Stratified turbulence

We describe a method for quantifying the effective numerical dissipation rate and the effective numerical viscosity in Computational Fluid Dynamics simulations. Differently from the previous approach that was formulated in spectral space, the proposed method is developed in a physical-space representation and allows for determining numerical dissipation rates and viscosities locally, i.e., at the individual cell level or for arbitrary subdomains of the computational domain. The method is self-contained using only results produced by the Navier-Stokes solver being investigated. Since no extraneous information is required, the method is suitable for a straightforward quantification of the numerical dissipation as a post-processing step. We demonstrate the method's capabilities on the example of implicit large-eddy simulations of three-dimensional Taylor-Green vortex flows that exhibit laminar, transitional, and turbulent flow behavior at different stages of time evolution. For validation, we compare the numerical dissipation rate obtained using this method with exact reference data obtained with an accurate, spectral-space approach.

We use direct numerical simulation (DNS) to study drag reduction in a lubricated channel, a flow instance in which a thin layer of lubricating fluid is injected in the near-wall region so as to favour the transportation of a primary fluid. In the present configuration, the two fluids have equal density but different viscosity, so that a viscosity ratio λ = η 1 /η 2 can be defined. To cover a meaningful range of possible situations, we consider five different λ in the range 0.25 ≤ λ ≤ 4. All DNS are run using the constant power input (CPI) approach, which prescribes that the flow rate is adjusted according to the actual pressure gradient so as to keep constant the power injected into the flow. The CPI approach has been purposely extended here for the first time to the case of multiphase flows. A phase-field method is used to describe the dynamics of the liquid-liquid interface. We unambiguously show that a significant drag reduction (DR) can be achieved for λ ≤ 2.00. Reportedly, the observed DR is a non-monotonic function of λ and, in the present case, is maximum for λ = 1.00 (13 % flow-rate increase). Upon a detailed analysis of the energy budgets, we are able to show the existence of two different DR mechanisms. For λ = 1.00 and λ = 2.00, DR is purely due to the effect of the surface tension-a localized elasticity element that separates the two fluids-which, decoupling the wall-normal momentum transfer mechanisms between the primary and the lubricating layer, suppresses turbulence in the lubricating layer (laminarization) and reduces the overall drag. For λ < 1.00, turbulence can be sustained in the lubricating layer, because of the increased local Reynolds number. In this case, DR is simply due to the smaller viscosity of the lubricating layer that acts to decrease directly the corresponding wall friction. Finally, we show evidence that an upper bound for λ exists, for which DR cannot be observed: for λ = 4.00, we report a slight drag enhancement, thereby indicating that the turbulence suppression observed in

A 2D numerical simulation of flow past rectangular cylinders has been carried out in a stratified fluid using lattice Boltzmann methods (LBM). Stratification is achieved using a conventional two-component LBM model. Simulations correspond to a range of Reynolds numbers ...

Submitted for the DFD15 Meeting of The American Physical Society Energy spectrum of stably-stratified and convective turbulent flows MAHENDRA VERMA, ABHISHEK KUMAR, IIT Kanpur, India-In the inertial range of fluid turbulence, the energy flux is constant, while the energy spectrum scales as k −5/3 (k=wavenumber). The buoyancy however could change the phenomenology dramatically. Bolgiano and Obukhov (1959) had conjectured that stably stratified flows (as in atmosphere) exhibits a decrease in the energy flux as k −4/5 due to the conversion of kinetic energy to the potential energy, consequently, the energy spectrum scales as k −11/5. We show using detailed numerical analysis that the stably stratified flows indeed exhibit k −11/5 energy spectrum for Froude numbers Fr near unity. The flow becomes anisotropic for small Froude numbers. For weaker buoyancy (large Fr), the kinetic energy follows Kolmogorov's spectrum with a constant energy flux. However, in convective turbulence, the energy flux is a nondecreasing function of wavenumber since the buoyancy feeds positively into the kinetic energy. Hence, the kinetic energy spectrum is Kolmogorov-like (k −5/3) or shallower. 1 We also demonstrate the above scaling using a shell model of buoyancydriven turbulence. 2

