Approximations to the scattering of water waves by steep topography

A new method is developed for approximating the scattering of linear surface gravity waves on water of varying quiescent depth in two dimensions. A conformal mapping of the fluid domain onto a uniform rectangular strip transforms steep and discontinuous bed profiles into relatively slowly varying, smooth functions in the transformed free-surface condition. By analogy with the mild-slope approach used extensively in unmapped domains, an approximate solution of the transformed problem is sought in the form of a modulated propagating wave which is determined by solving a second-order ordinary differential equation. This can be achieved numerically, but an analytic solution in the form of a rapidly convergent infinite series is also derived and provides simple explicit formulae for the scattered wave amplitudes. Small-amplitude and slow variations in the bedform that are excluded from the mapping procedure are incorporated in the approximation by a straightforward extension of the theory. The error incurred in using the method is established by means of a rigorous numerical investigation and it is found that remarkably accurate estimates of the scattered wave amplitudes are given for a wide range of bedforms and frequencies.

Testing the limits of quasi-geostrophic theory: application to observed laboratory flows outside the quasi-geostrophic regime

We compare laboratory observations of equilibrated baroclinic waves in the rotating two-layer annulus, with numerical simulations from a quasi-geostrophic model. The laboratory experiments lie well outside the quasi-geostrophic regime: the Rossby number reaches unity; the depth-to-width aspect ratio is large; and the fluid contains ageostrophic inertia–gravity waves. Despite being formally inapplicable, the quasi-geostrophic model captures the laboratory flows reasonably well. The model displays several systematic biases, which are consequences of its treatment of boundary layers and neglect of interfacial surface tension and which may be explained without invoking the dynamical effects of the moderate Rossby number, large aspect ratio or inertia–gravity waves. We conclude that quasi-geostrophic theory appears to continue to apply well outside its formal bounds.

Numerical study of 2D heat transfer in a scraped surface heat exchanger

A numerical study of fluid mechanics and heat transfer in a scraped surface heat exchanger with non-Newtonian power law fluids is undertaken. Numerical results are generated for 2D steady-state conditions using finite element methods. The effect of blade design and material properties, and especially the independent effects of shear thinning and heat thinning on the flow and heat transfer, are studied. The results show that the gaps at the root of the blades, where the blades are connected to the inner cylinder, remove the stagnation points, reduce the net force on the blades and shift the location of the central stagnation point. The shear thinning property of the fluid reduces the local viscous dissipation close to the singularity corners, i.e. near the tip of the blades, and as a result the local fluid temperature is regulated. The heat thinning effect is greatest for Newtonian fluids where the viscous dissipation and the local temperature are highest at the tip of the blades. Where comparison is possible, very good agreement is found between the numerical results and the available data. Aspects of scraped surface heat exchanger design are assessed in the light of the results. (C) 2003 Elsevier Ltd. All rights reserved.

The physics of baking: rheological and polymer molecular structure-function relationships in breadmaking

Molecular size and structure of the gluten polymers that make up the major structural components of wheat are related to their rheological properties via modem polymer rheology concepts. Interactions between polymer chain entanglements and branching are seen to be the key mechanisms determining the rheology of HMW polymers. Recent work confirms the observation that dynamic shear plateau modulus is essentially independent of variations in MW amongst wheat varieties of varying baking performance and is not related to variations in baking performance, and that it is not the size of the soluble glutenin polymers, but the structural and rheological properties of the insoluble polymer fraction that are mainly responsible for variations in baking performance. The rheological properties of gas cell walls in bread doughs are considered to be important in relation to their stability and gas retention during proof and baking, in particular their extensional strain hardening properties. Large deformation rheological properties of gas cell walls were measured using biaxial extension for a number of doughs of varying breadmaking quality at constant strain rate and elevated temperatures in the range 25-60 degrees C. Strain hardening and failure strain of cell walls were both seen to decrease with temperature, with cell walls in good breadmaking doughs remaining stable and retaining their strain hardening properties to higher temperatures (60 degrees C), whilst the cell walls of poor breadmaking doughs became unstable at lower temperatures (45-50 degrees C) and had lower strain hardening. Strain hardening measured at 50 degrees C gave good correlations with baking volume, with the best correlations achieved between those rheological measurements and baking tests which used similar mixing conditions. As predicted by the Considere failure criterion, a strain hardening value of I defines a region below which gas cell walls become unstable, and discriminates well between the baking quality of a range of commercial flour blends of varying quality. This indicates that the stability of gas cell walls during baking is strongly related to their strain hardening properties, and that extensional rheological measurements can be used as predictors of baking quality. (C) 2004 Elsevier B.V. All rights reserved.

