Comoving frames and symmetry-related motions in parallel shear flows

AbstractParallel shear flows have continuous symmetries of translation in the downstream and spanwise directions. As a consequence, flow states that differ in their spanwise or downstream location but are otherwise identical are dynamically equivalent. In the case of travelling waves, this trivial degree of freedom can be removed by going to a frame of reference that moves with the state, thereby turning the travelling wave in the laboratory frame into a fixed point in the comoving frame of reference. We here discuss a general approach, the method of comoving frames, by which the symmetry related motions can also be removed for more complicated and dynamically active states and demonstrate its application for several examples. For flow states in the asymptotic suction boundary layer (ASBL) we show that in the case of the long-period oscillatory edge state we can find local phase speeds which remove the fast oscillations and reveal the slow vortex dynamics underlying the burst phenomenon. For spanwise translating states we show that the method removes the drift but not the dynamical events that cause the big spanwise displacement. For a turbulent case we apply the method to the spanwise shifts and find slow components that are correlated over very long times. Calculations for plane Poiseuille flow show that the long correlations in the transverse motions are not specific to the ASBL.

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Transition in Homogeneously Heated Inclined Plane Parallel Shear Flows

Journal of Heat Transfer ◽

10.1115/1.1599370 ◽

2003 ◽

Vol 125 (5) ◽

pp. 795-803 ◽

Cited By ~ 8

Author(s):

S. Generalis ◽

M. Nagata

Keyword(s):

Shear Flows ◽

Numerical Study ◽

Fluid Layer ◽

Travelling Wave ◽

Inclined Plane ◽

Vertical Orientation ◽

Plane Parallel ◽

Wide Range ◽

Parallel Shear Flows ◽

Roll Type

The transition of internally heated inclined plane parallel shear flows is examined numerically for the case of finite values of the Prandtl number Pr. We show that as the strength of the homogeneously distributed heat source is increased the basic flow loses stability to two-dimensional perturbations of the transverse roll type in a Hopf bifurcation for the vertical orientation of the fluid layer, whereas perturbations of the longitudinal roll type are most dangerous for a wide range of the value of the angle of inclination. In the case of the horizontal inclination transverse roll and longitudinal roll perturbations share the responsibility for the prime instability. Following the linear stability analysis for the general inclination of the fluid layer our attention is focused on a numerical study of the finite amplitude secondary travelling-wave solutions (TW) that develop from the perturbations of the transverse roll type for the vertical inclination of the fluid layer. The stability of the secondary TW against three-dimensional perturbations is also examined and our study shows that for Pr=0.71 the secondary instability sets in as a quasi-periodic mode, while for Pr=7 it is phase-locked to the secondary TW. The present study complements and extends the recent study by Nagata and Generalis (2002) in the case of vertical inclination for Pr=0.

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Travelling Wave Control of Rotating Discs and Analysis of its Spillover Effect

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/pime_proc_1988_202_097_02 ◽

1988 ◽

Vol 202 (2) ◽

pp. 119-127 ◽

Cited By ~ 3

Author(s):

C-S Kim ◽

C-W Lee

Keyword(s):

Travelling Waves ◽

Spillover Effect ◽

Polynomial Equation ◽

Travelling Wave ◽

Rotating Disc ◽

System A ◽

Finite Dimensional ◽

Control Scheme ◽

Actuator System ◽

Wave Control

A modal control scheme for rotating disc systems is developed based upon the finite-dimensional sub-system model including a few lower backward travelling waves important to the disc response. For the single discrete sensor and actuator system, a polynomial equation, which determines the closed-loop system poles, is derived and the spillover effect is analysed, providing a sufficient condition for stability. Finally, simulation studies are performed to show the effectiveness of the travelling wave control scheme proposed.

