Enstrophy Budget Research Articles

A Multifaceted Isoneutral Eddy Transport Diagnostic Framework and Its Application in the Southern Ocean

AbstractWe propose an isoneutral eddy transport diagnostic framework and apply it in the Southern Ocean to analyze eddy potential vorticity (PV) transport dynamics. We leverage a thickness‐weighted spatiotemporal scale separation to define the generalized eddy flux. A local sampling method is employed and validated to estimate the transport tensor. We introduce Leonard's decomposition in Large Eddy Simulations (LES) to split the process into the Reynolds and non‐Reynolds parts. We also use the stationary‐transient decomposition. The non‐Reynolds term and the stationary effect contribute significantly to the transport properties. The Reynolds and transient parts have distinctions, that show oceanic mesoscale dynamics are not synergistic in time and spatial scales. We investigate the spatial scale dependence of each part and find that the transient and stationary components of the Reynolds (non‐Reynolds) term have consistent (dissimilar) scale dependence. The scale dependence of the generalized eddy flux and the transport eigenvalues is controlled by the non‐Reynolds term. We further investigate the eigenvalues in terms of the potential enstrophy and anisotropy. The potential enstrophy in the Southern Ocean holistically has a forward cascade on the researched scales. The Reynolds and transient effects efficiently promote the forward enstrophy cascade. The non‐Reynolds and stationary effects stimulate drastically anisotropic eddy transport. The PV gradient barrier mechanism dominates the drastic anisotropic case. When the anisotropy is weak, a special anisotropic case with a locally balanced potential enstrophy budget overwhelms the case of real isotropy. We also introduce several LES concepts, hoping to enlighten any potential ocean modeling and parameterization.

Entrainment at multi-scales in shear-dominated and Rayleigh–Taylor turbulence

A partially turbulent flow continuously incorporates irrotational fluid into the turbulent region, a phenomenon known as entrainment. Although entrainment locally acts at viscous scales, the thin interface separating the turbulent from the irrotational region is extremely convoluted, and twisted in such a way that renders the global entrainment flux scale-independent. Despite turbulent entrainment being widely recognized as a multi-scale process, the theoretical basis for quantifying the entrainment flux at multi-scales is lacking. In this paper we derive an equation that allows us to quantify the local entrainment velocity at multi-scales. This is done by defining the local entrainment velocity as the propagation speed of an iso-surface of filtered enstrophy relative to the coarse-grained velocity field, and using the filtered enstrophy budget to split the total velocity into its individual components, i.e. viscous, inviscid, baroclinic and sub-filter. The equation is used to investigate the entrainment at multi-scales in simulated turbulent mixing layers, where turbulence is sustained by either a mean shear or an unstable buoyancy gradient (Rayleigh–Taylor turbulence).

Exploring role of aspect ratio for compressible flow in a rectangular lid-driven cavity with a vertical temperature gradient

Numerical investigation of a compressible fluid in a two-dimensional rectangular lid-driven cavity (LDC) with a vertical temperature gradient is performed by solving the compressible Navier–Stokes equation. Here, we explore the role of aspect ratio (AR) (width/height) on the vorticity dynamics and redistribution by considering three ARs of 1:1, 2:1, and 3:1. The onset and propagation of the instability are explored via time-resolved and instantaneous distributions of vorticity, time-series of streamwise velocity, and its associated spectra. The flow physics reveal that the precessing vortical structures in certain square sub-cells of the rectangular LDC resemble that of orbital motion with a primary core eddy surrounded by gyrating satellite vortices, typical of a supercritical flow in a square LDC. Upon increasing the AR, there is a major shift in the vorticity transfer from the top right corner (acting as the source of maximum vorticity generation) toward the left square sub-cells in the domain. This is further aided by the convective motion due to the imposed destabilizing vertical thermal gradient. The spectra demonstrate that a multi-periodic, chaotic flow is the consistent flow feature for the rectangular LDC for Re = 5500, irrespective of the AR. The compressible enstrophy budget of the rectangular LDC with varying AR is computed for the first time. This shows the dominance of the baroclinic vorticity over the viscous diffusion terms, which was conceived of as the major contributor to the creation of rotational flow structures.

