Three-dimensional Disturbances Research Articles

Linear instability in thermally stratified quasi-Keplerian flows

Quasi-Keplerian flow, a special regime of Taylor–Couette co-rotating flow, is of great astrophysical interest for studying angular momentum transport in accretion disks. The well-known magnetorotational instability (MRI) successfully explains the flow instability and generation of turbulence in certain accretion disks, but fails to account for these phenomena in protoplanetary disks where magnetic effects are negligible. Given the intrinsic decrease of the temperature in these disks, we examine the effect of radial thermal stratification on three-dimensional global disturbances in linearised quasi-Keplerian flows under radial gravitational acceleration mimicking stellar gravity. Our results show a thermo-hydrodynamic linear instability for both axisymmetric and non-axisymmetric modes across a broad parameter space of the thermally stratified quasi-Keplerian flow. Generally, a decreasing Richardson or Prandtl number stabilises the flow, while a reduced radius ratio destabilises it. This work also provides a quantitative characterisation of the instability. At low Prandtl numbers $Pr$ , we observe a scaling relation of the linear critical Taylor number $Ta_c\propto Pr^{-6/5}$ . Extrapolating the observed scaling to high $Ta$ and low $Pr$ may suggest the relevance of the instability to accretion disks. Moreover, even slight thermal stratification, characterised by a low Richardson number, can trigger the flow instability with a small axial wavelength. These findings are qualitatively consistent with the results from a traditional local stability analysis based on short wave approximations. Our study refines the thermally induced linearly unstable transition route in protoplanetary disks to explain angular momentum transport in dead zones where MRI is ineffective.

Three-dimensional eigensolutions of the unsteady compressible boundary-region equations

A previous paper of the authors (Duck & Stephen, J. Fluid Mech., vol. 917, 2021, A56) considered the effect of three-dimensional, temporally periodic, linear and incompressible disturbances on a Blasius boundary layer, in particular when the disturbance wavelength is both comparable to and longer than the boundary-layer thickness. This previous study revealed that, unlike the two-dimensional counterpart, a mode exists that exhibits regimes of downstream spatial growth. In this paper we extend the analysis to the compressible regime, based on the boundary-region equations methodology. The aforementioned unstable mode is seen to persist into the compressible regime, and is studied using a combination of numerical and asymptotic methods. The paper adopts several approaches. First is a numerical approach in which the spatial development of the disturbances is assessed. This then leads to a consideration of the far-downstream behaviour, using (several) asymptotic limits. Of some note, in addition to unstable modes found in the incompressible case, is the existence of a further class of instability, not found in the incompressible case (which is also analysed asymptotically), corresponding to what amounts to an inviscid instability. The far-downstream analysis enables a (sub-)classification into entropy and non-entropy modes. The former, according to this analysis, are spatially damped, with one caveat, as revealed by our marching procedure, which highlights how spatial development of disturbances can be important.

Stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances

An analysis is presented for the stability of a viscous liquid film flowing down an inclined plane with respect to three-dimensional disturbances under the action of gravity and surface tension. Using momentum-integral method, the nonlinear free surface evolution equation is derived by introducing the self-similar semiparabolic velocity profiles along the flow (x- and y-axis) directions. A normal mode technique and the method of multiple scales are used to obtain the theoretical (linear and nonlinear stability) results of this flow problem, which conceive the physical parameters: Reynolds number Re, Weber number We, angle of inclination of the plane θ and the angle of propagation of the interfacial disturbances ϕ. The temporal growth rate ωi+ and second Landau constant J2, based on which various (explosive, supercritical, unconditional, subcritical) stability zones of this flow problem are categorized, contain the shape factors B and β owing to the non-zero steady basic flow along the y-axis direction. A novel result which emerges from the linear stability analysis is that for any given value of Re, We and θ, any stability that arises in two-dimensional disturbances (ϕ = 0) must also be present in three-dimensional disturbances. For ϕ = 0, there exists a second explosive unstable zone (instead of unconditional stable zone) after a certain value of Re (or θ) due to the involvement of B and β in the expression of J2. This explosive unstable zone vanishes after a certain value of ϕ depending upon the values of Re, We and θ, which confirms the stabilizing influence of ϕ on the thin film flow dynamics irrespective of the values of Re, We and θ.

