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Turbulent Parallel Flow in a Channel with Adjacent Fluid and Porous Regions
Introduction
This verification example reproduces the characteristics of turbulent flow in a channel modified by the presence of an adjacent porous region. Asymmetric profiles of velocity, turbulence kinetic energy, and shear stress, as well as higher turbulence levels and higher friction coefficients both at the solid wall and the interface between the clear-flow and porous regions, are observed. High-permeability cases reveal excellent agreement between COMSOL Multiphysics simulations, DNS, and experimental results without the need for specifically calibrated empirical functions. The turbulent flow field in both the clear-flow and porous domains is computed using the Turbulent Flow, Low Re k-ε interface.
Model Definition
Figure 1: Geometry with clear (2) and porous (1) domains, boundary conditions, mesh with boundary layer regions (at the solid wall and at the porous interface) zoomed in.
Turbulent flows in porous media can occur in many cases of academic and practical interest: filters, catalytic converters in exhaust systems, chemical reactors, oil transport and recovery in wells, transport of contaminants in groundwater, porous river beds, plant canopies and dense urban areas, thermal power plants, heat exchangers with porous-metal inserts, and even propagation of forest fires. Here, a simple model of parallel flows in adjacent free and porous domains is considered.
Figure 1 presents the model’s geometry, domain properties, boundary conditions, and mesh. A Periodic Flow Condition is employed to avoid simulating an upstream region with developing flow in the streamwise direction. A mapped mesh is used, and the region near porous interface (the boundary between the clear flow and porous domains) is meshed using a finer boundary layers mesh than that near the no-slip walls since the friction velocity at the porous interface, uτp, is expected to be significantly higher than the friction velocity at the solid walls, uτs.
The channel Reynolds number based on the average (bulk) velocity and the height of the clear channel, Re = UbH/ν, is higher than 5000 for the cases considered, which ensures that the flow is turbulent. The low Reynolds number k-ε turbulence model is chosen because it correctly predicts the friction coefficient in turbulent channel flow. The extension of the model to porous media is described in the theory section for the Turbulent Flow interfaces in the CFD Module User’s Guide.
In the Direct Numerical Simulations (DNS), Ref. 1, and experiments, Ref. 2, the average (bulk) velocity through the clear part of the channel was imposed. To obtain the required driving force, the Pressure difference option of the Periodic Flow Condition is combined with a Global Equations feature, as described in the Modeling Instructions section.
Here, only the cases with high permeability from both DNS and experiments are simulated, because at lower permeabilities the effect of increasingly “solid wall” behavior requires special treatment of the porous interface to correctly model pressure and momentum losses as well as turbulence generation at the interface (especially for nearly impermeable cases).
Note that before solving the problem of interest, with the porous domain activated, the cases are solved with the clear channel region only (applying a no-slip condition at the porous interface). A Parametric Sweep is used to facilitate the process of parameter variation.
Extrapolation of Experimental Results
Figure 2: Extrapolation of turbulence kinetic energy using only streamwise and wall-normal rms-velocities. The DNS results of  Ref. 1 are used. U.E. is the “Upper Estimate,” L.E. is the Lower Estimate and B.L. is the “Bottom Limit,” as defined in the lower figure and in the main text.
Flows in such “adjacent” configurations are referred to as flows with a “permeable wall” in Ref. 1 and Ref. 2. Indeed, a porous interface resembles a partial slip permeable wall since homogenization of the flow inside the porous region occurs within a few porous length scales ~κ1/2, where κ is the permeability. This is typically a quite small distance. Both Ref. 1 and Ref. 2 fitted the clear flow velocity near the porous interface by a log-law applicable to rough walls. However, as indicated in Ref. 1, the effect of permeability is intrinsically different from the effect of roughness and is of primary importance. To characterize it, the permeability Reynolds number was introduced:
(1)
where uτp is the friction velocity on the clear (fluid) side of the porous interface.
The experimental results in Ref. 2 contain only measurements of streamwise and wall-normal rms-velocities, u' and v', respectively. The DNS data of Ref. 1 is analyzed to find a reasonable way to extrapolate the spanwise rms-velocity, w', via the other two components, and to reconstruct the turbulence kinetic energy, k = (u'2 + v'2 + w'2)/2, consequently.
Figure 2 shows k for three different cases: E95 (high permeability, Reκ = 9.35), E80 (low permeability, Reκ = 1.06), and clear channel flow (zero permeability) as presented in Ref. 1. Near its lowest point k reaches the “bottom limit” (B.L.) based on w'2 = v'2 (it can be expected that unrestricted spanwise fluctuations are not smaller than restricted wall-normal fluctuations). Mostly, k is situated between a proposed “upper estimate” (U.E.) based on w'2 = 0.25 u'2 + v'2 and a proposed “lower estimate” (L.E.) based on w'2 = 0.1 u'2 + v'2. The lower estimate represents k very well not only for clear-channel flow with no-slip walls but also in more than half of the clear part of the channel (on the solid wall side) for both cases with a “permeable wall”. Near the porous interface, k approaches the upper estimate, the closer the higher the permeability Reynolds number Reκ. Note that these rules of thumb were formulated based on a limited number of cases and for the specific value Re = 5500. Nevertheless, these tentatively proposed upper and lower estimates are employed to extrapolate experimental values of k.
Results and Discussion
Table 1 contains parameters from DNS case E95 in Ref. 1, and from two experimental cases from series #06 in Ref. 2 (a bit corrected in Ref. 3). The Reynolds number is based on the average (bulk) velocity Ub and height of the clear channel H, Re = UbH/ν, while the Darcy number is defined as Da = κ/H2, where κ is the permeability of the porous medium. εp is the porosity, and cF is the Forchheimer parameter.
The numerical results of the COMSOL Multiphysics simulations are summarized in Table 2 for clear-channel flow and in Table 3 for the setup with adjacent clear and porous regions. The configurations are referred to as “clear” and “adjacent” (including the model plots). The following quantities are presented:
•
Friction coefficients at the solid wall and at the fluid side of the porous interface:
•
Friction velocities at the solid wall and at the fluid side of the porous interface:
•
Friction Reynolds number based on uτp and H (clear channel height):
where τws and τwp are the shear stress at the solid wall and at the porous interface, respectively:
Table 1: Parameters of the dns and experimental cases.
 
