Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows

A novel method is presented for obtaining rigorous upper bounds on the finite-amplitude growth of instabilities to parallel shear flows on the beta-plane. The method relies on the existence of finite-amplitude Liapunov (normed) stability theorems, due to Arnol'd, which are nonlinear generalizations of the classical stability theorems of Rayleigh and Fjørtoft. Briefly, the idea is to use the finite-amplitude stability theorems to constrain the evolution of unstable flows in terms of their proximity to a stable flow. Two classes of general bounds are derived, and various examples are considered. It is also shown that, for a certain kind of forced-dissipative problem with dissipation proportional to vorticity, the finite-amplitude stability theorems (which were originally derived for inviscid, unforced flow) remain valid (though they are no longer strictly Liapunov); the saturation bounds therefore continue to hold under these conditions.

An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems

Disturbances of arbitrary amplitude are superposed on a basic flow which is assumed to be steady and either (a) two-dimensional, homogeneous, and incompressible (rotating or non-rotating) or (b) stably stratified and quasi-geostrophic. Flow over shallow topography is allowed in either case. The basic flow, as well as the disturbance, is assumed to be subject neither to external forcing nor to dissipative processes like viscosity. An exact, local ‘wave-activity conservation theorem’ is derived in which the density A and flux F are second-order ‘wave properties’ or ‘disturbance properties’, meaning that they are O(a2) in magnitude as disturbance amplitude a [rightward arrow] 0, and that they are evaluable correct to O(a2) from linear theory, to O(a3) from second-order theory, and so on to higher orders in a. For a disturbance in the form of a single, slowly varying, non-stationary Rossby wavetrain, $\overline{F}/\overline{A}$ reduces approximately to the Rossby-wave group velocity, where (${}^{-}$) is an appropriate averaging operator. F and A have the formal appearance of Eulerian quantities, but generally involve a multivalued function the correct branch of which requires a certain amount of Lagrangian information for its determination. It is shown that, in a certain sense, the construction of conservable, quasi-Eulerian wave properties like A is unique and that the multivaluedness is inescapable in general. The connection with the concepts of pseudoenergy (quasi-energy), pseudomomentum (quasi-momentum), and ‘Eliassen-Palm wave activity’ is noted. The relationship of this and similar conservation theorems to dynamical fundamentals and to Arnol'd's nonlinear stability theorems is discussed in the light of recent advances in Hamiltonian dynamics. These show where such conservation theorems come from and how to construct them in other cases. An elementary proof of the Hamiltonian structure of two-dimensional Eulerian vortex dynamics is put on record, with explicit attention to the boundary conditions. The connection between Arnol'd's second stability theorem and the suppression of shear and self-tuning resonant instabilities by boundary constraints is discussed, and a finite-amplitude counterpart to Rayleigh's inflection-point theorem noted

Revisiting the ‘Venturi effect’ in passage ventilation between two non-parallel buildings

A recent study conducted by Blocken et al. (Numerical study on the existence of the Venturi effect in passages between perpendicular buildings. Journal of Engineering Mechanics, 2008,134: 1021-1028) challenged the popular view of the existence of the ‘Venturi effect’ in building passages as the wind is exposed to an open boundary. The present research extends the work of Blocken et al. (2008a) into a more general setup with the building orientation varying from 0° to 180° using CFD simulations. Our results reveal that the passage flow is mainly determined by the combination of corner streams. It is also shown that converging passages have a higher wind-blocking effect compared to diverging passages, explained by a lower wind speed and higher drag coefficient. Fluxes on the top plane of the passage volume reverse from outflow to inflow in the cases of α=135°, 150° and 165°. A simple mathematical expression to explain the relationship between the flux ratio and the geometric parameters has been developed to aid wind design in an urban neighborhood. In addition, a converging passage with α=15° is recommended for urban wind design in cold and temperate climates since the passage flow changes smoothly and a relatively lower wind speed is expected compared with that where there are no buildings. While for the high-density urban area in (sub)tropical climates such as Hong Kong where there is a desire for more wind, a diverging passage with α=150° is a better choice to promote ventilation at the pedestrian level.