Pseudo-spectral Direct Numerical Simulation Research Articles

A second-order-in-time, explicit approach addressing the redundancy in the low-Mach, variable-density Navier-Stokes equations

A novel algorithm for explicit temporal discretization of the variable-density, low-Mach Navier-Stokes equations is presented here. Recognizing there is a redundancy between the mass conservation equation, the equation of state, and the transport equation(s) for the scalar(s) which characterize the thermochemical state, and that it destabilizes explicit methods, we demonstrate how to analytically eliminate the redundancy and propose an iterative scheme to solve the resulting transformed scalar equations. The method obtains second-order accuracy in time regardless of the number of iterations, so one can terminate this subproblem once stability is achieved. Hence, flows with larger density ratios can be simulated while still retaining the efficiency, low cost, and parallelizability of an explicit scheme. The temporal discretization algorithm is used within a pseudospectral direct numerical simulation which extends the method of Kim, Moin, and Moser for incompressible flow [17] to the variable-density, low-Mach setting, where we demonstrate stability for density ratios up to ∼25.7.

Novel turbulence and coarsening arrest in active-scalar fluids.

We uncover a new type of turbulence - activity-induced homogeneous and isotropic turbulence - in a model that has been employed to investigate motility-induced phase separation (MIPS) in a system of microswimmers. The active Cahn-Hilliard-Navier-Stokes (CHNS) equations, also called active model H, provide a natural theoretical framework for our study. In this CHNS model, a single scalar order parameter ϕ, positive (negative) in regions of high (low) microswimmer density, is coupled with the velocity field u. The activity of the microswimmers is governed by an activity parameter ζ that is positive for extensile swimmers and negative for contractile swimmers. With extensile swimmers, this system undergoes complete phase separation, which is similar to that in binary-fluid mixtures. By carrying out pseudospectral direct numerical simulations (DNSs), we show, for the first time, that (a) this model develops an emergent nonequilibrium, but statistically steady, state (NESS) of active turbulence, for the case of contractile swimmers, if ζ is sufficiently large and negative, and (b) this turbulence arrests the phase separation. We quantify this suppression by showing how the coarsening-arrest length scale does not grow indefinitely, with time t, but saturates at a finite value at large times. We characterise the statistical properties of this active-scalar turbulence by employing energy spectra and fluxes and the spectrum of ϕ. For sufficiently high Reynolds numbers, the energy spectrum (k) displays an inertial range, with a power-law dependence on the wavenumber k. We demonstrate that, in this range, the flux Π(k) assumes a nearly constant, negative value, which indicates that the system shows an inverse cascade of energy, even though energy injection occurs over a wide range of wavenumbers in our active-CHNS model.

CCN activation in homogeneous isotropic turbulence: Response to particle characteristics and environmental conditions

Direct Numerical Simulation (DNS) of turbulent cloud parcels are presented where particles evolve in response to the local values in supersaturation (s) led by turbulent fluctuations. A pseudo-spectral DNS is modified to incorporate aerosol particles and Cloud Condensation Nuclei (CCN) activation applying a droplet growth equation based on κ-Köhler theory that works for the whole range of warm cloud particles, from un-activated deliquesced aerosol particles to activated cloud droplets, that grow by condensation. The Lagrangian microphysics applied ensures that each particle experiences a supersaturation (fluctuations added to the mean) corresponding to its near-particle gas phase surroundings, which differs from the uniform parcel view where an average value of supersaturation, s¯, is applied to every particle in the domain. A non-turbulent Lagrangian parcel model with similar mean thermodynamics and CCN properties is compared with turbulent DNS parcels of varying fluctuation intensities to distinguish the impact of turbulence on CCN activation. It is shown that, in a given distribution, a subset of aerosols respond to fluctuations rather than the mean thermodynamics allowing particles of similar dry size to co-exist as un-activated haze particles as well as cloud droplets. The cloud microphysical properties are analysed in a series of DNS experiments with contrasting thermodynamic and aerosol properties. DNS results agrees with the general understanding of aerosol activation and cloud droplets evolution in response to various background conditions such as pristine and polluted aerosol distributions, composition of CCN with respect to organic - inorganic components and magnitude of vertical velocity. DNS considers turbulence-microphysics interactions in such problems providing additional knowledge on the role of turbulence in clouds. It is argued that incorporating turbulent fluctuations in simulating the CCN activation and droplet growth is well reasoned and a constructive way forward.

