Reaction enhancement in an unsteady obstacle wake: Implications for broadcast spawning and other mixing-limited processes in marine environments

Journal of Marine Systems 143 (2015) 130–137

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Reaction enhancement in an unsteady obstacle wake: Implications for broadcast spawning and other mixing-limited processes in marine environments J.P. Crimaldi ⁎, T.R. Kawakami Department of Civil and Environmental Engineering, University of Colorado, Boulder, CO 80309-0428, USA

a r t i c l e

i n f o

Article history: Received 28 August 2014 Received in revised form 11 November 2014 Accepted 12 November 2014 Available online 21 November 2014 Keywords: Cylinder wake Mixing Reaction Fertilization

a b s t r a c t Structured wakes behind flow obstacles are shown to be regions that enhance mixing and reactions between initially distant scalars, with implications for a wide range of mixing-limited biogeochemical processes in marine systems (e.g., broadcast spawning, phytoplankton-nutrient interactions). Reaction of initially distant reactive scalars in the structured laminar wake of a round obstacle is quantified using direct numerical simulations of the 2D Navier–Stokes and reactive transport equations with Reynolds number of 100 and Schmidt number of 1. Scalars are released upstream of the obstacle, initially separated by ambient fluid that acts as a barrier to mixing and reaction. Reaction is computed using second-order kinetics in the low-Damkholer limit. Reaction enhancement is quantified by comparing the obstacle-wake reactions to those in a similar flow but without the obstacle. Integrated reaction rates are shown to be orders of magnitude larger in the obstacle wake for cases with significant initial separation between the scalars. The role of unsteady processes in the reaction enhancement is also investigated by quantifying the scalar covariance in different regions of the wake. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Fluid physics controls a wide range of biogeochemical processes in aquatic environments. In particular, fluid stirring serves to sharpen spatial gradients in transported scalars (e.g., chemicals, nutrients, microorganisms, gametes), enhancing diffusive mixing that promotes chemical reactions and other scalar interactions including nutrient uptake, respiration, and fertilization. In the absence of mechanical stirring, scalar mixing is accomplished only by molecular diffusion or microorganism motility, processes that can greatly limit interactions between initially distant scalar fields. Mixing-limited problems abound in natural and engineered aquatic systems. Physical stirring controls phytoplankton–nutrient interactions (Philippart et al., 2000), phytoplankton–zooplankton interactions (Fasham et al., 1990), interactions between nutrients and food-web structure (Polis et al., 1997), and nutrient uptake and chemotaxis by bacteria (Taylor and Stocker, 2012). In engineered systems, stirring and mixing are critical to wastewater treatment (Bagtzoglou et al., 2006) and disinfection (Liu and Ducoste, 2006). A particularly vivid example of a mixing-limited process in marine biology is broadcast spawning, the reproductive strategy used by a range of benthic invertebrates (Crimaldi and Zimmer, 2014). Most benthic invertebrates spawn their gametes into the ambient flow, and ⁎ Corresponding author. E-mail address: [email protected] (J.P. Crimaldi).

http://dx.doi.org/10.1016/j.jmarsys.2014.11.002 0924-7963/© 2014 Elsevier B.V. All rights reserved.

fertilization occurs externally to female reproductive tracts. Because male and female adults are typically separated by some distance on the seabed, successful fertilization relies on structured stirring to bridge the initial separation between sperm and eggs. The resulting fertilization is typically modeled as a second-order reaction. Stirring by the flow, however, has competing effects; while it can promote gamete aggregation, it can also act to dilute gamete concentrations. The details of physical–biological coupling in this process are still not understood. The exact contributions of larger-scale (stirring of gamete plumes) and smaller-scale (sperm–egg encounters) processes to fertilization success have not been established for even a single species. However, studies suggest that structured stirring by single vortices may enhance fertilization rates (Crimaldi and Browning, 2004; Crimaldi et al., 2008). It is likely that collections of vortices in obstacle wakes could induce similar, or greater, enhancements. Wake-producing obstacles could include (in decreasing scales) islands, reef structures, coral heads, coral branches, and various roughness elements on the seabed. Turbulence is ubiquitous in natural systems, and is typically cited as the key mechanism for promoting stirring, mixing, and reactions of biogeochemical agents. However, laminar flows can also be effective in this regard, and these laminar flows control biogeochemical processes at low Reynolds number, the regime of all aquatic processes at sufficiently small scales. In particular, a large body of literature exists on the topic of stirring, mixing, and reactions by laminar chaotic flows (Ottino, 1989; Muzzio and Liu, 1996; Károlyi et al., 1999; Tél et al., 2000; Szalai et al., 2003; Tél et al., 2005).

