Three-dimensional Disturbances Research Articles

Instability of stratified two-phase flows in inclined rectangular ducts

The effect of inclination on instability of two-phase stratified flow in inclined rectangular ducts is studied for the first time via a rigorous linear stability analysis that accounts for all possible infinitesimal three-dimensional disturbances. The results of stability analysis of concurrent downward, concurrent upward, and countercurrent air-water flows are presented in the form of stability diagrams in the coordinates of critical superficial velocities for each of the cases at a characteristic set of other governing parameters. The multiplicity of base-flow solutions predicted for inclined flows is addressed. This study is the first to show that within the rigorous formulation of the stability problem in a realistic duct geometry, the concurrent upward two-phase flows allow for multiple stable flow states. While the main bulk of results is obtained for ducts with a square cross-section, the effect of the duct aspect ratio on the onset of instability is studied for several representative examples. It is shown that in most cases the flow becomes unstable due to different modes of long-wave disturbances that are associated with either interface or shear instability mechanisms. The latter arises mainly due to the duct bottom and top walls. The results substantiate the premise that the simpler two-plate geometry can be used to estimate the effect of the system parameters on the flow stability. The three-dimensional finite-wavelength critical disturbances are visualized by a novel approach of divergence-free projections on the coordinate planes.

Non-Modal Three-Dimensional Optimal Perturbation Growth in Thermally Stratified Mixing Layers

A non-modal transient disturbances growth in a stably stratified mixing layer flow is studied numerically. The model accounts for a density gradient within a shear region, implying a heavier layer at the bottom. Numerical analysis of non-modal stability is followed by a full three-dimensional direct numerical simulation (DNS) with the optimally perturbed base flow. It is found that the transient growth of two-dimensional disturbances diminishes with the strengthening of stratification, while three-dimensional disturbances cause significant non-modal growth, even for a strong, stable stratification. This non-modal growth is governed mainly by the Holmboe modes and does not necessarily weaken with the increase of the Richardson number. The optimal perturbation consists of two waves traveling in opposite directions. Compared to the two-dimensional transient growth, the three-dimensional growth is found to be larger, taking place at shorter times. The non-modal growth is observed in linearly stable regimes and, in slightly linearly supercritical regimes, is steeper than that defined by the most unstable eigenmode. The DNS analysis confirms the presence of the structures determined by the transient growth analysis.

Linear stability analysis of three-dimensional natural convection at low Prandtl number in an annular enclosure in the presence of a toroidal magnetic field

The natural convection of a liquid metal in an annular enclosure with a square cross section in the presence of a toroidal static magnetic field was investigated by linear stability analysis. Three-dimensional steady disturbances were obtained in an annular enclosure where the walls parallel to the gravitational field were heated and cooled. The Prandtl number Pr was set to 0.025 and the radius ratio of the enclosure κ was set to 0.5, while the Rayleigh number Ra, the Hartmann number Ha, and the angular wavenumber m were considered as parameters. The dimensionless governing equations were discretized by the finite difference method. Since the newly developed dual staggered grid was employed, the interpolation in the outer product terms was not required. The linear growth rate for a standing wave mode was amplified by increasing Ra and attenuated by increasing Ha. For any integer m, the neutral Rayleigh number Ran at Ha = 0 and the neutral Hartmann number Han for Ran ≤ Ra ≤ 200 000 were identified. Based on these, the neutral lines for a given m were obtained as Han ∼ (Ra − Ran)1/3. Furthermore, the critical values Rac, Hac, and mc were obtained, where mc was not necessarily limited to an integer. The distribution of mc was approximated by the power of Ra. The disturbances had symmetry in the azimuthal direction and constituted a pair of vortices rotating in opposite directions. These vortices were elongated along the main stream.

EFFECT OF HEAT SUPPLY TO A NARROW BAND OF THE BOUNDARY LAYER ON ITS STABILITY

Stability of a subsonic boundary layer with heat supply into its narrow band is studied. The maximum length of the stable boundary layer in the case of heat supply is found to be approximately three times smaller as compared to the case of a thermally insulated wall without heat supply. The frequency range of disturbances and their growth rates are approximately doubled. In contrast to the case without heat supply, three-dimensional disturbances are the most rapidly growing disturbances. In the case of a heated wall, the growth rate of disturbances decreases after heat supply into the boundary layer, which may lead to an increase in the length of the laminar region of the boundary layer.

