C193a1 001205 this version not final

 

Comparing the RUNOFF kernel to laboratory experiments, including super-critical laminar flows.

William James

 

An attempt is made to validate the runoff kernel algorithm in SWMM against earlier findings on a laboratory-scale pavement. PCSWMM is used to calibrate the kernel and generate plots that are compared to the experimental results obtained from a laboratory rig described earlier.  The pavement was 100% impervious, with a 0.025 m/m slope over an area of 2.11 m2 and 1.171 m wide.  A small time step of 1 s was used. It is shown that supercritical laminar flow cases are included.

 

INTRODUCTION

 

In this paper the rain-runoff algorithm which forms the kernel of the RUNOFF module of SWMM is briefly reviewed, and then run against data obtained by Johanson (James and Johanson, 1999), since the data from the landmark studies by Izzard (Izzard, 1944, 1945; Izzard and Augustine, 1943) were not as readily available. The laboratory rig used is also described in an earlier monograph in this series (James et al., 1999). PCSWMM is used to calibrate and generate plots from the kernel routine. A manual method of storage routing with time lag (Falk and Niemczynowicz, 1979) is then used to demonstrate that the cases examined include super-critical laminar flow (a condition not reported elsewhere in the literature).
 
Overland flow routing in SWMM

This part has been abstracted from the SWMM-RUNOFF documentation (Huber et al., 1988).  Only the mathematics of overland flow routing is reviewed - snowmelt, evaporation, infiltration, and groundwater are not discussed in this chapter. 

In SWMM-RUNOFF all subcatchments are assumed to be rectangular and each subcatchment is schematized so that three or four subareas (depending on whether snowmelt is simulated) are used to represent different surface properties.  The slope of the idealized subcatchment is in the direction perpendicular to the width.  Flow from each subarea moves directly to an outlet and does not pass over any other subarea.  (Thus, it is not possible to route runoff from roofs over lawn surfaces, for instance). Subcatchments are subdivided into three subareas that simulate impervious areas with and without depression storage, and pervious area (with depression storage and infiltration).  These are areas A1, A3, and A2 respectively in Figure 1. The width of the pervious subarea, A2, is the entire subcatchment width, whereas the widths of the two impervious subareas, A1, and A3 are in proportion to the ratio of their area to the total impervious area, as indicated in Figure 1.  Of course, real subcatchments seldom exhibit the uniform rectangular geometries shown in Figure 1.  In our case, however the laboratory pavement met these conditions, corresponding to subarea A3.

 

Figure 1  Subcatchment schematization.  Each subarea flows instantaneously to an outlet node.  Flow from one subarea is not routed over another. (Adapted from Huber et al., 1988).  (Note: Figures are appended in order at the end of this paper)

Overland flow is generated from each of the three subareas by approximating them as non‑linear reservoirs, as sketched in Figure 2.  The width (and the slope and roughness) may generally be considered calibration parameters, but this approach was not adopted in this study.  Instead the initial abstractions and Manning’s n were used.

The non‑linear reservoir concept used in the kernel is established by coupling the continuity equation with Manning's equation (Huber et al., 1988).  Finally an equation for dd/dt is solved at each time step by means of a simple finite difference scheme.  For this purpose, the net inflow and outflow on the right hand side of the equation must be averaged over the time step.  The rainfall excess i* is given in the program as a time step average.  The average outflow is approximated by using the average between the old and new depths. 

 

Figure 2  Overland flow concept - depression storage is averaged over the area. (Adapted from Huber et al., 1988). 

 

If subscripts 1 and 2 denote the beginning and the end of a time step, respectively, the equation is:

 

                                        [1]

where

                           d = water depth, m,

                            t = time step, s.

