Growth Rate Of Disturbances Research Articles

Transitional Flow Dynamics Behind a Micro-Ramp

Micro-ramps are popular passive flow control devices which can delay flow separation by re-energising the lower portion of the boundary layer. We compute the laminar base flow, the instantaneous transitional flow, and the mean flow around a micro-ramp immersed in a quasi-incompressible boundary layer at supercritical roughness Reynolds number. Results of our Direct Numerical Simulations (DNS) are compared with results of BiLocal stability analysis on the DNS base flow and independent tomographic Particle Image Velocimetry (tomo-PIV) experiments. We analyse relevant flow structures developing in the micro-ramp wake and assess their role in the micro-ramp functionality, i.e., in increasing the near-wall momentum. The main flow feature of the base flow is a pair of streamwise counter-rotating vortices induced by the micro-ramp, the so-called primary vortex pair. In the instantaneous transitional flow, the primary vortex pair breaks up into large-scale hairpin vortices, which arise due to linear varicose instability of the base flow, and unsteady secondary vortices develop. Instantaneous vortical structures obtained by DNS and experiments are in good agreement. Matching linear disturbance growth rates from DNS and linear stability analysis are obtained until eight micro-ramp heights downstream of the micro-ramp. For the setup considered in this article, we show that the working principle of the micro-ramp is different from that of classical vortex generators; we find that transitional perturbations are more efficient in increasing the near-wall momentum in the mean flow than the laminar primary vortices in the base flow.

Effects of thermodiffusive instability on the spherical premixed flames anchored to a porous-plug burner

Steady state burning and thermodiffusive instability of spherical premixed flames stabilized on a spherical porous-plug burner are analyzed using large-activation-energy asymptotic approach. The steady-state solution and linear stability solution are validated by comparing them to the published theoretical studies since there is limited experiment result. Within the frame of the flame-sheet model, the analytic solutions depend on the temperature and mass fraction within the flame front. The coupling between the temperature and the stand-off distance for the steady state are analyzed in detail. The spherical flames show two types of dual flame behavior and are stabilized by either heat losses to the burner or flow divergence. Subsequently, the growth rates of disturbances superimposed on the spherical burner-stabilized flames depend on the wavenumbers and are resolved. Results show that the flames are stabilized, lost stability and being re-stabilized with increasing flow velocity. It is further shown that, with increasing Lewis number (Le), the instability of the spherical flame is suppressed and only the pulsating instability appears once the Le exceeds unity. Increasing the heat release parameter and the radius of the porous-plug can enhance flame instability. The porosity μ and thickness h of the porous-plug burner have the contradictive effect on flame stability. Moreover, with increasing the ratio of μ/h, the flame becomes more stable.

The evolution of wingtip vortex wandering: a stability analysis based on stereo-PIV experiment

Instability, as a potential mechanism for vortex flow control, is manifested typically in the phenomenon of wingtip vortex wandering. In this paper, the evolution of wingtip vortex generated by a NACA0015 rectangular wing within six chord lengths of its wake at Rec= 2.1 × 105–3.5 × 105 and AoA = 4°–10° is investigated by stereo particle image velocimetry experiment. The streamwise evolution of vortex wandering is clearly captured, which demonstrates strong anisotropy. The amplitude of vortex wandering increases under all conditions, which exhibits greater growth rates in the condition of larger AoA. To account for such phenomenon, spatial linear stability analysis is performed on the base flow obtained by SPIV experiment to investigate quantitatively the amplification of small disturbance and disturbance modes at the experiment conditions. It is discovered that the wingtip vortex in the experiment is marginally stable under all conditions. Furthermore, the spatial growth rates of instability by AoA correspond well with that of wandering amplitude, indicating that the quick enlargement of the amplitude of wandering at larger AoA condition is caused by larger spatial growth rate of disturbance. In addition, the least-stable disturbance mode by LSA reveals a non-zero directed velocity perturbation in the vortex core, which rotates periodically. Such perturbation developing in both time and space, which is stirred by vortex instability, leads to the behavior of wingtip vortex wandering.

