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Verification of sediment transport rate parameter on cross-shore sediment transport analysis
ARTICLE IN PRESS
Ocean Engineering 34 (2007) 1096–1103 www.elsevier.com/locate/oceaneng
Verification of sediment transport rate parameter on cross-shore sediment transport analysis U. Tu¨rkera,, M.S. Kabdas- lıb a
Near East University, Department of Civil Engineering, Northern Cyprus, Mersin 10, Turkey Istanbul Technical University, Department of Civil Engineering, 80626 Maslak, I˙stanbul, Turkey
b˙
Received 15 April 2006; accepted 16 August 2006 Available online 3 January 2007
Abstract The sediment transport parameter helps determining the amount of sediment transport in cross-shore direction. The sediment transport parameter therefore, should represent the effect of necessary environmental factors involved in cross-shore beach profile formation. However, all the previous studies carried out for defining shape parameter consider the parameter as a calibration value. The aim of this study is to add the effect of wave climate and grain size characteristics in the definition of transport rate parameter and thus witness their influence on the parameter. This is achieved by taking the difference in between ‘‘the equilibrium wave energy dissipation rate’’ and ‘‘the wave energy dissipation rate’’ to generate a definition for the bulk of sediment, dislocating within a given time interval until the beach tends reach an equilibrium conditions. The result yields that empirical definition of transport rate parameter primarily governs the time response of the beach profile. Smaller transport rate value gives a longer elapsed time before equilibrium is attained on the beach profile. It is shown that any significant change in sediment diameter or wave climate proportionally increases the value of the shape parameter. However, the effect of change in wave height or period on sediment transport parameter is not as credit to as mean sediment characteristics. r 2006 Published by Elsevier Ltd. Keywords: Beach profile; Equilibrium; Grain size; Transport rate parameter
1. Introduction Change in beach profile due to cross-shore sediment transport is one of the most important factors while dealing with nearshore dynamics. Beach profile change is generally treated in the framework of different approaches depending on the professions, like geologists, oceanographers and coastal engineers. There are different cross-shore sediment models that are developed for modeling beach profiles changes (Kriebel and Dean, 1985; Watanabe and Dibajnia, 1988; Larson and Kraus, 1989; Steetzel, 1990; Tu¨rker and Kabdas-lı, 2004). Some of these models are based on analytical grounds while the others on empirical results from experimental studies. Roelvink and Broker (1993), describe the state-of-the-art in modeling of cross shore profile evolution and classified the available models as; Corresponding author. Tel. +90 392 22 36 464; fax: +90 392 22 35 427.
E-mail address: [email protected] (U. Tu¨rker). 0029-8018/$ - see front matter r 2006 Published by Elsevier Ltd. doi:10.1016/j.oceaneng.2006.08.002
descriptive models (Lippmann and Holman, 1990), Equilibrium profile models (Bruun, 1954; Dean, 1977), Empirical profile evolution models (Kriebel and Dean, 1985) and Process Based Models (Broker Hedegaard et al., 1992). Although the fundamental assumptions used in these models are reasonable, the empirical coefficients involved in them cover so many processes that empirical coefficients must be determined through calibration for a given site (Roelvink and Broker, 1993). In order to provide uniqueness to the developed models, a definite characterization for empirical coefficients is necessary, rather than calibration. Among all these cross-shore profile evolution models the model proposed by Kriebel and Dean proved to be useful in cases of sandy beaches and dunes, where longshore transport gradients can be neglected, and where the profile is able to reach equilibrium. The model, however, uses an empirical coefficient called transport rate parameter, K. This transport rate parameter primarily governs the time response of the beach profile. Smaller
ARTICLE IN PRESS U. Tu¨rker, M.S. Kabdas- lı / Ocean Engineering 34 (2007) 1096–1103
value gives longer elapsed time before equilibrium is attained, whereas large value produces more rapid evolution. In other words, smaller K-value implies flatter equilibrium beach profile. K-value for Work and Dean (1995), in their solution, describing the response of an evolving beach nourishment project, is given as 0.2 106 m4/N. Kriebel and Dean (1985) refers to the study carried by Moore (1982) suggesting that K may either be a constant or a function of other wave and sediment parameters. Moore (1982) found K to be constant, equivalent to 2.2 106 m4/N. According to Moore (1982), the parameter K should also vary with the length scale of the system. For their analysis in SBEACH, Larson and Kraus (1989) used the average value of the transport rate coefficient as 1.1 106 m4/N, which is approximately half the value originally obtained by Moore (1982). In their studies, while comparing 10 cross shore sediment transport models Schoonees and Theron (1995) simulated a value of 8.75 106 m4/N. As it can be concluded, in all the above mentioned studies, the transport rate parameter is obtained by field calibration and is not entirely comparable between the models, in view of the fact that the structures of the models are different from each other. In this study, an attempt is made to generate a ratio, representing difference between ‘‘the equilibrium wave energy dissipation rate’’ and ‘‘the wave energy dissipation rate’’ to the bulk of sediment dislocated in cross shore direction. The result is incorporated with physical model based in the wave energy dissipation rates, thus, the more accurate values for transport rate parameter for any given climatic and sediment condition can be predicted. 2. Theoretical approach Cross-shore sediment transport relationships are generally categorized into two models, ‘Closed Loop and Open Loop’. ‘‘Closed Loop’’ models are those where the sediment transport relationships could converge to a target equilibrium profile, while in ‘‘open loop’’ models there is no priority to achieve final equilibrium profile. One of the first closed loop transport relationships proposed was that of Kriebel and Dean (1985). They have hypothesized that the cross-shore transport rate depends on the deviation of the dissipation rate per unit volume from its equilibrium, and thus improved a time dependent model that converges to Dean’s equilibrium profile. This concept was extended to barred profiles by Larson and Kraus (1989) and Larson (1993). Based on equilibrium profile considerations, it is reasonable to express the offshore transport rate of sediments at any point in the surf zone, qs, in terms of the difference between the actual and equilibrium levels of wave energy dissipation due to wave break. qs ¼ KðDw Deq Þ
(1)
in which Dw and Deq are the actual and equilibrium energy dissipation rates per unit volume and K is the transport
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rate parameter. Theoretically, qs represents the cross-shore sediment transport load resulting from destructive forces during breaking process. Namely, the expected occurrence of sediment transport mechanism depends on the level of wave energy dissipation, which should be above some equilibrium threshold, where no net transport occurs if not overestimated. Thus, above equation can be simplified to reflect the relation between the rate of energy required for sediment dislocation due to wave breaking per unit volume, Ds, to the energy difference between the equilibrium wave energy dissipation rate and the actual wave energy dissipation rate, (Tu¨rker and Kabdas-lı, 2004). Ds ¼ Dw Deq .