We present a new adaptive control strategy to isolate and stabilize turbulent states in transitional, stably-stratified plane Couette flow in which the gravitational acceleration (non-dimensionalised as the bulk Richardson number Ri) is adjusted in time to maintain the turbulent kinetic energy (TKE) of the flow. We demonstrate that applying this method at various stages of decaying stratified turbulence halts the decay process and allows a succession of intermediate turbulent states of decreasing energy to be isolated and stabilized. Once the energy of the initial flow becomes small enough, we identify a single minimal turbulent spot, and lower energy states decay to laminar flow. Interestingly, the turbulent states which emerge from this process have very similar time-averaged Ri, but TKE levels di↵erent by an order of magnitude. The more energetic states consist of several turbulent spots, each qualitatively similar to the minimal turbulent spot. This suggests that the minimal turbulent spot may well be the lowest energy turbulent state which forms a basic building block of stratified plane Couette flow. The fact that a minimal spot of turbulence can be stabilized so that it neither decays nor grows opens up exciting opportunities for further study of spatiotemporally intermittent stratified turbulence.

We present a new adaptive control strategy to isolate and stabilize turbulent states in transitional, stably-stratified plane Couette flow in which the gravitational acceleration (non-dimensionalised as the bulk Richardson number Ri) is adjusted in time to maintain the turbulent kinetic energy (TKE) of the flow. We demonstrate that applying this method at various stages of decaying stratified turbulence halts the decay process and allows a succession of intermediate turbulent states of decreasing energy to be isolated and stabilized. Once the energy of the initial flow becomes small enough, we identify a single minimal turbulent spot, and lower energy states decay to laminar flow. Interestingly, the turbulent states which emerge from this process have very similar time-averaged Ri, but TKE levels di↵erent by an order of magnitude. The more energetic states consist of several turbulent spots, each qualitatively similar to the minimal turbulent spot. This suggests that the minimal turbulent spot may well be the lowest energy turbulent state which forms a basic building block of stratified plane Couette flow. The fact that a minimal spot of turbulence can be stabilized so that it neither decays nor grows opens up exciting opportunities for further study of spatiotemporally intermittent stratified turbulence.

This dataset contains HDF5 files with three-dimensional flow fields from several key times in the reported model simulations. The initial condition is provided, and the code is open access, allowing the full time-dependent model simulations to be reproduced. In addition, several 3D fields from the subsequent model evolution are included which were used to make figures in the published manuscript. The HDF5 file format is self-describing with header metadata and is readable in a wide variety of available software packages.

Turbulence in a stratified fluid is a fundamental process in the atmosphere and oceans, responsible for mixing density and various tracers and dissipating kinetic and potential energy. Although turbulence is generally suppressed in very statically stable conditions, intermittent bursts of turbulence are still seen when the Reynolds number is sufficiently large. In this work, we study stratified turbulence in plane Couette flow using direct numerical simulations, focusing on the complexity arising from the spatio-temporal intermittency of the flow as the stabilizing stratification increases. Two external dimensionless parameters control the dynamics: the Reynolds number Re and the bulk Richardson number Ri b. We trace the boundary between laminar and turbulent states in the Re-Ri b plane and discuss the relevant dynamical quantities involved in the relaminarization process. We analyze the structures populating the intermittent regime and the coexistence between laminar and turbulent patches, focusing on similarities and differences between small-Re-small-Ri b and large-Re-large-Ri b intermittent dynamics. We conclude by discussing the applicability and breakdown of existing stratified turbulence theories, including the Monin-Obukhov self-similarity theory.

We present a new adaptive control strategy to isolate and stabilize turbulent states in transitional, stably stratified plane Couette flow in which the gravitational acceleration (non-dimensionalized as the bulk Richardson number$Ri$) is adjusted in time to maintain the turbulent kinetic energy (TKE) of the flow. We demonstrate that applying this method at various stages of decaying stratified turbulence halts the decay process and allows a succession of intermediate turbulent states of decreasing energy to be isolated and stabilized. Once the energy of the initial flow becomes small enough, we identify a single minimal turbulent spot, and lower-energy states decay to laminar flow. Interestingly, the turbulent states which emerge from this process have very similar time-averaged$Ri$, but TKE levels different by an order of magnitude. The more energetic states consist of several turbulent spots, each qualitatively similar to the minimal turbulent spot. This suggests that the minimal ...

