour of the condensation core. As expected, lower Euler values result in smaller cores. Notably, the values of the vortex Euler number are constrained by physical considerations. For instance, when
$Eu \gt 0.25$
, the pressure at the vortex centre
${p_f}( {r = 0} )$
drops below absolute zero, an impossible thermodynamic state in gases. Realistically, the limit of physically possible trailing vortices is significantly lower. The effect of ambient temperatures in the range
${T_0} = 274{{K}} - 290{{K}}$
is also explored and presented in Fig. 3. Smaller cores, corresponding to lower
${\rm{E}}{{\rm{u}}_v}$
or
$\phi $
values, are less influenced by temperature changes. However, given larger cores, the ambient temperature impact becomes more pronounced, though still relatively minor compared to other parameters. However, the air relative humidity is strongly temperature-dependent, indirectly influencing the characteristics of the condensation core.
3.2 Droplet dynamics
The Lagrangian governing equations presented in Section 2.2 suggest that one may characterise the complete dynamic and thermal behaviour of a droplet in the vicinity of idealised trailing vortices in terms of four state variables
${X_p} = \left[ {{\mathbf{x}_{\mathbf{p}}},{\mathbf{u}_{\mathbf{p}}},\mathbf{d}_{\mathbf{p}}^{\boldsymbol{2}},{\theta _{\mathbf{p}}}} \right]$
. The complex coupling between the external, vortex-induced forcing and the droplet’s response gives rise to a nonlinear dynamic system,
\begin{align}{\dot X_p} = F\!\left( {{X_p},{{\dot X}_p}, \ldots } \right).\end{align}
While aiming to investigate the system’s dynamic response, the nonlinear ODE system will be solved numerically using an adaptive time-stepping fourth-order Runge-Kutta scheme.
Since we seek to study the dynamics of water droplets within trailing vortices wherein condensation cores are forming, the air’s thermodynamic properties are fixed at
${p_0} = 1{{atm}}$
,
${T_0} = 280{{K}}$
and
$\phi = 0.8$
. Here we focus our discussion on the influence of the droplet’s properties, i.e. initial location
${\mathbf{x}_{\mathbf{p},\boldsymbol{0}}}$
and
${\rm{St}}{{\rm{k}}_0}$
, and thus set the vortex flow properties to be constant:
${\rm{E}}{{\rm{u}}_v} = 0.15$
,
$S = 1$
,
${C_D} = 0.05$
. Furthermore, the droplet area will be limited to
$d_p^2 \ge 0.1$
; i.e., the droplet diameter is limited to 1% of its initial diameter. Droplets crossing these thresholds are eliminated from the simulation; the mass equation, Equation (13), cannot capture the droplet’s complete drying without imposing an arbitrary threshold on the droplet’s size.
Figure 4 illustrates selected results of the Lagrangian model for droplets with various initial Stokes numbers: (a-b)
${\rm{St}}{{\rm{k}}_0} = 0.1$
, (c-d)
${\rm{St}}{{\rm{k}}_0} = 1$
, and (e-f)
${\rm{St}}{{\rm{k}}_0} = 10$
. In each case, 20 droplets are distributed along the
$x$
and
$y$
axes inside the condensation core at equal distances from each other, all placed at
$z = 0.1$
. An initial no-slip condition is assumed, as the droplet’s initial velocity is set to be equal to the local flow velocity. Similarly, the droplets are considered to maintain thermal equilibrium with the surrounding air,
${\theta _p}\!\left( {{\mathbf{x}_{\mathbf{p}}}} \right) = {\theta _{\mathbf{f}}}$
. The initial Stokes number is determined by both the flow regime and droplet size; the latter is dictated by the nucleation mechanism and is coupled to the thermodynamic conditions at nucleation inception. The present study does not aim to model the nucleation mechanisms of droplets from moist air, as its focus is on the growth and evaporation of pre-existing droplets within vortex flows.
Figure 4. Selected results of the Lagrangian model for droplets of various initial Stokes numbers: (a–b)
${\rm{St}}{{\rm{k}}_0} = 0.1$
, (c–d)
${\rm{St}}{{\rm{k}}_0} = 1$
, and (e–f)
${\rm{St}}{{\rm{k}}_0} = 10$
. The condensation core is illustrated by a blue ellipsoid, marking the edges of the region in which condensation occurs. The axial ordinate
${\rm{z}}$
is in a logarithmic scale. LHS panels (a),(c) and (e) present the three-dimensional trajectories of droplets placed at equal distances along
${\rm{x}}$
and
${\rm{y}}$
axes inside the condensation core. RHS panels (b),(d), and (f) present the radial location of the droplets as a function of their location along the vortex axis.
