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Procedia IUTAM 00 (2014) 000–000www.elsevier.com/locate/procedia

IUTAM ABCM Symposium on Laminar Turbulent Transition

Linear stability analysis of the flow over an open cavity controlledby a synthetic jet

Qiong Liua,∗, Francisco Gomezb, Vassilis Theofilisa

aScholl of Aeronautics, Universidad Politecnica de Madrid, Pza. Cardenal Cisneros 3, E-28040 Madrid, SpainbDepartment of Mechanical and Aerospace Engineering, Monash University, Victoria 3800, Australia

Abstract

Numerical simulation of incompressible laminar open cavity flow controlled by a synthetic jet is presented here. First, the directglobal mode and adjoint global mode of the open cavity flow are extracted from the solution of the linear and adjoint Navier-Stokesequations following the residual algorithm, the results show that as Reynolds number is increased, the location of wavemakermigrates from the rear part of cavity to shear layer. Secondly, we place the synthetic jet at the upstream of the cavity and studyby varying the amplitude and frequency of jet. In the conference, we will present how does the sensitive region change with theReynolds number increased in the cavity flow and the synthetic jet has an stabilization effect on the cavity flow.c© 2014 The Authors. Published by Elsevier B.V.Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering).

Keywords: cavity flow control; direct global modes; adjoint global modes

1. Introduction

The study of cavity flow is interesting from both academic researches and practical applications. From the view-point of theory, cavity flow consists of comprehensive flow characteristics: shear layer, compression and expansionwave, fluid acoustics interactions. For the practical perspectives, cavities are used to plenty of aircraft components,including weapon bays, landing system and instrumentation cavities. The cavity study began from 1950’s1, Rossiter2

predicted the resonant frequencies in semi-empirical formula. Later, a large number of noticeable studies for cavityflow spring up3 4 5. Regarding the instability analysis6, Theofilis et al.7 8 numerically studied the four classes of theviscous linear stability of incompressible flows inside rectangular containers, the results exhibited the first of the newmodes is stationary and the second is traveling with a critical Reynolds number and frequency. In the most situations,the cavity flow vibration is detrimental , it easily causes oscillation, fatigue and noise, so the cavity flow control be-come a main motivation. The flow control strategies include passive methods9 10 11 12 and active methods13. D. Sipp etal.14 present the results of the feedback configuration of open cavity flow and backward-facing step flow and give anoverview of the common tool for the design of control strategies, especially on linear control. D.C.Hill et al.15 used ageneral theoretical formalism to assess the base flow of cylinder modifications by placing a small cylinder, which prin-ciple we are going to use in the work. In order to make an appreciable control, the behavior of the flow system should

∗ Corresponding author. Tel.: +34-913364488 ; fax: +34 913363295.E-mail address: qiong@torroja.dmt.upm.es

2210-9838 c© 2014 The Authors. Published by Elsevier B.V.Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering).

2 Author name / Procedia IUTAM 00 (2014) 000–000

be analyzed firstly: initial conditions(stability), internal change(sensitivity) and external excitation(receptivity), allof these quantities involve the direct and adjoint global modes. Here we use the synthetic jet which is one of themost useful micro fluidic device to suppress the instabilities in the open cavity flow. The new ideal here is to use thedynamic mesh technique to generate the synthetic jet. From deforming the diaphragm by frequency and amplify, theflow around the slot of jet is changed and the goal of the cavity flow modification is achieved. As show in Fig. 1.

Fig. 1. Computational configuration of (OC+SJ) including boundary conditions and mesh details.

Fig. 2. Computational configuration of open cavity (OC).2. Problem formulation

2.1. Simulation configuration and fluid properties

The cavity and synthetic jet configuration and the fluid properties are defined as below: the ratio of the cavitylength and depth L/D = 2, the Reynolds number depend on the depth of cavity Re = UD∞

ν, the free stream velocity

U∞, the boundary layer momentum of thickness at the lip of the cavity θ, the displacement of the boundary layer δ,x means the distance from inlet to the cavity lip. In the configuration of synthetic jet: the ratio of depth and length isH/W, the slot length and width is h and d, the frequency and amplitude of diaphragm is f and A, the model detailsare plotted in the Fig.2. The flow conditions are shown in Table 1. The full incompressible N-S equations are solvedon a structure mesh, with clustering of points near the wall and the shear layer of the cavity. Spatial discretization iscarried out with finite volume method and time implicit integration.