 Re
 H
 εp
 cF
 Da
E95
5500
0.03 m
0.95
0.14
1.9·10-4
#06_5400
5400
0.029 m
0.8
0.095
1.04·10-4
#06_9500
9500
0.029 m
0.8
0.095
1.04·10-4
Table 2: Clear channel results.
 
 Cfs×103
 Cfp×103
 Reτ
 uτp/uτs
E95
8.28
8.25
353
0.998
#06_5400
8.32
8.28
348
0.998
#06_9500
7.26
7.23
571
0.998
Table 3: adjacent channel results: simulation versus DNS/experiment (bold).
 
 Cfs×103
 Cfp×103
 Reτ
 uτp/uτs
E95
11.03 (10.9)
31.2 (30.4)
687 (678)
1.68 (1.67)
#06_5400
10.59 (10.65)
26.5 (25.6)
622 (611)
1.58 (1.55)
#06_9500
9.21 (9.79)
22.3 (26)
1003 (1084)
1.56 (1.63)
For the case of clear channel flow at Re = 5500, the low Reynolds number k-ε model gives friction coefficients equal to 8.28·10−3 and 8.25·10−3 at the solid wall and at the porous interface, respectively (the difference is due to a finer resolution at the latter), while the DNS prediction is 8.18·10−3, so the accuracy of the model is near 1%. uτp/uτs deviates from 1 by 0.2%, which indicates that for clear-channel flow further refinement is not needed.
Figure 3 demonstrates the E95 case for which the computed profiles of velocity, turbulence kinetic energy, and shear stress are highly asymmetric. The velocity has a maximum magnitude close to the solid wall, a quarter of the clear-channel height from it. The minimum of the turbulence kinetic energy and the zero of the shear stress are situated approximately at the same location, while the maxima of k and the shear stress near the porous interface are much larger than the maxima near the solid wall. Note that even at large Reκ = 9.35, the flow gradients do not penetrate deep into the porous region; the velocity and k quickly attain small limiting values in the porous matrix while the shear stress effectively becomes zero.
Figure 3: Profiles of velocity, turbulence kinetic energy, and shear stress for the E95 case. The upper domain is a clear flow domain, whereas the lower domain is a porous domain. Note the quick damping of all the variables inside the latter.
For E95, Figure 4 demonstrates very good correspondence of u/Ub, k/(uτs)2, and −u'v'/(uτs)2 obtained in COMSOL Multiphysics simulations with the DNS results from Ref. 1. In the clear-channel flow, the maximum of k/(uτs)2 is underestimated by approximately 15%, and in the adjacent configuration the same underestimation near the solid wall is present, which seems to be characteristic for the low Reynolds number k-ε turbulence model. At the same time, the maximum of k/(uτs)2 near the porous interface is surprisingly correct, although both the maximum and minimum positions are somewhat shifted. The behavior of the quantities close to the porous interface is not captured flawlessly but even the absence of larger deviations is remarkable since we do not treat the porous interface specifically.
Figure 4: Velocity, turbulence kinetic energy and shear stress for E95.
Figure 5: Velocity, turbulence kinetic energy and shear stress for #06 Re = 5400.U.E. is the “Upper Estimate,” L.E. is the Lower Estimate and B.L. is the “Bottom Limit,” as defined in the main text.
Figure 6: Velocity, turbulence kinetic energy and shear stress for #06 Re  = 9500. U.E. is the “Upper Estimate,” L.E. is the Lower Estimate and B.L. is the “Bottom Limit,” as defined in the main text.
A comparison of simulations and experimental results from Ref. 2 for the case #06, Re = 5400 and Reκ = 6.23, is shown in Figure 5. The velocity profile and normalized shear stress are very well captured. The maximum of k/(uτs)2 is slightly shifted toward the “permeable wall” and seems to be underestimated by approximately 10%, while both the minimum value and its position are captured correctly.
Figure 6 presents a simulations versus experiment comparison for the case #06, Re = 9500 and Reκ = 11.05. The qualitative change from the “clear” profiles to the “adjacent” profiles is still very well captured, but now the computed velocity profile is a bit off the experimental one, and the profiles of turbulence kinetic energy and shear stress even more so: the maximum of −u'v'/(uτs)2 is 8% lower and the maximum of k/(uτs)2 is 13% lower, while its minimum is shifted and under predicted, too. Indeed, it is easy to see that the increase in Re from 5400 to 9500 does not result in a sufficiently significant shift in the computed profiles.These facts seem to be explained by the absence of additional treatment of the porous interface. Allegedly, a quantitative change of the influence of roughness occurs when increasing the velocity.
Finally, Figure 7 and Figure 8 graphically show the results presented in Table 2 and Table 3, the solution order 1-3 corresponds to Table 1. The remarkable accuracy of the calculations and excellent qualitative predictions, even in the last case when quantitative deviations become apparent, are visually emphasized.
Figure 7: Friction coefficients (×1000) at the solid wall (left) and at the porous interface (right).
Figure 8: Turbulence friction Reynolds number based on uτp and H (left panel). Ratio of friction velocities at the solid wall and the porous interface uτp/uτs (right panel).
References
1. W.P. Breugem, B.J. Boersma, and R.E. Uittenbogaard, “The Influence of Wall Permeability on Turbulent Channel Flow,” J. Fluid Mech., vol. 562, pp. 35–72, 2006.
2. K. Suga and others, “Effects of Wall Permeability on Turbulence,” Int’l J. Heat Fluid Flow, vol. 31, pp. 974–984, 2010.
3. Y. Kuwata, K. Suga, and Y. Sakurai, “Development and Application of a Multi-Scale k-ε Model for Turbulent Porous Medium Flows,” Int’l J. Heat Fluid Flow, vol. 49, pp. 135–150, 2014.
Application Library path: CFD_Module/Verification_Examples/turbulent_free_porous
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select Fluid Flow > Single-Phase Flow > Turbulent Flow > Turbulent Flow, Low Re k-ε (spf).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces > Stationary with Initialization.
6
Click  Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_parameters.txt.
Definitions
View 1
1
In the Model Builder window, expand the Component 1 (comp1) > Definitions node, then click View 1.
2
In the Settings window for View, locate the View section.
3
Select the Show geometry labels checkbox.
Geometry 1
Create a geometry that is short in the x (streamwise) direction because effectively fully developed flow will be modeled.
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type L.
4
In the Height text field, type H.
Rectangle 2 (r2)
1
Right-click Rectangle 1 (r1) and choose Duplicate.
2
In the Settings window for Rectangle, locate the Position section.
3
In the y text field, type -H.
Point 1 (pt1)
1
In the Geometry toolbar, click  Point.
2
In the Settings window for Point, locate the Point section.
3
In the x text field, type CL.
Point 2 (pt2)
1
Right-click Point 1 (pt1) and choose Duplicate.
2
In the Settings window for Point, locate the Point section.
3
In the y text field, type H.
4
Click  Build All Objects.
Definitions
Average 1 (aveop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Average.
2
In the Settings window for Average, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 9 only.
Domain Point Probe 1
1
In the Definitions toolbar, click  Probes and choose Domain Point Probe.
2
In the Settings window for Domain Point Probe, locate the Point Selection section.
3
In row Coordinates, set x to CL.
4
In row Coordinates, set y to H.
5
Select the Snap to closest boundary checkbox.
Friction at the solid wall
1
In the Model Builder window, expand the Domain Point Probe 1 node, then click Point Probe Expression 1 (ppb1).
2
In the Settings window for Point Probe Expression, type Friction at the solid wall in the Label text field.
3
In the Variable name text field, type friction_s.
4
Locate the Expression section. In the Expression text field, type -spf.nu*ppr(uy).
Domain Point Probe 2
1
