arXiv subjects

Gregory P. Chini

Publications and source records attributed to Gregory P. Chini.

14 recordsLinked to original sources

arXiv2026

Non-normal weakly nonlinear analysis: asymptotic consistency and non-universality

Non-normality can induce large transient growth in linearly stable systems. Determining whether this growth triggers a transition in the underlying nonlinear system, however, requires understanding the interaction between non-normality and nonlinearity. Here, we develop a weakly nonlinear theory for linearly-stable, non-normal systems subject to harmonic forcing, enabling a systematic analysis of this interaction. Following Ducimeti\`ere et al. (J. Fluid Mech., vol. 947, 2022, A43), we define a formal small parameter $\varepsilon$ as the reciprocal of the system's maximum linear amplification. However, we ensure asymptotic consistency by providing a framework that naturally adapts to the underlying structure of the system. The approach is applied to a harmonically forced channel flow and to a two-dimensional model mimicking the structure of the Orr-Sommerfeld-Squire equations. Unlike classical weakly nonlinear analysis near bifurcation points, the resulting amplitude equations are non-universal. In fact, a single linear mode amplified by the non-normality can nonlinearly excite a multi-modal and multi-frequency response at leading-order, which is system- or even regime-specific. Nevertheless, the method yields asymptotically consistent amplitude equations that capture this complexity provided a limit in which $\varepsilon\rightarrow0$ can be identified. As the forcing amplitude increases, the reduced equations capture stable nonlinear states emerging from the laminar flow, their subsequent bifurcations, and their eventual collision with the boundary of their basin of attraction. Thus, the amplitude equations can capture subcritical transitions driven by forcing and varied initial conditions and enable the identification of critical parameters beyond which no stable weakly nonlinear state exists.

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arXiv2025

Spatio-temporal characterization of nonlinear forcing and response in turbulent channel flow

The quadratic convection term in the incompressible Navier-Stokes equations is considered as a nonlinear forcing to the linear resolvent operator, and it is studied in the Fourier domain through the analysis of interactions between triadically compatible wavenumber-frequency triplets. A framework to quantify the triadic contributions to the forcing and response by each pair of triplets is developed and applied to data from direct numerical simulations of a turbulent channel at $Re_{\tau} \approx 550$. The linear resolvent operator is incorporated to provide the missing link from energy transfer between modes to the effect on the spectral turbulent kinetic energy. The coefficients highlight the importance of interactions involving large-scale structures, providing a natural connection to the modeling assumptions in quasi-linear (QL) and generalized quasi-linear (GQL) analyses. Specifically, it is revealed that the QL and GQL reductions efficiently capture important triadic interactions in the flow, especially when including of a small number of wavenumbers into the GQL large-scale base flow. Additionally, spatio-temporal analyses of the triadic contributions to a single mode representative of the near-wall cycle demonstrate the spatio-temporal nature of the triadic interactions and the effect of the resolvent operator, which selectively amplifies certain forcing profiles. The tools presented are expected to be useful for improving modeling of the nonlinearity, especially in QL, GQL, and resolvent analyses, and understanding the amplitude modulation mechanism relating large-scale fluctuations to the modulation of near-wall structures.

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arXiv2024

Aspect-ratio-dependent heat transport by baroclinic acoustic streaming

Standing acoustic waves have been known to generate Eulerian time-mean 'streaming' flows at least since the seminal investigation of Lord Rayleigh in the 1880s. Nevertheless, a recent body of numerical and experimental evidence has shown that inhomogeneities in the ambient density distribution lead to much faster flows than arise in classical Rayleigh streaming. The emergence of these unusually strong flows creates new opportunities to enhance heat transfer in systems in which convective cooling cannot otherwise be easily achieved. To assess this possibility, a theoretical study of acoustic streaming in an ideal gas confined in a rectangular channel with top and bottom walls maintained at fixed but differing temperatures is performed. A two time scale system of equations is utilized to efficiently capture the coupling between the fast acoustic waves and the slowly evolving streaming flow, enabling strongly nonlinear regimes to be accessed. A large suite of numerical simulations is carried out to probe the streaming dynamics, to highlight the critical role played by baroclinically generated wave vorticity and to quantify the additional heat flux induced by the standing acoustic wave. Proper treatment of the two-way coupling between the waves and mean flow is found to be essential for convergence to a self-consistent steady state, and the variation of the resulting acoustically enhanced steady-state heat flux with both the amplitude of the acoustic wave and the O(1) aspect ratio of the channel is documented. For certain parameters, heat fluxes almost two orders of magnitude larger than those realizable by conduction alone can be attained.

