Scalar absorption by particles advected in a turbulent flow
aa r X i v : . [ phy s i c s . f l u - dyn ] A p r Scalar absorption by particles advected in a turbulent flow
A. Sozza, ∗ M. Cencini, † F. De Lillo, and G. Boffetta Istituto dei Sistemi Complessi, CNR, via dei Taurini 19,00185 Rome, Italy and INFN, sez. Roma2 “Tor Vergata” Dipartimento di Fisica and INFN, Universit´a di Torino, via P. Giuria 1, 10125 Torino, Italy. (Dated: April 23, 2020)We investigate the effects of turbulent fluctuations on the Lagrangian statistics of absorp-tion of a scalar field by tracer particles, as a model for nutrient uptake by suspended non-motile microorganisms. By means of extensive direct numerical simulations of an Eulerian-Lagrangian model we quantify, in terms of the Sherwood number, the increase of the scalaruptake induced by turbulence and its dependence on the Peclet and Reynolds numbers. Nu-merical results are compared with classical predictions for a stationary shear flow extendedhere to take into account the presence of a restoring scalar flux. We find that mean fieldpredictions agree with numerical simulations at low Peclet numbers but are unable to de-scribe the large fluctuations of local scalar uptake observed for large Peclet numbers. Wealso study the role of velocity fluctuations in the local uptake by looking at the temporalcorrelation between local shear and uptake rate and we find that the latter follows fluidvelocity fluctuations with a delay given by Kolmogorov time scale. The relevance of ourresults for aquatic microorganisms is also discussed. ∗ Corresponding author; [email protected] † Corresponding author; [email protected]
I. INTRODUCTION
Mass or heat transfer in multi-phase systems is a problem of great interest for both theoryand applications. Several industrial processes involve fluids with suspended particles that undergochemical reactions, where particles exchange mass or heat with the surrounding fluid [1–3]. Fluidflows also mediate the uptake of nutrients and other biochemicals by suspended (unicellular) mi-croorganisms [4, 5]. Previous works have shown that small-scale turbulence enhances the transportof nutrients into the cell [4, 6]. Such increase in the nutrient flux is typically negligible for bacteriawhile it can be significant for larger cells such as eukaryotic phytoplankton. However, these studieshave considered turbulence as an average, time independent, shear flow while velocity and scalarfields in turbulent flows exhibit strong fluctuations and develop large gradients that are completelyoverlooked by a mean field description [7]. Consequently, the effects of turbulent fluctuationsand their intermittency on cellular uptake are largely unknown. By stirring the nutrient patches,turbulence creates inhomogeneities and complex landscapes of nutrient [8] that make the uptakeproblem non-trivial. Moreover, the variability induced by small-scale turbulence may affect theecological strategies in terms of growth and reproduction rates [9].Previous numerical simulations have studied the problem of cellular uptake using different ap-proaches. For instance, continuous (Eulerian) models for describing the concentration of bothnutrients and (a population of) absorbers have been used for studying phytoplankton dynamics[10, 11] and for quantifying the interaction between motility and turbulence in bacterial chemotaxis[12, 13], but do not provide information on the uptake by a single particle. Discrete, Lagrangianmodels have also been used for computing the uptake of nutrient in a still fluid or in simple laminarflows [14, 15], thus disregarding the unsteadiness of turbulent flows.In this work, we numerically study the effects of turbulent fluctuations on nutrient uptake byusing a mixed Lagrangian-Eulerian approach. The nutrient is represented by a continuous, passivescalar field, while the absorbers are represented by Lagrangian particles. Both the nutrient and theabsorbers are transported by a realistic turbulent flow obtained by the integration of the Navier-Stokes equations at high resolution. Scalar absorption is implemented by volumetric sinks centeredat the particle positions, which has been shown to give accurate results in the absence of a flowor in simple laminar flows [16]. In order to maintain a statistically stationary state in a finitevolume, nutrient is replenished by a uniform source (a chemostat, which models the upwellingfrom a nutrient rich reservoir [10, 17]). For an accurate comparison with simulations, we extendthe analytical results of the uptake enhancement by a shear flow [18] to the presence of a restoringflux.The remaining of the manuscript is organized as follows. In Sec. II, we introduce the model andbriefly describe the analytical derivation of nutrient uptake in the presence of a uniform restoringsource. In Sec. III, we summarize the numerical implementation of the model and the parametersused in the simulations. In Sec. IV, we discuss the numerical results and their comparison with theanalytical prediction based on the mean field model. Finally, Section V is devoted to conclusionsand discussions. The appendices detail the analytical results.
