Unsteady Free Surface Flows Research Articles

Parametric analysis of the performance of the SPH solutions of unsteady free-surface flow

ABSTRACT In this article, the performances of the SPH method to solve Shallow Water Equations SWEs with three investigation parameters were studied, such as the type of kernel functions, namely: cubic spline, Gaussian and quintic spline kernels, the number of particles used and the stabilization terms injected, specifically: Lax Friedrichs flux, artificial viscosity and two shocks Riemann solver. Three benchmarking tests make the subject of unsteady free surface flow in this study. It is 1D typical dam-break on wet and dry bottom; 2D partial dam-break on a dry floodplain and 2D partial dam-break on channel with 90° bend. The analysis of the different errors between the numerical and analytical solutions and/or the experimental data shows that the SPH method gives reliable values with the selected optimal parameters which are the cubic kernel function and the artificial viscosity term. The increase in the number of particles increases the precision but also the calculation time.

Unsteady free surface flow due to a line sink at an arbitrary location with surface tension

The unsteady flow generated when a line sink is located at an arbitrary location beneath the free surface of a fluid of finite depth is considered. Interest is in the evolution to a steady state flow, splashing flows and the drawdown of the surface. The location of the sink away from, but near to, a vertical barrier leads to some interesting new surface flows due to interactions and reflection, some of which have a profound impact on the nature of the critical drawdown flow parameters. Interesting new solutions with a delayed drawdown that are initiated by a reflected wave are presented. A linearized solution and a fully nonlinear solution are presented and the results discussed. Regions of steady, unsteady and supercritical flow are identified.

A 2D Hydraulic Simulation Model Including Dynamic Piping and Overtopping Dambreach

Numerical simulation of unsteady free surface flows using depth averaged equations that consider the presence of initial discontinuities has been often reported for situations dealing with dam break flow. The usual approach is to assume a sudden removal of the gate at the dam location. Additionally, in order to prevent any kind of dam risk in earthen dams, it is very interesting to include the possibility of a progressive dam breach leading to dam overtopping or dam piping so that predictive hydraulic models benefit the global analysis of the water flow. On the other hand, when considering a realistic large domain with complex topography, a fine spatial discretization is mandatory. Hence, the number of grid cells is usually very large and, therefore, it is necessary to use parallelization techniques for the calculation, with the use of Graphic Processing Units (GPU) being one of the most efficient, due to the leveraging of thousands of processors within a single device. The aim of the present work is to describe an efficient GPU-based 2D shallow water flow solver (RiverFlow2D-GPU) supplied with the formulation of internal boundary conditions to represent dynamic dam failure processes. The results obtained indicate that it is able to develop a transient flow analysis taking into account several scenarios. The efficiency of the model is proven in two complex domains, leading to >76× faster simulations compared with the traditional CPU computation.

Numerical Simulation of Three-Dimensional Free Surface Flows Using the K-BKZ-PSM Integral Constitutive Equation.

This work introduces a novel numerical method designed to address three-dimensional unsteady free surface flows incorporating integral viscoelastic constitutive equations, specifically the K-BKZ-PSM (Kaye-Bernstein, Kearsley, Zapas-Papanastasiou, Scriven, Macosko) model. The new proposed methodology employs a second-order finite difference approach along with the deformation fields method to solve the integral constitutive equation and the marker particle method (known as marker-and-cell) to accurately capture the evolution of the fluid's free surface. The newly developed numerical method has proven its effectiveness in handling complex fluid flow scenarios, including confined flows and extrudate swell simulations of Boger fluids. Furthermore, a new semi-analytical solution for velocity and stress fields is derived, considering fully developed flows of a K-BKZ-PSM fluid in a pipe.

