Olesen, John Forbes

Poulsen, Peter Noe

Skocek, Jan

Stang, Henrik

Svec, Oldrich

Self compacting concrete (SCC) is a promising material in the civil engineering industry. One of the benefits of the SCC is a fast and simplified casting followed by decreased labor costs. The SCC as any other type of concrete has a significantly lower tensile and shear strength in comparison to the compression strength and, therefore, it needs to be reinforced. Fiber reinforced concrete is an alternative to traditional stirrups reinforcement leading to lowered labor costs. To be able to access mechanical properties of the fiber reinforced concrete, knowledge of final spread and directions of fibers is necessary. Computational fluid dynamics (CFD) comes to play at this stage. Formulation of a possible CFD model that is able to solve multi-phase and multi component non-Newtonian flow with complex boundary conditions and fiber suspension and preferably in reasonable time brings a very challenging task. A relatively new group of models - Lattice Boltzmann Modeling (LBM) - is presented in this paper. The conventional LBM is modified to include fiber and particle suspensions and non-Newtonian rheology and is used to model the fiber reinforced self compacting concrete flow

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Downloaded from orbit.dtu.dk on: Dec 18, 2017Flow simulation of fiber reinforced self compacting concrete using Lattice BoltzmannmethodSvec, Oldrich; Skocek, Jan; Stang, Henrik; Olesen, John Forbes; Poulsen, Peter NoePublished in:DisseminationPublication date:2011Document VersionEarly version, also known as pre-printLink back to DTU OrbitCitation (APA):Svec, O., Skocek, J., Stang, H., Olesen, J. F., & Poulsen, P. N. (2011). Flow simulation of fiber reinforced selfcompacting concrete using Lattice Boltzmann method. In DisseminationFlow simulation of fiber reinforced self compacting concrete using Lattice Boltzmann method  1Oldřich Švec1* 1 Technical University of Denmark, Department of Civil Engineering, Lyngby, Denmark  2Jan Skoček 2 Technical University of Denmark, Department of Civil Engineering, Lyngby, Denmark  3Henrik Stang, 3John Forbes Olesen; 3Peter Noe Poulsen 3 Technical University of Denmark, Department of Civil Engineering, Lyngby, Denmark    Abstract Self compacting concrete (SCC) is a promising material in the civil engineering industry. One of the benefits of the SCC is a fast and simplified casting followed by decreased labor costs. The SCC as any other type of concrete has a significantly lower tensile and shear strength in comparison to the compression strength and, therefore, it needs to be reinforced. Fiber reinforced concrete is an alternative to traditional stirrups reinforcement leading to lowered labor costs. To be able to access mechanical properties of the fiber reinforced concrete, knowledge of final spread and directions of fibers is necessary. Computational fluid dynamics (CFD) comes to play at this stage. Formulation of a possible CFD model that is able to solve multi-phase and multi component non-Newtonian flow with complex boundary conditions and fiber suspension and preferably in reasonable time brings a very challenging task. A relatively new group of models - Lattice Boltzmann Modeling (LBM) - is presented in this paper. The conventional LBM is modified to include fiber and particle suspensions and non-Newtonian rheology and is used to model the fiber reinforced self compacting concrete flow.   Originality Application of Lattice Boltzmann Modeling to the field of fiber reinforced self compacting concrete is a novelty. It is the first paper dealing with multiphase flow with fiber shaped particle suspension and non-Newtonian rheology.   Chief contributions One of the main contributions presented by the paper is the ability to predict final dispersion and orientation of fibers in the fiber reinforced self compacting concrete and thus to allocate possible problematic places. Such a prediction is not possible in other conventional methods in a comparable computational time.   Keywords: Steel fiber reinforced concrete, Lattice Boltzmann method, Immersed Boundary method                                                  1 Corresponding author: Email olsv@byg.dtu.dk Tel +45 4525 1940 1 INTRODUCTIONIn the past, one of the main bottlenecks in application of steel fiber reinforced concrete for load carryingstructures has been the lack of design rules and design tools including the absence of safety factors tak-ing into account knowledge of expected variations in material properties and structural performance. Itis believed that fiber orientation and distribution plays an important role in the final properties of struc-tural elements and it is thus desirable to understand the underlying phenomena governing