Enstrophy Equations Research Articles

Kinetic energy and enstrophy transfer in compressible Rayleigh–Taylor turbulence

Abstract

Effects of forced frequency oscillations and free stream turbulence on the separation-induced transition in pressure gradient dominated flows

The individual and cumulative effects of forced frequency oscillations and free stream turbulence (FST), on the separation-induced transition caused by an adverse pressure gradient on a flat plate geometry, are numerically investigated by solving the Navier–Stokes equation. The flat plate is subjected to a streamwise pressure gradient via a contoured wall, representative of a low pressure turbine blade. The results are validated against existing numerical data, and insights into flow physics of the pre-transitional and turbulent regimes are gained using instantaneous, time-averaged, and phase-averaged flow fields, turbulent kinetic energy budgets, and a disturbance enstrophy-based nonlinear analysis based on direct numerical simulations. Oscillations are imposed at two values of reduced frequency (kosc). Higher reduced frequency oscillations are more effective in triggering early transition and reducing the separated region. This suppression of the separated region is more pronounced with FST. A reduction in the size of the separation bubble is noted along with enhanced near-wall mixing resulting in a shift in the inflection point of the velocity profile toward the wall, with FST and oscillations. The budget of the turbulent kinetic energy shows the dominance of the production term for the case with FST and forced oscillations. The disturbance enstrophy-based analysis provides insight into budget terms of the enstrophy equation, specifically on the role of vortex stretching. Finally, spatiotemporal linear receptivity of the case with oscillations near the separation bubble is reported. This provides a quantitative description of the instability triggered by the pressure gradient and oscillations.

A new mean‐field plasma edge transport model based on turbulent kinetic energy and enstrophy

AbstractThe predictive capabilities of mean‐field plasma edge transport codes are significantly reduced by the empirically determined radial diffusion coefficients commonly used to account for the turbulent transport. This paper aims to assess a two‐equation model including both turbulent kinetic energy and turbulent enstrophy for an improved description of the mean‐field turbulent transport. A turbulent enstrophy equation is analytically derived for isothermal, electrostatic, two‐dimensional, interchange‐dominated ExB turbulence in the scrape‐off layer. Evaluation of this equation shows that interchange is the dominant source, while the sheath loss and viscous terms provide the main sinks of turbulent enstrophy. Transport effects turn out to be of minor importance. A model for the combined turbulent kinetic energy and enstrophy equations and the particle transport is constructed starting from a one‐equation model for the turbulent kinetic energy. The resulting two‐equation model is shown to retain the dominant dynamics of the latter while improving its scalings.

Density effects on post-shock turbulence structure and dynamics

Turbulence structure resulting from multi-fluid or multi-species, variable-density isotropic turbulence interaction with a Mach 2 shock is studied using turbulence-resolving shock-capturing simulations and Eulerian (grid) and Lagrangian (particle) methods. The complex roles that density plays in the modification of turbulence by the shock wave are identified. Statistical analyses of the velocity gradient tensor (VGT) show that density variations significantly change the turbulence structure and flow topology. Specifically, a stronger symmetrization of the joint probability density function (PDF) of second and third invariants of the anisotropic VGT, PDF$(Q^{\ast },R^{\ast })$, as well as the PDF of the vortex stretching contribution to the enstrophy equation, are observed in the multi-species case. Furthermore, subsequent to the interaction with the shock, turbulent statistics also acquire a differential distribution in regions having different densities. This results in a nearly symmetric PDF$(Q^{\ast },R^{\ast })$ in heavy-fluid regions, while the light-fluid regions retain the characteristic tear-drop shape. To understand this behaviour and the return to ‘standard’ turbulence structure as the flow evolves away from the shock, Lagrangian dynamics of the VGT and its invariants is studied by considering particle residence times and conditional particle variables in different flow regions. The pressure Hessian contributions to the VGT invariants transport equations are shown to be not only affected by the shock wave, but also by the density in the multi-fluid case, making them critically important to the flow dynamics and turbulence structure.

