Proc. E. Boc. Lmil'. A 429, 119-140 (1990)

Printed in Great Britain

Obstacle dras in stratified flow

B v I . P . C e s r n o t , W . H . S N v n n n 2 t e N o P . G . B e r n n s 3rMechnnicq,l Engi'neeri,ng Department, Un'iuers'ity of Surrey, Guililford',

Surrey GAz \XH, A.K.2 Atmospheric Sc'iences Mocleling Diuision, Atmospheri,c Research and, ErTtosure

As sessment L abor atory, U S E nuir onmental P rotection Ag encg, ResearchTri,angle Park, North Carol'ina 27711, U.S.A.

3 C.B.I.R.O. Diuision oJ Atmospheric Research, Aspenilale, V'ictoria,Australia 3795

(Communicated, by T. B. Benjamin, F.R.S. - Eece'iued 13 July 1989)

[Plate 1]

This paper describes an experimental study of the drag of two- and three-dimensional bluff obstacles of various cross-stteam shapes when towedthrough a fluid having a stable, linear density gradient with Brunt-Vaisala frequency, N. Drag measurements were made directly using aforce balance, and effects of obstacle blockage (hf D, where h and D arethe obstacle height and the fluid depth, respectively) and Reynoldsnumber were effectively eliminated. It is shown that even in cases wherethe downstream lee waves and propagating columnar waves are of largeamplitude, the variation of drag with the parameter K (:NDlnU) isqualitatively close to that implied by linear theories, with drag minimaexisting at integral values of K. Under certain conditions large, steady,periodib variations in drag occur. Simultaneous drag measurements andvideo recordings of the wakes show that this unsteadiness is linkeddirectly with time-variations in the lee and columnar wave amplitudes.It is argued that there are, therefore, situations where the inviscid flowis always unsteady even for large times; the consequent implications foratmospheric motions are discussed.

1 . I n r n o n u c r r o N

There is an extensive literature on the now classical problem ofstratified flow overobstacles. Many of the early theoretical works, most of which were essentiallylinear theories of various kinds, have described the relationship between theobstacle drag and the amplitude of internal waves generated by the obstacle (see,for example, Drazirr& Moore r967; Miles r968;Trustrum r97r). The'wave'draghas components linked both to stationary lee waves and to 'columnar' waveswhich ca,n propagate upstream and downstream of the obstacle. Despite theimportance of drag in the meteorological context, there seems to have been lessemphasis on this link in more recent work, with attention being concentratedlargely on the nature and form of the wave field. For example, in one of the few

t On assignment from the National Oceanic and Atmospheric Administ'ration, USDepartment of Commerce.

t 1 1 9 1

120 I. P. Castro, 'W.

H. Snyder and P. G. Baines

recent, attempts to construct a nonlinear theory describing the flow over a two-dimensional barrier when the parameter 1{ is close to an integer (the 'near-

resonant' condition), Grimshaw & Smyth (r986) do not, mention drag at all. (K isdefined in the usual way as NDfntJ, where D is the channel depth and u is theupstream flow velocity, and note that ff is the ratio of the propagation speed ofthe lowest wave mode to the undisturbed flow velocity.) This is not intended as acriticism of such work; indeed, it would appear that the wave drag-waveamplitude links are most clearly formulated in works which are, arguably, ratherless mathematically rigorous (see, for example, Trustrum r97r; Janowitz l98r).

