Kinematic Shock Research Articles

The characteristics of ‘abrupt wave front’ floods on Pennine catchments, northern England, and their transmission downstream

Abstract Visible flood waves, described as abrupt wave front events (AWF), have been identified on rivers in northern England rising in the Pennines, from both historical and recent gauged data. The focus of this paper is on the characteristics of two gauged AWF events on the Rivers Wear and Tees in 1983 and their comparison to ‘normal’ floods. The description and analysis is based on contemporaneous photographs and observations and on digital level and flow records. The rapid 15-min rise in these events is compared with the maximum rate of rise in annual maximum peak floods by comparing flood hydrographs. The propagation of the flood wave downstream is illustrated. The 15-min increase in discharge is compared in relation to the peak flow for AWF and normal floods at different gauged locations down the catchments showing striking differences. The character of the AWF response in the vertical or near-vertical wave front and rapid increase in both level and discharge points to the occurrence of kinematic shock waves.

Using a proper orthogonal decomposition to elucidate features in granular flows

We apply proper orthogonal decomposition (POD) technique to analyze granular rheology in a laboratory-scale pulsed-fluidized bed. POD allows us to describe the inherent dynamics and energy budget in the dominant spatio-temporal modes in addition to identifying spatial coherence. This enables us to elucidate non-linear interactions between the different mechanisms which has been a shortcoming of conventional statistics-based approaches. The bubbling pattern is a result of interplay between the harmonic and sub-harmonic components. The mesoscopic flow features which contribute to the pattern are dependent on the modal energy budget which change with the pulsing frequency. It is also observed that the granular dynamics can be sufficiently reconstructed by the leading POD modes despite the presence of bubbles which represent kinematic shocks contributing to higher-order modes. In short, we highlight the utility of POD while analyzing fluidized granular flows, and pave the way for future analyses.Graphic abstractRaw tracks of experimental data (4Hz)

The elusive topic of rainfall-induced surge waves in rivers: lessons from canyon accidents

ABSTRACT Rainfall-induced surge waves in rivers are still today considered as an elusive and non-realistic phenomenon that has continued to mystify researchers and practitioners alike since the kinematic shock theory introduced in 1955. This is mainly due to scarce sightings of real cases, as well as physical and numerical uncertainties and sporadic calculations. In order to overcome such indefiniteness, a first set of documented real cases of surge waves in canyon rivers has been collected and their dreadful consequences highlighted. Then, the physical mechanism of surge wave development and propagation and the role of wave celerity differences are examined. A novel parameter (relating wave celerity and cross-section geometry) is introduced and used to highlight few relevant characteristics of rainfall-induced surge waves. The role of floods faster than ordinary flash floods (here defined as instant floods) is demonstrated. The conclusions are confirmed with a series of numerical tests for ideal and natural channels. Finally, differences from ordinary flash floods are identified in the framework of risk assessment, as well as the perspective to revisit real cases of surge waves in canyons and other streams.

Experimental study on void fraction, pressure drop and flow regime analysis in a large ID piping system

An experimental study of the characteristics of air-water two-phase flow in a piping system with an inner diameter of 152.4 mm is presented. Both horizontal and vertical pipes (with the lengths of 15.09 m and 6.42 m) are included in the system and two 90-degree elbows are used for their connections. 71 flow conditions are performed in this experiment, covering plug, slug, pseudo slug, and stratified flow for the horizontal flow, and bubbly, cap bubbly, churn turbulent, and falling film/annular flow for the vertical flow. Detailed flow regime analysis for the flow regimes of the horizontal and vertical pipe flow and the effects of the elbow on the flow regime transition are discussed. The flow regime maps proposed generally agree with the existing maps. A significant flow regime shift in the vertical section is also observed and discussed. The area averaged void fraction development is presented and a sharp drop of void fraction in vertical downward test section is observed which results from the kinematic shock phenomenon. The result of the frictional pressure drop is discussed, and it agrees with the Kim et al.’s correlation. With the database, the drift flux model for Goda et al.’s vertical downward flow is evaluated. Large difference has shown at low void fraction which may indicate further modeling investigation.