Properties of the hybrid solutions for a generalized (3 + 1)-dimensional KP equation

The Kadomtsev-Petviashvili (KP) equation is instrumental in characterizing nonlinear wave phenomena in fluid dynamics. With the incorporation of three-dimensional disturbances, the (3+1)-dimensional KP equation emerges. In this study, we employ the Hirota bilinear method to investigate a (3+1)-dimensional KP equation. We successfully derive both lump and lump-chain solutions for this equation. The characteristics of the resulting lump are thoroughly examined. It is observed that those lumps propagate along a specific ray direction, with the origin of these rays dependent on the imaginary part of σˆjk=σj−σk with 1≤j,k≤3 and j≠k. We have simultaneously obtained the propagation direction of these rays. The periodicity of those lumps is denoted as Tjk dependent on spectrum parameter. Furthermore, we explore two potential lines, including the characteristic line associated with the lumps, to gain a deeper understanding of the nonlinear waves controlled by this system.

Nonmodal stability analysis of Poiseuille flow through a porous medium

We unravel the nonmodal stability of a three-dimensional nonstratified Poiseuille flow in a saturated hyperporous medium constrained by impermeable rigid parallel plates. The primary objective is to broaden the scope of previous studies that conducted modal stability analysis for two-dimensional disturbances. Here, we explore both temporal and spatial transient disturbance energy growths for three-dimensional disturbances when the Reynolds number and porosity of the material are high, based on evolution equations with respect to time and space, respectively. Modal stability analysis reveals that the critical Reynolds number for the onset of shear mode instability increases as porosity increases. Moreover, the Darcy viscous drag term stabilizes shear mode instability, resulting in a delay in the transition from laminar flow to turbulence. In addition, it demonstrates the suppression of three-dimensional shear mode instability as the spanwise wavenumber increases, thereby confirming the statement of Squire’s theorem. By contrast, nonmodal stability analysis discloses that both temporal and spatial transient disturbance energy growths curtail as the effect of the Darcy viscous drag force intensifies. But their maximum values behave like O(Re2) for a fixed porous material, where Re is the Reynolds number. However, for different porous materials, the scalings for both temporal and spatial transient disturbance energy growths are different. Furthermore, increasing porosity also suppresses both temporal and spatial disturbance energy growths. Finally, we observe that temporal transient disturbance energy growth becomes larger for a spanwise perturbation, while spatial transient disturbance energy growth becomes larger for a steady perturbation when angular frequency vanishes. The initial disturbance that excites the largest temporal energy amplification generates two sets of alternating high-speed and low-speed elongated streaks in the streamwise direction.

Linear stability analysis of compressible boundary layer over an insulated wall using compound matrix method: Existence of multiple unstable modes for Mach number beyond 3

The linear spatial stability of a parallel two-dimensional (2D) compressible boundary layer on an adiabatic plate is investigated by considering both 2D and three-dimensional (3D) disturbances. The compound matrix method is employed here, for the first time, for compressible flows, which, unlike other conventional techniques, can efficiently eliminate the stiffness of the equations governing the spectral amplitudes. The method is first validated with published results in the literature corresponding to spatial and temporal instability of flows ranging from low subsonic to high supersonic Mach numbers (M), which shows a good match depending upon the proper choice of free-stream temperature and the wall dispersion relation. Subsequently, flow compressibility effects and the spanwise variation of disturbances are also investigated for M ranging from low subsonic to high supersonic cases (from M = 0.1 to 6). Mack (AGARD Report No. 709, 1984) reported the existence of two unstable modes for M > 3 from viscous calculations (the so-called “second mode”) that subsequently fuse to create only one unstable zone when M increases. Our calculations show a series of higher-order unstable modes for M > 3 in addition to the findings of Ma and Zhong [“Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions,” J. Fluid Mech. 488, 31–78 (2003)], where such higher-order modes for supersonic boundary layers are all noted to be spatially stable. The number and the frequency extent of the corresponding unstable zones increase with an increase in M beyond 3 while propagating downstream at a higher speed than those corresponding to incompressible, subsonic, and low supersonic (M < 2) cases.

