c Cambridge University Press 2014 J. Fluid Mech. (2014), vol. 750, pp. 5–32. doi:10.1017/jfm.2014.243
5
A numerical study of the dynamics of a particle settling at moderate Reynolds numbers in a linearly stratified fluid A. Doostmohammadi1 , S. Dabiri1,2 and A. M. Ardekani1,2, † 1 Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame,
IN 46556, USA
2 School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette,
IN 47907, USA
(Received 25 October 2012; revised 25 February 2014; accepted 28 April 2014; first published online 30 May 2014)
In this paper, the transient settling dynamics of a spherical particle sedimenting in a linearly stratified fluid is investigated by performing fully resolved direct numerical simulations. The settling behaviour is quantified for different values of Reynolds, Froude and Prandtl numbers. It is demonstrated that the transient settling dynamics is correlated to the induced Lagrangian drift of flow around the settling particle. A simplified model is provided to predict the maximum velocity of the settling particle in linearly stratified fluids. The peak velocity can be followed by the oscillation of the settling velocity and the particle can even reverse its direction of motion before reaching to its neutrally buoyant level. The frequency of oscillation of settling velocity scales with the Brunt–Väisälä frequency and the motion of the particle can lead to the formation of secondary and tertiary vortices following the primary vortex. Key words: particle/fluid flows, stratified flows
1. Introduction
Sedimentation of particles in the presence of background density gradients ubiquitously occurs in natural environments and plays a vital role in characterizing the geochemistry of the atmosphere and upper ocean (MacIntyre, Alldredge & Gotschalk 1995; Jacobson 1998). The gravitational settling of aerosols and dust particles in the stratified atmosphere has major effects on absorbing radiation and cloud formation, leading to both direct and indirect effects on climate changes (Jacobson 1998). In the ocean, the vertical settling of marine snow particles, composed of mostly organic matters, plays a pivotal role in nutrient transport rates and in connecting the pelagic and benthic food chains (Graf 1989). The fluid dynamics of these sedimentation processes is markedly affected by vertical variations of density, which commonly occur in both continuous and discontinuous forms, leading to locally linear and staircase stratification, respectively. Important environmental phenomena such as the accumulation of pollutants in the stratified layers (Kellogg 1980), the aggregation † Email address for correspondence:
[email protected]
6
A. Doostmohammadi, S. Dabiri and A. M. Ardekani
of marine snow particles, and the formation of algal blooms at thermoclines and haloclines (Alldredge & Gotschalk 1989; MacIntyre et al. 1995) are attributed to hydrodynamical effects of density stratification on the particle motion. Ardekani & Stocker (2010) introduced the smallest length scale at which a particle or swimmer is affected by density stratification and reported that it can be of the order of 1 mm. Recently, Doostmohammadi, Stocker & Ardekani (2012) investigated full nonlinear effects of density stratification and demonstrated that the vertical migration of small organisms is hydrodynamically affected due to the reduced flow signature and nutrient uptake as well as enhanced energy expenditure in stratified fluids. In this work, we present detailed information on settling rates and the dynamics of settling particles in stratified fluids, which is required to accurately predict the environmental impacts of stratification. The presence of a density gradient due to variation in salinity or temperature leads to significant modifications in the physics of fluid through which a particle settles. In a homogenous fluid with kinematic viscosity ν, the motion of a particle of diameter dp and characteristic velocity W is well described by the Reynolds number Re = Wdp /ν. When the particle settles in a stratified fluid, the buoyancy force and the diffusivity of the stratifying agent are also important. The buoyancy force is represented by the Froude number Fr = W/Ndp , where N = (γ g/ρ0 )1/2 is the Brunt–Väisälä frequency, the natural frequency of oscillation of a vertically displaced fluid parcel in a stratified fluid, ρ0 is a reference density, g is the acceleration due to gravity and γ is the background density gradient. The Prandtl number Pr = ν/κf describes the ratio of the momentum diffusivity to the diffusivity of the stratifying agent κf . In addition to introducing new dimensionless groups, the stratification leads to fundamental differences between flows in the horizontal and vertical directions. The horizontal flows of stratified fluids have been studied extensively (Hanazaki 1988; Greenslade 2000; Hunt & Snyder 2006). During the last few years, the role of stratification on the drag enhancement of a settling sphere has been investigated (Srdi´c-Mitrovi´c, Mohamed & Fernando 1999; Torres et al. 2000; Hanazaki, Konishi & Okamura 2009; Yick et al. 2009). The experimental study of a sphere settling through a sharp density interface by Srdi´c-Mitrovi´c et al. (1999) reported an order of magnitude drag increase compared with settling in a homogeneous fluid for 1.5 < Re < 15 and 3 < Fr < 10. The experiments of Abaid et al. (2004) demonstrated that for an order of magnitude larger Reynolds number, a sphere settling through a sharp density interface levitates momentarily and can even reverse its moving direction, i.e. ‘bounce’, for a short time. Camassa et al. (2009) have reported experimental observations on prolonged settling times for a low-Reynolds-number particle settling through discontinuous density variations and constructed a theoretical model describing the phenomena (Camassa et al. 2010). In a linearly stratified fluid, Torres et al. (2000) conducted an axisymmetric numerical simulation of the flow past a fixed buoyant sphere in the range of 10 < Re < 800 and 0.1 < Fr < 100 and demonstrated a significant increase in the drag force due to the collapse of rear vortices and the formation of a jet behind the sphere. In an extension to this work, Hanazaki et al. (2009) demonstrated the effects of larger molecular diffusion in forming a broader jet and lessening the stratification effects. Yick et al. (2009) extended the study to a low-Reynolds-number regime (0.1 6 Re 6 1) and reported up to 200 % drag increase due to viscous entrainment of lighter fluid behind the sphere. The remarkable drag enhancement of the settling sphere through stratified layers implies the importance of stratification on the sedimentation of particles in environmental fluids.
The dynamics of a particle settling in a linearly stratified fluid
7
z 1
2
y
x
0.99 1.00 1.01 1.02 1.03
g 3
F IGURE 1. (Colour online) Schematic of the problem for a particle settling in a linearly stratified fluid.
