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447419 Analysis of High-Speed Rotating Flow in Polar* (r-θ)* Coordinate

*(r-θ)*Coordinate

**Title: Analysis of High-Speed Rotating Flow in 2D Polar (r−θ) Coordinate**

**Author : Dr. Sahadev Pradhan **

**Affiliation : Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India**

**Abstract :**

The generalized analytical model for the radial boundary layer in a high-speed rotating cylinder is formulated for studying the secondary gas flow field due to insertion of mass, momentum and energy into the rotating cylinder [1, 2, 3, 4, 5, 6, 7] in the polar* (r - θ)* plane. The analytical model 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 ((Pradhan & Kumaran, *J. Fluid Mech*., vol. 686, 2011, pp. 109-159); (Kumaran & Pradhan, *J. Fluid Mech*., vol. 753, 2014, pp. 307-359)) 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 = √(mΩ*. This parameter

^{2}R^{2}/(2k_{B}T))*A*is the ratio of the peripheral speed,

*(ΩR)*, to the most probable molecular speed,

*√(2k*, the Reynolds number

_{B}T /m)*Re = (ρ*, where

_{w }Ω R^{2})/µ*m*is the molecular mass,

*Ω*and

*R*are the rotational speed and radius of the cylinder,

*k*is the Boltzmann constant,

_{B}*T*is the gas temperature,

*ρ*is the gas density at the wall, and

_{w}*µ*is the gas viscosity. 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.

The analytical solutions are compared with direct simulation Monte Carlo (DSMC) simulations [8]. The comparison shows that it is necessary to take certain precautions when making a quantitative comparison between analysis and simulations. It is necessary to ensure that the wall velocity slip and ’temperature slip’ (difference between gas temperature at the wall and wall temperature) are accurately incorporated in the analytical solutions ((Pradhan & Kumaran, *J. Fluid Mech*., vol. 686, 2011, pp. 109-159); (Kumaran & Pradhan,* J. Fluid Mech*., vol. 753, 2014, pp. 307-359)). It is also necessary to accurately enforce the homogeneous boundary conditions at the walls in the simulations. When these precautions are taken, we find that there is quantitative agreement (with a difference of less than 15%) between the predictions of the analytical model and the simulations even when the secondary radial flow is as large as the 20% of the base solid-body rotation, the stratification parameter is as low as 0.707, and the Reynolds number is as low as 100.

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)* increases progressively from zero, at the stagnation condition, to 0.024 as the equilibrium back pressure is reduced. Here, *H* is the total gas holdup and Ω is the rotational speed. The slow down of the circumferential 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 significant 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.

**Key words:** High speed rotating flows, Generalized analytical model in polar *(r - θ) *plane, DSMC Simulations, Rarefied gas flow.

**References** :

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

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

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

4. H. G. Wood and G. Sanders. Rotating compressible flows with internal sources and sinks. *J. Fluid Mech*., 1983, 127, 299 - 311.

5. H. G. Wood and R. J. Babarsky. Analysis of a rapidly rotating gas in a pie-shaped cylinder. *J. Fluid Mech*., 1992, 239, 249 - 271.

6. H. G. Wood and J. A. Jordan and M. D. Gunzburger. The effect of curvature on the flow field in rapidly rotating gas centrifuges. *J. Fluid Mech*., 1984, 140, 373 - 395.

7. D. R. Olander. The theory of uranium enrichment by the gas centrifuge. *Prog. Nucl. Energy.*, 1981, 8, 1 - 33.

8. G. A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. *Clarendon Press*, 1994.

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