Prediction of Flow and Heat Transfer through Stationary and Rotating Ribbed Ducts Using a Non-linear <Emphasis Type="Italic">k</Emphasis>−<Emphasis Type="Italic">ε</Emphasis> Model

The present paper deals with the prediction of three-dimensional fluid flow and heat transfer in rib-roughened ducts of square cross-section, which are either stationary, or rotate in orthogonal mode. The main objective is to assess how a recently developed variant of a cubic non-linear k−ε model (proposed by Craft et al. Flow Turbul Combust 63:59–80, 1999) can predict three-dimensional flow and heat transfer characteristics through stationary and rotating ribbed ducts. The present paper discusses turbulent air flow and heat transfer through two different configurations, namely: (I) a stationary square duct with “in-line” normal and (II) a square duct with normal ribs in a “staggered” arrangement under stationary and rotating conditions, with the axis of rotation normal to the flow direction and parallel to the ribs. In this paper the flow and thermal predictions of the linear k−ε model (EVM) are also included, as a set of baseline predictions. The mean flow predictions show that both linear and non-linear k−ε models can successfully reproduce most of the measured data for stream-wise and cross-stream velocity components. Moreover, the non-linear model is able to produce better results for the turbulent stresses. The heat transfer predictions show that both EVM and NLEVM2, the more recent variant of the non-linear k−ε, with the algebraic length-scale correction term, overestimate the measured Nusselt numbers for both geometries examined. While the EVM with the differential length-scale correction term underestimates heat transfer levels, the Nusselt number predictions with the NLEVM2 and the ‘NYP’ term are in close agreements with the measured data. Comparisons with our earlier work, Iacovides and Raisee (Int J Heat Fluid Flow, 20:320–328, 1999), show that the NLEVM2 thermal predictions are of similar quality to those of a second-moment closure.     

Entropy Solutions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript> for the Euler Equations in Nonlinear Elastodynamics and Related Equations

A compactness framework is established for approximate solutions to the Euler equations in one-dimensional nonlinear elastodynamics by identifying new properties of the Lax entropies, especially the higher order terms in the Lax entropy expansions, and by developing ways to employ these new properties in the method of compensated compactness. Then this framework is applied to establish the existence, compactness, and decay of entropy solutions in L for the Euler equations in nonlinear elastodynamics with a more general stress-strain relation than those for the previous existence results. This compactness framework is further applied to solving the Euler equations of conservation laws of mass, momentum, and energy for a class of thermoelastic media, and the equations of motion of viscoelastic media with memory.     

Adaptive <Emphasis Type="Italic">Q</Emphasis>–<Emphasis Type="Italic">S</Emphasis> synchronization of non-identical chaotic systems with unknown parameters

This work investigates the adaptive Q–S synchronization of non-identical chaotic systems with unknown parameters. The sufficient conditions for achieving Q–S synchronization of two different chaotic systems (including different dimensional systems) are derived, based on Lyapunov stability theory. By the adaptive control technique, the control laws and the corresponding parameter update laws are proposed such that the non-identical chaotic systems are to have Q–S synchronization. Finally, four illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.     

Convergence of Radially Symmetric Solutions for (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian Elliptic Equations with a Damping Term

This study considers the quasilinear elliptic equation with a damping term,

$$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$

where \({\mathbf {x}}\) is an N-dimensional vector in \(\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}\) for some \(\alpha > 0\) and \(N \in {\mathbb {N}}\setminus \{1\}\); \(D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}\) with \(1 < q \le p\); k is a nonnegative and locally integrable function on \([\alpha ,\infty )\); and \(\omega \) is a positive constant. A necessary and sufficient condition is given for all radially symmetric solutions to converge to zero as \(|{\mathbf {x}}|\rightarrow \infty \). Our necessary and sufficient condition is expressed by an improper integral related to the damping coefficient k. The case that k is a power function is explained in detail.

     

Analytical and numerical solutions of the problem of gravity-induced mixing of a light layer using the <Emphasis Type="Italic">k</Emphasis>–<Emphasis Type="Italic">ε</Emphasis> turbulence model

An analytical solution for the self-similar stage in the problem of gravity-induced turbulent mixing in a light (heavy) layer is obtained on the basis of the k–ε model equations. The solution obtained is compared with the results of a numerical investigation of the problem using both three-dimensional direct numerical simulation and the k–ε model. The calculations were performed using the two- and three-dimensional versions of the EGAK method. The results of all the calculations and the available experimental data are in reasonable agreement.     

