Whether or not 'separation' occurs in the airflow over wind-generated water ... is supported by the form drag of basic gravity waves, and that flow separation is.
A NOTE
ON ‘SEPARATION’ P. R. GENT* Dept. of Oceanography,
OVER
SHORT
WIND
WAVES
and P. A. TAYLOR** University of Southampton, U.K.
(Received 20 July, 1976) Whether or not ‘separation’ occurs in the airflow over wind-generated water waves is partly a question of semantics but also has an important bearing on the wave generation process. In the present paper we use a rather formal and perhaps narrow definition of ‘separation’ and show that it does not occur where the shear stress is zero but only in conjunction with wave breaking. This is unlikely to happen except in the presence of quite strong surface drift velocities in the water. A similar connection between separation, surface drift and wave breaking has recently been established by Banner and Melville (1976). The effects of increasing wave amplitude or steepness are investigated with a numerical model of the airflow over water waves. Variations in the depth of the closed streamline region are predicted. The model is also used to investigate the possible importance of surface drift velocities. Abstract.
1. Introduction In the study of wind wave generation, ‘separation’ has formed the basis for one theory (Jeffreys, 1925, 1926) and is a term frequently used by many experimentalists. Different authors have different criteria for assessingwhether ‘separation’ has occurred. For example, Chang et al. (1971) assumed separation had occurred because “the measured turbulent quantities consistently showed the characteristics of separated air flow” while Wu (1969) states: “After the occurrence of wave breaking (at high wind velocities), the similarity (in both trend and magnitude) between the average wave height and the roughness indicates that the wind stress is supported by the form drag of basic gravity waves, and that flow separation is likely to occur along the basic wave profile.” In addition Wu proposes that separation will occur for “waves having a phase velocity less than the shear (friction) velocity.” Shemdin (1969) states that “The deviation from simple harmonic of the velocity profile measured in the trough is used as a basis for establishing a criterion for flow separation over waves.” These three papers, in this context, are discussing short waves with phase speed c = uo, where u. is the friction velocity; and Chang et al. (1971) do suggest that for longer waves, with higher values of c/uo, separation need not occur at points where the surface shear stress G-is zero. This is proved for waves of all wavelengths in Section 3 of this note. We define separation, as does Longuet-Higgins (1973), as the occurrence, in a frame of reference moving with the waves, of a streamline leaving the water surface (which is itself a streamline). The essential distinction to be made is between separation as defined above and a thickening of the closed streamline * Present Address, NCAR, Boulder, Colorado, U.S.A. ** On leave 1976, Atmospheric Environment Service, Downsview, Ont., Canada. Boundary-Layer Meteorology 11 (1977) 65-87. All Rights Reserved Copyright @ 1977 by D. Reidel Publishing Company, Dordrecht-Holland
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region. This is clear from the work of Lighthill (1962), Phillips (1966, p. 134) and Stewart (1967). In Section 4 we use a numerical model to investigate the effect of increasing wave amplitude and in Section 5 we consider a case where true separation can occur as a result of high near-surface drift velocities in the water. 2. Flow Patterns and Definitions For simplicity and almost of necessity, we consider the airflow above a monochromatic, two-dimensional, uniform wave train. This is the traditional way in which waves have often been considered but we should note that, especially in the sea, a spectrum of waves is present and, as Stewart (1974) emphasises, threedimensional effects may be important. Even in this relatively straightforward situation, there is an underlying difficulty with the separation terminology due to the movement of the air-water interface. Let us first consider the case of a fixed, rigid, wavelike boundary. Early experimental studies of this situation by Stanton et al. (1932) and Motzfeld (1937) were used by Ursell (1956) to illustrate that some of the assumptions made by Jeffreys (1925) in his sheltering hypothesis were quantitatively incorrect. We will suppose the airflow to be basically a deep turbulent boundary layer with no thermal stratification which, for zero amplitude waves, would have a logarithmic mean velocity profile. We assume that the flow satisfies the boundary condition a = 0 on the (rough) surface. For steady flow above small-amplitude waves there would be perturbations to the mean flow induced by the waves but no separation and the streamlines would be as shown schematically in Figure 1. As the amplitude a, or maximum slope ak, increases, a limit (which depends upon k.q, where k is the wavenumber and z. the surface roughness) will be reached beyond which a totally different streamline pattern will emerge as in Figure 2. This flow will now possess: (a) A region of closed streamlines bounded by the surface and a separating streamline. (b) A portion of the lower boundary, z = zs, (between S and I?) where we would expect negative surface shear stresses, T$.If we suppose that some sort of gradient transfer relation holds for momentum, then the surface shear stress will necessarily be negative beneath the closed streamline region.
Fig. 1.
Flow above a small amplitude rigid wave.
