Buoyancy effects on turbulent swirling flows

A three-parameter model of turbulence applicable to free boundary layers has been developed and applied for the prediction of axisymmetric turbulent swirling flows in uniform and stagnant surroundings under the action of buoyancy forces. The turbulent momentum and heat fluxes appearing in the time-averaged equations for the mean motion have been determined from algebraic expressions, derived by neglecting the convection and diffusion terms in the differential transport equations for these quantities, which relate the turbulent fluxes to the kinetic energy of turbulence, k, the dissipation length scale of turbulence, L, and the temperature covariance, T ′². Differential transport equations have been used to determine these latter quantities. The governing equations have been solved using fully implicit finite difference schemes. The turbulence model is capable of reproducing the gross features of pure jet flows, buoyant flows and swirling flows for weak and moderate swirl. The behaviour of a turbulent buoyant swirling jet has been found to depend solely on exit swirl and Froude numbers. The predicted results indicate that the incorporation of buoyancy can cause significant changes in the behaviour of a swirling jet, particularly when the buoyancy strength is high. The jet exhibits similarity behaviour in the initial region for weak swirl and weak buoyancy strengths only, and the asymptotic case of a swirling jet under the action of buoyancy forces is a pure plume in the far field. The predicted results have been found to be in satisfactory agreement with the available experimental data and in good qualitative agreement with other predicted results.     

On Cascade Energy Transfer in Convective Turbulence

The paper is devoted to specificities of the cascade processes in developed turbulence existing on a background of the density (temperature) gradient either parallel (turbulence in a stably stratified (SS) medium) or antiparallel (convective turbulence (CT)) to the gravitational force. Our main attention is paid to the Obukhov–Bolgiano (OB) regime, which presumes a balance between the buoyancy and nonlinear forces in a sufficiently extensive part of the inertial interval. Up to now, there has been no reliable evidence of the existence of the OB regime, although fragments of spectra with slopes close to–11/5 and–7/5 were detected in some works on the numerical simulations of convective turbulence. The paper presents a critical comparison of these data with the results obtained in this work using the cascade model of convective turbulence, which makes it possible to consider a wide range of control parameters. The cascade model is new and was obtained by the generalization of the class of helical cascade models to the case of turbulent convection. It is shown that, in developed turbulence, which is characterized by an interval with a constant spectral flux of kinetic energy, the buoyancy force cannot compete with nonlinear interactions and has no essential effect on the dynamics of the inertial interval. It is the buoyancy force that supplies the cascade process with energy in convective turbulence but only in the maximum scales. Under the SS conditions, the buoyancy forces reduce the energy of turbulent pulsations. In the case of stable stratification, the buoyancy force reduces the turbulence pulsation energy. The OB regime arises in none of these cases, but, in the scales beyond the inertial interval, Kolmogorov’s turbulence with the “–5/3” law, in which temperature behaves like a passive admixture, is established. The observed deviations from the “–5/3” spectrum, erroneously interpreted as the OB regime, are manifested in the case of insufficient separation of the macroscale of turbulence and the dissipative scale.     

Accounting for Buoyancy Effects in the Explicit Algebraic Stress Model: Homogeneous Turbulent Shear Flows

Most explicit algebraic stress models are formulated for turbulent shear flows without accounting for external body force effects, such as the buoyant force. These models yield fairly good predictions of the turbulence field generated by mean shear. As for thermal turbulence generated by the buoyant force, the models fail to give satisfactory results. The reason is that the models do not explicitly account for buoyancy effects, which interact with the mean shear to enhance or suppress turbulent mixing. Since applicable, coupled differential equations have been developed describing these thermal turbulent fields, it is possible to develop corresponding explicit algebraic stress models using tensor representation theory. While the procedure to be followed has been employed previously, unique challenges arise in extending the procedure for developing the algebraic representations to turbulent buoyant flows. In this paper the development of an explicit algebraic stress model (EASM) is confined to the homogeneous buoyant shear flow case to illustrate the methodology needed to develop the proper polynomial representations. The derivation is based on the implicit formulation of the Reynolds stress anisotropy at buoyant equilibrium. A five-term representation is found to be necessary to account properly for the effect of the thermal field. Thus derived, external buoyancy effects are represented in the scalar coefficients of the basis tensors, and structural buoyancy effects are accounted for in additional terms in the stress anisotropy tensor. These terms will not vanish even in the absence of mean shear. The performance of the new EASM, together with a two-equation (2-Eq) model, the non-buoyant EASM of Gatski and Speziale (1993) and a full second-order model, is assessed against direct numerical simulations of homogeneous, buoyant shear flows at two different Richardson numbers representing weak and strong buoyancy effects. The calculations show that this five-term representation yields better results than the 2-Eq model and the EASM of Gatski and Speziale where buoyancy effects are not explicitly accounted for. Received 5 March 2001 and accepted 15 January 2002     

