Australian Journal of Basic and Applied Sciences, 5(7): 1213-1220, 2011 ISSN 1991-8178 Modeling Approach of the water/soil Interface in the Hole Erosion Test (HET) 1 Kissi Benaissa, 2 El Bakkali Larbi, 3 Parron Vera Miguel Angel and 4 Addaoudi Kaoutar 1,2 Modelling and Simulation of Mechanical Systems Laboratory, Faculty of Sciences at Tetouan, University Abdelmalek Essaadi, BP. 2121, M'hannech, 93002, Tetouan, Morocco. 3,4 Mechanical and Civil Engineering Department, Polytechnic High School of Algeciras, University of Cadiz, Ramon Pujol avenue, Algeciras 11202, Spain. Abstract: Soil erosion is a complex phenomenon which yields at its final stage insidious fluid leakages under the hydraulic infrastructures known as piping and which provokes their rupture. The Hole Erosion Test is commonly used to quantify the rate of piping erosion. In this work, The Hole Erosion Test is modelled by using Fluent package. This modelling makes it possible describing the effect of the clay concentration on erosion by using the RNG - based k-g turbulence model equations, and predicts a non uniform erosion along the hole length unlike the usual one dimensional models. In particular, the clay concentration is found to increase noticeably the erosion rate. Key words: Piping, Erosion, Turbulence, k-g model, Concentration of Clay INTRODUCTION Soil erosion is a natural process, occurring over geological time, and indeed it is a process that is essential for soil formation in the first place. With respect to soil degradation, most concerns about erosion are related to accelerated erosion, where the natural rate has been significantly increased mostly by human activity.this complex phenomenon yields at its final stage insidious fluid leakages under the hydraulic infrastructures known as piping and which provokes their rupture. Many dam ruptures have occurred throughout the world, some of these events have been reported in reference (Foster and Fell, 2000). Such catastrophic accidents can generate human casualties and material losses with dramatic consequences at the social and economic levels. Internal erosion is a progressive degradation of soils which is induced by the action of a flowing fluid through the porous medium. Many research activities related to the experimental and theoretical characterization of this phenomenon are reported in the literature such as (Wan and Fell, 2004a; Wan and Fell, 2004b) and (Bonelli et al., 2006). Several experiments were designed to reproduce this mechanism in laboratory conditions. Recently, the Hole Erosion Test (HET), figure 1, has been introduced. This test has been the subject of multiple investigations both experimentally and theoretically. Many HET experiments were carried out on several kind of soils, (Wan and Fell, 2004a; Wan and Fell, 2004b). Modeling of this test has also been presented (Bonelli et al., 2006). In all cases this test proved to be simple, fast, and well adapted to perform surface erosion characterization during piping development. A simplified one dimension modeling of the HET was introduced in (Bonelli et al., 2006). This modeling proved to be sufficient in explaining the erosion phenomenology related to piping problem. It yields a comprehensive description of the erosion initiation and kinetics for a given soil constitution. This rudimentary model enables also to evaluate the influence of the hydraulic conditions on the kinetics. Aspects associated to the two-dimensional nature of the HET are also present in the problem as it could be seen in figure 1. The objective of this research is to use the commercial CFD code Fluent to model the developing turbulent flow in the HET by using the RNG - based k-g turbulence model. The aim is to determine the shear stress and erosion rate taking place at the wall interface by considering the effect of clay concentration variations. Two-dimensional Modeling Approach of the HET: The turbulence modelling is achieved by means of Fluent software package. Fluent is a general purpose Computational Fluid Dynamics (CFD) code that has been applied to various problems in the fields of fluid mechanics and heat transfer. This code has been validated through numerous investigations. Fluent is especially Corresponding Author: Kissi Benaissa, Faculty of Sciences at Tetouan, Tetouan 93002, Phone: +212672866058, Fax: +212539994500, E-mail: kissifst@yahoo.fr, 1213