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Reefs and islands are common obstacles to incident flows in marine systems (albeit at relatively large scales and Reynolds numbers). Mixing in island wakes is associated with increased particle retention and nitrate concentrations (Coutis and Middleton, 2002; Hasegawa et al., 2004; Venchiarutti et al., 2008), phytoplankton blooms (Heywood et al., 1990; Signorini et al., 1999; Sandulescu et al., 2007; Hasegawa et al., 2008, 2009) and aggregations of coral gametes and larvae (Wolanski et al., 1989; Willis and Oliver, 1990). At the other end of the physical scale spectrum, the importance of obstacle wakes has been shown for benthic invertebrate sensory appendages (Koehl et al., 2001; Mead et al., 2003), and for swimming microorganisms (Yen and Strickler, 1996; Guasto et al., 2012). Despite the large range of studies about the importance of obstacle wakes on ecological function, very little is known about the quantitative reaction enhancement between initially distant scalars. To address this shortcoming, we consider an idealized scenario that can serve as a process-level model for a host of problems in marine science. In the present study, we consider stirring, mixing, and reaction of a pair of initially distant scalars (e.g., egg and sperm, plankton and nutrients, etc.) by one of the most basic and fundamental flows: the structured laminar wake behind an obstacle. We consider the specific case for a 2D flow around a round obstacle at Reynolds number 100, but many of the ideas that result from the study have general implications for similar flows around obstacles of other shapes, for other Reynolds numbers, and in 3D flows. The goal of the study is to show that the structured obstacle wake serves as an effective reactor vessel that enhances mixing and reactions between two initially distant scalars, and to quantify this reaction enhancement relative to a comparable flow without the obstacle.

We consider viscous 2D flow with density ρ and kinematic viscosity ν around a round obstacle with diameter ϕ as shown in Fig. 1. The incoming flow is steady, laminar and uniform with speed u0, and a quasi-steady periodic wake develops downstream of the obstacle. The viscous flow field u = [u, v] is governed by the incompressible Navier–Stokes equations ∂t u þ u  ∇u ¼ −∇p þ Reϕ ∇  u ¼ 0;

−1

2

∇ u;

ð2Þ

u0 ϕ : ν

ð3Þ

Fig. 1. Schematic of the flow and scalar release conditions. Steady laminar flow with speed u0 forms an unsteady wake behind a round obstacle with diameter ϕ. Scalars C1 and C2 are released upstream of the round obstacle and are stirred by the wake. Scalar filaments have initial width f and lateral separation s.

2

ð4Þ

2

ð5Þ

Reϕ Scð∂t C 1 þ u  ∇C 1 Þ ¼ ∇ C 1 −Da R; Reϕ Scð∂t C 2 þ u  ∇C 2 Þ ¼ ∇ C 2 −Da R;

where velocity is still scaled by u0, but length and time are now scaled by s and s/u0, respectively. The reaction rate R is scaled by kC 20, where k is the coefficient for the second-order reaction kinetics, giving R ¼ C1 C2 :

ð6Þ

The Schmidt number, Sc ¼

ν ; D

ð7Þ

is a ratio of the diffusivities of momentum and mass, and the Damköhler number, 2

s kC 0 ; D

ð8Þ

is defined here as a ratio of diffusive and reaction time-scales (small Da corresponds to reactions that proceed slowly relative to diffusion). The scalar boundary conditions are

C 1 ð x ¼ 0Þ ¼

ð1Þ

where length, time, velocity, and pressure have been scaled by ϕ, ϕ/u0, u0, and ρu20, respectively. The character of the resulting wake is parameterized solely by the Reynolds number Reϕ ¼

The flow is subject to boundary conditions u = (1, 0) for x → − ∞, and u = 0 on the obstacle boundary. The flow transports two reactive scalar filaments with initial width f and initial separation s. Filament concentrations are normalized by the source concentration C0, with the resulting non-dimensional concentration fields denoted C1(x, t) and C2(x, t). The concentration fields are governed by a coupled pair of 2D reactive transport equations

Da ¼

2. Problem Formulation

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C 2 ð x ¼ 0Þ ¼

8 <

1

:

0

8 <

1

:

