3D Disturbances Research Articles

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.

Dynamic response mechanism of the hole subjected to the 3D point source disturbance in an isotropic formation

Accurate estimation of the effects of dynamic disturbances on stress concentration is crucial for the stability of rock engineering and the corresponding analytical approaches are needed. This study presents an analytical approach to calculate the relative stress distribution with a point source inside and outside a hole using the linear theory of elasticity. The Helmholtz potentials and Sommerfeld integral are employed to describe the displacement and stress components, and then formulate the equilibrium equations to solve the equivalent stress distribution around the hole. Numerical examples demonstrate the impact of model parameters on the equivalent stress, such as frequency, hole radius, source location, etc. It is found that sometimes high frequencies can make the equivalent stress greater far from the source than that close to the source. Additionally, when the ratio of the distance between the source and the hole axis to the hole radius exceeds ten, the equivalent stress distribution around the hole remains nearly constant. This approach can be used for the design and assessment of underground engineering structures' stability under dynamic disturbances.

NUMERICAL THREE-DIMENSIONAL MODEL OF ULTRASONIC COAGULATION OF AEROSOL PARTICLES IN VORTEX ACOUSTIC STREAMING

Separation of highly dispersed systems with huge liquid-gas or liquid-solid interfaces is relevant for practical tasks of gas purification from the most highly dispersed and difficult-to-detect dispersed fraction PM2.5, and separation of nanoparticles (including their small agglomerates) in fine chemical technology processes. One of the most effective ways to separate highly dispersed systems with a large interface surface is to combine each of the closed subsurfaces (surfaces of individual dispersed particles) under the influence of hydrodynamic effects in the gas phase, arising both near the interface surfaces and at a considerable distance from them, due to the superposition of ultrasonic vibrations. Since the efficiency of ultrasonic coagulation decreases with a large distance between closed subsurfaces from each other in PM2.5 aerosol and the small size of these surfaces, it is necessary to create conditions for the emergence of new nonlinear effects that contribute to the local compaction of the dispersed fraction. In a resonant and significantly inhomogeneous ultrasonic field (with a scale of inhomogeneity on the order of the wavelength), vortex acoustic flows arise, which, due to inertial forces, locally compact the dispersed phase in the form of an increase in the concentration of aerosol particles. A numerical model of ultrasonic coagulation of PM2.5 aerosol particles in three-dimensional (3D) vortex acoustic streaming is proposed in this paper. The model is designed to identify the possibility of increasing the efficiency of ultrasonic coagulation in 3D streaming by virtue of the following mechanisms: (1) local increase in concentration caused by the inertial transfer of particles to the periphery of 3D vortices in the gas phase; (2) increase in the frequency of particle collisions due to 3D turbulent disturbances in ultrasonic fields; and (3) increase in productivity and ensuring uninterrupted implementation of the process in a flow mode owing to transfer of particles between the streamlines of the main vortices initiated by ultrasonic vibrations. The listed mechanisms for increasing the efficiency of coagulation in 3D streaming are taken into consideration by introducing two stream functions, considering turbulent chaotic disturbances of the flow resulting in dispersion of particle velocities. It was possible to establish based on numerical analysis of the model using the example of PM2.5 that laminar vortex flows begin to influence at sound pressure level from 160-165 dB, and turbulent disturbances make an additional contribution in the range of sound pressure levels from 140-170 dB. At the same time, as a result of 3D turbulent disturbances, the efficiency of coagulation reaches almost 100% at a sound pressure level 5 dB lower than with laminar flows (sound pressure amplitude, 3 times lower).

Meta-Learning-Based Incremental Nonlinear Dynamic Inversion Control for Quadrotors with Disturbances

This paper proposes an online meta-learning-based incremental nonlinear dynamic inversion (INDI) control method for quadrotors with disturbances. The quadrotor dynamic model is first transformed into linear form via an INDI control law. Since INDI largely depends on the accuracy of the control matrix, a method composed of meta-learning and adaptive control is proposed to estimate it online. The effectiveness of the proposed control framework is validated through simulation on a quadrotor with 3D wind disturbances.

