Viscous dissipation effect on heat transfer characteristics of rectangular microchannels under slip flow regime and H1 boundary conditions

Viscous dissipation effect on heat transfer characteristics of a rectangular microchannel is studied. Flow is governed by the Navier–Stokes equations with the slip flow and temperature jump boundary conditions. Integral transform technique is applied to derive the temperature distribution and Nusselt number. The velocity distribution is taken from literature. The solution method is verified for the case where viscous dissipation is neglected. It is found that, the viscous dissipation is negligible for gas flows in microchannels, since the contribution of this effect on Nu number is about 1%. However, this effect should be taken into account for much more viscous flows, such as liquid flows. Neglecting this effect for a flat microchannel with an aspect ratio of 0.1 for Br=0.04 underestimates the Nu number about 5%.     

Spray Penetration in a Turbulent Flow

Analytical expressions for mass concentration of liquid fuel in a spray are derived taking into account the effects of gas turbulence, and assuming that the influence of droplets on gas is small (intitial stage of spray development). Beyond a certain distance the spray is expected to be fully dispersed. This distance is identified with the maximum spray penetration. Then the influence of turbulence on the spray stopping distance is discussed and the rms spray penetration is computed from a trajectory (Lagrangian) approach. Finally, the problem of spray penetration is investigated in a homogeneous two-phase flow regime taking into account the dispersion of spray away from its axis. It is predicted that for realistic values of spray parameters the spray penetration at large distances from the nozzle is expected to be proportional to t ^2/3 (in the case when this dispersion is not taken into account this distance is proportional to t ^1/2). The t ^2/3 law is supported by experimental observations for a high pressure injector. This revised version was published online in July 2006 with corrections to the Cover Date.     

Computer simulation of turbulence in internal combustion engines

The paper presents two- and three-dimensional computations of the in-cylinder turbulent flow in a diesel engine. The mathematical formulation is presented first, with emphasis on the modifications made to the standard k-ε model of turbulence, to account for rapid compression/expansion, and on the k-w model also used in the computations. Then, the results of two-and three-dimensional transient calculations are presented and compared with experimental data. It is realized that two-dimensional computations may be of little value to real engines, which would probably require three-dimensional analyses. However, two-dimensional studies are still useful in allowing the testing of new ideas easily and economically. It is concluded that the standard k-ε model may lead to poor predictions when used for internal combustion (IC) engine simulations, and that the modified model leads to more reasonable length-scale distributions, and it improves significantly the overall agreement of velocity predictions with experiment. The effect of the k-ε modification is apparent in both the two- and three-dimensional simulations. It is also demonstrated that the k-w model provides better turbulence predictions than the unmodified k-ε model, for the cases considered, and that a similar modification of the k-w model, to account for rapid compression/expansion, might improve its predictions even further.     

Further study on large deflection of circular sandwich plates

In this paper, a nonlinear solution is first presented for a circular sandwich plate with the flexure rigidity of the face layers taken into account. In solving the nonlinear bending equations, a modified power series method is proposed. The uniformly distributed loading and the clamped but sliding boundary condition are also assumed. Then our results are compared with those from Liu Ren-huai and Shi-Yun-fang ^[15]. The present solution can be used as a more accurate basis in engineering applications.Project Supported by the Science Fund of the Chinese Academy of Sciences.     

Finite element computation of a turbulent flow over a two-dimensional backward-facing step

The present paper is devoted to the computation of turbulent flows by a Galerkin finite element method. Effects of turbulence on the mean field are taken into account by means of a (k-ε) turbulence model. The wall region is treated through wall laws and, more specifically, Reichardt's law. An inlet profile for ε is proposed as a numerical treatment for physically meaningless values of k and ε. Results obtained for a recirculating flow in a two-dimensional channel with a sudden expansion in width are presented and compared with experimental values.     

Finite element computation of a turbulent flow over a two-dimensional backward-facing step

This paper is devoted to the computation of turbulent flows by a Galerkin finite element method. Effects of turbulence on the mean field are taken into account by means of a k-? turbulence model. The wall region is treated through wall laws and, more specifically, Reichardt's law. An inlet profile for ? is proposed as a numerical treatment for physically meaningless values of k and ?. Results obtained for a recirculating flow in a two-dimensional channel with a sudden expansion in width are presented and compared with experimental values.     

The Post-Hopf-Bifurcation Response of a Loosely Supported Cylinder in an Array Subjected to Cross-Flow. Part II: Theoretical Model and Comparison with Experiments

A nonlinear quasi-steady model for the analysis of the dynamics of a loosely supported cylinder, which takes into account position-dependent nonlinear fluid forces as well as nonuniform flow, is formulated. The model includes an approximation for the equivalent viscous damping associated with energy dissipation on impact at the support. The nonlinear model shows reasonably good agreement with experiments, in predicting the observed bifurcations in the cylinder response. Comparison criteria include the standard orbital plots, time traces and response spectra. A borderline chaotic response is found to be predominant over the test velocity range. In this chaotic regime, the theoretical results were verified via attractor fractal-dimension calculations and saddle orbit distributions; theoretical values of these invariant measures compare reasonably well with their experimental counterparts. Two mechanisms leading to chaos have been identified for this system. The first is a switching mechanism , at the onset of impacting. The second, and more prevalent, is the type I intermittency route to chaos.     