Unravelling the existence of asymmetric bubbles in viscoelastic fluids

We study the motion and deformation of a single bubble rising inside a cylindrical container filled with a viscoelastic material. We solve numerically the mass and momentum balances along with the constitutive equation for the viscoelastic stresses, without assuming axial symmetry to allow the growth of three-dimensional disturbances. Hence, we may predict the emergence of the notorious knife-edge shape of the bubble, which is a result of a purely elastic instability triggered in the locality of the trailing edge. Our results compare well with existing experiments. We visualize, for the first time to the best of our knowledge, the flow kinematics and dynamics that arise downstream of the bubble. We propose two quantities, one kinematical and one geometrical, for the determination of the onset of the instability. We demonstrate that extension-rate thinning in the constitutive law is necessary for the emergence of the instability. Moreover, our results indicate that increasing (a) the deformability of the bubble and (b) the initial extension rate hardening of the viscoelastic material, prior to thinning, triggers the instability earlier. These novel findings help us formulate and propose a mechanism that controls the onset of the instability and explain why the knife-edge shape is not encountered as frequently.

The Performance Analysis of Grouting Repair Effect on the Accuracy of Disturbance Stress Test in Damaged Surrounding Rock Mass.

Disturbance stress assessment is crucial for ensuring the safety of deep engineering projects. Currently, the primary technique for continuously monitoring three-dimensional disturbance stress is the stress relief method, but its accuracy can be compromised by rock damage that occurs after excavation. To mitigate this issue, grouting is employed to repair damaged rock masses and enhance their mechanical properties. However, the impact of grouting techniques on improving the accuracy of disturbance stress testing is challenging to evaluate through laboratory and in situ experiments. To address this problem, numerical simulation technology is employed to investigate disturbance stress testing after the repair of damaged surrounding rock through grouting. The simulation results indicate that grouting repair significantly enhances the accuracy of stress testing. As the depth of damaged rock mass repair increases, the error in stress testing decreases. Achieving complete repair of the initial damage zone during grouting is essential to eliminate errors in stress testing. Expanding on the positive effects of grouting repair on stress testing, a segmented testing method for disturbance stress is proposed. The method involves separately testing the initial stress and stress changes, thereby reducing the stress level within the rock, minimizing rock failure, and enhancing the accuracy of disturbance stress testing. This study provides valuable reference methods, and the outcomes of this research will serve as a foundation for enhancing the accuracy of disturbance stress testing in deep hard rock engineering.

Theoretical study of the influence of gas phase turbulent disturbances on the kinetics of ultrasonic coagulation of PM2.5

A numerical model of ultrasonic coagulation of PM2.5 aerosol particles in three-dimensional vortex and turbulent acoustic flows is proposed in the article. The model is intended to identify the possibility of increasing the efficiency of the process. A numerical analysis of the model using the example of PM2.5 aerosol made it possible to establish that the presence of three-dimensional turbulent disturbances leads to the fact that the coagulation efficiency of PM2.5 reaches almost 100% at a sound pressure level of no more than 165 dB.

Spatiotemporal Evolution of Wind Turbine Wake Characteristics at Different Inflow Velocities

In this paper, the spatiotemporal evolution of wind turbine (WT) wake characteristics is studied based on lattice Boltzmann method-large eddy simulations (LBM-LES) and grid adaptive encryption at different incoming flow velocities. It is clearly captured that secondary flow occurs in the vortex ring under shear force in the incoming flow direction, the S-wave and the Kelvin–Helmholtz instability occur in the major vortex ring mainly due to the unstable vortex ring interface with small disturbance of shear velocity along the direction of flow velocity. The S-wave and Kelvin–Helmholtz instability are increasingly enhanced in the main vortex ring, and three-dimensional disturbances are inevitable along the mainstream direction when it evolves along the flow direction. With increasing incoming flow, the S-wave and Kelvin–Helmholtz instability are gradually enhanced due to the increasing shear force in the flow direction. This is related to the nonlinear growth mechanism of the disturbance. The analysis of the velocity signal, as well as the pressure signal with a fast Fourier transform, indicates that the interaction between the vortices effectively accelerates the turbulence generation. In the near-field region of the wake, the dissipation mainly occurs at the vortex at the blade tip, and the velocity distribution appears asymmetric around the turbine centerline under shear and the mixing of fluids with different velocities in the wake zone also leads to asymmetric distributions.