Experimental study of vertical and horizontal two-phase pipe flow through double 90 degree elbows

This paper presents an experimental study of the characteristics of the two-phase flow through vertical and horizontal pipes connected using 90-degree elbows with an inner diameter of 101.6 mm. 66 flow conditions are performed in this experiment, covering bubbly, plug, slug, pseudo slug, and stratified flow for horizontal pipe, and bubbly, cap bubbly, churn turbulent, and falling film/annular flow for vertical pipe. A detailed flow regime analysis for both horizontal and vertical pipe flow are discussed. Flow regime maps for both horizontal flow and vertical downward flow agree with the existing maps (Mandhane et al., 1974; Qiao et al., 2017). The effect of the elbow on the flow regime transition is primarily discussed based on the experimental results and observations. The effect of elbow observed in this experiment mainly contributes to the large bubble breakup and the change of void distribution. Area averaged void fraction development is presented and the sharp drop of void fraction in vertical downward test section is observed which results from kinematic shock phenomenon. A frictional pressure drop analysis is also studied using Lockhart-Martinelli correlation and it is found that C = 40, 100, and 125 are correlated with the frictional pressure drop data for horizontal, vertical, and whole test sections with minimum differences. With the database, the drift flux model for vertical downward flow (Goda et al., 2003) is validated and the relative difference between model and data is 15.95%.

On basic equations and kinematic-wave theory of separation processes in suspensions with gravity, centrifugal and Coriolis forces

Neglecting inertial and viscosity effects in the bulk flow is a common assumption in the analysis of separation processes in suspensions under the action of gravity or centrifugal and Coriolis forces. While there is a number of examples of particular solutions, the general form of the basic equations for three space dimensions, together with the appropriate boundary and initial conditions, is still uncertain and, with regard to certain aspects, even controversial. An essential point is a proper choice of the variables. Here it is proposed to introduce the mass density of the mixture, the mean mass velocity of the mixture and the total volume flux as a set of dependent variables. After some manipulations, a complete set of basic equations is obtained. It consists of two continuity equations, a generalized drift-flux relation, and two linearly independent components of a vector equation describing the total body force as irrotational. Then, by eliminating the mean mass velocity of the mixture from the set of unknowns, a generalized kinematic-wave equation is derived. It describes kinematic waves that are embedded in a bulk flow that may be one-, two- or three-dimensional. Concerning boundary conditions at solid walls, one has to ascertain whether the total body force at the wall points into the suspension or out of it. In the former case, a thin boundary layer of clear liquid is formed at the wall, whereas in the latter case a thin sediment layer may either stick at the wall or slide along it. Each of those three possibilities leads to a particular boundary condition for the bulk flow in terms of the dependent variables. In addition, initial conditions and kinematic shock relations are briefly discussed. Finally, the application of the kinematic-wave theory to the settling process in rotating tubes is outlined.

Hybrid WENO schemes for polydisperse sedimentation models

Polydisperse sedimentation models can be described by a strongly coupled system of conservation laws for the concentration of each species of solids. Typical solutions for the sedimentation model considered for batch settling in a column include stationary kinematic shocks separating layers of sediment of different composition. This phenomenon, known as segregation of species, is a specially demanding task for numerical simulation due to the need of accurate numerical simulations. Very high-order accurate solutions can be constructed by incorporating characteristic information, available due to the hyperbolicity analysis made in Donat and Mulet [A secular equation for the Jacobian matrix of certain multispecies kinematic flow models, Numer. Methods Partial Differential Equations 26 (2010), pp. 159–175.] But characteristic-based schemes, see Bürger et al. [On the implementation of WENO schemes for a class of polydisperse sedimentation models, J. Comput. Phys. 230 (2011), pp. 2322–2344], are very expensive in terms of computational time, since characteristic information is not readily available, and they are not really necessary in constant areas, where a less complex method can obtain similar results. With this idea in mind, in this paper we develop a hybrid finite difference WENO scheme that only uses the characteristic information of the Jacobian matrix of the system in those regions where singularities exist or are starting to develop, while it uses a component-wise approximation of the scheme in smooth regions. We perform some experiments showing the computational gains that can be achieved by this strategy.

Linear and nonlinear analysis of an unstable, but well-posed, one-dimensional two-fluid model for two-phase flow based on the inviscid Kelvin–Helmholtz instability

For nuclear reactor safety analysis, the one-dimensional two-fluid model equations are solved numerically with the first order upwind method because of its robust stability. In the present two-fluid model for horizontal stratified flow, surface tension is included because it makes the model well-posed. However, this is not done in industrial applications and numerical viscosity provides linear stabilization even when the model is ill-posed. It is now shown that numerical viscosity also provides nonlinear stabilization; meaning that the wave growth is bounded when the flow is unstable (e.g., in case of the Kelvin–Helmholtz instability). The formation of kinematic shocks in the presence of numerical viscosity provides the dissipation mechanism needed to stop the wave growth. However, numerical viscosity varies with the mesh size, which means that even though the unstable model is well-posed and meets von Neumann and nonlinear stabilization requirements, the solution does not converge for some short wavelengths greater than 2Δx. Furthermore, when the mesh size is large, significant artificial viscosity is added to the continuity and momentum equations and the model may become over-stabilized. A more scientific approach is proposed: to add a physical dissipation mechanism instead of numerical viscosity, i.e., Reynolds stresses. This approach has two advantages: first, the one-dimensional two-fluid model converges under severe dynamic conditions, such as the Kelvin–Helmholtz instability, and second, a higher order numerical method may be used instead of the first order upwind scheme without experiencing numerical excursions.

