On the consequences of material frame-indifference in algebraic stress models

The principle of material frame-indifference (MFI) is a fundamental and controversial principle of continuum mechanics that has been invoked recently to derive nonlinear algebraic models for stresses of viscoelastic liquids. The purpose of the present study is to identify regions of a flow field where MFI should be considered. Such regions are identified by computing the angular velocity of the principal directions of the rate-of-deformation tensor in order to obtain an Euclidean objective vorticity tensor. An analysis is carried out for uniform shear and extensional flows, and for a Couette flow. The method is then applied to the planar flow through an abrupt 4:1 contraction and to the two-dimensional stream past a circular cylinder. The main results are: (1) MFI should be taken into account in regions characterized by the transition between two different kinematics and a significant velocity magnitude, and (2) MFI can be safely ignored in regions of pure viscometric behaviour as well as in recirculation regions. The consequences of MFI being taken into account are then examined upon using the Euclidean objective vorticity tensor in a simple algebraic constitutive law for viscoelastic fluids.     

A practical implementation of turbulence models for the computation of three-dimensional separated flows

An upwind MUSCL-type implicit scheme for the three-dimensional Navier-Stokes equations is presented and details on the implementation for three-dimensional flows of a ‘diagonal’ upwind implicit operator are developed. Turbulence models for separated flows are also described with an emphasis on the numerical specificities of the Johnson-King non-equilibrium model. Good predictions of separated two- and three-dimensional flows are demonstrated.     

Statistical models for predicting the effect of bidisperse particle collisions on particle velocities and stresses in homogeneous anisotropic turbulent flows

The purpose of this paper is to present and compare two statistical models for predicting the effect of collisions on particle velocities and stresses in bidisperse turbulent flows. These models start from a kinetic equation for the probability density function (PDF) of the particle velocity distribution in a homogeneous anisotropic turbulent flow. The kinetic equation describes simultaneously particle–turbulence and particle–particle interactions. The paper is focused on deriving the collision terms in the governing equations of the PDF moments. One of the collision models is based on a Grad-like expansion for the PDF of the velocity distributions of two particles. The other model stems from a Grad-like expansion for the joint fluid–particle PDF. The validity of these models is explored by comparing with Lagrangian simulations of particle tracking in uniformly sheared and isotropic turbulent flows generated by LES. Notwithstanding the fact that the fluid turbulence may be isotropic, the particle velocity fluctuations are anisotropic due to the impact of gravitational settling. Comparisons of the model predictions and the numerical simulations show encouraging agreement.     

Self-consistency and the use of correlated stochastic separated flow models for prediction of particle-laden flows

Computational fluid dynamics (CFD) is being used increasingly in the design and analysis of particle-laden flows. A significant challenge of this work is in correctly predicting the interaction of the fluid turbulence with the particulate phase. Typically, Lagrangian tracking is used to calculate the particle trajectories with stochastic treatments used to provide an instantaneous turbulent flow field. The stochastic calculations are based on the mean velocities and turbulence quantities calculated by the CFD solver. The current work examines the correlated stochastic separated flow (SSF) model used to synthesize the instantaneous fluid velocity field. Two functional forms of the Eulerian spatial correlation are considered: exponential, and Frenkiel with loop parameter m equal to unity. It is well known that the use of a Frenkiel function is incorrect due to the Markovian nature of the model. Nonetheless, a literature review indicates that the Frenkiel function is still being used in the CFD community. In order to illustrate the implications of this, numerical predictions are compared to Taylor's analytical result for fluid particle dispersion in homogeneous isotropic turbulence. Excellent predictions are obtained with the exponential correlation and recommendations on timestep requirements are made. In contrast, predictions from the Frenkiel model are in poor agreement with Taylor's solution. This poor agreement results from an inconsistency between the effective correlation of fluid velocities arising from the model and the original intended correlation.     

Application of Reynolds stress transport turbulence closure models to flows affected by Lorentz and buoyancy forces

The effect of magnetic field strength and orientation on two types of electromagnetically influenced turbulent flows was studied numerically under the Reynolds averaged Navier–Stokes (RANS) framework. Previous work (Wilson et al., 2014) used an electromagnetically extended linear eddy-viscosity model, whilst the current paper focuses on the performance of a more advanced Reynolds stress transport type model both with and without electromagnetic modifications proposed by Kenjereš et al. (2004). First, a fully-developed 2D channel flow is considered with a magnetic field imposed in either the wall-normal or streamwise direction. Both forms of the RSM gave good agreement with the DNS data for the wall-normal magnetic field across the range of Hartmann numbers with the additional electromagnetic terms providing a small, but noticeable, difference. For the streamwise magnetic field, where electromagnetic influence is only through the turbulence, the electromagnetically extended RSM performed well at moderate Hartmann numbers but returned laminar flow at the highest Hartmann number considered, contrary to the DNS. The RSM results were, however, significantly better than the previous eddy-viscosity model predictions. The second case is that of unsteady 3D Rayleigh–Bénard convection with a magnetic field imposed in either a horizontal or vertical direction. Results revealed that a significant reorganization of the flow structures is predicted to occur. For a vertically oriented magnetic field, the plume structures increase in number and become thinner and elongated along the magnetic field lines, leading to an increase in thermal mixing within the core in agreement with Hanjalić and Kenjereš (2000). With a horizontal magnetic field, the structures become two-dimensional and a striking realignment of the roll cells’ axes with the magnetic field lines occurs. The results demonstrate the capability of the Reynolds stress transport approach in modelling MHD flows that are relevant to industry and offer potential for those wishing to control levels of turbulence, heat transfer or concentration without recourse to mechanical means.     