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TRAVELLING WAVES AND OSCILLATIONS IN SAL’NIKOV’S COMBUSTION REACTION IN A COMPRESSIBLE GAS

The ANZIAM Journal ◽

10.1017/s1446181114000479 ◽

2015 ◽

Vol 56 (3) ◽

pp. 233-247 ◽

Cited By ~ 3

Author(s):

RHYS A. PAUL ◽

LAWRENCE K. FORBES

Keyword(s):

Asymptotic Solution ◽

Multiple Scales ◽

Travelling Waves ◽

Method Of Lines ◽

Travelling Wave ◽

Reaction Scheme ◽

Compressible Viscous Gas ◽

The Method Of Lines ◽

Temperature Waves ◽

Parameter Values

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

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Nonlinear travelling internal waves with piecewise-linear shear profiles

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.679 ◽

2018 ◽

Vol 856 ◽

pp. 984-1013 ◽

Cited By ~ 2

Author(s):

K. L. Oliveras ◽

C. W. Curtis

Keyword(s):

Piecewise Linear ◽

Travelling Waves ◽

Tangential Velocity ◽

Travelling Wave ◽

Numerical Continuation ◽

Travelling Wave Solutions ◽

Water Wave Problem ◽

Two Fluids ◽

Wave Solutions ◽

Linear Shear

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129–148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.

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Quantitative diffusion tensor MRI-based fiber tracking of human skeletal muscle

Journal of Applied Physiology ◽

10.1152/japplphysiol.00290.2007 ◽

2007 ◽

Vol 103 (2) ◽

pp. 673-681 ◽

Cited By ~ 89

Author(s):

Drew A. Lansdown ◽

Zhaohua Ding ◽

Megan Wadington ◽

Jennifer L. Hornberger ◽

Bruce M. Damon

Keyword(s):

Muscle Fiber ◽

Diffusion Tensor ◽

Fiber Tracking ◽

Laboratory Frame ◽

Mean Value ◽

Pennation Angle ◽

Frame Of Reference ◽

Fiber Tracts ◽

Angle Measurements ◽

The Mean

Diffusion-tensor magnetic resonance imaging (DT-MRI) offers great potential for understanding structure-function relationships in human skeletal muscles. The purposes of this study were to demonstrate the feasibility of using in vivo human DT-MRI fiber tracking data for making pennation angle measurements and to test the hypothesis that heterogeneity in the orientation of the tibialis anterior (TA) muscle's aponeurosis would lead to heterogeneity in pennation angle. Eight healthy subjects (5 male) were studied. T1-weighted anatomical MRI and DT-MRI data were acquired of the TA muscle. Fibers were tracked from the TA's aponeurosis by following the principal eigenvector. The orientations of the aponeurosis and muscle fiber tracts in the laboratory frame of reference and the orientation of the fiber tracts with respect to the aponeurosis [i.e., the pennation angle (θ)] were determined. The muscle fiber orientations, when expressed relative to the laboratory frame of reference, did not change as functions of superior-to-inferior position. The sagittal and coronal orientations of the aponeurosis did not change in practically significant manners either, but the aponeurosis′ axial orientation changed by ∼40°. As a result, the mean value for θ decreased from 16.3 (SD 6.9) to 11.4° (SD 5.0) along the muscle's superior-to-inferior direction. The mean value of θ was greater in the deep than in the superficial compartment. We conclude that pennation angle measurements of human muscle made using DT-MRI muscle fiber tracking are feasible and reveal that in the foot-head direction, there is heterogeneity in the pennation properties of the human TA muscle.

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Nonuniform Continuity of the Osmosis K(2, 2) Equation

Abstract and Applied Analysis ◽

10.1155/2013/717042 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8

Author(s):

Aiyong Chen ◽

Yong Ding ◽

Wentao Huang

Keyword(s):

Differential Equations ◽

Small Amplitude ◽

Travelling Waves ◽

Travelling Wave ◽

Qualitative Theory ◽

Travelling Wave Solutions ◽

Solution Map ◽

Wave Solutions ◽

Periodic Travelling Wave ◽

Uniformly Continuous

The qualitative theory of differential equations is applied to the osmosis K(2, 2) equation. The parametric conditions of existence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

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FKPP equation with impulses on unbounded domain

