Abhishek Mukherjee, Amaresh Datta, Jayanta K Bhattacharjee
The first quantitative calculation of the hydraulic jump was carried out only a few years ago by Bohr et al. Since it is the only calculation of the jump, we have analysed it from a slightly different point of view.Our results are similar to those of Bohr et.al.
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Fluid dynamics video
This is a fluid dynamics video of two- and three-dimensional computational fluid dynamics simulations carried out at St. Anthony Falls Laboratory. A transient hydraulic jump is simulated using OpenFOAM, an open source numerical solver. A Volume of Fluid numerical method is employed with a realizable k-epsilon turbulence model. The goal of this research is to model the void fraction and bubble size in a transient hydraulic jump. This fluid dynamics video depicts the air entrainment characteristics and bubble behavior within a hydraulic jump of Froude number 4.82.
In this paper, we propose an analytical framework for internal hydraulic jumps. Density jumps or internal hydraulic jumps occur when a supper critical flow of water discharges into a stagnant layer of water with slightly different density. The approach used here is control volume method which is also used to analyze ordinary hydraulic jumps. The important difference here is that entrainment is taken into account. Using conservation equations with the aid of some simplifying assumptions we come to an equation that gives jump downstream height as function of jump upstream characteristics and the…
For more than a century, the consensus has been that the thin-film hydraulic jump that can be seen in kitchen sinks is created by gravity. However, we recently reported that these jumps are created by surface tension, and gravity does not play a significant role. In this paper, {we present experimental data for hydraulic jump experiments conducted in a micro-gravity environment ($\approx 2\%$ of Earth's gravity) (Avedisian \& Zhao 2000; Painter et al. 2007; Phillips et al. 2008). The existence of a hydraulic jump in micro-gravity unequivocally confirms that gravity is not the principal force c…
We carry out an analytical study of laminar circular hydraulic jumps, in generalized-Newtonian fluids obeying the two-parametric power-law model of Ostwald-de Waele. Under the boundary-layer approximation we obtained exact expressions determining the flow, an implicit relation for the jump radius is derived. Corresponding results for Newtonian fluids can be retrieved as a limiting case for the flow behavior index n=1, predictions are made for fluids deviating from Newtonian behavior.
We present a simple viscous theory of free-surface flows in boundary layers, which can accommodate regions of separated flow. In particular this yields the structure of stationary hydraulic jumps, both in their circular and linear versions, as well as structures moving with a constant speed. Finally we show how the fundamental hydraulic concepts of subcritical and supercritical flow, originating from inviscid theory, emerge at intermediate length scales in our model.
For more than a century, it has been believed that all hydraulic jumps are created due to gravity. However, we found that thin-film hydraulic jumps are not induced by gravity. This study explores the initiation of thin-film hydraulic jumps. For circular jumps produced by the normal impingement of a jet onto a solid surface, we found that the jump is formed when surface tension and viscous forces balance the momentum in the film and gravity plays no significant role. Experiments show no dependence on the orientation of the surface and a scaling relation balancing viscous forces and surface tens…
We present the results of some experiments on the circular hydraulic jump in normal and superfluid liquid helium. The radius of the jump and the depth of the liquid outside the jump are measured through optical means. Although the scale of the apparatus is rather small, the location of the jump is found to be consistent with the assumption that the jump can be treated as a shock, if the surface tension is taken into account. The radius of the jump does not change when going down in temperature through the lambda point; we think that the flow is supercritical. A remarkable feature of the experi…
When a falling jet of fluid strikes a horizontal fluid layer, a hydraulic jump arises downstream of the point of impact provided a critical flow rate is exceeded. We here examine a phenomenon that arises below this jump threshold, a circular deflection of relatively small amplitude on the free surface, that we call the hydraulic bump. The form of the circular bump can be simply understood in terms of the underlying vortex structure and its height simply deduced with Bernoulli arguments. As the incoming flux increases, a breaking of axial symmetry leads to polygonal hydraulic bumps. The relatio…