A NOTE
ON ‘SEPARATION’
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61
WAVES
Fig. 2. Flow above a rigid wave with separation (S) and re-attachment (R).
(c) A point of separation, S, where a separating, mean-flow streamline leaves the boundary, and a point of re-attachment, R. For a fixed lower boundary, these three features can only occur together. We now consider flow above moving water waves. If we choose a frame of reference stationary with respect to deep water (i.e., below the level affected by the waves), then streamlines and particle paths do not coincide and streamlines would intersect the instantaneous water surface almost everywhere. In a frame of reference moving with the phase speed, c, of water waves, however, we can, under appropriate circumstances, regard the airflow as steady and particle paths and streamlines will coincide. In this frame, which we will use, the wave surface is a streamline. If a> 0 and c>O (i.e., the wind and waves are in the same direction) then Lighthill (1962), Phillips (1966) and Stewart (1967) have shown that a region of closed streamlines centered on the critical level, where 0 = c, must occur if there is any periodic variation of the mean flow with position along the wave. This closed streamline region (Figure 3) will not extend right to the surface and there will be no separation or re-attachment points. The diagram given by Phillips (p, 91) is appropriate to rather longer waves (perhaps with c - 25 u. and Us/c = 1) than we will consider here and shows the critical level and closed streamline region considerably above the water surface. In many short-wave cases, we will have U,, the wind at 5 m, significantly greater than c. A typical value might be AIR
-
WATEd SURFACE
SlNGULAd POINTS OF STREAMLINE PATTERN
Fig. 3. Streamline pattern for flow over moving small-amplitude water waves. -..Water surface.
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Us/c = 2.5, and the critical level will be even closer to the surface than indicated in our Figure 3. We will show in Section 3 that when there are portions of the surface on which r CO it does not fundamentally alter the streamline pattern. In order for separation to occur there must be a genuine stagnation point at the surface in the frame of reference moving at speed c as suggested by Longuet-Higgins (1973). This is also used as the criterion for incipient breaking by Banner and Phillips (1974) and it requires points on the surface where the horizontal velocity of the air, u = c. If water and air particle velocities are matched at the surface, and water velocities correspond to those for irrotational waves, then the lower boundary condition is u-c = -c[l - ak cos kx], where a is the wave amplitude and k the wave number, and separation cannot then occur for ak < 1. This boundary condition cannot be expected to hold exactly but it is unlikely that it is so incorrect as normally to permit a stagnation point at the surface-however see Section 5. For water waves, therefore, we have three separate features: (a) A region of closed streamlines. This is always present if c >O and U is a function of x but the region does not extend down to the water surface unless separation occurs. This is shown in Figure 3. (b) There may be regions of negative surface shear stress but the streamline pattern is fundamentally unaltered from that in Figure 3. This will be shown in Section 3. (c) Separation. This requires stagnation points at the surface and that the closed streamline region intersects the surface (as in Figure 15). It does not normally occur except possibly for very short gravity and capillary waves.
-
(4 -
:r; ‘\‘.-I Fig. (hi 4.
Streamline patterns for wind blowing against the waves (c < 0). (a) Small-amplitude waves. (b) Possible flow pattern for very large amplitude waves; x marks elevated stagnation points.
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In the alternative, decaying swell, situation where the wind is blowing against the waves, c < 0 and the normal streamline pattern will be as illustrated in Figure 4a. This case has no region of closed streamlines. Note that very close to the surface the streamlines will be closer together in the trough than over the crest as a result of the boundary condition of matching the water velocities at the interface. The only alternative, provided us > c (negative), would be a closed mean streamline pattern, something like that illustrated in Figure 4b. We believe it highly unlikely that this would occur except possibly for very short, slowly propagating waves. Two stagnation points would need to be present but they would not be on the surface. Regions of negative surface shear stress can occur without the closed streamline pattern and are compatible with the streamline pattern of Figure 4a. 3. Heuristic Arguments We will not formally prove that the nature of the streamline patterns sketched in Figures 3 and 4 are correct in all the cases indicated but we can give heuristic arguments based on anticipated forms for the velocity profiles above the crest and trough of the waves. In Figure 5, velocity profiles (Z?- c) above the crest and trough of the wave are sketched as functions of 5 =ln ((z - z, + zO)/zo),where z. is the ‘local, small-scale’ value of the surface roughness length. Surface velocities are matched to the orbital velocities of irrotational waves in the water. The basic profile is assumed to be logarithmic and z. assumed constant. We have been influenced in some details of these sketches by our numerical solutions (Gent and Taylor, 1976) but the arguments only depend on their gross features. For the profiles, assuming ~,a(8 u//az),, the solid line corresponds to the situation with rs > 0 while the dotted line indicates what might happen if TV< 0 at either crest, T:, or trough, r:, (see Section 4). We can visualise integrating these profiles to obtain the stream function
The surface is set as $ = 0. Provided u, - c < 0, I+IJwill be negative close to the surface above the trough, the crest and at all other horizontal positions indicating the existence of continuous streamlines following the surface. The minimum value of $ above the crest is assigned the value -6 and occurs at the critical level 5:. Above the trough we can reasonably expect that IF,will attain lower values than -6 and we see that there must then be a region of closed streamlines with 9 < -8, centered on the critical level, l:, which does not extend as far as the crest. In fact the region of closed
70
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VELOCITY PROFILES
-c -c(l-ak)
0
-c(l+ak)
-c
STREAM FUNCTIONS
REGION OF CLOSED STREAMLINES NOT EXTENDING BEYOND i CREST
/I’
I 1\ \ ’ \\L‘Il -6 (NON-LINEAR SCALE) CREST
_L (NON-LINEAR
>
SCALE)
TROUGH
Fig. 5. Schematic diagram of velocity profiles and resulting stream function for flow over waves with Normal, small-amplitude situation, 7S> 0. - -- - modified profiles if 7S< 0. Critical levels; c>o. [:, If only marked for 7S> 0 cases.
