Nonlinear exact coherent structures in pipe flow and their instabilities

2019 ◽  
Vol 868 ◽  
pp. 341-368 ◽  
Author(s):  
Ozge Ozcakir ◽  
Philip Hall ◽  
Saleh Tanveer
Keyword(s):  
Unstable Mode ◽  
Travelling Waves ◽  
Wave Interaction ◽  
Reynolds Numbers ◽  
Large Reynolds Number ◽  
Interaction Theory ◽  
Asymptotic Results ◽  
Instability Modes ◽  
Viscous Core ◽  
The Waves

In this paper, we present computational results of some two-fold azimuthally symmetric travelling waves and their stability. Calculations over a range of Reynolds numbers ($Re$) reveal connections between a class of solutions computed by Wedin & Kerswell (J. Fluid Mech., vol. 508, 2004, pp. 333–371) (henceforth called the WK solution) and the $Re\rightarrow \infty$ vortex–wave interaction theory of Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). In particular, the continuation of the WK solutions to larger values of $Re$ shows that the WK solution bifurcates from a shift-and-rotate symmetric solution, which we call the WK2 state. The WK2 solution computed for $Re\leqslant 1.19\times 10^{6}$ shows excellent agreement with the theoretical $Re^{-5/6}$, $Re^{-1}$ and $O(1)$ scalings of the waves, rolls and streaks respectively. Furthermore, these states are found to have only two unstable modes in the large $Re$ regime, with growth rates estimated to be $O(Re^{-0.42})$ and $O(Re^{-0.92})$, close to the theoretical $O(Re^{-1/2})$ and $O(Re^{-1})$ asymptotic results for edge and sinuous instability modes of vortex–wave interaction states (Deguchi & Hall, J. Fluid Mech., vol. 802, 2016, pp. 634–666) in plane Couette flow. For the nonlinear viscous core states (Ozcakir et al., J. Fluid Mech., vol. 791, 2016, pp. 284–328), characterized by spatial a shrinking of the wave and roll structure towards the pipe centre with increasing $Re$, we continued the solution to $Re\leqslant 8\times 10^{6}$ and we find only one unstable mode in the large Reynolds number regime, with growth rate scaling as $Re^{-0.46}$ within the class of symmetry-preserving disturbances.

Linear instability of lid- and pressure-driven flows in channels textured with longitudinal superhydrophobic grooves

2021 ◽  
Vol 932 ◽  
Author(s):  
Samuel D. Tomlinson ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Reynolds Number ◽  
Unstable Mode ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Large Reynolds Number ◽  
Spanwise Direction ◽  
Channel Height ◽  
Critical Reynolds Numbers ◽  
Upper Lid ◽  
Pressure Driven

It is known that an increased flow rate can be achieved in channel flows when smooth walls are replaced by superhydrophobic surfaces. This reduces friction and increases the flux for a given driving force. Applications include thermal management in microelectronics, where a competition between convective and conductive resistance must be accounted for in order to evaluate any advantages of these surfaces. Of particular interest is the hydrodynamic stability of the underlying basic flows, something that has been largely overlooked in the literature, but is of key relevance to applications that typically base design on steady states or apparent-slip models that approximate them. We consider the global stability problem in the case where the longitudinal grooves are periodic in the spanwise direction. The flow is driven along the grooves by either the motion of a smooth upper lid or a constant pressure gradient. In the case of smooth walls, the former problem (plane Couette flow) is linearly stable at all Reynolds numbers whereas the latter (plane Poiseuille flow) becomes unstable above a relatively large Reynolds number. When grooves are present our work shows that additional instabilities arise in both cases, with critical Reynolds numbers small enough to be achievable in applications. Generally, for lid-driven flows one unstable mode is found that becomes neutral as the Reynolds number increases, indicating that the flows are inviscidly stable. For pressure-driven flows, two modes can coexist and exchange stability depending on the channel height and slip fraction. The first mode remains unstable as the Reynolds number increases and corresponds to an unstable mode of the two-dimensional Rayleigh equation, while the second mode becomes neutrally stable at infinite Reynolds numbers. Comparisons of critical Reynolds numbers with the experimental observations for pressure-driven flows of Daniello et al. (Phys. Fluids, vol. 21, issue 8, 2009, p. 085103) are encouraging.

scholarly journals From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow

2012 ◽  
Vol 709 ◽  
pp. 149-190 ◽  
Author(s):  
F. Mellibovsky ◽  
B. Eckhardt
Keyword(s):  
Invariant Manifolds ◽  
Travelling Waves ◽  
Path Following ◽  
Reynolds Numbers ◽  
Discrete Symmetry ◽  
Limited Range ◽  
Wide Range ◽  
Full State ◽  
Invariant States ◽  
The Waves

