An Asymptotic Self-Sustaining Process Theory for Uniform Momentum Zones and Internal Interfaces in Unbounded Couette Flow

Meinhart \& Adrian (Phys. Fluids, vol. 7, 1995, p 694) were the first investigators to document that the wall-normal ($y$) structure of the instantaneous streamwise velocity in the turbulent boundary layer exhibits a staircase-like profile: regions of quasi-uniform momentum are separated by internal shear layers across which the streamwise velocity jumps by an O(1) amount when scaled by the friction velocity $u_\tau$. This sharply-varying instantaneous profile differs dramatically from the well-known long-time mean profile, which is logarithmic over much of the boundary layer, and prompted Klewicki (Proc. IUTAM, vol. 9, 2013, p. 69--78) to propose that the turbulent boundary layer is singular in two distinct ways. Firstly, spanwise vorticity and mean viscous forces are concentrated in a near-wall region of thickness $\mathit{O}(h/\sqrt{Re_\tau})$, where $Re_\tau$ is the friction Reynolds number and $h$ is the boundary-layer height. Secondly, in a turbulent boundary layer, spanwise vorticity and viscous forces are also significant away from the wall (outboard of the peak in the Reynolds stress), but \emph{only} in spatially-localized regions, i.e. within the internal shear layers. This interpretation accords with Klewicki\u27s multiscale similarity analysis of the mean momentum balance for turbulent wall flows (J. Fluid Mech., vol. 522, 2005, pp. 303--327). The objective of the present investigation is to probe the governing Navier--Stokes equations in the limit of large $Re_\tau$ in search of a mechanistic self-sustaining process (SSP) that (i) can account for the emergent staircase-like profile of streamwise velocity in the inertial region and (ii) is compatible with the singular nature of turbulent wall flows. Plausible explanations for the formation and persistence of sharply-varying instantaneous streamwise velocity profiles all implicate quasi-coherent turbulent flow structures including streamwise roll motions that induce a cellular flow in the transverse (i.e. spanwise/wall-normal) plane. One proposal is that the large-scale structures result from the spontaneous concatenation of smaller--scale structures, particularly hairpin and cane vortices and vortex packets. A competing possibility, explored here, is that these large--scale motions may be \emph{directly} sustained via an inertial--layer SSP that is broadly similar to the near-wall SSP. The SSP theory derived in this investigation is related to the SSP framework developed by Waleffe (Stud. Appl. Math, vol. 95, 1995, p. 319) and, especially, to the closely-related vortex-wave interaction (VWI) theory derived by Hall \& Smith (J. Fluid Mech., vol. 227, 1991, pp. 641--666) and Hall \& Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178--205) in that a rational asymptotic analysis of the instantaneous Navier--Stokes equations is performed. Nevertheless, in this dissertation, it is argued that these theories cannot account for organized motions in the inertial domain, essentially because the roll motions are predicted to be viscously dominated even at large $Re_\tau$. The target of the present investigation is an inherently multiscale SSP, in which inviscid streamwise rolls differentially homogenize an imposed background shear flow, thereby generating uniform momentum zones and an embedded internal shear layer (or interface), and are sustained by Rayleigh instability modes having asymptotically smaller streamwise and spanwise length scales. The Rayleigh mode is supported by the inflectional wall-normal profile of the streamwise--averaged streamwise velocity. Because the thickness of the internal shear layer varies comparably slowly in the spanwise direction, the Rayleigh mode is refracted and rendered fully three--dimensional. This three--dimensional mode is singular, necessitating the introduction of a critical layer inside the shear layer within which the mode is viscously regularized. As in VWI theory, a jump in the spanwise Reynolds stress is induced across the critical layer, which ultimately drives the roll motions. This multiscale and three--region asymptotic structure is efficiently captured using a complement of matched asymptotic and WKBJ analysis. The resulting reduced equations require the numerical solution of both ordinary differential eigenvalue and partial differential boundary-value problems, for which pseudospectral and spectral collocation methods are employed. Crucially, in contrast to Waleffe\u27s SSP and to VWI theory, the rolls are sufficiently strong to differentially homogenize the background shear flow, thereby providing a plausible mechanistic explanation for the formation and maintenance of both UMZs and interlaced internal shear layers

Dynamics of Holographic Entanglement Entropy Following a Local Quench

We discuss the behaviour of holographic entanglement entropy following a local quench in 2+1 dimensional strongly coupled CFTs. The entanglement generated by the quench propagates along an emergent light-cone, reminiscent of the Lieb-Robinson light-cone propagation of correlations in non-relativistic systems. We find the speed of propagation is bounded from below by the entanglement tsunami velocity obtained earlier for global quenches in holographic systems, and from above by the speed of light. The former is realized for sufficiently broad quenches, while the latter pertains for well localized quenches. The non-universal behavior in the intermediate regime appears to stem from finite-size effects. We also note that the entanglement entropy of subsystems reverts to the equilibrium value exponentially fast, in contrast to a much slower equilibration seen in certain spin models.Comment: 27 pages, 12 figures. v2: added refs and fixed typos. v3: added clarifications, published versio

