Publications
37 results found
Walton A, Yu K, 2024, The linear stability of plane Couette flow with a compliant boundary, Journal of Engineering Mathematics, Vol: 144, ISSN: 0022-0833
The linear stability of plane Couette flow subject to one rigid boundary and one flexible boundary is considered at both finite and asymptotically large Reynolds number. The wall flexibility is modelled using a very simple Hooke-type law involving a spring constant K and is incorporated into a boundary condition on the appropriate Orr–Sommerfeld eigenvalue problem. This problem is analyzed at large Reynolds number by the method of matched asymptotic expansions and eigenrelations are derived that demonstrate the existence of neutral modes at finite spring stiffness, propagating with speeds close to that of the rigid wall and possessing wavelengths comparable to the channel width. A large critical value of K is identified at which a new short wavelength asymptotic structure comes into play that describes the entirety of the linear neutral curve. The asymptotic theories compare well with finite Reynolds number Orr–Sommerfeld calculations and demonstrate that only the tiniest amount of wall flexibility is required to destabilize the flow, with the linear neutral curve for the instability emerging as a bifurcation from infinity.
Chotai A, Walton A, 2023, A self-sustaining mechanism for plane Couette flow with a flexible boundary, Quarterly Journal of Mechanics and Applied Mathematics, Vol: 76, Pages: 371-403, ISSN: 0033-5614
The stability of plane Couette flow to travelling-wave disturbances is studied asymptotically at high Reynolds numbers Re when the lower boundary possesses a degree of flexibility modelled by a spring-backed plate. First it is shown that a three-dimensional linear instability exists, with streamwise and spanwise wavelengths comparable with the channel width. Building on this, nonlinear effects from the self-interaction of the wave are introduced, leading to a self-sustaining interaction between a roll/streak flow and the three-dimensional wave. Governing nonlinear vortex-waveinteraction (VWI) equations are derived and a perturbation analysis is carried out to guide a numerical investigation of the equations. The co-existence of two families of finite-amplitude solutions, each with different flow structures, is found. Numerical solutions of the VWI equations in each case show that a small wave amplitude of O(Re−1(log Re)−1/2) is all that is necessary to provoke an O(1) change to the basic Couette flow.
Kumar R, Walton A, 2021, Two-dimensional self-sustaining processes involving critical layer/Wall layer interaction, 9th IUTAM Symposium on Laminar-Turbulent Transition, Publisher: Springer International Publishing AG, Pages: 117-126, ISSN: 1875-3507
The nonlinear stability of plane Poiseuille-Couette flow is analyzed at large Reynolds number. An asymptotic self-sustaining structure is found which involves the interaction of two nonlinear critical layers with near-wall Stokes regions. A global property of the flow-field, involving the vorticity of the mean-flow distortion, is used to determine the amplitude dependence of the O(1) wavenumber and phasespeed and it is found that solutions exist for values of wall sliding speed considerably in excess of the linear cut-off. In the situation where the phasespeed is almost equal to the wall sliding speed, a new nonlinear structure arises involving critical layer/shear layer interaction. The numerical results from this latter interaction are found to compare well with full solutions of the Navier-Stokes equations.
Kumar R, Walton A, 2020, Self-sustaining critical layer/shear layer interaction in annular Poiseuille–Couette flow at high Reynolds number, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 476, ISSN: 1364-5021
The nonlinear stability of annular Poiseuille–Couette flow through a cylindrical annulus subjected to axisymmetric and helical disturbances is analysed theoretically at asymptotically large Reynolds number R based on the radius of the outer cylinder and the constant axial pressure gradient applied. The inner cylinder moves with a prescribed positive or negative velocity in the axial direction. A distinguished scaling for the disturbance size Δ = O(R−4/9) is identified at which the jump in vorticity across the fully nonlinear critical layer is in tune with that induced across a near-wall shear layer. The disturbance propagates at close to the velocity of the inner cylinder and possesses a wavelength comparable to the radius of the outer cylinder. The dynamics of the critical layer, shear layer and the Stokes layer adjacent to the stationary wall are discussed in detail. In the majority of the pipe, the disturbance is governed predominantly by inviscid dynamics with the pressure perturbation satisfying a form of Rayleigh’s equation. For a radius ratio δ in the range 0 < δ < 1 and a positive sliding velocity V, a numerical solution of the Rayleigh equation exists for sliding velocities in the range 0 < V < 1 − δ2 + 2δ2lnδ, whereas if V < 0, solutions exist for 1 − δ2 + 2lnδ < V < 0. The amplitude equations for both these situations are derived analytically, and we further find that the corresponding asymptotic structures break down when the maximum value of the basic flow becomes located at the inner and outer walls, respectively.