(2)
As the equilibrium in beach profile is attained, Dw ¼ Deq the net sediment transport is accepted as zero, Ds ¼ 0. Nevertheless, during the evolution of beach profile, the intensive sediment transport in the area enclosed within breaking zone and the coastline gives birth to high erosion rates in onshore–offshore direction. One of these changes is defined as storm profile or winter profile, where dominant sediment transport is in offshore direction. This transport mechanism continues until all the sediment grains accumulate some point offshore, forming an offshore bar. In the evolution of above physical event, it appears that there are destructive and constructive forces that tend to transport sand offshore and onshore, respectively. Gravity is one obvious destructive force, and it is evident that constructive forces, bottom shear stress etc., exist to counteract these destructive forces; otherwise; beaches cannot cease the sediment transport and thus the beach profile changes infinitely (Dean, 1995). Consider a unit area on a beach slope, locally inclined at an angle, b to the horizontal. The intensive wave breaking process conveying a sediment transport dissipates energy. When a certain transitional stage of bed movement is exceeded the fluid flow is considered as unaffected by the existence of stationary boundary. The whole resistance to flow, in the two-dimensional case, is that exerted by solid phase. Bagnold (1963) demonstrates the tangential thrust stress required to maintain the sediment in motion. rs r gm cos bðtan f tan bÞ, rs
(3)
where m is the whole mass of the displaced sediment per unit area, rs and r is the density of sediment and pervading fluid respectively, j is the angle of repose, which is a measure of the mean angle of contact between the sediments grains. The product of the load of grains and the mean speed u¯ of their travel is the rate of mass transport of the load per unit width. The multiplication of the tangential thrust stress, by mean speed of sediment grains results in the work done by the fluid per unit time. Hence the necessary fluid power per unit of bed area in transporting the sediment
ARTICLE IN PRESS U. Tu¨rker, M.S. Kabdas- lı / Ocean Engineering 34 (2007) 1096–1103
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load can be given as rs r gm cos bðtan f tan bÞu, rs
By inserting Eq. (10) into Eq. (8), we get, (4)
where the mean speed is the ratio between the average distance traveled by the sediment particles and the time scale that the event takes place. u ¼ L=T n ,
(5)
L is the length of displacement of bulk of sedimentary particles in cross-shore direction. This length is defined in terms of wave and sediment characteristics, and the term T* is fictive period representing the time interval starting by the incipient motion of sediment from foreshore until it settles at offshore (Tu¨rker and Kabdas- lı, 2004). Inserting Eq. (5) into (4) represents the energy that is dissipated in unit time per unit of bed area in the process of transporting the sediment grains: E s ¼ ðrs rÞ
g L m cos bðtan f tan bÞ n . rs T
(6)
The volume of sediment particles (m/rs) is directly related with the fluid energy required to maintain the incipient sediment dislocation. L . (7) Tn In Eq. (7), gs and gw are the specific weight of sediment and fluid, respectively, and Vg is the volume of sediment grains. Perhaps, the total volume of sediment grains, transported in offshore direction, qualifies the volume of the offshore bar. Therefore, the volume of offshore bar Vbar, can replace the volume of grains. Vbar is the optimum volume of sand contained within the bar per unit length of beach formed after the storm. In this form, Eq. (7) represents the amount of energy per unit of bed area that the sediment particles should overestimate in order to move from one place to another to form offshore bar. The representation of the energy dissipation per unit volume in transporting the sediment grains, Ds, can be therefore defined as
E s ¼ ðgs gw ÞV g cos bðtan f tan bÞ
Ds ¼ V bar ðgs gw Þ cos bðtan f tan bÞ
L 1 . Tn h
(8)
In Eq. (8), h is the local water depth. In the derivation of Eq. (8), it is assumed that the sea state and other boundary conditions are stationary over the duration of a morphological time step. Remembering the relation given in Eqs. (1) and (2), the term representing the energy dissipation per unit volume can also be defined in terms of offshore sediment transport rate, and transport rate parameter: Ds ¼ qs =K.
(9)
In the above equation, the sediment transport rate is the volume of sediment, forming the offshore bar, dislocating within a given time interval defined as fictive period: qs ¼ V bar =T n .
(10)
1 (11) Ds ¼ qs ðgs gw Þ cos bðtan f tan bÞL . h Eq. (11) shows that the necessary energy dissipation for the displacement of sediment grains is directly proportional with the rate of sediment transport. At the same time, as given in Eq. (1), the rate of sediment transport is also proportional with the difference between actual and equilibrium energy dissipation rates per unit volume. Combining Eqs. (1), (2) and (11) and choosing the sediment transport rate as subject of the formula, the model lately proposed by Kriebel and Dean (1985) can be written as qs ¼
1 ðDw Deq Þ. ðgs gw Þ cos bðtan f tan bÞL 1h
(12)
In the former expression, the first term at the right side of the equation represents the so called transport rate parameter, K in terms of wave climate and grain size characteristics. K¼
hb . ðgs gw Þ cos bðtan f tan bÞL
(13)
Dally et al. (1984) in their study suggest that water depths at the break and at stable conditions are equal. Therefore, using this suggestion, the considered water depth at the location of the event can also assumed to be equal to the water depth at the break. The validity of Eq. (13) can be checked by the help of experimental studies where the parameters evolved in the above equation can be used to find representative values for K parameter. 3. Experimental testing Model studies are conducted at I˙stanbul Technical University, Hydraulics Laboratory. The aim is to investigate a representative transport rate parameter under the attack of regular and irregular wave climates. Glass wave flume with dimensions 24 1 1 m is used to generate irregular waves (Fig. 1). Irregular waves were generated by computer controlled wave maker capable to generate waves practicing Pierson–Moskowitz wave spectrum. The still water depth channel was 60 cm. During the studies three different grain sizes were used. The size of materials involved in the test was selected as 0.38, 0.5 and 0.7 mm. The slope of beach profile prior to each test was kept at 1:5. About 31 experiments for irregular waves were carried out. A wave monitor amplifier and a resistance type wave electrode were used to record wave data. The wave records, wave height and its period and average dislocation of sediment particles are then measured for each test. The results of experiments are summarized in Table 1. The analysis on wave data show that, for the irregular waves the highest one-tenth of wave height was dominating
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Fig. 1. Experimental setup of irregular wave channel.
Table 1 Laboratory results of irregular wave tests The code of sediment particles
Wave height H1/10 (cm)
Wave period T1/10 (s)
Length of displacement L (cm)
The mean diameter d50 (mm)
No of experiments accepted
A B C
12.0–16.0 11.0–17.0 12.0–16.0
1.2–1.9 1.5–2.0 1.3–1.9
85–112 76–115 75–93
0.38 0.50 0.70
11 9 6
the process. Perhaps, the reason for such a result lays in the fact that actually the offshore migration of bars is necessarily associated with break point processes. Thus, under random waves the small amplitude waves even cannot broke to dissipate their energy to initiate sediment motion. Finally, the experimental results are analyzed and considered. Whilst some of the experimental results are accepted as a good indicator for the sediment transport parameter relationship some did not comply with. Due to the unfavorable outcomes of five of these experiments, they are rejected from the analyses. Four of them were rejected from the experiments on 0.7 mm sized sediments whereas one of them was from 0.5 mm. The unfavorable results are mainly due to the defaults happened during the experimental studies, for example in two of the experiments carried over on 0.7 mm diameter sediments the electricity shortcut has shorten the timescale of the experiments. 4. Discussion and results Theoretically, an attempt is made to generate a definition representing the transport rate parameter in crossshore sediment transport studies. The resultant equation, Eq. (13), depends on specific weight, angle of repose, wave breaking depth and the length of displacement of sediment particles until the equilibrium conditions are maintained. The length of displacement of sedimentary particles are studied by Tu¨rker and Kabdas-lı (2004). In their results, it is shown that the average dislocation depends on both the wave climate and sediment characteristics. However, their theoretical definition involves too many limitations and variables, which are correlated with empirical definitions. Therefore, herein by using the
experimental results length of displacement of sedimentary particles is to be redefined. 4.1. Length of closure effect It is clear that all the experiments that practiced increment in wave energy dissipation rate are related with an increase in incoming wave height. As a result, convolution in offshore displacement of sediment particles consumes more time. The reason, without a doubt, is increase in displacement length of total volume of sediments while displacing offshore to form larger offshore bars. Furthermore, according to stability criteria as the size of sediment particles increases, necessary wave energy required to dislocate the particles also increases. Perhaps, the initial slope of beach profile also affects the natural evolution of the displacement length of sediment particles to form offshore bar. It is also evident from the experimental studies that the rate of profile evolution is decreasing consistent with an approach to equilibrium. The estimation of the length of displacement of sediment grains can be valid parameter for predicting cross-shore profile change up to which the seaward limit to which the depositional front has advanced. According to the previous studies there are too many recommendations for seaward limit of offshore depositions. Vellinga (1983) proposed the depth of closure as 75% of the deepwater significant wave height using linear wave theory. Based on Laboratory and field data Hallermeier (1978, 1981), developed a condition for sediment motion resulting from wave conditions. According to his studies the resultant equation is given as 2 H hc ¼ 2:28H 68:5 , (14) gT 2
ARTICLE IN PRESS U. Tu¨rker, M.S. Kabdas- lı / Ocean Engineering 34 (2007) 1096–1103
where H and T are significant wave height and significant wave period, respectively. Birkemeier (1985) evaluated Hallermeier’s relationship creating a simple approximation to the effective depth of closure providing a good fit to the data given as hc ¼ 1:57H.
