Axisymmetric Free-surface Flows Research Articles

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.

Axisymmetric free-surface flow simulation using the moving surface mesh particle method and application to drop formation

It is difficult to accurately simulate axisymmetric free-surface flows with surface tension and topological changes, such as drop formation through a nozzle. A promising numerical method that can achieve this goal is the moving surface mesh particle method. It incorporates a moving surface mesh to explicitly represent a free-surface boundary, which enables accurate free-surface tracking and surface tension calculation. In the present study, this method is extended to an axisymmetric coordinate system. In the axisymmetric simulation, particles moving in a Lagrangian fashion result in significantly non-uniform particle distributions, leading to numerical instabilities. To circumvent this issue, a new algorithm is developed for determining the particle movement in an ALE fashion by considering a two-dimensional continuity equation. Moreover, in order to deal with the topological change arising from breakup of liquid domain, a simple algorithm employing surface mesh splitting is developed. The proposed method is verified using fundamental test cases such as axisymmetric cavity flow, axisymmetric patch test, spherical droplet oscillation, and hanging droplet. Therein, precise agreements to reference solutions are confirmed. Furthermore, the proposed method is applied to the numerical simulation of dripping faucet. As a result, complex drop formation phenomena are successfully simulated with reasonable computational cost. The numerical results are also shown to agree well with the experimental measurement in the literature, from which the validity of the proposed method is demonstrated.

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.

An assessment of the PTT model on the impacting drop problem

This work presents a numerical method and simulations of axisymmetric free-surface flows of a polymeric fluid using the Phan–Thien–Tanner (PTT) constitutive relationship. The governing equations are solved by finite differences on a staggered grid that employs accurate boundary conditions on the free surface together with bounded, monotone, high-order upwinding approximation of the convection terms. The numerical method is partially verified by solving fully developed flow in a tube and comparing it to a known analytic solution. The problem of a drop impacting on a rigid plate is considered. The effect of varying the five parameters, characterizing the PTT model, is studied extensively, and the results are interpreted physically. The expansion–contraction phenomenon of the viscoelastic drop that occurs when it hits a flat surface is tracked, and the time evolution of the normal and shear components of the extra-stress tensor is analyzed. To further verify the code, the impacting drop of an Oldroyd-B fluid is simulated and the time evolution of the drop width is compared to that of other authors.

Extending Incompressible SPH framework for simulation of axisymmetric free-surface flows

Dynamics of a Newtonian fluid in a cylindrical symmetric domain can be conceptualized as axisymmetric flow. Full-scale 3D simulation of such problems requires large computational overhead. A new divergence-free Axisymmetric Incompressible Smoothed Particle Hydrodynamics (AxISPH) framework is proposed for modeling of Newtonian fluid flow with free-surface. The formulation employs the concept of variable axisymmetric volume based on current particle position to incorporate symmetry around the central axis. The framework utilizes a semi-implicit two-step approach to obtain hydrodynamic variables and minimizes errors in particle position. Laplace and gradient operators are modified to incorporate axisymmetric nature. A radial dam-break flow has been simulated with the proposed model and compared with three-dimensional ISPH result.

Singularities of Steady Axisymmetric Free Surface Flows with Gravity

We consider a steady axisymmetric solution of the Euler equations describing the irrotational flow without swirl of an incompressible inviscid fluid acted on by gravity and with a free surface. We analyze stagnation points as well as points on the axis of symmetry. At points on the axis of symmetry that are not stagnation points, constant velocity motion is the only blowup profile consistent with the invariant scaling of the equation. This suggests the presence of downward‐pointing cusps at those points. At stagnation points on the axis of symmetry, the unique blowup profile consistent with the invariant scaling of the equation is the Garabedian pointed bubble solution with water above air. Thus at stagnation points on the axis of symmetry with no water above the stagnation point, the invariant scaling of the equation cannot be the right scaling. A finer blowup analysis of the velocity field yields that in the case when the surface is described by an injective curve, the velocity field scales almost like and is asymptotically given by urn:x-wiley:10429832:media:cpa21514:cpa21514-math-0002 with a nonzero constant c0. The last result relies on a frequency formula in combination with a concentration compactness result for the axially symmetric Euler equations by Delort. While the concentration compactness result alone does not lead to strong convergence in general, we prove the convergence to be strong in our application. © 2014 Wiley Periodicals, Inc.

