Kinematic Shock Research Articles

Non-classical kinematic shocks in suspensions of particles in fluids

The properties of weakly nonlinear kinematic waves in suspensions are investigated under the assumption that the particle concentration deviates only slightly from the value at the inflexion point of the drift flux curve. Special emphasis is placed on the conditions for the existence of an internal dissipative-dispersive shock structure. The resulting shock admissibility criteria are found to be significantly different from those following from standard theories of kinematic waves. Most interesting, the analysis shows that non-classical shocks which emanate rather than absorb characteristics may be admissible under certain conditions.

Analysis of sedimentation/consolidation by finite element method

Abstract A low concentration of solid suspension material is transformed into a soil layer through two stages of sedimentation and consolidation, occurring simultaneously. During sedimentation in the uppermost part of the problem domain, in the rest of the domain, the soil particles form a continuous structure, the ‘Terzaghi soil’ so called. This soil structure consolidates under its own weight and the weight of the upper part. Because of the formation of this structure, a discrete boundary separating sedimentation from suspension is formed. On this boundary, termed a ‘shock boundary’, kinematic shocks occur. In order to analyze the phenomena of sedimentation/consolidation by finite element method, in this study, the sedimentation was considered as a solid flux problem from the standpoint of the Eulerian coordinate system; while the consolidation was considered as a fluid flux problem from the standpoint of Lagrangian coordinate system, with keeping the shock boundary between them. From this basic understanding, a program for one-dimensional analysis of sedimentation/consolidation was developed using the finite element method. In the program the governing equations were separately applied — continuity equation only for sedimentation; and continuity and force equilibrium equations for consolidation. Nevertheless, the interaction between them was continuously accounted for by keeping track of the shock boundary. From a comparison of the results obtained from the program with those from two previous studies, the performance of the developed FEM program was validated. To apply this program to real dredging fields, two data sets obtained from sedimentation tests for Korean marine clay were analyzed.

Statistics of shock waves in a two-dimensional granular flow.

We have investigated the dynamics of shock waves in a single layer of uniform balls in a small-angle two-dimensional funnel. When the funnel half-angle 0 degrees < or approximately beta < or approximately 2 degrees, the flow is intermittent and kinematic shock waves are observed to propagate against the flow. We have used fast video equipment and image analysis methods to study the statistics of the shock waves. It is found that their speed and frequency increase with the distance from the outlet. In particular, the shock speed scales as the ratio of the local funnel width to the width of the funnel outlet. Various kinds of interactions between shock waves are observed, including repulsion. New shock waves are only created at those sites where a close-packed triangular packing of the monodisperse balls fits across the funnel.

Kinematic wave modeling in water resources: surface-water hydrology

Water Resources Modeling. Spatial Representation of Watersheds. HYDRAULIC PRELIMINARIES. Hydraulic Equations for Surface Flow. Linearization of Hydraulic Equations. Flow Resistance. WATER WAVES. Shallow Water Waves. Kinematic Wave Theory. Diffusion Wave Theory. Accuracy of Kinematic Wave and Diffusion Wave Theories. OVERLAND FLOW. St. Venant Equations for Flow Over A Plane. Diffusion Wave Modeling. Kinematic Wave Modeling of Overland Flow on A Plane: Analytic Solutions. Kinematic Wave Modeling of Overland Flow on an Infiltrating Plane: Analytical Solutions. Kinematic Wave Modeling of Overland Flow on a Plane: Numerical Solutions. Kinematic Wave Modeling of Overland Flow on Converging Surfaces. Kinematic Wave Modeling of Overland Flow on Diverging Surfaces. Kinematic Shock. CHANNEL FLOW ROUTING. Dynamic Wave Modeling for Channel Flow Routing. Diffusion Wave Modeling for Channel Flow Routing. Kinematic Wave Flow Routing. Dam--Break Flood--Wave Routing. Appendices. References. Index.

