Energy dissipation rate limits for flow through rough channels and tidal flow across topography

The bound derived in the submission (subsequently published in J. Fluid Mechanics vol 808 p 562-575, 2016) with the above title is incorrect. This corrigendum explains why and also why there can not be any quick fix.Comment: This is a corrigendum (to be published in J. Fluid Mechanics

Effect of nonlinear interactions on p-mode frequencies and line widths

We calculate the effect of nonlinear interactions among solar acoustic modes upon the modal frequencies and energy loss rates (or line widths). The frequency shift for a radial p-mode of frequency 3 mHz is found to be about -0.5 µHz. The magnitude of nonlinear frequency shift increases more rapidly with frequency than the inverse mode mass (mode mass is defined as the ratio of energy in the mode to its surface velocity amplitude squared). This frequency shift is primarily due to nonresonant three-mode interactions and is dominated by high l surface gravity waves (ƒ-modes) and p-modes. The line width of a radial p-mode of frequency 3 mHz, due to resonant nonlinear interactions, is about 0.3 µHz. This result is consistent with that of Kumar & Goldreich (1989). We also find, in agreement with these authors, that the most important nonlinear interactions of trapped p-modes involve ƒ-modes and high-frequency p-modes (frequency greater than about 5 mHz) which propagate in the solar photosphere. Thus, using the arguments advanced by Kumar & Goldreich (1989), we conclude that nonlinear couplings cannot saturate the overstable solar p-modes at their small observed amplitudes. Both the nonlinear frequency shifts and line widths, at a fixed frequency, are proportional to the inverse of mode mass which for modes of degree greater than about 100 is ~ l^(0.8). Therefore, the frequency of an ƒ-mode of l = 1000, due to nonlinear interactions, is decreased by approximately 0.4%

Growth of Hydrodynamic Perturbations in Accretion Disks: Possible Route to Non-Magnetic Turbulence

We study the possible origin of hydrodynamic turbulence in cold accretion disks such as those in star-forming systems and quiescent cataclysmic variables. As these systems are expected to have neutral gas, the turbulent viscosity is likely to be hydrodynamic in origin, not magnetohydrodynamic. Therefore MRI will be sluggish or even absent in such disks. Although there are no exponentially growing eigenmodes in a hydrodynamic disk, because of the non-normal nature of the eigenmodes, a large transient growth in the energy is still possible, which may enable the system to switch to a turbulent state. For a Keplerian disk, we estimate that the energy will grow by a factor of 1000 for a Reynolds number close to a million.Comment: 4 pages; to appear in the Proceedings of COSPAR Colloquium "Spectra & Timing of Compact X-ray Binaries," January 17-20, 2005, Mumbai, India; prepared on the basis of the talk presented by Mukhopadhya

Conceptual models of the climate : 2002 program of study, Bounds on turbulent transport

The subject of "Bounds of Turbulent Transport" was introduced in a series of ten lectures. The six lecturers constitute almost all the contributors to this subject. The subject was introduced and foundations laid by five lectures by F. H. Busse. In the middle of the first week, L. Howard reviewed his historical first approach to this subject and described more recent advances. Additional lectures by P. Constantine, R. Kerswell, C. Caulfield and C. Doering provided modern advances. We trust that the lecture notes will constitute a timely review of this promising subject.Funding was provided by the Office of Naval Research under contract number N00014-97-1-0934 and The National Science Foundation under contract number OCE 98-10647

Stabilising pipe flow by a baffle designed by energy stability

Previous experimental (Ku¨hnen et al., Flow Turb. Combust., 100, 919-943, 2018) and numerical (Marensi et al. J. Fluid Mech. 863, 850-875, 2019) studies have demonstrated that a streamwise-localised baffle can fully relaminarise pipe flow turbulence at Reynolds numbers of O(104). Optimising the design of the baffle involves tackling a complicated variational problem built around time-stepping turbulent solutions of the Navier-Stokes equations which is difficult to solve. Here instead, we investigate a much simpler ‘spectral’ approach based upon maximising the energy stability of the baffle-modified laminar flow. The ensuing optimal problem has much in common with the variational procedure to derive an upper bound on the energy dissipation rate in turbulent flows (e.g. Plasting & Kerswell, J. Fluid Mech. 477, 363-379, 2003) so well-honed techniques developed there can be used to solve the problem here. The baffle is modelled by a linear drag force −F(x)u (with F(x) > 0 ∀x) where the extent of the baffle is constrained by a L norm with various choices explored in the range 1 6 α 6 2. An asymptotic analysis demonstrates that the optimal baffle is always axisymmetric and streamwise-independent retaining just radial dependence. The optimal baffle which emerges in all cases has a similar structure to that found to work in experiments: the baffle retards the flow in the pipe centre causing the flow to become faster near the wall thereby reducing the turbulent shear there. Numerical simulations demonstrate that the designed baffle can relaminarise turbulence efficiently at moderate Reynolds numbers (Re 6 3500), and an energy saving regime has been identified. Direct numerical simulation at Re = 2400 also demonstrates that the drag reduction can be realised by truncating the energy-stability-designed baffle to finite length.EPSR

Inertial enhancement of the polymer diffusive instability

Beneitez et al. (Phys. Rev. Fluids, 8, L101901, 2023) have recently discovered a new linear “polymer diffusive instability” (PDI) in inertialess rectilinear viscoelastic shear flow using the FENE-P model when polymer stress diffusion is present. Here, we examine the impact of inertia on the PDI for both plane Couette (PCF) and plane Poiseuille (PPF) flows under varying Weissenberg number W, polymer stress diffusivity , solvent-to-total viscosity ratio , and Reynolds number Re, considering the FENE-P and simpler Oldroyd-B constitutive relations. Both the prevalence of the instability in parameter space and the associated growth rates are found to significantly increase with Re. For instance, as increases with fixed, the instability emerges at progressively lower values of and than in the inertialess limit, and the associated growth rates increase linearly with when all other parameters are fixed. For finite , it is also demonstrated that the Schmidt number = 1/() collapses curves of neutral stability obtained across various and . The observed strengthening of PDI with inertia and the fact that stress diffusion is always present in time-stepping algorithms, either implicitly as part of the scheme or explicitly as a stabiliser, implies that the instability is likely operative in computational work using the popular Oldroyd-B and FENE-P constitutive models. The fundamental question now is whether PDI is physical and observable in experiments, or is instead an artifact of the constitutive models that must be suppressed.EPSR

Kelvin-Helmholtz billows above Richardson number 1/4

We study the dynamical system of a forced stratified mixing layer at finite Reynolds number $Re$, and Prandtl number $Pr=1$. We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications. The system is forced in such a way that these background profiles are a steady solution of the governing equations. As is well-known, if the minimum gradient Richardson number of the flow, $Ri_m$, is less than a certain critical value $Ri_c$, the flow is linearly unstable to Kelvin-Helmholtz instability in both cases. Using Newton-Krylov iteration, we find steady, two-dimensional, finite amplitude elliptical vortex structures, i.e. `Kelvin-Helmholtz billows', existing above $Ri_c$. Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying $Re$. In particular, when $Re$ is sufficiently high we find that finite amplitude Kelvin-Helmholtz billows exist at $Ri_m>1/4$, where the flow is linearly stable by the Miles-Howard theorem. For the uniform background stratification, we give a simple explanation of the dynamical system, showing the dynamics can be understood on a two-dimensional manifold embedded in state space, and demonstrate the cases in which the system is bistable. In the case of a hyperbolic tangent stratification, we also describe a new, slow-growing, linear instability of the background profiles at finite $Re$, which complicates the dynamics