what is more surprising, perhaps, is that so few attempts have been made tomeasure drag forces experimentally. Accurate determination of the drag of anobstacle under stratified flow conditions would, in principle, be of considerablehelp in determining the nature of the flow. The longstanding discussionssurrounding the question of upstream influence and its implications for towingtank experiments, for example, would have been helped if even qualitative dragdata had been available. As far as the authors are aware, the only publishedexperimental results on obstacle drag in stratified flow are those obtained byDavis (tg6g, immortalized in the text by Turner, 1973) and the data of Lofquist& Purtell (rq8+). (see -l/ote aild,eil in proof.) only Davis made any attempt tocompare his measurements with the wave drag deduced theoretically from the leewave amplitude measurements. Now a close examination of Davis's experimentraises certain difficulties. The obstacle used for the drag measurements - a thinvertical barrier, or 'fence'-was suspended beneath the surface (and notsymmetrically between the surface and the base of the tank). Vortex sheddingwould therefore almost certainly have occurred, although strongly affected nodoubt by the stratification. Reynolds numbers, based on barrier height and towspeed, were less than about 5000, so that substantial Reynolds number effectscould also be anticipated. These two features alone, neither of which are discussedby Davis, make an accurate interpretation of the drag measurements difficult. Itis also worth pointing out that the drag force never exceeded about !4g,whichmade it difficult to measure accurately. Furthermore, Davis noted that in manycases the lee wave structure was far from steady. This made estimation of the lee-wave generated drag uncertain but, more importantly, it implies an unsteady dragforce.

Despite these difficulties, some of Davis's conclusions have been broadlyconfirmed by the work of Lofquist & Purtell (rs8+), who made very carefulmeasurements of the drag of spheres. (Again, the Reynolds numbers were quitelow but the authors successfully separated the effects of stratification from theeffects of Reynolds number.) First, both found that drag can be substantiallyincreased or reduced as a result ofstratification. Secondly, at large values ofK (inexcess of 3, say), Davis found no organized wave structure and the very large dragcoefficients predicted by Drazin & Moore (ts6z) and Miles (r968) did not seem tooccur in either set of experiments. This was attributed by Davis to the breakdownof the waves to turbulence, and the consequent inapplicability of Long's model, onwhich the theories were based. Miles did, in fact, anticipate this conclusion bypointing out that, even if the predicted results are legitimate steady-state

Obstacle ilrag in strat'ifi,ed, fl,ow

solutions, they probably represent unstable motions. Davis also concluded thatthe drag associated with lee-wave generation \Mas not (for a thin barrier) thedominant effect, of stratifi.cation.

Now, over recent, years substantial evidence has accumulated that, for two-dimensional obstacles at least, columnar wave modes of significant, amplitudescan be generated and propagate upstream once K > I. Small-parameter 'weakly

nonlinear' theories (Mclntyre (tgZz) for small obstacle height, and Baines &Grimshaw OSlil for small stratification) indicate only second-order upstreamdisturbances. However, experiments have clearly identified first-order distur-bances (Wei et al. ry75; Baines rg77, tg79; Castro & Snyder 1988) and thesecan occur even in effectively infinite depth cases (Baines & Hoinka 1985;Pierrehumbert & Wyman 1985). The generation mechanism is almost certainly anonlinear one and there have as yet been no theories which explain the processadequately. None the less, it is evident that the columnar modes, as well as the lee-wave modes, must be associated with additional obstacle drag. It is interestingthat the Oseen type (linear) theories can give reasonable quantitative agreementwith experiment, at least as far as the column&r w&ve amplitudes are concerned(Castro & Snyder r988), although this requires specification ofthe obstacle dragcoefficient and has only been tested at large values of K (O(10)).

Our initial objectives in undertaking direct drag measurements were four-fold.X'irstly, we wanted to compare measured drag coefficients with those required bythe Oseen theories to give agreement with measured wave amplitudes. Secondly,it was of interest, to compare the variations in drag with ff (and,Fo : U/Nh) withthose implied by the linear theories and by Long's model. All these theories (andthe time-dependent ones based, alternatively, on the Oseen assumption) predictminimum wave drag at integral values of K, with maxima occurring atintermediate values. Bluff obstacles cannot have zero drag, of course, because ofthe inevitable pressure drag associated with flow separation and a turbulent wake,but if the wave drag component were significant, one might expect variationswhich qualitatively follow the variations given by inviscid theories. Thirdly, wewanted to assess the influence of stratification on obstacle drag in the regimewhere no w&ves are generated (1( < 1). Finally, we anticipated that direct dragmeasurements would provide both a more immediate and a clearer indication ofwhether the flow ever reached a steady state during the course of the tow. Theintention in all of this was to concentrate on surface-mounted obstacles, for whichclassical (Karman) vortex shedding could not occur; this seems most relevant toatmospheric flows. Direct, drag measurements have not previously been reportedfor such cases. It was also felt, important to minimize the possible complicatingeffects of Reynolds number by comparing results obtained with and without,stratification at the same Reynolds number.