The present study evaluates the prevalence of Holmboe waves in an intrusive gravity current (IGC) containing particles, employing large Eddy simulation (LES). Holmboe waves, a type of stratified shear layer-generated wave, are characterised by a relatively thin density interface compared to the thickness of the shear layer. The study demonstrates the occurrence of secondary rotation, wave stretching over time, and fluid ejection at the interface between the IGC and a lower gravity current (LGC). Results indicate that, aside from J and R, the density difference between the IGC and the LGC has an impact on Holmboe instability. However, a reduction in the density difference does not manifest consistently in the frequency, growth rate, and phase speed, though it does cause an increase in the wavelength. It is important to note that small particles do not affect the Holmboe instability of the IGC, while larger particles cause the current to become unstable and vary the characteristics of Holmboe instability. Moreover, an increase in the particle diameter size results in an increment in the wavelength, growth rate, and phase speed; but is accompanied by a decrease in frequency. Additionally, the enlargement of the bed slope angle makes the IGC more unstable, encouraging the growth of Kelvin-Helmholtz waves; however, this causes Holmboe waves to disappear on inclined beds. Finally, a range for the instabilities of both Kelvin-Helmholtz and Holmboe is provided.

In this work, a spectral model is derived to investigate numerically unstably stratified homogeneous turbulence (USHT) at large Reynolds numbers. The modeling relies on an earlier work for passive scalar dynamics [Briard et al., J. Fluid Mech. 799, 159 (2016)] and can handle both shear and mean scalar gradients. The extension of this model to the case of active scalar dynamics is the main theoretical contribution of this paper. This spectral modeling is then applied at large Reynolds numbers to analyze the scaling of the kinetic energy, scalar variance, and scalar flux spectra and to study as well the temporal evolution of the mixing parameter, the Froude number, and some anisotropy indicators in USHT. A theoretical prediction for the exponential growth rate of the kinetic energy, associated with our model equations, is derived and assessed numerically. Throughout the validation part, results are compared with an analogous approach, restricted to axisymmetric turbulence, which is more accurate in term of anisotropy description, but also much more costly in terms of computational resources [Burlot et al., J. Fluid Mech. 765, 17 (2015)]. It is notably shown that our model can qualitatively recover all the features of the USHT dynamics, with good quantitative agreement on some specific aspects. In addition, some remarks are proposed to point out the similarities and differences between the physics of USHT, shear flows, and passive scalar dynamics with a mean gradient, the two latter configurations having been addressed previously with the same closure. Moreover, it is shown that the anisotropic part of the pressure spectrum in USHT scales in k −11/3 in the inertial range, similarly to the one in shear flows. Finally, at large Schmidt numbers, a different spectral range is found for the scalar flux: It first scales in k −3 around the Kolmogorov scale and then further in k −1 in the viscous-convective range.

We present three-dimensional (3D) numerical simulations of the pairing of two vertical columnar vortices in a stably stratified fluid. Whereas in two dimensions, merging of two isolated vortices occurs on a diffusion time scale, in the three-dimensional stratified case we show that merging is a much faster process that occurs over an inertial time scale. The sequence of dynamical processes that leads to this accelerated pairing involves first a linear stage where the zigzag instability develops displacing vortices alternately closer and farther with a vertical periodicity scaling on the buoyancy length scale L B = F h b, where F h is the horizontal Froude number (F h = Γ /πa 2 N with a the core size of the vortices, Γ their circulation and N the Brunt-Väisälä frequency) and b is the separation distance between the vortices. In layers where the vortices have started to move closer, their distance decreases exponentially with the growth rate of the zigzag instability. Non-linearities do not seem to affect this process and the decrease only stops when the pairing is completed in that layer. At the same time, enstrophy that has also grown exponentially reaches a magnitude of the order of the Reynolds number Re = Γ /(π ν) (where ν is the kinematic viscosity of the fluid) if the Reynolds number is not too large, meaning that energy is then dissipated on the inertial time scale. This dissipation occurs in thin layers and the vortices that were originally moving away in the intermediate layer start slowing down and rapidly merge.