A NOTE
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streamlines above the trough will be slightly deeper than this if the singular points of the streamline pattern (Figure 3) do not lie immediately above the crests and have I/J> -6. Note that the scale for J, in Figure 5 is somewhat non-linear and that the argument is unaffected by the sign of rs at either the trough or crest. Similar arguments can be given to justify the streamlines sketched for the c < 0 case, the anticipated velocity and stream-function profiles being shown in Figure 6. Note in particular that the profile over the trough would need to be very highly distorted for the closed streamline pattern to develop when c < 0. 4. But What is the Effect of Increasing Wave Amplitude? Although we are convinced that there will be no discontinuous change in the streamline patterns for air flowing over waves with c > 0 as the amplitude of the wave increases, we certainly suspect that considerable changes will take place; and the intuitive feeling that “something like the separation over a fixed wave” should occur at large enough amplitude is a strong one. A possibility might be a large increase in the depth of the region of closed streamlines and a gradual?, or perhaps quite abrupt? transition to the type of streamline configuration, suggested by Stewart (1967, Figure 6), with a deep closed streamline region sketched in our Figure 7. In order to investigate this, we have adapted our numerical model of airflow above water waves (see Gent and Taylor (1976), subsequently referred to as G&T) to cope with increased wave amplitude and regions of negative shear stress and we have run it for a range of wave amplitudes. The number of cases considered is limited mainly because, with a 20 x 15 finite-difference mesh, each run takes about 30 min of CDC 7600 computer time. Results obtained are with R = -In (kz,J = 8 and c = 8u,, corresponding, with z. = 0.01 cm, to a 1.87-m wave for which c = 1.71 m s-l and, for small a, U5 = 5.8 m s-l and with R 2: +5 and c = 1.5uo corresponding approximately to wind-tunnel waves with, say, wavelength L = 30 cm, z. = 0.03 cm, c = 0.65 m s-l and u. = 0.43 m s-r. These wind-tunnel values are estimated approximately from Chang et al.‘s (1971) data although u. is perhaps rather low and z. rather high. Some results have also been obtained with R = 5 and c = 0. As in G&T, the friction velocity, uo, used for scaling purposes is based on the sum of shear stress and form drag on the waves which, for the equilibrium assumed, is equal to the total turbulent shear stress well above the waves. Keeping c/u0 fixed implies some slight decrease in Us/c with increasing wave amplitude. The flow is assumed to be steady and periodic in x, the horizontal coordinate. Both cases considered here are for quite short waves; and the results to be presented go beyond the steepnesseswe can expect to encounter either in the sea or in the wind tunnel. Short waves are studied in part because of computer limitations but also because, in nature, they are likely to be steeper than longer, faster waves, which will require a much larger input of energy to attain steep
72
P. R. GENT
-c(l-ak)
AND
P. A. TAYLOR
-c
5s POSSIBLE CLOSED
STREAM FUNCTIONS
(NON-LINEAR CREST
SCALE)
(NON- LINEAR SCALE) TROUGH
Fig. 6. Schematic diagram of velocity profiles and stream functions for c 0.
