Analysis of efficient preconditioned defect correction methods for nonlinear water waves

Robust computational procedures for the solution of non‐hydrostatic, free surface, irrotational and inviscid free‐surface water waves in three space dimensions can be based on iterative preconditioned defect correction (PDC) methods. Such methods can be made efficient and scalable to enable prediction of free‐surface wave transformation and accurate wave kinematics in both deep and shallow waters in large marine areas or for predicting the outcome of experiments in large numerical wave tanks. We revisit the classical governing equations are fully nonlinear and dispersive potential flow equations. We present new detailed fundamental analysis using finite‐amplitude wave solutions for iterative solvers. We demonstrate that the PDC method in combination with a high‐order discretization method enables efficient and scalable solution of the linear system of equations arising in potential flow models. Our study is particularly relevant for fast and efficient simulation of non‐breaking fully nonlinear water waves over varying bottom topography that may be limited by computational resources or requirements. To gain insight into algorithmic properties and proper choices of discretization parameters for different PDC strategies, we study systematically limits of accuracy, convergence rate, algorithmic and numerical efficiency and scalability of the most efficient known PDC methods. These strategies are of interest, because they enable generalization of geometric multigrid methods to high‐order accurate discretizations and enable significant improvement in numerical efficiency while incuring minimal storage requirements. We demonstrate robustness using such PDC methods for practical ranges of interest for coastal and maritime engineering, that is, from shallow to deep water, and report details of numerical experiments that can be used for benchmarking purposes. Copyright © 2014 John Wiley & Sons, Ltd.     

A Note on the Numerical Treatment of the k-epsilon Turbulence Model*

Numerical solution of the equations arising from the κ mdash; ε turbulence model has difficulties inherent to nonlinear convection-reaction-diffusion equations with strong reaction terms, resulting in that numerical schemes easily become unstable. We present a formulation that stresses on the robustness of the solution method, tackling common problems that produce instability. The main contribution concerns the loss of positivity of κ and ε, which is addressed by acting on the coefficients of the reaction and diffusion terms rather than on the turbulent variables themselves. In addition, a linearized implicit, non-iterative, treatment of the wall law is proposed.     

Numerical investigation of viscous gas secondary flows in a rotating cylinder with sources and sinks

A source-sink model of secondary flow excitation in a rotating cylinder, which describes the interaction between a circulator and a rotating gas, is proposed for a nonlinear system of Navier-Stokes equations, and the results of a numerical calculation of the resulting circulating flows are presented. The modified Newton's method employed in the numerical solution makes use of regularizing perturbations to ensure its stability and convergence at low Ekman numbers and high rates of rotation of the cylinder. The combined effect of mechanical and thermal means of flow excitation and the influence of viscous energy dissipation are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 39–44, July–August, 1989.     

Articulated Pipes Conveying Fluid Pulsating with High Frequency

Stability and nonlinear dynamics of two articulated pipes conveying fluid with a high-frequency pulsating component is investigated. The non-autonomous model equations are converted into autonomous equations by approximating the fast excitation terms with slowly varying terms. The downward hanging pipe position will lose stability if the mean flow speed exceeds a certain critical value. Adding a pulsating component to the fluid flow is shown to stabilize the hanging position for high values of the ratio between fluid and pipe-mass, and to marginally destabilize this position for low ratios. An approximate nonlinear solution for small-amplitude flutter oscillations is obtained using a fifth-order multiple scales perturbation method, and large-amplitude oscillations are examined by numerical integration of the autonomous model equations, using a path-following algorithm. The pulsating fluid component is shown to affect the nonlinear behavior of the system, e.g. bifurcation types can change from supercritical to subcritical, creating several coexisting stable solutions and also anti-symmetrical flutter may appear.     

Ransom test results from various two-fluid schemes: Is enforcing hyperbolicity a thermodynamically consistent option?

