Undular Jumps Research Articles

Challenges of Undular Jump Modelling

The study of open channel flow hydraulics extends beyond supercritical and subcritical flow states to include a distinct state known as near-critical flows. This category encompasses various hydraulic phenomena, with some of the most notable being solitary waves, cnoidal waves, and undular jumps. This paper specifically addresses the phenomenon of undular jumps, providing a brief overview of its characteristics and discussing instances of undular jump formation in natural rivers settings and during the operation of various hydraulic structures. When there is a sudden change in flow depth from a lower level to a higher one, it typically leads to a sudden increase in the water surface, known as a hydraulic jump. However, if the jump is relatively small, meaning the depth change is minor, the water does not rise noticeably and abruptly. Instead, it will traverse from the lower to the higher stage through a sequence of gradually diminishing undulations that extend over a considerable distance. Such phenomenon is called an undular jump.The occurrence of undular hydraulic jumps can be observed in various open channels such as irrigation and water supply channels, beneath vertical sluice gates, within estuaries during specific tide periods, and in narrow or shallow passages affected by strong currents. Additionally, this phenomenon often manifests downstream of low drop structures or in transitional zones from steep to gently sloping channels. In channels where undular jumps occur, waves with significant amplitudes develop and travel downstream of the jump. Accounting for the propagation of these downstream free-surface waves is essential for canal design and natural channel maintenance. The wave height serves as a crucial design parameter that dictates the necessary sidewall height of the canal. In natural channels, the embankment height must exceed the crest of the free-surface undulations to prevent overtopping and subsequent erosion, which could ultimately lead to bank destruction. Moreover, the propagation of free-surface waves may subject downstream canal structures, such as gates, locks, and weirs, to additional impact loads, perturbations, and vibrations.In recent years, different scientists have made various attempts to investigate and describe this phenomenon using physical, mathematical and computer modelling. However, this phenomenon has a number of peculiarities that are not always taken into account. An objective of this article is to present the particularities of different type of undular jump modelling and to compare the obtain results.

Circular undular hydraulic jumps in turbulent free-surface flows

Near-critical turbulent axisymmetric free-surface flow over a horizontal bottom is investigated. Assuming that undular hydraulic jumps occur at large radii, an asymptotic analysis of the governing equations is performed in the double limit of very large Reynolds numbers and Froude numbers close to the critical value 1. The results are kept free of turbulence modelling due to a specific coupling of the two limiting processes. The final result of the asymptotic analysis is a new steady-state version of an extended Korteweg–de Vries (KdV) equation for the free-surface elevation. The extended KdV equation is derived as a uniformly valid differential equation, describing the flow near the origin of the undular jump as well as far downstream. Numerical solutions of the extended KdV equation show that circular undular jumps can develop if the reference state is located in the region where the effect due to axisymmetry prevails over the effect of friction. In this case, the solution oscillates over a very long distance until the accumulating friction effects force a breakdown. The comparison between the theories of undular jumps in turbulent and inviscid flows shows that friction is of minor importance near the development of the undular jump. However, friction has to be taken into account to describe the flow further downstream. Remarkably, the extended KdV equation is valid and yields undular solutions for both turbulent source and sink flow.

Flow Patterns and Turbulent Kinetic Energy Budget of Undular Jumps in a Narrow Flume

AbstractThis paper presents results obtained from a detailed analysis of the experimental data of three hydraulic jumps with low Froude numbers (1.5–1.9) in a narrow flume. With the increasing Frou...

Decelerating bores in channels and estuaries

ABSTRACT Decelerating bores are commonly seen in shorelines, estuaries and rivers in forms of swash run-up, tidal bores, tsunami bores. A decelerating bore propagating upstream can gradually change its shape, finally becoming an arrested bore, i.e. a stationary hydraulic jump. New experiments on decelerating bores against an adverse slope were conducted. Observations highlighted various types of arrested bores: fully breaking jumps, partially breaking jumps and non-breaking undular jumps. Measurements were repeated at least 25 times to obtain ensemble-averaged data with regards to instantaneous median and fluctuations of free-surface elevation, velocity components and turbulent shear stresses. An abrupt rise of free-surface elevation and immediate decrease in stream-wise velocity were observed during the passage of a decelerating bore. The arrival of decelerating bores induced some drastic increase of instantaneous free-surface fluctuations and all velocity components. Large-amplitude Reynolds stresses and extreme Reynolds stress fluctuations occurred in the same phase during and after the passage of decelerating bores. Histogram analysis of instantaneous normal and tangential Reynolds stresses suggested a preponderance of relatively smaller amplitudes. The upstream propagation of decelerating bores increased the probability density of large normal and tangential Reynolds stresses, yielding extrema vastly exceeding the critical threshold for inception of sediment motion.

