A novel reduced-order model for nonlinear flows is presented. The model arises from a resolvent decomposition in which the nonlinear advection terms of the Navier-Stokes equation are considered as the input to a linear system in Fourier space. Results show that Taylor-Görtler-like vortices can be represented from a low-order resolvent decomposition of a nonlinear lid-driven cavity flow. The present approach provides an approximation of the fluctuating velocity given the time-mean and the time history of a single velocity probe.
We use resolvent analysis to determine an unsteady active control setup to attenuate pressure fluctuations in turbulent supersonic flow over a rectangular cavity with a length-to-depth ratio of 6 at a Mach number of 1.4 and a Reynolds number based on cavity depth of 10,000. Large-eddy simulations (LES) and dynamic modal decomposition (DMD) of the supersonic cavity flow reveal the dominance of two-dimensional Rossiter modes II and IV. These predominantly two-dimensional vortical structures generate high-amplitude unsteadiness over the cavity through trailing-edge impingement and create oblique …
A novel method to estimate unsteady aerodynamic force coefficients from pointwise velocity measurements is presented. The methodology is based on a resolvent-based reduced-order model which requires the mean flow to obtain physical flow structures and pointwise measurement to calibrate their amplitudes. A computationally-affordable time-stepping methodology to obtain resolvent modes in non-trivial flow domains is introduced and compared to previous existing matrix-free and matrix-forming strategies. The technique is applied to the unsteady flow around an inclined square cylinder at low Reynold…
In this article, we propose a methodology to reconstruct, in a single step, the mean- and unsteady properties of a flow from very few time-resolved measurements. The procedure is based on the {\it a priori} alignement of Fourier- and Resolvent-modes over energetic frequencies, which is a common feature in shear-dominated transitional flows. Hence, the Reynolds-stresses, which determine the mean-flow, may be approximated from a series of Resolvent modes, which discretize the fluctuation field in the frequency space and whose amplitudes can be tuned thanks to few measurements. In practice, we so…
We investigate the influence of unsteady partial cavitation on the fluid-structure interaction of a freely vibrating hydrofoil section at high Reynolds numbers. We consider an elastically-mounted NACA66 hydrofoil section that is free to vibrate in the transverse flow direction. Cavitating flow dynamics coupled with the transverse vibration are studied at low angles of attack. For the numerical study, we employ a recently developed unified variational fluid-structure interaction framework based on homogeneous mixture-based cavitation with hybrid URANS-LES turbulence modeling. We first validate …
The present work demonstrates the use of resolvent analysis to obtain physical insights for open-cavity flows. Resolvent analysis identifies the flow response to harmonic forcing, given a steady base state, in terms of the response and forcing modes and the amplification gain. The response and forcing modes reveal the spatial structures associated with this amplification process. In this study, we perform resolvent analysis on both laminar and turbulent flows over a rectangular cavity with length-to-depth ratio of $L/D=6$ at a free stream Mach number of $M_\infty=0.6$ in a spanwise periodic se…
The decomposition of hydrodynamic and acoustic components of cavity flows can aid the understanding of the flow-acoustic interaction, which produces a variety of adverse effects in applications. We apply the approach of combining the resolvent analysis and Doak's momentum potential theory to examine the input-output flow-acoustic characteristics of compressible flow over an open cavity at Re=10,000. The methodology can decompose hydrodynamic and acoustic components from an input-output framework. We further localize the forcing at the leading-edge and front wall of the cavity and filter with a…
This investigation aims to assess the effect of different types of actuator forcing on the feedback loop of an under-expanded Mach 1.27 planar impinging jet using a resolvent framework. To this end, we employ a Large Eddy Simulation database as a truth model. The time and spanwise-averaged mean flow is taken as an input to global stability and resolvent analyses with the purpose of examining both the intrinsic instability and input-output characteristics. The results show that the inherent instability and primary energy amplification are attributed to the Kelvin-Helmholtz (K-H) instability. Mo…
We combine three-dimensional (3D) large-eddy simulations (LES) and resolvent analysis to design active separation control techniques on a NACA 0012 airfoil. Spanwise-periodic flows over the airfoil at a chord-based Reynolds number of $23,000$ and a free-stream Mach number of $0.3$ are considered at two post-stall angles of attack of $6^\circ$ and $9^\circ$. Near the leading edge, localized unsteady thermal actuation is introduced in an open-loop manner with two tunable parameters of actuation frequency and spanwise wavelength. For the most successful control case that achieves full reattachmen…