Low Reynolds number suspension gravity currents

The extension of a gravity current in a lock-exchange problem, proceeds as square root of time in the viscous-buoyancy phase, where there is a balance between gravitational and viscous forces. In the presence of particles however, this scenario is drastically altered, because sedimentation reduces the motive gravitational force and introduces a finite distance and time at which the gravity current halts. We investigate the spreading of low Reynolds number suspension gravity currents using a novel approach based on the Lattice-Boltzmann (LB) method. The suspension is modeled as a continuous medium with a concentration-dependent viscosity. The settling of particles is simulated using a drift flux function approach that enables us to capture sudden discontinuities in particle concentration that travel as kinematic shock waves. Thereafter a numerical investigation of lock-exchange flows between pure fluids of unequal viscosity, reveals the existence of wall layers which reduce the spreading rate substantially compared to the lubrication theory prediction. In suspension gravity currents, we observe that the settling of particles leads to the formation of two additional fronts: a horizontal front near the top that descends vertically and a sediment layer at the bottom which aggrandises due to deposition of particles. Three phases are identified in the spreading process: the final corresponding to the mutual approach of the two horizontal fronts while the laterally advancing front halts indicating that the suspension current stops even before all the particles have settled. The first two regimes represent a constant and a decreasing spreading rate respectively. Finally we conduct experiments to substantiate the conclusions of our numerical and theoretical investigation.

Spectral WENO schemes with Adaptive Mesh Refinement for models of polydisperse sedimentation

AbstractThe sedimentation of a polydisperse suspension with particles belonging to N size classes (species) can be described by a system of N nonlinear, strongly coupled scalar first‐order conservation laws. Its solutions usually exhibit kinematic shocks separating areas of different composition. Based on the so‐called secular equation [J. Anderson, Lin. Alg. Appl. 246, 49–70 (1996)], which provides access to the spectral decomposition of the Jacobian of the flux vector for this class of models, Bürger et al. [J. Comput. Phys. 230, 2322–2344 (2011)] proposed a spectral weighted essentially non‐oscillatory (WENO) scheme for the numerical solution of the model. It is demonstrated that the efficiency of this scheme can be improved by the technique of Adaptive Mesh Refinement (AMR), which concentrates computational effort on zones of strong variation. Numerical experiments for the cases N = 4 and N = 7 are presented.

On the implementation of WENO schemes for a class of polydisperse sedimentation models

The sedimentation of a polydisperse suspension of small rigid spheres of the same density, but which belong to a finite number of species (size classes), can be described by a spatially one-dimensional system of first-order, nonlinear, strongly coupled conservation laws. The unknowns are the volume fractions (concentrations) of each species as functions of depth and time. Typical solutions, e.g. for batch settling in a column, include discontinuities (kinematic shocks) separating areas of different composition. The accurate numerical approximation of these solutions is a challenge since closed-form eigenvalues and eigenvectors of the flux Jacobian are usually not available, and the characteristic fields are neither genuinely nonlinear nor linearly degenerate. However, the flux vectors associated with the widely used models by Masliyah, Lockett and Bassoon (MLB model) and Höfler and Schwarzer (HS model) give rise to Jacobians that are low-rank perturbations of a diagonal matrix. This property allows to apply a convenient hyperbolicity criterion that has become known as the “secular equation” [J. Anderson, A secular equation for the eigenvalues of a diagonal matrix perturbation, Lin. Alg. Appl. 246 (1996) 49–70]. This criterion was recently applied [R. Bürger, R. Donat, P. Mulet, C.A. Vega, Hyperbolicity analysis of polydisperse sedimentation models via a secular equation for the flux Jacobian, SIAM J. Appl. Math. 70 (2010) 2186–2213] to prove that the MLB and HS models are strictly hyperbolic under easily verifiable conditions, that their eigenvalues interlace with the velocities of the species that form the flux vector (so the velocities are good starting values for a root finder), and that the corresponding eigenvectors can be calculated with acceptable effort. In the present work, the newly available characteristic information is exploited for the implementation of characteristic-wise (spectral) weighted essentially non-oscillatory (WENO) schemes for the MLB and HS models. Numerical examples illustrate that WENO schemes which use this spectral information are superior in resolution, and even in efficiency for the same overall resolution, to component-wise WENO schemes.

