Free-Surface Hydraulics

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title:Free-surface Hydraulicsauthor:Townson, John M.publisher:Taylor & Francis Routledgeisbn10 | asin:0046270094print isbn13:9780046270094ebook isbn13:9780203358764language:EnglishsubjectHydrodynamics, Water waves, larpcal--Mecanica e dinamica dos fluidos , Hydraulic engineeringpublication date:1991lcc:TC171.T68 1991ebddc:627/.042subject:Hydrodynamics, Water waves, larpcal--Mecanica e dinamica dos fluidos , Hydraulic engineering

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1The free surface at rest

when the dam broke, or, to be more exact, when everybody in town thought that the dam broke.

James Thurber, My life and hard times

Our planet may be said to be composed largely of three elemental materials, namely earth, water and air. The physical property that most easily distinguishes them is probably that of density. Different gravity forces tend to separate them into layers with fascinating phenomena at the interfaces. In the region close to the surface of the Earth, the three densities are about 2500, 1000 and 1.3 kg/m3 for earth (i.e. rock), water and air, respectively. The planets rotation causes relative motions to occur between each layer, which are strongly dependent on the difference between their densities. It has become customary to refer to the air/water interface as a free surface, in which the adjective free implies some degree of independent vertical motion. In other words, it has the capacity to exhibit waves (Fig. 1.1). Waves also occur at the other interfaces, earth/air and earth/water, in the form of dunes or ripples. However, those interfaces might be said to be less free as a consequence of the greater weight and

Figure 1.1 The Earths elemental interfaces and their capacity for waves.

Free-Surface_Hydraulics/0046270094/files/page_10.html Page 10

Figure 1.10 Force components on a circular cylinder whose axis is horizontal. FV in case (b) is less obvious than in (a).

Figure 1.11 Hydrostatic pressure on a doubly curved shape.


Figure 4.13 Characteristic slopes at entry for the flood-wave simulation of Figure 4.12.

Depths at entry for K=0.4 increase by 50% of normal and by 20% halfway down the channel (67% and 30% for K=0.6). The peak is lower but progresses more quickly for K=0.6 than for K=0.4, despite smaller depths, because the friction is less. Discharge variations are even more pronounced, with lower friction giving higher peak flows, which subside more slowly. It is interesting to calculate the discharges from depths by assuming local uniform flow. A comparison for K=0.4 after 20t shows that these predict only 50% of the actual increaseexcept towards the entry, where they overestimate. Finally, peak discharges seem to travel slightly faster than peak depths, a feature that is enhanced for lower K (i.e. more friction).

4.9 Direct finite differences

Solutions of the shallow-water wave equations have been and are still being carried out by a wide variety of numerical methods other than characteristic. This is so particularly for the calculation of tidal flows in estuaries. Here the primary disturbance is the changing water elevation at the channel entrancecontrasting with the discharge variations that control unsteady flows in upland rivers. In principle, the wave arising from either boundary condition may be traced using the same technique. Dronkers (1969, p. 42) suggested that tidal flow was a more regular process than the propagation of overland flow in a river except where surges and bores were likely. However, the latter are not unknown in upland flows. They may indeed arise from intense storms in small, steep catchments or from sudden out-flows at reservoirs. Nevertheless, and in both tide and river flows, it has remained attractive to replace the partial derivatives in the shallow-water equations directly by their finite-difference equivalents.

An early application of the direct finite-difference approach to upland flow seems to have been by Thomas (1934), as reported by Chow (1973).


Using the grid-point coding of Figure 4.14a for clarity, Thomas routine was

/x=[(21)+(43)]/2x

with

/t=[(31)+(42)]2t

(4.31)

being centred at the half-increment in both space and time. This leads to two simultaneous equations in (U, d) provided that both (U, d) are known at point 3 as well as at points 1 and 2. Undifferentiated U was replaced by the average U(14)/4. Chow seemed to have been dismayed by the difficulty of manipulating the general solution, whereas the real problem lay in the ill-posed specification of both U and d at the left boundary unless some other routine is implemented and unless the flow is supercritical. A corresponding situation occurs at the right boundary, of course.

