The role of gravity in the prediction of the circular hydraulic jump radius for high-viscosity liquids

2019 ◽  
Vol 862 ◽  
pp. 128-161 ◽  
Author(s):  
Yunpeng Wang ◽  
Roger E. Khayat
Keyword(s):  
Flow Rate ◽  
Hydraulic Jump ◽  
Free Surface Flow ◽  
High Viscosity ◽  
Surface Flow ◽  
Type I ◽  
Capillary Length ◽  
Flow Energy ◽  
Thin Film Equation ◽  
Downstream Flow

The free-surface flow formed by a circular jet impinging on a stationary disk is analysed theoretically. We develop a simple and coherent model to predict the location and height of the jump for high-viscosity liquids. The study explores the effect of gravity in the supercritical flow. The formulation reduces to a problem, involving only one parameter: $\unicode[STIX]{x1D6FC}=Re^{1/3}Fr^{2}$, where $Re$ and $Fr$ are the Reynolds and Froude numbers based on the flow rate and the jet radius. We show that the jump location coincides with the singularity in the thin-film equation when gravity is included, suggesting that the jump location can be determined without the knowledge of downstream flow conditions such as the jump height, the radius of the disk, which corroborates earlier observations in the case of type I circular hydraulic jumps. Consequently, there is no need for a boundary condition downstream to determine the jump radius. Our results corroborate well existing measurements and numerical simulation. Our predictions also confirm the constancy of the Froude number $Fr_{J}$ based on the jump radius and height as suggested by the measurements of Duchesne et al. (Europhys. Lett., vol. 107, 2014, 54002). We establish theoretically the conditions for $Fr_{J}$ to remain independent of the flow rate. The subcritical flow and the height of the hydraulic jump are sought subject to the thickness at the edge of the disk, comprising contributions based on the capillary length and minimum flow energy. The thickness at the edge of the disk appears to be negligibly small for high-viscosity liquids.

Hydraulic jump in circular pipes

1999 ◽  
Vol 26 (3) ◽  
pp. 368-373 ◽  
Author(s):  
Helmut Stahl ◽  
Willi H Hager
Keyword(s):  
Pipe Flow ◽  
Hydraulic Jump ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Main Flow ◽  
Hydraulic Jumps ◽  
Circular Pipes ◽  
Conduit Diameter ◽  
Sequent Depth ◽  
Downstream Flow

Hydraulic jumps in conduits containing free surface flow have received practically no attention. This project was conducted to investigate experimentally the main features of such jumps and to obtain limits for conduit choking. The sequent depth ratio is determined in terms of the approach Froude number based on the conventional momentum approach. The lengths of the surface recirculation and aeration zones are also considered. Two different appearances of jumps are discussed and it is demonstrated that jumps with a small approach depth differ from those with a depth larger than about 30% of the conduit diameter. A choking condition is proposed for which conduits are subjected to full pipe downstream flow. Photographs are used to describe the main flow pattern. The results of this study are readily applicable for design.Key words: aeration, conduit choking, hydraulic jump, pipe flow, sequent depths.

Impinging jet flow and hydraulic jump on a rotating disk

2018 ◽  
Vol 839 ◽  
pp. 525-560 ◽  
Author(s):  
Yunpeng Wang ◽  
Roger E. Khayat
Keyword(s):  
Film Thickness ◽  
Hydraulic Jump ◽  
Free Surface Flow ◽  
Rotating Disk ◽  
Surface Flow ◽  
Thin Film Flow ◽  
Capillary Length ◽  
Rotating Speed ◽  
Stationary Disk ◽  
Low Viscosity

The free-surface flow formed by a circular jet impinging on a rotating disk is analysed theoretically. The study explores the effects of rotation and inertia on the thin-film flow. Both boundary-layer height and film thickness are found to diminish with rotation speed. A maximum film thickness develops in the supercritical region, which reflects the competition between the convective and centrifugal effects. Unlike the flow on a stationary disk, an increase in the wall shear stress along the radial direction is predicted, at a rate that strengthens with rotating speed. Our results corroborate well existing measurements. The location and height of the hydraulic jump are determined subject to the value of the thickness at the edge of the disk, which is established first for a stationary disk based on the capillary length, and then for a rotating disk using existing analyses and measurements in spin coating. The case of a stationary is revisited in an effort to predict the location and height of the jump uniquely. The formulated value of the height at the edge of the disk seems to give excellent results for a jet at moderately high flow rate (or low viscosity) where the jump structure is well identifiable in reality.

scholarly journals Flow from a source above a sloping base

2020 ◽  
Vol 61 ◽  
pp. C75-C88
Author(s):  
Shaymaa Mukhlif Shraida ◽  
Graeme Hocking
Keyword(s):  
Free Surface ◽  
Flow Rate ◽  
Water Waves ◽  
Line Source ◽  
Free Surface Flow ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Surface Flow ◽  
Theoretical Research ◽  
Line Sink

