In Fig. 4 the authors present run-up data computed with values of beach roughness 8,! = 4 mm and 4 cm. These values differ by an order of magnitude.
WAVE REFLECTION AND R U N - U P ON ROUGH SLOPES" Discussion by Manas K. Deb4 The paper addresses a very relevant problem in the area of wave run-up on natural beaches and seawalls. On the whole, the work is commendable, and in this spirit the writer would like to make some comments. 1. In Fig. 4 the authors present run-up data computed with values of beach roughness 8,! = 4 mm and 4 cm. These values differ by an order of magnitude. The figure indicates that the run-up R is insensitive to the magnitude of 8^, "in this range." The same figure also compares these numerical results with Ahren's laboratory data, which were derived with values of 8^ between 19 cm and 31 cm. Does this suggest that R is independent of the beach roughness? 2. Although the authors write that the choice of/' is expected to affect the computed results, the reason for choosing/' as 0.3 is not clearly stated in the paper. An elaboration on the selection of/' would be helpful to those who might want to adopt the approach described. 3. In connection with the sensitivity analysis, the authors suggest that a critical evaluation of the most suitable boundary condition must be carried out. The authors write that cnoidal and Stokes' second-order wave theories were used for values of the Ursell number Ur > 26 and Ur < 26, respectively. It is not obvious how the authors arrived at this particular criterion. Did they compare numerical run-up results derived with cnoidal waves as a boundary condition and numerical results derived with Stokes' waves with laboratory data and then decided when to use Stokes' waves or cnoidal waves? Since recent analytical results (22) suggest that on plane sloping beaches cnoidal waves run up higher than sinusoidal waves with the same offshore wave height and wavelength, it would be extremely helpful if the authors provided the details of their comparison. The writer wishes again to thank the authors for an interesting study of wave run-up. APPENDIX.
REFERENCES
22. Synolakis, C. E., Deb, M. K., and Skjelbreia, J. E. (1987). "On the anomalous behavior of the runup of cnoidal waves." Phys. Fluids.
Discussion by Costas Emmanuel Synolakis5 First, I would like to congratulate the authors for their attempt to solve the nonlinear shallow water wave equations and calculate run-up on rough slopes. Having used the same algorithm, I can attest to the effort necessary to produce the type of results presented in the paper. However, I have se"May, 1987, Vol. 113, No. 4, by Nobuhisa Kobayashi, Ashwini K. Otta, and Indarajut Roy (Paper 21520). 4 Res. Asst., School of Engrg., Univ. of Southern California, Los Angeles, CA 90089-0242. 5 Asst. Prof, of Civ. and Aerospace Engrg., School of Engrg., Univ. of Southern California, Los Angeles, CA 90089-0242. 139 Downloaded 28 Apr 2010 to 128.125.52.199. Redistribution subject to ASCE license or copyright. Visit
3.00
2.75
2.50
2.25
2.00
1.75
1.50
-c i=fcr 1.25
1.00
-INITIAL SHORELINE POSITION
0.75
0.50
1.5
0.5
2.0
FIG. 11. Run-Up of Uniform Bore with Initial Strength H'Id', = 0.1 up Sloping Beach; A * = 0.01, ht = 0.04
rious reservations on the application of their method to any but to the simplest run-up problems. In the spirit of advancing the contributions of the paper, I would like to pose the following questions. 1. In previous work, the Lax and Wendroff scheme for the wave evolution and the Hibberd and Peregrine shoreline algorithm have been used to calculate the evolution of bores once they have formed in the flow field. To my knowledge, it has never been demonstrated that it is possible to take a wave through breaking and then calculate its run-up. For plunging waves, the state-of-the-art numerical calculations of wave breaking do break down once the wave curls over into itself. For spilling waves, the numerical filtering used to "reduce numerical oscillations in the vicinity of the front" raises questions about the interpretation of the results. For example, is the shape of the surface profile at t = 5.0 in Fig. 2 physically correct? Have the authors been able to actually calculate the wave evolution through breaking? If this is the case, this is a significant breakthrough that deserves the presentation of detailed surface profiles just before and after breaking and comparisons with laboratory profiles to substantiate it. [Such detailed comparisons have been presented in (24) and (25)]. 140 Downloaded 28 Apr 2010 to 128.125.52.199. Redistribution subject to ASCE license or copyright. Visit
~~
3.00 r——^
j-
j — —
2.75-
2.50 2.252.001.75CO
8
'1-50-
^nM
1.25 -
*\
1.00/ 0.75 -
/ - — I N I T I A L SHORELINE POSITION
/
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/
0.25 r\t— u 0
V . y '
/ I O
S
I TO
I L5
_J_ Z O
I 2
FIG. 12. Run-Up of Uniform Bore with Initial Strength H'/d! Beach; A * = 0.01, At = 0.04
5
3.0
= 0.3 up Sloping