Even though the environmental processes in nature, such as vertical fluxes of particulate matters in the atmosphere and nutrient transport through pycnoclines, are characterized by the time-dependent behaviour of the sedimenting particles, all of the numerical studies on rigid particles in stratified fluids have focused on the steady-state characteristics of the flow past a fixed rigid sphere (Torres et al. 2000; Yick et al. 2009), which is numerically less challenging to model. Recently, Blanchette & Shapiro (2012) performed a numerical study of a settling drop through a sharp density interface and documented a bouncing behaviour for a specific range of surface tension. The environmental situations described above comprise settling of rigid particles through miscible fluid layers in which surface tension effects are irrelevant. This paper investigates the time-dependent gravitational settling of a non-neutrally buoyant particle in the presence of background density gradients utilizing direct numerical simulation to fully resolve the particle–fluid interaction. To the best of the authors’ knowledge, this is the first three-dimensional numerical simulation of the unsteady settling of rigid particles in stratified fluids. We characterize the anomalous dynamics of the sedimentation in stratified fluids and represent the formation of buoyancy-induced secondary and tertiary vortices. We document the role of Reynolds, Froude and Prandtl numbers on the temporal evolution of the settling velocity. The results may have a direct impact on many environmental and industrial processes where detailed understanding of settling dynamics in stratified fluids is of considerable importance. 2. Governing equations
Let us consider an incompressible, viscous flow around a rigid particle settling due to gravity in a linearly stratified fluid, as illustrated in figure 1. Let Γ represent the fluid boundary that is not shared with the particle. Here P and ∂P represent the solid domain (particle) and its boundary, respectively. The computational domain Ω includes both the fluid and the particle. Under the Boussinesq approximation, the dimensional governing equations in the fluid domain are written as ρ0
Du = ∇ · σ + ρf g Dt
in Ω \ P,
(2.1)
8
A. Doostmohammadi, S. Dabiri and A. M. Ardekani ∇·u Dρf Dt u σ ·n ∂ρf ∂n u|t=0 ρf |t=0
= 0
in Ω \ P,
(2.2)
in Ω \ P, = W p + ωp × x − xp = t on ∂P,
(2.3)
= κf ∇ 2 ρf
= 0
on ∂P,
(2.4) (2.5)
on ∂P,
(2.6)
= ufb (x) in Ω \ P, = ρfb (x) in Ω \ P,
(2.7) (2.8)
in addition to the outer boundary conditions on Γ . Here, Ω \ P represents the fluid domain excluding the particle domain. In these equations, u is the velocity vector, W p is the particle velocity vector, ωp is the particle angular velocity, xp is the position of the centre of mass of the particle, x is the position vector, ρf is the fluid density that is dependent on salinity or temperature, ρ0 is the fluid reference density, n is the unit vector normal to ∂P and t is the traction vector on the particle surface. The initial background fluid velocity ufb satisfies the continuity equation. ρfb is the initial background fluid density, σ = −pI + τ is the stress tensor, p is the pressure field, I is the identity tensor, and τ is the viscous stress: τ = 2µD[u] = µ ∇u + (∇u)T , (2.9) where µ is the dynamic viscosity of the fluid. The governing equations in the particle domain can be written as Du Dt ∇·u Dρp Dt D[u] u σ ·n u|t=0
ρp
= ∇ · σ + ρp g
in P,
(2.10)
= 0
in P,
(2.11)
= 0
in P,
(2.12)
= = = =
0 in P, W p + ωp × x − xp in P, t on ∂P, in P, W p0 + ωp0 × x − xp0
(2.13) (2.14) (2.15) (2.16)
where ρp is the particle density, W p0 is the initial velocity of the particle, ωp0 is the initial angular velocity of the particle and xp0 is the initial location of the centre of the particle. Equation (2.13) satisfies the continuity equation; however, to facilitate numerical implementation, equation (2.11) is retained. As noted by Sharma & Patankar (2005) and Ardekani & Rangel (2008), the rigidity constraint produces a stress field inside the particle that is a function of three scalar Lagrange multipliers for a threedimensional configuration: σ = −pI + τ + D[λ],
(2.17)
where λ represents the Lagrange multipliers and τ is zero inside the particle due to the rigidity constraint. The governing equations in the entire domain can be combined as ρ˜
Du = −∇p + ∇ · τ + ρg + f Dt
in Ω,
(2.18)
The dynamics of a particle settling in a linearly stratified fluid ∇·u Dρ Dt u|t=0 ρ|t=0
= 0
in Ω,
9 (2.19)
in Ω,
= ∇ · (κ∇ρ)
(2.20)
= ub (x) in Ω, = ρb (x) in Ω,
(2.21) (2.22)
where ( ρf in Ω \ P ρ= ρp in P ( ρ0 ρ˜ = ρp ( f=
(
in Ω \ P in P
0 in Ω \ P ∇ · D[λ] in P
( ub =
ρfb in Ω \ P ρp in P
(2.23a,b)
( κf in Ω \ P κ= 0 in P
(2.23c,d)
ρb =
ufb in Ω \ P W p0 + ωp0 × x − xp0
in P
(2.23e,f )
in addition to the outer boundary conditions on Γ . In the above equations, f is zero everywhere except in the particle domain and leads to the rigid-body motion inside the particle. The continuity of heat/salinity flux on the surface of the particle implies that ∇ρf · n = 0 (2.26) on the surface of the particle. This boundary condition is equivalent to the adiabatic/impermeable boundary condition on the particle surface that is widely used in the literature on the flow over a rigid particle in a stratified fluid (Hanazaki 1988; Torres et al. 2000; Hanazaki et al. 2009). It is noteworthy that the no-flux boundary condition implemented at the surface of the particle is more appropriate for a salt stratified fluid and it is an idealized assumption for thermally stratified fluids. In the case of temperature-stratified fluids, the insulating boundary condition is only an approximation in situations where the heat diffusivity of the particle can be neglected compared with that of the ambient fluid. 3. Numerical method
Equations (2.18)–(2.23) are solved over the entire domain utilizing a finite volume method using a staggered grid by a conventional operator splitting method to enforce the continuity equation. A non-uniform fixed Cartesian grid is used. The time discretization is obtained using a second-order TVD (total variation diminishing) Runge–Kutta method (Gottlieb & Shu 1998). The QUICK (quadratic upstream interpolation for convective kinetics) scheme is used to evaluate the spatial derivatives in the convection term, leading to third-order accuracy on the non-uniform staggered grid (Leonard 1979) used here. The particle motion is projected onto the solid domain by defining the rigidity force f following the procedure introduced by Ardekani, Dabiri & Rangel (2008) step by step: (i) Step 1. Equations (2.18)–(2.20) are solved with an initial guess for f .
