Rotational and Quasiviscous Cold Flow Models for Axisymmetric Hybrid Propellant Chambers

Abstract

In this work, we present two simple mean flow solutions that mimic the bulk gas motion inside a full-length, cylindrical hybrid rocket engine. Two distinct methods are used. The first is based on steady, axisymmetric, rotational, and incompressible flow conditions. It leads to an Eulerian solution that observes the normal sidewall mass injection condition while assuming a sinusoidal injection profile at the head end wall. The second approach constitutes a slight improvement over the first in its inclusion of viscous effects. At the outset, a first order viscous approximation is constructed using regular perturbations in the reciprocal of the wall injection Reynolds number. The asymptotic approximation is derived from a general similarity reduced Navier–Stokes equation for a viscous tube with regressing porous walls. It is then compared and shown to agree remarkably well with two existing solutions. The resulting formulations enable us to model the streamtubes observed in conventional hybrid engines in which the parallel motion of gaseous oxidizer is coupled with the cross-streamwise (i.e., sidewall) addition of solid fuel. Furthermore, estimates for pressure, velocity, and vorticity distributions in the simulated engine are provided in closed form. Our idealized hybrid engine is modeled as a porous circular-port chamber with head end injection. The mathematical treatment is based on a standard similarity approach that is tailored to permit sinusoidal injection at the head end.

Compressible integral representation of rotational and axisymmetric rocket flow

“…Akiki and J. Majdalani the nozzleless chamber length. While the incompressible motion is relatively well understood (Taylor 1956;Culick 1966;Majdalani & Akiki 2010), recent advances have enabled us to account for the presence of arbitrary headwall injection (Majdalani & Saad 2007b;Saad & Majdalani 2009), wall regression (Majdalani, Vyas & Flandro 2002;Zhou & Majdalani 2002), grain taper (Saad, Sams & Majdalani 2006;Sams, Majdalani & Saad 2007), variable cross-section (Kurdyumov 2006), headwall singularity (Chedevergne, Casalis & Féraille 2006), viscous effects (Majdalani & Akiki 2010), and stability (Chedevergne, Casalis & Majdalani 2012). Furthermore, flow approximations exhibiting smoother or steeper profiles than the cold flow equilibrium state have been studied in connection with their energy content by Majdalani & Saad (2007a) and Saad & Majdalani (2008).…”

On the swirling Trkalian mean flow field in solid rocket motors

“…After determining a generic relationship between the vorticity and the streamfunction, which can satisfy the VTE identically, substitution into the vorticity equation leads to a second-order partial differential equation that can be solved for ψ. As further Swirling motion in rockets 271 confirmed in the analogous treatment of the problem with viscosity and headwall injection (Majdalani & Akiki 2010), the VTE may be fulfilled by taking Ω θ = rΞ (ψ). To ensure linearity of the ensuing vorticity equation, Culick (1966) then inserts Ξ = C 2 ψ into the vorticity equation, Ω = ∇ × u (which may be written in terms of ψ only), to the extent of arriving at…”

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

“…The Taylor-Culick solution was originally verified to be an adequate representation of the expected flowfield in SRMs both numerically by Sabnis et al [44] and experimentally by Dunlap et al [45,46], thereby confirming its character in a nonreactive chamber environment. It was extended by Majdalani and Akiki [4] to include effects of viscosity and headwall injection, by Saad et al [42] and Sams et al [43] to account for wall taper, by Kurdyumov [47] to capture effects of irregular cross sections, and by Majdalani and Saad [48] to allow for arbitrary headwall injection. Then, using variational calculus and the Lagrangian optimization principle, Saad and Majdalani [49] uncovered a continuous spectrum of Taylor-like solutions exhibiting increasing kinetic energy signatures, while ranging from the traditional Culick profile down to its predecessor, the irrotational mean flow known as the Hart-McClure profile [50,51].…”

“…= total number of points in the integration domain n p = point where G 0 n p 0 n Taylor = number of points for which the Taylor series expansion is applied Re = Reynolds number r = radial coordinate S = surface t = time U m = mean axial velocity u = velocity vector V = absolute injection velocity as seen by a stationary observer V = volume V w = injection velocity with respect to the moving wall z = axial coordinate α = wall expansion ratio β = scaled expansion ratio, α∕b γ i = ith coefficient in the Taylor series expansion ϵ = small real number used in numerical integration ε = small perturbation parameter η = radial transformation variable, 1∕2r 2 λ = scaling factor, 2∕Re ν = kinematic viscosity ξ = transformation variable, bη ξ max = domain integration size ξ p = coordinate location corresponding to n p ρ = density ψ = Stokes stream function Ω = vorticity vector Subscripts exit = denotes an exit or outlet section m = denotes a mean value max = denotes a maximum r, z = radial and axial component or partial derivative w = wall variable θ = tangential component 0 = fixed reference or initial value Superscript = dimensional variable I. Introduction V ISCOUS motion in cylindrical chambers with sidewall injection is of interest in a variety of applications, including mean flow modeling of solid [1][2][3] and hybrid rockets [4,5], sweat cooling [6,7], boundary-layer control [8][9][10], peristaltic pumping [11,12], gaseous diffusion, and isotope separation [13][14][15]. It is the latter group of studies by Berman [13,14] that has actually provided the impetus to develop the first similarity transformation of the Navier-Stokes equations into a fourth-order nonlinear ordinary differential equation (ODE).…”