The Bernoulli Integral for a Certain Class of Non‐Stationary Viscous Vortical Flows of Incompressible Fluid

It has been shown in our paper [1] that there is a wide class of 3D motions of incompressible viscous fluid which can be described by one scalar function dabbed the quasi‐potential. This class of fluid flows is characterized by three‐component velocity field having two‐component vorticity field; both these fields can depend of all three spatial variables and time, in general. Governing equations for the quasi‐potential have been derived and simple illustrative example of 3D flow has been presented. Here, we derive the Bernoulli integral for that class of flows and compare it against the known Bernoulli integrals for the potential flows or 2D stationary vortical flows of inviscid fluid. We show that the Bernoulli integral for this class of fluid motion possesses unusual features: it is valid for the vortical nonstationary motions of a viscous incompressible fluid. We present a new very nontrivial analytical example of 3D flow with two‐component vorticity which hardly can be obtained by any of known methods. In the last section, we suggest a generalization of the developed concept which allows one to describe a certain class of 3D flows with the 3D vorticity.     

Numerical Solution of Newtonian and Non-Newtonian Flows

This paper deals with the numerical solution of Newtonian and non-Newtonian flows. The flows are supposed to be laminar, viscous, incompressible and steady. The model used for non-Newtonian fluids is some variant of power-law. Governing equations in this model are incompressible Navier-Stokes equations. For numerical solution one could use artificial compressibility method with three stage Runge-Kutta finite volume method in cell centered formulation for discretization of space derivatives. Following cases of flows are solwed: flow through a bypass connected to main channel in 2D and 3D and non-Newtonian flow through branching channels in 2D. These results are presented for 2D and 3D case. This problem could have an application in the area of biomedicine. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heat Transfer in Rotor/Rotor and Rotor/Stator Cavity: Physics,Correlations and Numerical Modeling

The transitional and turbulent flow in the near wall sublayer is now mostly modeled based on the existing knowledge of simple 2D flows. To determine the effect of three dimensionality on the turbulent flow structures and turbulent heat transfer in the near wall areas the authors investigate numerically (SVV) turbulent flow in rotor/stator and rotor/rotor flows (with and without axial throughflow). These simple model flows contain most of the phenomena that are needed to understand more complex, 3D transitional and turbulent flows. Attention is focused on the turbulent characteristics which should have more universal character. To stabilize calculations for high Reynolds numbers (up to Re=800 000) the SVV operator is introduced into the Navier-Stokes and energy equations solver for cylindrical coordinate system without using complex numbers. Code optimization and parallelization have speeded up computations 20 times. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dense underflow into a lake or reservoir—Sub-critical flow.

Steady two-dimensional flow of a dense stream down a slight embankment into a lake or a reservoir is considered. The inflowing water is separated from the ambient lake water by a density interface. This work follows on from earlier work in which the flows down a steep incline with a relatively high flow rate were considered. Here, the flow is slow and the entry angle is small, resulting in waves on the interface. The fluid is assumed to be of finite depth and the incoming channel makes an angle α to the horizontal. Limiting flows are found when the fluid separates at a stagnation point or alternatively when the waves reach maximum steepness. The regions in parameter space where such solutions are obtained are delineated for different flow conditions.     

Adaptive control and synchronization of identical new chaotic flows with unknown parameters via single input

The single input linear feedback control for synchronizing two identical new 3D chaotic flows reported by Li et al. [X.F. Li, K.E. Chlouverakis, D.L. Xu, Nonlinear dynamics and circuit realization of a new chaotic flow: a variant of Lorenz, Chen and Lü, Nonlinear Analysis RWA 10 (4) (2009) 2357-2368] is proposed in this paper. Sufficient conditions of synchronization are obtained for both linear feedback and adaptive control approaches. The problem of adaptive synchronization between two nearly identical chaotic systems with unknown parameters is also studied. Based on the Lyapunov stability theory, two kinds of single input adaptive synchronization controllers are designed and the adaptive parameter update laws are developed.     