dimensionless dislocation parameter;
H o =wT;
fall speed parameter;
H o gT 2 ;
Froude parameter:
10.0 8.0
(15)
Perhaps, the depth of closure is the best point to define the offshore limits of sediment disturbance due to wave action. However, during the offshore bar formation the depth of water above the offshore bar is not remaining constant. Consequently, the depth of water is not such a precise location to define the limiting depth of closure. Thus, it is necessary to define the depth of closure by another referring, which can be defined as length of closure. The necessary length of closure actually defines the overall length of displacement of sediment particles during the formation of offshore bar. The findings of the experiments that are carried over in I˙stanbul Technical University, Hydraulics Laboratory, concluded that in forming the equilibrium beach profile, the seaward limit of beach profile depends on wave height, wave length and its period; the size of sediment grains and the slope of beach profile. However, since in this study the beach profile slope is considered as constant, the results of experiments brings about uncertainty in the definition of seaward limit of beach profile. For sure, the consideration of the initial beach slope has unexpectedly failed to improve the accuracy of the analysis. Harmonizing the parameter which are effective in defining the seaward limit of beach profile results in three non-dimensional parameters: H o =L;
12.0
Λ/ H
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In some sense, fall speed parameter is both a measure of wave steepness and a sediment-related descriptor, indicating both profile response and geometric properties of various major morphologic features. It is also related to mobilization and suspension of sediment and performs well individually and in combination as criteria for predicting direction of cross-shore sediment transport. The strong relationship between wave and sand characteristics and morphologic features suggests the possibility of quantitatively predicting the beach profile response and evolution period (Larson and Kraus, 1989). The dimensionless dislocation parameter indicates the dominance of wave height on the morphological changes in seaward limitations of beach profile. The Froude parameter is useful in defining the dominancy of gravity forces, which in turn affects the evolution time (transport rate parameter) of equilibrium beach profile. The relationship among the dimensionless dislocation parameter, fall speed parameter and the Froude parameter for irregular wave analysis is given in the
y = -1.2808 Ln (x) - 8.74 2 R = 0.85
6.0 4.0 2.0
1.0E-07
1.0E-06
1.0E-05
0.0 1.0E-04
(H/gT2)2.68 (H /wT)-2.05 Fig. 2. Logarithmic variances in the definition of transport rate parameter.
following equation. The trend between the parameters is simulated in Fig. 2. ! L H 2:68 H 2:05 ¼ 1:28 ln 8:74. (16) H wT gT 2 The logarithmic relation between the dimensionless dislocation parameter and its empirical definition can be represented graphically on a semi logarithmic paper resulting with a coefficient of determination equivalent to 0.79. The quality of fit analysis is applied to the predicted equation and the experimental data. It is seen that the resultant empirical equation is fitting the data quite well. The quality of fit is determined through the following equation: 2 P Lm Lp x¼ , (17) P 2 Lm where Lm is the length of closure measured during the experimental analyses, whereas Lp is the length of closure values predicted by Eq. (17). The value x ¼ 0 corresponds to a perfect fit between the two data, and increasing value of x refers to poorer fit. Eq. (17) proves to have good results with x ¼ 0.00345. Fig. 3, on the other hand, plots the predicted dimensionless dislocation parameter against the observed values for the irregular wave analysis. The figure indicates that the predicted method fits the data very well. The presented method has root mean square error of 3% where the root mean square error, erms is defined as "P 2 #1=2 Lm Lp , (18) erms ¼ N where N is the number of observations carried out during the experimental work. In effect, the root mean square error describes the probability of a predicted position variation from the observed location. Therefore, an estimation of the dislocation parameter in natural environments practicing Pierson–Moskowitz wave spectrum will yield 3% error. The results for the irregular wave analysis are outstanding such that, out of the 26 experiments carried out, the number of errors exceeding 10% is only 1.
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3.5E-05 transport rate parameter, K
Predicted (Λ/ H)
U. Tu¨rker, M.S. Kabdas- lı / Ocean Engineering 34 (2007) 1096–1103
3.0E-05 2.5E-05 2.0E-05 1.5E-05
D = 0.38mm D = 0.7mm D = 0.5mm
1.0E-05 5.0E-06 0.0E+00
0
10
20
30
40 50 60 Observed (Λ / H)
70
80
0
90
Fig. 3. The relation between the observed and predicted dislocation values.
4
6 8 wave height, H
10
12
Fig. 4. Change in wave height with respect to transport rate parameter for different grain size diameters.
4.2. Transport rate parameter estimation
3.5E -05 transport rate parameter, K
During the experiments, the behavior of sedimentary grains was thoroughly observed. The results show that the wave climate parameters are not as effective as the sediment characteristics while defining the transport rate parameter. The transport rate parameter is primarily governing the time response of beach profile. The time dependent evolution of offshore bar is one of the observed proofs of this physical event. The experimental investigations helped to correlate the relationship between time response of beach profile and the wave climate and sediment characteristics. The theoretical proof of sediment transport rate parameter, K is already given in Eq. (13). Experimental investigations help to find the variables involved in Eq. (13) as physical quantities. Thus, the observed results are inserted into Eq. (13) to evaluate the sediment transport rate parameter. What is discussed was the change in transport rate parameter with respect to incoming wave height, wave period and length of displacement of sediment particles. The results depicted the independency of transport rate parameter to the wave climate parameters such as wave height and period (Figs. 4 and 5). Also, it is worth to mention that, whatever the length of displacement of sedimentary particles is; the transport rate parameter is not altering. However, any change in sediment grain size is directly affecting the scale of transport rate parameter. What is concluded after the investigations on experimental studies is that, detailed survey must be proceeded to detail the relationship between transport rate parameter and the mean grain size. The earlier observations show that transport rate parameter considerably increases with increase in mean grain size. Further increase in mean grain size, however, keeps altering transport rate parameter, but not as effective as in preliminary phase; slight changes are observed. For a specific grain diameter the changes in the transport rate parameter are considerably small with respect to sharp changes in wave period and height. The plotted data given in Fig. 4 shows that for different types of sediment grain sizes (0.38, 0.5 and 0.7 mm) any change in wave height is not significantly altering
2
3.0E - 05 2.5E - 05 2.0E - 05 1.5E- 05 1.0E - 05
D = 0.38 mm D = 0.7 mm D = 0.5 mm
5.0E - 06 0.0E+ 00 0
0.2
0.4
0.6 0.8 wave period, T
1
1.2
Fig. 5. The trend of the relation between wave period and transport parameter for different sized sediment particles.
