Lee waves: new understanding of a classical problem Stuart B Dalziel1, Michael D. Patterson2, Colm-cille P. Caulfield3,1 & Stéphane Le Brun1,4 1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK 2. Department of Architecture and Civil Engineering, University of Bath, UK 3. BP Institute, University of Cambridge, UK 4. École Polytechnique, France
[email protected],
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Abstract We explore the classical problem of lee wave generation by a stationary isolated threedimensional obstacle in a uniform low-Froude-number stratified flow. By separating the permanent waves produced into three distinct categories, we compare detailed experimental measurements with theoretical predictions. We pay particular attention to the time-dependent establishment of the wave field and the orientation of the fan of lee waves that is established. As seen by previous authors, we find that the uniform flow can be divided into two regions: an essentially two-dimensional flow around the base of the obstacle and a wave-generating flow over the top portion of the obstacle. We show that quantitative agreement between experimental observations and predictions of linear wave theory is only possible if we take into account a small slope across the obstacle of the surface separating the quasi-twodimensional flow around the base of the obstacle and the wave-generating flow over the top of the obstacle. 1. Introduction In this article, we seek to understand better the classical problem of a uniform stratified flow over and past an isolated topographic feature. We concentrate on the geophysically important regime where, far from the obstacle, the velocity field U(x) = (U,0,0) is purely horizontal and uniform with depth, and the Boussinesq stratification 0(z) is described by a constant buoyancy frequency g d 0 , N 00 dz where g is the acceleration due to gravity and 00 a reference density. The corresponding dispersion relation for linear waves, in the frame of reference of the fluid, is
cos
, N where is the frequency of the wave and the angle between the wavenumber vector k and the horizontal. It is well known (e.g. Turner 1973, Lighthill 1978) that a steady stratified flow over an obstacle excites lee waves that carry energy as they propagate away from the obstacle and are carried downstream by the mean flow. The ‘principle of stationary phase’ (Lighthill 1978) applies to the wavefield, ensuring it is phase-locked with the topography. In two dimensions this leads to the relationship
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N2 (1) U2 between the horizontal (k) and vertical (m) wavenumber components of the wavenumber. k 2 m2
For a hemispherical obstacle of radius R, at low topographic Froude number U , (2) FR NR Greenslade (1994, 2000) extended the ideas of Drazin (1961) and suggested that only streamlines above (3) zs 1 AFR R , far from the obstacle, pass over the obstacle and make the vertical excursions that lead to internal waves. Streamlines for z < zs remain essentially horizontal and pass around the obstacle instead. Greenslade (2000) suggested the constant A ~ O(1) and is independent of FR. Experimental measurements by Vosper et al. (1999) confirmed A ~ 1. In this paper we explore this division of the flow into two regions and find that details of what happens close to the obstacle are critical to the structure of the resulting internal wave field. In §2 we describe the experimental setup before presenting our basic observations in §3. In §4 we employ linear wave theory, while in §5 this is modified to take into account the establishment of a small slope across the obstacle. Finally our conclusions are presented in §6. 2. Experimental setup
Experiments were performed in a stratified shear flume based on the original design of Odell & Kovasznay (1971). The flume, sketched in figure 1a, was filled to a depth of H 300 mm with saltwater using the normal double-bucket technique to produce a stable, linear, density stratification with buoyancy frequency N 1.7 rad s-1. The flow is driven around the 200 mm wide race-track-like circuit of the flume through the viscous boundary layers on two stacks of intermeshing disks rotating about vertical axes. Here, all 36 disks were driven at the same speed to produce a uniform velocity. Nt 0.0 25.0
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The internal waves were generated by a hemispherical obstacle of radius R = 65mm position on the floor of the channel in the 2.4m working section. Quantitative measurements were then obtained using Synthetic Schlieren (Dalziel et al. 1998, 2000, 2007). As demonstrated by Scase & Dalziel (2006), if the density perturbations are an even function of position about the central plane of the flume, then the signal recorded by Synthetic Schlieren is proportional to the cross-channel (line-of-sight) mean of the gradient in the density perturbation normal to the
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viewing axis. Thus, with the arrangement used here, Synthetic Schlieren returns directly the cross-channel average perturbation buoyancy gradients g g bx bx and bz bz , 0 x 0 z where the over-bar represents a cross-channel average. Here we concentrate on bx as linear wave theory relates this to the vertical velocity of the wave field by N2 bx w . (4) U 3
Basic observations
At low Froude numbers, the flow past the hemisphere can be divided into two regions. Greenslade (1994) suggested that for heights less than zs (1 FR)R, the flow is essentially horizontal with minimal vertical excursions. Here FR = U/(NR) is the obstacle Froude number. This approximately two-dimensional flow does not give rise to lee waves. Lee wave generation is confined to the top portion of the hemisphere, of height ~ FRR, where the flow is able to rise over the cap of the hemisphere. This division is shown clearly in figure 2 where dye streaks are generated from potassium permanganate crystals dropped into the flow.