Feedback control of boundary layer Tollmien-Schlichting waves using a simple model-based controller

The attenuation of spatially evolving instability Tollmien-Schlichting (T-S) waves in the boundary layer of a flat plate with zero pressure gradients using an active feedback control scheme is theoretically and numerically investigated. The boundary layer is excited artificially by various perturbations to create a three-dimensional field of instability waves. Arrays of actuators and sensors are distributed locally at the wall surface and connected together via a feedback controller. The key elements of this feedback control are the determination of the dynamic model of the flat plate boundary layer between the actuators and the sensors, and the design of the model-based feedback controller. The dynamic model is established based on the linear stability calculation which simulates the three-dimensional input-output behaviour of the boundary layer. To simplify the control problem, an uncoupled control mode of the dynamic model is made to capture only those dynamics that have greatest influences on the input-output behaviour. A Proportional-Integral-Derivative (PID) controller, i.e. a lead-lag compensator, combining with a standard Smith predictor is designed based on the system stability criterion and the specifications using frequency-response methods. Good performance of the feedback control with the uncoupled control mode is demonstrated by the large reduction of the three-dimensional disturbances in the boundary layer. This simple feedback control is realistic and competitive in a practical implementation of T-S wave cancellation using a limited number of localised sensors and actuators.

Linear stability of the flow of a second order fluid past a wedge

The linear stability analysis of Rivlin–Ericksen fluids of second order is investigated for boundary layer flows, where a semi-infinite wedge is placed symmetrically with respect to the flow direction. Second order fluids belong to a larger family of fluids called order fluids, which is one of the first classes proposed to model departures from Newtonian behavior. Second order fluids can model non-zero normal stress differences, which is an essential feature of viscoelastic fluids. The linear stability properties are studied for both signs of the elasticity number K, which characterizes the non-Newtonian response of the fluid. Stabilization is observed for the temporal and spatial evolution of two-dimensional disturbances when K > 0 in terms of increase of critical Reynolds numbers and reduction of growth rates, whereas the flow is less stable when K < 0. By extending the analysis to three-dimensional disturbances, we show that a positive elasticity number K destabilizes streamwise independent waves, while the opposite happens for K < 0. We show that, as for Newtonian fluids, the non-modal amplification of streamwise independent disturbances is the most dangerous mechanism for transient energy growth, which is enhanced when K > 0 and diminished when K < 0.

Instability of stratified two-phase flows in rectangular ducts

The linear stability of stratified two-phase flows in rectangular ducts is studied numerically. The linear stability analysis takes into account all possible infinitesimal three-dimensional disturbances and is carried out by solution of the associated eigenproblem. The neutral stability boundary and the corresponding critical wave number are obtained for liquid – liquid and air – water systems. Depending on the problem parameters, the instability sets in owing to short, intermediate, or long wave most unstable perturbations. Patterns of the most unstable disturbances are reported and discussed. It is shown that the instability arises due to shear, or interfacial mechanisms. Effects of the surface tension and of width/height aspect ratio are also studied. The results support the premise that the stability analysis of stratified two-phase flow in the simpler geometry of two-infinite plates can provide a reasonable estimation of the conditions for which this flow pattern can be considered to be linearly stable.

Stability of a non-Newtonian falling film due to three-dimensional disturbances

In this paper, we apply the normal modes method to study the linear stability of a liquid film flowing down an inclined plane, taking into account the complex rheology of the media. We consider generalized Newtonian liquids; the conditions of the Squire theorem do not hold for this case. We check if the flow is unstable due to three-dimensional (3D) disturbances that propagate at a certain angle to the flow direction but stable for the two-dimensional (2D) ones. We derived the generalized Orr–Sommerfeld equation, considered a long-wave approximation, and proved that 3D long-wave disturbances are less growing than the 2D ones for any rheological law. We solved the problem for finite wavenumbers numerically and found that for low inclination angles of the plane, instability due to 3D disturbances prevails. In this case, the shear mode of instability dominates, and the surface tension destabilizes the flow. For shear-thickening liquids, the critical Reynolds number decreases down to zero.

