Theoretical Study of Mach number and Compressibility Effect on the Slender Airfoils

Theoretical development of the velocity potential equation for compressible flow and its various consequences has been presented. The geometrical interpretation of potential equation and conformal mapping technique are discussed where the mappings link the flow around a circular cylinder of a slender airfoil. The lift and drag coefficients are determined for the slender airfoils based on the Mach number and compressibility effects. The calculated lift coefficients show that with the increasing of attack angle it increases linearly and a higher lift coefficient is found for a smaller Mach number for any certain attack angle. Similarly, the drag profiles are determined which are exponentially decreased with the increasing of Mach number for any fixed attack angle. The calculated and experimental data on the lift and drag coefficients over the slender airfoil surface are compared and found in good agreement

A Mathematical Model of Avian Influenza for Poultry Farm and its Stability Analysis

This paper aims to estimate the basic reproduction number for Avian Influenza outbreak in local and global poultry industries. In this concern, we apply the SEIAVR compartmental model which is developed based on the well-known SEIR model. The SEIAVR model provides the mathematical formulations of the basic reproduction number, final size relationship and a relationship between these two phenomena. The developed model Equations are solved numerically with the help of Range-Kutta method and the values of initial parameters are taken from the several literatures and reports. The calculated result of basic reproduction number shows that it is locally and globally stable if it is less than and greater than one at disease free equilibrium and at endemic equilibrium, respectively. Furthermore, we have compared among the calculated susceptive, expose, infective, removal, virus and asymptotic compartments where infection rate and expose period are observed very sensitive compared to other parameters. In addition, the model result of infective is compared with the field data and other’s model where the present model shows good performance against the field data

Air Bubble Entrainment by Breaking Waves and Associated Energy Dissipation

A simple mathematical model (air bubble model) that describes energy dissipation due to air bubble entrainment is proposed and applied to a series of laboratory experiments for plunging jet flows (steady) and surf zone waves (unsteady). This leads to a formulation wherein the rise velocity of bubbles is included. For the unsteady case, i.e. air entrainment by wave breaking, some parameters of the model have been estimated from the experiments and expressed in terms of local wave height and distance. Results obtained through the air bubble model are summarized in the following. Experiments in vertical circular plunging jets (steady) were performed for both in freshwater and seawater, which highlighted the distribution of void fraction, that follows closely analytical solutions derived by Chanson (1997). In addition, various properties of void fraction field were emphasized. Three scale models were used with freshwater for identical Froude numbers in the experiments, which highlighted significant scale effects when Weber number is less than 1000. Similar experiments were also performed with freshwater and seawater and the results showed lesser air entrainment in seawater plunging jets. The pseudo-bubble chord sizes obtained from the experimental data were in the range from less than 0.5 mm to more than 1.0 mm. The rate of energy dissipation due to entrained air was investigated by applying the air bubble model for three typical phenomena. The results demonstrated that the ratios of energy dissipation due to air bubble entrainment with respect to total energy loss were 25%, 1.4% and (2-4)% for hydraulic jump, 2-D vertical plunging jet and vertical circular jet, respectively. Experiments on unsteady air bubble entrainment by wave breaking were conducted in a wave channel. Maps of the evolution of the void fraction distribution in surf zone generated by various sizes breaking waves were presented. A significant fraction of the potential energy of entrained air was measured from the void fraction distributions provided by breaking waves. Measurements showed high void fraction up to 19% in plunging breakers at still water surface whereas 16% in spilling breakers. The ratio of energy dissipation due to entrained air to total energy loss was found (18-22)% and (17-19)% for spilling and plunging breaker, respectively. The characteristics of time averaged wave parameters (e.g. potential energy, kinetic energy, energy flux, radiation stress) for regular waves were discussed taking into account the air bubble effects. Analytical solutions were sought and explicit expressions were obtained for wave parameters under sinusoidal waves. Effects of the air entrainment on density, pressure and velocity fields were also discussed in detail. The conservation equations for energy and momentum were solved numerically using finite difference methods. Boundary conditions were used at the breaking point. Scale effects were discussed based on laboratory air entrainment in 2-D wave flume, which was believed to occur in small size models. The data were in good agreement with the basic assumption for vertical distribution of void fraction both in spilling and plunging breakers. The results of the air bubble model were compared with experimental data and found to give good agreement between them for the wave height and wave setup. Water level rise by entrained air was determined and found significant effects on surf zone hydrodynamics. In addition, wave run-up was measured and discussed

Similitude of Air Bubble Entrainment and Dispersion in Vertical Circular Plunging Jet Flows. An Experimental Study with Freshwater, Salty Freshwater and Seawater