Almost periodically forced circle flows

We study general dynamical and topological behaviors of minimal sets in skew-product circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and unforced (i.e., a circle map) or continuous in time but periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the possible loss of mean motion property. On one hand, we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere non-locally connected. But on the other hand, we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of Denjoy theory. For instance, an almost periodically forced circle flow can be positively transitive and its minimal sets can be Li-Yorke chaotic and non-almost automorphic. As an application of our results, we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic, sl(2,R)-valued, continuous or discrete cocycle.Continuous almost periodically forced circle flows are among the simplest non-monotone, multi-frequency dynamical systems. They can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through Mather theory in the damping-free cases. They also naturally arise in 2D almost periodic Floquet theory as well as in climate models. Discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear Schrödinger equations with almost periodic potentials. They have been widely used as models for studying strange, non-chaotic attractors and intermittency phenomena during the transition from order to chaos. Hence the study of these flows is of fundamental importance to the understanding of multi-frequency-driven dynamical irregularities and complexities in non-monotone dynamical systems.     

Asymptotic and numerical solutions of three‐dimensional boundary‐layer flow past a moving wedge

We consider a laminar boundary‐layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3‐dimensional boundary‐layer equations that contains 2 physical parameters: pressure gradient (β) and shear‐to‐strain‐rate ratio parameter (α). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller‐box numerical method for the full nonlinear system. The results show that the flow field is divided into near‐field region (mainly dominated by viscous forces) and far‐field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3‐dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary‐layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing α when the wedge is moving in the x‐direction, while the case is different when it is moving in the y‐direction. Further, both analysis show that 3‐dimensional boundary‐layer solutions exist in the range −1<α<∞. These are some interesting results linked to an important class of boundary‐layer flows.     

Long Eddies in Sheared Flows

The characteristic feature of the wide variety of hydraulic shear flows analyzed in this study is that they all contain a critical level where some of the fluid is turned relative to the ambient flow. One example is the flow produced in a thin layer of fluid, contained between lateral boundaries, during the passage of a long eddy. The boundaries of the layer may be rigid, or flexible, or free; the fluid may be either compressible or incompressible. A further example is the flow produced when a shear layer separates from a rigid boundary producing a region of recirculating flow. The equations used in this study are those governing inviscid hydraulic shear flows. They are similar in form to the classical boundary layer equations with the viscous term omitted. The main result of the study is to show that when the hydraulic flow is steady and contained between lateral boundaries, the variation of vorticity ω(ψ) cannot be prescribed at any streamline which crosses the critical level. This variation is, in fact, determined by (1) the vorticity distribution at all streamlines which do not cross the critical level, by (2) the auxiliary conditions which must be satisfied at the boundaries of the fluid layer, and by (3) the dimensions of the region containing the turned flow. If at some instant the vorticity distribution is specified arbitrarily at all streamlines, generally the subsequent flow will be unsteady. In order to emphasize this point, a class of exact solutions describing unsteady hydraulic flows are derived. These are used to describe the flow produced by the passage of a long eddy which distorts as it is convected with the ambient flow. They are also used to describe the unsteady flow that is produced when a shear layer separates from a boundary. Examples are given both of flows in which the shear layer reattaches after separation and of flows in which the shear layer does not reattach. When the shear layer vorticity distribution has the form ωαyⁿ, where y is a distance measure across the layer, the steady flows are of Falkner-Skan type inside, and adjacent to, the separation region. The unsteady flows described in this paper are natural generalizations of these Falkner-Skan flows. One important result of the analysis is to show that if the unsteady flow inside the separation region is strongly sheared, then the boundary of the separation region moves upstream towards the point of separation, forming large transverse currents. Generally, the assumption of hydraulic flow becomes invalid in a finite time. On the other hand, if the flow inside the separation region is weakly sheared, this region is swept downstream and the flow becomes self-similar.     

Exponentially fitted methods applied to fourth-order boundary value problems

Fourth-order boundary value problems are solved by means of exponentially fitted methods of different orders. These methods, which depend on a parameter, can be constructed following a six-step flow chart of Ixaru and Vanden Berghe. Special attention is paid to the expression for the error term and to the choice of the parameter in order to make the error as small as possible. Some numerical examples are given to illustrate the practical implementation issues of these methods.     

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.     