zero amplitude waves) will be 23.5.~~ or 0.24 cm for R = 8 and only 0.82~~ or 0.02 cm for R = 5. In the latter case this is clearly well down among the very short capillary waves which we have assumed are providing the surface roughness. Amplitudes of such waves would be about 0.15 and 0.45 cm in the two casesif we take 2a ~302~ (the usual rule of thumb for sand-grain roughness). We should note that the model assumes the near-surface flow to be aerodynamically rough and ignores details of the viscous sublayer - whose thickness (SV/U,) would be about 0.04 and 0.02 cm for the R = 8 and R = 5 examples, respectively. Strictly speaking, our R = 8 example, for which uOzO/v= 1.3 does not quite satisfy the criterion for aerodynamically rough flow (Ursell (1956) suggests uOzo/v> 3) but it is far from being aerodynamically smooth. The R = 5 example has uOzO/v2: 8.6. In both examples, but especially in the R = 5 case, the critical level above the crests will lie very close to the surface in a region which the model treats in a highly idealized manner. This is inevitable in any model which does not wish to consider the details of the flow around the individual ‘roughness elements’. Although we agree with Townsend (1972) that ‘. . . the critical level . . . is merely an unimportant part of an equilibrium layer if turbulent stressesare included . . . ,’ we must admit to some dissatisfaction over this feature of the model when applied to short waves. For longer waves and lower values of Us/c, the critical height and the region of closed streamlines can be much farther from the surface (at 5 m if Us/c = 1) and we would anticipate a rather different behaviour for increasing ak from that to be presented here. We hope to explore this region of our parameter space at a later time. In Figure 8 we show the streamline patterns computed for the R = 8, c = 8uo case with approximate maximum wave slopes, ak = 0.157, 0.314 and 0.471. The frame of reference is moving with the waves and the upper flow direction is from left to right. Note the different vertical exaggerations in these figures. Although not shown, all cases have continuous streamlines close to the surface. At ak = 0,157, the depth of the closed streamline region is quite small in comparison to the wave height but in the other cases it grows to fill and then extend above the trough region between the crests. For ak = 0.314 and 0.471, we see the upper $ = 0 streamline ‘breaking away’ from the near-surface region at the crest in much the same way as in the case of near-surface streamlines at a separation point for a fixed wall boundary layer with high curvature. The surface shear stress is, however, everywhere positive for ak = 0.3 14; and for ak = 0.471, there is only a
14
P. R. GENT
AND
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2.0
OUTER FLOW DIRECTION
E IO 2 a, H 0 .u t 3 40 0.0
-WATER
-IO
0.2
O-6
0.4
0.8
SURFACE
I.0
I.2
X/L CREST
TROUGH
CREST
64
2.0
% I.0 2 w 5 .o t s 40 0.0
-1.0
02
0.6
0.4
0.8
IO
I.2
X/L CREST
TROUGH
CREST
(b)
Fig. 8. Streamline patterns for R = 8, c = 8uo cases. (a) ak = 0.157 (b) ak = 0.314 (c) ak = 0.471. -.-.Water surface; - - -- Critical height; x and z are horizontal and vertical positions relative to the wave crest and mean water level. Stream function, 4, scaled w.r.t. ~a.
A NOTE
ON ‘SEPARATION’
I
I
0.2
0.4
OVER
SHORT
WIND
15
WAVES
I
I
I
06
0.8
I.0
I
1.2
X/L CREST
TROUGH
CREST
(4
Fig. 8 (conr'd).
small negative stress region just forward (in the + c direction) of the crest. Surface shear-stress and pressure results for these cases and for ak = 0.628 are shown in Figure 9, which also shows the changes in surface wave shape that we have assumed for increasing amplitude (see G&T for details). We can see that for these values of R and c, the shear-stress minima and negative stress regions for the larger wave amplitudes are near the crest rather than in the trough. They are closely linked to the surface velocity condition, U= cak cos kx and the negative stresses occur where, in a stationary frame, the wave-induced surface velocities are faster than the airflow just above. Velocity profiles above the crest and trough are shown for ak = 0.314 and 0.628 in Figure 10. They are plotted as functions of 5 which, in these positions, is In (1 +(n/z,,)) where 71= z ‘F aeLk’ (see G&T for details). The influence of the surface boundary condition on U is clearly dominant in the ak = 0.628 cases. “Kinks” in the surface pressure results (Figure 9) are possibly associated with this surface-velocity condition but have some similarities with Stanton et al.‘s (1932) results for very steep fixed waves. For the ak = 0.628 case, there was slight contamination from a two-grid-point instability, but otherwise we have confidence in the numerical aspects of the model. The turbulence closure hypothesis (see G&T (1976) for details) must be rather suspect,
P. R. GENT
AND
P. A. TAYLOR
I
I
I
h’ ,y A--,
00
p-
/
\
\ \
nk.n.TlA
,
I\ i
-It-
-4.0
-8.0 I.2
i-,
-.-.-,r
X/L TROUGH
CREST
TROUGH
A NOTE ON ‘SEPARATION’
OVER SHORT WIND WAVES
(D-cl/u,
t&d/u,
Fig. 10. Velocity profiles above the crest and trough in the frame moving inrith the waves. Crest Profile; ---- Trough Profile; R = 8, c = 8u u. Basic results with no surface drift. ak = 0.314, ak = 0.628.0 Results with q = 2uu. (See Section 5).
