Transformation of the Navier-Stokes Equation to the Cauchy Momentum Equation Using a Novel Mathematical Notation

A transformation way of the Navier-Stokes differential equation was presented. The obtained result is the Cauchy momentum equation. The transformation was performed using a novel shorten mathematical notation presented at the beginning of the transformation.

A Unified Momentum Equation Approach for Computing Flow-Induced Stresses in Structures with Arbitrarily-Shaped Stationary Boundaries

This article presents a novel monolithic numerical method for computing flow-induced stresses for problems involving arbitrarily-shaped stationary boundaries.A unified momentum equation for a continuum consisting of both fluids and solids is derived in terms of velocity by hybridizing the momentum equations of incompressible fluids and linear elastic solids.Discontinuities at the interface are smeared over a finite thickness around the interface using the signed distance function,and the resulting momentum equation implicitly takes care of the interfacial conditions without using a body-fitted grid.A finite volume approach is employed to discretize the obtained governing equations on a Cartesian grid.For validation purposes,this method has been applied to three examples,lid-driven cavity flow in a square cavity,lid-driven cavity flow in a circular cavity,and flow over a cylinder,where velocity and stress fields are simultaneously obtained for both fluids and structures.The simulation results agree well with the results found in the literature and the results obtained by COMSOL Multiphysicsr.

Continuum modeling for two-lane traffic flow

In this paper, we study the continuum modeling of traffic dynamics for two-lane freeways. A new dynamics model is proposed, which contains the speed gradient-based momentum equations derived from a car-following theory suited to two-lane traffic flow. The conditions for securing the linear stability of the new model are presented. Numerical tests are can＇ied out and some nonequilibrium phenomena are observed, such as small disturbance instability, stop-and-go waves, local clusters and phase transition.

STUDY ON VARIATION OF SETTING AND STOPPING PRESSURES OF SAFETY VALVE WITH STRUCTURAL MODIFICATION

The possibility of pressure control with the structural change of a safety valve is investigated Safety valve is commonly used as safety devices for numerous applications which include boilers,ships,industrial plants,and piping Setting and stopping pressures of a safety valve, p set and p sto ,are traditionally adjusted with a fine tuning of seat ring and valve ring heights, h sr and h vr However, it is not easy to achieve the proper setting and stopping pressures of a safety valve in practice The depth of inside and outside grooves in a valve, d i and d o are modified and their effects on setting and stopping pressures of a safety vlave are tested The most appropriate values appear 1 0 mm in d i and 0 5～1 0 mm in d o,respectively The valve ring height, h vr ,shows that the best results can be achieved at 2 3 mm for setting pressures of 0 1～0 4 MPa and 1 0 mm for setting pressures of 0 5～1 0 MPa The stopping pressures increases with the increase of seat ring height, h sr , upto certain h sr value and then becomes independent to the seat ring height This implies that there exists the optimum h sr ,which provides the largest flow rate and the proper stopping pressure Stopping pressures of a safety valve are adjusted with the seat ring and valve ring heights This study,however,demonstrated that the modification of value grooves also changes setting and stopping pressures of a safety valve Therefore,the proper selection in dimensions of the inside and outside grooves should be considered for the safety valve design

Wind Power Potential in Interior Alaska from a Micrometeorological Perspective

The wind power potential in Interior Alaska is evaluated from a micrometeorological perspective. Based on the local balance equation of momentum and the equation of continuity we derive the local balance equation of kinetic energy for macroscopic and turbulent systems, and in a further step, Bernoulli’s equation and integral equations that customarily serve as the key equations in momentum theory and blade-element analysis, where the Lanchester-Betz-Joukowsky limit, Glauert’s optimum actuator disk, and the results of the blade-element analysis by Okulov and Sorensen are exemplarily illustrated. The wind power potential at three different sites in Interior Alaska (Delta Junction, Eva Creek, and Poker Flat) is assessed by considering the results of wind field predictions for the winter period from October 1, 2008, to April 1, 2009 provided by the Weather Research and Forecasting (WRF) model to avoid time-consuming and expensive tall-tower observations in Interior Alaska which is characterized by a relatively low degree of infrastructure outside of the city of Fairbanks. To predict the average power output we use the Weibull distributions derived from the predicted wind fields for these three different sites and the power curves of five different propeller-type wind turbines with rated powers ranging from 2 MW to 2.5 MW. These power curves are represented by general logistic functions. The predicted power capacity for the Eva Creek site is compared with that of the Eva Creek wind farm established in 2012. The results of our predictions for the winter period 2008/2009 are nearly 20 percent lower than those of the Eva Creek wind farm for the period from January to September 2013.

On the Maximum of Wind Power Efficiency

In our paper we demonstrate that the filtration equation used by Gorban’ et al. for determining the maximum efficiency of plane propellers of about 30 percent for free fluids plays no role in describing the flows in the atmospheric boundary layer (ABL) because the ABL is mainly governed by turbulent motions. We also demonstrate that the stream tube model customarily applied to derive the Rankine-Froude theorem must be corrected in the sense of Glauert to provide an appropriate value for the axial velocity at the rotor area. Including this correction leads to the Betz-Joukowsky limit, the maximum efficiency of 59.3 percent. Thus, Gorban’ et al.’s 30% value may be valid in water, but it has to be discarded for the atmosphere. We also show that Joukowsky’s constant circulation model leads to values of the maximum efficiency which are higher than the Betz-Jow-kowsky limit if the tip speed ratio is very low. Some of these values, however, have to be rejected for physical reasons. Based on Glauert’s optimum actuator disk, and the results of the blade-element analysis by Okulov and S&oslashrensen we also illustrate that the maximum efficiency of propeller-type wind turbines depends on tip-speed ratio and the number of blades.

NUMERICAL SIMULATION OF FLOWS FOR SECOND-ORDER VISCOELASTIC FLUID COUPLED WITH HEAT TRANSFER BY DIFFERENTIAL QUADRATURE METHOD

The problem of two-dimensional steady flow of an incompressible second-order viscoelastic fluid coupled with heat transfer between parallel plates was considered. A viscous dissipation function was included in the energy equation. When the elastic property of the fluid is weaker, the zeroth-order and first-order approximate governing equations were obtained by means of the perturbation method. To understand the behavior of flow near the tube wall, the half-domain was divided into two sub-domains, in which one is a thin layer near the wall called the inner domain and the remainder is called the outer domain. The governing equations in the inner domain and in the outer domain were discretized respectively by using the Differential Quadrature Method (DQM). The matching conditions at the interface between the inner and outer domains were presented. An iterative method for solving these discretized equations was given in this paper. The numerical results obtained agree with existing results.