We conducted a numerical simulation of ventilated supercavitation from a forward-facing cavitator in unsteady flows generated by a gust generator under different gust angles of attack and gust frequencies. The numerical method is validated through the experimental results under specific steady and unsteady conditions. It has been shown that the simulation can capture the degree of cavity shape fluctuation and internal pressure variation in a gust cycle. Specifically, the cavity centerline shows periodic wavelike undulation with a maximum amplitude matching that of the incoming flow perturbatio…
This fluid dynamics video shows the release of a 1/15th scale model store into a Mach 1.5 free-stream flow from a rectangular cavity. The images were acquired with a high-speed schlieren photography system using a Photron Ultima APX camera. Acoustic measurements showed that the unsteady shear layer over the cavity had a prominent oscillation at approximately 810 Hz which corresponds to fluctuations seen in the cavity's leading edge shock structure in the video. The video demonstrates the similarity in the time scales of the flow field dynamics and the residence time of the store in the shear l…
In this article, we propose a passive boundary layer control method to control the vortex structure of the cavity on the suction side and wake region of the CAV2003 benchmark hydrofoil. This method may be used in different applications such as marine, turbomachinery and hydraulic machinery. First, we used a hybrid URANS model for turbulence to simulate the 3D unsteady cloud cavitating flow and validated it based on experimental data. We performed the numerical simulations using open source code OpenFOAM and an Euler-Euler cavitation model. Second, we studied the effect of passive boundary laye…
Our study provides the first experimental investigation of the internal flows of ventilated partial cavitation (VPC) formed by air injection behind a backward-facing step. The experiments are conducted using flow visualization and planar particle image velocimetry (PIV) with fog particles for two different cavity regimes of VPC, i.e., open cavity (OC) and two-branch cavity (TBC), under various range of free stream velocity (U) and ventilation rates (Q). Our experiments reveal similar flow patterns for both OC and TBC, including forward flow region near the air-water interface, reverse flow reg…
Three-dimensional direct numerical simulations of an incompressible open square cavity flow are conducted. Features of the permanent (non-linear) regime together with the linear stability analysis of a two-dimensional steady base flow are discussed. Spanwise boundary conditions are periodic and control parameters set such that the shear layer is stable against Kelvin-Helmholtz modes. Three branches of destabilising modes are found. The most destabilising branch is associated with steady modes, over a finite range of spanwise wavenumbers. The two other branches provide unsteady modes. Features …
Self-sustained oscillation in Mach 3 supersonic cavity with a length-to-depth ratio of 3 is investigated using wall modeled Large Eddy Simulation (LES) methodology for ReD = 3.39*10^5. The unsteady data obtained through computation is utilized to investigate the spatial and temporal evolution of the flow field, especially second invariant of velocity tensor; while the phase averaged data is analyzed over a feedback cycle to study the spatial structures. This analysis is accompanied by the Proper Orthogonal Decomposition (POD) data, which reveals the presence of discrete vortices along the shea…
We model the advective Marangoni spreading of insoluble surfactant at the free surface of a viscous fluid that is confined within a two-dimensional rectangular cavity. Interfacial deflections are assumed small, with contact lines pinned to the walls of the cavity, and inertia is neglected. Linearizing the surfactant transport equation about the equilibrium state allows a modal decomposition of the dynamics, with eigenvalues corresponding to decay rates of perturbations. Computation of the family of mutually orthogonal two-dimensional eigenfunctions reveals singular flow structures near each co…
Phase-based analysis and control of supersonic turbulent cavity flows
University of California, Los Angeles（加州大学洛杉矶分校）；Japan Agency for Marine-Earth Science and Technology（日本海洋地球科学技术机构）；University of Florida（佛罗里达大学）；Illinois Institute of Technology（伊利诺伊理工学院）
We present spectral analysis modal methods (SAMMs) to perform POD in the frequency domain using non-time-resolved Particle Image Velocity (PIV) data combined with unsteady surface pressure measurements. In particular, time-resolved unsteady surface pressure measurements are synchronized with non-time-resolved planar PIV measurements acquired at 15 Hz in a Mach 0.6 cavity flow. Leveraging the spectral linear stochastic estimation (LSE) method of Tinney et al. (2006), we first estimate the cross correlations between the velocity field and the unsteady pressure sensors via sequential time shifts,…