How the circulation and axial velocity deficit in Batchelor vortices affect their disturbance growth?

In this study, inviscid linear stability theory is used to investigate the compressibility effects of Batchelor vortices. The growth rates are obtained from asymptotic analysis with large wavenumbers, and it is shown that the growth of disturbances is affected largely by both the Mach number and the axial velocity deficit. From the maximum growth rates for various base flow conditions, the growth properties of Batchelor vortices are summarized reasonably by using a convective Mach number based on two relative Mach numbers for three-dimensional disturbances. In addition, the ratio of the circulation and velocity deficit at the maximum growth rate is closely related to the growth rate as a function of the convective Mach number because the correlation between the ratio and the growth rate is high when the Mach number is less than unity. Therefore, the compressibility effects, which are expressed as the relation between the growth property and the convective Mach number, are estimated simply from only the base flows (circulation and velocity deficit) in Batchelor vortices.

Experimental study on suppressing vortex-induced vibration of a long-span bridge by installing the wavy railings

It is known that vortex shedding behind bluff bodies can be suppressed by introducing some three-dimensional geometric disturbance. Therefore, a new passive control device named wavy railings is proposed to mitigate the vortex-induced-vibration (VIV) of a single box section bridge. The wavy railing is composed of vertical posts and wavy crossbars, which can provide three-dimensional perturbation to the flow field and then lead to the development of a three-dimensional free shear layer which does not roll up into a vortex street and hence suppress the VIV. In this study, the control effects of the wavy railings on mitigating VIV are investigated through wind tunnel tests. The results reveal that when the wavelength of the wavy bar is 0.5H or 1H (H is the height of the bridge deck), both vertical and torsional VIV can be completely suppressed, and the upper surface edge is the optimal installation position.

Development of linear unstable modes in supersonic streamwise vortices using a weighted compact nonlinear scheme

Growth processes in supersonic streamwise vortices with linear unstable modes at Mach numbers 2.5 and 5.0 were numerically investigated using a weighted compact nonlinear scheme (WCNS) with three different accuracies. In the evolution of the inviscid linear unstable mode m=−6 as a high-wavenumber mode, the growth rate and eigenfunction profiles of the mode numerically resolved were generally consistent with those obtained with the linear stability theory (LST) during the early transition stage, regardless of the computational accuracy. The numerical scheme used here can capture the growth of three-dimensional linear disturbance in structure. However, the numerical results obtained by nonlinear developments differed according to the accuracy. Among the different accuracies under the same grid resolution, the ninth-order accuracy scheme for the interpolation of primitive variables was able to capture small vortical structures at the downstream in the supersonic flow. In addition, the negative circulation generated and the total disturbance energy indicated that such high accuracy is effective in resolving the developed flow in supersonic vortices, even at moderate grid resolutions.

Two- and Three-Dimensional Absolute Instabilities in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow

A linear stability analysis for the onset of mixed convection in a saturated porous medium through an absolute instability of both two- and three-dimensional disturbances is performed. Relevant control parameters associated with the inclined temperature gradient and the vertical throughflow are the vertical and horizontal Rayleigh numbers, $$R_{\mathrm{v}}$$ and $$R_{\mathrm{h}}$$, and the vertical Peclet number, $$Q_{\mathrm{v}}$$, respectively. This work extends previous studies on the very same problem in two fronts. For two-dimensional disturbances, the present results do not agree with the literature for a few of the parametric conditions reported. This is caused by the collision of the convectively unstable downstream propagating branch with multiple upstream propagating branches, which generates several saddle points and, hence, makes the identification of the correct pinching point more difficult. In other words, literature results are all saddle points but not always pinching points. For three-dimensional disturbances, this issue is not present and the current results agree with the literature. On the other hand, due to the inherent difficulties associated with a three-dimensional absolute instability analysis, literature results have only been able to report the group velocities at the onset of convective instability. When their real parts are zero, transition occurs directly from stable to absolutely unstable. Otherwise, transition occurs from stable to convectively unstable first and nothing can be said about the onset of absolute instability. In this work, a novel technique recently developed by the authors allowed the identification of the onset of absolute instability under all parametric conditions investigated in the literature, extending earlier results. Doing so confirmed the dichotomy already observed in these earlier studies, i.e., the onset of absolute instability for two- and three-dimensional longitudinal modes indeed differs.