II. MATHEMATICAL MODELS AND THEORETICAL RESULTSA. Model equations
We consider the general problem of N discrete particles – the absorbers – transported byan incompressible velocity field u ( x , t ) together with a passive scalar field c ( x , t ) – the nutrientconcentration. Particles are considered as point tracers whose position X i evolves according to˙ X i = u ( X i , t ) , i = 1 , . . . , N . (1)The velocity field is a solution of the incompressible Navier-Stokes equation ∂ t u + u · ∇ u = − ∇ p + ν △ u + f , (2)where ν is the kinematic viscosity, p the pressure and f a body forcing injecting energy at rate ε onaverage equal to the energy dissipation rate, so to establish a statistically steady turbulent state.The nutrient is advected by the flow, diffuses with diffusivity D , is absorbed by the particles anduniformly restored at a rate µ to maintain, on average, a constant concentration c ; its evolutionreads ∂ t c + u · ∇ c = D △ c − N X i =1 β i δ ( x − X i ) c − µ ( c − c ) . (3)In applications to plankton into the ocean, the uniform nutrient flux models the vertical advectionfrom a deep reservoir – a chemostat – with constant concentration [17]. Following Ref. [16],absorption from the i th particle is modeled through a volumetric absorption rate, β i , and not viaabsorbing boundary conditions at the particle surface. According to this model, the instantaneousuptake of i th particle is given by κ i ( t ) = Z d x β i δ ( x − X i ) c ( x , t ) . (4)The numerical implementation of this model has been calibrated and tested by using configurationsof one, two or more absorbers in still fluid and a laminar shear flow for which analytical resultsare available [16]. Our main interest here is to quantify the effect of turbulence on the uptakerate at the level of the single particle. Specifically, by denoting with κ µ the asymptotic uptakerate obtained with the same chemostat but in the absence of the flow (i.e. when only diffusionis at play), we aim at quantifying the statistics of instantaneous Sherwood number, defined asSh( t ) = κ i ( t ) /κ µ and its average, and how they depend on the relevant parameters of the problem.In the absence of a flow (see App. A), the effect of the chemostat is to exponentially cut-offthe modification of the concentration field with a screening length ξ = p D/µ , therefore reducingdiffusive interactions between particles that without the chemostat are long ranged. As a conse-quence, the usual Smoluchovsky rate at µ = 0, κ s = 4 πDRc , for an absorbing spherical particleof radius R is modified into Eq. (A3), which we rewrite here κ µ = κ s (1 + R/ξ ) . (5)By inverting (5), we obtain that a particle absorbing the nutrient with rate κ µ has radius R = ξ (cid:18)r κ µ πDξc − (cid:19) , (6)which can be used to define an effective radius for the point-particle model.The instantaneous particle Peclet number, quantifying the importance of advection by the flowover diffusive transport, is then defined as Pe( t ) = γR /D where γ ( t ) measures the instantaneousturbulent shear rate at the particle position defined as γ = (2 S ) / , where S ij = ( ∂ i u j + ∂ j u i )is the symmetric velocity gradient tensor at the particle position. It is useful to consider alsothe nominal Peclet number Pe η = γ η R /D where γ η = 1 /τ η is the inverse of the Kolmogorovtime τ η = ( ν/ε ) / . We remark that, since γ is a concave function of the energy dissipation rate,due to Jensen inequality we have h γ i ≤ γ η and therefore h Pe i ≤ Pe η . We also notice that, byintroducing the Schmidt number Sc = ν/D and the Kolmogorov length η = ( ν /ε ) / , the nominalPeclet number can be expressed as Pe η = ( R/η ) Sc. The latter expression shows that, since themodel requires R ≤ η , the maximum attainable value of Pe η is given by Sc. In the following, withsome abuse of notation, when there is no ambiguity, we will often indicate the average Peclet andSherwood numbers as Pe and Sh, respectively. B. Theory of nutrient uptake in the presence of a chemostat
Classical results on the effect of a flow on nutrient uptake, obtained assuming a constant con-centration at infinity, predict two different regimes for small and large Pe [6],Sh = .
