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Bi-slant submanifolds of S-manifolds are introduced, and some examples of these submanifolds are presented.

Some properties of D_i-geodesic and D_i-umbilical bi-slant submanifolds are examined.

The Riemannian curvature invariants of these submanifolds are computed, and some results are discussed with the help of these invariants.

The topic is original, and the manuscript has not been submitted to any other journal.

The purpose of this review is to comprehensively consider the material properties and processing of additive titanium alloy and provide a new perspective for the robotic grinding and polishing of additive titanium alloy blades to ensure the surface integrity and machining accuracy of the blades.

At present, robot grinding and polishing are mainstream processing methods in blade automatic processing. This review systematically summarizes the processing characteristics and processing methods of additive manufacturing (AM) titanium alloy blades. On the one hand, the unique manufacturing process and thermal effect of AM have created the unique processing characteristics of additive titanium alloy blades. On the other hand, the robot grinding and polishing process needs to incorporate the material removal model into the traditional processing flow according to the processing characteristics of the additive titanium alloy.

Robot belt grinding can solve the processing problem of additive titanium alloy blades. The complex surface of the blade generates a robot grinding trajectory through trajectory planning. The trajectory planning of the robot profoundly affects the machining accuracy and surface quality of the blade. Subsequent research is needed to solve the problems of high machining accuracy of blade profiles, complex surface material removal models and uneven distribution of blade machining allowance. In the process parameters of the robot, the grinding parameters, trajectory planning and error compensation affect the surface quality of the blade through the material removal method, grinding force and grinding temperature. The machining accuracy of the blade surface is affected by robot vibration and stiffness.

This review systematically summarizes the processing characteristics and processing methods of aviation titanium alloy blades manufactured by AM. Combined with the material properties of additive titanium alloy, it provides a new idea for robot grinding and polishing of aviation titanium alloy blades manufactured by AM.

This study aims to develop an experimentally validated three-dimensional numerical model for predicting different flow patterns produced with a gas dynamic virtual nozzle (GDVN).

The physical model is posed in the mixture formulation and copes with the unsteady, incompressible, isothermal, Newtonian, low turbulent two-phase flow. The computational fluid dynamics numerical solution is based on the half-space finite volume discretisation. The geo-reconstruct volume-of-fluid scheme tracks the interphase boundary between the gas and the liquid. To ensure numerical stability in the transition regime and adequately account for turbulent behaviour, the k-ω shear stress transport turbulence model is used. The model is validated by comparison with the experimental measurements on a vertical, downward-positioned GDVN configuration. Three different combinations of air and water volumetric flow rates have been solved numerically in the range of Reynolds numbers for airflow 1,009–2,596 and water 61–133, respectively, at Weber numbers 1.2–6.2.

The half-space symmetry allows the numerical reconstruction of the dripping, jetting and indication of the whipping mode. The kinetic energy transfer from the gas to the liquid is analysed, and locations with locally increased gas kinetic energy are observed. The calculated jet shapes reasonably well match the experimentally obtained high-speed camera videos.

The model is used for the virtual studies of new GDVN nozzle designs and optimisation of their operation.

To the best of the authors’ knowledge, the developed model numerically reconstructs all three GDVN flow regimes for the first time.