magnitudes of transport rate parameter. However, an increment on grain size from 0.38 to 0.5 mm puts up the magnitude of transport rate parameter by 25%. Further increment in grain size of sedimentary particle, 0.5–0.7 mm, increases the magnitude of transport rate parameter by 10%, indeed, the necessary wave energy requirement for the mobility of sediment grain size increases. The result depicts the asymptotic approach of K-value as much as the grain diameter approaches to nonerodable (hard) sea bottoms. Fig. 5 reclaims the independency of transport rate parameter on wave period. Different grain sizes posses’ approximately same transport rate parameter, whilst wave period is changing in between 0.6 and 1.1 s. Likewise, the effect of wave height, while considering the effects of wave period, the rate of change in transport rate parameter is descending in the order of increasing grain diameter. All these results are supporting zero cross-shore sediment transport phenomena like in non-erodible sea bottoms. The length of sediment displacement described in Eq. (16) can be defined in terms of both the sediment characteristics and the wave climate. However, it is not directly affecting the magnitude of transport rate parameter. Fig. 6 repeats the minor importance of length of sediment displacement in order to dislocate from onshore to offshore. All the above discussions related with Figs. 4–6, also reveals the important fact that whilst the sediment grain
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1.8E-06
3.0E-05
K parameter (N/m4)
transport rate parameter, K
3.5E-05
2.5E -05 2.0E-05 1.5E-05 D = 0.38mm D = 0.7mm D = 0.5mm
1.0E-05 5.0E -06 0.0E+00 0.0
Hs = 1m Hs = 2m Hs = 3m Hs = 4m Hs = 5m
1.0E-06
6.0E-07 0.1
10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 average dislocation of sediments, Λ
Fig. 6. The trend of the relation between average distance traveled by sedimentary particles and transport parameter for different sized sediment particles.
1.4E-06
1 Grain diameter, (mm)
10
Fig. 8. The trend of the relation between grain diameter and transport parameter for different wave heights where wave period is 8 s.
1.8E-06
K parameter (N/m4)
K parameter (N/m4)
1.8E-06
1.4E-06
Hs =1m Hs = 2m Hs = 3m Hs = 4m Hs = 5m
1.0E-06
6.0E-07 0.1
1 Gain diameter, (mm)
1.4E-06
6.0E -07 0.1 10
Fig. 7. The trend of the relation between grain diameter and transport rate parameter for different wave heights where wave period is 6 s.
size is 0.38 mm, whatever the wave height, wave period and the length of displacement is, the transport rate parameter is not changing significantly. However, an increase in sediment size strengthens the beach profile preventing the dislocation of sediment particles under the applied impact forces. Under these circumstances, the displacement of sedimentary particles requires higher waves that result in longer elapsed time for the evolution of equilibrium beach profiles. The sediment characteristics therefore, dominantly determine the time scale that a beach profile reaches to an equilibrium conditions. The outcome of the necessary magnitudes of transport rate parameter versus the sediment grain size, depending on the changes in wave height is plotted through Figs. (7–9). Analyzing in more detail clarifies that each curve consists of two phases; first phase is enclosed between 0.1 and 1 mm grain sizes, where as, phase two indicates the changes in transport rate parameter for grain sizes larger than 1 mm. In the first phase, the change in sediment transport rate parameter is steep, whereas the change in second phase flattens. The results of each figure show that under the constant grain size conditions, the rate of change of transport rate parameter increases, as much as the wave height increases. The turning point, 1 mm diameter grain size, represents the willingness of beach profile while approaching to equilibrium conditions. As the grain size of sea bottom increases, the waves require more energy to
Hs = 1m Hs = 2m Hs = 3m Hs = 4m Hs = 5m
1.0E-06
1 Grain diameter, (mm)
10
Fig. 9. The trend of the relation between grain diameter and transport rate parameter for different wave heights where wave period is 10 s.
destabilize beach profile, reflecting the time response of beach profile while attaining equilibrium conditions. Transport rate parameter is the one, which will decide on the moment of equilibrium of a beach profile. 5. Conclusion The ratio of unit discharge of sediment moving in offshore–onshore direction, and the difference in between ‘‘the equilibrium wave energy dissipation rate’’ and ‘‘the wave energy dissipation rate’’ results in so called transport rate parameter. This parameter mainly governs the time response of beach profiles. It is compulsory that wherever the transport rate parameter is used in cross-shore modeling studies, it is accepted as a calibration value. However, in this study, by means of theoretical and experimental proofs the effect of wave climate and grain size characteristics are added to the definition of transport rate parameter. The transport rate parameter observed to be dependent on specific weight, angle of repose, wave breaking depth and the average distance traveled by sedimentary particles. After the analysis based on experimental studies carried over, the latter one is defined in terms of wave climate parameters. This result is achieved only when the laboratory results are observed as a macroscale feature. It is found that the distance traveled by sedimentary particles is well governed by dimensionless Froude and Fall speed parameters.
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The combination of above variables addressed to a reasonable definition for transport rate parameter. The results depicted the independency of parameter to the wave climate parameters. However, any change in sediment grain size directly affects the parameter. Also, an asymptotic relationship between the parameter and grain size is investigated, approaching to hard sea bottoms. The relationship between transport rate parameter and sediment grain size keeps on linearly increasing trend. However, as the sediment grain develops to a size of greater than 1 mm, the evolution time of equilibrium beach profile increases and approaches to hard sea bottom surface profiles. References Bagnold, R.A., 1963. Beach and nearshore processes; Part I: mechanics of marine sedimentation. In: Hill, M.N. (Ed.), The Sea: Ideas and Observations, vol. 3. Interscience, New York, pp. 507–528. Birkemeier, W.A., 1985. Field data on seaward limit of profile change. Journal of the Waterways, Port Coastal and Ocean Engineering 111 (3), 598–602. Broker Hedegaard, I., Roelvink, J.A., Southgate, H., Pechon, P., Nicholson, J., Hamm, L., 1992. Intercomparison of coastal profile models. In: Proceedings of the 23rd International Conference on Coastal Engineering, Venice. Bruun, P., 1954. Coast Erosion and the Development of Beach Profiles, Beach Erosion Board Technical Memorandum, 44, US Army Engineer Waterway. Experiment Station, Vicksburg, Mississippi. Dally, W.R., Dean, R.G., Dalrymple, R.A., 1984. A model for breaker decay on beaches. In: Proceedings of 19th International Conference on Coastal Engineering, Houston, TX ASCE, New York, pp. 82–98. Dean, R.G., 1977. Equilibrium beach profiles: US Atlantic and Gulf Coasts. Ocean Engineering Report 12, University of Delaware, Newark, DE. Dean, R.G., 1995. Cross-shore sediment transport processes. In: Liu, F., Philip, L. (Eds.), Advance Series on Ocean Engineering, vol. 1. World Scientific Publication, Singapore, pp. 159–220.
