Fluid simulation and particle-fluid interaction

A new and experimental addition to sphere is the ability to simulate a mixture of particles and a Newtonian fluid. The fluid is simulated using an Eulerian continuum approach, using a custom CUDA solver for GPU computation. This approach allows for fast simulations due to the limited need for GPU-CPU communications, as well as a flexible code base.

The following sections will describe the theoretical background, as well as the solution procedure and the numerical implementation.

Derivation of the Navier Stokes equations with porosity

Following the outline presented by Limache and Idelsohn (2006), the continuity equation for an incompressible fluid material is given by:

\nabla \cdot \boldsymbol{v} = 0

and the momentum equation:

\rho \frac{\partial \boldsymbol{v}}{\partial t} + \rho (\boldsymbol{v} \cdot \nabla \boldsymbol{v}) = \nabla \cdot \boldsymbol{\sigma} - \boldsymbol{f}^i + \rho \boldsymbol{g}

Here, \boldsymbol{v} is the fluid velocity, \rho is the fluid density, \boldsymbol{\sigma} is the Cauchy stress tensor, \boldsymbol{f}^i is the particle-fluid interaction vector and \boldsymbol{g} is the gravitational acceleration. For incompressible Newtonian fluids, the Cauchy stress is given by:

\boldsymbol{\sigma} = -p \boldsymbol{I} + \boldsymbol{\tau}

p is the fluid pressure, \boldsymbol{I} is the identity tensor, and \boldsymbol{\tau} is the deviatoric stress tensor, given by:

\boldsymbol{\tau} = \mu_f \nabla \boldsymbol{v} + \mu_f (\nabla \boldsymbol{v})^T

By using the following vector identities:

\nabla \cdot (p \boldsymbol{I}) = \nabla p \nabla \cdot (\nabla \boldsymbol{v}) = \nabla^2 \boldsymbol{v} \nabla \cdot (\nabla \boldsymbol{v})^T = \nabla (\nabla \cdot \boldsymbol{v})

the deviatoric component of the Cauchy stress tensor simplifies to the following, assuming that spatial variations in the viscosity can be neglected:

= -\nabla p + \mu_f \nabla^2 \boldsymbol{v}

Since we are dealing with fluid flow in a porous medium, additional terms are introduced to the equations for conservation of mass and momentum. In the following, the equations are derived for the first spatial component. The solution for the other components is trivial.

The porosity value (in the saturated porous medium the volumetric fraction of the fluid phase) denoted \phi is incorporated in the continuity and momentum equations. The continuity equation becomes:

\frac{\partial \phi}{\partial t} + \nabla \cdot (\phi \boldsymbol{v}) = 0

For the x component, the Lagrangian formulation of the momentum equation with a body force \boldsymbol{f} becomes:

\frac{D (\phi v_x)}{D t} = \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\sigma}) \right]_x - \frac{1}{\rho} f^i_x + \phi g

In the Eulerian formulation, an advection term is added, and the Cauchy stress tensor is represented as isotropic and deviatoric components individually:

\frac{\partial (\phi v_x)}{\partial t} + \boldsymbol{v} \cdot \nabla (\phi v_x) = \frac{1}{\rho} \left[ \nabla \cdot (-\phi p \boldsymbol{I}) + \phi \boldsymbol{\tau}) \right]_x - \frac{1}{\rho} f^i_x + \phi g_x

Using vector identities to rewrite the advection term, and expanding the fluid stress tensor term:

\frac{\partial (\phi v_x)}{\partial t} + \nabla \cdot (\phi v_x \boldsymbol{v}) - \phi v_x (\nabla \cdot \boldsymbol{v}) = \frac{1}{\rho} \left[ -\nabla \phi p \right]_x + \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x - \frac{1}{\rho} f^i_x + \phi g_x

Spatial variations in the porosity are neglected,

\nabla \phi := 0

and the pressure is attributed to the fluid phase alone (model B in Zhu et al. 2007 and Zhou et al. 2010). The divergence of fluid velocities is defined to be zero:

\nabla \cdot \boldsymbol{v} := 0

With these assumptions, the momentum equation simplifies to:

\frac{\partial (\phi v_x)}{\partial t} + \nabla \cdot (\phi v_x \boldsymbol{v}) = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x - \frac{1}{\rho} f^i_x + \phi g_x

The remaining part of the advection term is for the x component found as:

\nabla \cdot (\phi v_x \boldsymbol{v}) = \left[ \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right] \left[ \begin{array}{c} \phi v_x v_x\\ \phi v_x v_y\\ \phi v_x v_z\\ \end{array} \right] = \frac{\partial (\phi v_x v_x)}{\partial x} + \frac{\partial (\phi v_x v_y)}{\partial y} + \frac{\partial (\phi v_x v_z)}{\partial z}

The deviatoric stress tensor is in this case symmetrical, i.e. \tau_{ij} = \tau_{ji}, and is found by:

\frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x = \frac{1}{\rho} \left[ \left[ \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right] \phi \left[ \begin{matrix} \tau_{xx} & \tau_{xy} & \tau_{xz}\\ \tau_{yx} & \tau_{yy} & \tau_{yz}\\ \tau_{zx} & \tau_{zy} & \tau_{zz}\\ \end{matrix} \right] \right]_x = \frac{1}{\rho} \left[ \begin{array}{c} \frac{\partial (\phi \tau_{xx})}{\partial x} + \frac{\partial (\phi \tau_{xy})}{\partial y} + \frac{\partial (\phi \tau_{xz})}{\partial z}\\ \frac{\partial (\phi \tau_{yx})}{\partial x} + \frac{\partial (\phi \tau_{yy})}{\partial y} + \frac{\partial (\phi \tau_{yz})}{\partial z}\\ \frac{\partial (\phi \tau_{zx})}{\partial x} + \frac{\partial (\phi \tau_{zy})}{\partial y} + \frac{\partial (\phi \tau_{zz})}{\partial z}\\ \end{array} \right]_x = \frac{1}{\rho} \left( \frac{\partial (\phi \tau_{xx})}{\partial x} + \frac{\partial (\phi \tau_{xy})}{\partial y} + \frac{\partial (\phi \tau_{xz})}{\partial z} \right)

In a linear viscous fluid, the stress and strain rate (\dot{\boldsymbol{\epsilon}}) is linearly dependent, scaled by the viscosity parameter \mu_f:

\tau_{ij} = 2 \mu_f \dot{\epsilon}_{ij} = \mu_f \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right)

With this relationship, the deviatoric stress tensor components can be calculated as:

\tau_{xx} = 2 \mu_f \frac{\partial v_x}{\partial x} \qquad \tau_{yy} = 2 \mu_f \frac{\partial v_y}{\partial y} \qquad \tau_{zz} = 2 \mu_f \frac{\partial v_z}{\partial z} \tau_{xy} = \mu_f \left( \frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x} \right) \tau_{xz} = \mu_f \left( \frac{\partial v_x}{\partial z} + \frac{\partial v_z}{\partial x} \right) \tau_{yz} = \mu_f \left( \frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y} \right)

where \mu_f is the dynamic viscosity. The above formulation of the fluid rheology assumes identical bulk and shear viscosities. The derivation of the equations for the other spatial components is trivial.

Porosity estimation

The solid volume in each fluid cell is determined by the ratio of the a cell-centered spherical cell volume (V_c) and the sum of intersecting particle volumes (V_s). The spherical cell volume has a center at \boldsymbol{x}_i, and a radius of R_i, which is equal to half the fluid cell width. The nearby particles are characterized by position \boldsymbol{x}_j and radius r_j. The center distance is defined as:

d_{ij} = ||\boldsymbol{x}_i - \boldsymbol{x}_j||

The common volume of the two intersecting spheres is zero if the volumes aren’t intersecting, lens shaped if they are intersecting, and spherical if the particle is fully contained by the spherical cell volume:

V^s_{i} = \sum_j \begin{cases} 0 & \textit{if } R_i + r_j \leq d_{ij} \\ \frac{1}{12d_{ij}} \left[ \pi (R_i + r_j - d_{ij})^2 (d_{ij}^2 + 2d_{ij}r_j - 3r_j^2 + 2d_{ij} R_i + 6r_j R_i - 3R_i^2) \right] & \textit{if } R_i - r_j < d_{ij} < R_i + r_j \\ \frac{4}{3} \pi r^3_j & \textit{if } d_{ij} \leq R_i - r_j \end{cases}

Using this method, the cell porosity values are continuous through time as particles enter and exit the cell volume. The rate of porosity change (d\phi/dt) is estimated by the backwards Euler method by considering the previous and current porosity.