28 Pe / Pe ≪ .
55 Pe / Pe ≫ c ( r , t ) relative to the center of a particle of radius R . The boundary conditions are c ( R, t ) = 0 (perfect absorption on the particle surface) andlim r →∞ c ( r , t ) = c . The main effect of the velocity field u is to change the uptake rate by de-forming the shape of the concentration profile with respect to the purely diffusive case. FollowingRef. [18], we consider a particle smaller than the smallest scale in the flow (i.e. R < η ), so that thevelocity field around it can be expressed as a linear shear. We decompose the concentration fieldinto a mean profile and a fluctuating one c ′ ( r , t ) that represents the deviations from the diffusive,spherical symmetric solution c ( r , t ) = c (cid:18) − κ ( t )4 πDrc (1 + R/ξ ) e − ( r − R ) /ξ (cid:19) + c ′ ( r , t ) (8)where κ ( t ) represents the total (still unknown) flux to the particle.In the absence of a flow ( u = 0) we have c ′ = 0 and κ = κ µ , as given by Eq. (5). In the presenceof a flow, the relative increase of nutrient uptake, given by the Sherwood number Sh( t ) = κ ( t ) /κ µ ,is readily obtained imposing the condition c ( R, t ) = 0 in (8) which gives Sh( t ) = 1 + c ′ ( R, t ) /c .The asymptotic (and here averaged) value of the Sherwood number, in the limit t → ∞ can beobtained by extending the analysis of [18], which for Pe ≪ χ ( α )Pe / (9)where χ ( α ) = 1(4 π ) / Z ∞ dz e − αz √ z
24 + z , (10) Run
M µ ν E U T τ η L η ℓ B ξ Re λ Sc α η A1 128 0 . . × − .
59 0 .
63 5 . .
40 3 .
70 0 .
08 0 .
025 0 .
089 38 10 0 . . . × − .
65 0 .
66 6 . .
25 4 .
28 0 .
04 0 .
013 0 .
057 66 10 0 . . . × − .
67 0 .
67 6 . .
16 4 .
48 0 .
02 0 .
006 0 .
035 109 10 0 . . . × − .
67 0 .
67 6 . .
16 4 .
48 0 .
02 0 .
006 0 .
028 109 10 0 . . . × − .
64 0 .
65 6 . .
25 4 .
18 0 .
04 0 .
04 0 .
179 65 1 . . . . × − .
70 0 .
68 7 . .
06 4 .
78 0 .
005 0 .
005 0 .
045 287 1 . . TABLE I. Simulation parameters: Run index, resolution M , chemostat rate µ , kinematic viscosity ν , energy E = h| u |i /
2, rms velocity U = (2 E/ / , integral times cale T = E/ε , Kolmogorov time scale τ η =( ν/ε ) / , integral length scale L = U T , Kolmogorov length scale η = ( ν /ε ) / , Batchelor length scale ℓ B = η/ Sc / , screening length ξ = ( D/µ ) / , shear rate γ η = 1 /τ η , Taylor Reynolds number Re λ = U (15 /νε ) / ,Schimdt number Sc = ν/D , and α η = µτ η . In all runs the energy injection rate is fixed at ε = 0 .
1, and k f = 1 . with α = µτ η the relative time scale between stirring and nutrient supply. For α = 0, Eq. (10)recovers Batchelor’s result, χ (0) = √ π/ (6 / ≃ . α > α ,meaning that the effect of the chemostat is to reduce the contribution of stirring to the nutrientuptake. This is somehow expected since with a fast chemostat (with α = O (1)) the nutrient aroundthe particle is uniformly restored before the flow deforms the iso-concentration surfaces. III. DIRECT NUMERICAL SIMULATIONS
We solve Eqs. (2-3) via direct numerical simulation (DNS) on a triply periodic cubic domainof side L = 2 π using up to M = 1024 grid points with a 2 / nd order Runge-Kutta time marching. The forcing in Eq. (2), acting only at large scales (inthe wave number shell k ≤ k f ), is chosen in such way as to maintain the energy input ε constant.This is obtained by taking f ( x , t ) = ε u ( x , t ) / E k ≤ k f Θ( k f − k ), where Θ is the Heaviside stepfunction and E k ≤ k f the kinetic energy restricted to the wavenumbers ≤ k f [21, 22]. We ensure thatsmall-scale fluid motion is well resolved by imposing the Batchelor length scale ℓ B = η/ √ Sc (thesmallest scale in the problem since Sc ≥
1) to be at least of the same order of the grid spacing,( k max ℓ B > .
