Apparent Angle Research Articles

The aerodynamic performances of the wing-shaped poop sail

ABSTRACT This research proposes a wing-shaped poop which cleverly combines the poop and the wing sail. The wing-shaped poop with the length of 60 m and the height of 16.5 m is designed. The influence of the camber ratios (0.15–0.30) and the thickness ratios (0.10–0.20) on the performance of wing-shaped poop is analyzed. The lift coefficient of the poop increases first and then decreases with the increase of camber ratio and decreases with the increase of thickness ratio. The maximum lift coefficient (2.27) is obtained when the camber ratio is 0.25 and the thickness ratio is 0.1. Considering the influence of the rudder, when the apparent wind angle is 130°, the effective thrust of the poop reaches the maximum of 82 kN. Besides, when the wind comes from the bow, the shape of the poop is changed to reduce wind resistance caused by the wing-shaped poop.

Experimental study on propulsion performance of unmanned sailboat with arc sail

In recent years, the unmanned sailboats, which constantly obtain energy from the environment for propulsion, have attracted great attention from global researchers. The existing studies on unmanned sailboats mainly concentrate on navigation path planning, control system design, speed prediction and attitude optimization design, but little emphasis is put on the sail that provides the core propulsion. This study carries out orthogonal analysis and CFD simulations to analyze the influence of coupling parameters on the aerodynamic performance of arc sail, and determines an arc sail suitable for the existing Petrel unmanned sailboats. The structure and turning logic of the selected arc sail is elaborated in detail. The pool experiment with adjustable initial apparent wind angle of the sailboat is designed to verify the turning logic of the arc sail and the performance of the sailboat. Finally, an outdoor experiment is performed to validate the navigation performance of the Petrel unmanned sailboat equipped with the selected arc sail in actual sea conditions.

Experimental investigation of the aerodynamic performance of Flettner rotors for marine applications

Flettner rotors are becoming alternative wind-assisted propulsion systems for marine applications. This study experimentally investigated the aerodynamic performance of large-scale Flettner rotors with a diameter D = 350 mm, 0 = α ≤ 4.5, 3.5 = L/D ≤ 6.0, 1.0 ≤ De/D ≤ 2.0, and 9.33 × 104 ≤ Re ≤ 3.15 × 105, where α is the velocity ratio, L is the cylinder span, De is the diameter of the endplate, and Re is the Reynolds number based on D. The initial, increasing, decreasing, and stable areas were defined to describe the variation in the time-averaged lift-to-drag ratio K‾ with an increase in α. Both the aerodynamic performance of Flettner rotors and the range of the four areas were significantly influenced by L/D, De/D, and Re. Based on the measurement of the aerodynamic forces acting on the tested Flettner rotor under the influence of an interfering Flettner rotor with the same α, the effects of aerodynamic interaction for different wind incidence angles β and spacing ratios S/D were studied in detail. In addition, the driving and heeling force coefficients were presented for the combination of two Flettner rotors. Once the arrangement of the two Flettner rotors installed on the deck of a ship was known, both the driving and heeling forces provided by the combination of these rotors depended on α and the apparent wind angle.

Evidence for Ion Association at the Gas-Liquid Interface of the Mixture of 1-Butyl-3-methylimidazolium Hexafluorophosphate and Benzonitrile: A Sum Frequency Generation Spectroscopy and Surface Tension Study.

The gas-liquid interface for the mixtures of [BMIM][PF6] and benzonitrile is studied by sum frequency generation (SFG) spectroscopy and surface tension measurements as an important solute to reduce the viscosity of ionic liquids. Solvation of ionic compounds in bulk solvent is not necessarily the same as that at the surface due to the lower dielectric medium at the air/liquid interface. The results from the temperature-dependent SFG spectroscopy and surface tension study suggest that the ionic liquid in a benzonitrile solvent is associated as ion pairs at the surface rather than as dissociated─solvated─ions in the bulk solution. The influence of ionic liquids on the surface structure of benzonitrile is investigated from 0 to 1.0 mole fraction of benzonitrile. The CH stretching mode of vibration of benzonitrile in the SFG spectrum begins from a 0.2 mole fraction (x) of benzonitrile, and the intensity of the peak constantly increases on increasing the concentration of benzonitrile. However, the addition of benzonitrile does not result in extra peaks or shifting of the peak frequency to the spectra of [BMIM][PF6]. The surface tension measurements further support the presence of benzonitrile at the gas-liquid interface. The surface tension data of the mixture smoothly decrease as the benzonitrile concentration increases. The apparent tilt angle of the terminal methyl group of the cation of [BMIM][PF6] is calculated from SFG polarization spectra and shows an apparent lowering with the addition of benzonitrile. The effect of temperature on the surface structure of the binary mixture is also reported at four different temperatures between -15 and 40 °C for both the SFG spectroscopy and surface tension study. Benzonitrile shows different behavior in the mixture at higher temperatures than pure benzonitrile, as observed in the SFG spectra. In contrast, it does not show any CN peak in the mixture below 0.9 mole fraction. The temperature dependence of the interfacial tension is used to evaluate thermodynamic functions such as surface entropy and surface enthalpy. Both were found to be lowered with the increasing concentration of benzonitrile. Both spectroscopic and thermodynamic analyses indicate that the ionic liquid is highly associated as ion pairs and the benzonitrile is more ordered at the surface at concentration <0.4×.

