Governing equations and numerical solutions of tension leg platform with finite amplitude motion

Governing equations for finite amplitude wave propagation in stretched hyperelastic strings are given in recent papers, (Beatty and Haddow, 1985), along with similarity solutions for symmetrically plucked and impacted strings. The similarity solutions are valid until the first reflections at the fixed ends and in this paper we consider symmetrically plucked Mooney-Rivlin strings and investigate the response after reflections. The method of characteristics is applied to extend the results of the similarity solutions and to obtain solutions for the interaction of a reflected longitudinal shock and incident transverse shock and the reflection of an incident transverse shock. A deformed shape, which is not intuitively obvious, is predicted by the solution of the interaction problem and is confirmed by an experimental study. A finite difference scheme is used to obtain numerical solutions, which are valid after multiple wave interactions and reflections occur. Solutions obtained by the method of characteristics are used as a partial check on the numerical results.

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Thermal Post-Buckling Analysis of Functionally Graded Composite Plates with Nonlinear Aerodynamic Forces

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.123-125.280 ◽

2010 ◽

Vol 123-125 ◽

pp. 280-283

Author(s):

Chang Yull Lee ◽

Ji Hwan Kim

Keyword(s):

Material Properties ◽

Numerical Solutions ◽

Virtual Work ◽

Shear Deformation Theory ◽

Composite Plates ◽

Buckling Analysis ◽

Functionally Graded ◽

Governing Equations ◽

Functionally Graded Composite ◽

Post Buckling

The post-buckling of the functionally graded composite plate under thermal environment with aerodynamic loading is studied. The structural model has three layers with ceramic, FGM and metal, respectively. The outer layers of the sandwich plate are different homogeneous and isotropic material properties for ceramic and metal. Whereas the core is FGM layer, material properties vary continuously from one interface to the other in the thickness direction according to a simple power law distribution in terms of the volume fractions. Governing equations are derived by using the principle of virtual work and numerical solutions are solved through a finite element method. The first-order shear deformation theory and von-Karman strain-displacement relations are based to derive governing equations of the plate. Aerodynamic effects are dealt by adopting nonlinear third-order piston theory for structural and aerodynamic nonlinearity. The Newton-Raphson iterative method applied for solving the nonlinear equations of the thermal post-buckling analysis

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A more general model equation of nonlinear Rayleigh waves and their quasilinear solutions

Modern Physics Letters B ◽

10.1142/s0217984916500962 ◽

2016 ◽

Vol 30 (08) ◽

pp. 1650096 ◽

Cited By ~ 4

Author(s):

Shuzeng Zhang ◽

Xiongbing Li ◽

Hyunjo Jeong

Keyword(s):

Rayleigh Waves ◽

General Equation ◽

Wave Motion ◽

Numerical Solutions ◽

Second Harmonic ◽

Approximate Equation ◽

Finite Amplitude ◽

Sound Beam ◽

Quasilinear Theory ◽

Sound Beams

A more general two-dimensional wave motion equation with consideration of attenuation and nonlinearity is proposed to describe propagating nonlinear Rayleigh waves of finite amplitude. Based on the quasilinear theory, the numerical solutions for the sound beams of fundamental and second harmonic waves are constructed with Green’s function method. Compared with solutions from the parabolic approximate equation, results from the general equation have more accuracy in both the near distance of the propagation direction and the far distance of the transverse direction, as quasiplane waves are used and non-paraxial Green’s functions are obtained. It is more effective to obtain the nonlinear Rayleigh sound beam distributions accurately with the proposed general equation and solutions. Brief consideration is given to the measurement of nonlinear parameter using nonlinear Rayleigh waves.

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Numerical simulation of two-dimensional compressible convection

Journal of Fluid Mechanics ◽

10.1017/s0022112075002297 ◽

1975 ◽

Vol 70 (4) ◽

pp. 689-703 ◽

Cited By ~ 79

Author(s):

Eric Graham

Keyword(s):

Rayleigh Number ◽

Numerical Solutions ◽

Fluid Velocity ◽

Critical Rayleigh Number ◽

Finite Amplitude ◽

Two Dimensions ◽

Onset Of Convection ◽

Local Sound ◽

Constant Viscosity ◽

Convection Problem

A procedure for obtaining numerical solutions to the equations describing thermal convection in a compressible fluid is outlined. The method is applied to the case of a perfect gas with constant viscosity and thermal conductivity. The fluid is considered to be confined in a rectangular region by fixed slippery boundaries and motions are restricted to two dimensions. The upper and lower boundaries are maintained at fixed temperatures and the side boundaries are thermally insulating. The resulting convection problem can be characterized by six dimension-less parameters. The onset of convection has been studied both by obtaining solutions to the nonlinear equations in the neighbourhood of the critical Rayleigh number Rc and by solving the linear stability problem. Solutions have been obtained for values of the Rayleigh number up to 100Rc and for pressure variations of a factor of 300 within the fluid. In some cases the fluid velocity is comparable to the local sound speed. The Nusselt number increases with decreasing Prandtl number for moderate values of the depth parameter. Steady finite amplitude solutions have been found in all the cases considered. As the horizontal dimension A of the rectangle is increased, the length of time needed to reach a steady state also increases. For large values of A the solution consists of a number of rolls. Even for small values of A, no solutions have been found where one roll is vertically above another.