Instability and transition induced by wall roughness in a finite Stokes layer

Abstract The transition of a classic unsteady flow, namely, a Stokes layer, induced by wall roughness is investigated numerically. On the basis of numerical simulations using a semispectral method, a detailed analysis of the evolution characteristics of the disturbances is carried out. The results show that turbulence arises from the presence of large pre-existing two-dimensional waves accompanied by rapidly increasing three-dimensional waves. It is found that the 3D disturbances exhibit exponential growth prior to the onset of the transition, while the 2D disturbances reach a saturated state. The nonlinear growth rate of the 3D disturbances displays a linear relation to the spanwise wavenumber, with the fundamental spanwise wavenumber itself being connected to the appearance of the transition. The breakdown in the transition process is also analyzed, and similar to steady flows, an enhanced instability is found.

Influence of distributed heavy-gas injection on stability and transition of supersonic boundary-layer flow

This study performed a joint theoretical and experimental investigation of the influence of distributed normal-to-the-surface and inclined heavy-gas injection into the near-wall sublayer of a boundary layer through a permeable wall on the laminar-turbulent transition (LTT). Sulfur hexafluoride (SF6) is used as a foreign gas for injection into the boundary layer. It also assessed stability in relation to both natural and artificial (controlled) disturbances of a supersonic flat-plate boundary layer at a free-stream Mach number (M) of 2. It is established, theoretically, that the action of a large molecular weight gas injection on the boundary layer is similar to the action of wall cooling and leads to an increase in boundary layer stability and LTT delay. The influence of injection on the position of transition is estimated by means of the eN method. Principally, the analysis shows the possibility of increasing the transition Reynolds number by means of SF6 injection. Controlled disturbances are introduced in the model boundary layer by means of a point harmonic glow-discharge disturbance generator and are measured by using a hot-wire anemometer. For the first time, it is shown experimentally that distributed injection of the heavy SF6 gas leads to boundary layer stabilization. This is mostly due to the reduction in growth rates of disturbances at higher frequencies, while the LTT shifted to higher Reynolds number values. Good qualitative agreement is achieved between the experimental data obtained with artificially generated disturbances and computations based on linear stability theory.

Modulation instability and rogue waves for shear flows with a free surface

The effect of shear currents on rogue waves on a free surface is modeled by the nonlinear Schr\odinger equation. Curvature of velocity profiles (or vorticity gradient) and the relative motion of the wave packet and shear current play critical roles, through enhanced growth rates of disturbances.

Investigation on Asymmetric Instability of Cylindrical Power-Law Liquid Jets

An investigation has been performed to reveal the breakup mechanism of three-dimensional power-law cylindrical jets with different mode disturbances. It is observed experimentally that the asymmetric mode disturbances could prevail over the counterpart of symmetric mode under special conditions. The dispersion equation characterizing the instability of three-dimensional cylindrical jets of power-law fluids is deduced. The effects of the Weber number, generalized Reynolds number, power-law exponent, and gas–liquid density ratio on the jet instability are studied in detail. It is found that the maximum growth rates of asymmetric mode disturbances are usually larger than those of symmetric mode disturbances under high Weber numbers and low generalized Reynolds numbers, which implies that the former are more likely to be responsible for the breakup of power-law fluids. Meanwhile, the large gas–liquid interaction could trigger more short, unstable waves. Interestingly, with the increase of jet velocity, the interaction between liquid and gas phases plays an increasingly leading role on the breakup of power-law cylindrical jets, whereas the viscous force and the power-law exponent have less significant impacts. Theoretical analysis results give a better comprehensive understanding for the power-law jets.

Instability of an annular liquid sheet exposed to compressible gas flows

A temporal linear stability analysis was performed for a viscous annular liquid sheet exposed to compressible gas flows. An analytical dispersion relation illustrating the disturbance growth characteristics at the gas–liquid interfaces was formulated and validated, and a parametric study was performed to explore the mechanism that affects the sheet instability. The results reflect that para-sinuous disturbances on the liquid sheet always predominate over para-varicose disturbances under the considered flow conditions. The aerodynamic interaction at the gas–liquid interfaces of the liquid sheet is the decisive factor on sheet instability. As for the main factors affecting the aerodynamic effects, the contribution of the gas–liquid velocity difference to aerodynamic interactions is more significant than that of the gas–liquid density ratio. Neglecting the gas compressibility markedly underestimates the aerodynamic effects at the gas–liquid interfaces for flow conditions of high gas Mach number, which results in decreased disturbance growth rate and dominant wave number. Moreover, both the liquid viscous force and the surface tension force hold back the disturbance growing and stabilize the liquid sheet under the considered conditions. Note that the destabilizing effect of the viscous force is limited for high liquid Reynolds number cases. In general, it is very necessary to consider the compressibility of the gas flows in the instability study of practical twin-fluid atomization.

