Closure to “Dispersion in Varying-Geometry Rivers with Application to Methanol Releases” by Mirmosadegh Jamali, Gregory A. Lawrence, and Kevin Mahoney
Combined Formulation To demonstrate the complementary nature of the two approaches, we propose
冉 冊
K* = 共1 − e−3x/Lt兲 1 +
May 2005, Vol. 131, No. 5, pp. 390–396.
DOI: 10.1061/共ASCE兲0733-9429共2005兲131:5共390兲 1
2
Gregory A. Lawrence, M.ASCE ; Mirmosadegh Jamali ; and Kevin Maloney3 1
Professor, Dept. of Civil Engineering, Univ. of British Columbia, Vancouver BC, Canada V6T 1Z4. E-mail:
[email protected] 2 Associate Professor, Dept. of Civil Engineering, Sharif Univ. of Technology, P.O. Box 11365-9313, Teheran, Iran. E-mail: jamali@ sharif.edu 3 Infrastructure Manager, Emerging Markets, Methanex Corp., Vancouver, Canada. E-mail:
[email protected]
x L
c
共3兲
where we have set Lt = 3xD. This combined formulation reduces to Eq. 共1兲 in the far field and Eq. 共2兲 in the near field. Eq. 共3兲 contains both the length scale for variations in river properties, L, and the transverse mixing distance, Lt. Both the authors and the discusser use the same relationship to estimate the transverse mixing distance, namely Lt = 10␣
w2o do
共4兲
Substituting from Eq. 共3兲 of the original paper into Eq. 共4兲 gives Lt = 2650␣Q0.495 0
Introduction The authors thank the discusser for his interest in our paper. He has drawn attention to a number of apparent contradictions that stem from the fact that our paper focuses on longitudinal dispersion in the far-field, whereas Beltaos 共1980兲 deals with the near field. In this closure, we show that by combining our approach with that of Beltaos 共1980兲 the apparent contradictions disappear. The resulting formulation agrees surprisingly well with Jobson’s 共1996兲 data set, which is considerably more extensive than the data sets that were available to Beltaos 共1980兲. To illustrate the importance of variations in hydraulic geometry on longitudinal dispersion in the far field, we substituted hydraulic equations derived from the data of Keup 共1985兲 into an expression for the longitudinal dispersion coefficient from Fischer et al. 共1979兲. The resulting Eq. 共9兲 of the original paper can be written in the form
冉 冊
K* = 1 +
x c , L
x ⬎ Lt
共1兲
The nondimensional longitudinal dispersion coefficient K* = K / K0, where K0 is obtained by evaluating Eq. 共8兲 of the original paper at the spill location; L is the representative length scale for variations in river properties given by Eq. 共6兲 of the original paper; Lt is the transverse mixing distance 共mixing length兲; and Eqs. 共5兲 and 共8兲 of the original paper yield c = 1.08. Clearly, the above idealization is not appropriate for all rivers; nevertheless, it serves as a useful starting point for an analysis of the effects of variations in hydraulic geometry on longitudinal dispersion. For longitudinal dispersion in the near field Beltaos 共1980, Eq. 15兲 proposed Eq. 共25兲 共“Discussion” by Beltaos, this issue兲, which yields K* = 共1 − e−x/xD兲
共2兲
where the dispersion scale, xD ⬇ Lt / 3. Note that Beltaos 共1980兲 assumes constant river geometry, corresponding to c = 0 and L = ⬁, for which Eq. 共1兲 yields K* ⬇ 1 in far field 共x ⬎ Lt兲, which is the same as predicted by Eq. 共2兲 in the far field. Thus, the two formulations are complementary rather than contradictory.
共5兲
and substituting from Eq. 共6兲 of the original paper into Eq. 共5兲 gives Lt 0.217␣ = 0.082 L Q0
共6兲
which, if we set ␣ = 3, is equivalent to the result quoted in the last paragraph of the discussion 共assuming that Lt = 3xD兲. Thus, the discusser is using ␣ = 3; whereas Fischer et al. 共1979兲 recommend ␣ = 1 / 6 for a midriver spill and ␣ = 2 / 3 for a bank spill. A discussion of the appropriate values to use for ␣ is beyond the scope of this closure. To provide consistency with the discussion, we set ␣ = 3. Values calculated for L and Lt for a range of values of Q0 are given in Table 1. The predicted variation of K* with x / L for each of the three approaches outlined above is plotted in Fig. 1. The combined Eq. 共3兲 asymptotes to Eq. 共2兲 at low values of x / L and to Eq. 共1兲 at high values, as we would expect. We therefore hypothesize that the combined equation is applicable in both the near and far fields.