especially at high wave amplitude, but we consider that the streamline patterns and general features of the flow will not depend strongly upon them. Returning to Figure 8, it would appear that a good indication of the depth of the closed streamline region will be given by the height of the II, = 0 streamline above the surface. We have computed this height above the crest, trough and the centre of the closed streamline region or “cat’s eye” for a wide range of wave amplitudes at R = 8, c = 8uo and also for some caseswith R = 5, c = 1.5uo and for a fixed wave with R = 5. These results are given in Tables I, II and also in Figure 11. Results for a = 0 are simply obtained from the requirement that
z(G=O) u.lnz+z. J ( 0
K
--c
20
dz=O,
1
where K = 0.4 is von Karman’s constant, and are shown in Figure 11 as a limit if there were no flow deformation induced by the waves. For both R values, the increase in the height of the I(,= 0 streamline above its “no flow deformation” position at the trough appears smooth and continuous but is quite rapid between ak = 0.2 to 0.4 for R = 8, c = 8uo and between ak = 0.3 and 0.4 for R = 5 and c = 1.5~~. As can be seen from Figure 8c, although the I) = 0 streamline starts to +Rg. 9. Surface shape, z,, shear stress, rs, and pressure, pr R = 8, c = 814~. ak = 0.157, 0 Results with surface drift q = 2uo, ----_- ak = 0.314, 0 Results with surface drift q = 2u,, -.-.ak=0.471 (see Section 5). - - -ak=0.628
Note that this figure is centred on the crest. ab is wave amplitude when ak = 0.157.
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TABLE I Height of (lr = 0 streamline above wave surface R = 8, c = 8u0
z(JI=O)-2, ak a/z0 (L = 18,730zo) 0 29.8 149.1 234 298 468 674 937 1405 1873
0 0.01 0.05 0.0785 0.1 0.157 0.226 0.314 0.471 0.628
Crest
-
Trough
-
61.3~~ 59.6zo 51.6~~ 44.520 39.8zo 36.3~~ 33.0zo 67.4~~ 891z,, 1641~~
2.00a
0.35a 0.19a 0.13a 0.08a
0.05a 0.07a 0.63a 0.88a
‘Cat’s Eye Centre’
-
61.32, 65.0~~ 80.820 107zo 120zo 199zo 402~~ 11512, 37962, 5588~~
2.18a 0.54a 0.46a 0.40a 0.43a
0.60a 1.23a 2.70a
2.98a
2.19a
0.56a 0.50a 0.46a 0.50a 0.74a 1.44a
2.58a 2.76a
move farther away from the surface at the crest, its height above the trough still gives a good estimate of the depth of the closed streamline region for the high amplitude cases. The depth of the region appears to be approaching a limit at large ak for R = 8. For a given wave slope, ak, the relative depth of the closed streamline region is much higher for the short sea-wave case (R = 8) than for the wind-wave tunnel example where at ak = 0.314 as shown in Figure 12, the “cat’s eye” is quite small and would be difficult to observe experimentally. The two casesfor c = 0 listed in Table II show genuine separation for ak = 0.314 and 0.471. In the casesinvestigated, the closed streamline regions were smaller than those for c = 1.5~~ by about 0.4a. Surface shear-stress and pressure distributions for the R = 5, ak = 0.471 cases are shown in Figure 13. The surface shear stressesare low everywhere except just upstream of the crest where they peak sharply. This is in considerable contrast to
Height of JI = 0 streamline
TABLE II above wave surface R = 5.07 (L = 1000~~)
z($=O)-z, a/z0 ak c=1.5uo
Crest
Trough
0 12.5 25 37.5 50 75
0
-
1.7920
-
0.0785 0.157 0.235 0.314 0.471
O.lOa
1.25~~
0.04a 0.03a
1.0~~
0.20a 0.16a 0.19a
c=o 50 75
0.314 0.471
‘Cat’s Eye Centre’
-
1.7920 2.512, 3.892, 7.320 16.0~~
0.42a
0.21a 0.18a 0.24a
0.02a
1.0~~ 1.0~~
O.Ola
0.7~~
0.32a 0.99a
74.4~~
1.07a
0 0
0 0
0 0.54a
0 40.5~~
0.66a
0.03a
1.7920 2.61~~ 4.61~~ 9.1420 21.0zo 80.620 1.5zo 49.8zo
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ak
cm i 0.0
(b)
0.6
Fig. 11. Height of Q=O streamline above local water surface. 0 Trough, +‘Cat’s Eye’, XCrest. - Limit with no deformation of flow. - --- Distance between this line and no deformation curve gives indication of maximum depth of closed streamline region. (a) R =8, c =8uo; (b) R =5.07, c = 1.5u,.