Particle-fluid interaction

The momentum exchange of the granular and fluid phases follows the procedure outlined by Gidaspow 1992 and Shamy and Zhegal 2005. The fluid and particle interaction is based on the concept of drag, where the magnitude is based on semi-empirical relationships. The drag force scales linearly with the relative difference in velocity between the fluid and particle phase. On the base of Newton’s third law, the resulting drag force is applied with opposite signs to the particle and fluid.

For fluid cells with porosities (\phi) less or equal to 0.8, the drag force is based on the Ergun (1952) equation:

\bar{\boldsymbol{f}}_d = \left( 150 \frac{\mu_f (1-\phi)^2}{\phi\bar{d}^2} + 1.75 \frac{(1-\phi)\rho_f ||\boldsymbol{v}_f - \bar{\boldsymbol{v}}_p||}{\bar{d}} \right) (\boldsymbol{v}_f - \bar{\boldsymbol{v}}_p)

here, \bar{d} denotes the average particle diameter in the cell, \boldsymbol{v}_f is the fluid flow velocity, and \bar{\boldsymbol{v}}_p is the average particle velocity in the cell. All particles in contact with the previously mentioned cell-centered sphere for porosity estimation contribute to the average particle velocity and diameter in the fluid cell.

If the porosity is greater than 0.8, the cell-averaged drag force (\bar{\boldsymbol{f}}_d is found from the Wen and Yu (1966) equation, which considers the fluid flow situation:

\bar{\boldsymbol{f}}_d = \left( \frac{3}{4} \frac{C_d (1-\phi) \phi^{-2.65} \mu_f \rho_f ||\boldsymbol{v}_f - \bar{\boldsymbol{v}}_p||}{\bar{d}} \right) (\boldsymbol{v}_f - \bar{\boldsymbol{v}}_p)

The drag coefficient C_d is evaluated depending on the magnitude of the Reynolds number Re:

C_d = \begin{cases} \frac{24}{Re} (1+0.15 (Re)^{0.687} & \textit{if } Re < 1,000 \\ 0.44 & \textit{if } Re \geq 1,000 \end{cases}

where the Reynold’s number is found by:

Re = \frac{\phi\rho_f\bar{d}}{\mu_f} ||\boldsymbol{v}_f - \bar{\boldsymbol{v}}_p||

The interaction force is applied to the fluid with negative sign as a contribution to the body force \boldsymbol{f}. The fluid interaction force applied particles in the fluid cell is:

\boldsymbol{f}_i = \frac{\bar{\boldsymbol{f}}_d V_p}{1-\phi}

where V_p denotes the particle volume. Optionally, the above interaction force could be expanded to include the force induced by the fluid pressure gradient:

\boldsymbol{f}_i = \left( -\nabla p + \frac{\bar{\boldsymbol{f}}_d}{1-\phi} \right) V_p

Fluid dynamics solution procedure by operator splitting

The partial differential terms in the previously described equations are found using finite central differences. Modifying the operator splitting methodology presented by Langtangen et al. (2002), the predicted velocity \boldsymbol{v}^* after a finite time step \Delta t is found by explicit integration of the momentum equation.

\frac{\Delta (\phi v_x)}{\Delta t} + \nabla \cdot (\phi v_x \boldsymbol{v}) = - \frac{1}{\rho} \frac{\Delta p}{\Delta x} + \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x - \frac{1}{\rho} f^i_x + \phi g_x \Downarrow \phi \frac{\Delta v_x}{\Delta t} + v_x \frac{\Delta \phi}{\Delta t} + \nabla \cdot (\phi v_x \boldsymbol{v}) = - \frac{1}{\rho} \frac{\Delta p}{\Delta x} + \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x - \frac{1}{\rho} f^i_x + \phi g_x

We want to isolate \Delta v_x in the above equation in order to project the new velocity.

