One-point Closure Models Research Articles

A Model for Gradual-phase Heating Driven by MHD Turbulence in Solar Flares

Coronal flare emission is commonly observed to decay on timescales longer than those predicted by impulsively driven, one-dimensional flare loop models. This discrepancy is most apparent during the gradual phase, where emission from these models decays over minutes, in contrast to the hour or more often observed. Magnetic reconnection is invoked as the energy source of a flare, but should deposit energy into a given loop within a matter of seconds. Models which supplement this impulsive energization with a long, persistent ad hoc heating have successfully reproduced long-duration emission, but without providing a clear physical justification. Here we propose a model for extended flare heating by the slow dissipation of turbulent Alfvén waves initiated during the retraction of newly reconnected flux tubes through a current sheet. Using one-dimensional simulations, we track the production and evolution of MHD wave turbulence trapped by reflection from high-density gradients in the transition region. Turbulent energy dissipates through nonlinear interaction between counter-propagating waves, modeled here using a phenomenological one-point closure model. Atmospheric Imaging Assembly EUV light curves synthesized from the simulation were able to reproduce emission decay on the order of tens of minutes. We find this simple model offers a possible mechanism for generating the extended heating demanded by observed coronal flare emissions self-consistently from reconnection-powered flare energy release.

Mean-Lagrangian formalism and covariance of fluid turbulence.

Mean-field-based Lagrangian framework is developed for the fluid turbulence theory, which enables physically objective discussions, especially, of the history effect. Mean flow serves as a purely geometrical object of Lie group theory, providing useful operations to measure the objective rate and history integration of the general tensor field. The proposed framework is applied, on the one hand, to one-point closure model, yielding an objective expression of the turbulence viscoelastic effect. Application to two-point closure, on the other hand, is also discussed, where natural extension of known Lagrangian correlation is discovered on the basis of an extended covariance group.

Unsteady Reynolds averaged Navier–Stokes simulations of a buoyant plume above a cylinder

Unsteady Reynolds averaged Navier–Stokes simulations of a horizontal heated cylinder with Rayleigh number Ra = 9·4 × 107 were carried out and compared with experimental data. Different one-point closure models were used, both two-equation turbulence models and more sophisticated Reynolds stress models. The results approach the laminar solution regardless of the model used, and thus turbulent effects that occur downstream of the cylinder are not captured and the ensemble averaged results differ significantly from the experimental results. Nevertheless, the simulations are examples of mainstay engineering approaches and illustrate possible problems and pitfalls when using commercial general-purpose computational fluid dynamics codes without comparison against experimental results or other means of verification. Natural convection heat transfer from a single horizontal cylinder at intermediate Rayleigh numbers is a well-examined classical problem within heat transfer research. However, even though the geometry is very simple, the problem involves rather complex flow features that are hardly amendable by Reynolds averaged Navier–Stokes simulations or even large eddy simulations.

An Attempt to Derive the ɛ Equation from a Two-Point Closure

Abstract The goal of this paper is to derive the equation for the turbulence dissipation rate ɛ for a shear-driven flow. In 1961, Davydov used a one-point closure model to derive the ɛ equation from first principles but the final result contained undetermined terms and thus lacked predictive power. Both in 1987 (Schiestel) and in 2001 (Rubinstein and Zhou), attempts were made to derive the ɛ equation from first principles using a two-point closure, but their methods relied on a phenomenological assumption. The standard practice has thus been to employ a heuristic form of the ɛ equation that contains three empirical ingredients: two constants, c1,ɛ and c2,ɛ, and a diffusion term Dɛ. In this work, a two-point closure is employed, yielding the following results: 1) the empirical constants get replaced by c1, c2, which are now functions of K and ɛ; 2) c1 and c2 are not independent because a general relation between the two that are valid for any K and ɛ are derived; 3) c1, c2 become constant with values close to the empirical values c1,ɛ, c2,ɛ (i.e., homogenous flows); and 4) the empirical form of the diffusion term Dɛ is no longer needed because it gets substituted by the K–ɛ dependence of c1, c2, which plays the role of the diffusion, together with the diffusion of the turbulent kinetic energy DK, which now enters the new ɛ equation (i.e., inhomogeneous flows). Thus, the three empirical ingredients c1,ɛ, c2,ɛ, Dɛ are replaced by a single function c1(K, ɛ) or c2(K, ɛ), plus a DK term. Three tests of the new equation for ɛ are presented: one concerning channel flow and two concerning the shear-driven planetary boundary layer (PBL).

