Lagrangian analysis of multiscale particulate flows with the particle finite element method
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Abstract
We present a Lagrangian numerical technique for the analysis of flows incorporating physical particles of different sizes. The numerical approach is based on the particle finite element method (PFEM) which blends concepts from particlebased techniques and the FEM. The basis of the Lagrangian formulation for particulate flows and the procedure for modelling the motion of small and large particles that are submerged in the fluid are described in detail. The numerical technique for analysis of this type of multiscale particulate flows using a stabilized mixed velocitypressure formulation and the PFEM is also presented. Examples of application of the PFEM to several particulate flows problems are given.
Keywords
Lagrangian analysis Multiscale particulate flows Particle finite element method1 Introduction
The study of fluid flows containing particles of different sizes (hereafter called particulate flows) is relevant to many areas of engineering and applied sciences. In this work we are concerned with particulate flows containing small to large particles. This type of flows is typical in slurry flows originated by natural hazards such as floods, tsunamis and landslides, as well as in many processes of the biomedical and pharmaceutical industries, in the manufacturing industry and in the oil and gas industry (i.e. cuttings transport in boreholes), among other applications [1, 2, 6, 7, 13, 14, 16, 21, 22, 23, 26, 47, 50, 51, 55, 61, 62].
Our interest in this work is the modelling and simulation of free surface particulate quasiincompressible flows containing particles of different sizes using a particular class of Lagrangian FEM termed the Particle Finite Element Method (PFEM, www.cimne.com/pfem) [4, 5, 8, 11, 17, 18, 19, 20, 25, 27, 28, 35, 36, 38, 40, 42, 43, 44, 45, 46, 52]. The PFEM treats the mesh nodes in the fluid and solid domains as particles which can freely move and even separate from the main fluid domain. A mesh connects the nodes discretizing the domain where the governing equations are solved using a stabilized FEM.
In Lagrangian analysis procedures (such as the PFEM) the motion of fluid particles is tracked during the transient solution. Hence, the convective terms vanish in the momentum equations and no numerical stabilization is needed. Another source of instability, however, remains in the numerical solution of Lagrangian flows, that due to the treatment of the incompressibility constraint which requires using a stabilized numerical method.
In this work we use a stabilized Lagrangian formulation that has excellent mass preservation features. The success of the formulation relies on the consistent derivation of a residualbased stabilized expression of the mass balance equation using the Finite Calculus (FIC) method [29, 30, 31, 32, 33, 37, 38, 39].
The layout of the paper is the following. In the next section we present the basic equations for conservation of linear momentum and mass for a quasiincompressible particulate fluid in a Lagrangian framework. The treatment of the different force terms for micro, macro and large particles are explained. Next we derive the stabilized FIC form of the mass balance equation. Then, the finite element discretization using simplicial element with equal order approximation for the velocity and the pressure is presented and the relevant matrices and vectors of the discretized problem are given. Details of the implicit transient solution of the Lagrangian FEM equations for a particulate flow using a NewtonRaphson type iterative scheme are presented. The basic steps of the PFEM for solving freesurface particulate flow problems are described.
The efficiency and accuracy of the PFEM for analysis of particulate flows are verified by solving a set of free surface and confined fluid flow problems incorporating particles of small and large sizes in two (2D) and three (3D) dimensions. The problems include the study of soil erosion, landslide situations, tsunami and flood flows, soil dredging problems and particle filling of fluid containers, among others. The excellent performance of the numerical method proposed for analysis of particulate flows is highlighted.
2 Modelling of micro, macro and large particles
Large particles, on the other hand, can have any arbitrary shape and they can be treated as rigid or deformable bodies. In the later case, they are discretized with the standard FEM. The forces between the fluid and the particles and viceversa are computed via fluidstructure interaction (FSI) procedures [31, 60].
The following sections describe the basic governing equations for a Lagrangian particulate fluid and the computation of the forces for microscopic, macroscopic and large particles.
3 Basic governing equations for a Lagrangian particulate fluid [1, 22, 23, 61]
3.1 Conservation of linear momentum
The fluid volume fraction \(n_f\) in Eq. (1) is a continuous function that is interpolated from the nodal values in the finite element fashion [41, 58, 59, 60].
Summation of terms with repeated indices is assumed in Eq. (1) and in the following, unless otherwise specified.
Remark 1
3.2 Constitutive equations
3.3 Mass conservation equation
Remark 2
For \(n_f =1\) (i.e. no particles are contained in the fluid) the standard momentum and mass conservation equations for the fluid are recovered.
