海岸動(dòng)力學(xué)英文課件CoastalHydrodynamics-25

1、Coastal Hydrodynamics Chapter 2The pressure field associated with a progressive wave is determined from the unsteady Bernoulli equation. Pressure fieldThe pressure equation contains two terms: the hydrostatic pressure (靜水壓強(qiáng)) & the dynamic pressure (動(dòng)水壓強(qiáng)) Chapter 2The total energy consists of two kin

2、ds: the potential energy (勢(shì)能), resulting from the displacement of the free surface； the kinetic energy (動(dòng)能), due to the orbital motion of the water particles.It is worthwhile emphasizing that neither theaverage potential nor kinetic energy per unitarea depends on water depth or wave length,but each

3、is simply proportional to the squareof the wave height. Wave energy Chapter 2The rate at which the energy is transferred in the direction of wave propagation is called the energy flux (波能流), and it is the rate at which work is being done by the fluid on one side of a vertical section on the fluid on

4、 the other side. Energy fluxThe relationship for the energy flux is Chapter 2Group velocity (群速)If there are two trains of waves of the same height propagating in the same direction with a slightly different frequencies and wave numbers, the resulting profile, is modulated by an envelop that propaga

5、tes with speed of group velocity.2/36 Chapter 2The group velocity is defined as This derivative can be evaluated from the dispersion relationship 3/36 Chapter 24. Standing wavesStanding waves (立波) often occur when incoming waves are completely reflected by vertical walls.If a progressive wave were n

6、ormally incident on a vertical wall, it would be reflected backward without a change in height, thus giving a standing wave in front of the wall. Standing waves are also called clapotis (駐波). 4/36 Chapter 2If the wave heights of the incident wave and the reflected wave are different, the superpositi

7、on creates a partial standing wave. 5/36 Chapter 2The reflection coefficient (反射系數(shù)) based on the linear wave theory can be determined by measuring the amplitudes at the antinode and node of the composite wave train. 6/361. Stokes wave theory Chapter 22.4 Finite Amplitude Wave Theory 2. Trochoidal wa

8、ve theory 3. Cnoidal wave theory 4. Solitary wave theory 7/36 Chapter 2In 1847, Stokes considered waves of small but finite height progressing over still water of finite depth and presented a second-order theory.1. Stokes wavesThe method of Stokes has been extended to higher orders of approximation.

9、The tabulated solutions to third and fifth orders are very useful in applications. 8/36 Chapter 2The solution depends on the presumed small quantity ka, which will be defined as.Therefore we will decompose all quantities into a power series in. Perturbation approach (攝動(dòng)法) 9/36 Chapter 2Velocity pote

10、ntial isSecond-order solutionFor finite height waves there is an additional term added onto the equation obtained in the linear wave theory. 10/36 Chapter 2Water surface displacement isSecond-order solutionFor finite height waves there is an additional term added onto the basic sinusoidal shape. 11/

11、36 Chapter 2The added term enhances the crest amplitude and detracts from the trough amplitude, so that the Stokes wave profile has steeper crests separated by flatter troughs than does the sinusoidal wave. 12/36 Chapter 2The dispersion equation remains the same, that is,Second-order solutionIt is n

12、oted that a correction occurs to the dispersion equation at the third order, which would result in a slight increase in wave celerity. 13/36 Chapter 2Velocity components are Second-order solution14/36 Chapter 2The effect of the added term is to increase the magnitude but shorten the duration of the

13、velocity under the crest and decrease the magnitude but lengthen the duration of the velocity under the trough. For the horizontal velocity, the velocities are greater under the crest but are reduced under the trough when compared with those of the linear wave. 15/36 Chapter 2Comparison of bottom or

14、bital velocity under Stokes wave with that of linear wave of the same height and length (H=4m, h=12m, T=12sec)16/36 Chapter 2An interesting departure of the Stokes wave from the Airy wave is that the particle orbits are not closed. The crest position after one wave cycle progresses in the direction

15、of wave propagation. Second-order solution17/36 Chapter 2The net horizontal displacement within one wave period is Thus there results a so-call “mass transport” (質(zhì)量輸移) with an associated velocity 18/36 Chapter 2Wave energy Second-order solution19/36 Chapter 2Wave pressure Second-order solution20/36

16、Chapter 2The first solution for periodic waves of finite height was developed by Gerstner in 1802. From the developed equations, he concluded that the surface profile is trochoidal in form. His solution is limited to waves in water of infinite depth. 2. Trochoidal wavesIn 1935, Gaillard attempted to

17、 extend the theory to water of finite depth and introduced an elliptic trochoidal wave theory for shallow water waves. 21/36 Chapter 2For an angle of rotation, the surface depression below crest level is 22/36 Chapter 2This theory has been wide application by civil engineers and naval architects bec

18、ause the solutions are exact and the equations simple to use. However, mass transport is not predicted and the velocity field is rotational. Even worse, in the trochoidal wave the particles rotate in a sense opposite to the rotational that would be present in a wave generated by a wind stress on the

19、 water surface. 23/36 Chapter 2In 1895, Korteweg and de Vries developed a shallow water wave theory which allowed periodic waves to exits. The name cnoidal is derived from the fact that the wave profile is expressed by the cn() function of Jacobis elliptic functions. 3. Cnoidal wavesThe cnoidal wave

20、 is a periodic wave that may have widely spaced sharp crests separated by wide troughs and so could be applicable to wave description just outside the breaker zone. 24/36 Chapter 2The first-order approximate solution is given by the following equations: 25/36 Chapter 2It is worthwhile mentioning tha

21、t the cnoidal wave has the unique feature of reducing to a profile expressed in terms of cosines at one limit and to the solitary wave theory at the other limit. It is apparent then that we could simply apply the cnoidal wave and ignore other theories. Unfortunately, the mathematics of the cnoidal w

22、ave is difficult, so that in practice it is applied to as limited range as possible. 26/36 Chapter 2The solitary wave is a translation wave consisting of a single crest. There is therefore no wave period or wave length associated with the solitary wave. 4. Solitary wavesThe solitary wave was first d

23、escribed by Rusell in 1844, who produced it in a laboratory wave tank by suddenly releasing a mass of water at one end of the wave tank. The first theoretical consideration was that of Boussinesq. 27/36 Chapter 2The first-order approximation of the water surface in the solitary wave theory is28/36 C

24、hapter 2Because of its similarity to real waves and its simplicity, the solitary wave has been wide application to nearshore studies. It would appear that the solitary wave would not be particularly useful in describing the periodic wind waves we deal with in the ocean. 29/36 Chapter 22.5 Wave Theor

25、y Limits of Applicability How does one decide which of the wave theories is applicable to his particular problem? 30/36 Chapter 2This validity is composed of two parts: the mathematical validity, which is the ability of any given wave theory to satisfy the mathematically posed boundary value problem. the physical validity, which refers to how well the prediction of the various theories agrees with actual measurements. 31/36 Chapter 2Ranges of applications of the several wave theories as function of the rations H/gT2 and h/gT2 32/36 Chapter 2In general, the linear wave t