Journal articles on the topic 'Turbulence Closures'
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1
Pearson, Brodie C., Alan L. M. Grant, and Jeff A. Polton. "Pressure–strain terms in Langmuir turbulence." Journal of Fluid Mechanics 880 (October 7, 2019): 5–31. http://dx.doi.org/10.1017/jfm.2019.701.
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This study investigates the pressure–strain tensor ($\unicode[STIX]{x1D72B}$) in Langmuir turbulence. The pressure–strain tensor is determined from large-eddy simulations (LES), and is partitioned into components associated with the mean current shear (rapid), the Stokes shear and the turbulent–turbulent (slow) interactions. The rapid component can be parameterized using existing closure models, although the coefficients in the closure models are particular to Langmuir turbulence. A closure model for the Stokes component is proposed, and it is shown to agree with results from the LES. The slow component of $\unicode[STIX]{x1D72B}$ does not agree with existing ‘return-to-isotropy’ closure models for five of the six components of the Reynolds stress tensor, and a new closure model is proposed that accounts for these deviations which vary systematically with Langmuir number, $La_{t}$, and depth. The implications of these results for second- and first-order closures of Langmuir turbulence are discussed.
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Larsen, Bjarke Eltard, and David R. Fuhrman. "On the over-production of turbulence beneath surface waves in Reynolds-averaged Navier–Stokes models." Journal of Fluid Mechanics 853 (August 23, 2018): 419–60. http://dx.doi.org/10.1017/jfm.2018.577.
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In previous computational fluid dynamics studies of breaking waves, there has been a marked tendency to severely over-estimate turbulence levels, both pre- and post-breaking. This problem is most likely related to the previously described (though not sufficiently well recognized) conditional instability of widely used turbulence models when used to close Reynolds-averaged Navier–Stokes (RANS) equations in regions of nearly potential flow with finite strain, resulting in exponential growth of the turbulent kinetic energy and eddy viscosity. While this problem has been known for nearly 20 years, a suitable and fundamentally sound solution has yet to be developed. In this work it is demonstrated that virtually all commonly used two-equation turbulence closure models are unconditionally, rather than conditionally, unstable in such regions. A new formulation of the $k$–$\unicode[STIX]{x1D714}$ closure is developed which elegantly stabilizes the model in nearly potential flow regions, with modifications remaining passive in sheared flow regions, thus solving this long-standing problem. Computed results involving non-breaking waves demonstrate that the new stabilized closure enables nearly constant form wave propagation over long durations, avoiding the exponential growth of the eddy viscosity and inevitable wave decay exhibited by standard closures. Additional applications on breaking waves demonstrate that the new stabilized model avoids the unphysical generation of pre-breaking turbulence which widely plagues existing closures. The new model is demonstrated to be capable of predicting accurate pre- and post-breaking surface elevations, as well as turbulence and undertow velocity profiles, especially during transition from pre-breaking to the outer surf zone. Results in the inner surf zone are similar to standard closures. Similar methods for formally stabilizing other widely used closure models ($k$–$\unicode[STIX]{x1D714}$ and $k$–$\unicode[STIX]{x1D700}$ variants) are likewise developed, and it is recommended that these be utilized in future RANS simulations of surface waves. (In the above $k$ is the turbulent kinetic energy density, $\unicode[STIX]{x1D714}$ is the specific dissipation rate, and $\unicode[STIX]{x1D700}$ is the dissipation.)
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Gladskikh, Daria, Lev Ostrovsky, Yuliya Troitskaya, Irina Soustova, and Evgeny Mortikov. "Turbulent Transport in a Stratified Shear Flow." Journal of Marine Science and Engineering 11, no. 1 (January 6, 2023): 136. http://dx.doi.org/10.3390/jmse11010136.
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Within the framework of the theory of unsteady turbulent flows in a stratified fluid, a new parameterization of the turbulent Prandtl number is proposed. The parameterization is included in the k-ε-closure and used within the three-dimensional model of thermohydrodynamics of an enclosed water body where density distribution includes pycnocline. This allows us to describe turbulence in a stratified shear flow without the restrictions associated with the gradient Richardson number and justify the choice of closure constants. Numerical experiments, where the downward penetration of turbulence was considered, confirm the advantage of the developed approach in describing the effects neglected in the classical closures.
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Mauritsen, Thorsten, Gunilla Svensson, Sergej S. Zilitinkevich, Igor Esau, Leif Enger, and Branko Grisogono. "A Total Turbulent Energy Closure Model for Neutrally and Stably Stratified Atmospheric Boundary Layers." Journal of the Atmospheric Sciences 64, no. 11 (November 1, 2007): 4113–26. http://dx.doi.org/10.1175/2007jas2294.1.
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Abstract This paper presents a turbulence closure for neutral and stratified atmospheric conditions. The closure is based on the concept of the total turbulent energy. The total turbulent energy is the sum of the turbulent kinetic energy and turbulent potential energy, which is proportional to the potential temperature variance. The closure uses recent observational findings to take into account the mean flow stability. These observations indicate that turbulent transfer of heat and momentum behaves differently under very stable stratification. Whereas the turbulent heat flux tends toward zero beyond a certain stability limit, the turbulent stress stays finite. The suggested scheme avoids the problem of self-correlation. The latter is an improvement over the widely used Monin–Obukhov-based closures. Numerous large-eddy simulations, including a wide range of neutral and stably stratified cases, are used to estimate likely values of two free constants. In a benchmark case the new turbulence closure performs indistinguishably from independent large-eddy simulations.
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Harcourt, Ramsey R. "An Improved Second-Moment Closure Model of Langmuir Turbulence." Journal of Physical Oceanography 45, no. 1 (January 2015): 84–103. http://dx.doi.org/10.1175/jpo-d-14-0046.1.
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AbstractA prior second-moment closure (SMC) model of Langmuir turbulence in the upper ocean is modified by introduction of inhomogeneous pressure–strain rate and pressure–scalar gradient closures that are similar to the high Reynolds number, near-wall treatments for solid wall boundaries. This repairs several near-surface defects in the algebraic Reynolds stress model (ARSM) of the prior SMC by redirecting Craik–Leibovich (CL) vortex force production of turbulent kinetic energy out of the surface-normal vertical component and into a horizontal one, with an associated reduction in near-surface CL production of vertical momentum flux. A surface-proximity function introduces a new closure parameter that is tuned to previous results from large-eddy simulations (LES), and a numerical SMC model based on stability functions from the new ARSM produces improved comparisons with mean profiles of momentum and TKE components from steady-state LES results forced by aligned wind and waves. An examination of higher-order quasi-homogeneous closures and a numerical simulation of Langmuir turbulence away from the boundaries both show the near-surface inhomogeneous closure to be both necessary for consistency and preferable for simplicity.
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Mortikov, E. V., A. V. Glazunov, A. V. Debolskiy, V. N. Lykosov, and S. S. Zilitinkevich. "On the modelling of the dissipation rate of turbulent kinetic energy." Доклады Академии наук 489, no. 4 (December 10, 2019): 414–18. http://dx.doi.org/10.31857/s0869-56524894414-418.
