to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

We review flows of dense cohesionless granular materials, with a special focus on the question of constitutive equations. We first discuss the existence of a dense flow regime characterized by enduring contacts. We then emphasize that dimensional analysis strongly constrains the relation between stresses and shear rates, and show that results from experiments and simulations in different configurations support a description in terms of a frictional visco-plastic constitutive law. We then discuss the successes and limitations of this empirical rheology in light of recent alternative theoretical approaches. Finally, we briefly present depth-averaged methods developed for free surface granular flows.

Flows of Dense Granular Media

流体力学杂志“Journal of Fluid Mechanics”由剑桥大学教授George Batchelor在1956年5月创办，在国际流体力学界享有很高的学术声望，被公认为是流体力学最著名的学术刊物之一，2005年的影响因子为2．061，雄居同类期刊之首．在它创刊50周年之际，2006年5月JFM出版了第554卷的纪念特刊，其中刊登了现任主编（美国西北大学S．H．Davis教授和英国剑桥大学T．J．Pedley教授）合写的述评：“Editorial：JFM at50”，以JFM为背景，从独特的视角对近50年来流体力学的发展进行了简明的回顾和展望，并归纳了一系列非常有启发性的有趣统计数字．2006年7月21日在剑桥大学应用数学和理论物理研究所（DAMTP）举行了创刊50周年的庆祝会．下午2点，JFM的新老编辑和来宾会聚一堂，Pedley教授致开幕词，其后是5个精彩的报告：The mysterious rattleback and its fluid counterpart（Keith Moffatt），Developments in shear instabilities（Patrick Huerre），Falling clouds（Elisabeth Guazzelli），Ecotectural fluid mechanics（Paul Linden），The success of JFM（Herbert Huppert），最后由Davis教授致闭幕词．

Journal of Fluid Mechanics创刊50周年

Because of its relevance for the global climate the Atlantic meridional overturning circulation (AMOC) has been a major research focus for many years. Yet the question of which physical mechanisms ultimately drive the AMOC, in the sense of providing its energy supply, remains a matter of controversy. Here we review both observational data and model results concerning the two main candidates: vertical mixing processes in the ocean's interior and wind-induced Ekman upwelling in the Southern Ocean. In distinction to the energy source we also discuss the role of surface heat and freshwater fluxes, which influence the volume transport of the meridional overturning circulation and shape its spatial circulation pattern without actually supplying energy to the overturning itself in steady state. We conclude that both wind-driven upwelling and vertical mixing are likely contributing to driving the observed circulation. To quantify their respective contributions, future research needs to address some open questions, which we outline.

/pdf/on-the-driving-processes-of-the-atlantic-meridional-4ze95ojzss.pdf

On the driving processes of the Atlantic meridional overturning circulation

The problem of understanding the nature of fluid flow through a circular straight pipe remains one of the oldest problems in fluid mechanics. So far no explanation has been substantiated to rationalize the transition process by which the steady unidirectional laminar flow state gives way to a temporally and spatially disordered three-dimensional (turbulent) solution as the flow rate increases. Recently, new travelling wave solutions have been discovered which are saddle points in phase space. These plausibly represent the lowest level in a hierarchy of spatio-temporal periodic flow solutions which may be used to construct a cycle expansion theory of turbulent pipe flows. We summarize this success against the backdrop of past work and discuss its implications for future research.

http://iopscience.iop.org/article/10.1088/0951-7715/18/6/R01/pdf

Recent progress in understanding the transition to turbulence in a pipe

An experimental investigation is conducted into the collapse of granular columns inside rectangular channels. The final shape is documented for slumps inside relatively wide channels, and for collapses inside much narrower slots. In both cases, the collapse is initiated by withdrawing a swinging gate or sliding door, and the flow remains fairly two-dimensional. Four different granular media are used; the properties of the materials vary significantly, notably in their angles of friction for basal sliding and internal deformation. If ). In all cases, the numerical constant of proportionality in these scaling relations shows clear material dependence. In wide slots, there is no obvious universal scaling behaviour of the final profile, but such a behaviour is evident in narrow slots. The experimental results are compared with theoretical results based on a shallow granular-flow model. The qualitative behaviour of the slump in the wide slot is reproduced by the theoretical model. However, there is qualitative disagreement between theory and the experiments in the narrow slot because of the occurrence of secondary surface avalanching.

/pdf/granular-collapse-in-two-dimensions-4kpsk80y72.pdf

Granular collapse in two dimensions

The laminar-turbulent boundary Sigma is the set separating initial conditions which relaminarize uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cyl ...

Transition in pipe flow: the saddle structure on the boundary of turbulence

Linear transient growth analysis is commonly used to suggest the structure of disturbances which are particularly efficient in triggering transition to turbulence in shear flows. We demonstrate that the addition of nonlinearity to the analysis can substantially change the prediction made in pipe flow from simple two-dimensional streamwise rolls to a spanwise and cross-stream localized three-dimensional state. This new nonlinear optimal is demonstrably more efficient in triggering turbulence than the linear optimal indicating that there are better ways to design perturbations to achieve transition.

Using Nonlinear Transient Growth to Construct the Minimal Seed for Shear Flow Turbulence

New families of three-dimensional nonlinear traveling waves are discovered in pipe flow. In contrast with known waves [H. Faisst and B. Eckhardt, Phys. Rev. Lett. 91, 224502 (2003); H. Wedin and R. R. Kerswell, J. Fluid Mech. 508, 333 (2004), they possess no discrete rotational symmetry and exist at a significantly lower Reynolds numbers (Re). First to appear is a mirror-symmetric traveling wave which is born in a saddle node bifurcation at Re=773. As Re increases, "asymmetric" modes arise through a symmetry-breaking bifurcation. These look to be a minimal coherent unit consisting of one slow streak sandwiched between two fast streaks located preferentially to one side of the pipe. Helical and nonhelical rotating waves are also found, emphasizing the richness of phase space even at these very low Reynolds numbers.

Rich Kerswell

Papers

Recent progress in understanding the transition to turbulence in a pipe

Granular collapse in two dimensions

Transition in pipe flow: the saddle structure on the boundary of turbulence

Using Nonlinear Transient Growth to Construct the Minimal Seed for Shear Flow Turbulence

Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow.