Transient response of a compressible fluid in a rapidly rotating circular pipe

Abstract

The transient adjustment process of a compressible fluid in a rapidly rotating pipe is studied. The system Ekman number E is small, and the assumptions of small Mach number and the heavy-gas limit (γ = 1.0) are invoked. Fluid motion is generated by imposing a step-change perturbation in the temperature at the pipe wall Tw. Comprehensive analytical solutions are obtained by deploying the matched asymptotic technique with proper timescales O(E−1/2) and O(E−1). These analytical solutions are shown to be consistent with corresponding full numerical solutions. The detailed profiles of major variables are delineated, and evolution of velocity and temperature fields is portrayed. At moderate times, the entire flow field can be divided into two regions. In the inner inviscid region, thermo-acoustic compression takes place, and the process is isothermal–isentropic with the angular momentum being conserved. In the outer viscous region, diffusion of angular momentum occurs. The principal dynamic mechanisms are discussed, and physical rationalizations are offered. The essential differences between the responses of a compressible and an incompressible fluid are highlighted.The issue of stability of the analytically obtained flow is addressed by undertaking a formal stability analysis. It is illustrated that, within the range of parameters of present concern, the flow is stable when ε ∼ O(E).