A thermodynamic model of turbulent motions in a granular material

This paper is devoted to a thermodynamic theory of granular materials subjected to slow frictional as well as rapid flows with strong collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable is of a kinematic nature and enters the formulation via the balance law of the configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin [1], but modified here. Complemented by constitutive equations, the emerging field equations are postulated to be adequate for motions, be they laminar or turbulent, if the resolved length scales are sufficiently small. On large length scales the sub-grid motion may be interpreted as fluctuations, which manifest themselves in correspondingly filtered equations as correlation products, like in the turbulence theory. We apply an ergodic (Reynolds) filter to these equations and thus deduce averaged equations for the mean motions. The averaged equations comprise balances of mass, linear and configurational momenta, energy, and turbulent kinetic energy as well as turbulent configurational kinetic energy. They are complemented by balance laws for two internal fields, the dissipation rates of the turbulent kinetic energy and of the turbulent configurational kinetic energy. We formulate closure relations for the averages of the laminar constitutive quantities and for the correlation terms by using the rules of material and turbulent objectivity, including equipresence. Many versions of the second law of thermodynamics are known in the literature. We follow the Müller-Liu theory and extend Müllers entropy principle to allow the satisfaction of the second law of thermodynamics for both laminar and turbulent motions. Its exploitation, performed in the spirit of the Müller-Liu theory, delivers restrictions on the dependent constitutive quantities (through the Liu equations) and a residual inequality, from which thermodynamic equilibrium properties are deduced. Finally, linear relationships are proposed for the nonequilibrium closure relations.Received: 21 March 2003, Accepted: 1 September 2003, Published online: 11 February 2004PACS: 05.70.Ln, 61.25.Hq, 61.30.-vCorrespondence to: I. Luca     

Spectral structure and linear mechanisms in a rapidly distorted boundary layer

The aim of the present work is to investigate the spectral structure of a rapidly distorted boundary layer that develops on a flat plate in presence of a localised patch of roughness or/and grid-generated freestream turbulence. We observe that, at a certain distance downstream of the roughness patch the boundary layer exhibits a bimodal shape in the energy spectrum of the streamwise velocity fluctuations, similar to that found in a fully-turbulent boundary layer at relatively high Reynolds numbers. The physical mechanism that gives rise to the low-wavenumber peak in the spectrum, which represents long streamwise motions or “superstructures”, is identified to be the interaction of the broadband disturbances with the region of high shear near the wall in the boundary layer. We next show that the flat-plate boundary layer combined with surface roughness and grid turbulence can serve as building-block elements towards synthesising the wall-normal structure of a canonical turbulent boundary layer, in the context of large-scale streamwise motions. The rapidly distorted (or “synthetic”) boundary layer presents a simpler environment in which the coherent motions can evolve and therefore can enable a better characterisation of these motions. To further illustrate the utility of the present approach we compare results from our measurements with the predictions of the Rapid Distortion Theory (RDT). We show that the streamwise turbulence energy in the near-wall region of the rapidly distorted boundary layer grows linearly with time consistent with the RDT results on the effect of pure shear on an initially isotropic turbulence. Moreover close to the edge of the boundary layer the large-scale fluctuations experience an enhancement in the streamwise turbulence energy in accordance with the linear blocking model in the RDT framework. The present work thus highlights the importance of linear processes in wall turbulence and can help us identify aspects of it to which the linear theories can be meaningfully applied.     

Cellular structures in solid fuel combustion

The objective of this contribution is to investigate whether the mechanism of the thermal diffusion instability in gaseous flames causing cellular flame structures also occurs during the combustion of porous solid fuel. Based on conservation for mass and energy, the relevant set of differential equations was derived. Assuming thermal equilibrium between fuel and oxidiser, a global energy equation was valid for both solid and gaseous phase. The resulting set of differential equations was discretised by the Collocation method to arrive at a system of algebraic equations. In order to investigate into cellular flame structures, an infinitesimal disturbance was superimposed onto the plane conversion front. Carrying out a linear instability analysis, yielded eigenvalues dependent on the wave number of the disturbance. A critical wave number exists below which the real part of the eigenvalues is positive, thus, indicating a regime of instability. Within this region, eigenvalues with a not-vanishing imaginary part of the eigen value existed causing cellular flame structures. However, the growth rate of disturbances was found to be small, which may explain the difficulty to investigate this phenomena experimentally.     

