Constructing transient response probability density of non-linear system through complex fractional moments

The probability density function for transient response of non-linear stochastic system is investigated through the stochastic averaging and Mellin transform. The stochastic averaging based on the generalized harmonic functions is adopted to reduce the system dimension and derive the one-dimensional Itô stochastic differential equation with respect to amplitude response. To solve the Fokker–Plank–Kolmogorov equation governing the amplitude response probability density, the Mellin transform is first implemented to obtain the differential relation of complex fractional moments. Combining the expansion form of transient probability density with respect to complex fractional moments and the differential relations at different transform parameters yields a set of closed-form first-order ordinary differential equations. The complex fractional moments which are determined by the solution of the above equations can be used to directly construct the probability density function of system response. Numerical results for a van der Pol oscillator subject to stochastically external and parametric excitations are given to illustrate the application, the convergence and the precision of the proposed procedure.     

Ferromagnetic core coil hysteresis modeling using fractional derivatives

Nonlinear Dynamics - The modeling of a ferromagnetic core coil magnetic hysteresis has been considered. The measurement basis consisted of waveforms that have been recorded for various levels of...     

Simulating turbulence in complex geometries

We first review the state-of-the art in direct numerical simulation and present a new class of spectral methods on unstructured grids for handling complex-geometry domains. Subsequently, we concentrate on the classical problem of the turbulent wake behind a circular cylinder and compare the accuracy of spectral DNS versus other LES results available in the literature. We find that DNS provides consistent agreement with the experimental results, but that LES predictions are inconsistent and depend strongly on the interaction between numerical discretization and the subgrid model. We also demonstrate via a simple vorticity-based analysis of the turbulent near-wake that eddy-viscosity models are inappropriate for sudgrid modeling. In contrast, preliminary a priori tests suggest that scale-similarity models may be a good candidate. We close the paper by forecasting the use of dynamic DNS and comment on its role in simulating turbulence in complex geometries.     

Wave equation for generalized Zener model containing complex order fractional derivatives

We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The initial boundary value problem for such materials is formulated and solution is presented in the form of convolution. Two specific examples are analyzed.     

Multigrid flow solutions in complex two-dimensional geometries

A finite difference solution algorithm is described for use on two-dimensional curvilinear meshes generated by the solution of the transformed Laplace equation. The efficiency of the algorithm is improved through the use of a full approximation scheme (FAS) multigrid algorithm using an extended pressure correction scheme as smoother. The multigrid algorithm is implemented as a fixed V-cycle through the grid levels with a constant number of sweeps being performed at each grid level. The accuracy and efficiency of the numerical code are validated using comparisons of the flow over two backward step configurations. Results show close agreement with previous numerical predictions and experimental data. Using a standard Cartesian co-ordinate flow solver, the multigrid efficiency obtainable in a rectangular system is shown to be reproducible in two-dimensional body-fitted curvilinear co-ordinates. Comparisons with a standard one-grid method show the multigrid method, on curvilinear meshes, to give reductions in CPU time of up to 93%.     

Characterisation of blast loading in complex,confined geometries using quarter symmetry experimental methods

Explosions in confined spaces lead to complicated patterns of shock wave reflection and interactions which are best investigated by use of experimental tests or numerical simulations. This paper describes the design and outcome of a series of experiments using a test cell to measure the pressures experienced when structures were placed inside to alter the propagation of shock waves, utilising quarter symmetry to reduce the size of the required test cell and charge. An 80 g charge of PE4 (a conventional RDX-based plastic explosive) was placed at half height in one corner of the test cell, which represents the centre of a rectangular enclosure when symmetry is taken into consideration. Steel cylinders and rectangular baffles were placed within the test cell at various locations. Good reproducibility was found between repeated tests in three different arrangements, in terms of both the recorded pressure data and the calculated cumulative impulse. The presence of baffles within the test cell made a small difference to the pressures and cumulative impulse experienced compared to tests with no baffles present; however, the number and spacing of baffles was seen to make minimal difference to the experienced pressures and no noticeable difference to the cumulative impulse history. The paper presents useful experimental data that may be used for three-dimensional code validation.     

Dissipative particle dynamics for complex geometries using non‐orthogonal transformation

Dissipative particle dynamics (DPD) was applied to fluid flow in irregular geometries using non‐orthogonal transformation, where an irregular domain is transformed into a simple rectangular domain. Transformation for position and velocity was used to relate the physical and computational domains. This approach was described by simulating fluid flow inside a two‐dimensional convergent–divergent nozzle. The nozzle geometry is controlled by the contraction ratio (CR) in the middle of the channel. The range of Reynolds number and CR, in this paper, was Re = 10hbox??200 and CR = 0.8 and 0.6, respectively. The DPD results were validated against in‐house computational fluid dynamic (CFD) finite volume code based on the stream function vorticity approach. The results revealed an excellent agreement between DPD and CFD. The maximum deviation between the DPD and CFD results was within 2%. Local and average coefficients of friction was calculated and it compared well with the CFD results. Copyright © 2011 John Wiley & Sons, Ltd.     

