# Surface Runoff in Watershed Modeling—Turbulent or Laminar Flows?

## Abstract

**:**

^{2}/s and bedslopes of 10%–66% with varying sand roughness depths that all flow depths were predicted by laminar flow equations alone and that equivalent Manning’s n values were depth dependent and quite small relative to those used in watershed modeling studies. Even for overland flow rates greater than those typically measured or modeled and using Manning’s n values of 0.30–0.35, often assumed in physical watershed model applications for relatively smooth surface conditions, the laminar flow velocities were 4–5 times greater, while the laminar flow depths were 4–5 times smaller. This observation suggests that travel times, surface storage volumes and surface shear stresses associated with erosion across the landscape would be poorly predicted using turbulent flow assumptions. Filling the flume with fine sand and conducting runoff studies, we were unable to produce sheet flow, but found that subsurface flows were onflow rate, soil depth and slope dependent and drainable porosities were only soil depth and slope dependent. Moreover, both the sand hydraulic conductivity and drainable porosities could be readily determined from measured capillary pressure displacement pressure head and assumption of pore-size distributions (i.e., Brooks-Corey lambda values of 2–3).

## 1. Introduction

#### 1.1. Experience with Tahoe Rainfall & Runoff Simulations

^{2}/s) implies average flow depths of less than 0.1 mm under both turbulent and laminar flow assumptions, respectively, or Reynolds numbers less than one. Similarly, results from a 1-m wide runoff simulator on a similar slope of nearly 52%, but at a much greater onflow rate of 3.4 Lpm for an average surface flow per unit width hundred times greater of 57 mm

^{2}/s results in a proportionately greater Reynolds number of 52 for laminar flow, while for turbulent flow it remains far smaller than one. In addition, from a runoff modeling perspective in the Tahoe Basin (e.g., Homewood Creek watershed, [6]), during a wet-year calibrated simulation, the maximum daily runoff value was about 19 mm during a spring rain on snowmelt day. The 19 mm runoff across the approximately square mile basin occurred within about a 10-hour period such that the average runoff rate was at most 2 mm/h implying an areal average Reynolds number of <0.1. This ~2 mm/h snowmelt rate is roughly consistent with, though smaller than rates of ~5 mm/h measured using artificial rain on snow events at Soda Springs just outside of the Tahoe Basin by [19]; however, the measured snowmelt rates are still well below those from rainfall simulations noted above. In the integrated physical modeling conducted by [1] in both a relatively steep forested watershed of Oregon and the mildly sloping rangeland Oklahoma watershed [20], overland flow depths outside of concentrated flow channels were of similar magnitudes, typically on the order of 0.1–1 mm. While these modeling flow depth estimates depend in part on use of the Manning’s equation, when combined with observations from rainfall/runoff simulations, they underscore the concept that overland flow rates are such that Reynolds numbers are at least an order of magnitude below the value of 500 often assumed to differentiate laminar from turbulent overland flow. While at relatively small slope gradients of <10%, the difference in predicted flow depths and velocities using either laminar or turbulent (Manning’s) flow equations are relatively minor, use of Manning’s equation still requires assumption of a flow-depth dependent roughness value ‘n’ in watershed modeling efforts. At the much greater slopes commonly encountered (10%–100%) in forest watersheds, use of Manning’s equation may result in over-estimation of the shallow flow depths and under-estimation of average velocities that in turn affect estimated flow travel times and the shear stresses associated with sediment detachment and-transport rates (erosion).

#### 1.2. Research Hypotheses and Objectives

- (a)
- determination of flow depths and Reynolds (
**Re**) numbers for a range of slopes and flow rates and planar surface roughness conditions, - (b)
- determination of fine-sand surface runoff, interflow and drainage rates for the range of slopes and flow rates considered in (a) in an effort to determine the combination of slope and onflow rates required to generate infiltration excess overland flow, and
- (c)
- development of a simple hillslope runoff-interflow model that includes basic soil hydraulic properties (effective conductivity and drainable porosities or yields) readily assessed in the field.