3.4 Boundary conditions
At a free surface the Neumann boundary conditions typically apply.
4 Motion of microscopic and macroscopic particles
4.1 Selfweight forces
4.2 Contact forces
For microscopic particles the tangential forces \(\mathbf{F}_s^{ij}\) are neglected in Eq. (15).
Fluidtoparticle forces: \(\mathbf{F}_i^{fp} = \mathbf{F}_i^d + \mathbf{F}_i^b\), where \(\mathbf{F}_i^b\) and \(\mathbf{F}_i^d\) are, respectively, the buoyancy and drag forces on the \(i\)th particle. These forces are computed as:
4.3 Buoyancy forces
4.4 Drag forces
We note that the forces on the particles due to lift effects have been neglected in the present analysis. These forces can be accounted for as explained in [22].
5 Motion of large particles
As mentioned earlier, large particles may be considered as rigid or deformable bodies. In the first case the motion follows the rules of Eq. (11) for rigid Lagrangian particles. The contact forces at the particle surface due to the adjacent interacting particles are computed using a frictional contact interface layer between particles as in the standard PFEM (Sect. 10.2).
The fluid forces on the particles are computing by integrating the tangential (viscous) and normal (pressure) forces at the edges of the fluid elements surrounding the particles.
Large deformable particles, on the other hand, behave as deformable bodies immersed in the fluid which are discretized via the standard FEM. Their motion can be followed using a staggered FSI scheme, or else by treating the deformable bodies and the fluid as a single continuum with different material properties. Details of this unified treatment of the interaction between fluids and deformable solids can be found in [12, 18, 46].
6 Stabilized FIC form of the mass balance equation
The modelling of incompressible fluids with a mixed finite element method using an equal order interpolation for the velocities and the pressure requires introducing a stabilized formulation for the mass balance equation.
The derivation of Eq. (19) for an homogeneous quasiincompressible fluid is presented in [45].
7 Variational equations for the fluid
The variational form of the momentum and mass balance equations is obtained via the standard weighted residual approach [9, 60]. The resulting integral expressions after integration by parts and some algebra are:
7.1 Momentum equations
7.2 Mass balance equation
8 FEM discretization
We discretize the analysis domain containing \(N_p\) microscopic and macroscopic particles and a number of large particles into finite elements with \(n\) nodes in the standard manner leading to a mesh with a total number of \(N_e\) elements and \(N\) nodes. In our work we will choose simple 3noded linear triangles (\(n=3\)) for 2D problems and 4noded tetrahedra (\(n=4\)) for 3D problems with local linear shape functions \(N_i^e\) defined for each node \(i\) of element \(e\) [41, 58]. The velocity components, the weighting functions and the pressure are interpolated over the mesh in terms of their nodal values in the same manner using the global linear shape functions \(N_j\) spanning over the elements sharing node \(j\) (\(j=1,N\)) [41, 58].
Element form of the matrices and vectors in Eq. (27b) for 2D problems
