Generalized Inflection Point Research Articles

Modal and non-modal stability for Hagen–Poiseuille flow with non-ideal fluid

Modal and non-modal stability analyses are applied to Hagen–Poiseuille flow with a non-ideal fluid. The non-ideal fluid is defined as a fluid close to its vapor–liquid critical point. In this region, properties of the fluid deviate significantly from the assumptions of the ideal gas model. In this paper, the specific example of CO2 near the critical point is taken as a non-ideal fluid. We studied fluids at supercritical pressure and different wall temperatures so that the centerline temperatures can be lower, equal, and higher than the pseudo-critical temperature. Flow instability is characterized by the Reynolds number, and the product of the Prandtl and Eckert numbers. In modal stability analysis, we observe that there is no unstable mode in Hagen–Poiseuille flow with a non-ideal fluid. Regarding the growth rate, as the axial wavenumber increases, another mode becomes the least stable. The non-modal theory is employed to investigate the optimal response to harmonic external force and transient energy growth. The influence of axial and azimuthal wave numbers, Prandtl and Eckert numbers, and thermodynamic states are also taken into account. In this study, we identify an generalized inflection point in the transcritical base profile, causing the transcritical state to be the most unstable. In non-modal instability, we observe that the optimal response mainly occurs at time invariant axisymmetric disturbance. This suggests that the axisymmetric disturbance could potentially initiate the transition to turbulence.

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.

Stability of the Boundary Layer with Internal Heat Release and Gas Injection through a Porous Wall

The problem of stability of a subsonic boundary layer is solved under the conditions of heat supply inside the boundary layer with injection of a homogeneous gas through a porous plate that partially simulates the problem of stability of a boundary layer with diffuse combustion. Two-dimensional waves are the most growing waves over the entire range of investigated parameters. It is found that in the case of a fixed norm of heat supply the maximum temperature in the boundary layer increases with the Reynolds number, i.e., with increase in the distance from the leading plate edge. This is in agreement with available experiments and calculations of parameters of the boundary layer with diffuse combustion. In this case the dependence of the degrees of amplification on the Reynolds number, which are maximum in the frequency, is nonmonotonic. It is shown that gas injection with heat supply destabilizes the boundary layer, as it occurs without heat supply. On the other hand, the stabilizing role of heat supply is also shown under the conditions of gas injection through a porous wall. As the frequency of growing wave increases, the phase velocity tends to the velocity at the generalized inflection point. Despite fairly large degrees of growing, the Gaster relation is valid. In accordance with this relation the spatial degree of amplification is equal to the temporal degree of amplification divided by the group velocity.

Experimental validation of inviscid linear stability theory applied to an axisymmetric jet

We study the development of perturbations in a submerged air jet with a round cross-section and a long laminar region (five jet diameters) at a Reynolds number of 5400 by both inviscid linear stability theory and experiments. The theoretical analysis shows that there are two modes of growing axisymmetric perturbations, which are generated by three generalized inflection points of the jet's velocity profile. To validate the results of linear stability theory, we conduct experiments with controlled axisymmetric perturbations to the jet. The characteristics of growing waves are obtained by visualization, thermoanemometer measurements and correlation analysis. Experimentally measured wavelengths, growth rates and spatial distributions of velocity fluctuations for both growing modes are in good agreement with theoretical calculations. Therefore, it is demonstrated that small perturbations to the laminar jet closely follow the predictions of inviscid linear stability theory.

Linear, nonlinear and transitional regimes of second-mode instability

Abstract

Dense-gas effects on compressible boundary-layer stability

A study of dense-gas effects on the stability of compressible boundary-layer flows is conducted. From the laminar similarity solution, the temperature variations are small due to the high specific heat of dense gases, leading to velocity profiles close to the incompressible ones. Concurrently, the complex thermodynamic properties of dense gases can lead to unconventional compressibility effects. In the subsonic regime, the Tollmien–Schlichting viscous mode is attenuated by compressibility effects and becomes preferentially skewed in line with the results based on the ideal-gas assumption. However, the absence of a generalized inflection point precludes the sustainability of the first mode by inviscid mechanisms. On the contrary, the viscous mode can be completely stable at supersonic speeds. At very high speeds, we have found instances of radiating supersonic instabilities with substantial amplification rates, i.e. waves that travel supersonically relative to the free-stream velocity. This acoustic mode has qualitatively similar features for various thermodynamic conditions and for different working fluids. This shows that the leading parameters governing the boundary-layer behaviour for the dense gas are the constant-pressure specific heat and, to a minor extent, the density-dependent viscosity. A satisfactory scaling of the mode characteristics is found to be proportional to the height of the layer near the wall that acts as a waveguide where acoustic waves may become trapped. This means that the supersonic mode has the same nature as Mack’s modes, even if its frequency for maximal amplification is greater. Direct numerical simulation accurately reproduces the development of the supersonic mode and emphasizes the radiation of the instability waves.