Three-dimensional vortex dipole solitons in self-gravitating systems.

We derive the nonlinear equationsgoverning the dynamics of three-dimensional (3D) disturbances in a nonuniform rotating self-gravitating fluid under the assumption that the characteristic frequencies of disturbances are small compared to the rotation frequency. Analytical solutions of these equations are found in the form of the 3D vortex dipole solitons. The method for obtaining these solutions is based on the well-known Larichev-Reznik procedure for finding two-dimensional nonlinear dipole vortex solutions in the physics of atmospheres of rotating planets. In addition to the basic 3D x-antisymmetric part (carrier), the solution may also contain radially symmetric (monopole) or/and antisymmetric along the rotation axis (z-axis) parts with arbitrary amplitudes, but these superimposed parts cannot exist without the basic part. The 3D vortex soliton without the superimposed parts is extremely stable. It moves without distortion and retains its shape even in the presence of an initial noise disturbance. The solitons with parts that are radially symmetric or/and z antisymmetric turn out to be unstable, although, at sufficiently small amplitudes of these superimposed parts, the soliton retains its shape for a very long time.

Physical origin of vortex stretching and twisting: Viscous or inertial forces

In this paper, the physical origin of vortex stretching and twisting is theoretically investigated. The effects of inertial and viscous forces are mainly considered and discussed. Two key conditions, i.e., solid walls and three-dimensional (3D) disturbances, are adopted in three typical cases. Among them, the first two cases are straight and curved vortex lines at the initial time without any kind of disturbance. The third case is a straight vortex line at the initial time with introduced 3D natural disturbances. Through experimental observations, numerical simulations, and theoretical analysis in these cases, the first two cases illustrate that the straight or curved vortex lines are still straight or curved at the next time, respectively, regardless of whether solid walls are introduced. However, the third case clearly shows that once natural disturbances are introduced, the straight vortex lines near and at solid walls at the initial time are stretched and twisted mainly by viscous forces, instead of inertial forces, typically demonstrated by the 3D wake transition of a straight circular cylinder and the transition of the laminar boundary layer at a flat plate. Accordingly, based on definitions of generation and enhancement in vortex stretching and twisting, it is confirmed that the viscous forces with two key conditions are the generation mechanism, while the inertial forces alone are the enhancement mechanism.

Nonlinear evolutions of streaky structures in viscoelastic pipe flows

The dynamics of streaky structures in a viscoelastic pipe flow of FENE-P fluids is studied numerically. The values of Weissenberg number ( W i , polymer relaxation time over flow turn-over time) and viscosity ratios are varied to evaluate the effects of viscoelasticity in the pipe. For Reynolds numbers 3000 ∼ 3500 (based on pipe radius and maximum laminar velocity) near the transition boundary, we select finite-amplitude two-dimensional rolls as initial conditions and add infinitesimal three-dimensional perturbations to trigger the transition. The 2D streamwise rolls are highly unstable to small 3D disturbances, which helps to better understand the evolution of large scale structures. The stronger temporal intermittency in viscoelastic flows makes it more difficult to identify a successful transition; we found that quantities directly related to viscoelasticity can serve as better indicators of transition to turbulence. The role of polymer stress is studied through the kinetic energy budget and it is found that polymer torque opposes the vorticity and become more dominant for higher W i , consistent with previous studies. The results show that the secondary instability of the streaky structures can be enhanced by viscoelasticity at low W i , while at high W i viscoelasticity delays the transition to turbulence, sometimes leading to relaminarization. • Direct numerical simulations of viscoelastic pipe flow of FENE-P fluids are conducted. • Viscoelasticity changes the evolution of streaks and causes strong intermittency. • Polymer stress is studied via turbulent kinetic energy budget in streak breakdown. • Non-monotonic influence of Weissenberg number on threshold boundary of transitions.