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We consider a relaxation equation for turbulence wavenumber for use in semi-empirical turbulence closures. It is shown that the well-known phenomenological equation for the dissipation rate of turbulent kinetic energy can be related to this relaxation equation as a close approximation of the latter for stably stratified quasi-stationary flows. The proposed approach allows for more physically found definition of the empirical constants and improvement of atmospheric and oceanic boundary layer turbulence closures by using direct numerical and large eddy simulation data to define equilibrium states and relaxation mechanisms.
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GODEFERD, F. S., C. CAMBON, and J. F. SCOTT. "Two-point closures and their applications: report on a workshop." Journal of Fluid Mechanics 436 (June 10, 2001): 393–407. http://dx.doi.org/10.1017/s0022112001004359.
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This international scientific workshop was organized in Lyon, France, from 10 to 12 May 2000. Its focus was ‘Two-point closures and their applications’, with the understanding that the analysis and design of such models requires expert knowledge coming from a wide range of areas in turbulence research, e.g. experiments, numerical simulations, asymptotic models, etc.In the global challenge of turbulence modelling, two-point closures prove useful in many ways. Two-point correlations and spectra are useful measures of the distortion of the eddy structure of turbulence by stratification, large-scale strains, rotation, etc. In some cases, e.g. near boundaries, spectra can be drastically changed. In addition to the accurate characterization of turbulence, the explicit computation of two-point correlations or spectra shows how the internal dynamics of the various scales of motion are affected by such distortion, especially the cascade process on which the production/dissipation relationship depends. Distortion can be the cause of large departures from isotropic homogeneous turbulence, pulling turbulent flows far away from the local equilibrium that is often assumed. A rather weak departure can allow the use of linearized theories such as rapid distortion theory, for the applicability of which rational bounds may be estimated by comparisons with weakly nonlinear calculations. A different approach is necessary when dealing with larger departures, for instance due to growth of instabilities. In that case new physical or similarity arguments have to be employed to obtain a satisfactory description of the modification to the cascade process, which can even undergo reversal in the limit when three-dimensional turbulence becomes two-dimensional. Of course, significant changes in spectra have direct implications for one-point measures of turbulence – which can be explicitly derived by integration of two-point correlations – used in most industrial closure schemes. Such one-point models consequently need to be adapted when turbulence is strongly affected by distortion.
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Yang, S. L., B. D. Peschke, and K. Hanjalic. "Second-Moment Closure Model for IC Engine Flow Simulation Using Kiva Code1." Journal of Engineering for Gas Turbines and Power 122, no. 2 (August 31, 1999): 355–63. http://dx.doi.org/10.1115/1.483213.
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The flow and turbulence in an IC engine cylinder were studied using the SSG variant of the Reynolds stress turbulence closure model. In-cylinder turbulence is characterized by strong turbulence anisotropy and flow rotation, which aid in air-fuel mixing. It is argued that solving the differential transport equations for each turbulent stress tensor component, as implied by second-moment closures, can better reproduce stress anisotropy and effects of rotation, than with eddy-viscosity models. Therefore, a Reynolds stress model that can meet the demands of in-cylinder flows was incorporated into an engine flow solver. The solver and SSG turbulence model were first successfully tested with two different validation cases. Finally, simulations were applied to IC-engine like geometries. The results showed that the Reynolds stress model predicted additional flow structures and yielded less diffusive profiles than those predicted by an eddy-viscosity model. [S0742-4795(00)00101-0]
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Pacciani, Roberto, Michele Marconcini, Francesco Bertini, Simone Rosa Taddei, Ennio Spano, Yaomin Zhao, Harshal D. Akolekar, Richard D. Sandberg, and Andrea Arnone. "Assessment of Machine-Learned Turbulence Models Trained for Improved Wake-Mixing in Low-Pressure Turbine Flows." Energies 14, no. 24 (December 10, 2021): 8327. http://dx.doi.org/10.3390/en14248327.
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This paper presents an assessment of machine-learned turbulence closures, trained for improving wake-mixing prediction, in the context of LPT flows. To this end, a three-dimensional cascade of industrial relevance, representative of modern LPT bladings, was analyzed, using a state-of-the-art RANS approach, over a wide range of Reynolds numbers. To ensure that the wake originates from correctly reproduced blade boundary-layers, preliminary analyses were carried out to check for the impact of transition closures, and the best-performing numerical setup was identified. Two different machine-learned closures were considered. They were applied in a prescribed region downstream of the blade trailing edge, excluding the endwall boundary layers. A sensitivity analysis to the distance from the trailing edge at which they are activated is presented in order to assess their applicability to the whole wake affected portion of the computational domain and outside the training region. It is shown how the best-performing closure can provide results in very good agreement with the experimental data in terms of wake loss profiles, with substantial improvements relative to traditional turbulence models. The discussed analysis also provides guidelines for defining an automated zonal application of turbulence closures trained for wake-mixing predictions.
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Frederiksen, Jorgen S., and Terence J. O’Kane. "Statistical Dynamics of Mean Flows Interacting with Rossby Waves, Turbulence, and Topography." Fluids 7, no. 6 (June 9, 2022): 200. http://dx.doi.org/10.3390/fluids7060200.
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Abridged statistical dynamical closures, for the interaction of two-dimensional inhomogeneous turbulent flows with topography and Rossby waves on a beta–plane, are formulated from the Quasi-diagonal Direct Interaction Approximation (QDIA) theory, at various levels of simplification. An abridged QDIA is obtained by replacing the mean field trajectory, from initial-time to current-time, in the time history integrals of the non-Markovian closure by the current-time mean field. Three variants of Markovian Inhomogeneous Closures (MICs) are formulated from the abridged QDIA by using the current-time, prior-time, and correlation fluctuation dissipation theorems. The abridged MICs have auxiliary prognostic equations for relaxation functions that approximate the information in the time history integrals of the QDIA. The abridged MICs are more efficient than the QDIA for long integrations with just two relaxation functions required. The efficacy of the closures is studied in 10-day simulations with an easterly large-scale flow impinging on a conical mountain to generate rapidly growing Rossby waves in a turbulent environment. The abridged closures closely agree with the statistics of large ensembles of direct numerical simulations for the mean and transients. An Eddy Damped Markovian Inhomogeneous Closure (EDMIC), with analytical relaxation functions, which generalizes the Eddy Dampened Quasi Normal Markovian (EDQNM) to inhomogeneous flows, is formulated and shown to be realizable under the same circumstances as the homogeneous EDQNM.
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Kurowski, Marcin J., and João Teixeira. "A Scale-Adaptive Turbulent Kinetic Energy Closure for the Dry Convective Boundary Layer." Journal of the Atmospheric Sciences 75, no. 2 (February 1, 2018): 675–90. http://dx.doi.org/10.1175/jas-d-16-0296.1.