In the Model Builder window, under Component 1 (comp1) > Definitions right-click Domain Point Probe 1 and choose Duplicate.
2
In the Settings window for Domain Point Probe, locate the Point Selection section.
3
In row Coordinates, set y to 0.
Friction at the porous interface
1
In the Model Builder window, expand the Domain Point Probe 2 node, then click Friction at the solid wall (ppb2).
2
In the Settings window for Point Probe Expression, type Friction at the porous interface in the Label text field.
3
In the Variable name text field, type friction_p.
4
Locate the Expression section. In the Expression text field, type side(2,(spf.nuT+spf.nu)*pprint(uy)).
Here pprint() is employed to guarantee that near the porous interface the smoothing is made using only data from the clear flow domain.
Variables 1
1
In the Model Builder window, right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_variables.txt.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Air.
4
Click the Add to Component button in the window toolbar.
5
In the Home toolbar, click  Add Material to close the Add Material window.
6
Click the  Show More Options button in the Model Builder toolbar.
7
In the Show More Options dialog, click  Select All.
8
Click OK.
Turbulent Flow, Low Re k-ε (spf)
1
In the Settings window for Turbulent Flow, Low Re k-ε, locate the Physical Model section.
2
Select the Enable porous media domains checkbox.
3
Locate the Turbulence section. From the Wall treatment list, choose Low Re.
Fluid Properties 1
1
In the Model Builder window, under Component 1 (comp1) > Turbulent Flow, Low Re k-ε (spf) click Fluid Properties 1.
2
In the Settings window for Fluid Properties, locate the Fluid Properties section.
3
From the ρ list, choose User defined. In the associated text field, type rho_i.
4
From the μ list, choose User defined. In the associated text field, type mu_i.
5
Locate the Distance Equation section. From the lref list, choose Manual.
6
In the text field, type H.
The low Reynolds number k-\u03b5 model admits a solution with vanishing turbulence levels. To ensure that the solver does not converge to this spurious solution during the first iterations, a variable initial velocity profile is chosen.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
Specify the u vector as
Ub*cos(2*pi*y/H)
x
Porous Medium 1
1
In the Physics toolbar, click  Domains and choose Porous Medium.
2
Select Domain 1 only.
Fluid 1
1
In the Model Builder window, click Fluid 1.
2
In the Settings window for Fluid, locate the Fluid Properties section.
3
From the ρ list, choose User defined. In the associated text field, type rho_i.
4
From the μ list, choose User defined. In the associated text field, type mu_i.
Porous Matrix 1
1
In the Model Builder window, click Porous Matrix 1.
2
In the Settings window for Porous Matrix, locate the Matrix Properties section.
3
From the εp list, choose User defined. In the associated text field, type epsilon_p_i.
4
From the κ list, choose User defined. In the associated text field, type kappa_i.
5
In the cF text field, type cF_i.
Periodic Flow Condition 1
1
In the Physics toolbar, click  Boundaries and choose Periodic Flow Condition.
2
Select Boundaries 1, 3, 8, and 9 only.
3
In the Settings window for Periodic Flow Condition, locate the Flow Condition section.
4
From the Flow condition list, choose Mass flow.
5
In the Mass flow text field, type rho_i*Ub*H*1[m].
Pressure Point Constraint 1
1
In the Physics toolbar, click  Points and choose Pressure Point Constraint.
2
Select Point 4 only.
Delete the porous domain from the Interface selection to start with the computation of the clear channel flow.
3
In the Model Builder window, click Turbulent Flow, Low Re k-ε (spf).
4
In the Domain Selection window, clear Domain 1.
Mesh 1
Use Mapped to guarantee the mesh’s consistency with the Periodic Flow Condition.
Mapped 1
1
In the Mesh toolbar, click  Mapped.
2
In the Settings window for Mapped, locate the Domain Selection section.
3
From the Geometric entity level list, choose Entire geometry.
Distribution 1
1
Right-click Mapped 1 and choose Distribution.
2
Select Boundaries 4–7 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 8.
Distribution 2
1
In the Model Builder window, right-click Mapped 1 and choose Distribution.
2
Select Boundaries 3 and 9 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type Nmesh/2.
Distribution 3
1
Right-click Mapped 1 and choose Distribution.
2
Select Boundaries 1 and 8 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type Nmesh/2.
Boundary Layers 1
1
In the Mesh toolbar, click  Boundary Layers.
2
In the Settings window for Boundary Layers, click to expand the Transition section.
3
Clear the Smooth transition to interior mesh checkbox.
Boundary Layer Properties
1
In the Model Builder window, expand the Boundary Layers 1 node, then click Boundary Layer Properties.
2
Select Boundaries 2, 5, and 7 only.
3
In the Settings window for Boundary Layer Properties, locate the Layers section.
4
In the Number of layers text field, type 8.
5
In the Stretching factor text field, type 1.2.
6
From the Thickness specification list, choose All layers.
7
In the Total thickness text field, type 3*H/Nmesh.
Boundary Layers 2
In the Model Builder window, under Component 1 (comp1) > Mesh 1 right-click Boundary Layers 1 and choose Duplicate.
Boundary Layer Properties
1
In the Model Builder window, expand the Boundary Layers 2 node, then click Boundary Layer Properties.
2
Select Boundaries 4 and 6 only.
3
In the Settings window for Boundary Layer Properties, locate the Layers section.
4
In the Number of layers text field, type 12.
5
From the Thickness specification list, choose All layers.
6
In the Total thickness text field, type 4*H/Nmesh.
Note that the boundary layers mesh is finer near the porous interface where the turbulence level and the friction velocity are higher.
7
Click  Build All.
Study 1
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
ReH (Bulk Reynolds number in clear channel)
5500 5400 9500
1
5
Click  Add.
6
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
H (Clear channel height)
H_E95 H_06 H_06
m
7
In the Study toolbar, click  Compute.
Parametric Solutions 1 (sol3)
1
In the Model Builder window, expand the Study 1 > Solver Configurations node.
2
Right-click Parametric Solutions 1 (sol3) and choose Solution > Copy.
Clear channel flow
1
In the Model Builder window, under Study 1 > Solver Configurations click Parametric Solutions 1 - Copy 1 (sol7).
2
In the Settings window for Solution, type Clear channel flow in the Label text field.
Results
Create a centerline for plotting the results to avoid distortion effects near the source and destination boundaries.
CL clear
1
In the Results toolbar, click  Cut Line 2D.
2
In the Settings window for Cut Line 2D, type CL clear in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Clear channel flow (sol7).
4
Locate the Line Data section. In row Point 1, set x to CL.
5
In row Point 2, set x to CL.
6
In row Point 1, set y to -H.
7
In row Point 2, set y to H.
Turbulent Flow, Low Re k-ε (spf)
Include the porous domain back to the Interface selection.
1
In the Model Builder window, under Component 1 (comp1) click Turbulent Flow, Low Re k-ε (spf).
2
Click in the Graphics window and then press Ctrl+A to select both domains.
Global Equations 1 (ODE1)
1
In the Physics toolbar, click  Global and choose Global Equations.
Global equations is needed to compute the pressure gradient, P_driving, which imposes the prescribed average (bulk) velocity in the clear part of the channel, as was done in the DNS and the experiments.
2
In the Settings window for Global Equations, locate the Global Equations section.
3
In the table, enter the following settings:
Name
f(u,ut,utt,t) (1)
Initial value (u_0) (1)
Initial value (ut_0) (1/s)
Description
P_driving
aveop1(u)-Ub
0
0
 