Comparative analysis of numerical solutions of 2D unsteady dambreak waves using FVM and SPH method

Abstract This work presents a comparison of two-dimensional numerical solutions of unsteady free surface flow. This is a simulation of the dam-break wave with different configurations using based-mesh finite volume method and meshless smoothed particle hydrodynamics (SPH). Two well-known approaches, widely used in the computational fluid dynamics (CFD). These techniques have proven their robustness in the numerical treatment of such conservation laws. The main goal is to check the ability of the SPH method and the first order finite volume HLLC solver to reproduce the numerical solutions of the 2D shallow water equations. Based on many benchmark tests, one investigates the effect of the topographic variation along the x and y directions on behavior of the numerical solutions namely at the wet-dry front. The comparison between the simulated results, the analytical solutions and the experimental measurements shows a good correlation, although the finite volume approach remains more advantageous in terms of accuracy and the CPU time.

Review of Experimental Investigations of Dam-Break Flows over Fixed Bottom

Laboratory experiments of dam-break flows are extensively used in investigations of geophysical flows involving flood waves, to provide insight into relevant aspects of the physics of the process and collect experimental data for validating numerical models. A dam-break flow is a typical example of a highly unsteady free surface flow with high reproducibility. Indeed, dam-break experiments can be repeated several times under the same test conditions obtaining large amounts of different types of data (possibly using various measuring techniques) that can be combined in a single rich dataset. Moreover, laboratory tests on dam-break flows are widely considered a valuable benchmark for the validation of numerical models, since field data from historical events are scarce, sparse, and highly uncertain. However, no systematic review of laboratory investigations of dam-break flows and existing related datasets are available in the literature to provide a comprehensive overview of the test conditions considered, the measuring techniques used, and the experimental data collected. This review article aims to fill this gap, focusing on laboratory tests in schematic and idealized setups with a fixed, non-erodible bed. In particular, this review aims to help researchers and modelers to: (a) select the most appropriate laboratory tests for validating their numerical models; (b) facilitate access to databases by indicating relevant bibliographic references; (c) identify specific challenging aspects worthy of further experimental research; and (d) support the development of new or improved technologies for the mitigation of the impact of dam-break flood waves. The references reviewed are organized into tables according to the purposes of the laboratory investigation, and comprehensive information is provided on test conditions, datasets, and data accessibility. Finally, suggestions for future experimental research on dam-break flows are provided.

New formulation of the two-dimensional steep-slope shallow water equations. Part I: Theory and analysis

Two-dimensional (2D) depth-averaged shallow water equations (SWE) are widely used to model unsteady free surface flows, such as flooding processes, including those due to dam-break or levee breach. However, the basic hypothesis of small bottom slopes may be far from satisfied in certain practical circumstances, both locally at geometric singularities and even in wide portions of the floodable area, such as in mountain regions. In these cases, the classic 2D SWE might provide inaccurate results, and the steep-slope shallow water equations (SSSWE), in which the restriction of small bottom slopes is relaxed, are a valid alternative modeling option. However, different 2D formulations of this set of equations can be found in the geophysical flow literature, in both global horizontally-oriented and local bottom-oriented coordinate systems. In this paper, a new SSSWE model is presented in which water depth is defined along the vertical direction and flow velocity is assumed parallel to the bottom surface. This choice of the dependent variables combines the advantages of considering the flow velocity parallel to the bottom, as can be expected in gradually varied shallow flow, and handling vertical water depths consistent with elevation data, usually available as digital terrain models. The pressure distribution is assumed linear along the vertical direction and flow curvature effects are neglected. A new formulation of the 2D depth-averaged SSSWE is derived, in which the two dynamic equations represent momentum balances along two spatial directions parallel to the bottom, whose horizontal projections are parallel to two fixed orthogonal coordinate directions. The analysis of the mathematical properties of the new SSSWE equations shows that they are strictly hyperbolic for wet bed conditions and reduce to the conventional 2D SWE when bottom slopes are small. Finally, it is shown that the SSSWE predict a slower flow compared with the conventional SWE in the theoretical case of a 1D dam-break on a frictionless channel with fixed slope. The capabilities of the proposed model are demonstrated in a companion paper on the basis of numerical and experimental tests.