Validating a turbulence closure against estuarine microstructure measurements

A simple k– ε turbulence closure is introduced which has no stability functions but instead a Richardson number-dependent turbulent Prandtl number. Its free parameters are determined in a comparison with microstructure observations from a stratified and sheared tidal estuary and laboratory measurements. The closure is able to simulate observed turbulent dissipation rates ( ε) and turbulent length scales ( l th) in regions of strong mean shear and small gradient Richardson number ( R g) to within factors of 2–3. It fails in regions of small shear and large R g, presumably because of the dominance of internal wave-driven mixing. Additional simulations with a k– ε closure with stability functions taken from Canuto et al. [Canuto, V.M., Howard, A., Cheng, Y., Dubovikov, M.S., 2001. Ocean turbulence I: one-point closure model. Momentum and heat vertical diffusivities. J. Phys. Oceanogr. 31, 1413–1426] and with the closure of Baumert and Peters [Baumert, H., Peters, H., 2004. Turbulence closure, steady state, and collapse into waves. J. Phys. Oceanogr. 34, 505–512] show poor performance. Establishing a valid 1:1 comparison of simulated and observed ε and l th requires nudging the model velocity and density toward observed values because free model integrations quickly diverge from the observations. Steady state gradient Richardson numbers are constrained to a range of 0.18–0.25, while flux Richardson numbers are constrained to the range of 0.1–0.22. The closure output is rather insensitive to such parameter variations. The simulations are sensitive, however, to the treatment of the observed velocity and density used to nudge the model. Good closure performance requires averaging the measured tidal flow over about an hour, a time scale for which conventional numerical models of estuarine circulations should be able to match observed shears. In the closure simulations the TKE balance stays close to a production–dissipation balance. The time rate of change and vertical diffusion of TKE are small, of the same order of magnitude, and vary in magnitude relative to each other systematically across the water column.

A two-component phenomenology for homogeneous magnetohydrodynamic turbulence

A one-point closure model for energy decay in three-dimensional magnetohydrodynamic (MHD) turbulence is developed. The model allows for influence of a large-scale magnetic field that may be of strength sufficient to induce Alfvén wave propagation effects, and takes into account components of turbulence in which either the wave-like character is negligible or is dominant. This two-component model evolves energy and characteristic length scales, and may be useful as a simple description of homogeneous MHD turbulent decay. In concert with spatial transport models, it can form the basis for approximate treatment of low-frequency plasma turbulence in a variety of solar, space, and astrophysical contexts.

Study of the oscillatory regime of a turbulent plane jet impinging in a rectangular cavity

The present work considers the general case of a turbulent plane jet flowing into a rectangular cavity. The study is relevant to a wide range of practical applications including forced convection, renewal of fluid inside a cavity and flowmeters. The experimental study considers the effect of the confinement on the jet characteristics. All the measurements are made using hot wire anemometry and complemented by visualisation of the various observed flow patterns. Numerical modelling has been carried out using two statistical models of the turbulence: the standard k– ε model and a two-scale energy–flux model. The key feature of the multiple scale energy–flux model is the splitting of the turbulence spectrum devised for non-equilibrium turbulence modelling. Three flow regimes are observed depending on the location of the jet exit inside the cavity: oscillatory, transitional and steady. The oscillatory flow characteristics have been analysed from the structural and parametric points of view and the underlying mechanisms are discussed. The one-point closure models proved to be able to reproduce the measured unsteady behaviour of the flow and some improvements are obtained using the multiple scale concept.