On the basis of the viscous Saint-Venant equations, hydraulic jumps in laminar open channel flow are obtained as continuous shock structures. Thanks to the inclusion of viscosity, the jumps are not abrupt, rendering the classic patchwork via the Rankine-Hugoniot shock relations unnecessary. The jumps arise as stable stationary solutions of the governing equations and lend themselves excellently to a Dynamical Systems analysis, manifesting themselves as near-parabolic trajectories in phase space. Based on this, we derive an analytic expression for the jump length as a function of the Froude and…
The properties of a standard hydraulic jump depend critically on a Froude number Fr defined as the ratio of the flow velocity to the gravity waves speed. In the case of a horizontal circular jump, the question of the Froude number is not well documented. Our experiments show that Fr measured just after the jump is locked on a constant value that does not depend on flow rate Q, kinematic viscosity ν and surface tension γ. Combining this result to a lubrication description of the outer flow yields, under appropriate conditions, a new and simple law ruling the jump radius RJ : $R_J (ln (\frac{R_\…
Rotation of non-circular hydraulic jumps is a recent experimental observation that lacks a theory based on first principles. Here we furnish a basic theory of this phenomenon founded on the shallow-water model of the circular hydraulic jump. The breaking of the axial symmetry morphs the circular jump into a polygonal state. Variations on this state rotate the polygon in the azimuthal direction. The dependence of the rotational frequency on the flow rate and on the number of polygon vertices agrees with known experimental results. We also predict how the rotational frequency varies with viscosi…
It was recently claimed by Bhagat et al. (J. Fluid Mech. vol. 851 (2018), R5) that the scientific literature on the circular hydraulic jump in a thin liquid film is flawed by improper treatment and severe underestimation of the influence of surface tension. Bhagat {\em et al.} use an energy equation with a new surface energy term that is introduced without reference, and they conclude that the location of the hydraulic jump is determined by surface tension alone. We show that this approach is incorrect and derive a corrected energy equation. Proper treatment of surface tension in thin film flo…
The formation of a planar hydraulic jump has been analysed in the framework of a full depth-averaged thin film model (DAM) with surface tension effects included. We have demonstrated regular weak solutions of the full DAM and analysed surface tension effects. It has been shown that surface tension effects within the parameter range relevant to the recent experiments are expected to be very weak and practically negligible. The developed methodology can be used in the analysis of laminar flow regimes and as a benchmark in developing full scale hydrodynamic models.
Oscillations and cavity modes in the circular hydraulic jump
本研究通过实验发现圆形水跃存在自发振荡，其周期取决于圆盘半径，并证明振荡频率与空腔表面波本征频率匹配，表明振荡由水跃与表面波模式相互作用自激产生。
We impose a linearized Eulerian perturbation on a steady, shallow, radial outflow of a liquid (water), whose local pressure function includes both the hydrostatic and the Laplace pressure terms. The resulting wave equation bears the form of a hydrodynamic metric. A dispersion relation, extracted from the wave equation, gives an instability due to surface tension and the cylindrical flow symmetry. Using the dispersion relation, we also derive three known relations that scale the radius of the circular hydraulic jump in the outflow. The first two relations are scaled by viscosity and gravity, wi…
Hydraulic jumps are oftentimes encountered in natural and human-made environments. The transition from supercritical to subcritical flow involves large energy dissipation rates and substantial air entrainment, preventing the use of monophasic flow measurement instrumentation. This paper presents an experimental study of a stable hydraulic jump with a Froude number of 4.25, utilizing novel intrusive phase-detection probe techniques and image-based velocimetry from a side perspective. Turbulence estimations were obtained for the impinging region and the roller region of the jump including Reynol…
