Transient Free Surface Flow Research Articles

Acoustic Doppler Velocimetry in Transient Free-Surface Flows: Field and Laboratory Experience with Bores and Surges

Acoustic Doppler Velocimetry in Transient Free-Surface Flows: Field and Laboratory Experience with Bores and Surges

A Dimension-Reduced Line Element Method for 3D Transient Free Surface Flow in Porous Media

In order to reduce the numerical difficulty of the 3D transient free surface flow problems in porous media, a line element method is proposed by dimension reduction. Different from the classical continuum-based methods, homogeneous permeable pores in the control volume are conceptualized by a 3D orthogonal network of tubes. To obtain the same hydraulic solution with the continuum model, the equivalent formulas of flow velocity, continuity equation and transient free surface boundary are derivable from the principle of flow balance. In the solution space of transient free surface flow, the 3D problem is transformed into 1D condition, and then a finite element algorithm is simply deduced. The greatest advantage of the line element method is line integration instead of volume/surface integration, which has dramatically decreased the integration difficulty across the jump free surface. Through the analysis of transient free surface flow in the unconfined aquifer, trapezoidal dam, sand flume and wells, the transient free surface locations predicted from the proposed line element method generally agree well with the analytical, experimental and other numerical data in the available literatures, the numerical efficiency can also be well guaranteed. Furthermore, the hydraulic anisotropy has significant effect on the evolution of free surface locations and the shape of depression cones in spatial. The line element method can be expanded to model the 3D unsaturated seepage flow, two-phase flow and thermos problems in porous media because of the similarity between the similarity of Darcy’s law, Buckingham Law and Fourier’s law.

SPH simulations of transient non-isothermal viscoelastic flows with free surfaces

• Transient non-isothermal viscoelastic free surface flows are simulated by SPH. • SPH discretization of the temperature equation is derived. • The impacting droplet and the filling process of circular disc with core are studied. • Both Oldroyd-B and UCM fluid models are considered. • The work provides references to handle non-isothermal viscoelastic flows in an entirely meshfree framework. This study presents a smoothed particle hydrodynamics (SPH) method for simulating transient non-isothermal viscoelastic flows with free surfaces. The basic equations governing the non-isothermal free surface flow of an Oldroyd-B fluid are considered and approximated by SPH. To model non-isothermal fluid flows, a working SPH discretization of the temperature equation is derived, and the temperature-dependent viscosity and relaxation time are represented by the time–temperature superposition principle. The proposed SPH method is first validated by solving the viscoelastic Poiseuille flow problem under isothermal and non-isothermal conditions and comparing the SPH solutions with the analytical solutions or those obtained by other methods. Then, the SPH method is extended to deal with the impact of a non-isothermal viscoelastic droplet with a cold wall and the non-isothermal filling process of a circular disc with a core. The convergence of the method is verified by four particle spacings of different levels of refinement. The effects of some important parameters, including the temperature dependency coefficient, reference temperature of the fluid, and Péclet number, on the flow behavior are analyzed in detail. All the results demonstrate that the proposed SPH method is capable of accurately and stably simulating transient non-isothermal viscoelastic free surface flows in an entirely meshfree framework.

Godunov-type solutions for free surface transient flow in pipeline incorporating unsteady friction

AbstractA finite-volume second-order Godunov-type scheme (GTS) combining the unsteady friction model (UFM) is introduced to simulate free surface flow in pipelines. The exact solution to the Riemann problem calculates the mass and momentum fluxes while considering the Brunone unsteady friction factor. One simple boundary treatment with double virtual cells is proposed to ensure the whole computation domain with second-order accuracy. Results of various transient free-surface flows achieved by the proposed models are compared with exact solution, experimental data, the four-point implicit Preissmann scheme solution, as well as predictions by the classic Method of Characteristics (MOC). Results show that the proposed second-order GTS UFMs are accurate, efficient, and stable even for Courant numbers less than one and sparse grid. The four-point implicit Preissmann scheme may produce severe numerical attenuation in the case of large time steps and unsuitable weighting factors, while the MOC scheme may produce severe numerical attenuation in the case of a low Courant number and could not maintain mass conservation. The numerical simulations considering the unsteady friction factor are closer to the measured water depth variations. The effect of unsteady friction becomes more important as the initial water depth difference increases significantly.