appropriate for the complex physics involved in heat and mass transfer and considers mixtures by modeling each fluid species independently or as a homogenized medium, [6-7]. Fig. 1: Sample tested with the HET (Wan and Fell, 2004) Flow taking place inside the hole is turbulent. To perform realistic simulation of turbulence, the exact instantaneous Navier-Stokes governing equations are habitually time-averaged or ensemble-averaged. The obtained averaged equations contain further unknown variables, and turbulence models are introduced to determine them in terms of known quantities. Various turbulence models have been proposed in the literature; however there is no single turbulence model which could be universally applied for all classes of problems. The choice of a pertinent model for a given problem will depend on the actual physics of the flow, the degree of accuracy required and the computational cost tolerated. Reference (Fieldview Reference Manual, 2004)] gives a detailed discussion on how to perform at best the appropriate choice of a turbulence model. Among the various models, the standard k-g model which was proposed first by Launder and Spalding, (1972) has become the most popular when dealing with practical engineering flow calculations. This model relies on phenomenological considerations and integrates empiricism to perform closure of equations. Improvements of the standard k-g model such as the RNG k-g model have been made, (Yakhot and Orszag, 1986). This model was derived by using a rigorous statistical technique (called renormalization group theory). In comparison with the standard k-g model, RNG model includes refinements which significantly improve the accuracy for rapidly strained flow. In contrast with the standard k-g model which is rather designed for high-reynolds-numbers and for which the effective viscosity is constant, the RNG theory provides an analytically-derived differential formula for effective viscosity that accounts also for low-reynolds-number effects. Use of this feature requires however an appropriate treatment of the near-wall region. These advantages make the RNG k-g model more accurate and reliable for a wider class of flows than the standard k-g model. Among the family of k-g turbulence based models Fluent, (2005), provides the following choices: the standard k-g model, the Renormalization-group (RNG) k-g model and the Realizable k-g model. Because of its relevance which was stated through various investigations, use is made subsequently of the RNG k-g model. The governing equations of this model are recalled in the following before presenting the special treatment adopted in Fluent for the near-wall boundary conditions. The RNG based k-g turbulence model: The RNG k e model is derived from the instantaneous Navier Stokes equations using a mathematical technique known as the renormalization group (RNG) methods. The basic idea of the RNG method as applied to turbulence modeling is the elimination of small-scale eddies from governing equations by expressing their effects in terms of larger scale motions and a modified viscosity. The interaction of small-scale flows with large-scale perturbations has been studied analytically by Sivashinsky and Yakhot, (1985). For the particular case of axisymmetrical problems, the unsteady Reynolds averaged incompressible Navier- Stokes equations write: 1 ( ru) v 0 (1) r r z 2 u 1 ru ( uv) 1 p 1 u 2 u u r ( u') uv 2 t r r z r r r r r z z (2) 1214
2 1 ruv ( v ) 1 p v 2 1 v ( v') r uv r r z z z z r r r (3) wheret is the time, ρ the density of fluid, u the main velocity in the radial direction r, v the mean velocity in the axial direction z, p the pressure, μ the kinematics viscosity, ur the fluctuating component of radial velocity, vr the fluctuating component of axial velocity and the symbol bar denotes statistical averaging. The momentum equations (2) and (3) contain the Reynolds stresses which result from the averaging process; their determination can be expressed according to the Boussinesq hypothesis in terms of the mean velocity gradients as 2 u 2 1 ur v u' t k t t (4) r 3 r r z u v uv t z r 2 v 2 1 ur v v t kt t z 3 r r z With the turbulent kinetic energy k u u uv 2 2 ( ) ( ) 2 /2 (5) (6) and the effective kinematic viscosity μ t given by t Ck 2 /, where C μ is constant. The RNG k-g model differs from the standard model by the special form of the transport equations which contain the additional term R g. These equations write k 1 rku kv 1 k k k 2 tr t t (7) 2 ts t r r z r r r z z r 2 ru v k 1 1 1 r C C R r t t t 2 1 2 t r r z r r r z z k k Where a is the inverse effective Prandtl number for both k and g, and are constants. C1 C 2 The RNG k-g model constants have values derived analytically by the RNG theory. They are given by: C 1.42, C 1.68, C 0.0845, C 1.68 and 0.012 1 2 2 For further details, a complete description of RNG theory and its application to turbulence modeling can be found in (Choudhury, 1993; Pope, 2000) and Fluent User s Guide (Fluent, 2005). The classical linear erosion law states that erosion rate, which corresponds to the amount of mass departure (8) due to erosion per unit time and by unit surface area, is given by: c ( ) er er s where C er and τ s are constant depending on the considering soil material. The rate er is related to time variation of local radius dr / dt by where is the dry density of soil. The erosion law yields that is proportional to er d the amount of shear exceeding the shear threshold limit τ s. d RESULTS AND DICUSSIONS The fluid domain which is assumed to be axisymmetric extends over 170 mm in the axial z-direction and 30 mm in the radial r-direction. The domain is oriented such that the inlet section is at left and the outlet section is at right, figure 2. 1215 er
