Control Of Flow Instability In Typical Oscillator And Noise Amplifier Flows

In the wall-bounded flow or the flow over a bluff body,energy of perturbations can take exponential growth or transient growth due to some instability mechanisms,such as Kelvin-Helmholtz(KH)instability or elliptic instability et al.The growing perturbations make the flow unstable thereby inducing laminar-turbulent transition and increasing the drag of the bluff body or the wall.This phenomena make against with improving the performances of vehicles.So,some controls should be applied in these flow systems to suppress the growth of perturbations.Flow systems can be classified as oscillator and noise amplifier flows according to the characters of perturbation evolution:if the temporally growing global perturbation mode is present,and the spatial evolution of the unsteady flow rely on the growth of the initial disturbances,these flows behave as oscillator flows;if the growing perturbations are adveeted downstream,and spatial evolution of the unsteady flow is in large part determined by the character of the excitation,these flows behave as noise amplifier flows.In this paper,control of instability in some typical oscillator and noise amplifier flows utilizing Lorentz force are numerically studied via high accuracy Spectral Element Method(SEM).Three control strategies are adopted.The first control strategy is base flow modification which means the streamwise Lorentz force is applied to modify the base flow thus changing the growth rate of perturbations.In the second control strategy,the transfer functions between the inputs(upstream disturbance and wallnormalwise Lorentz force)and the output(downstream amplified perturbation)are identified by Auto-Regressive Moving-Aaverage Exogenous(ARMAX)model.The feedforward controller is designed based on the presumptive output signal.So,the linear control model is built to accomplish control target.The third control strategy combines the first and second strategy to control the flow instability.Flow over a cylinder belongs to the typical oscillator flows.In this paper,the first control strategy is conducted to reduce the growth rate of the three-dimensional perturbations in the cylinder wake at Re=300(Re=U?d/v,whereU? is the freestream velocity,d is the cylinder diameter and v is the kinematic viscosity coefficient).The electromagnetic actuator is applied on surface of the cylinder to generate the streamwise Lorentz force which modifies the base flow thus changing the grate rate of the three-dimensional perturbations.As the growth rate of perturbations is sensitive to the control regions,three different control regions:incident side,lee side and whole cylinder are considered.The growth rate of the three-dimensional perturbations under different interaction numbers N(representing the strength of electrodynamic force relative to inertial force of the fluid)is predicted by Floquet stability analysis.Compared with the growth rates of the most unstable perturbations with a small spanwise wavelength(mode B)and a large spanwise wavelength(mode A)under no control,there are minor variations of the growth rates under incident side control,while the growth rates decrease with the increasing interaction number under lee side and whole cylinder control.Besides,there only exists unstable mode A at N? 0.8.Analyses of the elliptic and hyperbolic instability clarify the elliptic instability induces the growth of perturbations in mode A while the hyperbolic instability induces the growth of perturbations in mode B.The variation trends of inviscid growth rates of the elliptic and hyperbolic instability regions under different control cases agree well with the variation trends of the growth rate of the most unstable mode A and mode B.Moreover,the deformation of the wake in a short interval is described via the particle tracking method.Due to the suppression of flow separation,the deformation magnitude is reduced under lee side and whole cylinder control,then the inviscid growth rates of elliptic and hyperbolic instability regions decrease.This is responsible for the reduction of the growth rates of mode A and mode B under lee side and whole cylinder control.Several three-dimensional direct numerical simulations are conducted to explore the nonlinear evolution of perturbations and the control effects.It is found the spanwise distance of three-dimensional vortex pair is around 3.4 cylinder diameters and the drag reduction of 15.2%and 14.4%is achieved at N=1.0 under lee side and whole cylinder control respectively.Flow over a square leading-edge flat plate belongs to typical noise amplifier flows because the perturbation energy is amplified when flowing through the separation bubble.The first strategy is adopted in the control of the instability in the flow at Re=400(Re=U?h/v,where h is the plate thickness).The electromagnetic actuator is applied near the leading-edge of the plate to change the separation bubble scale via the excited streamwise Lorentz force thus changing the growth of the perturbation energy.In the three-dimensional