In this study, the transverse vorticoacoustic wave in a circular cylinder is characterized for a variable velocity profile at the injector faceplate. This particular configuration mimics the conditions leading to the onset of traveling radial and tangential waves in a simple liquid rocket engine (LRE). To capture the unsteady behavior in this physical setting, we consider a short thrust chamber with an injecting headwall and combine the benefits of three techniques: regular perturbations, Helmholtz decomposition, and boundary layer theory. First, regular perturbations are leveraged to linearize the equations of motion and, in the process, help to identify the unsteady interaction equations. Second, a Helmholtz decomposition of the first-order disturbance equations gives rise to a compressible, inviscid, and acoustic set that is responsible for driving the unsteady motion. This is accomplished in conjunction with an essentially incompressible, viscous, and vortical set that materializes by virtue of coupling with the acoustic wave at the boundaries. After recovering the acoustic mode from the resulting wave equation, the last step is to solve for the vortical mode by applying boundary layer theory and a judicious expansion of the rotational set with respect to a small viscous parameter, . After some effort, an explicit formulation for variable headwall injection is obtained and validated by means of a limiting process verification that is based on two previously investigated cases, the uniform and bell-shaped injection profiles. The solution is then illustrated using two new configurations corresponding to laminar and turbulent profiles. In the process of comparing the four representative cases, the characteristics of the vorticoacoustic wave, including its penetration depth, spatial wavelength, and overshoot factor, are systematically explored and discussed. Most characteristics are found to depend on the penetration and Strouhal numbers along with the distance from the centerline. Along the axis of the chamber, the waves attributed to different injection profiles behave similarly to the extent that behavioral deviations among them increase as the sidewall is approached. This work also accounts for the presence of a downstream boundary that stands to produce left-traveling reflections whose pairing with the right-traveling waves promotes the establishment of a standing wave environment. The combined waves are formulated analytically and shown to be appreciable in view of their amplitudes that twice exceed those associated with traveling waves.
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
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