Saltation flow measurements relating to modeling of snowdrifting

Journal of Wind Engineering and Industrial Aerodynamics, 10 (1982) 89--102 Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

89

S A L T A T I O N FLOW M E A S U R E M E N T S R E L A T I N G TO M O D E L I N G OF SNOWDRIFTING

R.J. KIND and S.B. MURRAY*

Department of Mechanical and Aeronautical Engineering, Carleton University, Ottawa, Ontario K I S 5B6 (Canada) (Received September 28, 1979; accepted in revised form August 12, 1981)

Summary In scale-model studies of snowdrifting phenomena the similarity requirements cannot all be satisfied. The question of which of the conflicting requirements should be satisfied reduces to one of whether low-density or high-density particles should be used to simulate the snow. Experiments aimed at resolving this question are reported. Wind-tunnel experiments involving saltation of low-density (specific gravity 0.16) and high-density (specific gravity 2.7) particles were carried out. Velocity profiles, particle transport rates~ and "snowdrift" shapes produced by model snowfences were measured and compared with field data. The results show that when using the low-density particles the saltation process is not dynamically similar to that in actual snowdrifting; important differences in snowdrift shape and particle transport rates occur. The lack of similarity apparently stems from aerodynamic lift forces which are not negligible in comparison with the weight of the low-density particles. The use of high-density particles is recommended when modeling snowdrifting phenomena.

Notation CD Cf d D g G h k L u U u* Uth*

drag c o e f f i c i e n t skin-friction coefficient, r / 0 . 5 p U 2 particle d i a m e t e r drag f o r c e gravitational a c c e l e r a t i o n particle t r a n s p o r t rate (mass flux o f particles in a lane o f unit w i d t h ) saltation t r a j e c t o r y h e i g h t effective r o u g h n e s s height r e f e r e n c e length; also lift f o r c e local v e l o c i t y in b o u n d a r y layer v e l o c i t y o u t s i d e b o u n d a r y layer local shear velocity, x / r / p shear v e l o c i t y at t h r e s h o l d o f saltation

*Now at Defence Research Establishment Suffield, Suffield, Alberta, Canada. 0304-3908/82/0000--0000/$02.75

© 1982 Elsevier Scientific Publishing Company

90 V V*

VT~R VREL

W X

Y 6 v

p o T

reference velocity shear velocity in undisturbed flow upstream of model terminal falling velocity of particles relative velocity b e t w e e n particles and fluid particle weight distance from leading edge of particle bed height above surface boundary-layer thickness kinematic viscosity of fluid fluid density mass density of particle material shear stress in fluid near surface

1. Introduction Snowdrifting p h e n o m e n a are of substantial economic importance; for example, deep snow accumulations due to drifting can seriously affect railways~ highways and airports and often provide the critical loading case in the design o f r o o f structures. Drifting of s n o w occurs when winds o f speed ~ ~ 4 m s -1 or higher blow over an unconsolidated snow surface. The wind causes snow particles to move b y bouncing along or "saltating" over the snow surface. The moving particles remain within ~ 3 cm of the surface. Deposition of snow particles occurs where local reductions in windspeed are c a u s e d b y terrain features, buildings~ etc. Thus deep accumulations of snow, or "snowdrifts", are formed. Physical modeling can be very useful f o r finding ways o f dealing with problems due to snowdrifting. The requirements for correct model simulation were examined in an earlier paper [1] b y one of the present authors. Iversen [2] has also recently examined t h e requirements for similitude. Strom et al. [3], Odar [4] and Isyumov [5] have considered this problem in the past. Considerable differences exist b e t w e e n the similarity requirements proposed b y t h e various authors; this is a reflection of the fact that the problem is far from straightforward. The difficulties arise because it is quite impossible to match all t h e similarity criteria deduced b y a simple dimensional analysis. It is therefore necessary to discard unimportant criteria and to combine others to t h e m a x i m u m extent possible. In ref. i this was done on the basis of experimental evidence and our understanding o f the saltation process. Unfortunately the requirements deduced in ref. 1 include b o t h a F r o u d e number and a Reynolds-number limit, and it is impossible t o ~ b o t h these t~quirements in scale-model tests. This then raises the question: which of these t w o requirements can best be relaxed? The present paper pzesents the results o f experiments directed at answering this question. The similarity requirements proposed in ref. 1 can b e summarized as follows: (i) geometric similarity and the correct approaeh flow and particle transport rate are required;