Inferring rheology from free-surface observations

We develop direct inversion methods for inferring the rheology of a fluid from observations of its shallow flow. First, the evolution equation for the free-surface flow of an inertia-less current with general constitutive law is derived. The relationship between the volume flux of fluid and the basal stress, $\tau _b$ , is encapsulated by a single function $F(\tau _b)$ , which depends only on the constitutive law. The inversion method consists of (i) determining the flux and basal stress from the free-surface evolution, (ii) comparing the flux with the basal stress to constrain $F$ and (iii) inferring the constitutive law from $F$ . Examples are presented for both steady and transient free-surface flows demonstrating that a wide range of constitutive laws can be directly obtained. For flows in which the free-surface velocity is known, we derive a different method, which circumvents the need to calculate the flux.

Vortex Identification in a Free-Surface Flow Problem Solved Using SPH

This article focuses on vortex structure identification methods and their implementation into smoothed particle hydrodynamics (SPH) framework. The most common criteria based on the local velocity gradient tensor analysis like Q-criterion, Δ-criterion, and λ2-criterion are introduced together with their implementation for SPH data. A two-dimensional ‘double dam break’ problem was chosen as a test case because it results in a violent transient free-surface flow with emerging and vanishing eddies of various sizes and intensities. Q-criterion and Δ-criterion results were virtually identical, whereas λ2-criterion appeared to be the most restrictive in vortex identification, so it was the best in suppressing incorrect findings due to the numerical solution imperfection. The analysis of the problem was therefore conducted using the λ2-criterion.

Statistical analysis methods for transient flows – the dam-break case

The Authors developed an interesting discussion on measurements in highly-unsteady transient free-surface flows. Using the dam break wave propagation on a dry bed as an example, they discussed the ...

A one-dimensional line element model for transient free surface flow in porous media

A simple and physically sound model has been proposed to investigate the transient free surface flow behavior in porous media. Instead of the continuum-based model, the pore space between solid grains is conceptualized as a network of horizontal and vertical flow paths and then the flow paths are characterized by straight line elements. Based on the Darcy's law and flux equivalence between the continuum-based model and line element model, equivalent hydraulic parameters and continuity equation are derived through the control volume. By introducing a local coordinate system, unified governing equations and equivalent boundary conditions of the line element model are obtained in the form of one-dimensional formulations. The two-dimensional transient free surface flow problem is reduced to a one-dimensional problem of line element network and a finite-element algorithm is developed. The proposed one-dimensional line element model is verified by three illustrative examples, which indicate that the numerical results can satisfactorily match the analytical solutions and experimental observations from different sources. It is found that the solution accuracy has weak sensitivity of time-step size but sensitive to the spacing size, while numerical efficiency can be well guaranteed for different time-step sizes and spacing sizes. Based on the numerical results, the proposed model can give better performance for the evolution of free surfaces than the residual flow procedure, and is efficient to model transient surface flow with a complex drainage system.

Modified incompressible SPH method for simulating free surface problems using highly irregular multi-resolution particle configurations

In this study, the conventional incompressible smoothed particle hydrodynamics (I-SPH) formulations are modified to be applicable for highly irregular multi-resolution particle distributions for simulation of transient free surface flows. The modifications are based on distinguishing between particle density calculated by interpolation functions in SPH method and fluid density as a physical quantity. I-SPH is a fully Lagrangian method having great potential in dealing with large deformations of the free surface. Flow domain is discretized into a set of moving particles, and each particle carries mass, velocity and pressure during simulation time. Variations in these field parameters can be determined by temporal discretization of Navier–Stokes equation followed by a spatial discretization using I-SPH method. In addition to this, three pressure gradient models are employed for simulation of free surface flows to examine efficiency of the models coupled with the modifications. Furthermore, a simple and efficient method to find free surface particles which is compatible with highly irregular multi-resolution particle distributions is proposed. The modified approach is tested against different free surface problems using highly irregular particle configurations. The classic pressure gradient model employed in SPH method coupled with constant smoothing distance gives accurate results with lower computational times over highly irregular particle distributions. The method can be applied for adaptive refinement strategy to reduce computational times of transient free surface problems.