The apparently straightforward differencing scheme of Figure 4.14b

/x=(21)/2x/t=30)/t

(4.32)

was shown to be unstable by Richtmeyer (1957) with gas flows in mind. Liggett and Woolhiser (1967) later demonstrated its unsuitability for the shallow-water equations. Abbott (1979) used it as an adverse example in his explanation of linear stability, and it should clearly be avoided. Since the

Figure 4.14 Direct finite-difference grid variations: (a) Thomas (1934); (b) unstable without Lax modification Q0=0.5 (Q1+Q2); (c) typical staggered grid system.


characteristics tend to transmit information from 1 and 2 to the forward point 3, the involvement of point 0 at the outset is misconceived. Replacing 0 by the average of 1 and 2 was proposed by Lax (1954). This is a stable scheme but liable to be strongly dissipative, causing excessive attenuation of the wave profile. It is also called the diffusing method for that reason.

A differencing scheme that is fully centred on a single grid point at 0 (Fig. 4.14b) is like that of Thomas, except that now two time steps and thus three lines of data are involved. This is known as the leapfrog method. The differences may also be fully centred but on two grid points displaced diagonally by one half time and one half space step. Such a scheme avoids the boundary difficulty through its alternating nature, allowing one variable to be input, either U or d, at each end of the problem. Internal solutions for U and d then appear at points on a staggered grid (a term often used to describe the method)see Figure 4.14c. Tidal flows in the Ems at Hamburg were thus calculated by Hansen (1957), being shortly followed by those for the Thames by Otter and Day (1960). The inconveniences of the alternating solution points are compensated by the fact that it remains explicitone variable being irrevocably calculated at each forward grid point, independently of its neighbours. A compromise that combines the diffusing and leapfrog methods is known as the Lax-Wendroff scheme (Lax & Wendroff 1960). All explicit difference methods tend towards variants of the characteristic approach, as indicated by Abbott (1979) and are subject to the Courant limitation for their stability.

If time and space differences are centred only on the half time step (as opposed to half-steps in both space and time), the forward solution for U and d becomes implicit. This then requires the simultaneous solution of a set of N2 equations along the entire space line of N grid pointsat the boundaries both U and d are supposed to be known. In principle, the method provides unconditional stability, which removes the Courant time-step restriction. In so doing there still remain those reservations regarding the initial (or starting) and boundary conditions and regarding grid resolution (points per wavelength), which apply to all simulations. Also any solution being continuous throughout x, such as is found implicitly, tends to smear those disturbances just entering the problem boundaries. This does not necessarily reduce the advantages of unconditional stability, especially where the motion is linked to mixing processes with inherently longer timescales (e.g. sediment, salt or pollutant movements). The balance is a subtle one, and readers are referred to Abbott (1979) and to Abbott and Cunge (1982) for its elaboration.

The position becomes yet more acute with the introduction of the remaining space dimensions, y and z. An excellent review of the whole range of numerical techniques for tidal flowsa large subset of shallow-water wave theorywas given by Liu and Leendertse (1978).


4.10 Supercritical flow

Although subcritical flow probably constitutes the major area of application of the method of characteristics, it is important to establish the technique for supercritical flows. These occur over comparatively short lengths, but are more difficult to control because of the violence with which they interact. Ellis and Pender (1982) have made use of the concept in the design of spillway chute calculations.

In supercritical flow U exceeds c, so that both of Uc take the same sign, namely that of U. The flow region then consists entirely of either forward or backward pairs of characteristics, as shown in Figure 4.15. This has the effect of restricting the propagation of disturbances to the direction of Uso that for U positive, conditions at the right boundary have no influence elsewhere. Either of the previous interpolation techniques may be applied once the direction is known, since this determines the integration routeto left or right of a particular time line.