The table 1 give the modeling parameters of the hole erosion. Fig. 2: Geometry of the tube Table 1: Homogenized density and viscosity for water-clay mixtures as function of clay concentration Clay concentration (%) Density of the mixture (kg/m 3 ) Viscosity of the mixture (Pa.s) 0 1000 0.001 1.8 1005 0.00177 2.6 1011 0.00189 3.85 1020 0.00194 Figure 3 shows in the case of the three clay concentration the static pressure at the axis of symmetry as function of the axial coordinate for the Pressure 11177(Pas). Fig. 3: Pressure at the axis of symmetry as function of the axial coordinate Figure 4,5,6 shows shear stress contours inside the soil sample for. Fig. 4: Wall-shear stress obtained for pinlet 3726 Pas as function of clay concentration 1216
Fig. 5: Wall-shear stress obtained for pinlet 7451 Pas as function of clay concentration Fig. 6: Wall-shear stress obtained for pinlet Figure 7,8,9,10 gives the erosion rate (in 11177 Pas as function of clay concentration 6 10 / 1217 kg s ). This amount is obtained by integrating the erosion law over the whole length of the hole and by multiplying the result by the initial circumference of the hole. 4 The erosion constants used are: c 5.510 s/ m and 7 Pa. These correspond to a specific er soil sample containing 50% kaolinit clay and 50% of sand that was tested in (Pham, 2008). Conclusions: A two-dimensional modeling of the Hole Erosion Test was carried out in this work. Unlike the early models which are essentially one-dimensional, the two-dimensional modeling had shown that the wall-shear stress is not uniform along the hole wall. It was possible then through using a linear erosion law to predict non uniform erosion along the hole length. s
Fig. 7: Erosion rate as function of inlet pressure and clay concentration combination for P1 Fig. 8: Erosion rate as function of inlet pressure and clay concentration combination for P2 Fig. 9: Erosion rate as function of inlet pressure and clay concentration combination P3 1218
Fig. 10: Erosion rate as function of inlet pressure and clay concentration combination P123 Studying the effect of clay concentration has shown that it has not a negligible effect on the wall-shear stress and thus would affect in its turn surface erosion that develops at the fluid soil sample interface, particularly at the outlet extremity of the hole where it is maximal. This enabled qualitatively understanding why the eroded profile of the hole wall as observed during experiment is not uniform. ACKNOWLEDGEMENT The authors would like to thank the Spanish Agency of International Cooperation (AECID) and CNRST Moroccan which has supported financially this research. REFERENCES Bonelli, S., O. Brivois, R. Borghi, N. Benahmed, 2006.: On the modelling of piping erosion. C.R. Mécanique, 22: 225-244. Bonelli, S., N. Benahmed, O. Brivois, 2006.: On modelling of the hole erosion test. ICSE, 3rd International Conference on Scour and Erosion, 1-3, Amsterdam. Choudhury, D., 1993. Introduction to the Renormalization Group Method and Turbulence Modeling. Fluent Inc. Technical Memorandum TM-107. Escue, A., J. Cui, 2010. Comparison of turbulence models in simulating swirling pipe flows. Applied Mathematical Modelling. (2010) doi:10.1016/j.apm.2009.12.018. Fieldview Reference Manual, 2004. Software Release Version 10. Intelligent Light. Fluent, 2005. 6.2 Users Guide. Fluent Inc. Fluent, 2005. 6.2 Users Guide. Fluent Inc. Foster, M.A., R. Fell, M. Spannangle, 2000.: The statistics of embankment dam failures and accidents. Canadian geotechnical Journal 37(5): 100-1024. Launder, B.E., D.B. Spalding, 1972. Lectures in Mathematical Models of Turbulence. Academic Press, London, England. Pham, T.L., 2008. Erosion et dispersion des sols argileux par un fluide. In French. Ph.D Thesis, Order number: D.U. ED: 430. Ecole Nationale des Ponts et Chaussées, Paris, France. Pope, S.B., 2000. Turbulent flows. Cambridge University Press. Sivashinsky, G., V.J. Yakhot, 1985. Phys. Fluids., 28: 1040. Vijiapurapu, S., J. Cui, 2010. Perforamnce of turbulence models for flows through rough pipes. Applied Mathematical Modelling., 34: 1458-1466. Wan, C.F., R. Fell, 2004a: Investigation of rate of erosion of soils in embankment dams, Journal of Geotechnical and Geoenvironmental Engineering, 30(N 4): 373-380. 1219
Wan, C.F., R. Fell, 2004b: Laboratory Tests on the Rate of Piping Erosion of Soils in Embankment Dams, Journal of Geotechnical Testing Journal, 27(3). Yakhot, V., S.A. Orszag, 1986. Renormalization Group Analysis of Turbulence: I-Basic Theory. Journal of Scientific Computing, 1(1): 1-51. 1220