The mechanisms of energy transformation in sharp open-channel bends: Analysis based on experiments in a laboratory flume

In sharp open-channel bends, there exists nonlinear interactions between streamwise and cross-stream velocities, which leads to the adjustment of flow energy transport, increases the energy dissipation and eventually influences the development of river bends. However, previous studies have not yet fully understood the mechanisms underlying the transport and dissipation of energy in open-channel bends. In addition, energy dissipation is directly related to the evolution of enstrophy Ω·Ω/2. However, it has not yet received much attention in the study of the resistance in meandering rivers. To gain an insight into these mechanisms, with the aid of detailed experimental data collected in a sharp open-channel bend, this study quantified the impacts of the key terms in the kinetic energy equation and the enstrophy equation. Results show that the term related to the transport of kinetic energy by cross-stream circulation is at least an order of magnitude larger than the other terms in the kinetic energy equation, which contributes significantly to the redistribution of kinetic energy over the cross section. The term related to the production of turbulent kinetic energy in the kinetic energy equation shows the smallest order of magnitude and thus is almost negligible for the time-averaged flow field. Furthermore, over most of the measured area, the term related to the centrifugal force shows values of at least an order of magnitude larger than the other terms in the enstrophy equation, which means that this term exerts the dominant influence on the energy dissipation. The results also revealed that the transformation process is insensitive to the variations in Froude number. These results are expected to improve the understanding of the complete process of energy transformation, especially the resistance caused by secondary flow in meandering rivers.

Vorticity budgets in premixed combusting turbulent flows at different Lewis numbers

A direct numerical simulations database of statistically planar turbulent premixed flames using a simple Arrhenius type irreversible chemistry for different values of global Lewis numbers, Le, (0.34, 0.60, 0.80, 1.00, 1.20) has been examined to analyze the effects of Le on vorticity transport within the flame. To meet this objective, a general enstrophy conservation equation has been considered, which distinctly describes contributions from vortex-stretching, destruction by volumetric dilatation rates, baroclinic and viscous force torques, viscous transport, and dissipation. The average statistical behavior of the various contributions conditioned upon the value of the reaction progress variable, c, has been analyzed in the preheat and reacting regions of the flame. The mean values of enstrophy monotonically decays with c from fresh reactants toward hot products for Le equal to 0.8, 1.0, and 1.2; vortex-stretching and viscous dissipation are the leading contributors, while the remaining contributions are slightly smaller although non-negligible. By contrast, the mean value of enstrophy decreases from the leading edge before increasing up to the trailing edge of the flame; in these cases, the mean value of baroclinic torque is significantly greater than the other contributions in most of the preheat and reacting regions; vortex-stretching, destruction by volumetric dilatation rates and viscous transport, and dissipation remain comparable over most of the flame. An explanation for the significant qualitative and quantitative differences in the enstrophy transport, taking place for Le between 0.6 and 0.8 for the given turbulence intensity, is sought in terms of the alignments of vorticity and the gradients of density, pressure, temperature, and reaction progress variable. The transport statistics of the enstrophies of the vorticity vector components tangential and normal to iso-scalar surfaces, c(x, t) = constant, provide further insight into the mechanisms of the differences in the enstrophy transport in response to the changes in the global Lewis number.

Evaluation of mixture-fraction-based turbulent-reaction-rate model assumptions for high-pressure reactive flows

Several assumptions of atmospheric-pressure (atmospheric-p) single-phase turbulent reaction rate models are examined for high-p reactive flows having turbulent characteristics. The study uses a Direct Numerical Simulation (DNS) database described elsewhere (Bellan, 2017). This database was obtained with a model combining multi-species mixing under high- p conditions, a real-gas equation of state (EOS) and a single-step chemical reaction. The database, created in the configuration of a temporal mixing layer, probes the effect of the initial Reynolds number, Re0, of the initial pressure, p0, and of the initial composition of the two mixing-layer streams. The reaction is initiated in a turbulent flow and in each simulation the computations are pursued past a time when a maximum average-volumetric p is attained, tpp*. The examination of the vorticity and enstrophy equations at a time before reaction initiation and also at tpp* highlights the pivotal role of the high density-gradient magnitude regions (Bellan, 2017) in producing turbulence and the crucial role at tpp* of the baroclinic effect representing the misalignments of gradients of thermodynamic variables. Compared to the classical mixture fraction equation, the results show that the mixture fraction obeys an equation in which there is an additional diffusion term having a larger r.m.s. magnitude than that of the mixture-fraction typical diffusion term. An assessment of the Conditional Source-term Estimation (CSE) model assumptions found that, using an accurate mixture fraction probability density function (PDF), one can obtain a very good representation of the turbulent reaction rate in the most intense reaction regions, however the quality of the predictions of CSE deteriorated significantly when a common model for the PDF, the β -PDF, was used despite the β-PDF being constructed with the accurate moments extracted from the DNS. In the regions of minimal reaction, the CSE model using the exact PDF extracted from the DNS does not provide an accurate representation of the turbulent reaction rate, a fact which is attributed to the combined effect of lack of correlation between fluctuations of the reaction rate and of the mixture fraction, and to the strong correlation of the thermodynamic variables through the real-gas EOS in regions of colder and denser fluid.