0


 1 1 byb þ f =s ; 2 2 otherwise
 1 1 − −f =s b y b − 2 2: otherwise

ð9Þ

ð10Þ

Numerical simulations of reactive flow around the round obstacle were performed via finite-element discretization to the Navier–Stokes equations (Eqs. (1)–(3)) and the reactive transport equations (Eqs. (4)–(10)) for Reϕ = 100, Sc = 1, and Da = 0.01. The COMSOL Multiphysics package was used to generate the finite-element mesh and solve the resulting system of equations with the direct PARDISO solver (Schenk and Gärtner, 2004); details are given in (Crimaldi and Kawakami, 2013). Time-stepping was adaptive, and mesh refinement was performed to ensure solution convergence. For the Navier–Stokes equations, the inlet boundary condition was uniform normal flow u = − u0n, and the outflow boundary condition was zero viscous stress ν(∇u + (∇u)T)n = 0 with a Dirichlet condition on pressure p = p0. The side walls were slip boundaries subject to impermeability u ⋅ n = 0 and zero stress t ⋅ (− pI + ν(∇u + (∇u)T))n = 0, and the obstacle surface was impermeable no-slip u = 0. The initial condition was u = 0, and the simulation was run until a quasi-steady-state was achieved, as determined by ensuring periodicity within the domain. The resulting vortex shedding frequency ω is often expressed nondimensionally as the Strouhal number, given by St = ωϕ/u0; for the present study we obtained St = 0.18, which is within the range of previously calculated values (Deshmukh and Vlachos, 2005) at Reϕ = 100.

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(a)

(b)

Fig. 2. Streamlines of the computed periodic obstacle wake at a single phase for Reϕ = 100, shown over the entire computational domain. (a) Streamlines of (u, v), which are dominated by the mean streamwise flow. (b) Streamlines of (u − 0.95u0, v), which correspond to a translating perspective moving with the mean wake, elucidating the underlying vortex structure. The dashed box denotes the cropped subregion shown for clarity in all subsequent plots.

convective flux condition n ⋅ (− D∇C) = 0. The upstream boundary conditions and the initial condition were C1 = C2 = 0. The simulation was run long enough to wash out the transient from the domain, and one full period of the quasi-steady-state scalar solution was then computed. In the next section, we compare mixing and reaction rates in the obstacle wake to those in a flow without the obstacle. To permit this, we performed a complementary numerical simulation without the

Representative streamlines at a single phase for the quasi-steady-state solutions are shown in Fig. 2(a). To elucidate the underlying vortex structure responsible for scalar stirring and reaction enhancement, the streamlines as viewed from an observer translating with the wake at 0.95u0 are shown in Fig. 2(b). The reactive transport equations were then solved using the computed flow solution. The lateral scalar boundary conditions were no-flux n ⋅ (− D∇C + uC) = 0, and the outlet boundary had the

5

5

0

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25 1.0

0.00

0.0 0.0

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1.0

Fig. 3. Scalar concentrations C1 and C2 (left) at a single phase for the obstacle wake, along with corresponding reaction rates R = C1C2 (right). Results are shown for three different filament spacings s/ϕ (listed in right margin).

J.P. Crimaldi, T.R. Kawakami / Journal of Marine Systems 143 (2015) 130–137

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Fig. 4. Scalar concentrations C1 and C2 and reaction rates R corresponding to the cases in Fig. 3, but absent the flow obstacle and associated wake.

obstacle. The resulting flow field for this case is laminar and steady, and scalar mixing and reaction are driven solely by lateral diffusion. 3. Results 3.1. Effect of obstacle wake on reaction rates Computed scalar fields C1 and C2 in the obstacle wake are shown in the left column of Fig. 3, where each of the three rows corresponds to initial scalar separations s/ϕ = 0, 1, and 2, respectively. The twodimensional color scale for the concentrations (bottom left in the figure) indicates the local magnitudes of C1 and C2 as well as any possible mixture of the two. In all cases, even when the scalars are initially distant, the two scalar fields are drawn together by the coherent wake structures. This flow-induced coalescence is driven by lateral flow incursions that combine with scalar diffusion to promote subsequent scalar mixing and reaction. For the s/ϕ = 0 case, the scalars are entrained into the discrete vortices as they are shed, such that the scalars populate the vortex cores. As s/ϕ increases, the scalars do not have the opportunity to enter the cores, and are confined to the braid regions that connect the vortices. Corresponding reaction rates R for each s/ϕ are shown in the right column, coded according to the color scale shown in the lower right of the figure. Increases in s/ϕ have the effect of pushing the location of reaction onset further downstream, and decreasing local reaction rates due to diffusive dilution prior to