                          i* = rainfall excess,

                            = rainfall/snowmelt intensity -

                                 evaporation/infiltration rate, m/s

                          dp = depth of depression storage, m

                                                                       [2]

                          A =    surface area of subcatchment, m2,

                         W =    subcatchment width, m,

                         cm =    unit conversion constant

                               =    1 for SI (m-s) units

                               = 1.49 for U.S.  Customary (ft-s) units

                           n = Manning's roughness coefficient,

                           S = subcatchment slope, m/m.

 

The subcatchment outflow rate Q (m3/s) is generated using Manning's equation:

 

                                                                [3]

 

 

VALIDATION TESTS OF SWMM-RUNOFF

 

An attempt was made to validate the above kernel against the findings of the previous sections.  Four tests were run and plots were generated, calibrated, and compared to the experimental results using a laboratory rig described earlier (James and Johanson, 1999; James and Wylie, 1999; James et al., 1999).

Input data files

The input files were constructed based on the conditions applicable to overland flow in the laboratory rig.  The pavement was 100% impervious, with a 0.025 m/m slope over an area of 0.000211 ha, and 1.171 m width.  A small time step of 1 s was used since the flow rates were expected to be small.  Table 1 outlines the rainfall parameters used in the simulation. 

Each test was run in PCSWMM98, then overlaid upon the observed hydrograph points of Johanson (1967).  Tests 1 and 2 required calibration to better fit the data.  The difference was determined between observed flow rate and calculated flow rate, which was then applied to the rainfall intensity input in RUNOFF.  SWMM-RUNOFF was re-run and results compared again.  The resulting peak flow rates differed by less than 1% in each run.

 

Table 1  Summary of SWMM-RUNOFF rainfall parameters.

 

Parameter

Test #1

(Fig. 3)

Test #2

(Fig. 4)

Test #3

(Fig. 5)

Test #4

(Fig. 6)

Intensity

(mm/h)

117.5

120.3

a)      82.5

b)      121.2

a)      122.1

b)      83.1

Duration

(s)

120

60

a)      60

b)      60

a)      60

b)      60

Initial rainfall, a, is followed by b, at given intensity and duration.

 

Results

Figures 3 to 6 represent the results of the SWMM-RUNOFF runs for each test.  Each figure represents the generated SWMM-RUNOFF hydrographs superimposed on the original observed hydrographs of Johanson (1967).  It can be seen that SWMM-RUNOFF calculates similar peaks, but different overall hydrograph shapes from those observed.  Thus, validation of the program is not accurately attained.  Recall that the program does not account for induced upper surface shear stresses due to rain, rain induced turbulence, transition to laminar flow, induced lower surface shear stresses due to the transverse momentum of infiltration, or transition between super- and sub-critical flow.

 

Figure 3  Long-duration comparison.

 

Figure 4  Short-duration comparison.

 

Figure 5  Variable increasing rainfall comparison.

 

Figure 6  Variable decreasing rainfall comparison.

STORAGE ROUTING WITH TIME LAG

 

A storage-routing model was developed by Falk and Niemczynowicz (1979) and fitted to their observations of rain on and runoff from thirteen small paved catchments in the city of Lund in Sweden.  They introduced a time lag t into the dynamic equation:

 

                                                                                     [4]

where:

                           S  = storage volume (mm)

                          Q  = outflow (mm×hour-1)

              k, n, and t  = model parameters

 

The process uses a step by step procedure (Falk and Niemczynowicz, 1978).  To determine the model parameter t one must plot various relationships between runoff and storage for different values of t and n. The values chosen are those that yield the least hysteresis effect (where values of runoff are different for the rising and falling limb of the hydrograph, for the same storage).  The continuity equation is:

 

                                                                                       [5]

where:

                           I  = rain intensity, mm×hour-1

 

To give the catchment water budget for the time step (t-2t) to (t-t), equation 5 is changed to the discreet form:

 

                                     [6]

 

Qt can be determined by substituting the value of St-t  (the only unknown variable in equation 6) by the storage calculated in Equation 10.  By expressing Q and I in mm×hour-1 and S in mm, the equation becomes:

 

                           [7]

 

Falk and Niemczynowicz (1979) cite the following example:

Assume the following values for parameters k and n have been found for a specific catchment:

                                                 k = 0.215

                                                 n = 0.667

Knowing the rainfall intensities, the following steps will determine Qt :

Choose an appropriate time step, t, say one minute.