An ultraviolet dyegraph for measuring the chemical disturbances of sinking particles and swimming plankton

AbstractSinking particles and swimming plankton may deform ambient chemical gradients, leaving long trails of chemical disturbance that dissipate on a diffusive timescale. Imaging this phenomenon in a laboratory setting requires a large field of view, a broad depth of field, and a passive chemical substance that does not influence the hydrodynamics. These requirements obviate the use of traditional schlieren optical systems, which rely on density stratification and their consequent buoyancy forces, and modern planar laser‐induced fluorescence, which has a narrow depth of field. Further, the light used to induce plankton movement must be kept separate from the recorded signal. Here, we describe a new ultraviolet “dyegraph” technique that quantifies dye disturbance caused by sinking spheres and swimming zooplankton. An experimental tank is filled with seawater and a gradient of an ultraviolet‐absorbing pigment. Using modified shadowgraph optics, ultraviolet light spreads from a pinhole to a collimating lens, passes through the tank to a telecentric lens, and is recorded by a camera fitted with an ultraviolet bandpass filter. The dye is transparent in visible light but appears dark in the imaging system. The 3D chemical disturbance is reconstructed from images using tomographic inversion. After outlining the method, we present examples of chemical deformation trails generated by sinking spheres and live, swimming copepods.

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.

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.

Effect of frequency and wavenumber on the three-dimensional routes of transition by wall excitation

Three-dimensional (3D) routes of transition affected by the frequency of monochromatic wall excitation started impulsively are studied by performing direct numerical simulation. The computed results for the two frequencies of excitation reported here resemble the experimental setup of Klebanoff et al. [“The three-dimensional nature of boundary-layer instability,” J. Fluid Mech. 12(1), 1–34 (1962)], where a two-dimensional boundary layer is excited using 3D disturbances. Such a monochromatic wall excitation creates three-component disturbance field: a near-field followed by the Tollmien-Schlichting wave-packet and a spatiotemporal wave-front (STWF), which is responsible for eventual transition. It is noted that the case of moderate frequency of excitation shows a complete noninteracting nature of the near-field solution and the STWF. We report another route of transition computationally for a lower frequency of excitation. This case shows an interacting nature of the near-field solution and the STWF. While both frequency of excitations can cause transition for moderate spanwise wavelength (λz) disturbances, dependence of transition on λz is reported here for the first time. It is noted that doubling the spanwise wavenumber leads to the disappearance of STWF and no transition of the flow. We use the recently developed disturbance enstrophy transport equation in Sengupta et al. [“An enstrophy based linear and nonlinear receptivity theory,” Phys. Fluids 30, 054106 (2018)] for a better quantitative method to trace the evolution of disturbance field of the imposed 3D disturbances.

Non-symmetric branching of fluid flows in 3D vessels

Nonsymmetric branching flow through a three-dimensional (3D) vessel is considered at medium-to-high flow rates. The branching is from one mother vessel to two or more daughter vessels downstream, with laminar steady or unsteady conditions assumed. The inherent 3D nonsymmetry is due to the branching shapes themselves, or the differences in the end pressures in the daughter vessels, or the incident velocity profiles in the mother. Computations based on lattice-Boltzmann methodology are described first. A subsequent analysis focuses on small 3D disturbances and increased Reynolds numbers. This reduces the 3D problem to a two-dimensional one at the outer wall in all pressure-driven cases. As well as having broader implications for feeding into a network of vessels, the findings enable predictions of how much swirling motion in the cross-plane is generated in a daughter vessel downstream of a 3D branch junction, and the significant alterations provoked locally in the shear stresses and pressures at the walls. Nonuniform incident wall-shear and unsteady effects are examined. A universal asymptotic form is found for the flux change into each daughter vessel in a 3D branching of arbitrary cross-section with a thin divider.

