Improved mean-flow solution for slab rocket motors with regressing walls

Abstract

The Navier-Stokes equations are solved to obtain an exact description of the internal mean flow in a slab rocket motor having a rectangular cross section and two equally regressing porous walls. The scope is limited to two-dimensional incompressible and nonreactive laminar flow. In seeking an exact solution, similarity transformations are used in both space and time. Subsequently, the original problem is reduced to a single fourth-order differential equation with four boundary conditions. The exact similarity transformations involve a linearly varying axial velocity, an axially independent normal velocity, and a constant dimensionless regression ratio. The emerging nonlinear differential equation is solved both numerically and asymptotically, using regular perturbations in the injection Reynolds number R. Results are correlated and compared via variations in R and the dimensionless wall regression rate α. For hard blowing and moderate regression rates, the effect of varying α is found to be small. This justifies the use of Taylor's mean-flow profile in high Reynolds number applications. For small regression rates, the agreement between analytical and numerical solutions appears to be quite satisfactory. Since most physical problems correspond to small values of α, the analytical solution constitutes a practical equivalent to the numerical solution over a wide range of R. By way of verification, the governing equation we obtain is reducible to the classic Berman equation when regression is suppressed. Moreover, the asymptotic solution reduces to Taylor's classic expression for large injection. From the analytical formulation, closed-form expressions are produced for the velocity, pressure and shear stress distributions.

On the rotational compressible Taylor flow in injection-driven porous chambers

“…To date, Taylor's incompressible model remains one of the most cited and used in solid rockets. It has been considered by Traineau et al (1986), Balakrishnan et al (1991), Beddini & Roberts (1992), Apte & Yang (2001), Balachandar, Buckmaster & Short (2001), Zhou & Majdalani (2002), Najjar et al (2003) and many others. Two key factors justify its ubiquitous use: the suitability of non-reactive models to simulate the idealized environment in a rocket motor and the validity of using a non-deformable permeable chamber with non-regressing walls.…”

Some Recent Developments in Rocket Core Dynamics

“…Nevertheless, porosity does not elude theoretical idealizations. It is recognized and described by Zhou and Majdalani 36 as the agent in control of the wall injection coefficient. As for the oscillating pressure forces, only in the case of resonant burning does the fluid-structure interaction become a concern.…”

“…Aside from these sequential contributions, it must be noted that recent efforts have been successful in producing analytical core flow solutions that are sensitive to the movement of transpiring boundaries. The reader is referred, in that regard, to the planar and axisymmetric solutions presented by Zhou and Majdalani 36,44 or Majdalani, Vyas, and Flandro. 45,46 Therein, wall regression is accounted for alongside viscous and rotational features.…”

“…In the case of wall regression that was particularly pointed out, recent findings have indicated its small influence on the overall acoustic field. 36,44,45 This could perhaps justify its dismissal in many laboratory, computational, and theoretical investigations.…”

Energy Steepened States of the Taylor-Culick Profile

“…[15][16][17] Its accuracy has been verified in several investigations including computational, 18-20 experimental, [20][21][22] and theoretical pursuits carried out in both cylindrical 14 and planar (slab) configurations. 23,24 Recently, it has been extended to noncircular cross-sections by Kurdyumov. 25 It is clearly one of the most ubiquitous cold flow approximations for a full-length cylindrical motor.…”