Evaluation of Dam Break Flow of Thixotropic Fluids by Smoothed Particle Hydrodynamics
Dam-break flows can be described as the flow caused by the sudden release of a contained portion of fluid. Many environmental flows can be modeled as dam-break flows. In this study, the unsteady 2D dam break problem is solved by weakly compressible SPH with water as a Newtonian fluid and a thixotropic gel as a non-Newtonian fluid. In this method, the flow domain is replaced by several representative particles and the mass and momentum conservation equations are solved in a Lagrangian frame work for each representative particle. First, the Newtonian case is verified with previous numerical and experimental published results. Then the thixotropic gel is modeled by Moore rheological model and several simulations are performed to investigate the effects of the model constants. Furthermore, the differences of Newtonian and thixotropic fluid flow including free surface shape and leading edge position are mentioned.
At a glance: Figures
Keywords: dam break, thixotropic fluid, numerical simulation, SPH
Journal of Mechanical Design and Vibration, 2014 2 (1),
Received May 20, 2014; Revised May 24, 2014; Accepted May 25, 2014Copyright © 2014 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Kootiani, Reza Cheraghi. "Evaluation of Dam Break Flow of Thixotropic Fluids by Smoothed Particle Hydrodynamics." Journal of Mechanical Design and Vibration 2.1 (2014): 31-34.
- Kootiani, R. C. (2014). Evaluation of Dam Break Flow of Thixotropic Fluids by Smoothed Particle Hydrodynamics. Journal of Mechanical Design and Vibration, 2(1), 31-34.
- Kootiani, Reza Cheraghi. "Evaluation of Dam Break Flow of Thixotropic Fluids by Smoothed Particle Hydrodynamics." Journal of Mechanical Design and Vibration 2, no. 1 (2014): 31-34.
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The Free surface flows are present in a lot of natural phenomena such as volcanic lava and coastal waves or industrial process such as casting, food industries, and concrete production . Thus, correct understanding of the free surface flow is useful. The complication of free surface flow is that the free surface shape is a part of solution and not a prior known parameter. Moreover, most of industrial and physiological fluids indicate thixotropic behaviors, namely their viscosity decreases with the passing of time even exposed to a constant stress; thus, investigation of free surface flow problems in thixotropic fluids is as vital as in Newtonian ones. One of the most famous benchmark problems of free surface flows is dam break problem. Dam break problem is imputed to sudden collapse of fluid column and spread on a plate. This problem is studied numerically, experimentally [1, 2] and analytically  by numerous of researchers particularly with Newtonian fluids. The Newtonian dam break is numerically simulated by variety of Eulerian approaches such as VOF and MAC . However, these method uses fixed Eulerian grid which is not suitable for moving material interfaces specially when this movement is very large because it may lead to numerical dissipation arising due to advection nonlinear terms of NS equations . Another numerical method which is applied to dam break problem is smoothed particle hydrodynamics (SPH) which uses pure Lagrangian frame work. The method is originally introduced for three dimensional open astrophysics problems first by Lucy and then developed by Gingold and Monaghan in 1977 . The essence of the method is that the fluid flow is comprised of single particles interactions which can be tracked in a Lagrangian frame work. The field variables are computed by integral interpolants which are converted to summation over neighboring particles. The original idea treated the incompressible flow as partially compressible flow, but later some researchers developed strictly incompressible SPH (I-SPH) model by solving pressure Poisson equation in a two-steps algorithm . One of the advantages of SPH is its inherent unsteady nature which makes it a suitable candidate for fluids with time dependent viscosity i.e. thixotropic fluids. The dam break problem is investigated by SPH method in some literatures. Shao et al. (2003) studied two dimensional dam break flows of water as a Newtonian and mud as a non- Newtonian Cross model by I-SPH method. Roubtsova et al. (2006) used a Riemann solver in SPH calculation and modeled in viscid dam break flow. Dalrymple et al. (2006) summarizedthe SPH works on John Hopkins University containing a 3D dam break turbulent flow with an obstacle in the flow path. Hosseini et al. (2006) proposed a three-step SPH method and examined their model with a 2D dam break flow with three kind of rheological model including power-law, Bingham plasticand Herschel-Bulkley. Molteni et al. (2009) introduced an artificial density diffusion term to the SPH and applied their new algorithm to dam break flow and showed that this artificial density removed the pressure noises and retrieved the energy level lost due to artificial viscosity. In the present study, first we use weakly compressible XSPH method to simulate Newtonian dam break flow. After parameter setting and validation with previous published results, we simulate dam break problem with a thixotropic fluid which obeys Moore model. Thixotropic dam break is analytically investigated by Chanson et al. (2006) with another model. However, this may be the first time that SPH is used for simulating a thixotropic fluid.
The governing equations of the problem are continuity and conservation of momentum, which is stated in Lagrangian frame work as follows:
Where ρ, t, V, σ, and g density, time, velocity vector, stress tensor and gravitational acceleration, respectively. Stress tensor is containing pressure (p) and shear stress tensor () such as below:
In which is Dirac's delta. Shear stress tensor in Newtonian fluids is related to the strain rate through dynamic viscosity (μ); thus, each component of the shear stress tenser is written in the following form:
Where the strain rate is defined as equation (5):
The rheological model of thixotropic gel is the Moore model which defines the viscosity of the fluid as a function of time and shear rate corresponding to next equation :
Note that the viscosity depends on the shear rate through a structural parameter (s) as below:
Where a and b are constant associated with rebuilding and destruction of the fluid chain structure, respectively and the shear rate term is equal to ( is second invariant of strain tensor). Note that in equation (7) the absolute value of shear rate is used because both compression and elongation lead to breakdown of the structure.