ReD Reθ x θ δ ReD Reθ x θ δ

1400 47.8 4.70 0.03848 0.289 1900 53.84 3.46 0.0283 0.2134

Table 1. Parameters characterizing in the OC system2.2. BiGlobal linear stability theory

In the approach of linear stability analysis, the transient solutions can be decomposed to base flow which is an O(1)steady or time periodic laminar base flow and perturbation: q(x, y, z, t) = q(x, y) + εq(x, y, z, t), ε << 1. Assume theperturbation is a sinusoidal, depending the homogeneous Z direction with periodic length Lz = 2π/β, the perturbationcan be simplified as q(x, y, z, t) = q(x, y)ei(βz−ωt) + CC., where both ω and q can be complex while q is always real.ω = ωr +ωi, ωr is the frequency of the perturbation and ωi means damping/growth rate, if the ωi > 0, the perturbationwill grow exponential in time and the flow is unstable.if the ωi = 0, the flow is neutrally stable and ωi < 0 the flow isstable.2.3. Adjoint problem

The adjoint framework appears very natural in optimization theory since it yields a very efficient way to determinethe sensitivity of an objective with respect to control variables. As corresponding to the linear N-S operator, wedefine q∗(u∗, v∗, w∗, p∗) adjoint perturbation variables. The inner products in the space-time domain are defined,< q∗, q >=

∫ T0

∫Ω

q∗H qdVdt. In general, for any linear operator L, its adjoint operator L∗ is defined by the adjointidentity, so transpose the linearized operator by the inner product.

< q∗, Lq >=< L∗q∗, q > +B(q∗, q) (1)

Author name / Procedia IUTAM 00 (2014) 000–000 3

Where B(q∗, q) involves the boundary conditions in the adjoint system.2.4. Residual methods

Theofilis et al.(2003)7 analyzed the behavior of numerical residuals near convergence, which aims at extractingthe steady-state and amplitude functions of the BiGlobal eigenmodes by simple algebraic operations in the flowsimulations with the least-damped eigenmodes. When the steady-state solution exists, we can get the damping rate ofthe least-stable eigemode from the residual of the velocity, it decays like the line and the slope of the decay is equalto the damping rate. Considering the transient solution from two time levels, t1 and t2, the steady-state flow and theeigenmodes can be get from the algebraic equations.

q(x, y, z, t1) = q(x, y) + q(x, y)e−iωt1 , (2)

q(x, y, z, t2) = q(x, y) + q(x, y)e−iωt2 . (3)3. Stability and sensitivity results

The open cavity flow without synthetic jet has been studied here in order to locate the centrifugal instability regionin the cavity. At the aid of the adjoint method, we calculate the optimal location to set the synthetic jet. Table 2 showsthe damping/growth rate and frequency of the direct global mode and the adjoint mode at Re = 1400 and Re = 1900.At Re=1400, the perturbation decays with the exponential behavior. At Re = 1900, the leading eigenmode is atraveling one, and its structural sensitivity changes in base flow lies outside of the cavity. The contrast of two casesspatially localized feedbacks are depicted Fig. 3. Further numerical simulations are on progress and conclusions willbe presented at time of the conference.

Formulation ωr ωi (Re=1400) ωr ωi (Re=1900)Full N-S equations - −0.0172 3.1424 −0.0123

Linear N-S equations - −0.0172 3.1427 −0.0124Adjoint Linear N-S equations - 0.0172 3.1429 0.0124

Table 2. The frequencies and damping/growth rates extracted from three simulations of open cavity flow

Fig. 3. The receptivity to spatially localized feedback of open cavity at Re=1400 (first) and Re=1900 (second) respectively.References1. K. Krishnamurty, Acoustic radiation from two dimensional rectangular coutouts in aerodynamic surfaces. NACA TN 1955; 3487.2. J. Rossiter. Wind-tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. Aeronautical research council

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