Kumar R, Walton A, 2019, Self-sustaining dual critical layer states in plane Poiseuille–Couette flow at large Reynolds number, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 475, ISSN: 1364-5021
The nonlinear stability of plane Poiseuille–Couette flow subjected to three-dimensional disturbances is studied asymptotically at large Reynolds number R. By analysing the nature of the instability for increasing disturbance size Δ, the scaling Δ = O(R−1/3) is identified at which a strongly nonlinear neutral wave structure emerges, involving the interaction of two inviscid critical layers. The striking feature of this structure is that the travelling wave disturbances have both streamwise and spanwise wavelengths comparable to the channel width, with an associated phase speed of O(1). An alternative method to the classical balancing of phase shifts is proposed, involving vorticity jumps, that uses a global property of the flow-field and enables the amplitude-dependence of the neutral modes to be determined in terms of the wavenumbers and the properties of the basic flow. Numerical computation of the Rayleigh equation which governs the flow outside of the critical layers shows that neutral solutions exist for non-dimensional wall sliding speeds in the range 0 ≤ V < 2. It transpires that the critical layers merge and the asymptotic structure referred to above breaks down both in the large-amplitude limit and the limit V→2 when the maximum of the basic flow becomes located at the upper wall.
Kumar R, Walton A, 2019, Amplitude-dependent three-dimensional neutral modes in plane Couette-Poiseuille flow at large Reynolds number, Quarterly Journal of Mechanics and Applied Mathematics, Vol: 72, Pages: 87-130, ISSN: 0033-5614
The non-linear stability of plane Poiseuille–Couette flow to three-dimensional disturbances is investigated asymptotically at large values of the Reynolds number R based on channel half-width and the maximum velocity of the Poiseuille component. The asymptotic theory, aimed at a detailed understanding of the physical mechanisms governing the amplitude-dependent stability properties of the flow, shows that the phase shifts induced across the critical layer and a near-wall shear layer are comparable when the disturbance size Δ=O(R−4/9). In addition, it emerges that at this crucial size both streamwise and spanwise wavelengths of the travelling wave disturbance are comparable with the channel width, with an associated phasespeed of O(1). Neutral solutions are found to exist in the range 0<V<2 with c0=V to leading order, where c0 and V are non-dimensional quantities representing the dominant phasespeed of the non-linear travelling waves and the wall sliding speed respectively. Moreover, these instability modes exist at sliding speeds well in excess of the linear instability cut-off. The amplitude equation governing these modes is derived analytically and we further find that this asymptotic structure breaks down in the limit V→2 when the disturbance streamwise wavelength decreases to O(R−1/3) and the maximum of the basic flow becomes located at the upper wall.
Deguchi K, Walton AG, 2018, Bifurcation of nonlinear Tollmien-Schlichting waves in a high-speed channel flow, Journal of Fluid Mechanics, Vol: 843, Pages: 53-97, ISSN: 0022-1120
Plane Poiseuille flow has long served as the simplest testing ground for Tollmien–Schlichting wave instability. In this paper, we provide a comprehensive comparison of equilibrium Tollmien–Schlichting wave solutions arising from new high-resolution Navier–Stokes calculations and the corresponding predictions of various large-Reynolds-number asymptotic theories developed in the last century, such as double-deck theory, viscous nonlinear critical layer theory and strongly nonlinear critical layer theory. In the relatively small to moderate amplitude regime, the theories excellently predict the behaviour of the numerical solutions at Reynolds numbers of order and above, whilst for larger amplitudes our computations suggest the need for further asymptotic theories to be developed.
