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

http://basin.earth.ncu.edu.tw/download/courses/seminar_MSc/2012/1018_1_1.pdf

Marine hyperpycnal flows: initiation, behavior and related deposits. A review

Terror in the Mind of God: The Global Rise of Religious Violence

Pyoclastic density currents are awesome volcanic phenomena that can wreak destruction on a regional scale and can impact global climate. They deposit ignimbrites, which include vast impact lansdscape-modifying sheets with volumes exceeding 1000 km3.This book takes stock of our understanding of pyroclastic density currents and presents a new conceptual framework for investigating how ignimbrites are deposited. It integrates the results of field-based studies, laboratory experiments and numerical modelling, including work on clastic sedimentologym and industrial particle transport. Topics covered include the behaviour or particulate currents, mechanisms of clast support and segregation, interpreting ignimbrite lithofacies and architectures, and future research directions. The new approach focuses on processes and conditions within the lower flow-boundary zone of currents. Superb diagrams explain many new concepts, while the 95 photographs make an explanatiry atlas of deposit types. This is essential reading for workers investigating volcanic hazards, and for anyone wishing to interpret modern or ancient ignimbrites, as well as other catastrophically emplaced sediments.

“Given the depth of scholarship that they have brought to the subject, the power of their arguments, and the degree of synthesis with other fields, this would seemto qualify as a seminal work… I think that this will be the paper on the topic that others will have to contend with for many years to come.” Marcus Bursik, State University of New York

Pyroclastic density currents and the sedimentation of ignimbrites

A high-order Godunov-type scheme based on MUSCL variable extrapolation and slope limiters is presented for the resolution of 2D free-surface flow equations. In order to apply a finite volume technique of integration over body-fitted grids, the construction of an approximate Jacobian (Roe type) of the normal flux function is proposed. This procedure allows conservative upwind discretisation of the equations for arbitrary cell shapes. The main advantage of the model stems from the adaptability of the grid to the geometry of the problem and the subsequent ability to produce correct results near the boundaries. Verification of the technique is made by comparison with analytical solutions and very good agreement is found. Three cases of rapidly varying two-dimensional flows are presented to show the efficiency and stability of this method, which contains no terms depending on adjustable parameters. It can be considered well suited for computation of rather complex free-surface two-dimensional problems.

A high-resolution godunov-type scheme in finite volumes for the 2d shallow-water equations

A mathematical model of the stream flow in border irrigation is presented in the context of negligible accelerations everywhere in the stream. During the advance phase, numerical solution of the governing equations is achieved on an oblique grid in the x-t plane. The equations of motion are integrated over each oblique cell formed by joining the node points at constant times and distances by diagonals. The resulting nonlinear algebraic equations for depth and discharge at the upper corners of a cell (on the unknown time lines) are linearized with respect to the known values at the lower corners. The resultant set of linear algebraic equations in the incremental changes of depth and discharge that occur over the time interval are solved by a double-sweep technique. During runoff and recession, the grid is changed to a rectangular net. Solutions for advance recession and runoff volume compare very favorably with the results of models based on the complete hydrodynamic equations and with field tests, at but a fraction of the expense.

Border-Irrigation Hydraulics with Zero Inertia

A mathematical model of two-dimensional unsteady flow through a breached dam is developed. The numerical method of characteristics in three independent variables is used to construct an algorithm correct to second order with respect to time. The solution progresses at specified time intervals. At any time, the flow conditions are obtained by extending characteristic conoids back to a previously computed time plane and allowing information to propagate along the curvilinear rays of these conoids. Both the negative wave propagating into the reservoir and the jet emerging from the breach and eventually inundating the valley downstream of the dam are modeled in detail. Arbitrary dam geometry and channel topography can be taken into account. Propagation of the wave fronts is simulated by a computational grid moving with time. The exact computed results of the model are presented graphically as axonometric projects for the depth, and as plan-view, two-dimensional vector fields for the flow velocities.

Computing Two-Dimensional Dam-Break Flood Waves

A model is presented for the simulation of shallow water flow and, specifically, flood waves propagating on a dry bed. The governing equations are transformed to an equivalent system valid on a deforming coordinate system and are solved by a dissipative finite-element technique. A second-order difference scheme is employed for the integration in time. The implicit nonlinear equations resulting from the weak formulations are solved by the Newton-Raphson method, and the set of linear algebraic equations generated is solved by a frontal algorithm. A deforming grid generation scheme is introduced in the dissipative finite-element formulation to account for the effects of the propagating or receding wave fronts on dry land. The accuracy and stability of the model is examined by comparing the model results with observed data from an experimental field test. Results of trial runs for the simulation of overland flow are also presented.

Closure of "Model for Flood Propagation on Initially Dry Land"

The most common type of submarine fan is created by the passage of a succession of depositional turbidity currents. It is thus of interest to note that even in this essentially depositional environment the fan surface is often intensely channelized. Here a two-dimensional numerical model describing the formation of a submarine fan by a spreading turbidity current is presented. Layer-averaged governing equations describing the turbidity current are presented and solved numerically in conjunction with the Exner equation of bed sediment continuity. The formulation describes fan evolution from an initially flat surface. The bed is allowed to evolve in response to the exchange of sediment with the turbidity current by means of simultaneous erosion and deposition of suspended sediment. The upstream boundary condition is chosen so as to approximate a submarine canyon debouching upon the fan. Under some but not all conditions the early stage of the depositional process is accompanied by the formation of incipient natural levees that act to partially channelize the flow, indicating the formation of a channel-levee system. Dimensionless parameters are used to describe conditions for optimal channelization. The model can explain some of the basic mechanisms that account for the persistent tendency for channelization of the fan surface under a wide range of conditions.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/97JC01721

Nikolaos D. Katopodes

Papers

Border-Irrigation Hydraulics with Zero Inertia

Computing Two-Dimensional Dam-Break Flood Waves

Closure of "Model for Flood Propagation on Initially Dry Land"

A numerical model of channel inception on submarine fans

Coupled modeling of hydrologic and hydrodynamic processes including overland and channel flow