On a Numerical Model for Free Surface Flows of a Conductive Liquid Under an Electrostatic Field

In this paper, a numerical model is developed that can simulate the unsteady axisymmetric free-surface flow of a perfectly conductive liquid under an electrostatic field. The effect of the electrostatic field is modeled by a force distributed on the liquid free surface. Assuming the liquid as a perfect conductor makes it possible to reduce the general electromagnetic equations to electrostatic equations. The Navier–Stokes equations are solved to find the velocity and pressure fields. The free surface advection and reconstruction are performed based on the volume-of-fluid method using Youngs’ algorithm. To evaluate the effect of the electric field on the free surface, the electrostatic potential is first solved for the entire computational domain. Next, the electric field intensity and the surface density of the electric charge are calculated on the free surface after which the electric force can be determined. The computational method for treating this force is similar to that of the surface tension using the continuum surface force method. The developed model is validated by a comparison between the calculated results with those of the analytics as well as experiments for an electrowetting scenario.

Integral methods for flow in a conical centrifuge

A set of through-thickness averaged equations for momentum and strain rate are derived for the problem of axi-symmetric free-surface flow within a spinning cone. The expressions are independent of the choice of constitutive law and can therefore be used for modeling the flow of a variety of materials within an industrial conical centrifuge. By assuming a through-thickness velocity profile the distribution of flow thickness and average velocity over the internal surface of the cone can be obtained. The approach has been validated for thin Newtonian viscous flow by comparison with full three-dimensional solutions of the Navier–Stokes equations obtained with a commercial CFD package. The averaged equations provide an accurate prediction of the flow thickness, velocity and the length of the zones of influence of inlet and outlet boundary conditions.

Die-swell, splashing drop and a numerical technique for solving the Oldroyd B model for axisymmetric free surface flows

This work deals with the development of a numerical method for simulating viscoelastic axisymmetric free surface flow of an Oldroyd B fluid. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries and on the symmetry axis. The full free surface stress conditions are employed. The resulting governing equations are solved by finite differences on a Marker-and-cell (MAC) type grid. Validation is provided by simulating a pipe flow problem. The classical die-swell problem is solved and swelling ratios are provided. The height of the splash caused by a falling liquid drop for various Reynolds and Weissenberg numbers is then studied, and the height of the splash is shown to diminish with increasing viscoelasticity.

A second-order boundary-fitted projection method for free-surface flow computations

This paper describes a new approach to the high-fidelity simulation of axisymmetric free-surface flows. A boundary-fitted grid is coupled with a new projection method for the solution of the Navier–Stokes equations with second-order accuracy in space and time. Two variants of this new method are developed by adapting two existing algorithms, suitable for prescribed velocity boundary conditions, to the case of prescribed normal and tangential stresses at the free boundary. A normal-mode analysis for a fixed-boundary problem confirms the second-order accuracy of the algorithms. The approach is validated by comparison with a Rayleigh–Plesset solution for an oscillating spherical bubble, with an analysis of shape oscillations, and with existing results for the buoyant rise of a deforming bubble for Reynolds numbers up to 200 and Weber numbers up to 12. In addition to the simulation of axisymmetric free-surface flows of intrinsic interest, the present approach is suitable for the validation of genuinely three-dimensional calculations.

A low‐dimensional spectral approach for transient axisymmetric free‐surface flow inside thin cavities of arbitrary shape

AbstractA low‐dimensional spectral method is used to solve the transient axisymmetric free surface flow inside thin cavities of arbitrary shape. The flow field is obtained on the basis of the lubrication equations, which are expanded in terms of orthonormal functions over the cavity gap. The formulation accounts for nonlinearities stemming from inertia and front location. The work is of close relevance to the filling stage during die casting, and injection molding, or the flow inside annular (extrusion) dies. Both flows under an imposed flow rate, and an imposed pressure at the cavity entrance are examined. The influence of inertia, aspect ratio, gravity, and wall geometry on the evolution of the front, flow rate, and pressure is assessed particularly in the early stage of flow, when a temporal behavior of the ‘boundary‐layer’ type develops. The multiple‐scale method is applied to obtain an approximate solution at small Reynolds number, Re. Comparison with the exact (numerical) solution indicates a wide range of validity for the multiple‐scale approach, including the moderately small Re range. Copyright © 2002 John Wiley & Sons, Ltd.