4
Locate the Units section. Click  Define Dependent Variable Unit.
5
In the Dependent variable quantity table, enter the following settings:
Dependent variable quantity
Unit
Custom unit
Pa
6
Click  Define Source Term Unit.
7
In the Source term quantity table, enter the following settings:
Source term quantity
Unit
Custom unit
m/s
The Mass Flow option of the Periodic Flow Condition should be replaced by the Pressure difference with the pressure difference equal to P_driving.
Periodic Flow Condition 1
1
In the Model Builder window, click Periodic Flow Condition 1.
2
In the Settings window for Periodic Flow Condition, locate the Flow Condition section.
3
From the Flow condition list, choose Pressure difference.
4
In the Δp text field, type P_driving.
Study 1
Parametric Sweep
1
In the Model Builder window, under Study 1 click Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_parametric_study.txt.
Solution 1 (sol1)
1
In the Model Builder window, right-click Solver Configurations and choose Show Default Solver.
2
In the Study toolbar, click  Compute.
Adjacent free-porous flow
In the Settings window for Solution, type Adjacent free-porous flow in the Label text field.
Results
CL adjacent free-porous flow
1
In the Model Builder window, right-click CL clear and choose Duplicate.
2
In the Settings window for Cut Line 2D, type CL adjacent free-porous flow in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Adjacent free-porous flow (sol3).
Import the DNS and the experimental results.
U_E95
1
In the Results toolbar, click  Table.
2
In the Model Builder window, under Results click Tables.
3
Analogously, create 7 more instances of Table.
4
In the Settings window for Table, type U_E95 in the Label text field.
5
Locate the Data section. Click  Import.
6
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_U_E95.txt.
k_E95
1
In the Model Builder window, under Results > Tables click Table 3.
2
In the Settings window for Table, type k_E95 in the Label text field.
3
Locate the Data section. Click  Import.
4
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_k_E95.txt.
uv_E95
1
In the Model Builder window, under Results > Tables click Table 4.
2
In the Settings window for Table, type uv_E95 in the Label text field.
3
Locate the Data section. Click  Import.
4
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_uv_E95.txt.
U_06_5400
1
In the Model Builder window, under Results > Tables click Table 5.
2
In the Settings window for Table, type U_06_5400 in the Label text field.
3
Locate the Data section. Click  Import.
4
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_U_#06_5400.txt.
uv_06_5400
1
In the Model Builder window, under Results > Tables click Table 6.
2
In the Settings window for Table, type uv_06_5400 in the Label text field.
3
Locate the Data section. Click  Import.
4
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_uv_#06_5400.txt.
U_06_9500
1
In the Model Builder window, under Results > Tables click Table 7.
2
In the Settings window for Table, type U_06_9500 in the Label text field.
3
Locate the Data section. Click  Import.
4
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_U_#06_9500.txt.
uv_06_9500
1
In the Model Builder window, under Results > Tables click Table 8.
2
In the Settings window for Table, type uv_06_9500 in the Label text field.
3
Locate the Data section. Click  Import.
4
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_uv_#06_9500.txt.
u-rms and v-rms #06
1
In the Model Builder window, under Results > Tables click Table 9.
2
In the Settings window for Table, type u-rms and v-rms #06 in the Label text field.
3
Locate the Data section. Click  Import.
4
Browse to the model’s Application Libraries folder and double-click the file turbulent_free_porous_rms_#06.txt.
Definitions
Build Interpolation functions for the u-rms and v-rms velocities taken from the experimental #06-series. The aim is to reconstruct the w-rms velocity by extrapolation as explained in the main text.
u_rms #06 5400
1
In the Definitions toolbar, click  Interpolation.
2
In the Settings window for Interpolation, type u_rms #06 5400 in the Label text field.
3
Locate the Definition section. From the Data source list, choose Result table.
4
From the Table from list, choose u-rms and v-rms #06.
5
Locate the Data Column Settings section. In the table, click to select the cell at row number 2 and column number 1.
6
In the Name text field, type u1.
v_rms #06 5400
1
Right-click u_rms  #06 5400 and choose Duplicate.
2
In the Settings window for Interpolation, type v_rms #06 5400 in the Label text field.
3
Locate the Data Column Settings section. In the table, enter the following settings:
Columns
Type
Settings
u_5400
Ignored column
 