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Abstract A pragmatic scale-adaptive turbulent kinetic energy (TKE) closure is proposed to simulate the dry convective boundary layer for a variety of horizontal grid resolutions: from 50 m, typical of large-eddy simulation models that use three-dimensional turbulence parameterizations/closures, up to 100 km, typical of climate models that use one-dimensional turbulence and convection parameterizations/closures. Since parameterizations/closures using the TKE approach have been frequently used in these two asymptotic limits, a simple method is proposed to merge them with a mixing-length-scale formulation for intermediate resolutions. This new scale-adaptive mixing length naturally increases with increasing grid length until it saturates as the grid length reaches mesoscale-model resolution. The results obtained using this new approach for dry convective boundary layers are promising. The mean vertical profiles of potential temperature and heat flux remain in good agreement for different resolutions. A continuous transition (in terms of resolution) across the gray zone is illustrated through the partitioning between the model-resolved and the subgrid-scale transports as well as by documenting the transition of the subgrid-scale TKE source/sink terms. In summary, a natural and continuous transition across resolutions (from 50 m to 100 km) is obtained, for dry convection, using exactly the same atmospheric model for all resolutions with a simple scale-adaptive mixing-length formulation.
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Lin, F. B., and F. Sotiropoulos. "Strongly-Coupled Multigrid Method for 3-D Incompressible Flows Using Near-Wall Turbulence Closures." Journal of Fluids Engineering 119, no. 2 (June 1, 1997): 314–24. http://dx.doi.org/10.1115/1.2819136.
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An efficient artificial compressibility algorithm is developed for solving the three-dimensional Reynolds-averaged Navier-Stokes equations in conjunction with the low-Reynolds number k-ω turbulence model (Wilcox, 1994). Two second-order accurate central-differencing schemes, with scalar and matrix-valued artificial dissipation, respectively, and a third-order accurate flux-difference splitting upwind scheme are implemented for discretizing the convective terms. The discrete equations are integrated in time using a Runge-Kutta algorithm enhanced with local time stepping, implicit residual smoothing, and V-cycle multigrid acceleration with full- and semi-coarsening capabilities. Both loosely and strongly-coupled strategies for solving the turbulence closure equations are developed and their relative efficiency is evaluated. Calculations are carried out for turbulent flow through a strongly-curved 180 deg pipe bend discretized with fine, highly-stretched and skewed meshes. It is shown that the strongly-coupled multigrid algorithm, with semi-coarsening in the transverse plane, is an efficient approach for simulating flows of practical interest with advanced near-wall turbulence closures.
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Yoo, G. J., R. M. C. So, and B. C. Hwang. "Calculation of Developing Turbulent Flows in a Rotating Pipe." Journal of Turbomachinery 113, no. 1 (January 1, 1991): 34–41. http://dx.doi.org/10.1115/1.2927735.
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Internal rotating boundary-layer flows are strongly influenced by large circumferential strain and the turbulence field is anisotropic. This is especially true in the entry region of a rotating pipe where the flow is three dimensional, the centrifugal force due to fluid rotation is less important, and the circumferential strain created by surface rotation has a significant effect on the turbulence field near the wall. Consequently, viscous effects cannot be neglected in the near-wall region. Several low-Reynolds-number turbulence closures are proposed for the calculation of developing rotating pipe flows. Some are two-equation closures with and without algebraic stress correction, while others are full Reynolds-stress closures. It is found that two-equation closures with and without algebraic stress correction are totally inadequate for this three-dimensional flow, while Reynolds-stress closures give results that are in good agreement with measurements over a wide range of rotation numbers.
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de Divitiis, Nicola. "Statistical Lyapunov Theory Based on Bifurcation Analysis of Energy Cascade in Isotropic Homogeneous Turbulence: A Physical–Mathematical Review." Entropy 21, no. 5 (May 23, 2019): 520. http://dx.doi.org/10.3390/e21050520.
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This work presents a review of previous articles dealing with an original turbulence theory proposed by the author and provides new theoretical insights into some related issues. The new theoretical procedures and methodological approaches confirm and corroborate the previous results. These articles study the regime of homogeneous isotropic turbulence for incompressible fluids and propose theoretical approaches based on a specific Lyapunov theory for determining the closures of the von Kármán–Howarth and Corrsin equations and the statistics of velocity and temperature difference. While numerous works are present in the literature which concern the closures of the autocorrelation equations in the Fourier domain (i.e., Lin equation closure), few articles deal with the closures of the autocorrelation equations in the physical space. These latter, being based on the eddy–viscosity concept, describe diffusive closure models. On the other hand, the proposed Lyapunov theory leads to nondiffusive closures based on the property that, in turbulence, contiguous fluid particles trajectories continuously diverge. Therefore, the main motivation of this review is to present a theoretical formulation which does not adopt the eddy–viscosity paradigm and summarizes the results of the previous works. Next, this analysis assumes that the current fluid placements, together with velocity and temperature fields, are fluid state variables. This leads to the closures of the autocorrelation equations and helps to interpret the mechanism of energy cascade as due to the continuous divergence of the contiguous trajectories. Furthermore, novel theoretical issues are here presented among which we can mention the following ones. The bifurcation rate of the velocity gradient, calculated along fluid particles trajectories, is shown to be much larger than the corresponding maximal Lyapunov exponent. On that basis, an interpretation of the energy cascade phenomenon is given and the statistics of finite time Lyapunov exponent of the velocity gradient is shown to be represented by normal distribution functions. Next, the self–similarity produced by the proposed closures is analyzed and a proper bifurcation analysis of the closed von Kármán–Howarth equation is performed. This latter investigates the route from developed turbulence toward the non–chaotic regimes, leading to an estimate of the critical Taylor scale Reynolds number. A proper statistical decomposition based on extended distribution functions and on the Navier–Stokes equations is presented, which leads to the statistics of velocity and temperature difference.
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Yu, Li-ren, and Jun Yu. "ENVIRONMENTAL FLOW AND CONTAMINANT TRANSPORT MODELING IN THE AMAZONIAN WATER SYSTEM BY USING Q3DRM1.0 SOFTWARE." International Journal of Research -GRANTHAALAYAH 5, no. 12 (July 3, 2020): 377–91. http://dx.doi.org/10.29121/granthaalayah.v5.i12.2017.525.
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This paper reports a fine numerical simulation of environmental flow and contaminant transport in the Amazonian water system near the Anamã City, Brazil, solved by the Q3drm1.0 software, developed by the Authors, which can provide the different closures of three depth-integrated two-equation turbulence models. The purpose of this simulation is to refinedly debug and test the developed software, including the mathematical model, turbulence closure models, adopted algorithms, and the developed general-purpose computational codes as well as graphical user interfaces (GUI). The three turbulence models, provided by the developed software to close non-simplified quasi three-dimensional hydrodynamic fundamental governing equations, include the traditional depth-integrated two-equation turbulence model, the depth-integrated two-equation turbulence model, developed previously by the first Author of the paper, and the depth-integrated two-equation turbulence model, developed recently by the Authors of this paper. The numerical simulation of this paper is to solve the corresponding discretized equations with collocated variable arrangement on the non-orthogonal body-fitted coarse and fine two-levels’ grids. With the help of Q3drm1.0 software, the steady environmental flows and transport behaviours have been numerically investigated carefully; and the processes of contaminant inpouring as well as plume development, caused by the side-discharge from a tributary of the south bank (the right bank of the river), were also simulated and discussed in detail. Although the three turbulent closure models, used in this calculation, are all applicable to the natural rivers with strong mixing, the comparison of the computational results by using the different turbulence closure models shows that the turbulence model with larger turbulence parameter provides the possibility for improving the accuracy of the numerical computations of practical problems.