Using the time-line interpolation for left-to-right flow gives =(U2c) as

3=0(4,50)G1,2

while

4,5=(U2c)4,5=(U2c)1+R1,2t on G1,2=(Uc)1

(4.33)

Figure 4.15 Forward characteristics for simulating unsteady, supercritical flow from a control at the left boundary.


Figure 4.16 Limiting the integration of subcritical flow as it approaches a transition to supercritical through the condition U=c.

In the above the subscripts are paired with the appropriatesigns, as before. Adding and subtracting (U2c)3 gives U3 and c3 as

U3=U0(1c1)+3U1c1U1[2c0t(R1R2)] c1[U1t(R1+R2)]

c3=c0(1c1)+1.5U1c10.25U1[2U0t(R1+R2) 0.25c1[4c0t(R1R2)]

(4.34)

Clearly (U,c)1 must be specified independently at the left-hand boundary. At the right-hand boundary both (U,c)3 are integrated from the internal flow.

How is the development of supercritical flow from subcritical flow recognized in the calculation routine? In the case of time-line interpolations, as U increases from left to right, the intersection at 5 of the backward characteristic from 2 becomes greatly delayed. When U=c, dx/dt=0 and point 5 is at infinite timethe corresponding position for space interpolation is that point 2 approaches point 0. Simultaneously the forward characteristic gradient approaches dx/dt=2cand also is the limiting influence on the time step. This is shown in Figure 4.16. Since the critical flow condition U=c is in any case physically unstable, it seems reasonable to restrict the numerical approach to it. Thus within a band abs(dx/dt)1, which implies only that . Since

(4.44)

it is clear that we may have U1/(gd1)1/2(gd1)1/2 implies U2>(1D1)(gd1)1/2

(4.45)

with 01/ and H/d>2

Both of these figures overestimate what really takes place by a factor of rather more than 2. This is because, as wave height increases, the prime Airy assumptions of relatively small particle speeds, linear free-surface condition and irrotational flow no longer apply. The breaking conditions usually quoted are

H/L>0.14 and H/d>0.78

(5.39)

The first of these was an interpretation by Michell (1893) of a condition already obtained by Stokes (1847). Stokes solved the equations of motion subject to both the complete free-surface condition and irrotational flow. His method was approximate and led to expressions for the Laplacian potential function as power series. These could be truncated so as to give .


Figure 5.22 Profiles of Stokes finite waves for different steepness in deep water.

a hierarchy of progressively dependent and ordered solutionsthe first of which coincides with Airys simple harmonic wave. The second-order solution for displacements contains characteristically sharper peaks and flatter troughs (Fig. 5.22). The waves progress with the Airy celerity but particles exhibit a net drift (Fig. 5.23), during one orbit, known as mass transport. Stokes found that, for particle speed to equal celerity, the deep-water wave peak contained an angle of 120. This was later converted to a steepness of 0.142 by Michell.

The notion of deep-water breaking is less easily envisaged than breaking at a beach. It has been argued by Silvester (1974) that deep-water breaking is an essential component of the energy transfer from wind to water, allowing a redistribution between waves of different frequencies. Thus Stokes assumption of irrotational flow might be questioned, since the absence of vorticity inhibits the development of surface shear effects. Nevertheless, Stokes wave calculations have been carried out by Longuet-Higgins and Cokelet (1976), showing realistic overturning of the steepest wave (Fig. 5.24).

It has been shown that waves progressing into shallow water contain a substantially greater horizontal particle motion than in the verticaland which is more or less uniform with depth. One might say that the oscillation pattern changes from local rotation in deep water to a mass

Figure 5.23 Showing the change of mean level and net particle displacement in the highest Stokes wave.


