Simulating two-dimensional turbulent flow by using the κ–ϵ model and the vorticity–streamfunction formulation

A numerical procedure to solve turbulent flow which makes use of the κ–? model has been developed. The method is based on a control volume finite element method and an unstructured triangular domain discretization. The velocity-pressure coupling is addressed via the vorticity-streamfunction and special attention is given to the boundary conditions for the vorticity. Wall effects are taken into account via wail functions or a low-Reynolds-number model. The latter was found to perform better in recirculation regions. Source terms of the κ and ε transport equations have been linearized in a particular way to avoid non-realistic solutions. The vorticity and streamfunction discretized equations are solved in a coupled way to produce a faster and more stable computational procedure. Comparison between the numerical predictions and experimental data shows that the physics of the flow is correctly simulated.     

A method for deriving hydrodynamic equations for complicated systems

A general algorithm is proposed for constructing a uniformly valid asymptotic solution of the kinetic equations under conditions when the number of slowly varying macroscopic variables is greater than the number of integral invariants of the collision operator. The case of a chemically reacting gas mixture is considered, and a method for constructing the asymptotic solution for this case is described. The hydrodynamic equations for reacting and relaxing gas mixtures are described in general form and it is noted that consistent allowance for the disequilibrium of the reaction and relaxation processes leads to the appearance in the hydrodynamic equations of a number of additional terms, which describe the dependence of the rates of these processes on the spatial derivatives of the hydrodynamic variables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 96–105, May–June, 1981.     

A Field Monte Carlo Formulation for Calculating the Probability Density Function of a Single Scalar in a Turbulent Flow

The probability density function (PDF) formulation of one scalar field undergoing diffusion, turbulent convection and chemical reaction is restated in terms of stochastic fields. These fields are smooth in space as they have a length scale similar to that of the PDF. Their evolution is described by a set of stochastic partial differential equations, which are solved using a finite volume scheme with a stochastic source term. The application of this methodology to a particular flow is shown first for a linear source term, with exact analytical solution for the mean and standard deviation, and then for a nonlinear reaction.     

Uniqueness of the Bistable Traveling Wave for Mutualist Species

For a system of reaction–diffusion equations that models the interaction of n mutualist species, the existence of the bistable traveling wave solution has been proved where the nonlinear reaction terms possess a certain type of monotonicity. However the problem of whether there can be two distinct traveling waves remains open. In this paper we use a homotopy approach incorporated with the Liapunov–Schmidt method to show that the bistable traveling wave solution is unique. Our method developed in this paper can also be applied to study the existence and uniqueness of traveling wave solutions for some competition models.     

On the numerical solution of the turbulence energy equations for wave and tidal flows

This paper deals with the numerical solution, using finite difference methods, of the hydrodynamic and turbulence energy equations which describe wind wave and tidally induced flow. Calculations are performed using staggered and non-staggered finite difference grids in the vertical, with various time discretizations of the production and dissipation terms in the turbulence energy equations. It is shown that the time discretization of these terms can significantly influence the stability of the solution. The effect of time filtering on the numerical stability of the solution is also considered. The form of the mixing length is shown to significantly influence the bed stress in wind wave problems. A no-slip condition is applied at the sea bed, and the associated high-shear bottom boundary layer is resolved by transforming the equations onto a logarithmic or log-linear co-ordinate system before applying the finite difference scheme. A computationally economic method is developed which remains stable even when a very fine vertical grid (over 200 points) is used with a time step of up to 30 min.     

Fundamental solutions of the streamfunction–vorticity formulation of the Navier–Stokes equations

A new boundary element procedure is developed for the solution of the streamfunction–vorticity formulation of the Navier–Stokes equations in two dimensions. The differential equations are stated in their transient version and then discretized via finite differences with respect to time. In this discretization, the non-linear inertial terms are evaluated in a previous time step, thus making the scheme explicit with respect to them. In the resulting discretized equations, fundamental solutions that take into account the coupling between the equations are developed by treating the non-linear terms as in homogeneities. The resulting boundary integral equations are solved by the regular boundary element method, in which the singular points are placed outside the solution domain.     

Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes

A 2D, depth-integrated, free surface flow solver for the shallow water equations is developed and tested. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov-type second-order upwind finite volume formulation, whereby the inviscid fluxes of the system of equations are obtained using Roe's flux function. The eigensystem of the 2D shallow water equations is derived and is used for the construction of Roe's matrix on an unstructured mesh. The viscous terms of the shallow water equations are computed using a finite volume formulation which is second-order-accurate. Verification of the solution technique for the inviscid form of the governing equations as well as for the full system of equations is carried out by comparing the model output with documented published results and very good agreement is obtained. A numerical experiment is also conducted in order to evaluate the performance of the solution technique as applied to linear convection problems. The presented results show that the solution technique is robust. © 1997 John Wiley & Sons, Ltd.     