Numerical Hydrodynamic Performance of the Stepped Planing Craft and Its Step Height Effect

One of the most efficient methods of reducing drag on planing craft is the use of transverse step on the bottom of a hull. Applying steps on the hull reduces the contact area with water and as a result, it reduces drag of the craft. Planing craft are able to have one or two transverse steps. In this paper, numerical hydrodynamic performance of the stepped planing craft and its step height effect is investigated by making use of finite volume method (FVM). The Reynolds-Averaged Navier-Stokes (RANS) equations are coupled with the standard k- e turbulence model and volume of fluid equations are solved to simulate transient turbulent free surface flow surrounding the hull by ANSYS-CFX. In order to predict the motion of the craft, equations of two degrees of freedom for rigid body are coupled with governing equations of fluid flow. In order to validate the numerical model presented in this paper, the obtained numerical results are compared with the available experimental data. The numerical results obtained for drag, dynamic trim, rising of center of mass and the pressure distribution on the body at different speeds and different heights of the steps are presented and discussed.

A three-dimensional non-hydrostatic coupled model for free surface – Subsurface variable – Density flows

A three dimensional coupled numerical model has been developed for incompressible density-driven free surface and saturated porous media flows. For free surface, a time-averaged Navier-Stokes equation has been used whereas Darcy's law has been utilized in porous media flow. The algorithm has been based upon a staggered finite volume scheme on a Cartesian grid that solves the 3D non-hydrostatic density-dependent Darcy equation in one step and complete 3D Navier–Stokes equations in two major steps based on a projection method. The 3D system is decomposed into a series of 2D vertical plane sub-systems which have been solved individually by a direct matrix solver. An iteration procedure can be deployed to achieve the fully 3D implicitness of the solution where high density gradients or sharp variation of free surface elevation is present. An efficient, simple and stable algorithm has been proposed to track the free surface elevation in a Cartesian coordinate system within which the water surface position has not been restricted to a specific layer.The model has been validated using five test cases to simulate integrated transient free surface and porous media flows where fluid density effects as well as the water surface gradient have a considerable effect on the velocity field. Two of the examples involve modeling free surface flow (Wave Reflection test) and a phreatic line prediction (Cone of depression test). Two other test cases involve significant contrasts in fluid density including 3D density-driven flow in porous media and 3D lock exchange tests. The final test of a salinity interface involved a complicated scenario consisting of a density contrast in both saturated porous media and free surface flow, subjected to injection and pumping simulated simultaneously using an integrated domain. Close agreement between numerical results and experimental data demonstrates the capability of the model for the coupled simulation of 3D density-driven flow in both integrated free surface and saturated porous media including freshwater recharge, saltwater discharge, hydrodynamic dispersion and turbulence effects.

Meshfree numerical simulation of free surface thermal flows in mould filling processes using the Finite Pointset Method

The purpose of this work is to carry out a meshfree implementation for the numerical simulation of two-dimensional transient incompressible free surface flows coupled with heat transfer. The Finite Pointset Method is applied in order to solve the involved partial differential equations where the corresponding classical or strong formulation is directly used instead of the corresponding weak form as needed for some other meshfree approaches. The incorporation of the boundary conditions is done in a direct and simple manner. The simplicity and efficiency of this numerical method are demonstrated on two complex two-dimensional mould filling processes.