Effects of stabilizing and destabilizing thermal gradients on reversed shear-stratified flows: Combined Kelvin–Helmholtz Rayleigh–Taylor instability

Numerical investigation of the coupled Kelvin–Helmholtz Rayleigh–Taylor instability (KHRTI) is presented here by solving the compressible Navier–Stokes equations for two air streams differentially heated in two halves of a three-dimensional (3D) box. Here, we explore the role of a stabilizing and destabilizing thermal gradient and that of reversing the direction of the air streams considered for Atwood numbers of ±0.1567 and dimensionless tangential shear of ΔU=0.68 and 4.1. The onset of the KHRTI and development of the turbulent mixing layer are explored via time-resolved and instantaneous distributions of temperature and vorticity. Early stages of the KHRTI with reversed air streams follows a Kelvin–Helmholtz (KH) mechanism, with Rayleigh–Taylor (RT) dynamics becoming important at later times. This leads to an earlier development of the turbulent mixing layer. The KHRTI with stabilizing or destabilizing thermal gradients shows a dominance of the buoyancy-driven mechanism, right from the onset. The transition from laminar to turbulent mixing layer involves the creation of coherent structures of spikes, bubbles, and KH whirls for the destabilizing, stabilizing thermal gradient, and reversed shear cases, respectively. The spectra of the turbulent signals reveal a −5/3 scaling when the shear-driven mechanism is prevalent in the flow and −11/5 scaling when the buoyancy-driven effects become prominent. The compressible enstrophy budget of the KHRTI shows that the onset process is dominated by vortex stretching or compressibility effects, followed by a sharp rise in baroclinic torque contribution once the buoyancy effects become relevant.

Pressure gradient tailoring effects on vorticity dynamics in the near-wake of bluff-body premixed flames

We investigate the role of mean streamwise pressure gradients in the development of a bluff-body-stabilized premixed flame in the near-wake of the bluff body. To this end, a triangular prism flame holder is situated in three different channel geometries: a nominal case with straight walls, a nozzle with a stronger mean pressure gradient, and a diffuser with a comparatively weaker mean pressure gradient. All geometries are implemented using embedded boundaries, and adaptive mesh refinement is used to locally resolve all relevant thermal (i.e., flame) and fluid-mechanical (i.e., vorticity) scales. A premixed propane flame, modeled using a 66-step skeletal mechanism, interacts with vorticity in the boundary layer of the triangular bluff body in the presence of each mean pressure gradient. Analysis of flame-related enstrophy budget terms reveals key differences in the behavior of baroclinic torque between cases, the specifics of which are tied to larger variations in the mean flow structure, recirculation zone structure, and confinement effects. Our results show that the baroclinic torque changes significantly among the configurations, with the nozzle exhibiting the largest baroclinic torque production. However, these differences are shown to be only a secondary consequence of the background pressure gradient, with the primary consequence being the change in the recirculation zone length resulting from the different channel configurations. These results are relevant for flame stabilization with bluff bodies, where clear understanding of the sensitivities to global mean pressure gradient is important to engineering design.

On the physical nature of the turbulent/turbulent interface

The existence of a turbulent/turbulent interface (TTI) has recently been verified in the far wake of a circular cylinder exposed to free-stream turbulence (Kankanwadi & Buxton, J. Fluid Mech., vol. 905, 2020, p. A35). This study aims to understand the physics within the TTI. The wake boundary, approximately 40 diameters downstream of a circular cylinder subjected to grid-generated turbulence, was investigated through simultaneous cinematographic, stereoscopic particle image velocimetry and planar laser induced fluorescence experiments. With no grid placed upstream of the cylinder, the behaviour of the resultant interface, our closest approximation to a turbulent/non-turbulent interface, exactly matched what is observed in existing literature. When background turbulence is present, viscous action is no longer the only method by which enstrophy is transported to the background fluid, unlike for turbulent/non-turbulent interfaces. The presence of rotational fluid on both sides of the TTI allows the vorticity stretching term of the enstrophy budget equation to be the dominant actor in this process. The role of viscosity within a TTI is greatly diminished as the vorticity stretching term takes over responsibilities for enstrophy production. The turbulent strain rate normal to the TTI was found to be enhanced in the interfacial region. Decomposing the vorticity stretching term into components aligned with the three principal strain-rate directions, it was found that the term most aligned with the interface-normal direction contributed to the largest share of enstrophy production. This indicates that better ‘organised’ vorticity on the wake side of the interface yields the enstrophy amplification leading to the previously discovered enstrophy jump across the TTI by Kankanwadi & Buxton (J. Fluid Mech., vol. 905, 2020, p. A35).