3. SPH Method
First the problem domain is substitute with N arbitrary distributed particles. These particles transport the flow properties such as density, velocity and pressure. Then the integral representation is applied to field functions. Because the domain is represented by some particles, these integral forms of field functions are further converted to algebraic summation over all the corresponding parameters of the neighboring particles in a local domain (support domain). This step is called particle approximation. By substituting density in the place of field function, the so-called summation density approach is evaluated which is usually (as here) substitutes the mass conservation in the partially compressible SPH.
Where W is a kernel function. In the present study, cubic spline function known as B-spline function is used which is defined in . Furthermore, the momentum conservation equation in the particle form and tensor notation may be written as follows by some algebraic work:
Owing to particles movements, this particle summation changes over time and must reconstruct in every time steps. Besides we know that the particle velocity is material derivative of its position. The evaluated system of ODE could be solved by any time integration algorithms. After that, particles are moved to their new places and the paradigm is repeated. Note that strain tensor in the SPH formulation is written as below:
By assuming that the fluid is partially compressed, the following state equation could be considered between the density and pressure :
Where B and are usually set , respectively . This state equation may lead to a sound speed equal to 10 ( is the initial height of fluid). One of the proposed improvements on the original SPH is XSPH . In this technique, the particles velocity are modified by their neighboring particles velocity according to Equation (12), so the particle velocities are closer to the average velocity of neighboring particles and the particles adhere to each other.
The SPH method does not need any additional work for free surfaces. Also, the walls are replaced by some fix particles which has constant positions and properties over the whole simulation. These boundary particles imposed a repulsive force on the inner particles which are closer than a throats hold distance. The imposed force is usually written in the Lennard-Jones form. Furthermore, in each time step some dummy particles are put out of the domain in the mirror position of each near boundary inner particle. Pressure and density of a dummy particle is dictated from its twin inner particle. In this study, we consider dummy particle velocity is inversed only in the normal direction to the wall and the tangential velocity is set to zero.
4. Results and Discussion
A fluid column of =0.2m height is held between a vertical wall and a dam plane with a distance of =0.1m. The dam plate is removed at the start time and the fluid spreads on the horizontal dry bed because of gravity force. First we use weakly compressible XSPH method to simulate dam break flow with water as a Newtonian fluid. Good agreement with presented experimental and numerical results in  is achieved as seen in Figure 1. Also, we observe that small value of ε (i.e. 0.01) could remove the numerical noise caused by small sound speed selection. Besides, the small value of it does not decay the flow dispersion, unrealistically.
Now we consider a Thixotropic gel as a non-Newtonian case which is well defined by Moore thixotropic model. The initial dynamic viscosity of this mixture is pa.s while its ultimate viscosity when all of its internal structures are destroyed is 0.001pa.s . Material constants (a and b) are both set to 0.1. Initial condition for the structural parameter is s(t=0) = 1because the structure is complete everywhere at the beginning. The release leads the fluid structure at the front free surface is destroyed, so the structural parameter of those particles are set to zero. Furthermore, the settlement of the fluid near the walls permits the fluid structure to be rebuilt. Thus, we set the structural parameter of the particles which are at rest equal to 1. By exploiting 40×80 inner particles, the spread of fluid column on a horizontal dry bed is simulated and presented in Figure 2. It is evident that the bulge shape in the leading edge is somewhat different from Newtonian case. Furthermore, the spread process of the gel is much slower than the water.
Furthermore, in Figure 3. non-dimensional leading edge of the thixotropic gel is compared with two Newtonian fluids with viscosity of 1pa.s and 0.001pa.s (water). As shown, in the initial time ranges the gel behaves similar to high viscous fluid but by pasting time, the gel behaves similar to water. The reason is that in the introductory time, the gel is fully structured but as the time going on, the gel structures are destroyed.
Also, the effects of structural parameter constants are investigated through some additional simulations. As expected by increasing ‘a’, the structure breakdown in the spreading process is decreased and the viscosity remains constant near its initial value; thus the fluid acts like a high viscous fluid (Figure 4.). As shown in Figure 5, by increasing ‘b’ the structure breakdown in the spreading process is increased and the viscosity is decreased to its ultimate value. Note that the results are more sensitive to variation of ‘b’ than ‘a’. This is logical, because a function is always more sensitive to its derivative rather than to its own value. Also, it is obvious that the fluid with a small rebuild coefficient cannot act as a high viscous fluid.
The dam break problem is simulated by weakly compressible XSPH. First, it is shown that the water results are in good agreement with previous published results. Then the dam break of Moore thixotropic gel is simulated. As expected, the gel is much slower than the water and the shapes of free surfaces are somewhat different. Also, it is evident that the gel behaves similar to high viscous fluid in the elementary time steps and its behavior changes to the water as the internal structure is destroyed. It is concluded that the thixotropic behavior of fluids must be considered because it could dramatically affect the free surface shape and position. Finally, the effects of model constants are investigated and it is shown that the selected model obeys the rule of thumb that the destruction is more rapid than the rebuild.
I would like to appreciate UniversitiTeknologi Malaysia and Isfahan University of Medical Sciences for their continual support during the course of this paper.
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