NONSYMMETRIC BRANCHING OF FLUID FLOWS IN 3D VESSELS

Nonsymmetric branching flow through a three-dimensional (3D) vessel is considered at medium-to-high flow rates. The branching is from one mother vessel to two or more daughter vessels downstream, with laminar steady or unsteady conditions assumed. The inherent 3D nonsymmetry is due to the branching shapes themselves, or the differences in the end pressures in the daughter vessels, or the incident velocity profiles in the mother. Computations based on lattice-Boltzmann methodology are described first. A subsequent analysis focuses on small 3D disturbances and increased Reynolds numbers. This reduces the 3D problem to a two-dimensional one at the outer wall in all pressure-driven cases. As well as having broader implications for feeding into a network of vessels, the findings enable predictions of how much swirling motion in the cross-plane is generated in a daughter vessel downstream of a 3D branch junction, and the significant alterations provoked locally in the shear stresses and pressures at the walls. Nonuniform incident wall-shear and unsteady effects are examined. A universal asymptotic form is found for the flux change into each daughter vessel in a 3D branching of arbitrary cross-section with a thin divider.

Application of Arnoldi method to boundary layer instability**Project supported by the National Natural Science Foundation of China (Grant Nos. 11202147, 11332007, 11172203, and 91216111) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120032120007).

The Arnoldi method is applied to boundary layer instability, and a finite difference method is employed to avoid the limit of the finite element method. This modus operandi is verified by three comparison cases, i.e., comparison with linear stability theory (LST) for two-dimensional (2D) disturbance on one-dimensional (1D) basic flow, comparison with LST for three-dimensional (3D) disturbance on 1D basic flow, and comparison with Floquet theory for 3D disturbance on 2D basic flow. Then it is applied to secondary instability analysis on the streaky boundary layer under spanwise-localized free-stream turbulence (FST). Three unstable modes are found, i.e., an inner mode at a high-speed center streak, a sinuous type outer mode at a low-speed center streak, and a sinuous type outer mode at low-speed side streaks. All these modes are much more unstable than Tollmien–Schlichting (TS) waves, implying the dominant contribution of secondary instability in bypass transition. The modes at strong center streak are more unstable than those at weak side streaks, so the center streak is ‘dangerous’ in secondary instability.

CURRENT SINGULARITIES IN LINE-TIED THREE-DIMENSIONAL MAGNETIC FIELDS

This paper considers the current distributions that derive from finite amplitude perturbations of line-tied magnetic fields comprising hyperbolic field structures. The initial equilibrium on which we principally focus is a planar magnetic X-point threaded by a uniform axial field. This field is line-tied on all surfaces but subject to three-dimensional (3D) disturbances that alter the initial topology. Results of ideal relaxation simulations are presented which illustrate how intense current structures form that can be related, through the influence of line-tying, to the quasi-separatrix layers (QSLs) of the initial configuration. It is demonstrated that the location within the QSL that attracts the current, and its scaling properties, are strongly dependent on the relative dimensions of the QSL with respect to the line-tied boundaries. These results are contrasted with the behavior of a line-tied 3D field containing an isolated null point. In this case, it is found that the dominant current always forms at the null, but that the collapse is inhibited when the null is closer to a line-tied boundary.

Precursor of transition to turbulence: spatiotemporal wave front.

To understand transition to turbulence via 3D disturbance growth, we report here results obtained from the solution of Navier-Stokes equation (NSE) to reproduce experimental results obtained by minimizing background disturbances and imposing deterministic excitation inside the shear layer. A similar approach was adopted in Sengupta and Bhaumik [Phys. Rev. Lett. 107, 154501 (2011)], where a route of transition from receptivity to fully developed turbulent stage was explained for 2D flow in terms of the spatio-temporal wave-front (STWF). The STWF was identified as the unit process of 2D turbulence creation for low amplitude wall excitation. Theoretical prediction of STWF for boundary layer was established earlier in Sengupta, Rao, and Venkatasubbaiah [Phys. Rev. Lett. 96, 224504 (2006)] from the Orr-Sommerfeld equation as due to spatiotemporal instability. Here, the same unit process of the STWF during transition is shown to be present for 3D disturbance field from the solution of governing NSE.