An overview is given on direct numerical simulations and on large eddy simulations of homogeneous turbulence under the impact of shear and stable stratification. We describe the methods used and report on the results of various studies. In particular, the vortex structure of turbulent motions is discussed. Moreover, the dynamics of statistical mean quantities are investigated. The mean variances are compared with experimental data. The turbulent diffusivity tensor for passive species in a stratified shear flow is computed. Moreover, a simple model is described which allows us to estimate the vertical diffusivities for heat and momentum for such flows when the vertical velocity variance or the dissipation rate are known.

A simple spectral model is used to examine what is required to determine the energy and integral scale in homogeneous isotropic turbulence. The problem is that these are determined in part by the largest scales of the turbulence which are either not simulated at all by DNS or experiments, or cannot be estimated because of an insufficient statistical sample. The absence of scales an order of magnitude below the peak in the energy spectrum is shown to affect the determination significantly. Since this energy peak shifts to lower wavenumbers as the flow evolves, the problem becomes progressively worse during decay. It is suggested that almost all reported integral scales for isotropic decaying turbulence are questionable, and that the power laws fitted to them are seriously in error. Approximate correction using the spectral model shows that recent DNS data which decay as u2 ∝ tn with constant n, are also consistent with L ∝ t1/2.

Numerical simulations are made for forced turbulence at a sequence of increasing values of Reynolds number, R, keeping fixed a strongly stable, volume-mean density stratification. At smaller values of R, the turbulent velocity is mainly horizontal, and the momentum balance is approximately cyclostrophic and hydrostatic. This is a regime dominated by so-called pancake vortices, with only a weak excitation of internal gravity waves and large values of the local Richardson number, Ri, everywhere. At higher values of R there are successive transitions to (a) overturning motions with local reversals in the density stratification and small or negative values of Ri; (b) growth of a horizontally uniform vertical shear flow component; and (c) growth of a large-scale vertical flow component. Throughout these transitions, pancake vortices continue to dominate the large-scale part of the turbulence, and the gravity wave component remains weak except at small scales.

Stratified flows, flows where density varies in one direction, have wide applications in some of the phenomena occurring in the atmospheric and ocean. Direct numerical simulations (DNS) of transition to turbulence in a stably stratified Boussinesq fluid are presented for the three-dimensional Taylor-Green vortex problem at different stratification and turbulence intensities measured in terms of different Froude (Fr) ($\infty$ and $10^{-2}-10^{-1}$) and Reynolds numbers (Re) (800 and 1600), respectively. Features investigated include temporal variations of the energy spectrum cascade, local Froude numbers, vertical shearing of the velocities, and dissipation of kinetic and potential energy. The results from these simulations demonstrated forward cascade of energy for high Re and revealed the strong anisotropic structure of turbulence and suppression of vertical motion under stratification.

Stratification due to salt or heat gradients greatly affects the distribution of inert particles and living organisms in the ocean and the lower atmosphere. Laboratory studies considering the settling of a sphere in a linearly stratified fluid confirmed that stratification may dramatically enhance the drag on the body, but failed to identify the generic physical mechanism responsible for this increase. We present a rigorous splitting scheme of the various contributions to the drag on a settling body, which allows them to be properly disentangled whatever the relative magnitude of inertial, viscous, diffusive and buoyancy effects. We apply this splitting procedure to data obtained via direct numerical simulation of the flow past a settling sphere over a range of parameters covering a variety of situations of laboratory and geophysical interest. Contrary to widespread belief, we show that, in the parameter range covered by the simulations, the drag enhancement is generally not primari...

We present numerical, theoretical and experimental studies of a new instability of two corotating vertical vortices in a vertically stratified fluid. This instability induces the formation of thin horizontal layers with a thickness inversely proportional to the Brünt-Väisälä frequency. This three-dimensional instability is believed to make stratified turbulence depart from two-dimensional turbulence since it alters the pairing of vortices.

We investigate the spectral properties of the turbulence generated during the nonlinear evolution of a Lamb–Chaplygin dipole in a stratified fluid for a high Reynolds number $Re= 28\hspace{0.167em} 000$ and a wide range of horizontal Froude number ${F}_{h} \in [0. 0225~0. 135] $ and buoyancy Reynolds number $\mathscr{R}= Re{{F}_{h} }^{2} \in [14~510] $. The numerical simulations use a weak hyperviscosity and are therefore almost direct numerical simulations (DNS). After the nonlinear development of the zigzag instability, both shear and gravitational instabilities develop and lead to a transition to small scales. A spectral analysis shows that this transition is dominated by two kinds of transfer: first, the shear instability induces a direct non-local transfer toward horizontal wavelengths of the order of the buoyancy scale ${L}_{b} = U/ N$, where $U$ is the characteristic horizontal velocity of the dipole and $N$ the Brunt–Väisälä frequency; second, the destabilization of the Kelv...

