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Curves with various profiles have been demonstrated to be more attractive and decorative than the straight lines by William Hogarth. Among all kinds of curves, Hogarth proposed seven serpentine lines as the most beautiful curves, i.e., Hogarth curves. Those seven Hogarth curves are subsequently applied in a wide range of fields, e.g., sculpture, painting, architecture and fashion design, indicating their significance to the development of the formal beauty. Recently, the beauty of Hogarth curves has been suspected to be induced by their special-designed curvature, which could have the potential relationship with the Golden Ratio. The purpose of this paper is to explore the relationship between the Hogarth curves and golden ratio by comparing the curvature of curves with the Fibonacci sequence.

Each of the Hogarth curves was fully restored and divided into two parts according to the turning point of the curvature; the ratios of span, curvature and angles between these two parts were compared with the Fibonacci sequence.

The experimental results disclosed that the ratio of the fourth Hogarth curve, which was considered as the most beautiful line by Hogarth, was infinitely approaching the golden ratio. Based on the relationship between the fourth Hogarth curve and the golden ratio, the ratios of each curve were employed to define and normalize these curves, providing a quantitative way to redraw the Hogarth curves.

This research work unlocked the information of the relationship between the Hogarth curves and golden ratio, and proposed an effective and convenient mathematic way to quantify the Hogarth curves. The experimental findings disclosed the underlying mechanisms of the beauty of the forth Hogarth curves. Such a fundamental study will promote the establishment of the normalized methods for evaluating the beauty of arts and provide novel ideas for researchers and industrial technologists to use the Hogarth curves.

The purpose of this paper is to develop a general mathematical model for laminated curved structure of different geometries using higher-order shear deformation theory to evaluate in-plane and out of plane shear stress and strains correctly. Subsequently, the model has to be validated by comparing the responses with developed simulation model (ANSYS) as well as available published literature. It is also proposed to analyse thermal buckling load parameter of laminated structures using Green–Lagrange type non-linear strains for excess thermal distortion under uniform temperature loading.

Laminated structures known for their flexibility as compared to conventional material and the deformation behaviour are greatly affected due to combined thermal/aerodynamic environment. The vibration/buckling behaviour of shell structures are very different than that of the plate structures due to their curvature effect. To model the exact behaviour of laminated structures mathematically, a general mathematical model is developed for laminated shell geometries. The responses are evaluated numerically using a finite element model-based computer code developed in MATLAB environment. Subsequently, a simulation model has been developed in ANSYS using ANSYS parametric design language code to evaluate the responses.

Vibration and thermal buckling responses of laminated composite curved panels have been obtained based on proposed model through a customised computer code in MATLAB environment and ANSYS simulation model using ANSYS parametric design language code. The convergence behaviour are tested and compared with those available in published literature and ANSYS results. Finally, the investigation has been extended to examine the effect of different parameters (thickness ratios, curvature ratios, modular ratios, number of layers and support conditions) on the free vibration and thermal buckling responses of laminated curved structures.

The present paper intends to give sufficient amount of numerical experimentation, which may lead to help in designing of finished product made up of laminated composites. Most of the aerospace, space research and defence organisation intend to develop low cost and high durable products for real hazard conditions by taking combined loading and environmental conditions. Further, case studies might lead to a lighter design of the laminated composite panels used in high-performance systems, where the weight reduction is the major parameter, such as aerospace, space craft and missile structures.

In this analysis, the geometrical distortion due to temperature is being introduced through Green–Lagrange sense in the framework of higher-order shear deformation theory for different types of laminated shells (cylindrical/spherical/hyperboloid/elliptical). A simulation-based model is developed using ANSYS parametric design language in ANSYS environment for different geometries and loading condition and compared with the numerical model.

Numerical simulations are performed to determine the heat transfer characteristics of slot jet impingement of air on a concave surface. The purpose of this paper is to investigate the effect of protrusions on the heat transfer by placing semi-circular protrusions on the concave surface at several positions. After identifying appropriate locations where the heat transfer is a maximum, multiple protrusions are placed at desired locations on the plate. The gap ratio, curvature ratio (d/D) and the dimensions of the plate are varied so as to obtain heat transfer data. The curvature ratio is varied first, keeping the concave diameter (D) fixed followed by a fixed slot width (d). A surrogate model based on an artificial neural network is developed to determine optimum locations of the protrusions that maximize the heat transfer from the concave surface.

The scope and objectives of the present study are two-dimensional numerical simulations of the problem by considering all the geometrical parameters (H/d, d_p, Re, θ) affecting heat transfer characteristics with the help of networking tool and numerical simulation. Development of a surrogate forward model with artificial neural networks (ANNs) with a view to explore the full parametric space. To quantitatively ascertain if protrusions hurt or help heat transfer for an impinging jet on a concave surface. Determination of the location of protrusions where higher heat transfer could be achieved by using exhaustive search with the surrogate model to replace the time consuming forward model.