0, where k max = M/ c . We explored a range of Taylor-scale Reynolds number(Re λ ≈ − , N particles uniformly in the domain and let them move accordingto Eq. (1). The fluid velocity and its gradients (needed to estimate the shear rate) at particlepositions are obtained via a 3 d order interpolation scheme. The regularized δ -function in Eq. (3)has support over a cube with a side of 4 grid points centered on the particle position, as detailedin [16].In order to explore different values of the Peclet number, we selected different effective radii ofthe particles by tuning the absorption rate β . In particular, the radius is calibrated by performing,for each set of parameters, a diffusive simulation without flow and with static particles. For each β , the asymptotic uptake rate κ µ is measured and Eq. (6) is used to define the particle radius [16].To optimize the computational costs, several particles were integrated in each run. We remarkthat, in general, the presence of many particles in a finite domain induces diffusive interactionswhich tend to reduce the single particle uptake rate [16, 23, 24]. Although this effect is relevant toand interesting for applications [25, 26], in this work we focus on the single particle absorption, andtherefore on dilute concentrations such that diffusive interactions are negligible. In this respect,the chemostat, inducing a screening length ξ , reduces the interactions among particles.We can exploit the knowledge of the screening length to estimate the number of particles to beused to minimize the diffusive interactions. The flow being incompressible, the particle distributionremains uniform. By assuming a random uniform distribution of N particles in a cube domain ofside L , the probability density function (PDF) of nearest-neighbors distance, for small r takes theform [27] P ( r ) = 4 πr ρ exp (cid:18) − πr ρ (cid:19) . (11)The mean inter-particle distance is h r i = a/ρ / p , with a = Γ(1 / / (36 π ) / , where ρ p = N/ L isthe particle number density. By choosing, e.g., h r i = 8 ξ one has that the probability to find twoparticles at distance less than 2 ξ is only 1%. IV. RESULTS
We start by showing in Fig. 1 a typical example of the concentration field in a two-dimensionalsection of the computational box. Due to the absorption, small depletion zones are created aroundthe particles, which are then stretched by turbulence leading to filament-like structures. The pres-ence of these structures reflects how turbulence locally increases scalar gradients, thus impactingparticle uptake. - c ( x , y ) / c FIG. 1. (color online) Fluctuations of the concentration field 1 − c ( x , t ) /c in a two-dimensional slab of 16grid points. Resolution M = 1024 (Run B2). -5 -4 -3 -2 -1
0 10 20 30 40 50 Re λ =38 Re λ =65 Re λ =109 Re λ =287 P ( γ ) γ 〈 γ 〉 Re λ γ η 〈γ〉 FIG. 2. (color online) Probability density function of shear rate γ = (2 S ) / for different Re λ . Inset: meanshear rate h γ i (filled circles with solid line) and its root mean square (area in gray) compared with thedimensional estimation γ η = τ − η (empty circles with dashed line), due to the Jensen inequality h γ i < γ η . -2 -1 -3 -2 -1 S h - Pe 〈 Sh 〉 -1 R/ η FIG. 3. (color online) Local gain of Sherwood number Sh − R as inlabel. The solid line with filled circles represents the behavior of h Sh i − h Pe i . In Fig. 2, we plot the PDF of the shear rate γ for different values of the Reynolds numberRe λ . The form of this distribution has been widely studied in previous works and is characterizedby non-Gaussian tails [28, 29], which become wider and wider at increasing Re λ , the hallmark ofintermittency in the statistics of the velocity gradients.Strong gradients are expected to cause local modification of the absorption. Indeed variationsof γ along the particle path modify the instantaneous value of the Peclet number. To understandand characterize these variations and their effect on the uptake we measure the instantaneousindividual uptake κ , by using Eq. (4), and shear rate γ along each particle trajectories, in thisway we can compute the local Peclet and Sherwood numbers. In Fig. 3, we plot the instantaneousvalue of Sh −