哈密尔顿系统的连续有限元法及守恒性

利用常微分方程的连续有限元法,证明了线性哈密尔顿系统的连续一、二、三次有限元法为辛算法;对非线性哈密尔顿系统,本文证明了连续一次有限元在3阶量意义下近似保辛,且保持能量守恒,并在数值计算上探讨了守恒性和近似程度,结果与理论相吻合.     

On the linearized disturbance dynamic equations for buckling and free vibration of cylindrical helical coil springs under combined compression and torsion

In this study a set of twelve linearized disturbance dynamic equations in canonical form is derived systematically and in a comprehensive manner based on the first order shear deformation theory to study the buckling and vibration analysis of helical coil springs made of isotropic linear materials. Those complete equations comprise the axial and shear deformation effects together with rotatory inertia effects. The special case of these equations corresponds also to the equations for straight and circular rods. The main differences among the existing formulations based on the same approach are discussed briefly. The resulting equations are used for numerical buckling and free vibration analyses to show its soundness.     

The effect of transient viscoelastic properties on interfacial instabilities in superposed pressure driven channel flows

In a recent study, Ganpule and Khomami (submitted to J. Non-Newtonian Fluid Mech.) have shown that in order to accurately describe the experimentally observed interfacial instability phenomenon in superposed channel flow of viscoelastic fluids, a constitutive equation that can accurately depict not only the steady viscometric properties of the experimental test fluids, but also their transient viscoelastic properties must be used in the analysis. In the present study, the effect of differences in transient viscoelastic properties which can arise either due to the differences in the predictive capabilities of various constitutive models or from the presence of multiple modes of relaxation on the interfacial instabilities of the superposed pressure driven channel flows has been investigated. Specifically, a linear stability analysis is performed using nonlinear constitutive equations which predict identical steady viscometric properties but different transient viscoelastic properties. It is shown that different nonlinear constitutive equations give rise to the same mechanism of interfacial instability, but the boundaries of the neutral stability contours and the magnitudes of the growth/decay rates, especially at intermediate and shortwaves, are shifted due to the overshoots in the transient viscoelastic responses predicted by the constitutive equations. In addition, the effect of the presence of multiple modes of relaxation on interfacial stability is studied using single and multiple mode upper convected Maxwell (UCM) fluids and it is shown that pronounced differences in the intermediate and shortwave linear stability predictions arise due to the fact that the increase in the number of modes gives rise to additional fast as well as slow relaxation modes and the presence of these additional relaxation modes gives rise to differences in the transient viscoelastic response of the fluids in the absence of any overshoots. The effect of fluid inertia on the interfacial stability of viscoelastic liquids is examined and it is shown that at longwaves, inertia has a pronounced effect on the stability of the interface, whereas at shortwaves, elastic and viscous effects dominate. Furthermore, the mechanism of viscoelastic interfacial instabilities is studied by a careful examination of disturbance eigenfunctions as well as performing a disturbance energy analysis. The results indicate that the mechanism of viscoelastic interfacial instabilities can be described in terms of interaction of mechanisms of purely viscous and purely elastic instabilities. However, since more than one mechanism for the instability is at work, the disturbance energy analysis can not clearly distinguish between them due to the fact that the eigenfunctions used in the energy analysis contain the information regarding both viscous and elastic effects. Hence, the mechanism of the instability must be determined by a careful examination of disturbance eigenfunctions.     