We utilize resolvent and weakly nonlinear analyses in combination with direct numerical simulations (DNS) to identify mechanisms for oblique transition in a Mach 5 hypersonic flow over an adiabatic slender double-wedge. Even though the laminar separated flow is globally stable, resolvent analysis demonstrates significant amplification of unsteady external disturbances to the linearized flow equations. These disturbances are introduced upstream of the separation zone and they lead to the appearance of oblique waves further downstream. We demonstrate that large amplification of oblique waves ari…
This work is dedicated to providing a detailed account of the flow dynamics in the closure region of an internal ship air cavity. A geometrically simple multiwave test cavity is considered, and a simulation of the flow is conducted using large-eddy simulation coupled with an algebraic Volume of Fluid interface capturing method. Results reveal that the flow in the closure region is highly unsteady and turbulent. The main cause of this is established to be the pressure gradient occurring due to the stagnation of the flow on the beach wall of the cavity. The pressure gradient leads to a steep inc…
The impinging shock of varying strengths on the free shear layer in a confined supersonic cavity flow is studied numerically using the detached-eddy simulation. The resulting spatiotemporal variations are analyzed between the different cases using unsteady statistics, $x-t$ diagrams, spectral analysis, and modal decomposition. A cavity of length to depth ratio $[L/D]=2$ at a freestream Mach number of $M_\infty = 1.71$ is considered to be in a confined passage. Impinging shock strength is controlled by changing the ramp angle ($θ$) on the top-wall. The static pressure ratio across the impinging…
A new application of subspaces interpolation for the construction of nonlinear Parametric Reduced Order Models (PROMs) is proposed. This approach is based upon the Riemannian geometry of the manifold formed by the quotient of the set of full-rank N-by-q matrices by the orthogonal group of dimension q. By using a set of untrained parametrized Proper Orthogonal Decomposition (POD) subspaces of dimension q, the subspace for a new untrained parameter is obtained as the generalized Karcher barycenter which solution is sought after by solving a simple fixed point problem. Contrary to existing PROM a…
We consider the unsteady regimes of an acoustically-driven jet that forces a recirculating flow through successive reflections on the walls of a square cavity. The specific question being addressed is to know whether the system can sustain states of low-dimensional chaos when the acoustic intensity driving the jet is increased, and, if so, to characterise the pathway and underlying physical mechanisms. We adopt two complementary approaches, both based on data extracted from numerical simulations: (i) We first characterise successive bifurcations through the analysis of leading frequencies. Two…
The study of cavity flow is one of the most important research topics of unsteady aerodynamics. Supersonic turbulent flows over a cavity are mostly encountered in missiles, turbomachinery, and high-speed aircraft. The turbulence inside the cavity gives rise to excess drag, acoustic waves, pressure fluctuations, and vibrations which may lead to numerous problems like excess fuel consumption, failure of missile trajectory or mechanical parts, and aerodynamic heating. We conduct numerical simulations to investigate the flow and compare our results to existing experimental data to show quantitativ…
Turbulent mixing layers over cavities can couple with acoustic waves and lead to undesired oscillations. To understand the nonlinear aspects of this phenomenon, a turbulent mixing layer over a deep cavity at Reynolds number 150 000 is considered and its response to harmonic forcing is analysed with large-eddy simulations (LES) and linearised Navier-Stokes equations (LNSE). As a model of incoming acoustic perturbations, spatially uniform time-harmonic forcing is applied at the cavity end, with amplitudes in the wide range 0.045-8.9% of the bulk velocity. Compressible LES provide reference nonli…
We present a parametric study of the unsteady phenomena associated with the flow of elongated gas bubbles travelling through liquid-filled square capillaries under high Weber number conditions. These conditions consistently induce the formation of a re-entrant jet at the back of the bubble that commonly gives way to a deep liquid cavity. Subsequent steps include pinch-off events in the cavity to generate one or multiple encapsulated drops which may coalesce, in conjunction with the bursting of the bubble-liquid interface by either the cavity or the drops. Some of these interfacial instabilitie…
In this paper, we have proposed a modified Marker-And-Cell (MAC) method to investigate the problem of an unsteady 2-D incompressible flow with heat and mass transfer at low, moderate, and high Reynolds numbers with no-slip and slip boundary conditions. We have used this method to solve the governing equations along with the boundary conditions and thereby to compute the flow variables, viz. $u$-velocity, $v$-velocity, $P$, $T$, and $C$. We have used the staggered grid approach of this method to discretize the governing equations of the problem. A modified MAC algorithm was proposed and used to…