1, i.e. the deviation from the diffusive uptake induced by turbulence, as a functionof Pe for particles with 9 different radii (each represented by a different color) transported by aturbulent flow at Re λ = 109. Although a clear correlation between uptake rate and local shear isobserved, as indeed the solid line shows that at changing the local Peclet number the Sherwoodnumber changes according to the prediction valid for the average, we also observe large fluctuationsof these values on a single particle (i.e. at fixed R ).The average values of the Peclet number h Pe i and of the Sherwood number h Sh i for the differentsimulations are shown in Fig. 4a together with the classical theoretical prediction (7a). Averagesare computed over all particles having the same size and over time. Different symbols code the runssummarized in Table I. The h Pe i / behavior of Eq. (7a) is clearly observable for small values of h Pe i . We do not observe the h Pe i / scaling of Eq. (7b), which is expected at larger h Pe i , however0 -2 -1 -3 -2 -1 (a) 〈 S h - 〉 〈 Pe 〉 Run A1Run A2Run A3Run A4Run B1Run B2 -1 -3 -2 -1 (b) 〈 ( S h - ) / χ ( α ) 〉 〈 Pe 〉 Run A1Run A2Run A3Run A4Run B1Run B2
FIG. 4. (color online) Dependence of the mean Sherwood number on the mean Peclet number: (a) h Sh i− h Pe i for all runs. Average is taken over all the particles with the same radius and over time, after discardinga transient. The dashed line displays the h Pe i / behavior of Eq. (7a). (b) h Sh i− χ ( α ), see Eq. (10). The dashed line represents the prediction (9). the points at largest h Pe i of run A1 show a transition to a flatter scaling. As one can see, while thescaling Eq. (7a) is well reproduced, data obtained with different values of µ and Re λ are not on thesame master curve. The reason for this is the presence of the chemostat that modifies the constantin front of the h Pe i / scaling. In Figure 4b we plot the Sherwood number rescaled with thecoefficient χ ( α ) (with α = µτ η ) given by (10), which generalizes Batchelor’s result (correspondingto α = 0). As one can see, now we find a good collapse for all the runs characterized by differentvalues of Re λ and α . For h Pe i . .
5, the analytical prediction (9) provides an accurate descriptionof the effect of turbulence on the uptake, which is mainly controlled by the Peclet number. Weobserve also some small difference between runs A and B, indicating a possible dependence on theSchmidt number which is not fully captured by the theoretical analysis. Remarkably, the collapseof the different curves is observed for all the available values of h Pe i , even beyond the range ofvalidity of (9).Given the intense fluctuations that characterize turbulent gradients, we expect the local uptakerate to be subjected to strong variations with respect to the mean. In order to characterize thosefluctuations we study the PDF of the local Sherwood number Sh for different values of the controlparameters Re λ , Pe η , Sc. In Fig. 5, the distribution of Sh is plotted, at fixed Re λ , Sc and α , fordifferent values of Pe η obtained by changing the effective particle radius R . For very small valuesof Pe η (hence, small R ), the observed values of Sh are confined to a narrow interval around 1. Thisconfirms that the local uptake rate by small particles is mildly influenced by turbulence both in1 -5 -4 -3 -2 -1 -0.5 0 0.5 1 1.5 2 2.5 3(a) P ( S h - ) Sh-1 Pe η -5 -4 -3 -2 -1 -1 0 1 2 3 4 5(b) P ( S h - )( 〈 S h 〉 - ) (Sh-1)/( 〈 Sh 〉 -1) FIG. 5. (color online) PDF of the instantaneous Sherwood number Sh at varying the nominal Pe η number,i.e. for various particle radius: (a) Pdf of the deviation Sh − η and fixed Re λ = 109, Sc = 10 and α = 0 .