DNS Scrutiny of the <Emphasis Type="Italic">ζ</Emphasis>-<Emphasis Type="Italic">f</Emphasis> Elliptic-Relaxation Eddy-Viscosity Model in Channel Flows with a Moving Wall

In this paper we present a Direct Numerical Simulations (DNS) of channel flow with stationary and moving walls. Three cases, Poiseuille-type with U_W/U_b = 0.75, intermediate-type with U_W/U_b = 1.215, and Couette-type with U_W/U_b = 1.5 (U_W and U_b are the wall and the bulk velocity), were compared with the pure Poiseuille U_W/U_b = 0, at a bulk Reynolds number equal to 4,800 corresponding to Re _\uptau = 288_{\uptau} =288. The DNS results were used to scrutinize the capabilities of ζ-f eddy viscosity model (based on the elliptic relaxation concept) in reproducing the near-wall turbulence in non conventional flows where the shear stress structures are strongly different with respect to the cases used for models calibration. The ζ-f model (also in its basic formulation) demonstrated to have good prospects to reproduce the main phenomenology of such class of flows due to its built-in capabilities to account separately for the different (and opposite) near wall effects on turbulence: the damping due to viscosity and pressure reflection. The results of the computations demonstrated that standard ζ-f model can reasonably reproduce the phenomenology of these flows in terms of velocity and turbulent kinetic energy profiles and budgets.     

Almost Periodicity in an Ecological Model with <Emphasis Type="Italic">M</Emphasis>-Predators and <Emphasis Type="Italic">N</Emphasis>-Preys by “Pure-Delay Typ” System

By using comparison theorem and constructing suitable Lyapunov functional, we study the following periodic Lotka–Volterra model with M-predators and N-preys by pure-delay type

Roughness effects in low-<Emphasis Type="Italic"> Re</Emphasis><Subscript><Emphasis Type="Italic"> θ</Emphasis></Subscript> open-channel turbulent boundary layers

This paper reports velocity measurements obtained on a smooth and two geometrically different types of rough surfaces in an open channel. The measurements were obtained using a laser-Doppler anemometer. The recent boundary layer theory proposed by George and Castillo (1997) and conventional scaling laws are used to analyze the data. The present flow shows a strong structural similarity to a canonical turbulent boundary layer in the inner layer. The results demonstrate that surface roughness increases the wake parameter. Surface roughness also enhances the levels of turbulence intensities, Reynolds shear stress and triple correlations over most of the boundary layer, but decreases the stress anisotropy.     

Two-Phase Problems for Linear Elliptic Operators with Variable Coefficients: Lipschitz Free Boundaries are <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">1,γ</Emphasis></Superscript>

We prove C ^1, regularity of Lipschitz free boundaries of two-phase problems for linear elliptic operators with Hölder continuous coefficients.     

The Navier-Stokes Equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with Linearly Growing Initial Data

Consider the equations of Navier-Stokes on ⁿ with initial data U₀ of the form U₀(x)=u₀(x)–Mx, where M is an n×n matrix with constant real entries and u₀ L^p(ⁿ). It is shown that under these assumptions the equations of Navier-Stokes admit a unique local solution in L^p(ⁿ). Moreover, if ||e^tM||1 for all t0, then this mild solution is even analytic in x. This is surprising since the underlying semigroup of Ornstein-Uhlenbeck type is not analytic, in contrast to the Stokes semigroup.Acknowledgement It is our pleasure to thank G. METAFUNE, E. PRIOLA and A. RHANDI for fruitful discussions on the Ornstein-Uhlenbeck     

The Singular Set of <Emphasis Type="Italic">ω</Emphasis>-minima

We consider ω-minima of convex variational integrals in the vectorial case n,N≥2, and we provide estimates for the Hausdorff dimension of their singular sets.     

Development of an Analytical <Emphasis Type="Italic">β</Emphasis>-Function PDF Integration Algorithm for Simulation of Non-premixed Turbulent Combustion

Among many presumed-shape pdf approaches for modeling non-premixed turbulent combustion, the presumed β-function pdf is widely used in the literature. However, numerical integration of the β-function pdf may encounter singularity difficulties at mixture fraction values of Z = 0 or 1. To date, this issue has been addressed by few publications. The present study proposes the Piecewise Integration Method (PIM), an efficient, robust and accurate algorithm to overcome these numerical difficulties with the added benefit of improving computational efficiency. Comparison of this method to the existing numerical integration methods shows that the PIM exhibits better accuracy and greatly increases computational efficiency. The PIM treatment of the β-function pdf integration is first applied to the Burke–Schumann solution in conjunction with the k − ε turbulence model to simulate a CH₄/H₂ bluff-body turbulent flame. The proposed new method is then applied to the same flow using a more complex combustion model, the laminar flamelet model. Numerical predictions obtained by using the proposed β-function pdf integration method are compared to experimental values of the velocity field, temperature and species mass fractions to illustrate the efficiency and accuracy of the present method.     

