A thermodynamic model of turbulent motions in a granular material

This paper is devoted to a thermodynamic theory of granular materials subjected to slow frictional as well as rapid flows with strong collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable is of a kinematic nature and enters the formulation via the balance law of the configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin [1], but modified here. Complemented by constitutive equations, the emerging field equations are postulated to be adequate for motions, be they laminar or turbulent, if the resolved length scales are sufficiently small. On large length scales the sub-grid motion may be interpreted as fluctuations, which manifest themselves in correspondingly filtered equations as correlation products, like in the turbulence theory. We apply an ergodic (Reynolds) filter to these equations and thus deduce averaged equations for the mean motions. The averaged equations comprise balances of mass, linear and configurational momenta, energy, and turbulent kinetic energy as well as turbulent configurational kinetic energy. They are complemented by balance laws for two internal fields, the dissipation rates of the turbulent kinetic energy and of the turbulent configurational kinetic energy. We formulate closure relations for the averages of the laminar constitutive quantities and for the correlation terms by using the rules of material and turbulent objectivity, including equipresence. Many versions of the second law of thermodynamics are known in the literature. We follow the Müller-Liu theory and extend Müllers entropy principle to allow the satisfaction of the second law of thermodynamics for both laminar and turbulent motions. Its exploitation, performed in the spirit of the Müller-Liu theory, delivers restrictions on the dependent constitutive quantities (through the Liu equations) and a residual inequality, from which thermodynamic equilibrium properties are deduced. Finally, linear relationships are proposed for the nonequilibrium closure relations.Received: 21 March 2003, Accepted: 1 September 2003, Published online: 11 February 2004PACS: 05.70.Ln, 61.25.Hq, 61.30.-vCorrespondence to: I. Luca     

Effect of particles on turbulent thermal field of channel flow with different Prandtl numbers

The direct numerical simulation(DNS) of heat transfer in a fully developed non-isothermal particle-laden turbulent channel flow is performed.The focus of this paper is on the modulation of the particles on turbulent thermal statistics in the particle-laden flow with three Prandtl numbers(P r = 0.71,1.5,and 3.0) and a shear Reynolds number(Reτ = 180).Some typical thermal statistics,including normalized mean temperature and their fluctuations,turbulent heat fluxes,Nusselt number and so on,are analyzed.The results show that the particles have less effects on turbulent thermal fields with the increase of Prandtl number.Two reasons can explain this.First,the correlation between fluid thermal field and velocity field decreases as the Prandtl number increases,and the modulation of turbulent velocity field induced by the particles has less influence on the turbulent thermal field.Second,the heat exchange between turbulence and particles decreases for the particle-laden flow with the larger Prandtl number,and the thermal feedback of the particles to turbulence becomes weak.     

Analytical determination of the heat transfer coefficient for gas,liquid and liquid metal flows in the tube based on stochastic equations and equivalence of measures for continuum

The stochastic equations of continuum are used for determining the heat transfer coefficients. As a result, the formulas for Nusselt (Nu) number dependent on the turbulence intensity and scale instead of only on the Reynolds (Peclet) number are proposed for the classic flows of a nonisothermal fluid in a round smooth tube. It is shown that the new expressions for the classical heat transfer coefficient Nu, which depend only on the Reynolds number, should be obtained from these new general formulas if to use the well-known experimental data for the initial turbulence. It is found that the limitations of classical empirical and semiempirical formulas for heat transfer coefficients and their deviation from the experimental data depend on different parameters of initial fluctuations in the flow for different experiments in a wide range of Reynolds or Peclet numbers. Based on these new dependences, it is possible to explain that the differences between the experimental results for the fixed Reynolds or Peclet numbers are caused by the difference in values of flow fluctuations for each experiment instead of only due to the systematic error in the experiment processing. Accordingly, the obtained general dependences of Nu for a smooth round tube can serve as the basis for clarifying the experimental results and empirical formulas used for continuum flows in various power devices. Obtained results show that both for isothermal and for nonisothermal flows, the reason for the process of transition from a deterministic state into a turbulent one is determined by the physical law of equivalence of measures between them. Also the theory of stochastic equations and the law of equivalence of measures could determine mechanics which is basis in different phenomena of self-organization and chaos theory.     

Computation of Gas–Dynamic Parameters and Heat Transfer in Supersonic Turbulent Separated Flows near Backward–Facing Steps

Computation results of plane turbulent flows in the vicinity of backward–facing steps with leeward–face angles = 8, 25, and 45° for Mach numbers M_infin = 3 and 4 are presented. The averaged Navier—Stokes equations supplemented by the Wilcox model of turbulence are used as a mathematical model. The boundary–layer equations were also used for the case of an attached flow ( = 8°). The computed and experimental distributions of surface pressure and skin friction, the velocity and pressure fields, and the heat–transfer coefficients are compared.     