\(\displaystyle \mathbf M _{0_{ij}}^e =\int _{V^e}\rho {N}_{i}^e {N}_{j}^e \mathbf{I}_2 dV ~,~\mathbf K ^e_{ij} =\int _{V^e} \mathbf B _i^{eT} \mathbf D \mathbf B _j^e dV ~,~\mathbf Q ^e_{ij} =\int _{V^e} \mathbf B _i^{eT} \mathbf m {N}_j^e dV\) 
\(\displaystyle {M}_{1_{ij}}^e =\int _{V^e} \frac{1}{\kappa }{N}_i^e {N}_j^e dV ~,~ {M}_{b_{ij}}^e = \int _{\varGamma _t^e} \frac{2\tau }{h_n} {N}_i^e{N}_j^e d\varGamma \) 
\(\displaystyle {L}^e_{ij}= \int _{V^e} \tau ({\pmb \nabla }^T {N}_i^e) {\pmb \nabla } N_j^e dV ~,~ \mathbf f ^e_{v_i}= \int _{V^e}\mathbf N _{i}^e \left( \mathbf b + \frac{1}{n_f}\mathbf{f}^{pf}\right) dV + \int _{\varGamma _t^e} \mathbf N _{i}^e \mathbf{t}^p d\varGamma \) 
\(\displaystyle {f}^e_{p_i}=\int _{\varGamma _t^e}\tau N_i^e \left[ \rho \frac{Dv_n}{Dt}\frac{2}{h_n} (2\mu {\partial v_n \over \partial n} t_n)\right] d\varGamma  \int _{V^e} \left( \tau {\pmb \nabla }^T {N}_i^e \mathbf{b}  {N}_i^e \frac{1}{n_f} \frac{Dn_f}{Dt}\right) dV\) 
\(\hbox {with } i,j=1,n.\) 
\(\displaystyle \mathbf{D}= 2\mu \begin{bmatrix} 2/3&1/3&0\\ 1/3&2/3&0\\ 0&0&1/2 \end{bmatrix}~~,~~ \displaystyle \mathbf{B}_i^e = \left[ \begin{array}{ll} \displaystyle {\partial N_i^e \over \partial x_1} &{}0\\ \displaystyle 0&{} \displaystyle {\partial N_i^e \over \partial x_2}\\ \displaystyle 0&{}0\\ \displaystyle {\partial N_i^e \over \partial x_2}&{}\displaystyle {\partial N_i^e \over \partial x_1}\\ \end{array}\right] \) 
\(\mathbf N _{i}^e = N_i^e \mathbf{I}_2 \quad \hbox {and} \quad {\pmb \nabla } = \left\{ \begin{array}{l} \displaystyle {\partial \over \partial x_1}\\ \displaystyle {\partial \over \partial x_2} \end{array}\right\} , \mathbf{m}= \left\{ \begin{array}{l} 1\\ 1\\ 0\end{array}\right\} \) 
\(N_i^e\): Local shape function of node \(i\) of element \(e\) [41, 58, 59, 60] 
Remark 3
The boundary terms of vector \(\mathbf{f}_p\) can be incorporated in the matrices of Eq. (27b). This, however, leads to a non symmetrical set of equations. For this reason we have chosen to compute these boundary terms iteratively within the incremental solution scheme.
9 Incremental solution of the discretized equations

Initialize variables: \(({}^{n+1}\mathbf x ^1,{}^{n+1}\bar{\mathbf{v}}^1,{}^{n+1}\bar{\mathbf{p}}^1,{}^{n+1}n_f^i, {}^{n+1}\bar{\mathbf{r}}^1_m)\equiv \left\{ {}^{n}\mathbf{x},{}^{n}\bar{\mathbf{v}},{}^{n}\bar{\mathbf{p}},{}^{n} n_f ,{}^{n}\bar{\mathbf{r}}_m \right\} \).

Iteration loop: \(k=1,\cdots , NITER\). For each iteration.
 Step 1. Compute the nodal velocity increments \(\Delta \bar{\mathbf{v}}\) From Eq. (27ba), we deducewith the momentum residual \(\bar{\mathbf{r}}_m\) and the iteration matrix \(\mathbf{H}_v\) given by$$\begin{aligned} {}^{n+1}\mathbf{H}_v^i \Delta \bar{\mathbf{v}} =  {}^{n+1}\bar{\mathbf{r }}_m^k \rightarrow \Delta \bar{\mathbf{v}} \end{aligned}$$(28a)with$$\begin{aligned} \bar{\mathbf{r }}_m\! = \mathbf M _0 {\dot{\bar{\mathbf{v}}}} \!+ \mathbf K \bar{\mathbf{v}}\!+\mathbf Q \bar{\mathbf{p}}\!\mathbf f _v, \quad \mathbf H _v = \frac{1}{\Delta t} \mathbf M _0 + \mathbf K \! + \mathbf K _v\nonumber \\ \end{aligned}$$(28b)$$\begin{aligned} \mathbf K _v^e = \int \limits _{{}^nV^e} \mathbf B ^T \mathbf m \theta \Delta t\kappa \mathbf m ^T \mathbf B dV \end{aligned}$$(28c)