A fourth order approach to relativistic extended thermodynamics

Relativistic Extended Thermodynamics is a very important scientific achievement of the last decades, and has inspired many subsequent authors to apply its methodology to lots of other possible applications. In short it furnishes field equations which are “closed” by imposing the relativity principle and the entropy principle up to second order, with respect to equilibrium; the linear closure is explicitly reported. Here these principles are imposed up to fourth order; the second order closure is explicitly reported, while the subsequent ones are reported as implicit functions. It is also proved that no internal inconsistencies are generated by the theory. In fact, in the process of computations many complicated conditions on the lower order terms are deduced, but they turn out to be identically satisfied. Received January 20, 1998     

Pressure–Strain Correlation Modeling: Towards Achieving Consistency with Rapid Distortion Theory

In turbulence closure modeling, it is widely accepted that the rapid pressure–strain correlation (RPSC) model be consistent with the rapid distortion theory (RDT). It is desirable to achieve this consistency with a closure model that is computationally tractable and satisfies the requisite mathematical constraints of realizability and linearity in the appropriate variables. In this investigation, starting from a detailed modal analysis of two-dimensional mean flows, we identify important flow features to be incorporated into the model. However, the dynamical system analysis shows that the suggested physics cannot be embodied in a model with all desired computational and mathematical attributes. To resolve this conflict, we propose a slight compromise in the physical requirement and ease one of the linearity constraints leading to a “best possible” tractable model. Overall, the present work provides important insight into RPSC closure modeling challenges—arising from the interplay among physical fidelity, computational viability and mathematical constraints—and proposes avenues for future improvement.     

Model-Based Feedback Control of Acoustic Radiation from Vibrating Structures by Means of Structural Control

In this work, it is investigated how classical techniques of linear feedback control design can be applied to the problem of the reduction of acoustic radiation from vibrating structures for cases where the disturbance is broadband and where no reference is available. Much of the work carried out to date in the field of active noise and vibration control has concentrated on applications where either the disturbance to be cancelled is periodic (propeller noise in aircraft,...) or a reference signal, highly correlated with the disturbance, is available (air conditioning duct noise,...) such that a feedforward control approach can be used. When the disturbance is broadband and where no reference is available, feedforward control cannot be used and feedback control must instead be used. Feedback control theory is well established and a vast amount of analytical tools are available to the feedback control designer. However, due to the inherent delays associated with the propagation of sound waves, feedback control of acoustic fields is prone to being unstable. In this paper, a controller is presented which feeds back a measure of the structural response (vibration) of the system in order to determine the control force that needs to be applied to the vibrating structure in order to reduce the total acoustic energy radiated by the vibrating structure. This revised version was published online in July 2006 with corrections to the Cover Date.     

On non-linear Beltrami-Mitchell equations

Beltrami-Mitchell equations for non-linear elasticity theory are derived using the first Piola-Kirchhoff stress and the deformation gradient tensors as field variables so as to yield linear equilibrium and compatibility equations, respectively. In the derivation it is assumed that a strain energy density and, correspondingly, a complementary strain energy density exist, and satisfy the axiom of objectivity. Substitution for the deformation gradient in the compatibility equations yields non-linear differential equations in terms of the first Piola-Kirchhoff stress tensor which may be regarded as the Beltrami-Mitchell equations of non-linear elasticity. The equations are also derived for “semi-linear” isotropic elastic materials and the theory is illustrated by three simple examples.     

Dual-plane stereoscopic particle image velocimetry: system set-up and its application on a lobed jet mixing flow