AbstractWe study numerically a succession of transitions in pipe Poiseuille flow that lead from simple travelling waves to waves with chaotic time-dependence. The waves at the origin of the bifurcation cascade are twofold azimuthally periodic, shift–reflect symmetric, and have a non-dimensional axial wavelength of $1. 927$ diameters. As the Reynolds number is increased, successive transitions result in a wide range of time-dependent solutions that include spiralling, modulated travelling, modulated spiralling, doubly modulated spiralling and mildly chaotic waves. Numerical evidence suggests that the latter spring from heteroclinic tangles of the stable and unstable invariant manifolds of two shift–reflect symmetric, modulated travelling waves. The chaotic set thus produced is confined to a limited range of Reynolds numbers, bounded by the occurrence of manifold tangencies. The subspace of discrete symmetry to which the states studied here belong makes many of the bifurcation and path-following investigations presented readily accessible. However, we expect that most of the phenomenology carries over to the full state space, thus suggesting a mechanism for the formation and break-up of invariant states that can give rise to chaotic dynamics.

Centre modes in pipe flow

2019 ◽  
Vol 84 (5) ◽  
pp. 854-872
Author(s):  
Ozge Ozcakir ◽  
Philip Hall ◽  
Saleh Tanveer
Keyword(s):  
Reynolds Number ◽  
Coherent Structures ◽  
Pipe Flow ◽  
Travelling Waves ◽  
Wave Interaction ◽  
Boundary Layer Transition ◽  
Asymptotic State ◽  
Layer Transition ◽  
Viscous Core ◽  
High Reynolds

Abstract In two previous papers, Ozcakir, Tanveer, Hall, & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328) and Ozcakir, Hall & Tanveer (2019, Nonlinear exact coherent structures in pipe flow and their instabilities, J. Fluid Mech., 868, 341–368) investigated numerically and asymptotically high Reynolds number exact coherent structures in pipe flow. It was found that, in addition to the structures described by the vortex–wave interaction theory by Hall & Smith (1991, On strongly nonlinear vortex/wave interactions in boundary layer transition. J. Fluid Mech., 227, 641–666), there exists vortical structures localized near the centre of the pipe with a core of size $O(Re^{-1/4})$ convected downstream at a speed that deviates from the pipe centreline speed by $O(Re^{-1/2})$, where $Re$ is the Reynolds number. In the finite Reynolds number calculations by Ozcakir, Tanveer, Hall & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328), asymptotic state was referred to as a nonlinear viscous core state (NVC). However the reduced asymptotic equations were not solved and only limited confirmation of the theory was found numerically. Here, in order to conclusively confirm the existence of the NVC state we first describe direct numerical calculations on the asymptotically reduced $Re>>1$ equations for such state states. The results are then compared in detail to the finite $Re$ calculations up-to $Re=10^6$; the latter regime is at much higher values of the Reynolds number than those reported in Ozcakir, Tanveer, Hall & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328). The results are found to be in excellent agreement with the finite $Re$ calculations in a region between $Re=10^5$ and $10^6$, thereby confirming that the structure observed by Ozcakir, Tanveer, Hall & Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328) is indeed a finite Reynolds number realization of an asymptotic NVC state.

scholarly journals High-speed shear-driven dynamos. Part 1. Asymptotic analysis

2019 ◽  
Vol 868 ◽  
pp. 176-211 ◽  
Author(s):  
Kengo Deguchi
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Solar Phys ◽  
Wave Interaction ◽  
Resonant Absorption ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Alfven Wave ◽  
Hydrodynamic Theory ◽  
Large Reynolds Number

Rational large Reynolds number matched asymptotic expansions of three-dimensional nonlinear magneto-hydrodynamic (MHD) states are the concern of this contribution. The nonlinear MHD states, assumed to be predominantly driven by a unidirectional shear, can be sustained without any linear instability of the base flow and hence are responsible for subcritical transition to turbulence. Two classes of nonlinear MHD states are found. The first class of nonlinear states emerged out of a nice combination of the purely hydrodynamic vortex/wave interaction theory by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and the resonant absorption theories on Alfvén waves, developed in the solar physics community (e.g. Sakurai et al. Solar Phys., vol. 133, 1991, pp. 227–245; Goossens et al. Solar Phys., vol. 157, 1995, pp. 75–102). Similar to the hydrodynamic theory, the mechanism of the MHD states can be explained by the successive interaction of the roll, streak and wave fields, which are now defined both for the hydrodynamic and magnetic fields. The derivation of this ‘vortex/Alfvén wave interaction’ state is rather straightforward as the scalings for both of the hydrodynamic and magnetic fields are identical. It turns out that the leading-order magnetic field of the asymptotic states appears only when a small external magnetic field is present. However, it does not mean that purely shear-driven dynamos are not possible. In fact, the second class of ‘self-sustained shear-driven dynamo theory’ shows a magnetic generation that is slightly smaller in size in the absence of any external field. Despite its small size, the magnetic field causes the novel feedback mechanism in the velocity field through resonant absorption, wherein the magnetic wave becomes more strongly amplified than the hydrodynamic counterpart.