Geometrical statistics of the vorticity vector and the strain rate tensor in rotating turbulence

We report results on the geometrical statistics of the vorticity vector obtained from experiments in electromagnetically forced rotating turbulence. A range of rotation rates Ω is considered, from non-rotating to rapidly rotating turbulence with a maximum background rotation rate of Ω = 5 rad/s (with Rossby number much smaller than unity). Typically, the Taylor-scale Reynolds number in our experiments is around Reλ ≈ 100. The measurement volume is located in the centre of the fluid container above the bottom boundary layer, where the turbulent flow can be considered locally statistically isotropic and horizontally homogeneous for the non-rotating case, see L.J.A. van Bokhoven, H.J.H. Clercx, G.J.F. van Heijst, and R.R. Trieling, Experiments on rapidly rotating turbulent flows, Phys. Fluids 21 (2009) 096601. Based on the full set of velocity derivatives, measured in a Lagrangian way by three-dimensional (3D) particle tracking velocimetry, we have been able to quantify statistically the effect of system rotation on several flow properties. For the range of rotation rates considered, the experimental results show how the turbulence evolves from almost isotropic 3D turbulence (Ω ≲ 0.2 rad/s) to quasi-two-dimensional turbulence (Ω ≈ 5.0 rad/s), and how this is reflected by several statistical quantities. In particular, we have studied the orientation of the vorticity vector with respect to the three eigenvectors of the local strain rate tensor and with respect to the vortex stretching vector. Additionally, we have quantified the role of system rotation on the self-amplification terms of the enstrophy and strain rate equations and the direct contribution of the background rotation on these evolution equations. The main effect of background rotation is the strong reduction of extreme events and related (strong) reduction of the skewness of PDFs of several quantities, for example, the intermediate eigenvalue of the strain rate tensor and the enstrophy self-amplification term.

Evolution of enstrophy in shock/homogeneous turbulence interaction

AbstractInteraction of turbulent fluctuations with a shock wave plays an important role in many high-speed flow applications. This paper studies the amplification of enstrophy, defined as mean-square fluctuating vorticity, in homogeneous isotropic turbulence passing through a normal shock. Linearized Navier–Stokes equations written in a frame of reference attached to the unsteady shock wave are used to derive transport equations for the vorticity components. These are combined to obtain an equation that describes the evolution of enstrophy across a time-averaged shock wave. A budget of the enstrophy equation computed using results from linear interaction analysis and data from direct numerical simulations identifies the dominant physical mechanisms in the flow. Production due to mean flow compression and baroclinic torques are found to be the major contributors to the enstrophy amplification. Closure approximations are proposed for the unclosed correlations in the production and baroclinic source terms. The resulting model equation is integrated to obtain the enstrophy jump across a shock for a range of upstream Mach numbers. The model predictions are compared with linear theory results for varying levels of vortical and entropic fluctuations in the upstream flow. The enstrophy model is then cast in the form of$k$–$\epsilon $equations and used to compute the interaction of homogeneous isotropic turbulence with normal shocks. The results are compared with available data from direct numerical simulations. The equations are further used to propose a model for the amplification of turbulent viscosity across a shock, which is then applied to a canonical shock–boundary layer interaction. It is shown that the current model is a significant improvement over existing models, both for homogeneous isotropic turbulence and in the case of complex high-speed flows with shock waves.