10
A. Doostmohammadi, S. Dabiri and A. M. Ardekani
(ii) Step 2. The particle linear velocity W p and angular velocity ωp are calculated: Z 1 ρp udV, (3.1) Wp = Mp P Z I p ωp = ρp (x − xp ) × udV, (3.2) P
where Mp , I p and ρp are the mass, moment of inertia and density of the solid body, respectively, and dV is a differential volume element. (iii) Step 3. The rigidity force is corrected using f = f∗ +
αρφ [W p + ωp × (x − xp ) − u], 1t
(3.3)
where α is the relaxation parameter, φ is a phase indicator parameter that is unity inside the particle and zero elsewhere in the computational domain and f ∗ is the force from the previous iteration. Steps 1–3 are repeated until the maximum of Euclidean norm of (f − f ∗ )/f and the normalized residual falls below the specified tolerance of 10−3 . The normalized residual is defined as Z |W p + ωp × (x − xp ) − u|φdV P , (3.4) WVp where Vp is the volume of the particle and W denotes the characteristic velocity of the particle. It is straightforward to demonstrate that the converged solution is the equivalent of Z Z d2 xp Mp 2 = σ · ndS + Mp g = ∇ · σ dV + Mp g, (3.5) dt ∂P P where dS is the surface differential element. Integrating (2.10) over the particle volume and using Reynolds transport theorem along with (2.14) lead to Z Z d ρp [W p + ωp × (x − xp )]dV = ∇ · σ dV + Mp g in P, (3.6) dt P P which is equivalent to (3.5). This method has been used extensively for studying particle motion in homogenous fluids and has been verified in previous publications (Ardekani & Rangel 2008; Ardekani et al. 2008; Ardekani, Dabiri & Rangel 2009). The current numerical tools for particle motion in a stratified fluid are verified and validated in appendix A. In this paper, the sphere is released from rest in a quiescent stratified fluid, i.e. ub |t=0 = 0, inside a computational domain of size Ω = 20dp × 20dp × 80dp . The particle is initially placed at the centre of the x–y plane and 0.25l3 below the top boundary of the domain, where l3 is the length of the computational domain along the z direction, as illustrated in figure 1. To ensure that the role of the outer boundary of the domain Γ on the particle settling velocity is negligible, we doubled the domain size to Ω = 40dp × 40dp × 160dp . We observed that in the range of 3.53 < Re < 353, considered in this study, the change in the settling velocity of the particle is less
The dynamics of a particle settling in a linearly stratified fluid
11
than 1 % when the domain size is doubled. The following boundary conditions are used on the outer boundaries of the domain: periodic boundary conditions for density and velocity components are used on the side boundaries of the outer domain. The normal gradient of the velocity field is set to zero on the top and bottom boundaries of the domain, i.e. ∂u/∂n = 0. The normal gradient of density on the top and bottom boundaries of the computational domain is specified as dρ/dz = −γ to maintain the linear background density gradient corresponding to the linear stratification. We have monitored the mass flux for different portions of the outer boundary of the computational domain separately. The maximum mass flux occurs for the bottom boundary of the domain and its dimensionless value is smaller than 10−7 . The initial background density is set to ρfb = ρ0 − γ z, where z is the vertical component of the position vector measured from the initial location of the particle and ρ0 is the background density at the initial location of the particle. 4. Results and discussion
In this section, we present the settling behaviour of a particle in a linearly stratified fluid. The role of Reynolds, Froude, Prandtl numbers and density ratio ρp /ρ0 will be quantified. Unless otherwise stated, ρp /ρ0 = 1.14 and Pr = 700. Since the velocity of the particle is not known a priori, the characteristic velocity W is defined as the Stokes terminal velocity: 1 (ρp − ρ0 )gdp2 W= . (4.1) 18 µ In the numerical simulations reported in this paper, the grid spacing is selected to resolve both the density and momentum boundary layers for the range of Reynolds and Prandtl numbers associated with the settling of the particle. The momentum and density boundary-layer thicknesses can be estimated as dp δm ∼ O √ (4.2) Re and δd ∼ O
√
dp RePr
,
(4.3)
respectively (Schlichting 1968). For example, for the cases with Re = 14.1 and 127, Fr = 1.62 and Pr = 700, the particle Reynolds number calculated based on the maximum velocity of the particle is 5.4 and 25.0, respectively. These values correspond to the minimum density boundary-layer thicknesses of δd ∼ O(0.016dp ) and δd ∼ O(0.008dp ) and the minimum momentum boundary-layer thicknesses of δm ∼ O(0.43dp ) and δm ∼ O(0.2dp ), respectively. It should be noted that these estimates are obtained based on the boundary-layer theory requiring that Re approaches to infinity and may not be accurate for finite values of Reynolds number discussed in this paper. As indicated by Torres et al. (2000), the actual thickness of the density boundary layer in a moderate Reynolds number regime is larger than the values predicted by (4.3). For example, for Re = 200, the authors reported δd to be four times thicker than the prediction based on (4.3). In our calculations for a sphere settling in salt stratified fluids (Pr = 700) at Re = 14.1 and Re = 127, we use non-uniform grids with minimum grid sizes of hΩ = 0.02dp and hΩ = 0.01dp , respectively. For the temperature stratified fluid, the Prandtl number is two orders of magnitude smaller, which results in a thicker boundary layer. The accuracy of these
12
A. Doostmohammadi, S. Dabiri and A. M. Ardekani (a) 1.4
(b) 0.6 0.5
1.2
0.4 1.0 0.3
w* 0.8
0.2 0.6
0.4
0.1
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0
0.1
0.2
0.3
0.4
0.5
0.6
(d) 0.8
(c) 0.5
0.7 0.6
0.4
0.5
w * 0.3
0.4 0.3
0.2
0.2 0.1
0.1
0
0.2
0.4
0.6
0.8
1.0
r/(dp /2) – 1
1.2
1.4
0
0.1
0.2
0.3
0.4
r/(dp /2) – 1
F IGURE 2. (Colour online) Variations of the dimensionless vertical velocity w∗ = w/W and the dimensionless perturbed density ρ ∗ = (ρ − ρfb |z=zpeak )/(γ dp ) near the sphere on the z = zpeak plane, when the particle has reached its peak velocity. For (a,b) Re = 14.1, Pr = 700, Fr = 1.62 and for (c,d) Re = 127, Pr = 700, and Fr = 1.62. The distance from the centre of the sphere is represented by r.
grid sizes in capturing the local behaviour is examined by comparing the density and vertical fluid velocity near the particle for three different grid sizes at the time corresponding to the peak velocity. As illustrated in figure 2, the difference between the results of the two finer resolutions are quantitatively negligible and the normalized root-mean-square deviation is less than 0.5 %. However, it is worth mentioning that the high computational cost prevents us from further refinement of the grid and examining the density profiles very close to the sphere’s surface. Nevertheless, the shape of the isopycnals and velocity field outside the boundary layer and trajectory and velocity of the sphere remain unaffected. 4.1. The governing role of stratification We first examine the effects of the background density gradient on the settling dynamics. Figure 3 illustrates the temporal evolution of the particle settling velocity p Wp for different values of the Froude number, where time is normalized by τ = dp /g. The particle starts from rest, where its density is higher than the local density of
The dynamics of a particle settling in a linearly stratified fluid
13
0 –0.2 –0.4
Wp / W
–0.6 Homogeneous Fr = 16.2 Fr = 6.49 Fr = 3.25 Fr = 1.62 Fr = 1.15 Fr = 0.829
–0.8 –1.0 –1.2 –1.4
0
10
20
30
40
50
60
F IGURE 3. (Colour online) Stratification affects the time-dependent behaviour of a settling sphere. The particle settling velocity Wp normalized by the terminal velocity in a homogeneous fluid W is shown (Pr = 700 and Re = 14.1).
the background fluid. Thus, the particle initially accelerates until it reaches the point of maximum velocity; thereafter, the balance of drag and gravitational forces acting on the sphere causes the vertical velocity to decrease. The maximum velocity can be markedly smaller than the terminal velocity of the particle in the absence of stratification (γ = 0). As the stratification increases (the Froude number decreases), the buoyancy effects more effectively oppose the vertical motion and the particle maximum velocity is reduced. The suppression of the vertical motion is consistent with previous studies and is a known phenomenon in the sedimentation through density interfaces (Srdi´c-Mitrovi´c et al. 1999; Camassa et al. 2009). Below, we provide a simple model to estimate this peak velocity. The maximum velocity of the particle can be estimated using a force balance at the time corresponding to the velocity peak. The buoyancy corrected weight, added mass force, history force and steady drag force acting on the settling sphere in a homogenous fluid are defined, respectively, as π 3 d (ρp − ρfb )g, 6 p 1 dWp FA = − mf , 2 dt Z t 3 2 W˙ p ds 1/2 FH = − dp ρf (πν) , 1/2 2 −∞ (t − s) B =
π DH (Wp ) = − dp2 CDH ρf |Wp |Wp , 8
CDH =
24ν 6 p + + 0.4. dp |Wp | 1 + dp |Wp |/ν
(4.4) (4.5) (4.6) (4.7a,b)
Here, mf is the mass of the displaced fluid and CDH is the drag coefficient of a sphere in a homogeneous fluid. Equation (4.7) is valid in the range of 0 < Re < 2 × 105 with less than 10 % error (White 2006). The steady drag force acting on a sphere settling
14
A. Doostmohammadi, S. Dabiri and A. M. Ardekani 0.15 DS + DH B FH
0.12
0.09
F* 0.06 1.5 1.0
0.03
0.5 0
0 0
10 20 30 40 50 60
10
20
30
40
50
60
F IGURE 4. (Colour online) The temporal evolution of the magnitude of forces acting on the particle at Re = 14.1, Fr = 3.25 and Pr = 700. The forces on the abscissa are scaled with Mp g.