Morphological Features of Fractures Developed by Direct Shearing of Intact Granites after Water Cooling Cycles.

Rock fractures are considered as favorable objects for enhanced geothermal development. The fracture morphologies play an important role in enhanced geothermal development. Therefore, the study of fracture morphologies has a certain guiding significance for the geothermal reservoir. Water cooling and water cooling cycles can change the morphology of fracture surfaces formed by the shear failure of intact granites. To date, however, there is little work on the effect of water cooling and water cooling cycles on the morphology of fracture surfaces formed by direct shearing of intact granites. In this study, the direct shear tests of intact granites treated by water cooling cycles at different temperatures were conducted, and the variations in the laws of shear strength of intact granites and morphologies of fracture surfaces with temperature or cycle times were analyzed. Test results showed that the shear strength and shear stiffness of intact granites decreased nonlinearly with the increase of temperature or cycle times, but the height and apparent dip angle of asperities on the fracture surface increased with the increase of temperature or cycle times, and the overall uniformity of the fracture surface was improved. The height distribution frequency of asperities on fracture surfaces can be divided into four types: right-biased peak type, left-biased peak type, left-biased middle peak, and left-biased flat peak. The asperities on the fracture surface formed by the shearing of intact granites have asymmetric characteristics. The maximum apparent dip angle and average apparent dip angle in the reverse shear direction are larger than those in the shear direction, and the initial contact area ratio between the shear direction and reverse shear direction is in the range of fluctuation between 1.4 and 2.

Effect of Self-Focusing on Propagation of a Laser Beam in the Atmosphere Along a Slant Path

The equivalent B-integral phase screen (EBPS) method of numerical simulation is proposed to study the propagation of a high-power laser beam in the atmosphere along a slant path, and the computational accuracy and efficiency of the EBPS method are confirmed. It is shown that a decrease of the self-focusing caused by the atmosphere refraction effect. However, the refraction correction for the elevation angle is hardly affected by the self-focusing effect. On the other hand, the analytical expressions of the B integral and the optimal initial beam power of a high-power laser beam propagating in the atmosphere along a slant path are derived. It is found that the optimal beam power and the target maximum intensity decrease as the apparent elevation angle decreases, while the optimal beam power decreases but the target maximum intensity increases as the wavelength decreases. A method to optimize the beam quality on the target is presented.

The apparent focal depth, emergence angle, and take-off angle of seismic wave measured by YRY-4-type borehole strainmeter as one kind of strain seismograph

Introduction: In theory, the observation objects and principles of strain seismograph and traditional pendulum seismograph are different, and the characteristics of observed signals should also be dissimilar. The observation results of pendulum seismograph show that seismic waves in inhomogeneous media will undergo refraction, reflection, and attenuation. Then, what signal characteristics can be detected by strain seismograph is great significance for understanding and explaining the observation results.Methods: Using YRY-4 type four-gauge borehole strainmeter as one kind of strain seismograph to detect the strain tensor change of the plane seismic wave emitted from the surface, a five-site strain seismograph observation network was built in Shanxi Province, with continuous observation for 2 years at a sampling rate of 100 Hz. In this paper, two local events occurring in the area covered by the strain seismograph observation network are taken as examples. We systematically studied the characteristics of seismic wave signals recorded by strain seismographs at five sites, inverted for the focal depth of the two local earthquakes and the relationship between the wave velocity and the wave velocity gradient of the focal depth, and calculated the apparent focal depth, the emergence angle and the take-off angle of seismic waves.Results: These results show stable uniqueness and apparent regularity, especially since the inverted focal depths are basically consistent with the seismic solutions based on those traditional pendulum seismographs. The observations from this study show that the strain seismograph can be used as an effective supplement to the pendulum seismograph.Discussion: In the future, we will continue to study the rupture process and focal mechanism of moderate-strong earthquakes and teleseismic earthquakes by combining two kinds of observations.