Three-dimensional direct numerical simulation of Rayleigh–Taylor instability triggered by acoustic excitation

Rayleigh–Taylor instability (RTI) occurs when the interface between two fluids of different densities is removed, with the heavier (cold) fluid resting on top of the lighter (hot) fluid in the equilibrium state. This arrangement is unstable due to buoyancy, in the absence of any other forces. RTI is noted across a range of length scales from very small in nuclear fusion to supernova explosion at astrophysical scales. RTI is viewed as a baroclinic instability if viscous actions are ignored. An accurate non-overlapping parallel algorithm is used to solve a three-dimensional RTI problem, employing more than 4 × 109 points and a refined time step (7.69×10−8s) for the direct numerical simulation. Air masses at two different temperatures are initially separated by a non-conducting partition inside a box (with a temperature difference of 200 K). The impermeable partition is removed impulsively at t = 0, and the ensuing instability is triggered by an acoustic mechanism involving infra to ultrasonic pulses that travel to either side of the interface. Present high precision petascale computations enable one to capture acoustic disturbances with unprecedented accuracy without any additional interfacial disturbances. The creation of the vorticity is studied by performing enstrophy budget for the compressible flow for RTI, which shows that the viscous terms are dominant compared to the baroclinic one.

Assessing diffusion model impacts on enstrophy and flame structure in turbulent lean premixed flames

Diffusive transport of mass occurs at small scales in turbulent premixed flames. As a result, multicomponent mass diffusion, which is often neglected in direct numerical simulations (DNS) of premixed combustion, has the potential to impact both turbulence and flame characteristics at small scales. In this study, we evaluate these impacts by examining enstrophy dynamics and the internal structure of the flame for lean premixed hydrogen-air combustion, neglecting secondary Soret and Dufour effects. We performed three-dimensional DNS of these flames by implementing the Stefan–Maxwell equations in the code NGA to represent multicomponent mass transport, and we simulated statistically planar lean premixed hydrogen-air flames using both mixture-averaged and multicomponent models. The mixture-averaged model underpredicts the peak enstrophy in the multicomponent simulation by up to 13% in the flame front. Comparing the enstrophy budgets of these flames, the multicomponent simulation yields larger peak magnitudes compared to the mixture-averaged simulation in the reaction zone, showing differences of 17% and 14% in the normalised vortex stretching and viscous effects terms. In the super-adiabatic regions of the flame, the mixture-averaged model overpredicts the viscous effects by up to 13%. To assess the effect of these differences on flame structure, we reconstructed the average local internal structure of the turbulent flame through statistical analysis of the scalar gradient field. Based on this analysis, we show that large differences in viscous effects contribute to significant differences in the average local flame structure between the two models.

Linear temporal stability analysis on cross sheared flow: The stabilization effects via cross shear

The stability features of the inviscid, homogenous, and free cross sheared flow, with base flow velocities U=tanh(z) in the primary direction, which is unstable, and V=ξz3 in the orthogonal direction, which is stable, are thoroughly examined with linear temporal stability analysis, where z is the transverse coordinate perpendicular to the flow and ξ is the cross shear ratio which is the ratio of the characteristic magnitudes of V to U. The map of the unstable regions directly related to χ=ξ(β/α) is obtained, where (β/α) is the ratio between the orthogonal and primary wavenumbers. Further examination of the eigenfunctions shows that the eigenfunction structures divide into the orthogonal wavenumber (OW) mode where (β/α) dominates and the cross shear (CS) mode where ξ dominates. The cross shear is found necessary for stabilization in spite of different fashions for the OW and CS modes. The transition from the OW mode to the CS mode shows that the developments of the two modes inherently compete with each other, so that when ψ=(β/α)/ξ decreases the enhanced cross shear needs to deteriorate the OW mode before it helps the growth of the CS mode. Based on the magnitudes of the associated eigenfunctions in the enstrophy budget, the map of the OW, CS, and hybrid modes, which includes the mixed features of both the OW and CS modes, is produced and discussed.