Walton AG, Dempsey LJ, 2017, Vortex/tollmien-schlichting wave interaction states in the asymptotic suction boundary layer, Quarterly Journal of Mechanics and Applied Mathematics, Vol: 70, Pages: 187-213, ISSN: 1464-3855
A self-sustaining interaction between a roll/streak structure and a three-dimensional Tollmien–Schlichting wave is considered at high-Reynolds-number within the asymptotic suction boundary layer. Strongly nonlinear governing equations, taking the form of a vortex–wave interaction (VWI) are derived and solved numerically. Finite amplitude travelling wave states, bifurcating from the lower branch of the linear neutral curve, are obtained. These states exhibit spanwise focusing, developing steep wall-shear gradients at specific spanwise locations as the wave amplitude rises. A spanwise-local analytic analysis reveals explicitly how the solution gradually loses regularity as the nonlinearity of the VWI system is increased.
Dempsey LJ, Deguchi K, Hall P, et al., 2016, Localized vortex/Tollmien-Schlichting wave interaction states in plane Poiseuille flow, Journal of Fluid Mechanics, Vol: 791, Pages: 97-121, ISSN: 0022-1120
Strongly nonlinear three-dimensional interactions between a roll-streak structure and aTollmien-Schlichting wave in plane Poiseuille flow are considered in this study. Equationsgoverning the interaction at high Reynolds number originally derived by Bennett, Hall& Smith (J. Fluid Mech, vol. 223, 1991, pp. 475–495) are solved numerically. Travellingwave states bifurcating from the lower branch linear neutral point are tracked to finiteamplitudes, where they are observed to localize in the spanwise direction. The nature ofthe localization is analysed in detail near the relevant spanwise locations, revealing thepresence of a singularity which slowly develops in the governing interaction equationsas the amplitude of the motion is increased. Comparisons with the full Navier-Stokesequations demonstrate that the finite Reynolds number solutions gradually approach thenumerical asymptotic solutions with increasing Reynolds number.
Deguchi K, Walton AG, 2013, A swirling spiral wave solution in pipe flow, Journal of Fluid Mechanics, Vol: 737, ISSN: 0022-1120
Deguchi K, Hall P, Walton AG, 2013, The emergence of localized vortex-wave interaction states in plane Couette flow, Journal of Fluid Mechanics, Vol: 721, Pages: 58-85
The recently understood relationship between high Reynolds number vortex-wave interactiontheory and computationally-generated self-sustaining processes provides a possibleroute to an understanding of some of the underlying structures of fully turbulent flows.Here vortex-wave interaction theory, which we now refer to as VWI, is used in the longstreamwise wavelength limit to continue the development found at order one wavelengthsby Hall and Sherwin (2010). The asymptotic description given reduces the Navier-Stokesequations to the so-called boundary-region equations for which we find equilibrium statesdescribing the change in the VWI as the wavelength of the wave increases from O(h) toO(Rh) where R is the Reynolds number and 2h is the depth of the channel. The reducedequations do not include the streamwise pressure gradient of the perturbation or theeffect of streamwise diffusion of the wave-vortex states. The solutions we calculate havean asymptotic error proportional to R−2 when compared to the full Navier-Stokes equations.The results found correspond to the minimum drag configuration for VWI statesand might therefore be of relevance to the control of turbulent flows. The key feature ofthe new states discussed here is the thickening of the critical layer structure associatedwith the wave part of the flow to completely fill the channel so that the roll part ofthe flow is driven throughout the flow rather than as in Hall and Sherwin (2010) as astress discontinuity across the critical layer. We identify a critical streamwise wavenumberscaling which when approached causes the flow to localise and take on similaritieswith computationally-generated or experimentally-observed turbulent spots. In effect theidentification of this critical wavenumber for a given value of the assumed high Reynoldsnumber fixes a minimum box length necessary for the emergence of localized structures.Whereas nonlinear equilibrium states of the Navier-Stokes equations are thought to forma bac