For initial Stokes numbers below unity (
${\rm{St}}{{\rm{k}}_0} \lt 1$
), Fig. 4(a–b) reveals that the droplets retain scattered distribution without notable clustering. Their spread radius slightly exceeds the initial condensation core radius, indicating a relatively even dispersion throughout the vortex flow. However, clustering phenomena become more apparent as the initial Stokes number reaches unity (
${\rm{St}}{{\rm{k}}_0} = 1$
). Figure 4(d) highlight that this clustering primarily occurs when droplets exit the condensation core region, typically between
$z = 1$
and
$z = 5$
. Beyond
$z = 10$
, the droplets’ radial position stabilises as they steadily swirl within the vortex, ultimately undergoing complete evaporation. The initial growth by condensation, followed by subsequent evaporation, is illustrated in Fig. 5, which shows the square diameter of four selected droplets. Two droplets are positioned just outside the vortex condensation core and thus undergo only evaporation. In contrast, droplets initially located on the vortex axis remain within it and initially grow in size. Once transported out of the condensation core, the mass transfer mechanism reverses, leading to evaporation at a rate similar to that of droplets initially outside the core. As expected, smaller droplets respond more sharply to thermodynamic conditions, both during the initial condensation phase and the subsequent evaporation phase.
Figure 5. Droplet square diameter as a function of its position along the vortex axis for two initial Stokes numbers,
${\rm{S}}{{\rm{t}}_0} = 0.1$
and
$1$
. Droplets are initially located either on the vortex axis (
${{\rm{r}}_0} = 0$
) or at the edge of the condensation core (
${{\rm{r}}_0} = {{\rm{r}}_{{\rm{eq}}}}$
).
With further increases in the initial Stokes number and surpassing
${\rm{St}}{{\rm{k}}_0} \gt 1$
, clustering of droplets emerges. Such droplets aggregate around specific radial locations, forming dense clusters with distinct spatial patterns. In the case of
${\rm{St}}{{\rm{k}}_0} = 10$
, Fig. 4(f) reveals that the clustering radius is expanding, surpassing the initial condensation core radius. The resulting pattern resembles an annular formation surrounding the core of the trailing vortex, with droplets accumulating around a specific radius.
Further investigation into the clustering phenomena was conducted through statistical analysis of
${10^5}$
monodisperse droplets, as illustrated in Fig. 6. Initially, the particles were uniformly distributed across the cross-sectional area of the condensation core, simulating the aftermath of droplet nucleation. This simplified model aims to elucidate the underlying physical mechanisms governing droplet behaviour. The variability in droplet sizes is represented by different initial Stokes numbers.
Figure 6. Statistics from Lagrangian simulations of
${10^5}$
monodispersed droplets. Droplets are initially randomly distributed within the condensation core at the axial plane
${\rm{z}} = 0.1$
. (a–c) Radial probability density functions at axial positions
${\rm{z}} = 0.1,1,5,10$
for droplets with initial Stokes numbers: (a)
${\rm{St}}{{\rm{k}}_0} = 0.1$
, (b)
${\rm{St}}{{\rm{k}}_0} = 1$
, and (c)
${\rm{St}}{{\rm{k}}_0} = 10$
. (d) Mean droplet diameter variation as a function of the axial location for different initial Stokes numbers.
For small droplets (
${\rm{St}}{{\rm{k}}_0} \le 0.1$
), the radial probability density functions remain largely consistent across different axial positions. A slight decrease in droplet density near the axis is observed, attributable to centrifugal effects. As evident in Fig. 6(d), beyond
$z \gt 10$
, these small droplets begin to evaporate as the condensation core diminishes. The formation of annular structures becomes evident for
${\rm{St}}{{\rm{k}}_0} \ge 1$
. Figure 6(b) demonstrates this phenomenon, revealing a notable shift in the droplet location probability for
$z \ge 5$
, coinciding with the droplets’ exit from the condensation region. For larger droplets (
${\rm{St}}{{\rm{k}}_0} = 10$
), this shift is more pronounced. The droplets aggregate within a relatively narrow radial band for
$z \ge 5$
, as shown in Fig. 6(c). The annular formation exhibits a slight radial expansion as the droplets evaporate far from the condensation region.
