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The dynamics of cloud cavitation about a three-dimensional hydrofoil are investigated experimentally in a cavitation tunnel with depleted, sparse and abundant free-stream nuclei populations. A rectangular planform, NACA 0015 hydrofoil was tested at a Reynolds number of $1.4\\times 10^{6}$, an incidence of $6^{\\circ }$ and a range of cavitation numbers from single-phase flow to supercavitation. High-speed photographs of cavitation shedding phenomena were acquired simultaneously with unsteady force measurement to enable identification of cavity shedding modes corresponding to force spectral peaks. The shedding modes were analysed through spectral decomposition of the high-speed movies, revealing different shedding instabilities according to the nuclei content. With no active nuclei, the fundamental shedding mode occurs at a Strouhal number of 0.28 and is defined by large-scale re-entrant jet formation during the growth phase, but shockwave propagation for the collapse phase of the cycle. Harmonic and subharmonic modes also occur due to local tip shedding. For the abundant case, the fundamental shedding is again large-scale but with a much slower growth phase resulting in a frequency of $St=0.15$. A harmonic mode forms in this case due to the propagation of two shockwaves; an initial slow propagating wave followed by a second faster wave. The passage of the first wave causes partial condensation leading to lower void fraction and consequent increase in the speed of the second wave along with larger-scale condensation. For a sparsely seeded flow, coherent fluctuations are reduced due to intermittent, disperse nuclei activation and cavity breakup resulting in an optimal condition with maximum reduction in unsteady lift.

The stability of steady convective exchange flow with a rectangular planform in an unbounded three-dimensional porous medium is explored. The base flow comprises a balance between vertical advection with amplitude $A$ in interleaving rectangular columns with aspect ratio $\\unicode[STIX]{x1D709}\\leqslant 1$ and horizontal diffusion between the columns. Columnar flow with a square planform ( $\\unicode[STIX]{x1D709}=1$ ) is found to be weakly unstable to a large-scale perturbation of the background temperature gradient, irrespective of $A$ , but to have no stronger instability on the scale of the columns. This result provides a stark contrast to two-dimensional columnar flow (Hewitt et al., J. Fluid Mech., vol. 737, 2013, pp. 205–231), which, as $A$ is increased, is increasingly unstable to a perturbation on the scale of the columnar wavelength. For rectangular planforms with $\\unicode[STIX]{x1D709}<1$ , a critical aspect ratio is identified, below which a perturbation on the scale of the columns is the fastest growing mode, as in two dimensions. Scalings for the growth rate and the structure of this mode are identified, and are explained by means of an asymptotic expansion in the limit $\\unicode[STIX]{x1D709}\\rightarrow 0$ . The difference between the stabilities of two-dimensional and three-dimensional exchange flow provides a potential explanation for the apparent difference in dominant horizontal scale observed in direct numerical simulations of two-dimensional and three-dimensional statistically steady ‘Rayleigh–Darcy’ convection at high Rayleigh numbers.

Here, we present a Navier close form solution method for some type of the higher-order theories for elastic shells of revolution developed using the CUF approach. The higher-order models of elastic shells of revolution are developed using the variational principle of virtual power for 3-D equations of the linear theory of elasticity and generalized series in the coordinates of the shell thickness. The higher-order cylindrical supported on the edges and axisymmetric shells, as well as the shallow spherical shells with rectangular planform, are considered. Numerical calculations were performed using the computer algebra software Mathematica. The resulting equations can be used for theoretical analysis and calculation of the stress–strain state, as well as for modeling thin-walled structures used in science, engineering, and technology. The numerical results can be used as benchmark examples for finite element analysis of the higher-order elastic shells.

Free vibrations of shallow sandwich shells resting on elastic foundations are investigated. It is assumed that the shell consists of three layers of defined thickness. The core is made of ceramics or metal, while the upper and lower layers are made of functionally graded material (FGM). The volume fractions of metal and ceramics are described by the power law. Hence, estimation methods for higher accuracy remain a challenge. The elastic foundation is described by two-parameter Pasternak’s model. Both higher-order shear deformation shell theory (HSDT), which includes interactions with elastic foundation, and the R-functions theory combined with the variational Ritz method are used to study shells with arbitrary planforms. The numerical study is carried out in the framework of the refined third-order theory. The proposed method and developed numerical techniques have been validated on test problems for FG shell with rectangular planform. Furthermore, new results for shallow shells with a cut-out of the complex form are obtained. The effects of the gradient index, boundary conditions, thickness of core and face sheet layers, as well as elastic foundations on fundamental frequencies, are investigated. In addition, we demonstrate how the divergence of the results varies with the gradient index. The considered studied cases show that the natural frequencies depend on the foundation parameters, the thickness of the layers, boundary conditions, and the thickness arrangement.

This paper investigates an improvement of the aerodynamic performance of a wing at high, including post-stall angles of attack by re-designing its camber line to control the separation of its boundary layer. This is experimentally implemented using an Aluminum secondary skin on the wing surface, which aligns itself to the separated boundary layer at high angles to attack, such that the flow remains attached to it, which otherwise would have separated on the baseline configuration. The shape of the skin, which is now regarded as the active flow surface, is essentially a morphed version of the baseline shape of the wing and is predicted numerically using an in-house code based on a ‘decambering’ technique that accounts for the local deviation of camber by accounting for the difference in the coefficients of lift and pitching moment predicted by viscous and potential flows. This technique is tested on a rectangular planform using different wing sections, NACA0012, NACA4415, and NRELS809. The effective morphed flow surface is also used for the baseline wing to operate at a design local 2D Cl, which is obtained by incrementing the baseline Cl by a user defined percentage at design pre and post-stall angles of attack. Aerodynamic characteristics of the effective morphed configurations using numerical analysis, CFD, and wind tunnel experiments are reported.