05 (run A3). (b) thesame PDF of panel (a) normalized by their average. Note that the increased uptake by larger particles isalso accompanied by more intense fluctuations. terms of its average and of its fluctuations. Increasing Pe η (and consequently, the particle radius),the distribution moves towards larger values of Sh and develops wider right tails. This changeof shape in the distribution is made more evident in Fig. 5b, where the PDFs are normalizedwith the average value h Sh i . The small left tails for Sh < λ . In Figure 6a, we showthe PDF of Sh for three cases in which the nutrient replenishment rate µ of the chemostat is keptconstant while varying Re λ . By definition, for fixed µ , α decreases as turbulence becomes moreintense and this produces a shift of the PDF towards larger values of Sh. An increase in Re λ ,however, does not seem to affect the shape of the distribution, as one can appreciate from Fig. 6b,where the PDFs are normalized with their mean values. In these three cases both the nominalPeclet, Pe η , and the Schmidt, Sc, numbers are kept constant, which physically speaking meansthat the ratio of the particle radius to the Kolmogorov length is also constant, as R/η = p Pe η / Sc.In Fig. 6c, we show the PDFs for three different values of
R/η . As one can see, the rescaled PDFs2 -4 -3 -2 -1 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 (a) P ( S h - ) Sh-1 Re λ -4 -3 -2 -1 -1 -0.5 0 0.5 1 1.5 2 2.5 3(b) P ( S h - )( 〈 S h 〉 - ) (Sh-1)/( 〈 Sh 〉 -1) 10 -4 -3 -2 -1 -1 0 1 2 3 4 5R/ η (c) P ( S h - )( 〈 S h 〉 - ) (Sh-1)/( 〈 Sh 〉 -1) Re λ FIG. 6. (color online) PDF of Sh − λ : (a) holding fixed Pe η = 3, Sc = 10. The chemostatrate µ is also fixed so that larger Re λ corresponds to smaller α ; (b) same as (a), but normalizing the PDFsof Sh − h Sh i increases with Re λ ,the shape of the distribution is not strongly affected by the turbulence intensity. (c) same rescaling as in(b) repeated for three sets of particles, with different radii, namely R/η = p Pe η / Sc = 0 .
57 (red symbols),0 . .
70 (empty symbols) respectively. collapse fairly well, implying that
R/η dominates the overall shape of the distributions, especiallythe behaviour of the right tail, with more intense fluctuations in uptake measured when the effectiveradius reaches the Kolmogorov scale. Residual effects in Re λ , however, cannot be completely ruledout. It is worth emphasizing that R ∼ η constitutes the upper limit for the particle size withinour model, therefore the details of the statistics close to this limit should be taken with caution.However, the consistency of the behaviour through about a factor three in radius seems to supportthe robustness of the observation.We conclude by briefly discussing the relevance of time correlations in the uptake process.Indeed, it is reasonable to expect that a local fluctuation of the velocity gradient at a given timeshould produce a corresponding fluctuation in the scalar uptake with some delay time. This isqualitatively confirmed by inspecting the temporal signals of the instantaneous Sherwood andPeclet numbers, shown in Fig. 7a. In order to quantify this delay we compute the connectedcross correlation between Pe and Sh normalized by their standard deviation, which is shown inFigs. 7(b) and (c) at varying the relevant parameters. As one can see, the correlation functionsattain their maximum value for a delay time of the order of τ η . The Kolmogorov times scale isindeed the time scale for the deformation of the nutrient field to take place around the particle inthe viscous-diffusive regime of scalar transport, which is the one relevant to the problem [30].3 -2-1 0 1 2 3 40 45 50 55 60(a) S h ~ ( t ) , P e ~ ( t ) t Sh~ (t)Pe~ (t) C ( τ ) τ / τ η Re λ =38Re λ =109Re λ =287 C ( τ ) τ / τ η Pe η FIG. 7. (color online) Correlation between local strain and uptake rates: (a) Temporal signal of the normal-ized variables f Sh( t ) = (Sh( t ) − h Sh i ) /σ Sh (black solid line) and f Pe( t ) = (Pe( t ) − h Pe i ) /σ Pe (blue dashed line)for Re λ = 66 (Run A2) and P e η = 2. (b) Temporal cross correlation C ( τ ) = h f Pe( t ) f Sh( t + τ ) i for different Re λ (Run A1,A3 and B2) while taking µ = 0 . α , with P e η = 1. Time is rescaledby τ η to make clear that the maximum is attained at τ ≈ τ η . Without this rescaling the long time decay ofthe different curves is basically the same, meaning that it is mainly controlled by the chemostat rate µ (notshown). (c) for different Pe η , as in label, at fixed Re λ = 66 and Sc = 10 (for clarity, not all data points arerepresented). V. CONCLUSIONS
