Proceedings of the Seventh International Conference on Hydroscience and Engineering, Philadelphia, PA, September 2006.  http://hdl.handle.net/1860/732The dynamics of diffusion in laminar flow constructed stormwater wetlands are presently not fully
understood. A field experiment conducted at Villanova University in 2006 compared field diffusion
coefficients with those of the laboratory experiments of Nepf et al. (1997) and Serra et al. (2004); all
of these studies examined the effect of plant density on diffusion coefficients. The field conditions
at Villanova included several additional factors that were not present in the laboratory experiments:
non-uniform flow, plant debris and additional bed shear stress. The results of this study show that
these field factors significantly affect the diffusion coefficients and that a new model is needed to
predict the diffusion coefficients in field conditions

Burke, Erin

Wadzuk, Bridget M.

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Proceedings of the 7th International Conference on HydroScience and Engineering 
Philadelphia, USA September 10-13, 2006 (ICHE 2006) 
ISBN: 0977447405 
 
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The 7th Int.  Conf. on Hydroscience and Engineering (ICHE-2006),Sep 10 –Sep 13,Philadelphia,USA 1 
 
 
 
 
 
 
A DIFFUSION MODEL FOR LOW REYNOLDS NUMBER FLOWS THROUGH A 
CONSTRUCTED STORMWATER WETLAND 
 
 
Bridget M. Wadzuk1 and Erin Burke2 
 
 
ABSTRACT 
 
The dynamics of diffusion in laminar flow constructed stormwater wetlands are presently not fully 
understood.  A field experiment conducted at Villanova University in 2006 compared field diffusion 
coefficients with those of the laboratory experiments of Nepf et al. (1997) and Serra et al. (2004); all 
of these studies examined the effect of plant density on diffusion coefficients.  The field conditions 
at Villanova included several additional factors that were not present in the laboratory experiments: 
non-uniform flow, plant debris and additional bed shear stress.  The results of this study show that 
these field factors significantly affect the diffusion coefficients and that a new model is needed to 
predict the diffusion coefficients in field conditions. 
 
 
1. INTRODUCTION 
 
Stormwater runoff is a major factor in non-point source pollution.  Contaminants from lawns and 
parking lots including fertilizers and hydrocarbons enter surface water systems through storm drains 
and direct runoff.  Over the past decade several best management practices (BMPs) have been 
developed to reduce the quantity and improve the quality of stormwater runoff; one popular BMP is 
a constructed stormwater wetland (CSW).  CSWs serve three main purposes: reduce peak flow, treat 
water quality and recharge ground water.  In addition to these main functions, CSWs are 
aesthetically pleasing and increase biodiversity.  CSWs have been included in many stormwater 
management plans throughout the country recently, however the science lags behind 
implementation; there is little known about the hydrodynamic properties of a CSW.  This study will 
focus on one hydrodynamic process: diffusion during laminar flow conditions.  While the present 
application focuses on the diffusion in a CSW, the basic physics of diffusion have applications to 
many environmental fluid flows (i.e. pesticides through crops).   
 
2. NUMERICAL BACKGROUND 
 
Fluid flow mechanics depend strongly on the flows tendency to be inertially or viscous dominated, 
that is turbulent or laminar flow as described by the Reynolds number (Re).  The characteristic 
length used in the Re for vegetated flows is the stem diameter, 
                                                 
1 Assistant Professor, Department of Civil and Environmental Engineering, Villanova University, Villanova, PA 19085, 
USA (bridget.wadzuk@villanova.edu) 
2 Graduate Student, Department of Civil and Environmental Engineering, Villanova University, Villanova, PA 19085, 
USA (erin.burke@villanova.edu) 
  2 
  
 Re udν=  (1) 
 
where u is the average velocity, d is the stem diameter and ν is the kinematic viscosity.  In a CSW, 
the change between laminar and turbulent flow occurs around a Reynolds number of approximately 
200 (Nepf et al, 1997), although there is some contention that this limit may be lower or higher 
(Ayaz and Pedley, 1999; Hill and Koch, 2002; Leonard and Luther, 1995).  Natural wetlands generally 
occur in tidal regions with larger flow rates, currents and turbulent flow conditions.  Conversely, 
CSWs have high and low flow conditions.  High flow conditions occur after rainfall and are well 
within the turbulent flow regime.  Low flow conditions (i.e. baseflow) occur during dry periods and 
the flow is often in the laminar flow regime. 
 Diffusion in a CSW is comprised of both turbulent and mechanical diffusion (note: the term 
diffusion is often used interchangeably with dispersion in literature). Turbulent diffusion is the result 
of wakes and eddies while mechanical diffusion is the result of the random walk of a fluid particle 
when interacting with plant stems.  Equation (2) gives the dimensionless lateral diffusion coefficient 
(Dy/ud) developed by Nepf (1999), where the first term on the right hand side represents turbulent 
diffusion and the second term represents mechanical diffusion. 
 