Particle image velocimetry measurements in complex geometries

 One of the advantages of Particle Image Velocimetry (PIV) is its ability to determine the instantaneous flow field over two- or three-dimensional domains. Yet PIV has had limited application to complex flow passages because of the difficulty in replicating these geometries with optically transparent materials. In this work, we describe a method for overcoming this difficulty using rapid prototyping techniques. As an illustrative example, the technique has been used to characterize flow in a model of the human nasal cavity. Received: 12 January 1999/Accepted: 30 September 1999     

The flow of fiber suspensions in complex geometries

A continuum theory for dilute suspensions of large-aspect-ratio particles is applied to the flow of fiber suspensions through contractions. The theory, which incorporates the statistical orientation distribution function into the stress equation, predicts that the flow of dilute suspensions will differ qualitatively from the flow of the suspending fluid. The theory is in excellent agreement with experiments on the flow of suspensions of chopped-glass fibers through axisymmetric contractions, where substantial enlargement of the recirculating corner vortex is observed at volume fractions of 0.1% and less.     

Local mass transfer measurements for corals and other complex geometries using gypsum dissolution

A new method of measuring local mass transfer for highly complex geometries is demonstrated. The method combines gypsum dissolution and X-ray computed tomography (CT). An object coated in gypsum is CT scanned before and after exposure to fluid flow. Digital three-dimensional pre- and post-flow geometries are created using the CT data, and the local dissolution thickness is determined by subtracting the post-flow from the pre-flow object. The method is first demonstrated for cylinders in cross-flow and validated with mass and heat transfer data. The measurements agree with values from correlations reported in the literature when scaled using Reynolds, Sherwood, and Schmidt numbers. The method is then applied to measure local mass transfer for the complex geometry of a 0.75-scale branching scleractinian coral Stylophera pistillata. Local Sherwood numbers vary between nearly zero on the backward facing surfaces of the downstream branches of the coral and nearly 200 at the tips of the branches at the top of the coral. The upstream facing surfaces at radii between 20 and 70 % of the overall radius of the coral experience Sherwood numbers close to 100. The local measurements are integrated to produce a bulk mass transfer coefficient that lies within the range of previous bulk measurements in the literature for similar coral species. With this method, local mass transfer rates can be measured for complex objects in laboratory or natural in situ flow environments. These are geometries for which only bulk measurements were previously possible.     

High performance parallel computing of flows in complex geometries

Efficient numerical tools taking advantage of the ever increasing power of high-performance computers, become key elements in the fields of energy supply and transportation, not only from a purely scientific point of view, but also at the design stage in industry. Indeed, flow phenomena that occur in or around the industrial applications such as gas turbines or aircraft are still not mastered. In fact, most Computational Fluid Dynamics (CFD) predictions produced today focus on reduced or simplified versions of the real systems and are usually solved with a steady state assumption. This article shows how recent developments of CFD codes and parallel computer architectures can help overcoming this barrier. With this new environment, new scientific and technological challenges can be addressed provided that thousands of computing cores are efficiently used in parallel. Strategies of modern flow solvers are discussed with particular emphases on mesh-partitioning, load balancing and communication. These concepts are used in two CFD codes developed by CERFACS: a multi-block structured code dedicated to aircrafts and turbo-machinery as well as an unstructured code for gas turbine flow predictions. Leading edge computations obtained with these high-end massively parallel CFD codes are illustrated and discussed in the context of aircrafts, turbo-machinery and gas turbine applications. Finally, future developments of CFD and high-end computers are proposed to provide leading edge tools and end applications with strong industrial implications at the design stage of the next generation of aircraft and gas turbines.     