On separated shear layers and the fractal geometry of turbulent scalar interfaces at large Reynolds numbers

2009 ◽  
Vol 624 ◽  
pp. 389-411 ◽  
Author(s):  
FAZLUL R. ZUBAIR ◽  
HARIS J. CATRAKIS
Keyword(s):  
Fractal Dimension ◽  
Scalar Field ◽  
Reynolds Numbers ◽  
Shear Layers ◽  
Large Reynolds Number ◽  
Geometrical Properties ◽  
Counting Approach ◽  
Box Counting ◽  
Wide Range ◽  
Reference Areas

This work explores fractal geometrical properties of scalar turbulent interfaces derived from experimental two-dimensional spatial images of the scalar field in separated shear layers at large Reynolds numbers. The resolution of the data captures the upper three decades of scales enabling examination of multiscale geometrical properties ranging from the largest energy-containing scales to inertial scales. The data show a −5/3 spectral exponent over a wide range of scales corresponding to the inertial range in fully developed turbulent flows. For the fractal aspects, we utilize two methods as it is known that different methods may lead to different fractal aspects. We use the recently developed method for fractal analysis known as the Multiscale-Minima Meshless (M3) method because it does not require the use of grids. We also use the conventional box-counting approach as it has been frequently employed in various past studies. The outer scalar interfaces are identified on the basis of the probability density function (p.d.f.) of the scalar field. For the outer interfaces, the M3 method shows strong scale dependence of the generalized fractal dimension with approximately linear variation of the dimension as a function of logarithmic scale, for interface-fitting reference areas, but there is evidence of a plateau near a dimension D ~ 1.3 for larger reference areas. The conventional box-counting approach shows evidence of a plateau with a constant dimension also of D ~ 1.3, for the same reference areas. In both methods, the observed plateau dimension value agrees with other studies in different flow geometries. Scalar threshold effects are also examined and show that the internal scalar interfaces exhibit qualitatively similar behaviour to the outer interfaces. The overall range of box-counting fractal dimension values exhibited by outer and internal interfaces is D ~ 1.2–1.4. The present findings show that the fractal aspects of scalar interfaces in separated shear layers at large Reynolds number with −5/3 spectral behaviour can depend on the method used for evaluating the dimension and on the reference area. These findings as well as the utilities and distinctions of these two different definitions of the dimension are discussed in the context of multiscale modelling of mixing and the interfacial geometry.

scholarly journals 3D turbulent reconnection driven current-sheet dynamics: solar applications

2010 ◽  
Vol 6 (S274) ◽  
pp. 458-460
Author(s):  
Lapo Bettarini ◽  
Giovanni Lapenta
Keyword(s):  
Current Sheet ◽  
Characteristic Time ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Anomalous Effect ◽  
Instability Modes ◽  
Diffusive Dynamics ◽  
Three Dimensional Configuration ◽  
A Current ◽  
Dimensional Picture

AbstractWe provide a complete three-dimensional picture of the reconnecting dynamics of a current-sheet. Recently, a two-dimensional non-steady reconnection dynamics has been proved to occur without the presence of any anomalous effect (Lapenta, 2008, Skender & Lapenta, 2010, Bettarini & Lapenta, 2010) but such a picture must be confirmed in a full three-dimensional configuration wherein all instability modes are allowed to drive the evolution of the system, i.e. to sustain a reconnection dynamics or to push the system along a different instability path. Here we propose a full-space analysis allowing us to determine the longitudinal and, possibly, the transversal modes driving the different current-sheet disruption regimes, the corresponding characteristic time-scales and to study system's instability space- parameter (plasma beta, Lundquist and Reynolds numbers, system's aspect ratio). The conditions leading to an explosive evolution rather then to a diffusive dynamics as well as the details of the reconnection inflow/outflow regime at the disruption phase are determined. Such system embedded in a solar-like environment and undergoing a non-steady reconnection evolution may determine the formation both of jets and waves influencing the dynamics and energetic of the upper layers and of characteristic down-flows as observed in the low solar atmosphere.

