ON THE VALIDATION OF THE PARTICLE FINITE ELEMENTS …

European Conference on Computational Fluid DynamicsECCOMAS CFD 2006

P. Wesseling, E. Onate and J. Periaux (Eds)c© TU Delft, The Netherlands, 2006

ON THE VALIDATION OF THE PARTICLE FINITEELEMENTS METHOD (PFEM) FOR COMPLEX

ENGINEERING FLUID FLOW PROBLEMS

A. Larese De Tetto, R. Rossi, S.R. Idelsohn and E.Onate

International Center for Numerical Methods in Engineering (CIMNE)Campus Norte UPC, Edifici C1, Gran Capitan, 08034, Barcelona

e-mail: [email protected] page: http://www.cimne.com

Key words: PFEM, Fluid Dynamics

Abstract. Several comparisons between experiments and computational models are pre-sented in the following pages. The objective is to verify the ability of Particle FiniteElements Methods (PFEM) [1] [2] to reproduce hydraulic phenomena involving large de-formation of the fluid domain [4].

1 INTRODUCTION

The simulation of complex fluid flows involving large variations of the computationaldomain, constitutes an open challenge using most numerical techniques. The ParticleFinite Element Method allows to merge the advantages of the “standard” FEM withthe ones of meshless methods and it is naturally well suited to address this category ofphenomena [3], [6]. At the current stage it still remains open the aspect of its validationin application to real problems. Current work aims to fill this gap by providing someexperimental comparison to real flow cases.

First of all the behavior of a jet after a flip bucket is analyzed both for a 2D and a 3Dcase. The parameters compared are in this case the trajectory and the values of pressureon the “invert”. It follows the analysis of the opposite phenomenon: the under seal flowunder a planar sluice gate. Finally the flux over a stepped spillway is briefly analyzed.

2 FLIP BUCKET

Flip buckets are energy dissipators used at the end of ski jump spillway of big dam:the purpose of this structure is to throw water well clear of the dam. The jet of a skijump spillway leaves horizontally whereas the jet of a flip bucket is deflected upwards toinduce disintegration in the air.

Both a 2D and a 3D model are considered in order to reproduce the experimental setupused by W. H. Hager [7] and R. Juon at the Zurich University. The original aim of the

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investigation was to propose a simple theory for the behavior of a flux over flip buckets.This included the creation of fitting curves for the experimental data which can be takenfor a comparison with the PFEM numerical solution.

Figure 1: Photo of the experimental set-up at th Zurich University [7]

(a) Geometry of the experimental setting (b) Schematic representation of a flip bucket

Figure 2: 2D model

Figure 3: 2D model

A simple 2D model which reproduced the geometry was built. All the fluid used in theanalysis is progressively inserted in a Lagrangian way imposing the water depth at theinlet and the discharge [5].

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Figure 4: Jet development

The two parameters analyzed are the jet trajectory and the pressure along the “invert”(the reversed curve which makes the fluid to “jump”) of the incoming channel. Differentscenarios are considered by varying the discharge and consequently the Froude Numberwhile preserving the geometry of the invert and the depth at the inlet. For each casedifferent meshes are used to verify the convergence to the real solution.

All the details on the theoretical and empirical functions used in the comparison canbe found in [7].

Figure 5: Theoretical trajectory

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(a) 1cm mesh

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(b) 0.5cm mesh

Figure 6: Comparison between theoretical and computational output: mesh variation

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The solutions are improving as expected when the mesh is refined as can be seen infig. 9.

The results seem also to improve for the increasing of Froude Numbers. This can beexplained by the reduced importance of the viscosity effects which cannot be resolved onthe coarse meshes used.

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(a) Froude Number= 3

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(b) Froude Number= 5

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(c) Froude Number= 7

Figure 7: Comparison between theoretical and computational output: velocity variation

Analogous considerations can be made in the case of the analysis of pressure head onthe invert that can be compared with an empirical function given in [7].

The 3D model was then built to reproduce the effect of the introduction of a deflectoras shown in fig 10 and 11, the planar and side developments of this wave were comparedwith photos of the experiment and it is qualitatively respected as shown in fig.12 and 13.

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Figure 8: Empirical pressure head development above the invert

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a]

(b) 0.5cm mesh

Figure 9: Comparison between empirical values and computational output: mesh variation

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Figure 10: Effect of the insertion of a deflector

Figure 11: 3D Model

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Figure 12: Fr5-Side

Figure 13: Fr5-Plane

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3 SLUICE GATE

The behavior of an under seal flow is the second analyzed hydraulic phenomenon. Aplanar sluice gate creates a regular and controlled discharge of fluid: this is controlledonly by the geometrical characteristics and by the depth of water of the upstream tank.The data are taken from an experiment made at the hydraulic division of the Universityof Padua. The under seal flow is governed by:

Q = a · Cc

√2gh (1)

where Cc is the contraction coefficient that, for a planar thin gate is 0.611, a is the sluicegate elevation from the bottom of the channel and h is the water level in the upstreamtank.