Floquet-based analysis on three-dimensional non-axisymmetric instabilities in oscillatory-driven Taylor–Couette flows and their low-frequency asymptotic behavior using Wentzel–Kramers–Brillouin method

This paper revisits the linear stability analysis of oscillatory-driven flows between two oscillating cylinders against non-axisymmetric disturbances. This study is motivated by the lack of a sufficiently reliable theoretical analysis giving insight into the experimentally observed spiral-like non-axisymmetric patterns when the cylinders are counter-oscillating. A new generalized time-dependent algebraic eigenvalue problem is constructed from the linearized set of the three-dimensional Navier–Stokes equations around the purely azimuthal basic state. Numerical evaluation of the critical eigenvalues combining both Floquet theory and spectral method reveals the existence of frequency ranges where this basic state becomes unstable against three-dimensional non-axisymmetric disturbances before it does so for two-dimensional axisymmetric ones. Indeed, as the oscillation frequency of the cylinders increases, the azimuthal wave number of the critical eigensolution is found to change from 0 to 2 to 1 and then back to 0. The primary bifurcation exchange between two instability modes with different azimuthal wave numbers occurs via different types of codimension-2 bifurcation points giving rise to discontinuities in the critical axial wave number where reversing and non-reversing non-axisymmetric Taylor vortex flows are identified. In addition, by extending our numerical calculations to the co-oscillating case, we show that the axisymmetric disturbances are the most unstable confirming thus existing experimental findings. Furthermore, a Wentzel–Kramers–Brillouin (WKB) analysis is performed to shed light on the asymptotic behavior of these time-dependent flows in the low-frequency limit when the cylinders are slowly oscillating.

Intermittent disturbance mechanical behavior and fractional deterioration mechanical model of rock under complex true triaxial stress paths

Mechanical excavation, blasting, adjacent rockburst and fracture slip that occur during mining excavation impose dynamic loads on the rock mass, leading to further fracture of damaged surrounding rock in three-dimensional high-stress and even causing disasters. Therefore, a novel complex true triaxial static-dynamic combined loading method reflecting underground excavation damage and then frequent intermittent disturbance failure is proposed. True triaxial static compression and intermittent disturbance tests are carried out on monzogabbro. The effects of intermediate principal stress and amplitude on the strength characteristics, deformation characteristics, failure characteristics, and precursors of monzogabbro are analyzed, intermediate principal stress and amplitude increase monzogabbro strength and tensile fracture mechanism. Rapid increases in microseismic parameters during rock loading can be precursors for intermittent rock disturbance. Based on the experimental result, the new damage fractional elements and method with considering crack initiation stress and crack unstable stress as initiation and acceleration condition of intermittent disturbance irreversible deformation are proposed. A novel three-dimensional disturbance fractional deterioration model considering the intermediate principal stress effect and intermittent disturbance damage effect is established, and the model predicted results align well with the experimental results. The sensitivity of stress states and model parameters is further explored, and the intermittent disturbance behaviors at different f are predicted. This study provides valuable theoretical bases for the stability analysis of deep mining engineering under dynamic loads.

Three-dimensional trajectory tracking guidance against near-space maneuvering targets with multiple constraints under wind field

This paper presents a three-dimensional disturbance observer-based adaptive tracking guidance scheme for the trajectory tracking problem of a missile intercepting a hypersonic vehicle with the wind field, autopilot lag dynamics, and input saturation. Unlike most of the existing missile trajectory tracking guidance systems without velocity controllability and external wind field model, the wind field model is added to the dynamic model, and the velocity control channel is introduced into the three-dimensional trajectory tracking guidance problem to prevent impractical scenarios and increase tracking precision. The proposed tracking guidance methodology incorporates the sliding mode and adaptive control techniques, in which the sliding mode motion is finite time stable independently of the tracking guidance system's initial conditions. Also, the super twisting switch control law is added to cope with the chattering problem. In this context, external uncertainty disturbances and unmeasured states are lumped into lump disturbance and estimated using the backpropagation neural network extended state observer. To this end, numerical simulations demonstrate the tracking to reach zero between 3 s and 5 s against various uncertainties in engineering practical, and position tracking errors are within 0.15 m, superior to the existing tracking guidance methods.