1/3 24 4
2
y
n d
D P n PC
ud
βα π π
⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
 (2) 
 
The values αn and β are constants, Cd is the drag coefficient and P is the percent of the plan area 
occupied by stem stalks.  This study focuses only on laminar flow conditions, so only mechanical 
diffusion is evaluated assuming the turbulent contribution is negligible.  
There are numerous studies on flow through wetlands, however most field experiments are 
in naturally occurring, turbulent wetlands and the few studies that do examine laminar flow are 
laboratory experiments.  Nepf et al. (1997) and Serra et al. (2004) (herein referred to as Nepf and 
Serra, respectively) have conducted laboratory studies on laminar flow through vegetated conditions 
to model the relationship between the diffusion coefficient and plant density.  The results of their 
studies are shown in Figure 1.  Nepf’s diffusion coefficients (open triangles) are for Re of 114 and 
lower.  Nepf reported dimensionless diffusion coefficients of 0.2 – 0.35 for Re = 192, which are 
characterized by the total diffusion (Dtotal) model and are neglected here as they are considered 
nearly turbulent.  Serra’s diffusion coefficients (open circles) are an order of magnitude greater than 
Nepf’s and lie near values for the theoretical total diffusion.  This indicates that at large plant 
densities, mechanical diffusion (Dm) is the dominant mechanism in the total diffusion, but plays a 
minor role for more sparse densities.  The goal of this study is to investigate the diffusion 
coefficients for low Re flows with plant densities between Nepf and Serra. 
 
  3 
100 101 102
10
-3
10-2
10-1
100
101
P%
D
y/
ud
 
 
Nepf
Serra
Field
Dm,β=1
Dm,β=2
Dt
Dtotal
Figure 1: Dimensionless diffusion coefficient (Dy/ud) versus plant density (P) for the 
laboratory experiments of Nepf (open triangles) and Serra (open circles) and the field data 
(pluses).  The diffusion coefficient was calculated from equation for Nepf and Serra and 
by equation for the field data.  Nepf et al (1999) developed a mechanical diffusion model 
(the dashed and dotted line for β = 1 and 2, respectively), turbulent diffusion (dash-dot 
line) and a total diffusion model (solid line, for D-m with β = 2). 
D
y/
ud
 
 
 
3. METHODS 
 
3.1 Experiment 
A dye tracer study to determine the diffusion coefficient in a laminar flow CSW was conducted at 
Villanova University.  The dye tracer study occurred over a 1.9 m x 1.9 m area of the CSW, as seen 
in Figure 2.  Rhodamine WT was injected in the center of the upstream cross-section in the 
longitudinal direction at the in situ velocity regulated by a variable speed pump.  The dye solution 
concentration was 100 mg/L and released through a ¼ inch nozzle at mid-depth.  The dye dispersed 
through the wetland and was measured 1.9 m downstream (at the rope in Figure 2) every 10 cm 
across the cross-section.  The dye concentration was measured using a YSI Flourometer.  Velocity 
measurements were taken using a SonTek Velocimeter to 1) establish the velocity at the dye 
injection position and 2) monitor the velocity distribution across the downstream cross-section for 
the duration of the experiment.  The vegetation within the CSW test area was Phragmites australis, 
which has a simple cylindrical structure (similar to the cylindrical rods used in Nepf’s and Serra’s 
laboratory experiments).  Fifty stems were collected from the test area to determine the average stem 
  4 
Figure 2: Test setup.  The table in the background of 
the picture is where the dye was injected.  The rope 
in the foreground is where the dye concentration was 
measured along the cross-section.  
 
diameter of 1.52 ± 1.1 cm.    The test site was weeded in order to create different plant densities. 
Nepf and Serra calculated the plant distribution using a computer-generated random 
distribution, and then calculated the plant density as a ratio of the plant area to a unit area.  Both 
laboratory studies used consistent diameter dowels to represent the plant stems (Nepf – 0.6 cm and 
1.2 cm in different experimental runs; Serra – 1 cm).  The plant density for the Villanova University 
field experiments was calculated using field measurements.  A 9.8 cm x 9.8 cm plant frame was 
used to isolate an area (A), the number of stem stalks (n) contained in the box was recorded, and the 
plant density (P) was calculated as 
 
( )214 %n dP
A
π=  (3) 
 
Nepf had plant densities between 0 and 5.3%, while Serra had plant densities between 10 to 35 %.  
The Villanova field experiment began with a natural plant density of 10.2% and was modified to 
have densities of 6.8 and 3.3%. 
  
