4 results
Cause-and-effect of linear mechanisms sustaining wall turbulence
- Adrián Lozano-Durán, Navid C. Constantinou, Marios-Andreas Nikolaidis, Michael Karp
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- Journal:
- Journal of Fluid Mechanics / Volume 914 / 10 May 2021
- Published online by Cambridge University Press:
- 05 March 2021, A8
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Despite the nonlinear nature of turbulence, there is evidence that part of the energy-transfer mechanisms sustaining wall turbulence can be ascribed to linear processes. The different scenarios stem from linear stability theory and comprise exponential instabilities, neutral modes, transient growth from non-normal operators and parametric instabilities from temporal mean flow variations, among others. These mechanisms, each potentially capable of leading to the observed turbulence structure, are rooted in simplified physical models. Whether the flow follows any or a combination of them remains elusive. Here, we evaluate the linear mechanisms responsible for the energy transfer from the streamwise-averaged mean flow ($\boldsymbol {U}$) to the fluctuating velocities ($\boldsymbol {u}'$). To that end, we use cause-and-effect analysis based on interventions: manipulation of the causing variable leads to changes in the effect. This is achieved by direct numerical simulation of turbulent channel flows at low Reynolds number, in which the energy transfer from $\boldsymbol {U}$ to $\boldsymbol {u}'$ is constrained to preclude a targeted linear mechanism. We show that transient growth is sufficient for sustaining realistic wall turbulence. Self-sustaining turbulence persists when exponential instabilities, neutral modes and parametric instabilities of the mean flow are suppressed. We further show that a key component of transient growth is the Orr/push-over mechanism induced by spanwise variations of the base flow. Finally, we demonstrate that an ensemble of simulations with various frozen-in-time $\boldsymbol {U}$ arranged so that only transient growth is active, can faithfully represent the energy transfer from $\boldsymbol {U}$ to $\boldsymbol {u}'$ as in realistic turbulence. Our approach provides direct cause-and-effect evaluation of the linear energy-injection mechanisms from $\boldsymbol {U}$ to $\boldsymbol {u}'$ in the fully nonlinear system and simplifies the conceptual model of self-sustaining wall turbulence.
The Nusselt numbers of horizontal convection
- Cesar B. Rocha, Navid C. Constantinou, Stefan G. Llewellyn Smith, William R. Young
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- Journal:
- Journal of Fluid Mechanics / Volume 894 / 10 July 2020
- Published online by Cambridge University Press:
- 07 May 2020, A24
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In the problem of horizontal convection a non-uniform buoyancy, $b_{s}(x,y)$, is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, $\boldsymbol{J}$, defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that $\overline{\boldsymbol{J}\boldsymbol{\cdot }\unicode[STIX]{x1D735}b_{s}}=-\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$; the overbar denotes a space–time average over the top surface, angle brackets denote a volume–time average and $\unicode[STIX]{x1D705}$ is the molecular diffusivity of buoyancy $b$. This connection between $\boldsymbol{J}$ and $\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$ justifies the definition of the horizontal-convective Nusselt number, $Nu$, as the ratio of $\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of $Nu$ over other definitions of horizontal-convective Nusselt number. We investigate transient effects and show that $\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$ equilibrates more rapidly than other global averages, such as the averaged kinetic energy and bottom buoyancy. We show that $\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$ is the volume-averaged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux through the top surface. This leads to an equivalent ‘surface Nusselt number’, defined as the surface average of vertical buoyancy flux through the top surface times the imposed surface buoyancy $b_{s}(x,y)$. In experimental situations it is easier to evaluate the surface entropy flux, rather than the volume integral of $|\unicode[STIX]{x1D735}b|^{2}$ demanded by $\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$.