Figure 5.24 A numerically calculated succession of surface profiles showing the overturning of the steepest Stokes wave. After Longuet-Higgins and Cokelet (1976).

reciprocation in shallow water. The effects of a single, impulsive translation on the water mass, in a confined channel, were first recognized and investigated by Scott Russell (1840, 1845) and are well known. They constitute the solitary wave or, as he called it, the great wave of translation. The solitary wave may be regarded as the limit of shallow-water waves having a continuous profile and finite height. Complex harmonic waves in deep water may yet maintain a smoothly connected sequence. In shallow water this sequence tends towards a series of solitary waves. The features of the solitary wave are:

(a) profile entirely above the initial or equilibrium water level;

(b) non-oscillatory displacements; and

(c) a celerity close to [g(d+H)]1/2.

Since Scott Russells time, the wave has been much studiedas reported by Ippen (1966). Boussinesq (1877) was responsible for the most frequently quoted profile. Interestingly, this followed from a first approximation to the effects of surface curvature on the hydrostatic pressure assumption for long waves. McCowan (1891) applied Stokes notion for breaking to obtain the maximum relative height of 0.78 (Fig. 5.25).

It seems necessary to recall that the above treatments of finite-height waves apply strictly to water of constant depth. Other waveforms have been proposed, and tables of their properties have recently been given by Williams (1986). In practice, the uncertainty of estimating wave height itself from the wind, quoted by Torum (1983) as 20%, usually masks the discrepancies between the various finite wave theories. Furthermore, the


rolling axis. Once BM is known, the relative positions of M and G may be found and the stability position assessed.

A rectangular pontoon, S by L in plan, floating with displacement (or draught) D, has IA=S3L/12, while vol=SLD, so that BM=S2/12D. B is clearly D/2 below the water surface, and thus the maximum height PM of any additional load w, which raises G to M, can be found. The moment of w about M is then equal to that of W located at G. Usually, w is small compared with W (Fig. 1.14b).

EXAMPLE

As an example of the above, it is interesting to consider the barge used for transporting components of the Foyle Bridge by sea from Belfast to Londonderry in Northern Ireland. The voyage length was about 200 km around the North Antrim coastwhich is subject to strong wave and tide action. Details of the planning and execution of this manoeuvre are given by Hunter and McKeowen (1984), from which the following broad calculations were made (Photo 1.4).

The bridge unit shown in the photograph has a mass of 950 tonnes. The barge dimensions of 27.5 m91.5 m6 m give a displaced mass of 15000 tonnes. Thus

BM=S2/12D=(27.5)2/72=10m

(1.8)

Photo 1.4 A 180 m long, 950 t box-girder unit, for the sidespans of the Foyle Bridge, Londonderry. The unit is supported on a barge in Belfast harbour, prior to a voyage of some 150 km to site. Photograph by courtesy of John Graham Ltd, Dromore, Co. Down, NI.


Figure 5.25 The shape of the highest solitary wave, its celerity and the approximate particle displacement. Note that horizontal and vertical scales are in the ratio of about 1 to 2.

statistical treatment of wave occurrence seems presently to rely strongly upon linear wave interactionsand thus still on Airys theory!

5.9 Wave-height prediction from wind

The general approach to this problem is rather similar to that for streamflow prediction. One may attempt to prescribe the meteorology and thus to determine the consequences at the land (or sea) surface. Alternatively, one may proceed more directly via the records of occurrence of streamflow (or wave height). Often the latter are much less extensive than records of the most significant meteorological parameters, i.e. rainfall and wind speed, respectively. Both of these are correspondingly easier to measure than streamflow and wave height, although the extent of their influences is more difficult to assess in view of their remoteness. The position is changing, however, and global wave-height records are now available, e.g. British Maritime Technology (1987). For any particular locality, it remains likely that a joint approach will be appropriate.