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Yan, Chao, and James G. McDonald. "Hyperbolic turbulence models for moment closures." Journal of Computational Physics 422 (December 2020): 109753. http://dx.doi.org/10.1016/j.jcp.2020.109753.
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Fitch, A. C. "An Improved Double-Gaussian Closure for the Subgrid Vertical Velocity Probability Distribution Function." Journal of the Atmospheric Sciences 76, no. 1 (January 1, 2019): 285–304. http://dx.doi.org/10.1175/jas-d-18-0149.1.
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Abstract The vertical velocity probability distribution function (PDF) is analyzed throughout the depth of the lower atmosphere, including the subcloud and cloud layers, in four large-eddy simulation (LES) cases of shallow cumulus and stratocumulus. Double-Gaussian PDF closures are examined to test their ability to represent a wide range of turbulence statistics, from stratocumulus cloud layers characterized by Gaussian turbulence to shallow cumulus cloud layers displaying strongly non-Gaussian turbulence statistics. While the majority of the model closures are found to perform well in the former case, the latter presents a considerable challenge. A new model closure is suggested that accounts for high skewness and kurtosis seen in shallow cumulus cloud layers. The well-established parabolic relationship between skewness and kurtosis is examined, with results in agreement with previous studies for the subcloud layer. In cumulus cloud layers, however, a modified relationship is necessary to improve performance. The new closure significantly improves the estimation of the vertical velocity PDF for shallow cumulus cloud layers, in addition to performing well for stratocumulus. In particular, the long updraft tail representing the bulk of cloudy points is much better represented and higher-order moments diagnosed from the PDF are also greatly improved. However, some deficiencies remain owing to fundamental limitations of representing highly non-Gaussian turbulence statistics with a double-Gaussian PDF.
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Frederiksen, Jorgen S., and Terence J. O’Kane. "Markovian inhomogeneous closures for Rossby waves and turbulence over topography." Journal of Fluid Mechanics 858 (October 31, 2018): 45–70. http://dx.doi.org/10.1017/jfm.2018.784.
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Manifestly Markovian closures for the interaction of two-dimensional inhomogeneous turbulent flows with Rossby waves and topography are formulated and compared with large ensembles of direct numerical simulations (DNS) on a generalized $\unicode[STIX]{x1D6FD}$-plane. Three versions of the Markovian inhomogeneous closure (MIC) are established from the quasi-diagonal direct interaction approximation (QDIA) theory by modifying the response function to a Markovian form and employing respectively the current-time (quasi-stationary) fluctuation dissipation theorem (FDT), the prior-time (non-stationary) FDT and the correlation FDT. Markov equations for the triad relaxation functions are derived that carry similar information to the time-history integrals of the non-Markovian QDIA closure but become relatively more efficient for long integrations. Far from equilibrium processes are studied, where the impact of a westerly mean flow on a conical mountain generates large-amplitude Rossby waves in a turbulent environment, over a period of 10 days. Excellent agreement between the evolved mean streamfunction and mean and transient kinetic energy spectra are found for the three versions of the MIC and two variants of the non-Markovian QDIA compared with an ensemble of 1800 DNS. In all cases mean Rossby wavetrain pattern correlations between the closures and the DNS ensemble are greater than 0.9998.
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Fisher, Alexander W., Lawrence P. Sanford, and Malcolm E. Scully. "Wind-Wave Effects on Estuarine Turbulence: A Comparison of Observations and Second-Moment Closure Predictions." Journal of Physical Oceanography 48, no. 4 (April 2018): 905–23. http://dx.doi.org/10.1175/jpo-d-17-0133.1.
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AbstractObservations of turbulent kinetic energy, dissipation, and turbulent stress were collected in the middle reaches of Chesapeake Bay and were used to assess second-moment closure predictions of turbulence generated beneath breaking waves. Dissipation scaling indicates that the turbulent flow structure observed during a 10-day wind event was dominated by a three-layer response that consisted of 1) a wave transport layer, 2) a surface log layer, and 3) a tidal, bottom boundary layer limited by stable stratification. Below the wave transport layer, turbulent mixing was limited by stable stratification. Within the wave transport layer, where dissipation was balanced by a divergence in the vertical turbulent kinetic energy flux, the eddy viscosity was significantly underestimated by second-moment turbulence closure models, suggesting that breaking waves homogenized the mixed surface layer to a greater extent than the simple model of TKE diffusing away from a source at the surface. While the turbulent transport of TKE occurred largely downgradient, the intermittent downward sweeps of momentum generated by breaking waves occurred largely independent of the mean shear. The underprediction of stress in the wave transport layer by second-moment closures was likely due to the inability of the eddy viscosity model to capture the nonlocal turbulent transport of the momentum flux beneath breaking waves. Finally, the authors hypothesize that large-scale coherent turbulent eddies played a significant role in transporting momentum generated near the surface to depth.
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Borello, Domenico, Kemal Hanjalic, and Franco Rispoli. "Prediction of Cascade Flows With Innovative Second-Moment Closures." Journal of Fluids Engineering 127, no. 6 (July 11, 2005): 1059–70. http://dx.doi.org/10.1115/1.2073267.
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We report on the performances of two second-moment turbulence closures in predicting turbulence and laminar-to-turbulent transition in turbomachinery flows. The first model considered is the one by Hanjalic and Jakirlic (HJ) [Comput. Fluids, 27(2), pp. 137–156 (1998)], which follows the conventional approach with damping functions to account for the wall viscous and nonviscous effect. The second is an innovative topology-free elliptic blending model, EBM [R. Manceau and K. Hanjalic, Phys. Fluids, 14(3), pp. 1–11 (2002)], here presented in a revised formulation. An in-house finite element code based on a parallel technique is used for solving the equation set [Borello et al., Comput. Fluids, 32, pp. 1017–1047 (2003)]. The test cases under scrutiny are the transitional flow on a flat plate with circular leading edge (T3L ERCOFTAC-TSIG), and the flow around a double circular arc (DCA) compressor cascade in quasi-off-design condition (i=−1.5°) [Zierke and Deutsch, NASA Contract Report 185118 (1989)]. The comparison between computations and experiments shows a satisfactory performance of the HJ model in predicting complex turbomachinery flows. The EBM also exhibits a fair level of accuracy, though it is less satisfactory in transition prediction. Nevertheless, in view of the robustness of the numerical formulation, the relative insensitivity to grid refinement, and the absence of topology-dependent parameters, the EBM is identified as an attractive second-moment closure option for computation of complex 3D turbulent flows in realistic turbomachinery configurations.
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Umlauf, Lars. "The Description of Mixing in Stratified Layers without Shear in Large-Scale Ocean Models." Journal of Physical Oceanography 39, no. 11 (November 1, 2009): 3032–39. http://dx.doi.org/10.1175/2009jpo4006.1.
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Abstract Large-scale geophysical flows often exhibit layers with negligible vertical shear and infinite gradient Richardson number Ri. It is well known that these layers may be regions of active mixing, even in the absence of local shear production of turbulence because, among other processes, turbulence may be supplied by vertical turbulent transport from neighboring regions. This observation is contrasted by the behavior of most turbulence parameterizations used in ocean climate modeling, predicting the collapse of mixing of mass and matter if the Richardson number exceeds a critical threshold. Here, the performance of a simple model without critical Richardson number is evaluated, taking into account the diffusion of turbulence into layers without shear production and therefore avoiding the suppression of mixing at large values of Ri. The model is based on the framework of second-moment turbulence closures, focusing on the consistent modeling of the turbulent length scale for strongly stratified turbulence. Results are compared to eddy-resolving simulations of stratified shear flows that have recently become available. The model is simple enough for inclusion in ocean climate models.