Updated frequency-dependent directivity indices for large solid rocket motors

For many years, empirical models for rocket noise radiation relied on the directivity indices published in NASA SP-8072 (K. Eldred, 1971). Because these indices had known issues, NASA led a campaign to update the indices using a Space Shuttle reusable solid rocket motor (RSRM). The RSRM measurements, involving a polar arc at a radius of about 80 nozzle exit diameters, resulted in updated directivity indices (Haynes and Kenny, AIAA 2009–3160). However, because the arc origin was placed at the nozzle exit plane, James et al. [Proc. Mtgs. Acoust., 18, 040008 (2012)] corrected the low-frequency indices using an estimated dominant source position for each frequency and assumed spherical spreading. This paper revisits that correction effort by using near-field vector intensity measurements from a similar, but smaller diameter and lower thrust, GEM-60 motor to determine the frequency-dependent origin to adjust the apparent angles and distances of the measurements. Additionally, an effort is made to account for RSRM plume impingement downstream that likely resulted in lower high-frequency levels than would have been measured if the plume had been entirely free. This analysis results in updated, frequency-dependent directivity indices for a large solid rocket motor. Their applicability to other rockets will be discussed.

Study on the Prediction of Debris Flow Risk Using LS-RAPID

In this study, the LS-RAPID program was applied to evaluate the debris flow deposition area using the field surveyed topographical data in the areas of debris flow occurrence. In 2020, aerial photogrammetry was performed using an unmanned aerial vehicle in Seonseri, Osan-myeon, and Gokseong-gun. The erosion and deposition areas owing to debris flow were estimated using the orthographic images generated from photogrammetry and high-resolution Digital Surface Mode (DSM), and the extent of debris flow was accurately estimated. For debris flow simulation, the LS-RAPID program, with the apparent friction angle of the Coulomb's friction model as an important factor, was applied. The simulation results were significantly affected by the quality of the topographical data. It is essential to apply DSM data for obtaining a highly accurate simulation of the deposition area where damage generally occurs. To evaluate the performance of the simulation, the LSSI index with the deposition area as a factor was applied; it was found to be 65%, showing a superior predictive performance. In conclusion, it is possible to predict the debris-flow damage using LS-RAPID simulation, and the DSM data on deposition areas are required to generate a reliable disaster map.

Joint Inversion of Receiver Functions and Apparent Incidence Angles to Determine the Crustal Structure of Mars

AbstractRecent estimates of the crustal thickness of Mars show a bimodal result of either ∼20 or ∼40 km beneath the InSight lander. We propose an approach based on random matrix theory applied to receiver functions (RFs) to further constrain the subsurface structure. Assuming a spiked covariance model for our data, we first use the phase transition properties of the singular value spectrum of random matrices to detect coherent arrivals in the waveforms. Examples from terrestrial data show how the method works in different scenarios. We identify three previously undetected converted arrivals in the InSight data, including the first multiple from a deeper third interface. We then use this information to jointly invert RFs with the absolute S‐wave velocity information in the polarization of body waves. Results show a crustal thickness of 43 ± 5 km beneath the lander with two mid‐crustal interfaces at depths of 8 ± 1 and 21 ± 3 km.

Failure mechanism and movement process of three loess landslides due to freeze-thaw cycle in the Fangtai village, Yongjing County, Chinese Loess Plateau