91 (ii) the parameters V/U*th, VTER/V and V:/Lg must be matched in the model and p r o t o t y p e cases; (iii) either U*th3/2gv should be greater than ~ 3 0 for b o t h model and protot y p e or this Reynolds number should be matched for the model and protot y p e (this is a form of roughness-height Reynolds number, since u*2/2g is a measure of the trajectory height of the saltating particles); (iv) the particles used for simulation of snowdrifting should lie within a fairly narrow size range; (v) the angle of repose of model and p r o t o t y p e materials should, if possible: be equal. Here U*th is the shear velocity at which the particles just begin to saltate (threshold condition), VTER is the terminal falling velocity of the particles, g is gravitational acceleration, v is the kinematic viscosity of the fluid and L and V are a reference length and velocity, respectively, for the flow. Matching the Froude number V:/Lg requires that V = x/L; since U*th/V must also be matched, this implies that U*th3/2gv ~ L 3/2. In snowdrifting the p r o t o t y p e value of this Reynolds number is only ~ 2 0 . Unfortunately, therefore, it is impossible to satisfy simultaneously b o t h the Froude-number and Reynolds-number requirements, because L of the model is normally many times less than L of the prototype. Experiments have been conducted to assess the consequences of violating the Froude-number requirement in order to satisfy the Reynolds-number requirement, and vice versa; all the other main similarity requirements were satisfied in both cases. The use of high-density and low-density particle material in the first and second cases, respectively, is essential, as can be seen b y considering the particle selection procedures presented in ref. 1. In effect, therefore, the experiments are to examine whether low- or high-density particles are best used in physical modeling of snowdrifting. 2. Description of experiments Two different particle types were used to simulate snow in an open-return wind tunnel having a working section 0.91 m × 0.91 m × 10 m long. The entire floor o f the working section was covered with a layer of particles 2.5 cm deep. Drift profiles produced b y model snowfences for each particle t y p e were compared with one another and with field data [6, 7]. Flow velocity profiles and particle transport rates were also examined. Table 1 summarizes the particle properties and typical test conditions. Using sand particles the Reynolds-number requirement is satisfied, b u t the Froude-number requirement is violated; the reverse is approximately true using polystyrene particles. All the other major similarity requirements are satisfied b y b o t h particle types. The shear velocity V* in the undisturbed flow is used as the reference velocity in Table 1 ; snowfence height is used as the reference length L. The values given for U*th are those measured in the present tests. The geometric scale ratio was 1 : 20 for most of the model tests, with 1 : 40 being used in a few supple-

92 1

TABLE

Particle p r o p e r t i e s and typical test conditions Parameter

Mean diameter Specific gravity Angle o f r e p o s e U*th ( m s -1)

u*th3/2gv V*i/Lg VTER/V* V*/u*th

Expanded polystyrene

Silica

Field data

sand

[6,

0.6 m m 0.16 30 ° 0.09 2.0 0.03 3.0 2.0

0.2 m m 2.7 35 ° 0.15 13.0 0.19 2.0 2.7

snow snow snow ~0.15 ~15 ~0.01 ~ 2.5 ~ 2

7]

m e n t a r y runs. The silica sand particles had the irregular shape typical o f natural sand; the polystyrene particles, on the other hand, were essentially spherical in shape. Some supplementary runs were conducted using spherical glass beads (mean diameter 0.2 mm, specific gravity 2.7) to simulate snow. The model snow-fences spanned the full width of t h e wind tunnel and were oriented normal to the tunnet centreline, so that the tests were essentially twodimensional. Zero static-pressure gradient prevailed all along the tunnel working section. The model fences were m o u n t e d 5.5 m downstream o f the start of the working section on a short "ground plane" consisting of foam-backed short-pile carpet material whose surface was 2.5 cm above the tunnel floor, that is, flush with t h e surface of the undisturbed particle layer. This ground plane absorbed t h e kinetic energy of impacting particles so that initial particle deposition and erosion proceeded in a realistic way near the model snowfence; a hard ground plane was found to be unsatisfactory in this respect. A row of slender graduated pins was placed along the tunnel centreline near the snowfence models to allow measurement of snowdrift depth profiles. A schematic diagram of the experimental set-up is shown in Fig. 1. ~PARTICLE TRAPS

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DEPTH MEASURING~

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tom

Fig. 1. S c h e m a t i c d i a g r a m o f e x p e r i m e n t a l Set-up.