Dynamics of a rigid cylinder near a plane boundary in the radiation field of an acoustic wave

An approach is described for investigation of the interaction between a rigid body and a viscous fluid boundary under acoustic wave propagation. The influence of the liquid on the rigid body is determined as a mean force, which is a constant in the time component of the hydrodynamic force. This enables the use of a previously developed technique for calculation of pressure in a compressible viscous liquid. The technique takes into account the second-order terms with respect to the wave field parameters and is based on investigation of a system of initially nonlinear hydromechanics equations that can be simplified with respect to the wave motion parameters of the liquid. It has proven possible to retain the second-order terms for determination of stresses in the liquid without having to solve the system of nonlinear equations. The stresses can be expressed in terms of parameters found in the solution of the linearized equations of the compressible viscous liquid. In this way, the solution of linearized equations is expressed in terms of a scalar and vector potentials. The problem statement is derived for a rigid cylinder located near a rigid flat wall under the effects of a wave propagating perpendicular to the wall. The solution for this particular example is obtained.     

Reaction surface (flame-sheet) model in the theory of combustion of unmixed fuels. Molecular-kinetic solution in the reaction region

In the first part [1] of this investigation, a study was made of the main characteristic features of the flow resulting from the mixture of two flows with components that enter into an irreversible chemical reaction. The main errors in the solution of the corresponding gas-dynamic problem resulting from the use of various assumptions were also analyzed. It was found that the influence of the usually ignored Burnett and super-Burnett terms in the transport properties (i.e., in the stress tensor and in the heat flux and diffusion vectors) can be much stronger in this problem than in ordinary mixing problems. In the present, second part, a molecular-kinetic study is made of the flow in the thin reaction region. For the finding of the external solution of the problem in the mixing layer or when the classical flame-sheet model is used, this region is treated as a surface in the flow field on which all the chemical transformations of the components of the gas mixture take place. Gas-dynamic equations are obtained that describe the main and some of the subsequent approximations in the reaction region; these equations confirm the arguments put forward in the first part concerning the part played by the non-Navier-Stokes terms in the transport properties.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 77–90, March–April, 1979.     

Solving the three-dimensional equations of the linear theory of elasticity

The three-dimensional Lamé equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically     

Treatment of non-linearities in the numerical solution of the incompressible Navier–Stokes equations

The solution of the full non-linear set of discrete fluid flow equations is usually obtained by solving a sequence of linear equations. The type of linearization used can significantly affect the rate of convergence of the sequence to the final solution. The first objective of the present study was to determine the extent to which a full Newton–Raphson linearization of all non-linear terms enhances convergence relative to that obtained using the ‘standard’ incompressible flow linearization. A direct solution procedure was employed in this evaluation. It was found that the full linearization enhances convergence, especially when grid curvature effects are important. The direct solution of the linear set is uneconomical. The second objective of the paper was to show how the equations can be effectively solved by an iterative scheme, based on a coupled-equation line solver, which implicitly retains all the inter-equation couplings. This solution method was found to be competitive with the highly refined segregated solution methods that represent the current state-of-the-art.     

Laminar boundary layer between two planes perpendicular to each other

In this paper, we obtain a third-order approximate solution for the laminar boundary layer between two planes perpendicular to each other.In boundary layer equations, the viscous and the inertial terms have the same quantity step. In this paper, at first, supposing that the inertial terms are bigger than the viscous terms, we solve the boundary layer equations, and then we suppose that the viscous terms are bigger than the inertial terms. At last, we take the mean value as the valid solution of the boundary layer equations.The first- and the second-order approximate solutions obtained in this paper coincide with the results in ref. [1], while the third-order solution obtained in this paper is better than that in ref. [1].     

Thermoplastic strain of circular sandwich plates on an elastic base

We consider thermomechanical bending of an elastoplastic circular (solid or annular) light-filler sandwich plate resting on an elastic base. The hypotheses of broken normal are used to describe the kinematics of the plate stack nonsymmetric along the thickness. The base reaction is described by the Winkler model. We obtain the system of equilibrium equations and its exact solution in terms of displacements. We also present numerical results for a sandwich annular metal-polymer plate.     