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BROCCHINI, M., and D. H. PEREGRINE. "The dynamics of strong turbulence at free surfaces. Part 2. Free-surface boundary conditions." Journal of Fluid Mechanics 449 (December 12, 2001): 255–90. http://dx.doi.org/10.1017/s0022112001006024.
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Strong turbulence at a water–air free surface can lead to splashing and a disconnected surface as in a breaking wave. Averaging to obtain boundary conditions for such flows first requires equations of motion for the two-phase region. These are derived using an integral method, then averaged conservation equations for mass and momentum are obtained along with an equation for the turbulent kinetic energy in which extra work terms appear. These extra terms include both the mean pressure and the mean rate of strain and have similarities to those for a compressible fluid. Boundary conditions appropriate for use with averaged equations in the body of the water are obtained by integrating across the two-phase surface layer.A number of ‘new’ terms arise for which closure expressions must be found for practical use. Our knowledge of the properties of strong turbulence at a free surface is insufficient to make such closures. However, preliminary discussions are given for two simplified cases in order to stimulate further experimental and theoretical studies.Much of the turbulence in a spilling breaker originates from its foot where turbulent water meets undisturbed water. A discussion of averaging at the foot of a breaker gives parameters that may serve to measure the ‘strength’ of a breaker.
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Zhou, Bowen, and Fotini Katopodes Chow. "Nested Large-Eddy Simulations of the Intermittently Turbulent Stable Atmospheric Boundary Layer over Real Terrain." Journal of the Atmospheric Sciences 71, no. 3 (February 27, 2014): 1021–39. http://dx.doi.org/10.1175/jas-d-13-0168.1.
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Abstract The nighttime stable atmospheric boundary layer over real terrain is modeled with nested high-resolution large-eddy simulations (LESs). The field site is located near Leon, Kansas, where the 1999 Cooperative Atmosphere–Surface Exchange Study took place. The terrain is mostly flat with an average slope of 0.5°. The main topographic feature is a shallow valley oriented in the east–west direction. The night of 5 October is selected to study intermittent turbulence under prevailing quiescent conditions. Brief turbulent periods triggered by shear-instability waves are modeled with good magnitude and temporal precision with a dynamic reconstruction turbulence closure. In comparison, conventional closures fail to excite turbulent motions and predict a false laminar flow. A plausible new intermittency mechanism, previously unknown owing to limited spatial coverage of field instruments at this site, is unveiled with the LESs. Turbulence can be generated through gravity wave breaking over a stagnant cold-air bubble in the valley upwind of the main tower. The bubble is preceded by the formation of a valley cold-air pool due to down-valley drainage flows during the evening transition. The bubble grows in depth by entraining cold down-valley and downslope flows from below and is eroded by shear-induced wave breaking on the top. The cyclic process of formation and erosion is repeated during the night, leading to sporadic turbulent bursting.
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Lai, Y. G., R. M. C. So, M. Anwer, and B. C. Hwang. "Calculations of a Curved-Pipe Flow Using Reynolds Stress Closure." Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science 205, no. 4 (July 1991): 231–44. http://dx.doi.org/10.1243/pime_proc_1991_205_115_02.
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It has been observed that as a fully developed turbulent flow enters a curved bend the anisotropy of the normal stresses near the outer bend (furthest from the centre of the bend curvature) increases. According to the arguments of vorticity generation, a sudden increase in the anisotropy of the normal stresses may lead to the formation of a secondary flow of the second kind. If this secondary motion is to be calculated, then a near-wall Reynolds stress closure that can mimic the anisotropic turbulence behaviour near a wall has to be used. This study presents the results of just such an attempt. In addition, two high Reynolds number closures assuming wall functions in the near-wall region are tested for their ability to replicate the behaviour of the normal stresses in a curved-pipe flow. These two closures differ in their modelling of the pressure-strain terms. Consequently, the effects of near-wall and pressure-strain modelling on curved-pipe flow calculations can be examined. Comparisons are also made with recent curved-pipe flow measurements. The results show that pressure-strain modelling alone is not sufficient to predict the rapid rise of the anisotropy of the normal stresses near the outer bend, and hence the formation of the secondary flow of the second kind. Overall, the near-wall Reynolds stress closure gives a more accurate prediction of the measured mean flow and turbulence statistics, and a realistic calculation of the secondary flow of the second kind near the outer bend.
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Bertoglio, Jean-Pierre, Françoise Bataille, and Jean-Denis Marion. "Two-point closures for weakly compressible turbulence." Physics of Fluids 13, no. 1 (January 2001): 290–310. http://dx.doi.org/10.1063/1.1324005.
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Elkhoury, M. "Assessment and Modification of One-Equation Models of Turbulence for Wall-Bounded Flows." Journal of Fluids Engineering 129, no. 7 (January 25, 2007): 921–28. http://dx.doi.org/10.1115/1.2743666.
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This work assesses the performance of two single-equation eddy viscosity transport models that are based on Menter’s transformation of the k-ε and the k-ω closures. The coefficients of both models are set exactly the same and follow directly from the constants of the standard k-ε closure. This in turn allows a cross-comparison of the effect of two different destruction terms on the performance of single-equation closures. Furthermore, some wall-free modifications to production and destruction terms are proposed and applied to both models. An assessment of the baseline models with and without the proposed modifications against experiments, and the Spalart-Allmaras turbulence model is provided via several boundary-layer computations. Better performance is indicated with the proposed modifications in wall-bounded nonequilibrium flows.
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Fu, S., P. G. Huang, B. E. Launder, and M. A. Leschziner. "A Comparison of Algebraic and Differential Second-Moment Closures for Axisymmetric Turbulent Shear Flows With and Without Swirl." Journal of Fluids Engineering 110, no. 2 (June 1, 1988): 216–21. http://dx.doi.org/10.1115/1.3243537.
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Computations are reported for three axisymmetric turbulent jets, two of which are swirling and one containing swirl-induced recirculation, obtained with two models of turbulence: a differential second-moment (DSM) closure and an algebraic derivative thereof (ASM). The models are identical in respect of all turbulent processes except that, in the ASM scheme, stress transport is represented algebraically in terms of the transport of turbulence energy. The comparison of the results thus provides a direct test of how well the model of stress transport adopted in ASM schemes simulates that of the full second-moment closure. The comparison indicates that the ASM hypothesis seriously misrepresents the diffusive transport of the shear stress in nonswirling axisymmetric flows, while in the presence of swirl the defects extend to all stress components and are aggravated by a failure to account for influential (additive) swirl-related stress-transport terms in the algebraic modelling process. The principal conclusion thus drawn is that in free shear flows where transport effects are significant, it is advisable to adopt a full second-moment closure if turbulence modelling needs to proceed beyond the eddy-viscosity level.