Horizontal dam-break flow past a blocking-drag region

A one-dimensional model framework of unsteady free-surface flow through a blocking-drag region is developed, tested, and applied to understand the dam-break flow past a rectangular building. This is achieved by studying the steady-state, adjustment to steady-state, and the unsteady response of a blocking-drag region. Three steady flow regimes are identified based on the Froude number upstream and downstream of a blocking-drag region: a subcritical state, a choked state with a subcritical-supercritical transition, and a supercritical state. The interaction between a dam-break flow and a blocking-drag region can be mostly understood from the quasi-steady analysis using the variation of the Froude number with time, and comparing the upstream and downstream Froude number scatter plots against the steady curve. The force time-series depends on the height of the precursor layer and the position of the blocking-drag region relative to the lock-length. This model provides considerable insight into the types of flow characteristics observed at low/high Froude numbers and goes some way to clarifying the relationship between the drag force and the dam-break flow properties.

A Study on Numerical Methodologies in Solving Fluid Flow and Heat Transfer Problems

Numerical methods are described as techniques by which several mathematical problems are formulated, because they may be easily solved with arithmetic operations. These methodologies have a great impact on the current development of finite element theory and other areas. We have given a short study of numerical methodologies applied in fluid flow and heat and mass transfer problems in mechanical engineering which includes finite difference method, Finite element method, Boundary value problems (general), Lattice Boltzmann’s methods, Crank-Nicolsan scheme methods, boundary integral method, Runge-Kutta method, Taylor series method and so on. We have discussed some phenomena taking place in fluids such as surface tension, coning, water scattering, Stokes law, gravity-capillary, and unsteady free-surface flows, swirling, and so on. We have also analyzed boundary value problems on boundary problems, eigenvalue problems and found a numerical way to solve these problems. We have presented different numerical methods applied to different fundamental modeling approaches in heat transfer and the performance of the mechanisms (modes) vary concerning the methods applied. The paper is dedicated to demonstrating how the methods are beneficial in solving real-life heat transfer problems in engineering applications. Results of the parameters like thermal conductivity, energy flux, entropy, temperature, etc. have been compared with the existing methods

An Oldroyd-B solver for vanishingly small values of the viscosity ratio: Application to unsteady free surface flows

This work is concerned with time-dependent axisymmetric free surface flows of Oldroyd-B fluids for any value of ▪, the ratio of solvent to total viscosities. The Oldroyd-B constitutive equation is dealt with by employing a novel technique to calculate the conformation tensor while an EVSS transformation allows the solution of the momentum equations coupled with the free surface stress conditions: this avoids numerical instabilities that can arise when using small values of ▪. The convergence of this new methodology is verified on pipe flow and also by comparing results from the literature for the time-dependent impacting drop problem. This approach is then used to predict the time-dependent free surface flow after a viscoelastic drop impacts a solid surface for ▪ values in the range [0, 1]. The impacting drop problem is investigated for polymer solutions containing a small solvent contribution (▪) or without any solvent viscosity (▪). In addition, a study of the bouncing drop problem for different values of ▪, Weissenberg and Reynolds numbers is undertaken.

Undular and broken surges in dam-break flows: a review of wave breaking strategies in a Boussinesq-type framework

