Pavel P. Popov

Two-Dimensional Model for Liquid-Rocket Transverse Combustion Instability

Nonlinear, transverse-mode, liquid-propellant-rocket-motor combustion instability is examined with a two-dimensional model. The three-dimensional equations are integrated over the axial direction, for a multi-orifice short nozzle. Nonlinear transverse-wave oscillations in the circular combustion chamber are examined with the primary flow in the axial direction. Turbulent mixing of methane and gaseous oxygen with coaxial injection is analyzed. The combustion has two characteristic times, one for mixing and the other for chemical kinetics, producing a time lag in the energy-release rate relative to pressure. Then, the coupled combustion process and wave dynamics are calculated for a 10-injector chamber with methane and gaseous-oxygen propellants. The linear first tangential mode is imposed initially. Nonlinear triggering occurs; above a critical initial amplitude, the amplitude grows; otherwise, it decays with time. The second tangential mode also develops, and the nonlinear resonance creates a subharmonic ...

Weak second-order splitting schemes for Lagrangian Monte Carlo particle methods for the composition PDF/FDF transport equations

We study a class of methods for the numerical solution of the system of stochastic differential equations (SDEs) that arises in the modeling of turbulent combustion, specifically in the Monte Carlo particle method for the solution of the model equations for the composition probability density function (PDF) and the filtered density function (FDF). This system consists of an SDE for particle position and a random differential equation for particle composition. The numerical methods considered advance the solution in time with (weak) second-order accuracy with respect to the time step size. The four primary contributions of the paper are: (i) establishing that the coefficients in the particle equations can be frozen at the mid-time (while preserving second-order accuracy), (ii) examining the performance of three existing schemes for integrating the SDEs, (iii) developing and evaluating different splitting schemes (which treat particle motion, reaction and mixing on different sub-steps), and (iv) developing the method of manufactured solutions (MMS) to assess the convergence of Monte Carlo particle methods. Tests using MMS confirm the second-order accuracy of the schemes. In general, the use of frozen coefficients reduces the numerical errors. Otherwise no significant differences are observed in the performance of the different SDE schemes and splitting schemes.

Propellant Injector Influence on Liquid-Propellant Rocket Engine Instability

The avoidance of acoustic instabilities, which may cause catastrophic failure, is demanded for liquid-propellant rocket engines. This occurs when the energy released by combustion amplifies acoustic disturbances; it is therefore essential to avoid such positive feedback. Although the energy addition mechanism operates in the combustion chamber, the propellant injector system may also have considerable influence on the stability characteristics of the overall system, with pressure disturbances in the combustion chamber propagating back and forth in the propellant injectorchannels.Theintroducedtimedelaymayaffectstability,dependingontheratioofthewavepropagationtime throughtheinjectortotheperiodofthecombustionchambersacousticmodes.Thisstudyfocusesontransverse-wave liquid-propellant rocket engine instabilities using a two-dimensional polar coordinate solver (with averaging in the axial direction) coupled to one-dimensional solutions in each of the coaxial oxygen–methane injectors. A blockage in one (or more) of the injectors is analyzed as a stochastic event that may cause an instability. A properly designed temporaryblockage of oneor more injectors can also be used for control of an oscillation introduced by any physical event.Thestochasticanddesignvariablesparameterspaceis exploredwiththepolynomial chaosexpansionmethod.

Specific volume coupling and convergence properties in hybrid particle/finite volume algorithms for turbulent reactive flows

We investigate the coupling between the two components of a Large Eddy Simulation/Probability Density Function (LES/PDF) algorithm for the simulation of turbulent reacting flows. In such an algorithm, the Large Eddy Simulation (LES) component provides a solution to the hydrodynamic equations, whereas the Lagrangian Monte Carlo Probability Density Function (PDF) component solves for the PDF of chemical compositions. Special attention is paid to the transfer of specific volume information from the PDF to the LES code: the specific volume field contains probabilistic noise due to the nature of the Monte Carlo PDF solution, and thus the use of the specific volume field in the LES pressure solver needs careful treatment. Using a test flow based on the Sandia/Sydney Bluff Body Flame, we determine the optimal strategy for specific volume feedback. Then, the overall second-order convergence of the entire LES/PDF procedure is verified using a simple vortex ring test case, with special attention being given to bias errors due to the number of particles per LES Finite Volume (FV) cell.

Implicit and explicit schemes for mass consistency preservation in hybrid particle/finite-volume algorithms for turbulent reactive flows

This work addresses the issue of particle mass consistency in Large Eddy Simulation/Probability Density Function (LES/PDF) methods for turbulent reactive flows. Numerical schemes for the implicit and explicit enforcement of particle mass consistency (PMC) are introduced, and their performance is examined in a representative LES/PDF application, namely the Sandia-Sydney Bluff-Body flame HM1. A new combination of interpolation schemes for velocity and scalar fields is found to better satisfy PMC than multilinear and fourth-order Lagrangian interpolation. A second-order accurate time-stepping scheme for stochastic differential equations (SDE) is found to improve PMC relative to Euler time stepping, which is the first time that a second-order scheme is found to be beneficial, when compared to a first-order scheme, in an LES/PDF application. An explicit corrective velocity scheme for PMC enforcement is introduced, and its parameters optimized to enforce a specified PMC criterion with minimal corrective velocity magnitudes.

