Turbularization of an acoustic boundary layer (Stokes layer) on impermeable and permeable surfaces is analytically considered. The theoretical approach utilizes a second-order closure model of turbulence. Both an approximate, closed-form solution and a more comprehensive finite-difference solution of the time-dependent, parabolic, one-dimensional governing equations are obtained. For simple acoustic boundary layers on impermeable surfaces, both the approximate solution and the numerical results for the critical acoustic Mach number required for turbulent transition are qualitatively confirmed by experiment. The calculations for acoustic boundary layers with transpiration (injection) indicate a substantial reduction of the acoustic Mach number required for transition, up to a limiting injection velocity that is frequency dependent. The results may provide a mechanism for flow-related combustion instability in practical systems, particularly solid propellant rockets, since turbularization of the near-surface combustion zone could result at relatively low-acoustic Mach numbers.
Nomenclaturea = sonic speed c p = specific heat at constant pressure / = frequency, Hz h = specific sensible enthalpy k = thermal conductivity k s = equivalent sand roughness height p = static pressure ___ q = turbulence intensity, (u ri u • ) I/2 q m = maximum value of q R = inner radius of a cylindrical duct Re c = axial-flow Reynolds number, p c ii c d/jji c Re s = injection Reynolds number, p s v s d/jJi s Re t = turbulence Reynolds number, pqA/p R u = universal gas constant t -time T = static temperature Uj = velocity vector (u, v, w) Xj = coordinate vector (x,r,z) x 0 = axial distance at which computational initial conditions are specified y = distance from surface 4> -characteristic length scale A = turbulence macrolength scale //, = viscosity v -kinematic viscosity p = density a =k/c o v = r = turbulence shear stress < > = time mean of variable Superscripts -= average of variable over turbulent fluctuations = turbulent fluctuating value of variable " = acoustic fluctuating value of variable