Period 1 min:

                                     

Period 2 min:

                        

Period 3 min:

                 

The following equations can be used to determine and :

                                                                                [8]

                                                                              [9]

The two storage terms, DS and St , can be determined using the following equations:

                                                                                       [10]

                                                                                 [11]

Table 2 shows calculations for storage St  for the first 5 of 20 minutes.

Table 2: Storage and runoff calculations (first 5 minutes only).

Time

Intensity

I bar

Qt

Q bar

delta S

St

min

mm/hour

mm/min

mm/hour

mm/min

mm

mm

0

0

0

0.00

0.00

0.00

0.00

1

60

0.5

0.00

0.00

0.50

0.50

2

60

1

3.54

0.03

0.97

1.47

3

120

1.5

17.86

0.18

1.32

2.79

4

60

1.5

46.71

0.54

0.96

3.75

5

0

0.5

72.80

1.00

-0.50

3.26

 

 

 

 

 

 

 

 

Shown in Figure 7 is the rating curve, the plot of storage (mm) vs. outflow (mm×hour-1) for their example.  The net rainfall used in this model is obtained from observed rainfall for the specified area by subtracting losses.  In their catchment the water stored in depression storage is the only loss.  They assume that only after the depression storage is filled can runoff occur - the model does not incorporate other losses such as evaporation or infiltration.  By choosing Dt = t =1 minute, their method is evidently equivalent to using backward differences for storage, central differences for outflow, and t = 0.

 

Application to our experimental results:

We now apply the method to the observations of Johanson (1967) that we have used above.  Storage is computed in two ways: the first is the same cumulative method as in the Falk and Niemczynowicz example, the second uses equation 7 directly (k is obtained from the slope of the outflow vs. storage plot, created using the storage determined by the cumulative method).  For the second method, a value of n is chosen.  The two storage values calculated are compared by determining the percent difference, given by equation 12. 

                                   [12]

The value of n is changed in the second method, and the new computed storage is again compared with the cumulative storage.

Figure 7  Rating curve for Falk and Niemczynowicz example.

Figure 8 depicts the rating curve for the cumulative storage vs. outflow for Johanson’s experimental data shown in Figure 9.  Table 3 compares the two storage methods for the first 100 of 200 seconds.  Results indicate that n=1 is better than n=0.667. The table also facilitates depiction of the flow transitions during shallow overland flow on pavements.

Figure 8  Rating curve for Johanson’s data.

DETERMINATION OF THE REYNOLDS AND THE FROUDE NUMBERS

 

The tables of computations above allow further investigation of the concepts and effects of flow transitions discussed in the previous sections. Both the Reynolds number and the Froude number were calculated:

 

                                                                         [13]

 

                                                                           [14]

where

                          V = runoff velocity (runoff flow/runoff area) m/s

                           d = storage depth (m)

n   = kinematic viscosity (m2/s)

g = gravitational acceleration (m/s2)

 

Figures 9 and 10 show plots of the Reynolds number and the Froude number for the rainfall input used in Johanson’s data.  In Figure 9, the Reynolds number indicates laminar flow, while the Froude number indicates sub-critical flow.  This is expected due to both the low volume and the low velocity of the runoff.

 

 

Table 3  Calculations of storage for Johanson’s data (first 100 secs).