Aspect-ratio dependence of transient Taylor vortices close to threshold

We perform a detailed numerical study of transient Taylor vortices arising from the instability of cylindrical Couette flow with the exterior cylinder at rest for radius ratio η = 0.5 and variable aspect ratio Γ. The result of Abshagen et al. (J Fluid Mech 476:335–343, 2003) that onset transients apparently evolve on a much smaller time–scale than decay transients is recovered. It is shown to be an artefact of time scale estimations based on the Stuart–Landau amplitude equation which assumes frozen space dependence while full space–time dependence embedded in the Ginzburg–Landau formalism needs to be taken into account to understand transients already at moderate aspect ratio. Sub-critical pattern induction is shown to explain the apparently anomalous behaviour of the system at onset while decay follows the Stuart–Landau prediction more closely. The dependence of time scales on boundary effects is studied for a wide range of aspect ratios, including non-integer ones, showing general agreement with the Ginzburg–Landau picture able to account for solutions modulated by Ekman pumping at the disks bounding the cylinders.      

Strain energy density prediction of crack initiation and direction in cracked T-beams and pipes

In this paper, the S-theory is applied to determine crack initiation and direction for cracked T-beams and circumferentially cracked pipes. It makes use of a parameter called strain energy density factor, S, which is a function of the stress intensity factors. The strain energy density theory provides a more general treatment of fracture mechanics problems by virtue of its ability in describing the multiscale feature of material damage and in dealing with mixed mode crack propagation problem. A simple method for obtaining approximate stress intensity factors is also applied. It takes into account the elastic crack tip stress singularity while using the elementary beam theory. Some basic loading conditions in beams and pipes are studied.     

The Modeling of Variable Density Turbulent Flows. A review of first-order closure schemes

The paper is intended as a review of where eddy-viscosity turbulence models have reached in accounting for the several distinct effects of variable density and high-Mach number behaviour on the development of turbulent shear flows. Three generations of ``compressible models' are depicted: a first one based on variable density and compressible adjustments of ``incompressible' schemes, a second one in which it is presumed that explicit dilatational terms can account for variable density and compressibility effects, and a third one, where such effects are taken implicitly in association with structural changes of the turbulent field. The latter are hardly tractable when using first-order closure schemes, but more reliable than the former in accounting for most observed variable density and compressibility effects in turbulent shear flows. This revised version was published online in August 2006 with corrections to the Cover Date.     

A New Low-Reynolds-Number One-Equation Model of Turbulence

In this study, we propose a new Low-Reynolds-Number (LRN)one-equation model, which is derived from an LRN two-equation(k-ε) model. The derivation of the transport equation, in principle, is based on the assumption that the turbulent structure parameter remains constant. However, the relation for the turbulent structure parameter a ₁(=|− |/k) is modified to account for near-wall turbulence. As a result, the present one-equation model contains a term which takes the near-wall limiting behavior explicitly into account. Thus, the present model provides the correct wall-limiting behavior of turbulence in the vicinity of the wall and can be applied to the analysis of heat transfer. The validity of the present model is tested in channel flows, boundary layer flows with and without pressure gradient, plane wall jet, and flow with separation and reattachment. The calculated results showed good agreement with the direct numerical simulation (DNS) and experimental data. This revised version was published online in July 2006 with corrections to the Cover Date.     

Impact of a bead on a rotating wall

We study experimentally the impact of a plastic bead on a rotating wall made of steel (velocity Ω; radial position x₀). The results show that the restitution coefficient is directly function of the impact velocity x₀Ω and is invariant by changing frame reference. The influence of the height of release of the particle on its angular velocity after impact is also studied. We observe an increase of the angular velocity with height followed by a saturation. We propose an interpretation for this evolution considering that the particle may roll without sliding during all the impact. This physical feature is not always taken into account in existing models of impact between rigid bodies. To cite this article: F. Rioual et al., C. R. Mecanique 336 (2008).     

Well-posedness and Stability of a Multi-dimensional Tumor Growth Model

We study a moving boundary problem modeling the growth of in vitro tumors. This problem consists of two elliptic equations describing the distribution of the nutrient and the internal pressure, respectively, and a first-order partial differential equation describing the evolution of the moving boundary. An important feature is that the effect of surface tension on the moving boundary is taken into account. We show that this problem is locally well-posed for a large class of initial data by using analytic semi-group theory. We also prove that if the surface tension coefficient γ is larger than a threshold value γ _* then the unique flat equilibrium is asymptotically stable, whereas in the case γ  < γ _* this flat equilibrium is unstable.     