On the mechanism by which nose bluntness suppresses second-mode instability

ABSTRACT A physical mechanism by which nose bluntness suppresses second-mode instability is proposed. Considered are 7 degree half-angle straight cones with nose bluntness radii of 0.15 mm, 3.556 mm, 5 mm, 9.525 mm, 12.7 mm and 25.4 mm at tunnel conditions relevant to the AFOSR-Notre Dame Large Mach 6 Quiet Tunnel. It is shown that second-mode suppression is achieved via entropy layer modulation of the basic state density gradient. A weakening of the density gradient disrupts the acoustic resonance necessary to sustain second-mode growth. These results are consistent with the thermoacoustic interpretation which posits that second-mode instability can be modeled as thermoacoustic resonance of acoustic energy trapped within an acoustic impedance well. Furthermore, the generalized inflection point criterion of Lees and Lin is applied to develop a criterion for the existence of second-mode instability based on the strength of the basic state density gradient.

First-mode-induced nonlinear breakdown in a hypersonic boundary layer

The mechanisms underlying transition of a Mach 6 flat plate boundary layer with adiabatic wall conditions are explored with Direct Numerical Simulations (DNS) using a high-order hybrid upwind-centered method. The forcing employed is obtained from linear stability theory (LST), which is used to identify Tollmien–Schlichting (T-S) or first-mode instability in the hypersonic boundary layer (HBL). At small amplitudes, the DNS displays linear growth, and matches those obtained from LST, including the spanwise structure of the most amplified oblique wave in the first-mode as well as the spreading angle of the turbulent wedge induced by the oblique wave under transitional scenarios. The main emphasis of the paper is on the nonlinear breakdown process, obtained by forcing the DNS at a higher amplitude so that the linear amplification region yields to nonlinear saturation and breakdown of the first-mode. The oblique waves are seen to first evolve into lambda-vortices with peak perturbation amplitudes around the generalized inflection point (GIP) of the HBL. The fundamental forcing frequency gives rise to progressive superharmonics: These first appear within the foot region of the lambda-vortices, and spread in the spanwise direction into the ring-shaped vortices that develop along the symmetry plane of the oblique waves. Transition and loss of symmetry is initiated upon saturation of the fundamental and harmonics. A modal analysis used to extract each frequency component yields further insight into streak-formation in these modes, resulting in localized regions of peak skin friction near the transition location. The near-wall boundary layer in the transitional zone displays coherent spanwise structures, which represent compressed regions of fluid and induce high gradients over the surface. Towards the end of the breakdown region, the flow displays a broadband spectrum, and the anticipated inertial sub-range appears inside the boundary layer, as a prelude to self-similar turbulent boundary layer evolution.

Control of oblique-type breakdown in a supersonic boundary layer employing streaks

The effectiveness of streak modes in controlling the oblique-type breakdown in a supersonic boundary-layer at Mach 2.0 is investigated using direct numerical simulations. Investigations in the literature have shown the effectiveness of streak modes in delaying the onset of transition dominated by two-dimensional waves, but in oblique breakdown, three-dimensional waves and a strong streak mode dominate the transition process. Paredes et al. (J. Fluid Mech., vol. 831, 2017, pp. 524–553) discussed the possible stabilization of supersonic boundary layers by optimally growing streaks using parabolized stability equations. However, no study has as yet been reported regarding direct nonlinear control of oblique breakdown. This study deals with the effects of large-amplitude decaying streak modes generated by a blowing–suction strip at the wall to control full breakdown in a reference case. Modes with four to five times the fundamental wavenumber are found to be beneficial for controlling the transition. In the first region after the control-mode forcing, the beneficial mean-flow distortion (MFD), generated by inducing the control mode, is solely responsible for hampering the growth of the fundamental-mode. On the whole, the MFD and the three-dimensional part of the control contribute equally towards controlling the oblique breakdown. The results show significant suppression of transition, and substantial improvements have been observed in the levels of the skin-friction coefficient and wall-temperature in comparison to the uncontrolled case. Moreover, refreshing the control using an additional downstream control strip increases the gain. However, the forcing amplitude must be carefully chosen in order not to introduce a generalized inflection point in the spanwise averaged mean flow invoking enhanced disturbance growth.

