Anand B. Vyas

Higher mean-flow approximation for solid rocket motors with radially regressing walls

The bulk gas motion in a circular-port rocket motor is described using a rotational, incompressible, and viscous flow model that incorporates the effect of wall regression. The mathematical idealization developed is also applicable to semi-open porous tubes with expanding walls. Based on mass conservation, a linear variation in the mean axial velocity is ascertained. This relationship suggests investigating a spatial transformation of the Proudman-Johnson form. With the use of similar arguments, a temporal transformation is also introduced. When these transformations are applied in both space and time, the Navier-Stokes equations are reduced to a single, nonlinear, fourth-order differential equation. Following this exact Navier-Stokes reduction, the resulting problem is solved using variation of parameters and small-parameter perturbations. The asymptotic solutions for the velocity, pressure, vorticity, and shear are obtained as function of the injection Reynolds number Re and the dimensionless regression ratio a. By way of verification, it is shown that, as α/Re → 0, Yuan and Finkelsteins solutions can be restored from ours. Similarly, as α/Re → 0, Culicks inviscid profile is recovered. It is demonstrated that, for a range of small α/Re, inviscid solutions are practical. However, for fast burning propellants under development, the inviscid assumption deteriorates. Because it is applicable over a broader range of operating parameters, the current analysis leads to a closed-form mean-flow solution that can be used, instead of the inviscid profile, to 1) prescribe an adjusted aeroacoustic field, 2) describe the so-called acoustic boundary layer, 3) evaluate the viscous and rotational contributions to the acoustic stability growth rate factor, 4) track the evolution of hydrodynamic instability, and 5) accurately simulate the internal gasdynamics in rapidly regressing motors and cold-flow experiments with medium-to-high levels of injection.

Exact Solution of the Bidirectional Vortex

In this paper, we present an inviscid solution that describes the cyclonic motion of a bidirectional vortex in a cylindrical chamber. The study is prompted by the need to characterize the flowfield inside a swirl-driven thrust chamber. This chamber has the advantage of confining mixing and combustion to an inner vortex tube that remains separated from the chamber walls by virtue of an outer stream of swirling, low temperature oxidizer. Our model is based on nonreactive, steady, rotational, axisymmetric, incompressible, and inviscid flow conditions. Unlike other studies of columnar vortices where the axial dependence is not considered, the present model accounts for the chambers finite body length. In fact, it incorporates the inlet and headwall conditions associated with a swirl-driven cyclone. Based on the resulting formulation, several flow features are captured. Among them is the location of the inner-outer vortex interface where the axial velocity vanishes.

The Bidirectional Vortex. Part 1: An Exact Inviscid Solution

In this paper, we derive an exact solution that describes the bulk fluid motion of a bidirectional coaxial vortex appropriate of a liquid propellant combustion chamber. The study is prompted by the need to characterize the flow inside a laboratory-scale thrust chamber. This chamber has the advantage of confining mixing and combustion to an inner vortex tube that remains separated from the chamber walls by an outer stream of swirling, low temperature oxidizer. Our mathematical model is based on steady, rotational, axisymmetric, incompressible, and inviscid flow conditions. In contrast to other studies of columnar vortices (where the axial dependence is ignored), our model accounts for the chamber’s finite body length. In fact, it incorporates the proper inlet and head-end flow conditions associated with a bipolar swirl-driven combustor. Based on the exact solution, several important flow attributes are illuminated. Among them is the location of the nontranslating vortex layer known as the mantle. This cylindrical layer separates the outer and inner vortex tubes (i.e., the updraft and the downdraft) and is confirmed using computational fluid dynamics and flow visualization.

The Bidirectional Vortex. Part 2: Viscous Core Corrections

This article focuses on the viscous core of the bidirectional flowfield arising in a swirldriven thrust chamber. By regularizing the momentum equation in the tangential direction, the boundary layer equation that controls the forced vortex near the chamber axis is obtained. After identifying the coordinate transformation needed to resolve the rapid changes near the core, an inner expansion is arrived at. This expansion is then matched with the outer solution associated with the free vortex; the latter is known to prevail in the outer region. By combining inner and outer expansions, uniformly valid approximations are obtained for the swirl velocity, vorticity, and pressure. These are shown to be strongly influenced by a dynamic similarity parameter that combines the mean flow Reynolds number and the chamber aspect ratio. Referred to as the vortex Reynolds number V, this dimensionless grouping enables us to quantify the characteristic features of the bidirectional vortex. Among them is the thickness of the viscous core which is found to decrease with the square root of V. The converse can be said of the maximum swirl velocity. In the same vein, the angular frequency of the rigid-body rotation of the forced vortex near the core is found to be linearly proportional to V. The form of the swirl velocity is reminiscent of the Burgers vortex; here, it is based on the aspect ratio of the thrust chamber. The resulting theoretical predictions compare favorably with experimental measurements and computational results over the length of the chamber.

