Kıyı bölgesinde düzenli ve karışık dalgaların kırılmalarından ötürü enerji kaynaklarının incelenmesi

Abstract

Bu çalışmada, Battjes (1986) ve Battjes ile Janssen (1978) 'in sahile doğru ilerleyen dalgaların kırıldıktan sonraki dalga yüksekliklerini hesaplamaya yarayan iki model tanıtılmıştır. Bu modellerden ilki periyodik ve yükselti dalgaları için verilen (Battjes, 1986) ve monoton olarak azalan dip eğimleri için uygundur. Ardından Battjes ve Janssen (1978) 'in hem sabit eğimli hem de tepe-çukur (bar- trough) tipi profiller için 8Px/8x+D=0 enerji denge denklemindeki enerji kaybını ifade eden kayıp fonksiyonu, D değerini hesaplayan ikinci bir model açıklanmıştır. İkinci modelden faydalanılarak kırılan dalgaların kırılma sonrası dalga yüksekliklerinin hesabı ve momentum denge denkleminin kullanılması ile sakin su seviyesinde ki yükselme (set-up) veya alçalmalar (set-down) bir bilgisayar programı (Ek A) yapılarak hesaplanmıştır. Bu ikinci model için Battjes ve Janssen (1978) 'in elde ettiği deney sonuçları ile bilgisayar programından elde edilen sonuçlar karşılaştırılmış ve bunlarla ilgili şekiller Ek B'de sunulmuştur. Battjes ve Janssen (1978) 'in kıyıya gelen karışık dalgaların kırıldıktan sonraki dalga yüksekliklerini ve su seviyesindeki değişimi veren teorik modelin gayet iyi sonuçlar verdiği görülmüştür.