We consider the outflow of water from the peak of a triangular ridge into a channel of finite depth. Solutions are computed for different flow rates and bottom angles. A numerical method is used to compute the flow from the source for small values of flow rate and it is found that there is a maximum flow rate beyond which steady solutions do not seem to exist. Limiting flows are computed for each geometrical configuration. One application of this work is as a model of saline water being returned to the ocean after desalination. References Craya, A. ''Theoretical research on the flow of nonhomogeneous fluids''. La Houille Blanche, (1):22–55, 1949. doi:10.1051/lhb/1949017 Dun, C. R. and Hocking, G. C. ''Withdrawal of fluid through a line sink beneath a free surface above a sloping boundary''. J. Eng. Math. 29:1–10, 1995. doi:10.1007/bf00046379 Hocking, G. ''Cusp-like free-surface flows due to a submerged source or sink in the presence of a flat or sloping bottom''. ANZIAM J. 26:470–486, 1985. doi:10.1017/s0334270000004665 Hocking, G. C. and Forbes, L. K. ''Subcritical free-surface flow caused by a line source in a fluid of finite depth''. J. Eng. Math. 26:455-466, 1992. doi:10.1007/bf00042763 Hocking, G. C. ''Supercritical withdrawal from a two-layer fluid through a line sink", J. Fluid Mech. 297:37–47, 1995. doi:10.1017/s0022112095002990 Hocking, G. C., Nguyen, H. H. N., Forbes, L. K. and Stokes,T. E. ''The effect of surface tension on free surface flow induced by a point sink''. ANZIAM J., 57:417–428, 2016. doi:10.1017/S1446181116000018 Landrini, M. and Tyvand, P. A. ''Generation of water waves and bores by impulsive bottom flux'', J. Eng. Math. 39(1–4):131-170, 2001. doi:10.1023/A:1004857624937 Lustri, C. J., McCue, S. W. and Chapman, S. J. ''Exponential asymptotics of free surface flow due to a line source''. IMA J. Appl. Math., 78(4):697–713, 2013. doi:10.1093/imamat/hxt016 Stokes, T. E., Hocking, G. C. and Forbes, L.K. ''Unsteady free surface flow induced by a line sink in a fluid of finite depth'', Comp. Fluids, 37(3):236–249, 2008. doi:10.1016/j.compfluid.2007.06.002 Tuck, E. O. and Vanden-Broeck, J.-M. ''A cusp-like free-surface flow due to a submerged source or sink''. ANZIAM J. 25:443–450, 1984. doi:10.1017/s0334270000004197 Vanden-Broeck, J.-M., Schwartz, L. W. and Tuck, E. O. ''Divergent low-Froude-number series expansion of nonlinear free-surface flow problems". Proc. Roy. Soc. A., 361(1705):207–224, 1978. doi:10.1098/rspa.1978.0099 Vanden-Broeck, J.-M. and Keller, J. B. ''Free surface flow due to a sink'', J. Fluid Mech, 175:109–117, 1987. doi:10.1017/s0022112087000314 Yih, C.-S. Stratified flows. Academic Press, New York, 1980. doi:10.1016/B978-0-12-771050-1.X5001-3

scholarly journals A modified hydrodynamic model for routing unsteady flow in a river having piedmont zone

2017 ◽  
Vol 65 (1) ◽  
pp. 60-67
Author(s):  
Sudarshan Patowary ◽  
Arup Kumar Sarma
Keyword(s):  
Unsteady Flow ◽  
Hydrodynamic Model ◽  
River Flow ◽  
Free Surface Flow ◽  
Flood Management ◽  
Surface Flow ◽  
Flow Equations ◽  
Piedmont Zone ◽  
River Bed ◽  
Downstream Flow

Abstract Existence of piedmont zone in a river bed is a critical parameter from among numerous variations of topographical, geological and geographical conditions that can significantly influence the river flow scenario. Downstream flow situation assessed by routing of upstream hydrograph may yield higher flow depth if existence of such high infiltration zone is ignored and therefore it is a matter of concern for water resources planning and flood management. This work proposes a novel modified hydrodynamic model that has the potential to accurately determine the flow scenario in presence of piedmont zone. The model has been developed using unsteady free surface flow equations, coupled with Green-Ampt infiltration equation as governing equation. For solution of the governing equations Beam and Warming implicit finite difference scheme has been used. The proposed model was first validated from the field data of Trout Creek River showing excellent agreement. The validated model was then applied to a hypothetical river reach commensurate with the size of major tributaries of Brahmaputra Basin of India. Results indicated a 10% and 14% difference in the maximum value of discharge and depth hydrograph in presence and absence of piedmont zone respectively. Overall this model was successfully used to accurately predict the effect of piedmont zone on the unsteady flow in a river.