\phi \frac{\Delta v_x}{\Delta t} = - \frac{1}{\rho} \frac{\Delta p}{\Delta x} + \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x - \frac{1}{\rho} f^i_x + \phi g_x - v_x \frac{\Delta \phi}{\Delta t} - \nabla \cdot (\phi v_x \boldsymbol{v}) \Delta v_x = - \frac{1}{\rho} \frac{\Delta p}{\Delta x} \frac{\Delta t}{\phi} + \frac{1}{\rho} \left[ \nabla \cdot (\phi \boldsymbol{\tau}) \right]_x \frac{\Delta t}{\phi} - \frac{\Delta t}{\rho\phi} f^i_x + \Delta t g_x - v_x \frac{\Delta \phi}{\phi} - \nabla \cdot (\phi v_x \boldsymbol{v}) \frac{\Delta t}{\phi}

The term \beta is introduced as an adjustable, dimensionless parameter in the range [0;1], and determines the importance of the old pressure values in the solution procedure (Langtangen et al. 2002). A value of 0 corresponds to Chorin’s projection method originally described in Chorin (1968).

v_x^* = v_x^t + \Delta v_x v_x^* = v_x^t - \frac{\beta}{\rho} \frac{\Delta p^t}{\Delta x} \frac{\Delta t}{\phi^t} + \frac{1}{\rho} \left[ \nabla \cdot (\phi^t \boldsymbol{\tau}^t) \right]_x \frac{\Delta t}{\phi} - \frac{\Delta t}{\rho\phi} f^i_x + \Delta t g_x - v^t_x \frac{\Delta \phi}{\phi^t} - \nabla \cdot (\phi^t v_x^t \boldsymbol{v}^t) \frac{\Delta t}{\phi^t}

Here, \Delta x denotes the cell spacing. The velocity found (v_x^*) is only a prediction of the fluid velocity at time t+\Delta t, since the estimate isn’t constrained by the continuity equation:

\frac{\Delta \phi^t}{\Delta t} + \nabla \cdot (\phi^t \boldsymbol{v}^{t+\Delta t}) = 0

The divergence of a scalar and vector can be split:

\phi^t \nabla \cdot \boldsymbol{v}^{t+\Delta t} + \boldsymbol{v}^{t+\Delta t} \cdot \nabla \phi^t + \frac{\Delta \phi^t}{\Delta t} = 0

The predicted velocity is corrected using the new pressure (Langtangen et al. 2002):

\boldsymbol{v}^{t+\Delta t} = \boldsymbol{v}^* %- \frac{\Delta t}{\rho} \nabla \epsilon - \frac{\Delta t}{\rho \phi^t} \nabla \epsilon \quad \text{where} \quad \epsilon = p^{t+\Delta t} - \beta p^t

The above formulation of the future velocity is put into the continuity equation:

\Rightarrow \phi^t \nabla \cdot \left( \boldsymbol{v}^* - \frac{\Delta t}{\rho \phi^t} \nabla \epsilon \right) + \left( \boldsymbol{v}^* - \frac{\Delta t}{\rho \phi^t} \nabla \epsilon \right) \cdot \nabla \phi^t + \frac{\Delta \phi^t}{\Delta t} = 0

\Rightarrow \phi^t \nabla \cdot \boldsymbol{v}^* - \frac{\Delta t}{\rho \phi^t} \phi^t \nabla^2 \epsilon + \nabla \phi^t \cdot \boldsymbol{v}^* - \nabla \phi^t \cdot \nabla \epsilon \frac{\Delta t}{\rho \phi^t} + \frac{\Delta \phi^t}{\Delta t} = 0

\Rightarrow \frac{\Delta t}{\rho} \nabla^2 \epsilon = \phi^t \nabla \cdot \boldsymbol{v}^* + \nabla \phi^t \cdot \boldsymbol{v}^* - \nabla \phi^t \cdot \nabla \epsilon \frac{\Delta t}{\rho \phi^t} + \frac{\Delta \phi^t}{\Delta t}

The pressure difference in time becomes a Poisson equation with added terms:

\Rightarrow \nabla^2 \epsilon = \frac{\nabla \cdot \boldsymbol{v}^* \phi^t \rho}{\Delta t} + \frac{\nabla \phi^t \cdot \boldsymbol{v}^* \rho}{\Delta t} - \frac{\nabla \phi^t \cdot \nabla \epsilon}{\phi^t} + \frac{\Delta \phi^t \rho}{\Delta t^2}

The right hand side of the above equation is termed the forcing function f, which is decomposed into two terms, f_1 and f_2:

f_1 = \frac{\nabla \cdot \boldsymbol{v}^* \phi^t \rho}{\Delta t} + \frac{\nabla \phi^t \cdot \boldsymbol{v}^* \rho}{\Delta t} + \frac{\Delta \phi^t \rho}{\Delta t^2} f_2 = \frac{\nabla \phi^t \cdot \nabla \epsilon}{\phi^t}

During the Jacobi iterative solution procedure f_1 remains constant, while f_2 changes value. For this reason, f_1 is found only during the first iteration, while f_2 is updated every time. The value of the forcing function is found as:

f = f_1 - f_2

Using second-order finite difference approximations of the Laplace operator second-order partial derivatives, the differential equations become a system of equations that is solved using iteratively using Jacobi updates. The total number of unknowns is (n_x - 1)(n_y - 1)(n_z - 1).