In this study, two different models due respectively to Battjes (1986) and Battjes and Janssen (1978) have been re examined for determining heights of breaking waves in nearshore zone. Battjes arid Janssen' s experimental data are compared with the model predections computed by using a computer program. Estimation of the the breaking and/or broken wave heights of random waves in nearshore is quite important in coastal engineering. The modelling of the energy dissipation rate due to wave breaking leads to determination of the decay of the wave height in surf zone. It is well known that on gentle slopes the wave height after breaking does not decay in proportion to the mean depth. The hypothesis H(x)=y.h(x) (H:wave height, hrwater depth) is not applicable in regions where the depth is constant or increasing in the propagation direction, such as in a bar-trough profile because the wave height after breaking does not increase with increasing depth of water. Taking into account the above points, a more general model will be explained in this study after the first model, where the periodic and solitary wave breaking over monotonically decreasing depth are considered. Le Mehaute (1962), Divoky (1970), Hwang and Divoky (1970), have calculated the energy dissipation rate per unit distance in a spilling breaker by taking into account a bore with the same height of foam region. The energy dissipation in the spilling wave breakers can be modelled by establishing an analogy with the propagation of a bore with horizontal velocities ux and u2 and corres ponding water depths hx and h2. If Q denotes the volume per unit breadth we have 0=1^=1/2/22 (1) Due to the difference in water levels at right and left sides, a pressure difference arises: ~KP=P2-PX (3) Change in momentum across the bore, pQC^-Uj), must be equal to the net force due to pressure differences. Thus, after some manipulations (see the thesis) Q^gh&ihL+hJ (4) Change in energy across the bore is given as the difference of kinetic and potential energy and this can be expressed as follows A£Tr=-|p {ul-ul) +pSr(h1-lia) (5) which may be put into the form a,-, 1 (İ22-İ2,)3 lcx The rate of energy dissipation in a bore connecting two regions of uniform flow can then be written as VI Bat t j es (1986) assumed that h2-h1=P and that h!=h2=h which simplified (7) as D^ipgBsmi£ (8) 4 h where c=(gh)1/2 and both B and P are coefficients supposed to be 0(1). Battjes (1986) introduced the following hypothesis in which yh is a local breaker height, where y may be calculated from empirical relationships including the bottom slope S and perhaps the wave steepness (Miche). For instance, y=0.7+5S, (O.OliS^O.l) (10) for periodic waves (Battjes, 1986). Eq. (8) is re-written using (9) as 4 y3 II 4 y3 Î1 in which B=0(1) as proposed by Battjes (1986), independent of bottom slope. The dissipation function, D denoting the energy lost in breaking wave can only be used in momentum balance equation. The unknown coefficient of dissipation function, B, can be determined by comparing the results obtained from solution of energy balance equation and experiments. Battjes (1986) obtained an estimate of the parameter B from a comparison with Street and Camfield's (1966) data on solitary waves breaking on a 1:100 slope. Choosing B=2 he gave a curve which in most of the decay region agrees vi x remarkably well with the data, in fact better than might have been expected. The model however does not predict the observed surviving wave height at the shore line. Quantitative predictions of wave induced mean sea level variations and currents in the near shore region require a specification of the mean wave energy density (E) in that region. For this reason, Battjes (1972) and Goda (1975) have presented methods to that effect applicable to random and breaking waves. Any approach based on the energy balance is more physically realistic, and is needed in applications to profiles where the depth is not monotonically decreasing shoreward, such as commonly occurring bar-trough profiles. For these reasons an attempt was made to develop a model for the dissipation of wave energy in random waves, breaking on a beach. Here Battjes and Janssen's dissipation model for random waves will be described in detail. They consider a two dimensional situation of waves normally incident on a beach with straight and parallel depth contours. For given incident wave parameters and beach profile, the variation of mean wave energy density (E), with distance to the shore line can in principle be calculated form the energy balance, written as 3P °^+£>=0 (12) Bx in which Px is the x component of the time mean energy flux per unit length, x is a horizontal coordinate, normal to the still water line, and D is the time mean dissipated power per unit area. The use of (12), rather than H(x) =y.h(x), is in prin ciple much to be preferred for a number of reasons, the most fundamental one being that (12) has a sound physical, whereas it is assumed ad hoc. Associated with this are the following items : i) Assumption H(x)=y.h(x) relates the local wave height to the local mean depth. This introduces an unrealistically large dependence of dissipation rate on local bottom slope (Battjes, 1986). In (12), the local wave height is found form an integration, so that it not only depends on the local V1XX depth but also on those further seaward, which is more realistic. ii) If necessary, other dissipation mechanisms than that due to breaking can be incorporated in (12) in a straight forward manner, whereas this is not the case for H(x)=y.h(x). iii) H(x)=y.h(x) is restricted to profiles in which the depth decreases monotonically in the shoreward direction. This is not the case for (12), which in principle can be applied in bar-trough profiles as well. One such model which has been developed will be described in the following. i) An abrupt upper cut-off of the wave height distribution. ii) Linear approximation for energy flux P. iii) Dissipation only due. to breaking 8P/8x=-D. The first assumption stated above is written in terms of the probability distribution of the wave heights, F(H). The shape of F(H) for the lower, non-broken wave heights is assumed to be the same as it is in absence of wave breaking, of the Rayleigh type, with modal value H. This leads to F(H) =Pr lH*H] =l-exp ( -±H2/82) for Q*H*Ha ( 13 } =1 for Hm*H Eq. (13) ^represents a probability distribution with two parameters, H and H". All the statistics of the wave heights can therefore be expressed in terms of (H,Hm). Among those are the rms (Hmg), defined by i Hrms={fH2dF(H))2 <14> 1"Qb__ / HjmsjZ /15) XX Eq. (15) is a key element in Battjes and Janssen's (1978) model. It expresses the fraction of waves which at any one point are breaking or broken, in terms of the ratio of the H^g actually present, to the maximum wave height which the given depth can sustain. The local value of H^ is not known a priori; it is found by integrating the differential equation (12). The importance of Qb for this equation is due to the fact that the average local energy dissipation rate D is proportional to it, at least in the dissipation model to be described below. It is mainly through Qb that this model reacts to changes in depth. Hm's are computed fromMiche's criterion for the maximum height of periodic waves of constant form: fl_,«0.14L tanM2ıtiî/I.)»0.88Jc-1 tanhkh (16) Eq. (16) as it stands would predict H^O.88h in shallow water. In application to waves on beaches Battjes and Janssen (1978) use a similar functional relationship as in (16) but also allow some freedom of the transformation to random waves, such that in shallow water their expression for Hm reduces to Hm=y.h, in which y is a slightly adjustable coefficient. In order that the deep water limit shall not be influenced by the bottom slope they finally adopt the following form Hm=O.BQk-r tanh(yJch/0.88) (17) Using B/y3 is constant and Hm/h«l (at time of breaking) the following dissipation function is obtained D~±fpgH* (18) or written as an equality n^^Q^pgHl (19) in which a is a constant of order one. x At first sight, it might seem from (19) as if D is decreasing with decreasing ^ (decreasing h), but this is not normally the case. As long as a random wave train of low or moderate steepness is in relatively deep water, Qb will be virtually zero, and therefore also D. When, upon approaching the beach, the depth becomes less than 2 to 3 times the rms wave height, Qb increases strongly, such that its increase in fact more than compensates for the reduction in Hm2, so that D also increases. Having established a dependence of the dissipation rate D on the rms wave height, it remains to do the same for the energy flux P in order to be able to integrate the energy balance, eg. (12) A simple linear approximation will be used, Px=P=E.Cff (20) in which E=±pgH2rma (21) _ r 2%£ / 1 kh » ¦[.22. This closes the system of equations for H^. For given depth profile h(x), given incident wave parameters, and a suitable choice of the model parameters a and y, eq. (12) can be integrated to find Hms (x). The height of the mean water level above the reference plane, written as T|(x) (set-up), can be determined from the mean momentum balance. Following Lonquet-Higgins and Stewart (1962) the mean balance of x-momentum is written as pgh^Q (23) - +r-^- dx dx XX in which h=d+l\ (24) and s~'lı+ım]B (25) the component of the radiation stress tensor normal to planes x (=constant). As explained earlier, the energy density is important in surf zone. Since the stress of radiation is proportional to E, and the gradients of these stresses provide driving forces for the mean flow. To check the validity of the model Battjes and Janssen (1978) conducted a series of experiments to measure both wave-height decay and the wave- induced set up. Comparisons of the computations with their experimental data are reproduced here and found to agree well. The computer routine used for these computations are also included in the appendix.