Numerical study of laminar, standing hydraulic jumps in a planar geometry.

We solve the two-dimensional, planar Navier-Stokes equations to simulate a laminar, standing hydraulic jump using a Volume-of-Fluid method. The geometry downstream of the jump has been designed to be similar to experimental conditions by including a pit at the edge of the platform over which liquid film flows. We obtain jumps with and without separation. Increasing the inlet Froude number pushes the jump downstream and makes the slope of the jump weaker, consistent with experimental observations of circular jumps, and decreasing the Reynolds number brings the jump upstream while making it steeper. We study the effect of the length of the domain and that of a downstream obstacle on the structure and location of the jump. The transient flow which leads to a final steady jump is described for the first time to our knowledge. In the moderate Reynolds number regime, we obtain steady undular jumps with a separated bubble underneath the first few undulations. Interestingly, surface tension leads to shortening of wavelength of these undulations. We show that the undulations can be explained using the inviscid theory of Benjamin and Lighthill (Proc. R. Soc. London, Ser. A, 1954). We hope this new finding will motivate experimental verification.

Depth-averaged model for undular hydraulic jump

ABSTRACTAn undular hydraulic jump corresponds to the weak transition from super- to subcritical-flow in the form of steady free surface undulations. Previous models on undular hydraulic jumps employed the potential flow theory, i.e., the solitary and cnoidal wave theories. Experimental observations indicate the inadequacy of this theory, which motivated the development of more advanced approximations. Basic flow features including friction effects on the velocity profile, modelling of the bed-shear stress, and Reynolds stresses are considered. However, none of the models currently available include all these aspects. In this study, a general depth-averaged model is developed based on the k-ϵ turbulence closure. The general depth-averaged equations are applied to the undular jump problem, introducing a suitable time-averaged velocity distribution based on a composite power-law model, in which both streamline curvature and vorticity are accounted for. The bed-shear stress closure is included by a boundary layer method. Predictions of the depth-averaged Reynolds averaged Navier-Stokes (RANS) model are shown to be close to the 2D RANS solution and to experimental observations.

3D SPH modelling of hydraulic jump in a very large channel

The formation of different undular hydraulic jumps in a very large channel is investigated and reproduced using a weakly-compressible XSPH scheme which includes a mixing-length turbulence model. An analysis of the ability and of the limits of the SPH method to reproduce undular hydraulic jumps is preliminarily performed on reference two-dimensional cases. The numerical description of the three-dimensional jump in a very large channel, where the hydraulic-jump front is trapezoidal and the lateral shock waves induce a large recirculation region along the side walls, is compared with experiments in a laboratory flume on two undular jumps at upstream Froude number equal to 3.9 and 8.3. Acoustic Doppler velocity measurements were compared with SPH instantaneous and time-averaged flow fields in order to evaluate whether the numerical method could help in having a clearer understanding of both hydraulic-jump development and lateral shockwave formation. The predicted free-surface elevations and velocity profiles show a satisfactory agreement with measurements and most of the peculiar features of the flow, such as the trapezoidal shape of the wave front and the flow separations at the toe of the oblique shock wave along the side walls, are qualitatively and quantitatively reproduced.

Numerical simulation of undular jumps on graveled bed using volume of fluid method

Undular hydraulic jumps are characterized by a smooth rise of the free surface, followed by a train of stationary waves. These jumps sometimes occur in natural waterways and rivers. Numerical difficulties are especially distinct when the flow condition is close to the critical value because of the high sensitivity of the near-critical flow field to flow and channel conditions. Furthermore, the free surface has a wavy shape, which may indicate the occurrence of several transitions from supercritical to subcritical states and vice versa (i.e., undular hydraulic jumps). In this study, a flow model is used to predict an undular hydraulic jump in a rectangular open channel. The model is based on the general two-dimensional, Reynolds-averaged, Navier-Stokes flow equations. The resulting set of partial differential equations is solved using the FLOW-3D solver. The results are compared with the experimental data to validate the model. The comparative analysis shows that the proposed model yields good results. Several types of undular hydraulic jumps occurring in different situations are then simulated to prove the potential application of the model.

Undular hydraulic jumps arising in non-developed turbulent flows

Turbulent plane flow over a bottom of constant slope is considered for very large Reynolds numbers, very small slopes of the bottom, and Froude numbers close to the critical value 1. In contrast to a previous work (Grillhofer and Schneider, Phys Fluids 15:730–735, 2003), it is not assumed that the flow far upstream is fully developed. The first-order perturbation equations contain unknown functions that are determined from a solvability condition of the second-order equations. Without making use of turbulence modeling or empirical parameters, a third-order ordinary differential equation is obtained for the shape of the free surface. Slow changes of amplitudes and wave lengths, respectively, associated with a small damping parameter are described by a multiple-scales solution, which also reveals the source of peculiarities of numerical solutions. A universal diagram of solutions and a universal map of the initial conditions that lead to undular jumps are given. Both numerical and multiple-scales solutions are compared with experimental data.

