Acoustic boundary layers in solid propellant rocket motors using Navier-Stokes equations

Abstract

The numerical solution of laminar, two-dimensional, compressible, and unsteady Navier-Stokes equations is aimed at a complete description of acoustic boundary layers that develop above a burning pro pell ant. Such acoustic boundary layers can be responsible for the so-called flow turning losses. They can govern the local unsteady flow conditions that are seen by the burning propellant to which it finally responds. In those respects, a complete understanding of such acoustic boundary layers is essential to improve existing solid rocket stability prediction codes. The full numerical solution of the Navier-Stokes equations incorporates into the analysis all the features of two-dimensional rocket chamber mean flowfield in a natural manner. After a standing wave pattern is established through forcing at a given frequency, a special Fourier treatment is used to transform the numerical results in a form directly comparable to available linear acoustic data. The presented results indicate that the acoustic boundary layer is substantially thinner than predicted by simplified models. Moreover, its acoustic admittance is found to vary significantly along the chamber, a result that is of major importance to stability predictions. Finally, the acoustic field is found to be rotational over a significant volume of the chamber. Nomenclature a -speed of sound A = excitation amplitude A () = amplitude, or modulus of a complex quantity (2 = grid cell area C p9 C v = constant pressure, volume specific heats E = total specific internal energy, E = C V T + (u 2 + v 2 )/2 FT = complex Fourier component at excitation frequency of a time varying quantity / = frequency h = chamber half-height / = complex number, / = V -1 k -thermal conductivity L = chamber length M = Mach number n -unit normal vector Pr = Prandtl number, Pr = \iC p /k p = pressure r = perfect gas constant (Re)o = reference Reynolds number, (Re) 0 = p 0 a^L/^Q T = temperature or time period of an oscillation t = time U = column vector of conservative flow variables u = axial flow velocity v = transverse or normal flow velocity x = axial coordinate Y = dimensionless acoustic admittance, Y = p 0 a 0 v'/yp' y = transverse coordinate 7 = ratio of specific heats X = coefficient of bulk viscosity fi = dynamic viscosity v = kinematic viscosity p = density Received June 2, 1988; presented as Paper 88-2940 at the AIAA/ ASME/SAE/ASEE 24th Joint Propulsion Conference, a x ,a y = normal stresses T xy> T yx -shear stresses $() = argument of a complex quantity or phase angle (expressed in degrees) co = angular frequency 0 = vorticity Subscripts, Superscripts ()o = reference state Q = steady-state value ()' = unsteady component ()/", = injection ()/ = stagnation value

Longitudinal acoustic instabilities in slender solid propellant rockets: linear analysis

“…Notice that near the chamber axis, once the vorticity and entropy waves are damped, we recover the pure acoustic solution. Numerical studies and experiments (see Avalon & Comas 1991;Vuillot 1991;Vuillot & Avalon 1991) rapid radial oscillations near the chamber wall, recovering the classical acoustic plane wave solution near the chamber axis. According to (97), the oscillations in the solid rocket motor become linearly unstable if Re (A b 0 ) > (γ + 1)/γ.…”

“…After the introduction by Culick (1975) of this idea much research has been carried out to try to clarify this phenomenon; see Baum & Levine (1986), Kuentzmann (1991), Van Moorhem (1982) and Vuillot & Avalon (1991). The technique of Culick (1970Culick ( , 1973Culick ( , 1975 and Culick & Yang (1992), which is based on the integral form of the equations, is useful for the physical understanding of the stability mechanisms.…”

“…The wavelength in the radial direction of the oscillations is of the order l 0 = V b L/c b = a l f . These waves have been demonstrated experimentally and numerically; see, for example, Vuillot & Kuentzmann (1986), Vuillot (1991), Vuillot & Avalon (1991) and the review by Flandro (1995). There are similar radial temperature oscillations due to the heating of the fluid particles under compression by the oscillating pressure field, because they come out of the thin reaction zone with nearly constant temperature T b -and thus with an oscillating entropy -a consequence of the small value of the characteristic response time of the reaction layer (of order l f /V b in the gas phase) compared with L/c b .…”

Vorticity generation and acoustic resonance of simulated solid rocket motor chamber with high wave number wall injection

“…Vuillot and Avalon [16] predicted the presence of vorticity in an injected internal flow using a computational solution to the Navier-Stokes equation. Similar results were obtained by Smith et al [17].…”

Lessons Learned In Solid Rocket Combustion Instability

“…There have been numerous papers on the subject covering both experimental observations and modeling. 4,30,31,32,33,40,41,45,46,49,50,51,52,53,59,60,61,66,68,71,72,73 The details of this work is well beyond the scope of this report. Some important conclusions and comments about non-linear combustion instability can be made.…”