The water waves resulting from the collapse of a dam are important unsteady free surface flows in civil and environmental engineering. Considering the basic case of ideal dam break waves in a horizontal and rectangular channel the wave patterns observed experimentally depends on the initial depths downstream (hd) and upstream (ho) of the dam. For r = hd/ho above the transition domain 0.4–0.55, the surge travelling downstream is undular, a feature described by the dispersive Serre–Green–Naghdi (SGN) equations. In contrast, for r below this transition domain, the surge is broken and it is well described by the weak solution of the Saint–Venant equations, called Shallow Water Equations (SWE). Hybrid models combining SGN–SWE equations are thus used in practice, typically implementing wave breaking modules resorting to several criteria to define the onset of breaking, frequently involving case-dependent calibration of parameters. In this work, a new set of higher-order depth-averaged non-hydrostatic equations is presented. The equations consist in the SGN equations plus additional higher-order contributions originating from the variation with elevation of the velocity profile, modeled here with a Picard iteration of the potential flow equations. It is demonstrated that the higher-order terms confer wave breaking ability to the model without using any empirical parameter, such while, for r > 0.4–0.55, the model results are essentially identical to the SGN equations but, for r < 0.4–0.55, wave breaking is automatically accounted for, thereby producing broken waves as part of the solution. The transition from undular to broken surges predicted by the high-order equations is gradual and in good agreement with experimental observations. Using the solution of the new higher-order equations it was further developed a new wave breaking index based on the acceleration at the free surface to its use in hybrid SGN–SWE models.

Vortex-dynamical implications of nonmonotonic viscous dissipation of off-center droplet bouncing

In this paper, vortex-dynamical perspectives were adopted to interpret the recently reported observation that the total viscous dissipation of off-center droplet bouncing varies nonmonotonically with the impact parameter [C. He, X. Xia, and P. Zhang, “Non-monotonic viscous dissipation of bouncing droplets undergoing off-center collision,” Phys. Fluids 31, 052004 (2019)]. The particular interest of this study is on analyzing the velocity and vorticity vector fields and their correlations, such as helicity and enstrophy. The helicity analysis identifies a strong interaction between the “ring-shaped” vortices and the “line-shaped” shear layers in the non-axisymmetric droplet internal flow. A general relation between the total enstrophy and the total viscous dissipation rate for an unsteady free-surface flow was theoretically derived and numerically verified. It shows that the equality between the total enstrophy and the total viscous dissipation rate holds for a single-phase flow confined by stationary boundaries but is not satisfied for a gas–liquid two-phase flow due to the interfacial movement. Both the total enstrophy and a defined “half-domain” helicity show the nonmonotonic variation with the impact parameter, implying their interrelation with the nonmonotonic viscous dissipation.

Application of BRAHMA hydrodynamic model for flood forecasting

Flood forecasting models are of paramount importance to predict the flood and thereby warning the community living near the floodplains or the vulnerable areas. In the quest for understanding the complex processes of a river, scientist and researchers have given importance to river modeling both analytically and physically by conducting experiments in the laboratory. The numerical model which is basically a mathematical model has gained popularity in the last few decades. BRAHMA (Braided River Aid: Hydro-Morphological Analyzer) is a hydrodynamic model for simulating the spatial variation in water depth, flows, and velocities in the unsteady free surface flow. It is developed by IIT Guwahati in collaboration with Brahmaputra Board, Govt. of India. The BRAHMA 1-D model is capable of simulating both prismatic as well as non-prismatic channel reach. It is also capable of simulating channels with varying width and varying slope. In this paper, an attempt has been made to use the BRAHMA 1-D model for determining the water surface elevation at certain flood-prone areas of Ranganadi River which covers Assam and parts of Arunachal Pradesh, North-East India. The Ranganadi hydroelectric dam is situated in Arunachal Pradesh, India. In this study, the length of the river reach considered is 77.09 km from the Ranganadi Hydroelectric power project. The cross-sectional width of the river in different sections is extracted from Google Earth and depth was obtained by approximate field measurement. The peak release from the dam, the downstream water level and initial water levels are given as input to the model. The water levels at the particular sections of interest are computed as output from the model. From the generated output data set, it is possible to assess the vulnerable areas by the water levels which will be generated from the model.

