Instability Analysis of Incompressible Open Cavity Flows

El problema del flujo sobre una cavidad abierta ha sido estudiado en profundidad en la literatura, tanto por el interés académico del problema como por sus aplicaciones prácticas en gran variedad de problemas ingenieriles, como puede ser el alojamiento del tren de aterrizaje de aeronaves, o el depósito de agua de aviones contraincendios. Desde hace muchos a˜nos se estudian los distintos tipos de inestabilidades asociadas a este problema: los modos bidimensionales en la capa de cortadura, y los modos tridimensionales en el torbellino de recirculación principal dentro de la cavidad. En esta tesis se presenta un estudio paramétrico completo del límite incompresible del problema, empleando la herramienta de estabilidad lineal conocida como BiGlobal. Esta aproximación permite contemplar la estabilidad global del flujo, y obtener tanto la forma como las características de los modos propios del problema físico, sean estables o inestables. El estudio realizado permite caracterizar con gran detalle todos los modos relevantes, así como la envolvente de estabilidad en el espacio paramétrico del problema incompresible (Mach nulo, variación de Reynolds, espesor de capa límite incidente, relación altura/profundidad de la cavidad, y longitud característica de la perturbación en la dirección transversal). A la luz de los resultados obtenidos se proponen una serie de relaciones entre los parámetros y características de los modos principales, como por ejemplo entre el Reynolds crítico de un modo, y la longitud característica del mismo. Los resultados numéricos se contrastan con una campaña experimental, siendo la principal conclusión de dicha comparación que los modos lineales están presentes en el flujo real saturado, pero que existen diferencias notables en frecuencia entre las predicciones teóricas y los experimentos. Para intentar determinar la naturaleza de dichas diferencias se realiza una simulación numérica directa tridimensional, y se utiliza un algoritmo de DMD (descomposición dinámica de modos) para describir el proceso de saturación. ABSTRACT The problem of the flow over an open cavity has been studied in depth in the literature, both for being an interesting academical problem and due to the multitude of industrial applications, like the landing gear of aircraft, or the water deposit of firefighter airplanes. The different types of instabilities appearing in this flow studied in the literature are two: the two-dimensional shear layer modes, and the three-dimensional modes that appear in the main recirculating vortex inside the cavity. In this thesis a parametric study in the incompressible limit of the problem is presented, using the linear stability analysis known as BiGlobal. This approximation allows to obtain the global stability behaviour of the flow, and to capture both the morphological features and the characteristics of the eigenmodes of the physical problem, whether they are stable or unstable. The study presented here characterizes with great detail all the relevant eigenmodes, as well as the hypersurface of instability on the parameter space of the incompressible problem (Mach equal to zero, and variation of the Reynolds number, the incoming boundary layer thickness, the length to depth aspect ratio of the cavity and the spanwise length of the perturbation). The results allow to construct parametric relations between the characteristics of the leading eigenmodes and the parameters of the problem, like for example the one existing between the critical Reynolds number and its characteristic length. The numerical results presented here are compared with those of an experimental campaign, with the main conclusion of said comparison being that the linear eigenmode are present in the real saturated flow, albeit with some significant differences in the frequencies of the experiments and those predicted by the theory. To try to determine the nature of those differences a three-dimensional direct numerical simulation, analyzed with Dynamic Mode Decomposition algorithm, was used to describe the process of saturation.

Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile

In this work, various turbulent solutions of the two-dimensional (2D) and three-dimensional compressible Reynolds averaged Navier?Stokes equations are analyzed using global stability theory. This analysis is motivated by the onset of flow unsteadiness (Hopf bifurcation) for transonic buffet conditions where moderately high Reynolds numbers and compressible effects must be considered. The buffet phenomenon involves a complex interaction between the separated flow and a shock wave. The efficient numerical methodology presented in this paper predicts the critical parameters, namely, the angle of attack and Mach and Reynolds numbers beyond which the onset of flow unsteadiness appears. The geometry, a NACA0012 profile, and flow parameters selected reproduce situations of practical interest for aeronautical applications. The numerical computation is performed in three steps. First, a steady baseflow solution is obtained; second, the Jacobian matrix for the RANS equations based on a finite volume discretization is computed; and finally, the generalized eigenvalue problem is derived when the baseflow is linearly perturbed. The methodology is validated predicting the 2D Hopf bifurcation for a circular cylinder under laminar flow condition. This benchmark shows good agreement with the previous published computations and experimental data. In the transonic buffet case, the baseflow is computed using the Spalart?Allmaras turbulence model and represents a mean flow where the high frequency content and length scales of the order of the shear-layer thickness have been averaged. The lower frequency content is assumed to be decoupled from the high frequencies, thus allowing a stability analysis to be performed on the low frequency range. In addition, results of the corresponding adjoint problem and the sensitivity map are provided for the first time for the buffet problem. Finally, an extruded three-dimensional geometry of the NACA0012 airfoil, where all velocity components are considered, was also analyzed as a Triglobal stability case, and the outcoming results were compared to the previous 2D limited model, confirming that the buffet onset is well detected.