Assessment of the performance of different classes of turbulence models in a wide range of non-equilibrium flows

The performance of thirteen benchmark turbulence models within the RANS framework has been assessed in classical non-equilibrium flows. Linear and non-linear eddy-viscosity schemes, Reynolds stress transport models and single- and two-time-scale approaches have been considered in the investigation. Among the test cases studied are homogeneous shear and normally strained flows, adverse-pressure-gradient, favourable-pressure-gradient and oscillatory boundary layer flows, fully developed oscillatory and ramp up pipe flows and steady and pulsated backward-facing-step flows. The main advantages and drawbacks of the models in each of the test cases are discussed. These discussions provide a reasonably wide understanding of the expected behaviour of the models for future applications in non-equilibrium flows, and also result in suggestions on how the effectiveness of existing models can be further improved.     

Differential and algebraic models for the second moments of particle velocity and temperature fluctuations in turbulent flows

Differential and algebraic models are constructed for the dispersed-phase turbulent stresses and heat fluxes and for the mixed moments of the velocity and temperature fluctuations in the continuous and dispersed phases. The models are based on a kinetic equation for the joint probability density of the particle velocity and temperature in an anisotropic turbulent flow. The results are compared with the available direct numerical simulation (DNS) data.     

Finite element implementation of two‐equation and algebraic stress turbulence models for steady incompressible flows

The main purpose of this paper is to describe a finite element formulation for solving the equations for k and ε of the classical k–ε turbulence model, or any other two‐equation model. The finite element discretization is based on the SUPG method together with a discontinuity capturing technique to deal with sharp internal and boundary layers. The iterative strategy consists of several nested loops, the outermost being the linearization of the Navier–Stokes equations. The basic k–ε model is used for the implementation of an algebraic stress model that is able to account for the effects of rotation. Some numerical examples are presented in order to show the performance of the proposed scheme for simulating directly steady flows, without the need of reaching the steady state through a transient evolution. Copyright © 1999 John Wiley & Sons, Ltd.     

Progress in the extension of a second-moment closure for turbulent environmental flows

An advanced second-moment closure for the double-averaged turbulence equations of porous medium and vegetation flows is proposed. It treats three kinds of second moments which appear in the double-averaged momentum equation. They are the dispersive covariance, the volume averaged (total) Reynolds stress and the micro-scale Reynolds stress. The two-component-limit pressure–strain correlation model is applied to model the total Reynolds stress equation whilst a novel scale-similarity non-linear $k''–''\epsilon$ two-equation eddy viscosity model is employed for the micro-scale turbulence. For the dispersive covariance, an algebraic relation is applied. Model validation in several fully developed homogeneous porous medium flows, porous channel flows and aquatic vegetation canopy flows is performed with satisfactory agreement with the data.     

Optimal treatment of stratified Carreau and Casson nanofluids flows in Darcy-Forchheimer porous space over porous matrix

A comparative three-dimensional(3 D) analysis for Casson-nanofluid and Carreau-nanofluid flows due to a flat body in a magnetohydrodynamic(MHD) stratified environment is presented. Flow is estimated to be suspended in a Darcy-Forchheimer medium. Soret and Dufour responses are also accommodated in the flow field. A moving(rotating) coordinate system is exercised to examine the bidirectionally stretched flow fields(flow, heat transfer, and mass transfer). Nanofluid is compounded by taking ethylene...     

Effectiveness of Darcy-Forchheimer and nonlinear mixed convection aspects in stratified Maxwell nanomaterial flow induced by convectively heated surface

The effect of nonlinear mixed convection in stretched flows of rate-type non-Newtonian materials is described. The formulation is based upon the Maxwell liquid which elaborates thermal relation time characteristics. Nanofluid properties are studied considering thermophoresis and Brownian movement. Thermal radiation, double stratification, convective conditions, and heat generation are incorporated in energy and nanoparticle concentration expressions. A boundary-layer concept is implemented for the simplification of mathematical expressions. The modeled nonlinear problems are computed with an optimal homotopy scheme. Moreover, the Nusselt and Sherwood numbers as well as the velocity, nanoparticle concentration, and temperature are emphasized. The results show opposite impacts of the Deborah number and the porosity factor on the velocity distribution.     

