This work seeks to provide a closed-form analytical solution for the transverse vortical wave generated at the sidewall of a circular cylinder with headwall injection. This particular configuration mimics the conditions leading to the onset of traveling radial and tangential waves in an idealized liquid rocket engine (LRE) chamber. Assuming a short cylindrical enclosure with an axisymmetric injection model, regular perturbations are used to linearize the problem's conservation equations. Flow decomposition is subsequently applied to the first-order disturbance equations, thus giving rise to a compressible, inviscid, acoustic set responsible for driving the unsteady motion, and to an incompressible, viscous, vortical set driven by virtue of coupling with the acoustic mode along both the sidewall and headwall.
While the acoustic mode is readily recovered from the wave equation, the induced vortical mode is resolved using boundary layer theory and an expansion of the rotational equations with respect to a small viscous parameter, δ [delta]. Subsequently, an explicit formulation for the leading-order vortical field is derived and verified numerically. A radial penetration number akin to the Stokes or Womersley numbers is identified and found to control the penetration depth of the viscous boundary layer forming above the inert sidewall. This parameter is based on the transverse oscillation mode frequency and scales with the squared ratio of the Stokes layer and the chamber's characteristic radius.
Nomenclature0 a = speed of sound of incoming flow, 12 0 () T R L = chamber length b M = average blowing Mach number at the headwall Pr = Prandtl number, ratio of kinematic viscosity to thermal diffusivity p = pressure R = chamber radius a Re = acoustic Reynolds number, 00 aR k Re = kinetic Reynolds number, 2 0 / mn R ,, rz = radial, tangential, and axial coordinates r S = radial penetration number T = temperature t = time U = mean flow velocity vector () b Ur = blowing velocity profile at the headwall u = total velocity vector Greek = Womersley number, 1/2 0 ( / ) mn R = viscous parameter, 1/2 a Re d = dilatational parameter, 2 BL = boundary layer thickness = wave amplitude = ratio of specific heats = bulk viscosity S = Stokes number, 0 / (2 ) mn R = dynamic viscosity = kinematic viscosity, = density = mean vorticity = unsteady vorticity mn = circular frequency, 0 / mn a k R Subscripts 0 = mean chamber properties Superscripts * = dimensional variables = unsteady flow variable = steady flow variable