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arXiv2024

Regimes of stratified turbulence at low Prandtl number

Quantifying transport by strongly stratified turbulence in low Prandtl number ($Pr$) fluids is critically important for the development of better models for the structure and evolution of stellar and planetary interiors. Motivated by recent numerical simulations showing strongly anisotropic flows suggestive of scale-separated dynamics, we perform a multiscale asymptotic analysis of the governing equations. We find that, in all cases, the resulting slow-fast systems take a quasilinear form. Our analysis also reveals the existence of several distinct dynamical regimes depending on the emergent buoyancy Reynolds and Péclet numbers, $Re_b = α^2 Re$ and $Pe_b = Pr Re_b$, respectively, where $α$ is the aspect ratio of the large-scale turbulent flow structures, and $Re$ is the outer scale Reynolds number. Scaling relationships relating the aspect ratio, the characteristic vertical velocity, and the strength of the stratification (measured by the Froude number $Fr$) naturally emerge from the analysis. When $Pe_b \ll α$, the dynamics at all scales is dominated by buoyancy diffusion, and our results recover the scaling laws empirically obtained from direct numerical simulations by Cope et al. (2020). For $Pe_b \ge O(1)$, diffusion is negligible (or at least subdominant) at all scales and our results are consistent with those of Chini et al. (2022) for strongly stratified geophysical turbulence at $Pr = O(1)$. Finally, we have identified a new regime for $α\ll Pe_b \ll 1$, in which slow, large scales are diffusive while fast, small scales are not. We conclude by presenting a map of parameter space that clearly indicates the transitions between isotropic turbulence, non-diffusive stratified turbulence, diffusive stratified turbulence and viscously-dominated flows, and by proposing parameterisations of the buoyancy flux, mixing efficiency and turbulent diffusion coefficient.

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arXiv2024

Following marginal stability manifolds in quasilinear dynamical reductions of multiscale flows in two space dimensions

A two-dimensional extension of a recently developed formalism for slow-fast quasilinear (QL) systems subject to fast instabilities is derived. Prior work has demonstrated that the emergent dynamics of these systems is characterized by a slow evolution of mean fields coupled to marginally stable, fast fluctuation fields. By exploiting this emergent behavior, an efficient fast-eigenvalue/slow-initial-value solution algorithm can be developed in which the amplitude of the fast fluctuations is slaved to the slowly evolving mean fields to ensure marginal stability (and temporal scale separation) is maintained. For 2D systems that are spatially-extended in one direction, the fluctuation eigenfunctions are labeled by their wavenumbers characterizing spatial variability in that direction, and the marginal mode(s) also must coincide with the fastest-growing mode(s) over all admissible wavenumbers. Here, we introduce two equivalent procedures for deriving an ordinary differential equation governing the slow evolution of the wavenumber of the fastest-growing fluctuation mode that simultaneously must be slaved to the mean dynamics to ensure the mode has zero growth rate. We illustrate the procedure in the context of a 2D model partial differential equation that shares certain attributes with the equations governing strongly stratified shear flows. The slaved evolution follows one or more marginal stability manifolds, which constitute select state-space structures that are not invariant under the full flow dynamics yet capture quasi-coherent states in physical space in a manner analogous to invariant solutions identified in, e.g., transitionally-turbulent shear flows. Accordingly, we propose that marginal stability manifolds are central organizing structures in a dynamical systems description of certain classes of multiscale flows where scale separation justifies a QL approximation of the dynamics.