 Step 2. Update the velocities$$\begin{aligned}&\hbox {Fluid nodes:}~~{}^{n+1}\bar{\mathbf{v}}^{k+1}= {}^{n+1}\bar{\mathbf{v}}^k + \Delta \bar{\mathbf{v}}\end{aligned}$$(29a)$$\begin{aligned}&\hbox {Rigid particles:}~~\left\{ \begin{array}{l} {}^{n+1/2}\dot{\mathbf{u}}_{j} = {}^{n1/2}\dot{\mathbf{u}}_{j} + {}^{n}\ddot{\mathbf{u}}_{j}^{k+1} \Delta t\\ \dot{\mathbf{u}}_{j} = \frac{1}{m_j} {}^{n}\mathbf{F}_{j}^{k+1} ~~,~~j=1,N_p\end{array}\right. \nonumber \\ \end{aligned}$$(29b)
 Step 3. Compute the nodal pressures \( {}^{n+1}\bar{\mathbf{p}}^{k+1}\) From Eq. (27b) we obtainwith$$\begin{aligned}&{}^{n+1} \mathbf H _p^i {}^{n+1}\bar{\mathbf{p}}^{k+1} = \frac{1}{\Delta t} \mathbf M _1 {}^{n+1}\bar{\mathbf{p}}^i\nonumber \\&\quad +\, \mathbf{Q}^T {}^{n+1}\bar{\mathbf{v}}^{k+1}+ {}^{n+1}\bar{\mathbf{f}}^{i}_p \rightarrow {}^{n+1}\bar{\mathbf{p}}^{k+1} \end{aligned}$$(30a)$$\begin{aligned} \mathbf H _p = \frac{1}{\Delta t} \mathbf M _1 + \mathbf L + \mathbf M _b \end{aligned}$$(30b)
 Step 4. Update the coordinates of the fluid nodes and particles$$\begin{aligned}&\hbox {Fluid nodes:}~~{}^{n+1}\mathbf{x}^{k+1}_i= {}^{n+1}\mathbf{x}^k_i + \frac{1}{2} ({}^{n+1}\bar{\mathbf{v}}^{k+1}_i + {}^{n}\bar{\mathbf{v}}_i) \,\Delta t \quad ,\nonumber \\&i=1,N\end{aligned}$$(31a)$$\begin{aligned}&\hbox {Rigid particles:}~~\left\{ \begin{array}{l} {}^{n+1}\mathbf{u}_{i}^{k+1} = {}^{n}\mathbf{u}_{i}^{k+1} + {}^{n+1/2}\dot{\mathbf{u}}_{i}^{k+1}\Delta t\\ {}^{n+1}\mathbf{x}^{k+1}_i = {}^{n}\mathbf{x}_i + {}^{n+1}\mathbf{u}^{k+1}_i \quad ,~~ i=1,N_p \end{array}\right. \nonumber \\ \end{aligned}$$(31b)

Step 5. Compute the fluid volume fractions for each node \({}^{n+1}{n}^{k+1}_{f_i}\) via Eq. (2)

Step 6. Compute forces and torques on particles: \({}^{n+1}\mathbf{F}^{k+1}_i,{}^{n+1}\mathbf{T}^{k+1}_i~ ,~i=1,N_p\)

Step 7. Compute particletofluid nodes: \(({}^{n+1}\mathbf{f}^{pf}_i)^{k+1} =  ({}^{n+1}\mathbf{f}^{fp}_i)^{k+1} ~ ,~ i=1,N\) with \(\mathbf{f}^{fp}_i\) computed by Eq. (18)

Step 8. Check convergence
If both conditions (32) are satisfied then make \(n \leftarrow n+1 \) and proceed to the next time step.
Otherwise, make the iteration counter \(k \leftarrow k+1 \) and repeat Steps 1–8.
Remark 5
In Eq. (28)–(32) \({}^{n+1}(\cdot )\) denotes the values of a matrix or a vector computed using the nodal unknowns at time \(n+1\). In our work the derivatives and integrals in the iteration matrices \(\mathbf{H}_v\) and \(\mathbf{H}_p\) and the residual vector \(\bar{\mathbf{r}}_m\) are computed on the discretized geometry at time \(n\) (i.e. \(V^e = {}^{n}V^e\)) while the nodal force vectors \(\mathbf{f}_v\) and \(\mathbf{f}_p\) are computed on the current configuration at time \(n+1\). This is equivalent to using an updated Lagrangian formulation [3, 12, 44, 59].
Remark 6
The tangent “bulk” stiffness matrix \(\mathbf K _v\) in the iteration matrix \(\mathbf H _v\) of Eq. (28b) accounts for the changes of the pressure due to the velocity. Including matrix \(\mathbf K _v\) in \(\mathbf H _v\) has proven to be essential for the fast convergence, mass preservation and overall accuracy of the iterative solution [11, 45, 48].