Vortex Instability and Transient Growth

The dynamics of vortex flow is studied theoretically and numerically. Starting from a local analysis, where the perturbation in the vortex flow is Fourier decomposed in both radial and azimuthal directions, a modified Chebyshev polynomial method is used to discretize the linearized governing operator. The spectrum of the operator is divided into three groups: discrete spectrum, free-stream spectrum and potential spectrum. The first can be unstable while the latter two are always stable but highly non-normal. The non-normality of the spectra is quantitatively investigated by calculating the transient growth via singular value decomposition of the operator. It is observed that there is significant transient energy growth induced by the non-normality of continuous spectra. The non-normality study is then extended to a global analysis, in which the perturbation is decomposed in the radial or azimuthal direction. The governing equations are discretized through a spectral/hp element method and the maximum energy growth is calculated via an Arnoldi method. In the azimuthally-decomposed case, the development of the optimal perturbation drives the vortex to vibrate while in the stream-wise-decomposed case, the transient effects induce a string of bubbles along the axis of the vortex. A further transient growth study is conducted in the context of a co-rotating vortex pair. It is noted that the development of optimal perturbations accelerates the vortex merging process. Finally, the transient growth study is extended to a sensitivity analysis of the vortex flow to inflow perturbations. An augmented Lagrangian function is built to optimize the inflow perturbations which maximize the energy inside the domain over a fixed time interval

Stability of exact force-free electrodynamic solutions and scattering from spacetime curvature

Recently, a family of exact force-free electrodynamic (FFE) solutions was given by Brennan, Gralla and Jacobson, which generalizes earlier solutions by Michel, Menon and Dermer, and other authors. These solutions have been proposed as useful models for describing the outer magnetosphere of conducting stars. As with any exact analytical solution that aspires to describe actual physical systems, it is vitally important that the solution possess the necessary stability. In this paper, we show via fully nonlinear numerical simulations that the aforementioned FFE solutions, despite being highly special in their properties, are nonetheless stable under small perturbations. Through this study, we also introduce a three-dimensional pseudospectral relativistic FFE code that achieves exponential convergence for smooth test cases, as well as two additional well-posed FFE evolution systems in the appendix that have desirable mathematical properties. Furthermore, we provide an explicit analysis that demonstrates how propagation along degenerate principal null directions of the spacetime curvature tensor simplifies scattering, thereby providing an intuitive understanding of why these exact solutions are tractable, i.e. why they are not backscattered by spacetime curvature.Comment: 33 pages, 21 figures; V2 updated to match published versio

Algebraically diverging modes upstream of a swept bluff body

Classical stability theory for swept leading-edge boundary layers predicts eigenmodes in the free stream with algebraic decay far from the leading edge. In this article, we extend the classical base flow solution by Hiemenz to a uniformly valid solution for the flow upstream of a bluff body, which includes a three-dimensional boundary layer, an inviscid stagnation-point flow and an outer parallel flow. This extended, uniformly valid base flow additionally supports modes which diverge algebraically outside the boundary layer. The theory of wave packet pseudomodes is employed to derive analytical results for the growth rates and for the eigenvalue spectra of this type of mode. The complete spectral analysis of the flow, including the algebraically diverging modes, will give a more appropriate basis for receptivity studies and will more accurately describe the interaction of perturbations in the free stream with disturbances in the boundary laye

Algebraically diverging modes upstream of a swept bluff body

International audienceClassical stability theory for swept leading-edge boundary layers predicts eigenmodes in the free stream with algebraic decay far from the leading edge. In this article, we extend the classical base flow solution by Hiemenz to a uniformly valid solution for the flow upstream of a bluff body, which includes a three-dimensional boundary layer, an inviscid stagnation-point flow and an outer parallel flow. This extended, uniformly valid base flow additionally supports modes which diverge algebraically outside the boundary layer. The theory of wave packet pseudomodes is employed to derive analytical results for the growth rates and for the eigenvalue spectra of this type of mode. The complete spectral analysis of the flow, including the algebraically diverging modes, will give a more appropriate basis for receptivity studies and will more accurately describe the interaction of perturbations in the free stream with disturbances in the boundary layer. © Cambridge University Press 2011