Integration of the Equilibrium Equations for Inhomogeneous, Transversely Isotropic Plates

A system of differential equilibrium equations for inhomogeneous transversely isotropic plates is derived based on the Fourier series in terms of Legendre polynomials. It is assumed that Poisson's ratios are constant and the elastic moduli are linear functions of the transverse coordinate. A method of finding the general solution to the system of equations derived is set forth     

Averaging method for the solution of non-linear differential equations with periodic non-harmonic solutions

While Krylov and Bogolyubov used harmonic functions in their averaging method for the approximate solution of weakly non-linear differential equations with oscillatory solution, we apply a similar averaging technique using Jacobi elliptic functions. These functions are also periodic and are exact solutions of strongly non-linear differential equations. The method is used to solve non-linear differential equations with linear and non-linear small dissipative terms and/or with time dependent parameters. It is also shown that quite general dissipative terms can be transformed into time-dependent parameters. As a special example, the Langevin (collisional) equation of motion of electrons in a neutralizing ion background under the influence of a time and space-dependent electric field is presented. The method may also be used for non-linear control theory, dynamic and parametric stabilization of non-linear oscillations in plasma physics, etc.     

Efficient algorithm for modeling transport in porous media with mass exchange between mobile fluid and reactive stationary media

We compare two approaches to numerically solve the mathematical model of reactive mass transport in porous media with exchange between the mobile fluid and the stationary medium. The first approach, named the “monolithic algorithm,” is the approach in which a standard finite-difference discretization of the governing transport equations yields a single system of equations to be solved at each time step. The second approach, named the “system-splitting algorithm,” is here applied for the first time to the problem of transport with mass exchange. The system-splitting algorithm (SSA) solves two separate systems of equations at each time step: one for transport in the mobile fluid, and one for uptake and reaction in the stationary medium. The two systems are coupled by a boundary condition at the mobile– immobile interface, and are solved iteratively. Because the SSA involves the solution of two smaller systems compared to that of the monolithic algorithm, the computation time may be greatly reduced if the iterative method converges rapidly. Thus, the main objective of this paper is to determine the conditions under which the SSA is superior to the monolithic algorithm (MA) in terms of computation time. We found that the SSA is superior under all the conditions that we tested, typically requiring only 0.3–50% of the computation time required by the MA. The two methods are indistinguishable in terms of accuracy. Further advantages to the SSA are that it employs a modular code that can easily be modified to accommodate different mathematical representations of the physical phenomena (e.g., different models for reaction kinetics within the stationary medium), and that each module of the code can employ a different numerical algorithm to optimize the solution.     

The decay of weak,homogeneous turbulence in the presence of a vertical body force and concentration gradient with a first-order reaction

The effects of buoyancy, produced by a uniform vertical concentration gradient and body force, on a homogeneous turbulent field accompanied by a first-order chemical reaction, are analysed by considering a simplified model. A system of two-point correlation equations, which contains mean concentration gradient and body force terms, is constructed from the Navier-Stokes, convective diffusion and continuity equations. By well-known methods, these equations are converted into equations for the spectrum functions in the wave-number space and solutions for different spectral tensors are obtained by neglecting the contributions of the triple correlation terms. For carrying out the numerical calculations, it is assumed that the turbulence is initially isotropic and the concentration fluctuations initially zero. It turns out that the turbulence decays with time, although the buoyancy forces do alter the rate of decay. The buoyancy forces can either extract energy from the turbulent field or feed energy into it, depending upon the direction of the body force and the concentration gradient. Spectra are displayed graphically for several values of the reaction rate parameter for stabilizing, as well as destabilizing, buoyancy forces.     

A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term

We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering integro-differential equations, neutral differential equations and general delay differential equations within a unified framework. We show that reasonable differential equations lead to causal solution operators.     

The cumulant method for the space-homogeneous Boltzmann equation

In this work we give a comparison of the exact Bobylev/Krook-Wu solution to the space-homogeneous Boltzmann equation and numerical results obtained by a implementation of the cumulant method for the space-homogeneous case. We find excellent agreement of the numerical solution to the cumulant equations with the exact solution of the space-homogeneous Boltzmann equation as long as the exact, non-linear production terms are used. If a linearized variant of the production terms is used, relaxation rates may be underestimated due to convergence to the solution of the linearized equations.Received: 3 April 2004, Accepted: 3 September 2004, Published online: 22 February 2005PACS: 51.10. + y, 51.30. + i, 47.11 + j, 47.45.-n Correspondence to: K.H. Hoffmann     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.