Pradhan, S (2016) Analysis of high-speed rotating flow in 2D polar (r � θ) coordinate. In: 14th International Energy Conversion Engineering Conference, 2016, 25-27 July 2016, Salt Lake City; United States.
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Abstract
The generalized analytical model for the radial boundary layer in a high-speed rotating cylinder is formulated for studying the secondary radial flow due to insertion of mass, momentum and energy into the rotating cylinder in the polar (r�θ) plane. The analytical solution includes the sixth order differential equation for the radial boundary layer at the cylindrical surface in terms of master potential (�), which is derived from the equations of motion in a polar (r�θ) plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional assumptions in the analytical model include constant temperature in the base state (isothermal condition), and high Reynolds number, but there is no limitation on the stratification parameter. In this limit, the gas flow is restricted to a boundary layer of thickness (Re�1/3R) at the wall of the cylinder. Here, the stratification parameter A = (formula presented). This quantity(A) is the ratio of the peripheral speed (ΩR) to the most probable molecular speed (formula presented), the Reynolds number Re = (�wΩR2)/μ, where m is the molecular mass, and R are the rotational speed and radius of the cylinder, kB is the Boltzmann constant, T is the gas temperature, �wis the gas density at wall, and μ is the gas viscosity. For the case of mass/momentum/energy insertion into the flow, the separation-of-variables procedure is used, and the appropriate homogeneous boundary conditions are specified so that the linear operators in the meridional and radial directions are self-adjoint. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order and second-order in the radial and meridional directions for the generalized analytical equation) are obtained. The solutions for the secondary radial flows are determined in terms of these eigenvalues and eigenfunctions. The major advantage of the present formulation is that it is not restricted to the asymptotic limit of high stratification parameter, and since we have used the conservative form of the compressible mass, momentum and energy conservation equations in deriving the generalized analytical model, obviously, the shock type solutions (Rankine-Hugoniot relations) are automatically satisfied. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations. The comparison shows that it is necessary to take certain precautions when making a quantitative comparison between analysis and simulations. The commonly used�diffuse reflection� boundary conditions at solid walls in DSMC simulations result in a non-zero slip ((Pradhan & Kumaran, J. Fluid Mech., vol. 686, 2011, pp. 109-159); (Kumaran & Pradhan, J. Fluid Mech., vol. 753, 2014, pp. 307-359)). These have to be incorporated in the analysis in order to make quantitative predictions. In the case of mass/momentum/energy sources within the flow, it is necessary to ensure that the homogeneous boundary conditions (zero temperature perturbation) are accurately satisfied in the simulations. When these precautions are taken, there is quantitative agreement (with a difference of less than 10) between the predictions of the analytical model and the DSMC simulations, even when stratification parameter is as low as 0.707, and the Reynolds number is as low as 100. The predictions of the generalized analytical model is also significantly better than that of the analyti- cal model, which is restricted to thin boundary layers in the limit of high stratification parameter. In a high speed rotating field we examine the mass flow rate through the stationary intake tube. The simulations show that the scaled mass flow rate m/(HΩ) in- creases progressively from zero, at the stagnation condition, to 0.024 as the equilibrium back pressure is reduced. Here, H is the total gas holdup. The slow down of the circum- ferential velocity of the bulk of the rotating fluid due to the presence of stationary intake tube is studied for stratification parameter in the range 0.707-3.535, and found signif- icant slow down (between 8 to 28), which induces the secondary radial flow towards the axis, and it further excites the secondary axial flow, which could be very important for the centrifugal gas separation processes. An important finding is that the stagnation pressure (no mass flow through the intake tube) is strongly affected by the wall gap, as well as with stratification parameter indicating a strong coupling between the local temperature, density, pressure and velocity fields. We have further carried out the DSMC simulations for a binary gas mixture with large mass difference (εa= (2�m/(m1+m2)) = 0.5), and compare the simulation results with those of single-component gas incorpo- rating the homogeneous boundary conditions. Here, �m = (m1�m2) is the difference in the molecular masses of the two species. An important point to note is that in all cases, internal momentum source, mass source/sink, and stationary intake tube, the peak value of the secondary radial flow for a binary gas mixture with εa= 0.5 increases by (19 - 30) compared to the single-component gas, as stratification parameter A is increased from 0.707 to 3.535, for Re = 102. © 2016, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.
| Item Type: | Conference Paper |
|---|---|
| Publication: | 14th International Energy Conversion Engineering Conference, 2016 |
| Publisher: | American Institute of Aeronautics and Astronautics Inc, AIAA |
| Additional Information: | cited By 0; Conference of 14th International Energy Conversion Engineering Conference, 2016 ; Conference Date: 25 July 2016 Through 27 July 2016; Conference Code:179019 |
| Keywords: | Analytical models; Boundary conditions; Boundary layers; Cylinders (shapes); Density of gases; Eigenvalues and eigenfunctions; Energy conversion; Equations of motion; Flow of gases; Forecasting; Gas mixtures; Mass transfer; Mathematical operators; Molecular mass; Momentum; Monte Carlo methods; Radial flow; Reynolds number; Rotational flow; Velocity, Direct simulation Monte Carlo; DSMC simulation; Energy conservation equations; High-speed rotating; Homogeneous boundary condition; Linearization approximation; Rankine-Hugoniot relations; Rarefied gas flow, Gases |
| Department/Centre: | Division of Mechanical Sciences > Chemical Engineering |
| Date Deposited: | 19 Nov 2020 09:26 |
| Last Modified: | 19 Nov 2020 09:26 |
| URI: | http://eprints.iisc.ac.in/id/eprint/65946 |
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