## 2. Experimental Methods

#### 2.1. Theory—Laminar & Turbulent Overland Flow Equations

_{o}/h = ρgh sin θ

_{o}is the maximum velocity such that the mean velocity, u

_{m}from the parabolic vertical velocity distribution is simply u

_{m}= (2/3)u

_{o}. Then, the total flow rate per unit width, Q, is then given by

_{m}h = (ρgh

^{3}/3μ)sin θ

**Re**= Qρ/μ.

_{m}= (ρgh

^{2}/3μ)sin θ ≅ (0.7524ρgh

^{2}/3μ)S

^{0.983}

^{1.667}S

^{0.5}

_{m}= (1/n)h

^{0.667}S

^{0.5}

#### 2.2. Experimental Apparatus and Measurement Methods

## 3. Results and Discussion

#### 3.1. Laminar or Turbulent Sheet Flows, the Distinction Appears to be Important

^{2}/s at 16 C (equivalent to

**Re**= 18). Note that at slopes less than about 40%, the dependence of mean velocity on slope is practically equivalent between the laminar and turbulent flow conditions, especially for n values between 0.03 and 0.04; however, at larger or very small (e.g., glass n < 0.01) n values, Manning’s equation underestimates, or overestimates, respectively, the velocities by a factor of 2–3 times at any slope. Also shown in Figure 4 and Figure 5, use of an n value of approximately 0.035 in the Manning’s equation at slopes less than ~40% is required to obtain flow depths and mean velocities roughly equivalent to that determined by the laminar flow equation. An even smaller ‘n’ value (~0.02) is required to match turbulent and laminar flow celerities (see Figure 3). The range of surface runoff rates considered and listed in Table 1 readily exceed those commonly found in the field (e.g., Tahoe Basin; [40]), or predicted in rainfall-runoff modeling, suggesting that assumption of laminar flow conditions for all but the defined channel flows are likely more appropriate in hillslope modeling efforts rather than assuming application of the Manning’s equation and guesstimation of ‘appropriate’ roughness ‘n’ values.

#### 3.2. Can Hillslope Drainage be Predicted from Simple Laboratory Measurements?

_{s}was ~900 mm/h; these values are consistent with other sands [44]. Andesitic and granitic soils in the basin are characterized as sands to loamy sands with relatively high hydraulic conductivity [7]. From RS studies, Grismer [40] measured surface soil effective hydraulic conductivities averaging about 70 mm/h for both soils, while field permeameter-based measurements (bore-hole methods) of saturated hydraulic conductivity, K

_{s}, were about an order of magnitude larger (700–900 mm/h).

_{s}of 1.7 m/h based on pore-size distribution considerations [44]. Despite efforts to pack the sandbox to a similar bulk density as the columns, steady subsurface flows in the sandbox experiments (e.g., onflow rate of 3 Lpm on a 36.6% slope) implied K

_{s}values greater than 5 m/h. This discrepancy was due in part to the lower bulk density of the fine sand that could be achieved in the sandbox (1590 kg/m

^{3}) as compared to the laboratory columns (1700 kg/m

^{3}), so a second set of column tests were conducted in which the sand was only loosely packed, which resulted in an equivalent pore-size distribution (~3.5) as the original tests, and a slightly greater porosity of 39%, but a smaller displacement pressure head of ~80 mm. The latter implied a K

_{s}of 6.6 m/h based on pore-size distribution considerations for fine sands [40]. This second set of columns resulted in more-or-less the same water contents as listed in Table 2 with the exception that all the water contents were shifted up such that for capillary pressure heads of 100 and 120 mm, for example, the water contents were 0.34 and 0.33, respectively.