v_5400
Function values
Function name=v1
4
In the Name text field, type v1.
u_rms #06 9500
1
Right-click v_rms  #06 5400 and choose Duplicate.
2
In the Settings window for Interpolation, type u_rms #06 9500 in the Label text field.
3
Locate the Data Column Settings section. In the table, enter the following settings:
Columns
Type
Settings
v_5400
Ignored column
 
u_9500
Function values
Function name=u2
4
In the Name text field, type u2.
v_rms #06 9500
1
Right-click u_rms  #06 9500 and choose Duplicate.
2
In the Settings window for Interpolation, type v_rms #06 9500 in the Label text field.
3
Locate the Data Column Settings section. In the table, enter the following settings:
Columns
Type
Settings
u_9500
Ignored column
 
v_9500
Function values
Function name=v2
4
In the Name text field, type v2.
Results
Grid Exp
1
In the Results toolbar, click  More Datasets and choose Grid > Grid 1D.
2
In the Settings window for Grid 1D, type Grid Exp in the Label text field.
3
Locate the Data section. From the Source list, choose Function.
4
From the Function list, choose All.
U_E95
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type U_E95 in the Label text field.
3
Locate the Data section. From the Dataset list, choose CL adjacent free-porous flow.
4
From the Parameter selection (ReH, H, epsilon_p_i, cF_i, Da_i) list, choose From list.
5
In the Parameter values (ReH,H (m),epsilon_p_i,cF_i,Da_i) list box, select 1: ReH=5500, H=0.03 m, epsilon_p_i=0.95, cF_i=0.14, Da_i=1.9E-4.
6
Click to expand the Title section. From the Title type list, choose Manual.
7
In the Title text area, type Velocity E95 Breugem and Others (2006).
8
Locate the Plot Settings section.
9
Select the x-axis label checkbox. In the associated text field, type y/H.
10
Select the y-axis label checkbox. In the associated text field, type U/Ub.
11
Locate the Axis section. Select the Manual axis limits checkbox.
12
In the x minimum text field, type -0.2.
13
In the y minimum text field, type 0.
14
In the y maximum text field, type 1.3.
15
Locate the Legend section. From the Position list, choose Lower middle.
Adjacent
1
Right-click U_E95 and choose Line Graph.
2
In the Settings window for Line Graph, type Adjacent in the Label text field.
3
Locate the y-Axis Data section. In the Expression text field, type u/Ub.
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type y_H.
6
Click to expand the Coloring and Style section. From the Color list, choose Red.
7
From the Width list, choose 3.
8
Click to expand the Legends section. Select the Show legends checkbox.
9
Find the Include subsection. Clear the Solution checkbox.
10
Select the Label checkbox.
Clear
1
Right-click Adjacent and choose Duplicate.
2
In the Settings window for Line Graph, type Clear in the Label text field.
3
Locate the Data section. From the Dataset list, choose CL clear.
4
From the Parameter selection (ReH, H) list, choose From list.
5
In the Parameter values (ReH,H (m)) list box, select 1: ReH=5500, H=0.03 m.
6
Locate the Coloring and Style section. From the Color list, choose Green.
7
From the Width list, choose 2.
DNS
1
In the Model Builder window, right-click U_E95 and choose Table Graph.
2
In the Settings window for Table Graph, type DNS in the Label text field.
3
Locate the Data section. From the Table list, choose U_E95.
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose None.
5
From the Color list, choose Black.
6
Find the Line markers subsection. From the Marker list, choose Circle.
7
Click to expand the Legends section. Select the Show legends checkbox.
8
Find the Include subsection. Clear the Headers checkbox.
9
Select the Label checkbox.
k_E95
1
Right-click U_E95 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type k_E95 in the Label text field.
3
Locate the Title section. In the Title text area, type Turbulence kinetic energy E95 Breugem et al (2006).
4
Locate the Plot Settings section. In the y-axis label text field, type k/u<sup>s</sup><sub>\tau</sub> <sup>2</sup>.
5
Locate the Axis section. In the y maximum text field, type 8.5.
6
Locate the Legend section. From the Position list, choose Upper right.
Adjacent
1
In the Model Builder window, expand the k_E95 node, then click Adjacent.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type k/friction_s.
Clear
1
In the Model Builder window, click Clear.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type k/friction_s.
DNS
1
In the Model Builder window, click DNS.
2
In the Settings window for Table Graph, locate the Data section.
3
From the Table list, choose k_E95.
4
Locate the Coloring and Style section. Find the Line markers subsection. From the Positioning list, choose Interpolated.
5
In the Number text field, type 100.
uv_E95
1
In the Model Builder window, right-click U_E95 and choose Duplicate.
2
In the Settings window for 1D Plot Group, locate the Title section.
3
In the Title text area, type Shear stress E95 Breugem et al (2006).
4
In the Label text field, type uv_E95.
5
Locate the Plot Settings section. In the y-axis label text field, type -u<sup>\prime</sup>v<sup>\prime</sup>/u<sup>s</sup><sub>\tau</sub> <sup>2</sup>.
6
Locate the Axis section. In the y minimum text field, type -0.8.
7
In the y maximum text field, type 2.7.
8
Locate the Legend section. From the Position list, choose Upper right.
Adjacent
1
In the Model Builder window, expand the uv_E95 node, then click Adjacent.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type shear_stress/friction_s.
Clear
1
In the Model Builder window, click Clear.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type shear_stress/friction_s.
DNS
1
In the Model Builder window, click DNS.
2
In the Settings window for Table Graph, locate the Data section.
3
From the Table list, choose uv_E95.
U_#06_Re=5400
1
In the Model Builder window, right-click U_E95 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type U_#06_Re=5400 in the Label text field.
3
Locate the Data section. In the Parameter values (ReH,H (m),epsilon_p_i,cF_i,Da_i) list box, select 2: ReH=5400, H=0.029 m, epsilon_p_i=0.8, cF_i=0.095, Da_i=1.04E-4.
4
Locate the Title section. In the Title text area, type Velocity #06 Re=5400 Suga et al (2010).
Clear
1
In the Model Builder window, expand the U_#06_Re=5400 node, then click Clear.
2
In the Settings window for Line Graph, locate the Data section.
3
In the Parameter values (ReH,H (m)) list box, select 2: ReH=5400, H=0.029 m.
Exp
1
In the Model Builder window, under Results > U_#06_Re=5400 click DNS.
2
In the Settings window for Table Graph, type Exp in the Label text field.
3
Locate the Data section. From the Table list, choose U_06_5400.
k_#06_Re=5400
1
In the Model Builder window, right-click k_E95 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type k_#06_Re=5400 in the Label text field.
3
Locate the Data section. In the Parameter values (ReH,H (m),epsilon_p_i,cF_i,Da_i) list box, select 2: ReH=5400, H=0.029 m, epsilon_p_i=0.8, cF_i=0.095, Da_i=1.04E-4.
4
Locate the Title section. In the Title text area, type Turbulence kinetic energy #06 Re=5400 Suga et al (2010).
Clear
1
In the Model Builder window, expand the k_#06_Re=5400 node, then click Clear.
2
In the Settings window for Line Graph, locate the Data section.
3
In the Parameter values (ReH,H (m)) list box, select 2: ReH=5400, H=0.029 m.
Clear 1
Right-click Clear and choose Duplicate.
DNS
In the Model Builder window, right-click DNS and choose Delete.