in a stratified fluid is, however, modified and the steady drag enhancement can be approximated as r Ndp dp |Wp | π 2 DS (Wp ) = − dp CDS ρf |Wp |Wp , CDS = c˜ CDH , (4.8a,b) 8 |Wp | ν
where c˜ is a constant equal to 0.67 based on the empirical correlation of Yick et al. (2009). For an inviscid flow past a particle in a weak density gradient, Eames & Hunt (1997) evaluated the drag force acting on the particle moving parallel and perpendicular to the density gradient. For a particle moving parallel to the density gradient, their results can be written as (see also Magnaudet & Eames (2000)) DA = −
1γ mf |Wp |Wp . 4 ρf
(4.9)
This added-mass contribution is, however, two orders of magnitude smaller than the steady drag correlation of Yick et al. (2009) for weak stratifications considered in the present manuscript, γ dp /ρf 1, and can be neglected. As evident from figure 4, the history force is negligible compared with the steady drag force of a sphere in a stratified fluid, DH + DS , for the parameters studied in this paper. Consequently, the force balance at the time corresponding to the velocity peak Wpeak can be written as π 3 d (ρp − ρfb |z=zpeak )g = DH (Wpeak ) + DS (Wpeak ). (4.10) 6 p The added mass force is not included in (4.10) because the acceleration is zero at the moment the particle reaches its peak value, and the history force is neglected. The unknown velocity Wpeak can be calculated by solving (4.10) after evaluating the vertical displacement of the particle as Z tpeak zpeak = Wp (t)dt, (4.11) B|t=tpeak =
0
The dynamics of a particle settling in a linearly stratified fluid
15
0.8
Wpeak / W
0.6
0.4
0.2 Numerical results Simple model 0
5
10
15
20
Fr
F IGURE 5. (Colour online) A comparison of the peak velocity as a function of the Froude number using numerical results and the simple model (Pr = 700 and Re = 14.1).
where tpeak is the time when the particle reaches the maximum velocity and Wp (t) is the instantaneous settling velocity of the particle. As observed in figure 3, the deviation of the settling velocity in a stratified fluid from its counterpart in a homogeneous fluid (ρfb = ρ0 ) is relatively small before the particle reaches the peak velocity. Thus, we exploit the analytical description of the settling velocity for a homogeneous fluid (Mordant & Pinton 2000) to calculate the integral in (4.11): " √ !# −3 2t , (4.12) Wp (t) = WH 1 − exp τ where WH is the terminal settling velocity of the particle in the homogeneous fluid. After solving (4.10)–(4.12) at t = tpeak , the peak velocity can be estimated as a function of the Froude number, as demonstrated in figure 5. The normalized root-mean-square deviation between the fully resolved numerical simulation and the estimated value from the simple model is 8.5 % and can be attributed to the use of the velocity profile of the homogenous fluid in (4.11). It is instructive to test the sensitivity of the results of the simple model to the value of constant c˜ in the empirical drag law. As evident from figure 6, 10 % change in the value of c˜ leads to less than 2 % change in the value of the peak velocity predicted by the simple model. As evident from figure 3, after the settling velocity of the particle in a linearly stratified fluid reaches a maximum, the settling particle decelerates monotonically for a weak stratification. This continuous deceleration, however, is observed over the time range of 0 < t/τ < 60 and may not persist for all times as the concavity of the velocity plot may change for times beyond those used in numerical simulations. Further extension of the time is prone to limitations by computational costs since longer domain is needed and the simulation time increases. As we decrease the Froude number, anomalous oscillations are observed in the settling velocity of the sphere. Further decrease in the Froude number, leads to larger amplitude of oscillations and change in the sign of the velocity of the settling sphere.
16
A. Doostmohammadi, S. Dabiri and A. M. Ardekani 0.6
Wpeak / W
0.4
0.2
0
Fr = 16.2 Fr = 6.49 Fr = 3.25 Fr = 1.62 Fr = 0.829 0.60
0.65
0.70
0.75
F IGURE 6. (Colour online) A test of the sensitivity of the peak velocity, obtained from the simple model, to the choice of the constant in the empirical drag law of Yick et al. (2009) (Pr = 700 and Re = 14.1).
Consequently, the sphere levitates momentarily and can even reverse its direction of motion. In this case, after the sphere reaches its maximum velocity, it decelerates to a complete stop and then starts to rise to lower density levels. The ascent continues until the buoyancy corrected weight dominates the drag force acting on the sphere. At this point, the sphere can levitate again or experience oscillation in the settling velocity, depending on the strength of the background density gradient. The motion of the spherical particle can be best visualized in a phase–space diagram with the sphere displacement and velocity on the abscissa and ordinate, respectively. Here, we replace the displacement with a dimensionless density difference between the sphere and the background fluid as doing so unveils more information about the characteristics of the settling dynamics of the spherical particle. The dimensionless density difference is linearly proportional to the displacement of the sphere and is defined as ρp − ρfb , (4.13) ρˆ = 1 − ρp − ρ0
where its value is zero at the point of release and unity at the neutrally buoyant level of the sphere. The loops in figure 7 correspond to the change in the direction of the motion. A surprising feature of the levitation in a linearly stratified fluid can be inferred from figure 7, which reveals that the density level at which levitation occurs is smaller than the neutrally buoyant density level of the particle (ρfb = ρp ). It should be noted that although levitation phenomena have not been observed in experimental studies of particle sedimentation in linearly stratified fluids, Abaid et al. (2004) previously reported an experimental observation of the levitation of a sedimenting bead during penetration into a sharp density interface. However, when the particle size is much smaller than the length scale of the stratification, the continuous stratified model is a more appropriate representation of the physical conditions in oceans and lakes. Figure 8 presents the normalized settling velocity variation as a function of dimensionless settling time for a range of Reynolds number 3.53 < Re < 127. As
The dynamics of a particle settling in a linearly stratified fluid
17
0
–0.2
Wp / W –0.4 Fr = 3.25 Fr = 1.62 Fr = 1.15 Fr = 0.829 –0.6
0
0.2
0.4
0.6
0.8
F IGURE 7. (Colour online) The effect of buoyancy on the variation of the normalized settling velocity with respect to the normalized difference between the particle density and the density of the background stratified fluid ρfb at Re = 14.1 and Pr = 700. The value of ρˆ = 0 represents the initial location of the sphere, while ρˆ = 1 corresponds to the density level at which the sphere is neutrally buoyant. 0.1 0 –0.1 –0.2
Wp / W –0.3 –0.4
Re = Re = Re = Re =
–0.5 –0.6
0
20
40
60
80
3.53 14.1 56.5 127 100
F IGURE 8. (Colour online) Temporal evolution of normalized settling velocity at Pr = 700 and Fr = 1.62. The terminal velocity of the settling particle at the corresponding Reynolds number in a homogeneous fluid is used to scale the settling velocity.
the Reynolds number decreases, larger viscous effects suppress the oscillation of the sphere velocity during the sedimentation. It should be noted that in the range of parameters studied in the present paper 3.53 < Re < 353, we did not observe vortex shedding and results remained axisymmetric. Unfortunately, we cannot extend the range of the results to higher Reynolds numbers to study vortex shedding.
