Direct Numerical Simulation (DNS) of turbulent flow

This protocol is extracted from research article:

Chain formation can enhance the vertical migration of phytoplankton through turbulence

**
Sci Adv**,
Oct 16, 2019;
DOI:
10.1126/sciadv.aaw7879

Chain formation can enhance the vertical migration of phytoplankton through turbulence

Procedure

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$$\mathrm{\rho}[\frac{\mathrm{\partial}\mathbf{u}}{\mathrm{\partial}t}+(\mathbf{u}\cdot \nabla )\mathbf{u}]=-\nabla P+\mathrm{\mu}{\nabla}^{2}\mathbf{u}+\mathbf{f}$$(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 ${k}_{\text{max}}=\frac{1}{3}{M}^{1/3}$. We used a Fourier pseudospectral code to simulate the flow using *M* = 128^{3} mesh points, which ensures that the smallest eddies in the flow are adequately resolved (*k*_{max}η_{K} ≤ 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$$\mathrm{\epsilon}={\displaystyle {\int}_{0}^{{k}_{\text{max}}}}D(k)\mathit{dk}=2\mathrm{\nu}{\displaystyle {\int}_{0}^{{k}_{\text{max}}}}{k}^{2}E(k)\mathit{dk}$$(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, *Re*_{λ} = *u*_{rms}λ/ν, where *u*_{rms} is the root mean square flow velocity, ν is the kinematic viscosity, and $\mathrm{\lambda}=\sqrt{15\mathrm{\nu}{u}_{\text{rms}}^{2}/\mathrm{\epsilon}}$ is the Taylor length scale. Our simulations used *Re*_{λ} = 62, which previous work has shown to be sufficient to resolve the interactions of gyrotactic swimmers with turbulent flow (*20*).

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