Beta-plane turbulence above monoscale topography
- Navid C. Constantinou, William R. Young
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- Journal:
- Journal of Fluid Mechanics / Volume 827 / 25 September 2017
- Published online by Cambridge University Press:
- 24 August 2017, pp. 415-447
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Using a one-layer quasi-geostrophic model, we study the effect of random monoscale topography on forced beta-plane turbulence. The forcing is a uniform steady wind stress that produces both a uniform large-scale zonal flow $U(t)$ and smaller-scale macroturbulence characterized by standing and transient eddies. The large-scale flow $U$ is retarded by a combination of Ekman drag and the domain-averaged topographic form stress produced by the eddies. The topographic form stress typically balances most of the applied wind stress, while the Ekman drag provides all of the energy dissipation required to balance the wind work. A collection of statistically equilibrated numerical solutions delineate the main flow regimes and the dependence of the time average of $U$ on parameters such as the planetary potential vorticity (PV) gradient $\unicode[STIX]{x1D6FD}$ and the statistical properties of the topography. We obtain asymptotic scaling laws for the strength of the large-scale flow $U$ in the limiting cases of weak and strong forcing. If $\unicode[STIX]{x1D6FD}$ is significantly smaller than the topographic PV gradient, the flow consists of stagnant pools attached to pockets of closed geostrophic contours. The stagnant dead zones are bordered by jets and the flow through the domain is concentrated into a narrow channel of open geostrophic contours. In most of the domain, the flow is weak and thus the large-scale flow $U$ is an unoccupied mean. If $\unicode[STIX]{x1D6FD}$ is comparable to, or larger than, the topographic PV gradient, then all geostrophic contours are open and the flow is uniformly distributed throughout the domain. In this open-contour case, there is an ‘eddy saturation’ regime in which $U$ is insensitive to large changes in the wind stress. We show that eddy saturation requires strong transient eddies that act effectively as PV diffusion. This PV diffusion does not alter the kinetic energy of the standing eddies, but it does increase the topographic form stress by enhancing the correlation between the topographic slope and the standing-eddy pressure field. Using bounds based on the energy and enstrophy power integrals, we show that as the strength of the wind stress increases, the flow transitions from a regime in which the form stress balances most of the wind stress to a regime in which the form stress is very small and large transport ensues.
A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow
- Brian F. Farrell, Petros J. Ioannou, Javier Jiménez, Navid C. Constantinou, Adrián Lozano-Durán, Marios-Andreas Nikolaidis
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- Journal:
- Journal of Fluid Mechanics / Volume 809 / 25 December 2016
- Published online by Cambridge University Press:
- 09 November 2016, pp. 290-315
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The perspective of statistical state dynamics (SSD) has recently been applied to the study of mechanisms underlying turbulence in a variety of physical systems. An SSD is a dynamical system that evolves a representation of the statistical state of the system. An example of an SSD is the second-order cumulant closure referred to as stochastic structural stability theory (S3T), which has provided insight into the dynamics of wall turbulence, and specifically the emergence and maintenance of the roll/streak structure. S3T comprises a coupled set of equations for the streamwise mean and perturbation covariance, in which nonlinear interactions among the perturbations has been removed, restricting nonlinearity in the dynamics to that of the mean equation and the interaction between the mean and perturbation covariance. In this work, this quasi-linear restriction of the dynamics is used to study the structure and dynamics of turbulence in plane Poiseuille flow at moderately high Reynolds numbers in a closely related dynamical system, referred to as the restricted nonlinear (RNL) system. Simulations using this RNL system reveal that the essential features of wall-turbulence dynamics are retained. Consistent with previous analyses based on the S3T version of SSD, the RNL system spontaneously limits the support of its turbulence to a small set of streamwise Fourier components, giving rise to a naturally minimal representation of its turbulence dynamics. Although greatly simplified, this RNL turbulence exhibits natural-looking structures and statistics, albeit with quantitative differences from those in direct numerical simulations (DNS) of the full equations. Surprisingly, even when further truncation of the perturbation support to a single streamwise component is imposed, the RNL system continues to self-sustain turbulence with qualitatively realistic structure and dynamic properties. RNL turbulence at the Reynolds numbers studied is dominated by the roll/streak structure in the buffer layer and similar very large-scale structure (VLSM) in the outer layer. In this work, diagnostics of the structure, spectrum and energetics of RNL and DNS turbulence are used to demonstrate that the roll/streak dynamics supporting the turbulence in the buffer and logarithmic layer is essentially similar in RNL and DNS.