Considerations for Pressure Coupling Calculations in Combustion Instability

Abstract

In solid rockets, it is well understood that an unsteady pressure field excites an unsteady burning response. Often this leads to a positive net work on the fluid flow. This manifests as “pressure coupling:” a significant combustion instability driving mechanism. This mechanism can be expressed mathematically as a volumetric work integral describing the divergence of steady and unsteady field variables. The divergence theorem allows for volume integrals to be expressed as surface integrals. In solid rocket motors, it is customary to express the unsteady work (and therefore the pressure coupling) as an integral over the burning surface. However, a disparity arises if the control volume surface does not coincide with the burning surface. For instance, if the control volume surface is taken to be the inner surface of the motor case, no flux is present and the unsteady work is zero. A similar problem arises in liquid rocket engines when complicated feed system and injector configurations are included. Ideally, the control volume in question is the total volume contained within the rigid, physical hardware of feed system (where applicable) and combustion chamber. This is most appropriate because it contains all fluid dynamic and chemical processes while remaining constant in time. However, since no flux occurs through any of the physical hardware surfaces, pressure coupling can only be correctly described as a volumetric effect. In this paper, the volumetric form of the unsteady work term is considered. As a result, a consistent definition and procedure for pressure coupling calculations can be applied to both solid and liquid rockets. Moreover, a volumetric formulation facilitates the calculation of pressure coupling resulting from droplet/acoustic field interactions. This appears as a new driving mechanism in liquid propulsion devices. Nomenclature a0 = speed of sound a = chamber radius D = energy system source terms H = heat release H = total enthalpy h = enthalpy E = system energy E2 m = stored energy L = chamber length m = mass flow rate m = integer mode number p = pressure shape function Q = heat generation q = heat flux Rm = acoustic amplitude function T = temperature U = quasi-steady velocity vector, Urer + Uθeθ + Uzez u = velocity shape function W = energy system work terms ∗Stability Engineer, Member AIAA. †Stability Engineer, Member AIAA. 1 American Institute of Aeronautics and Astronautics Greek Φ = dissipation function μ = dynamic viscosity Ω = quasi-steady vorticity ω = frequency of oscillation, ωr + iωi ω1 = unsteady vorticity ψ = viscous stress terms ρ = density Subscripts m = denotes the mode index 0, 1 = denotes a quasi-steady and unsteady variable, respectively Superscripts i = denotes the imaginary part of a complex number r = denotes the real part of a complex number