Dynamics of an electrified confined planar jet in a zero gravity environment

The present work deals with temporal electro-aerodynamics instability properties of an inviscid confined planar jet in a zero gravity environment and the existence of electric field. The considered system is composed of a horizontal liquid sheet surrounded by two identical moving layers of fluid. The electric field acts uniformly normal to the interfaces. The governing equations with the boundary conditions are solved for fluid dynamics and electric fields. The dispersion relation and the integrated solution for the model are obtained for both varicose and sinuous modes. The influence of different parameters governing the flow on instability behavior of the system such as the confinement, basic velocity ratio, and electric voltage is discussed in detail. We conclude that increasing the thickness of the outer fluids for small range of unstable wave numbers decreases the disturbance growth rate and increases the number of saddle points. In addition, when the basic velocity ratio is less than one the electric field has a stabilizing effects, but when it exceeds one it has a destabilizing effect.

Stability of a thin film flow on a weakly wavy wall

We consider a thin viscous liquid film flow down an inclined plane with sinusoidal topography. We use the integral boundary layer approach to derive governing equations. By use of Floquet theory, we study linear stability of steady-state flow. For small values of the topography amplitude, perturbation analysis gives an analytical expression for corrections to the growth rate of a normal mode disturbance. This correction is proportional to the second power of the topography amplitude. The topography can both stabilize and destabilize the flow, depending on the corrugation period and the plane inclination angle. We study the topology of neutral curves and the critical Reynolds number dependence on topography parameters.

Characteristics of the Instability Developing in the Turbulent Flow in a Plane Channel

Disturbance evolution in developed turbulent flows in a plane channel is numerically investigated at the Reynolds numbers up to Reτ = 586. Steady turbulent flows calculated on the basis of the solution of the incompressible Navier—Stokes equations are then used as the baseline flows in studying the disturbance development process. The values of the highest Lyapunov exponent (HLE) λ1 are found and the instantaneous and statistical properties of the corresponding leading Lyapunov vector (LLV) are determined. Under arbitrary initial conditions the regime of the exponential disturbance growth ~ exp(λ1t) is reached for a time Δt+ < 50. It is found that the HLE value increases with the Reynolds number from λ1+ ≈ 0.021 at Reτ = 180 to λ1+ ≈ 0.026 at Reτ = 586. The LLV exhibits itself in the form of time- and space-localized spots of highly intense pulsations, concentrated in the buffer layer region. The distributions of the r.m.s. intensities of the velocity and vorticity pulsations in the LLV are qualitatively similar with the corresponding distributions in the main flow with near-wall streaks artificially extracted from it. The difference is a large disturbance concentration in the vicinity of the buffer layer, y+ = 10–30, and a relatively high (about 80% higher) vorticity pulsations amplitude. Basing upon the energy spectra of the velocity and vorticity pulsations we determined the integral spatial scales of the structures in the LLV field. It is found that the LLV structures are on average half as wide and long as the corresponding structures in the main flow. The contributions of all the terms included in the expression for the production of the kinetic energy of disturbances are determined. It is shown that the process of disturbance development is essentially controlled by the main flow inhomogeneity and the occurrence of transverse motion in it. Neglecting these factors leads to a considerable underestimation of the disturbance growth rate. Contrariwise, the presence of near-wall streaks in the main flow does not play a considerable role in the LLV disturbance development. The artificial extraction of the streaks from the main flow field does not change the nature of disturbance growth.

Investigation of the primary breakup length and instability of non-Newtonian viscoelastic liquid jets

The temporal instability and primary breakup length of a non-Newtonian viscoelastic liquid jet moving in an inviscid gaseous environment were carried out by solving a set of linearized Navier-Stokes equations and employing the linear viscoelastic model, respectively. The dimensionless dispersion equation that governs the instability was derived and solved by a numerical method. The effects of fluid properties on the instability and primary breakup length of viscoelastic liquid jets were carried out. It could be seen that by increasing the growth rate, the instability range and the primary breakup length of the viscoelastic liquid jets could result in an increase in the liquid Weber number and the ratio of gas to liquid density. Moreover, the significant findings are that an increase in the time constant ratio, and also the Ohnesorge number reduced both of the growth rates of disturbances and primary breakup length. Though, increasing the elasticity number resulted in a higher growth rate of disturbances and enhanced the breakup mechanism.

Molecular Dynamics Simulation of a Jet in a Binary System at Supercritical Environment.