The Scottish lighthouse engineer, Thomas Stevenson, actually attempted to measure wave force in his early years. This was followed by studies of wave height and, only much later, of wind speed. Remarkably, his first wave-height formula, in Stevenson (1852), took no account of wind speed, but only of the fetch distance (see Townson 1980):

H(feet)=1.5[F(miles)]1/2

(5.40)

Townson (1981) concluded that the measurements leading to this expression were probably made under similar wind speeds, of about 30 mph, over semi-enclosed areas of water. Since then the statistical mechanics of wave generation and growth have developed massively. They are well described by Leblond and Mysak (1978) and also by Kinsman (1965). There follows


here a rather rudimentary derivation of a parametric relationship for the fully arisen sea. This may help towards a hydraulic understanding of the physical system involved. However, it is not claimed to provide other than a general magnitude indication.

5.9.1 Concept of a fully arisen sea

The fully arisen sea is based on the idea that, if wind blows for an indefinite duration at one speed over a sea of limited extent, an equilibrium wave condition develops. This may be typified by a significant wave height and frequency, which together represent, in a limited way, a spectral distribution of the energy being transferred to the sea surface. Such spectra have already been derived for the Atlantic by Pierson and Moskowitz (1964) and also for the North Sea by Hasselmann et al. (1973). These show the spread of wave height among a range of frequencies.

Suppose that air moves steadily with a mean velocity U over still water. Unless U is very small, the air flow is turbulent and contains pressure fluctuations. These disturb the water surface and create ripples. If U sufficiently exceeds the celerity of these ripples, energy is transferred by mutual shear action and exchange of vorticity between air and water surface. The ripples then become progressively larger waves until their celerity approaches the wind speed and no further growth takes place. Since the water particles at the wave peak are then also moving with the wave celerity, a breaking condition exists, which limits the wave steepness.

Whatever the wind speed, the wave height remains zero at the windward shore. In the seaward direction there develops an air boundary layer whose mean velocity distribution in the vertical determines the energy exchange. The thickness of the layer at any point (Fig. 5.26) depends on the distance X from the windward shore, i.e. the fetch. Furthermore, the time taken to

Figure 5.26 Depicting the growth of an atmospheric boundary layer alongside a hypothetical development of the steepest monochromatic waves.


reach such a thickness depends on the time taken by the wind to travel over the fetch, and is at least X/U. A fully arisen sea is one in which the wind duration has been sufficient for maximum wave steepness to be reached consistent with fetch and wind speed. The energy flux lost by the wind is by then equal to the energy flux carried by the waves, both through a vertical plane at the end of the fetch. This assumes both that the wind always blows and that the waves remain progressive in the X direction only. For a large sea area, this is far from the reality. It is also assumed that the waves are always in deep water, height variations towards the coast being assessed otherwise, via shoaling and refraction, etc.

As in calculations for wall friction, some knowledge of mean velocity distribution is required to proceed further. First, suppose that this is a smooth turbulent one-seventh power law, as for a flat plate, so that

u/U=(z/)1/7

The final rate of energy loss in the air boundary layer is

a(U2u2)u dz=7aU3/40

(5.41)

The rate of energy transmission for waves of average height H and travelling at group celerity (0.5 of wave celerity) is

P=wgH2c/16

(5.42)

For waves at maximum height and steepness in deep water, with period T,

H/L=1/7L=gT2/2c=gT/2

so that

P=wg4T5/6272

(5.43)

Equating 5.42 and 5.43 gives

T5=4.8(U/10)3

(5.44)

Following the Blasius flat-plate momentum theory for turbulent boundary-layer growth, suppose that

/X=0.375(UX/v)1/5

(5.45)

Taking v=1.5106 m/s gives

=0.025(X0.8/U0.2)T5=12(U2.8X0.8)105


i.e.

T=0.16(U0.56X0.16)

(5.46)

and if

H=L/7=gT2/14=0.22T2

then finally

H=5.63(U1.12X0.32)103

(5.47)

with

T=(4.48H)1/2

(5.48)

This height and period may be reduced if the sea state is not fully developed, because either:

(a) the wave celerity is less than the surface wind speed Us, defined as

Us=(H/)1/7U

(5.49)

i.e. wave age c/Us

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