A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equation

The three main approaches to exploring fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics. Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author embarked on a doctoral research on the refinement of CPT. He introduced the Refined Potential Theory (RPT) that provides the Kwasu function as a physically consistent solution to the NSE problem. The Kwasu function is a viscous scalar potential function that captures known and observable unsteady features of experimentally observed wall bounded flows including flow separation, wake formation, vortex shedding, compressibility effects, turbulence and Reynolds-number-dependence. It is appropriately defined to combine the properties of a three-dimensional potential function to satisfy the inertia terms of the NSE and the features of a stream function to satisfy the continuity equation, the viscous vorticity equation and the viscous terms of the NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere and spheroid. It is concluded that the Kwasu function is a physically consistent and closed-form analytical solution to the NSE problem.

An Analytical Solution for Seismic Interaction between Urban River-Canyon Topography and Nearby Buildings

The interaction between urban river-canyon topography and the river-side building is investigated by using a whole analytic model of a semicircle river-canyon and a shear wall supported by a semicircle rigid foundation embedded in a homogenous half-space. The closed-form analytical solution for system response is presented based on the wave function expansion method. The analysis focuses on the effects of the canyon-building interaction on system response. The strength of the interaction between the river-canyon topography and the building changes periodically as the distance between the canyon and the structure increases, leading to the interaction having beneficial or harmful effects on the building’s seismic response. The foundation peak response of the building can be amplified by about 10%, and the peak of the building relative response can be amplified by about 40%. The distribution of canyon-structure spacing with strong or weak interaction is closely related to the dynamic characteristics of the building and the incident angle of the wave. When designing buildings along the river, the building and canyon should be analyzed as a whole model to determine whether the location of the building is in a position with strong interaction with the river-canyon. The model in this paper may be useful for obtaining insight into the effects of canyon-structure interaction and interpreting the observed response in buildings and seismic response estimation in general.

Dynamic Interaction of Two Independent SDOF Oscillators Supported by a Flexible Foundation Embedded in a Half-Space: Closed-Form Analytical Solution for Incident Plane SH-Waves

This paper presents a closed-form analytical solution for the dynamic response of two independent SDOF oscillators standing on one flexible foundation embedded in an elastic half-space and excited by plane SH waves. The solution is obtained by the wave function expansion method and is verified by comparison with the results of the special cases of a rigid foundation and the published research result of a flexible foundation. The model is utilized to investigate how the foundation stiffness influences the system response. The results show that there will be a significant interaction between the two independent structures on one flexible foundation and the intensity of the interaction is mainly dependent on foundation stiffness and structural stiffness. For a system with more flexible foundation, strong interaction will exist between the two structures; larger structural stiffness will also lead to a strong interaction between the two structures. When the structural mass and the structural stiffness are all larger, the flexible foundation cannot be treated as a rigid foundation even if the foundation stiffness is many times larger than that of soil. This model may be useful to get insight into the effects of foundation flexibility on the interaction of two independent structures standing on one flexible foundation.

A closed form for Jacobian reconstruction from time series and its application as an early warning signal in network dynamics

The Jacobian matrix of a dynamical system describes its response to perturbations. Conversely, one can estimate the Jacobian matrix by carefully monitoring how the system responds to environmental noise. We present a closed-form analytical solution for the calculation of a system’s Jacobian from a time series. Being able to access the Jacobian enables a broad range of mathematical analyses by which deeper insights into the system can be gained. Here we consider in particular the computation of the leading Jacobian eigenvalue as an early warning signal for critical transitions. To illustrate this approach, we apply it to ecological meta-foodweb models, which are strongly nonlinear dynamical multi-layer networks. Our analysis shows that accurate results can be obtained, although the data demand of the method is still high.

Elastic-Plastic Analytical Solution for SEMI’s Horizontal Brace with a Circumferential Through Crack under Tension and Bending

The elastic-plastic behavior of semi-submersible’s horizontal brace with a circumferential through crack which lies at its boundary was studied. Both tension and bending were considered to investigate the closed-form analytical solution. The results indicate that the tensile plastic zone and crack tip opening displacement (CTOD) on the cracked section increase sharply after a smoothly increment when loads became larger. The cracked horizontal brace with a greater initial circumferential through crack has a larger tensile plastic zone and earlier compressive plastic zone appearance on the cracked section. Compared with the load of tension, the bending load has larger effect on the plastic zones of the cracked section and CTOD of the crack.

Nonlinear Dynamic Behavior of Porous and Imperfect Bernoulli-Euler Functionally Graded Nanobeams Resting on Winkler Elastic Foundation

Nonlinear free vibrations of functionally graded porous Bernoulli–Euler nano-beams resting on an elastic foundation through a stress-driven nonlocal elasticity model are studied taking into account von Kármán type nonlinearity and initial geometric imperfection. By using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency is then established using the Hamiltonian approach to nonlinear oscillators. Several comparisons with existing models in the literature are performed to validate the accuracy and reliability of the proposed approach. Finally, a numerical investigation is developed in order to analyze the effects of the gradient index coefficient, porosity volume fraction, initial geometric imperfection, and the Winkler elastic foundation coefficient, on the nonlinear flexural vibrations of metal–ceramic FG porous Bernoulli–Euler nano-beams.

Pipeline Walking due to Temperature Transient: Enhanced Analytical Calculations

Pipeline walking induced by transient temperature profiles as the pipeline heats up is assessed. Firstly, an integrated, closed-form solution is given for a problem for which only an ‘incremental’ solution was available [1]. This involves a short pipeline unrestrained at both ends subject to a linearly ramping temperature front. Secondly, the range of validity of this solution is significantly extended, whilst still presenting it in closed-form. Results are compared with previously published FEA results, presenting remarkable agreement. The closed-form analytical solution is then compared with FEA of more realistic transients (non-linear temperature front). Results show that the linear simplification can introduce excessive conservatism. The FEA results are then examined, and the reason for the excessive conservatism is found to be associated to early expansion of the cold end, which is not observed with the simplified linear front. A simplified incremental solution for non-linear transients is proposed. This is shown to be simple and effective in improving prediction for the range over which where the closed form solution is most conservative.