ONE-POINT CLOSURE MODELS FOR BUOYANCY-DRIVEN TURBULENT FLOWS

▪ Abstract Turbulent flows driven by thermal buoyancy are featured by phenomena that pose a special challenge to conventional one-point closure models. Inherent unsteadiness, energy nonequilibrium, counter-gradient diffusion, strong pressure fluctuations, and lack of universal scaling, all believed to be associated with distinct large-scale coherent eddy structures, are hardly tractable by Reynolds-type averaging. Second-moment closures, though inadequate for providing information on eddy structure, offer better prospects than eddy-viscosity models for capturing at least some of the phenomena. For some configurations (e.g., with heating from below), unsteady computational solutions of ensemble-averaged equations, using a one-point closure as the subscale model, may be unavoidable for accurate prediction of flow details and wall heat transfer. This article reviews the rationale and some specific modeling issues related to buoyant flows within the realm of one-point closures. The inadequacy of isotropic eddy-diffusivity models is discussed first, followed by the rationale of the second-moment modeling and its term-by-term scrutiny based on direct numerical simulations (DNS). Algebraic models based on a rational truncation of the differential second-moment closure are proposed as the minimum closure level for complex flows. These closures are also recommended as subscale models for transient statistical modeling (T-RANS) and very large eddy simulations (VLES). Examples of applications illustrate some recent achievements.

Ocean Turbulence. Part I: One-Point Closure Model—Momentum and Heat Vertical Diffusivities

Since the early forties, one-point turbulence closure models have been the canonical tools used to describe turbulent flows in many fields. In geophysics, Mellor and Yamada applied such models using the 1980 state-of-the art. Since then, no improvements were introduced to alleviate two major difficulties: 1) closure of the pressure correlations, which affects the correct determination of the critical Richardson number Ri(sub cr) above which turbulent mixing is no longer possible and 2) the need to express the non-local third-order moments (TOM) in terms of lower order moments rather than via the down-gradient approximation as done thus far, since the latter seriously underestimates the TOMs. Since 1) and 2) are still being dealt with adjustable parameters which weaken the credibility of the models, alternative models, not based on turbulence modeling, have been suggested. The aim of this paper is to show that new information, partly derived from the newest 2-point closure model discussed, can be used to solve these shortcomings. The new one-point closure model, which in its simplest form is algebraic and thus simple to implement, is first shown to reproduce a variety of data. Then, it is used in a Ocean-General Circulation Model (O-GCM) where it reproduces well a large variety of ocean data. While phenomenological models are specifically tuned to ocean turbulence, the present model is not. It is first tested against laboratory data on stably stratified flows and then used in an O-GCM. It is more general, more predictive and more resilient, e.g., it can incorporate phenomena like wave-breaking at the surface, salinity diffusivity, non-locality, etc. One important feature that naturally comes out of the new model is that the predicted Richardson critical value Ri(sub cr) is Ri (sub cr approx. = 1) in agreement with both Large Eddy Simulations (LES) and empirical evidence while all previous models predicted Ri (sub cr approx. = 0.2) which led to a considerable underestimate of the extent of turbulent mixing and thus to an incorrect mixed layer depth. The predicted temperature and salinity profiles (vs. depth) are presented and compared with those of the Kolmogorov-Petruvsky-Piskunuv (KPP) model and Levitus data.