Coherent structure modification by a shear acting at the surface of a turbulent open channel

Recent experiments in the river Seine have revealed the presence of persistent large rolls. This finding has motivated the present numerical study of a turbulent open channel flow. Our pseudo-spectral direct numerical simulations are aimed at understanding how the shear at the surface, for instance produced by an external wind, can change vortical structures in the bulk. Simulations are run at a Reynolds number in the range [5times 10^3div 10^4], in line with similar laboratory experiments. We investigate how flow structures located near the bottom wall, near the surface or inside the bulk are modified by a surface shear. Statistical signatures are extracted through Fourier analysis of the simulated fields, and typical wavelengths of vortices or streaks are computed. Moreover, instantaneous fields are used to show the existence of upwelling and downwelling motions. These motions can be hindered by the presence of a large enough surface shear. This condition turns out to be necessary for the existence of streaks at the surface as well. Our investigation clarifies the dynamics of vortices in an open channel flow at moderate Reynolds number, indicating that unsteady vortex structures are indeed present, but the existence of long coherent rolls cannot be accounted for by the sole presence of shear at the surface. Numerical studies with a much longer domain and possibly at higher Reynolds numbers are needed to provide a firm answer to the question.

Inertial particles in superfluid turbulence: Coflow and counterflow

We use pseudospectral direct numerical simulations to solve the three-dimensional (3D) Hall–Vinen–Bekharevich–Khalatnikov (HVBK) model of superfluid helium. We then explore the statistical properties of inertial particles, in both coflow and counterflow superfluid turbulence (ST) in the 3D HVBK system; particle motion is governed by a generalization of the Maxey–Riley–Gatignol equations. We first characterize the anisotropy of counterflow ST by showing that there exist large vortical columns. The light particles show confined motion as they are attracted toward these columns, and they form large clusters; by contrast, heavy particles are expelled from these vortical regions. We characterize the statistics of such inertial particles in 3D HVBK ST: (1) The mean angle Θ(τ) between particle positions, separated by the time lag τ, exhibits two different scaling regions in (a) dissipation and (b) inertial ranges, for different values of the parameters in our model; in particular, the value of Θ(τ), at large τ, depends on the magnitude of Uns. (2) The irreversibility of 3D HVBK turbulence is quantified by computing the statistics of energy increments for inertial particles. (3) The probability distribution function (PDF) of energy increments is of direct relevance to recent experimental studies of irreversibility in superfluid turbulence; we find, in agreement with these experiments, that, for counterflow ST, the skewness of this PDF is less pronounced than its counterparts for coflow ST or for classical fluid turbulence.

Statistical properties of three-dimensional Hall magnetohydrodynamics turbulence

The three-dimensional (3D) Hall magnetohydrodynamics (HMHD) equations are often used to study turbulence in the solar wind. Some earlier studies have investigated the statistical properties of 3D HMHD turbulence by using simple shell models or pseudospectral direct numerical simulations (DNSs) of the 3D HMHD equations; these DNSs have been restricted to modest spatial resolutions and have covered a limited parameter range. To explore the dependence of 3D HMHD turbulence on the Reynolds number Re and the ion-inertial scale di, we have carried out detailed pseudospectral DNSs of the 3D HMHD equations and their counterparts for 3D MHD (di = 0). We present several statistical properties of 3D HMHD turbulence, which we compare with 3D MHD turbulence by calculating (a) the temporal evolution of the energy-dissipation rates and the energy; (b) the wave-number dependence of fluid and magnetic spectra; (c) the probability distribution functions of the cosines of the angles between various pairs of vectors, such as the velocity and the magnetic field; and (d) various measures of the intermittency in 3D HMHD and 3D MHD turbulence.

Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation.

We develop a Fourier-Chebyshev pseudospectral direct numerical simulation (DNS) to examine a potentially singular solution of the radially bounded, three-dimensional (3D), axisymmetric Euler equations[G. Luo and T.Y. Hou, Proc. Natl. Acad. Sci. USA 111, 12968 (2014)0027-842410.1073/pnas.1405238111]. We demonstrate that (a) the time of singularity is preceded, in any spectrally truncated DNS, by the formation of oscillatory structures called tygers, first investigated in the one-dimensional (1D) Burgers and two-dimensional (2D) Euler equations; (b) the analyticity-strip method can be generalized to obtain an estimate for the (potential) singularity time.

Statistical mechanics of the Euler equations without vortex stretching

We consider the relaxation to thermal equilibrium of the Galerkin-truncated Euler equations in three dimensions, from which vortex stretching is removed. We prove that helicity and enstrophy are conserved by the system. Using statistical mechanics, we derive analytical predictions for the equilibrium energy and helicity spectra. Results are verified using pseudo-spectral direct numerical simulations. Results show that if the initial condition contains helicity, the system relaxes to a force-free large-scale structure akin to an Arnold–Beltrami–Childress (ABC) flow.

Machine learning strategies for path-planning microswimmers in turbulent flows.

We develop an adversarial-reinforcement learning scheme for microswimmers in statistically homogeneous and isotropic turbulent fluid flows, in both two and three dimensions. We show that this scheme allows microswimmers to find nontrivial paths, which enable them to reach a target on average in less time than a naïve microswimmer, which tries, at any instant of time and at a given position in space, to swim in the direction of the target. We use pseudospectral direct numerical simulations of the two- and three-dimensional (incompressible) Navier-Stokes equations to obtain the turbulent flows. We then introduce passive microswimmers that try to swim along a given direction in these flows; the microswimmers do not affect the flow, but they are advected by it. Two nondimensional control parameters play important roles in our learning scheme: (a) the ratio V[over ̃]_{s} of the microswimmer's bare velocity V_{s} and the root-mean-square (rms) velocity u_{rms} of the turbulent fluid and (b) the product B[over ̃] of the microswimmer-response time B and the rms vorticity ω_{rms} of the fluid. We show that the average time required for the microswimmers to reach the target, by using our adversarial-reinforcement learning scheme, eventually reduces below the average time taken by microswimmers that follow the naïve strategy.

One-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: Limit distributions.

Tracy-Widom and Baik-Rains distributions appear as universal limit distributions for height fluctuations in the one-dimensional Kardar-Parisi-Zhang (KPZ) stochastic partial differential equation (PDE). We obtain the same universal distributions in the spatiotemporally chaotic, nonequilibrium, but statistically steady state of the one-dimensional Kuramoto-Sivashinsky (KS) deterministic PDE, by carrying out extensive pseudospectral direct numerical simulations to obtain the spatiotemporal evolution of the KS height profile h(x,t) for different initial conditions. We establish, therefore, that the statistical properties of the one-dimensional (1D) KS PDE in this state are in the 1D KPZ universality class.