Influence of inertia on the transient axisymmetric free-surface flow inside thin cavities of arbitrary shape

The influence of inertia is examined for transient axisymmetric free surface flow inside a thin cavity of arbitrary shape. The flow field is obtained by solving the lubrication equations, which are averaged over the cavity gap by expanding the velocity in terms of Chandrasekhar functions and using the Galerkin projection method. The formulation accounts for the transverse flow, as well as nonlinearities stemming from inertia and front location. Both flows under an imposed flow rate, and an imposed pressure at the cavity entrance are examined. The influence of inertia, aspect ratio, gravity, and cavity geometry on the evolution of the front, flow rate, and pressure is assessed particularly in the early stage of flow. Comparison with existing results shows full qualitative agreement for cavities of various geometries and flow conditions. Inertia is found to have a significant influence on early transient behavior, leading to the development of a flow of the “boundary-layer” type upon inception. The effect of inertia is further explored by developing a multiple-scale analysis to obtain an approximate solution at small Reynolds number, Re. Comparison with the exact (numerical) solution indicates a wide range of validity for the multiple-scale approach, even in the moderately small Re range.

COMPUTATIONS OF MULTIPHASE FLOWS WITH SURFACE TENSION USING AN ADAPTIVE FINITE ELEMENT METHOD

An adaptive finite element method for solving incompressible steady-state axisymmetric free surface flow problems is presented. While the methodology is applicable to most types of multiphase flows, we are particularly interested in modeling laminar free surface flows such as free and impinging jets, which are usually modeled in a Lagrangian framework. An Eulerian strategy is used to capture the interface. Surface tension is included in the model in order to study its influence on the topology of the free surface. A stabilized finite element discretization is used to help solve the coupled system of partial differential equations. An adaptive methodology helps optimize the accuracy of the computed solution. The proposed adaptive free surface capturing methodology is competitive with interface tracking techniques, while being more flexible. The verification and validation of the methodology completes the article.

Annular liquid jets and other axisymmetric free-surface flows at high Reynolds numbers

A perturbation method based on a long wavelength approximation is used to obtain the leading order equations governing the fluid dynamics of laminar, annular, round and compound liquid jets and liquid films on convex and concave cylindrical surfaces. An approximate, integral balance method is also used to determine the inviscid core and the thickness of the boundary layers of annular liquid jets near the nozzle exit. The steady state equations are transformed into parabolic ones by means of the von Mises transformation and solved in an adaptive, staggered grid to determine the axial velocity distribution and the location of the free surfaces. It is shown that, for free surface flows subject to inertia, gravity and surface tension, there is a contraction near the nozzle which increases as the Reynolds and Froude numbers are decreased, and is nearly independent of the Weber number for Weber numbers larger than about one hundred. It is also shown that this contraction depends on the flow considered, and is larger for films on convex surfaces. It is also shown that, for round jets, the acceleration of the jet's free surface is larger than that of the jet's centerline, although, sufficiently far from the nozzle exit, the axial velocity is uniform across the jet.

Effects of Variable Properties and Viscous Dissipation During Optical Fiber Drawing

The axisymmetric free-surface flow and thermal transport of fused silica during optical fiber drawing was considered with variable properties, prescribed heat flux, and neck shape. Experimental data from previous researchers were adapted or used as the basis for assumptions in order to enable a realistic analysis. The main objectives were to model the neck-down process in order to clarify the effects of the variable properties and the associated viscous dissipation. Due to the large changes in dimension and viscosity, this system poses severe nonlinearities, and a new solution algorithm was necessarily developed. Validation was achieved and several important results were obtained. Among these, it was shown that the viscous dissipation has considerable impact on the fiber temperature due to its localization to a small volume near the fiber section. Also, it was shown that a variable viscosity generated vorticity, which was localized to the region where the preform radius undergoes large changes.

Universal pinching of 3D axisymmetric free-surface flow.

We consider the viscous motion of an axisymmetric column of fluid with a free surface. The Navier-Stokes equation forms a singularity as the height of the fluid neck goes to zero. Close to pinchoff, the solutions have a scaling form characterized by a set of universal exponents. The shape of the neck and its velocity field is described by scaling functions, which we predict without adjustable parameters.

The cauchy‐poisson waves in an inviscid rotating stratified liquid

Based upon the Boussinesq approximation, an initial value investigation is made of the axisymmetric free surface flows generated in an inviscid rotating stratified liquid of infinite depth by the prescribed free surface disturbance. The asymptotic analysis of the integral solution is carried out by the stationary phase method to describe the solution for large time and large distance from the source of the disturbance. The asymptotic solution is found to consist of the classical free surface gravity waves and the internal‐inertial waves.