In the experiments #06 the spanwise rms-velocity was not measured. Here its "upper estimate" and "lower estimate", as well as "bottom limit", are reconstructed (extrapolated) using the streamwise and the wall-normal rms-velocities according to the main text.
Exp Upper Estimate
1
In the Settings window for Line Graph, type Exp Upper Estimate in the Label text field.
2
Locate the Data section. From the Dataset list, choose Grid Exp.
3
Locate the y-Axis Data section. In the Expression text field, type ((1+cUu)*u1(x)^2+(1+cUv)*v1(x)^2)/2.
4
Locate the x-Axis Data section. In the Expression text field, type x.
5
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
6
From the Color list, choose Black.
7
From the Width list, choose 1.
8
Locate the Legends section. From the Legends list, choose Manual.
9
In the table, enter the following settings:
Legends
Exp U.E.
Exp Lower Estimate
1
Right-click Exp Upper Estimate and choose Duplicate.
2
In the Settings window for Line Graph, type Exp Lower Estimate in the Label text field.
3
Locate the y-Axis Data section. In the Expression text field, type ((1+cLu)*u1(x)^2+(1+cLv)*v1(x)^2)/2.
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Solid.
5
Locate the Legends section. In the table, enter the following settings:
Legends
Exp L.E.
Exp Bottom Limit
1
Right-click Exp Lower Estimate and choose Duplicate.
2
In the Settings window for Line Graph, type Exp Bottom Limit in the Label text field.
3
Locate the y-Axis Data section. In the Expression text field, type (u1(x)^2+2*v1(x)^2)/2.
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dotted.
5
Locate the Legends section. In the table, enter the following settings:
Legends
Exp B.L.
uv_#06_Re=5400
1
In the Model Builder window, right-click uv_E95 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type uv_#06_Re=5400 in the Label text field.
3
Locate the Data section. In the Parameter values (ReH,H (m),epsilon_p_i,cF_i,Da_i) list box, select 2: ReH=5400, H=0.029 m, epsilon_p_i=0.8, cF_i=0.095, Da_i=1.04E-4.
4
Locate the Title section. In the Title text area, type Shear stress #06 Re=5400 Suga et al (2010).
Clear
1
In the Model Builder window, expand the uv_#06_Re=5400 node, then click Clear.
2
In the Settings window for Line Graph, locate the Data section.
3
In the Parameter values (ReH,H (m)) list box, select 2: ReH=5400, H=0.029 m.
Exp
1
In the Model Builder window, under Results > uv_#06_Re=5400 click DNS.
2
In the Settings window for Table Graph, type Exp in the Label text field.
3
Locate the Data section. From the Table list, choose uv_06_5400.
U_#06_Re=9500
1
In the Model Builder window, right-click U_#06_Re=5400 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type U_#06_Re=9500 in the Label text field.
3
Locate the Data section. In the Parameter values (ReH,H (m),epsilon_p_i,cF_i,Da_i) list box, select 3: ReH=9500, H=0.029 m, epsilon_p_i=0.8, cF_i=0.095, Da_i=1.04E-4.
4
Locate the Title section. In the Title text area, type Velocity #06 Re=9500 Suga et al (2010).
Clear
1
In the Model Builder window, expand the U_#06_Re=9500 node, then click Clear.
2
In the Settings window for Line Graph, locate the Data section.
3
In the Parameter values (ReH,H (m)) list box, select 3: ReH=9500, H=0.029 m.
Exp
1
In the Model Builder window, click Exp.
2
In the Settings window for Table Graph, locate the Data section.
3
From the Table list, choose U_06_9500.
k_#06_Re=9500
1
In the Model Builder window, right-click k_#06_Re=5400 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type k_#06_Re=9500 in the Label text field.
3
Locate the Data section. In the Parameter values (ReH,H (m),epsilon_p_i,cF_i,Da_i) list box, select 3: ReH=9500, H=0.029 m, epsilon_p_i=0.8, cF_i=0.095, Da_i=1.04E-4.
4
Locate the Title section. In the Title text area, type Turbulence kinetic energy #06 Re=9500 Suga et al (2010).
Clear
1
In the Model Builder window, expand the k_#06_Re=9500 node, then click Clear.
2
In the Settings window for Line Graph, locate the Data section.
3
In the Parameter values (ReH,H (m)) list box, select 3: ReH=9500, H=0.029 m.
Exp Upper Estimate
1
In the Model Builder window, click Exp Upper Estimate.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type ((1+cUu)*u2(x)^2+(1+cUv)*v2(x)^2)/2.
Exp Lower Estimate
1
In the Model Builder window, click Exp Lower Estimate.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type ((1+cLu)*u2(x)^2+(1+cLv)*v2(x)^2)/2.
Exp Bottom Limit
1
In the Model Builder window, click Exp Bottom Limit.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type (u2(x)^2+2*v2(x)^2)/2.
uv_#06_Re=9500
1
In the Model Builder window, right-click uv_#06_Re=5400 and choose Duplicate.
2
In the Settings window for 1D Plot Group, type uv_#06_Re=9500 in the Label text field.
3
Locate the Data section. In the Parameter values (ReH,H (m),epsilon_p_i,cF_i,Da_i) list box, select 3: ReH=9500, H=0.029 m, epsilon_p_i=0.8, cF_i=0.095, Da_i=1.04E-4.
4
Locate the Title section. In the Title text area, type Shear stress #06 Re=9500 Suga et al (2010).
Clear
1
In the Model Builder window, expand the uv_#06_Re=9500 node, then click Clear.
2
In the Settings window for Line Graph, locate the Data section.
3
In the Parameter values (ReH,H (m)) list box, select 3: ReH=9500, H=0.029 m.
Exp
1
In the Model Builder window, click Exp.
2
In the Settings window for Table Graph, locate the Data section.
3
From the Table list, choose uv_06_9500.
Solid Wall Friction Coefficient (x1000)
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Solid Wall Friction Coefficient (x1000) in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Solid Wall Friction Coefficient (x1000).
5
Locate the Axis section. Select the Manual axis limits checkbox.
6
In the x minimum text field, type 0.9.
7
In the x maximum text field, type manual_x_max.
8
In the y minimum text field, type 0.
9
In the y maximum text field, type 12.
10
Locate the Grid section. Select the Manual spacing checkbox.
11
In the y spacing text field, type 0.25.
Adjacent
1
Right-click Solid Wall Friction Coefficient (x1000) and choose Global.
2
In the Settings window for Global, type Adjacent in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Adjacent free-porous flow (sol3).
4
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
Cs1000
1
Friction coefficient (x1000) at the solid wall
5
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose None.
6
From the Color list, choose Black.
7
Find the Line markers subsection. From the Marker list, choose Plus sign.
8
Click to expand the Legends section. Find the Include subsection. Clear the Description checkbox.
9
Clear the Solution checkbox.
10
Select the Label checkbox.
Clear
1
Right-click Adjacent and choose Duplicate.
2
In the Settings window for Global, type Clear in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Clear channel flow (sol7).
4
Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Square.
DNS E95
1
In the Model Builder window, right-click Adjacent and choose Duplicate.
2
In the Settings window for Global, type DNS E95 in the Label text field.
3
Locate the Data section. From the Parameter selection (ReH, H, epsilon_p_i, cF_i, Da_i) list, choose From list.
4
In the Parameter values (ReH,H (m),epsilon_p_i,cF_i,Da_i) list box, select 1: ReH=5500, H=0.03 m, epsilon_p_i=0.95, cF_i=0.14, Da_i=1.9E-4.
5
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
10.9
1
 