Whither tube theory: From believing to measuring

This short contribution examines the difficulties that have not yet been fully overcome in the many developments made from the simplest (and original) tube model for entangled polymers. It is concluded that many more length scales have to be considered sequentially when deriving a continuum rheological model from molecular considerations than have been considered in the past. In particular, most unresolved issues of the tube theory are related to the length scales of tube diameter, and molecular dynamics simulations is the perfect route to resolve them. The power of molecular simulations is illustrated by two examples: stress contributions from bonded and non-bonded interaction, and the inter-chain coupling, which is usually neglected in the tube theory.

A novel X-ray rheometer for in situ studies of polymer melts and solutions during shear flow

A novel X-ray rheometer based on a parallel plate geometry is described. This system allows time-resolved X-ray scattering intensity data to be obtained from polymeric samples subjected to shear flow. The range of quantitative structural parameters, such as molecular orientation and inter chain correlations, which can be obtained from the data is highlighted. Examples of the utility of X-ray scattering in examining optically opaque samples and the extraction of 〈P2〉 and 〈P4〉 orientation parameters are given using anisotropic hydroxypropylcellulose solutions as the sample.

The wind in the willows: flows in forest canopies in complex terrain

Forest canopies are important components of the terrestrial carbon budget, which has motivated a worldwide effort, FLUXNET, to measure CO2 exchange between forests and the atmosphere. These measurements are difficult to interpret and to scale up to estimate exchange across a landscape. Here we review the effects of complex terrain on the mean flow, turbulence, and scalar exchange in canopy flows, as exemplified by adjustment to forest edges and hills, including the effects of stable stratification. We focus on the fundamental fluid mechanics, in which developments in theory, measurements, and modeling, particularly through large-eddy simulation, are identifying important processes and providing scaling arguments. These developments set the stage for the development of predictive models that can be used in combination with measurements to estimate exchange at the landscape scale.

Available potential energy and exergy in stratified fluids

Lorenz’s theory of available p otential energy (APE) remains the main framework for studying the atmospheric and oceanic energy cycles. Because the APE generation rate is the volume integral of a thermodynamic efficiency times the local diabatic heating/cooling rate, APE theory is often regarded as an extension of the theory of heat engines. Available energetics in classical thermodynamics, however, usually relies on the concept of exergy, and is usually measured relative to a reference state maximising entropy at constant energy, whereas APE’s reference state minimises p otential energy at constant entropy. This review seeks to shed light on the two concepts; it covers local formulations of available energetics, alternative views of the dynamics/thermodynamics coupling, APE theory and the second law, APE production/dissipation, extensions to binary fluids, mean/eddy decomp ositions, APE in incompressible fluids, APE and irreversible turbulent mixing, and the role of mechanical forcing on APE production.

Turbulence dynamics near a turbulent/non-turbulent interface

The characteristics of the boundary layer separating a turbulence region from an irrotational (or non-turbulent) flow region are investigated using rapid distortion theory (RDT). The turbulence region is approximated as homogeneous and isotropic far away from the bounding turbulent/non-turbulent (T/NT) interface, which is assumed to remain approximately flat. Inviscid effects resulting from the continuity of the normal velocity and pressure at the interface, in addition to viscous effects resulting from the continuity of the tangential velocity and shear stress, are taken into account by considering a sudden insertion of the T/NT interface, in the absence of mean shear. Profiles of the velocity variances, turbulent kinetic energy (TKE), viscous dissipation rate (epsilon), turbulence length scales, and pressure statistics are derived, showing an excellent agreement with results from direct numerical simulations (DNS). Interestingly, the normalized inviscid flow statistics at the T/NT interface do not depend on the form of the assumed TKE spectrum. Outside the turbulent region, where the flow is irrotational (except inside a thin viscous boundary layer), epsilon decays as z^{-6}, where z is the distance from the T/NT interface. The mean pressure distribution is calculated using RDT, and exhibits a decrease towards the turbulence region due to the associated velocity fluctuations, consistent with the generation of a mean entrainment velocity. The vorticity variance and epsilon display large maxima at the T/NT interface due to the inviscid discontinuities of the tangential velocity variances existing there, and these maxima are quantitatively related to the thickness delta of the viscous boundary layer (VBL). For an equilibrium VBL, the RDT analysis suggests that delta ~ eta (where eta is the Kolmogorov microscale), which is consistent with the scaling law identified in a very recent DNS study for shear-free T/NT interfaces.