FINITE ELEMENT SOLUTIONS OF LAMINAR AND TURBULENT MIXED CONVECTION IN A DRIVEN CAVITY

This paper presents laminar and turbulent mixed convection solutions of a driven cavity flow using the finite element method. For the laminar flow, distributions of velocity and temperature with and without the effect of buoyancy force are presented and compared. For the turbulent flow, governing partial differential equations of the thermal turbulence two-equati on model and kinetic turbulence two-equation model are used. Corresponding results such as kinetic eddy diffusivity, kinetic eddy energy, thermal eddy energy and their dissipations are presented.     

Turbulence Budgets in Buoyancy-affected Vertical Backward-facing Step Flow at Low Prandtl Number

The paper investigates buoyancy impact on the vertical flow over a backward-facing step at low Prandtl number by Direct Numerical Simulation. In particular, the very low Prandtl number of liquid sodium, 0.0088, is considered in the regime of mixed convection, i.e. for Richardson numbers below unity. The effects of buoyancy on mean flow, heat transfer and turbulence are assessed. Buoyancy is found to attenuate recirculation and, consequently, increase heat transfer. Turbulence is decreased in the attached boundary layer for moderate buoyancy impact but surpasses the levels found in forced convection at the largest Richardson number investigated. Beyond the mean flow and second moments, the budgets of turbulent kinetic energy, Reynolds shear stress, temperature variance, and turbulent heat flux components are studied and related to the alterations in the mean field quantities. Due to scale separation, production and dissipation nearly balance for temperature variance while this is not the case for turbulent kinetic energy. Similar findings for the turbulent heat fluxes show that the correlation between temperature and pressure gradient is the most important contribution to the budget aside from production and dissipation. In addition to the physical insight into this flow, the data presented may be used for the validation and improvement of turbulence models for liquid metal flows.     

Development of lattice turbulence in a stream with a constant velocity gradient

On the basis of the equations for the Reynolds stresses and the equation for the scale of the turbulence, an analysis is made of the development of lattice turbulence in a stream with a constant velocity gradient. The constants in the equations are determined under the assumption that, far from the lattice and with large Reynolds numbers, the structure of the turbulence tends toward a limiting state with constant values of the correlation coefficient, the degree of anisotropy, and the dimensionless velocity gradient. The constants in terms containing the viscosity are determined from a consideration of the flow beyond the lattice without a velocity gradient in the final stage of decay of the turbulence. The equations obtained were solved in an electronic computer. The calculation is in satisfactory agreement with the existing experimental data. For calculating flows with a variable velocity gradient, instead of the equation of the scale, it is proposed to use an equation for the frequency of the turbulent pulsations obtained in the present work. The computer calculations were made by S. I. Bekritskaya.     

Similarity solutions of vertical plane wall plume based on finite analytic method

The turbulent flow of vertical plane wall plume with concentration variation was studied with the finite analytical method. The k-epsilon model with the effect of buoyancy on turbulent kinetic energy and its dissipation rate was adopted. There were similarity solutions in the uniform environment for the system of equations including the equation of continuity, the equation of momentum along the flow direction and concentration, and equations of k, epsilon. The finite analytic method was applied to obtain the similarity solution. The calculated data of velocity, relative density difference, the kinetic energy of turbulence and its dissipation rate distribution for vertical plane plumes are in good agreement with the experimental data at the turbulent Schmidt number equal to 1.0. The variations of their maximum value along the direction of main flow were also given. It shows that the present model is good, i.e., the effect of buoyancy on turbulent kinetic energy and its dissipation rate should be taken into account, and the finite analytic method is effective.     