18
A. Doostmohammadi, S. Dabiri and A. M. Ardekani (a) 100 90 80
(b) 100 Levitation–levitation Levitation–oscillation Oscillation Monotonic deceleration
90 80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
1
2
Fr
3
0 0.5
1.0
1.5
2.0
2.5
3.0
3.5
Fr
F IGURE 9. (Colour online) Classification of different dynamical behaviours during the descent of a particle in a linearly stratified fluid for (a) Re = 14.1, Pr = 700 and (b) Re = 353, Pr = 700.
In figure 9, the settling dynamics of the sphere is classified in a phase diagram, illustrating the range of the density ratios and Froude numbers for which different dynamics is observed. For the lower Reynolds number (Re = 14.1), the occurrence of levitation is associated with strong stratification and small density ratios (figure 9a). For the larger Froude number and density ratio, monotonic deceleration occurs. As the Reynolds number increases (figure 9b), particle levitation is observed at larger Froude numbers and density ratios, and the dimensionless particle displacement before reaching its neutrally buoyant level can be as large as 15. This displacement refers to the distance that the particle moves from its initial position to the density level where it becomes neutrally buoyant. 4.2. Induced Lagrangian drift and vorticity generation The deflection of constant density lines (isopycnals) along with vorticity contours during the gravitational settling of the sphere is shown in figure 10. As the sphere settles in a stratified fluid, the motion of the sphere induces disturbances in the adjacent fluid and causes the distortion of isopycnals. However, deflected isopycnals are forced by buoyancy effects to return to their neutrally buoyant levels. In the following, we determine the induced Lagrangian drift of flow surrounding the sphere as it settles by quantitatively tracking the trajectories of 500 tracer particles. At the initial time, these Lagrangian tracer particles are introduced at several planes initially above and below the sphere, located at distance ZL = −0.6dp , 0.6dp , dp , 2dp , 3dp from the sphere. The volume VD between the distorted Lagrangian plane and its original position is calculated by assuming an axisymmetric geometry. Figure 11(a) shows two frames corresponding to the settling of the sphere through a plane of Lagrangian particles initially located one diameter upstream of the sphere. The temporal displacement of the centroid of the drift volume Zc (t) and drift volume VD are plotted in figure 11(b,c). The peaks of the oscillation of Zc (t) and VD considering a Lagrangian plane initially at ZL = −0.6 downstream of the sphere approximately
The dynamics of a particle settling in a linearly stratified fluid
19
t=0
2.1 0.7 –0.7 –2.1
F IGURE 10. Snapshots of the settling of a sphere for Re = 14.1, Pr = 700, and Fr = 1.62. All frames are taken with a dimensionless time increment of 1t/τ = 2.5. The density difference between two adjacent black solid lines is 1ρ/γ dp = 0.577. The contour map shows dimensionless vorticity contours.
correspond to the peaks of the oscillation of the sphere settling velocity. The frequency of oscillations in the sphere velocity scales with the Brunt–Väisälä frequency as shown in figure 12. Thus, it would be of interest to investigate the generation of internal waves due to the sedimentation of the sphere in linearly stratified fluids. Here, we employ the linear wave theory to predict the internal wave patterns and compare them with the results of numerical simulations. We follow the analysis of Mowbray & Rarity (1967) who used the Lamb–Lighthill principle and the method of stationary waves to predict the curves of constant phase Φ corresponding to internal wave patterns. The coordinate of the lines of constant phase (˜y, z˜) can be described as 1/2 σ −4 + σ −2 − 1 , 1 + σ 2 (2 − σ 2 )2 (1 − σ 2 )−3 1/2 z˜ zp 1 σ −4 + σ −2 − 1 2 2 −3/2 = + FrΦ(2 − σ )(1 − σ ) , dp dp 2 1 + σ 2 (2 − σ 2 )2 (1 − σ 2 )−3 yp 1 y˜ = + FrΦ dp dp 2
(4.14) (4.15)
where (yp , zp ) are the components of the position vector of the sphere and σ = χ dp Fr is a variable parameter in which χ denotes the wavenumber. As noted by Mowbray & Rarity (1967), |σ | 6 1 and as σ → 0, y˜ − yp → 0 and z˜ − zp → ∞; while as σ → 1, y˜ − yp → Φdp Fr and z˜ − zp → 0. Figure 13 illustrates a comparison of the internal wave patterns obtained by numerical simulations and lines of constant phase (Φ = (n + 1/4)π, n = 0, 1, . . .) obtained from the linear wave theory. It should be noted that the linear wave theory is applied to large Reynolds numbers and is applicable only to the far field. Consequently, as evident from figure 13 only a qualitative agreement is observed between numerical simulations and linear wave theory.
20
A. Doostmohammadi, S. Dabiri and A. M. Ardekani
(a)
ZL
Zc
(b)
(c) 0
2.5 2.0
VD / Vp
–0.2
Zc /dp
3.0
–0.4
–0.8 0
10
20
30
1.0 0.5
ZL = –0.6dp ZL = 0.6dp ZL = 1dp ZL = 2dp ZL = 3dp
–0.6
1.5
0 –0.5 40
50
60
0
10
20
30
40
50
60
F IGURE 11. (Colour online) (a) Deflection of a plane of Lagrangian particles initially located one diameter upstream of the sphere. (b) and (c) The time evolution of Zc and VD , respectively, for Re = 14.1, Pr = 700 and Fr = 1.62. Negative values in (c) correspond to the Lagrangian reflux of the fluid.
The vorticity contours for a settling sphere at Re = 127 are presented in figure 14. The primary vortex is formed on the sides of the sphere as it begins to settle. The primary vortex increases in size as the sphere accelerates and decreases as it decelerates later. The rapid descent of the sphere in the beginning leads to the entrainment of a relatively large shell of light fluid in the wake of the sphere. As the tail of the perturbed isopycnals returns to the neutrally buoyant level, the secondary vortex starts to form in a counter direction compared with the primary vortex. Meanwhile, the sphere decelerates due to the increase in the buoyancy force. The rising wake passes over its neutrally buoyant level creating an oscillatory motion accompanied with the generation of a tertiary vortex. It is worth noting that the oscillatory motion of the particle and the generation of secondary and tertiary vorticities are due to internal wave generation discussed earlier.
21
The dynamics of a particle settling in a linearly stratified fluid 0.30 0.24 0.18 0.12 0.06 0 0.8
1.0
1.2
1.4
1.6
1.8
Fr
F IGURE 12. (Colour online) The normalized frequency of oscillation of settling velocity of the sphere as a function of Froude number at Re = 14.1 and Pr = 700. (b)
(z – zp)/dp
(a)
(c)
4
4
4
3
3
3
2
2
2
1
1
1
0
0
0
–1
–2
–1
0
1
(y – yp)/dp
2
–1
–2
–1
0
1
(y – yp)/dp
2
–1
0.6 0.4 0.2 0
–2
–1
0
1
2
(y – yp)/dp
F IGURE 13. (Colour online) Comparison of internal wave patterns obtained from numerical simulations with the lines of constant phase obtained from the linear wave theory (Mowbray & Rarity 1967) for Re = 127, Pr = 700: (a) Fr = 1.31; (b) Fr = 1.47; and (c) Fr = 1.97. Dashed lines show the lines of constant phase and contour maps illustrate the contours of velocity magnitude.