Numerical study on suppression of vortex-induced vibration of circular cylinder by helical wires

The mechanism of suppression for the vortex-induced vibration (VIV) of circular cylinder by the helical wires is investigated using LES turbulence model. Numerical models for the free-vibrating circular cylinders with and without the helical wires are proposed and validated by the experiments. Simulations are carried out for a large mass ratio of 248 and a small damping ratio of 0.00257 with the Reynolds numbers ranging from 16000 to 24500 and the reduced velocities ranging from 4.5 to 6.8. It is noticed that the helical wires of diameters ratio d/D ​= ​0.1 effectively suppress the amplitude of VIV by nearly 80% and avoid the “lock in”. The steady and unsteady aerodynamic forces of circular cylinders with and without the helical wires as well as the flow patterns are also examined to clarify the mechanism of VIV suppression. It is found that the fluctuating lift forces and their spanwise correlation for the wired cylinder are significantly reduced comparing with those for the bare cylinder, due to the enhancement of three-dimensional disturbance to the wake caused by helical wires. The aerodynamic damping for the oscillated bare cylinder is negative, while that for the wired cylinder is positive with a vibration amplitude A/D ​= ​0.1 ​at the resonance velocity of bare cylinder.

David J. Benney: Nonlinear Wave and Instability Processes in Fluid Flows

David J. Benney (1930–2015) was an applied mathematician and fluid dynamicist whose highly original work has shaped our understanding of nonlinear wave and instability processes in fluid flows. This article discusses the new paradigm he pioneered in the study of nonlinear phenomena, which transcends fluid mechanics, and it highlights the common threads of his research contributions, namely, resonant nonlinear wave interactions; the derivation of nonlinear evolution equations, including the celebrated nonlinear Schrödinger equation for modulated wave trains; and the significance of three-dimensional disturbances in shear flow instability and transition.

Analysis of azimuthal magnetorotational instability of rotating magnetohydrodynamic flows and Tayler instability via an extended Hain-Lüst equation.

We consider a differentially rotating flow of an incompressible electrically conducting and viscous fluid subject to an external axial magnetic field and to an azimuthal magnetic field that is allowed to be generated by a combination of an axial electric current external to the fluid and electrical currents in the fluid itself. In this setting we derive an extended version of the celebrated Hain-Lüst differential equation for the radial Lagrangian displacement that incorporates the effects of the axial and azimuthal magnetic fields, differential rotation, viscosity, and electrical resistivity. We apply the Wentzel-Kramers-Brillouin method to the extended Hain-Lüst equation and derive a comprehensive dispersion relation for the local stability analysis of the flow to three-dimensional disturbances. We confirm that in the limit of low magnetic Prandtl numbers, in which the ratio of the viscosity to the magnetic diffusivity is vanishing, the rotating flows with radial distributions of the angular velocity beyond the Liu limit, become unstable subject to a wide variety of the azimuthal magnetic fields, and so is the Keplerian flow. In the analysis of the dispersion relation we find evidence of a new long-wavelength instability which is caught also by the numerical solution of the boundary value problem for a magnetized Taylor-Couette flow.

Stability of Detonation Waves Propagating in Plane and Rectangular Channels

Stability of detonation waves (DWs) propagating in a plane or rectangular channel with respect to two-dimensional and three-dimensional disturbances is considered. Accepting a simple hypothesis that the most unstable mode of the linear theory continues to dominate even in the nonlinear regime, one can derive a number of fairly definite predictions of the developed DW structure from the linear stability theory. In particular, the theory predicts the number of detonation cells formed in a channel of a specified size and the cell size, the minimum size of the channel in which the multifront DW structure can still exist, and the parameters at which the number of cells changes in a jump-like manner. All these predictions are qualitatively consistent with available experimental data and numerical results.