To improve the understanding of mud suspension in estuaries, a parametric stability analysis of a two-dimensional shear flow was carried out with a model of two miscible fluid layers of different mass density and dynamic viscosity. The direct numerical simulation code JADIM of IMFT (Institut de Mecanique des Fluides de Toulouse) is used to compute the temporal evolution of these flows. This study is completed by a linear stability study realized with the code LiSa developed at IMFT. A critical Richardson number close to 0.25 is observed. Results show that the characteristics of the most unstable modal wave number and the location of the interface instability, depend on the ratio of two fluid viscosity values. The model is initialized with continuous erf(z) vertical profiles for all quantities. Likely outcomes of this stability study are new parameterizations for realistic modelling of estuaries.

Kurzfassung: Turbulence modeling and the numerical discretization of the Navier-Stokes equations are strongly coupled in large-eddy simulations. The truncation error of common approximations for the convective terms can outweigh the effect of a physically sound ...

In this paper we show how the stability of Prandtl boundary layers is linked to the stability of shear flows in the incompressible Navier–Stokes equations. We then recall classical physical instability results, and give a short educational presentation of the construction of unstable modes for Orr–Sommerfeld equations. We end the paper with a conjecture concerning the validity of Prandtl boundary layer asymptotic expansions.

To reduce the computational costs of numerical studies of gravity wave breaking in the atmosphere, the grid resolution has to be reduced as much as possible. Insufficient resolution of small-scale turbulence demands a proper turbulence parameterization in the framework of a large-eddy simulation (LES). The authors validate three different LES methods—the adaptive local deconvolution method (ALDM), the dynamic Smagorinsky method (DSM), and a naïve central discretization without turbulence parameterization (CDS4)—for three different cases of the breaking of well-defined monochromatic gravity waves. For ALDM, a modification of the numerical flux functions is developed that significantly improves the simulation results in the case of a temporarily very smooth velocity field. The test cases include an unstable and a stable inertia–gravity wave as well as an unstable high-frequency gravity wave. All simulations are carried out both in three-dimensional domains and in two-dimensional domai...

Density stratification has a strong impact on turbulence in geophysical flows. Stratification changes the spatial turbulence spectrum and the energy transport and conversion within the spectrum. We analyze these effects based on a series of direct numerical simulations (DNS) of stratified turbulence. To facilitate simulations of real-world problems, which are usually beyond the reach of DNS, we propose a subgrid-scale turbulence model for large eddy simulations of stratified flows based on the Adaptive Local Deconvolution Method (ALDM). Flow spectra and integral quantities predicted by ALDM are in excellent agreement with direct numerical simulation. ALDM automatically adapts to strongly anisotropic turbulence and is thus a suitable tool for studying turbulent flow phenomena in atmosphere and ocean.

The analysis of the spectral eddy viscosity is a handy tool to analyze the performance of LES methods. It reflects the cumulated effect of numerical discretization and turbulence subgrid-scale model on the spectral energy transfer. We compute this quantity by filtering and truncating data from direct numerical simulations of neutrally and stably stratified homogeneous turbulence. The results are used for testing of implicit and explicit LES methods. We find indications that for stably stratified turbulence it is necessary to use different subgrid-scale models for the horizontal and vertical velocity components.

We consider the influence of transverse confinement on the instability properties of velocity and density distributions reminiscent of those pertaining to exchange flows in stratified inclined ducts, such as the recent experiment of Lefauve et al. (J. Fluid Mech. 848, 508-544, 2018). Using a normal mode streamwise and temporal expansion for flows in ducts with various aspect ratios B and non-trivial transverse velocity profiles, we calculate two-dimensional (2D) dispersion relations with associated eigenfunctions varying in the ‘crosswise’ direction, in which the density varies, and the spanwise direction, both normal to the duct walls and to the flow direction. We also compare these 2D dispersion relations to the so-called one-dimensional (1D) dispersion relation obtained for spanwise invariant perturbations, for different aspect ratios B and bulk Richardson numbers Rib. In this limited parameter space, the presence of lateral walls has a stabilizing effect, in that the 1D growth-r...

We investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebio et al. (J. Fluid Mech., vol. 781, 2015, pp. 298–329) expanding the range of Prandtl number $Pr$ examined by two orders of magnitude from 0.7 up to 70. Significant effects of $Pr$ on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile are observed. These effects can be described through a mixing length model coupling Monin–Obukhov (M–O) similarity theory and van Driest damping functions. We then employ M–O theory to formulate similarity scalings for various flow diagnostics for the stratified turbulence in the gap interior. The midchannel gap gradient Richardson number $Ri_{g}$ is determined by the length scale ratio $h/L$ , where $h$ is the half-channel gap depth and $L$ is the Obukhov length scale. As $h/L$ approaches very large values, $Ri_{g}$ asymptotes to a maximum characteristic val...

The performance of commonly used subgrid-scale (SGS) models is evaluated for large-eddy simulation (LES) of turbulent katabatic flow. The very stable stratification and strong low-level shear in this flow provide a stringent test for SGS models. Using an a posteriori testing procedure, the SGS models' performance in reproducing turbulence statistics and spectra in katabatic flow is investigated.

For this study a spatially high-order, shock capturing non-oscillatory finite volume method is combined with a weakly compressible flow modeling. As an alternative to methods based on the incompressibility assumption this weakly compressible high-resolution approach is both robust to underresolution and spatially highly accurate. The implicit subgrid-scale (SGS) model permits physically consistent underresolved simulations of incompressible, isotropic turbulent flows at very high Reynolds numbers. Underresolved three-dimensional Taylor-Green vortex (TGV) simulations at finite Reynolds numbers are compared to reference data. Hereby, direct numerical simulation (DNS) data for Re ≤ 3000 is used to assess the accuracy and physical consistency. Large eddy simulation (LES) predictions with two explicit as well as one implicit SGS model help to benchmark the SGS modeling capabilities. The weakly compressible high-resolution approach gives most accurate predictions for the viscous TGV even when resolution is very low. In contrast to the LES our implicit LES predict the laminar-turbulent transition physically consistently. The dissipation rates compare to those of the reference implicit LES, however, at much lower computational costs and mathematical complexity. As our weakly compressible high-resolution approach is designed for the physically consistent simulation of very high Re turbulent flows, an infite Re TGV is studied for an extended period of time. Thereby, the evolution at times beyond the obviously temporary quasi-isotropic state are of particular interest. For the high and infinite Re TGV flows, transition to the isotropic state is observed. Its onset and end are identifiable from a macroscopic energy redistribution within the low-modes. Subsequently, the inertial subrange scales according to E(k) ∝ k −5/3 and is self-similar in time.

The temporal stability of a parallel shear flow of miscible fluid layers of different density and viscosity is investigated through a linear stability analysis and direct numerical simulations. The geometry and rheology of this Newtonian fluid mixing can be viewed as a simplified model of the behavior of mudflow at the bottom of estuaries for suspension studies. In this study, focus is on the stability and transition to turbulence of an initially laminar configuration. A parametric analysis is performed by varying the values of three control parameters, namely the viscosity ratio, the Richardson and Reynolds numbers, in the case of initially identical thickness of the velocity, density and viscosity profiles. The range of parameters has been chosen so as to mimic a wide variety of real configurations. This study shows that the Kelvin-Helmholtz instability is controlled by the local Reynolds and Richardson numbers of the inflection point. In addition, at moderate Reynolds number, viscosity stratification has a strong influence on the onset of instability, the latter being enhanced at high viscosity ratio, while at high Reynolds number, the influence is less pronounced. In all cases, we show that the thickness of the mixing layer (and thus resuspension) is increased by high viscosity stratification, in particular during the non-linear development of the instability and especially pairing processes. This study suggests that mud viscosity has to be taken into account for resuspension parameterizations because of its impact on the inflection point Reynolds number and the viscosity ratio, which are key parameters for shear instabilities.