3.2  Modeling 
The diffusivity coefficients in Nepf and Serra were calculated from the solution of the one-
dimensional advection-diffusion equation: 
 
2
max
exp
4 y
C y u
C D X
⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠
 (4) 
 
Where Dy is the diffusion coefficient in the lateral direction, X is the longitudinal distance traveled 
by the dye, u is the average velocity over the entire cross section, y is lateral distance from the center 
of the cross-sectional span, C and Cmax are the local and maximum concentrations.  The method of 
least squares is used to determine a single diffusion coefficient and error value (e).  
 The flow within the CSW was not uniform in the x-direction, as assumed in the laboratory 
experiment.  Thus, an advection-diffusion equation that considered velocity fluctuation and 
concentration changes in the x- and y-directions was used 
 ( ) ( ) ( ) ( )2 22 2 '' ' ' 'y CC C u u C C D C Ct x x y⎡ ⎤∂ ∂ ∂ ∂+ + + + = + +⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦  (5) 
  5 
         
where 'C  is the concentration perturbation from the mean concentration (C ) and 'u  is the velocity 
perturbation from the mean free-stream velocity (u).  The diffusion coefficient was determined using 
the method of least squares.     
 
4. RESULTS AND DISCUSSION 
 
There were several dye tracer injections in the CSW over a two month period during 2006.  All field 
experiments, regardless of plant density, had non-uniform flow conditions where the flow velocity 
was greatest in the middle of the cross-section and smallest at the sides, as seen in one example run 
in Figure 3a.  The non-uniform velocity is a result of a channel bottom that is deepest at the center 
and shallow at the edges.  Along the same cross-section, the dye concentration was measured and in 
general showed a distribution that was skewed to the left of the cross-section (Figure 3b).  The skew 
is most likely due to the lateral velocities and preferential stream paths.  The left tail of the 
concentration curve is cutoff due to the natural boundary condition of the wetland bank.  
Additionally, non-trivial amounts of plant debris (twigs and dead stems and stalks) were observed 
through the test area.  The plant debris was not removed as this is the “natural” system, however the 
debris created channels, or preferential paths, within the test area for the flow. 
 
-100 -80 -60 -40 -20 0 20 40 60 80 100
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
(y-ycenter)/dy
V
/V
m
ax
3a
-100 -80 -60 -40 -20 0 20 40 60 80 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(y-ycenter)/dy
C
/C
m
ax
3b
Figure 3a and Figure 3b: Raw data of an example run from the Villanova field experiment.  a) velocity 
measured across the downstream cross-section; b) Concentration of Rhodamine WT measured across the 
downstream cross-section. 
V
/V
m
ax
C
/C
m
ax
V
/V
m
ax
C
/C
m
ax
 
  
 
The dimensionless diffusion coefficient for the field experiments for the three plant densities 
are seen in Figure 1 and Table 1.  The field experiment had diffusion coefficients about one order of 
magnitude higher than Nepf’s, while of similar values to Serra’s results.  In the field experiment, for 
the two larger plant densities, the diffusion coefficient coincides with the total diffusion model of 
Nepf; the diffusion coefficient is greater than the total diffusion for the lowest plant density.  This 
result is counterintuitive since it is assumed laminar flow characteristics, that is mechanical 
diffusion mechanisms, dominate for the CSW and the diffusion coefficient would increase as plant 
density increased.   
The differences between the results of the laboratory and field experiments point to several 
major differences in the experiments.  The laboratory experiments were conducted in a smooth, 
rectangular channel with uniform velocities along the longitudinal axis, with a negligible 
contribution from the lateral velocity; the field experiment had an uneven channel bottom, non-
uniform longitudinal velocities and significant lateral velocities.  The lateral velocities, along with 
the plant debris, led to the dye preferentially moving to one area of the test section as opposed to the 
  6 
entire test section.  In fact, Koch and Ladd (1997) found that the drag of flow around the plant stems 
is not insignificant when the flow approaches the stem on an angle.  The bed drag associated with 
the plant debris may contribute to the turbulence and larger diffusion coefficients.  Lastly, the 
widely varying plant stem diameter (d = 1.52 ± 1.1 cm) may impact the flow’s turbulence level; as 
the stem diameter increases, the Reynolds number increases.  While, on average, the flow may be 
characterized as laminar, locally around larger plant stems the flow may actually behave turbulently.  
If the flow is turbulent, then total diffusion is due to the mechanical and turbulent diffusion.  In the 
field experiment, the flow may be laminar on average and so it is assumed mechanical diffusion is 
the governing diffusion mechanism.  However, there may be pockets of turbulent flow due to larger 
stem diameters, so both mechanical and turbulent diffusion will contribute to the total diffusion.  
This may be one explanation for the field experiment to fit better to the total diffusion model.  In the 
future, the calculation for Dy for field experiments should include lateral velocities and a more 
thorough estimation of bed drag coefficients for the diffusion model.   
 