Interactions between vortical, acoustic and thermal components during hypersonic transition

Discrete unstable modes of hypersonic laminar boundary layers, obtained from an eigenvalue analysis, provide insight into key transition scenarios. The character of such modes near the leading edge is often identified with the corresponding asymptotic free-stream behaviour of acoustic, vortical or entropic (thermal) content, which we designate fluid-thermodynamic (FT) components. In downstream regions, however, this direct one-to-one correspondence between discrete modes and FT components does not hold, since FT components interact in well-defined ways with the basic state and with each other (even under linear scenarios). In the present work, we perform an FT decomposition of discrete modes using momentum potential theory, to yield a physics-based analysis that complements linear stability theory in the linear regime, and seamlessly extends to the nonlinear domain where direct numerical simulations are appropriate. Linear and nonlinear saturated disturbance effects, different forcing types and wall thermal conditions are considered, with emphasis on phenomena occurring near stability-mode synchronization locations. The results show that, in the linear regime, each discrete mode contains all FT components, whose relative amplitudes vary with streamwise distance. Vortical components are always the largest, followed by thermal and acoustic components. These latter two show distinct fore and aft signatures near mode synchronization. The vortical component displays a series of rope-shaped recirculation-cell patterns across the generalized inflection point. However, both acoustic and thermal components display ‘trapped’ structures. The former contains an alternating monopole array between the wall and the critical layer, while the latter is confined to an undulating region between the wall and a wavy locus straddling the generalized inflection point. Nonlinear saturation in the region of Mack-mode growth further strengthens the rope-shaped structures in the vortical component and higher harmonics appear, whose form and location depend on the specific component. Wall cooling modifies the eigenfunctions such that the acoustic component accounts for more of its composition, consistent with its destabilization. Analysis of energy interactions among the FT components indicates that, even though the vorticity component is the largest, the thermal component induces the most significant source term for the growth of acoustic perturbations, possibly due to the trapped nature of both.

Single-mode flutter of an elastic plate in the presence of the boundary layer

The flutter of an elastic plate in a viscous supersonic gas flow is investigated. The influence of viscous perturbations of the boundary layer on the single-mode flutter is studied considering two different types of a boundary layer profile: the generalized convex boundary layer profile and the profile with a generalized inflection point. It is shown that in the case of the convex layer for thick boundary layers the plate is fully stabilized. In the case of a profile with the generalized inflection point, the thickening of the layer first yields the increase of the growth rates of the perturbations, and the growth is greater than in the inviscid approximation. Numerical simulation of two-dimensional supersonic gas flows with boundary layers (that can have a destabilizing effect on the flutter of an elastic surface) over different curved surfaces is performed. The flow patterns over them are studied: regions of these surfaces, over which stable boundary-layer profiles with a generalized inflection point are formed, are found, and their boundaries are obtained.

Shape Analysis of C-B-splines

The shape features of planar C-B-spline segments are analyzed. The necessary and sufficient conditions are derived for the curve having one or two inflection points, a loop or a cusp, or be locally or globally convex. All conditions are completely characterized by the relative position of the control polygon’s side vectors, and are summarized in three kinds of shape diagrams in terms of the linear independence of the three side vectors. Moreover, it is proved that a spatial C-B-spline segment has no singularities and generalized inflection points.

Interaction of panel flutter with inviscid boundary layer instability in supersonic flow

AbstractWe investigate the stability of an elastic plate in supersonic gas flow. This problem has been studied in many papers regarding panel flutter, where uniform flow is usually considered. In this paper, we take the boundary layer on the plate into account and investigate its influence on plate stability. Three problem formulations are studied. First, we investigate the stability of travelling waves in an infinite-length plate. Second, the nature of the instability (absolute or convective instability) is examined. Finally, by using solutions of the first two problems, instability of a long finite-length plate is studied by using Kulikovskii’s global instability criterion. The following results are obtained. All the eigenmodes of a finite-length plate are split into two types, which we call subsonic and supersonic. The influence of the boundary layer on these eigenmodes can be of two kinds. First, for a generalized convex boundary layer profile (typical for accelerating flow), supersonic eigenmodes are stabilized by the boundary layer, whereas subsonic disturbances are destabilized. Second, for a profile with a generalized inflection point (typical for constant and decelerating flows), supersonic eigenmodes are destabilized in a thin boundary layer and stabilized in a thick layer; subsonic eigenmodes are damped. The correspondence between the influence of the boundary layer on panel flutter and the stability of the boundary layer over a rigid wall is established. Examples of stable boundary layer profiles of both types are given.