The vorticity equation under the Boussinesq approximation, which can be directly derived by taking the curl of the momentum equation (2.1), can be used to explain this observation: Dω g ˆ = (ω · ∇)u + ν∇ 2 ω − ∇ρf × k. (4.16) Dt ρ0 The last term on the right-hand side of the above equation leads to the vorticity generation during the particle descent in a stratified fluid due to the distortion of density levels and local non-alignment of the density gradient vectors and the vertical direction. This vorticity generation is specific to the settling in stratified fluids, as demonstrated in figure 15, where the velocity fields around a settling particle in the homogenous and stratified fluids are compared. It should be noted that with the Boussinesq approximation the vorticity source term (the last term on the right-hand side of (4.16)) includes the hydrostatic pressure contribution alone, as opposed to the usual baroclinic term proportional to ∇ρ × ∇p. We have compared
22
A. Doostmohammadi, S. Dabiri and A. M. Ardekani
t=0
2.1 0.7 –0.7 –2.1
F IGURE 14. Snapshots of the settling of a sphere at Re = 127, Pr = 700 and Fr = 1.62. All frames are taken with a dimensionless time increment of 1t/τ = 4.5. The density difference between two adjacent black solid lines is 1ρ/γ dp = 0.610. The contour map shows dimensionless vorticity contours.
F IGURE 15. The flow field around the settling particle at Re = 127 in a homogeneous fluid (a) and a stratified fluid (b) with Fr = 1.62 and Pr = 700. The plots are represented at the same instant of time (t/τ = 45) when the particle in the homogeneous fluid has reached the terminal velocity.
the vorticity contours with and without the Boussinesq approximation. As evident from figure 16, the difference between the contours is indistinguishable and we observed a normalized root-mean-square deviation of 0.56 %. To further investigate the validity of the Boussinesq approximation, we have performed a set of numerical simulations without using the Boussinesq approximation for the range of parameters
The dynamics of a particle settling in a linearly stratified fluid
23
–0.14
–0.42 –0.69 0.14 0.42 0.69 0.97
1.25
F IGURE 16. (Colour online) The comparison of the vorticity contours with and without using the Boussinesq approximation at Re = 127, Fr = 1.62, Pr = 700 and dimensionless time t/τ = 38.45. The dashed black lines represent the contours associated with using the Boussinesq approximation and the solid lines (shown in red online) show the results obtained without using the Boussinesq approximation. The lines are indistinguishable. The numbers on the contours represent the magnitude of the dimensionless vorticity ωτ .
considered in this manuscript. We found that the difference between the results with and without the Boussinesq approximation is quantitatively negligible and the normalized root-mean-square deviation is less than 0.46 %. More detail is given in appendix B. In the following, we provide a discussion on the forces acting on a spherical particle settling in a linearly stratified fluid to delineate the occurrence of oscillations in the sphere velocity. The particle equation of motion can be written as Mp
dWp = −B + DH + FA + FH + Fs . dt
(4.17)
In addition to the steady drag, added mass and Basset history forces, there exists an additional history force that acts on a settling particle in a stratified fluid. This history force is present due to the time delay it takes for the deflection of isopycnals behind the particle to reach a steady state and is associated with the buoyancy force acting on the developing wake of the dragged-down light fluid behind the sphere. The unsteady stratified force Fs is calculated using (4.17) and numerical results of figure 3 and is plotted in figure 17. As the sphere accelerates from rest, the density wake begins to form and the stratified force increases. Its value is initially smaller than the steady drag enhancement Ds since the density wake behind the sphere has not developed yet. Later, the rapid settling of the sphere creates a wake larger than the steady wake leading to a larger stratified drag force. The maximum value of the stratified drag force occurs after the velocity has reached maximum. The sphere reaches its
24
A. Doostmohammadi, S. Dabiri and A. M. Ardekani –MpdWp /dt B DH Ds Fs FA FH
0.10
F* 0.05
0 0
10
20
30
40
50
60
F IGURE 17. (Colour online) The temporal evolution of forces acting on the particle for Re = 14.1, Fr = 1.62, ρp /ρ0 = 1.14 and Pr = 700. The force on the abscissa is scaled by Mp g.
minimum velocity approximately when the stratified drag force and secondary vortex (see figure 10) are maximum. When the sphere velocity approaches zero (t/τ > 25), particle inertial force, added mass, history and steady drag forces are close to zero. At this time the unsteady stratified force balances the buoyancy corrected weight of the sphere. The unsteady stratified force slowly approaches zero due to the diffusion of the stratifying agent; consequently, the decaying time scale is of the order of t/τ ∼ O(PrRe) ∼ 9800 which is much larger than the simulation time. 4.3. The role of Prandtl number In oceans and lakes, vertical density variations are induced by temperature or salinity gradients in the water column, while the stratified layers in the atmosphere are formed due to temperature variations in the air. The difference in diffusion of salt and temperature as well as the different diffusivity in water and air can lead to distinct settling behaviours in these stratified fluids. We characterize the relative importance of diffusion by varying the Prandtl number. Figure 18 presents the temporal evolution of the settling velocity for three different Prandtl numbers. The value of Pr = 0.7 corresponds to the temperature stratification in the atmosphere, and Pr = 7 and 700 are associated with temperature- and salinity-induced stratifications in water, respectively. The sphere velocity does not experience a notable change before reaching the peak velocity (see figure 19). At lower Prandtl numbers, the density diffusion hinders the formation of buoyancy-induced vortices and weakens the stratification effects. This result can be associated with the tendency of higher diffusion to prevent the piling up of isopycnals near the settling sphere. As isopycnals accumulate near the sphere surface, the stronger diffusion more rapidly restores the distorted density layers. 5. Conclusions
The appearance of vertical density gradients due to the variation in temperature or salinity is ubiquitously observed in aquatic environments and the presence of stratified
The dynamics of a particle settling in a linearly stratified fluid
25
0
–0.1
–0.2
–0.3
Pr = 0.7 Pr = 7 Pr = 700
–0.4
–0.5
0
20
40
60
F IGURE 18. (Colour online) The variation of settling velocity of the sphere as a function of time for different Prandtl numbers at Re = 14.1 and Fr = 1.62.
layers can have a considerable effect on the gravitational settling of particles. Here, we have demonstrated that the settling dynamics of a sphere in a linearly stratified fluid can be affected significantly. We have provided a simple model to estimate the maximum velocity that the sphere experiences during its vertical descent. After reaching the peak velocity, the stratification effects can lead to the oscillation of the settling velocity and even particle levitation. We have illustrated that the frequency of oscillations in the settling velocity scales with the Brunt–Väisälä frequency. In addition, we have documented the formation of buoyancy-induced secondary and tertiary vortices following the primary vortex behind the settling particle in linearly stratified fluids. Acknowledgements
This work is supported by NSF grants CBET-1066545 and CBET-1414581. Appendix A. Validation of the numerical results
The numerical algorithm is tested against different known experimental and numerical results of settling of particles in homogeneous and stratified fluids. To evaluate the performance of the solver at finite Re, we first compare our results with the numerical work published by Torres et al. (2000) on the drag coefficient of a sphere in a linearly stratified fluid. Table 1 reports the value of the drag coefficient for Pr = 700 and various Reynolds and Froude numbers. The results indicate a relative difference of less than 3 % compared with the numerical results of Torres et al. (2000), who studied the flow past a fixed sphere in linearly stratified fluids. The focus of the present paper is on the transient dynamics of a particle sedimenting in a linearly stratified fluid because in many natural circumstances, the size of particles is much smaller than the length scale of the density variation and linear stratification provides a much better representation of the natural conditions than two-layer sharp
26
A. Doostmohammadi, S. Dabiri and A. M. Ardekani
t=5
t = 30 2.1 0.7 –0.7 –2.1
t = 50
F IGURE 19. Snapshots of the settling of a sphere at Re = 14.1 and Fr = 1.62. The first column represents the flow field at Pr = 0.7. The second and third columns correspond to Pr = 7 and 700, respectively. The density difference between two adjacent black solid lines is 1ρ/γ dp = 0.6. The contour map shows dimensionless vorticity contours.