0.131.462.83.2
0.10.675.23.2
0.10.759.33.2
0.10.494.76.8
0.10.6089.26.8
0.20.6079.26.8
0.00.666.66.8
0.30.162.510.2
0.200.469.510.2
0.401.973.010.2
Error 
(e)
Dy/ud
Reynolds 
Number
(Re)
Plant 
Density
(P%)
Table 1: Summary of Villanova field experiment results.  
The dimensionless diffusion coefficient is calculated by 
equation 5. 
 
 
 
 A final analysis was to compare the Re to the diffusion coefficients (Figure 4).  There is no 
significant correlation between Re and the diffusion coefficients for Nepf, Serra or Villanova’s 
results (r2 = 0.45, 0.12, 0.04, respectively).  However, Serra shows a positive relationship where the 
diffusion coefficient increases with Re; Nepf and Villanova show a slight negative relationship. The 
relationship between Reynolds number and diffusion coefficient is presently inconclusive.   
  7 
 
0 20 40 60 80 100 120
10-2
10-1
100
101
Re
D
y/u
d
 
 
Nepf
Serra
Field
Figure 4: Effect of the diffusion coefficient with Reynolds number.  Serra (circle) 
shows a positively correlated set of data, whereas the data for Nepf (triangle) and 
field (cross) do not show any significant correlation between the dimensionless 
diffusion coefficient and Reynolds number. 
D
y/u
d
 
 
5. CONCLUSIONS 
 
The diffusivity coefficients are an important part of the hydrodynamic properties of a stormwater 
wetland; they can help develop more sophisticated models for detention time and water quality.  
This field study compared its results to laboratory data produced by Nepf and Serra.  The diffusivity 
coefficients from Villanova’s calculations were in the same range or one order of magnitude higher 
than Serra and Nepf, respectively.  This indicates that the field conditions such as non-uniform and 
lateral flows, debris and bed shear stress effect diffusion and should be considered when computing 
the diffusivity coefficient.  A new model incorporating field conditions such as lateral flow 
velocities must be developed in order to determine the diffusion coefficients from field data.  
Additionally, the findings here may be used to improve existing hydrodynamic models’ ability to 
simulate the diffusion and mixing mechanisms in vegetated flows. 
 
 
ACKNOWLEDGEMENTS 
 
This work was partially funded by Villanova University’s Summer Research Fellowship and a 
Research Support Grant.  
 
 
  8 
REFERENCES 
 
Ayaz, F. and T.J. Pedley (1999) “Flow through and particle interception by an infinite array of 
closely-spaced circular cylinders”, European Journal of Mechanics B/Fluids, Vol 18, No. 2, 
pp. 173-196. 
Hill, R.J. and D.L. Koch (2002) “Moderate-Reynolds-number flow in a wall-bounded porous 
medium”, Journal of Fluid Mechanics, Vol. 453, pp. 315-344. 
Koch, D.L. and A.J.C. Ladd (1997) “Moderate Reynolds number flows through a periodic and 
random arrays of aligned cylinders”, Journal of Fluid Mechanics, Vol. 349, pp. 31-66. 
Leonard, L.A. and M.E. Luther (1995)  “Flow hydrodynamics in tidal marsh canopies”, Limnology 
and Oceanography, Vol. 40, No. 8, pp. 1474-1484.  
Nepf, H.M. (1999) “Drag, turbulence, and diffusion in flow through emergent vegetation”, Water 
Resources Research, Vol. 35, No. 2, pp. 279-289. 
Nepf H.M., J.A. Sullivan, and A. Zavistoski (1997) “A model for diffusion within emergent 
vegetation”, Limnology and Oceanography, Vol. 42, No. 8, pp. 1735-1745. 
Serra, T., H.J.S. Fernando, and R.V. Rodrígues “Effects of emergent vegetation on lateral diffusion 
in wetlands”, Water Research, Vol. 38, pp. 139-147.   
 


A diffusion model for low reynolds number flows through a constructed stormwater wetland

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A diffusion model for low reynolds number flows through a constructed stormwater wetland

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