Remark 7
The parameter \(\theta \) in \(\mathbf{K}_v\) (\(0< \theta \le 1\)) has the role of preventing the illconditioning of the iteration matrix \(\mathbf{H}_v\) for very large values of the speed of sound in the fluid that lead to a dominant role of the terms of the tangent bulk stiffness matrix \(\mathbf{K}_v\). An adequate selection of \(\theta \) also improves the overall accuracy of the numerical solution and the preservation of mass for large time steps. Details of the derivation of Eq. (28c) can be found in [45].
Remark 8
The iteration matrix \(\mathbf H _v\) in Eq. (28a) is an approximation of the exact tangent matrix in the updated Lagrangian formulation for a quasi/fully incompressible fluid [44]. The simplified form of \(\mathbf H _v\) used in this work has yielded very good results with convergence achieved for the nodal velocities and pressure in 3–4 iterations in all the problems analyzed.
Remark 9
The time step within a time interval \([n,n+1]\) has been chosen as \(\Delta t =\min \left( \frac{^n l_{\min }^e}{\vert {}^n\mathbf{v}\vert _{\max }},\Delta t_b\right) \) where \(^n l_{\min }^e\) is the minimum characteristic distance of all elements in the mesh, with \(l^e\) computed as explained in Sect. 6, \(\vert ^n\mathbf{v}\vert _{\max }\) is the maximum value of the modulus of the velocity of all nodes in the mesh and \(\Delta t_b\) is the critical time step of all nodes approaching a solid boundary [45].
10 About the particle finite element method (PFEM)
10.1 The basis of the PFEM
Let us consider a domain \(V\) containing fluid and solid subdomains. Each subdomain is characterized by a set of points, hereafter termed virtual particles. The virtual particles contain all the information for defining the geometry and the material and mechanical properties of the underlying subdomain. In the PFEM both subdomains are modelled using an updated Lagrangian formulation [3, 44, 59].
 1.The starting point at each time step is the cloud of points \(C\) in the fluid and solid domains. For instance \(^{n} C\) denotes the cloud at time \(t={}^n t \) (Fig. 5).
 2.
Identify the boundaries defining the analysis domain \(^{n} V\), as well as the subdomains in the fluid and the solid. This is an essential step as some boundaries (such as the free surface in fluids) may be severely distorted during the solution, including separation and reentering of nodes. The Alpha Shape method [10] is used for the boundary definition. Clearly, the accuracy in the reconstruction of the boundaries depends on the number of points in the vicinity of each boundary and on the Alpha Shape parameter. In the problems solved in this work the Alpha Shape method has been implementation as described in [17, 35].
 3.
Discretize the the analysis domain \(^{n} V\) with a finite element mesh \({}^{n} M.\)We use an efficient mesh generation scheme based on an enhanced Delaunay tesselation [17, 35].
 4.
Solve the Lagrangian equations of motion for the overall continuum using the standard FEM. Compute the state variables in at the next (updated) configuration for \(^n t+\Delta t\): velocities, pressure and viscous stresses in the fluid and displacements, stresses and strains in the solid.
 5.
Move the mesh nodes to a new position \(^{n+1} C\) where \(n+1\) denotes the time \({}^n t +\Delta t\), in terms of the time increment size.
 6.
Go back to step 1 and repeat the solution for the next time step to obtain \({}^{n+2} C\).
The CPU time required for meshing grows linearly with the number of nodes. As a general rule, meshing consumes for 3D problems around 15 % of the total CPU time per time step [43].
Application of the PFEM in fluid and solid mechanics and in fluidstructure interaction problems can be found in [4, 5, 8, 11, 17, 18, 19, 20, 25, 27, 28, 35, 36, 38, 40, 42, 43, 44, 45, 46, 52], as well in www.cimne.com/pfem.
10.2 Treatment of contact conditions
Known velocities at boundaries in the PFEM are prescribed in strong form to the boundary nodes. These nodes might belong to fixed external boundaries or to moving boundaries linked to the interacting solids. Surface tractions are applied to the Neumann part of the boundary, as usual in the FEM.
Contact between fluid particles and fixed boundaries is accounted for by adjusting the time step so that fluid nodes do not penetrate into the solid boundaries [45].
This algorithm allows us to model complex frictional contact conditions between two or more interacting bodies moving in water in an a simple manner. The algorithm has been used to model frictional contact situations between rigid and elastic solids in structural mechanics applications, such as soil/rock excavation problems [4, 5]. The frictional contact algorithm described above has been extended by Oliver et al. [27, 28] for analysis of metal cutting and machining problems.