Direct Numerical Simulation of a Warm Cloud Top Model Interface: Impact of the Transient Mixing on Different Droplet Population

Turbulent mixing through atmospheric cloud and clear air interface plays an important role in the life of a cloud. Entrainment and detrainment of clear air and cloudy volume result in mixing across the interface, which broadens the cloud droplet spectrum. In this study, we simulate the transient evolution of a turbulent cloud top interface with three initial mono-disperse cloud droplet population, using a pseudo-spectral Direct Numerical Simulation (DNS) along with Lagrangian droplet equations, including collision and coalescence. Transient evolution of in-cloud turbulent kinetic energy (TKE), density of water vapour and temperature is carried out as an initial value problem exhibiting transient decay. Mixing in between the clear air and cloudy volume produced turbulent fluctuations in the density of water vapour and temperature, resulting in supersaturation fluctuations. Small scale turbulence, local supersaturation conditions and gravitational forces have different weights on the droplet population depending on their sizes. Larger droplet populations, with initial 25 and 18 μ m radii, show significant growth by droplet-droplet collision and a higher rate of gravitational sedimentation. However, the smaller droplets, with an initial 6 μ m radius, did not show any collision but a large size distribution broadening due to differential condensation/evaporation induced by the mixing, without being influenced by gravity significantly.

Acceleration statistics of prolate spheroidal particles in turbulent channel flow

ABSTRACTComputation of a dilute suspension of prolate spheroidal particles in a turbulent channel flow is undertaken to study the influence of inertia and shape on acceleration statistics. A pseudo-spectral direct numerical simulation is coupled with Lagrangian tracking under the one-way coupling assumption. Simulations are carried out at friction Reynolds number , for three aspect ratios , 3 and 10, and two Stokes numbers St=5 and 30. Results indicate that, as a consequence of the filtering effect of inertia, particle acceleration RMS decreases with increasing inertia. In addition, the normalised streamwise acceleration PDFs depart from that of the conditional fluid and their tails become narrower as inertia is increased. Furthermore, acceleration statistics show that particle elongation has a significant effect on the mean drag and on the particle selective sampling. Moreover, the classification of elongated particle behaviour in turbulent channel flow based on a global Stokes number is questioned. The zero-crossing acceleration autocorrelation timescales presented points out the need for a local dimensionless number estimation adapted to the case of prolate spheroids.

Local and nonlocal energy spectra of superfluidHe3turbulence

Below the phase transition temperature $Tc \simeq 10^{-3}$K He-3B has a mixture of normal and superfluid components. Turbulence in this material is carried predominantly by the superfluid component. We explore the statistical properties of this quantum turbulence, stressing the differences from the better known classical counterpart. To this aim we study the time-honored Hall-Vinen-Bekarevich-Khalatnikov coarse-grained equations of superfluid turbulence. We combine pseudo-spectral direct numerical simulations with analytic considerations based on an integral closure for the energy flux. We avoid the assumption of locality of the energy transfer which was used previously in both analytic and numerical studies of the superfluid He-3B turbulence. For T<0.37 Tc, with relatively weak mutual friction, we confirm the previously found "subcritical" energy spectrum E(k), given by a superposition of two power laws that can be approximated as $E(k)~ k^{-x}$ with an apparent scaling exponent 5/3 <x(k)< 3. For T>0.37 Tc and with strong mutual friction, we observed numerically and confirmed analytically the scale-invariant spectrum $E(k)~ k^{-x}$ with a (k-independent) exponent x > 3 that gradually increases with the temperature and reaches a value $x\simeq 9$ for $T\approx 0.72 Tc$. In the near-critical regimes we discover a strong enhancement of intermittency which exceeds by an order of magnitude the corresponding level in classical hydrodynamic turbulence.