coalescence. But, in all cases, the reaction rate field mimics the spatial structure of the obstacle wake seen in Fig. 2(b). Note that the maximum possible reaction rate R = 0.25 corresponds to the case of mutual mixing of the two scalars without the inclusion of any ambient fluid (in which case C1 = C2 = 0.5, since each scalar is diluted by the other). For comparison, the scalar fields and reaction rates for the same conditions, but absent the flow obstacle and associated wake structure, are shown in Fig. 4. In this case, lateral spreading and mixing of the scalars are accomplished by scalar diffusion alone.1 For the s/ϕ = 0 case, the reaction rate magnitudes are comparable to those in the obstacle wake because there is no intervening barrier of ambient fluid to curtail scalar mixing and reaction. However, as s/ϕ increases, the reaction rates decrease dramatically due to the lack of lateral incursions associated with the wake.

1 For consistency with Fig. 3, the horizontal axis is again nondimensionalized as x/ϕ, rather than the more traditional scaling u0x/D for diffusive spreading. However, the latter scaling can be easily computed from the former via

u0 x x ¼ Reϕ Sc ; D ϕ where, for this study, Reϕ = 100 and Sc = 1.

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Fig. 5. Scalar concentrations and reaction rates for the obstacle wake (see Fig. 3), averaged over one full period. As discussed in the text, the average reaction rate (right column) is not equal to the product of the average concentration plumes in the left column.

The reaction rates for the obstacle wake are unsteady and periodic, whereas those without the obstacle wake are steady. In order to make a quantitative comparison between the reaction rates with and without the obstacle wake, it is convenient to first time-average the reaction rates for the unsteady wake as shown in Fig. 5.This figure is analogous to Fig. 3 except that the concentration fields and reaction rates are now averaged over one full period of the flow (indicated, here and elsewhere, by the overbar). As will be discussed in the following section, the average reaction rate shown in the right column R ¼ C 1 C 2 is not equal to the product of the average concentrations C 1 C 2 . Instead, the average reaction rate is computed as the average over one period of the unsteady reaction rates (right column in Fig. 3). Because the reaction rates with and without the wake (Figs. 5 and 4) vary spatially in the x and y directions, a direct quantitative comparison between the rates is further facilitated by integrating in space to get a total reaction rate given by

RTotal ðxÞ ¼

Z xZ 0

þ∞

−∞




1 T0

Z 0

T0

 0  R x ; y; t dt



0

dy dx :

ð12Þ

outer integrals sum the reaction rate over the entire lateral domain, and then sum in the streamwise direction from the origin to any given streamwise location x (the x′ in the streamwise integral is a dummy variable of integration). Thus, RTotal gives the total, cumulative average reaction rate in the system up to location x. Comparisons of the integrated reaction rate RTotal with and without the obstacle wake are shown in Fig. 6. In all cases the total reaction rate with the obstacle wake (solid lines) exceeds that without (dotted lines). As s/ϕ increases, the magnitudes of RTotal generally decrease (for clarity, the axis scale for RTotal decreases by a factor of two in each successive plot). The decrease in RTotal is due to the buffering effect of the ambient fluid between the scalars. However, the relative difference between RTotal for the obstacle and no-obstacle cases grows significantly as s/ϕ increases. The ratio of each pair of RTotal curves in Fig. 6 is shown in Fig. 7. The reaction rates for the obstacle wake are orders of magnitude larger than those without, especially for larger values of s/ϕ and smaller values of x. These results demonstrate that the obstacle wake serves as an effective reactor vessel that can provide significant enhancement of reactions between initially distant scalars. 3.2. Role of unsteady processes in the wake

The expression within the parentheses is the time-averaging step performed for Fig. 5, and is used only for the unsteady wake cases. The

We now turn to an investigation of the role of unsteady processes in determining the average reaction rate in the unsteady obstacle wake. To

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6

135

Rewriting the equation for the reaction kinetics R = C1C2 (Eq. (6)) in terms of Eqs. (13) and (14), and then time-averaging the result over one period gives (Crimaldi et al., 2006)

4 2

0

0

R ¼ C 1 C 2 ¼ C 1 C 2 þ c1 c2 :

