Corner flow of a suspension of rigid rods

Stress singularities in the neighbourhood of sharp corners can be a source of severe problems in the numerical simulation of non-Newtonian flows leading to loss of convergence with grid refinement (G.G. Lipscombe, R. Kennings and M.M. Denn, J. Non-Newtonian Fluid Mech., 24 (1987) 85 [1]). For Newtonian flows the nature of this singularity is given by the analysis of Dean and Montagnon (W.R. Dean and P.E. Montagnon, Phil. Trans. R. Soc. London, Ser. A., 308 (1949) 199 [2]) in terms of similarity solutions. In this paper we extend this similarity analysis to a suspension of rigid rods. In the limit of nearly full extension the FENE constitutive model has the same behaviour as such a suspension. Our analysis predicts the possibility of lip vortices but their behaviour is somewhat inconsistent with those observed experimentally.     

用子波变换检测壁湍流信号局部标度指数

本文用子波变换检测了刻画壁湍流脉动信号自相似性的局部标度指数，研究了不同尺度的湍流结构的自相似性，发现在湍流边界层猝发过程中，喷射和扫掠发生时刻小尺度脉动速度信号的局部标度指数为负值，说明在大尺度猝发事件发生的时刻小尺度结构具有奇异的自相似性，在猝发过程中其作用不仅仅是对湍能的耗散．     

Effects of the Similarity Model in Finite-Difference LES of Isotropic Turbulence Using a Lagrangian Dynamic Mixed Model

A Lagrangian dynamic formulation of the mixed similarity subgrid (SGS) model for large-eddy simulation (LES) of turbulence is proposed. In this model, averaging is performed over fluid trajectories, which makes the model applicable to complex flows without directions of statistical homogeneity. An alternative version based on a Taylor series expansion (nonlinear mixed model) is also examined. The Lagrangian models are implemented in a finite difference code and tested in forced and decaying isotropic turbulence. As comparison, the dynamic Smagorinsky model and volume-averaged formulations of the mixed models are also tested. Good results are obtained, except in the case of low-resolution LES (32³) of decaying turbulence, where the similarity coefficient becomes negative due to the fact that the test-filter scale exceeds the integral scale of turbulence. At a higher resolution (64³), the dynamic similarity coefficient is positive and good agreement is found between predicted and measured kinetic energy evolution. Compared to the eddy viscosity term, the similarity or the nonlinear terms contribute significantly to both SGS dissipation of kinetic energy and SGS force. In order to dynamically test the accuracy of the modeling, the error incurred in satisfying the Germano identity is evaluated. It is found that the dynamic Smagorinksy model generates a very large error, only 3% lower than the worst-case scenario without model. Addition of the similarity or nonlinear term decreases the error by up to about 50%, confirming that it represents a more realistic parameterization than the Smagorinsky model alone.     

Algebraic tensegrity form-finding

This paper concerns the form-finding problem for general and symmetric tensegrity structures with shape constraints. A number of different geometries are treated and several fundamental properties of tensegrity structures are identified that simplify the form-finding problem. The concept of a tensegrity invariance (similarity) transformation is defined and it is shown that tensegrity equilibrium is preserved under affine node position transformations. This result provides the basis for a new tensegrity form-finding tool. The generality of the problem formulation makes it suitable for the automated generation of the equations and their derivatives. State-of-the-art numerical algorithms are applied to solve several example problems. Examples are given for tensegrity plates, shell-class symmetric tensegrity structures and structures generated by applying similarity transformation.     

The functional similarity between faxén relations and singularity solutions for fluid-fluid, fluid-solid and solid-solid dispersions

The functional similarity between Faxén relations (for moments of the appropriate transport flux) and singularity solutions has been noted in the past. For rigid particles (and perfect conductors etc.) it has been noted in the literature that the root of this similarity is linked to the Lorentz reciprocal theorem. However, the duality applies even to more general two-phase problems such as a viscous drop in another solvent, with the relevant singularity distribution taken from the exterior solution. Although two-phase Faxén relations are available for various particle shapes, until now, the root of this duality has not been demonstrated explicitly. The application of the duality is illustrated by the derivation of new Faxén relations for ellipsoidal inclusions.     