THE RADIO JET INTERACTION IN NGC 5929: DIRECT DETECTION OF SHOCKED GAS

We report the discovery of kinematic shock signatures associated with a localized radio jet interaction in the merging Seyfert galaxy NGC 5929. We explore the velocity-dependent ionization structure of the gas and find that low ionization gas at the interaction site is significantly more disturbed than high ionization gas, which we attribute to a local enhancement of shock ionization due to the influence of the jet. The characteristic width of the broad low-ionization emission is consistent with shock velocities predicted from the ionization conditions of the gas. We interpret the relative prominence of shocks to the high density of gas in nuclear environment of the galaxy and place some constraints on their importance as a feedback mechanism in Seyferts.

On the settling of a bidisperse suspension with particles having different sizes and densities

The settling of a bidisperse suspension with small particles having different sizes and densities can be described by an initial value problem for a system of two non-linear, first-order conservation laws. Solutions to this problem are in general discontinuous and exhibit kinematic shocks that separate areas of different composition. The solution requires the construction of so-called elementary curves in phase space, which are determined from eigenvector fields of the Jacobian of the flux function. Differences in solution behavior to the previously analyzed equal-density case are due to an umbilic point, which appears for different densities only. The initial value problem is eventually solved by the front tracking method, which generates a series of Riemann problems. It turns out that the solution of the problem predicts a fairly complex process of sediment formation, and that the stationary solution can consist of non-constant smooth transitions. This observation is of interest for manufacturing of functionally graded materials.

Turbine vibration with a pulsed kinematic perturbation

A basic algorithm for calculation the rotor and bearing vibrations in a turbine due to a kinematic shock is outlined. Some results are discussed for linear vibrations of a rotor on three bearings with a determinate pulsed perturbation.

Adaptive multiresolution WENO schemes for multi-species kinematic flow models

Multi-species kinematic flow models lead to strongly coupled, nonlinear systems of first-order, spatially one-dimensional conservation laws. The number of unknowns (the concentrations of the species) may be arbitrarily high. Models of this class include a multi-species generalization of the Lighthill–Whitham–Richards traffic model and a model for the sedimentation of polydisperse suspensions. Their solutions typically involve kinematic shocks separating areas of constancy, and should be approximated by high resolution schemes. A fifth-order weighted essentially non-oscillatory (WENO) scheme is combined with a multiresolution technique that adaptively generates a sparse point representation (SPR) of the evolving numerical solution. Thus, computational effort is concentrated on zones of strong variation near shocks. Numerical examples from the traffic and sedimentation models demonstrate the effectiveness of the resulting WENO multiresolution (WENO-MRS) scheme.

Shallow Water Wave Propagation in Convectively Accelerating Open-Channel Flow Induced by the Tailwater Effect

Propagation of shallow water waves in viscous open-channel flows that are convectively accelerating or decelerating under gradually varying water surface profiles is theoretically investigated. Issues related to the hydrodynamics of wave propagation in a rectangular open channel are studied: the effect of viscosity in terms of the Manning coefficient; the effect of gravity in terms of the Froude number; wave translation and attenuation characteristics; nonlinearity and wave shock; the role of tailwater in wave propagation; and free surface instability. A uniformly valid nonlinear solution to describe the unsteady gradually varying flow throughout the complete wave propagation domain at and away from the kinematic wave shock as well as near the downstream boundary that exhibits the tailwater effect is derived by employing the matched asymptotic method. Different scenarios of hydraulically spatially varying surface profiles such as M1, M2, and S1 type profiles are discussed. Results from the nonlinear wave analysis are further interpreted and the influence of the tailwater effect is identified. In addition to the nonlinear wave analysis, a linear stability analysis is introduced to quantify the impact from such water surface profiles on the free surface instability. It is shown that the asymptotic flow structure is composed of three distinct regions: an outer region that is driven by gravity and channel resistance; a near wave shock region dominated by the convective inertia, pressure gradient, gravity and channel resistance; and a downstream boundary impact region where the convective inertia, pressure gradient, gravity and channel resistance terms are of importance. The tailwater effect is demonstrated influential to the flow structure, free surface stability, wave transmission mechanism, and hydrostatic pressure gradient in flow.

Shock waves in two-dimensional granular flow: effects of rough walls and polydispersity.