Linear temporal stability analysis on non-parallel free cross sheared flow with a primary hyperbolic velocity and an orthogonal Bickley jet velocity

A linear temporal stability analysis is carried out on non-parallel free cross sheared flow consisting of the unstable hyperbolic velocity U=tanh(z) in the primary direction and the comparable unstable Bickley jet velocity V=sech2(z) in the orthogonal direction, where z is the coordinate perpendicular to the plane of the primary and orthogonal directions. The cross sheared flow involves the non-parallel effects such as twisted flow and cross flow. The linearized perturbation equations are derived which are subsequently used to examine the stability features. It is found that the instability associated with U=tanh(z) and V=sech2(z) transitions asymptotically to each other as the combined factor χ=ξ(β/α) varies, where ξ=||V||/||U|| is the cross shear ratio and β/α is the ratio between the orthogonal and primary wavenumbers with || || representing the characteristic magnitudes of the velocity components. In addition to the hyperbolic flow (HF) mode and the Bickley jet flow (BJF) mode associated with U=tanh(z) and V=sech2(z), respectively, the orthogonal wavenumber (OW) mode where β/α dominates and the cross shear (CS) mode where ξ dominates are found in the eigenfunction structures. The physical mechanisms of the four eigenfunctions modes are examined with the kinetic energy and enstrophy budgets. The map of the unstable regions influenced and dominated by the OW, CS, HF, and BJF eigenfunction modes is also obtained and discussed.

Turbulent characteristics and energy transfer in the far field of active-grid turbulence

Turbulent characteristics in the far field of active-grid turbulence have been investigated through wind tunnel experiments using hot-wire anemometry. Two forcing protocols are employed following previous studies: one is the double-random mode and the other is the open mode with the grid remaining static with minimum blockage. The integral length scale L for the double-random modes slightly decreases with streamwise distance in the far field as observed in the near field of the active-grid turbulence. The nondimensional dissipation rate Cε for the double-random modes is around 0.5. This asymptotic value is different from those reported in previous active-grid turbulence experiments and could be nonuniversal. The equilibrium scaling L/λ=CεReλ/15 (λ is the Taylor microscale and Reλ is the turbulent Reynolds number) with a constant Cε is established in the far field of the double-random modes regardless of active-grid motions. The sum of production and destruction terms in the enstrophy budget equation for homogeneous and isotropic turbulence S+2G/Reλ (S is the skewness of the longitudinal velocity derivative and G is the destruction coefficient) is proportional to Reλ−1 and close to zero in the present active-grid turbulence, suggesting that the equilibrium scaling is possibly related to the balance between the production and destruction of the enstrophy.

Lagrangian analysis of enstrophy dynamics in a highly turbulent premixed flame

A Lagrangian analysis approach is used to examine the effects of heat release on the dynamics of the enstrophy during highly turbulent premixed combustion. The analysis is performed using data from a direct numerical simulation of a statistically planar premixed methane–air flame at a Karlovitz number of 100. Through cumulative, conditional, and correlation analyses, we show, consistent with prior studies, that vortex stretching and baroclinic torque both increase enstrophy at these highly turbulent conditions, while viscous transport and dilatation both lead to enstrophy destruction. However, although vortex stretching and viscous transport are individually an order of magnitude greater than all other terms in the enstrophy budget, the cumulative and combined effect of these two terms along Lagrangian trajectories is roughly only twice as large as the combined cumulative effect of dilatation and baroclinic torque. Moreover, trajectories that exhibit an increase in enstrophy through the flame are found to frequently have cumulative contributions from budget terms outside a single standard deviation of the mean contribution, indicating that enstrophy production at such highly turbulent conditions is associated with relatively infrequent but large values of dynamical terms. Lagged correlations further reveal a small but measurable contribution of baroclinic torque in enstrophy production, but these increases are overwhelmed, on average, by concurrent decreases in enstrophy due to viscous transport and dilatation. Taken together, these results provide further understanding of enstrophy dynamics in highly turbulent premixed flames.