the final fiberdistribution and orientation. Extensive experimental work becomes relatively expensive and time con-suming and therefore simulations capable describing the nature of the problem might be an appropriatealternative.In this paper a relatively new Lattice Boltzmann Method (LBM) (Aidun & Clausen, 2010) is beingused as a Computational Fluid Dynamics (CFD) solver to simulate a non-Newtonian flow of the selfcompacting concrete (SCC). Particles of any shape such as round or elongated are implemented in theform of a force field using a modified Immersed Boundary Method (IBM) with direct forcing (Feng& Michaelides, 2005). A new non-linear dynamics algorithm has been developed and is being appliedto ensure stability of the simulation (Skocek et al., 2011). Free surface is implemented using a masstracking algorithm (Ko¨rner et al., 2005).2 LATTICE BOLTZMANNIn the traditional CFD tools such as the Finite Volume Method, the problem is formulated in the formof macroscopic quantities, namely spatially and time dependent velocity and pressure fields. The LBMon the other hand, with its roots in the kinetic theory of gases, treats fluid as particle distributions wherethe particles can be seen as e.g. molecules. Discrete particle distribution functions are used to discretizethe particle distributions. The LBM provides rules for mutual collisions and propagation of the particledistributions as well as for the computation of the macroscopic quantities.In the framework of the LBM, space is usually discretized by a set of Eulerian cells of a uniformsize. Similarly, time is discretized into uniform time steps. Usually, the space and time spacing, aswell as a reference density of the fluid are set to unity and the real problem is scaled accordingly.Dimensionless lattice units are considered if not stated otherwise. Each Eulerian cell has one latticenode in its center. The lattice node holds a set of N discrete particle distributions fα,α = 1...N fordifferent lattice velocities.The LBM equation can then be written asfα(x+ cα∆t, t+∆t) = fα(x, t)+Ωα(x, t), (1)where cα is a microscopic lattice velocity of the particle distribution and Ωα(x, t) is a collision operator.The lattice velocities are in the order of the speed of sound i.e. in the order of 1. The LBM for an incom-pressible fluid is valid only for low Mach numbers and therefore u f (x) 1, where u f is a macroscopicvelocity of the fluid. Local density can be computed as a sum of particle distributions in one nodeρ(x, t) =N∑α=1fα(x, t) (2)and local velocity as a sum of particle distributions in one node multiplied by their respective latticevelocities cαu f (x, t) =1ρ(x, t)N∑α=1fα(x, t)cα. (3)The equations presented above comprise almost all ingredients required for the fully working LBM.Some parts such as the collision operator Ωα(x, t) were omitted for brevity. The LBM therefore has arelatively simple implementation and reasoning for the given problem. The code is also easily paral-lelized since most of the parts of the LBM core require local information only.Further information regarding the LBM may be obtained from (Sukop & Thorne, 2005; Wolf-Gladrow, 2000).3 IMMERSED BOUNDARY METHOD MODIFICATIONThe Immersed Boundary Method (IBM) will in short be introduced via an stationary example of a singlesolid body immersed in a flow as presented in Figure 1. In the framework of the IBM, the fluid observesthe body in the form of a force field. Traditionally one would measure momentum exchange caused byparticle distributions hitting the body for a unit time. In the IBM, the particle distributions are allowed topenetrate the solid body. At the end of each time step the penetrated particle distributions are measuredand corresponding forces F are exerted both back on the fluid and on the body. The total force exertedon the body is defined asF =∫Vρ f (x)(u f (x)−ub(x))dV, (4)where ρ f (x), u f (x) and ub(x) stand for the local density of the fluid, local velocity of the fluid and localvelocity of the immersed body. V is the volume of the body.Measuring the amount of particle distributions penetrating the solid body is a crucial part and, as willbe shown, it restricts the usage of the IBM to bodies of the minimum size larger than√2 lattice units.Outer particle distributions are schematically shown in Figure 1 A as hollow dots. Inner particle distri-butions are not displayed for simplicity. All the particle distributions can be measured macroscopicallyby a local velocity as presented in Eq. ( 3 ).A) B) C) D)Figure 1: Representation of particle distributions penetrating a solid bodyHaving u f (xb) = 0, where xb ∈ V , suggests, that no outer particle distributions penetrated the sta-tionary solid body in that region. Contrary to this, a non-zero velocity u f (xb) 6= 0 suggests that particledistributions