Transport of a sediment layer due to a laminar, stratified flow

It is known that a bed of settled particles resuspends under the action of shear caused by a laminar flow. Earlier papers investigated such a resuspension in a fully-developed Hagen-Poiseuille channel flow and in a gravity-driven film flow along an inclined plate. The theoretical results were compared with experiments which, however, disagree for larger flow-rates. In this paper we study the propagation of a change of height of a resuspended sediment, e.g., the beginning (onset) or the end of a particle layer. We apply the previously developed model for the base-state flow and find by means of the theory of kinematic waves that both a sudden onset and a sudden end of a resuspended layer in either a fully-developed Hagen-Poiseuille channel flow or a two-dimensional gravity-driven film flow move as a combination of kinematic shocks and kinematic waves. Experimental observations of the motion of a sudden end of a resuspended layer in a channel flow correspond well with the theory if the flow-rate is small but show distinct discrepancies from the theoretical predictions for larger flow-rates.

Kinematic Wave Controversy

Kinematic and diffusion waves are reviewed prompted by the continuing controversy regarding their nature and applicability. Kinematic waves are shown to be nondiffusive, but to undergo change in shape due to nonlinearity. This latter feature gives kinematic waves the capability of steepening, eventually leading to the formation of the kinematic shock. Kinematic wave solutions using finite differences are shown to possess intrinsic amounts of numerical diffusion and dispersion. These numerical effects are artificial and tend to disappear as the grid size is refined, making the solution dependent on the choice of grid size. Kinematic wave theory can be improved by extending it to the realm of diffusion waves. In this way, the diffusion inherent in many practical runoff computations can be accounted for directly in the modeling, rather than as an afterthought. The use of a kinematic wave method is indicated for small catchments, in cases where it is possible to resolve the physical detail without compromisin...

Finite Element Watershed Modeling: One‐Dimensional Elements

A procedure for calculating the spatial and temporal distribution of a direct runoff hydrograph from an ungauged watershed is presented. Nodal values of hydraulic roughness and slope are used to avoid kinematic shock. This formulation allows the simulation of overland flow over spatially variable surfaces typical of natural watersheds. The procedure is compared to an analytic solution for a simple case of a watershed composed of two planes. Broader application to watersheds with spatially variable parameters is possible because of the flexibility of the finite element. The Galerkin formulation of the finite element method is used to solve the kinematic wave equations. The method of characteristic and the finite element method were compared on a two‐plane system with an abrupt change in slope. The numerical and analytical considerations are presented to enable practicing engineers to adopt this method for the calculation of flow depth and discharge rate in a distributed manner throughout a watershed domina...

On the flow of a rotating mixture in a sectioned cylinder

We consider the centrifugal separation of an initially homogeneous mixture in an asymmetric geometry. The mixture is shown to acquire a uniform and negative relative vorticity, manifested as a retrograde circulation, as the heavier phase is forced outwards by the centrifugal acceleration. The perturbation theory formulated accounts for inertial effects and endwall boundary layers as well as slight deviations of endwall shape from a level plane. The time-dependent separation process is described and the flow field, including kinematic shocks, is calculated in some cases of technological interest.

A perturbation solution of the flood-routing problem

Singular perturbation techniques are used to obtain an approximate solution of the flood routing problem. The first-order outer solution is given by the kinematic-wave approximation. Singularities in this outer solution then lead to two inner solutions: one that follows a kinematic shock as it moves downstream and one either at the downstream boundary in subcritical flow or at the upstream boundary in supercritical flow. Solutions calculated in each of these subregions allow conclusions to be drawn about the relative importance of various terms in the momentum equation. The inner and outer solutions are combined to obtain composite solutions for flow depths and flow rates, and a numerical example is worked.

Kinematic shock waves on curved surfaces and application to the cascade approximation

An analytical solution for kinematic flow on a curved surface is presented for a specific rainfall. This solution is used to analyse the solutions obtained on a kinematic cascade. An illustration is given using a simple curved surface which allows algebraic manipulation. The formation of a shock wave, and the determination of its correct position are also illustrated. The present solution can be used to improve the choice of a cascade to minimise the errors introduced by this approximation.

