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Abstract The need for operational models describing the friction factorfin streams remains undisputed given its utility across a plethora of hydrological and hydraulic applications concerned with shallow inertial flows. For small-scale roughness elements uniformly covering the wetted parameter of a wide channel, the Darcy-Weisbachf = 8(u*/Ub)2is widely used at very high Reynolds numbers, whereu*is friction velocity related to the surface kinematic stress,Ub = Q/Ais bulk velocity,Qis flow rate, andAis cross-sectional area orthogonal to the flow direction. In natural streams, the presence of vegetation introduces additional complications to quantifyingf, the subject of the present work. Turbulent flow through vegetation are characterized by a number of coherent vortical structures: (i) von Karman vortex streets in the lower layers of vegetated canopies, (ii) Kelvin-Helmholtz as well as attached eddies near the vegetation top, and (iii) attached eddies well above the vegetated layer. These vortical structures govern the canonical mixing lengths for momentum transfer and their influence onfis to be derived. The main novelty is that the friction factor of vegetated flow can be expressed asfv = 4Cd(Uv/Ub)2whereUvis the spatially averaged velocity within the canopy volume, andCdis a local drag coefficient per unit frontal area derived to include the aforemontioned layer-wise effects of vortical structures within and above the canopy along with key vegetation properties. The proposed expression is compared with a number of empirical relations derived for vegetation under emergent and submerged conditions as well as numerous data sets covering a wide range of canopy morphology, densities, and rigidity. It is envisaged that the proposed formulation be imminently employed in eco-hydraulics where the interaction between flow and vegetation is being sought.
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How Do the Purcell Factor, the Q ‐Factor, and the Beta Factor Affect the Laser Threshold?
Abstract As lasers get more and more miniaturized and their dimensions become comparable to the wavelength, two interconnected phenomena take place: the fraction of spontaneous radiation going into a specific laser mode (β‐factor) increases and can ultimately reach unity, while the radiative lifetime gets shortened by the Purcell factorFp. Often it is assumed that an increase of these two factors, along with the quality factor (Q‐factor), almost invariably causes reduction of the lasing threshold. This assumption is tested on various photonic and plasmonic lasers, demonstrating that, while there is obvious correlation between the aforementioned factors and the laser threshold, the dependence is far from being straightforward and omnipresent. Depending on specific laser material and geometry, the threshold can decrease, increase, or stay unchanged whenβ‐factor,Q‐factor, andFpincrease. For the most part, the reduction of threshold is achieved simply by reducing the laser volume and this volume reduction can concurrently cause the increase inβ‐factor and/or Purcell factor, but it would be imprudent to say that the increase in either of these factors is the cause of the threshold reduction.
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- PAR ID:
- 10385388
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Laser & Photonics Reviews
- Volume:
- 15
- Issue:
- 3
- ISSN:
- 1863-8880
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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