Analysis of turbulent flows in fixed and moving permeable media

The ability to realistically model flows through heterogeneous domains, which contain both solid and fluid phases, can benefit the analysis and simulation of complex real-world systems. Environmental impact studies, as well as engineering equipment design, can both take advantage of reliable modelling of turbulent flow in permeable media. Turbulence models proposed for such flows depend on the order of application of volume-and time-average operators. Two methodologies, following the two orders of integration, lead to distinct governing equations for the statistical quantities. This paper reviews recently published methodologies to mathematically characterize turbulent transport in permeable media. A new concept, called double-decomposition, is here discussed and instantaneous local transport equations are reviewed for clear flow before the time and volume averaging procedures are applied to them. Equations for turbulent transport follow, including their detailed derivation and a proposed model for suitable numerical simulations. The case of a moving porous bed is also discussed and transport equations for the mean and turbulent flow fields are presented.     

Mathematical modeling of the transportation competency of an ice-covered flow

A two-dimensional model was developed to study the effect of ice cover on the transportation competence of ice-covered flows. The model is based on the equations of motion, impurity transfer, turbulence energy, and the ratio of turbulent energy to turbulent kinetic energy with allowance made to the effect of turbulent energy bursts on solid surfaces bounding the flow. The turbulent bursts on solid boundaries of open and ice-covered flows are shown to have no effect on the structure of flows, characterized by their aver-aged characteristics, but considerably change the distribution of suspended sediment concentrations.     

Numerical Modeling of Wind-Induced Flows in Stratified Water Bodies

Two-dimensional (in the vertical plane) wind-induced flows in no-flow-through reservoirs are considered. A numerical algorithm in the flow function–vortex variables is proposed based on the equations of slow stratified flows in the Boussinesq and boundary layer approximations with variable coefficient of vertical turbulent exchange. An analytical solution is given for a simplified problem.     

A model of the wave and current boundary‐layer structure

A model of the wave and current boundary‐layer structure was developed using the k–ε turbulent closure model. The finite‐difference method was used to solve the governing equations. Vertical logarithmic grids and equal time steps were adopted. The following modelled simulations were obtained: (1) vertical profiles of wave velocity amplitude, eddy viscosity coefficient and turbulent kinetic energy with waves only; (2) vertical profiles of wave velocity amplitude, mean current velocity, eddy viscosity coefficient and turbulent kinetic energy with waves having a following current. To test the validity and the rationality of the present model, vertical profiles of modelled wave velocity amplitude and mean velocity were compared with corresponding experimental results available in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Vortex evolution

Abstract

Friedmann's equation and the potential vorticity equation are generalised for turbulent motion. The generalised equations incorporate some new phenomena connected with turbulent transport of mass. It is proved that, if ?×[S×Ω+S(?·S)]≠0 where Ω is the absolute vorticity of the velocity and S is the turbulent density flux, then the Helmholtz-Kelvin theorem concerning the conservation of the velocity circulation around a closed path is violated and the potential vorticity is not a Lagrangian adiabatic invariant. The effects of this turbulent transport of mass on the creation or dissipation of vorticity discussed here is not equivalent to effects of baroclinicity or viscosity. Some possible implications of the new circulation theorem in geophysical and astrophysical fluid dynamics are discussed.     

Problem of transient turbulent convection and a semiempirical profile of turbulent heat exchange coefficient in freshwater bodies

A problem of convective cooling of both freshwater and saltwater lake is formulated. A transient one-dimensional turbulent model with variable layer depth is proposed to describe the process of freezing. The dependence of layer depth on time is determined by a universal equation describing the propagation of Deardorff convective layer. A combination of the main form of equations of Kolmogorov–Obukhov theory and available experimental data is shown to allow the construction of a universal profile of turbulent exchange coefficient, depending on the height.     

A SIMPLIFIED APPROACH TO MODELING 3D SEDIMENT-LADEN TURBULENT FLOWS

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Hydromagnetic energy balance equations for the solar atmosphere

Summary Following the pattern established in meteorology, a system of energy balance equations for the solar atmosphere is presented. Since both hydromagnetic and thermodynamic processes are considered, the system includes kinetic, potential, thermal and magnetic forms of energy. Ionization energy is indirectly included in the treatment. The spatial distribution of the energy forms and the processes transforming them are separated into zonal means and departures from such means in order to depict turbulent eddy effects. Where available data concerning solar processes are used in order to appraise certain of the terms which arise.     