stratification. However, there are no experimental results on the settling of particles in linearly stratified fluids in the inertial regime. The only experimental work on the settling dynamics of a particle in linearly stratified fluids is the study by Yick et al. (2009), which focuses on the low-Reynolds-number regime (Re < 1). On the other hand, there are several works on the settling of a particle in a two-layer stratified fluid for different Reynolds number regimes (Srdi´c-Mitrovi´c et al. 1999; Abaid et al. 2004; Camassa et al. 2009). Consequently, to validate our numerical simulation, we have compared our results with the existing experiments on the settling dynamics of a particle in a two-layer stratified fluid. We first compare the time-dependent settling velocity of a sphere settling through a sharp density interface with the experimental results of Srdi´c-Mitrovi´c et al. (1999), who studied the sedimentation of spherical particles of different materials and diameters in salt-stratified water. The sphere is
The dynamics of a particle settling in a linearly stratified fluid Re
Fr
CD , Torres et al.
CD , present study
50 100 200
5 0.5 1
1.89 6.22 4.43
1.92 6.36 4.55
27
TABLE 1. Drag coefficient of a sphere in a density-stratified fluid at different Reynolds and Froude numbers. The Reynolds and Froude numbers are calculated based on the freestream velocity.
released from rest in a computational domain of size Ω = 10dp × 10dp × 160dp . The particle is initially placed at the centre of the x–y plane and 4dp below the top boundary of the domain. The interface is located 68dp below the top boundary of the domain. The following boundary conditions are used on the outer boundaries of the domain. The normal gradient of the velocity field is set to zero on top and bottom boundaries of the domain, i.e. ∂u/∂n = 0. The density is set to the density of the lighter fluid ρtop and the density of the heavier fluid ρbottom on the top and bottom boundaries, respectively. Periodic boundary conditions are used on the side boundaries of the outer domain. The following initial background density profile is used z˜ (ρtop + ρbottom ) (ρtop − ρbottom ) + tanh , (A 1) ρfb = 2 2 L where L determines the length scale over which the density changes and z˜ is the vertical component of the position vector measured from the density interface. Figure 20 presents a comparison of a typical density profile used in the experiments of Srdi´c-Mitrovi´c et al. (1999) and the density profile given by the hyperbolic tangent function. The hyperbolic tangent profile fits the density profile used in the experiment very well. The numerical parameters and the specified background density are selected such as to ensure that the density ratio ρp /ρ0 = 1.048, buoyancy jump 1ρ/ρ0 = 6.47 × 10−2 and Reynolds number based on the terminal velocity in the upper homogeneous fluid Re = 22.9 are the same as the values used in the experiments. The simulation is performed on a non-uniform structured grid with the smallest mesh size of hΩ = 1 × 10−2 dp close to the sphere, and the CFL = 0.25. For the results reported in this study, we observed that increasing the grid spacing by a factor of five leads to less than a 1 % difference in the settling velocity of the particle. Figure 21 illustrates the variation of the vertical velocity of the sphere at the density interface versus the travelled distance from its initial position. The velocity of the sphere drops from the terminal velocity in the upper layer to a minimum value inside the density interface and then starts to rise again to the terminal velocity in the lower heavier layer. The normalized root-mean-square deviation between the numerical simulation and the experiment of Srdi´c-Mitrovi´c et al. (1999) is 3.0 %. In addition, table 2 provides a quantitative comparison of the terminal velocity of the particle in upper and lower homogeneous layers with the values obtained by experiments of Srdi´c-Mitrovi´c et al. (1999). The numerical scheme has also been validated against the experimental results of Camassa et al. (2010), who studied the settling dynamics of particles in two-layer stratified fluids settling at the low-Reynolds-number regime in a circular tube (see figure 22). The dimensionless parameters are set to match those of the experiments. Thus, the Reynolds number based on the terminal velocity in the upper layer is
28
A. Doostmohammadi, S. Dabiri and A. M. Ardekani 1.04 Srdic et al. (1999) Present work 1.03
1.02
1.01
1.00
0.99
0
3
6
9
12
15
z (cm)
F IGURE 20. (Colour online) A typical density profile used in the experiments of Srdi´cMitrovi´c et al. (1999) fitted to a hyperbolic tangent function.
0
–0.2
Wp /WH
–0.4
–0.6
–0.8 Srdic et al. (1999) Present study
–1.0 70
80
90
100
110
z/d p
F IGURE 21. (Colour online) Settling velocity of a spherical particle in a sharply stratified fluid as a function of the fallen distance z. The vertical velocity w is normalized by the terminal settling velocity of the particle in the homogeneous fluid of the upper layer WH .
Srdi´c-Mitrovi´c et al. Present study Difference
Ut (cm s−1 )
Ul (cm s−1 )
2.052 2.037 0.73 %
0.621 0.616 0.80 %
TABLE 2. The comparison of the terminal velocity of the sphere in upper Ut and lower Ul homogeneous fluids.
The dynamics of a particle settling in a linearly stratified fluid
29
0 Present study Camassa et al. (2010)
Wp /WH
–0.2
–0.4
–0.6
–0.8
–1.0 0
5
10
15
20
z/d p
F IGURE 22. (Colour online) The variation of the settling velocity as a function of the normalized fallen distance z/dp for a spherical particle settling through a sharp density interface in the Stokes regime. The vertical velocity Wp is normalized by the terminal settling velocity of the particle in the homogeneous fluid of the upper layer WH . 0.40 0.35 0.30 0.25
Wp /W 0.20 0.15 0.10 Present study Mordant & Pinton (2000)
0.05 0
5
10
15
20
F IGURE 23. (Colour online) The transient variation of the settling velocity of a sphere in a homogenous fluid.
1.66 × 10−2 , the buoyancy jump is 1ρ/ρ0 = 1.73 × 10−3 and the density ratio is ρp /ρ0 = 1.47. The particle is initially located at 8dp distance from the interface. The minimum grid size of hΩ = 0.05dp is used near the particle and the time step is adjusted by using CFL = 0.25. Here, we use no-slip boundary conditions on the side boundaries to model the effect of the container’s walls. Camassa et al. (2010) showed that results are domain dependent for settling spheres in circular tubes of diameter 4.9dp , 8.5dp and 15dp in such a low-Reynolds-number regime. We can only numerically model rectangular tanks with our code; consequently, we compared our results with their larger domain size and we used a domain size of 15dp × 15dp × 60dp ,
30
A. Doostmohammadi, S. Dabiri and A. M. Ardekani (a)
(b) 0
0.1 0 –0.1
–0.2
–0.2
Wp / W –0.3 –0.4
–0.6
–0.4
Fr = 3.250 Fr = 1.620 Fr = 1.150 Fr = 0.829 0
10
20
30
40
50
Re = 3.53 Re = 56.50 Re = 127.00
–0.5 60
–0.6
0
20
40
60
80
100
F IGURE 24. (Colour online) The comparison of the temporal evolution of the settling velocity with and without using the Boussinesq approximation. The continuous curves and symbols represent the results with and without using the Boussinesq approximation, respectively. For (a) Re = 14.1, Pr = 700 and for (b) Fr = 1.62, Pr = 700.