Mechanical realization of the medium imitating electromagnetic fields and particles

A turbulent fluid exhibits elastic properties. Turbulence may generate in the medium a body force. Small perturbations of averaged ideal turbulence reproduce the electromagnetic field. The averaged fluid velocity corresponds to the magnetic vector-potential, the perturbation of the averaged pressure to the scalar potential, and the body force due to nonuniformity of Reynolds stresses corresponds to the electric field. Discontinuities of the medium model particles and electric charges. A vapor bubble can be taken as a model of the neutron. Under the action of turbulent fluctuations the bubble turns into the inhomogeneity of the elastic medium. The region of rarefaction of the medium thus formed produces in the turbulent fluid the field of positive perturbation of the turbulence energy. This medium defect may serve as a model of the proton. In order to maintain the energy balance the respective field of negative perturbation of the turbulence energy should be formed. This field can be viewed as generated by an isle of quiescent fluid. The latter singularity models the electron. Electromagnetic interactions may be concerned with the entrainment of the pressure center in the turbulent aether. The magnetic force is due to the entrainment by the fluid stream, and the electric force is due to the entrainment by the turbulence.     

A boundary element formulation for natural convection problems

This paper presents a boundary element formulation employing a penalty function technique for two-dimensional steady thermal convection problems. By regarding the convective and buoyancy force terms in Navier-Stokes equations as body forces, the standard elastostatics analysis can be extended to solve the Navier-Stokes equations. In a similar manner, the standard potential analysis is extended to solve the energy transport equation. Finally, some numerical results are included, for typical natural convection problems, in order to demonstrate the efficiency of the present method.     

A structure-based model for transport in stably stratified homogeneous turbulent flows

We present an extension that allows a recently proposed structure-based model for turbulent scalar transport to account for buoyancy effects. The proposed model is based on a generalization of the Interactive Particle Representation Model (IPRM) and is accompanied by a four-equation transport model that provides the turbulence scales needed for the closure of the complete structure-based model (SBM). The structure tensors and their invariants are used to model the additional buoyancy terms that emerge in the four-equation transport equations. Model parameters are set by matching the asymptotic decay exponents in decaying turbulence. The validity of the model is considered for a large number of different types of stably stratified flows at different Richardson numbers (Ri), showing encouraging results. The complete structure-based model achieves fair agreement with LES and DNS predictions for vertical shear in the presence of vertical mean stratification, while the structure tensors are shown to be suitable for use as diagnostic tools for the morphology of highly anisotropic turbulent structures. Additionally, the proposed model is shown to be sensitive to the variation of the inclination angle θ between the direction of the mean velocity gradient and the orientation of the mean scalar gradient. Furthermore, the model correctly predicts that the evolution of the inverse shear parameter is insensitive to the choice of inclination angle, yielding a turbulent Prandtl number close to unity, in accordance with DNS results.     

Mathematical modelling of bubble driven flows in metallurgical processes

The complex fluid dynamics of two-phase bubbly flows in metallurgical reactors is modelled numerically by using a k–e turbulence model for the liquid phase, with a driving force determined by considering the motion of the bubbles. The latter are affected by the buoyancy forces and the drag caused by their relative motion with the mean and turbulent motions of the liquid, the turbulent component being obtained by random sampling to give an ensemble of bubble trajectories. The two-way coupling between the two phases is resolved by an iterative procedure which converges on a stable overall solution. The results are compared with measurements carried out on an air-water model and show good overall agreement.     

Direct numerical simulation of turbulent heat transfer in a wall-normal rotating channel flow

Investigations into the characteristics of turbulent heat transfer and coherent flow structures in a plane-channel subjected to wall-normal system rotation are conducted using direct numerical simulation (DNS). In order to investigate the influence of system rotation on the temperature field, a wide range of rotation numbers are tested, with the flow pattern transitioning from being fully turbulent to being quasilaminar, and eventually, fully laminar. In response to the Coriolis force, secondary flows appear as large vortical structures, which interact intensely with the wall shear layers and have a significant impact on the distribution of turbulence kinetic energy (TKE), turbulence scalar energy (TSE), temperature statistics, and turbulent heat fluxes. The characteristic length scales of turbulence structures responsible for the transport of TSE are the largest at the quasilaminar state, which demands a very large computational domain in order to capture the two-dimensional spectra of temperature fluctuations. The effects of the Coriolis force on the turbulent transport processes of the temperature variance and turbulent heat fluxes are thoroughly examined in terms of their respective budget balances.     