The amplitude of turbulent shear flow

A new formulation of the problem of the statistical stability of fully turbulent shear flow is proposed, in which one seeks mean fields that bound the observed flow from the stable side. In the spirit of maximum transport theory, this formulation admits a larger set of “flows” than are dynamically possible. A sequence of constraints derived from the equations of motion can narrow this set, permitting at each step the determination of a “most stable” field free of any empirical elements. Turbulent channel flow is proposed as the first application and test of this quantitative theory. Past deductive theories for this flow, from “mean field” to “transport upper bounds,” are assessed. It is shown why these theories do not retain the significant destabilizing mechanisms of the actual flow. The implications for turbulent flow of recent work on the nonlinear and three-dimensional instability of laminar shearing flow are described. In first exploration of the “decoupled mean” stability theory proposed here, approximate analytical and numerical stability methods are used to find an amplitude and structure for the averaged flow propoerties. The quantitative results differ by considerably less than two from the observed values, providing an incentive for a more complete numerical study and for further constraints on the admitted class of flows. In the language now current for nonlinear stability theory, evidence is advanced here that anN-dimensional central manifold is adjacent to the realized turbulent flow, whereN has the largest possible value compatible with the dynamical relations.     

Turbulent flow through porous media

The pressure driving flow through porous media must be equal to the viscous resistance plus the inertial resistance. Formulas are developed for both the viscous resistance and the inertial resistance. The expression for the coefficient of permeability consists of parameters which describe the characteristics of the porous medium and the permeating fluid and which, for unconsolidated isotropic granular media, are all measurable. A procedure is proposed for testing for the occurrence of turbulence and calculating the effective permeability when it occurs. The formulas are applied to a set of data from 588 permeameter runs ranging from laminar to highly turbulent. The equations fit the data from the permeameter closely through the laminar flow conditions and quite closely through the turbulent conditions. In the turbulent range, the plotting of the data separates into three distinct lines for each of the three shapes of particles used in the tests. For the porous medium and fluid of these tests, turbulence begins at a head gradient of about 0.1.     

湍流输送的非线性热力学性质

湍流输送是一种热力学不可逆过程,本文利用非线性热力学研究了湍流输送的特征. 将热力学流对热力学力以平衡态作为参考态进行Taylor展开,可以得到湍流输送系数是系统宏观参量梯度的Taylor级数. 线性湍流输送系数是Reynolds湍流闭合方案的K闭合湍流输送系数;而湍流输送系数非线性项则是系统偏离热力学平衡态所造成的热力学非线性效应. 湍流输送系数这一热力学性质提供了一种热力学湍流闭合方案. 线性湍流输送系数是正定的,湍流输送只能使系统宏观参量均匀化;而在远离平衡态的热力学非线性区,可能导致湍流输送系数负黏性现象. 在最小熵产生态的条件下,热力学流对热力学力Taylor展开的各级系数间存在一种递推关系. 利用这种递推关系大大减少了由实验确定的Taylor级数的系数个数.     

Computation of shallow water waves with hybrid finite elements

The numerical computation of two-dimensional incompressible long-period shallow water waves using the method of finite elements is presented. It is shown that the set of equations having been solved in each time step is reduced to a third of the usual size by satisfying the equations of motion on the element level only (hybrid finite elements). The water levels in all nodes are the sole parameters computed from the set of equations. The velocities are subsequently calculated on the element level in each time step. Special emphasis is placed on reproducing eddies behind corners using an anisotropic turbulent exchange tensor.     

Equation de transport et de diffusion pour l'atmosphère turbulente avec des masses d'air différemment échauffées

Summary Generalization of the diffusion and transport equation are given. They refer to an atmosphere in which in addition to irregular turbulent motion exists also regular motion of the heated and cooled air masses. The equations are briefly discussed.     

Lateral friction in the oceans as an effect of potential vorticity mixing

Abstract

Applying a mixing-length calculation to potential vorticity rather than to momentum a new type of lateral friction appears in the oceanic mass transport equations. This friction is evaluated for the special case of horizontally homogeneous, quasi-geostrophic turbulence. The main effect is a westward force arising from the so-called β-term. This produces an additional southward interior transport and a strengthening of the western boundary current. A turbulent exchange coefficient K^H = 10⁸ em²s^?1 is sufficient to give a Gulf Stream transport twice that obtained by the classical Sverdrup model.     

Dynamics of aircraft exhaust plumes in the jet-regime

A computational model describing the two-dimensional, turbulent mixing of a single jet of exhaust gas from aircraft engines with the ambient atmosphere is presented. The underlying assumptions and governing equations are examined and supplemented by a discussion of analytical solutions. As an application, the jet dynamics of a B747-400 aircraft engine in cruise and its dependence on key parameters is investigated in detail. The computer code for this dynamical model is computationally fast and can easily be coupled to complex chemical and microphysical models in order to perform comprehensive studies of atmospheric effects from aircraft exhaust emissions in the jet regime.     