The basic elliptic ill-posedness of physical models and numerical schemes for two-fluid flows is a recurring issue that has motivated the introduction of numerous possible correction strategies. In practical applications physical terms are generally present and regularize the models (viscosity, drag, surface tension, etc.). Yet, many numerical schemes were developed with the stringent and self-imposed constraint that the convective part of the models to be solved had to be hyperbolic, regardless of the type and magnitude of the particular physical regularizing terms. This leads to consider the simplest possible two-fluid “backbone” models corrected with the simplest “universal” terms to ensure hyperbolicity.Among the proposed corrections is the introduction of an interfacial pressure, either closed by algebraic relations or by supplementary evolution equations. Concurrently with the shift to hyperbolic behavior, these techniques also affect other features of systems: Kelvin–Helmholtz type instabilities are notably quenched at all scales, a highly undesirable effect in many practical situations. Less commonly recognized are also distortions in the transfers between kinetic, reversible, and irreversible energies, sometimes up to thermodynamic inconsistency.The present work aims at comparing on the standard Ransom-faucet test the results from various available hyperbolic and elliptic schemes and models against an explicit double Lagrange-plus-remap discretization of the basic elliptic, one-pressure, compressible, six-equations system (i.e. with energy equations). Four features are examined on this test: the entropy preservation, the stretched stream profile, the volume fraction discontinuity, and the unstable character of the analytical solution for the simplest backbone model.The paper highlights the fact that the convective part of two-fluid models might not be necessarily hyperbolic provided that it is physically consistent and numerically robust. Observation of published results for Ransom’s test shows that by enforcing hyperbolicity regardless of thermodynamical consistency, numerical models remove instabilities at the volume fraction discontinuity, but at the expense of distorted profiles of the stretched stream due to excessive numerical diffusion and to spurious forces in the momentum equation. The present approach provides a form of neutral starting point before including dissipative terms: robust but not excessively diffusive, with accurate capture of the stretched stream and volume fraction discontinuity for any practical mesh refinement. Moreover, and consistently with the chosen elliptic model, this numerical scheme eventually generates the elliptic instabilities for late times or fine meshes (but remains robust under the appropriate time step restrictions). It can be supplemented by any kind of small-scale regularization term in order to introduce a cut-off under which physical or numerical stability may be necessary.     

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.     

A fully implicit scheme for simulating ionized gas flows using the gas dynamics electrodynamics coupled system

The flow of ionized gases under the influence of electromagnetic fields is governed by the coupled system of the compressible flow equations and the Maxwell equations. In this system, coupling of the flow with the electromagnetic field is obtained through nonlinear and stiff source terms, which may cause difficulties with the numerical solution of the coupled system. The discontinuous Galerkin finite element method is used for the numerical solution of this system. For the magnetic field vector, discontinuous Galerkin discretization is performed using a divergence‐free vector base for the magnetic field to preserve zero divergence in the element and retain the implicit constraint of a divergence‐free magnetic field vector down to very low level both globally and locally. To circumvent difficulties resulting from the presence of the stiff source terms, implicit time marching is used for the fully coupled system to avoid wrong wave shapes and propagation speeds that are obtained when the coupling source terms are lagged in time or by using splitting iterative schemes. Numerical solutions for benchmark problems computed on collocated meshes for the flow and electromagnetic field variables with this fully coupled monolithic approach showed good agreement with other numerical solutions and exact results. Copyright © 2014 John Wiley & Sons, Ltd.     

A THEORETICAL MODEL FOR NONLINEAR PLANAR MOTIONS OF ROTORS UNDER FLUID CONFINEMENT

Following previous papers by Axisa, Antunes and co-workers, the authors address a theoretical model for immersed rotors, under moderate confinement, using simplified flow equations on the gap-averaged fluctuating quantities. However, in contrast to our previous efforts, the nonlinear terms of the flow equations are here fully accounted. Because such nonlinear analysis is quite involved, this paper will focus on the simpler case of planar motions, in order to emphasize the main aspects of our approach. A direct integration of the continuity and momentum equations leads to extremely lengthy formulations. Here, in order to solve the flow equations, we perform an exact integration of the continuity equation and an approximate solution of the momentum equation, based on a Fourier representation of the azimuthal pressure gradient. Then, an exact formulation for the dynamic flow force can be obtained. Our solution is discussed in connection with physical phenomena. Numerical simulations of the nonlinear rotor-flow coupled system are presented, showing that the linearized and the fully nonlinear models produces similar results when the eccentricity and the spinning velocity are low. However, if such conditions are not met, the qualitative dynamics stemming from these models are quite distinct. Experimental results indicate that the nonlinear flow model leads to better predictions of the rotor dynamics when the eccentricity is significant, when approaching instability and for linearly unstable regimes.     