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arXiv2023

Spatio-temporal characterization of non-linear forcing in turbulence

The quadratic convection term in the incompressible Navier-Stokes equations is considered as a non-linear forcing to the linear operator, and it is studied in the Fourier domain through the analysis of interactions between triadically compatible wavenumber-frequency triplets. Interaction coefficients are proposed to quantify the contribution to the forcing by each pair of triplets and are computed using data from direct numerical simulations of a turbulent channel at $Re_τ \approx 550$. The coefficients show the importance of interactions involving streamwise large scales. The regions of non-linear interactions permitted under the quasi-linear (QL) and generalized quasi-linear (GQL) assumptions are shown to be significant contributors to the forcing and increasing the number of GQL-large scales is shown to monotonically increase the total forcing captured, providing a possible reason for the success of QL and GQL simulations. The tools presented are expected to be useful for improving modeling of the nonlinearity, especially in QL, GQL, and resolvent analyses, and understanding the amplitude modulation mechanism relating large-scale fluctuations to the modulation of near-wall structures.

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arXiv2022

Steady Rayleigh--B\'enard convection: strongly nonlinear high-wavenumber rolls

In Rayleigh--B\'{e}nard convection, two-dimensional steady rolls bifurcate supercritically at a Rayleigh number $Ra$ that depends on their horizontal-to-vertical aspect ratio $\Gamma$, and they exist at all larger $Ra$ despite being unstable. Heat transport by certain rolls---quantified by the Nusselt number $Nu$---closely resembles turbulent transport, yet $Nu$ scalings of rolls are understood only for specific boundary conditions and $\Gamma$--$Ra$ limits. Here we investigate the high-wavenumber limit $\Gamma = O(Ra^{-1/4})$ as $Ra \to \infty$, using numerics and matched asymptotic analysis. We compute steady rolls between stress-free boundaries for Prandtl numbers $10^{-1} \leq Pr \leq 10^{3/2}$ and $Ra$ reaching $10^{19}$. While the $\Gamma = O(Ra^{-1/4})$ limit gives smaller $Nu$ than when $\Gamma = O(1)$, we identify prefactors $c$ in $\Gamma = c\,Ra^{-1/4}$ that locally maximize $Nu$. These locally $Nu$-maximizing rolls display approximate scalings $Nu \propto Ra^{0.29}$ and $Re \propto Ra^{0.40}$, with the Reynolds number $Re$ defined using root-mean-square velocity. Our asymptotic analysis reveals a vertically stacked four-layer structure near each boundary, predicting $Nu = O(Ra^{3/10})$ and $Re = O(Ra^{2/5})$. This asymptotic construction largely follows that of Taylor vortices by \cite{Deguchi2023}, but we identify a thin plume region within the middle boundary layer whose inclusion eliminates the logarithmic factors in Deguchi's predictions. Asymptotic arguments and numerics suggest the same scalings for stress-free or no-slip boundaries, unlike in other $\Gamma$--$Ra$ limits. Our asymptotics extend the weakly nonlinear analysis of Blennerhassett \& Bassom (1994) into the strongly nonlinear regime and complement the asymptotics of Chini \& Cox (2009) for $\Gamma = O(1)$ rolls.

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arXiv2021

Exploiting self-organized criticality in strongly stratified turbulence

A multiscale reduced description of turbulent free shear flows in the presence of strong stabilizing density stratification is derived via asymptotic analysis of the Boussinesq equations in the simultaneous limits of small Froude and large Reynolds numbers. The analysis explicitly recognizes the occurrence of dynamics on disparate spatiotemporal scales, yielding simplified partial differential equations governing the coupled evolution of slow large-scale hydrostatic flows and fast small-scale isotropic instabilities and internal waves. The dynamics captured by the coupled reduced equations is illustrated in the context of two-dimensional strongly stratified Kolmogorov flow. A noteworthy feature of the reduced model is that the fluctuations are constrained to satisfy quasilinear (QL) dynamics about the comparably slowly-varying large-scale fields. Crucially, this QL reduction is not invoked as an ad hoc closure approximation, but rather is derived in a physically relevant and mathematically consistent distinguished limit. Further analysis of the resulting slow-fast QL system shows how the amplitude of the fast stratified-shear instabilities is slaved to the slowly-evolving mean fields to ensure the marginal stability of the latter. Physically, this marginal stability condition appears to be compatible with recent evidence of self-organized criticality in both observations and simulations of stratified turbulence. lgorithmically, the slaving of the fluctuation fields enables numerical simulations to be time-evolved strictly on the slow time scale of the hydrostatic flow. The reduced equations thus provide a solid mathematical foundation for future studies of three-dimensional strongly stratified turbulence in extreme parameter regimes of geophysical relevance and suggest avenues for new sub-grid-scale parameterizations.