Kinematic Shock: Sensitivity Analysis

A series of numerical experiments is performed to determine the flow and channel characteristics that are most conducive to kinematic wave steepening and associated kinematic shock phenomena. Relevant flow and channel characteristics are identified at the outset, and a program of 80 computer runs is completed, varying the inflow hydrograph peak Froude number, time‐to‐peak, and base‐to‐peak flow ratio, and the channel cross‐sectional shape. It is found that all have a definite effect on kinematic shock development. The size of the wave is perhaps mostly responsible for the occurrence or nonoccurrence of the shock.

Morphological instability of poly(ethylene terephthalate) cyclic oligomer crystals

The mechanism of formation of depressions and cavities in the middle of the basal faces of hexagonal plates of poly(ethylene terephthalate) oligomer single crystals upon growth during heat treatment is investigated. It is proposed that this effect is caused by two simultaneously occurring processes: supersaturation inhomogeneity in the centre of the plates and formation of the kinematic (shock) waves at the edges of the plates.

Perturbation Solution for Dam‐Break Floods

A solution for the complete and instantaneous collapse of a dam in a prismatic, infinitely‐long sloping channel is found by using the method of matched asymptotic expansions. The outer solution is a kinematic‐wave approximation, and an inner solution that includes a depth‐gradient term in the momentum equation is calculated for the region near the moving kinematic shock front. The reservoir is modeled with a point mass source, and the solution is believed to become asymptotically valid after the shock front has advanced about four reservoir lengths downstream. A comparison with experiment shows good qualitative agreement, but more comparisons with experiment should be made in order to assess the qualitative accuracy of the solution.

On centrifugal separation of particles of two different sizes

We consider flow in a centrifugal force field of a non-dilute suspension with particles or droplets of two sizes. The volume fraction and the velocity fields are determined assuming small convection and shear terms. The resulting flow field is quite different from that in a gravitational settling of a similar mixture. In particular, the volume fraction is a function of time and radius in the sectors separated by kinematic shocks and the settling velocity is a non-monotonic function of the particle size.

Closure to “ On Zero‐Inertia and Kinematic Waves ” by Nikolaos D. Katopodes (November, 1982)

Small perturbation analysis of the linearized open-channel flow equations leads to the conclusion that zero-inertia waves propagate in the downstream direction only with a speed equal to that of kinematic waves. In addition, such an analysis indicates that kinematic waves travel with no attentuation in amplitude. In this work, it is shown both by theory and nonlinear numerical experiments that zero-inertia waves propagate both in the downstream and upstream direction. In fact, the zero-inertia wave characteristics are horizontal lines, and the problem requires two-point boundary data, which extends the applicability of zero-inertia models to cases of great practical interest. It is also shown the nonlinear kinematic-wave characteristics are inclined relative to each other and that they may intersect to create a kinematic shock. This shock is always present in wave motion on a dry bed and leads to significant subsidence of the depth and discharge in the flow behind it, a behavior which is not due to amplitude attentuation, as in the case of dynamic waves. Merely it is due to a consequence of the presence of the kinematic-shock front.

On centrifugal separation of a mixture

An exact solution of the equations of motion of a two-phase fluid is given which describes separation in a centrifugal force field. A retrograde rotation is always produced in the settling process, which is characterized by propagating kinematic shocks and a time-dependent volume fraction of the mixture.