On the nonlinear interactions and existence of breathers in a chain of coupled pendulums

The paper considers a chain of linearly coupled pendulums. Continues first order system equations are treated via time and space multiple scale method which lead to nonlinear Schrödinger equation. Further investigations on the nonlinear Schrödinger equation detects systems responses in the form of propagated nonlinear waves as functions of their envelope and phases. This provides information about localization of nonlinear waves and their directions in space and time.     

Numerical investigation of variable viscosities and thermal stratification effects on MHD mixed convective heat and mass transfer past a porous wedge in the presence of a chemical reaction

An analysis is presented to investigate the effects of variable viscosities and thermal stratification on the MHD mixed convective heat and mass transfer of a viscous, incompressible, and electrically conducting fluid past a porous wedge in the presence of a chemical reaction. The wall of the wedge is embedded in a uniform nonDarcian porous medium in order to allow for possible fluid wall suction or injection. The governing boundary layer equations are written into a dimensionless form by similarity transformations. The transformed coupled nonlinear ordinary differential equations are solved numerically with finite difference methods. Numerical calculations up to the thirdorder level of truncation are carried out for different values of dimensionless parameters. The results are presented graphically, and show that the flow field and other quantities of physical interest are significantly influenced by these parameters. The results are compared with those available in literature, and show excellent agreement.     

Localized solutions to the nonlinear wave equation for flexural motions of an elastic beam traveling in an air-filled tube

Steady-progressive-wave solutions are sought to the nonlinear wave equation derived previously [J. Fluids Struct. 16 (2002) 597] for flexural motions of an elastic beam traveling in an air-filled tube along its center axis at a subsonic speed. Fluid-structure interactions are taken into account through aerodynamic loading on the lateral surface of the beam subjected to small but finite deflection but end effects and viscous effects are neglected. Linear dispersion characteristics are first examined by exploiting the small ratio of the induced mass to the mass of the beam per unit length. Centered around the traveling speed of the beam, there exists such a narrow range of propagation velocity that the linear steady propagation is prohibited. In this range, it is revealed that some interesting nonlinear solutions exist. The periodic wavetrain is found to exist as the exact solution. Asymptotic analysis is then made by applying the method of multiple scales and the stationary nonlinear Schrödinger equation is derived for a complex amplitude. A monochromatic solution to this equation corresponds to the exact periodic solution. Imposing undisturbed boundary conditions at infinity, it is revealed that the localized solution exists as a result of balance between the linear instability and the nonlinearity. This solution is checked by solving the nonlinear equation numerically. It is further revealed that the amplitude-modulated wavetrain exists not only in the range of the velocity mentioned above but also outside of it.     

An analysis of perturbation based methods for the treatment of parameter uncertainty in numerical groundwater models

Monte Carlo methods are robust approaches for the estimation of prediction uncertainty in groundwater flow and transport modelling under uncertain model parameters. However Monte Carlo procedures estimate the prediction statistics by generating a population of solutions from random realisations of the model parameters which are consistent with the parameter statistics, and as a result are computationally demanding. Taylor series based procedures offer an alternative to Monte Carlo methods for calculating the prediction statistics. Two such approaches, the first-order second moment and McLaughlin and Wood's perturbation method, are based on using a Taylor series to derive approximate expressions for the model predictions first and second statistical moments. In this paper the perturbation method presented by McLaughlin and Wood is rederived using Vetter matrix notation. This is compared with the first-order second moment (FOSM) method and while the steady state expressions for these two approaches are shown to be equivalent, the transient forms are considerably different. A new form of the FOSM is derived, which is simpler and has a lower computational burden. However, the transient McLaughlin and Wood expression is found to have a significantly lower computational overhead than either of the FOSM methods presented.     

First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow

This article deals with a review and critical analysis of first order hydrodynamic models of vehicular traffic flow obtained by the closure of the mass conservation equation. The closure is obtained by phenomenological models suitable to relate the local mean velocity to local density profiles. Various models are described and critically analyzed in the deterministic and stochastic case. The analysis is developed in view of applications of the models to traffic flow simulations for networks of roads. Some research perspectives are derived from the above analysis and proposed in the last part of the paper. To cite this article: N. Bellomo, V. Coscia, C. R. Mecanique 333 (2005).     

Modelling and numerical methods for the dynamics of impurities in a gas

The dynamics of a mixture of impurities in a gas can be represented by a system of linear Boltzmann equations for hard spheres. We assume that the background is in thermodynamic equilibrium and that the polluting particles are sufficiently few (in comparison with the background molecules) to admit that there are no collisions among couples of them. In order to derive non‐trivial hydrodynamic models, we close the Euler system around local Maxwellian's which are not equilibrium states. The kinetic model is solved by using a Monte Carlo method, the hydrodynamic one by implicit–explicit Runge–Kutta schemes with weighted essentially non‐oscillatory reconstruction (J. Sci. Comput. 2005; 25 (1–2):129–155). Several numerical tests are then computed in order to compare the results obtained with the kinetic and the hydrodynamic models. Copyright © 2007 John Wiley & Sons, Ltd.