originating outside of the body have reached the region inside the body and therefore shouldbe accounted for by the immersed boundary force. This is schematically shown in Figure 1 B, wherefilled dots are outer particle distributions that penetrated the body in the previous time step. Decreasingthe minimum size of the body, outer particle distributions penetrating the solid body will stream throughthe whole body and leave the body on the other side during one time step as presented in Figure 1 C, D.Thus, the local velocity field no longer describes penetrating particle distributions correctly. The moreparticle distributions leave the body, i.e. the smaller the body is with respect to spatial discretization,the higher inaccuracy is observed as shown in Figure 3. Therefore, the minimum size of the element isequal to the speed of particle distributions and is in the range of√2 lattice units.Fibers are bodies with traditionally high aspect ratio. For the case of steel fibers the aspect ratioranges between 30 - 80. The discretized length of the fiber would have to be 40 - 110 lattice units tosatisfy the above mentioned condition. This would significantly restrict the usage of the method andhence a correction of the method is explored in the following.3.1 Fitting the Correction termDrag force on an infinite circular cylinder midway between two parallel plates has been measured usingan uncorrected version of the IBM and compared to an empirical solution as described in (Benrichou,2004). Geometry of the problem is shown in Figure 2. The flow is driven by a Poiseuille velocity profilexybdLFigure 2: Geometry of the Poiseuille flow modelplaced both at the inflow and at the outflow. An empirical drag force is computed asFx = µUx,maxpi(4k2−8)1.9362−3.7520k2 +2ln(k) (5)where Fx is the drag force,Ux,max is the maximum speed of the defined Poiseuille flow and k= d/b is thefraction of the cylinder diameter and the channel width. The empirical solution in Eq. ( 5 ) is valid onlyfor a Newtonian flow at very low Reynolds numbers and for k< 0.4. In all the simulations, the Reynoldsnumber has been set to 0.1. The length of the channel has been set to value of L = max(30d,30), thewidth of the channel to b= 20 and the diameter of the cylinder ranges from 0.3 up to 5 lattice units. ANewton - Raphson method has been used for each of the diameters to find a correction term C, such asto equalize the simulation drag force F with the empirical one Fx, assumingFx = F =C∫Ωρ f (x)(u f (x)−up(x))dV. (6)It has been found out that the correction term can be approximated byC = 1.8d−1.2, d < 1.5 (7)1, d > 1.5. (8)Figure 3 presents an error of the drag force as a function of the cylinder diameter both for a correctedand uncorrected version of the IBM. It is clearly seen, that such a simple fitting function significantlyreduces the error from 80 % to 3 % for the diameter 0.3.The Couette flow test has been run to support the suggested correction term.4 COUETTE FLOWA suspension of steel fibers of various volume fractions immersed in a Bingham fluid were subjectedto the Couette flow. The geometry of the problem can be seen in Figure 4. The following values were-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.5  1  1.5  2  2.5  3  3.5  4  4.5  5Error [-]Cylinder diameter [lattice units]Corrected IBMUncorrected IBMFigure 3: Error of the IBM simulation before and after correctionconsidered: Lp = 0.06 m corresponding to 12 lattice units, aspect ratio of the fiber 37.5, Lx = Lz = 2Lp,Ly = 4Lp, ρp = 7000 kgm−3, ρ f = 2160 kgm−3, τy = 5 Pa and µpl = 10 Pas. The simulation wasperformed for three shear rates equal to 0.4 s−1, 4.0 s−1 and 10.0 s−1 respectively, and for volumefractions ranging between 0.001 and 0.04.LxLyLzLp ρf , τy, µplu/2u/2ρpFigure 4: Couette flow geometry.Resulting effective viscosities µe f f were computed from the stresses acting on the boundary walls.Shear rates were obtained by a volume averaging of the fluid between the walls excluding zones closerthan 0.5Lp to walls. An assumption that the resulting effective fluid is a Bingham fluid has been applied.Figure 5 shows a relation of the relative viscosity µe f f /µpl for different volume concentrations offibers. The computed values are compared with a relation by (Ghanbari & Karihaloo, 2009) both forthe case of the corrected and uncorrected version of the IBM represented as filled and hollow marksrespectively. A simulation of fibers with no rotation allowed has been run (circular marks) togetherwith simulations of fibers with allowed rotation (square and triangular marks). Simulations of fiberswith allowed rotations were further divided into two simulations, one with lubrication forces deactivated(square marks) (Yamane et al., 1994) and the other with lubrication forces activated (triangular marks).Looking at hollow and filled marks, one can observe that the IBM correction term apparently helpsto catch correctly the underlying physics and therefore results in more realistic values of the dynamicviscosity comparable to (Ghanbari & Karihaloo, 2009). The IBM correction plays an important roleeven for fibers with allowed rotation as shown by square and triangular marks. Increase of the viscositydue to lubrication forces can be observed in the figure comparing square and triangular hollow marks.The lubrication forces, however, play a less significant role in comparison to the IBM correction termfor the investigated volume fractions. 