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Heat Transfer in a Laminar Channel Flow Generated by Injection Through Porous Walls

Journal of Fluids Engineering ◽

10.1115/1.2746908 ◽

2007 ◽

Vol 129 (8) ◽

pp. 1048-1057 ◽

Cited By ~ 4

Author(s):

Clarisse Fournier ◽

Marc Michard ◽

Françoise Bataille

Keyword(s):

Channel Flow ◽

Numerical Solutions ◽

Scaling Law ◽

Similarity Solutions ◽

Temperature Profiles ◽

Governing Equations ◽

Temperature Boundary ◽

Laminar Channel Flow ◽

Porous Walls ◽

Uniform Fluid

Steady state similarity solutions are computed to determine the temperature profiles in a laminar channel flow driven by uniform fluid injection at one or two porous walls. The temperature boundary conditions are non-symmetric. The numerical solution of the governing equations permit to analyze the influence of the governing parameters, the Reynolds and Péclet numbers. For both geometries, we deduce a scaling law for the boundary layer thickness as a function of the Péclet number. We also compare the numerical solutions with asymptotic expansions in the limit of large Péclet numbers. Finally, for non-symmetric injection, we derive from the computed temperature profile a relationship between the Nusselt and Péclet numbers.

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Spray Cloud Formation Over Marine Vessels by a Water Breakup Model

Volume 7A: Ocean Engineering ◽

10.1115/omae2017-62193 ◽

2017 ◽

Author(s):

Saeed R. Dehghani ◽

Greg F. Naterer ◽

Yuri S. Muzychka

Keyword(s):

Numerical Solutions ◽

Reynolds Numbers ◽

Cloud Formation ◽

Sea Spray ◽

Governing Equations ◽

Fishing Vessel ◽

Wave Impact ◽

Droplet Sizes ◽

Marine Vessels ◽

Breakup Model

Water breakup affects the variety of droplet sizes and velocities in a cloud of spray resulting from a sea wave striking a vessel bow. The Weber and Reynolds numbers of droplets are the main parameters for water breakup phenomena. “Stripping breakup” is a faster phenomenon than “bag breakup” and occurs at higher velocities and with larger diameters of droplets. A water breakup model employs droplet trajectories to develop a predictive model for the extent of spray cloud. The governing equations of breakup and trajectories of droplets are solved numerically. Stripping breakup is found as the major phenomenon in the process of the formation of wave-impact sea spray. Bag breakup acts as a complementary phenomenon to the stripping breakup. The extent of the spray as well as wet heights, for a Mediumsized Fishing Vessel (MFV), are obtained by numerical solutions. The results show that bag breakup occurs at higher heights. In addition, there is no breakup when droplets move over the deck.

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An Elastic-Plastic Stress Analysis of the Specimen Used in the Torsional Kolsky Bar

Journal of Applied Mechanics ◽

10.1115/1.3153655 ◽

1980 ◽

Vol 47 (2) ◽

pp. 278-282 ◽

Cited By ~ 10

Author(s):

Eric K. C. Leung

Keyword(s):

Numerical Solutions ◽

Elastic Stress ◽

Dynamic Testing ◽

Kolsky Bar ◽

The Finite Element Method ◽

Governing Equations ◽

Elastic Plastic ◽

Applied Torque ◽

Torsional Kolsky Bar ◽

Tubular Specimens

This paper examines the stress concentration, the yielding process, and the growth of the elastic-plastic boundary as a function of applied torque in tubular specimens with a short thin-walled section. Although the analysis is entirely quasi-static, it can, under the proper circumstances, be applied to the deformation of short specimens as generally used for dynamic testing in the torsional Kolsky bar. In the analysis, the governing equations for both elastic and elastic-plastic analyses are presented, the latter taking into account work hardening. Numerical solutions of these equations employ the finite-element method. The elastic stress distribution in the specimen and the elastic-plastic enclaves are presented for various loading stages.