Our initial measurements have been partially discussed already (Castro &Snyder 1987, hereafter referred to as CS), where it was shown that under somecircumstances large, periodic oscillations in drag could occur. This was anunexpected finding; it was not caused by wave reflections from the tank end-wallsand there seemed to be no obvious explanation for the phenomenon. In this paperwe first summarize the more important results of this initial work (in $3) and

12t

Proc. R. Soc. Lond,. A, uolume 429 Castro et al., plate I

Freunl 8. Photographs taken lrhen the drag was (a) maximum and (b) minimum(times of 25 and 46 s, respectively, on figure 6).

(IaciW tt. 132)

Obstacle d,rag in strat'if,ed, f,ow 133

time/s

Frcunn 9. Raw drag trace for two-dimensional fence. K: I.4, In: 1.05.

r.0 1.5 2.0 2.5 3.0K

Frcunn 10. Variation in (normalized) drag with 1{. a, two-dimensional fence (various -Q) ;4, two-dimensional fence, Fn: I.O; r, Witch of Agnesi (various .fl).

decreasing /, corresponds to increasing blockage (hlD) and figure 11 includes anhf D scale. Evidently the unsteadiness only occurs over a limited range of .F'u andthis range will presumably depend, to some extent at least, on the particular valueof 1(. Note that the fence results for ff > 2 in figure 10 were all obtained at Eo :1.0; oscillations might well occur at different Fr,bat we have no data to confirmthat. Also included in figure 11 is wave-amplitude data obtained from the

2 .0

1 . 5

ooV

b r .o6

0.5

100804020

" 3rS

l ) 2

r34 I. P. Castro, W. H. Snyder and P. G. Baines

corresponding video recording. As before, the results represent the amplitude asdeduced from a single dye streamer. X'or {2 < I.25, the high drag statecorresponded to the occurrence of wave breaking in the lee, and wave amplitudemeasurements were only possible during those times when the drag was near itsminimum. At the very low end of the Fo range, where oscillations again did notoccur, wave breahing occurred steadily throughout, the tow.

1.0 0.4 o .2h/D

0.15 0.1 0.08 0.06

rS

l : '

4

3

2

l F - -

1.5

'e l.o

0.5

0.5 1.0 1.5 2.0 2.5 3.0Eh

X'reunn 11. Variation of (a) drag and (b) lee-wave amplitude with { at fixed K:I.7.- denotes wave breaking.,( and hfD scales apply to both (a) and (b).

These periodic oscillations in wake structure and obstacle drag might beexplained, first, by reference to the flow variations with K in cases whereoscillations do not occur. As discussed in $3.3, both drag and wave amplitude havea maximum in between integral values of K. Recall that K: ND/vU. Afterstarting a tow the lee and upstream propagating columnar w&ves will graduallyincrease in strength. (Note, incidentally , that a number of specially designed tostsdemonstrated that the final flow was essentially independent, of how the tow wasstarted.) The upstream w&ves will cause an effective reduction in the upstreamvelocity ahead of the obstacle, with a corresponding increase in the effective valueof K. In cases for which this effective J( is higher than the value corresponding tothe maximum w&ve amplitude condition (i.e. 1.5 < K < 2.0) this will lead in turnto a reduction in the wave amplitude (because dl ldK is negative), which will thencause a reduction in the effective value ofK and a consequent increase in the waveamplitude. There seems no reason why this cycle of increasing and decreasingeffective K (wit'h the consequent decreasing and increasing wave amplitude and

Obsto,cle d,rag 'in stratif,ed, fl,ow 135

drag) could not repeat continuously. The inherent instability arises becausedl|dK is negative. X'or 1.0 < K < 1.5, dl/dk is positive and no such instabilityexists. Note that this explanation does not essentially depend on nonlinearprocesses near the obstacle. The requirement is only that dl/dK has a turningpoint, which many steady state linear theories yield. Whether oscillations actuallydo occur in cases when, in principle, they can, will depend on the amplitude of thecolumnar w&ve. If this is sufficiently small, viscous effects can presumably dampout the small oscillations that would otherwise occur.