In this paper, we seek an adequate macroscopic model for a hydraulic jump in Bingham fluid. The formulas for conjugate depths, sequent bottom shear stress and critical depth are established. Since no exact analytical solution in closed form is available for conjugate depths, an approximate formula is developed. This formula can provide good results with an error less than 4%. The analytical results have revealed that the critical depth and the ratio of conjugate depths increase until bottom shear stress exceeds a certain value and then decrease afterwards. The bottom shear stress downstream of…
Hydraulic jumps occur commonly in natural channels and energy dissipation systems of hydraulic structures in the violent transition from supercritical to subcritical flows. They are characterised by large flow aeration, high turbulence and strong fluctuations of the free surface and the jump toe. For free surface measurements, fast-sampling, fixed-point instruments such as acoustic displacement meters (ADMs) and wire gauges (WGs) are commonly used, while LIDAR technology is a relatively new method for recording instantaneous free surface motions of aerated flows. While each of these instrument…
Numerical Modeling of Flow and Air Entrainment in Hydraulic Jumps for a Wide Range of Froude Numbers
本文提出基于URANS框架的三相混合模型，用于模拟十二个弗鲁德数从1.98到8.48的水跃流体和空气混入，验证了该模型在计算效率与精度上的平衡。
Recently, an unusual scaling law has been observed in circular hydraulic jumps and has been attributed to a supposed missing term in the local energy balance of the flow [\cite{bhagat_2018}]. In this paper, we show that - though the experimental observation is valuable and interesting - this interpretation is presumably not the good one. When transposed to the case of a axial sheet formed by two impinging liquid jets, the assumed principle leads in fact to a velocity distribution in contradiction with the present knowledge for this kind of flows. We show here how to correct this approach by ke…
Free surface, axially symmetric shallow flow is analysed in both the centrifugal and centripetal directions. Referring to the inviscid steady flow over a flat plate characterised by a unique value of specific energy, the analytical sub- and supercritical solutions are determined. Furthermore, the sub- and supercritical steady solutions for the flow with friction over a flat plate are determined, provided that inertial terms are important compared with frictional terms. As the inviscid case, the sub- and supercritical solutions over a bottom topography are determined, provided that the bottom u…
We investigated numerically the relation between a roller and the pressure distribution to clarify the dynamics of the roller in circular hydraulic jumps. We found that a roller which characterizes a type II jump is associated with two high pressure regions after the jump, while a type I jump (without the roller) is associated with only one high pressure region. Our numerical results show that building up an appropriate pressure field is essential for a roller.
Fully conservative hydraulic jumps and solibores in two-layer Boussinesq fluids
研究两层Boussinesq流体中完全守恒的水跃与孤立波，发现两种守恒跃变组合并验证其与DNS结果一致，指出波色散是内波损耗的主要机制。
When a fluid jet strikes an inclined solid surface at normal incidence, gravity creates a flow pattern with a thick outer rim resembling a parabola and reminiscent of a hydraulic jump. There appears to be little theory or experiments describing simple aspects of this phenomenon, such as the maximum rise height of the fluid above the impact point, and its dependence on jet velocity and inclination angle. We address this with experiments, and present a simple theory based on horizontal hydraulic jumps which accounts for the rise height and its scaling, though without describing the shape of the …
We explore the complex dynamics of a non-coalescing drop of moderate size inside a circular hydraulic jump of the same liquid formed on a horizontal disk. In this situation the drop is moving along the jump and one observes two different motions: a periodic one (it orbitates at constant speed) and an irregular one involving reversals of the orbital motion. Modeling the drop as a rigid sphere exchanging friction with liquid across a thin film of air, we recover the orbital motion and the internal rotation of the drop. This internal rotation is experimentally observed.
Hydraulic jumps created by gravity are seen every day in the kitchen sink. We show that at small scales a circular hydraulic jump can be created in the absence of gravity, by surface tension. The theory is motivated by our experimental finding of a height discontinuity in spreading submicron molten metal droplets created by pulsed-laser ablation. By careful control of initial conditions, we show that this leads to solid femtolitre cups of gold, silver, copper, niobium and tin.
