Centre modes in inviscid swirling flows and their application to the stability of the Batchelor vortex

2007 ◽  
Vol 576 ◽  
pp. 325-348 ◽  
Author(s):  
C. J. HEATON
Keyword(s):  
Continuous Spectrum ◽  
Single Point ◽  
Stability Boundary ◽  
Infinite Family ◽  
Neutral Curve ◽  
Swirling Flows ◽  
Mean Flow ◽  
Linearized Euler Equations ◽  
Numerical Studies ◽  
The Stability

We identify a family of centre-mode disturbances to inviscid swirling flows such as jets, wakes and other vortices. The centre modes form an infinite family of modes, increasingly concentrated near to the symmetry axis of the mean flow, and whose frequencies accumulate to a single point in the complex plane. This asymptotic accumulation allows analytical progress to be made, including a theoretical stability boundary, inO(1) parameter regimes. The modes are located close to the continuous spectrum of the linearized Euler equations, and the theory is closely related to that of the continuous spectrum. We illustrate our analysis with the inviscid Batchelor vortex, defined by swirl parameterq. We show that the inviscid instabilities found in previous numerical studies are in fact the first members of an infinite set of centre modes of the type we describe. We investigate the inviscid neutral curve, and find good agreement of the neutral curve predicted by the analysis with the results of numerical computations. We find that the unstable region is larger than previously reported. In particular, the value ofqabove which the inviscid vortex stabilizes is significantly larger than previously reported and in agreement with a long-standing theoretical prediction.

Numerical studies of the stability of inviscid stratified shear flows

1972 ◽  
Vol 51 (1) ◽  
pp. 39-61 ◽  
Author(s):  
Philip Hazel
Keyword(s):  
Richardson Number ◽  
Shear Flows ◽  
Stability Boundary ◽  
Phase Speed ◽  
Neutral Curve ◽  
Free Shear Layer ◽  
Density Profiles ◽  
Complex Phase ◽  
Numerical Studies ◽  
The Stability

The infinitesimal stability of inviscid, parallel, stratified shear flows to two-dimensional disturbances is described by the Taylor-Goldstein equation. Instability can only occur when the Richardson number is less than 1/4 somewhere in the flow. We consider cases where the Richardson number is everywhere non- negative. The eigenvalue problem is expressed in terms of four parameters,Ja ‘typical’ Richardson number, α the (real) wavenumber andcthe complex phase speed of the disturbance. Two computer programs are developed to integrate the stability equation and to solve for eigenvalues: the first findscgiven α andJ, the second finds α andJwhenc≡ 0 (i.e. it computes the stationary neutral curve for the flow). This is sometimes,but not always, the stability boundary in the α,Jplane. The second program works only for cases where the velocity and density profiles are antisymmetric about the velocity inflexion point. By means of these two programs, several configurations of velocity and density have been investigated, both of the free-shear-layer type and the jet type. Calculations of temporal growth rates for particular profiles have been made.

The onset of meandering in a barotropic jet

2001 ◽  
Vol 449 ◽  
pp. 85-114 ◽  
Author(s):  
N. J. BALMFORTH ◽  
C. PICCOLO
Keyword(s):  
Nonlinear Theory ◽  
Stability Boundary ◽  
Wave Model ◽  
Inviscid Limit ◽  
Reduced Model ◽  
Mean Flow ◽  
Single Wave ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
The Stability

This study explores the dynamics of an unstable jet of two-dimensional, incompressible fluid on the beta-plane. In the inviscid limit, standard weakly nonlinear theory fails to give a low-order description of this problem, partly because the simple shape of the unstable normal mode is insufficient to capture the structure of the forming pattern. That pattern takes the form of ‘cat's eyes’ in the vorticity distribution which develop inside the modal critical layers (slender regions to either side of the jet's axis surrounding the levels where the modal wave speed matches the mean flow). Asymptotic expansions furnish a reduced model which is a version of what is known as the single-wave model in plasma physics. The reduced model predicts that the amplitude of the unstable mode saturates at a relatively low level and is not steady. Rather, the amplitude evolves aperiodically about the saturation level, a result with implications for Lagrangian transport theories. The aperiodic amplitude ‘bounces’ are intimately connected with sporadic deformations of the vortices within the cat's eyes. The theory is compared with numerical simulations of the original governing equations. Slightly asymmetrical jets are also studied. In this case the neutral modes along the stability boundary become singular; an extension of the weakly nonlinear theory is presented for these modes.