A Theoretical Model of Hysteresis and Dynamic Effects in the Capillary Relation for Two-phase Flow in Porous Media

It is well known that the relationship between capillary pressure and saturation, in two-phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit representation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that P ^c–S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist P ^c–S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in P ^c–S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.     

Transient pressure effects in the evolution equation for premixed flame fronts

A nonlinear evolution equation for a scalar field G(x, t) is derived, whose level surface G ₀=const. represents the interface of a thin premixed flame propagating in a flow field. The derivation is an extended version of an equation already proposed by Markstein [1]. It was reconsidered by Williams [2] as a basis for theoretical and numerical analysis and takes, in addition to flame curvature and flame stretch time variations of the bulk pressure, heat loss and nonconstant transport coefficients into account. The equation is an extension of earlier analyses where a flame evolution equation was derived for slightly wrinkled flames such that the front can be described by a single-valued function of a normal coordinate. That formulation excluded situations where the mean flame front has an arbitrary shape in space. Here the more general situation is analysed by using a two-length-scale asymptotic analysis. The leading-order solution of this analysis is equivalent to the equation originally derived by Markstein [1]. In addition to nonconstant properties and heat-loss effects, that had already been considered by Clavin and Nicoli [3], the influence of transient changes of the bulk pressure is analysed. All these effects are combined into a unified formulation which will serve as a basis for a new flamelet concept for premixed turbulent combustion.     

Toward accurate hybrid prediction techniques for cavity flow noise applications

A large variety of hybrid computational aeroacoustics (CAA) approaches exist differing from each other in the way the source region is modeled, in the way the equations are used to compute the propagation of acoustic waves in a non-quiescent medium, and in the way the coupling between source and acoustic propagation regions is made. This paper makes a comparison between some commonly used numerical methods for aeroacoustic applications. The aerodynamically generated tonal noise by a flow over a 2D rectangular cavity is investigated. Two different cavities are studied. In the first cavity (L/D=4, M=0.5), the sound field is dominated by the cavity wake mode and its higher harmonics, originating from a periodical vortex shedding at the cavity leading edge. In the second cavity (L/D=2, M=0.6), shear-layer modes, due to flow-acoustic interaction phenomena, generate the major components in the noise spectrum. Source domain modeling is carried out using a second-order finite-volume large eddy simulation. Propagation equations, taking into account convection and refraction effects, are solved using high-order finite-difference schemes for the linearized Euler equations and the acoustic perturbation equations. Both schemes are compared with each other for various coupling methods between source region and acoustic region. Conventional acoustic analogies and Kirchhoff methods are rewritten for the various propagation equations and used to obtain near-field acoustic results. The accuracy of the various coupling methods in identifying the noise-generating mechanisms is evaluated. In this way, this paper provides more insight into the practical use of various hybrid CAA techniques to predict the aerodynamically generated sound field by a flow over rectangular cavities. Copyright © 2009 John Wiley & Sons, Ltd.     

A new physics-based γ–kL transition model

A new physics-based γ–k_L transition model is proposed for the first time in this paper where γ is the intermittency and k_L is the laminar kinetic energy. Unlike the correlation-based γ–Re_θ model, the transport equations of the γ–k_L model are constructed based on basic physical mechanisms and their interactions. The relationship among γ, k_L and k enhances the coupling mechanism between transition and turbulence. The derivation of the γ-equation, following the definition of γ in terms of k_L and k, is presented here in detail. The shear-sheltering effect is also taken into account to damp or promote the influence of bypass transition mechanism. To account for the transitional effects on the mean flow, the γ–k_L model is readily coupled to the shear stress transport k–ω turbulence model via the production and destruction terms of the k-equation without any modification to the turbulence model. The ERCOFTAC test cases of T3AM, T3A and T3B are employed to validate this γ–k_L model. It is found that the γ–k_L model can predict the natural and bypass transitions better than the k_L model and as accurately as the γ–Re_θ model.     

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.     

On the thermodynamics of thermoelastic materials with additional scalar degrees of freedom

This work deals with the thermodynamic formulation of a model for a class of materials containing microstructure which evolves or changes relative to\/ the (global) bulk material. The approach taken here is based on a generalization of the total energy, total energy flux, and total energy supply to take into account the corresponding additional degrees of freedom involved. Restricting attention for simplicity to thermoelastic materials with scalar-valued such degrees of freedom, the thermodynamically-consistent forms of the remaining balance and corresponding constitutive relations for this material class are obtained in the context of the Müller-Liu entropy principle. In particular, the thermodynamically-consistent form of the evolution relation for the additional scalar-valued degrees of freedom obtained in this fashion contains in part\/ the well-known generalized Euler-Lagrange or Ginzburg-Landau relations established in many previous work, with the remaining terms accounting for effects associated with microinertia, or with non-equilibrium processes. Received September 14, 1998