10.3 Treatment of surface erosion
Prediction of bed erosion and sediment transport in open channel flows are important tasks in many areas of river and environmental engineering. Bed erosion can lead to instabilities of the river basin slopes. It can also undermine the foundation of bridge piles thereby favouring structural failure. Modeling of bed erosion is also relevant for predicting the evolution of surface material dragged in earth dams in overspill situations. Bed erosion is one of the main causes of environmental damage in floods.
Oñate et al. [36] have proposed an extension of the PFEM to model bed erosion. The erosion model is based on the frictional work at the bed surface originated by the shear stresses in the fluid.
 1.
Compute at every point of the bed surface the tangential stress \(\tau \) induced by the fluid motion.
 2.
Compute the frictional work \(W_f\) originated by the tangential stress at the bed surface.
 3.
The onset of erosion at a bed point occurs when \({}^nW_f\) exceeds a critical threshold value \(W_c\).
 4.
If \({}^nW_f > W_c\) at a bed node, then the node is detached from the bed region and it is allowed to move with the fluid flow. As a consequence, the mass of the patch of bed elements surrounding the bed node vanishes in the bed domain and it is transferred to the adjacent fluid node. This mass is subsequently transported with the fluid as an immersed macroscopic particle.
 5.
Sediment deposition can be modeled by an inverse process to that described in the previous step. Hence, a suspended node adjacent to the bed surface with a velocity below a threshold value is attached to the bed surface.
11 A nodal algorithm for transporting microscopic and macroscopic particles within a finite element mesh
The fact that in the PFEM a new mesh is regenerated at each time step allows us to make microscopic and macroscopic particles to be coincident with fluid nodes. An advantage of this procedure is that it provides an accurate definition of the particles at each time step which is useful for FSI problems.
The algorithm to compute the position of the particles using the nodal algorithm is as follows.
 1.Compute the converged value of the position of the fluid nodes (\({}^{n+1}\mathbf{x}_i,~i = 1,\ldots ,N\)) and the particles (\({}^{n+1}\mathbf{x}_j,~j = 1,\ldots ,N_p\)) using the algorithm of Sect. 9. The \(N_p\) particles coinciding with \(N_p\) fluid nodes (\(N_p \le N\)) will typically move to a different position than that of the corresponding fluid nodes (Fig. 9).
 2.
Regenerate the mesh discretizing the fluid domain treating the \(N_p\) particles as fluid nodes and ignore the fluid nodes originally coinciding with the \(N_p\) particles at \({}^{n+1} t\).
 3.
Interpolate the velocity of the fluid nodes at the position of the \(N_p\) particles surrounding the fluid nodes.
Other examples of application of this algorithm are shown in the next section.
12 Examples
We present the study of a several problems involving the transport of macroscopic and large particles in fluid flows. The problems are schematic representations of particulate flows occurring in practical problems of civil and environmental engineering and industrial problems.
Most problems studied have been solved with the PFEM using the nodal algorithm for the transport of macroscopic particles described in the previous section. An exception are the problems in Sect. 12.6 dealing with the vertical transport of spherical particles in a cylinder where the standard immersed approach for the transport of macroparticles described in Sects. 1–4 was used and the fluid equations were solved with an Eulerian flow solver implemented in the Kratos opensource software platform of CIMNE [24].
12.1 Erosion of a slope adjacent to the shore due to sea waves
This example, although still quite simple and schematic, evidences the possibility of the PFEM for modeling FSSI problems involving soil erosion, transport and deposition of soil particles, free surface waves and rigid/deformable structures.
12.2 Landslide falling on houses
12.3 Dragging of rocks by a water stream
12.4 Suction of cohesive material submerged in water
Figure 19 shows a similar 3D problem. The suction in the tube erodes the surface of the soil bed in the right hand container. The mixture of water and eroded particles is transported to the adjacent containers.
12.5 Filling of a water container with particles
12.6 Transport of spherical particles in a tube filled with water
12.7 Dragging of large objects and small particles in a tsunami type flow
13 Concluding remarks
We have presented a Lagrangian numerical technique for analysis of flows incorporating physical particles of different sizes. The numerical approach is based on the PFEM and a stabilized Lagrangian mixed velocitypressure formulation. The examples presented in the paper evidence the possibilities of the PFEM for analysis of practical multiscale particulate flows in industrial and environmental problems.
Notes
Acknowledgments
This research was supported by the Advanced Grant project SAFECON of the European Research Council.
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