A single protrusion has nearly no effect on the heat transfer. For a fixed diameter of concave surface, a smaller jet possesses high turbulence kinetic energy with greater heat transfer. ANN is a powerful tool to not only predict impingement heat transfer characteristics by considering multiple parameters but also to determine the optimum configuration from many thousands of candidate solutions. A maximum increase of 8 per cent in the heat transfer is obtained by the best configuration constituting of multiple protrusions, with respect to the baseline smooth configuration. Even this can be considered as marginal and so it can be concluded that first cut results for heat transfer for an impinging jet on a concave surface with protrusions can be obtained by geometrically modeling a much simpler plain concave surface without any significant loss of accuracy.

The heat transfer during impingement cooling depends on various geometrical parameters but, not all the pertinent parameters have been varied comprehensively in previous studies. It is known that a rough surface may improve or degrade the amount of heat transfer depending on their geometrical dimensions of the target and the rough geometry and the flow conditions. Furthermore, to the best of authors’ knowledge, scarce studies are available with inclusion of protrusions over a concave surface. The present study is devoted to development of a surrogate forward model with ANNs with a view to explore the full parametric space.

Although mechanical behavior of rigid frame pier has been clearly recognized, their time-varying seismic performance are yet to be well characterized due to some offshore piers that are eroded by chloride ion and located in earthquake-prone area. In this study, the time-variant seismic fragility analysis was conducted to evaluate seismic performance of rigid frame pier under four damage states with considering the time-varying characteristics of the material.

This paper establishes the nonlinear finite element model for the investigated offshore reinforcement concrete (RC) pier with considering the time-varying durability damage of the materials and defines the damage state, damage position and damaged index of the offshore RC pier. It also analyzes the time-varying seismic fragility of the offshore RC pier by using the capacity demand ratio method in the whole life cycle.

The results show that chloride induced corrosion has a significant effect on the rigid frame pier and bending capacity of top section is less than that of bottom section. The rate of decline accelerates after the service life reaching 30 years under the coupling of the earthquake and the environmental erosion. In the early years of service, the seismic fragility of the structure changed slowly.

This paper analyzes the influencing factors of seismic performance of rigid structure pier, and analyzes the seismic capacity and seismic performance of rigid structure pier under different service periods.

The purpose of this paper is to develop a general mathematical model for the evaluation of the theoretical flexural responses of the functionally graded carbon nanotube-reinforced composite doubly curved shell panel using higher-order shear deformation theory with thermal load. It is well-known that functionally graded materials are a multidimensional problem, and the present numerical model is also capable of solving the flexural behaviour of different shell panel made up of carbon nanotube-reinforced composite with adequate accuracy in the absence of experimentation.

In this current paper, the responses of the single-walled carbon nanotube-reinforced composite panel is computed numerically using the proposed generalised higher-order mathematical model through a homemade computer code developed in MATLAB. The desired flexural responses are computed numerically using the variational method.

The validity and the convergence behaviour of the present higher-order model indicate the necessity for the analysis of multidimensional structure under the combined loading condition. The effect of various design parameters on the flexural behaviour of functionally graded carbon nanotube doubly curved shell panel are examined to highlight the applicability of the presently proposed higher-order model under thermal environment.

In this paper, for the first time, the static behaviour of functionally graded carbon nanotube-reinforced composite doubly curved shell panel is analysed using higher-order shear deformation theory. The properties of carbon nanotube and the matrix material are considered to be temperature dependent. The present model is so general that it is capable of solving various geometries from single curve to doubly curved panel, including the flat panel.

The heat transfer within a heat exchanger is highly influenced by geometry of the components especially those with hollow structures like tubes. This paper aims to intend toward the study of efficient and optimized heat transfer in the bends of superheater tubes, with different curvature ratio at constant Reynolds Number.

The effect of changing curvature ratio on enthalpy of the fluid passing through the superheater tubes for multi-pass system has been studied with the aid of computational fluid dynamics (CFD) using ANSYS 14.0. Initially a superheater tube with two pass system has been examined with different curvature ratios of 1.425, 1.56, 1.71, 1.85 and 1.99. An industry specified curvature ratio of 1.71 with two pass is investigated, and a comparative assessment has been carried out. This is intended toward obtaining an optimized radius of curvature of the bend for enhancement of heat transfer.

The results obtained from software simulation revealed that the curvature ratio of 1.85 provides maximum heat transfer to the fluid flowing through the tube with two pass. This result has been found to be consistent with higher number of passes as well. The effect of secondary flow in bends of curvature has also been illustrated in the present work.

