Linear to turbulent Görtler instability transition

High-resolution simulations are performed for gravity currents propagating on a no-slip boundary to study the merging and splitting processes in the lobe-and-cleft structure at a gravity current head. The simulations reproduce the morphological features observed in the laboratory and provide more detailed flow information to elucidate the merging and splitting processes. Our mean lobe width $\tilde {b}$ and mean maximum lobe width $\tilde {b}_{max}$ satisfy the empirical relationships $\tilde {b}/\tilde {d}=7.4 Re^{-0.39}_f$ and $\tilde {b}_{max}/\tilde {d}=12.6 Re^{-0.38}_f$ , respectively, over the front Reynolds number in the range $383 \le Re_f \le 3267$ , where the front Reynolds number is defined as $Re_f= \tilde {u}_f \tilde {d} / \tilde {\nu }, \tilde {u}_f$ is the front velocity, $\tilde {d}$ is the height of the gravity current head and $\tilde {\nu }$ is the fluid kinematic viscosity. When measured in terms of the viscous length scale $\tilde {\delta }_\nu = \tilde {\nu }/\tilde {u}^*$ , where $\tilde {u}^*$ is the shear velocity at the gravity current head, the mean lobe width and the mean maximum lobe width increase with increasing front Reynolds number and asymptotically approach $126 \tilde {\delta }_\nu$ and $230 \tilde {\delta }_\nu$ at $Re_f=3267$ , respectively. The vortical structure inside a lobe has an elongated tooth-like shape and a pair of counter-rotating streamwise vortices are positioned on the left- and right-hand sides of each cleft. For the merging process, it requires the interaction of three tooth-like vortices and the middle tooth-like vortex breaks up and reconnects with the two neighbouring tooth-like vortices. Therefore, a cleft may continually merge with another neighbouring cleft but may never disappear. For the splitting process, even before the new cleft appears, a new born streamwise vortex is created by the parent vortex of opposite orientation and the parent vortex can be either the left part or the right part of the existing tooth-like vortex inside the splitting lobe. The new born streamwise vortex then induces the other counter-rotating streamwise vortex as the new cleft develops. The initiation of the splitting process can be attributed to the Brooke–Hanratty mechanism reinforced by the baroclinic production of vorticity. Depending on the orientation of the parent vortex, the resulting new cleft after the splitting process can shift laterally in the positive or negative spanwise direction along the leading edge of the gravity currents as the lobe-and-cleft structure moves forward in the streamwise direction. For gravity currents propagating on a no-slip boundary, the lobe-and-cleft structure is self-sustaining and the manifestations of the merging and splitting processes are in accord with reported laboratory observations.

Section: Resultsmentioning

confidence: 99%

On the merging and splitting processes in the lobe-and-cleft structure at a gravity current head

Dai

Huang

2021

“…They are pairs of counter-rotating and streamwise-aligned vortices, which originate because of an imbalance between the centrifugal force and the radial pressure gradient. For details on the TGVs and the related instability, we refer the interested reader to Floryan (1991), Saric (1994), Luchini & Bottaro (1998) and to the more recent works of Xu, Liu & Wu (2020) and Dagaut et al (2021). Whether TGVs are present in turbulent flows with localised curvature is still unclear.…”

Section: Introductionmentioning

confidence: 99%

“…For details on the TGVs and the related instability, we refer the interested reader to Floryan (1991), Saric (1994), Luchini & Bottaro (1998) and to the more recent works of Xu, Liu & Wu (2020) and Dagaut et al. (2021). Whether TGVs are present in turbulent flows with localised curvature is still unclear.…”

Section: Introductionmentioning

confidence: 99%

Curvature effects on the structure of near-wall turbulence

Selvatici,

Quadrio,

Chiarini

2023

The interaction between near-wall turbulence and wall curvature is described for the incompressible flow in a plane channel with a small concave–convex–concave bump on the bottom wall, with height comparable to the wall-normal location of the main turbulent structures. The analysis starts from a database generated by a direct numerical simulation and hinges upon the anisotropic generalised Kolmogorov equations, i.e. the exact budget equations for the second-order structure function tensor. The influence of the bump on the wall cycle and on the energy production, redistribution and transfers is described in the physical and scale spaces. Over the upstream side of the bump, the energy drained from the mean flow to sustain the streamwise fluctuations decreases, and the streaks of high and low streamwise velocity weaken and are stretched spanwise. After the bump tip, instead, the production of streamwise fluctuations grows and the streaks intensify, progressively recovering their characteristic spanwise scale. The wall-normal fluctuations, and thus the quasi-streamwise vortices, are sustained by the mean flow over the upstream side of the bump, while energy flows from the vertical fluctuations to the mean field over the downstream side. On the concave portion of the upstream side, the near-wall fluctuations form structures of spanwise velocity which are consistent with Taylor–Görtler vortices at an early stage of development. Their evolution is described by analysing the scale-space pressure–strain term. A schematic description of the bump flow is presented, in which various regions are identified according to the signs of curvature and streamwise pressure gradient.

Low-dimensional analysis and modelling of the flow over a forward-facing step

Podvin,

Fraigneau

2024

We consider the direct numerical simulation of the flow over a forward-facing step protruding in a turbulent boundary layer. Proper orthogonal decomposition (POD) is applied to the velocity field in different regions using Fourier modes in the spanwise direction. The upstream flow is characterized by a structure with a spanwise modulation of the order of the step height, the origin of which is consistent with a centrifugal instability. The structure is associated with ejections over the step of low-speed fluid from the upstream recirculation, and organized into streaks through the action of strong spanwise motions along the step face. The spanwise-averaged instantaneous momentum deficit created by the ejections is directly related to the maximal shear rate at the edge of the step, and is well correlated with the dynamics of the downstream recirculation. The most energetic patterns consist of three-dimensional motions with a large spanwise wavelength located in the shear layer developing at the edge of the step, as well as two-dimensional fluctuations downstream of the reattachment. A linear model based on the interaction of the mean flow with the dominant POD modes is able to recover the main frequencies of the fluctuations at these wavenumbers. Including the time variations of the ejections into the model yields temporal spectra that resemble qualitatively those computed from the simulation. The results suggest that the global dynamics of the flow are at least partly driven by linear mechanisms and depend on the characteristic structure identified in the upstream region close to the step.