which is close but not identical to theirs. The results of the present study shows normalized root-mean-square deviation of 4.9 % with the experimental results of Camassa et al. (2010) which may be due to the difference in the geometry of the outer boundaries. It should be noted that our numerical code is most efficient in the finite-Reynolds-number regime, due to the use of explicit time integration, as opposed to the Stokes regime where a month-long simulation was performed for this validation test. We have also added a comparison between our numerical results and experimental measurements of the settling velocity of a spherical particle in a homogenous fluid by Mordant & Pinton (2000) (see figure 23). The physical parameters are set in such a way as to ensure that the density ratio ρp /ρ0 = 2.56 and the Reynolds number Re = 134 match the experiments. The computational domain of size 8dp × 8dp × 64dp is used with a grid resolution of 12 points across the particle diameter. Increasing the domain size to 16dp × 16dp × 128dp leads to less than 0.8 % variation in the settling velocity of the particle. The comparison of the settling velocity of the particle as a function of time for experimental and numerical studies shows that the normalized root-mean-square deviation is 0.89 %. In all of the above verification and validation cases, the Engauge Digitizer software has been used to extract the data from the previously published results of Srdi´c-Mitrovi´c et al. (1999), Mordant & Pinton (2000), Torres et al. (2000) and Camassa et al. (2010). Appendix B. Validity of the Boussinesq approximation
To investigate the validity of the Boussinesq approximation, we have performed a set of numerical simulations without using the Boussinesq approximation for the range of parameters considered in this study. Figure 24 illustrates that the difference between the results with and without the Boussinesq approximation is quantitatively negligible (the normalized root-mean-square deviation is less than 0.46 %).
The dynamics of a particle settling in a linearly stratified fluid
31
REFERENCES
A BAID , N., A DALSTEINSSON , D., AGYAPONG , A. & M C L AUGHLIN , R. M. 2004 An internal splash: levitation of falling spheres in stratified fluids. Phys. Fluids 16, 1567–1580. A LLDREDGE , A. L. & G OTSCHALK , C. C. 1989 Direct observations of the mass flocculation of diatom blooms: characteristics, settling velocities and formation of diatom aggregates. Deep-Sea Res. A 36, 159–171. A RDEKANI , A. M., DABIRI , S. & R ANGEL , R. H. 2008 Collision of multi-particle and general shape objects in a viscous fluid. J. Comput. Phys. 227, 10094–10107. A RDEKANI , A. M., DABIRI , S. & R ANGEL , R. H. 2009 Deformation of a droplet in a particulate shear flow. Phys. Fluids 21, 093302. A RDEKANI , A. M. & R ANGEL , R. H. 2008 Numerical investigation of particle–particle and particle– wall collisions in a viscous fluid. J. Fluid Mech. 596, 437–466. A RDEKANI , A. M. & S TOCKER , R. 2010 Stratlets: low Reynolds number point-force solutions in a stratified fluid. Phys. Rev. Lett. 105, 084502. B LANCHETTE , F. & S HAPIRO , A. 2012 Drops settling in sharp stratification with and without Marangoni effects. Phys. Fluids 24, 042104. C AMASSA , R., FALCON , C., L IN , J., M C L AUGHLIN , R. M. & M YKINS , N. 2010 A first-principle predictive theory for a sphere falling through sharply stratified fluid at low Reynolds number. J. Fluid Mech. 664, 436–465. C AMASSA , R., FALCON , C., L IN , J., M C L AUGHLIN , R. M. & PARKER , R. 2009 Prolonged residence times for particles settling through stratified miscible fluids in the Stokes regime. Phys. Fluids 21, 031702. D OOSTMOHAMMADI , A., S TOCKER , R. & A RDEKANI , A. M. 2012 Low-Reynolds number swimming at pycnoclines. Proc. Natl Acad. Sci. U.S.A. 109, 3856–3861. E AMES , I. & H UNT, J. C. R. 1997 Inviscid flow around bodies moving in weak density gradients without buoyancy effects. J. Fluid Mech. 353, 331–355. G OTTLIEB , S. & S HU , C. W. 1998 Total variation diminishing Runge–Kutta schemes. Maths Comput. 67 (221), 73–85. G RAF , G. 1989 Benthic–pelagic coupling in a deep-sea benthic community. Nature 341, 437–439. G REENSLADE , M. D. 2000 Drag on a sphere moving horizontally in a stratified fluid. J. Fluid Mech. 418, 339–350. H ANAZAKI , H. 1988 A numerical study of three-dimensional stratified flow past a sphere. J. Fluid Mech. 192, 393–419. H ANAZAKI , H., K ONISHI , K. & O KAMURA , T. 2009 Schmidt-number effects on the flow past a sphere moving vertically in a stratified diffusive fluid. Phys. Fluids 21, 026602. H UNT, J. C. R. & S NYDER , W. H. 2006 Experiments on stably and neutrally stratified flow over a model three-dimensional hill. J. Fluid Mech. 96, 671–704. JACOBSON , M. Z. 1998 Aerosol scattering and absorption. In Fundamentals of Atmospheric Modelling, Cambridge University Press. K ELLOGG , W. W. 1980 Aerosols and climate. In Interactions of Energy and Climate (ed. W. Bach, J. Pankrath & J. Williams), pp. 281–296. Kluwer Academic. L EONARD , B. P. 1979 A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Meth. Appl. Mech. Engng 19, 59–98. M AC I NTYRE , S., A LLDREDGE , A. L. & G OTSCHALK , C. C. 1995 Accumulation of marine snow at density discontinuities in the water column. Limnol. Oceanogr. 40, 449–468. M AGNAUDET, J. & E AMES , I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32 (1), 659–708. M ORDANT, N. & P INTON , J. F. 2000 Velocity measurement of a settling sphere. Eur. Phys. J. B 18, 343–352. M OWBRAY, D. E. & R ARITY, B. S. H. 1967 The internal wave pattern produced by a sphere moving vertically in a density stratified liquid. J. Fluid Mech. 30 (03), 489–495. S CHLICHTING , H. 1968 Boundary-Layer Theory. McGraw-Hill. S HARMA , N. & PATANKAR , N. A. 2005 A fast computation technique for direct numerical simulation of rigid particulate flows. J. Comput. Phys. 205, 439–457.
32
A. Doostmohammadi, S. Dabiri and A. M. Ardekani
S RDI C´ -M ITROVI C´ , A. N., M OHAMED , N. A. & F ERNANDO , H. J. S. 1999 Gravitational settling of particles through density interfaces. J. Fluid Mech. 381, 175–198. T ORRES , C. R., H ANAZAKI , H., O CHOA , J., C ASTILLO , J. & VAN W OERT, M. 2000 Flow past a sphere moving vertically in a stratified diffusive fluid. J. Fluid Mech. 417, 211–236. W HITE , F. M. 2006 Viscous Fluid Flow. McGraw-Hill Higher Education. Y ICK , K. Y., T ORRES , C. R., P EACOCK , T. & S TOCKER , R. 2009 Enhanced drag of a sphere settling in a stratified fluid at small Reynolds numbers. J. Fluid Mech. 632, 49–68.