In the presence of viscosity the hydraulic jump in one dimension is seen to be a first-order transition. A scaling relation for the position of the jump has been determined by applying an averaging technique on the stationary hydrodynamic equations. This gives a linear height profile before the jump, as well as a clear dependence of the magnitude of the jump on the outer boundary condition. The importance of viscosity in the jump formation has been convincingly established, and its physical basis has been understood by a time-dependent analysis of the flow equations. In doing so, a very close …
The influence of surface tension in thin-film hydrodynamics: gravity free planar hydraulic jumps
本研究提出无重力条件下表面张力主导的薄液膜水跃理论，通过主导平衡分析建立无参数控制方程，并揭示韦伯数奇点作为水跃控制准则。
An experimental and numerical study of the circular hydraulic jump
本文通过实验和数值模拟研究圆形水跃，测量液膜厚度并估算韦伯数、弗劳德数和雷诺数，发现厨房水槽尺度下表面张力控制水跃，并定义了重力起作用的临界射流流量。
In the geometry of the circular hydraulic jump, the velocity of the liquid in the interior region exceeds the speed of capillary-gravity waves (ripplons), whose spectrum is `relativistic' in the shallow water limit. The velocity flow is radial and outward, and thus the relativistic ripplons cannot propagating into the interior region. In terms of the effective 2+1 dimensional Painleve-Gullstrand metric appropriate for the propagating ripplons, the interior region imitates the white hole. The hydraulic jump represents the physical singularity at the white-hole horizon. The instability of the va…
A wave equation for a time-dependent perturbation about the steady shallow-water solution emulates the metric an acoustic white hole, even upon the incorporation of nonlinearity in the lowest order. A standing wave in the sub-critical region of the flow is stabilised by viscosity, and the resulting time scale for the amplitude decay helps in providing a scaling argument for the formation of the hydraulic jump. A standing wave in the super-critical region, on the other hand, displays an unstable character, which, although somewhat mitigated by viscosity, needs nonlinear effects to be saturated.…
We investigate the numerical artifact known as a carbuncle, in the solution of the shallow water equations. We propose a new Riemann solver that is based on a local measure of the entropy residual and aims to avoid carbuncles while maintaining high accuracy. We propose a new challenging test problem for shallow water codes, consisting of a steady circular hydraulic jump that can be physically unstable. We show that numerical methods are prone to either suppress the instability completely or form carbuncles. We test existing cures for the carbuncle. In our experiments, only the proposed method …
Results from large-eddy simulations of a classical hydraulic jump at inlet Froude number 2 are reported. The computations are performed using the general-purpose finite-volume based code OpenFOAM, and the primary goal is to evaluate the influence of modelling parameters on the predictive accuracy, as well as establish associated best-practice guidelines. A benchmark simulation on a dense computational mesh is conducted, and good agreement with existing reference data is found. The remaining simulations cover different selections of modelling parameters: geometric vs algebraic interface capturi…
We propose a phenomenological model for the polygonal hydraulic jumps discovered by Ellegaard et al., based on the known flow structure for the type II hydraulic jumps with a "roller" (separation eddy) near the free surface in the jump region. The model consists of mass conservation and radial force balance between hydrostatic pressure and viscous stresses on the roller surface. In addition, we consider the azimuthal force balance, primarily between pressure and viscosity, but also including non-hydrostatic pressure contributions from surface tension in light of recent observations by Bush et …
The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct as a model for continuously-forced density-stratified flows such as those found in estuaries. Experiments have shown that the emergence of interfacial waves and their transition to turbulence as the tilt angle is increased appears linked to a threshold in the exchange flow rate given by inviscid two-layer hydraulics. We uncover these hydraulic mechanisms with (i) recent direct numerical simulations (DNS) providing full flow data in the key flow regimes (Zhu & Atoufi et al., arXiv:2301.09773, 2023), (ii)…
We study laminar thin film flows with large distortions in the free surface using the method of averaging across the flow. Two concrete problems are studied: the circular hydraulic jump and the flow down an inclined plane. For the circular hydraulic jump our method is able to handle an internal eddy and separated flow. Assuming a variable radial velocity profile like in Karman-Pohlhausen's method, we obtain a system of two ordinary differential equations for stationary states that can smoothly go through the jump where previous studies encountered a singularity. Solutions of the system are in …
An exact solution to the lock-exchange problem, which is a two-layer analogue of the classical dam-break problem, is obtained in the shallow-water (SW) approximation for two immiscible fluids with slightly different densities. The problem is solved by the method of characteristics using analytic expressions for the Riemann invariants. The obtained solution, which represents an inviscid approximation to the high-Reynolds-number limit, is in general discontinuous containing up to three hydraulic jumps which are due to either multivaluedness or instability of the continuous SW solution. Hydraulic…
Self-contained two-layer shallow water theory of strong internal bores
本文提出含自由参数α的局部守恒两层浅水动量方程，无需额外锋面条件即可描述强内涌，其Rankine-Hugoniot条件等价于经典锋面条件，并由实验和数值结果验证。
This paper presents the coupled model of a hydraulic fracturing and proppant transport. The former is described in terms of enhanced pseudo-3D model that considers height growth across two symmetric stress barriers, while the latter is given by two-dimensional transport model, stemming from the solution of an elliptical equation for fluid pressure and advection equation for the proppant transport. These two sub-modules are solved numerically using implicit time integration in hydraulic part and explicit time stepping in the transport part. In addition to that, interpolation is used to couple t…
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