Analytical and numerical studies of the structure of steady separated flows

1966 ◽  
Vol 24 (1) ◽  
pp. 113-151 ◽  
Author(s):  
Odus R. Burggraf
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Reynolds Numbers ◽  
Total Power ◽  
Large Reynolds Number ◽  
Boundary Velocity ◽  
Displacement Thickness ◽  
Low Reynolds

The viscous structure of a separated eddy is investigated for two cases of simplified geometry. In § 1, an analytical solution, based on a linearized model, is obtained for an eddy bounded by a circular streamline. This solution reveals the flow development from a completely viscous eddy at low Reynolds number to an inviscid rotational core at high Reynolds number, in the manner envisaged by Batchelor. Quantitatively, the solution shows that a significant inviscid core exists for a Reynolds number greater than 100. At low Reynolds number the vortex centre shifts in the direction of the boundary velocity until the inviscid core develops; at large Reynolds number, the inviscid vortex core is symmetric about the centre of the circle, except for the effect of the boundary-layer displacement-thickness. Special results are obtained for velocity profiles, skin-friction distribution, and total power dissipation in the eddy. In addition, results of the method of inner and outer expansions are compared with the complete solution, indicating that expansions of this type give valid results for separated eddies at Reynolds numbers greater than about 25 to 50. The validity of the linear analysis as a description of separated eddies is confirmed to a surprising degree by numerical solutions of the full Navier–Stokes equations for an eddy in a square cavity driven by a moving boundary at the top. These solutions were carried out by a relaxation procedure on a high-speed digital computer, and are described in § 2. Results are presented for Reynolds numbers from 0 to 400 in the form of contour plots of stream function, vorticity, and total pressure. At the higher values of Reynolds number, an inviscid core develops, but secondary eddies are present in the bottom corners of the square at all Reynolds numbers. Solutions of the energy equation were obtained also, and isotherms and wall heat-flux distributions are presented graphically.

Instability in Stratified Shear Flow: Review of a Physical Interpretation Based on Interacting Waves

2011 ◽  
Vol 64 (6) ◽  
Author(s):  
Jeffrey R. Carpenter ◽  
Edmund W. Tedford ◽  
Eyal Heifetz ◽  
Gregory A. Lawrence
Keyword(s):  
Shear Flow ◽  
Physical Interpretation ◽  
Shear Flows ◽  
Flow Instability ◽  
Phase Locking ◽  
River Estuary ◽  
Wave Interaction ◽  
The Stability ◽  
Interaction Approach ◽  
The Waves

Instability in homogeneous and density stratified shear flows may be interpreted in terms of the interaction of two (or more) otherwise free waves in the velocity and density profiles. These waves exist on gradients of vorticity and density, and instability results when two fundamental conditions are satisfied: (I) the phase speeds of the waves are stationary with respect to each other (“phase-locking“), and (II) the relative phase of the waves is such that a mutual growth occurs. The advantage of the wave interaction approach is that it provides a physical interpretation to shear flow instability. This paper is largely intended to purvey the basics of this physical interpretation to the reader, while both reviewing and consolidating previous work on the topic. The interpretation is shown to provide a framework for understanding many classical and nonintuitive results from the stability of stratified shear flows, such as the Rayleigh and Fjørtoft theorems, and the destabilizing effect of an otherwise stable density stratification. Finally, we describe an application of the theory to a geophysical-scale flow in the Fraser River estuary.

A falling film down a slippery inclined plane

2011 ◽  
Vol 684 ◽  
pp. 353-383 ◽  
Author(s):  
A. Samanta ◽  
C. Ruyer-Quil ◽  
B. Goyeau
Keyword(s):  
Travelling Waves ◽  
Falling Film ◽  
Inclined Plane ◽  
Local Flow ◽  
Navier Slip ◽  
Averaged Equations ◽  
Gravity Driven ◽  
The Waves ◽  
Navier Slip Condition ◽  
Good Agreement

AbstractA gravity-driven film flow on a slippery inclined plane is considered within the framework of long-wave and boundary layer approximations. Two coupled depth-averaged equations are derived in terms of the local flow rate $q(x, t)$ and the film thickness $h(x, t)$. Linear stability analysis of the averaged equations shows good agreement with the Orr–Sommerfeld analysis. The effect of a slip at the wall on the primary instability has been found to be non-trivial. Close to the instability onset, the effect is destabilising whereas it becomes stabilising at larger values of the Reynolds number. Nonlinear travelling waves are amplified by the presence of the slip. Comparisons to direct numerical simulations show a remarkable agreement for all tested values of parameters. The averaged equations capture satisfactorily the speed, shape and velocity distribution in the waves. The Navier slip condition is observed to significantly enhance the backflow phenomenon in the capillary region of the solitary waves with a possible effect on heat and mass transfer.

Sign in / Sign up

Export Citation Format

Share Document