 The technical basis and system set-up of a dual-plane stereoscopic particle image velocimetry (PIV) system, which can obtain the flow velocity (all three components) fields at two spatially separated planes simultaneously, is summarized. The simultaneous measurements were achieved by using two sets of double-pulsed Nd:Yag lasers with additional optics to illuminate the objective fluid flow with two orthogonally linearly polarized laser sheets at two spatially separated planes, as proposed by Kaehler and Kompenhans in 1999. The light scattered by the tracer particles illuminated by laser sheets with orthogonal linear polarization were separated by using polarizing beam-splitter cubes, then recorded by high-resolution CCD cameras. A three-dimensional in-situ calibration procedure was used to determine the relationships between the 2-D image planes and three-dimensional object fields for both position mapping and velocity three-component reconstruction. Unlike conventional two-component PIV systems or single-plane stereoscopic PIV systems, which can only get one-component of vorticity vectors, the present dual-plane stereoscopic PIV system can provide all the three components of the vorticity vectors and various auto-correlation and cross-correlation coefficients of flow variables instantaneously and simultaneously. The present dual-plane stereoscopic PIV system was applied to measure an air jet mixing flow exhausted from a lobed nozzle. Various vortex structures in the lobed jet mixing flow were revealed quantitatively and instantaneously. In order to evaluate the measurement accuracy of the present dual-plane stereoscopic PIV system, the measurement results were compared with the simultaneous measurement results of a laser Doppler velocimetry (LDV) system. It was found that both the instantaneous data and ensemble-averaged values of the stereoscopic PIV measurement results and the LDV measurement results agree well. For the ensemble-averaged values of the out-of-plane velocity component at comparison points, the differences between the stereoscopic PIV and LDV measurement results were found to be less than 2%. Received: 18 April 2000/Accepted: 2 February 2001     

Cosserat (micropolar) elasticity in Stroh form

The theory of defects in Cosserat continua is sketched out in strict analogy to the theory of line defects in anisotropic elasticity (Stroh theory). This rewrite of the second order equilibrium equations of elasticity in a 3-dimensional space as first order equations in a 6-dimensional space is analogous to replacing the Laplace equation by the Riemann–Cauchy equations. For generalized plane strain of anisotropic micropolar (Cosserat) elasticity one obtains a 14-dimensional coupled linear system of differential equations of first order and for plane strain of anisotropic micropolar (Cosserat) elasticity we obtain a 6-dimensional coupled linear system of differential equations of first order.     

Interface stabilization in two-layer channel flow by surface heating or cooling

The linear stability of plane Poiseuille flow of two immiscible Newtonian liquids in a differentially heated channel is considered. The equations of motion and energy are fully coupled via temperature-dependent fluid-viscosity coefficients. A long-wave asymptotic formulation of the stability problem is presented together with numerical results for disturbances of arbitrary wavelength. Two combinations of immiscible liquids are analyzed: silicone/water and oil/water (water at the bottom layer in both cases). It is shown that an imposed wall temperature difference can be stabilizing or destabilizing depending on the disturbance wavenumber and layer thickness ratio. Interfacial tension has a stabilizing effect on the interface. Stabilizing influence of interfacial tension is observed at intermediate and large wavenumbers. Most importantly, for certain ranges of the controlling dimensionless parameters, stable interfaces at all disturbance wavelengths can be attained.     

Nonlinear random coupled motions of structural elements with quadratic nonlinearities

The second-order closure method is used to analyze the nonlinear response of two-degree-of-freedom systems with quadratic nonlinearities. The excitation is assumed to be the sum of a deterministic harmonic component and a random component. The case of primary resonance of the second mode in the presence of a two-to-one internal (autoparametric) resonance is investigated. The method of multiple scales is used to obtain four first-order ordinary-differential equations that describe the modulation of the amplitudes and phases of the two modes. Applying the second-order closure method to the modulation equations, we determine the stationary mean and mean-square responses. For the case of a narrow-band random excitation, the results show that the presence of the nonlinearity causes multi-valued mean-square responses. The multi-valuedness is responsible for a jump phenomenon. Contrary to the results of the linear analysis, the nonlinear analysis reveals that the directly excited second mode takes a small amount of the input energy (saturates) and spills over the rest of the input energy into the first mode, which is indirectly excited through the autoparametric resonance.     