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Hallermeier, R.J., 1978. Uses for a calculated limit depth to beach erosion. In: Proceedings of 16th International Conference on Coastal Engineering, American Society of Civil Engineers, Hamburg, pp. 1493–1512. Hallermeier, R.J., 1981. A profile zonation for seasonal sand beaches from wave climate. Coastal Engineering 4, 253–277. Kriebel, D.L., Dean, R.G., 1985. Numerical simulation of time dependent beach and dune erosion. Coastal Engineering 9, 221–245. Larson, M., Kraus, N.C., 1989. SBEACH: Numerical model for simulating storm-induced beach change. Report 1, Empirical Foundation and Model Development, US Army Engineer Waterway Experiment Station Technical Report, CERC-89-9, Coastal Engineering Research Center, Vicksburg, Mississippi. Larson, M., 1993. Model for decay of random waves in the surf zone. Lund Institute of Technology Technical Report 3167, Lund University, Lund, Sweden. Lippmann, T.C., Holman, R.A., 1990. The spatial and temporal variability of sand bar morphology. Journal of Geophysical Research 95 (C7), 11575–11590. Moore, B., 1982. Beach profile evolution in response to changes in water level and wave height. M.Sc. Thesis, University of Delaware. Roelvink, J.A., Broker, I., 1993. Cross-shore profile models. Coastal Engineering 21, 163–191. Schoonees, J.S., Theron, A.K., 1995. Evaluation of 10 cross-shore sediment transport morphological models. Coastal Engineering 25, 1–41. Steetzel, H.J., 1990. Cross-shore transport during storm surges. In: Proceedings of the 22nd International Conference on Coastal Engineering, Delft. ASCE, New York, pp. 1922–1934. Tu¨rker, U., Kabdas-lı, M.S., 2004. Average sediment dislocation analysis for barred profiles. Ocean Engineering 31, 1741–1756. Vellinga, P., 1983. Predictive computational model for beach and dune erosion during storm surges. In: Proceedings of the American Society of Civil Engineers Specialty Conference on Coastal Structures, vol. 83, pp. 806–819. Watanabe, A., Dibajnia, M., 1988. Numerical modeling of nearshore waves, cross shore sediment transport and beach profile change. In: Proceedings of the IAHR Symposium on Mathematical Modeling of Sediment transport in the coastal zone, Copenhagen, pp. 166–174. Work, A.P., Dean, R.G., 1995. Assessment and prediction of beachnourishment evolution. Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE 121, 182–189.
Ocean Engineering 34 (2007) 1096–1103 www.elsevier.com/locate/oceaneng
Verification of sediment transport rate parameter on cross-shore sediment transport analysis U. Tu¨rkera,, M.S. Kabdas- lıb a
Near East University, Department of Civil Engineering, Northern Cyprus, Mersin 10, Turkey Istanbul Technical University, Department of Civil Engineering, 80626 Maslak, I˙stanbul, Turkey
b˙
Received 15 April 2006; accepted 16 August 2006 Available online 3 January 2007
Abstract The sediment transport parameter helps determining the amount of sediment transport in cross-shore direction. The sediment transport parameter therefore, should represent the effect of necessary environmental factors involved in cross-shore beach profile formation. However, all the previous studies carried out for defining shape parameter consider the parameter as a calibration value. The aim of this study is to add the effect of wave climate and grain size characteristics in the definition of transport rate parameter and thus witness their influence on the parameter. This is achieved by taking the difference in between ‘‘the equilibrium wave energy dissipation rate’’ and ‘‘the wave energy dissipation rate’’ to generate a definition for the bulk of sediment, dislocating within a given time interval until the beach tends reach an equilibrium conditions. The result yields that empirical definition of transport rate parameter primarily governs the time response of the beach profile. Smaller transport rate value gives a longer elapsed time before equilibrium is attained on the beach profile. It is shown that any significant change in sediment diameter or wave climate proportionally increases the value of the shape parameter. However, the effect of change in wave height or period on sediment transport parameter is not as credit to as mean sediment characteristics. r 2006 Published by Elsevier Ltd. Keywords: Beach profile; Equilibrium; Grain size; Transport rate parameter
1. Introduction Change in beach profile due to cross-shore sediment transport is one of the most important factors while dealing with nearshore dynamics. Beach profile change is generally treated in the framework of different approaches depending on the professions, like geologists, oceanographers and coastal engineers. There are different cross-shore sediment models that are developed for modeling beach profiles changes (Kriebel and Dean, 1985; Watanabe and Dibajnia, 1988; Larson and Kraus, 1989; Steetzel, 1990; Tu¨rker and Kabdas-lı, 2004). Some of these models are based on analytical grounds while the others on empirical results from experimental studies. Roelvink and Broker (1993), describe the state-of-the-art in modeling of cross shore profile evolution and classified the available models as; Corresponding author. Tel. +90 392 22 36 464; fax: +90 392 22 35 427.
E-mail address: [email protected] (U. Tu¨rker). 0029-8018/$ - see front matter r 2006 Published by Elsevier Ltd. doi:10.1016/j.oceaneng.2006.08.002
descriptive models (Lippmann and Holman, 1990), Equilibrium profile models (Bruun, 1954; Dean, 1977), Empirical profile evolution models (Kriebel and Dean, 1985) and Process Based Models (Broker Hedegaard et al., 1992). Although the fundamental assumptions used in these models are reasonable, the empirical coefficients involved in them cover so many processes that empirical coefficients must be determined through calibration for a given site (Roelvink and Broker, 1993). In order to provide uniqueness to the developed models, a definite characterization for empirical coefficients is necessary, rather than calibration. Among all these cross-shore profile evolution models the model proposed by Kriebel and Dean proved to be useful in cases of sandy beaches and dunes, where longshore transport gradients can be neglected, and where the profile is able to reach equilibrium. The model, however, uses an empirical coefficient called transport rate parameter, K. This transport rate parameter primarily governs the time response of the beach profile. Smaller
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value gives longer elapsed time before equilibrium is attained, whereas large value produces more rapid evolution. In other words, smaller K-value implies flatter equilibrium beach profile. K-value for Work and Dean (1995), in their solution, describing the response of an evolving beach nourishment project, is given as 0.2 106 m4/N. Kriebel and Dean (1985) refers to the study carried by Moore (1982) suggesting that K may either be a constant or a function of other wave and sediment parameters. Moore (1982) found K to be constant, equivalent to 2.2 106 m4/N. According to Moore (1982), the parameter K should also vary with the length scale of the system. For their analysis in SBEACH, Larson and Kraus (1989) used the average value of the transport rate coefficient as 1.1 106 m4/N, which is approximately half the value originally obtained by Moore (1982). In their studies, while comparing 10 cross shore sediment transport models Schoonees and Theron (1995) simulated a value of 8.75 106 m4/N. As it can be concluded, in all the above mentioned studies, the transport rate parameter is obtained by field calibration and is not entirely comparable between the models, in view of the fact that the structures of the models are different from each other. In this study, an attempt is made to generate a ratio, representing difference between ‘‘the equilibrium wave energy dissipation rate’’ and ‘‘the wave energy dissipation rate’’ to the bulk of sediment dislocated in cross shore direction. The result is incorporated with physical model based in the wave energy dissipation rates, thus, the more accurate values for transport rate parameter for any given climatic and sediment condition can be predicted. 2. Theoretical approach Cross-shore sediment transport relationships are generally categorized into two models, ‘Closed Loop and Open Loop’. ‘‘Closed Loop’’ models are those where the sediment transport relationships could converge to a target equilibrium profile, while in ‘‘open loop’’ models there is no priority to achieve final equilibrium profile. One of the first closed loop transport relationships proposed was that of Kriebel and Dean (1985). They have hypothesized that the cross-shore transport rate depends on the deviation of the dissipation rate per unit volume from its equilibrium, and thus improved a time dependent model that converges to Dean’s equilibrium profile. This concept was extended to barred profiles by Larson and Kraus (1989) and Larson (1993). Based on equilibrium profile considerations, it is reasonable to express the offshore transport rate of sediments at any point in the surf zone, qs, in terms of the difference between the actual and equilibrium levels of wave energy dissipation due to wave break. qs ¼ KðDw Deq Þ
(1)
in which Dw and Deq are the actual and equilibrium energy dissipation rates per unit volume and K is the transport
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rate parameter. Theoretically, qs represents the cross-shore sediment transport load resulting from destructive forces during breaking process. Namely, the expected occurrence of sediment transport mechanism depends on the level of wave energy dissipation, which should be above some equilibrium threshold, where no net transport occurs if not overestimated. Thus, above equation can be simplified to reflect the relation between the rate of energy required for sediment dislocation due to wave breaking per unit volume, Ds, to the energy difference between the equilibrium wave energy dissipation rate and the actual wave energy dissipation rate, (Tu¨rker and Kabdas-lı, 2004). Ds ¼ Dw Deq .