Renormalization Group Model of Large-Scale Turbulence in Porous Media

Renormalization group methods are used to develop a macroscopic (large-scale) turbulence model for incompressible flow in porous media. The model accounts for the large-distance and large-time behavior of velocity correlations generated by the momentum equation for a randomly stirred, incompressible flow. Utilizing the renormalization procedure, the transport equations for the large-scale modes and expressions for effective transport coefficients are obtained. Expressions for renormalized turbulent viscosity, which accounts for the ultraviolet subrange of the turbulent kinetic energy spectrum, are also obtained.     

A Priori Tests of RANS Models for Turbulent Channel Flows of a Dense Gas

Dense gas effects, encountered in many engineering applications, lead to unconventional variations of the thermodynamic and transport properties in the supersonic flow regime, which in turn are responsible for considerable modifications of turbulent flow behavior with respect to perfect gases. The most striking differences for wall-bounded turbulence are the decoupling of dynamic and thermal effects for gases with high specific heats, the liquid-like behavior of the viscosity and thermal conductivity, which tend to decrease away from the wall, and the increase of density fluctuations in the near wall region. The present work represents a first attempt of quantifying the influence of such dense gas effects on modeling assumptions employed for the closure of the Reynolds-averaged Navier–Stokes equations, with focus on the eddy viscosity and turbulent Prandtl number models. For that purpose, we use recent direct numerical simulation results for supersonic turbulent channel flows of PP11 (a heavy fluorocarbon representative of dense gases) at various bulk Mach and Reynolds numbers to carry out a priori tests of the validity of some currently-used models for the turbulent stresses and heat flux. More specifically, we examine the behavior of the modeled eddy viscosity for some low-Reynolds variants of the \(k-\varepsilon \) model and compare the results with those found for a perfect gas at similar conditions. We also investigate the behavior of the turbulent Prandtl number in dense gas flow and compare the results with the predictions of two well-established turbulent Prandtl number models.     

A Simple Nonlinear Thermodynamic Theory of Arbitrary Elastic Beams

A ‘classical’ theory of beams (i.e., a theory in which the basic kinetic variables are a stress resultant and a stress couple) undergoing elastic, thermodynamic processes is developed by first deriving exact beamlike (one-dimensional) equations of motion and a beamlike Second Law (Clausius–Duhem inequality) by descent from three-dimensions. Then what may be considered as the three basic assumptions of a classical theory are introduced: an assumed form of the First Law (conservation of energy), a relaxed form of the Second Law, and a general form of the constitutive relations. Throughout, detailed specification of geometry, kinematics, or constitution is minimized. It is shown how the kinematic Kirchhoff hypothesis may be avoided by first introducing a mixed-energy density and then imposing a logically more satisfying constitutive Kirchhoff hypothesis. Mathematics Subject Classifications (2000) 74A15, 74B20, 74K10     

Concentration fluctuations in a turbulent jet

Using Spalding's model of turbulence in a turbulent shear flow, we have calculated the root-mean-square value of the concentration fluctuations inside a turbulent jet. Although we used the same equations and the same solution technique as Spalding, we have not been able to find precisely his numerical results derived for a jet issuing into a fluid at rest with the same density as the jet. The differences between our numerical results, Spalding's numerical results and the experimental data of Becker, Hottel and Williams are fairly small only if the initial values of the turbulence energy and the mixing length inside the jet and the turbulence in the ambient fluid are taken into account in the model. For a turbulent jet issuing into a turbulently flowing surrounding stream of different density, we found that the relative concentration fluctuations can increase considerably. This brings out the importance of taking into account property variables in analysing turbulent mixing processes.     

Direct simulation of two‐dimensional turbulent flow over a surface‐mounted obstacle

Two‐dimensional turbulent flow over a surface‐mounted obstacle is studied as a numerical experiment that takes place in a wind tunnel. The transient Navier–Stokes equations are solved directly with Galerkin finite elements. The Reynolds number defined with respect to the height of the wind tunnel is 12 518. Instantaneous streamline patterns are shown that give a complete picture of the flow phenomena. Energy and enstrophy spectra yield the dual cascade of two‐dimensional turbulence and the ?1 power law decay of enstrophy. Mean values of velocities and root mean square fluctuations are compared with the available experimental results. Other statistical characteristics of turbulence such as Eulerian autocorrelation coefficients, longitudinal and lateral coefficients are also computed. Finally, oscillation diagrams of computed velocity fluctuations yield the chaotic behaviour of turbulence. Copyright © 2007 John Wiley & Sons, Ltd.     