Input

Parameter n = 1

Parameter n = 0.667

1

2

3

4

5

6

7

8

9

0

0

0.00

0.00

0.00

0.00

0.00

0.00

0.00

5

120.78

0.69

0.08

0.01

7.64

0.08

0.01

7.56

10

120.78

0.69

0.25

0.01

24.32

0.25

0.01

24.23

15

120.78

0.69

0.42

0.01

41.00

0.42

0.01

40.91

20

120.78

20.44

0.57

0.20

36.56

0.57

0.07

49.52

25

120.78

61.33

0.68

0.61

6.77

0.68

0.16

52.53

30

120.78

91.63

0.74

0.92

17.38

0.74

0.20

53.90

35

120.78

102.92

0.78

1.03

25.40

0.78

0.22

55.52

40

120.78

108.55

0.80

1.09

28.95

0.80

0.23

56.81

45

120.78

102.21

0.82

1.02

20.47

0.82

0.22

59.85

50

0.00

79.65

0.78

0.80

2.15

0.78

0.19

58.96

55

0.00

55.67

0.68

0.56

12.43

0.68

0.15

53.50

60

0.00

39.46

0.61

0.39

22.03

0.61

0.12

49.89

65

0.00

29.60

0.57

0.30

27.10

0.57

0.10

47.12

70

0.00

25.37

0.53

0.25

27.52

0.53

0.09

44.24

75

0.00

22.56

0.50

0.23

27.00

0.50

0.08

41.56

80

0.00

17.62

0.47

0.18

29.15

0.47

0.07

39.99

85

0.00

15.50

0.44

0.16

28.97

0.44

0.06

38.24

90

0.00

13.39

0.42

0.13

29.07

0.42

0.06

36.82

95

0.00

11.27

0.41

0.11

29.48

0.41

0.05

35.72

100

0.00

11.98

0.39

0.12

27.15

0.39

0.05

33.89

1: Time (s)                                2: Rainfall intensity (mm/hour)

3: Observed runoff (mm/hr)       4, 7: Accumulated storage (Falk and Niemczynowicz method)

5, 8: Computed storage              6, 9 % Difference between storage

 

 

Figure 9  Reynolds Number for Johanson’s data.

Figure 10  Plot Froude number for Johanson’s data.

Further work done by Johanson included varying the rainfall intensities during the same rainfall duration as shown in Figures 5 and 6.  Figures 11 and 12 show plots of the Reynolds and Froude numbers for the same increasing and decreasing rainfall intensities.  Table 1 provides the intensities and the durations for test numbers 3 and 4.

 

 

Figure 11  Re and Fr for Johanson’s increasing rain intensity data (test 3).

Figure 12  Re and Fr using Johanson’s decreasing rain intensity data (test 4).

In Figures 11 and 12, the Reynolds Number indicates laminar flow, whereas the Froude Number indicates super-critical flow.  Laminar super-critical flow evidently is not commonly reported in the literature.  It’s appearance here helps explain conditions giving rise to the anomalous hump.  Recall that, for a given flow condition, laminar flow is more efficient than turbulent flow.  Thus the flow will appear to accelerate at the cessation of rain, when the rain-induced turbulence disappears.  Calculated Froude numbers are high, but even higher Froude numbers are obtained at the tail of the runoff recession (not shown).

The rating curves for the varying rainfall intensity data are plotted in Figures 13 and 14.

Figure 13  Rating curve for Johanson’s increasing rain intensity data (test 3).

Figure 14  Rating curve for Johanson’s decreasing rain intensity data (test 4).

   How are these results to be interpreted? Thin sheet flow under light rain on a steep impervious asphalt road is shown in Figure 15. The RUNOFF kernel does not explicitly include mathematics for most of the processes evident in Figure 15, such as (1) induced shear stresses at the upper surface due to the vertical transfer of rain momentum, (2) turbulence induced by raindrop impacts, (3) flow transition to relatively efficient laminar flow, (4) induced shear stresses at the lower surface due to the transverse momentum of infiltration, (5) flow transition between super- and sub-critical flow. Flow transitions are  usually accompanied by unstable depths, such as those seen in Figure 15.