Destabilization characteristics of three dimensional Rayleigh–Taylor mechanism on a cylindrical interface

The stability of a cylindrical interface separating two incompressible, inviscid and immiscible fluids under the action of radial motion is studied. Linear stability analysis is employed to understand the destabilization characteristics of the two fluid interface. Three-dimensional as well as two-dimensional (axisymmetric and azimuthal) disturbances are separately considered in the presence of surface tension. A dispersion relation governing this problem is taken from literature and analyzed. The relevant dimensionless parameters governing this study are Bond number, radial Weber number and density ratio. This dispersion relation permits the consideration of two different scenarios: (1) density of the inner fluid being greater than the outer fluid and (2) density of the inner fluid being lesser than the outer fluid. It is found out that the surface tension restricts and aids the destabilization for the former and latter case, respectively. It is also observed that the interface is unstable when even at a constant radial velocity. This is contrary to our common understanding of Rayleigh–Taylor instability where acceleration is required. Three destabilization modes are identified namely Taylor (axial) mode, flute (azimuthal) mode and helical (three-dimensional) mode for a range of parameters. It is found that three-dimensional modes are more susceptible to destabilization than the two-dimensional modes when the radial Weber number is close to zero. Regime maps are created in the Bond number, radial Weber number and density ratio space to establish the regimes where different modes occur. For a given density ratio, the destabilization starts from one mode to another two-dimensional disturbance through three-dimensional disturbances. The second and utmost finding of this study reveals that radial Weber number alone (albeit the Bond number) is sufficient to destabilize a cylindrical interface.

Polymer infused composite metal foam as a potential aircraft leading edge material

The leading edge of aircraft wings must be free from three-dimensional disturbances caused by insect adhesion, ice accretion, and particle wear in order to improve flight performance, safety, and fuel efficiency of the aircraft. An innovative solution was explored in this work by infusing stainless steel composite metal foam (S-S CMF) with a hydrophobic epoxy resin system. S-S CMF was made with 100% stainless steel using a powder metallurgy technique. The infused epoxy filled the macro- and microporosities, unique to S-S CMF’s structure, creating a product with a density similar to that of aluminum. The contact angle, wear rate, erosion resistance, and insect adhesion of the novel infused composite metal foam were measured and compared to aluminum, epoxy and stainless steel. The infusion process was determined to fill up to 88% of the pores within the S-S CMF and was found to reduce wettability and insect residue accretion. The contact angle of the infused S-S CMF was 43% higher than its parent material, stainless steel, and 130% higher than aluminum. Insect residue maximum height and areal coverage were reduced by 60 and 30%, respectively, compared to aluminum. Grit blast experiments to simulate erosion resulted in a greater roughness increase for aluminum than for the parent epoxy resin or the resin-infused S-S CMF. These results suggest that the durability and performance of infused S-S CMF was superior compared to aluminum, which is the current leading edge material of choice. Based on the promising results under relevant wear and erosion conditions, it is concluded that the infused S-S CMF can offer a potential tailored replacement to aluminum leading edge material.

Fully 3D Rayleigh-Taylor instability in a Boussinesq fluid

Rayleigh–Taylor instability occurs when a heavier fluid overlies a lighter fluid, and the two seek to exchange positions under the effect of gravity. We present a linearized theory for arbitrary three-dimensional (3D) initial disturbances that grow in time, and calculate the evolution of the interface for early times. A new spectral method is introduced for the fully 3D nonlinear problem in a Boussinesq fluid, where the interface between the light and heavy fluids is approximated with a smooth but rapid density change in the fluid. The results of large-scale numerical calculation are presented in fully 3D geometry, and compared and contrasted with the early-time linearized theory.
 
 doi:10.1017/S1446181119000087

Temperature Factor Effect on the Disturbance Propagation in Hypersonic Flow past a Yawed Flat Plate

The flow in the three-dimensional laminar boundary layer on a yawed flat plate of finite length is studied in the regime of strong viscous-inviscid interaction with a hypersonic flow. In the vicinity of the leading edge the flow functions are expanded in series under the assumption that the base pressure dependent on the transverse coordinate is given at the trailing edge of the plate. It is established that the expansions obtained include an indefinite function and its derivative with respect to the transverse coordinate. The corresponding boundary value problems are formulated and numerically solved, the eigenvalues are found, and it is shown that the exponent in the third term of the expansion differs from that in the second term only by one. The plate surface temperature effect on the flow parameters and the initiation of three-dimensional disturbances is investigated.