Single-mode plate flutter taking the boundary layer into account

The stability of an elastic plate in supersonic gas flow is investigated using asymptotic methods and taking the boundary layer formed on the plate surface into account. It is shown that the effect of the boundary layer can be of two types depending on its profile. In the case of generalized convex profiles (characteristic of accelerated flow) supersonic and subsonic plate oscillations are stabilized and destabilized, respectively. In the case of profiles with a generalized inflection point located in the subsonic part of the layer (characteristic of homogeneous and decelerated flows) supersonic perturbations are destabilized in the thin boundary layer and stabilized when the layer is fairly thick; subsonic perturbations are damped.

Generalized inflection points of very general effective divisors on smooth curves

Let E be a very general effective divisor of degree d on a smooth curve C of genus g. We study inflection points on linear systems |aE | for an integer a ≥ 1. They are called generalized inflection points of the invertible sheaf \({\mathcal{O}_C(E)}\). In case \({P\notin E}\) is a generalized inflection point of \({\mathcal{O}_C(E)}\) then it is a normal generalized inflection point. In case \({P\in E}\) then P has minimal vanishing sequences for E.

Stability analysis of reacting mixing layers with density gradient and wake deficit

This paper investigates the linear stability of both uniform and non-uniform density plane mixing layers with special emphasis on the effect of the wake component in the velocity profile. Velocity and density profiles for laminar flows are obtained from analytic profiles. Mixing layers with wakes have two generalized inflection points and two unstable modes-sinuous and varicose modes. For uniform density mixing layers, sinuous modes are more unstable than varicose modes, which shows wakes will be destabilized by sinuous modes. For non-uniform density mixing layers with high density in high speed flows, sinuous modes are more unstable than varicose modes. For non-uniform density mixing layers with high density in low speed flows, varicose modes can be more unstable than sinuous modes.

Stability analysis of wakes with density gradient

This paper investigates the linear stability of wakes with special emphasis on the difference of velocity and density. Velocity and density profiles for laminar flows have gaussian profiles. Incompressible wakes have two generalized inflection points and two unstable modes-sinuous and varicose modes. Sinuous modes are more unstable than varicose modes irrespective of density variation, which shows wakes will be destabilized by sinuous modes. Large velocity difference and density difference lead to more unstable wakes due to large momentum difference.

Generalized inflection points of very general line bundles on smooth curves

Let L be an invertible sheaf on a smooth curve C. A generalized inflection point of L is an inflection point of $$L^{\otimes n}$$ for some integer n > 0. A generalized inflection point P of L is called strongly normal if there is a unique integer n > 0 such that P is an inflection point of $$L^{\otimes n}$$ and moreover its inflection weight is equal to 1. In case L is a very general invertible sheaf of degree x on C then all generalized inflection points of L are strongly normal.

Multiple Unstable Modes in the Reacting Mixing Layer

This paper investigates the linear stability of reacting mixing layers with special emphasis on the existence of multiple unstable modes. The governing equations for laminar flows are from two-dimensional compressible boundary-layer equations. The chemistry is a finite rate single step irreversible reaction with Arrhenius kinetics. For the incompressible reacintg mixing layer with variable density. A necessary condition for instability has been derived. The condition requires that the angular momentum, not the vorticity, to have a maximum in the flow domain. New inflectional modes of instability are found to exist in the outer part of the mixing layer. For the compressible reacting mixing layer, supersonic unstable modes may exist in the abscence of a generalized inflection point. The outer modes at high Mach numbers in the reacting mixing layer are continuations of the inflectional modes of low Mach number flows. However, the generalized inflection point is less important at supersonic flows.

Effect of density gradients in confined supersonic shear layers. I. Two-dimensional disturbances

The effect of density gradients on two-dimensional (2-D) supersonic wall modes (acoustic modes) of a 2-D confined compressible shear layer are investigated using linear analysis. Due to the inadequacies of the hyperbolic tangent velocity profile, the boundary layer basic flow profiles are used. First, a test case is taken with the same parameters as in a previous analysis by Tam and Hu [J. Fluid Mech. 203, 51 (1989)], who used a hyperbolic tangent profile. For the boundary layer profiles, three generalized inflection points are found giving rise to three modes. The first two show similar properties to those found by Tam and Hu, whereas the third is a new mode that can have a higher growth rate than the others. As the density ratio is increased above that of the test case, the smallest of the three neutral phase speeds tends toward the speed of the lower-velocity stream, and the other two eventually coalesce and then disappear. These two effects lead to a linear resonance between some of the modes that increases the cutoff frequency and growth rate of the lowest mode. In fact, growth rates of two to four times the test case were found as the density ratio was increased to 7. A similar trend (linear resonance) is observed as the density ratio is decreased from the test case, but the growth rate is only slightly changed from the test case.