In the seasonal freeze-thaw region of China, the long-term freeze-thaw cycle (FTC) weakens the strength of the soil's structure leading to a series of geohazard events such as landslides and collapses. In order to understand the formation mechanism of the landslide during periods of freeze-thaw in the Chinese Loess Plateau (CLP). In this study, we analyzed the evolution process of the microstructure and the strength combined with triaxial test, resistivity experiments and CT technology. Furthermore, we collected meteorological, irrigation, groundwater level and landslide data to discuss landslide inducing factors. The results show that (1) the FTC has a significant damage effect on the strength of the original loess. After 10 FTCs, the strength of the original loess with various moisture content (ωmc) is reduced by at least 50%. The higher the ωmc of the original loess sample, the greater the strength reduction and the more apparent the deterioration characteristics. (2) FTC causes the expansion of the pore structure, that gradually presents non-uniformity and complexity, resulting in the continuous increase of resistivity. The pore area of macropores and mesopores are the most significant feature after 10 FTCs, and the increase of pore structure leads to the gradual loosening of the soil structure, which reflects the essence of strength deterioration. (3) Fangtai 1# landslide has the typical characteristics of high-speed and long-distance. The primary movement period of the landslide was from 4 s to 30s, the maximum speed of the landslide was 20 m/s, and the apparent friction angle was 0.34. (4): The most direct inducing factor of the three consecutive landslides in the Fangtai village was a temperature change lasting >10 days. Through mechanical and microstructure analysis, we found that the increase of FTCs leading to soil strength damage is the trigger factor, especially in the slope softening zone, which promotes the frequent occurrence of landslides. The rise of groundwater level caused by irrigation is the fundamental condition for the formation of landslide, and the irrigation plan of the Heifangtai tableland needs to be paid reasonable attention.

A preliminary study of hydro-mechanical properties for bentonite-sand mixture

Radioactive waste is a consequence of treatment radioactive material in industrial, medical, military and research applications for remaining electrical energy. For high-level waste (HLW), spent fuel (SF) and long-lived waste categories, geological disposal at depths of much hundreds of maters is considered in safety design. This study has aim to appear that swelling pressures due to hydration effort, hydraulic conductivity and stress-strain properties associated to evaluation to stress paths using two different bentonite-sand mixture, which are Sodium type bentonite and Calcium type bentonite. These materials have key feature in hydro-mechanical properties established in many reported obtained from laboratory test and underground investigations. Swelling pressure test, hydraulic conductivity test and triaxial compression test are conducted out, and developed apparatus are useful to results for series of some testing programs. Particularly, salinity concentration with increasing and decreasing on considering salinity effort is so interesting that further changing is applied to hydraulic conductivity for bentonite. While large lateral pressures are applied the bentonite at triaxial compression test, depending on lateral pressure is obviously to strength parameters that are apparent cohesion and angle of internal friction.

Deflection of charged signals in a dipole magnetic field in a Schwarzschild background using the Gauss-Bonnet theorem

This paper studies the deflection of charged particles in a dipole magnetic field in Schwarzschild spacetime background in the weak field approximation. To calculate the deflection angle, we use Jacobi metric and Gauss-Bonnet theorem. Since the corresponding Jacobi metric is a Finsler metric of Randers type, we use both the osculating Riemannian metric method and generalized Jacobi metric method. The deflection angle up to fourth order is obtained and the effect of the magnetic field is discussed. It is found that the magnetic dipole will increase (or decrease) the deflection angle of a positively charged signal when its rotation angular momentum is parallel (or antiparallel) to the magnetic field. It is argued that the difference in the deflection angles of different rotation directions can be viewed as a Finslerian effect of the nonreversibility of the Finsler metric. The similarity of the deflection angle in this case with that for the Kerr spacetime allows us to directly use the gravitational lensing results in the latter case. The dependence of the apparent angles on the magnetic field suggests that by measuring these angles the magnetic dipole might be constrained.

Putting the micro into the macro: a molecularly augmented hydrodynamic model of dynamic wetting applied to flow instabilities during forced dewetting

We report a molecularly augmented continuum-based computational model of dynamic wetting and apply it to the displacement of an externally driven liquid plug between two partially wetted parallel plates. The results closely follow those obtained in a recent molecular dynamics (MD) study of the same problem (Fernández-Toledano et al., J. Colloid Interface Sci., vol. 587, 2021, pp. 311–323), which we use as a benchmark. We are able to interpret the maximum speed of dewetting $U^*_{{crit}}$ as a fold bifurcation in the steady phase diagram and show that its dependence on the true contact angle $\theta _{{cl}}$ is quantitatively similar to that found using MD. A key feature of the model is that the contact angle is dependent on the speed of the contact line, with $\theta _{{cl}}$ emerging as part of the solution. The model enables us to study the formation of a thin film at dewetting speeds $U^*&gt;U^*_{{crit}}$ across a range of length scales, including those that are computationally prohibitive to MD simulations. We show that the thickness of the film scales linearly with the channel width and is only weakly dependent on the capillary number. This work provides a link between matched asymptotic techniques (valid for larger geometries) and MD simulations (valid for smaller geometries). In addition, we find that the apparent angle, the experimentally visible contact angle at the fold bifurcation, is not zero. This is in contrast to the prediction of conventional treatments based on the lubrication model of flow near the contact line, but consistent with experiment.