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The objective of this paper is the analysis and the control of local truncation errors in Large Eddy Simulations. We show that physical reasoning can be incorporated into the design of discretization schemes. Using systematic procedures, a nonlinear discretization method is developed where numerical and turbulence-theoretical modeling are fully merged. The truncation error itself functions as an implicit turbulence model which accurately represents the effects of unresolved turbulence.

The purpose of this paper is to simulate a two-dimensional Rayleigh-Taylor instability problem using the diffuse-interface formulation of the incompressible Navier-Stokes equations. The governing equations consist of a system of coupled nonlinear partial differential equations for conservation of mass, momentum and phase transport. The Boussinesq approximation is introduced in the momentum equation to relax the complexity of variable density formulation. Due to the simplicity, this approximation can be used for small density variations in simulating two-phase flows. The numerical scheme is based on an artificial compressibility formulation with a finite difference scheme for the space discretization. To validate the method, the penetration of a heavier fluid into the lighter one is computed and illustrated graphically.

Density stratification has a strong impact on turbulence in geophysical flows. Stratification changes the spatial turbulence spectrum and the energy transport and conversion within the spectrum. We analyze these effects based on a series of direct numerical simulations (DNS) of stratified turbulence. To facilitate simulations of real-world problems, which are usually beyond the reach of DNS, we propose a subgrid-scale turbulence model for large eddy simulations of stratified flows based on the Adaptive Local Deconvolution Method (ALDM). Flow spectra and integral quantities predicted by ALDM are in excellent agreement with direct numerical simulation. ALDM automatically adapts to strongly anisotropic turbulence and is thus a suitable tool for studying turbulent flow phenomena in atmosphere and ocean.

We have performed fully resolved three-dimensional numerical simulations of a statically unstable monochromatic inertia–gravity wave using the Boussinesq equations on an $f$-plane with constant stratification. The chosen parameters represent a gravity wave with almost vertical direction of propagation and a wavelength of 3 km breaking in the middle atmosphere. We initialized the simulation with a statically unstable gravity wave perturbed by its leading transverse normal mode and the leading instability modes of the time-dependent wave breaking in a two-dimensional space. The wave was simulated for approximately 16 h, which is twice the wave period. After the first breaking triggered by the imposed perturbation, two secondary breaking events are observed. Similarities and differences between the three-dimensional and previous two-dimensional solutions of the problem and effects of domain size and initial perturbations are discussed.

An overview is given on direct numerical simulations and on large eddy simulations of homogeneous turbulence under the impact of shear and stable stratification. We describe the methods used and report on the results of various studies. In particular, the vortex structure of turbulent motions is discussed. Moreover, the dynamics of statistical mean quantities are investigated. The mean variances are compared with experimental data. The turbulent diffusivity tensor for passive species in a stratified shear flow is computed. Moreover, a simple model is described which allows us to estimate the vertical diffusivities for heat and momentum for such flows when the vertical velocity variance or the dissipation rate are known.

The objective of this project was the analysis and the control of local truncation errors in large eddy simulations. We show that physical reasoning can be incorporated into the design of discretization schemes. Using systematic procedures, a non-linear discretization method is developed where numerical and turbulence-theoretical modeling are fully merged. The truncation error itself functions as an implicit turbulence model which accurately represents the effects of unresolved turbulence.

The dynamics of three-dimensional turbulence under the influence of density stratification can be classified in different regimes that differ fundamentally; see Brethouwer et al. (2007) for a good review. Up to now, strongly stratified turbulent flows were mainly studied by direct numerical simulations (DNS). The available computer resources, however, limit the application of DNS to Reynolds numbers that are too small for studying the regime of real scale atmospheric problems. Large Reynolds number flows in general can be simulated efficiently by large- eddy simulation (LES). However, most subgrid-scale (SGS) models for LES are not suitable for stratified flows, as local isotropy of the SGS turbulence is assumed, and therefore require ad-hoc modifications. We propose an implicit subgrid-scale model that handles anisotropic turbulence in a straight forward way without any limiting assumptions. We solve the non-linear Boussinesq equations for a fluid with a constant background stratif...

