4 results
Instability of sheared density interfaces
- T. S. Eaves, N. J. Balmforth
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- Journal:
- Journal of Fluid Mechanics / Volume 860 / 10 February 2019
- Published online by Cambridge University Press:
- 03 December 2018, pp. 145-171
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Of the canonical flow instabilities (Kelvin–Helmholtz, Holmboe-wave and Taylor–Caulfield) of stratified shear flow, the Taylor–Caulfield instability (TCI) has received relatively little attention, and forms the focus of the current study. First, a diagnostic of the linear instability dynamics is developed that exploits the net pseudomomentum to distinguish TCI from the other two instabilities for any given flow profile. Second, the nonlinear dynamics of TCI is studied across its range of unstable horizontal wavenumbers and bulk Richardson numbers using numerical simulation. At small bulk Richardson numbers, a cascade of billow structures of sequentially smaller size may form. For large bulk Richardson numbers, the primary nonlinear travelling waves formed by the linear instability break down via a small-scale, Kelvin–Helmholtz-like roll-up mechanism with an associated large amount of mixing. In all cases, secondary parasitic nonlinear Holmboe waves appear at late times for high Prandtl number. Third, a nonlinear diagnostic is proposed to distinguish between the saturated states of the three canonical instabilities based on their distinctive density–streamfunction and generalised vorticity–streamfunction relations.
Multiple instability of layered stratified plane Couette flow
- T. S. Eaves, C. P. Caulfield
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- Journal:
- Journal of Fluid Mechanics / Volume 813 / 25 February 2017
- Published online by Cambridge University Press:
- 17 January 2017, pp. 250-278
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We present the linear stability properties and nonlinear evolution of two-dimensional plane Couette flow for a statically stable Boussinesq three-layer fluid of total depth $2h$ between two horizontal plates driven at constant velocity $\pm \unicode[STIX]{x0394}U$. Initially the three layers have equal depth $2h/3$ and densities $\unicode[STIX]{x1D70C}_{0}+\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$, $\unicode[STIX]{x1D70C}_{0}$ and $\unicode[STIX]{x1D70C}_{0}-\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$, such that $\unicode[STIX]{x1D70C}_{0}\gg \unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$. At finite Reynolds and Prandtl number, we demonstrate that this flow is susceptible to two distinct primary linear instabilities for sufficiently large bulk Richardson number $Ri_{B}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}h/(\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x0394}U^{2})$. For a given bulk Richardson number $Ri_{B}$, the zero phase speed ‘Taylor’ instability is always predicted to have the largest growth rate and to be an inherently two-dimensional instability. An inherently viscous instability, reminiscent of the ‘Holmboe’ instability, is also predicted to have a non-zero growth rate. For flows with Prandtl number $Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}=1$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity, and $\unicode[STIX]{x1D705}$ is the diffusivity of the density distribution, we find that the most unstable Taylor instability, maximized across wavenumber and $Ri_{B}$, has a (linear) growth rate which is a non-monotonic function of Reynolds number $Re=\unicode[STIX]{x0394}Uh/\unicode[STIX]{x1D708}$, with a global maximum at $Re=700$ over 50 % larger than the growth rate as $Re\rightarrow \infty$. In a fully nonlinear evolution of the flows with $Re=700$ and $Pr=1$, the two interfaces between the three density layers diffuse more rapidly than the underlying instabilities can grow from small amplitude. Therefore, we investigate numerically the nonlinear evolution of the flow at $Re=600$ and $Pr=300$, and at $Re=5000$ and $Pr=70$ in two-dimensional domains with streamwise extent equal to two wavelengths of the Taylor instability with the largest growth rate. At both sets of parameter values, the primary Taylor instability undergoes a period of identifiable exponential ‘linear’ growth. However, we demonstrate that, unlike the so-called ‘Kelvin–Helmholtz’ instability that it superficially resembles, the Taylor instability’s finite-amplitude state of an elliptical vortex in the middle layer appears not to saturate into a quasiequilibrium state, but is rapidly destroyed by the background shear. The decay process reveals $Re$-dependent secondary processes. For the $Re=600$ simulation, this decay allows the development to finite amplitude of the co-existing primary ‘viscous Holmboe wave instability’, which has a substantially smaller linear growth rate. For the $Re=5000$ simulation, the Taylor instability decay induces a non-trivial modification of the mean velocity and density distributions, which nonlinearly develops into more classical finite-amplitude Holmboe waves. In both cases, the saturated nonlinear Holmboe waves are robust and long-lived in two-dimensional flow.