With the development of large-thrust liquid rocket engines, the behavior of liquid in supercritical conditions arouses increasing public interest. Due to the high pressure and temperature of the combustion chamber, fuel reaches its critical point much more easily, and enters supercritical conditions. Due to the drastic changes in the physical properties of the fluid near the critical point, it is usually difficult to simulate the fluid motion using traditional computational fluid dynamic methods; but molecular dynamics (MD) can simulate fluid motion at the molecular level. In view of the engineering application, the physical properties of a binary system consisting of argon and nitrogen, and the stability of subcritical jets sprayed into supercritical environment, has been studied here using the MD method. First, the molecular dynamic simulation of the equation of state (EOS) of the mixture was put forward. Four conditions, with different mixing ratios of nitrogen, were designed. The results showed that the mixing ratio of nitrogen noticeably affected the results; these results were compared with the Soave-Redich-Kwong (SRK) EOS. Second, a simulation was conducted of subcritical nitrogen jet sprayed into a supercritical argon environment. After analyzing the results, the jet density and temperature distributions were obtained and the disturbance growth rate of the shear layer was analyzed.

On the growth rate of disturbances in the turbulent Couette flow

On the growth rate of disturbances in the turbulent Couette flow

Conservation relation of generalized growth rate in boundary layers

The elementary task is to calculate the growth rates of disturbances when the eN method in transition prediction is performed. However, there is no unified knowledge to determine the growth rates of disturbances in three-dimensional (3D) flows. In this paper, we study the relation among the wave parameters of the disturbance in boundary layers in which the imaginary parts of wave parameters are far smaller than the real parts. The generalized growth rate (GGR) in the direction of group velocity is introduced, and the conservation relation of GGR is strictly deduced in theory. This conservation relation manifests that the GGR only depends on the real parts of wave parameters instead of the imaginary parts. Numerical validations for GGR conservation are also provided in the cases of first/second modes and crossflow modes. The application of GGR to the eN method in 3D flows is discussed, and the puzzle of determining growth rates in 3D flows is clarified. A convenient method is also proposed to calculate growth rates of disturbances in 3D flows. Good agreement between this convenient method and existing methods is found except the condition that the angle between the group velocity direction and the x-direction is close to 90° which can be easily avoided in practical application.

The Growth and Saturation of Submesoscale Instabilities in the Presence of a Barotropic Jet

AbstractMotivated by recent observations of submesoscales in the Southern Ocean, we use nonlinear numerical simulations and a linear stability analysis to examine the influence of a barotropic jet on submesoscale instabilities at an isolated front. Simulations of the nonhydrostatic Boussinesq equations with a strong barotropic jet (approximately matching the observed conditions) show that submesoscale disturbances and strong vertical velocities are confined to a small region near the initial frontal location. In contrast, without a barotropic jet, submesoscale eddies propagate to the edges of the computational domain and smear the mean frontal structure. Several intermediate jet strengths are also considered. A linear stability analysis reveals that the barotropic jet has a modest influence on the growth rate of linear disturbances to the initial conditions, with at most a ~20% reduction in the growth rate of the most unstable mode. On the other hand, a basic state formed by averaging the flow at the end of the simulation with a strong barotropic jet is linearly stable, suggesting that nonlinear processes modify the mean flow and stabilize the front.

Molecular Dynamics Study on the Mechanism of Nanoscale Jet Instability Reaching Supercritical Conditions

This paper investigates the characteristics of a nitrogen jet (the thermodynamic conditions ranging from subcritical to supercritical) ejected into a supercritical nitrogen environment using the molecular dynamics (MD) simulation method. The thermodynamic properties of nitrogen obtained by molecular dynamics show good agreement with the Soave-Redlich-Kwong (SRK) equation of state (EOS). The agreement provides validation for this nitrogen molecular model. The molecular dynamics simulation of homogeneous nitrogen spray is carried out in different thermodynamic conditions from subcritical to supercritical, and a spatio-temporal evolution of the nitrogen spray is obtained. The interface of the nitrogen spray is determined at the point where the concentration of ejected fluid component reaches 50%, since the supercritical jet has no obvious vapor-liquid interface. A stability analysis of the transcritical jets shows that the disturbance growth rate of the shear layer coincides very well with the classical theoretical result at subcritical region. In the supercritical region, however, the growth rate obtained by molecular dynamics deviates from the theoretical result.