Subgrid-scale characterization and asymptotic behavior of multidimensional upwind schemes for the vorticity transport equations

We establish subgrid-scale (SGS) characteristics of a finite volume vorticity-transport-based approach for large-eddy simulations. Modified equation analysis indicates that dissipation can be controlled locally via nonlinear limiting of the gradient employed for the vorticity reconstruction. The enstrophy budget highlights the remarkable ability of the truncation terms to mimic the true SGS dissipation and diffusion. Numerical dissipation in under-resolved simulations can be characterized by diffusion terms discovered in the modified equation analysis.

Dependence of Enstrophy Transport and Mixed Mass on Dimensionality and Initial Conditions in the Richtmyer–Meshkov Instability Induced Flows1

AbstractThis paper presents a comparative study of the enstrophy budget and mixed mass between two- and three-dimensional flows induced by Richtmyer–Meshkov instability (RMI). Specifically, the individual contributions to the enstrophy budget due to the production from baroclinicity and from vortex stretching (which vanishes in two-dimensional (2D) flow) are delineated. This is enabled by a set of two- and three-dimensional computations at Atwood 0.5 having both narrow- and broad-band perturbations. A further three-dimensional (3D) computation is conducted at Atwood 0.9 using an identical narrowband perturbation to the Atwood 0.5 case to examine the sensitivity to density ratio. The mixed mass is also considered with the goal to obtain insight on how faithfully a simplified calculation performed in two dimensions can capture the mixed mass for an inertial confinement fusion (ICF) or other practical application. It is shown that the late time power law decay of variable density enstrophy is substantially different in two and three dimensions for the narrowband initial perturbation. The baroclinic production term is negligible in three dimensions (aside from the initial shock interaction), as vortex stretching is larger by two orders of magnitude. The lack of vortex stretching considerably reduces the decay rate in both narrowband and broadband perturbations in two dimensions. In terms of mixed mass, the lack of vortex stretching reduces the mixed mass in two dimensions compared to three in all cases. In the broadband cases, the spectral bandwidth in the 2D case is wider; hence, there is a longer time period of sustained linear growth which reduces the normalized mixed mass further.

Log-Normal Turbulence Dissipation in Global Ocean Models.

Data from turbulent numerical simulations of the global ocean demonstrate that the dissipation of kinetic energy obeys a nearly log-normal distribution even at large horizontal scales O(10 km). As the horizontal scales of resolved turbulence are larger than the ocean is deep, the Kolmogorov-Yaglom theory for intermittency in 3D homogeneous, isotropic turbulence cannot apply; instead, the down-scale potential enstrophy cascade of quasigeostrophic turbulence should. Yet, energy dissipation obeys approximate log-normality-robustly across depths, seasons, regions, and subgrid schemes. The distribution parameters, skewness and kurtosis, show small systematic departures from log-normality with depth and subgrid friction schemes. Log-normality suggests that a few high-dissipation locations dominate the integrated energy and enstrophy budgets, which should be taken into account when making inferences from simplified models and inferring global energy budgets from sparse observations.

Fully resolved simulations of turbulence modulation by high-inertia particles in an isotropic turbulent flow

We performed direct numerical simulations of decaying isotropic turbulence laden with finite-size particles using an immersed boundary method combined with a soft-sphere collision model. The particle diameter is about 25 times the Kolmogorov length scale, and the particle-to-fluid density ratio ranges from 10 to 1000. The turbulence is enhanced by heavy particles but suppressed by light particles. The radial and spectral distributions of the main quantities are examined. The enstrophy budget is analyzed to reveal how the particles enhance the dissipation rate. The dependence of the two-way coupling term on the particle Reynolds number is loosely predicted. The relation between the particle-induced dissipation rate and particle Reynolds number is also estimated. The results indicate that the work done by particles outweighs the particle-induced dissipation at high particle Reynolds number, thereby enhancing the turbulence. Thus the particle Reynolds number should be a key parameter for turbulence modulation by finite-size and high-inertia particles.