On Cost Allocation in Hub-Like Networks

We consider telecommunication network design in which each pair of nodes can communicate via a direct link and the communication flow can be delivered through any path in the network. The cost of flow through each link is discounted if and only if the amount of flow exceeds a certain threshold. This exploitation of economies of scale encourages the concentration of flows and use of relatively small number of links. We will call such networks hub-like networks. The cost of services delivered through a hub-like network is distributed among its users who may be individuals or organizations with possibly conflicting interests. The cooperation of these users is essential for the exploitation of economies of scale. Consequently, there is a need to find a fair distribution of the cost of providing the service among users of such network. In order to describe this cost allocation problem we formulate the associated cooperative game, to be referred to as the hub-like game. Special attention is paid to users' contribution to economies of scale. We then demonstrate that certain cost allocation solutions (the core and the nucleolus of the hub-like game), which provide users with the incentive to cooperate, can be efficiently characterized.     

Three-dimensional numerical simulation of nonisotropic thermal density flow in a strongly curved open channel

The results from a 3D nonisotropic algebraic stress/flux turbulence model are presented to investigate the structure of thermal density flow and the temperature distribution in a strongly curved open channel (180° bend). The numerically simulated results show that (i) several secondary flows take place at the bend cross-section 90° of the curved open channel, the feature which is not found for the isothermal flows and thermal density flow in a straight channel, and (ii) the thermocline in a curved channel is thicker than that in a straight channel due to the secondary flows-induced strong mixing process taking place in the former. Such features may be ascribed to the complex interaction of the buoyant force, the centrifugal force and the Reynolds stresses taking place only in curved channels. The simulated results are in good agreement with available experimental data, which indicates that the developed model can be applied for predicting the motion of the nonisotropic thermal density flow in the curved open channel.     

Comparative study of 1D, 2D and 3D simplified gas kinetic schemes for simulation of inviscid compressible flows

With assumption that all the particles in the phase velocity space are concentrated on a circle and on a sphere, the circular function-based gas kinetic scheme and sphere function-based gas kinetic scheme have been developed by Shu and his coworkers [21], [22], [23]. These schemes are simpler than the Maxwellian function-based gas kinetic schemes. The simplicity is due to the fact that the integral domain of phase velocity of circular function and sphere function is a finite region while the integral domain of Maxwellian distribution function is infinite. In this work, the 1D delta function-based gas kinetic scheme is also developed to form a complete set of the simplified gas kinetic schemes. The 1D, 2D and 3D simplified gas kinetic schemes can be viewed as the truly 1D, 2D and 3D flux solvers since they are based on the multi-dimensional Boltzmann equation. On the other hand, to solve the 3D flow problem, the tangential velocities are needed to be approximated by some ways for the 1D and 2D simplified gas kinetic schemes, and to solve the 1D flow problem, the tangential velocities should be taken as zero for the 2D and 3D simplified gas kinetic schemes. The performances of these three schemes for simulation of inviscid compressible flows are investigated in this work by their application to solve the test problems from 1D to 3D cases. Numerical results showed that the efficiency of the delta function-based gas kinetic scheme is slightly superior to that of the circular function- and sphere function-based gas kinetic schemes, while its stability is inferior significantly to the latter. For simulation of the 3D hypersonic flows, the sphere function-based gas kinetic scheme could be the best choice.     

A note on the parametric maximum flow problem and some related reoptimization issues

In this paper, we will extend the results about the parametric maximum flow problem to networks in which the parametrization of the arc capacities can involve both the source and the sink, as in Gallo, Grigoriadis, and Tarjan (1989), and also an additional node. We will show that the minimum cuts of the investigated networks satisfy a relaxed form of the generalized nesting property (Arai, Ueno, and Kajitani, 1993). A consequence is that the corresponding parametric maximum flow value function has at most n −1 breakpoints. All the minimum cut capacities can therefore be computed by O(1) maximum flow computations. We will show then that, given O(n) increasing values of the parameter, it is possible to compute the corresponding maximum flows by O(1) maximum flow computations, by suitably extending Goldberg and Tarjan’s maximum flow algorithm.     