Characterization of the Tangential Boundary Layers in the Bidirectional Vortex Thrust Chamber

We consider the tangential boundary layers of the bidirectional vortex, specifically, those forming near the core and sidewall of a swirl-driven cyclonic chamber. The analysis is based on the regularized, tangential momentum equation. The latter is rescaled in a manner to capture the forced vortex near the chamber axis and the no slip requirement at the sidewall. After identifying the coordinate transformations needed to resolve the rapid changes in the regions of nonuniformity, two inner expansions are arrived at. These expansions are then matched with the outer, free vortex solution. By combining inner and outer expansions, uniformly valid approximations are obtained for the swirl velocity, vorticity, and pressure. These are shown to be strongly influenced by the vortex Reynolds number, V. This key parameter appears as a ratio of the mean flow Reynolds number and the product of the swirl number and chamber aspect ratio. Based on V, several fundamental features of the bidirectional vortex are quantified. Among them are the thicknesses of the viscous core and sidewall boundary layers; these decrease with V 1/2 and V, respectively. The converse may be said of the maximum swirl velocity which increases with V 1/2 . In the same vein, the angular speed of the rigid-body rotation characteristic of the forced vortex is found to be linearly proportional to V. The form of the swirl velocity is reminiscent of Sullivans vortex; here, it is based on the aspect ratio of the chamber. The resulting theoretical predictions are found to be in good agreement with PIV measurements and Navier-Stokes simulations.

Inviscid Models of the Classic Hybrid Rocket

In this work, we derive two analytical solutions that mimic the bulk gas motion corresponding to the classic full-length, cylindrical hybrid rocket engine with circular bore. Our approach is based on steady, axisymmetric, incompressible, and inviscid flow conditions. Two exact solutions are presented starting from Euler’s equations. The first is rotational, assumes normal sidewall mass addition, and employs a harmonic injection profile at the head end wall. The second is irrotational but allows uniform head end injection. 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. The 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 accept either sinusoidal or uniform injection at the head end.

Rotational Axisymmetric Mean Flow for the Vortex Injection Hybrid Rocket Engine

In this work, we derive an approximate solution that describes the mean flow motion of the bidirectional coaxial vortex that characterizes NASA’s Hybrid Injection Vortex Rocket Engine (VIHRE). Our mathematical model is based on steady, rotational, axisymmetric, incompressible, and inviscid flow conditions. The resulting Euler-type solution is obtained using variation of parameters; it enables us to reproduce the bipolar motion observed in laboratory-scale VIHRE engines. Other predictions include pressure distributions, axial and radial velocity extrema, vorticity formation, and the dynamic mantle location that separates the outflow from the inflow. By relating fundamental variables to the bidirectional swirl number and wall regression rate, essential flow characteristics are captured. For example, our findings provide an explicit relation between the mantle location and the solid propellant wall regression rate. Nomenclature a = chamber radius i A = inlet area of the incoming swirl flow b = chamber discharge radius, a β l = chamber aspect ratio, / La p = normalized pressure, 2 /( ) pU ρ i Q = inlet volumetric flow rate at the base i Q = normalized flow rate, 21 /( ) i QU a σ −

Estimation of the laminar premixed flame temperature and velocity in injection-driven combustion chambers

Several laminar flame theories have been proposed in the past, the objective of each being the determination of fundamental flame attributes. Classification of these theories has been based on the degree of realism associated with their attendant assumptions, and these are carefully described by Kuo [1] and others [2–4]. Our approach proceeds along the lines of Mallard and Le Châtelier [5], yet differs in some aspects. Whereas Mallard and Le Châtelier divide the flame region into a preheat and a reaction zone, our premixed flame will be treated in a single zone. Furthermore, the chemical reaction rates will be either prescribed or simulated before being introduced into the analytical model as a spatially distributed heat source. The main objective of this effort is to obtain a closed-form approximation for the steady temperature distribution in a premixed laminar flame inside a planar chamber with porous walls. A premixed flame is often used to simulate solid-propellant burning in core flow studies of injection-driven combustion chambers (see [6] and the companion paper [7]). Our main objective stems from the need to accurately capture the thermal trends reported in the flame zone formed above the surface following propellant pyrolysis. The simplified model to be described will be particularly created to facilitate the merger between the chamber gas dynamics and the heat addition process. The temperature distribution thus approximated will provide the means to further study the decay or growth mechanisms affecting acoustical and vortical waves near the propellant surface. The current effort is also motivated by the inability of other flame theories to mimic the temperature trends in solid rocket motors (e.g., [8,9]). A second and equally compelling objective here is to obtain the thermally enhanced velocity vf directly above the reactive flame zone. In [7], it is shown that vf, must be used instead of the wall-injection speed vw in realistic models of the injection-driven combustion field based on nonreactive gas mixtures. A similar but more detailed model was developed by T’ien [10]. However, the temperature distribution obtained by T’ien was found numerically using Runge-Kutta integration. At present, an asymptotic solution will be derived for a model that follows similar lines to those described by T’ien. In view of the small size of the reactive flame thickness f relative to the radius of an idealized rocket motor chamber [7], curvature effects seem unimportant inside the flame zone. Provided that f remains relatively small, a planar model can arguably provide an adequate approximation of the analogous problem in a chamber with circular cross section. In the interest of simplicity, the planar model will be developed here.

Technical Brief: Asymptotic Temperature Distribution in a Simulated Combustion Chamber

In an axisymmetric model of a solid rocket motor, a cylindrical combustion chamber with porous walls is considered. For a posited range of operating parameters, the energy equation is perturbed and linearized using the dimensionless Peclet number. The possibility of circumventing chemical reactions while retaining the essential physics of the problem is explored. This is accomplished by artificially introducing a distributed heat source above the propellant surface. The resulting energy equation is then solved to zeroth order. The analytical solution and corresponding temperature maps are verified qualitatively using comparisons with numerical simulations of the combustion chamber.