Air bubble entrainment at plunging jet takes place when the jet impact velocity exceeds a critical velocity Ve which is a function of the jet turbulence. Several researchers proposed an analogy between plunging jet flows and plunging breakers. Most works considered steady plunging jet results obtained with freshwater and there have been suggestions that air entrainment at breaking waves in the sea might be an entirely different process. This study investigates scale effects affecting air entrainment and bubble dispersion at vertical circular plunging jets. Three scale models were used and detailed air-water measurements were performed systematically for identical Froude numbers. For one scale, three water solutions were tested. All experimental results are presented in Appendices B and C. In freshwater, the results highlight significant scale effects when We1 = rw*V1 2*d1/s < 103 or V1/ur < 10. Identical experiments were performed with freshwater, salty freshwater and seawater. The results show lesser air entrainment in seawater plunging jets for identical inflow conditions. Results with saltwater are intermediate between freshwater and seawater results. Bubble chord times tch were also measured. The data were analysed in terms of pseudo-bubble chord length chab = V1*tch, which was found to overestimate real bubble chords by 10 to 30%. The data show that pseudo-bubble chord sizes range from less than 0.5 mm to more than 10 mm. The average pseudo-chord sizes are between 4 and 6 mm for all water solutions. Comparatively, however, more fine bubbles were detected in seawater than in freshwater. Fine bubbles (less than 0.5 mm) have a slower rise velocity and their underwater residence time is larger in seawater. They give a visual, misleading appearance to the air-water flow suggesting that very fine bubbles are predominantly entrained in seawater plunging jets. Overall the study contributes to a better understanding of air entrainment and bubble dispersion in vertical circular plunging jets. Differences in air entrainment mechanisms between freshwater and seawater are described. Scales effects are believed to occurs in laboratory wave flumes when the wave height at breaking is less than 0.25 to 0.35 m. Experimental results in seawater and saltwater suggest further that classical dimensional analysis of air entrainment is incomplete, and that further physical, chemical and biological fluid properties must be taken into account

Scaling Bubble Entrainment And Dispersion In Vertical Circular Plunging Jet Flows: Freshwater Versus Seawater

At a plunging jet, air entrainment may take place downstream of the impingement perimeter. This study investigates air entrainment and bubble dispersion at vertical circular plunging jets in freshwater, salty freshwater and seawater. The results show lesser air entrainment in seawater plunging jets for identical inflow conditions. The average pseudo-chord sizes are between 4 and 6 mm for all water solutions, although more fine bubbles were detected in seawater. It is believed that surfactants and bio-chemicals harden the induction trumpet at plunge point and diminish air entrapment at impingement in seawater

Similitude of air entrainment at vertical circular plunging jets

Air bubble entrainment at plunging jet takes place when the jet impact velocity exceeds a critical velocity function of the inflow conditions. This study investigates scale effects affecting air entrainment and bubble dispersion at vertical circular plunging jets. Three scale models were used and detailed air-water measurements were performed systematically for identical Froude numbers. The results highlight significant scale effects when We1 < 1E+3 or V1/ur < 10. Bubble chord times were also measured and presented in terms of pseudo-bubble chord length which was found to overestimate real bubble chords by 10 to 30%. The data show pseudo-bubble chord sizes ranging from less than 0.5 mm to more than 10 mm. The average pseudo-chord sizes were between 5 and 7 mm

Air entrainment by breaking waves: A theoretical study

17-23The variation of static pressure, density, potential energy and kinetic energy due to air bubble entrainment in the surf zone and their physical mechanisms have been studied theoretically. An averaging technique has been used to investigate the behavior of every single bubble in the bubble cloud. The above parameters have been found to be affected significantly by the entrained air bubbles

Air Bubble Entrainment by Breaking Waves and Associated Energy Dissipation

豊橋技術科学大

Development of a cylindrical polar coordinates shallow water storm surge model for the coast of Bangladesh

The coast of Bangladesh is funnel shaped. The narrowing of the Meghna estuary along with its peculiar topography creates a funneling effect that has a large impact on surge response. In order to have an accurate estimation of surge levels, the impacts of the estuary should be treated with due importance. To represent in detail the real complexities of the estuary, a very high resolution is required, which in turn necessitates more computational cost. Considering the facts into account, a location specific vertically integrated shallow water model in cylindrical polar coordinates is developed in this study to foresee water levels associated with a storm. A one-way nested grid technique is used to incorporate coastal complicities with minimum cost. In specific, a fine mesh scheme (FMS) capable of incorporating coastal complexities with acceptable accuracy is nested into a coarse mesh scheme (CMS) covering up to 15°N latitude in the Bay of Bengal. The coastal and island boundaries are approximated through appropriate stair step representation and the model equations are solved by a conditionally stable semi-implicit finite difference technique using a structured C-grid. Numerical experiments are performed using the model to estimate water levels due to surge associated with the April 1991 and AILA, 2009 cyclones, which struck the coast of Bangladesh. Time series of tidal level is generated from an available tide table through a cubic spline interpolation method. The computed surge response is superimposed linearly with the generated time series of tidal oscillation to obtain the time series of total water levels. The model results exhibit a good agreement with observation and reported data