(2)
As the equilibrium in beach profile is attained, Dw ¼ Deq the net sediment transport is accepted as zero, Ds ¼ 0. Nevertheless, during the evolution of beach profile, the intensive sediment transport in the area enclosed within breaking zone and the coastline gives birth to high erosion rates in onshore–offshore direction. One of these changes is defined as storm profile or winter profile, where dominant sediment transport is in offshore direction. This transport mechanism continues until all the sediment grains accumulate some point offshore, forming an offshore bar. In the evolution of above physical event, it appears that there are destructive and constructive forces that tend to transport sand offshore and onshore, respectively. Gravity is one obvious destructive force, and it is evident that constructive forces, bottom shear stress etc., exist to counteract these destructive forces; otherwise; beaches cannot cease the sediment transport and thus the beach profile changes infinitely (Dean, 1995). Consider a unit area on a beach slope, locally inclined at an angle, b to the horizontal. The intensive wave breaking process conveying a sediment transport dissipates energy. When a certain transitional stage of bed movement is exceeded the fluid flow is considered as unaffected by the existence of stationary boundary. The whole resistance to flow, in the two-dimensional case, is that exerted by solid phase. Bagnold (1963) demonstrates the tangential thrust stress required to maintain the sediment in motion. rs r gm cos bðtan f tan bÞ, rs
(3)
where m is the whole mass of the displaced sediment per unit area, rs and r is the density of sediment and pervading fluid respectively, j is the angle of repose, which is a measure of the mean angle of contact between the sediments grains. The product of the load of grains and the mean speed u¯ of their travel is the rate of mass transport of the load per unit width. The multiplication of the tangential thrust stress, by mean speed of sediment grains results in the work done by the fluid per unit time. Hence the necessary fluid power per unit of bed area in transporting the sediment
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load can be given as rs r gm cos bðtan f tan bÞu, rs
By inserting Eq. (10) into Eq. (8), we get, (4)
where the mean speed is the ratio between the average distance traveled by the sediment particles and the time scale that the event takes place. u ¼ L=T n ,
(5)
L is the length of displacement of bulk of sedimentary particles in cross-shore direction. This length is defined in terms of wave and sediment characteristics, and the term T* is fictive period representing the time interval starting by the incipient motion of sediment from foreshore until it settles at offshore (Tu¨rker and Kabdas- lı, 2004). Inserting Eq. (5) into (4) represents the energy that is dissipated in unit time per unit of bed area in the process of transporting the sediment grains: E s ¼ ðrs rÞ
g L m cos bðtan f tan bÞ n . rs T
(6)
The volume of sediment particles (m/rs) is directly related with the fluid energy required to maintain the incipient sediment dislocation. L . (7) Tn In Eq. (7), gs and gw are the specific weight of sediment and fluid, respectively, and Vg is the volume of sediment grains. Perhaps, the total volume of sediment grains, transported in offshore direction, qualifies the volume of the offshore bar. Therefore, the volume of offshore bar Vbar, can replace the volume of grains. Vbar is the optimum volume of sand contained within the bar per unit length of beach formed after the storm. In this form, Eq. (7) represents the amount of energy per unit of bed area that the sediment particles should overestimate in order to move from one place to another to form offshore bar. The representation of the energy dissipation per unit volume in transporting the sediment grains, Ds, can be therefore defined as
E s ¼ ðgs gw ÞV g cos bðtan f tan bÞ
Ds ¼ V bar ðgs gw Þ cos bðtan f tan bÞ
L 1 . Tn h
(8)
In Eq. (8), h is the local water depth. In the derivation of Eq. (8), it is assumed that the sea state and other boundary conditions are stationary over the duration of a morphological time step. Remembering the relation given in Eqs. (1) and (2), the term representing the energy dissipation per unit volume can also be defined in terms of offshore sediment transport rate, and transport rate parameter: Ds ¼ qs =K.
(9)
In the above equation, the sediment transport rate is the volume of sediment, forming the offshore bar, dislocating within a given time interval defined as fictive period: qs ¼ V bar =T n .
(10)
1 (11) Ds ¼ qs ðgs gw Þ cos bðtan f tan bÞL . h Eq. (11) shows that the necessary energy dissipation for the displacement of sediment grains is directly proportional with the rate of sediment transport. At the same time, as given in Eq. (1), the rate of sediment transport is also proportional with the difference between actual and equilibrium energy dissipation rates per unit volume. Combining Eqs. (1), (2) and (11) and choosing the sediment transport rate as subject of the formula, the model lately proposed by Kriebel and Dean (1985) can be written as qs ¼
1 ðDw Deq Þ. ðgs gw Þ cos bðtan f tan bÞL 1h
(12)
In the former expression, the first term at the right side of the equation represents the so called transport rate parameter, K in terms of wave climate and grain size characteristics. K¼
hb . ðgs gw Þ cos bðtan f tan bÞL
(13)
Dally et al. (1984) in their study suggest that water depths at the break and at stable conditions are equal. Therefore, using this suggestion, the considered water depth at the location of the event can also assumed to be equal to the water depth at the break. The validity of Eq. (13) can be checked by the help of experimental studies where the parameters evolved in the above equation can be used to find representative values for K parameter. 3. Experimental testing Model studies are conducted at I˙stanbul Technical University, Hydraulics Laboratory. The aim is to investigate a representative transport rate parameter under the attack of regular and irregular wave climates. Glass wave flume with dimensions 24 1 1 m is used to generate irregular waves (Fig. 1). Irregular waves were generated by computer controlled wave maker capable to generate waves practicing Pierson–Moskowitz wave spectrum. The still water depth channel was 60 cm. During the studies three different grain sizes were used. The size of materials involved in the test was selected as 0.38, 0.5 and 0.7 mm. The slope of beach profile prior to each test was kept at 1:5. About 31 experiments for irregular waves were carried out. A wave monitor amplifier and a resistance type wave electrode were used to record wave data. The wave records, wave height and its period and average dislocation of sediment particles are then measured for each test. The results of experiments are summarized in Table 1. The analysis on wave data show that, for the irregular waves the highest one-tenth of wave height was dominating
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Fig. 1. Experimental setup of irregular wave channel.