2’o-1
I
-
I
0.2
0.4
0.6
0.8
I.0
1,
X/L CREST
TROUGH
Fig. 12. Streamline patterns with R = 5.07, c = 1.5~~; ak = 0.314. - .-.
CREST
-
Water surface.
P. R. GENT
-04
-0.2
AMI
p. A, ‘,-AmGR
0.0
0.2
o-4
X/L TROUGH
CREST
TROUGH
Fig. 13. Surface shape, z,, shear-stress, T,, and pressure, ps. distributions, R = 5.07, ak = 0.471 -c = 1.5t.40, -- -- c = 0. Figure centred on crest.
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81
the R = 8, c = 8uo, cases shown in Figure 9. The pressure results, which are far from the ideal sinusoidal form predicted by linear theory, show a sharp minimum at the crest with a maximum on the upwind side. Again there are considerable differences between these results and those in Figure 9. For the fixed-wave case (c = 0) at ak = 0.471, we note that the region of negative surface shear stress is much longer than for c = 1.5~~ and covers more than one half of the wavelength. The closed streamline region is, however, larger for the moving water wave. Velocity profiles above the crest and trough are shown in Figure 14 for ak = 0.314 and 0.471. These fail, except for ak =0.471 and c=O, to show velocity maxima of the type observed by Chang et al. (1971) at ak-0.31. This can probably be attributed to the different outer flow conditions prevailing in our model compared to those in their experiment. To summarize briefly, we can say that for the waves studied here, increasing wave amplitude causes an increase in the relative depth of the closed streamline region. For ak < 1.0, there is no separation in the sense used in this paper. The surface shear-stress distributions for the R = 8, c = 8uo case show very short negative regions but they are only just downstream of the crest for ak = 0.471 and even extend upstream of the crest at ak = 0.628. In this respect, the results for R = 5, c = 1.5~~ are closer to the behaviour for a fixed wave and, when negative surface shear stresses occur (at ak = 0.471), they are spread over a larger region on the downwind slope of the wave. In many respects, the behaviour at the low values of c/u0 (1.5) is distinctly different from that at higher values, indicating that considerable caution is needed in relating wind-wave tunnel observations to sea waves.
‘i
1
/
1
A
ak= 0314
Fig. 14. Velocity profiles above the crest and trough in the frame moving with the waves. Crest Profiles, ---- Trough Profiles. R = 5, c = 1.5~ ,,. Basic results are with no surface drift, ak = 0.314, 0.471. 0 Results with q = 0.5~~. XResults with c = 0.
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5. Surface Wind Drift Effects and Separation Provided we insist that air and water velocities are matched at the surface, then our condition for separation- a stagnation point at the surface in a frame of reference moving with the waves-is also the condition for incipient breaking of the waves used by Banner and Phillips (1974). Under these circumstances, we can sketch the streamlines in both air and water in a frame of reference moving with the wave as in Figure 15. Note that the surface separation and re-attachment points for the air and water coincide. Banner and Phillips consider the near-surface drift produced in the water by the action of the wind stress at the surface and obtain conditions under which the waves would break. In particular, they show that for non-breaking waves, the tangential surface speed in a frame of reference moving with the waves is given by
where U(s) is the irrotational motion (-c + irrotational wave orbital velocities) as a function of position, s, on the wave and q is the wind drift at the point where the wave profile crosses the mean water level. Typical values for q are reported as 3-4% of the wind speed at 10 m; the requirement that u, G 0 at the crest gives a condition for incipient breaking as (l-ak)* 0.47 m s-l. This could well occur for a wind speed of about 12 m s-r. If ak = 0.157, the corresponding limit is q > 0.79 m s-l which is less likely. To investigate the effects of surface wind drift, without breaking, on the airflow above the waves, we have rerun two cases with R = 8, c = 8u0, ak = 0.157 and SEPARATION POINT
RE-ATTACHMENT POINT
Fig. 15. Schematic diagram of mean-flow streamlines in the air and water in a frame moving with the waves. Surface drift velocities large enough to lead to separation. --+-- Streamlines in air, lowest streamline is water surface. -- f - - Streamlines in water.