Unsteady free surface flow above a moving circular cylinder

A problem on non-stationary free surface flow of infinitely deep ideal fluid generated by the motion of a submerged body is considered. The water wave problem is reduced to the integral–differential system of equations for the functions defining free surface shape, normal, and tangential velocity components on the free boundary. Small-time asymptotic solution is constructed for the case of circular cylinder that moves with constant acceleration from rest. The role of non-linearity is clarified by analysis of this approximate solution which describes the formation of added mass layers, splash jets, and finite amplitude surface waves.

Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.

A similarity solution of time dependent MHD liquid film flow over stretching sheet with variable physical properties

Abstract An analysis is performed for the fluid dynamics incorporating the variation of viscosity and thermal conductivity on an unsteady two-dimensional free surface flow of a viscous incompressible conducting fluid taking into account the effect of a magnetic field. Surface tension quadratically vary with temperature while fluid viscosity and thermal conductivity are assumed to vary as a linear function of temperature. The boundary layer partial differential equations in cartesian coordinates are transformed into a system of nonlinear ordinary differential equations (ODEs) by similarity transformation. The developed nonlinear equations are solved analytically by Homotopy Analysis Method (HAM) while numerically by using the shooting method. The Effects of natural parameters such as the variable viscosity parameter A, variable thermal conductivity parameter N, Hartmann number Ma, film Thickness, unsteadiness parameter S, Thermocapillary number M and Prandtl number Pr on the velocity and temperature profiles are investigated. The results for the surface skin friction coefficient f ″ ( 0 ) , Nusselt number (heat flux) - θ ′ ( 0 ) and free surface temperature θ ( 1 ) are presented graphically and in tabular form.

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

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.

Time-dependent conformal mapping of doubly-connected regions

This paper examines two key features of time-dependent conformal mappings in doubly-connected regions, the evolution of the conformal modulus Q(t) and the boundary transformation generalizing the Hilbert transform. It also applies the theory to an unsteady free surface flow. Focusing on inviscid, incompressible, irrotational fluid sloshing in a rectangular vessel, it is shown that the explicit calculation of the conformal modulus is essential to correctly predict features of the flow. Results are also presented for fully dynamic simulations which use a time-dependent conformal mapping and the Garrick generalization of the Hilbert transform to map the physical domain to a time-dependent rectangle in the computational domain. The results of this new approach are compared to the complementary numerical scheme of Frandsen (J. Comput. Phys. 196:53---87, 14) and it is shown that correct calculation of the conformal modulus is essential in order to obtain agreement between the two methods.

HLLC法による彦山川の平面２次元洪水流解析とFDS法との比較

Flood flows analysis in the Hikosan river basin in July 2012 were performed by numerical models for the 2D unsteady free-surface flow based on HLLC(Harten-Lax-van Leer and Contact). A new procedure for a dry/wet boundary over a topography is incorporated into the model based on HLLC solver. The HLLC model was verified against the observed water level. The computation accuracy and speed were compared between HLLC model and FDS(Flux-Difference Splitting) model. It shows that (1) the HLLC model reproduce the behavior of flood flows with reasonable accuracy, and that (2) the calculation efficiency is high compared with FDS model, and that (3) the treatment of a dry/wet boundary needs to be improved for the HLLC model to conserve the mass.

Operation of an irrigation canal by means of the passive canal control

Modern irrigation techniques involve large spatial and temporal demand variations in distribution networks. This makes the flow unsteady and generates perturbations that travel upstream along the network. Perturbations can also be generated by variable water inflows. This is the case when water is pumped into the network under variable energy rates, generating perturbations that travel downstream on the network. The passive canal control is a design criteria and a flow distribution method that make most of the storage capacity needed in any irrigation project, in order to mitigate the perturbations coming from both directions. In this paper, the passive canal control is applied to the design and operation of the Xerta-Senia Canal Irrigation Project considering an unsteady free-surface flow model. The key aspect of the project is the location of irrigation reservoirs in-line with the canals at the same level, allowing water flow from canals to reservoir and vice versa. Three performance scenarios are evaluated, and the results of a simulation model are presented.