Vorticity equation for the streamline and the velocity profile

Water flow over short hydraulic transitions is usually modelled by the irrotational flow theory, for which the energy head remains constant over the entire flow domain. Velocity and pressure distributions are then obtained based on the stream and potential functions of the flow net. However, experimental evidence may dictate non-uniform energy profiles resulting either from vorticity, viscous effects, or both. For rapidly-varied flows, Hunter Rouse (1906–1996) proposed in the early 1930s the inviscid Euler flow equations. Most of the current curvilinear flow models, however, resort to irrotational flow. In this work, a generalized curvilinear method for real fluid flow is proposed. Based on the streamline vorticity equation, a practical approach is presented for inviscid, rotational flow. The model is verified by data pertaining to the free overfall and further tested with data on viscous flow in undular jumps, thereby resulting in good agreement.

Review of Wastewater Hydraulics: Theory and Practice, 2nd Ed., by Willi H. HagerWastewater Hydraulics: Theory and PracticeSpringer-Verlag978-3-642-11382-6

Review of Wastewater Hydraulics: Theory and Practice, 2nd Ed., by Willi H. HagerWastewater Hydraulics: Theory and PracticeSpringer-Verlag978-3-642-11382-6

Weakly undular hydraulic jump: effects of friction

The undular hydraulic jump is a transitional flow phenomenon where both streamline curvature and frictional effects are important. In the past, streamline curvature effects were treated with the Boussinesq equations for the potential flow approach, thereby overlooking the real flow features. If friction is included, additional terms appear, because the specific energy variation along the undular jump is related to the boundary shear stress. However, these effects are neither addressed in the literature, nor compared with the classical Boussinesq-type solutions. Also, neither information on how boundary layer methods for adverse pressure gradients behave in undular flows is available, nor experimental data. Herein the frictional effects on the Boussinesq equations are systematically analysed using boundary layer methods for adverse pressure gradients. Based on these results, a new simplified approach is proposed to reasonably reproduce the oscillatory boundary layer characteristics in weakly undular hydraulic jumps under a steadily changing pressure gradient from adverse to favourable, and vice versa.

Upstream Internal Jumps in Stratified Sill Flow: Observations of Formation, Evolution, and Release

Abstract The time-dependent response of upstream undular bores and internal hydraulic jumps from initial formation to eventual release is documented. Two events, characterized by qualitatively different responses, are discussed. In the first case, an undular bore develops upstream of the sill crest. This disturbance remains upstream through the ebb tidal flow but is transformed to a hydraulic jump as its amplitude increases. Toward the end of ebb tide, it is released and subsequently disperses into a group of solitary-like waves. During the second event, an upstream jump also develops at an early stage of the tide. However, it is subsequently swept downstream by the tidal flow such that the upstream region then appears featureless. Approaching slack tide, as an exchange flow becomes established, a large bore or gravity current is emitted. The different responses seen in these two events are interpreted in terms of the Froude number associated with the near-surface stratification.

The circular internal hydraulic jump

Circular hydraulic jumps are familiar in single layers. Here we report the discovery of similar jumps in two-layer flows. A thin jet of fluid impinging vertically onto a rigid horizontal plane surface submerged in a deep layer of less-dense miscible fluid spreads radially, and a near-circular internal jump forms within a few centimetres from the point of impact with the plane surface. A jump is similarly formed as a jet of relatively less-dense fluid rises to the surface of a deep layer of fluid, but it appears less stable or permanent in form. Several experiments are made to examine the case of a downward jet onto a horizontal plate, the base of a square or circular container. The inlet Reynolds numbers, Re, of the jet range from 112 to 1790. Initially jumps have an undular, laminar form with typically 2–4 stationary waves on the interface between the dense and less-dense layers but, as the depth of the dense layer beyond the jump increases, the transitions become more abrupt and turbulent, resulting in mixing between the two layers. During the transition to a turbulent regime, single and sometimes moving multiple cusps are observed around the periphery of jumps. A semi-empirical model is devised that relates the parameters of the laboratory experiment, i.e. flow rate, inlet nozzle radius, kinematic viscosity and reduced gravity, to the layer depth beyond the jump and the radius at which an undular jump occurs. The experiments imply that surface tension is not an essential ingredient in the formation of circular hydraulic jumps and demonstrate that stationary jumps can exist in stratified shear flows which can be represented as two discrete layers. No stationary circular undular jumps are found, however, in the case of a downward jet of dense fluid when the overlying, less-dense, fluid is stratified, but a stationary turbulent transition is observed. This has implications for the existence of stationary jumps in continuously stratified geophysical flows: results based on two-layer models may be misleading. It is shown that the Froude number at which a transition of finite width occurs in a radially diverging flow may be less than unity.