Comparison with Field Measurements To test the preceding hypothesis, we compare the predictions of Eq. 共3兲 with the data set compiled by Jobson 共1996兲. This data set is an extension of the compilation of 51 dye injections by Nordin and Sabol 共1974兲, to which Jobson 共1996兲 added results from a further 58 injections to create an impressive data set. The discusser expresses concern that many of the tests may not extend into the far-field. Although this may be true of some of the tests compiled by Nordin and Sabol 共1974兲, all the tests added by Jobson 共1996兲 appear to extend well into the far field. Some of the more notable tests are listed in Table 2. Consider the injection into the Sabine River: the peak of the dye cloud passed the first sampling point 17 km downstream after 67 h and passed the final sampling point 121 km downstream after 303 h. Between the first and last sampling points, the discharge increased from 1 m3 / s to 6.9 m3 / s. An injection into the Kaskasia River passed the first sampling point 3.2 km downstream after 4.4 h and passed the final sampling point 60 km downstream after 62 h. The flow increased from 0.12 m3 / s at the first sampling point to 8.6 m3 / s at the last sampling point, a 72-fold increase.
1094 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2007
Table 1. Comparison of Lt and L as a Function of Q0 Assuming ␣ = 3 Q0 共m3 / s兲
0.01
0.1
1
10
100
1000
Lt 共km兲 L 共km兲 Lt / L
0.81 0.86 0.95
2.5 3.2 0.79
8.0 12.2 0.65
25 46 0.54
78 174 0.45
243 657 0.37
To enable comparison with the data of Jobson 共1996兲, we use Eqs. 共16, 21兲 of the original paper to obtain ¯兲 = Cup共x
106
冑4
¯兲 u共x
冑冕
¯x
0
K共x兲 dx u共x兲
共7兲
where ¯x⫽location of the concentration peak, and Cup⫽unit peak concentration 共Jobson 1997兲. Predictions of Cup for Q0 = 10 m3 / s are plotted in Fig. 2 together with the field data. In general the near-field formula of Eq. 共2兲 matches the data in the near field, the far-field formula of Eq. 共1兲 matches the data in the far field, and the combined formula of Eq. 共3兲 matches the data in both the near field and far field, as intended. The only exception is that after about 100 h, the measured concentrations drop more rapidly than predicted. One possible reason for the disparity after 100 h is that, in general, the recovery ratio drops the longer that an experiment has been running 共Jobson 1996兲. If we adjust the measurements of Cup to account for the imperfect recovery, the
Fig. 2. Comparison of data of Jobson 共1996兲 with the predictions obtained by using the near-field, far-field, and combined formulas
comparison is better 共Fig. 3兲. The curve calculated by using the combined formula more closely matches the adjusted data cloud. There is room for further improvement, for instance, increasing the value of the exponent c will improve the fit at later times but will not affect the fit at earlier times. It should also be noted that the calculations have been performed for Q0 = 10 m3 / s; whereas in the experiments, Q0 varied over a very wide range. However, further calculations 共not shown兲 reveal that the results are relatively insensitive to variations in Q0. Of course there are serious limitations to a formula that makes predictions that are based solely on the elapsed time since an injection. The scatter about the prediction is large, as it should be. No attempt has been made to account for the effects of dead zones, channel curvature, bed material, bed slope, or any number of other factors that could affect dispersion. Also, while the formulation based on Keup’s 共1985兲 data provides a useful guide for assessing the effects of changes in hydraulic geometry on dispersion, the hydraulic geometry of each river is different and should ideally be calculated separately. Finally, the expression of Fischer
Fig. 1. Variation of K* with x / L for ␣ = 3 Table 2. Summary of Selected Tests of Jobson 共1996兲 River
Inj #
a
t1 tn Q1 Qn x 1b x nb 共km兲 共km兲 共hour兲 共hour兲 共m3 / s兲 共m3 / s兲
Apple 84 1.9 58 3.7 106 0.59 3.1 Bayou Bartholomew 21 3.2 118 5.8 234 4.1 8.1 Elkhorn 89 1.6 65 3.3 87 0.18 1.9 Kaskaskia 94 3.2 60 4.4 62 0.12 8.6 Sabine River 33 17.1 121 67 303 1.0 6.9 Shenandoah 70 2.3 95 4.0 193 1.0 10.6 Shenandoah 73 12.7 131 17 292 9.5 18.1 a The injection number corresponds to that assigned in Jobson 共1996兲 b The subscripts 1 and n refer to the first and last sampling stations, respectively
Fig. 3. Comparison of data of Jobson 共1996兲, adjusted by the recovery ratio, with the predictions obtained by using the near-field, farfield, and combined formulas
JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2007 / 1095
xD =
␣ w 2u 3 du*
K=
3␥ ux ␣
into Eq. 共9兲 yields 共10兲
which is equivalent to the relationship proposed by Hunt 共1999兲 with =
Fig. 4. Variation of the parameter  with x / L for the three formulations
et al. 共1979兲 for the longitudinal dispersion coefficient is very approximate, as is the assumption that u* = 0.1 u. Given all these limitations, it is surprising that the combined formulation matches Jobson’s 共1996兲 data as well as it does.