Disruption of $\text{SSP}/\text{VWI}$ states by a stable stratification
- T. S. Eaves, C. P. Caulfield
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- Journal:
- Journal of Fluid Mechanics / Volume 784 / 10 December 2015
- Published online by Cambridge University Press:
- 06 November 2015, pp. 548-564
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We identify ‘minimal seeds’ for turbulence, i.e. initial conditions of the smallest possible total perturbation energy density $E_{c}$ that trigger turbulence from the laminar state, in stratified plane Couette flow, the flow between two horizontal plates of separation $2H$, moving with relative velocity $2{\rm\Delta}U$, across which a constant density difference $2{\rm\Delta}{\it\rho}$ from a reference density ${\it\rho}_{r}$ is maintained. To find minimal seeds, we use the ‘direct-adjoint-looping’ (DAL) method for finding nonlinear optimal perturbations that optimise the time-averaged total dissipation of energy in the flow. These minimal seeds are located adjacent to the edge manifold, the manifold in state space that separates trajectories which transition to turbulence from those which eventually decay to the laminar state. The edge manifold is also the stable manifold of the system’s ‘edge state’. Therefore, the trajectories from the minimal seed initial conditions spend a large amount of time in the vicinity of some states: the edge state; another state contained within the edge manifold; or even in dynamically slowly varying regions of the edge manifold, allowing us to investigate the effects of a stable stratification on any coherent structures associated with such states. In unstratified plane Couette flow, these coherent structures are manifestations of the self-sustaining process (SSP) deduced on physical grounds by Waleffe (Phys. Fluids, vol. 9, 1997, pp. 883–900), or equivalently finite Reynolds number solutions of the vortex–wave interaction (VWI) asymptotic equations initially derived mathematically by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666). The stratified coherent states we identify at moderate Reynolds number display an altered form from their unstratified counterparts for bulk Richardson numbers $\mathit{Ri}_{B}=g{\rm\Delta}{\it\rho}H/({\it\rho}_{r}{\rm\Delta}U^{2})=O(\mathit{Re}^{-1})$, and exhibit chaotic motion for larger $\mathit{Ri}_{B}$. We demonstrate that at hith Reynolds number the suppression of vertical motions by stratification strongly disrupts input from the waves to the roll velocity structures, thus preventing the waves from reinforcing the viscously decaying roll structures adequately, when $\mathit{Ri}_{B}=O(\mathit{Re}^{-2})$.
Genetic effects on the variation and covariation of attention deficit-hyperactivity disorder (ADHD) and oppositional-defiant disorder/conduct disorder (ODD/CD) symptomatologies across informant and occasion of measurement
- T. S. NADDER, M. RUTTER, J. L. SILBERG, H. H. MAES, L. J. EAVES
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- Journal:
- Psychological Medicine / Volume 32 / Issue 1 / January 2002
- Published online by Cambridge University Press:
- 05 February 2002, pp. 39-53
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Background. Previous studies have shown that the presence of conduct disorder may contribute to the persistence of attention deficit-hyperactivity disorder (ADHD) symptomatology into adolescence; however, the aetiological relationship between the two phenotypes remains undetermined. Furthermore, studies utilizing multiple informants have indicated that teacher ratings of these phenotypes are more valid than maternal reports.
Methods. The genetic structure underlying the persistence of ADHD and oppositional-defiant disorder/conduct disorder (ODD/CD) symptomatologies as rated by mothers and teachers at two occasions of measurement was investigated on a sample of 494 male and 603 female same sex adolescent twin pairs participating in the Virginia Twin Study of Adolescent Behavioral Development (VTSABD).
Results. Using structural modelling techniques, one common genetic factor was shown to govern the covariation between the phenotypes across informants and occasion of measurement with additional genetic factors specific to ODD/CD symptomatology and persistence of symptomatology at reassessment. Genetic structures underlying the phenotypes were, to some extent, informant dependent.
Conclusions. The findings indicate that it is unlikely that the co-morbidity between ADHD and ODD/CD is due to environmental influences that are independent of ADHD. Rather it is likely to be due to a shared genetic liability either operating directly, or indirectly through gene–environment correlations or interactions. The covariation between phenotypes across informants and time is governed by a common set of genes, but it seems that ODD/CD is also influenced by additional genetic factors. Developmentally, different forms of genetic liability control ADHD in males and inattention in females.