FlowDict computes the fluid flow through the 3D structure model.
All experiments in FlowDict assume a steady flow regime, i.e., do not allow for time-dependent behavior such as turbulence. In practice, this means that the velocity and pressure drop cannot be arbitrarily high. FlowDict therefore searches for a stationary solution and the partial differential equations that need to be solved do not contain derivatives regarding time.
Other assumptions are that the fluid is a Newtonian fluid and it is incompressible.
If the geometry of the pores is resolved by the computational grid, all voxels are either pore space or solid material. The fluid flow in the pores is then described by the stationary Navier-Stokes equations:
(269) Momentum balance
(270) Mass conservation
Here, is the fluid density, is the dynamic viscosity, is the velocity vector, is the pressure and is a force density.
The Navier-Stokes momentum balance equation includes inertia. For such flows, the resulting relation between pressure drop and mean velocity is nonlinear.
When the flow is slow and laminar, the Navier-Stokes equations can be simplified by neglecting the term that describes inertial effects. The resulting equations are the Stokes equations, which read as follows:
(271)
(272)
Here, is the dynamic viscosity, is the velocity vector, is the pressure and is aforce density.
For such flows, the resulting relation between pressure drop and mean velocity is linear and Darcy's law can be applied. The sketch below illustrates this behavior: when the pressure drop is small, Stokes and Navier-Stokes equation give identical results, and the flow velocity depends linearly on the applied pressure drop. When the pressure drop gets larger, the results of Stokes and Navier-Stokes differ - in this case, only the Navier-Stokes equation describes the physically correct solution and the velocity depends non-linear on the applied pressure drop.
If the geometry of the pores is not fully resolved by the computational grid, a voxel may describe pore space, solid material, or porous material. An additional term, called Brinkman term, allows to describe a Darcy-type flow in an unresolved porous medium. The Navier-Stokes-Brinkman equations are given as follows:
(273)
(274)
Here, is the velocity vector, is the pressure, is aforce density, and is the dynamic viscosity. The local viscous flow resistivity of a voxel is related to the local permeability by
(275)
where is the isotropic permeability of the voxel. In the pore space, the flow resistivity is zero and the Brinkman term disappears.
Under the assumption of a slow and laminar flow, the Navier-Stokes-Brinkman equations can be simplified by neglecting the term that describes inertial effects. The resulting equations read as follows:
(276)
(277)
Here, is the velocity vector, is the pressure, is aforce density, and is the dynamic viscosity. The local viscous flow resistivity of a voxel is related to the local permeability by
(278)
where is the isotropic permeability of the voxel. In the pore space, the flow resistivity is zero and the Brinkman term disappears.
When we analyze the asymptotic behavior of the terms of the Stokes-Brinkman equation, we find that
(279) Viscous term
(280) Brinkman term
where is the voxel length.
Therefore, when the permeability is small, and the voxel length is relatively large, the Brinkman term dominates, and the viscous term can be neglected. This leads to
(281) Darcy flow
Numerically, finding a solution for the Darcy flow equation is much faster that finding a solution for the Stokes-Brinkman or Navier-Stokes-Brinkman equation. However, this simplified equation only holds true when the assumptions used in its derivation it are valid:
Creeping flow, i.e., very small Reynolds number.
Porous voxels with small permeabilities dominate, all through pores have diameters smaller than the voxel length.
The following sketch illustrates which equation must be solved. The decision is based on the points:
Creeping flow or fast flow (gray boxes): In the creeping flow case, the Reynolds number is small, inertia can be neglected, and a linear relationship between pressure and velocity exists. In the fast flow case, the Reynolds number is large, inertial effects cannot be neglected, and a non-linear relationship between pressure and velocity exists.
Resolution (blue boxes): If all pores are resolved, each voxel either represents fluid or solid material. If some pores are unresolved, a voxel may also represent porous material. In the extreme case, all through-pores are unresolved.
All five derived momentum balance equations are shown in the table below:
Navier-Stokes-Brinkman Equations