FlowDict solves the partial differential equation for the defined boundary values with the selected flow solver. The direct result of the computation is a 3D data field consisting of a vector-valued velocity and a scalar-valued pressure for each fluid or porous voxel that describes the resulting steady-state flow.
This field can be visualized to illustrate the flow. Additionally, a number of quantities describing the flow characteristics or the material properties are derived from the result:
For each layer in the selected computational direction, the average pressure and the average flow velocity in the computational direction are computed. One of these quantities is typically fixed by the selected boundary conditions:
If the pressure drop is set as boundary condition, the average flow velocity is reported as result.
If the mean flow velocity or the flow rate is set as boundary condition, the pressure drop (the pressure difference between top and bottom layer) is reported as result.
There are two possibilities to compute an average flow velocity. The superficial flow velocity reported here averages over all grid cells, including the solid cells which are included in the averaging using a velocity of 0.0. The interstitial flow velocity averages only over the grid cells that contain a fluid. This velocity is found in a plot in the result file.
The Reynolds number is a dimensionless value used to predict flow patterns. It represents the ratio of inertial forces to viscous forces within a fluid. The Reynolds number is an indicator if the flow is
slow and linear (Stokes), or
fast and non-linear (Navier Stokes), or
turbulent (no stationary solution exists).
The Reynolds number (Re) is defined as
(291) Reynolds Number
where is the density of the fluid (SI units: kg/m3), is the velocity of the fluid (m/s), is the dynamic viscosity of the fluid (Pa·s or N·s/m2 or kg/m·s), and is a characteristic length (m). Depending on the different types of porous media and different applications, there are different choices for characteristic length. Four options are provided:
Numerical Length: The square root of the permeability is taken as characteristic length, (Zheng and Grigg, 2006). This option is used for Stokes by default.
Maximum Pore Diameter: The characteristic length is the diameter of the largest through pore. From structure analysis the maximum pore diameter is found. This option is used for Navier-Stokes by default.
Maximum Fiber/Particle Diameter: The characteristic length is the largest fiber diameter for fibrous media and the largest particle size for grain structures. The largest fiber/particle diameter is found by an analysis of the 3D structure.
Manually specified: When none of the previous three options fits, you can set a characteristic length manually.
Permeability
If the flow is slow and laminar, a linear relationship between the average flow velocity and the pressure drop is observed. This is known as Darcy's law
(292) Darcy's law (1D)
where is the average (superficial) flow velocity, is the permeability, is the dynamic fluid viscosity, is the pressure drop, and is the thickness of the media.
The resulting permeability is independent of the applied pressure drop, as well as from the used fluid viscosity. Thus, the permeability is considered a material property. For isotropic materials, the permeability is a scalar value. For anisotropic materials, the permeability depends on the flow direction and can be described as a 3x3 tensor as described below.
Important! The permeability can only be determined from the solution of the Stokes equation. For fast flows, Darcy's law no longer holds true and the relation between pressure drop and flow velocity is no longer described by the material's permeability alone.
The structure’s permeability tensor is found from the three-dimensional version of Darcy's law:
(293) Darcy's law (3D)
Here is the averaged velocity vector (averaged flux with i=1 corresponding to a pressure drop in the X-direction, i=2 corresponding to a pressure drop in the Y-direction and i=3 corresponding to a pressure drop in the Z-direction). denotes the fluid viscosity, and
(294)
is the pressure gradient (or pressure difference) in the ith direction. , , and represent the pressure drop in the corresponding direction in Pa.
(295)
are the physical lengths of the computational domain in the directions of interest. is the physical length of a voxel, and NX, NY and NZ are the numbers of voxels in the three coordinate directions (X, Y, and Z).
The diagonal entries of the permeability tensor describe the measurable one-dimensional permeabilities in the x, y, and z direction.
The off-diagonal entries of the permeability tensor describe if a pressure drop in one direction will cause some flux in another, perpendicular direction. This might happen, e.g., for diagonal pores. If symmetric boundary conditions are used in perpendicular (or transverse) directions, no net flow is possible over the sides of the domain, and the off-diagonal entries will be zero.
Important! Note that the length definition in (295) includes the complete thickness of the 3D grid. This is an appropriate definition if the 3D model shows a representative inner part of the material, and does not include any inflow or outflow regions.
If the 3D model includes empty regions as inflow or outflow region, this definition is no longer correct and the length used for the permeability computation has to be adjusted. You can do that during the post-processing, and define the material thickness in flow direction, which is then used to compute the permeability.
For some boundary conditions (e.g., constant velocity at the inflow), the solver needs an empty inflow area to be able to compute a solution. These contradicting demands can be handled in two different ways:
Explicit inflow/outflow region: The inlet or outlet is part of the 3D structure model. You can, e.g., use ProcessGeo - Embed to add this areas explicitly to the structure. You then have to enter the material thickness (without inflow and outflow) manually in the post-processing to compute the correct permeability.
Implicit inflow/outflow region: The inlet or outlet is not part of the 3D structure model. The flow solver internally adds empty regions to the domain and solves the boundary value problem. The solver then removes these regions from the result files and does not use them for the post-processing. With this, the permeability computation uses the correct thickness by default. The disadvantage is that you cannot visualize the flow in the implicit regions as they are not part of the results.
The following two properties are derived from the flow permeability. Thus, they can also only be computed from the solution of the Stokes or Stokes-Brinkman equation, and not for fast flows.
In experiments, the mean velocity in the direction of the pressure drop is measured. The measured pressure drop is then divided by the measured mean velocity to obtain the measured flow resistivity.
The flow resistivity in one space direction is therefore given by
(296)
where is the permeability and is the dynamic fluid viscosity. In 3D, a flow resistivity tensor would be defined as . Thus, flow resistivity is NOT a material property, because it always depends on the fluid viscosity.
In FlowDict, the flow resistivity is not determined as a 3D tensor, but reported for each computational direction. In X-direction, is computed, and similar in Y- and Z-direction.
The Gurley second or Gurley unit describes the number of seconds required for 100 cubic centimeters (1 deciliter) of air to pass through 1.0 square inch of a given material at a pressure differential of 4.88 inches of water (0.176 psi) (ISO 5636-5:2003).
With these values gathered in the table:
Δp
= 0.176 psi
= 1213.48 Pa
A
= 1 in2
= 0.00064516 m2
V
= 100 cm3
= 10-4 m3
the Gurley value can be obtained from Darcy’s law with:
(297)
The Gurley value depends on the permeability and the physical length of the computational domain in the directions of interest.
Average (superficial) flow velocity and pressure drop