Turbulent Convection: Old and New Models

view Abstract Citations (40) References (22) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Turbulent Convection: Old and New Models Canuto, V. M. Abstract This paper contains (1) a physical argument to show that the one-eddy MLT model underestimates the convective flux Fc in the high-efficiency regime, while it overestimates Fc in the low-efficiency regime, and (2) a new derivation of the Fc(MLT) using a turbulence model in the one-eddy approximation. (3) We forsake the one-eddy approximation and adopt the Kolmogorov spectrum to represent the turbulent energy spectrum. The resulting Fc > Fc(MLT) in the high-efficiency regime, and Fc <Fc(MLT) in the low-efficiency case, are in agreement with the physical arguments concerning the one-eddy MLT model. (4) By forsaking the Kolmogorov model and solving a two-point closure model, one obtains the CM model. The Fc(CM) satisfies (1). Fc(CM) corresponds to a "tilt" in efficiency space of Fc(MLT), an effect that cannot be achieved by changing α. We discuss the astrophysical tests of the CM model. (5) Concerning the laboratory turbulent convection, we show that the CM model provides a better fit than the MLT to recent high Rayleigh number (Ra) laboratory data on convection. (6) Concerning nonlocal convection, the most complete model available is the one-point closure model (Reynolds stress model), which entails five differential equations for the five second-order moments. We present the solution corresponding to the local, stationary case. The results are expressed analytically in terms of Ko (Kolmogorov constant), Pe (Peclet number), and S (convective efficiency). (7) We find that the superadiabatic temperature gradient is given by - ∂T/∂r - cp-1gr where the renormalized gr = g(1 + g-1p-1dpt/dz) and Pt is the turbulent pressure. This result, which follows naturally from the Reynolds stress approach, contrasts with previous empirical suggestions to include Pt. (8) We derive new expressions for the turbulence pressure using two different turbulence models and (9) we show that the often used Kolmogorov-Prandtl expression for the turbulent diffusivity is valid only in the high convective efficiency limit. We derive a new expression valid for arbitrary Peclet numbers. (10) We derive an expression for the flux conservation law, which includes F(KE), the flux of turbulent kinetic energy, a third-order moment for which we provide a new expression. (11) No model has thus far accounted for the influence on Fc due to the presence of a stable layer (radiative layer) bordering the convective zone. We work out the first such model, and (12) we discuss topics for future research. Publication: The Astrophysical Journal Pub Date: August 1996 DOI: 10.1086/177613 Bibcode: 1996ApJ...467..385C Keywords: CONVECTION; STARS: INTERIORS; TURBULENCE full text sources ADS |

Advanced turbulence closure models: a view of current status and future prospects

This report presents the status and reviews some of the current activities in the development of advanced one-point turbulence closure models with special reference to the modeling and computation of turbulent wall flows. It gives a brief retrospective of the development of the one-point closure model and highlights major achievementss. It also recalls some of the pertinent issues that are still open and that pose challenges to modelers. Some pertaining deficiencies, which have been obvious since the very beginning of the development and are still present, are discussed, together with some marked differences in the views of various groups of modelers. The review is limited to conventional one-point two-equation Eddy Viscosity Models (EVMs) and Differential Second-Moment (Re-stress) Models (DSMs). It does not deal with any spectral approach, nor partial field models, such as subgrid modeling or similar methods used within the framework of other simulation techniques. Some new approaches, such as the Renormalization group theory (RNG), are mentioned briefly and only insofar as they may inspire modifications and improvements of conventional models. The report deals with general aspects of modeling complex turbulent flows, but is restricted to incompressible (or mildly compressible) fluid, with particular focus on computation of wall flows.

Evaluation of Simple Subgrid-Scale Models for the Numerical Simulation of Homogeneous Turbulence

Several subgrid-scale models are evaluated in a 163 large eddy simulation (16 discretization points in each space direction) of isotropic and anisotropic turbulence and will be used in a 643 numerical simulation of homogeneous turbulence submitted to two successive plane strains. An inexpensive numerical code, implemented on a parallel structure system, has been used with simple eddy-viscosity subgrid-scale models. Decaying isotropic turbulence has been simulated with a 16 3 version of the code. Comparisons with experiment show the necessity for the eddy viscosity to allow both a strong local energy transfer near the cutoff wave number and a time-dependent nonlocal transfer. In the case of homogeneous turbulence submitted to two successive plane strains, time evolutions of turbulence energy and anisotropy obtained with 643 full-turbulence simulations (FTS), 163 large eddy simulations (LES), and classical one-point closure models compare well with experiment. The reorientation process of the Reynolds stress tensor during and after the second deformation is well simulated. The reorientation process of the slow-pressure/strain tensor (not accessible by experiment) is also described: it shows, as proposed by some authors, that the quantity to be modeled is a combination of the dissipation and slow-pressure/strain tensors. Some numerical difficulties due to the presence of strong deformation rates are identified and minimized by a judicious choice of the initial geometry of the predistorted grid.