A k–$${\varepsilon}$$ turbulence closure model of an isothermal dry granular dense matter

The turbulent flow characteristics of an isothermal dry granular dense matter with incompressible grains are investigated by the proposed first-order k–\({\varepsilon}\) turbulence closure model. Reynolds-filter process is applied to obtain the balance equations of the mean fields with two kinematic equations describing the time evolutions of the turbulent kinetic energy and dissipation. The first and second laws of thermodynamics are used to derive the equilibrium closure relations satisfying turbulence realizability conditions, with the dynamic responses postulated by a quasi-linear theory. The established closure model is applied to analyses of a gravity-driven stationary flow down an inclined moving plane. While the mean velocity decreases monotonically from its value on the moving plane toward the free surface, the mean porosity increases exponentially; the turbulent kinetic energy and dissipation evolve, respectively, from their minimum and maximum values on the plane toward their maximum and minimum values on the free surface. The evaluated mean velocity and porosity correspond to the experimental outcomes, while the turbulent dissipation distribution demonstrates a similarity to that of Newtonian fluids in turbulent shear flows. When compared to the zero-order model, the turbulent eddy evolution tends to enhance the transfer of the turbulent kinetic energy and plane shearing across the flow layer, resulting in more intensive turbulent fluctuation in the upper part of the flow. Solid boundary as energy source and sink of the turbulent kinetic energy becomes more apparent in the established first-order model.     

Investigation of thermal dispersion and intra-pore turbulent heat flux in porous media

In the present study, the importance of the thermal dispersion and the turbulent heat flux in porous media and their effects on the macroscopic distribution of thermal energy are investigated. To this end, turbulent flow and heat transfer within five unit-cells mimicking porous media are solved using large eddy simulation. It is shown that the thermal dispersion and the turbulent heat flux are negligible as compared to the convection term in the macroscopic energy equation. When further scrutinizing this equation, it is revealed that except for the longitudinal components of the thermal dispersion, the other components of thermal dispersion and turbulent heat flux may be neglected away from the boundaries as compared to the interfacial heat transfer. Visualizations of vortices show that the size of the turbulence structures within the cells is of the same order as the size of the pores; therefore, the turbulent heat flux is limited to the intra-pore level. Finally, a discussion is provided on the accuracy of the gradient type diffusion model commonly used for turbulent heat flux in porous media in the absence of macroscopic turbulence. It is shown that the intra-pore turbulence does not affect the macroscopic transport of thermal energy within the porous media studied.     

Use of turbulent energy balance equation in the theory of turbulent flows along walls

The turbulent flow of an incompressible fluid is considered in a plane channel, a circular tube, and the boundary layer on a flat plate. The system of equations describing the motion of the fluid consists of the Reynolds equations and the mean kinetic energy balance equation for turbulent fluctuations. On the basis of an analysis of experimental data, hypotheses are formulated with respect to the eddy kinematic viscosity and lengthl entering into the expression for specific dissipation of turbulent energy into heat. It is assumed that in the central (outer) region of the flow in a channel, andl are constants, and expressions are taken for them which are used for a free boundary layer; near the walll varies linearly and almost linearly. Results of calculations of the turbulent energy distribution, the mean velocity, and the drag coefficient are in good agreement with the existing experimental data. The values of two empirical coefficients, which enter into the system of equations as the result of the hypotheses, are close to those obtained for a free boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–33, May–June, 1973.     

Cone-and-plate apparatus: a compact system for studying well-characterized turbulent flow fields

We have investigated the turbulent flow field inside a cone-and-plate Couette flow apparatus (CPA). The CPA presents a compact, easily varied and easily measured flow system with well-characterized turbulent and laminar flow fields and a good full turbulent spectrum. The CPA can also produce the same time-average stress for both laminar and turbulent regimes. The apparatus consisted of a shallow 2° cone of 0.75 m radius rotating above a flat plate in air. A high-resolution X-wire probe was used to measure the turbulent and time-mean flow properties. A single parameter , which is a measure of the centrifugal forces acting on the moving fluid, governs the degree of secondary flow and turbulence in the device. In the fully turbulent region, R > 4, the mean velocity profile has steep gradients near each surface and is nearly uniform in the central core; and the normalized azimuthal velocity closely follows the ubiquitous law of the wall near the plate. The turbulence intensity exhibits two peaks at the edges of the core flow, and the frequency spectrum of velocity fluctuations near the plate resembles that found in isotropic turbulence. This study also suggests that the rotational symmetry with the stationary bottom plate is a much more convenient experimental geometry than a circular Couette flow.     