Reduced African Easterly Wave Activity with Quadrupled CO2 in the Superparameterized CESM

Abstract African easterly wave (AEW) activity is examined in quadrupled CO2 experiments with the superparameterized CESM (SP-CESM). The variance of 2–10-day filtered precipitation increases with warming over the West African monsoon region, suggesting increased AEW activity. The perturbation enstrophy budget is used to investigate the dynamic signature of AEW activity. The northern wave track becomes more active associated with enhanced baroclinicity, consistent with previous studies. The southern track exhibits a surprising reduction of wave activity associated with less frequent occurrence of weak waves and a slight increase in the occurrence of strong waves. These changes are connected to changes in the profile of vortex stretching and tilting that can be understood as interconnected consequences of increased static stability from the lapse rate response, weak temperature gradient balance, and the fixed anvil temperature hypothesis.

Scalings of heat transport and energy spectra of turbulent Rayleigh-Bénard convection in a large-aspect-ratio box

Direct Numerical Simulations of turbulent convection in a large aspect-ratio box are carried out in the range of Rayleigh number 7 × 104 ≤ Ra ≤ 2 × 106 at Prandtl number Pr=0.71. A strong correlation between the vertical velocity and temperature is observed in the turbulent regime at almost all the length scales. Frequency spectra of all the velocities and temperature show a −5/3 law for a wide band of frequencies. The variances of horizontal velocities at different points in the flow yield a single power-law. Probability density functions of velocities and temperature are close to Gaussian only at higher Rayleigh numbers. The mean and variance of temperature clearly show boundary layers, surface layers and a near-homogeneous bulk region. The boundary layer thickness decreases and bulk-homogeneity is enhanced on increasing the Rayleigh numbers. The wave number spectra of the turbulent kinetic energy exhibit Kolmogorov like (E(k)∼k−5/3) and Bolginao-Obukhov like (E(k)∼k−11/5) behaviour respectively in the central and near-wall regions of the container. An approximate balance between the production due to buoyancy and the dissipation is found in the turbulent kinetic energy budget. Taylor’s approximate equation of the production due to turbulent stretching and the dissipation of turbulent enstrophy is modified by the inclusion of buoyancy production in the enstrophy budget. The present results support the previously proposed 2/7 power-law dependence of the average Nusselt number on the Rayleigh number by yielding an exponent of 0.272, but do not necessarily support the proposed classification of “soft” and “hard” turbulence on the basis of this exponent.

Optimal Wall-to-Wall Transport by Incompressible Flows.

We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields u. Given an enstrophy budget ⟨|∇u|^{2}⟩≤Pe^{2} we construct steady two-dimensional flows that transport at rates Nu(u)≳Pe^{2/3}/(logPe)^{4/3} in the large enstrophy limit. Combined with the known upper bound Nu(u)≲Pe^{2/3} for any such enstrophy-constrained flow, we conclude that maximally transporting flows satisfy Nu∼Pe^{2/3} up to possible logarithmic corrections. Combined with known transport bounds in the context of Rayleigh-Bénard convection, this establishes that while suitable flows approaching the "ultimate" heat transport scaling Nu∼Ra^{1/2} exist, they are not always realizable as buoyancy-driven flows. The result is obtained by exploiting a connection between the wall-to-wall optimal transport problem and a closely related class of singularly perturbed variational problems arising in the study of energy-driven pattern formation in materials science.

Optimal convection cooling flows in general 2D geometries

We generalize a recent method for computing optimal 2D convection cooling flows in a horizontal layer to a wide range of geometries, including those relevant for technological applications. We write the problem in a conformal pair of coordinates which are the pure conduction temperature and its harmonic conjugate. We find optimal flows for cooling a cylinder in an annular domain, a hot plate embedded in a cold surface, and a channel with a hot interior and cold exterior. With a constraint of fixed kinetic energy, the optimal flows are all essentially the same in the conformal coordinates. In the physical coordinates, they consist of vortices ranging in size from the length of the hot surface to a small cutoff length at the interface of the hot and cold surfaces. With the constraint of fixed enstrophy (or fixed rate of viscous dissipation), a geometry-dependent metric factor appears in the equations. The conformal coordinates are useful here because they map the problems to a rectangular domain, facilitating numerical solutions. With a small enstrophy budget, the optimal flows are dominated by vortices that have the same size as the flow domain.