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Mishra, Aashwin A., and Sharath S. Girimaji. "Intercomponent energy transfer in incompressible homogeneous turbulence: multi-point physics and amenability to one-point closures." Journal of Fluid Mechanics 731 (August 28, 2013): 639–81. http://dx.doi.org/10.1017/jfm.2013.343.
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AbstractIntercomponent energy transfer (IET) is a direct consequence of the incompressibility-preserving action of pressure. This action of pressure is inherently non-local, and consequently its modelling must address multi-point physics. However, in second moment closures, pragmatism mandates a single-point closure model for the pressure–strain correlation, that is, the source of IET. In this study, we perform a rapid distortion analysis to demonstrate that for a given mean-flow gradient, IET is strongly dependent on fluctuation modes and critically influences the flow stability, asymptotic states and their bifurcations. The inference is that multi-point physics must be characterized and appropriately incorporated into pressure–strain correlation closures. To this end, we analyse and categorize various multi-point characteristics such as: (i) the fluctuation mode wavevector dynamics; (ii) the spectral space topology of dominant modes; and (iii) the range of IET behaviour and statistically most likely (SML) outcomes. Thence, this characterization is used to examine the validity and limitations of current one-point closures and to propose directions for improving the fidelity of future models.
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SEENI, Aravind. "Effect of Turbulence Models in Performance Characterization of a Low Reynolds Number UAV Propeller." INCAS BULLETIN 13, no. 4 (December 5, 2021): 151–66. http://dx.doi.org/10.13111/2066-8201.2021.13.4.13.
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The advancement of computer technology has given the necessary impetus to perform numerical modelling and simulation in engineering. Turbulence modelling in Computational Fluid Dynamics is characterized by non-physics based modelling and there are several developments in this area that also has contributed to the growing rise in empiricism. Typically, turbulence models are chosen based on expert knowledge and experience. In this paper, the problem of selecting a turbulence closure is addressed for a small Unmanned Aerial Vehicle propeller rotating at a low Reynolds number. Using scientific approaches, verification and validation of performance data against experimental results have been performed for a selected number of turbulence model candidates available in the well-known finite-volume solver Fluent. Modified bivariate plots of performance data error reveal a few numbers of strong candidates of turbulence closures for this problem. After performing a series of checks for consistency, accuracy and computational cost, the two-equation standard k-ω is selected as the preferred model for further propeller simulations.
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Straatman, Anthony G. "A Modified Model for Diffusion in Second-Moment Turbulence Closures." Journal of Fluids Engineering 121, no. 4 (December 1, 1999): 747–56. http://dx.doi.org/10.1115/1.2823532.
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A study has been carried out to determine the relative roles of the three diffusion sub-processes contained in the Lumley (1978) diffusion model. The three sub-processes are described as being the production of turbulent transport, the third-order pressure-velocity process, which regulates the relative magnitudes of the turbulent transport components, and the pressure-diffusion. The present work describes a unique method for calibrating the model based on an analysis of zero-mean-shear turbulence. On the basis of the analysis, and using recent direct numerical simulation and experimental data, the coefficients in the Lumley (1978) model are modified such that the model gives the correct behavior in the diffusive limit. The modified model was then validated by carrying out CFD predictions for three benchmark flows of engineering interest. The modified Lumley (1978) diffusion model has two clear advantages over the more commonly used Daly and Harlow (1970) model. First, unlike the Daly and Harlow (1970) model, the modified Lumley (1978) model is mathematically correct and second, it was shown in the present validation that the modified Lumley (1978) model gives the most consistently reasonable predictions.
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Larson, Vincent E., and Jean-Christophe Golaz. "Using Probability Density Functions to Derive Consistent Closure Relationships among Higher-Order Moments." Monthly Weather Review 133, no. 4 (April 2005): 1023–42. http://dx.doi.org/10.1175/mwr2902.1.
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Parameterizations of turbulence often predict several lower-order moments and make closure assumptions for higher-order moments. In principle, the low- and high-order moments share the same probability density function (PDF). One closure assumption, then, is the shape of this family of PDFs. When the higher-order moments involve both velocity and thermodynamic scalars, often the PDF shape has been assumed to be a double or triple delta function. This is equivalent to assuming a mass-flux model with no subplume variability. However, PDF families other than delta functions can be assumed. This is because the assumed PDF methodology is fairly general. This paper proposes closures for several third- and fourth-order moments. To derive the closures, the moments are assumed to be consistent with a particular PDF family, namely, a mixture of two trivariate Gaussians. (This PDF is also called a double Gaussian or binormal PDF by some authors.) Separately from the PDF assumption, the paper also proposes a simplified relationship between scalar and velocity skewnesses. This PDF family and skewness relationship are simple enough to yield simple, analytic closure formulas relating the moments. If certain conditions hold, this set of moments is specifically realizable. By this it is meant that the set of moments corresponds to a real Gaussian-mixture PDF, one that is normalized and nonnegative everywhere. This paper compares the new closure formulas with both large eddy simulations (LESs) and closures based on double and triple delta PDFs. This paper does not implement the closures in a single-column model and test them interactively. Rather, the comparisons are diagnostic; that is, low-order moments are extracted from the LES and treated as givens that are input into the closures. This isolates errors in the closures from errors in a single-column model. The test cases are three atmospheric boundary layers: a trade wind cumulus layer, a stratocumulus layer, and a clear convective case. The new closures have shortcomings, but nevertheless are superior to the double or triple delta closures in most of the cases tested.
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Fiori, Elisabetta, Antonio Parodi, and Franco Siccardi. "Turbulence Closure Parameterization and Grid Spacing Effects in Simulated Supercell Storms." Journal of the Atmospheric Sciences 67, no. 12 (December 1, 2010): 3870–90. http://dx.doi.org/10.1175/2010jas3359.1.
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Abstract Many meteorological organizations plan to substantially increase the resolution of the limited-area models used for severe weather prediction. Such an approach does not guarantee a priori the reduction of the uncertainty of the decision maker in the prediction of severe weather impact. A deep moist convective process, a supercell, is studied in a simplified atmospheric scenario by means of high-resolution numerical simulations with the Consortium for Small-Scale Modeling (COSMO) model. Different turbulence closure models and their impact on the spatiotemporal properties of storm processes are discussed. In the range of grid spacing between 1 km and 100 m, also termed “terra incognita,” the simulations of a supercell converge with respect to flow field structure, transport properties, and precipitation fields when a turbulence closure derived from large-eddy simulation (LES) is used. In contrast, more simplified turbulence closures such as 1D (vertical) boundary layer approximations yield substantially worse results than the 0.2-km LES reference simulation.
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Viollet, P. L., and O. Simonin. "Modelling Dispersed Two-Phase Flows: Closure, Validation and Software Development." Applied Mechanics Reviews 47, no. 6S (June 1, 1994): S80—S84. http://dx.doi.org/10.1115/1.3124445.
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Closure for the Eulerian modelling of two-phase flows have been developed, based upon extensions of the theory of Tchen of the dispersion of particles in homogeneous turbulence. This model has been validated using large-eddy simulation of homogeneous turbulence, jets loaded with particles, and bubbly flows. In addition with k-epsilon model for the continuous phase, and closures for the Reynolds stresses of the dispersed phase, this theory has been implemented in 2D and 3D software solving the Eulerian two-phase equations (Me´lodif in 2D, as a research code, and ESTET-ASTRID in 3D). These softwares have been applied to complex situations of industrial interest.