A piecewise bearing capacity method of unstructured terrain considering characteristics of soil mechanic and wheel geometry

Bearing capacity of the unstructured terrain considering the effects of the wheel geometry and soil mechanic properties is analyzed in this paper. Two-dimensional pressure-sinkage simulations are conducted to evaluate the degrees of similarity between the flat plate and wheel in terms of their ultimate bearing characteristics of Terzaghi theory. The results show that these degrees of similarity are mainly reflected in the soil in-depth direction and the corresponding failure behaviors. Based on the approximation of the ultimate bearing capacity between the wheel and flat plate, a piecewise bearing capacity evaluation method with the effects of the soil mechanic properties and three-dimensional wheel geometry is proposed. The pressure-sinkage values of the proposed model show a satisfactory agreement with the experimental ones. The proposed model performs better than the semi-empirical models, as it considers more soil bearing features and needs less fitting parameters to assess the unstructured terrain.     

Painlevé integrability, similarity reductions, new soliton and soliton-like similarity solutions for the (2+1)-dimensional BKP equation

The (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation of B-type (BKP) is hereby investigated. New soliton solutions and soliton-like similarity solutions are constructed for the (2+1)-dimensional BKP equation. The similarity solutions are not travelling wave solutions when the arbitrary functions involved are chosen appropriately. Painlevé test shows that there are two solution branches, one of which has the resonance ?2. And four similarity reductions for the BKP equation are given out through nontrivial variable transformations. Moreover, abundant soliton behaviour modes of the solutions, such as soliton fusion and soliton reflection, are discussed in detail.     

Note on similarity solutions for viscous flow over an impermeable and non-linearly (quadratic) stretching sheet

Similarity solutions for flow over an impermeable, non-linearly (quadratic) stretching sheet were studied recently by Raptis and Perdikis (Int. J. Non-linear Mech. 41 (2006) 527-529) using a stream function of the form ψ=αxf(η)+βx²g(η). A fundamental error in their problem formulation is pointed out. On correction, it is shown that similarity solutions do not exist for this choice of ψ.     

Calibration of discrete element modeling: Scaling laws and dimensionless analysis

In this paper, dynamic similarity conditions are derived for discrete element simulations by non-dimensionalising the governing equations. These conditions must be satisfied so that the numerical model is a good representation of the physical phenomenon. For a pure mechanical system, if three independent ratios of the corresponding quantities between the two models are set, then the ratios of other quantities must be chosen according to the similarity principles. The scalability of linear and non-linear contact laws is also investigated. Numerical tests of 3D uni-axial compression are carried out to verify the theoretical results. Another example is presented to show how to calibrate the model according to laboratory data and similarity conditions. However, it is impossible to reduce computer time by scaling up or down certain parameters and continue to uphold the similarity conditions. The results in this paper provide guidelines to assist discrete element modelers in setting up the model parameters in a physically meaningful way.     

Particle-driven gravity currents: asymptotic and box model solutions

We employ shallow water analysis to model the flow of particle-driven gravity currents above a horizontal boundary. While there exist similarity solutions for the propagation of a homogeneous gravity current, in which the density difference between the current and ambient is constant, there are no such similarity solutions for particle-driven currents. However, because the settling velocity of the particles is often much less than the initial velocity of propagation of these currents, we can develop an asymptotic series to obtain the deviations from the similarity solutions for homogeneous currents which describe particle-driven currents. The asymptotic results render significant insight into the dynamics of these flows and their domain of validity is determined by comparison with numerical integration of the governing equations and also with experimental measurements. An often used simplification of the governing equations leads to `box' models wherein horizontal variations within the flow are neglected. We show how to derive these models rigorously by taking horizontal averages of the governing equations. The asymptotic series are then used to explain the origin of the scaling of these `box' models and to assess their accuracy.     

Experimental study of the Kolmogorov refined similarity variable

Recently Thoroddsen and Van Atta (1992, Phys. Fluids A4, 2592) showed that Kolmogorov's refined similarity hypothesis (1962, J. Fluid Mech. 13, 77; 82) is supported by experimental data from a wind-tunnel study of a cylinder wake, at Re_λ of 550. We show here that the probability density of the Kolmogorov similarity variable approaches a Gaussian distribution. The data also suggest that it may obey an even more demanding conditional similarity, which leads to important conclusions regarding the scaling exponents.     

