Acoustic-Rotational Internal Flow Caused by Transient Sidewall Mass Addition

Abstract

Asymptotic and numerical methods are used to describe thermal transients in an internal flow caused by time-dependent, spatially distributed sidewall mass addition. Solutions are obtained for the temperature distribution and wall heat transfer, as well as the vorticity, in a high Reynolds number (Re), low Mach number (M), compressible flow in a cylinder. A multiple-scale analysis, valid in the limit of $M\to 0$ and $Re\to\infty$, is used to derive an equation for O(M) acoustic disturbances arising from transient injection. The limit process is also used to obtain reduced equations for the rotational axial velocity field and the nonacoustic temperature variation arising from a balance of convection and transverse diffusion. The convection-diffusion equations are characterized by viscous and conductive transport on an O(M) radial scale relative to the O(1) nondimensional cylinder radius. These small-scale diffusive effects are pervasive throughout the entire cylinder in this large Re, injected flow. The thermal analysis presented here shows that the transient temperature disturbance consists of an O(M) acoustic component and an O(M) radially dependent "rotational" component arising from the convection and diffusion of large radial gradients. The latter are generated on the injection surface and then gradually fill the cylinder as time elapses. The radial gradient of temperature is O(1), although the temperature disturbance is only O(M). Results for the radial and axial variations of the instantaneous temperature distribution imply that the O(M) acoustic and "rotational" temperature components make the largest contributions to the total energy transient. Smaller kinetic energy effects appear only at O(M2 ). These results emphasize the importance of modeling intense thermal transients, including the sidewall heat transfer, in addition to the more familiar vorticity distributions.