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Abou-Arab, T. W., and M. C. Roco. "Solid Phase Contribution in the Two-Phase Turbulence Kinetic Energy Equation." Journal of Fluids Engineering 112, no. 3 (September 1, 1990): 351–61. http://dx.doi.org/10.1115/1.2909411.
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This paper presents a multiphase turbulence closure employing one transport equation, namely, the turbulence kinetic energy equation. The proposed form of this equation is different from the earlier formulations in some aspects. The power spectrum of the carrier fluid is divided into two regions, which interact in different ways and at different rates with the suspended particles as a function of the particle-eddy size ratio and density ratio. The length scale is described algebraically. A double-time averaging approach for the momentum and kinetic energy equations is adopted. The resulting turbulence correlations are modeled under less restrictive assumptions comparative to the previous work. The closures for the momentum and kinetic energy equations are given. Comparisons of the predictions with experimental results on liquid-solid jet and gas-solid pipe flow show satisfactory agreement.
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Chaibina, F., G. Bellakhal, and J. Chahed. "First and Second Order Turbulence Closures Applied to Homogeneous Turbulent Bubbly Flows." Journal of Applied Fluid Mechanics 12, no. 6 (November 1, 2019): 1813–23. http://dx.doi.org/10.29252/jafm.12.06.29756.
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Zhou, Ye, and George Vahala. "Aspects of subgrid modelling and large-eddy simulation of magnetohydrodynamic turbulence." Journal of Plasma Physics 45, no. 2 (April 1991): 239–49. http://dx.doi.org/10.1017/s0022377800015671.
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Subgrid-scale closures for magnetohydodynamic (MHD) turbulence are examined using the filtering technique. From the similarities between incompressible MHD turbulence and its hydrodynamic counterpart, as well as ideas from dynamo theory, a subgrid model is constructed from the large-eddy simulation (LES) of MHD turbulence. This model should find applicability in treating LES of the reversed-field pinch.
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Tjernström, Michael, Ben B. Balsley, Gunilla Svensson, and Carmen J. Nappo. "The Effects of Critical Layers on Residual Layer Turbulence." Journal of the Atmospheric Sciences 66, no. 2 (February 1, 2009): 468–80. http://dx.doi.org/10.1175/2008jas2729.1.
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Abstract The authors report results of a study of finescale turbulence structure in the portion of the nocturnal boundary layer known as the residual layer (RL). The study covers two nights during the Cooperative Atmosphere–Surface Exchange Study 1999 (CASES-99) field experiment that exhibit significant differences in turbulence, as indicated by the observed turbulence dissipation rates in the RL. The RL turbulence sometimes reaches intensities comparable to those in the underlying stable boundary layer. The commonly accepted concept of turbulence generation below critical values of the gradient Richardson number (Rig) is scale dependent: Ri values typically decrease with decreasing vertical scale size, so that critical Rig values (≈0.25) occur at vertical scales of only a few tens of meters. The very small scale for the occurrence of subcritical Ri poses problems for incorporating experimentally determined Rig -based methods in model closures in models with poor resolution. There appear to be two distinct turbulence “regimes” in the RL: a very weak but ever-present background turbulence level with minimal temporal and spatial structure and a more intense intermittent regime during which turbulent intensity can approach near-surface nighttime turbulent intensities. It is hypothesized that the locally produced RL turbulence can be related to upward-propagating atmospheric gravity waves generated by flow over the low-relief terrain. The presence of critical layers in the RL, caused by wind turning with height, results in the generation of intermittent turbulence.
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Batten, Paul, Uriel Goldberg, and Sukumar Chakravarthy. "Interfacing Statistical Turbulence Closures with Large-Eddy Simulation." AIAA Journal 42, no. 3 (March 2004): 485–92. http://dx.doi.org/10.2514/1.3496.
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So, R. M. C., Y. G. Lai, H. S. Zhang, and B. C. Hwang. "Second-order near-wall turbulence closures - A review." AIAA Journal 29, no. 11 (November 1991): 1819–35. http://dx.doi.org/10.2514/3.10807.
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Younis, Bassam A., and Ye Zhou. "Accounting for mean-flow periodicity in turbulence closures." Physics of Fluids 18, no. 1 (January 2006): 018102. http://dx.doi.org/10.1063/1.2166458.
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Rubinstein, Robert, and Timothy T. Clark. "Reassessment of the classical closures for scalar turbulence." Journal of Turbulence 14, no. 2 (February 2013): 71–98. http://dx.doi.org/10.1080/14685248.2013.769685.
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Speziale, Charles G., Ridha Abid, and Paul A. Durbin. "On the realizability of reynolds stress turbulence closures." Journal of Scientific Computing 9, no. 4 (December 1994): 369–403. http://dx.doi.org/10.1007/bf01575099.
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Maulik, Romit, and Omer San. "Explicit and implicit LES closures for Burgers turbulence." Journal of Computational and Applied Mathematics 327 (January 2018): 12–40. http://dx.doi.org/10.1016/j.cam.2017.06.003.
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Duvigneau, Régis, Jérémie Labroquère, and Emmanuel Guilmineau. "Comparison of turbulence closures for optimized active control." Computers & Fluids 124 (January 2016): 67–77. http://dx.doi.org/10.1016/j.compfluid.2015.10.011.
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Speziale, Charles G., and Nessan Mac Giolla Mhuiris. "On the prediction of equilibrium states in homogeneous turbulence." Journal of Fluid Mechanics 209 (December 1989): 591–615. http://dx.doi.org/10.1017/s002211208900323x.
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A comparison of several commonly used turbulence models (including the K–ε model and three second-order closures) is made for the test problem of homogeneous turbulent shear flow in a rotating frame. The time evolution of the turbulent kinetic energy and dissipation rate is calculated for these models and comparisons are made with previously published experiments and numerical simulations. Particular emphasis is placed on examining the ability of each model to predict equilibrium states accurately for a range of the parameter Ω/S (the ratio of the rotation rate to the shear rate). It is found that none of the commonly used second-order closure models yield substantially improved predictions for the time evolution of the turbulent kinetic energy and dissipation rate over the somewhat defective results obtained from the simpler K–ε model for the unstable flow regime. There is also a problem with the equilibrium states predicted by the various models. For example, the K–ε model erroneously yields equilibrium states that are independent of Ω/S while the Launder, Reece & Rodi model and the Shih-Lumley model predict a flow relaminarization when Ω/S > 0.39 - a result that is contrary to numerical simulations and linear spectral analyses, which indicate flow instability for at least the range 0 [les ] Ω/S [les ] 0.5. The physical implications of the results obtained from the various turbulence models considered herein are discussed in detail along with proposals to remedy the deficiencies based on a dynamical systems approach.
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Lance, M., J. L. Marie´, and J. Bataille. "Homogeneous Turbulence in Bubbly Flows." Journal of Fluids Engineering 113, no. 2 (June 1, 1991): 295–300. http://dx.doi.org/10.1115/1.2909495.