High performance Python for direct numerical simulations of turbulent flows

Direct Numerical Simulations (DNS) of the Navier Stokes equations is an invaluable research tool in fluid dynamics. Still, there are few publicly available research codes and, due to the heavy number crunching implied, available codes are usually written in low-level languages such as C/C++ or Fortran. In this paper we describe a pure scientific Python pseudo-spectral DNS code that nearly matches the performance of C++ for thousands of processors and billions of unknowns. We also describe a version optimized through Cython, that is found to match the speed of C++. The solvers are written from scratch in Python, both the mesh, the MPI domain decomposition, and the temporal integrators. The solvers have been verified and benchmarked on the Shaheen supercomputer at the KAUST supercomputing laboratory, and we are able to show very good scaling up to several thousand cores.A very important part of the implementation is the mesh decomposition (we implement both slab and pencil decompositions) and 3D parallel Fast Fourier Transforms (FFT). The mesh decomposition and FFT routines have been implemented in Python using serial FFT routines (either NumPy, pyFFTW or any other serial FFT module), NumPy array manipulations and with MPI communications handled by MPI for Python (mpi4py). We show how we are able to execute a 3D parallel FFT in Python for a slab mesh decomposition using 4 lines of compact Python code, for which the parallel performance on Shaheen is found to be slightly better than similar routines provided through the FFTW library. For a pencil mesh decomposition 7 lines of code is required to execute a transform.

Direct numerical simulation of turbulent channel flow over porous walls

We perform direct numerical simulations (DNS) of a turbulent channel flow over porous walls. In the fluid region the flow is governed by the incompressible Navier–Stokes (NS) equations, while in the porous layers the volume-averaged Navier–Stokes (VANS) equations are used, which are obtained by volume-averaging the microscopic flow field over a small volume that is larger than the typical dimensions of the pores. In this way the porous medium has a continuum description, and can be specified without the need of a detailed knowledge of the pore microstructure by independently assigning permeability and porosity. At the interface between the porous material and the fluid region, momentum-transfer conditions are applied, in which an available coefficient related to the unknown structure of the interface can be used as an error estimate. To set up the numerical problem, the velocity–vorticity formulation of the coupled NS and VANS equations is derived and implemented in a pseudo-spectral DNS solver. Most of the simulations are carried out at$Re_{{\it\tau}}=180$and consider low-permeability materials; a parameter study is used to describe the role played by permeability, porosity, thickness of the porous material, and the coefficient of the momentum-transfer interface conditions. Among them permeability, even when very small, is shown to play a major role in determining the response of the channel flow to the permeable wall. Turbulence statistics and instantaneous flow fields, in comparative form to the flow over a smooth impermeable wall, are used to understand the main changes introduced by the porous material. A simulation at higher Reynolds number is used to illustrate the main scaling quantities.

Turbulence modification by dispersion of large deformable droplets

Dispersion of large deformable droplets in turbulent channel flow is investigated: the fluids are considered of same density and viscosity for a broad range of plausible values of surface tension. Droplets much larger than the Kolmogorov length scale are released in a channel flow; the turbulent flow is computed with pseudo-spectral Direct Numerical Simulations (DNS), while phase interactions are described with a Phase Field Model (PFM). It is shown that droplets segregation toward the center of the channel is promoted by deformability in particular, for the considered Weber numbers (ratio between inertia and surface tension), droplets do not deposit on the walls. The turbulent flow is modified by the presence of the droplets: for small values of We the wall drag is increased with respect to the single phase flow. The wall drag enhancement reduces increasing We and its time evolution depends on the droplets coalescence and breakup rates.

Coalescence and breakup of large droplets in turbulent channel flow

Coalescence and breakup of large deformable droplets dispersed in a wall-bounded turbulent flow are investigated. Droplets much larger than the Kolmogorov length scale and characterized by a broad range of surface tension values are considered. The turbulent field is a channel flow computed with pseudo-spectral direct numerical simulations, while phase interactions are described with a phase field model. Within this physically consistent framework, the motion of the interfaces, the capillary effects, and the complex topological changes experienced by the droplets are simulated in detail. An oil-water emulsion is mimicked: the fluids are considered of same density and viscosity for a range of plausible values of surface tension, resulting in a simplified system that sets a benchmark for further analysis. In the present conditions, the Weber number (We), that is, the ratio between inertia and surface tension, is a primary factor for determining the droplets coalescence rate and the occurrence of breakups. Depending on the value of We, two different regimes are observed: when We is smaller than a threshold value (We &amp;lt; 1 in our simulations), coalescence dominates until droplet-droplet interactions are prevented by geometric separation; when We is larger than the threshold value (We &amp;gt; 1), a permanent dynamic equilibrium between coalescence and breakup events is established.