In this work we have studied the effect of turbulence on the scalar uptake by spherical absorbingparticles advected by the flow. By means of realistic, direct numerical simulations at differentturbulent intensities, we computed the instantaneous absorption of a scalar field, and its gain withrespect to a purely diffusive process. We used a point particle method with volumetric absorptionof the nutrient scalar field calibrated to represent particles of different sizes (i.e. Peclet numbers).The scalar uptake relative to a purely diffusive process, quantified by the Sherwood number Sh,is found to depend on the average particle Peclet number Pe, in agreement with classical predictionsbased on a mean-field representation of turbulence. Nonetheless, we observe strong fluctuationsin the instantaneous and local value of Sh, whose PDF develops wide tails in particular for largevalues of Peclet (i.e. large radii) and Reynolds numbers.By analyzing the time series of Sh and Pe along a particle trajectory, we observed a delaybetween the two signals which was quantified by computing the cross correlation function. Thisdelay is found to be of the order of the Kolmogorov time of the flow and depends weakly on thePeclet number.We conclude by discussing the possible relevance of our findings for the nutrient uptake by smallnon-motile microorganisms transported by fluid flows. In the ocean [31, 32], using as referenceparameters ε = 10 − − − m s − , ν = 10 − m s − , and D ∼ − m s − (for the most important4nutrient like phosphate and nitrate) one has τ η = 0 . − s and Sc = 10 . For a phytoplanktoncell of size R = 10 − m (e.g., a bacterium), the Peclet number ranges between Pe = 10 − − − and, therefore, the effect of turbulence on nutrient uptake is expected to be negligible. Conversely,for cells of size R = 10 − m we have Pe = 1 −
100 and the average cellular uptake, according toour results, can be substantially affected by turbulence with an increase up to about two timeswith respect a purely diffusive environment [6]. In particular, our results show that, in this regime,the local value of the uptake can be much larger than the mean, with a PDF which developsvery large tails. To the best our knowledge, the effect of fast and strong nutrient fluctuations onphytoplankton growth is not known. Our findings suggest that it would be interesting to investigatethis issue experimentally.The present analysis focused on the situation in which the nutrient is continuously replenishedin time and uniformly in space by a constant chemostat. Of course, in many applications, nutrientsources are often distributed in patches [33]. Thus, a natural extension of this model would be toconsider a forcing for the nutrient not uniform in space and time, representing the variability presentin nature. Other possible extensions of the model are in the direction of a better representationof the absorption mechanism which could take into account more accurately of the local effects ofthe flow such as, for example, rotation of the cell due to local vorticity and its effect on the uptake[34].
ACKNOWLEDGMENTS
We acknowledge HPC CINECA for computing resources (INFN-CINECA grant no. INFN19-fldturb). FD acknowledges PRACE for awarding access to GALILEO at CINECA through projectLiLiPlaTE. GB and FD acknowledge support by the Departments of Excellence grant (MIUR). ASacknowledges support from grant MODSS (Monitoring of space debris based on intercontinentalstereoscopic detection) ID 85-2017-14966, research project funded by
Lazio Innova/Regione Lazio according to Italian law L.R. 13/08. GD.