scholarly journals The impedance boundary condition for acoustics in swirling ducted flow

2018 ◽  
Vol 849 ◽  
pp. 645-675 ◽  
Author(s):  
Vianney Masson ◽  
James R. Mathews ◽  
Stéphane Moreau ◽  
Hélène Posson ◽  
Edward J. Brambley
Keyword(s):  
Boundary Condition ◽  
Surface Waves ◽  
Swirling Flow ◽  
Swirling Flows ◽  
Impedance Boundary Condition ◽  
Mean Flow ◽  
Thin Boundary Layer ◽  
Impedance Boundary ◽  
Unstable Surface ◽  
The Stability

The acoustics of a straight annular lined duct containing a swirling mean flow is considered. The classical Ingard–Myers impedance boundary condition is shown not to be correct for swirling flow. By considering behaviour within the thin boundary layers at the duct walls, the correct impedance boundary condition for an infinitely thin boundary layer with swirl is derived, which reduces to the Ingard–Myers condition when the swirl is set to zero. The correct boundary condition contains a spring-like term due to centrifugal acceleration at the walls, and consequently has a different sign at the inner (hub) and outer (tip) walls. Examples are given for mean flows relevant to the interstage region of aeroengines. Surface waves in swirling flows are also considered, and are shown to obey a more complicated dispersion relation than for non-swirling flows. The stability of the surface waves is also investigated, and as in the non-swirling case, one unstable surface wave per wall is found.

Stability of spatially developing boundary layers in pressure gradients

1995 ◽  
Vol 300 ◽  
pp. 117-147 ◽  
Author(s):  
Rama Govindarajan ◽  
R. Narasimha
Keyword(s):  
Boundary Layer ◽  
Stability Boundary ◽  
Streamwise Velocity ◽  
Present Approach ◽  
Pressure Gradients ◽  
Mean Flow ◽  
Rational Analysis ◽  
Displacement Effect ◽  
The Stability ◽  
New Formulation

A new formulation of the stability of boundary-layer flows in pressure gradients is presented, taking into account the spatial development of the flow and utilizing a special coordinate transformation. The formulation assumes that disturbance wavelength and eigenfunction vary downstream no more rapidly than the boundary-layer thickness, and includes all terms nominally of orderR−1in the boundary-layer Reynolds numberR. In Blasius flow, the present approach is consistent with that of Bertolottiet al.(1992) toO(R−1) but simpler (i.e. has fewer terms), and may best be seen as providing a parametric differential equation which can be solved without having to march in space. The computed neutral boundaries depend strongly on distance from the surface, but the one corresponding to the inner maximum of the streamwise velocity perturbation happens to be close to the parallel flow (Orr-Sommerfeld) boundary. For this quantity, solutions for the Falkner-Skan flows show the effects of spatial growth to be striking only in the presence of strong adverse pressure gradients. As a rational analysis toO(R−1) demands inclusion of higher-order corrections on the mean flow, an illustrative calculation of one such correction, due to the displacement effect of the boundary layer, is made, and shown to have a significant destabilizing influence on the stability boundary in strong adverse pressure gradients. The effect of non-parallelism on the growth of relatively high frequencies can be significant at low Reynolds numbers, but is marginal in other cases. As an extension of the present approach, a method of dealing with non-similar flows is also presented and illustrated.However, inherent in the transformation underlying the present approach is a lower-order non-parallel theory, which is obtained by dropping all terms of nominal orderR−1except those required for obtaining the lowest-order solution in the critical and wall layers. It is shown that a reduced Orr-Sommerfeld equation (in transformed coordinates) already contains the major effects of non-parallelism.

The rapid-rotation limit of the neutral curve for Taylor–Couette flow

2016 ◽  
Vol 808 ◽  
Author(s):  
Kengo Deguchi
Keyword(s):  
Couette Flow ◽  
Asymptotic Theory ◽  
Stability Boundary ◽  
Outer Cylinder ◽  
Stability Result ◽  
Neutral Curve ◽  
Navier Stokes ◽  
Taylor Couette ◽  
The Stability ◽  
Taylor Couette Flow

An asymptotic theory is developed for the linear stability curve of rapidly rotating Taylor–Couette flow. The analytic curve obtained by the theory excellently explains the limiting Navier–Stokes stability result for general disturbances. When the cylinders are corotating, the asymptotic theory describes the gap between the neutral curve and the Rayleigh stability criterion. For the case when the cylinders are counter-rotating, it is found that, along the stability boundary, the Reynolds number based on the inner cylinder speed is proportional to that based on the outer cylinder speed to the power of $3/5$.