The optimal exponentially-fitted Numerov method for solving two-point boundary value problems

Second order boundary value problems are solved by means of exponentially-fitted Numerov methods. These methods, which depend on a parameter, can be constructed following a six-step flow chart of Ixaru and Vanden Berghe [L.Gr. Ixaru, G. Vanden Berghe, Exponential Fitting, Kluwer Academic Publishers, Dordrecht, 2004]. Special attention is paid to the expression of the error term of such methods. An algorithm concerning the choice of the best suited method and its parameter is discussed. Several numerical examples are given to sustain the theory.     

On the stability of the upper airways system

Simple models for upper pharyngeal obstruction, describing the sleep apnea syndrome are proposed. Stability is discussed, of two and three individualized elements, with and without elastic connections, interacting with the steady flow. Considering the flow as the controlling parameter, critical steady state flows are located and their post-critical behavior is discussed for various models. It is pointed out that non-linear constitutive elastic laws are necessary contrary to the linear ones introduced by Fodil (1998) [15]. Finally the three element model will be presented and studied with non-linear constitutive relations and side connections. Applications of the theory will be performed and discussion for the three models will be presented. It is pointed out that the sleep apnea syndrome is due to the instability of the upper pharyngeal region.     

A one- and two-dimensional generalized Pareto model for a river flow

The main objective of this paper is to find a two-dimensional model for the flow of the Romaine River in Québec, Canada, which could be used to forecast the flow one day after the currently observed flow. The 2D density function proposed must be such that the correlation coefficient between the two variables can be chosen close to 1, since the river flows on two consecutive days are very highly correlated. We find that a generalized Pareto distribution provides a good fit to the data. We then propose 2D versions of this distribution. Finally, a linear combination of two such 2D distributions is used to obtain the required model. In the case of the Romaine River, the model considered works very well. It could be used with or modified for other rivers.     

A two scale cellular automaton for flow dynamics modeling (2CAFDYM)

In this paper we propose a Two scale Cellular Automaton for Flow DYnamics Modeling (2CAFDYM) in a lowland region. Cells are terrain meshes with a predefined size, arranged in a bi-dimensional hexagonal lattice. The state of the cell consists of two scales: groundwater and surface water, in order to combine flows over saturated soil (Dunne flow) and flows exceeding the infiltration capacity (Hortonian flow). This allows for survey flood events and water resources. Each cell has intrinsic terrain attributes: altitude, soil type and land use. The obtained slopes are considered towards all the neighboring cells such that water flows simultaneously in multiple directions during the same time step. This helps us characterize laminar and turbulent flows. The model is subjected to climatic constraints: rainfall and temperature. The flow dynamics are regulated by mass conservation laws on hydraulic balance sheets (received, evaporated, infiltrated and drained water). Using Java Object Oriented Programming we have designed decision-aided software for the real-time monitoring of flow processes in 2D or 3D scenes through 2CAFDYM. We give some simulations for a basin in northern Morocco covering 34.3 km², including some areas that are potentially vulnerable to flooding. Digital terrain models, geological maps and satellite images are used to extract input data.     

Numerical Investigations of Relaxation Phenomena in Cavitating Nozzle Flows

Numerical simulations of cavitating flows are frequently performed by applying simple law of state‐models. In this study an advanced law of state‐model on the basis of a Landau‐type approach is used that focusses on the physical treatment of relaxation phenomena. Relaxation phenomena or phase non‐equilibrium effects occur within the scope of two‐phase fluid dynamics if the time scale of the flow problem is small. This appears e.g. in the case of cavitating flow in injector nozzles of diesel engines. The aim of this study is the determination of the relaxation parameter of the advanced law of state‐model. For this reason a theoretical approach is presented as well as simulations of unsteady cavitating nozzle flows that are compared with experimental data. Concerning the calculation of 2‐D unsteady cavitating flow the evolution equation for the vapor fraction is solved by a modified Volume‐of‐Fluid algorithm.     

Continuous approximation for a 1D analog of the Gol’dshtik model for separated flows of an incompressible fluid

A modification of a 1D analog of the Gol’dshtik mathematical model for separated flows of an incompressible fluid is considered. The model is a nonlinear differential equation with a boundary condition. Nonlinearity in the equation is continuous and depends on a small parameter. When this parameter tends to zero, we have a discontinuous nonlinearity. The results of the solutions are in agreement with the results obtained for the 1D analog of the Gol’dshtik model for separated flows of an incompressible fluid.