A Torpedo Target Recognition Method Based on the Correlation between Echo Broadening and Apparent Angle

As acoustic decoys can simulate the scale of the target through orderly control of the echo delay, simulated acoustic decoys have scale characteristics similar to those of the scaled target. Consequently, simulated acoustic decoys make it difficult for active acoustic homing torpedoes to recognize acoustic decoys through traditional echo broadening or apparent angle. This will lead to a decrease in the anti-interference capability of torpedoes. In combat, acoustic decoys deceive torpedoes and deviate from the tracking course so that torpedoes cannot find the real target, or waste the range, eventually failing to strike the target and failing in combat. The accurate underwater target scale recognition of active acoustic homing torpedoes is considered a difficult technique. In this paper, we propose a target recognition method based on the correlation between target echo broadening and apparent angle. This specific simulation example shows that conventional target scale recognition methods cannot distinguish between suspended and homing acoustic decoys with virtual scale. By contrast, the target scale recognition method proposed in this paper can accurately distinguish between suspended and homing acoustic decoys with virtual scale at close range, under non-positive transverse port angle conditions. This method improves the anti-interference capability of torpedoes. In addition, it can improve the accuracy of active sonar recognition scale targets of ships, which guide active sonar target recognition.

Bimodal distribution of amphibole-melt apparent dihedral angles in the product of the early crystallization of hydrous basaltic melts

• Amp-Melt-Amp apparent dihedral angle distribution shows bimodality. • Bimodality becomes obvious after annealing for more than 1 h at 0.6 GPa and 970 °C. • Median-standard deviation evolution obeys no longer a unidirectional change. • The high anisotropy of amphibole is the main cause of the bimodality. The dihedral angle of mineral-melt is a very important parameter for further understanding the textural evolution of magma crystallization. In this study, we have investigated the amphibole-melt dihedral angle distribution of four previous experimental samples at 0.6 GPa and 970 °C with annealing time of 1 h, 10 h, 60 h, and 100 h, as well as two additional experiments with annealing time of 5 h and 30 h. In this experiment, we have achieved the assemblage of amphibole + granodioritic melt ± traces of other minerals in the process of melt crystallization under the above experimental conditions. The apparent dihedral angles of amphibole-melt of the experimental products were obtained by randomly measuring the dihedral angles in the backscattered images. The results indicate that the apparent dihedral angle presents a gradually obvious bimodal distribution with the increasing annealing time. The high anisotropy of amphibole is believed to be the main cause of the bimodal characteristics of the apparent dihedral angle distribution. Consequently, two Gauss functions are used to fit one distribution. As the annealing time increases, the median-standard deviation relationships of the apparent dihedral angle for these two Gauss functions show great differences. Therefore, our experiments proved that in the amphibole-melt system, the median-standard deviation with the growth of textural equilibrium in the early crystallization is no longer a unidirectional change. Furthermore, combining the results of this study and the previous work, whether the dihedral angle distribution in the early crystallization is unimodal or bimodal may have great relationship with the anisotropy of minerals.

Does elastic stress modify the equilibrium corner angle?

We consider the influence of elasticity and anisotropic surface energy on the energy-minimizing shape of a two-dimensional void under biaxial loading. In particular, we consider void shapes with corners for which the strain energy density is singular at the corner. The elasticity problem is formulated as a boundary integral equation using complex potentials. By incorporating the asymptotic behavior of the singular elastic fields at corners of the void, we develop a numerical spectral method for determining the stress for a class of arbitrary void shapes and corner angles. We minimize the total energy of surface energy and elastic potential energy using calculus of variations to obtain an Euler-Lagrange equation on the boundary that is coupled to the elastic field. The shape of the void boundary is determined using a numerical spectral method that simultaneously determines the equilibrium void shape and singular elastic fields. Our results show that the precise corner angles that minimize the total energy are not affected by the presence of the singular elastic fields. However, the stress singularity on the void surface at the corner must be balanced by a singularity in the curvature at the corner that effectively changes the macroscopic geometry of the shape and effectively changes the apparent corner angle. These results reconcile the apparent contradiction regarding the effect of elasticity on equilibrium corner angles in the results of Srolovitz and Davis (2001) and Siegel, Miksis and Voorhees (2004), and identify an important nontrivial singular behavior associated with corners on free-boundary elasticity problems.