An analysis of the mechanism of flow in ice-covered rivers

The paper presents a mechanism of flow of water in an ice-covered river in the case of movable bottom. The analysis is based upon the principal hydrodynamics equations of turbulent flow in the case of steady uniform motion. It leads to the conclusion of linear distribution of the turbulent shear stress with depth. It allows to obtain the vertical distribution of velocity of flowing water under the assumption that at the boundaries (movable bottom and ice) the viscosity of water is greater than the kinematics viscosity. The relations describing the vertical distribution of velocity of flowing water, as well as the eddy viscosity coefficient under these conditions, are given.     

Indirect air–sea interactions simulated with a coupled turbulence-resolving model

A turbulence-resolving parallelized atmospheric large-eddy simulation model (PALM) has been applied to study turbulent interactions between the humid atmospheric boundary layer (ABL) and the salt water oceanic mixed layer (OML). The most energetic three-dimensional turbulent eddies in the ABL–OML system (convective cells) were explicitly resolved in these simulations. This study considers a case of shear-free convection in the coupled ABL–OML system. The ABL–OML coupling scheme used the turbulent fluxes at the bottom of the ABL as upper boundary conditions for the OML and the sea surface temperature at the top of the OML as lower boundary conditions for the ABL. The analysis of the numerical experiment confirms that the ABL–OML interactions involve both the traditional direct coupling mechanism and much less studied indirect coupling mechanism (Garrett Dyn Atmos Ocean 23:19–34, 1996). The direct coupling refers to a common flux-gradient representation of the air–sea exchange, which is controlled by the temperature difference across the air–water interface. The indirect coupling refers to thermal instability of the Rayleigh–Benard convection, which is controlled by the temperature difference across the entire mixed layer through formation of the large convective eddies or cells. The indirect coupling mechanism in these simulations explained up to 45 % of the ABL–OML co-variability on the turbulent scales. Despite relatively small amplitude of the sea surface temperature fluctuations, persistence of the OML cells organizes the ABL convective cells. Water downdrafts in the OML cells tend to be collocated with air updrafts in the ABL cells. The study concludes that the convective structures in the ABL and the OML are co-organized. The OML convection controls the air–sea turbulent exchange in the quasi-equilibrium convective ABL–OML system.     

Importance of subgrid-scale parameterization in numerical simulations of lake circulation

Two subgrid-scale modeling techniques––Smagorinsky’s postulation for the horizontal eddy viscosity and the Mellor–Yamada level-2 model for the vertical eddy viscosity––are applied as turbulence closure conditions to numerical simulations of resolved-scale baroclinic lake circulations. The use of the total variation diminishing (TVD) technique in the numerical treatment of the advection terms in the governing equations depresses numerical diffusion to an acceptably low level and makes stable numerical performances possible with small eddy viscosities resulting from the turbulence closure parameterizations. The results show that, with regard to the effect of an external wind stress, the vertical turbulent mixing is mainly restricted to the topmost epilimnion with the order of magnitude for the vertical eddy viscosity of 10⁻³ m² s⁻¹, whilst the horizontal turbulent mixing may reach a somewhat deeper zone with an order of magnitude for the horizontal eddy viscosity of 0.1–1 m² s⁻¹. Their spatial and temporal variations and influences on numerical results are significant. A comparison with prescribed constant eddy viscosities clearly shows the importance of subgrid-scale closures on resolved-scale flows in the lake circulation simulation. A predetermination of the eddy viscosities is inappropriate and should be abandoned. Their values must be determined by suitable subgrid-scale closure techniques.     

Linear mechanism by which internal gravity waves are generated and intensified in the ionosphere when interacting with an inhomogeneous zonal wind: 2. Internal gravity wave generation and intensification during the linear evolution stage

The linear mechanism by which internal gravity waves (IGWs) are generated and subsequently intensified in a stably stratified dissipative ionosphere in the presence of an inhomogeneous zonal wind (shear flow) has been studied. In the case of shear flows, the operators of linear problems are nonself-adjoint and the corresponding eigenfunctions are nonorthogonal; a canonical approach can hardly be used to study such motions. It is more adequate to apply the so-called nonmodal calculation. Dynamic equations and equations of energy transfer of IGW disturbances in the ionosphere with a shear flow have been obtained based on a nonmodal approach. Exact analytical solutions for the constructed dynamic equations have been found. The growth rate of the IGW shear instability has been determined. It has been established that IGW disturbances are intensified in an algebraically power manner rather than exponentially in the course of time. The effectiveness of the linear mechanism by which IGWs are intensified when interacting with an inhomogeneous zonal wind is analyzed. It has been indicated that IGWs effectively obtain the shear flow energy during the linear evolution stage and substantially increase (by an order of magnitude) their amplitude and energy. The frequency and the wave vector of generated IGW modes depend on time; therefore, a wide spectrum of wavelike disturbances, depending on the linear, rather than nonlinear, turbulent effects, is formed in the ionosphere with a shear flow. Thereby, a new degree of freedom appears, and the turbulent state of atmospheric—ionospheric layers can be formed on IGW disturbances.