A technique to remove the tensile instability in weakly compressible SPH

When smoothed particle hydrodynamics (SPH) is directly applied for the numerical simulations of transient viscoelastic free surface flows, a numerical problem called tensile instability arises. In this paper, we develop an optimized particle shifting technique to remove the tensile instability in SPH. The basic equations governing free surface flow of an Oldroyd-B fluid are considered, and approximated by an improved SPH scheme. This includes the implementations of the correction of kernel gradient and the introduction of Rusanov flux into the continuity equation. To verify the effectiveness of the optimized particle shifting technique in removing the tensile instability, the impacting drop, the injection molding of a C-shaped cavity, and the extrudate swell, are conducted. The numerical results obtained are compared with those simulated by other numerical methods. A comparison among different numerical techniques (e.g., the artificial stress) to remove the tensile instability is further performed. All numerical results agree well with the available data.

The “avalanche effect” of an elasto-viscoplastic thixotropic material on an inclined plane

The so-called avalanche effect is one of the fingerprints of thixotropic materials. This self-reinforcing process where the decrease in viscosity, due to a rejuvenation process triggered by a stress field, induces a motion which in turn contributes to decrease the viscosity again, is well exemplified by the inclined plane problem. In this situation, the material in its fully-structured state is placed on an inclined plane with respect to the gravity force which is responsible for the beginning of the breakdown process. These thixotropic systems generally have a yield stress, a strength that must be overcome in order to induce rejuvenation. In addition, they exhibit elastic features, especially in the pre-yield state. In the present work we numerically solve the transient evolution of an elasto-viscoplastic thixotropic material subjected to the action of gravity on an inclined plane. In order to handle with the moving free-surface boundary condition encountered in the avalanche effect, we have used a combination of the Marker-And-Cell (MAC) method with the front-tracking scheme. This formulation was successfully employed for this kind of material in the recent paper of Oishi et al. (2016) [28]. In the present work, we have adapted our finite difference formulation to analyze the effects associated with an extended Herschel–Bulkley model in the simulation of a transient complex free surface flow. Concerning the parameters of the flow curve, it is shown that the dimensionless yield stress (plastic number) is the most significant one. However, for a fixed plastic number, different combinations of dimensionless consistency index and dimensionless Newtonian viscosity plateau can lead to a diversity of responses. The thixotropic equilibrium time had a significant impact on shifting the instant when the flow regime changes from an accelerating (when the front part of the material accelerates) to a retardation one (when this front part decelerates). Higher elasticity, as captured by the Weissenberg number, led to longer distances covered by the material.

Three‐dimensional modelling and simulation of sequential co‐injection moulding with application to a toothbrush handle

Sequential co‐injection moulding (SCIM) is a promising technique in industrial production. Because of the complex hydrodynamic effect of air on polymer melts in a moulding process, it is difficult to accurately design and control the process. Corresponding theoretical investigations are very limited, especially for the transient free surface flows in the filling stage. In this paper, a three‐dimensional (3‐D) unified model is proposed for simulating fluids flow in SCIM. The melted polymer and air in the cavity can be regarded as a continuous fluid. The evolution of the melt front interface and skin/core melt interface are captured simultaneously at any moment by the level set method; this method has been widely‐used in two‐phase flow for its ability to track interfaces. The finite volume method is combined with a domain extension technique to deal with 3‐D flows in an irregular cavity. The model is validated by simulating the SCIM process for a cavity with a block insert, and a distinct corner effect is observed. For a widely‐used plastic toothbrush made by SCIM, the evolutions of transient free surfaces in the filling stage are investigated. All numerical results are consistent with corresponding experimental results, and demonstrate the capability of the model with the domain extension technique to simulate multiphase flows in SCIM.