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arXiv2020

Steady Rayleigh--Bénard convection between stress-free boundaries

Steady two-dimensional Rayleigh--Bénard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios $π/5\leΓ\le4π$, where $Γ$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\le Ra\le10^{11}$, and four orders of magnitude in the Prandtl number, $10^{-2}\le Pr\le10^2$. At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra \rightarrow \infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly in $Pr$. The prefactor of this scaling depends on $Γ$ and is largest at $Γ\approx 1.9$. The Reynolds number $Re$ for large-$Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $Γ\approx 4.5$. All of these large-$Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini \& Cox (2009). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $Γ$ is smaller.

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arXiv2020

A self-sustaining process theory for uniform momentum zones and internal shear layers in high Reynolds number shear flows

Many exact coherent states (ECS) arising in wall-bounded shear flows have an asymptotic structure at extreme Reynolds number Re in which the effective Reynolds number governing the streak and roll dynamics is O(1). Consequently, these viscous ECS are not suitable candidates for quasi-coherent structures away from the wall that necessarily are inviscid in the mean. Specifically, viscous ECS cannot account for the singular nature of the inertial domain, where the flow self-organizes into uniform momentum zones (UMZs) separated by internal shear layers and the instantaneous streamwise velocity develops a staircase-like profile. In this investigation, a large-Re asymptotic analysis is performed to explore the potential for a three-dimensional, short streamwise- and spanwise-wavelength instability of the embedded shear layers to sustain a spatially-distributed array of much larger-scale, effectively inviscid streamwise roll motions. In contrast to other self-sustaining process theories, the rolls are sufficiently strong to differentially homogenize the background shear flow, thereby providing a mechanistic explanation for the formation and maintenance of UMZs and interlaced shear layers that respects the leading-order balance structure of the mean dynamics.

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arXiv2018

Strong wave-mean-flow coupling in baroclinic acoustic streaming

The interaction of an acoustic wave with a stratified fluid can drive strong streaming flows owing to the baroclinic production of fluctuating vorticity, as recently demonstrated by Chini et al. (J. Fluid Mech., 744, 2014, pp. 329). In the present investigation, a set of wave/mean-flow interaction equations is derived that governs the coupled dynamics of a standing acoustic wave mode of characteristic (small) amplitude ε and the streaming flow it drives in a thin channel with walls maintained at differing temperatures. Unlike classical Rayleigh streaming, the resulting mean flow arises at O(ε) rather than at O({ε^2}). Consequently, fully two-way coupling between the waves and the mean flow is possible: the streaming is sufficiently strong to induce O(1) rearrangements of the imposed background temperature and density fields, which modifies the spatial structure and frequency of the acoustic mode on the streaming time scale. A novel Wentzel-Kramers-Brillouin-Jeffreys analysis is developed to average over the fast wave dynamics, enabling the coupled system to be integrated strictly on the slow time scale of the streaming flow. Analytical solutions of the reduced system are derived for weak wave forcing and are shown to reproduce results from prior direct numerical simulations (DNS) of the compressible Navier Stokes and heat equations with remarkable accuracy. Moreover, numerical simulations of the reduced system are performed in the regime of strong wave mean flow coupling for a fraction of the computational cost of the corresponding DNS. These simulations shed light on the potential for baroclinic acoustic streaming to be used as an effective means to enhance heat transfer.