Texts in Biomathematics ◽

10.11145/texts.2017.11.157 ◽

2017 ◽

Vol 1 ◽

pp. 1 ◽

Cited By ~ 1

Author(s):

Valaire Yatat ◽

Yves Dumont

Keyword(s):

Travelling Waves ◽

Reaction Diffusion ◽

Bare Soil ◽

Travelling Wave ◽

Reaction Diffusion Equations ◽

Spreading Speed ◽

Systems Dynamics ◽

Travelling Wave Solutions ◽

Minimal Speed ◽

Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

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Mistuning Effects on the Nonlinear Friction Saturation of Flutter Vibration Amplitude

Volume 10A: Structures and Dynamics ◽

10.1115/gt2020-15442 ◽

2020 ◽

Author(s):

Carlos Martel ◽

Salvador Rodríguez

Keyword(s):

Vibration Amplitude ◽

Gas Flow ◽

Travelling Waves ◽

Computational Cost ◽

Travelling Wave ◽

Nonlinear Friction ◽

Blade Vibration ◽

Bladed Disk ◽

Spring System ◽

Mass Spring

Abstract The blade vibration level of an aerodynamically unstable rotor is a quantity of crucial importance to correctly estimate the blade fatigue life. This amplitude is the result of the balance between the energy pumped into the blades by the gas flow, and the nonlinear dissipation at the blade-disk contact interfaces. In a tuned configuration, the blade displacements can be described as a travelling wave consisting of one fundamental nodal diameter and frequency and its higher harmonics, and the problem can be reduced to the computation of a time periodic solution in just one sector. This simplification is no longer valid for a mistuned bladed disk. The resulting nonlinear vibration of the mistuned system is a combination of several travelling waves with different number of nodal diameters, coupled through mistuning. In this case, the complete bladed disk has to be considered, which requires an extremely high computational cost, and, for this reason, reduced order models (ROM) are required to analyze this situation. In this work, we use a 3 DOF/sector mass-spring system to describe the nonlinear friction saturation of the flutter vibration amplitude of a realistic mistuned bladed disk. The convergence of the solution of the mass-spring system is still quite slow because of the presence of many unstable modes with very similar growth rates. In order to speed-up the simulations a simpler asymptotic ROM is derived from the mass-spring model, which allows for much faster integration times. The simulations of the asymptotic ROM are compared with the measurements obtained in the European project FUTURE, where an aerodynamically unstable LPT rotor was tested with different intentional mistuning patterns.

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Asymmetric travelling waves in a square duct

Journal of Fluid Mechanics ◽

10.1017/jfm.2011.455 ◽

2012 ◽

Vol 693 ◽

pp. 57-68 ◽

Cited By ~ 6

Author(s):

Shinya Okino ◽

Masato Nagata

Keyword(s):

Cross Sections ◽

Mirror Symmetry ◽

Symmetric Solution ◽

Travelling Waves ◽

Edge State ◽

Duct Flow ◽

Streamwise Vortices ◽

Square Duct ◽

Geometrical Difference ◽

Asymmetric Solution

AbstractTwo types of asymmetric solutions are found numerically in square-duct flow. They emerge through a symmetry-breaking bifurcation from the mirror-symmetric solutions discovered by Okino et al. (J. Fluid Mech., vol. 657, 2010, pp. 413–429). One of them is characterized by a pair of streamwise vortices and a low-speed streak localized near one of the sidewalls and retains the shift-and-reflect symmetry. The bifurcation nature as well as the flow structure of the solution show striking resemblance to those of the asymmetric solution in pipe flow found by Pringle & Kerswell (Phys. Rev. Lett., vol. 99, 2007, A074502), despite the geometrical difference between their cross-sections. The solution seems to be embedded in the edge state of square-duct flow identified by Biau & Bottaro (Phil. Trans. R. Soc. Lond. A, vol. 367, 2009, pp. 529–544). The other solution deviates slightly from the mirror-symmetric solution from which it bifurcates: the shift-and-rotate symmetry is retained but the mirror symmetry is broken.

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