ð15Þ

0 0

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10

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20

25

In general,C 1 C 2 ≠ C 1 C 2, since the two terms differ by the covariance 0

3 2 1 0 0

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0

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1.5 1.0 0.5 0.0

Fig. 6. Integrated reaction rate RTotal as a function of x for with the obstacle wake (solid lines) and without (dotted lines). RTotal accumulates as a function of streamwise distance x, and is shown for three different initial scalar separation distances s/ϕ.

this end, we employ a Reynolds decomposition to split each of the scalar concentrations into the sum of a mean and a fluctuating component. Thus, for C1, the decomposition is 0

C 1 ðt Þ ¼ C 1 þ c1 ðt Þ;

ð13Þ

where C 1 is the local mean concentration, and c'1(t) is the fluctuating component relative to the mean. Likewise, the decomposition for C2 is 0

C 2 ðt Þ ¼ C 2 þ c2 ðt Þ:

ð14Þ

1000

100

10

1

0

5

10

15

20

0

c1 c2 (which may or may not be zero). The left and right columns in Fig. 8 show the contributions to R from each of the two terms on the right side of Eq. (15). Summing the two columns within a row gives the average reaction rates shown previously in the right column of Fig. 5. The contribution to the average reaction 0 0 rate from the covariance term c1 c2 can be negative (blue), zero (white), or positive (red). The negative covariances occur in regions where the lateral scalar incursions produced by the wake are largely exclusionary; one scalar displaces the other without significant mixing. The positive regions correspond to regions in the vortex braids where the two scalars converge due to strain along the braid axis. The net effect of the covariance contribution in a laterally integrated sense is shown in Fig. 9. Each curve corresponds to the y-direction integral of the covariance contributions from Fig. 8. In the near-field of the wake, the net contribution from the covariance is negative, with this effect increasing with s/ϕ (note the scale change between plots). In the far-field, the net contribution is positive for s/ϕ = 0, but asymptotes to zero in all cases. For the obstacle wake, the product of the mean scalar plumes C 1 C 2 will tend to overestimate the average reaction C 1 C 2 due to the generally negative contributions from the covariance.

25

Fig. 7. Ratio of RTotal values with and without the obstacle wake from Fig. 6, showing the reaction rate enhancement provided by the wake.

4. Summary The obstacle wake used in this study is shown to be effective at enhancing reaction rates between initially distant scalars. The enhancement relative to the no-obstacle case can be orders of magnitude for cases where the initial scalar separation s/ϕ is greater than unity. For these larger separations, the intervening barrier of ambient fluid is too large for diffusion alone to bridge, and by the time (or distance) that diffusion does bridge the gap, dilution is so large that the resulting reaction rate R is trivial. The obstacle wake overcomes this diffusive limitation by promoting lateral scalar incursions that enhance scalar mixing and reaction. The moderate Reynolds number Reϕ = 100 used in the study was chosen to be large enough to produce a well-defined periodic wake structure, but small enough to produce a laminar wake, facilitating numerical computation of the flowfield. While wakes in natural marine environments will span a large range of Reynolds numbers, the general periodic wake-shedding character of most of these flows is reasonably well captured by this single Reynolds number. At larger Reynolds numbers, the wake begins to have small-scale turbulent fluctuations in addition to the large-scale periodic vortex shedding. This small-scale turbulence has the effect of increasing the local dispersion and dilution of the scalar fields, which is often modeled as an effective increase in scalar diffusivity. The relatively large Schmidt number Sc = 1 used in this study was chosen to reflect the larger effective scalar diffusivities that would be seen in many natural flows. For broadcast spawning, a prototypical example of a mixing-limited reaction process in marine systems, sperm and egg released by distant adult males and females face a daunting challenge. At large scales, they are advected passively by the ambient flow, during which time flow-induced dilution limits fertilization rates of sperm that succeed in locating eggs. However, structured wakes behind local obstacles (e.g., coral heads, reef structures) are predicted to serve as local incubators for fertilization enhancement. The results of this study may even be relevant to engineered solutions to promoting fertilization success for imperiled coral populations.

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Fig. 8. Contributions of C 1 C 2 (left) and c1 c2 (right) to R ¼ C 1 C 2 , as given by Eq. (15). The two columns within a row sum to the R fields in Fig. 5.

0.2

Aknowledgments

0.0 -0.2

This work was supported by the National Science Foundation under grants OCE-0849695 and PHY-1205816.

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References

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Fig. 9. Lateral integration of the covariances from Fig. 8. Note that the scale of the vertical axis changes in each successive plot.

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