On compressible flow in a rotating cylinder

Summary

Axisymmetric steady flow of a perfect gas in a rotating cylinder is studied by applying a linearised analysis to a small perturbation about isothermal rigid body rotation. Motivated by present day gas centrifuges, special attention is focussed on the effect of a length-to-radius ratio which increases from unit magnitude to infinity and on the effect of a strong radial density gradient associated with the isothermal rigid body rotation. The Ekman number E _*based on the small radial density scale and the density at the cylinder wall is taken to be small. It appears that the flow outside Ekman boundary layers at the end caps consists of three types. These correspond to 1 ≪ L _* ≪ E ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]}_* E ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]}_* ∼ L _*, ≪ E ⁻¹_* andE ⁻¹_* ∼L _* where L _*is the ratio of the cylinder-length to the radial density scale. For 1 ≪ L _* ≪E ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]}_* an inviscid flow in a region of limited thickness near the cylinder wall is found. Due to the strong decrease of the density, radial diffusion is not confined to Stewartson boundary layers at the wall (typical for incompressible flow) but extends in the core. This finds expression in two layers in the centre of the cylinder, parallel to the rotation axis, having a structure similar to both Stewartson layers and adjusting the inviscid flow near the wall to a flow dominated by radial diffusion near the rotation axis. For L _* ∼ E ^{% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% qaaSqaaiaaigdaaeaacaaIYaaaaaaa!386D!\[ - \tfrac{1}{2}\]}_* and L _* ∼ E ⁻¹_* both Stewartson layers become successively of the same thickness as the density scale. At the same time the corresponding layers in the core go to the wall and join. As a result, for L _* ≥ E ⁻¹_* radial diffusive processes are significant in the entire cylinder, a situation also known from studies of flows in semi-infinite gas centrifuges.