Appendix A: Smoluchowski rate in a chemostat
Consider a spherical particle immersed in a quiescent nutrient concentration c ( x , t ) sustainedby a chemostat, i.e. ruled by the equation ∂ t c = D ∆ c − µ ( c − c ) , (A1)5with initial condition c ( x ,
0) = c and boundary conditions c ( R, t ) = 0 at the surface of the sphereand c ( ∞ , t ) = c , where we used the spherical symmetry of the problem. At stationarity, therelative concentration, ψ = 1 − c/c , satisfies the equation ψ ′′ + 2 ψ ′ /r − µψ/D = 0. Solving for theconcentration yields c ( r , t ) = c (cid:20) − Rr e − ( r − R ) /ξ (cid:21) , (A2)where the long-range behavior of the solution without source term is exponentially damped with screening length ξ = p D/µ . The uptake rate is obtained by integrating the nutrient flux J = − D∂ r c over the surface of the sphere κ µ = I J · ˆ n dS = κ s (cid:18) Rξ (cid:19) (A3)with κ s = 4 πDRc being the usual Smoluchowski rate, which is recovered in the limit µ → ξ → ∞ ). Appendix B: Generalization of the Batchelor calculations for a chemostat
We report here the details of the analytical derivation of the Sherwood number behavior in thePe ≪ c ( x , t ), advected by thevelocity field u ( x , t ) and replenished by a chemostat with rate µ , ∂ t c + u · ∇ c = D △ c − µ ( c − c ) . (B1)Absorption by the particle is modeled by the boundary conditions: c = 0 at the particle surface,i.e. for r = R (where r = | r | ) and c = c in the far away distance, r → ∞ .As a first approximation, we assume a time-independent linear shear flow, i.e. u i = G ij x j ,with G ij = ∂ j u i constant. As usual, the gradient tensor G ij can be written as G ij = S ij + Ω ij ,with the symmetric, S ij = ( ∂ j u i + ∂ i u j ), and anti-symmetric, Ω ij = ( ∂ j u i − ∂ i u j ), componentrepresenting straining motion and rigid-body rotation, respectively.We begin by searching for a solution in Fourier space for the concentration field in the case ofan instantaneous source with uptake rate κ . In order to study the mass transfer in the proximityof the particle, it is convenient to adopt the comoving coordinates r = x − X i . We also rewrite6Eq. (B1) for the relative concentration ψ = 1 − c/c , with boundary condition ψ ( ∞ , t ) = 0 and ψ ( R, t ) = 1. Considering the Fourier transform b ψ ( q , t ) = R ∞−∞ ψ ( r , t ) e − i q · r d r , Eq. (B1) reads ∂ b ψ∂t − G ij q i ∂ b ψ∂q j = − Dq b ψ − µ b ψ (B2)with q = | q | .The concentration field for a sustained source is obtained as the time integral of the solution ofthe instantaneous source, i.e. ˆ ψ ( q , t ) = κ Z t ds e − Dq i B ij q j − µs (B3)with B ij ( t ) a time-dependent symmetric matrix that incorporates the effects of the shear. In theabsence of shear, it reduces to a diagonal matrix that describes isotropic diffusion with a Gaussiansolution [35, 36]. Plugging the solution in Eq. (B2) yields the equation for the tensor B ij that,after some algebra, reads dB ij dt = δ ij + G il B jl + G jl B il (B4)Now anti-transforming, the concentration field becomes ψ ( r , t ) = κ Z t ds Z ∞−∞ d q (2 π ) e i q · r − D q · Bq − µs , (B5)which can be easily solved by Gaussian integration, yielding in physical space, ψ ( r , t ) = κ (4 πD ) / Z t ds p det( B ) e − r · B − r / (4 D ) − µs . (B6)The steady state solution, which corresponds to the diffusive approximation around the absorb-ing particle, is obtained taking the limit t → ∞ , and approximating the integrand with its Taylorexpansion in r = 0 and s = 0 [19]. At the lowest order we can assume B ij = s δ ij , det( B ij ) = s and B − ij = s − δ ij , obtaining ψ ( r ) ≈ κ (4 πD ) / Z ∞ dss / e − r Ds − µs = κe − r/ξ πDr (B7)The Sherwood number, defined as Sh( t ) = 1 + c ′ ( R, t ) /c is computed subtracting the steady statesolution to the global solution as c ′ ( r , t ) /c = ψ ( r ) − ψ ( r , t ). We then perform the average overtime by taking the limit t → ∞ , and then evaluate the integral at particle surface by taking thelimit r →
0. Since κ differs weakly from κ s , we can approximate the integral asSh = 1 + R (4 πD ) / Z ∞ (cid:16) s − / − det( B ) − / (cid:17) e − µs ds (B8)7An approximation valid for a generic linear shear flows can be found by expanding B ij as powerseries in t : B ij = δ ij t + B (2) ij t + B (3) ij t + . . . . Hence, substituting in Eq. (B4) we can determinethe first coefficients B (2) ij = ( G ij + G ji ) = S ij B (3) ij = S il S jl + ( S il Ω jl + S jl Ω il ) (B9)Now we consider the axes of reference to coincide with the principal axes of the rate of strain tensor S ij to obtain a simple shear flow along a preferential direction G = γ , S = S = γ/
2, so thatEq. (B9) is satisfied by B = t (1 + γ t ) , B = t, B = tB = γt , B = B = B = 0 . (B10)The determinant of B ij is given by (cid:18) det( B ) t (cid:19) / = 1 + γ t
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