Erratum on Higher Mean-Flow Approximation for a Solid Rocket Motor with Radially Regressing Walls

Majdalani, Joseph; Vyas, Anand B.; Flandro, Gary A.

doi:10.2514/1.40061

Cited by 22 publications

(27 citation statements)

References 2 publications

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“…Although the problem in a tube with expanding or contracting surfaces was treated by Uchida and Aoki [12] in the late 1970s, the effects of an injecting sidewall were incorporated more than a decade later by Goto and Uchida [55], in the context of a pulsating porous tube and by Majdalani et al [1,2] in the context of a rocket chamber. In their work, the effect of wall regression was prescribed by a dimensionless wall expansion ratio, α, written as a viscous Reynolds number based on the radial regression speed of the sidewall.…”

Section: Doi: 102514/1j055949mentioning

confidence: 99%

“…In this article, we revisit the problem of viscous motion in a porous tube and allow the radius to transiently expand or contract [1,2]. As before, we employ a dual similarity transformation in space and time to reduce the Navier-Stokes equations into a nonlinear fourth-order ODE that can be solved both asymptotically and numerically.…”

Section: Doi: 102514/1j055949mentioning

confidence: 99%

“…= 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).…”

Section: Doi: 102514/1j055949mentioning

confidence: 99%

“…As before, we employ a dual similarity transformation in space and time to reduce the Navier-Stokes equations into a nonlinear fourth-order ODE that can be solved both asymptotically and numerically. After reconstructing the analytical solution of Majdalani et al [1,2] for large injection, we employ two perturbation parameters, the injection Reynolds number Re and the wall expansion ratio α to formulate an asymptotic series for the case of small Reynolds number and smallto-moderate α. We then introduce an efficient numerical approach that overcomes the singularity encountered in this model, specifically in the form of an intrinsically satisfied limit at the origin.…”

Section: Doi: 102514/1j055949mentioning

confidence: 99%

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Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

Saad

Majdalani

2017

AIAA Journal

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The mean gaseous motion in solid rocket motors has been traditionally described using an inviscid solution in a porous tube of fixed radius and uniform wall injection. This model, usually referred to as the Taylor-Culick profile, consists of a rotational solution that captures the bulk gaseous motion in a frictionless rocket chamber. In practice, however, the port radius increases as the propellant burns, thus leading to time-dependent effects on the mean flow. This work considers the related problem in the context of viscous motion in a porous tube and allows the radius to be time dependent. By implementing a similarity transformation in space and time, the incompressible Navier-Stokes equations are first reduced to a nonlinear fourth-order ordinary differential equation with four boundary conditions. This equation is then solved both numerically and asymptotically, using the injection Reynolds number Re and the dimensionless wall regression ratio α as primary and secondary perturbation parameters. In this manner, closedform analytical solutions are obtained for both large and small Reynolds number with small-to-moderate α. The resulting approximations are then compared with the numerical solution obtained for an equivalent third-order ordinary differential equation in which both shooting and the irregular limit that affects the fourth-order formulation are circumvented. This code is found to be capable of producing the stable solutions for this problem over a wide range of Reynolds numbers and wall regression ratios.

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Section: Doi: 102514/1j055949mentioning

confidence: 99%

Section: Doi: 102514/1j055949mentioning

confidence: 99%

Section: Doi: 102514/1j055949mentioning

confidence: 99%

Section: Doi: 102514/1j055949mentioning

confidence: 99%

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Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

Saad

Majdalani

2017

AIAA Journal

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“…In the same vein, Erdogan & Imrak (2008) presented a laminar solution for the flow in a porous tube. Their solution was obtained by expanding the velocity field as a series of modified Bessel functions of order n. As for the problem involving wall regression, it was tackled by Dauenhauer & Majdalani (2003), Zhou & Majdalani (2002), and Majdalani & Zhou (2003) for the slab with regressing sidewall, and by Goto & Uchida (1990) and Majdalani et al (2002) for the internal burning cylinder with expanding walls (see also Majdalani et al, 2009, for an error-free form). The next noteworthy improvement in this area consists of the compressible Taylor-Culick profile that was first presented in multiple dimensions by Majdalani (2007b).…”

Section: Generalizedmentioning

confidence: 99%

Internal Flows Driven by Wall-Normal Injection

Majdalani¹,

Saad²

2012

Advanced Fluid Dynamics

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Direct numerical simulation and biglobal stability investigations of the gaseous motion in solid rocket motors

2012

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In this article, a biglobal stability approach is used in conjunction with direct numerical simulation (DNS) to identify the instability mode coupling that may be responsible for triggering large thrust oscillations in segmented solid rocket motors (SRMs). These motors are idealized as long porous cylinders in which a Taylor-Culick type of motion may be engendered. In addition to the analytically available steadystate solution, a computed mean flow is obtained that is capable of securing all of the boundary conditions in this problem, most notably, the no-slip requirement at the chamber headwall. Two sets of unsteady simulations are performed, static and dynamic, in which the injection velocity at the chamber sidewall is either held fixed or permitted to vary with time. In these runs, both DNS and biglobal stability solutions converge in predicting the same modal dependence on the size of the domain. We find that increasing the chamber length gives rise to less stable eigenmodes. We also realize that introducing an eigenmode whose frequency is sufficiently spaced from the acoustic modes leads to a conventional linear evolution of disturbances that can be accurately predicted by the biglobal stability framework. While undergoing spatial amplification in the streamwise direction, these disturbances will tend to decay as time elapses so long as their temporal growth rate remains negative. By seeding the computations with the real part of a specific eigenfunction, the DNS outcome reproduces not only the imaginary part of the disturbance, but also the circular frequency and temporal growth rate associated with its eigenmode. For radial fluctuations in which the vorticoacoustic wave contribution is negligible in relation to the hydrodynamic stability part, excellent agreement between DNS and biglobal stability predictions is ubiquitously achieved. For axial fluctuations, however, the DNS velocity will match the corresponding stability eigenfunction only when properly augmented by the vorticoacoustic solution for axially travelling waves associated with the Taylor-Culick profile. This analytical approximation of the vorticoacoustic mode is found to be quite accurate, especially when modified using a viscous dissipation function that captures the decaying envelope of the inviscid acoustic wave amplitude. In contrast, pursuant to both static and dynamic test cases, we find that when the frequency of the introduced eigenmode falls close to (or crosses over) an acoustic mode, a nonlinear mechanism is triggered that leads to the emergence of a † Instability waves in solid rocket motors 191 secondary eigenmode. Unlike the original eigenmode, the latter materializes naturally in the computed flow without being artificially seeded. This natural occurrence may be ascribed to a nonlinear modal interplay in the form of internal, eigenmode-toeigenmode coupling instead of an external, eigenmode pairing with acoustic modes. As a result of these interactions, large amplitude oscillations are induced.

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Erratum on Higher Mean-Flow Approximation for a Solid Rocket Motor with Radially Regressing Walls

Cited by 22 publications

References 2 publications

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

Internal Flows Driven by Wall-Normal Injection

Direct numerical simulation and biglobal stability investigations of the gaseous motion in solid rocket motors

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