High-order WENO simulations of three-dimensional reshocked Richtmyer–Meshkov instability to late times: dynamics, dependence on initial conditions, and comparisons to experimental data

The dynamics of the reshocked multi-mode Richtmyer–Meshkov instability is investigated using 513 × 2572 three-dimensional ninth-order weighted essentially nonoscillatory shock-capturing simulations. A two-mode initial perturbation with superposed random noise is used to model the Mach 1:5 air/SF6 Vetter–Sturtevant shock tube experiment. The mass fraction and enstrophy isosurfaces, and density cross-sections are utilized to show the detailed ow structure before, during, and after reshock. It is shown that the mixing layer growth agrees well with the experimentally measured growth rate before and after reshock. The post-reshock growth rate is also in good agreement with the prediction of the Mikaelian model. A parametric study of the sensitivity of the layer growth to the choice of amplitudes of the short and long wavelength initial interfacial perturbation is also presented. Finally, the amplification effects of reshock are quantified using the evolution of the turbulent kinetic energy and turbulent enstrophy spectra, as well as the evolution of the baroclinic enstrophy production, buoyancy production, and shear production terms in the enstrophy and turbulent kinetic transport equations.

Low‐frequency characteristics of wave group–forced vortices

The dynamics of vorticity motions forced by wave groups incident on an alongshore‐uniform barred beach are analyzed. For both normally and obliquely incident wave groups, the potential vorticity and enstrophy equations reveal that the temporal variability of wave group–forced vortices is directly linked to the variability in the incoming wave groups rather than bottom friction, as previously hypothesized. Analysis of the lifespan of individual vortices further shows that the wave group forcing is responsible for not only the temporal variations of the vortices but also their eventual demise. Vortices in the simulations persist for 5 to 45 min, which is consistent with recent field observations. For oblique wave groups, the resulting vortices are advected by the mean current, yielding a signature in the frequency–wave number spectrum that is similar to that usually attributed to shear instabilities of the alongshore current. These results may explain previous observations of alongshore‐propagating vorticity motions in the presence of a stable alongshore current. For simulations involving an unstable alongshore current, we find that the inclusion of wave group forcing results in velocity spectra that are much broader compared to the simulations that neglect wave grouping, which could explain discrepancies between previously observed and modeled spectral widths of propagating vorticity motions. Finally, the potential enstrophy balance shows that vorticity production due to wave groups may be as important as that due to the instability process and that not all low‐frequency vortical motions observed during oblique wave incidence should be attributed to shear instabilities of the alongshore current.

Curvature Effect on Compressible Turbulent Flow over a WavyWall

A fully developed compressible turbulent flow in a channel with alower wavy wall and a upper plane wall is studied using large eddysimulation. We mainly attempt to deal with the curvature effect oncompressible turbulent flow over the wavy wall. Some typicalquantities including the mean turbulence statistics, dilatation andbaroclinic terms in the enstrophy equation, turbulent kinetic energybudgets and the near-wall turbulent structures are analysed. Theresults obtained in this study provide physical insight into theunderstanding of the effects of curvature and compressibility onwall-bounded compressible turbulent flow.

Cascade Inequalities for Forced–Dissipated Geostrophic Turbulence

AbstractAnalysis of spectral kinetic energy fluxes in satellite altimetry data has demonstrated that an inverse cascade of kinetic energy is ubiquitous in the ocean. In geostrophic turbulence models, a fully developed inverse cascade results in barotropic eddies with large horizontal scales. However, midocean eddies contain substantial energy in the baroclinic mode and in compact horizontal scales (scales comparable to the deformation radius Ld). This paper examines the possibility that relatively strong bottom friction prevents the oceanic cascade from becoming fully developed. The importance of the vertical structure of friction is demonstrated by contrasting numerical simulations of two-layer quasigeostrophic turbulence forced by a baroclinically unstable mean flow and damped by bottom Ekman friction with turbulence damped by vertically symmetric Ekman friction (equal decay rates in the two layers). “Cascade inequalities” derived from the energy and enstrophy equations are used to interpret the numerical results. In the symmetric system, the inequality formally requires a cascade to large-scale barotropic flow, independent of the stratification. The inequality is less strict when friction is in the bottom layer only, especially when stratification is surface intensified. Accordingly, model runs with surface-intensified stratification and relatively strong bottom friction retain substantial small-scale baroclinic energy. Altimetric data show that the symmetric inequality is violated in the low- and midlatitude ocean, again suggesting the potential impact of the “bottomness” of friction on eddies. Inequalities developed for multilayer turbulence suggest that high baroclinic modes in the mean shear also enhance small-scale baroclinic eddy energy. The inequalities motivate a new interpretation of barotropization in weakly damped turbulence. In that limit the barotropic mode dominates the spatial average of kinetic energy density because large values of barotropic density are found throughout the model domain, consistent with the barotropic cascade to large horizontal scales, while baroclinic density is spatially localized.