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83
WAVES
IO.0
50
/
N, 5 k2
00
2 2 g
-5.0
Y iI3 3 -00 co -15.0 IO
No z
5.0
s 2 0.0 2 iif 8 -5.0 i2 [r 54 -10.0
-15.0
/
c
-0.4
A -0.2
0.2
0.0
0.6
0.4
X/L TROUGH
CREST
TROUGH
Fig. 16. Surface pressure distributions with and without surface drift. R = 8, c = 8uo. ---- q = 2uo. (a) ak = 0.157. (b) ak = 0.314.
q=o,
0.314 with q = 2u,, or 0.25~. The effects were quite pronounced and indicate that surface wind drift may well be an important factor in determining some features of the airflow above the waves. There were only small changes in the predicted surface shear-stress results which are shown in Figure 9. Surface pressures, however, were modified quite dramatically. Figure 16 shows that for ak = 0.157, there is a marked increase in amplitude (from about 15.4~~; to 25.6~~:) and slight change in phase but no large change in shape while for ak = 0.314, the change in amplitude is accompanied by a change in the shape of the surface-pressure curve to one with a sharp minimum at the crest not unlike those obtained in the c = 1.5uo, R = 5 case.
84
P. R. GENT
AND
P. A. TAYJ.OR
These changes in surface pressure distribution are balanced in such a way as to leave the form-drag contributions to total horizontal stress on the lower boundary essentially unchanged. Values are 19 and 55% compared to 19 and 57% when q = 0, and so there will be no increase in the energy input to the waves due to the working of the pressure. There will, however, now be the possibility of a significant energy transfer, to both mean drift and wave energy as a result of the working of the surface shear stresses. Velocities are increased relative to the deep water but, as shown in Figure 10, the shape of the profile is not changed qualitatively. The main change in the velocity fields comes directly as a result of the change in lower boundary condition on the velocity and, as we would expect, the region of closed streamlines is considerably reduced (see for example Figure 17). Its depth, Table III, is reduced by about 50% in the R = 8, c = 8u0 case for q = 2~~. The critical level and JI = 0 streamline come very close to the surface near the crest but, formally, there are still no separation points. For R = 5.07, c = 1.5~~ and with ak = 0.314 and 0.157, the critical values of 0.272~ and 0.462~ for q correspond, in our example, to surface drifts of about 0.18 m s-l and 0.30 m s-l. In Chang et al’s (1971) experiments with a wind speed in the tunnel of about 10 m s-l, it is possible that the near-surface drift could attain values of up to 0.3 m s-l and that true separation would occur as a result. Although Banner and Phillips’ analysis leading to our Equation (1) is only valid for non-breaking situations, we have chosen, perhaps recklessly?, to apply it to a breaking wave situation by setting U, = 0 if V”(s) < q(2c - q). The case considered is for c = 1.5uo, q = 0.5uo, R = 5.07 and ak = 0.314 corresponding approximately to Chang et ul.‘s wind-tunnel situation. There is now genuine separation of the flow in a frame moving with the waves but, as in the longer wave case, the closed
-1.0 0.0
0.2
0.4
0.6
0.8
I.0
I.2
X/L CREST
TROUGH
CREST
Fig. 17. Streamline patterns with surface drift. R = 8, c = 8~0, ak = 0.314, q =2uo. ---- Critical level, - . -. - Water surface.
A NOTJ?
ON ‘SEPARATION’
OVER
SHORT
WIND
85
WAVES
TABLE III Height of JI = 0 streamline above wave surface-computations
with surface drift
z(JI=O)-2, ak
4
Crest
R=8, c=8uo 0.157 0 0.08a 36.3~~ 0.157 2us 0.02a 11.22s 0.314 0 0.07a 67,420 0.314 2uu 0.005a 4.72, R = 5.07, c = 1.5~~ 0.157 0 0.04a 1.02, 0.52s 0.157 0.5us 0.02a l.Ozs 0.314 0 0.02a 0 0.314 o.5uo 0
Trough
“Cat’s Eye Centre” Comment
0.43a 199ze 91.7z, 0.20a 1.23~ 1151~~ 0.57a 533~~
0.50a 23520 0.24a lllzu 1.44a 134420 0.72a 677~~
0.16a 3.92, O.lla 2.7~~ 0.32a 16.0~~ 0.23a 11.4~~
O.lSa O.lla 0.42a 0.30a
No separation No separation
4.6~~ 2.8~~ No separation 21.ozu 14.8~~ Separation
streamline region is reduced in depth as a result of the surface drift. As shown in Table III, the reduction is of order 30% both in this case and in a non-separating case with ak = 0.157. Mean velocity profiles are again shifted to the right (seeFigure 14) but with no major change in shape while surface shear stresses and, for these values of R and c/uO, surface pressures are only slightly modiii;d (Figure 18). Thus for values of c/u0 near 1.0, the effects of surface drift velocity, even though they may lead to separation of the airflow, are not very pronounced with the assumptions used in the present model. It is, however, conceivable that the associated wave breaking would modify the flow in other ways, for example, by producing changes in surface shape or variations of surface roughness with position on the wave. (See Gent and Taylor (1976) and Banner and Melville (1976)J 6. Conclusions Our initial intentions were to write a short note to try to clarify what had been and what should be termed ‘separation’ for airflow over water waves and to show that the Jeffreys (1925, p. 192) analogy between separation over water waves and of flow past a fixed solid sphere (the sheltering hypothesis) is inappropriate. The question of exactly what does happen to the airflow above steep waves then naturally arose and our note grew as a result of applying our numerical model to this situation. We would, however, like to return to our original aim and make the following statements: (a) Separation in a frame of reference moving with the waves (which is the only sensible frame to use for streamlines) can only occur in conjunction with wave breaking. If ak < 1, then surface drift velocities are needed to enable this to happen. (b) For c# 0, negative surface shear stresses and separation are distinct phenomena and neither implies the other.