Deguchi K, Walton AG, 2013, Axisymmetric travelling waves in annular sliding Couette flow at finite and asymptotically large Reynolds number, Journal of Fluid Mechanics, Vol: 720, Pages: 582-617
The relationship between numerical finite-amplitude equilibrium solutions of the fullNavier-Stokes equations and nonlinear solutions arising from a high Reynolds numberasymptotic analysis is discussed for a Tollmien-Schlichting wave type two-dimensionalvortical flow structure. The specific flow chosen for this purpose is that which arises fromthe mutual axial sliding of co-axial cylinders for which nonlinear axisymmetric travellingwavesolutions have been discovered recently by Deguchi & Nagata (J. Fluid Mech., vol.678, 2011, pp. 156–178). We continue this solution branch to a Reynolds number R = 108and confirm that the behaviour of its so-called lower branch solutions, which typicallyproduce a smaller modification to the laminar state than the other solution branches,quantitatively agrees with the axisymmetric asymptotic theory developed in this paper.We further find that this asymptotic structure breaks down when the disturbance wavelengthis comparable with R. The new structure which replaces it is investigated and thegoverning equations are derived and solved. The flow visualization of the resultant solutionsreveals that they possess a streamwise localized structure, with the trend agreeingqualitatively with full Navier-Stokes solutions for relatively long wavelength disturbances.
Wong AWH, Walton AG, 2012, Axisymmetric Travelling Waves in Annular Couette-Poiseuille Flow, Quarterly Journal of Mechanics and Applied Mathematics, Vol: 65, Pages: 293-311
A computational study of the stability of the axial pressure-gradient-driven viscous flow between concentric sliding cylinders is presented. Nonlinear axisymmetric travelling wave solutions, which bifurcate from the linear neutral curve, are found for a range of radius ratio, sliding velocity and disturbance amplitude. Solutions which come into existence at finite amplitude and are not connected to the linear state in wavenumber–Reynolds number space are also found. The critical Reynolds numbers associated with these solutions are found to be substantially lower than those predicted by linear theory. For some combinations of parameters, two nonlinear solutions with different wavelengths are found to coexist and appear to merge into one as the amplitude is increased.
Walton AG, 2011, The stability of developing pipe flow at highReynolds number and the existence of nonlinearneutral centre modes, Journal of Fluid Mechanics, Vol: 684, Pages: 284-315
The high-Reynolds-number stability of unsteady pipe flow to axisymmetricdisturbances is studied using asymptotic analysis. It is shown that as the disturbanceamplitude is increased, nonlinear effects first become significant within the criticallayer, which moves away from the pipe wall as a result. It is found that the flowstabilizes once the basic profile has become sufficiently fully developed. By tracingthe nonlinear neutral curve back to earlier times, it is found that in addition tothe wall mode, which arises from a classical upper branch linear stability analysis,there also exists a nonlinear neutral centre mode, governed primarily by invisciddynamics. The centre mode problem is solved numerically and the results show theexistence of a concentrated region of vorticity centred on or close to the pipe axisand propagating downstream at almost the maximum fluid velocity. The connectionbetween this structure and the puffs and slugs of vorticity observed in experiments isdiscussed.