The parameters controlled in this case are:- The pressure along the gate;- The outing discharge;- The analysis of the free surface of the downstream water;

13sec

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Y [

m]

Pressure onthe gate

hydrostaticdistribution

Numericalmodel

(a) After 13sec

52sec

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(b) After 52sec

Figure 14: Comparison between empirical functions, computational output and hydrostatic distribution

Different models have been built for the creation of a system that represented the realsetting of an upstream tank with constant level of water. The inflow is given again ina Lagrangian way which originates a perturbation in the level of the reservoir [5]. Thepressure head value in function of the vertical coordinate is compared with the hydrostaticdistribution and the experimental values, as can be seen in fig. 14.

The discharge, obtained by the integration of the velocity diagram viewed in differentsections for a same time instance, presents oscillations with an error that arrives at the

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0.0000.0050.0100.0150.0200.0250.0300.0350.040

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Experimental Data40.96sec

(a) After 40sec

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m]

Experimental Data50sec

(b) After 50sec

Figure 15: Contraction of the free surface after the sluice gate in two different instances

10% of the expected values. This can be explained by the oscillations in the level of theupstream tank.

The contraction of the under seal flow, on the contrary, is well reproduced in fact theoscillation which is present is of the same order than the dimension of the mesh, as canbe seen in fig.15.

Figure 16: Hydraulic jump, velocity output

Inserting an high step, a slow downstream discharge is created. From the clash betweenan upstream fast discharge and a downstream slow one, an hydraulic jump has to occur,this is what is shown in fig. 16; the development of the free surface is the parameter whichis compared with experimental data: the manual measurement of the free surface in adissipation phenomenon like the hydraulic jump can only be qualitative and can be madewith a low precision in some points whereas the computational datum is a continuum one.

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22.75sec

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Y [

m]

Figure 17: Hydraulic jump. Blue line: free surface in the computational model. Pink line: interpolationbetween experimental measurements. Pink points: experimental measurements

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4 STEPPED SPILLWAY

The last problem considered, which is currently under investigation, is the flux over astepped spillway. This is a category of structure which is nowadays becoming commonbecause of the introduction of roller compacted concrete (RCC) that made it to becomeeconomically competitive with traditional spillway with dissipation pools. Basically, whatis analyzed is the flux over a stair. The experimental data are taken from a Phd thesisdone by Prof M.Sanchez-Juni and A.Taboas Amador at the Universitat Politecnica deCatalunya [8], [9].

Figure 18: Example of a stepped spillway

Many and precise informations are available to reproduce accurate simulations of thedevelopment of the phenomenon. 2D models are right now created to analyze the devel-opment of velocity and pressure over the steps in the upper part of the stair where air isnot present.

Figure 19: The 2D model

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(a) Computational results

(b) Experimental results

Figure 20: Velocity distribution after 3.4sec

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REFERENCES

[1] E. Onate , S. R. Idelsohn, F. Del Pin and R. Aubry. The Particle Finite ElementMethod. An overview. International Journal of Computational Method., 2,267–307,2004 .

[2] S. R. Idelsohn, E. Onate, F. del Pin and N. Calvo. Fluid-Structure Interaction Usingthe Particle Finite Element Method. Computer Method in Applied Mechanics andEngineering., 195,2100–2123, 2006.

[3] S. R. Idelsohn, E. Onate and F. Del Pin. The Particle Finite Element Method: apowerful tool to solve incompressible flows with free-surfaces and breaking waves.International Journal for Numerical Methods in Engineering., 61,964–984, 2004.

[4] E. Onate, J. Rojek, M. Chiumenti, S. R. Idelsohn, F. del Pin and R. Aubry. Ad-vances in stabilized finite element and particle methods for bulk forming processes.Submitted to: Computer Methods in Applied Mechanics and Engineering., 2004.

[5] E. Onate, J. Garcıa, S. R. Idelsohn and F. Del Pin. FIC formulations for finite elementanalysis of incompressible flows. Eulerian, ALE and Lagrangian approaches. In Pressin: Computer Methods in Applied Mechanics and Engineering., 2004.

[6] R. Aubry, S. R. Idelsohn and E. Onate. Particle Finite Element Method inFluid Mechanics including Thermal Convection-Diffusion. Computer and Structures.,83,1459–1475, 2005.

[7] R. Juon , W. H. Hager. Flip Bucket Without And With Deflector. Journal of Hy-draulic Engineering., 126,837–845, 2000 .

[8] M. Sanchez-Juni. Comportaminento hidraulico de los haliviadero escalonados en presade hormigon compactado. Analis del campo de presiones, Phd Thesis of the Univer-sitat Politecnica de Catalunya, (2001).

[9] A. Taboas Amador. Comportaminento hidraulico de los haliviadero escalonadosen presa de hormigon compactado, Phd Thesis of the Universitat Politecnica deCatalunya, (2004).

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ON THE VALIDATION OF THE PARTICLE FINITE ELEMENTS …