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Numerical Solutions of Natural Convection Flow of a Dusty Nanofluid About a Vertical Wavy Truncated Cone

Journal of Heat Transfer ◽

10.1115/1.4034815 ◽

2016 ◽

Vol 139 (2) ◽

Cited By ~ 9

Author(s):

Sadia Siddiqa ◽

Naheed Begum ◽

M. A. Hossain ◽

Rama Subba Reddy Gorla

Keyword(s):

Natural Convection ◽

Mass Transport ◽

Numerical Solutions ◽

Convection Flow ◽

Governing Equations ◽

Natural Convection Flow ◽

Two Phase ◽

Heat And Mass Transport ◽

Implicit Finite Difference ◽

Control Skin

This paper reports the numerical results for the natural convection flow of a two-phase dusty nanofluid along a vertical wavy frustum of a cone. The general governing equations are transformed into parabolic partial differential equations, which are then solved numerically with the help of implicit finite difference method. Comprehensive flow formations of carrier and dusty phases are given with the aim to predict the behavior of heat and mass transport across the heated wavy frustum of a cone. The effectiveness of utilizing the nanofluids to control skin friction and heat and mass transport is analyzed. The results clearly show that the shape of the waviness changes when nanofluid is considered. It is shown that the modified diffusivity ratio parameter, NA, extensively promotes rate of mass transfer near the vicinity of the cone, whereas heat transfer rate reduces.

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Experiments on Laminar Flow about a Rotating Sphere in an Air Stream

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/pime_proc_1987_201_146_02 ◽

1987 ◽

Vol 201 (6) ◽

pp. 427-438 ◽

Cited By ~ 2

Author(s):

M A I El-Shaarawi ◽

M M Kemry ◽

S A El-Bedeawi

Keyword(s):

Boundary Layer ◽

Reynolds Number ◽

Laminar Flow ◽

Numerical Solutions ◽

Separation Angle ◽

Governing Equations ◽

Rotating Sphere ◽

Axis Of Rotation ◽

Air Stream ◽

Uniform Stream

Laminar flow about a rotating sphere which is subjected to a uniform stream of air in the direction of the axis of rotation is investigated experimentally. Measurements of the velocity components within the boundary layer and the separation angle were performed at a Reynolds number, Re, of 10 000 and Ta/Re 2 of 0, 1 and 5. These measurements are compared with the numerical solutions of the same problem where either theoretical potential or actual experimental boundary conditions are imposed on the governing equations.

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Forced Flow of Vapor Condensing Over a Horizontal Plate (Problem of Cess and Koh): Steady and Unsteady Solutions of the Full 2D Problem

Journal of Heat Transfer ◽

10.1115/1.4001636 ◽

2010 ◽

Vol 132 (10) ◽

Cited By ~ 3

Author(s):

S. Kulkarni ◽

A. Narain ◽

S. Mitra

Keyword(s):

Numerical Solutions ◽

Saturated Vapor ◽

Threshold Value ◽

Steady Solution ◽

Forced Flow ◽

Film Condensation ◽

Horizontal Plate ◽

Governing Equations ◽

Long Time ◽

Steady Solutions

Accurate steady and unsteady numerical solutions of the full 2D governing equations—which model the forced film condensation flow of saturated vapor over a semi-infinite horizontal plate (the problem of Cess and Koh)—are obtained over a range of flow parameters. The results presented here are used to better understand the limitations of the well-known similarity solutions given by Koh. It is found that steady/quasisteady filmwise solution exists only if the inlet speed is above a certain threshold value. Above this threshold speed, steady/quasisteady film condensation solutions exist and their film thickness variations are approximately the same as the similarity solution given by Koh. However, these steady solutions differ from the Koh solution regarding pressure variations and associated effects in the leading part of the plate. Besides results based on the solutions of the full steady governing equations, this paper also presents unsteady solutions that characterize the steady solutions’ attainability, stability (response to initial disturbances), and their response to ever-present minuscule noise on the condensing-surface. For this shear-driven flow, the paper finds that if the uniform vapor speed is above a threshold value, an unsteady solution that begins with any reasonable initial-guess is attracted in time to a steady solution. This long time limiting solution is the same—within computational errors—as the solution of the steady problem. The reported unsteady solutions that yield the steady solution in the long time limit also yield “attraction rates” for nonlinear stability analysis of the steady solutions. The attraction rates are found to diminish gradually with increasing distance from the leading edge and with decreasing inlet vapor speed. These steady solutions are generally found to be stable to initial disturbances on the interface as well as in any flow variable in the interior of the flow domain. The results for low vapor speeds below the threshold value indicate that the unsteady solutions exhibit nonexistence of any steady limit of filmwise flow in the aft portion of the solution. Even when a steady solution exists, the flow attainability is also shown to be difficult (because of waviness and other sensitivities) at large downstream distances.

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