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The present study is devoted to the interaction between a swarm of bubbles and a turbulent field in a linear shear flow. The transversal and longitudinal evolutions of the void fraction and of the Reynolds stress tensor have been measured. When the air bubbles are blown uniformly into the shear, the void fraction profiles exhibit a strong gradient which can be explained by kinematical effects. No void migration has been observed. The behavior of the Reynolds tensor indicates that the nonisotropy induced by the mean velocity gradient decreases when the void fraction increases. A simple mechanism is proposed to interpret this fact, and a turbulence model based on one-point closures is compared to the experimental data.
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Rani, Sarma L., Vijay K. Gupta, and Donald L. Koch. "Clustering of rapidly settling, low-inertia particle pairs in isotropic turbulence. Part 1. Drift and diffusion flux closures." Journal of Fluid Mechanics 871 (May 22, 2019): 450–76. http://dx.doi.org/10.1017/jfm.2019.204.
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In this two-part study, we present the development and analysis of a stochastic theory for characterizing the relative positions of monodisperse, low-inertia particle pairs that are settling rapidly in homogeneous isotropic turbulence. In the limits of small Stokes number and Froude number such that $Fr\ll St_{\unicode[STIX]{x1D702}}\ll 1$, closures are developed for the drift and diffusion fluxes in the probability density function (p.d.f.) equation for the pair relative positions. The theory focuses on the relative motion of particle pairs in the dissipation regime of turbulence, i.e. for pair separations smaller than the Kolmogorov length scale. In this regime, the theory approximates the fluid velocity field in a reference frame following the primary particle as locally linear. In this part 1 paper, we present the derivation of closure approximations for the drift and diffusion fluxes in the p.d.f. equation for pair relative positions $\boldsymbol{r}$. The drift flux contains the time integral of the third and fourth moments of the ‘seen’ fluid velocity gradients along the trajectories of primary particles. These moments may be analytically resolved by making approximations regarding the ‘seen’ velocity gradient. Accordingly, two closure forms are derived specifically for the drift flux. The first invokes the assumption that the fluid velocity gradient along particle trajectories has a Gaussian distribution. In the second drift closure, we account for the correlation time scales of dissipation rate and enstrophy by decomposing the velocity gradient into the strain-rate and rotation-rate tensors scaled by the turbulent dissipation rate and enstrophy, respectively. An analytical solution to the p.d.f. $\langle P\rangle (r,\unicode[STIX]{x1D703})$ is then derived, where $\unicode[STIX]{x1D703}$ is the spherical polar angle. It is seen that the p.d.f. has a power-law dependence on separation $r$ of the form $\langle P\rangle (r,\unicode[STIX]{x1D703})\sim r^{\unicode[STIX]{x1D6FD}}$ with $\unicode[STIX]{x1D6FD}\sim St_{\unicode[STIX]{x1D702}}^{2}$ and $\unicode[STIX]{x1D6FD}<0$, analogous to that for the radial distribution function of non-settling pairs. An explicit expression is derived for $\unicode[STIX]{x1D6FD}$ in terms of the drift and diffusion closures. The $\langle P\rangle (r,\unicode[STIX]{x1D703})$ solution also shows that, for a given $r$, the clustering of $St_{\unicode[STIX]{x1D702}}\ll 1$ particles is only weakly anisotropic, which is in conformity with prior observations from direct numerical simulations of isotropic turbulence containing settling particles.
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SPEZIALE, C. G., B. A. YOUNIS, and S. A. BERGER. "Analysis and modelling of turbulent flow in an axially rotating pipe." Journal of Fluid Mechanics 407 (March 25, 2000): 1–26. http://dx.doi.org/10.1017/s0022112099007600.
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The analysis and modelling of the structure of turbulent flow in a circular pipe subjected to an axial rotation is presented. Particular attention is paid to determining the terms in various turbulence closures that generate the two main physical features that characterize this flow: a rotationally dependent axial mean velocity and a rotationally dependent mean azimuthal or swirl velocity relative to the rotating pipe. It is shown that the first feature is well represented by two-dimensional explicit algebraic stress models but is irreproducible by traditional two-equation models. On the other hand, three-dimensional frame-dependent models are needed to predict the presence of a mean swirl velocity. The latter is argued to be a secondary effect which arises from a cubic nonlinearity in standard algebraic models with conventional near-wall treatments. Second-order closures are shown to give a more complete description of this flow and can describe both of these features fairly well. In this regard, quadratic pressure–strain models perform the best overall when extensive comparisons are made with the results of physical and numerical experiments. The physical significance of this problem and the implications for future research in turbulence are discussed in detail.
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Dussauge, J. P., and J. Gaviglio. "The rapid expansion of a supersonic turbulent flow: role of bulk dilatation." Journal of Fluid Mechanics 174 (January 1987): 81–112. http://dx.doi.org/10.1017/s0022112087000053.
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The rapid expansion of a turbulent boundary layer in supersonic flow is studied analytically and experimentally. Emphasis is placed on the effect of bulk dilatation on turbulent fluctuations. The hypotheses made in the analysis are similar to those in the rapid distortion theory and are used to simplify second-order closures. By assuming that the fluctuating velocity is solenoidal an extension of classical subsonic models is proposed. A new variable is defined, which takes into account the mean density variations, and behaves like the Reynolds stress tensor in subsonic flows with weak inhomogeneities and a weak dissipation rate. The results of the analysis are compared with turbulence measurements performed in a supersonic boundary layer subjected to an expansion fan. The proposed approximations describe correctly the evolution of turbulence intensities: bulk dilatation contributes predominantly to the Reynolds stress evolution. The boundary layer is ‘relaminarized’ by the expansion. Downstream of the latter, the layer returns to equilibrium. Measurements show that the turbulence decays slowly in the outer layer and increases rapidly in the inner layer.
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McTaggart-Cowan, Ron, and Ayrton Zadra. "Representing Richardson Number Hysteresis in the NWP Boundary Layer." Monthly Weather Review 143, no. 4 (March 31, 2015): 1232–58. http://dx.doi.org/10.1175/mwr-d-14-00179.1.
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Abstract Turbulence in the planetary boundary layer (PBL) transports heat, momentum, and moisture in eddies that are not resolvable by current NWP systems. Numerical models typically parameterize this process using vertical diffusion operators whose coefficients depend on the intensity of the expected turbulence. The PBL scheme employed in this study uses a one-and-a-half-order closure based on a predictive equation for the turbulent kinetic energy (TKE). For a stably stratified fluid, the growth and decay of TKE is largely controlled by the dynamic stability of the flow as represented by the Richardson number. Although the existence of a critical Richardson number that uniquely separates turbulent and laminar regimes is predicted by linear theory and perturbation analysis, observational evidence and total energy arguments suggest that its value is highly uncertain. This can be explained in part by the apparent presence of turbulence regime-dependent critical values, a property known as Richardson number hysteresis. In this study, a parameterization of Richardson number hysteresis is proposed. The impact of including this effect is evaluated in systems of increasing complexity: a single-column model, a forecast case study, and a full assimilation cycle. It is shown that accounting for a hysteretic loop in the TKE equation improves guidance for a canonical freezing rain event by reducing the diffusive elimination of the warm nose aloft, thus improving the model’s representation of PBL profiles. Systematic enhancements in predictive skill further suggest that representing Richardson number hysteresis in PBL schemes using higher-order closures has the potential to yield important and physically relevant improvements in guidance quality.