Figure 2: Dye from potassium permanganate crystals showing (a) separation height (b) flow on surface of obstacle and (c) wake region below z = zs downstream of obstacle.
We consider here an experiment with N = 1.75 rad/s and U = 14.8 mm/s giving FR = 0.13 and explore the internal wave field at t = 388s, shortly before the fluid initially located by the disks reaches our observation window. By this time the uniform flow velocity is within a few percent of its asymptotic value. The bx and bz fields (figures 3a & b) give clear evidence of an internal wave field propagating upwards, away from the obstacle, as the waves are swept down stream. These waves are reflected from the free surface, to propagate back towards the base of the tank as they continue to be carried downstream. The different components of this wave field may be separated by considering the orientation of the wavenumber vector. In particular, we separate the waves into upward propagating (figures 3c & d), downward propagating (figures 3e & f) and essentially horizontally propagating (figures 3g & h) waves. In each case the insert shows the corresponding part of the power spectrum.
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(g) (h) Figure 3: The gradient fields for a typical experiment. The left column shows bx and the right column shows bz. (a,b) The full fields; (c,d) the upward propagating waves; (e,f) the downward propagating waves; (g,h) the essentially horizontally propagating waves.
For issues associated with the establishment of this wave field, and the impact of the gradual acceleration of flow around the channel, the reader is referred to Dalziel et al. (2011). 4
Linear theory
Since FR is small, then zs is close to R and so height of the spherical cap responsible for generating the waves is small compared with its length. We may this linearise the boundary condition of no normal flow on the spherical cap and express it as the prescribed velocity w = U h/x at z = zs. It is then a simple matter to take the Fourier transform of the boundary of the upper flow (the greater of zs and h(x,y)) and thus construct the solution for bx in an steady inviscid flow. In particular, the Fourier transform of bx is given by N 2 imz N khˆ k , l e , k 0, ˆ bx k , l , z (5) U 0, otherwise,
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which may then be inverted and cross-tank averaged to yield the wave field shown in figure 4. -0.1 2.0
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Figure 4: Wave field for steady inviscid flow over spherical cap.
While the wave field shown in figure 4 captures some of the key details of the experimental observations in figure 3c, there are also distinct differences. In particular, the wave crests and troughs are described by smooth arcs above and behind the obstacle, whereas the experimental observations indicate that there is a phase jump in these features. Additionally, the dominant orientation of the waves appears to be steeper in the linearised theory than the ~45 (diagonal line on figure 4) seen in the experiments. Incorporating causality and viscosity into the theoretical solution (see Dalziel et al. 2011) improves the visual match between the solution and the experiments, but these two quantitative differences (orientation and phase jumps) remain. 5
Cross-obstacle slope
The cause of the discrepancy in the structure of the lee waves between experiments and theory is hinted at by figure 2. In particular, careful inspection of the dye streaks on the surface of the sphere show that there is a front-to-back asymmetry in the flow over the top of the cap. The cause of this is obvious when one recognises the clear presence of a separated wake in the lee of the obstacle. For a given z in the nearly horizontal flow around the base of the obstacle, the pressure at the front stagnation point is greater than that at the rear stagnation point. This causes the flow along each streamline in this region to be slightly lower at the rear of the obstacle than the front, and so forces the height of the surface zs (dividing the region where the flow is around the obstacle from that where the flow is over the obstacle) to adopt a slight tilt across the obstacle. By noting that this angle is small, we can anticipate that the change in the inclination of the mean flow as a function of height can be ignored, even though the mean streamlines must return to horizontal at the free surface of the flow in the channel. This approximation then suggests that we should consider over a spherical cap located on a gradually sloping boundary, rather than one on a horizontal boundary, as would be the classical assumption. The impact of this gradually sloping boundary is surprisingly strong. To understand its origins, the condition for steady waves (the ‘principle of stationary phase’) has to be recast in terms of the sloping flow, leading to the condition
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1 N m (6) cos sin k 2 m 2 2 U k in two-dimensions (see Dalziel et al. 2011 for a more complete discussion). Here, is the (small) angle between the surface and the horizontal. Using (6) to determine the group velocity then allows us to draw the corresponding causality shells for waves released from a point source at fixed time intervals in the past. The result of this is shown in figure 5 for a range of slopes.