DNS of the Turbulent Channel Flow of a Dilute Polymer Solution

A direct numerical simulation of the turbulent channel flow of a dilute polymer solution has been performed in order to compare its turbulence statistics with those obtained in a Newtonian channel flow. The viscoelastic flow has been simulated by solving the whole set of continuity, momentum and constitutive equations for the six independent components of the extra-stress tensor induced by polymer addition. The Finitely Extensible Nonlinear Elastic dumbbell model was adopted in order to simulate a non-linear modulus of elasticity and a finite extendibility of the polymer macromolecules. Simulations were carried out under the narrow channel assumption at a Reynolds number of 169 based on the channel half height and on the friction velocity; they showed a significant reduction in drag, dependent on the influence of the elastic properties of the chains. A qualitative comparison with experiments at a higher Reynolds number has shown that the model here adopted is capable of reproducing all the main features of the polymer solution flow. Analysis of the turbulence statistics suggests that a dilute polymer solution can affect the intensity of the streamwise vortices, leading to an increase in the spacing between low speed streaks and eventually to a turbulent shear stress reduction.     

Predicting Turbulent Spectra in Drag-reduced Flows

In the present work we describe how turbulent skin-friction drag reduction obtained through near-wall turbulence manipulation modifies the spectral content of turbulent fluctuations and Reynolds shear stress with focus on the largest scales. Direct Numerical Simulations (DNS) of turbulent channels up to Re _τ = 1000 are performed in which drag reduction is achieved either via artificially removing wall-normal turbulent fluctuations in the vicinity of the wall or via streamwise-travelling waves of spanwise wall velocity. This near-wall turbulence manipulation is shown to modify turbulent spectra in a broad range of scales throughout the whole channel. Above the buffer layer, the observed changes can be predicted, exploiting the vertical shift of the logarithmic portion of the mean streamwise velocity profile, which is a classic performance measure for wall roughness or drag-reducing riblets. A simple model is developed for predicting the large-scale contribution to turbulent fluctuation and Reynolds shear stress spectra in drag-reduced turbulent channels in which a flow control acts at the wall. Any drag-reducing control that successfully interacts with large scales should deviate from the predictions of the present model, making it a useful benchmark for assessing the capability of a control to affect large scales directly.     

A Gas-Kinetic Scheme for Turbulent Flow

RANS simulations may not provide accurate results for all flow conditions. The interaction between a shock wave and a turbulent boundary layer is an example which may still be difficult to simulate accurately. Beside the inability to reproduce physical phenomena such as shock unsteadiness, the argument is put forward that the conventional numerical schemes, based on the Navier-Stokes equations, may be unable to generate a physically consistent turbulent stress tensor in the presence of large unresolved scales of motion. A large ratio between unresolved and resolved scales of motion, a sort of Knudsen number based on turbulent fluctuations, might introduce inaccuracies for which the turbulence model is not accountable. In order to improve the accuracy of RANS simulations, researchers have suggested various ad-hoc modifications to standard turbulence models which limit eddy viscosity or the turbulent stress tensor in the presence of strong gradients. Gas-kinetic schemes might be able to improve RANS predictions in shocklayers by removing or limiting the errors caused by the large scales ratio. These schemes are a class of their own; in the framework of a finite-volume or finite-elements discretizations, they model the numerical fluxes on the basis of the Boltzmann equation instead of the Navier-Stokes equations as is conventionally done. In practical terms, these schemes provide a higher accuracy and, more importantly, an in-built “multiscalar” mechanism, i.e. the ability to adjust to the size of unresolved scales of motion. This property makes them suitable for shock-capturing and rarefied flow. Gas-kinetic scheme may be coupled to a conventional RANS turbulence model; it is shown that the turbulent stress tensor is naturally adjusted as a function of the unresolved-to-resolved scales ratios and achieves a higher physical consistency than conventional schemes. The simulations shown - well-known benchmark cases with strong shock-boundary layer interactions - have been obtained with a standard two-equation turbulence model (k- ω). It is shown that the gas-kinetic scheme provides good quality predictions, where conventional schemes with the same turbulence model are known to fail.     

Computation of Compressible Flows using a Two-equation Turbulence Model on Unstructured Grids

This paper presents a numerical method for solving compressible turbulent flows using a k - l turbulence model on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multi-stage Runge-Kutta time stepping scheme, while the turbulence equations are advanced using a multi-stage point-implicit scheme. The positivity of turbulence variables is achieved using a simple change of dependent variables. The developed method is used to compute a variety of turbulent flow problems. The results obtained are in good agreement with theoretical and experimental data, indicating that the present method provides a viable and robust algorithm for computing turbulent flows on unstructured meshes.     

A K-ε TWO-EQUATION TURBULENCE MODEL FOR THE SOLID-LIQUID TWO-PHASE FLOWS

A two-equation turbulence model has been dereloped for predicting two-phase flow the two equations describe the conserration of turbulence kinetic energy and dissipation rate of that energy for the incompressible carrier fluid in a two-phase flow The continuity, the momentum, K and ε equations are modeled. In this model,the solid-liquid slip veloeites, the particle-particte interactions and the interactions between two phases are considered,The sandy water pipe turbulent flows are sueeessfuly predicted by this turbulince model.