6
Locate the Coloring and Style section. From the Color list, choose Red.
7
Find the Line markers subsection. From the Marker list, choose Circle.
Exp #06 Re=5400
1
Right-click DNS E95 and choose Duplicate.
2
In the Settings window for Global, type Exp #06 Re=5400 in the Label text field.
3
Locate the Data section. In the Parameter values (ReH,H (m),epsilon_p_i,cF_i,Da_i) list box, select 2: ReH=5400, H=0.029 m, epsilon_p_i=0.8, cF_i=0.095, Da_i=1.04E-4.
4
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
C_help_var*(6.23/1.55)^2
1
 
5
Locate the Coloring and Style section. From the Color list, choose Blue.
Exp #06 Re=9500
1
Right-click Exp #06 Re=5400 and choose Duplicate.
2
In the Settings window for Global, type Exp #06 Re=9500 in the Label text field.
3
Locate the Data section. In the Parameter values (ReH,H (m),epsilon_p_i,cF_i,Da_i) list box, select 3: ReH=9500, H=0.029 m, epsilon_p_i=0.8, cF_i=0.095, Da_i=1.04E-4.
4
Locate the Coloring and Style section. From the Color list, choose Green.
5
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
C_help_var*(11.05/1.63)^2
1
 
Annotation 1
1
In the Model Builder window, right-click Solid Wall Friction Coefficient (x1000) and choose Annotation.
2
In the Settings window for Annotation, locate the Annotation section.
3
In the Text text field, type Sol 1: E95 \\ Re=5500 \\ $\epsilon_p=0.95$ \\ Da=1.9e-4 \\ $c_F=0.14$ \\ ~ \\ Sol 2: #06 Re=5400 \\ Sol 3: #06 Re=9500 \\ $\epsilon_p$=0.8 \\ Da=1.04e-4 \\ $c_F=0.095$.
4
Select the LaTeX markup checkbox.
5
Locate the Coloring and Style section. From the Anchor point list, choose Upper right.
6
Locate the Position section. In the x text field, type 3.9.
7
In the y text field, type 5.
8
Locate the Coloring and Style section. Clear the Show point checkbox.
9
Select the Show frame checkbox.
10
From the Background color list, choose White.
Porous Interface Friction Coefficient (x1000)
1
Right-click Solid Wall Friction Coefficient (x1000) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Porous Interface Friction Coefficient (x1000) in the Label text field.
3
Click to expand the Title section. In the Title text area, type Porous Interface Friction Coefficient (x1000).
4
Locate the Axis section. In the y maximum text field, type 32.
Adjacent
1
In the Model Builder window, expand the Porous Interface Friction Coefficient (x1000) node, then click Adjacent.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
Cp1000
1
Friction coefficient (x1000) at the porous interface
Clear
1
In the Model Builder window, click Clear.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
Cp1000
1
Friction coefficient (x1000) at the porous interface
DNS E95
1
In the Model Builder window, click DNS E95.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
30.4
1
 
Exp #06 Re=5400
1
In the Model Builder window, click Exp #06 Re=5400.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
C_help_var*(6.23)^2
1
 
Exp #06 Re=9500
1
In the Model Builder window, click Exp #06 Re=9500.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
C_help_var*(11.05)^2
1
 
Annotation 1
1
In the Model Builder window, click Annotation 1.
2
In the Settings window for Annotation, locate the Position section.
3
In the y text field, type 14.
Porous Friction Reynolds Number
1
In the Model Builder window, right-click Porous Interface Friction Coefficient (x1000) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Porous Friction Reynolds Number in the Label text field.
3
Click to expand the Title section. In the Title text area, type Porous Friction Reynolds Number Re<sub>\tau</sub> <sup>p</sup>.
4
Locate the Axis section. In the y maximum text field, type 1100.
5
Locate the Grid section. In the y spacing text field, type 50.
Adjacent
1
In the Model Builder window, expand the Porous Friction Reynolds Number node, then click Adjacent.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
Retp
1
 
Clear
1
In the Model Builder window, click Clear.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
Retp
1
 
