^{1}, Gary A. Flandro

^{1}and Joseph Majdalani

^{1,a)}

### Abstract

This study considers a simplified model of a liquid rocket engine in which uniform injection is imposed at the faceplate. The corresponding cylindrical chamber has a small length-to-diameter ratio with respect to solid and hybrid rockets. Given their low chamber aspect ratios, liquid thrust engines are known to experience severe tangential and radial oscillation modes more often than longitudinal ones. In order to model this behavior, tangential and radial waves are superimposed onto a basic mean-flow model that consists of a steady, uniform axial velocity throughout the chamber. Using perturbation tools, both potential and viscousflowequations are then linearized in the pressure wave amplitude and solved to the second order. The effects of the headwall Mach number are leveraged as well. While the potential flow analysis does not predict any acoustic streamingeffects, the viscous solution carried out to the second order gives rise to steady secondary flow patterns near the headwall. These axisymmetric, steady contributions to the tangential and radial traveling waves are induced by the convective flow motion through interactions with inertial and viscous forces. We find that suppressing either the convective terms or viscosity at the headwall leads to spurious solutions that are free from streaming. In our problem, streaming is initiated at the headwall, within the boundary layer, and then extends throughout the chamber. We find that nonlinear streaming effects of tangential and radial waves act to alter the outer solution inside a cylinder with headwall injection. As a result of streaming, the radial wavevelocities are intensified in one-half of the domain and reduced in the opposite half at any instant of time. Similarly, the tangential waves are either enhanced or weakened in two opposing sectors that are at 90° angle to the radial velocity counterparts. The second-order viscous solution that we obtain clearly displays both an oscillating and a steady flow component. The steady part can be an important contributor to wave steepening, a mechanism that is often observed during the onset of acoustic instability.

This project is sponsored partly by the National Science Foundation, through Grant No. CMMI-0928762, and partly by the University of Tennessee Space Institute, through institutional cost sharing.

I. INTRODUCTION

II. FORMULATION

A. Unsteady flowequations

B. Headwall injection flow field

III. POTENTIAL FLOW SOLUTION

A. First-order potential solution

B. Second-order potential solution

IV. VISCOUSFLOW

A. First-order viscous solution

1. Solution for the first order tangential velocity

2. Solution for the first order radial velocity

3. Solution for the first order axial velocity

4. Solution for the complete first order velocity

B. Second-order viscous solution

V. CONCLUSIONS

### Key Topics

- Viscosity
- 38.0
- Potential flows
- 30.0
- Mach numbers
- 20.0
- Radial velocities
- 15.0
- Acoustic waves
- 11.0

## Figures

Chamber geometry and coordinate system.

Chamber geometry and coordinate system.

First-order approximations for (a) radial and (b) tangential velocities. The scale on the left-hand side is for injection Mach numbers of 0.3 and 0.03. The scale on the right-hand side is for .

First-order approximations for (a) radial and (b) tangential velocities. The scale on the left-hand side is for injection Mach numbers of 0.3 and 0.03. The scale on the right-hand side is for .

Steady second-order approximations for (a) radial and (b) tangential velocities. The scale on the left axis corresponds to an injection Mach number of 0.3 while that on the right is set to display the and 0.003 cases.

Steady second-order approximations for (a) radial and (b) tangential velocities. The scale on the left axis corresponds to an injection Mach number of 0.3 while that on the right is set to display the and 0.003 cases.

First-order traveling wave vector plot at and three headwall injection Mach numbers of (a) , (b) 0.03, and (c) 0.003.

First-order traveling wave vector plot at and three headwall injection Mach numbers of (a) , (b) 0.03, and (c) 0.003.

Steady second-order velocity vector plot at and three headwall injection Mach numbers of (a) , (b) 0.03, and (c) 0.003.

Steady second-order velocity vector plot at and three headwall injection Mach numbers of (a) , (b) 0.03, and (c) 0.003.

Total vector plot in the outer region illustrating the behavior of (a) the purely inviscid potential approximation up to the second order and [(b) and (c)] the same total potential solution augmented by the streaming contribution. Results are shown for , , , , and 0, (b) 0.006 47, and (c) 0.0647.

Total vector plot in the outer region illustrating the behavior of (a) the purely inviscid potential approximation up to the second order and [(b) and (c)] the same total potential solution augmented by the streaming contribution. Results are shown for , , , , and 0, (b) 0.006 47, and (c) 0.0647.

Streaming velocity in the radial direction shown at several injection Mach and acoustic Reynolds numbers. Results correspond to , , , and an injection Mach number of (a) 0.003, (b) 0.03, and (c) 0.3. The scale on the left axis is specified for the case while that on the right corresponds to the and 0.0647 cases.

Streaming velocity in the radial direction shown at several injection Mach and acoustic Reynolds numbers. Results correspond to , , , and an injection Mach number of (a) 0.003, (b) 0.03, and (c) 0.3. The scale on the left axis is specified for the case while that on the right corresponds to the and 0.0647 cases.

Streaming velocity in the tangential direction shown at several injection Mach and acoustic Reynolds numbers. Results correspond to , , , and an injection Mach number of (a) 0.003, (b) 0.03, and (c) 0.3.

Streaming velocity in the tangential direction shown at several injection Mach and acoustic Reynolds numbers. Results correspond to , , , and an injection Mach number of (a) 0.003, (b) 0.03, and (c) 0.3.

Sectors in which oscillatory waves are enhanced or weakened by virtue of streaming. These illustrate the outcome of interactions between (a) radial and (b) tangential velocities with the streaming motion. For example, in part (a) the radially outward streaming contributions act to decelerate the radial wave in the right half while accelerating it in the left half. In part (c) the main regions of interest are delineated along with their pertinent equations.

Sectors in which oscillatory waves are enhanced or weakened by virtue of streaming. These illustrate the outcome of interactions between (a) radial and (b) tangential velocities with the streaming motion. For example, in part (a) the radially outward streaming contributions act to decelerate the radial wave in the right half while accelerating it in the left half. In part (c) the main regions of interest are delineated along with their pertinent equations.

## Tables

Boundary conditions for both potential and viscous flow analyses.

Boundary conditions for both potential and viscous flow analyses.

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