The Measurement of Shear Characteristics of Paddy Soil in Poyang Lake Area

Understanding the shear characteristics of paddy fields is of great significance for the design of high-trafficability paddy wheels and to improve the trafficability of the tiller in paddy fields. Taking the paddy soil in Poyang Lake area of Jiangxi Province as the research object, a simple, self-designed bevameter was constructed to measure the apparent cohesion c and apparent internal friction angle φ of paddy soil with water content of 37% by weight using an annular shear plate. The effects of different shear speeds, shear grouser heights, and shear plate diameters on the measured c and φ were investigated. The measured values of c and φ were 1.89–2.35 kPa and 10.2–11.4°(deg), respectively. With the increase in shear speed, shear plate diameter, and grouser height, the value of c increased. Shear plate diameter and shear speed had small effect on c, whose change rate did not exceed 7.8%. However, shear grouser height had a greater influence on c, and the measured c increased by 24.5%. The influence of the three factors on φ followed the order: shear speed &gt; shear plate diameter &gt; shear grouser height, but the variation in φ did not exceed 1.2°(deg). The optimal operational conditions were: shear plate diameter of 250 mm with inner diameter of 150 mm, shear grouser height of 10 mm, with a shear speed of π/120 rad/s, and the apparent cohesion and apparent internal friction angle are 1887.8 pa and 11.06° (deg), respectively.

Dewetting acceleration by evaporation

Dewetting of a substrate, i.e. the liquid film retraction under partial wetting conditions, has been studied extensively over the past decades. This theoretical work deals with the dewetting phenomenon in the presence of liquid evaporation into the pure vapour atmosphere driven by substrate superheating with respect to the saturation temperature corresponding to the vapour pressure. The dynamic profile of the vapour–liquid interface is analysed numerically by using the generalised lubrication approximation that can be applied to an interface slope larger than its conventional counterpart. This approximation uses a phenomenological argument to increase the precision of calculations. For small slopes, it reduces to the conventional lubrication approach. Several nanoscale effects are included in the theory to obtain more precise results: the Kelvin effect, the hydrodynamic slip, the Marangoni effect, the vapour recoil and the interfacial thermal resistance. Their relative importance is discussed. The results include the film shape evolution, the apparent contact angle and the contact line speed, all defined by the substrate superheating and its wetting properties. These dependencies agree with existing experimental results. It is found that the dewetting is accelerated by evaporation. The numerical results are compared to those of the multiscale theory that uses two main parameters, the Voinov angle (the apparent angle obtained within the microregion problem) and the Voinov length. The latter is studied for several fluids as a function of superheating, and an approximate expression for it is proposed. A good agreement between the numerics and the multiscale theory shows the validity of the latter. This means that the dewetting speed in the presence of evaporation can be calculated with an analytical expression that requires only the Voinov angle as a numerical input. Such results are important for many applications, for instance, bubble growth in boiling or film dynamics in heat pipes.

Deflection and gravitational lensing of null and timelike signals in the Kiselev black hole spacetime in the weak field limit

In this work we study the deflection and gravitational lensing of null and timelike signals in the Kiselev spacetime in the weak field limit, to investigate the effects of the equation of state parameter ω and the matter amount parameter α. In doing this, we extend a perturbative method previously developed for asymptotically flat spacetimes whose metric functions have integer-power asymptotic expansions to the case that may or may not be asymptotically flat but with non-integer power expansions. It is found that in the asymptotically flat case (−1/3 < ω < 0) the deflection angles are expressable as quasi-power series of the dimensionless quantities M/b, b/r s,d and α/M 1+3ω where M, b, r s,d are respectively the lens mass, impact parameter and source/detector radius. A similar series exists for the non-asymptotically flat case of (−1 < ω < −1/3), but with the closest radius r 0 replacing b. In the asymptotically flat (or non-flat) case, the increase of α or decrease of ω will increase (or increase) the deflection angle. Since the obtained deflection angles naturally take into account the finite distance effect of the source and the detector, we can establish an exact gravitational lensing equation, from which the apparent angles of the images and their magnifications are solved. It is found that generally for the asymptotically flat case, increasing α or decreasing ω will increase the apparent angles of the images. While for the non-asymptotically flat case, increasing α or ω will both lead to smaller apparent angles.