flow at Re=400,there exists Pattern B:the hairpin-shaped vortices offsetting by 1800 from one row to the next.Though former researches demonstrate the elliptic instability mechanism and transient growth of perturbations induce the transition,it is found subharmonic resonance is the transition mechanism of this flow in this paper.The primary instability is detected by proper orthogonal decomposition(POD)of the two dimensional unsteady flow and the convectively unstable secondary instability which is a subharmonic perturbat:ion mode is found by Floquet stability analysis of the two dimensional unsteady flow.Optimal transient growth analysis of perturbation in the two dimensional steady flow illustrate the optimal perturbation mode which contains the maximum energy amplification magnitude always occurs near the reattachment point.Moreover,the similarity between the optimal perturbation mode and the first POD mode which represents the major components of the primary instability means transient growth of perturbations induce the primary instability thus forming the KH vortices.Pairing instability of adjacent KH vortices generates the secondary instability thereby subharmonic resonance transition occurs.The streamwise Lorentz force decrease the scale of the separation bubble.The maximum energy amplification magnitude of perturbations takes a linear attenuation with the interaction number,thus the primary instability is reduced under control.The reduced primary instability is not strong enough to induce the secondary instability,so the flow is globally stable under control.Three-dimensional direct numerical simulation explores the nonlinear evolution of perturbation and the control effects.Though the growth rate of the convectively unstable secondary instability is limited by the flow field scale,the feedback loop of energy transfer promotes the resonance transition.However,as the separation bubble scale is reduced and the feedback loop is broken by the streamwise Lorentz force,the three-dimensional transition is suppressed,and a skin-friction drag reduction of 31.3%and 2G.8%at N = 0.04 and TV = 0.1 is achieved respectively.Finally,three control strategies are adopted to perform the control of the instability in the backward-facing step flow which belongs to the noise amplifier flows,and compare the control effects between the three strategies.The Reynolds numerber of the flow is Re = 500(Re=U?h/v where h is the hight of the step)and the expansion ratio is 2.There exists a primary separation bubble on the lower wall and a secondary separation bubble on the upper wall.In the first control strategy,the streamwise Lorentz force is applied in the primary and secondary bubble which called lower wall control and upper wall control respectively.The interrelationship between the primary and secondary separation bubbles is decrease of ons's scale can make a increase of the other one's scale.The initial wall-normalwise perturbations with a gaussian distribution near the step can achieve the minimum peak value of energy magnification at TV = 0.15 under upper wall control.Optimal transient growth analysis of perturbations demonstrates the peak value of the the optimal energy magnification is minimum at TV = 0.1,0.15 under upper wall control.This means the magnification of perturbation energy can be reduced by adopting the first control strategy.In the second control strategy,the sensor upstream the step,the control actuator(electromagnetic actuator)and the sensor downstream the reattachment point of the primary separation bubble are placed in the flow system.The upstream noise source is the wall-normalwise perturbation with the gaussian distribution.The wall-normalwise Lorentz force is formed by the actuator.Both sensors measure the wall-normalwise gradient of the streamwise perturbation velocity.The gradient describes the skin-friction of the measured regions.The inputs of the system contain the signal from the upstream sensor and the control signal which represents the magnitude of the Lorentz force,and the output signal comes from the downstream sensor.The ARMAX model is built based on the numerical data to carry out system identification.The goodness of fit between the sampling signal and the predicted signal by ARMAX model reaches to 91.2%.After the closed-loop control system containing a feedforward controller is applied in direction numerical simulation,the perturbation energy of the whole flow is reduced,but the perturbation energy also oscillates with a minor amplitude.In the third strategy,the base flow is modified with N=0.15 under upper wall control.The goodness of fit between the sampling signal and the predicted signal by ARMAX model is 86.4%,and there is no oscillations of the perturbation energy in the direct numerical simulation.This illustrates the amplification of perturbation anergy can be completely suppressed by adopting the third control strategy.Researches in this paper broaden the applications of the Lorentz force in flow control,and provide the references for the application of the Lorentz force in engineering.