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(c) (d) Figure 5: Causality curves due to a flow with a vertical component for (a) tan = 0, (b) tan = 0.01, (c) tan = 0.05 and (d) tan = 0.1. In each case the flow is from left to right. Lines are also drawn showing the loci of the maximum vertical velocity and the maximum velocity normal to the slope.
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Figure 6: Prediction of wave field bx/N2 produced with a slope of = 2 over the obstacle. The insert shows the corresponding experimental field.
As figure 5 shows, the domain in which steady waves can exist changes dramatically with the slope. If the source has been propagating for ever, then for a horizontal boundary (tan = 0), the waves can occupy the entire quadrant above and behind the source. However, as soon as
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the slope departs from zero, the causality front becomes inclined, dramatically reducing the part of the domain in which permanent waves can exist. Dalziel et al. (2011) have demonstrated that this change, as quantified by (6), is visible in the two-dimensional bx spectra from the experimental measurements. As shown by Dalziel et al. (2011, see figure 6), we can incorporate this slope into a linearised boundary calculation to produce a dramatically improved prediction of the wave field that both recovers the orientation of the waves and recovers the observed phase jumps in bx. Here, we have introduced an assumed slope of = 3 across the obstacle, leading to a substantial rotation of the causality front and agreement to within 2 for the dominant angle of propagation of the waves. Dalziel et al. (2011) also demonstrate that the dominant angle is an approximately linear function of the topographic Froude number FR, as one might expect by considering the formation of a separation bubble behind the hemispherical obstacle. 6
Conclusions
The principle of stationary phase and linearised boundary conditions have been used extensively, and successfully, both separately and in combination, for modelling internal lee waves. Care must be taken, however, when applying these approximations. In particular, for linearised boundary calculations it is almost inevitably assumed that the mean flow is horizontal in the region of interest. However, as we have shown here, even a very small slope can have a dramatic impact on shape of the domain in which permanent waves can form and on the structure of the wave field within this domain. An asymptotic analysis of the limit of this domain shows it varies as 1/3, accounting for the high sensitivity to small departures from horizontal and suggesting that the simple approximation of a perturbation to a horizontal boundary may often be misleading. Motivation for this work has been establishing the cause of a mismatch between the theoretical and experimentally observed structure of lee waves across a hemispherical object. Features such as the orientation of the dominant waves and the phase jumps in the lee of the obstacle cannot be accounted for by simply locally modifying the geometry of the forcing of a linearised calculation. During the process we have confirmed that the Drazin-Greenslade division of the flow into two regions (a quasi-horizontal flow around the base and a wavegenerating flow over the cap) yields a good agreement for the amplitude of the waves, but if quantitative match of the structure of the wave field is required then the establishment of a small slope in the dividing surface across the obstacle must be taken into account. References
Dalziel, S.B., Hughes, G.O. & Sutherland, B.R. 1998 Synthetic schlieren; Proceedings of the 8th International Symposium on Flow Visualization, ed. Carlomagno & Grant. ISBN 0 9533991 0 9, paper 062. Dalziel, S.B., Hughes, G.O. & Sutherland, B.R. 2000 Whole field density measurements by ‘synthetic schlieren’; Experiments in Fluids 28, 322-335. Dalziel, S.B., Carr, M., Sveen, K.J. & Davies, P.A. 2007 Simultaneous Synthetic Schlieren and PIV measurements for internal solitary waves. Meas. Sci. Tech. 18, 533-547. Dalziel, S.B., Patterson, M.D., Caulfield, C.P. & Le Brun, S. 2011 The structure of low Froude number lee waves over an isolated obstacle. Submitted to J. Fluid Mech.
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Drazin, P.G. 1961 On the steady flow of a fluid of variable density past an obstacle. Tellus 13, 239-251. Greenslade, M.D. 1994 Strongly stratified airflow over and around mountains. In Stably Stratified Flows: Flow and Dispersion over Topography. Ed. I.P. Castro & N.J. Rockcliff. Oxford. Greenslade, M.D. 2000 Drag on a sphere moving horizontally through a stratified fluid. J. Fluid Mech. 418, 339-350. Lighthill, M.J. 1978 Waves in fluids. Cambridge University Press, UK, 504pp. Odell, G.M. & Kovasznay, L.S.G. 1971 A new type of water channel with density stratification. J. Fluid Mech. 50, 535-543. Scase, M.M. & Dalziel, S.B. 2006 Internal wave fields generated by a translating body in a stratified fluid: an experimental comparison. J. Fluid Mech. 564, 305-331. Turner, J.S. 1973 Buoyancy effects in fluids. Cambridge University Press, UK, 367pp. Vosper, S.B., Castro, I.P., Snyder, W.H. & Mobbs, S.D. 1999 Experimental studies of strongly stratified flow past three-dimensional orography. J. Fluid Mech. 390, 223-249.
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