An ensemble-variational approach for open-loop flow control
ONERA, Institut Polytechnique de Paris（法国航空航天研究院，巴黎理工学院）；CNAM（国立工艺学院）；Centre National d’Etudes Spatiales CNES（法国国家空间研究中心）；Sorbonne Université, CNRS, Institut Jean Le Rond d’Alembert（索邦大学，法国国家科学研究中心，让·勒朗·达朗贝尔研究所）
In this paper, 14-velocity and 18-velocity multiple-relaxation-time (MRT) lattice Boltzmann (LB) models are proposed for three-dimensional incompressible flows. These two models are constructed based on the incompressible LBGK model proposed by He et al. (Chin. Phys., 2004, 13: 40-46) and the MRT LB model proposed by d'Humières et al. (Philos. Trans. R. Soc., A, 2002, 360: 437-451), which have advantages in the computational efficiency and stability, respectively. Through the Chapman-Enskog analysis, the models can recover to three-dimensional incompressible Navier-Stokes equations in the low …
We show how to accelerate the direct solution of the Boltzmann equation using Graphics Processing Units (GPUs). In order to fully exploit the computational power of the GPU, we choose a method of solution which combines a finite difference discretization of the free-streaming term with a Monte Carlo evaluation of the collision integral. The efficiency of the code is demonstrated by solving the two-dimensional driven cavity flow. Computational results show that it is possible to cut down the computing time of the sequential code of two order of magnitudes. This makes the proposed method of solu…
In this paper, sparsity-promoting regression techniques are employed to automatically identify from data relevant triadic interactions between modal structures in large Galerkin-based models of two-dimensional unsteady flows. The approach produces interpretable, sparsely-connected models that reproduce the original dynamical behaviour at a much lower computational cost, as fewer triadic interactions need to be evaluated. The key feature of the approach is that dominant interactions are selected systematically from the solution of a convex optimisation problem, with a unique solution, and no a …
Airflow through the nasal cavity exhibits a wide variety of fluid dynamicsbehaviour due to the intricacy of the nasal geometry. The flow is naturallyunsteady and perhaps turbulent, despite CFD in the literature that assumesa steady laminar flow. Time-dependent simulations can be used to generatedetailed data with the potential to uncover new flow behaviour, although theyare more computationally intensive compared with steady-state simulations. Furthermore, verification of CFD results has relied on reported pressure drop(e.g. nasal resistance) across the nasal airway although the geometries use…
A major goal for reduced-order models of unsteady fluid flows is to uncover and exploit latent low-dimensional structure. Proper orthogonal decomposition (POD) provides an energy-optimal linear basis to represent the flow kinematics, but converges slowly for advection-dominated flows and tends to overestimate the number of dynamically relevant variables. We show that nonlinear correlations in the temporal POD coefficients can be exploited to identify the underlying attractor, characterized by a minimal set of driving modes and a manifold equation for the remaining modes. By viewing these nonli…
Incompressible, inviscid, irrotational, and unsteady flows with circulation $Γ$ around a distorted toroidal bubble are considered. A general variational principle that determines the evolution of the bubble shape is formulated. For a two-dimensional (2D) cavity with a constant area $A$, exact pseudo-differential equations of motion are derived, based on variables that determine a conformal mapping of the unit circle exterior into the region occupied by the fluid. A closed expression for the Hamiltonian of the 2D system in terms of canonical variables is obtained. Stability of a stationary drif…
Many unsteady flows exhibiting complex dynamics are nevertheless characterized by emergent large-scale coherence in space and time. Reduced-order models based on Galerkin projection of the governing equations onto an orthogonal modal basis approximate the flow as a low-dimensional dynamical system with linear and quadratic terms. However, these Galerkin models often fail to reproduce the true dynamics, in part because they ignore important nonlinear interactions with unresolved flow scales. Here, we use a separation of time scales between the resolved and subscale variables to derive a reduced…
A methodology to generate sparse Galerkin models of chaotic/unsteady fluid flows containing a minimal number of active triadic interactions is proposed. The key idea is to find an appropriate set of basis functions for the projection representing elementary flow structures that interact minimally one with the other and thus result in a triadic interaction coefficient tensor with sparse structure. Interpretable and computationally efficient Galerkin models can be thus obtained, since a reduced number of triadic interactions needs to be computed to evaluate the right hand side of the model. To f…