Numerical Analysis of Flux Reconstruction

High-order methods have become of increasing interest in recent years in computational physics. This is in part due to their perceived ability to, in some cases, reduce the computational overhead of complex problems through both an efficient use of computational resources and a reduction in the required degrees of freedom. One such high-order method in particular – Flux Reconstruction – is the focus of this thesis. This body of work relies and expands on the theoretical methods that are used to understand the behaviour of numerical methods – particularly related to their real-world application to industrial problems. The thesis begins by challenging some of the existing dogma surrounding computational fluid dynamics by evaluating the performance of high-order flux reconstruction. First, the use of the primitive variables as an intermediary step in the construction of flux terms is investigated. It is found that reducing the order of the flux function by using the conserved rather than primitive variables has a substantial impact on the resolution of the method. Critically, this is supported by a theoretical analysis, which shows that this mechanism of error generation becomes increasing important to consider as the order of accuracy increases. Next, the analysis of Flux Reconstruction was extended by analytically and numerically exploring the impact of higher dimensionality and grid deformation. It is found that both expanding and contracting grids – essential components of real-world domain decomposition – can cause dispersion overshoot in two dimensions, but that FR appears to suffer less that comparable Finite Difference approaches. Fully discrete analysis is then used to show that, depending on the correction function, small perturbations in incidence angle can cause large changes in group velocity. The same analysis is also used to theoretically demonstrate that Discontinuous Galerkin suffers less from dispersion error than Huynh’s FR scheme – a phenomenon that has previously been observed experimentally, but not explained theoretically. This thesis concludes with the presentation of a robust theoretical underpinning for determining stable correction functions for FR. Three new families of correction functions are presented, and their properties extensively explored. An important theoretical finding is introduced – that stable correction functions are not defined uniquely be a norm. As a result, a generalised approach is presented, which is able to recover all previously defined correction functions, but in some instances via a different norm to their original derivation. This new super-family of correction functions shows considerable promise in increasing temporal stability limits, reducing dispersion when fully discretised, and increasing the rate of convergence. Taken altogether, this thesis represents a considerable advance in the theoretical characterisation and understanding of a numerical method – one which, it has been shown, has enormous potential for forming the heart of future computational physics codes

Mathematical modelling of dune formation

This study is concerned with the mathematical modelling of the formation and subsequent evolution of sand dunes, both beneath rivers (fluvial) and in deserts (Aeolian). Dunes are observed in the environment in many different shapes and sizes; we begin by discussing qualitatively how and why the different forms exist. The most important aspect of a successful model is the relationship between the bed shape and the shear stress that the flow exerts on the bed. We first discuss a simple model for this stress applied to fluvial dunes, which is able to predict dune-like structures, but does not predict the instability of a flat bed which we would hope to find. We therefore go on to look at improved models for the shear stress based on theories of turbulent flow and asymptotic methods, using assumptions of either a constant eddy viscosity or a mixing length model for turbulence. Using these forms for the shear stress, along with sediment transport laws, we obtain partial integrodifferential equations for the evolution of the bed, and we study these numerically using spectral methods. One important feature of dunes which is not taken into account by the above models is that of the slip face - a region of constant slope on the downwind side of the dune. When a slip face is present, there is a discontinuity in the slope of the bed, and hence it is clear that flow separation will occur. Previous studies have modelled separated flow by heuristically describing the boundary of the separated region with a cubic or quintic polynomial which joins smoothly to the bed at each end. We recreate this polynomial form for the wake profile and demonstrate a method for including it into an evolution system for dunes. The resulting solutions show an isolated steady-state dune which propagates downstream. From the asymptotic framework developed earlier with a mixing length model for turbulence, we are able, using techniques of complex analysis, to model the shape of the wake region from a purely theoretical basis, rather than the heuristic one used previously. We obtain a Riemann-Hilbert problem for the wake profile, which can be solved using well-known techniques. We then use this method to calculate numerically the wake profile corresponding to a number of dune profiles. Further, we are able to find an exact solution to the wake profile problem in the case of a sinusoidally shaped dune with a slip face. Having found a method to calculate the shear stress exerted on the dune from the bed profile in the case of separated flow, we then use this improved estimate of the shear stress in an evolution system as before. In order to do this efficiently, we consider an alternative method for calculating the wake profile based on the spectral method used for solving the evolution system. We find that this system permits solutions describing an isolated dune with a slip face which propagates downstream without changing shape. All of the models described above are implemented in two spatial dimensions; the wind is assumed to blow in one direction only, and the dunes are assumed to be uniform in a direction perpendicular to the wind flow. While this is adequate to explain the behaviour of transverse dunes, other dune shapes such as linear dunes, barchans, and star dunes are by nature three-dimensional, so in order to study the behaviour of such dunes, the extension of the models to three dimensions is essential. While most of the governing equations generalize easily, it is less obvious how to extend the model for separated flow, due to its reliance on complex variables. We implement some three-dimensional evolution models, and discuss the possibility of modelling three-dimensional flow separation