Damping of three-dimensional waves on coating films dragged by moving substrates

Paints and coatings often feature interfacial defects due to disturbances during the deposition process, which, if they persist until solidification, worsens the product quality. In this article, we investigate the stability of a thin liquid film dragged by a vertical substrate moving against gravity, a fundamental flow configuration in various coating processes. The receptivity of the liquid film to three-dimensional disturbances is analyzed with Direct Numerical Simulations (DNS) and an in-house Integral Boundary Layer (IBL) film model. The latter was used for linear stability analysis and nonlinear wave propagation analysis. The numerical implementation of the IBL film model combines a finite volume formulation with a pseudo-spectral approach for the capillary terms that allows one to investigate non-periodic surface tension-dominated flows. The numerical model was successfully validated with DNS computations. The combination of these numerical tools allows one to describe the mechanisms of capillary and nonlinear damping and identify the instability threshold of the coating processes. The results show that transverse modulations can be beneficial for damping two-dimensional waves within the range of operational conditions considered in this study, which are relevant to air-knife and slot-die coating.

Research on a Three-Dimensional Fuzzy Active Disturbance Rejection Controller for the Mechanical Arm of an Iron Roughneck

In the position control of the mechanical arm of an iron roughneck (MAIR), a controller with high responsiveness, high accuracy, and high anti-interference capability is necessary. An MAIR consists of two proportional-valve-controlled single-extension (PVCSE) hydraulic cylinders, and a traditional proportional–integral–derivative (PID) controller cannot easily achieve the accuracy and robustness requirements of the hydraulic cylinders. In this paper, a three-dimensional fuzzy active disturbance rejection controller (TF-ADRC) is proposed for an MAIR, which adds a three-dimensional fuzzy module to a classical active disturbance rejection controller (ADRC) to adjust the controller output according to the tracking of differential deviation, deviation change rate, and deviation change acceleration rate. Firstly, the trajectory planning of the MAIR was carried out using the quintic polynomial interpolation method to improve the smoothness of the target trajectory. Then, the reliability of the established model was verified by experiments. Finally, the comprehensive performance of a PID controller, fuzzy PID controller, ADRC, and TF-ADRC were compared based on the AMESim-Simulink model. The system with the TF-ADRC exhibits higher position control accuracy and better anti-interference capability than the system with a PID or Fuzzy PID controller, and accuracy is higher compared with the common ADRC.

An optimized stability framework for three-dimensional Hartman flow via Chebyshev collocation simulations

Hydrodynamics instability is studied for an electrical conducting fluid against small disturbance between the channels by using normal magnetized force. Chebyshev collocation technique is used to determine the stability of three-dimensional Hartmann flow problem. The distinctive case of the perturbations is calculated. It is also considered that perturbations intensity be determined by just on quantified by different parameters. We have considered one of the flow stabilities conditions to analyze our problem. QZ (Qualitat and Zuverlassigkeit) technique is applied to investigate the problem to draw stability curves. αc,βc are critical wave numbers in streamwise and span-wise respectively and for a big range of Hartmann value (Ha), we obtained critical Reynolds number (Rec). It is found that Couette flow is destabilized for a specific value of Magnetize force while with greater or lesser magnitude than the particular one will stabilize the flow. Disturbances with particular oblique angle θ will grow while the others will decay for the three-dimensional disturbance. We observed from over results that for Ha(>2.0) Rec, increases steadily and when Ha more than 3.886, Rec decreases fast to the minimum. It is also established that for drop down from Ha = 3.886 and Rec becomes very large. For distinct Hartmann values, there exist two Rec numbers due to closed contour. The outcomes of current study are utilized in drug-delivery systems and photodynamic therapy.

Superposition of AC-DBD plasma actuator outputs for three-dimensional disturbance production in shear flows

This investigation explores the utility of Alternating Current, Dielectric Barrier Discharge (AC-DBD) plasma actuators for producing three-dimensional disturbances of a desired spanwise wavelength via superposition. The technique utilizes two pairs of exposed and covered electrodes on a single dielectric layer arranged in streamwise succession. Two-dimensional forcing is achieved through operation of the upstream, spanwise uniform electrode pair, while three-dimensional forcing at a prescribed spanwise wavelength is attained by operating both electrode pairs simultaneously, with the downstream actuator spanwise modulating the upstream, two-dimensional output. The ability to produce disturbances of different spanwise wavelengths but with equal streamwise wavelength, frequency and total momentum is established through a combined characterization effort that considers quiescent and in-flow conditions. A demonstration of the technique in an exemplary wall-bounded shear flow, a laminar separation bubble, is provided, revealing spanwise wavelength dependent disturbance growth in the flow that could be exploited for performance gains in future flow control endeavours.Graphical abstract