scholarly journals Stability and energy budget of pressure-driven collapsible channel flows

2011 ◽  
Vol 705 ◽  
pp. 348-370 ◽  
Author(s):  
H. F. Liu ◽  
X. Y. Luo ◽  
Z. X. Cai
Keyword(s):  
Pressure Drop ◽  
Kinetic Energy ◽  
Energy Budget ◽  
Neutral Curve ◽  
Channel Flows ◽  
Mean Flow ◽  
Mode 2 ◽  
The Mean ◽  
The Stability ◽  
Pressure Driven

AbstractAlthough self-excited oscillations in collapsible channel flows have been extensively studied, our understanding of their origins and mechanisms is still far from complete. In the present paper, we focus on the stability and energy budget of collapsible channel flows using a fluid–beam model with the pressure-driven (inlet pressure specified) condition, and highlight its differences to the flow-driven (i.e. inlet flow specified) system. The numerical finite element scheme used is a spine-based arbitrary Lagrangian–Eulerian method, which is shown to satisfy the geometric conservation law exactly. We find that the stability structure for the pressure-driven system is not a cascade as in the flow-driven case, and the mode-2 instability is no longer the primary onset of the self-excited oscillations. Instead, mode-1 instability becomes the dominating unstable mode. The mode-2 neutral curve is found to be completely enclosed by the mode-1 neutral curve in the pressure drop and wall stiffness space; hence no purely mode-2 unstable solutions exist in the parameter space investigated. By analysing the energy budgets at the neutrally stable points, we can confirm that in the high-wall-tension region (on the upper branch of the mode-1 neutral curve), the stability mechanism is the same as proposed by Jensen & Heil. Namely, self-excited oscillations can grow by extracting kinetic energy from the mean flow, with exactly two-thirds of the net kinetic energy flux dissipated by the oscillations and the remainder balanced by increased dissipation in the mean flow. However, this mechanism cannot explain the energy budget for solutions along the lower branch of the mode-1 neutral curve where greater wall deformation occurs. Nor can it explain the energy budget for the mode-2 neutral oscillations, where the unsteady pressure drop is strongly influenced by the severely collapsed wall, with stronger Bernoulli effects and flow separations. It is clear that more work is required to understand the physical mechanisms operating in different regions of the parameter space, and for different boundary conditions.

The stability of a trailing line vortex. Part 2. Viscous theory

1974 ◽  
Vol 65 (4) ◽  
pp. 769-779 ◽  
Author(s):  
Martin Lessen ◽  
Frederick Paillet
Keyword(s):  
Critical Reynolds Number ◽  
Swirling Flow ◽  
Reynolds Numbers ◽  
Neutral Curve ◽  
Swirling Flows ◽  
Velocity Defect ◽  
The Stability ◽  
Far Wake ◽  
Disturbance Growth ◽  
Swirl Parameter

In a previous paper, the inviscid stability of a swirling far wake was investigated, and the superposition of a swirling flow on the axisymmetric wake was shown to be initially destabilizing, although all modes investigated eventually become more stable at sufficiently large swirl. The most unstable disturbances were non-axisymmetric modes with negative azimuthal wavenumber n representing helical wave paths opposite in sense to the wake rotation. The disturbance growth rate appeared to increase continuously with |n|, while all modes with |n| > 1 represented disturbances which are completely stable for the non-swirling wake. In the present analysis, both timewise and spacewise growth rates are calculated for the lowest three negative non-axisymmetric modes (n = −1, −2 and −3). Vortex intensity is characterized by a swirl parameter q proportional to the ratio of the maximum swirling velocity to the maximum axial velocity defect. The large wavenumbers associated with the disturbances at large |n| allow the n = −1 mode to have the minimum critical Reynolds number of 16 (q ≃ 0·40). The other two modes investigated have minimum Reynolds numbers on the neutral curve of 31 (n = −2, q = 0·60) and 57 (n = −3, q = 0·80). For each mode, the neutralstability curve is shown to shift rapidly towards infinite Reynolds numbers once the swirl becomes sufficiently large. Some of the most unstable swirling flows are shown to possess spacewise amplification factors almost ten times that for the most unstable wavenumber for the non-swirling wake at moderate Reynolds numbers.