Synchronization of unified chaotic system by sliding mode/mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

This paper deals with the synchronization of uncertain unified chaotic system in the presence of two kinds of disturbances, white noise and bounded power signal. A sliding mode controller (SMC) is established to guarantee the sliding motion. Moreover, a proportional-integral (PI) switching surface is used to determine the performance of the system in the sliding motion. Also, by using a mixed H ₂/H _∞ approach, the effect of external disturbances on the sliding motion is reduced. The necessary parameters of constructing controller and switching surface are found via semidefinite programming (SDP) which can be solved effectively by a standard software. Finally, a numerical simulation is presented to show the effectiveness of the proposed method.     

A Global Foliation of Einstein–Euler Spacetimes with Gowdy-Symmetry on <Emphasis Type="Italic">T</Emphasis><Superscript>3</Superscript>

We investigate the initial value problem for the Einstein–Euler equations of general relativity under the assumption of Gowdy symmetry on T ³, and we construct matter spacetimes with low regularity. These spacetimes admit both impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularities propagating at light speed) and shock waves in the fluid (that is, discontinuities propagating at about the sound speed). Given an initial data set, we establish the existence of a future development, and we provide a global foliation in terms of a globally and geometrically defined time-function, closely related to the area of the orbits of the symmetry group. The main difficulty lies in the low regularity assumed on the initial data set which requires a distributional formulation of the Einstein–Euler equations.     

Multivalued penalty method for evolution variational inequalities with <Emphasis Type="Italic">λ</Emphasis><Subscript>0</Subscript>-pseudomonotone multivalued maps

We consider evolution variational inequalities with λ ₀-pseudomonotone maps. The main properties of these maps are investigated. By using the finite-difference method, we prove the property of strong solvability for the class of evolution variational inequalities with λ ₀-pseudomonotone maps. Using the penalty method for multivalued maps, we show the existence of weak solutions of evolution variational inequalities on closed convex sets. The class of multivalued penalty operators is constructed. We also consider a model example to illustrate this theory.     

<Emphasis Type="Italic">p</Emphasis>th moment exponential synchronization for stochastic delayed Cohen–Grossberg neural networks with Markovian switching

This paper is a contribution to the analysis of the pth moment exponential synchronization problem for a class of stochastic delayed Cohen–Grossberg neural networks with Markovian switching. The jumping parameters are determined by a continuous-time, discrete-state Markov chain, and the delays are time-varying delays.     

Explicit equations for transient heat conduction in finite solids for <Emphasis Type="Italic">Bi</Emphasis> > 2

A set of correlations is developed for transient heat conduction in finite solids (plates, cylinders and spheres) exposed to high Biot number convection boundary conditions. For Bi ≥ 3, the error of the approximate solutions is well below 1% of the initial temperature difference driving the transient.     

Linear <Emphasis Type="Italic">Γ</Emphasis>-limits of multiwell energies in nonlinear elasticity theory

We derive linearized theories from nonlinear elasticity theory for multiwell energies. Under natural assumptions on the nonlinear stored energy densities, the properly rescaled nonlinear energy functionals are shown to Γ-converge to the relaxation of a corresponding linearized model. Minimizing sequences of problems with displacement boundary conditions and body forces are investigated and found to correspond to minimizing sequences of the linearized problems. As applications of our results we discuss the validity and failure of a formula that is widely used to model multiwell energies in the regime of linear elasticity. Applying our convergence results to the special case of single well densities, we also obtain a new strong convergence result for the sequence of minimizers of the nonlinear problem.      

Neglected transport equations: extended Rankine–Hugoniot conditions and <Emphasis Type="Italic">J</Emphasis> -integrals for fracture

Transport equations in integral form are well established for analysis in continuum fluid dynamics but less so for solid mechanics. Four classical continuum mechanics transport equations exist, which describe the transport of mass, momentum, energy and entropy and thus describe the behaviour of density, velocity, temperature and disorder, respectively. However, one transport equation absent from the list is particularly pertinent to solid mechanics and that is a transport equation for movement, from which displacement is described. This paper introduces the fifth transport equation along with a transport equation for mechanical energy and explores some of the corollaries resulting from the existence of these equations. The general applicability of transport equations to discontinuous physics is discussed with particular focus on fracture mechanics. It is well established that bulk properties can be determined from transport equations by application of a control volume methodology. A control volume can be selected to be moving, stationary, mass tracking, part of, or enclosing the whole system domain. The flexibility of transport equations arises from their ability to tolerate discontinuities. It is insightful thus to explore the benefits derived from the displacement and mechanical energy transport equations, which are shown to be beneficial for capturing the physics of fracture arising from a displacement discontinuity. Extended forms of the Rankine–Hugoniot conditions for fracture are established along with extended forms of J -integrals.     

Superconvergence analysis of bi-<Emphasis Type="Italic">k</Emphasis>-degree rectangular elements for two-dimensional time-dependent Schrödinger equation

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schrödinger equation with the finite element method. The error estimate and superconvergence property with order O(h^k+1) in the H¹ norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(h^k+1 + τ²) in the H¹ norm can be obtained in the Crank-Nicolson fully discrete scheme.