On the instability of the magnetohydrodynamic pipe flow subject to a transverse magnetic field

The linear stability of a fully developed liquid–metal magnetohydrodynamic pipe flow subject to a transverse magnetic field is studied numerically. Because of the lack of axial symmetry in the mean velocity profile, we need to perform a BiGlobal stability analysis. For that purpose, we develop a two-dimensional complex eigenvalue solver relying on a Chebyshev–Fourier collocation method in physical space. By performing an extensive parametric study, we show that in contrast to the Hagen–Poiseuille flow known to be linearly stable for all Reynolds numbers, the magnetohydrodynamic pipe flow with transverse magnetic field is unstable to three-dimensional disturbances at sufficiently high values of the Hartmann number and wall conductance ratio. The instability observed in this regime is attributed to the presence of velocity overspeed in the so-called Roberts layers and the corresponding inflection points in the mean velocity profile. The nature and characteristics of the most unstable modes are investigated, and we show that they vary significantly depending on the wall conductance ratio. A major result of this paper is that the global critical Reynolds number for the magnetohydrodynamic pipe flow with transverse magnetic field is Re = 45 230, and it occurs for a perfectly conducting pipe wall and the Hartmann number Ha = 19.7.

High-fidelity computational study of roughness effects on high pressure turbine performance and heat transfer

While blade surface roughness arising from in-service wear and/or the manufacturing process greatly affects aero-thermal performance, the detailed underlying physical mechanisms remain far from fully understood. In this study, a series of highly-resolved Large-Eddy Simulations of compressible flow past a high-pressure turbine vane with systematically varied levels of blade surface roughness have been performed, along with a smooth-blade simulation at matched flow conditions for comparison. Three non-dimensional roughness amplitudes have been investigated, namely, ks/c={1.0,2.0,3.0}×10−3, where ks is an equivalent value of Nikuradse’s sandgrain roughness for an irregular, multi-scale near-Gaussian height distribution, and c is the axial blade chord. All simulations have been performed at an axial chord Reynolds number of 0.59 ×106 and a Mach number of 0.9, based on the exit conditions of the reference smooth vane, and with synthetic inflow turbulence to mimic unsteady, three-dimensional disturbances from an upstream combustion chamber. The present investigation highlights the profound impact that blade surface roughness can have upon boundary-layer transition mechanisms, wall shear stress and blade surface heat flux, as well as the levels of turbulence kinetic energy and total pressure losses incurred in the wake. While blade surface roughness leads to major aero-thermal differences between the suction-side of the smooth and rough vanes, the pressure-side surface remains relatively unaffected — even for the largest roughness amplitude investigated here.

On the origin of spanwise vortex deformations during the secondary instability stage in compressible mixing layers

The three-dimensionality of turbulence initiates with spanwise vortex deformations associated with the amplification of three-dimensional disturbance modes. However, the origin of spanwise vortex deformations is still not well understood. In this paper, compressible mixing layers are performed via direct numerical simulation (DNS). Two typical types of secondary instabilities producing spanwise vortex deformations are of consideration: fundamental instability and subharmonic instability. Based on the fast Fourier transform and DNS data, a low-rank velocity model v0 is obtained to demonstrate that spanwise vortex deformations are originated from a linear superposition of fundamental norm mode, a pair of fundamental or subharmonic oblique modes, and the mean mode. Through observing flow structures of the above norm and oblique modes, a striking feature is found that the velocity model v0 containing deformed spanwise vortices can be decomposed into three new velocity models v1, v2, and v3 containing relatively simplified counterparts (spanwise or oblique vortices). Then, the instability mechanism of the latter vortices is explored by analyzing the position relationship between the function of the generalized inflection points and cores of relatively simplified vortices. We find that an inviscid inflectional instability mechanism is responsible for the formation of spanwise and oblique vortices. Based on the above findings, a view is first proposed that spanwise vortex deformations with aligned and staggered patterns are a joint result of the parametric resonant mechanism and the inviscid inflectional instability mechanism.