Instability of pressure-driven gas–liquid two-layer channel flows in two and three dimensions

We study unstable waves in gas–liquid two-layer channel flows driven by a pressure gradient, under stable stratification, not assumed to be set in motion impulsively. The basis of the study is direct numerical simulation (DNS) of the two-phase Navier–Stokes equations in two and three dimensions for moderately large Reynolds numbers, accompanied by a theoretical description of the dynamics in the linear regime (Orr–Sommerfeld–Squire equations). The results are compared and contrasted across a range of density ratios $r=\unicode[STIX]{x1D70C}_{liquid}/\unicode[STIX]{x1D70C}_{gas}$. Linear theory indicates that the growth rate of small-amplitude interfacial disturbances generally decreases with increasing $r$; at the same time, the cutoff wavenumbers in both streamwise and spanwise directions increase, leading to an ever-increasing range of unstable wavenumbers, albeit with diminished growth rates. The analysis also demonstrates that the most dangerous mode is two-dimensional in all cases considered. The results of a comparison between the DNS and linear theory demonstrate a consistency between the two approaches: as such, the route to a three-dimensional flow pattern is direct in these cases, i.e. through the strong influence of the linear instability. We also characterize the nonlinear behaviour of the system, and we establish that the disturbance vorticity field in two-dimensional systems is consistent with a mechanism proposed previously by Hinch (J. Fluid Mech., vol. 144, 1984, p. 463) for weakly inertial flows. A flow-pattern map constructed from two-dimensional numerical simulations is used to describe the various flow regimes observed as a function of density ratio, Reynolds number and Weber number. Corresponding simulations in three dimensions confirm that the flow-pattern map can be used to infer the fate of the interface there also, and show strong three-dimensionality in cases that exhibit violent behaviour in two dimensions, or otherwise the development of behaviour that is nearly two-dimensional behaviour possibly with the formation of a capillary ridge. The three-dimensional vorticity field is also analysed, thereby demonstrating how streamwise vorticity arises from the growth of otherwise two-dimensional modes.

Stability Analysis of a Falling Film Flow Down a Plane with Sinusoidal Corrugations

The liquid viscous film falling down a vertical wall with sinusoidal relief is considered. The linear stability of steady-state flow with respect to time-periodic disturbances is studied using the Floquet theory. It is shown that in the case of applying corrugations the variation in the disturbance growth rate is proportional to the second power of their undulations. Depending on the relief parameters there exist two possibilities: the instability domain can expand or certain disturbances can be stabilized. The growth rates are obtained numerically and analytically in the approximation of low-amplitude corrugations. The development of waves from small disturbances is simulated within the framework of nonlinear equations and the formation of structures whose wavelength is significantly greater than the space relief period is found out.

Linear and weakly nonlinear analysis of Rayleigh–Bénard convection of perfect gas with non-Oberbeck–Boussinesq effects

The influences of non-Oberbeck–Boussinesq (NOB) effects on flow instabilities and bifurcation characteristics of Rayleigh–Bénard convection are examined. The working fluid is air with reference Prandtl number $Pr=0.71$ and contained in two-dimensional rigid cavities of finite aspect ratios. The fluid flow is governed by the low-Mach-number equations, accounting for the NOB effects due to large temperature difference involving flow compressibility and variations of fluid viscosity and thermal conductivity with temperature. The intensity of NOB effects is measured by the dimensionless temperature differential $\unicode[STIX]{x1D716}$. Linear stability analysis of the thermal conduction state is performed. An $\unicode[STIX]{x1D716}^{2}$ scaling of the leading-order corrections of critical Rayleigh number $Ra_{cr}$ and disturbance growth rate $\unicode[STIX]{x1D70E}$ due to NOB effects is identified, which is a consequence of an intrinsic symmetry of the system. The influences of weak NOB effects on flow instabilities are further studied by perturbation expansion of linear stability equations with regard to $\unicode[STIX]{x1D716}$, and then the influence of aspect ratio $A$ is investigated in detail. NOB effects are found to enhance (weaken) flow stability in large (narrow) cavities. Detailed contributions of compressibility, viscosity and buoyancy actions on disturbance kinetic energy growth are identified quantitatively by energy analysis. Besides, a weakly nonlinear theory is developed based on centre-manifold reduction to investigate the NOB influences on bifurcation characteristics near convection onset, and amplitude equations are constructed for both codimension-one and -two cases. Rich bifurcation regimes are observed based on amplitude equations and also confirmed by direct numerical simulation. Weakly nonlinear analysis is useful for organizing and understanding these simulation results.