Variation of the Decay Exponent  The variation of the exponent  in Eq. 共22兲 of the original paper as a function of x / L is plotted in Fig. 4 for each of the formulations for K*. Note that the symbol  is used for a different parameter in Beltaos 共1980兲. This figure illustrates the apparent contradiction highlighted by the discusser, that the near-field formula of Beltaos 共1980兲 yields  increasing from 0.5 to 1, whereas the far-field formula proposed by the authors does the opposite. However, no contradiction occurs if these formulas are applied in their appropriate range. The combined formula provides a smooth transition between the near and far-field formulas. With the combined formula,  drops from 1 in the near field, to a value between 0.5 and 1 in the intermediate field and then rises toward =
1 + c − 3e 2共1 − e兲
共8兲
in the far field, where e is the exponent for u in Eq. 共5兲 of the original paper. Substituting c = 1.08 and e = 0.22 in Eq. 共8兲 gives  = 0.91.
Near-Field Comparisons For x Ⰶ xD, both Eqs. 共2兲 and 共3兲 yield the following expression for the dispersion coefficient: K=
K0 x xD
Substituting K0 = ␥ and
w 2u 2 du*
共9兲
3␥ ␣
共11兲
Hunt 共1999兲 has calculated from the data of Day 共1974, 1975兲. The estimates range from 0.044 to 0.064 共Bealey River兲, from 0.012 to 0.086 共Bruce River兲, and from 0.010 to 0.116 共Thomas River兲. Using ␥ = 0.011 共Fischer et al. 共1979兲 and ␣ = 3 gives = 0.011 which is at the low end of the values obtained by Hunt 共1999兲. Nevertheless, these results are encouraging since ␣ and ␥ are by no means precisely known and almost certainly vary from river to river, as well as with time and position in a given river.
Conclusions We have developed and tested a formulation for the variation in longitudinal dispersion that resolves the apparent contradictions raised by the discusser. We have combined the near field formulation of Beltaos 共1980兲 with our far-field formulation. The combined formulation provides a good fit to the extensive data set of Jobson 共1996, 1997兲, which is notable since the predictions are based on the empirical formulas derived from Keup 共1980兲. The combined formulation clearly shows that there are two reasons that previous investigators have not been able to reconcile field data with theories that are based on a constant longitudinal dispersion coefficient. The first, emphasized by the discusser, is that in the near field the flow is not one-dimensional, so there is no reason to expect a constant coefficient; the second, the focus of the authors’ investigation, is the downstream variation of river properties.
References Beltaos, S. 共1980兲 “Longitudinal dispersion in revers.” J. Hydr. Div., 106共1兲, 157–172. Day, T. J. 共1974兲. Dispersion in natural channels, Ph.D. thesis, Dep. of Geography, Univ. of Canterbury, Christchurch, New Zealand. Day, T. J. 共1975兲. “Longitudinal dispersion in natural channels.” Water Resour. Res., 11共6兲, 908–918. Fischer, H. B., List, E. J., Koh, R. Y. C., Imberger, J., and Brooks, N. H. 共1979兲. Mixing in inland and coastal waters, Academic, San Diego, Calif. Hunt, B. 共1999兲. “Dispersion model for mountain streams.” J. Hydraul. Eng., 125共2兲, 99–105. Jobson, H. E. 共1996兲. “Prediction of travel time and longitudinal dispersion in rivers and streams.” Water-Resources Investigations Rep. 894133, U.S. Geological Survey. Jobson, H. E. 共1997兲. “Predicting travel time and dispersion in rivers and streams.” J. Hydraul. Eng., 123共11兲, 971–978. Keup, L. E. 共1985兲. “Flowing water resources.” Water Resour. Bull., 21共2兲, 291–296. Nordin, C. F., Jr., and Sabol, G. V. 共1974兲. “Empirical data on longitudinal dispersion in rivers.” Water-Resour, Investigations, 20–74, U.S. Geological Survey.
1096 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2007
