Egg-shaped Sections Research Articles

Egg-shaped sections of fluid structures around black holes

Astrophysical fluids subjected to the strong gravity of black holes can form rotating toroidal-like structures known as thick accretion discs. In order to facilitate the study of such structures, we introduce approximate empirical formulae for a relatively simple analytical description of their poloidal cross-sections that are usually described only numerically. Our comparisons of the approximate and exact descriptions of the cross-sections of the structures rotating around Schwarzschild and Kerr central black holes, together with the determination of the associated total masses of these structure, indicate a high degree of accuracy for these introduced formulae.

Modern vision for critical flow in an egg-shaped section (Part II)

The main aim of this research was to examine the simultaneous effects of all flow parameters on the critical regime occurrence frequency in an egg-shaped conduit. Furthermore, one may first study the impacts of the conduit's generator diameter, then the influence of the generator diameter/longitudinal slope, and finally the effect of all parameters: the generator diameter, longitudinal slope, roughness and viscosity. It has been concluded that all these parameters, combined, have limit values that can generate a single critical state from which a diagram can be drawn showing different states of the critical flow that can be observed. Notably, this diagram is established according to the following practical ranges: relative roughness 0 ≤ ε/D ≤ 5.10-2, generator diameter 0.25 m ≤ D ≤ 4 m and Galileo number 2.105 ≤ √Ga ≤ 4.107. This revealed that the critical regime can develop smooth, rough and transition domains. Critical flow is very unstable; therefore, it is to be avoided in practice, which means that the dimensions and the slope of the conduit must be such that they do not cause such a state of flow. This explains why knowledge of the characteristics of the critical flow, including the frequency of its occurrence for a given installation, is essential.

Modern vision for critical flow in an egg-shaped section

Abstract The critical regime plays a primordial role in the study of gradually varying flows by classifying flow regimes and slopes. Through this work, a new approach is proposed to analyze critical flow regime in an egg-shaped channel. Based on both the definition of Froude number and Achour and Bedjaoui general discharge relationship, a relation between critical and normal depths is derived and then graphically represented for the particular case of a smooth channel characterized by a generating diameter equal to 1 m. The results show the influence of the slope on the frequency of occurrence of the critical regime. At the same time and independently of the flow rate, a very advantageous approach for the calculation of the Froude number has been proposed. The study shows that there are six zones to differentiate the various flow states, namely: on the one hand for steep slopes two subcritical zones interspersed by a supercritical zone and on the other hand for mild slopes a zone corresponding to uniform flow, an area where the flow is probably gradually varied and finally an area where the flow is abruptly varied. Based on the specific energy equation, a validation process concluded that the proposed relationships were reliable.

Improvements to solutions for normal depth in multiple sections of tunnels

The normal depth is an essential parameter for the design, operation, and maintenance of open channels. The circular, city-gate crossing and egg-shaped sections are often used in non-pressure tunnels in hydraulic engineering, agricultural irrigation, and sewerage works. However, the equations governing the normal depth in various sections are complicated implicit transcendental equations. In order to improvements the solutions for normal depth in multiple sections of tunnels, a function model is first established for the geometric features of various sections using the mathematical transform method and while considering non-dimensional parameters. Then implement the revised PSO algorithms in MATLAB, and establish three right solution's formulas for the normal depths in different non-pressure tunnel sections through optimization. The error analysis results and project cases show that the established formula has broad applicability. The maximum relative errors of the formula for normal depths are less than 0.07%, 0.04%, and 0.07% in circular, city-gate crossing, and egg-shaped sections, respectively, which are more accurate than the existing formulas. The results of this research may be useful in design, operation, and maintenance in tunnel engineering.

Explicit Solution for Critical Depth in Closed Conduits Flowing Partly Full

Critical depth is an essential parameter for the design, operation, and maintenance of conduits. Circular, arched, and egg-shaped sections are often used in non-pressure conduits in hydraulic engineering, irrigation, and sewerage works. However, equations governing the critical depth in various sections are complicated implicit transcendental equations. The function model is established for the geometric features of multiple sections using the mathematical transform method and while considering non-dimensional parameters. Then, revised PSO algorithms are implemented in MATLAB, and the right solution’s formula for the critical depths in various non-pressure conduit sections is established through optimization. The error analysis results show that the established formula has broad applicability. The maximum relative errors of the formula for critical depths are less than 0.182%, 0.0629%, and 0.170% in circular, arched, and egg-shaped sections, respectively, which are more accurate than those of existing formulas; the form of the formula proposed in this work is also more compact than that of the existing formulas. The results of this research may be useful in design, operation, and maintenance in conduit engineering.