An improved weakly compressible SPH method for simulating free surface flows of viscous and viscoelastic fluids

In this paper, an improved weakly compressible smoothed particle hydrodynamics (SPH) method is proposed to simulate transient free surface flows of viscous and viscoelastic fluids. The improved SPH algorithm includes the implementation of (i) the mixed symmetric correction of kernel gradient to improve the accuracy and stability of traditional SPH method and (ii) the Rusanov flux in the continuity equation for improving the computation of pressure distributions in the dynamics of liquids. To assess the effectiveness of the improved SPH algorithm, a number of numerical examples including the stretching of an initially circular water drop, dam breaking flow against a vertical wall, the impact of viscous and viscoelastic fluid drop with a rigid wall, and the extrudate swell of viscoelastic fluid have been presented and compared with available numerical and experimental data in literature. The convergent behavior of the improved SPH algorithm has also been studied by using different number of particles. All numerical results demonstrate that the improved SPH algorithm proposed here is capable of modeling free surface flows of viscous and viscoelastic fluids accurately and stably, and even more important, also computing an accurate and little oscillatory pressure field.

Transient Free Surface Flow Past a Two-dimensional Flat Stern

A transient free surface flow past a two-dimensional semi-infinite flat plate in the fluid of a finite depth is considered in the linear approximation. It is assumed that the fluid is inviscid and incompressible and the flow is irrotational. The plate is suddenly submerged at relatively small depth below the free surface into the fluid uniformly moving with a constant velocity. The linearized problem is solved for relatively small Froude numbers F < 1 using the Laplace and Fourier transforms, as well as the Wiener–Hopf technique. It is shown that eventually at large time, the transient solution approaches asymptotically the steady-state solution. Peculiarities of the solution obtained are discussed and illustrated graphically.

Numerical simulation of transient flow in horizontal drainage systems

A numerical simulation model based on the characteristic-based finite-difference method with a time-line interpolation scheme was developed for predicting transient free surface flow in horizontal drainage systems. The fundamental accuracy of the numerical model was first clarified by comparison with the experimental results for a single drainage pipe. Boundary conditions for junctions and bends, which are often encountered in drainage systems, were studied both experimentally and numerically. The numerical model was applied to an actual drainage system. Comparison with a full-scale model experiment indicates that the model can be used to accurately predict flow characteristics in actual drainage networks.

Transient free-surface seepage in three-dimensional general anisotropic media by BEM

Kinematic boundary condition is usually used when dealing with transient free-surface flow problems in isotropic media. When dealing with anisotropic problems, a transformation can transform the anisotropic media to an equivalent isotropic media for seepage analysis, but the kinematic boundary condition cannot be used directly in the transformed media. A generalization of the kinematic boundary condition along any arbitrary direction is derived for use in the transformed domain for general three-dimensional anisotropic problems. A boundary element method for solving transient free-surface seepage problems is developed and the treatment of the proposed kinematic boundary condition in the boundary element method is given. Three examples have been solved to show the reliability and flexibility of the model. Examples are verified with some available experimental and numerical cases to show the accuracy of the model for predicting the phreatic surface and it is shown that anisotropy has a very important and non-neglecting effect in the behavior and the shape of phreatic surface.

A SPH-based particle method for simulating 3D transient free surface flows of branched polymer melts

In this article we present a particle method based on smoothed particle hydrodynamics (SPH) for simulating three-dimensional (3D) transient free surface flows of branched polymer melts which are governed by the single equation version of the eXtended Pom-Pom (XPP) model. In order to remedy the so-called tensile instability which results in particle clustering and unphysical fracture in the state of fluid stretching, both artificial stress and artificial viscosity are incorporated into the momentum equation. For facilitating the implementation of the wall boundary condition in 3D space, an enhanced treatment of solid boundaries is proposed in this paper. The proposed SPH algorithm is validated by the excellent agreement between the numerical and the analytical solutions of Hagen-Poiseuille flow of an Oldroyd-B fluid. To demonstrate the ability of the numerical method in simulating free surface flows of branched polymer melts, the fall and impact on a rigid plate of a XPP fluid droplet is investigated for a wide range of the relevant parameters. Specifically, the convergence of the numerical results is discussed by three particle sizes of different levels of refinement. To provide some additional applications of 3D free surface flows of branched polymer melts, the challenging jet buckling and rod-climbing effect of viscoelastic fluids are further simulated and analyzed. All numerical results establish the capability of the SPH-based particle method to simulate 3D transient free surface flows of branched polymer melts in an entirely mesh-free framework.