Use of the generalized similarity method for calculating the inner zone of a turbulent boundary layer

The use of the generalized similarity method for calculating laminar boundary layers has been fully justified (see [1, §113, 114, 148]). The replacement of the partial differential equations by ordinary differential equations, their universality and the possibility of physically interpreting the solutions in the first, parametric stage of the calculations, which distinguish the generalized similarity method from direct numerical integration methods, are preserved in the case of a turbulent boundary layer also. A comparison of the calculated and experimental velocity profiles in the inner zone of the turbulent boundary layer suggests that the generalized similarity method could be used for calculating the turbulent layer as a whole.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 25–34, September–October, 1990.     

Boundary turbulence in weak polymer solutions

The construction of a semiempirical method for determining boundary turbulence close to a wall (based on the theory of similarity, dimensional analysis, and certain hypotheses regarding mixing paths) is considered; also considered is the generalization of this method to the case of the turbulent flow of weak polymer solutions in the neighborhood of both smooth and rough surfaces.     

Comments on “Effects of temperature dependent fluid properties and variable Prandtl number on the transient convective flow due to a porous rotating disk by M. S. Alam,S. M. Chapal Hossain,M. M. Rahman [Meccanica (2014) 49:2439–2451]”

In this letter, we submit our comments on recently published paper titled “Effects of temperature dependent fluid properties and variable Prandtl number on the transient convective flow due to a porous rotating disk by Alam et al. (Meccanica 49: 2439–2451, 2014)”. Authors of this paper have attempted to present similarity solutions in the paper. We comment in this letter is that the similarity transformations considered in Alam et al. (Meccanica 49: 2439–2451, 2014) are not correct and thus results are leading to invalid conclusions.     

Similarity research of anomalous dynamic response of ship girder subjected to near field underwater explosion

The final anomalous sag distortion of the ship girder subjected to the near field underwater explosion (undex) below the middle ship is studied. The sinking exercise of Spruance class destroyer DD973 sunk by one MK48 torpedo is first presented, and a simulation model is established. The exponential attenuation phase, the reciprocal attenuation phase, the post reciprocal attenuation phase, and the negative pressure phase of the undex load are precisely applied in this model. The fluid-solid interaction, the added water mass, the gravity, and the change of buoyancy are also taken into account. The similarity theory is then used to analyze the dynamic response of the ship girder. Similarity parameters and theory prediction formulae of the dynamic response of the ship girder are presented. The physical meaning and influences of these similarity parameters are analyzed.     

Predicting the behaviour of structures under impact loads using geometrically distorted scaled models

When a scaled structure (model or replica) is used to predict the response of a full-size compound (prototype), the model geometric dimensions should relate to the corresponding prototype dimensions by a single scaling factor. However, owing to manufacturing technical restrictions, this condition cannot be accomplished for some of the dimensions in real structures. Accordingly, the distorted geometry will not comply with the overall geometric scaling factor, infringing the Π theorem requirements for complete dynamic similarity. In the present study, a method which takes geometrical distortions into account is introduced, leading to a model similar to the prototype. As a means to infer the performance of this method, three analytical problems of structures subjected to dynamic loads are analysed. It is shown that the replica developed applying this technique is able to accurately predict the full-size structure behaviour even when the studied models have some of their dimensions severely distorted.     

Similarity solutions of Prandtl mixing length modelled two dimensional turbulent boundary layer equations

The exact similarity solutions of two dimensional laminar boundary layer were obtained by Blasius in 1908, however, for two dimensional turbulent boundary layers, no Blasius type similarity solutions (special exact solutions) have ever been found. In the light of Blasius’ pioneer works, we extend Blasius similarity transformation to the two dimensional turbulent boundary layers, and for a special case of flow modelled by Prandtl mixing-length, we successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation. The ordinary differential equation is numerically solved and some useful quantities are produced. For numerical calculations, a complete Maple code is provided.