DNS E95
1
In the Model Builder window, click DNS E95.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
678
1
 
Exp #06 Re=5400
1
In the Model Builder window, click Exp #06 Re=5400.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
6.23/sqrt(Da_i)
 
 
Exp #06 Re=9500
1
In the Model Builder window, click Exp #06 Re=9500.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
11.05/sqrt(Da_i)
 
 
Annotation 1
1
In the Model Builder window, click Annotation 1.
2
In the Settings window for Annotation, locate the Position section.
3
In the y text field, type 600.
Ratio of Friction Velocities
1
In the Model Builder window, right-click Porous Friction Reynolds Number and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Ratio of Friction Velocities in the Label text field.
3
Click to expand the Title section. In the Title text area, type Ratio of Friction Velocities u<sub>\tau</sub> <sup>p</sup>/u<sub>\tau</sub> <sup>s</sup>.
4
Locate the Axis section. In the y maximum text field, type 1.8.
5
Locate the Grid section. In the y spacing text field, type 0.1.
Adjacent
1
In the Model Builder window, expand the Ratio of Friction Velocities node, then click Adjacent.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
utau_p/utau_s
1
 
Clear
1
In the Model Builder window, click Clear.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
utau_p/utau_s
1
 
DNS E95
1
In the Model Builder window, click DNS E95.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
sqrt(30.4/10.9)
1
 
Exp #06 Re=5400
1
In the Model Builder window, click Exp #06 Re=5400.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
1.55
1
 
Exp #06 Re=9500
1
In the Model Builder window, click Exp #06 Re=9500.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
1.63
1
 
Annotation 1
1
In the Model Builder window, click Annotation 1.
2
In the Settings window for Annotation, locate the Position section.
3
In the y text field, type 1.
Arrow Surfaces E95
1
In the Home toolbar, click  Add Plot Group and choose 2D Plot Group.
2
In the Settings window for 2D Plot Group, type Arrow Surfaces E95 in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Adjacent free-porous flow (sol3).
4
From the Parameter value (ReH,H (m),epsilon_p_i,cF_i,Da_i) list, choose 1: ReH=5500, H=0.03 m, epsilon_p_i=0.95, cF_i=0.14, Da_i=1.9E-4.
5
Click to expand the Title section. From the Title type list, choose Manual.
6
In the Title text area, type Velocity, turbulence kinetic energy and shear stress.
7
Locate the Color Legend section. Clear the Show legends checkbox.
8
Click to expand the Plot Array section. From the Array type list, choose Linear.
9
In the Relative padding text field, type 4.
Velocity
1
Right-click Arrow Surfaces E95 and choose Arrow Surface.
2
In the Settings window for Arrow Surface, type Velocity in the Label text field.
3
Locate the Expression section. In the x-component text field, type u/Ub.
4
In the y-component text field, type 0.
5
Locate the Arrow Positioning section. Find the x grid points subsection. In the Points text field, type 1.
6
Find the y grid points subsection. In the Points text field, type 500.
7
Locate the Coloring and Style section. From the Arrow type list, choose Cone.
8
Select the Scale factor checkbox. In the associated text field, type 1e-2.
Color Expression 1
1
Right-click Velocity and choose Color Expression.
2
In the Settings window for Color Expression, locate the Expression section.
3
From the Color data list, choose Arrow length.
Turbulence kinetic energy
1
In the Model Builder window, right-click Velocity and choose Duplicate.
2
In the Settings window for Arrow Surface, type Turbulence kinetic energy in the Label text field.
3
Locate the Expression section. In the x-component text field, type k/utau_s^2.
4
Locate the Coloring and Style section. In the Scale factor text field, type 2e-3.
Color Expression 1
1
In the Model Builder window, expand the Turbulence kinetic energy node, then click Color Expression 1.
2
In the Settings window for Color Expression, locate the Coloring and Style section.
3
From the Color table list, choose Disco.
Shear stress
1
In the Model Builder window, right-click Turbulence kinetic energy and choose Duplicate.
2
In the Settings window for Arrow Surface, type Shear stress in the Label text field.
3
Locate the Expression section. In the x-component text field, type shear_stress/utau_s^2.
4
Locate the Coloring and Style section. In the Scale factor text field, type 5e-3.
Color Expression 1
1
In the Model Builder window, expand the Shear stress node, then click Color Expression 1.
2
In the Settings window for Color Expression, locate the Coloring and Style section.
3
From the Color table list, choose Plasma.
Annotation 1
1
In the Model Builder window, right-click Arrow Surfaces E95 and choose Annotation.
2
In the Settings window for Annotation, locate the Annotation section.
3
Select the LaTeX markup checkbox.
4
In the Text text field, type \Huge$\frac{u}{U_b}$.
5
Locate the Position section. In the x text field, type L.
6
In the y text field, type -2*L.
7
Locate the Coloring and Style section. Clear the Show point checkbox.
8
From the Anchor point list, choose Middle left.
9
Click to expand the Plot Array section. Clear the Belongs to array checkbox.
Annotation 2
1
Right-click Annotation 1 and choose Duplicate.
2
In the Settings window for Annotation, locate the Annotation section.
3
In the Text text field, type \Huge$\frac{k}{{(u_\tau^s})^2}$.
4
Locate the Position section. In the x text field, type 6*L.
Annotation 3
1
Right-click Annotation 2 and choose Duplicate.
2
In the Settings window for Annotation, locate the Annotation section.
3
In the Text text field, type \Huge$\frac{-\overline{u^\prime\,v^\prime}}{{(u_\tau^s})^2}$.
4
Locate the Position section. In the x text field, type 11*L.
5
Click the  Zoom Extents button in the Graphics toolbar.
Evaluate the friction coefficients, friction velocities, and Reynolds numbers and compare the results with Table 2 and Table 3.
Clear channel
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, type Clear channel in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Clear channel flow (sol7).
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
Cs1000
1
Friction coefficient (x1000) at the solid wall
Cp1000
1
Friction coefficient (x1000) at the porous interface
utau_p*H/nu_i
1
Reynolds number based on the porous interface friction velocity
utau_p/utau_s
1
Ratio of friction velocities (porous interface over solid wall)
5
Click  Evaluate. Compare with Table 2.
Adjacent
1
Right-click Clear channel and choose Duplicate.
2
In the Settings window for Global Evaluation, type Adjacent in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Adjacent free-porous flow (sol3).
4
Click  Evaluate. Compare with Table 3.