Table 1 Laboratory results of irregular wave tests The code of sediment particles
Wave height H1/10 (cm)
Wave period T1/10 (s)
Length of displacement L (cm)
The mean diameter d50 (mm)
No of experiments accepted
A B C
12.0–16.0 11.0–17.0 12.0–16.0
1.2–1.9 1.5–2.0 1.3–1.9
85–112 76–115 75–93
0.38 0.50 0.70
11 9 6
the process. Perhaps, the reason for such a result lays in the fact that actually the offshore migration of bars is necessarily associated with break point processes. Thus, under random waves the small amplitude waves even cannot broke to dissipate their energy to initiate sediment motion. Finally, the experimental results are analyzed and considered. Whilst some of the experimental results are accepted as a good indicator for the sediment transport parameter relationship some did not comply with. Due to the unfavorable outcomes of five of these experiments, they are rejected from the analyses. Four of them were rejected from the experiments on 0.7 mm sized sediments whereas one of them was from 0.5 mm. The unfavorable results are mainly due to the defaults happened during the experimental studies, for example in two of the experiments carried over on 0.7 mm diameter sediments the electricity shortcut has shorten the timescale of the experiments. 4. Discussion and results Theoretically, an attempt is made to generate a definition representing the transport rate parameter in crossshore sediment transport studies. The resultant equation, Eq. (13), depends on specific weight, angle of repose, wave breaking depth and the length of displacement of sediment particles until the equilibrium conditions are maintained. The length of displacement of sedimentary particles are studied by Tu¨rker and Kabdas-lı (2004). In their results, it is shown that the average dislocation depends on both the wave climate and sediment characteristics. However, their theoretical definition involves too many limitations and variables, which are correlated with empirical definitions. Therefore, herein by using the
experimental results length of displacement of sedimentary particles is to be redefined. 4.1. Length of closure effect It is clear that all the experiments that practiced increment in wave energy dissipation rate are related with an increase in incoming wave height. As a result, convolution in offshore displacement of sediment particles consumes more time. The reason, without a doubt, is increase in displacement length of total volume of sediments while displacing offshore to form larger offshore bars. Furthermore, according to stability criteria as the size of sediment particles increases, necessary wave energy required to dislocate the particles also increases. Perhaps, the initial slope of beach profile also affects the natural evolution of the displacement length of sediment particles to form offshore bar. It is also evident from the experimental studies that the rate of profile evolution is decreasing consistent with an approach to equilibrium. The estimation of the length of displacement of sediment grains can be valid parameter for predicting cross-shore profile change up to which the seaward limit to which the depositional front has advanced. According to the previous studies there are too many recommendations for seaward limit of offshore depositions. Vellinga (1983) proposed the depth of closure as 75% of the deepwater significant wave height using linear wave theory. Based on Laboratory and field data Hallermeier (1978, 1981), developed a condition for sediment motion resulting from wave conditions. According to his studies the resultant equation is given as 2 H hc ¼ 2:28H 68:5 , (14) gT 2
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where H and T are significant wave height and significant wave period, respectively. Birkemeier (1985) evaluated Hallermeier’s relationship creating a simple approximation to the effective depth of closure providing a good fit to the data given as hc ¼ 1:57H.
dimensionless dislocation parameter;
H o =wT;
fall speed parameter;
H o gT 2 ;
Froude parameter:
10.0 8.0
(15)
Perhaps, the depth of closure is the best point to define the offshore limits of sediment disturbance due to wave action. However, during the offshore bar formation the depth of water above the offshore bar is not remaining constant. Consequently, the depth of water is not such a precise location to define the limiting depth of closure. Thus, it is necessary to define the depth of closure by another referring, which can be defined as length of closure. The necessary length of closure actually defines the overall length of displacement of sediment particles during the formation of offshore bar. The findings of the experiments that are carried over in I˙stanbul Technical University, Hydraulics Laboratory, concluded that in forming the equilibrium beach profile, the seaward limit of beach profile depends on wave height, wave length and its period; the size of sediment grains and the slope of beach profile. However, since in this study the beach profile slope is considered as constant, the results of experiments brings about uncertainty in the definition of seaward limit of beach profile. For sure, the consideration of the initial beach slope has unexpectedly failed to improve the accuracy of the analysis. Harmonizing the parameter which are effective in defining the seaward limit of beach profile results in three non-dimensional parameters: H o =L;
12.0
Λ/ H
1100
In some sense, fall speed parameter is both a measure of wave steepness and a sediment-related descriptor, indicating both profile response and geometric properties of various major morphologic features. It is also related to mobilization and suspension of sediment and performs well individually and in combination as criteria for predicting direction of cross-shore sediment transport. The strong relationship between wave and sand characteristics and morphologic features suggests the possibility of quantitatively predicting the beach profile response and evolution period (Larson and Kraus, 1989). The dimensionless dislocation parameter indicates the dominance of wave height on the morphological changes in seaward limitations of beach profile. The Froude parameter is useful in defining the dominancy of gravity forces, which in turn affects the evolution time (transport rate parameter) of equilibrium beach profile. The relationship among the dimensionless dislocation parameter, fall speed parameter and the Froude parameter for irregular wave analysis is given in the
y = -1.2808 Ln (x) - 8.74 2 R = 0.85
6.0 4.0 2.0
1.0E-07
1.0E-06
1.0E-05
0.0 1.0E-04
(H/gT2)2.68 (H /wT)-2.05 Fig. 2. Logarithmic variances in the definition of transport rate parameter.
following equation. The trend between the parameters is simulated in Fig. 2. ! L H 2:68 H 2:05 ¼ 1:28 ln 8:74. (16) H wT gT 2 The logarithmic relation between the dimensionless dislocation parameter and its empirical definition can be represented graphically on a semi logarithmic paper resulting with a coefficient of determination equivalent to 0.79. The quality of fit analysis is applied to the predicted equation and the experimental data. It is seen that the resultant empirical equation is fitting the data quite well. The quality of fit is determined through the following equation: 2 P Lm Lp x¼ , (17) P 2 Lm where Lm is the length of closure measured during the experimental analyses, whereas Lp is the length of closure values predicted by Eq. (17). The value x ¼ 0 corresponds to a perfect fit between the two data, and increasing value of x refers to poorer fit. Eq. (17) proves to have good results with x ¼ 0.00345. Fig. 3, on the other hand, plots the predicted dimensionless dislocation parameter against the observed values for the irregular wave analysis. The figure indicates that the predicted method fits the data very well. The presented method has root mean square error of 3% where the root mean square error, erms is defined as "P 2 #1=2 Lm Lp , (18) erms ¼ N where N is the number of observations carried out during the experimental work. In effect, the root mean square error describes the probability of a predicted position variation from the observed location. Therefore, an estimation of the dislocation parameter in natural environments practicing Pierson–Moskowitz wave spectrum will yield 3% error. The results for the irregular wave analysis are outstanding such that, out of the 26 experiments carried out, the number of errors exceeding 10% is only 1.
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1101
3.5E-05 transport rate parameter, K
Predicted (Λ/ H)
U. Tu¨rker, M.S. Kabdas- lı / Ocean Engineering 34 (2007) 1096–1103
3.0E-05 2.5E-05 2.0E-05 1.5E-05
D = 0.38mm D = 0.7mm D = 0.5mm
1.0E-05 5.0E-06 0.0E+00
0
10
20
30
40 50 60 Observed (Λ / H)
70
80
0
90
Fig. 3. The relation between the observed and predicted dislocation values.