2. The authors use the Hibberd and Peregrine (7,10) algorithm and its derivative versions to calculate wave run-up. Although this algorithm works rather well when calculating the evolution of a wave up a sloping beach, when the wave reaches the shoreline, the algorithm reverts to a shoreline procedure whose details are very dependent on the particular wave running up the beach (24). Figs. 11, 12, and 13 illustrate this point. Fig. 11 shows the evolution of a uniform bore up a sloping beach using the original shoreline algorithm (7). Figs. 12 and 13 present numerical results derived using the same code as in Fig. 11 to calculate the evolution of stronger bores up the same beach; a conspicuous numerical instability appears near the shoreline tip in Fig. 12 and a more serious instability is seen evolving in Fig. 13. In a personal communication, Peregrine acknowledged that when calculating the run-up of "strong" bores, the shoreline algorithm was modified. In the modified procedure, when negative depths are predicted by the algorithm they are not suppressed. The authors also acknowledge that their shoreline procedure is "somewhat intuitive." The question then arises, whether a sensitive numerical procedure—where modifications in the procedure are necessary to account even for small changes in the boundary conditions—can be relied upon to produce consistent results. 141 Downloaded 28 Apr 2010 to 128.125.52.199. Redistribution subject to ASCE license or copyright. Visit
4.0i
1
1
1
1
1
1
1
1
1
1
1
1
r
3.5
3.0
•ie|=e
FIG. 13. Run-Up of Uniform Bore with Initial Strength H'/d', = 0.6 up Sloping Beach; Ax = 0.01, A? = 0.04
3. The authors indicate that the friction factor/' used in their numerical simulation has to be determined empirically. Based on a "sensitivity analysis" they propose a value of/' = 0.3. How was the sensitivity analysis performed? Is this value o f / ' that the value that produces the best comparisons with laboratory experiments? Does Fig. 4 suggest that the relative run-up does not depend on /'? 4. To evaluate the applicability of the model further, it would be extremely helpful to compare the numerical results of the authors with the prediction based on Stoa's work (23), which is widely used by the Corps of Engineers (21). Also, in order to evaluate the importance of scale effects, it would be helpful to quote the roughness used in the simulations in dimensionless form. 5. Given the need to calibrate the numerical model even in a laboratory setting, what is the justification for advocating its use to "supplement the present design practices" in the prototype? Again, I would like to emphasize that these comments are in no way intended to put down an otherwise interesting and significant attempt to understand wave run-up in a more realistic setting than elsewhere (25). In fact, were it not for the authors' comparisons with the laboratory data and their frankness in discussing the details of their work and its limitations, it would not have been possible to write this discussion. 142 Downloaded 28 Apr 2010 to 128.125.52.199. Redistribution subject to ASCE license or copyright. Visit
APPENDIX.
REFERENCES
23. Stoa, P. N. (1979). "Wave runup on-rough slopes." CETA 79-2, U.S. Army Corps of Engrs., Fort Belvoir, Va. 24. Synolakis, C. E. (1986). "The runup of long waves," thesis presented to the California Institute of Technology, Pasadena, Calif., in partial fulfillment of the requirements for the degree of Doctor of Philosophy. 25. Synokalis, C. E., (1987). "The runup of solitary waves." /. Fluid Mech., ASCE, 523-547.
Closure by Nobuhisa Kobayashi,6 Member, ASCE The writers would like to thank Synolakis and Deb for their interest in the paper and for their comments. The points raised by the discusser may be separated into: (1) Seaward boundary location outside the breakpoint; (2) shoreline computation; (3) friction factor/' and water depth 8^; and (4) further evaluation of the numerical model and practical applications. In the following, an attempt is made to clarify these points. It should be noted that Kobayashi and Greenwald (1986) critically reviewed the mathematical and numerical backgrounds of the model presented in the paper. SEAWARD BOUNDARY LOCATION OUTSIDE BREAKPOINT
As stated in the paper, Hibberd and Peregrine (1979) prescribed the value of a associated with an incident uniform bore, while Packwood (1980) used the measured water depth inside the breakpoint as input at the seaward boundary. These seaward boundary conditions are more rigorous mathematically than the approximate seaward boundary condition employed in the paper. However, these boundary conditions are too restrictive and not suited for practical applications. Moreover, Packwood (1980) noticed spurious long period oscillations in his computed results. Since existing wave theories such as cnoidal and Stokes wave theories can not describe the asymmetry of the wave profile about the crest inside the breakpoint (Flick et al. 1981), the seaward boundary needs to be located outside the breakpoint to specify the incident wave train using existing wave theories. Since incident design waves are generally specified, it is appropriate to specify the normalized incident wave train T|,-(f) at the seaward boundary and compute the normalized wave train T|r(0 reflected from the computation domain. Comparisons of the numerical model with available monochromatic tests have been made using cnoidal or Stokes second-order wave theory depending on whether the Ursell number Ur a 26 or Ur < 26. This criterion is reasonably well known as was used in Figs. 7—75 in the Shore Protection Manual (U.S. Army Coastal Engineering Research Center 1984), although use could also be made of higher-order Stokes wave theories or numerical wave theories such as that proposed by Rienecker and Fenton (1981). Any appropriate wave theory may be used to specify the periodic variation of t),(0 where the height and period of the normalized variation are unity. This procedure is approximate because existing nonlinear wave theories do not account for the effects of reflected waves. It should be mentioned that the open boundary condition 5
Assoc. Prof., Dept. of Civ. Engrg., Univ. of Delaware, Newark, DE 19716. 143 Downloaded 28 Apr 2010 to 128.125.52.199. Redistribution subject to ASCE license or copyright. Visit