Simulation and Modelling of a Skewed Turbulent Channel Flow

A time-dependent three-dimensionally skewed flow is investigated using direct numerical simulations of the incompressible Navier-Stokes equations. The effect on the instantaneous and mean turbulent field is investigated. Instantaneous flowfields reveal that the skewing has the effect of initially reducing the strength and height of quasi-streamwise vortices of both signs of rotation with respect to the skewing. A mechanism for this process is put forward. The mean flowfields show drops in turbulence quantities such as turbulence kinetic energy. In addition to this, two-equation turbulence modelling of the flow is carried out. This highlights a deficiency, in that the standard turbulence models are unable to capture the drop in turbulence intensity due to the skewing. A modification based on the exact dissipation equation is found to significantly improve the model behaviour for this flow. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effect of thermo‐solutal stratification on recirculation flow patterns in a backward‐facing step channel flow

This paper presents results on the combined effect of thermo‐solutal buoyancy forces on the recirculatory flow behavior in a horizontal channel with backward‐facing step and the ensuing impact on heat and mass transfer phenomena. The governing equations for double diffusive mixed convection are represented in velocity–vorticity form of momentum equations, velocity Poisson equations, energy and concentration equations. Galerkin's finite‐element method has been employed to solve the governing equations. Recirculatory flow fields with heat and mass transfer are simulated for opposing and aiding thermo‐solutal buoyancy forces by assuming suitable boundary conditions for energy and concentration equations. The effect of Richardson number (0.1?Ri?10) and buoyancy ratio (?10?N?10) on the recirculation bubble and Nusselt and Sherwood numbers are studied in detail. For Richardson number greater than unity, distinct variations in the gradients of Nusselt number and Sherwood number with buoyancy ratio are observed for flow regimes with opposing and aiding buoyancy forces. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of unsteady gas–liquid flows in a rectangular tank: Comparison of Euler–Eulerian and Euler–Lagrangian simulations

We study the dynamics of gas–liquid flows experimentally and computationally in a rectangular bubble column where the gas source is introduced at the corner. The flow in this reactor is complex and inherently unsteady in nature. The two-dimensional liquid phase velocity field is calculated by an Eulerian approach solving the unsteady Reynolds Averaged Navier Stokes equations. The conservation equations are closed using a two parameter turbulence model. The two-way coupling was accounted for by adding source terms in the conservation equations of the continuous phase to take into account the interaction with the dispersed phase. Bubble tracking is achieved through a Lagrangian approach. Here the equations of motion are solved taking into account the drag, pressure, buoyancy and gravity forces. The time-averaged flows along with the variables which characterize turbulence are analyzed for a wide range of gas flow-rates using Euler–Lagrangian simulations. These simulation predictions are validated with Euler–Eulerian simulations where the gas-phase distribution is captured as a void fraction and PIV experiments. The motion of bubbles induces turbulence in the flow. The applicability of two parameter models for turbulence like the standard k–ε model on time-averaged flow properties is addressed. From the results of the time averaged velocity field, turbulence intensity, turbulent viscosity and gas hold-up profiles, it is concluded that the Euler–Lagrangian model is applicable at lower gas flow-rates. The Euler–Eulerian approach was found to be valid at lower as well as higher gas flow-rates.     

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     

Turbulent natural convection in vertical parallel-plate channels

The problem of buoyancy driven turbulent flow in parallel-plate channels is investigated. The investigation is limited to vertical channels of uniform cross-section with different modes of heating. The details of the flow and thermal fields are obtained from the solution of the conservation equations of mass, momentum, and energy in addition to equations of the low Reynolds number turbulence model. The study covers Rayleigh number ranging from 10⁵ to 10⁷ and focuses on the effect of channel geometry on the characteristic of the flow and thermal fields as well as the local and average Nusselt number variation. A Nusselt number correlation has been developed in terms of a modified Rayleigh number and channel aspect ratio for the cases of symmetrically heated isothermal and isoflux conditions.     