Based on a boundary layer theory of shell buckling, the semi-analytical solutions for nonlinear stability analysis of anisotropic laminated composite doubly-curved shells with rectangular planform subjected to lateral pressure are derived. A new shell model of arbitrary constant curvature and fibre stacking sequences but constant thickness is developed. The governing equations are based on an extended higher-order shear deformation shell theory with von Kármán-type of kinematic nonlinearity and including the effect on stiffness couplings. The nonlinear deformation and initial deflection of shells are both taken into account. The boundary layer equations of buckling for doubly-curved shells are introduced to match the asymptotic solutions satisfying the clamped or simply-supported boundary condition. The closed-form solutions for buckling and postbuckling analysis of an anisotropic shear deformable laminated doubly-curved panel are obtained by the two-step perturbation methods and the boundary layer theory for shell buckling, which is employed to determine interactive buckling loads and postbuckling equilibrium paths. At the same time, the internal quantitative relationship in the asymptotic sense between deflection and rotations of the normal to the middle surface is for the first time obtained. The influences of anisotropic lay-up, change in the stacking sequence, temperature variation, different types of elastic foundation and boundary condition on nonlinear stability behaviour are analysed and discussed. The study provides a good theoretical method for the load-carrying capacity design of composite shell structures.

The paper discusses the process of non-linear deformation of shell structures made of reinforced concrete. A mathematical model of deformation in the form of the functional of full potential deformation energy is provided. The model is based on the Kirchhoff–Love hypotheses, and allows accounting for structure reinforcement with stiffeners. An orthogonal network of stiffeners, located from the concave side, is considered as the structure support. Type of load — external, uniformly distributed. The Ritz method is applied to the functional to reduce the variational problem of the functional minimum to a system of nonlinear algebraic equations. Then, for each load value, the problem is solved using iterative methods. Analysis of strength and stability of shallow shells of double curvature and rectangular planform is performed. Values of critical loads, deflection and stress fields are obtained. Curves of deflection depending on load are provided. All results are given in dimensionless parameters. The Mohr–Coulomb criterion was used to analyze concrete strength, and the Lyapunov criterion was used for stability analysis. Influence of the number of stiffeners reinforcing the shell on the resulting stress values is shown. It has been revealed that with account for physical non-linearity of concrete, when the dependence of stresses and deformations is curvilinear, deformations (and deflections as well) of shells increase in comparison with the linear-elastic solution. It has been also found that when nonlinearity is taken into account, redistribution of stresses over the shell field occurs (the maximum stresses shift towards the shell contour).

This paper studies steady waves in fluid and in semi-infinite ice cover generated by a constant pressure distribution with a rectangular planform moving uniformly along the edge of ice cover at fixed distance. This load simulates the air-cushion vehicle (ACV). We consider two cases: (i) the surface of fluid is free outside of ice sheet, (ii) fluid is bounded by a solid vertical wall and the edge of ice cover can be either clamped or free. The fluid is assumed to be ideal incompressible and of finite depth. The ice sheet is modelled by elastic thin plate. The solution of linear hydroelastic problem is obtained by two methods: the Wiener-Hopf technique and matched eigenfunction expansions. The deflection of ice sheet and free surface elevation, as well as wave forces acting on ACV are investigated for different speeds of motion.

A flexible flapping wing with a rectangular planform was designed to investigate the influence of flexible deformation. This planform is more convenient and easier to define and analyzed its deforming properties in the direction of spanwise and chordwise. The flapping wings were created from carbon fiber skeleton and polyester membrane with similar size to medium birds. Their flexibility of deformations was tested using a pair of high-speed cameras, and the 3D deformations were reconstructed using the digital image correlation technology. To obtain the relationship between the flexible deformation and aerodynamic forces, a force/torque sensor with 6 components was used to test the corresponding aerodynamic forces. Experimental results indicated that the flexible deformations demonstrate apparent cyclic features, in accordance with the flapping cyclic movements. The deformations in spanwise and chordwise are coupled together; a change of chordwise rib stiffness can cause more change in spanwise deformation. A certain lag in phase was observed between the deformation and the flapping movements. This was because the deformation was caused by both the aerodynamic force and the inertial force. The stiffness had a significant effect on the deformation, which in turn, affected the aerodynamic and power characteristics. In the scope of this study, the wing with medium stiffness consumed the least power. The purpose of this research is to explore some fundamental characteristics, as well as the experimental setup is described in detail, which is helpful to understand the basic aerodynamic characteristics of flapping wings. The results of this study can provide an inspiration to further understand and design flapping-wing micro air vehicles with better performance.

The implementation of the B-spline Wavelet on the Interval (BSWI) for curved shell elements with rectangular planform is presented in this paper. By aid of the general shell theory, cylinder shells, doubly-curved shallow shells and hyperbolic paraboloidal shells BSWI elements are formulated. Instead of traditional polynomial interpolation, scaling functions at certain scale have been adopted to form the shape functions and construct wavelet-based elements. Because of the good character of BSWI scaling functions, the BSWI curved shell elements combine the accuracy of wavelet-based elements approximation and the character of B-spline functions for structural analysis. Different from the flat shell elements, the curved shell elements obtain a better geometrical fitting property in idealizing the practical curved structures. This paper focuses on the dynamic analysis of shell. The study covers wide combinations of boundaries such as cantilever, simply supported and clamped boundary. Numerical results have been established to validate the efficiency and accuracy of the presented elements through comparison with published data from the open literature and some commercial finite element method software.