As a second and alternative explanation for the flow and drag variations, it ispossible that the nonlinear processes in the region of the obstacle are, in fact,crucial. Apart from Long's model, two dynamical models have been advanced todescribe the nonlinear flow of stratified fluid of finite depth over topography.Grimshaw & Smyth (1986) considered the self-interaction of the critical mode(when, in the present context, K is near an integer) and showed that the modeamplitude satisfies a forced Korteweg-de-Vries equation. In the case of uniformstratification, however, the nonlinear interaction term vanishes so the theory isnot applicable directly to the present case. Baines & Guest ( r 986) described a modelin which it is assumed that the flow reaches a steady state. The model requires thatthe flow become critical over the obstacle to control the nonlinear character and.for this to occur with uniform stratification, the Long's model solution mustbecome unstable. However, in the present experiments drag variations areobserved for cases in which the Long's model solution is stable (no verticalstreamlines anywhere), so that this model is also inapplicable here.

Recent theoretical studies, to be reported elsewhere, indicate that a possiblenonlinear explanation is along the following lines. For strictly hydrostatic flow,the linear solution for the perturbation streamfunction when the flow is initiatedfrom rest contains a steady part (which becomes Long's model for finite D) plustwo time-dependent terms for each mode (Baines & Guest 1988). One of thesepropagates against the stream and the other propagates with it. The steady-statepart and the upstream propagating part become large as K approaches an integer.As the wave propagates slowly through the steady state part of the solution,however, its propagation speed may be affected by the changed flow; it may evenchange direction, so tha,t the leading part of the wave may propagate upstreambut the trailing part may never esc&pe from the region of the obstacle. Aperturbation expansion in disturbance amplitude to obtain the effects on wavespeed does give a form containing terms which modify the usual linear wave speed.Now for non-hydrostatic flow with shorter obstacles, the equation for the amplitudeof the upstream propagating mode is expected to contain a w&ve dispersion term,giving an equation of the Korteweg-de-Vries or a similar form. By analogy withthe work of Grimshaw & Smyth (1986) such a system may be expgcted to giveperiodic solutions in some parameter ranges and it is these which may correspondto the drag variations observed in our experiments.

Whatever the explanation for the observed drag variations, this distinctionbetween cases in which towing-tank experiments on two-dimensional obstaclesma,y or may not be steady, quite independently of the effects of columnar-wavereflections, a,ppears not to haye been recognized previously. A number of authors

136 I. P. Castro, W. H. Snyder and P. G. Baines

have noted unsteady behaviour, including Davis (rg6g), but the possibility ofcolumnar mode reflections, particularly in cases where the tank was relativelyshort, tends to obscure matters. However, there have been previous results which,with hindsight, would appear to confirm the present conclusions. A recentexample is provided by the work of Boyer & Tao fig87a), who studied stratifredflow over triangular and cosine-shaped obstacles. Careful inspection oftheir dataindicates that the flows which were most clearly unsteady were those for which1.5<K <2.0 and 2.5<K <3.0. They also found that, , for certain parameters(K : I.3, Fn: 0.7), the flow was relatively insensitive to the starting conditions,while for others (K:1.6, Fn:0.76), the final flow depended critically on thetranslational history of the obstacle (Boyer & Tao Qg87b), although presumablysuch dependence would disappear if the tank were sufficiently long). This isprecisely what would be expected on the basis of the present work.