聚变等离子体理想动力系统的非线性热不稳定性

本文讨论氖氖聚变等离子体理想动力系统的非线性热不稳定性问题，这个系统与外界有物质与能量交换。采用几何上零维近似的演化方程。对自治系统，讨论了在各种不同粒子约束定律和能量约束定律下的非线性热不稳定性问题，给出稳定性判据。研究了自治系统不稳定解的非线性演化。     

A finite element method for two‐dimensional water‐wave problems

In this paper, the authors treat the free‐surface waves generated by a moving disturbance with a constant speed in water of finite and constant depth. Specifically, the case when the disturbance is moving with the critical speed is investigated. The water is assumed inviscid and its motion irrotational. The surface tension is neglected. It is well‐known that the linear theory breaks down when a disturbance is moving with the critical speed. As a remedy to overcome the invalid linear theory, approximate non‐linear theories have been applied with success in the past, i.e. Boussinesq and Korteweg de Vries equations, for example. In the present paper, the authors describe a finite element method applied to the non‐linear water‐wave problems in two dimensions. The present numerical method solves the exact non‐linear formulation in the scope of potential theory without any additional assumptions on the magnitude of the disturbances. The present numerical results are compared with those obtained by other approximate non‐linear theories. Also presented are the discussions on the validity of the existing approximate theories applied to two types of the disturbances, i.e. the bottom bump and the pressure patch on the free‐surface at the critical speed. Copyright © 1999 John Wiley & Sons, Ltd.     

Optimally growing boundary layer disturbances in a convergent nozzle preceded by a circular pipe

We report the findings from a theoretical analysis of optimally growing disturbances in an initially turbulent boundary layer. The motivation behind this study originates from the desire to generate organized structures in an initially turbulent boundary layer via excitation by disturbances that are tailored to be preferentially amplified. Such optimally growing disturbances are of interest for implementation in an active flow control strategy that is investigated for effective jet noise control. Details of the optimal perturbation theory implemented in this study are discussed. The relevant stability equations are derived using both the standard decomposition and the triple decomposition. The chosen test case geometry contains a convergent nozzle, which generates a Mach 0.9 round jet, preceded by a circular pipe. Optimally growing disturbances are introduced at various stations within the circular pipe section to facilitate disturbance energy amplification upstream of the favorable pressure gradient zone within the convergent nozzle, which has a stabilizing effect on disturbance growth. Effects of temporal frequency, disturbance input and output plane locations as well as separation distance between output and input planes are investigated. The results indicate that optimally growing disturbances appear in the form of longitudinal counter-rotating vortex pairs, whose size can be on the order of several times the input plane mean boundary layer thickness. The azimuthal wavenumber, which represents the number of counter-rotating vortex pairs, is found to generally decrease with increasing separation distance. Compared to the standard decomposition, the triple decomposition analysis generally predicts relatively lower azimuthal wavenumbers and significantly reduced energy amplification ratios for the optimal disturbances.     

Relationship between multibody dynamics computation and nonlinear vibration theory

This paper presents the ground-work of implementing the multibody dynamics codes to analyzing nonlinear coupled oscillators. The recent developments of the multibody dynamics have resulted in several computer codes that can handle large systems of differential and algebraic equations (DAE). However, these codes cannot be used in their current format without appropriate modifications. According to multibody dynamics theory, the differential equations of motion are linear in the acceleration, and the constraints are appended into the equations of motion through Lagrange's multipliers. This formulation should be able to predict the nonlinear phenomena established by the nonlinear vibration theory. This can be achieved only if the constraint algebraic equations are modified to include all the system kinematic nonlinearities. This modification is accomplished by considering secondary nonlinear displacements which are ignored in all current codes. The resulting set of DAE are solved by the Gear stiff integrator. The study also introduced the concept of constrained flexibility and uses an instantaneous energy checking function to improve integration accuracy in the numerical scheme. The general energy balance is a single scalar equation containing all the energy component contributions. The DAE solution is then compared with the solution predicted by the nonlinear vibration theory. It also establishes new foundation for the use of multibody dynamics codes in nonlinear vibration problems. It is found that the simulation CPU time is much longer than the simulation of the original equations of the system.