Investigation of three dimensional air flow characteristic and effect of heated obstruction within room

This study reports the results of a numerical investigation of three-dimensional turbulent buoyant recirculating flow within rooms with heated obstruction. The study involves the solution of partial differential equations for the conservation of mass, momentum, energy, concentration, turbulent energy and its dissipation rate. These equations were solved together with algebraic expressions for the turbulent viscosity and heat diffusivity using k-ε turbulence model by performing simulations on FLUENT 6.3. The CFD method was validated via comparing with the available experimental data. A comparison with experimental results shows good agreement. This means that the present computer code has a good capability to simulate 3D airflow and effect of obstruction within room. The present study demonstrates the flow behavior, thermal distribution and CO₂ concentration inside the room in the presence of heat flux obstruction. Two different configurations of ventilation system have been studied. Mixing and Displacement ventilation system have been used in two geometries depending on location of opening inlet. The ventilation effectiveness for heat removal (E_T) is used to evaluate the indoor climate and average temperature is an important parameter in designs the ventilation systems. Two notable points are presented; first, mixing ventilation is depending on throw of jet. CO₂ concentration and temperature distribution have been effected in upper zone more than occupied zone with presence the obstruction. Second notable points are presented; in displacement ventilation buoyancy effect is considerable. Vertical temperature gradient above the obstruction implies that both fresh air and CO₂ concentration.     

Effect of particles suspended in a fluid flow on the decay of isotropic turbulence

Dynamic equations have been obtained for the two-point double correlations of the fluctuation velocities of a fluid and the particles suspended in it at low volume concentrations of the solid phase. In the case of uniform isotropic turbulence these equations can be considerably simplified. The final period of decay of isotropic turbulence has been studied in detail. At this stage in the case of high-inertia particles the inhomogeneous-fluid turbulence is similar to the turbulence of a homogeneous fluid (without particles) in the sense that the presence of the particles affects only the fluctuation energy but leaves unchanged the spatial scales of turbulence and the spatial energy spectrum function. The suspended particles lead to exponential damping of the turbulent pulsations.Little theoretical information is available on the hydrodynamics of a suspension of fine particles in a turbulent liquid or gas. Research has been mainly confined to the behavior of the individual particles in a given turbulence field [1]. The problem of the turbulent motion of the mixture as a whole has been examined by Barenblatt [2], who derived the equations of motion of the mixture, using Kolmogorov's hypothesis to close them. Hinze [3] has also attempted to derive equations for turbulent pulsations of the mixture. However, as Murray showed [4], Hinze' s equations contradict Newton' s third law.The effect of suspended particles on the turbulence of a two-phase flow is governed by the noncorrespondence of the local velocities of the particles and the medium. The forces of resistance to the motion of the particles relative to the fluid lead to additional dissipation of fluctuation energy and decay of turbulence [2]. On the other hand, if the averaged velocities of particles and medium do not correspond, the suspended particles may also have a destabilizing effect [5, 6], causing energy transfer from the averaged to the pulsating motion. Below we shall consider the case where the averaged velocities of the two phases coincide, i.e., we shall deal only with the first of the two above-mentioned effects.The authors thank G.I. Barenblatt for his useful advice.     

Transport and deposition of neutral particles in magnetohydrodynamic turbulent channel flows at low magnetic Reynolds numbers

The effect of Lorentz force on particle transport and deposition is studied by using direct numerical simulation of turbulent channel flow of electrically conducting fluids combined with discrete particle simulation of the trajectories of uncharged, spherical particles. The magnetohydrodynamic equations for fluid flows at low magnetic Reynolds numbers are adopted. The particle motion is determined by the drag, added mass, and pressure gradient forces. Results are obtained for flows with particle ensembles of various densities and diameters in the presence of streamwise, wall-normal or spanwise magnetic fields. It is found that the particle dispersion in the wall-normal and spanwise directions is decreased due to the changes of the underlying fluid turbulence by the Lorentz force, while it is increased in the streamwise direction. The particle accumulation in the near-wall region is diminished in the magnetohydrodynamic flows. In addition, the tendency of small inertia particles to concentrate preferentially in the low-speed streaks near the walls is strengthened with increasing Hartmann number. The particle transport by turbophoretic drift and turbulent diffusion is damped by the magnetic field and, consequently, particle deposition is reduced.