Figure 6(d) quantifies the mean droplet diameter variation as a function of axial position for different initial Stokes numbers, providing insight into the droplets’ growth and evaporation patterns along their trajectories. These results suggest that a comprehensive analysis of droplet dynamics in such scenarios necessitates incorporating models accounting for droplet-droplet interactions, including collision, coalescence and grouping phenomena. The observed clustering behaviour indicates that the dispersed phase assumption may be invalid under certain conditions, particularly for larger droplets.
The onset of droplet evaporation as they are transported away from the condensation region was previously illustrated in Fig. 6(d). To further elucidate this phenomenon, we extended our simulations to determine the average axial location at which complete droplet evaporation occurs, effectively defining the length of the condensation trail. Figure 7 presents the trail length as a function of the initial droplet Stokes number (
${\rm{St}}{{\rm{k}}_0}$
) under various thermodynamic and flow conditions. As anticipated, smaller droplets (low
${\rm{St}}{{\rm{k}}_0}$
) evaporate rapidly, resulting in shorter trails. Generally, an increase in the initial Stokes number corresponds to an extension of the trail length.
Figure 7. Comparison of condensation trail length normalised by the chord length, as a function of the initial droplet Stokes number
${\rm{St}}{{\rm{k}}_0}$
at different thermodynamic and flow conditions. (a) Effects of changes in the air relative humidity
$\phi $
. (b) Effects of changes in the vortex Euler number
${\rm{E}}{{\rm{u}}_{\rm{v}}}$
.
The influence of thermodynamic states on trail length is significant. Figure 7(a) demonstrates that an increase in relative humidity (
$\phi $
) suppresses evaporation, thereby extending the trail length across all
${\rm{St}}{{\rm{k}}_0}$
values. Conversely, as shown in Fig. 7(b), the effect of changes in the relative pressure drop within the vortex is not linear. For small droplets (
${\rm{St}}{{\rm{k}}_0} \lt 0.1$
), an increase in
${\rm{E}}{{\rm{u}}_v}$
leads to a substantial elongation of the trail. However, for
${\rm{St}}{{\rm{k}}_0}\parallel 0.1$
, the trail length remains relatively unaffected by changes in
${\rm{E}}{{\rm{u}}_v}$
. This suggests that droplets with larger initial diameters are less influenced by condensation within the vortex core, as their initial size governs the dynamics. Interestingly, for low initial Stokes numbers
${\rm{St}}{{\rm{k}}_0} \approx {10^{ - 4}}$
combined with high
${\rm{E}}{{\rm{u}}_v} = 0.25$
values, a slight decline in trail length is observed. This behaviour may be attributed to enhanced entrapment of smaller droplets within the vortex. In such conditions, the droplets rapidly gain mass due to intense condensation inside the vortex core, altering their trajectory and thus extending their lifetime as compared with initially larger droplets.
4.0 Conclusions
A mathematical analysis of discrete droplet dynamics within a Batchelor vortex was conducted, revealing the complex coupling between the droplet motion and the thermodynamic gradients generated by the vortex. A Lagrangian approach is used to analyse the coupling between droplet motion and the flow field generated by the vortex. Under certain thermodynamic and hydrodynamic conditions, droplets may undergo evaporation and condensation when circulating the vortex core due to sharp changes in the environmental conditions induced by the vortex. We evaluated the pressure drop due to the vortical flow, quantifying it using a non-dimensional vortex Euler number
${\rm{E}}{{\rm{u}}_v}$
. The resultant gradients may be large enough to initiate condensation within the vortex core. The onset of condensation was studied by defining a mass transfer coefficient
${C_m}$
, indicating the direction and extent of mass transfer to the Lagrangian water droplet. Our study uncovered a distinct clustering phenomenon linked to the initial Stokes number, with droplets showing a tendency to aggregate at higher Stokes numbers. In summary, the presented model offers valuable insights into droplet dynamics within trailing vortices, contributing to improved modelling and prediction of droplet transport phenomena across various fields, from atmospheric science to engineering applications.
Funding statements
This research was supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1762/20).
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