^{2}/s), only surface runoff fingers developed on the sand surface; the fingers rarely extended more than 100–200 mm before submerging or branching and always remained behind the average subsurface wetting front. Formation of overland flow appeared to be related to aspects of the microtopography and wettability of the sand as well as, of course, its relatively large hydraulic conductivity. When a surface runoff finger reached the end of the sandbox, subsurface flows were already nearly equivalent to the onflow rate and the sand had destabilized within the box. At the steeper 51.8% sandbox slope, onflow rates less than 2 Lpm did not result in surface fingered flow, though at greater flow rates, incipient seepage faces appeared in the sand prior to the sand slumping in the box. As a result of the limited surface runoff achieved in any of the sandbox experiments, the research focus shifted towards developing the data necessary for the subsurface flow modeling effort of research objective (c) and determination of effective drainable porosities.

_{d}= 80 mm), the drainable volume prisms were calculated at the different bedslopes assuming originally saturated sand. These were converted to drainable porosities using the total sand volume and compared with those measured from the ratios of the sandbox total drainage volumes to sand volumes at different slopes as shown in Figure 10. As with the larger hydraulic conductivity associated with the smaller bulk density of the sand within the sandbox experiments as compared to that of the laboratory columns, the simple correction to the smaller displacement pressure head resulted in the predicted and measured drainable being practically equivalent to and having the same trends as the bedslope. Note also that in the sandbox experiments, consistent with the notion of equivalent saturated sand thickness from Figure 8, the drainable porosities of the 36.6% and 51.8% slopes are the same at ~27%, a value quite similar to the maximum predicted possible of ~29%.

## 4. Summary and Conclusions—Impacts on Watershed Modeling Efforts

^{2}/s) and bed slopes (10%–66%) with varying sand roughness depths, we found that all flow depths were predicted by laminar flow equations and that equivalent Manning’s n values were depth dependent and quite small as compared to those used in watershed modeling studies. For example, as indicated in Figure 3 and Figure 4, for a range of large overland flow rates and n = 0.35 typically assumed in the InHM model applications for relatively smooth surface conditions, the laminar flow velocities were 4–5 times greater, while the laminar flow depths were 4–5 times smaller as compared to those estimated from the Manning’s equation. This gross under-estimation of the overland flow velocities of course would result in an over-estimation of travel times for surface flows to reach defined channels and under-estimation of associated shear stresses for estimation of erosion rates using stream power concepts. Without the rainfall and runoff simulation to provide insight into actual flow depths and velocities, this work would not have been possible, which enables determination of the importance of clarifying flow regimes in watershed modeling.

## Conflicts of Interest

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**Figure 1.**Diagram of undisturbed steady laminar flow down an inclined plane with parameter definitions.

**Figure 3.**Dependence of the wave celerity on laminar or turbulent flow assumptions for 0.2 mm (

**a**) and 0.5 mm (

**b**) flow depth and Manning’s n = 0.30.

**Figure 4.**Dependence of overland flow mean velocities (Q = 20 mm

^{2}/s) on slope for laminar and turbulent (Manning’s equation) flow and different roughness values.

**Figure 5.**Dependence of the ratio of Manning’s number to laminar predicted flow depths (or velocities) on overland flow rate for a 20% slope and different roughness values.

**Figure 6.**Sandbox subsurface outflow rate at 2 m downslope as it depends on elapsed time after onset of onflow at rates listed for 20.6% bed slope and 143 mm depth sand. All tests under conditions of previous ‘natural drainage’ except as indicated for the ‘dry’ sand case at 2.9 Lpm.

**Figure 7.**Sandbox subsurface outflow rate at 2 m downslope as it depends on elapsed time after onset of onflow at rates listed for 51.8% bed slope and 143 mm depth sand. All tests under conditions of previous ‘natural drainage’.

**Figure 8.**Sandbox subsurface outflow rate as it depends on elapsed time after beginning at 2 m downslope for bed slope of 51.8% and 143 mm depth sand.

**Figure 9.**Time to maximum subsurface outflow rate as it depends on onflow rate for 143 mm depth sand at four different slopes.