Turbulent breakage of ductile aggregates.

In this paper we study breakage rate statistics of small colloidal aggregates in nonhomogeneous anisotropic turbulence. We use pseudospectral direct numerical simulation of turbulent channel flow and Lagrangian tracking to follow the motion of the aggregates, modeled as sub-Kolmogorov massless particles. We focus specifically on the effects produced by ductile rupture: This rupture is initially activated when fluctuating hydrodynamic stresses exceed a critical value, σ>σ(cr), and is brought to completion when the energy absorbed by the aggregate meets the critical breakage value. We show that ductile rupture breakage rates are significantly reduced with respect to the case of instantaneous brittle rupture (i.e., breakage occurs as soon as σ>σ(cr)). These discrepancies are due to the different energy values at play as well as to the statistical features of energy distribution in the anisotropic turbulence case examined.

Nonlinear dynamics and anisotropic structure of rotating sheared turbulence

Homogeneous turbulence in rotating shear flows is studied by means of pseudospectral direct numerical simulation and analytical spectral linear theory (SLT). The ratio of the Coriolis parameter to shear rate is varied over a wide range by changing the rotation strength, while a constant moderate shear rate is used to enable significant contributions to the nonlinear interscale energy transfer and to the nonlinear intercomponental redistribution terms. In the destabilized and neutral cases, in the sense of kinetic energy evolution, nonlinearity cannot saturate the growth of the largest scales. It permits the smallest scale to stabilize by a scale-by-scale quasibalance between the nonlinear energy transfer and the dissipation spectrum. In the stabilized cases, the role of rotation is mainly nonlinear, and interacting inertial waves can affect almost all scales as in purely rotating flows. In order to isolate the nonlinear effect of rotation, the two-dimensional manifold with vanishing spanwise wave number is revisited and both two-component spectra and single-point two-dimensional energy components exhibit an important effect of rotation, whereas the SLT as well as the purely two-dimensional nonlinear analysis are unaffected by rotation as stated by the Proudman theorem. The other two-dimensional manifold with vanishing streamwise wave number is analyzed with similar tools because it is essential for any shear flow. Finally, the spectral approach is used to disentangle, in an analytical way, the linear and nonlinear terms in the dynamical equations.

Anisotropy in pair dispersion of inertial particles in turbulent channel flow

The rate at which two particles separate in turbulent flows is of central importance to predict the inhomogeneities of particle spatial distribution and to characterize mixing. Pair separation is analyzed for the specific case of small, inertial particles in turbulent channel flow to examine the role of mean shear and small-scale turbulent velocity fluctuations. To this aim an Eulerian-Lagrangian approach based on pseudo-spectral direct numerical simulation (DNS) of fully developed gas-solid flow at shear Reynolds number Reτ = 150 is used. Pair separation statistics have been computed for particles with different inertia (and for inertialess tracers) released from different regions of the channel. Results confirm that shear-induced effects predominate when the pair separation distance becomes comparable to the largest scale of the flow. Results also reveal the fundamental role played by particles-turbulence interaction at the small scales in triggering separation during the initial stages of pair dispersion. These findings are discussed examining Lagrangian observables, including the mean square separation, which provide prima facie evidence that pair dispersion in non-homogeneous anisotropic turbulence has a superdiffusive nature and may generate non-Gaussian number density distributions of both particles and tracers. These features appear to persist even when the effects of shear dispersion are filtered out, and exhibit strong dependency on particle inertia. Application of present results is discussed in the context of modelling approaches for particle dispersion in wall-bounded turbulent flows.