The discrete Laplacian (approximation of the Laplace operator) can be obtained by a finite-difference seven-point stencil in a three-dimensional, cubic grid with cell spacing \Delta x, \Delta y, \Delta z, considering the six face neighbors:

\nabla^2 \epsilon_{i_x,i_y,i_z} \approx \frac{\epsilon_{i_x-1,i_y,i_z} - 2 \epsilon_{i_x,i_y,i_z} + \epsilon_{i_x+1,i_y,i_z}}{\Delta x^2} + \frac{\epsilon_{i_x,i_y-1,i_z} - 2 \epsilon_{i_x,i_y,i_z} + \epsilon_{i_x,i_y+1,i_z}}{\Delta y^2} + \frac{\epsilon_{i_x,i_y,i_z-1} - 2 \epsilon_{i_x,i_y,i_z} + \epsilon_{i_x,i_y,i_z+1}}{\Delta z^2} \approx f_{i_x,i_y,i_z}

Within a Jacobi iteration, the value of the unknowns (\epsilon^n) is used to find an updated solution estimate (\epsilon^{n+1}). The solution for the updated value takes the form:

\epsilon^{n+1}_{i_x,i_y,i_z} = \frac{-\Delta x^2 \Delta y^2 \Delta z^2 f_{i_x,i_y,i_z} + \Delta y^2 \Delta z^2 (\epsilon^n_{i_x-1,i_y,i_z} + \epsilon^n_{i_x+1,i_y,i_z}) + \Delta x^2 \Delta z^2 (\epsilon^n_{i_x,i_y-1,i_z} + \epsilon^n_{i_x,i_y+1,i_z}) + \Delta x^2 \Delta y^2 (\epsilon^n_{i_x,i_y,i_z-1} + \epsilon^n_{i_x,i_y,i_z+1})} {2 (\Delta x^2 \Delta y^2 + \Delta x^2 \Delta z^2 + \Delta y^2 \Delta z^2) }

The difference between the current and updated value is termed the normalized residual:

r_{i_x,i_y,i_z} = \frac{(\epsilon^{n+1}_{i_x,i_y,i_z} - \epsilon^n_{i_x,i_y,i_z})^2}{(\epsilon^{n+1}_{i_x,i_y,i_z})^2}

Note that the \epsilon values cannot be 0 due to the above normalization of the residual.

The updated values are at the end of the iteration stored as the current values, and the maximal value of the normalized residual is found. If this value is larger than a tolerance criteria, the procedure is repeated. The iterative procedure is ended if the number of iterations exceeds a defined limit.

After the values of \epsilon are found, they are used to find the new pressures and velocities:

\bar{p}^{t+\Delta t} = \beta \bar{p}^t + \epsilon

\bar{\boldsymbol{v}}^{t+\Delta t} = \bar{\boldsymbol{v}}^* - \frac{\Delta t}{\rho\phi} \nabla \epsilon

Boundary conditions

The lateral boundaries are periodic. This cannot be changed in the current version of sphere. This means that the fluid properties at the paired, parallel lateral (x and y) boundaries are identical. A flow leaving through one side reappears on the opposite side.

The top and bottom boundary conditions of the fluid grid can be either: prescribed pressure (Dirichlet), or prescribed velocity (Neumann). The (horizontal) velocities parallel to the boundaries are free to attain other values (free slip). The Dirichlet boundary condition is enforced by keeping the value of \epsilon constant at the boundaries, e.g.:

\epsilon^{n+1}_{i_x,i_y,i_z = 1 \vee n_z} = \epsilon^{n}_{i_x,i_y,i_z = 1 \vee n_z}

The Neumann boundary condition of no flow across the boundary is enforced by setting the gradient of \epsilon perpendicular to the boundary to zero, e.g.:

\nabla_z \epsilon^{n+1}_{i_x,i_y,i_z = 1 \vee n_z} = 0

Numerical implementation

Ghost nodes

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