Application of the theory of kinematic waves to the centrifugation of suspensions

The settling of solid particles in a liquid due to centrifugal force is described by using the equations of one-dimensional unsteady two-phase flow. By neglecting the acceleration terms the momentum equation can be reduced to a functional relationship between the dependant variables of the problem (volumetric concentration α of the solid particles and volumetric flux j). As an additional relation for the unknown quantities, a first order partial differential equation is obtained from the equation of continuity for the solid particles. Concentration jumps (e.g. discontinuities between suspension and clear liquid or sediment) are described as kinematic shock waves. Analytical solutions are obtained for the kinematic wave fronts and for certain cases of shock waves. The results for the centrifugation process with uniform particle size show that several cases are to be distinguished. Under certain conditions the concentration of the suspension depends on time only and not on the radial coordinate of the rotating system. Other initial conditions give additional discontinuities within the suspension.

On Zero-Inertia and Kinematic Waves

Small perturbation analysis of the linearized open-channel flow equations leads to the conclusion that zero-inertia waves propagate in the downstream direction only with a speed equal to that of kinematic waves. In addition, such an analysis indicates that kinematic waves travel with no attentuation in amplitude. In this work, it is shown both by theory and nonlinear numerical experiments that zero-inertia waves propagate both in the downstream and upstream direction. In fact, the zero-inertia wave characteristics are horizontal lines, and the problem requires two-point boundary data, which extends the applicability of zero-inertia models to cases of great practical interest. It is also shown the nonlinear kinematic-wave characteristics are inclined relative to each other and that they may intersect to create a kinematic shock. This shock is always present in wave motion on a dry bed and leads to significant subsidence of the depth and discharge in the flow behind it, a behavior which is not due to amplitude attentuation, as in the case of dynamic waves. Merely it is due to a consequence of the presence of the kinematic-shock front.

Kinematic-wave theory of sedimentation beneath inclined walls

The two-phase flow in settling vessels with walls that are inclined to the vertical is investigated. By neglecting inertial effects and the viscosity of the suspension i t is shown that the particle concentration remains constant on kinematic-wave fronts. The wave fronts are horizontal and propagate in a quasi-one-dimensional manner, but are imbedded in a two-dimensional or three-dimensional basic flow which, in turn, depends on the waves via the boundary conditions. Concentration discontinuities (interfaces) are described by kinematic-shock theory. The kinematic shocks are shown to be horizontal, with the possible exception of discontinuities that separate the suspension from the sediment.At downward-facing inclined walls conservation of mass enforces the existence of a boundary-layer flow with relatively large velocity. As G/R2→∞ and G/R4→ 0, where G and R are respectively a sedimentation Grashof number and a sedimentation Reynolds number, the entrainment of suspended particles into the boundary-layer flow of clear liquid is negligibly small. This provides an appropriate boundary condi- tion for the basic flow of the suspension. Thus, in the double limit considered, a kine- matic theory suffices to determine the convective flow of the suspension due to the presence of inclined walls.As an example batch sedimentation in vessels with inclined plane or conical walls is investigated. The settling process is terminated after a time that can be considerably smaller than the time required in a vertical vessel under the same conditions.Depending on the initial particle concentration, there are centred kinematic waves that are linked to a continuous increase of the particle concentration in the suspension. In an appendix, the flow in the boundary layer at a downward facing, inclined wall is investigated. With G/R2→∞ and G/R4→ 0, the boundary layer consists of an inviscid particle-free main part, a viscous sublayer at the wall, and a free shear sublayer at the liquid/particle interface.

Multiple-valued and non-convergent solutions in kinematic cascade models

The Lax-Wendroff finite-difference solution of the equations of motion with the kinematic flow approximation is now widely used in hydrology in cascade models of overland flow. The physical relevance of kinematic shocks and the stability and convergence of the finite-difference solution are problems that are often undetected in contemporary applications due to the complexity of model inputs. Past criteria, developed for discerning either the presence of a shock or the adequacy of finite-difference solutions, are inadequate for complex cascade models. A general method is devised for locating the point along a cascade segment where the solution first becomes multiple-valued, for the lateral inflow situation. An example comparison with an exact solution reveals that stable, finite-difference solutions apparently converge to single-valued hydrographs and water surface profiles with spurious peaks that can go undetected in complex models. Finally, shock propagation and the potential for error growth in cascade models are discussed.