Analysis of High-Speed Rotating Flow in 2D Polar (r - θ) Coordinate
Authors: Pradhan Dr. Sahadev 
Published: February 2020

The generalized analytical model for the radial boundary layer in a high-speed rotating cylinder is formulated for studying the gas flow field 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 curved surface in terms of master potential (χ), which is derived from the equations of motion in a polar (r - θ) plane. The linearization approximation (Wood & Morton (JFM-1980); Pradhan & Kumaran (JFM-2011); Kumaran & Pradhan (JFM-2014)) 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^{−1/3} R) at the wall of the cylinder. Here, the stratification parameter A = √((m Ω^2 R^2)/(2 k_B T)). This parameter A is the ratio of the peripheral speed, ΩR, to the most probable molecular speed, √(2 k_B T/m), the Reynolds number Re = (ρ_w Ω R^2/μ), where m is the molecular mass, and Ω and R are the rotational speed and radius of the cylinder, k_B is the Boltzmann constant, T is the gas temperature, ρ_w is the gas density at wall, and μ is the gas viscosity.

The analytical solutions are compared with Direct Simulation Monte Carlo (DSMC) simulations. The comparison reveals that the boundary conditions in the simulations and analysis have to be matched with care. The commonly used ’diffuse reflection’ boundary conditions at solid walls in DSMC simulations result in a non-zero slip velocity as well as a ‘temperature slip’ (gas temperature at the wall is different from wall temperature) (Pradhan & Kumaran (JFM-2011); Kumaran & Pradhan (JFM-2014)). 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 are accurately satisfied in the simulations. When these precautions are taken, there is excellent agreement between analysis and simulations, to within 15%. The major advantage of the present formulation is that it is not restricted to the asymptotic limit of high stratification parameter, even though we have assumed that the Reynolds number is high.

In a high speed rotating field, we examine the mass flow rate through the stationary insert (intake tube). The simulations show that it varies significantly due to the equilibrium back pressure maintained at the rare end of the intake tube. An important finding is that the stagnation pressure (no mass flow through the intake tube) is significantly affected by the wall gap, as well as with stratification parameter, indicating a strong coupling between the local temperature, density, pressure and velocity fields.

Keywords: Rotating flows, generalized analytical model, polar (r - θ) plane, DSMC simulations, Rarefied gas flow.

References:

[1] Pradhan, S. & Kumaran, V. 2011 The generalized Onsager model for the secondary flow in a high-speed rotating cylinder. J. Fluid Mech. 686, 109 - 159.

[2] Kumaran, V. & Pradhan, S. 2014 The generalized Onsager model for a binary gas mixture. J. Fluid Mech. 753, 307 - 359.

[3] Wood, H.G. & Morton, J.B. 1980 Onsager’s pancake approximation for the fluid dynamics of a gas centrifuge. J. Fluid Mech. 101, 1- 31.