During preparation of the final version of this paper, the recent work ofHanazaki (rS8S) came to our notice. He has undertaken a numerical study ofstratified flow past a square plate, and found similar periodic variations in thedrag, which were explained as arising from 'successive release of the upstreamcolumnar disturbance (or the'eddy') of the first internal wave mode'. InHanazaki's work no distinction was made between cases for which 1{ was less thanor greater than 1.5, although the numerical results clearly indicate much largerfluctuations for K> 1.5, in line with the present work. Other authors have alsotalked in terms of 'detaching upstream eddies'. We prefer to think simply in termsof periodic variations in the strength of the columnar mode, arising from changesin the effective value of 1{. tr'or 1( < L.5, such variations will (usually) eventuallydie away, so that no further apparent'perturbation eddies'will 'detach'.Hanazaki's computations did not continue long enough for a steady state to bereached in those cases. It is also worth noting that these computations, forRe: 50, demonstrate variations in drag wilh K which are very similar to thepresent experimental results. Indeed, Cu/CuoatK : 1 (0.71) andK : 1.5 (1.11) areremarkably close to our results for the n&rrow three-dimensional fence (figure 2c),which - taking the water surface as a symmetry plane - has the same aspect ratio.This is a further confirmation that the wave-induced motions and pressure fielddominate the influences of turbulence.

It remains to discuss the mechanisms which determine the timescale of theperiodic oscillations in drag. X'igure 12 shows the period, 7, of these oscillations,normalized by towing speed and water depth, as a function of (a) K and (b) Fuforthe various obstacles used. In every case the period falls with increasing J(,starting from a maximum at the value of K for which oscillations first begin (1.5and 2.5). According to our first possible explanation for the drag variations,fluctuations in the apparent value of K drive the instability and these will onlyoccur once the wave amplitudes have grown sufficiently. Then the period willdepend essentially on the time required to develop a significant lee (and columnar)wave amplitude. If this were a function mainly of the lee-wave wavelength(2nU/N), lhen TU/D would be inversely proportional to K. The data certainlybehave, qualitatively, in that w&y, as do the results of Hanazaki's (rg89)computations, referred to above. However, TUID would then also be independent

Obstacle d,rag 'in strati,fi,ed, fl,ow r37

15 (a)

Fr

t .5 2.0 2.5 3.0K

r.0

0.3 0.2 0.15 0.1h/D

X'rcunn 12. Period of unsteadiness, as a function of (a) R or (b) Fo. r,, Witch of Agnesi,Fn:1.0-1.2 (4 Fu:1.7-I.9; \ Fn:0.7--0.9). n, wide three-dimensional fence,

4 : 0.8-1.0; l, two-dimensional fence, K : I.7.

of Fr; frgtre 12 b shows that this is not, the case. A dependence of the period on theamplitude of the columna,r wave is consistent with the results in both figure I2aand b, since the wave amplitudes are largest near K: 1.5 (and 2.5), fall with Kincreasing towards the next integral value, and also fall with increasing F, forconstant K (compare figures 11b and I2b). If seems reasonable to conclude thatthe period of the oscillation depends primarily on the wave amplitude that wouldoccur if steady flow could be maintained; this is a characteristic of nonlinearprocesses, so could be regarded as supporting the second possible explanation forthe drag variations. In the case of inherently steady flow (1.0 <K< 1.5), itpresumably takes longer for the wave field to develop its full amplitude in caseswhere the latter is larger.

Note that the timescale is much larger than any scale based on towing speed andsome typical longitudinal length scale charactefizing the flow. The lee-wavewavelength, i., is in many cases the longest such length scale, and ? can be morethan an order-of-magnitude larger lhan l/U.If the towing tank is too short (ormore strictly, tank length/water depth is too small) then the oscillation periodmay be of the sa,me order as, or even less than, the tow time. This was, in fact, thecase for some of the earlier full-depth tows discussed by CS; it may also have beentrue for other cases in the literature reported as unsteady.

5 . C o N c r , u s r o N S

The major conclusions of the work can be summarized as follows.L For bodies of any shape but sufficiently bluff to promote separation in the

lee, stratification first acts so as to reduce the obstacle drag. This is largely aconsequence of the damping of vertical motions and the reduction of verticalmixing and shear layer entrainment, and occurs in both laminar and turbulentflows.