Analysis of the velocity field in a large rectangular channel with lateral shockwave

In this work the authors describe the main characteristics of the velocity field of hydraulic jumps in a very large channel where lateral shockwaves occur. Experiments were carried out at the Coastal Engineering Laboratory of the Water Engineering and Chemistry Department of the Technical University of Bari (Italy). Extensive flow velocity measurements were investigated in order to have a clearer understanding of both hydraulic jump development and lateral shockwave formation in a very large channel. Eight experiments were performed in a 4m wide rectangular channel; the experiments differed in the inlet Froude number F0 and the jump type. Seven tests were carried out with undular jumps and one with a roller jump. The flow velocity and the flow free surface measurements were taken using a two-dimensional Acoustic Doppler Velocimeter (ADV) and an ultrasonic profiler, respectively. The experimental results can be summarized as follow: (i) the formation of well developed lateral shockwaves similar to those of oblique jumps were observed; (ii) the comparison of the experimental and theoretical data shows that the classic shockwave theory is sufficiently confirmed in the analyzed range of Reynolds number, taking into account the experimental errors and the difference between the theoretical and experimental assumptions; (iii) the transversal flow velocity profiles in the recirculating zone show a good agreement with the numerical simulations presented in literature in the case of a separated turbulent boundary layer over a flat plate. This conclusion enables us to confirm the hypothesis that the lateral shockwaves in the channel are the result of a boundary layer which, as observed, forms on the channel sidewalls.

Discussion of “Particle Image Velocity Measurements of Undular and Hydraulic Jumps” by J. M. Lennon and D. F. Hill

Discussion of “Particle Image Velocity Measurements of Undular and Hydraulic Jumps” by J. M. Lennon and D. F. Hill

Dimensions of standing waves at steps in mountain rivers

Oscillations of the water surface are common features of flows around critical conditions (Fr ∼ 1) and of both natural and structure‐induced undular hydraulic jumps. The paper describes experimental tests and field data on the dimensions of standing waves created by artificial drops in steep gravel bed rivers. The purpose of the research is to analyze what parameters affect the phenomenon, to provide semiempirical formulations for wave dimensions, and to assess whether previous models for undular hydraulic jumps and antidunes are reasonably predictive. The dimensionless wavelength and wave amplitude were correlated to the Froude number at the sill. The first wavelength appears linearly related to the Froude parameter, whereas the amplitude of the first wave shows a less defined behavior, possibly indicating that a maximum value occurs for sill Froude numbers around 1.6–1.7. The corresponding wave steepness (i.e., wave amplitude/length) results between 0.15–0.16, and this range also characterizes the point at which the dimensionless trough depth below the sill achieves a local maximum, probably related to hydrodynamic conditions just before wave breaking. The transition between roller and undular jumps at drops is then argued to likely represent a critical stage in step pool streams, and the hydraulic and morphological implications of this finding are also discussed, with particular emphasis on step stability and step wavelength.

Open Channel Waves that Form after the Removal of a Screen before Uneven Bed

The results of an experimental study of the gravity waves that form in an open channel at the decomposition of an initial discontinuity in the free surface level above an uneven bed in the form of a shelf are discussed. Self-similar solutions obtained in the context of the first approximation of the shallow-water theory are tested. It is shown that more complex mathematical models must be used to adequately describe the waves of level drop in the upper pool and undular hydraulic jumps.

波状跳水の流況特性に対する水路勾配の影響

波状跳水の流況特性に対する水路勾配の影響

Flow Conditions of Undular Hydraulic Jumps in Horizontal Rectangular Channels

The flow conditions of undular jumps for fully developed inflow condition have been investigated systematically. If the inflow Froude number is larger than 1.2, an undular jump has lateral shock waves near the toe of the jump. For a narrow channel, the shock waves cross upstream of the first wave crest, and the flow conditions of undular jumps depend on the aspect ratio and the inflow Froude number. In contrast, for a wide channel the shock waves do not cross upstream of the first wave crest, and the flow conditions of undular jumps are independent of the aspect ratio. The flow conditions of undular jumps are classified by considering the cross position of the lateral shock waves and the inflow Froude number. Also, the hydraulic conditions for the formation of nonbreaking and breaking undular jumps are determined. The effect of the Reynolds number on undular-jump formations is discussed, and changes of the flow conditions with the Reynolds number are described.