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arXiv2018

Reduced modeling of porous media convection in a minimal flow unit at large Rayleigh number

Direct numerical simulations (DNS) indicate that at large values of the Rayleigh number ($Ra$) convection in porous media self-organizes into narrowly-spaced columnar flows, with more complex spatiotemporal features being confined to boundary layers near the top and bottom walls. In this investigation of high-$Ra$ porous media convection in a minimal flow unit, two reduced modeling strategies are proposed that exploit these specific flow characteristics. Both approaches utilize the idea of decomposition since the flow exhibits different dynamics in different regions of the domain: small-scale cellular motions generally are localized within the thermal and vorticity boundary layers near the upper and lower walls, while in the interior, the flow exhibits persistent large-scale structures and only a few low (horizontal) wavenumber Fourier modes are active. Accordingly, in the first strategy, the domain is decomposed into two near-wall regions and one interior region. Our results confirm that suppressing the interior high-wavenumber modes has negligible impact on the essential structural features and transport properties of the flow. In the second strategy, a hybrid reduced model is constructed by using Galerkin projection onto a fully \emph{a priori} eigenbasis drawn from energy stability and upper bound theory, thereby extending the model reduction strategy developed by Chini \emph{et al.} (\emph{Physica~D}, vol. 240, 2011, pp. 241--248) to large $Ra$. The results indicate that the near-wall upper-bound eigenmodes can economically represent the small-scale rolls within the exquisitely-thin thermal boundary layers. Relative to DNS, the hybrid algorithm enables over an order-of-magnitude increase in computational efficiency with only a modest loss of accuracy.

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arXiv2014

Reduced description of exact coherent states in parallel shear flows

Exact coherent states of a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force (a flow first introduced by F. Waleffe, Phys. Fluids 9, 883 (1997)) are computed using equations obtained from a large Reynolds-number asymptotic reduction of the Navier-Stokes equations. The reduced equations employ a decomposition into streamwise-averaged (mean) and streamwise-varying (fluctuation) components and are characterized by an effective order one Reynolds number in the mean equations along with a formally higher-order diffusive regularization of the fluctuation equations. A robust numerical algorithm for computing exact coherent states is introduced. Numerical continuation of the lower branch states to lower Reynolds numbers reveals the presence of a saddle-node; the saddle-node allows access to upper branch states that, like the lower branch states, appear to be self-consistently described by the reduced equations. Both lower and upper branch states are characterized in detail.

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arXiv2014

Wall to Wall Optimal Transport

The calculus of variations is employed to find steady divergence-free velocity fields that maximize transport of a tracer between two parallel walls held at fixed concentration for one of two constraints on flow strength: a fixed value of the kinetic energy or a fixed value of the enstrophy. The optimizing flows consist of an array of (convection) cells of a particular aspect ratio Gamma. We solve the nonlinear Euler-Lagrange equations analytically for weak flows and numerically (and via matched asymptotic analysis in the fixed energy case) for strong flows. We report the results in terms of the Nusselt number Nu, a dimensionless measure of the tracer transport, as a function of the Peclet number Pe, a dimensionless measure of the energy or enstrophy of the flow. For both constraints the maximum transport Nu_{MAX}(Pe) is realized in cells of decreasing aspect ratio Gamma_{opt}(Pe) as Pe increases. For the fixed energy problem, Nu_{MAX} \sim Pe and Gamma_{opt} \sim Pe^{-1/2}, while for the fixed enstrophy scenario, Nu_{MAX} \sim Pe^{10/17} and Gamma_{opt} \sim Pe^{-0.36}. We also interpret our results in the context of certain buoyancy-driven Rayleigh-Benard convection problems that satisfy one of the two intensity constraints, enabling us to investigate how the transport scalings compare with upper bounds on Nu expressed as a function of the Rayleigh number \Ra. For steady convection in porous media, corresponding to the fixed energy problem, we find Nu_{MAX} \sim \Ra and Gamma_{opt} \sim Ra^{-1/2}$, while for steady convection in a pure fluid layer between free-slip isothermal walls, corresponding to fixed enstrophy transport, Nu_{MAX} \sim Ra^{5/12} and Gamma_{opt} \sim Ra^{-1/4}.

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