Viscoelastic effects on higher order statistics and on coherent structures in turbulent channel flow

In this work we study, using the results of direct numerical simulations [Housiadas and Beris, “Polymer-induced drag reduction: Viscoelastic and inertia effects of the variations in viscoelasticity and inertia,” Phys. Fluids 15, 2369 (2003)], the effects of changes in the flow viscoelasticity and the friction Reynolds number on several higher order statistics of turbulence, such as the Reynolds stress, the enstrophy, the averaged equations for the conformation tensor, as well as on the coherent structures through a Karhunen–Loeve (K-L) analysis and selected flow and conformation visualizations. In particular, it is shown that, as the zero friction Weissenberg number Weτ0 increases (for a constant zero friction Reynolds number Reτ0) dramatic reductions take place in many terms in the averaged equations for the Reynolds stresses and in all terms of the averaged enstrophy equations. From a Karhunen–Loeve analysis of the eigenmodes of the flow we saw that the presence of viscoelasticity increases significantly the coherence and energy content of the first few modes. The K-L dimension of the flow at Weτ0=125 is fully one order of magnitude lower than its Newtonian counterpart. As far as the effect of viscoelasticity is concerned, it is manifested primarily by changes in the boundary layer, which are mostly accomplished by Weτ0=50–62.5. In comparison, increasing the Reτ0, from 125 to 590, induces significant changes in various terms in the budgets, despite the fact that the drag reduction remains practically the same over that range. However, the near the wall region seems to change significantly only up to Reτ0=395, with few changes observed upon a further increase to Reτ0=590.

Three‐dimensional effects in shear waves

Most studies on shear waves to date have assumed the flow is depth uniform (two dimensional). In the present study, we utilize the quasi‐three‐dimensional (quasi‐3D) nearshore circulation model SHORECIRC to study shear waves. Our results show that shear wave flow is more organized in the quasi‐3D simulation than in the 2D simulation. In the 2D simulation, the vortices are moving farther offshore of the bar, while in the quasi‐3D simulation, they are more confined to the shoreward side of the bar. Moreover, the shear waves in the quasi‐3D simulation are much less energetic than in the 2D simulation, though the total momentum mixing for the two cases is not significantly different. To understand which mechanisms cause the differences in the 2D and the quasi‐3D simulation, the momentum, kinetic energy, and enstrophy equations for the mean flow and the shear waves are derived. The momentum, energy, and enstrophy balances are discussed using the numerical results from the idealized SUPERDUCK topography and the wave conditions on October 16, 1986. The effects of the quasi‐3D dispersion due to the depth varying currents on shear waves are illustrated. Analysis of the mean momentum balance shows that both the shear waves and the quasi‐3D current pattern contribute to the momentum transfer, and the momentum transfer provided by the shear waves is sometimes larger than that by the quasi‐3D dispersive terms. The kinetic energy balance of the shear waves shows that the quasi‐3D dispersive terms will extract kinetic energy from the depth‐averaged shear waves. Furthermore, the enstrophy equation demonstrates that the quasi‐3D dispersion terms provide vortex tilting, which allows three‐dimensional vortex interactions.

A simple model for self-sustained oscillations in homogeneous shear flow

Generation of the large-scale coherent vortical structures in homogeneous shear flow couples dynamical processes of energy and enstrophy production. In the large rate of strain limit, the simple estimates of the contributions to the energy and enstrophy equations result in a dynamical system, describing experimentally and numerically observed self-sustained nonlinear oscillations of energy and enstrophy. It is shown that the period of these oscillations is independent of the box size and the energy and enstrophy fluctuations are strongly correlated.