Transonic flow around bodies of revolution with a passage in the presence of a jet issuing from the passage

Similarity laws are obtained for the transonic flow around axisymmetric bodies with a passage of the nacelle type air-breathing engine (or its root section) in the presence of a jet issuing from a nozzle. The validity of the laws established is verified by the results of a numerical solution of the problem of the transonic flow around the bodies investigated, executed by integrating the complete nonlinear system of flow equations.     

Instability of fully developed mixed convection with viscous dissipation in a vertical porous channel

Flow driven by an externally imposed pressure gradient in a vertical porous channel is analysed. The combined effects of viscous dissipation and thermal buoyancy are taken into account. These effects yield a basic mixed convection regime given by dual flow branches. Duality of flow emerges for a given vertical pressure gradient. In the case of downward pressure gradient, i.e. upward mean flow, dual solutions coincide when the intensity of the downward pressure gradient attains a maximum. Above this maximum no stationary and parallel flow solution exists. A nonlinear stability analysis of the dual solution branches is carried out limited to parallel flow perturbations. This analysis is sufficient to prove that one of the dual solution branches is unstable. The evolution in time of a solution in the unstable branch is also studied by a direct numerical solution of the governing equation.     

Shear flow over a rotating plate

A shear flow interacts with a rotating boundary. The three dimensional Navier-Stokes equations reduce to a set of ninth order, nonlinear, ordinary differential equations which are partially decoupled. Universal similarity velocity profiles are found by numerical integration. If the shear is high enough, reverse flow occurs and the mean drag may be negative. The solution is a rare exact similarity solution of the Navier-Stokes equations.     

Buckling of a twisted and compressed rod

We consider the problem of determining the stability boundary for an elastic rod under thrust and torsion. The constitutive equations of the rod are such that both shear of the cross-section and compressibility of the rod axis are considered. The stability boundary is determined from the bifurcation points of a single nonlinear second order differential equation that is obtained by using the first integrals of the equilibrium equations. The type of bifurcation is determined for parameter values. It is shown that the bifurcating branch is the branch with minimal energy. Finally, by using the first integral, the solution for one specific dependent variable is expressed in terms of elliptic integrals. The solution pertaining to the complete set of equilibrium equations is obtained by numerical integration.     

On the need of mode interpolation for data-driven Galerkin models of a transient flow around a sphere

We present a low-dimensional Galerkin model with state-dependent modes capturing linear and nonlinear dynamics. Departure point is a direct numerical simulation of the three-dimensional incompressible flow around a sphere at Reynolds numbers 400. This solution starts near the unstable steady Navier–Stokes solution and converges to a periodic limit cycle. The investigated Galerkin models are based on the dynamic mode decomposition (DMD) and derive the dynamical system from first principles, the Navier–Stokes equations. A DMD model with training data from the initial linear transient fails to predict the limit cycle. Conversely, a model from limit-cycle data underpredicts the initial growth rate roughly by a factor 5. Key enablers for uniform accuracy throughout the transient are a continuous mode interpolation between both oscillatory fluctuations and the addition of a shift mode. This interpolated model is shown to capture both the transient growth of the oscillation and the limit cycle.     

Unstable manifold computations for the two-dimensional plane Poiseuille flow

We follow the unstable manifold of periodic and quasi-periodic solutions in time for the Poiseuille problem, using two formulations: holding a constant flux or mean pressure gradient. By means of a numerical integrator of the Navier–Stokes equations, we let the fluid evolve from an initially perturbed unstable solution until the fluid reaches an attracting state. Thus, we detect several connections among different configurations of the flow such as laminar, periodic, quasi-periodic with two or three basic frequencies, and more complex sets that we have not been able to classify. These connections make possible the location of new families of solutions, usually hard to find by means of numerical continuation of curves, and show the richness of the dynamics of the Poiseuille flow. PACS 05.45.-a, 47.11.+j, 47.20.-k, 47.20.Ft     