Structural Evaluation of Masonry Sewer Pipes Under Transverse Settlement Due to Tunneling

The intention of this research is to perform structural evaluation on masonry sanitary sewer pipes impacted by tunnel excavation. Until about 1900, it was common to use several layers of brick to construct municipal sanitary sewer lines, and currently thousands of miles of brick sewer lines are still in service. The construction of new underground facilities inevitably disturbs the soil above and will have an impact on the existing masonry sanitary sewer pipes within the zone of influence of the excavation. Masonry pipes are typically susceptible to the longitudinal interaction failure mechanism at the hogging zone of the settlement trough due to their low tensile capacity. In this research standard egg-shaped section for brick sewer were subjected to ground settlements obtained using Attewell method. The greenfield condition is assumed to be applicable with homogeneous ground conditions. The impact of the depth to the tunnel axis and trough width parameter on the settlement trough and associated strains developed in the pipes are subjected to investigation. Tensile strain developed due to axial strain and bending moment are calculated for five different section sizes. Mitigation measures and recommendations are provided

Structural Evaluation of Masonry Sewer Pipes Under Transverse Settlement Due to Tunneling

The intention of this research is to perform structural evaluation on masonry sanitary sewer pipes impacted by tunnel excavation. Until about 1900, it was common to use several layers of brick to construct municipal sanitary sewer lines, and currently thousands of miles of brick sewer lines are still in service. The construction of new underground facilities inevitably disturbs the soil above and will have an impact on the existing masonry sanitary sewer pipes within the zone of influence of the excavation. Masonry pipes are typically susceptible to the longitudinal interaction failure mechanism at the hogging zone of the settlement trough due to their low tensile capacity. In this research standard egg-shaped section for brick sewer were subjected to ground settlements obtained using Attewell method. The greenfield condition is assumed to be applicable with homogeneous ground conditions. The impact of the depth to the tunnel axis and trough width parameter on the settlement trough and associated strains developed in the pipes are subjected to investigation. Tensile strain developed due to axial strain and bending moment are calculated for five different section sizes. Mitigation measures and recommendations are provided

Eco-technique of sewer renovation using composite shells: structural analysis

An eco-technical renovation of the sewage system is developed in this paper; this technique involves incorporating into the existing sewer a series of jointed prefabricated sandwich or composite shells. The purpose of his study is to determine the structural shell deflection, the high displacement areas and to validate the non-failure criterion for each ply constituting the inner and outer laminate facings. The numerical results were obtained at low cost by using the finite element method. Studies have focused on structural analysis of a typical shell unit with an ovoid form (egg-shaped section) when it is subjected, during annular space filling operation, to pressure forces generated by wet concrete. To ensure the safety of the composite shell structure, Tsai-Hill criterion function is applied and results are presented for the most stressed plies Keywords: sanitation; sustainable development; composite collector en composite; sandwich shell; finite element approach

The Direct Solution on Normal Depth of an Egg-Shaped Section Tunnel

An egg-shaped is consisted of four parts: One arc bottom, two side circular arc arch and one circular arc roof umbrella arch. It is complex shape. Aiming at traditional methods there are complicated and the defects of inaccurate. on the basis of summarizing the predecessors' research, Based on the normal water depth basic equation mathematical transformation of egg-shaped sections, the interrelation between dimensionless water depth and the dimensionless parameter were analyzed, We use the power function as the formula form, with minimum residual standard deviation as the goal, the coefficients of the formula can be calculated based on the theory of optimization and regression. Then, we get calculation formula for normal water depth. This formula is not a piecewise function, and need not to choose the suitable formula according to the dividing flow. The results can be directly calculated. Results of the example calculation and error analysis showed that the maximum relative error of normal water depth was 0.675% respectively in the utility range (The ratio of water depth divided by height of tunnel is from 0.05 to 0.85). Using this algorithm, above 92 % of the computation error on normal water depth was less than 0.25%. The average error of the formulas was not more than 0.118% .The hydraulic calculation formula will be widely used in engineering design and project management due to its high accuracy and simple form.

Buckling of Stiffened Cylindrical Shells of Circular-Arc Cross Section

SummaryCylinders of non-circular cross-section which can be constructed by circular arcs and straight line segments are analysed. Differential equations and boundary conditions for the buckling of eccentrically stiffened, isotropic or orthotropic, circular panels of zero Gaussian curvature are derived through variation of the total potential. Eight equations of continuity along the generators of adjacent arc-segments are enforced simultaneously through the use of Lagrangian multipliers which, with the potential minimised with respect to the admissible displacement functions, constitute an approximate solution of the cylinder problem. Numerical examples include stiffened cylinders with oval and egg-shaped sections. The buckling strengths of these non-circular cylindrical shells under bending are compared with their circular counterparts on the basis of equal circumferential length.

Design and Construction Table for Egg-Shaped Sewers

The design for the cross-section of the first brick-sewers constructed in Cleveland (except for steep grades) was substantially the usual egg-shaped section common in England and in this country. T...