Transverse Combustion Instability in a Rectangular Rocket Motor

A computational analysis of transverse acoustic instability is presented for an experimental combustion chamber with rectangular cross section. The analysis is shown to be efficient and accurate. The governing equations are solved on multiple, coupled grids, which are two-dimensional in the combustion chamber and nozzle and one-dimensional in the injector port. Thus, they allow for a fast simulation, even in a serial run. Because of the lengthscale difference, the jet flame behavior at the injectors (including effects of turbulence) can be decoupled from the acoustic effects and solved on a local grid for each jet flame emerging from an injector. Wave propagation through the injector feed ports is evaluated on additional, one-dimensional grids for each injector port. The overall algorithm is used to simulate the Purdue seven-injector rocket engine; good quantitative agreement between simulations and experiment is achieved. All simulations that are predicted to be unconditionally unstable are confirmed by ...

An accurate time advancement algorithm for particle tracking

We describe a particle position time advancement algorithm that is designed for use with several subgrid velocity reconstruction schemes used in LES/FDF methods, and potentially in other applications. These reconstruction schemes yield a subgrid velocity field with desirable divergence properties, but also with discontinuities across cell faces. Therefore, a conventional time advancement algorithm, such as second-order Runge-Kutta (RK2), does not perform as well as it does with a smooth velocity field. The algorithm that we describe, called Multi-Step RK2 (MRK2), builds upon RK2 by breaking up the time step into two or more substeps whenever a particle crosses one or more velocity discontinuities. When used in conjunction with the parabolic edge reconstruction method, MRK2 performs considerably better than RK2: both the final position of an advected particle, and the final area of a 2D infinitesimal area element are second-order accurate in time (as opposed to first-order accurate for RK2). Furthermore, MRK2 has the theoretical advantage that it better preserves the continuity of the mapping between initial and final particle positions.

Triggering and Restabilization of Combustion Instability with Rocket Motor Acceleration

The probability of a liquid-propulsion rocket motor to develop screeching instability is studied computationally. The combustion chamber is accelerated as a rigid body using a prescribed acceleration time history; it is found that accelerations of proper magnitude, duration, and frequency induce a pressure wave inside the combustion chamber that grows to a screeching acoustic wave limit cycle. For a rectangular rocket motor, a reciprocating transverse acceleration leads to the development of a transverse pressure wave limit cycle; for a cylindrical rocket motor, the limit cycle may be either a standing wave, for a transverse reciprocating acceleration, or a spinning wave, for a transverse rotating acceleration. It is found that a limit cycle may be induced by either a large acceleration pulse of short duration or a smaller acceleration pulse of a longer duration. The polynomial chaos expansion method is used to study the probability of growth to a limit-cycle oscillation when the amplitude and frequency o...

Low-Probability Events Leading to Rocket Engine Combustion Instability

A new method is proposed for calculations of rare-event rocket combustion instabilities. Acceleration of the combustion chamber is modeled as a stochastic process of long duration and moderate amplitude. Using a simplified model for the effect of acceleration on the evolution of the first tangential mode of pressure within the chamber, a modified sampling distribution is obtained that yields higher occurrence of the rare event: in this case, growth to instability. Statistics are then calculated for the original distribution of the stochastic process using an importance sampling procedure. Knowledge of the likelihood ratios between the real and modified probability density functions allows a low-cost computation of the probability of the rare event of triggering the combustion instability by low-amplitude acceleration fluctuations. There are two distinct regimes of high and low probabilities of triggering, with a critical acceleration amplitude threshold between them; in the low-probability regime, the pro...

Triggering and Re-Stabilization of Combustion Instability with Rocket Motor Acceleration

The probability of a liquid-propulsion rocket motor to develop screeching instability is studied computationally. The combustion chamber is accelerated as a rigid body using a prescribed acceleration time history; it is found that accelerations of proper magnitude, duration, and frequency induce a pressure wave inside the combustion chamber that grows to a screeching acoustic wave limit cycle. For a rectangular rocket motor, a reciprocating transverse acceleration leads to the development of a transverse pressure wave limit cycle; for a cylindrical rocket motor, the limit cycle may be either a standing wave, for a transverse reciprocating acceleration, or a spinning wave, for a transverse rotating acceleration. It is found that a limit cycle may be induced by either a large acceleration pulse of short duration or a smaller acceleration pulse of a longer duration. The polynomial chaos expansionmethod is used to study the probability of growth to a limit-cycle oscillation when the amplitude and frequency of the transverse acceleration pulse are random.