   Derived Reynolds Numbers were rather low, indicating laminar flow, whereas the simultaneous Froude Numbers were rather high, indicating super-critical flow.  A literature search was unsuccessful in finding other examples of laminar super-critical flow, which evidently is not commonly reported. 

   In terms of the SWMM-RUNOFF rain-runoff kernel algorithm, however, no particular significance is imputed to these revelations, other than that the performance of the kernel nevertheless remains reasonable.

 

CONCLUSIONS

 

This chapter has described an investigation into the performance of the rain-runoff kernel algorithm in SWMM-RUNOFF. An input datafile was run to replicate earlier findings on a laboratory-scale pavement. PCSWMM is used to calibrate the kernel and generate plots that are compared to the experimental results obtained from a laboratory rig described earlier.  The pavement was 100% impervious, with a 0.025 m/m slope over an area of 2.11 m2 and 1.171 m wide.  A small time step of 1 s was used.

Results demonstrated that SWMM-RUNOFF calculates similar peak flows, but different overall hydrograph shapes than those observed on the laboratory rig.  Validation of the program is not proven for the rising and falling limbs of the hydrographs. However readers should recall that the program does not account for induced upper surface shear stresses due to the vertical transfer of rain momentum, rain induced turbulence, flow transition to relatively efficient laminar flow, induced lower surface shear stresses due to the transverse momentum of infiltration, or transition between super- and sub-critical flow, usually accompanied by unstable depths.

Further calculations were carried out to elucidate these flow transitions. A storage-lag method of routing was used to derive information for calculating transient Reynold’s and Froude numbers for the overland flow process.  Derived Reynolds Numbers were rather low, indicating laminar flow, whereas the simultaneous Froude Numbers were rather high, indicating super-critical flow.  A literature search was unsuccessful in finding other examples of laminar super-critical flow, which evidently is not commonly reported.  In terms of the SWMM-RUNOFF rain-runoff kernel algorithm, however, no particular significance is imputed to these revelations, other than that the performance of the kernel nevertheless remains reasonable.

 

REFERENCES

 

Falk, J. and Niemczynowicz, J. 1979. Modelling of Runoff From Impermeable Surfaces. Report No. 3024 for Dept. of Water Resources Engn’rg, Lund Inst. of Tech., U. of Lund.

Huber, W.C., Dickinson, R.E, Roesner, L.A. and Aldrich, J.A.  1988.  Stormwater management model user’s manual, Version 4.  EPA/600/3-88-001, US Environmental Protection Agency, Athens, GA.

Izzard, C.F.  1944.  The surface profile of overland flow.  Trans.  Am.  Geophys.  Union, 25:959-968.

Izzard, C.F.  1946.  Hydraulics of runoff from developed surfaces.  Proc.  Highway Research Board.  26:129-150.

Izzard, C.F. and Augustine, M.T. 1943. Prelim. report on analysis of runoff from simulated rainfall on a paved plot. Trans. AGU. 2:500-509.

James, W. and Johanson, R.C.  1999.  A note on an inherent difficulty with the unit hydrograph method.  Chapter 1, New Applications in Modelling Urban Water Systems (monograph 7 in the series).  CHI, Guelph, ON.  ISBN 0-9697422-9-0.  pp.  1-21.

James, W. and Wylie, S.C.  1999.  Numerical techniques for overland flow from pavement.  Chapter. 5, Applied Modelling of Urban Water Systems (monograph 8 in the series).  CHI, Guelph, ON. ISBN 0-9683681-3-1. pp. 77-112.

James, W., Wylie, S.C. and Johanson, R.C.  1999.  A laboratory rig for testing runoff from paved surfaces.  Chapter 6, Applied Modelling of Urban Water Systems (monograph 8 in the series).  CHI, Guelph, ON.  ISBN 0-9683681-3-1. pp. 113-122.

Johanson, R.C.  1967.  System analysis of the rainfall-runoff process.  M.Sc.Eng.  thesis, University of Natal, Durban, South Africa.