The DRBEM solution of incompressible MHD flow equations

This paper presents a dual reciprocity boundary element method (DRBEM) formulation coupled with an implicit backward difference time integration scheme for the solution of the incompressible magnetohydrodynamic (MHD) flow equations. The governing equations are the coupled system of Navier‐Stokes equations and Maxwell's equations of electromagnetics through Ohm's law. We are concerned with a stream function‐vorticity‐magnetic induction‐current density formulation of the full MHD equations in 2D. The stream function and magnetic induction equations which are poisson‐type, are solved by using DRBEM with the fundamental solution of Laplace equation. In the DRBEM solution of the time‐dependent vorticity and current density equations all the terms apart from the Laplace term are treated as nonhomogeneities. The time derivatives are approximated by an implicit backward difference whereas the convective terms are approximated by radial basis functions. The applications are given for the MHD flow, in a square cavity and in a backward‐facing step. The numerical results for the square cavity problem in the presence of a magnetic field are visualized for several values of Reynolds, Hartmann and magnetic Reynolds numbers. The effect of each parameter is analyzed with the graphs presented in terms of stream function, vorticity, current density and magnetic induction contours. Then, we provide the solution of the step flow problem in terms of velocity field, vorticity, current density and magnetic field for increasing values of Hartmann number. Copyright © 2010 John Wiley & Sons, Ltd.     

A numerical method for 3D viscous incompressible flows using non-orthogonal grids

This paper presents a numerical method for fluid flow in complex three-dimensional geometries using a body-fitted co-ordinate system. A new second-order-accurate scheme for the cross-derivative terms is proposed to describe the non-orthogonal components, allowing parts of these terms to be treated implicitly without increasing the number of computational molecules. The physical tangential velocity components resulting from the velocity expansion in the unit tangent vector basis are used as dependent variables in the momentum equations. A coupled equation solver is used in place of the complicated pressure correction equation associated with grid non-orthogonality. The co-ordinate-invariant conservation equations and the physical geometric quantities of control cells are used directly to formulate the numerical scheme, without reference to the co-ordinate derivatives of transformation. Several two- and three-dimensional laminar flows are computed and compared with other numerical, experimental and analytical results to validate the solution method. Good agreement is obtained in all cases.     

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I

We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo–Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper.A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues–Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given.Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo–Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass.The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.     

Stability analysis of fluid flow over a nonlinearly stretching sheet

We discuss the stability of solutions to a class of nonlinear third-order ordinary differential equations arising in the viscous flow over a nonlinearly stretching sheet. In particular, we consider solutions over the semi-infinite interval [0, ∞). These results complement the available existence and uniqueness results in the literature. We find that, in general, there is one stable solution branch and one unstable solution branch. Furthermore, it is observed that the stable solution becomes more stable with an increase in the nonlinearity due to the stretching sheet, while the unstable solution branch becomes more unstable given such an increase in the nonlinearity. The stable solution is the physically meaningful solution.     

Dynamics of axially accelerating beams with multiple supports

This study represents the transverse vibrations of an axially accelerating Euler–Bernoulli beam resting on multiple simple supports. This is one of the examples of a system experiencing Coriolis acceleration component that renders such systems gyroscopic. A small harmonic variation with a constant mean value for the axial velocity is assumed in the problem. The immovable supports introduce nonlinear terms to the equations of motion due to stretching of neutral axis. The method of multiple scales is directly applied to the equations of motion obtained for the general case. Natural frequency equations are presented for multiple support case. Principal parametric resonances and combination resonances are discussed. Solvability conditions are presented for different cases. Stability analysis is conducted for the solutions; approximate stable and unstable regions are identified. Some numerical examples are presented to show the effects of axial speed, number of supports, and their locations.     

Variational iteration method for two-dimensional steady slip flow in micro-channels

In this paper, the two-dimensional steady slip flow in microchannels is investigated. Research on micro flow, especially on micro slip flow, is very important for designing and optimizing the micro electromechanical system (MEMS). The Navier-Stokes equations for two-dimensional steady slip flow in microchannels are reduced to a nonlinear third-order differential equation by using similarity solution. The variational iteration method (VIM) is used to solve this nonlinear equation analytically. Comparison of the result obtained by the present method with numerical solution reveals that the accuracy and fast convergence of the new method.