Automatic grid generation of complex geometries in Cartesian co-ordinates

A method of automatic grid generation for complex boundaries in Cartesian co-ordinates is proposed in this paper. In addition to the Cartesian grid lines the diagonal segments are used for the approximations of complex geometries in Cartesian co-ordinates. A structured Cartesian grid is employed for the sake of the numerical simplicity and the potential of automatic grid generation. The automatic grid generation is achieved by this diagonal Cartesian method and the accuracy estimations of geometry approximations are given. The approximations of a few complex geometries, such as the multibody system in porous media, lake banks, grooved channels and spheres are shown and analyzed. The proposed method is verified by the numerical solutions of a rotated cavity flow. It is shown that the diagonal Cartesian method improves both the accuracy of geometry approximations and the numerical solution of a rotated cavity flow, comparing with the traditional saw-tooth method in which only Cartesian grid lines are utilized for geometry approximations. The stability and convergence of the proposed method is demonstrated. Finally, the application of the diagonal Cartesian method for the prediction of a grooved channel flow is presented. © 1998 John Wiley & Sons, Ltd.     

Hysteresis cycles and fatigue criteria using anelastic models based on fractional derivatives

The constitutive equation and the fatigue of anelastic media are described by using fractional order derivatives. The stress–strain relation, based on a generalization of the Kelvin–Voigt model, describes typical hysteresis cycles with the stress increasing as the number of cycles increases, a phenomenon known as cyclic hardening and observed in many materials such as, for instance, steel. Criteria are established to find the number of cycles which may cause fatigue for a strain with a given amplitude and frequency. They are based on the yield and fatigue stresses, on the melting temperature through the dissipated energy, and on the strain energy. In all the cases, it is seen that the number of cycles to failure is inversely proportional to the amplitude and to the frequency of the applied strain. Comparison to experimental data indicates that the model satisfies, at least qualitatively, the behavior of real materials under cyclic loading.     

Parallel computation of incompressible flows with complex geometries

We present our numerical methods for the solution of large-scale incompressible flow applications with complex geometries. These methods include a stabilized finite element formulation of the Navier–Stokes equations, implementation of this formulation on parallel architectures such as the Thinking Machines CM-5 and the CRAY T3D, and automatic 3D mesh generation techniques based on Delaunay–Voronoi methods for the discretization of complex domains. All three of these methods are required for the numerical simulation of most engineering applications involving fluid flow. We apply these methods to the simulation of airflow past an automobile and fluid–particle interactions. The simulation of airflow past an automobile is of very large scale with a high level of detail and yielded many interesting airflow patterns which help in understanding the aerodynamic characteristics of such vehicles. © 1997 John Wiley & Sons, Ltd.     

Efficient preprocessing of complex geometries for CFD simulations

Preprocessing remains one the main bottlenecks in the computational fluid Dynamics simulations of flows involving complex geometries as advances in the algorithm development, turbulence modelling and parallel computing are made. To this end, two approaches are presented here to efficiently deal with complex geometries in order to reduce the preprocessing time and manual effort. First, a hybrid blocking approach, combining the medial axis-based method with level set iso-surface is presented to aid the block topology generation for subsequent structured meshing of complex 3D external flow domains. Secondly, a hierarchical geometry handling approach is demonstrated which makes use of the lower order modelling, overset meshes and zonal blocking for efficient preprocessing. Typical external aerodynamics cases have been showcased to describe how such techniques can be used to address modern challenges in the preparation of complex geometries for flow simulations.     

Modeling of complex fluids using micro-macro approach with transient network dynamics

In this work, the micro-macro approach is used to simulate the flow of dilute polymer solutions by means of a kinetic model coupled with the dynamics of a transient network. The transient network modeling is based on the original formulation, in which the kinetics of microstates describes the complexity of interactions among the macromolecules suspended in a Newtonian solvent (Rincón et al, J Non-Newton Fluid 131:64–77, 2005). The average concentration of microstates, at a given time, defines a variable maximum segment length (variable extensibility) of the molecular FENE model. The non-Newtonian contribution to the extra stress tensor is computed according to the Brownian configuration-fields method. Comparisons with the Oldroyd-B model validates its limiting behavior. Numerical results show the influence of the solvent to total viscosity ratio and shear rate, on the transient and steady rheological phenomena of complex fluids with microstates.     

On the definition of fractional derivatives in rheology

During the last two decades fractional calculus has been increasingly applied to physics, especially to rheology. It is well known that there are obivious differences between Riemann-Liouville (R-L) definition and Caputo definition, which are the two most commonly used definitions of fractional derivatives. The multiple definitions of fractional derivatives have hindered the application of fractional calculus in rheology. In this paper, we clarify that the R-L definition and Caputo definition are both rheologically unreasonable with the help of the mechanical analogues of the fractional element model. We also find that to make them more reasonable rheologically, the lower terminals of both definitions should be put to ?∞. We further prove that the R-L definition with lower terminal ?∞ and the Caputo definition with lower terminal ?∞ are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points. Thus we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal ?∞ (or, equivalently, the Caputo derivatives with lower terminal ?∞ ) not only for steady-state processes, but also for transient processes.