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Direct Numerical Simulation (DNS) of turbulent flow
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Chain formation can enhance the vertical migration of phytoplankton through turbulence

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We modeled a homogeneous, isotropic turbulent flow field by direct integration of the Navier-Stokes equations without any closure model. This method, known as DNS, is computationally expensive but yields the most accurate representation of small-scale turbulence possible. The Navier-Stokes equations are given by$ρ[∂u∂t+(u⋅∇)u]=−∇P+μ∇2u+f$(3)where u is the fluid velocity; P is the pressure; μ is the dynamic fluid viscosity; f is a zero-mean, temporally uncorrelated Gaussian forcing to sustain statistically steady turbulence; and flow is incompressible such that · u = 0. The computational domain is a cubic box with triply periodic boundary conditions. The solution of Eq. 3 is time-advanced using a second-order Adams-Bashforth scheme, while the Poisson equation for the pressure is solved with a spectral method. The unsteady forcing is spread over a narrow shell of small wave numbers to generate large-scale flows. To eliminate aliasing errors, we used the 2/3 dealiasing technique, which sets the largest 1/3 of all wave numbers to zero after each computation of the nonlinear terms in the Navier-Stokes equations such that the largest resolved wave number is $kmax=13M1/3$. We used a Fourier pseudospectral code to simulate the flow using M = 1283 mesh points, which ensures that the smallest eddies in the flow are adequately resolved (kmaxηK ≤ 1).

At steady state, the average rate of kinetic energy dissipation per unit volume, ε, is equal to the average rate of energy injected into the system through the large-scale forcing. The total dissipation rate can be directly calculated by integrating the dissipation spectrum D(k) over the entire range of wave numbers(4)where E(k) is the energy spectrum.

A measure of the diversity of length scales within the turbulent flow is given by the Taylor Reynolds number, Reλ = urmsλ/ν, where urms is the root mean square flow velocity, ν is the kinematic viscosity, and $λ=15νurms2/ε$ is the Taylor length scale. Our simulations used Reλ = 62, which previous work has shown to be sufficient to resolve the interactions of gyrotactic swimmers with turbulent flow (20).

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