2. Once the stratification has become strong enough to allow strong stationary

138 I. P. Castro, W. H. Snvder and P. G. Baines

lee wayes and propagating columna,r waves (K > I), the subsequent variation ofdrag with increasing stratification is qualitatively similar to that predicted bylinear theories, whether these are based on Long's model or on the time-dependent, oseen equations. The drag follows the wave amplitudes in havingminima at integral values of K with maxima in between. Changes in the dragassociated with separation and the turbulent, wake are small and, while such dragis certainly not negligible, the total drag is dominated by the effects of the wavemotions. Since the large-amplitude column&r waves are probably a direct result ofnonlinear effects, it would seem that even a full nonlinear theory would give a dragvariation qualitatively similar to that implied by linear theories.

3. For two-dimensional obstacles in conditions such that K is in the upper halfof the range between two integral values, strong oscillations in the wave-field and,consequently the obstacle drag, can occur. This phenomenon has nothing to dowith the arrival of wave reflections from the tank end-walls. The relativemagnitude of the oscillation depends both on obstacle shape and on Flr, via theinfluence of these parameters on the wave amplitudes. Three-dimensionalobstacles can also generate these oscillations, provided they are sufficiently longin the spanwise direction.

Since large-amplitude columnar modes can occur even in infinite depthsituations (Pierrehumbert & Wyman r985;Baines & Hoinka 1985)the instabilitycould, in principle, arise in the atmosphere. The authors are not aware of anyatmospheric observations which indicate this behaviour, but in view of itsinherently long timescale it may be a rather rare occurrence.

All the data reported here and in previous papers on obstacle drag measurementsare in the context of relatively bluff obstacles which lead to separation in the lee.I{owever, there seems no rea,son to suppose that such oscillatory motions couldnot be generated with obstacles of rather lower slope. Separation is not arequirement for the generation of strong columnar modes and it is the latter thatdrive the instability through their influence on the apparent value of 1{.

4' At extreme levels of stratification (very low obstacle X'roude number, U/Lh)the wake flow and the drag are more likely to be steady even if lee-wave breakingoccurs. Lee-wave drag is then relatively small, but the presence of a spectrum ofrapidly propagating columnar waves leads, in towing tank experiments, tomultiple reflections from end walls which imply, in the limit of zero Fu, anasymptotically large drag coefficient. fn this respect our drag measurements atK > g agree qualitatively with the theory of x'oster & saffman (rszo). They donot agree with the linear, time-dependent theory of Janowitz (r98r), in that themeasured Cu values are much higher than required by the theory to match thecolumnar wave amplitudes measured in our earlier experiments (Castro & Snydert9B8). Since the generation of first-order amplitude waves is a strongly nonlinearphenomenon, this is not, perhaps, surprising.

5. As a final comment, it is worth pointing out that in cases where the obstacleeither spans the towing tank completely or is sufficiently wide to lead to significantupstream motions (under appropriate circumstances), wave reflections candrastically reduce the effective towing time. X'or a case where the distance betweenthe starting position of the obstacle and the upstream end wall is L andK > 1, the

Obstacle ilra,g 'i,n stratifi,eil f,ow 139

reflection of the fastest moving mode arrives back at the obstacle after a timegiven by Ut / D : 12 I $ + K)l L / D. Only if the wave amplitude is small can the towcontinue $ntil Ut /d, : L lD at the very most) without serious distortion of the flowfield.

We are grateful to the staff of the X'luid Modeling X'acility and to Mr L. Marshin particular. I. P. C. acknowledges support from North Carolina State Universityas a Visiting Associate Professor under Cooperative Agreements (no. CR,814346and no. CR807854) and with the U.S. Environmental Protection Agency. P.G.B.acknowledges support from the same source. Although the research described inthis paper has been supported by the U.S. Environmental Protection Agency, ithas not been subjected to Agency review and therefore does not necessarily reflectthe views of the Agency and no official endorsement should be inferred. Mentionof trade names or commercial products does not constitute endorsement orrecommendation for use.

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r40 I. P. Castro, W.H. Snyder and P. G. Baines

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Note add,ed, in proof (23 Eebru,ary 1990). We recently became aware of the work ofMason (Geophys. Astrophys. Fluid, Dyn. 8, 137*154 (lg77D, who measured bodyforces on spheres in rotati,ng stratified fluid.