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The incompressible flow equations are function of the pressure gradients and not the pressure. The most important issue in solution of flow equations of incompressible fluid is the pressure gradient vector which is appearing as a source term in the momentum equation, but does not have any obvious equation coupling it with other dependent variables. Accurate numerical solutions are obtained for the incompressible Navier Stokes equations in primitive variables. Explicit finite difference scheme computer code is developed to solve incompressible flow equations.In this study, consistent with the physics of incompressible flows, the velocity and pressure gradient vectors are considered as the dependent variables. In this case, that satisfies continuity equation to machine zero, the pressure gradient vector increases the number of dependent variables which requires additional equations to close the system of governing equations. Additional equations are obtained by reformulating the continuity equation and adding a time derivative term for the pressure gradient.Upon, convergence of the numerical solution, the continuity equation will be satisfied to an arbitrary constant. To enforce that constant to be zero, the continuity equation is set to be zero on the boundary of the solution domain. It is important to note that the curl of the reformulated continuity equation automatically satisfies the curl of the pressure gradient identity. This scheme is applicable for two and three dimensions, inviscid and viscous flows. Multistage axial compressor has an advantage of lower stage loading as compared to a single stage. Several stages with low pressure ratio are linked together which allows for multiplication of pressure to generate high pressure ratio in an axial compressor.Since each stage has low pressure ratio they operate at a higher efficiency and the efficiency of multi-stage axial compressor as a whole is very high.Although, single stage centrifugal compressor has higher pressure ratio compared with an axial compressor but multistage centrifugal compressors are not as efficient because the flow has to be turned from radial at outlet to axial at inlet for each stage. The present study explores the advantages of extending the axial compressor efficient flow path that consist of rotor stator stages to the centrifugal compressor stage. In this invention, two rotating rows of blades are mounted on the same impeller disk, separated by a stator blade row attached to the casing. A certain amount of turning can be achieved through a single stage centrifugal compressor before flow starts separating, thus dividing it into multiple stages would be advantageous as it would allow for more flow turning.Flow characteristics of the novel multistage design are compared with a single stage centrifugal compressor. The flow path of the baseline and multi-stage compressor are created using 3DBGB tool and DAKOTA is used to optimize the performance of baseline as well novel design. The optimization techniques used are Genetic algorithm followed by Numerical Gradient method. The multi-stage compressor is more efficient with a higher pressure ratio compared with the base line design for the same work input and initial conditions

Conservation of atmospheric mass is one of the fundamental concepts used in the study of meteorology. Any time precipitation occurs, however, atmospheric mass is not conserved. As precipitation is removed from the atmosphere, the hydrostatic pressure in the precipitating region is reduced. In a tropical cyclone, the heaviest precipitation occurs in the eyewall, and this localized region of heavy precipitation leads to mass and moisture convergence towards the center of the tropical cyclone. The Coriolis force deflects the converging air, thereby contributing to vorticity generation and increasing the cyclonic wind speed. Moisture convergence can also enhance precipitation. In addition, potential vorticity (PV) increases as a result of the precipitation mass sink. To assess the significance of the precipitation mass sink, several hypothesis tests are performed. The MM5 model is used to create a physically realistic dataset of Hurricane Lili (2002) from which mass and PV budgets can be performed. The mass budget reveals that in a 100-km radius cylinder around the model storm center, the 5-h total and 1-h average pressure-equivalent mass loss due to precipitation during forecast hours 30 to 35 are -7.25 hPa and -1.45 hPa h⁻¹, respectively, while the 5-h total and 1-h average model surface pressure change during that time are -2.29 hPa and -0.46 hPa h⁻¹, respectively. Although the surface pressure change in the cylinder is mostly due to large cancellation between strong low-level convergence and stronger upper-level divergence, the continuous removal of mass via precipitation represents a non-negligible effect in the mass budget. The PV budget reveals that in the same cylinder, the average hourly instantaneous mass sink PV tendency during forecast hours 30 to 35 is 0.42 PVU day-1, while the average hourly instantaneous diabatic PV tendency is -2.19 PVU day⁻¹. The diabatic PV tendency exhibits large spatial and temporal cancellation, so the small but continuously positive PV contribution from the precipitation mass sink has a non-negligible effect on the PV budget. In addition, according to the PV tendency terms, an air parcel rising through the troposphere in the eyewall should experience nearly continuous PV generation via the precipitation mass sink but both PV generation and destruction via latent heat release, leading to large cancellation from the latter. Therefore, the parcel upon reaching the upper troposphere can likely attribute a non-negligible amount of its PV to the precipitation mass sink, but trajectory computations would be needed to quantify the relative contributions of the PV tendency terms on a given parcel. In addition to the mass and PV budgets, the workstation version of the Eta model is used to perform multiple sensitivity experiments with and without the precipitation mass sink. The most realistic of these sensitivity experiments reveals that the precipitation mass sink generally reduces the central pressure by ~5-7 hPa, increases the wind field by ~5-15 kt, and increases the precipitation rate by ~5-25 mm h⁻¹, but the rainfall rate difference in particular exhibits large spatial and temporal variation. PV and geopotential height cross sections show maximum precipitation mass sink-induced PV increase (>5 PVU) and geopotential height reduction (>4 dam) near the surface and near the melting layer. The results of this study suggest that the precipitation mass sink should not be neglected in tropical cyclones. Further research possibilities include detailed analysis of the impact of the precipitation mass sink on the tropical cyclone track, as well as the importance of the precipitation mass sink in heavily precipitating systems other than tropical cyclones.