Extended Squire’s transformation and its consequences for transient growth in a confined shear flow

AbstractThe classical Squire transformation is extended to the entire eigenfunction structure of both Orr–Sommerfeld and Squire modes. For arbitrary Reynolds numbers $\mathit{Re}$, this transformation allows the solution of the initial-value problem for an arbitrary three-dimensional (3D) disturbance via a two-dimensional (2D) initial-value problem at a smaller Reynolds number $\mathit{Re}_{2D}$. Its implications for the transient growth of arbitrary 3D disturbances is studied. Using the Squire transformation, the general solution of the initial-value problem is shown to predict large-Reynolds-number scaling for the optimal gain at all optimization times $t$ with ${t}/{\mathit{Re}}$ finite or large. This result is an extension of the well-known scaling laws first obtained by Gustavsson (J. Fluid Mech., vol. 224, 1991, pp. 241–260) and Reddy & Henningson (J. Fluid Mech., vol. 252, 1993, pp. 209–238) for arbitrary $\alpha \mathit{Re}$, where $\alpha $ is the streamwise wavenumber. The Squire transformation is also extended to the adjoint problem and, hence, the adjoint Orr–Sommerfeld and Squire modes. It is, thus, demonstrated that the long-time optimal growth of 3D perturbations as given by the exponential growth (or decay) of the leading eigenmode times an extra gain representing its receptivity, may be decomposed as a product of the gains arising from purely 2D mechanisms and an analytical contribution representing 3D growth mechanisms equal to $1+ \left (\beta \mathit{Re}/\mathit{Re}_{2D}\right )^2 \mathcal{G}$, where $\beta $ is the spanwise wavenumber and $\mathcal{G}$ is a known expression. For example, when the leading eigenmode is an Orr–Sommerfeld mode, it is given by the product of respective gains from the 2D Orr mechanism and an analytical expression representing the 3D lift-up mechanism. Whereas if the leading eigenmode is a Squire mode, the extra gain is shown to be solely due to the 3D lift-up mechanism. Direct numerical solutions of the optimal gain for plane Poiseuille and plane Couette flow confirm the novel predictions of the Squire transformation extended to the initial-value problem. These results are also extended to confined shear flows in the presence of a temperature gradient.

Nonlinear evolution of Klebanoff type second mode disturbances in supersonic flat-plate boundary layer

Studying the evolution of 3D disturbances is of crucial theoretical importance for understanding the transition process. The present study concerns the nonlinear evolution of second mode unstable disturbances in a supersonic boundary layer by the numerical simulation, and discusses the selectivity of 3D disturbances and possibility to transition. The results indicate that a Klebanoff type nonlinear interaction between 2D and 3D disturbances with the same frequency may amplify a band of 3D disturbances centered at a finite spanwise wavenumber. That is, certain 3D disturbances can be selectively and rapidly amplified by the unstable 2D disturbances, and certain small-scale 3D structures will appear.

Analysis on secondary instability of shear layer based on the concept of phase synchronization

A concept of phase synchronization point is proposed, and then a model is built using this concept to explain secondary instabilities. This model has been used to determine the conditions of K- and H-type secondary instabilities, which are coincident with the conditions published in literatures. It also can be used to analyze other secondary instability phenomena. For example, the numerical results validate the analysis results in the case of 1/3rd subharmonic mode secondary instability. Furthermore, the numerical results indicate that the spanwise wave number of 3D disturbance has significant effect on the secondary instability.

Onset of instability of a flag in uniform flow

This paper numerically and analytically studies the onset of instability of a flag in uniform flow. The three-dimensional (3D) simulation is performed by using an immersed-boundary method coupled with a nonlinear finite element method. The global stability, bistability and instability are identified in the 3D simulations. The Squire's theorem is extended to analyze the stability of the fluid-flag system with 3D initial perturbations. It is found that if a parallel flow around the flag admits an unstable 3D disturbance for a certain value of the flutter speed, then a two-dimensional (2D) disturbance at a lower flutter speed is also admitted. In addition, the growth rate of 2D disturbance is larger than that of the 3D disturbance.