The study of heat transfer in thermodynamic systems is a never-ending process and has to be continued for the upliftment of power plant performances. This study has been conducted on steady flow behavior of the fluid which may be upgraded by carrying out the same in transient mode. The impact of different curvature ratios on some important parameters such as heat transfer coefficients will certainly upgrade the value of research.

This computational study provided comprehensive information on fluid flow behavior and its effect on heat transfer in bends of curvature of superheater tubes inside the boiler. It also provides information on optimized bend of curvature for efficient heat transfer process.

The purpose of this paper is to study the effect of curvature on the magnetic field distribution and no-load rotor eddy current losses in electric machines, particularly in high-speed permanent magnet (PM) machines.

The magnetic field distribution is obtained using conformal mapping, and the eddy current losses are obtained using a cylindrical multilayer model. The analytical results are validated using a two-dimensional finite element analysis. The analytical method is based on a proportional-logarithmic conformal transformation that maps the cylindrical geometry of a rotating electric machine into a rectangular configuration without modifying the length scale. In addition, the appropriate transformation of PM cylindrical domains into the rectangular domain is deduced. Based on this conformal transformation, a coefficient to quantify the effect of curvature is proposed.

Neglecting the effect of curvature can produce significant errors in the calculation of no-load rotor losses when the ratio between the air-gap length and the rotor diameter is large.

The appropriate transformation of PM cylindrical domains into the rectangular domain is deduced. The proportional-logarithmic transformation proposed provides an insight into the effect of curvature on the magnetic field distribution in the air-gap and no-load rotor losses. Furthermore, the proposed curvature coefficient gives a notion of the effect of curvature for any particular geometry without the necessity of any complicated calculation. The case study shows that neglecting the effect of curvature underestimates the rotor eddy-current losses significantly in machines with large gap-to-rotor diameter ratios.

The steady laminar quasi-3D fully developed magnetohydrodynamic (MHD) flow of a liquid metal in a curved annular channel is studied in order to determine the effect of the magnetic field on the velocity distribution. The paper aims to discuss this issue.

Due to the fluid motion under the effect of the applied transverse external magnetic field, an additional magnetic field and an electric current density are induced. A hybrid formulation is used for the induced electric current density, implementing for its axial component the Ohm’s law and for its transverse components the Ampere’s law. The suggested formulation (denominated h-formulation) is combined with the extended Continuity Vorticity Pressure numerical variational method for MHD flows.

Results are obtained for different values of curvature ratios and Hartmann numbers. It is proved that as the strength of the magnetic field increases, two side regions of velocity jets are formulated in the right and in the left half sides of the inner cylinder in a direction parallel to the external magnetic field. The magnitude of the axial velocity at each region is determined by the balance of the centrifugal and the electromagnetic forces. The results help to better understand the MHD flow in toroidal ducts.

The results aim to help to a better understanding of the MHD flow in curved ducts.

In this paper, XIANGRONG XI and ISAO AJIKI define an actual overfeed ratio depending on the recovered length of an overfed seam. In particular, the effect of the actual overfeed ratio on the bending properties of the overfed seam was measured and the effect of fabric recovery on the fabric overfeed ratio was investigated. The results show that, for all samples, the natural curvature of an overfed seam increases proportionally as the actual overfeed ratio increase, which is different from the situation with the overfeed ratio. The two definitions of actual overfeed ratio and overfeed ratio can be connected by factor η, which is affected by stitch length, and by the recovery and compressive properties of fabric. The results also show that the factor provided the same varying tendency as the fabric recovery ratio, and that the differences between the factor and the recovery ratio depended on the kind of fabric involved.

To investigate the effect of aspect ratio on the quantitative analogy between developing laminar flows in orthogonally rotating straight ducts and stationary curved ducts

A fractional step method is used to obtain the numerical solution of the governing equations by decoupling the solution of the momentum equations from the solution of the continuity equation. In order to clarify the similarity of the two flows, dimensionless parameters K_LR and Rossby number, Ro, in a rotating straight duct were used as a set corresponding to Dean number, K_LC, and curvature ratio, λ, in a stationary curved duct.

Under the condition that the aspect ratio was larger than one and that the magnitude of Ro or λ was large enough to satisfy the “asymptotic invariance property” the quantitative analogy between the two flows was established clearly.

As the aspect ratio decreased below one, the difference between the secondary flow intensities of these two flows increased, and therefore, the analogy between the two flows was not as evident as that for the larger aspect ratios.

Based on this methodology, the characteristics of the developing flow in orthogonally rotating ducts of higher aspect ratio can be predicted by considering the flow in stationary curved ducts, and vice versa.

The results obtained in this study will suggest an optimal criterion for the application of this approach to the flow similarity analysis in rectangular ducts with arbitrary aspect ratios.