4
6 8 wave height, H
10
12
Fig. 4. Change in wave height with respect to transport rate parameter for different grain size diameters.
4.2. Transport rate parameter estimation
3.5E -05 transport rate parameter, K
During the experiments, the behavior of sedimentary grains was thoroughly observed. The results show that the wave climate parameters are not as effective as the sediment characteristics while defining the transport rate parameter. The transport rate parameter is primarily governing the time response of beach profile. The time dependent evolution of offshore bar is one of the observed proofs of this physical event. The experimental investigations helped to correlate the relationship between time response of beach profile and the wave climate and sediment characteristics. The theoretical proof of sediment transport rate parameter, K is already given in Eq. (13). Experimental investigations help to find the variables involved in Eq. (13) as physical quantities. Thus, the observed results are inserted into Eq. (13) to evaluate the sediment transport rate parameter. What is discussed was the change in transport rate parameter with respect to incoming wave height, wave period and length of displacement of sediment particles. The results depicted the independency of transport rate parameter to the wave climate parameters such as wave height and period (Figs. 4 and 5). Also, it is worth to mention that, whatever the length of displacement of sedimentary particles is; the transport rate parameter is not altering. However, any change in sediment grain size is directly affecting the scale of transport rate parameter. What is concluded after the investigations on experimental studies is that, detailed survey must be proceeded to detail the relationship between transport rate parameter and the mean grain size. The earlier observations show that transport rate parameter considerably increases with increase in mean grain size. Further increase in mean grain size, however, keeps altering transport rate parameter, but not as effective as in preliminary phase; slight changes are observed. For a specific grain diameter the changes in the transport rate parameter are considerably small with respect to sharp changes in wave period and height. The plotted data given in Fig. 4 shows that for different types of sediment grain sizes (0.38, 0.5 and 0.7 mm) any change in wave height is not significantly altering
2
3.0E - 05 2.5E - 05 2.0E - 05 1.5E- 05 1.0E - 05
D = 0.38 mm D = 0.7 mm D = 0.5 mm
5.0E - 06 0.0E+ 00 0
0.2
0.4
0.6 0.8 wave period, T
1
1.2
Fig. 5. The trend of the relation between wave period and transport parameter for different sized sediment particles.
magnitudes of transport rate parameter. However, an increment on grain size from 0.38 to 0.5 mm puts up the magnitude of transport rate parameter by 25%. Further increment in grain size of sedimentary particle, 0.5–0.7 mm, increases the magnitude of transport rate parameter by 10%, indeed, the necessary wave energy requirement for the mobility of sediment grain size increases. The result depicts the asymptotic approach of K-value as much as the grain diameter approaches to nonerodable (hard) sea bottoms. Fig. 5 reclaims the independency of transport rate parameter on wave period. Different grain sizes posses’ approximately same transport rate parameter, whilst wave period is changing in between 0.6 and 1.1 s. Likewise, the effect of wave height, while considering the effects of wave period, the rate of change in transport rate parameter is descending in the order of increasing grain diameter. All these results are supporting zero cross-shore sediment transport phenomena like in non-erodible sea bottoms. The length of sediment displacement described in Eq. (16) can be defined in terms of both the sediment characteristics and the wave climate. However, it is not directly affecting the magnitude of transport rate parameter. Fig. 6 repeats the minor importance of length of sediment displacement in order to dislocate from onshore to offshore. All the above discussions related with Figs. 4–6, also reveals the important fact that whilst the sediment grain
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1102
1.8E-06
3.0E-05
K parameter (N/m4)
transport rate parameter, K
3.5E-05
2.5E -05 2.0E-05 1.5E-05 D = 0.38mm D = 0.7mm D = 0.5mm
1.0E-05 5.0E -06 0.0E+00 0.0
Hs = 1m Hs = 2m Hs = 3m Hs = 4m Hs = 5m
1.0E-06
6.0E-07 0.1
10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 average dislocation of sediments, Λ
Fig. 6. The trend of the relation between average distance traveled by sedimentary particles and transport parameter for different sized sediment particles.
1.4E-06
1 Grain diameter, (mm)
10
Fig. 8. The trend of the relation between grain diameter and transport parameter for different wave heights where wave period is 8 s.
1.8E-06
K parameter (N/m4)
K parameter (N/m4)
1.8E-06
1.4E-06
Hs =1m Hs = 2m Hs = 3m Hs = 4m Hs = 5m
1.0E-06
6.0E-07 0.1
1 Gain diameter, (mm)
1.4E-06
6.0E -07 0.1 10
Fig. 7. The trend of the relation between grain diameter and transport rate parameter for different wave heights where wave period is 6 s.
size is 0.38 mm, whatever the wave height, wave period and the length of displacement is, the transport rate parameter is not changing significantly. However, an increase in sediment size strengthens the beach profile preventing the dislocation of sediment particles under the applied impact forces. Under these circumstances, the displacement of sedimentary particles requires higher waves that result in longer elapsed time for the evolution of equilibrium beach profiles. The sediment characteristics therefore, dominantly determine the time scale that a beach profile reaches to an equilibrium conditions. The outcome of the necessary magnitudes of transport rate parameter versus the sediment grain size, depending on the changes in wave height is plotted through Figs. (7–9). Analyzing in more detail clarifies that each curve consists of two phases; first phase is enclosed between 0.1 and 1 mm grain sizes, where as, phase two indicates the changes in transport rate parameter for grain sizes larger than 1 mm. In the first phase, the change in sediment transport rate parameter is steep, whereas the change in second phase flattens. The results of each figure show that under the constant grain size conditions, the rate of change of transport rate parameter increases, as much as the wave height increases. The turning point, 1 mm diameter grain size, represents the willingness of beach profile while approaching to equilibrium conditions. As the grain size of sea bottom increases, the waves require more energy to
Hs = 1m Hs = 2m Hs = 3m Hs = 4m Hs = 5m
1.0E-06
1 Grain diameter, (mm)
10
Fig. 9. The trend of the relation between grain diameter and transport rate parameter for different wave heights where wave period is 10 s.
destabilize beach profile, reflecting the time response of beach profile while attaining equilibrium conditions. Transport rate parameter is the one, which will decide on the moment of equilibrium of a beach profile. 5. Conclusion The ratio of unit discharge of sediment moving in offshore–onshore direction, and the difference in between ‘‘the equilibrium wave energy dissipation rate’’ and ‘‘the wave energy dissipation rate’’ results in so called transport rate parameter. This parameter mainly governs the time response of beach profiles. It is compulsory that wherever the transport rate parameter is used in cross-shore modeling studies, it is accepted as a calibration value. However, in this study, by means of theoretical and experimental proofs the effect of wave climate and grain size characteristics are added to the definition of transport rate parameter. The transport rate parameter observed to be dependent on specific weight, angle of repose, wave breaking depth and the average distance traveled by sedimentary particles. After the analysis based on experimental studies carried over, the latter one is defined in terms of wave climate parameters. This result is achieved only when the laboratory results are observed as a macroscale feature. It is found that the distance traveled by sedimentary particles is well governed by dimensionless Froude and Fall speed parameters.
ARTICLE IN PRESS U. Tu¨rker, M.S. Kabdas- lı / Ocean Engineering 34 (2007) 1096–1103
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