0 2 4 6 8 10 12 14 16 0  0.01  0.02  0.03  0.04µ eff / µpl [-] Volume concentration of fibers [-] Ghanbari, KarihalooCorrected IBM, No RotationUncorrected IBM, No RotationCorrected IBM, No LubricationUncorrected IBM, No LubricationCorrected IBM, LubricationUncorrected IBM, LubricationFigure 5: Relative viscosity comparison for different volume concentrations of fibers5 SLUMP TESTFigure 6: Slump test time snaps. Color stands for viscosity.The slump test has been modeled to present the capabilities of the method. A Cone of height 0.3 m,base radius 0.1 m and top radius 0.05 m has been filled with the Bingham fluid of dynamic plasticviscosity 10 Pas and yield stress 5 Pa. Steel fibers of the length = 0.05 m, aspect ratio = 37.5 and fibervolume fraction = 0.2 % have been used in the simulation. Length of fibers corresponds to 12.5 latticeunits and diameter to 0.33 lattice units. A correction term from Eq. ( 7 ) has been applied. The shape ofthe cone and distribution of fibers at 3 different time snaps are shown in Figure 6.As for the efficiency of the code, it takes ca. 30 minutes using Intel Core 2 Due E8400 CPU with 3GB of RAM to run 0.1 second of the simulation corresponding to 407 time steps. The fibers would haveto be at minimum 54 lattice units long without the correction term to obtain correct immersed boundaryforces. 0.1 second of the simulation would then take ≈ 1400× longer i.e. 1 month and the memoryconsumption would also grow significantly.6 CONCLUSIONSThe Lattice Boltzmann method together with the corrected Immersed boundary method and mass track-ing algorithm has been applied to simulate a non-Newtonian liquid together with a particle inclusion.A correction term of the IBM has been proposed and determined by fitting, to allow for particles of aminimum size less than√2 to be included in the modeling. Fibers of a diameter not less than 0.3 latticeunits have been used so far and some of the results such as Couette flow or Slump test are presented inthis paper.The presented model is based on well established methods and provides an efficient tool that can beapplied to a range of engineering problems on different length scales yielding results matching favorablytheoretical or experimental findings in a reasonable amount of time and using a reasonable amountcomputation power.7 ACKNOWLEDGMENTSSupport from the Innovation Consortium: ”Sustainable Concrete Structures with Steel Fibres - TheSFRC Consortium” funded by the Danish Agency for Science Technology and Innovation is acknowl-edged.ReferencesC. K. Aidun & J. R. Clausen (2010). ‘Lattice-Boltzmann Method for Complex Flows’. Annual Review of FluidMechanics 42(1):439–472.A. Benrichou (2004). ‘Drag force on a circular cylinder midway between two parallel plates at very low Reynoldsnumbers - Part 1: Poiseuille flow (numerical)’. Chemical Engineering Science 59(15):3215–3222.Z. Feng & E. Michaelides (2005). ‘Proteus: a direct forcing method in the simulations of particulate flows’.Journal of Computational Physics 202(1):20–51.A. Ghanbari & B. L. Karihaloo (2009). ‘Prediction of the plastic viscosity of self-compacting steel fibre reinforcedconcrete’. Cement and Concrete Research 39(12):1209–1216.C. Ko¨rner, et al. (2005). ‘Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming’. Journal ofStatistical Physics 121(1-2):179–196.J. Skocek, et al. (2011). ‘Modeling of flow of flow of particles in non-Newtonian fluid using lattice Boltzmannmethod’. In Proceedings of ICCC.M. C. Sukop & D. T. Thorne (2005). Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engi-neers. Springer.D. A. Wolf-Gladrow (2000). Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction(Lecture Notes in Mathematics). Springer.Y. Yamane, et al. (1994). ‘Numerical simulation of semi-dilute suspensions of rodlike particles in shear flow’.Journal of Non-Newtonian Fluid Mechanics 54:405–421.

Flow simulation of fiber reinforced self compacting concrete using Lattice Boltzmann method

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Flow simulation of fiber reinforced self compacting concrete using Lattice Boltzmann method

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