We have studied the two-dimensional flow of balls in a small-angle funnel, when either the side walls are rough or the balls are polydisperse. As in earlier work on monodisperse flows in smooth funnels, we observe the formation of kinematic shock waves (density waves). We find that for rough walls the flows are more disordered than for smooth walls and that shock waves generally propagate more slowly. For rough wall funnel flow, we show that the shock velocity and frequency obey simple scaling laws. These scaling laws are consistent with those found for smooth wall flow, but here they are cleaner since there are fewer packing-site effects and we study a wider range of parameters. For pipe flow (parallel side walls), rough walls support many shock waves, while smooth walls exhibit fewer or no shock waves. For funnel flows of balls with varying sizes, we find that flows with weak polydispersity behave qualitatively similar to monodisperse flows. For strong polydispersity, scaling breaks down and the shock waves consist of extended areas where the funnel is blocked completely.

Application of kinematic wave theory for predicting flash flood hazards on coupled alluvial fan–piedmont plain landforms

AbstractA rainstorm that caused a severe flash flood on the piedmont plain at the toe positions of two alluvial fans located to the west of the Organ Mountains in Dona Ana County, New Mexico, USA, is analysed. The space–time distributions of rainfall are evaluated from the Next Generation Weather Radar (NEXRAD) and overland flow is modelled as kinematic wave. The spatial distribution of rainfall shows a topographic control. The greatest rainfall depth, duration, and intensity occurred at the higher elevation mountain slopes and decreased with decreasing elevation from the alluvial fans to the piedmont plain. The alluvial fan–piedmont plain system is modelled by coupling divergent and rectangular overland flow planes. Explicit finite difference approximations, hybridized with the analytical method of characteristics, are made to the kinematic wave equations to account for the spatial and temporal distribution of the rainfall and variable boundary conditions. Simulation results indicate that sheet‐flow floodwater elevations rise (1) in a nonlinear fashion from the apex to toe positions of the alluvial fans, and (2) near linearly from the toe positions of the alluvial fans onto the piedmont plains with the formation of kinematic shocks near the middle to the upstream end of the plane at times between the initiation of the rainstorm and the time of concentration of the plane. Thus, the maximum flooding occurs at the middle or upstream sections of the piedmont plains regardless of the pattern of space–time variability of rainfall. These results are in agreement with observed geomorphologic features suggesting that piedmont plains are naturally flood‐prone areas. This case study demonstrates that flood hazards on piedmont plains can exceed those on alluvial fans. The models presented in this study suggest that the flood hazard zones on coupled alluvial fan–piedmont plain landforms should be delineated transverse to the flow directions, as opposed to the flood hazard zones with boundaries in the longitudinal direction of the axis of an alluvial fan. Copyright © 2003 John Wiley & Sons, Ltd.

Monte Carlo simulation of nearly kinematic shock fronts in rarefied gases

The effect of inter particle collisions on a “kinematic” shock wave produced by a velocity gradient in a cold atomic gas is studied, to the end of establishing a connection between such kinematic shocks and the “traditional” fluid dynamic shock front. In order to include the collisional effects we use both a DSMC (Direct Simulation Monte Carlo) model and a Monte Carlo solution of the Bhatnagar-Gross-Krook equation (BGK/MC). The substantial agreement of DSMC and BGK/MC results for a sufficiently high value of the collision frequency proves that the kinematic and thermodynamic descriptions of the shock front can be smoothly joined by scaling this parameter.

Use of resistance coefficients derived from single planes to estimate time of concentration of two-plane systems

By coupling the Darcy-Weisbach friction formula with the kinematic wave time of concentration formula, a kinematic-Darcy-Weisbach time of concentration formula for a series of planes is derived. The formula is applicable to a cascade of planes, to planes of different roughnesses, to planes of different flow regimes, to planes of different soil types and infiltration rates resulting in different net rainfall intensities, to planes subject to different rainfall intensities, and to planes with a combination of all these variables. By applying the observed times of concentration to the formula for single planes, the Darcy-Weisbach resistance coefficient for a concrete surface and for an artificial grass surface are derived. Based on the derived resistance coefficients, they indicate that the flow regime on concrete was transitional and the flow regime on artificial grass was laminar. Further, by applying the derived resistance coefficients to two-plane systems (i.e. planes with combinations of concrete surface and artificial grass surface in series), the comparisons of the estimated times of concentration from the formula with the observed values show very good agreement. In one particular two-plane system, even with the occurrence of kinematic shocks, the agreement is still good. This study shows that the resistance coefficients that are derived from uniform planes can be applied to two-plane systems.