The hydrodynamic stability of the flow in a solid rocket motor is revisited using a general linear stability approach. A harmonic perturbation is introduced into the linearized Navier-Stokes equations leading to an eigenvalue problem posed as a system of partial differential equations with respect to the spatial coordinates. The system is discretized by a spectral collocation method applied to each spatial coordinate and the eigenvalues are determined using Arnoldi’s procedure. A special emphasis is placed on the boundary conditions. The main result is the discrete nature of the eigenvalue set. According to the present theory and the obtained results, only some discrete frequencies may exist in the motor (as eigenmodes). These frequencies only depend on the Reynolds number based on the injection velocity and the radius of the pipe flow. They are compared to measurements that have been performed at ONERA in one case with a cold-gas setup and in another case with a reduced scale live motor. Due to the agreement obtained with both experiments, the biglobal stability approach seems to offer new insight into the unresolved thrust oscillations problem.
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.
For several years, a promising Plasma Synthetic Jet actuator for high-speed flow control has been under development at ONERA. So far, its confined geometry and small space-time scales at play have prevented its full experimental characterization. Complementary accurate numerical simulations are then considered in this study in order to provide a complete aerothermodynamic description of the actuator. Two major obstacles have to be overcome with this approach: the modeling of the energy deposited by the electric arc and the accurate computation of the transient response of the cavity generating the pulsed jet. To solve the first problem, an Euler solver coupled with an electric circuit model was used to evaluate the energy deposition in the cavity. Such a coupling is performed by considering the electric field between the two electrodes. The second issue was then addressed by injecting these source terms in large Eddy simulations of the entire actuator. Aerodynamic results were finally compared with Schlieren visualizations. Using the proposed methodology, the temporal evolution of the jet front is remarkably well predicted. C 2014 AIP Publishing LLC. [http://dx.
The Discrete Element Method (DEM) for modeling of flow over rough walls is revised in the framework of the DANS (Double Averaged Navier-Stokes) equations, with a special focus on the drag term appearing in the mean momentum equation as a key to the robustness of the model. A set of 14 Direct Numerical Simulations (DNS) of channel flows with systematically varying roughness topographies is considered to assess the performances of different drag coefficient closures. While the standard model of Taylor et al. [J. Fluids Eng. 107 (1985), 251257] is found not to be successful in reproducing the distribution of the drag force, a new model is derived. The new model along with the model recently proposed by Kuwata et al. [ Int. J. Heat Fluid Flow 77 (2019), 186201] are employed for the solution of channel flow along with a simple mixing length model. Both models are shown to be successful in prediction of roughness function as long as a constant in the latter model is readjusted. The velocity profiles are also well recovered and in particular the roughness sublayers are accurately reproduced.
The discrete element (roughness) method developed a few decades ago is revisited using the double averaging technique applied to the Navier-Stokes equation. A k − ω based DANS turbulence model is thus derived to be able to account for roughness effects. Several closure relations are proposed to model all terms induced by the use of the double averaging. The momentum and energy equations are considered in their simplified forms adapted to a 1D channel code in accordance with the DNS results used for the validation. To reconcile the discrete element (roughness) method with the double averaged Navier-Stokes equations the notion of representative elementary roughness is introduced. A large validation dataset coming from various DNS configurations is then used to assess the predictions of the proposed DANS model. Yet not fully complete, especially regarding the dispersive terms due to a lack of data, the performed validation already proves the overall excellent behavior of the DANS model and demonstrates the relevance of the present methodology based on the representative elementary roughness.
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