The H1-compact global attractor for the solutionsto the Navier-Stokes equations in two-dimensional unbounded domains

We extend previous results obtained by Rosa (1998 Nonlinear Anal. 32 71-85) on the existenceof the global attractor for the two-dimensional Navier-Stokes equations on someunbounded domains. We show that if the forcing term is in the naturalspace H, then the global attractor is compact not only in the L2norm but also in the H1 norm, and it attracts all bounded sets inH in the metric of V.The proof is based on the concept of asymptotic compactness andthe use of the enstrophy equation. As compared with the work of Rosa, whichproved the compactness and the attraction in the L2 norm, the newdifficulty comes from the fact that the nonlinear term of theNavier-Stokes equations does not disappear from the enstrophy equation,while it does disappear in the energy equation due to its antisymmetryproperty.

Simulation of turbulent square-duct flow - Dissipation and small-scale motion

Results and the database from a direct numerical simulation of turbulent flow in a square duct are used to investigate the dissipation processes in a flow with two inhomogeneous directions. First, the effect of the corner on small-scale topological patterns is identified, and second, information for the assessment and development of turbulence closure models is provided by calculating all of the terms in the transport equations for the turbulent dissipation rate and enstrophy. It is shown that, with the exception of two production terms, these budgets have the same dynamics down to the wall. The effects of the corner are manifested by vanishing turbulence production in the viscous sublayer very close to the corner, and reduced dissipation of turbulent vorticity along the diagonal, farther away from the corner. IRECT numerical simulation (DNS) of simple shear flows has proven to be an effective and important tool for studying turbulence structures and near-wall effects.1 The present paper extends this idea to complex turbulent flows by using the database from a recent DNS of turbulent square-duct flow2 to investigate the influence of the corner on the structure of small-scale turbulence. These effects are manifested by vanishing turbulence production as well as reduced turbulence energy and dissipation along the diagonal (corner bisector).2 The imbalance between the turbulence along the corner bisector and the midchannel (wall bisector) creates the stress-driven secondary flow, defined as secondary flow of the second kind.24 Of particular interest in this paper is the turbulence dissipationrate (and enstrophy) budget because the dissipation rate contributes significantly to the Reynolds stress balance.3 An order-of-magnitude analysis of the dissipation process is provided by Tennekes and Lumley5 by estimating the terms in the turbulence enstrophy budget. Mansour et al.6 presented all of the terms in the dissipation-rate budget from the DNS of turbulent channel flow and demonstrated the significance of these terms in a wall-bounded flow. The work of Rodi and Mansour7 also indicates the importance of all terms in the dissipation-rate budget near the wall, including the transport and diffusion. Because of their connection to the dissipation processes,8 the small-scale topological structures also are considered. By displaying scatter plots of the invariants of the local deformation tensor obtained in the homogeneous direction and in time, we can deduce the preferred small-scale topologies and dissipation production processes in the square duct. The present results document all of the terms in the turbulence dissipation rate and enstrophy equations that are essential for both the modeling of Reynolds stress transport equations and the development of subgrid-scale models.

Modeling of the turbulent enstrophy equation

Though the turbulent enstrophy and turbulent dissipation rate equations have many similarities, the former lacks expressions involving the pressure whose modeling has proved to be largely intractable. Moreover, it will be shown here that several of the production terms in the enstrophy equation may be analyzed through formal vorticity transport analysis. The result is a more careful accounting of production effects than has thus far been possible in the dissipation rate equation. When used with the conventional dissipation model, the new production terms significantly affect the enstrophy equation budget, but not the predicted mean velocity and turbulent kinetic energy. Thus, improved modeling of the enstrophy dissipation term is also necessary before the new production terms can have a beneficial impact on the computed flow fields. For low Reynolds number channel flow, a disspation model is constructed which demonstrates how accurate representations of the individual terms in the enstrophy balance can lead to improved predictions of turbulent quantities.

Enstrophy Generation in a Shock-Dominated Turbulence

A mechanism of enstrophy generation is investigated numerically in a shock-dominated turbulence driven by a random external force which has only the compressible component. Enstrophy is generated, especially on collision of shock, as a pair of vortex tube of opposite sense of rotation behind curved shocks. The roles of various terms in enstrophy equation are clarified in enstrophy generation process. Generation of enstrophy is enhanced by strong alignment of each term of the enstrophy equation with the vorticity vector.