Simulation of Solid-Liquid Interaction in Presence of a Free Surface

2011 ◽  
Author(s):  
Iman Mirzaii ◽  
Mohammad Passandideh-Fard
Keyword(s):  
Free Surface ◽  
Rigid Motion ◽  
Free Fall ◽  
Free Surface Flow ◽  
High Viscosity ◽  
Surface Flow ◽  
Computational Domain ◽  
Solid Object ◽  
Solid Liquid Interface ◽  
Solid Liquid

In this study, a numerical algorithm is developed for simulating the interactions between a liquid and solid object in presence of a free-surface flow. The presented model is that of the fast-fictitious-domain method integrated into the volume-of-fluid (VOF) technique used for tracking the free surface motion. The developed model considers the solid object as a fluid with a high viscosity resulting in a rigid motion of the object and solves the governing equations everywhere in the computational domain including the solid object. In this methodology, the application of the no-slip condition on the solid-liquid interface and the evaluation of the acting forces on the solid object are performed implicitly. The developed model is validated by a comparison of the simulation results with those of the available experiments in the literature for the free fall of one and two circular disks in a liquid domain and a sphere during its entry into a more dense liquid through a free surface. For all cases considered, the results are in good agreement with those of the experiments and other numerical studies. The model is then used to simulate the complex liquid-solid interaction during the entry of a spinning disk into a liquid free surface.

Free Surface Flow Profile and Fluctuations of a Circular Hydraulic Jump Formed by an Impinging Jet

1995 ◽  
Vol 117 (4) ◽  
pp. 677-682 ◽  
Author(s):  
J. W. Stevens
Keyword(s):  
Free Surface ◽  
Hydraulic Jump ◽  
Surface Profile ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Flow Profile ◽  
Flow Depth ◽  
Supercritical Flow ◽  
Jump Size ◽  
Free Surface Profile

A fine wire probe was used to make quantitative measurements of the free surface profile and surface fluctuations around the hydraulic jump formed by a normally impinging free liquid jet. Representative magnitudes of both radial and axial fluctuations were presented for two nozzle sizes and several jet Reynolds numbers and subcritical flow depths. The results were compared to previous measurements of the supercritical flow depth and to theoretical predictions of the circular hydraulic jump size. The agreement appeared reasonable for the supercritical flow depth while the analytical expressions predicted a shorter hydraulic jump than that found by the measurements for the same supercritical flow conditions.

scholarly journals CFD Software Applications for Transcritical Free Surface Flow

2009 ◽  
Author(s):  
Saira F. Pineda ◽  
Armando J. Blanco ◽  
Luis Rojas-Solo´rzano
Keyword(s):  
Free Surface ◽  
Hydraulic Jump ◽  
Stokes Equations ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Commercial Software ◽  
Flow Equations ◽  
Software Packages ◽  
Commercial Package ◽  
Benchmark Solutions

Open flow channel is very common in engineering applications. Traditional approaches solve shallow-water flow equations, known as Saint-Venant equations, when one or two dimension solutions can be adequate for obtaining most of the important flow characteristics. However, complex situations can require solving Navier-Stokes equations. The arrival of high performance computers and commercial software packages offers new possibilities in the field of numerical hydraulics. However, commercial software packages should be tested on some specific cases; so that these can be used with confidence. In this paper we solve several cases of free surface flow that consider subcritical, supercritical, critical, oscillatory depth profiles and hydraulic jumps using a commercial package, CFX™. Most of these cases are proposed as benchmark solutions for non-prismatic cross section, non-uniform bed slope and transition between subcritical and supercritical flow. Other cases as Hydraulic jump consist of experimental data of hydraulics jumps for incident flow with Froude numbers up to 4.23. Both types of cases allow us to perform the verification and validation of the commercial package used. Results obtained with CFX™ show excellent agreement with analytical solutions, for subcritical, supercritical, transitional and hydraulic jump cases. Special care with grid selection and entrance boundary condition is crucial to simulate with accuracy these types of flows. In particular, when a proper structured mesh is used, quality results are highly improved. Finally, results show to be insensitive to entrance turbulence conditions.

Leveling of a Line of Paint Droplets on a Surface

2014 ◽  
Author(s):  
J. Esmaeelpanah ◽  
A. Dalili ◽  
S. Chandra ◽  
J. Mostaghimi
Keyword(s):  
Liquid Film ◽  
Time Scale ◽  
Water Solution ◽  
Three Dimensional ◽  
Free Surface Flow ◽  
High Viscosity ◽  
Surface Flow ◽  
Experimental Measurements ◽  
Thickness Change ◽  
Model Free

When liquid droplets are sprayed onto a solid surface during spray painting and coating, defects may be created due to non-uniformity of the liquid film. The time scale required for leveling the liquid film is of utmost importance. In this paper, the leveling of a droplet line was modeled numerically using a three-dimensional parallel code, based on a two-step projection method with a volume-of-fluid technique used to model free surface flow. Simulations and experiments were performed with a high viscosity liquid (87%wt glycerin in water solution and commercial paint) where a droplet line was deposited with varying center-to-center distance between the droplets and the time scale for leveling was measured. Predictions from the numerical model for droplet line thickness and variation in the thickness agreed well with experimental measurements. Simulation and experimental measurements clearly showed that the drastic portion of thickness change happened in the first 30 seconds. When the droplets were dispensed close to one another (initially thicker films), there was a lot of interaction between the droplets and therefore there was a huge variation in the thickness of the droplet line until the variation died down and the line was level. On the other hand, when droplets were dispensed further apart (initially thinner films), there was less interaction between the droplets and therefore less variation of the film thickness from the beginning and the leveling occurred much sooner.

Sign in / Sign up

Export Citation Format

Share Document