scholarly journals A Geometric Interpretation of Zonostrophic Instability

2020 ◽  
Vol 50 (9) ◽  
pp. 2759-2779
Author(s):  
Georgios Kontogiannis ◽  
Nikolaos A. Bakas
Keyword(s):  
Stress Tensor ◽  
Stability Boundary ◽  
Geometric Interpretation ◽  
Mean Flow ◽  
Ginzburg Landau ◽  
Zonal Jets ◽  
Beta Plane ◽  
Geometric Decomposition ◽  
Eddy Dynamics ◽  
The Stability

AbstractThe zonostrophic instability that leads to the emergence of zonal jets in barotropic beta-plane turbulence is analyzed through a geometric decomposition of the eddy stress tensor. The stress tensor is visualized by an eddy variance ellipse whose characteristics are related to eddy properties. The tilt of the ellipse principal axis is the tilt of the eddies with respect to the shear, and the eccentricity of the ellipse is related to the eddy anisotropy, and its size is related to the eddy kinetic energy. Changes of these characteristics are directly related to the vorticity fluxes forcing the mean flow. The statistical state dynamics of the turbulent flow closed at second order is employed as it provides an analytic expression for both the zonostrophic instability and the stress tensor. For the linear phase of the instability, the stress tensor is analytically calculated at the stability boundary. For the nonlinear equilibration of the instability the tensor is calculated in the limit of small supercriticality in which the amplitude of the jet velocity follows Ginzburg–Landau dynamics. It is found that, dependent on the characteristics of the forcing, the jet is accelerated either because the jet primarily anisotropizes the eddies so as to produce upgradient fluxes, or because the jet changes the eddy tilt. The instability equilibrates as these changes are partially reversed by the nonlinear jet–eddy dynamics.

scholarly journals Calculation of the Thermoacoustic Stability of a Main Stage Thrust Chamber Demonstrator

2020 ◽  
pp. 235-247
Author(s):  
Alexander Chemnitz ◽  
Thomas Sattelmayer
Keyword(s):  
Transfer Function ◽  
Injection Pressure ◽  
Scattering Matrix ◽  
Main Stage ◽  
Stability Prediction ◽  
Mean Flow ◽  
Linearized Euler Equations ◽  
Thrust Chamber ◽  
Flame Response ◽  
The Stability

Abstract The stability behavior of a virtual thrust chamber demonstrator with low injection pressure loss is studied numerically. The approach relies on an eigenvalue analysis of the Linearized Euler Equations. An updated form of the stability prediction procedure is outlined, addressing mean flow and flame response calculations. The acoustics of the isolated oxidizer dome are discussed as well as the complete system incorporating dome and combustion chamber. The coupling between both components is realized via a scattering matrix representing the injectors. A flame transfer function is applied to determine the damping rates. Thereby it is found that the procedure for the extraction of the flame transfer function from the CFD solution has a significant impact on the stability predictions.

Wave breakdown in stratified shear flows

1977 ◽  
Vol 79 (3) ◽  
pp. 481-497 ◽  
Author(s):  
M. T. Landahl ◽  
W. O. Criminale
Keyword(s):  
Shear Flows ◽  
Flow Instability ◽  
Stability Boundary ◽  
Small Scale ◽  
Mean Flow ◽  
Order Unity ◽  
Stable Regime ◽  
Density Interface ◽  
Density Interfaces ◽  
The Stability

The wave-mechanical condition (Landahl 1972) for breakdown of an unsteady laminar flow into strong small-scale secondary instabilities is applied to some simple stratified inviscid shear flows. The cases considered have one or two discrete density interfaces and simple discontinuous or continuous velocity profiles. A primary wavelike disturbance to such a flow produces a perturbation velocity that is discontinuous at a density interface. The resulting instantaneous system, defined as the sum of the mean flow and the primary oscillation, develops a local secondary shear-flow instability that has a group velocity equal to the arithmetic mean of the instantaneous velocities on the two sides of the interface. According to the breakdown criterion, the disturbed flow will become critical whenever this velocity reaches a value equal to the phase velocity of the primary wave. The calculations show that for a single density interface breakdown may occur for low to moderate wave amplitudes in a fairly narrow range of Richardson numbers on the stable side of the stability boundary. On the other hand, in the unstable regime maximum wave slopes of order unity may be reached before breakdown occurs; this conclusion is in qualitative agreement with experiments. When the system includes two density interfaces, it is found that there exists a range of high Richardson numbers far into the stable regime for which breakdown may take place even for very small and zero wave interface deflexions.

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