**Figure 10.**Predicted and measured drainable porosities as they depend on bedslope for 143 mm sand depth.

Slope (%) | Q (mm^{2}/s) | Re | Predicted h_{laminar} (mm) | Bare Aluminum h_{meas} (mm) | #60 ^{1} sandpaper h_{meas} (mm) | Apparent n |
---|---|---|---|---|---|---|

10.7 | 26.7 | 24 | 0.44 | 0.43–0.45 | 0.41–0.44 | 0.024 |

37.5 | 34 | 0.49 | 0.48–0.50 | 0.46–0.49 | 0.024 | |

46.7 | 42 | 0.53 | 0.52–0.54 | 0.50–0.53 | 0.024 | |

57.2 | 51 | 0.57 | 0.56–0.58 | 0.54–0.57 | 0.023 | |

20.6 | 26.7 | 24 | 0.36 | 0.35–0.37 | 0.33–0.36 | 0.029 |

37.5 | 34 | 0.40 | 0.39–0.41 | 0.37–0.39 | 0.026 | |

46.7 | 42 | 0.43 | 0.42–0.44 | 0.4–0.43 | 0.024 | |

57.2 | 51 | 0.46 | 0.45–0.47 | 0.44–0.47 | 0.022 | |

36.6 | 26.7 | 24 | 0.30 | 0.29–0.31 | 0.27–0.30 | 0.033 |

37.5 | 34 | 0.33 | 0.32–0.34 | 0.30–0.33 | 0.029 | |

46.7 | 42 | 0.36 | 0.35–0.37 | 0.33–0.36 | 0.026 | |

57.2 | 51 | 0.38 | 0.34–0.39 | 0.35–0.38 | 0.023 | |

51.8 | 26.7 | 24 | 0.27 | 0.26–0.28 | NA ^{2} | 0.028 |

37.5 | 34 | 0.30 | 0.29–0.30 | NA | 0.027 | |

46.7 | 42 | 0.32 | 0.31–0.33 | NA | 0.025 | |

57.2 | 51 | 0.35 | 0.34–0.36 | NA | 0.023 | |

66.2 | 26.7 | 24 | 0.25 | 0.22–0.26 | NA | 0.031 |

37.5 | 34 | 0.28 | 0.27–0.30 | NA | 0.029 | |

46.7 | 42 | 0.31 | 0.30–0.32 | NA | 0.028 | |

57.2 | 51 | 0.33 | 0.32–0.35 | NA | 0.025 |

^{1}Results for finer 100 and 150 grit sandpaper were the same as those for bare aluminum;

^{2}Uniform width flow could not be established for the steep slopes on the sandpaper (i.e., fingered flow).

Capillary Pressure Head (mm) | Volumetric Water Content ^{1} (m^{3}/m^{3}) | Apparent Specific Yield (%) |
---|---|---|

10–50 | 0.40 ^{2} | 0 |

60–80 | 0.36 | 0 |

100 | 0.36 | 0 |

120 | 0.36 | 0 |

140 | 0.34 | 0.02 |

160 | 0.33 | 0.42 |

180 | 0.33 | 0.74 |

200 | 0.29 | 1.3 |

220 | 0.22 | 2.5 |

240 | 0.18 | 3.8 |

260 | 0.15 | 5.1 |

280 | 0.13 | 6.4 |

300 | 0.10 | 7.7 |

320 | 0.09 | 8.9 |

340 | 0.07 | 10.1 |

360 | 0.06 | 11.2 |

380 | 0.05 | 12.2 |

400 | 0.04 | 13.2 |

^{1}Five column average values at average bulk density of 1700 kg/m

^{3};

^{2}Exceeds average porosity and value reflects “free-water” in column ring.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Grismer, M.E.
Surface Runoff in Watershed Modeling—Turbulent or Laminar Flows? *Hydrology* **2016**, *3*, 18.
https://doi.org/10.3390/hydrology3020018

**AMA Style**

Grismer ME.
Surface Runoff in Watershed Modeling—Turbulent or Laminar Flows? *Hydrology*. 2016; 3(2):18.
https://doi.org/10.3390/hydrology3020018

**Chicago/Turabian Style**

Grismer, Mark E.
2016. "Surface Runoff in Watershed Modeling—Turbulent or Laminar Flows?" *Hydrology* 3, no. 2: 18.
https://doi.org/10.3390/hydrology3020018