P. R. GENT
AND
P. A. TAYLOR
,I -04
1 -0.2
02
0.0
1 06
04
X/L TROUGH
CREST
TROUGH
Fig. 18. Surface shear-stress and pressure distributions with and without surface drift, R = 5.07, q = 0.0 q = OSU@ c = lSu,, ak = 0.157, 0.314. -
A NOTE
ON ‘SEPARATION’
OVER
SHORT
WIND
WAVES
87
(c) For c > 0, the effect of increasing wave slope is to produce a thickening of the closed streamline region. While this may lead to a flow which, for the short waves considered here, looks rather like a separated eddy behind a fixed wave and for which the 1,5= 0 streamline may come very close to the crest, the two phenomena should be clearly distinguished. Acknowledgments The first draft of this paper was written during a visit by P. R. Gent to the Atmospheric Environment Service in Downsview, Canada. This visit was funded by the Air Sea Interaction Committee of NATO. P. A. Taylor must thank the Atmospheric Environment Service for a rather broad interpretation of the scope of his contract there and for the use of a considerable amount of computer time. The work has been partially supported under NERC (U.K.) grant GR3/1932. It is a pleasure to acknowledge helpful discussions with Dr F. Dobson, Dr J. Elliott and Prof. 0. M. Phillips during NATO-supported visits by P. R. Gent to the Bedford Institute and Johns Hopkins University. References Banner, M. L. and Phillips, 0. M.: 1974, ‘On the Incipient Breaking of Small Scale Waves’, J. Fluid Mech. 65, 647-656.
Banner, M. L. and Melville, W. K.: 1976, ‘On the Separation of Airflow over Water Waves’, to appear in J. Fluid Me& Chang, P. C., Plate, E. J. and Hidy, G. M.: 1971, ‘Turbulent Air Flow over the Dominant Component of Wind Generated Water Waves’, J. Fluid Mech. 47, 183-208. Gent, P. R. and Taylor, P. A.: 1976, ‘A Numerical Model of the Airflow above Water Waves’, J. Fluid Mech. 77, 105-128. Jeffreys, H.: 1925, ‘On the Formation of Water Waves by Wind’. Proc. Roy. Sot. A, 107, 189-206. Jeffreys, H.: 1926, ‘On the Formation of Water Waves by Wind, (second paper)‘, Proc. Roy. Sot. A,
110,241-247. Lighthill, M. J.: 1962, ‘Physical Interpretation of the Mathematical Theory of Wave Generation by Wind’, J. Fluid Mech. 14, 385-398. Longuet-Higgins, M. S.: 1973, ‘A Model of Flow Separation at a Free Surface’, J. Fluid Mech. 57, 129-148. Motzfeld, H.: 1937, ‘Die turbulente Stromung an Welligen Wanden’, Z. angew Math. Mech. 17, 193-212. Phillips, 0. M.: 1966, Dynamics of the Upper Ocean, Cambridge University Press, 261 pp. Shemdin, 0. H.: 1969, ‘Instantaneous Velocity and Pressure Measurements above Propagating Waves’, Technical Report No. 4. Coastal and Engineering Lab., University of Florida. Stanton, T. E., Marshall, D. and Houghton, R.: 1932, ‘The Growth of Waves on Water due to the Action of the Wind’, Proc. Roy. Sot. A, 137, 238-293. Stewart, R. W.: 1967, ‘Mechanics of the Air-Sea Interface’, Phys. of Fluids, lOS, S47-S55. Stewart, R. W.: 1974, ‘The Air-Sea momentum exchange’, Boundary-Layer Meteorol. 6, 151-167. Townsend, A. A.: 1972, ‘Flow in a Deep Turbulent Boundary-Layer over a Surface Distorted by Water Waves’, J. Fluid Mech. 55, 719-735. Ursell, F.: 1956, ‘Wave Generation by Wind’, in G. K. Batchelor and R. M. Davies (eds.), Surueys in Mechanics, Cambridge University Press, pp. 216-249. Wu, J.: 1969, ‘A Criterion for Determining Air-Flow Separation from Wind Waves’, Tellus, 21,
707-713.