Walton AG, Labadin J, Ping YS, 2010, Axial Flow Between Sliding, Non-Concentric Cylinders with Applications to Thread Injection, Quarterly Journal of Mechanics and Applied Mathematics, Vol: 63, Pages: 315-334, ISSN: 0033-5614
An investigation is conducted into the nature of the pressure-driven axial flow between cylinders with the inner cylinder moving in the axial direction. Such a flow is often referred to in the literature as a ‘thread-annular’ flow and is relevant to the procedure of thread injection: a surgical technique that allows the injection of porous medical implants, consisting of synthetic biocompatible materials, into the body in a minimally invasive way. A perturbation solution for the fluid flow is derived under the assumption that the inner cylinder is displaced slightly from a concentric position within the outer cylinder. From the basic solution, expressions for the force on the thread and the friction factor for the flow are derived. Our results are compared with the concentric case, experimental results and also the exact solution to the problem. It is found that reported discrepancies between experimental measurements and theory based on a concentric flow model can be explained by our inclusion of thread eccentricity.
Labadin J, Walton AG, 2006, Modeling of axial flow between eccentric cylinders, 2nd IMT-GT Regional Conference on Mathematics, Statistics and their Applications, Pages: 1-7
Labadin J, Walton AG, 2006, Theory and computation of high Reynolds number flow near ground corner of a tall building, First International Conference on Computational Methods
Walton AG, 2005, The linear and nonlinear stability of thread-annular flow, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 363, Pages: 1223-1233, ISSN: 1364-503X
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- Citations: 13
Walton AG, 2005, The stability of nonlinear neutral modes in Hagen-Poiseuille flow, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 461, Pages: 813-824, ISSN: 1364-5021
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- Citations: 6
Walton AG, 2004, Stability of circular Poiseuille-Couette flow to axisymmetric disturbances, JOURNAL OF FLUID MECHANICS, Vol: 500, Pages: 169-210, ISSN: 0022-1120
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- Citations: 24
Walton AG, 2003, The nonlinear instability of thread-annular flow at high Reynolds number, JOURNAL OF FLUID MECHANICS, Vol: 477, Pages: 227-257, ISSN: 0022-1120
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- Citations: 23
Walton AG, 2002, The temporal evolution of neutral modes in the impulsively started flow through a circular pipe and their connection to the nonlinear stability of Hagen-Poiseuille flow, JOURNAL OF FLUID MECHANICS, Vol: 457, Pages: 339-376, ISSN: 0022-1120
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- Citations: 11
Walton AG, 2001, Existence of neutral Rayleigh waves in the Hagen-Poiseuille flow through a pipe of circular cross section, STUDIES IN APPLIED MATHEMATICS, Vol: 106, Pages: 315-335, ISSN: 0022-2526
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- Citations: 3
Walton AG, Patel RA, 1999, Singularity formation in the strongly nonlinear wide-vortex/Tollmien-Schlichting-wave interaction equations, Journal of Fluid Mechanics, Vol: 400, Pages: 265-293, ISSN: 0022-1120
Smith FT, Walton AG, 1998, Flow past a two- or three-dimensional steep-edged roughness, Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences, Vol: 454, Pages: 31-69, ISSN: 1364-5021
Walton AG, Patel RA, 1998, On the neutral stability of spanwise-periodic boundary-layer and triple-deck flows, Quarterly Journal of Mechanics and Applied Mathematics, Vol: 51, Pages: 311-328, ISSN: 0033-5614
Walton AG, Smith FT, 1997, Concerning three-dimensional flow past a tall building on flat ground, QUARTERLY JOURNAL OF MECHANICS AND APPLIED MATHEMATICS, Vol: 50, Pages: 97-128, ISSN: 0033-5614
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- Citations: 7
Walton AG, 1996, On the non-existence of periodic neutral-wave solutions to a complex-valued periodic differential equation, MATHEMATIKA, Vol: 43, Pages: 371-381, ISSN: 0025-5793
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- Citations: 1
Walton AG, 1996, Strongly nonlinear vortex-Tollmien-Schlichting-wave interactions in the developing flow through a circular pipe, Journal of Fluid Mechanics, Vol: 319, Pages: 77-107, ISSN: 0022-1120
Walton AG, 1995, Recent developments in the theory of the nonlinear stability of high Reynolds number flows, Berlin, Publisher: Springer-Verlag, Pages: 93-98
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