In this paper we analyze the data that was collected at the British Haley Station in Antarctica on June 22, 1986. This data contains measurements of the temperature and wind velocity at three heights (5 m,16 m and 32 m). Using the Karahunen-Loeve algorithm we decompose this ``raw&quot; data into mean flow, waves and turbulent residuals. We then apply three tests to find if the turbulent field might represent ``two-dimensional turbulence&quot;. The first of these tests was devised by Dewan (see Radio Sci., 20 (1985) 1301), while the second relates to the scaling of the structure function (see Lindborg E., J. Fluid Mech., 388 (1999) 259). To confirm further the results of these two tests, we show that around a frequency of 0.5 rad/s most of the spectral plots for the raw data exhibit a slope of -3. We also construct a scaling model in an attempt to interpret part of the high-frequency spectrum of this data which is almost flat and discuss its possible relation to Bolgiano ``buoyancy r...

This paper builds upon the investigation of Augier et al. (Phys. Fluids, vol. 26 (4), 2014) in which a strongly stratified turbulent-like flow was forced by 12 generators of vertical columnar dipoles. In experiments, measurements start to provide evidence of the existence of a strongly stratified inertial range that has been predicted for large turbulent buoyancy Reynolds numbers $\mathscr{R}_{t}={\it\varepsilon}_{\!K}/({\it\nu}N^{2})$, where ${\it\varepsilon}_{\!K}$ is the mean dissipation rate of kinetic energy, ${\it\nu}$ the viscosity and $N$ the Brunt–Väisälä frequency. However, because of experimental constraints, the buoyancy Reynolds number could not be increased to sufficiently large values so that the inertial strongly stratified turbulent range is only incipient. In order to extend the experimental results toward higher buoyancy Reynolds number, we have performed numerical simulations of forced stratified flows. To reproduce the experimental vortex generators, columnar di...

We investigate the spectral properties of the turbulence generated during the nonlinear evolution of a Lamb–Chaplygin dipole in a stratified fluid for a high Reynolds number $Re= 28\hspace{0.167em} 000$ and a wide range of horizontal Froude number ${F}_{h} \in [0. 0225~0. 135] $ and buoyancy Reynolds number $\mathscr{R}= Re{{F}_{h} }^{2} \in [14~510] $. The numerical simulations use a weak hyperviscosity and are therefore almost direct numerical simulations (DNS). After the nonlinear development of the zigzag instability, both shear and gravitational instabilities develop and lead to a transition to small scales. A spectral analysis shows that this transition is dominated by two kinds of transfer: first, the shear instability induces a direct non-local transfer toward horizontal wavelengths of the order of the buoyancy scale ${L}_{b} = U/ N$, where $U$ is the characteristic horizontal velocity of the dipole and $N$ the Brunt–Väisälä frequency; second, the destabilization of the Kelv...

Recently, Deloncle, Billant & Chomaz (J. Fluid Mech., vol. 599, 2008, p. 229) and Waite & Smolarkiewicz (J. Fluid Mech., vol. 606, 2008, p. 239) have performed numerical simulations of the nonlinear evolution of the zigzag instability of a pair of counter-rotating vertical vortices in a stratified fluid. Both studies report the development of a small-scale secondary instability when the vortices are strongly bent if the Reynolds number Re is sufficiently high. However, the two papers are at variance about the nature of this secondary instability: it is a shear instability according to Deloncle et al. (J. Fluid Mech., vol. 599, 2008, p. 229) and a gravitational instability according to Waite & Smolarkiewicz (J. Fluid Mech., vol. 606, 2008, p. 239). They also profoundly disagree about the condition for the onset of the secondary instability: ReF2h &gt; O(1) according to the former or ReFh &gt; 80 according to the latter, where Fh is the horizontal Froude number. In order to understand...

Following the Kolmogorov technique, an exact relation for a vector third-order moment $\mathbi{J}$ is derived for three-dimensional incompressible stably stratified turbulence under the Boussinesq approximation. In the limit of a small Brunt–Väisälä frequency, isotropy may be assumed which allows us to find a generalized $4/ 3$-law. For strong stratification, we make the ansatz that $\mathbi{J}$ is directed along axisymmetric surfaces parameterized by a scaling law relating horizontal and vertical coordinates. An integration of the exact relation under this hypothesis leads to a generalized Kolmogorov law which depends on the intensity of anisotropy parameterized by a single coefficient. By using a scaling relation between large horizontal and vertical length scales we fix this coefficient and propose a unique law.