/ \

SCREENS ~

A

/

~(\

93 Three types of snowfence were used in the model tests: a Norwegian-type fence having horizontal slats, a Canadian-type fence having vertical slats, and a solid fence. Details o f these model snowfences are shown in Fig. 2. The dimensions shown in Fig. 2 for the Norwegian fence, for which field data are available [6], are 1/20 of full scale; a 1/40 scale model, having the same proportions as those in Fig. 2(a), was also tested. 20 ° l

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• ALL DIMENSIONS IN mm • SOUDITY= 0 3 4 .

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• SOLID FENCE FORMED BY TAPING OVER (b) C A N A D I A N - T Y P E

Fig. 2. D i a g r a m s o f m o d e l s n o w f e n e e s .

FENCE

SLATS

94 It was desired that t h e particle transport rate be in equilibrium with the flow approaching the models. Therefore no artificial devices such asspires or roughness elements were used to thicken the b o u n d a r y layer in the wind-tunnel flow. Consequently t h e ratio of boundary-layer thickness to fence height in the m o d e l tests was only ~ 2 : 1, which is of course much lower than in the p r o t o t y p e case, where the atmospheric b o u n d a r y layer would be 300 m or more thick. Nevertheless, the particle bed extended 54 fence heights upstream of the model in the 1 : 20 scale tests, so that it is plausible to assume that the model b o u n d a r y layer would be reasonably similar to the "internal" b o u n d a r y layer that would develop over a similar fetch of snow surface in the p r o t o t y p e case. There would of course be differences in the velocity profile shape and in the turbulence structure, b u t these are expected to be of secondary importance. In particular large~scale or low-frequency eddies will be lacking in the model simulation, b u t the effects of these should be of a quasi-steady nature which would n o t affect time~vemged results. These arguments are supported b y the fact that the results of the 1 : 20 and the 1 : 40 scale tests were virtually identical. The experimental procedure consisted of running the wind tunnel at constant speed and recording the depth profiles of the drifts formed near the model fences at various time intervals. In addition, velocity profiles over the saltating sand and polystyrene particle beds were measured with the model fences removed. Profiles were measured at four streamwise positions, at the saltation threshold speed and at the speeds used in the model test runs. A pitot t u b e and sensitive manometer were used for the sand, while a carefully calibrated hot-film anemometer was used for the polystyrene particles. The main purpose of the velocity profile measurements was twofold: first to allow determination of the shear velocity V*, and second, to determine whether the velocity profiles over the saltating polystyrene particle beds exhibited any unusual features due to the relatively tow values of the Reynolds numbers u*~/2g~, and V*3/2g~,. Previously, data for velocity profiles in saltation b o u n d a r y layers were available only for values of these Reynolds numbers in excess of 30. The data showed that the flow is aerodynamically rough and independent of Reynolds number in such cases [8]. 3. Results

Substantial differences b e t w e e n the sand and polystyrene particles were apparent with respect to the snowdrift profile shapes, velocity profiles, particle transport rates and saltation trajectory heights and lengths. Typical comparisons of drift profile shapes are shown in Fig. 3(a)--(c) for the Norwegian, Canadian, and solid fences, respectively. Figure 3(a) includes field data from ref. 6. The polystyrene and sand drift shapes are essentially the same for the solid fence, b u t t h e y are not very realistic because b o t h particle types are unable to duplicate the steep slopes produced b y solid fences in the field. Both Fig. 3(a) and (b) show substantial differences between the

95 -,,,i--- W I N D

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(a)

NORWEGIAN-

TYPE

fence

SNOWFENCE

WIND

j ~ J r

(b)

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CANADIAN - T Y P E

SNOWFENCE

polystyrene

WIND

$Gnd field data [6]

~model

(c)

SOLID

fence

SNOWFENCE

Fig. 3. Typical drift profile shapes produced by model and prototype snowfences.

polystyrene and sand drift shapes; the sand results agree well with the field data in Fig. 3(a). Negligible changes in the sand drift shapes were observed over a range of u*/u*th from 1.5 to 3.5; the polystyrene drift shapes, on the other hand, were somewhat sensitive to windspeed, becoming more elongated when u*/u*th was changed from 1.5 to 2. The velocity profile data were plotted in the form u--U versus log y and compared with the well-known velocity-defect law [9] for zero pressuregradient turbulent boundary layers. This defect law is k n o w n to apply to both smooth- and rough-wall boundary layers; it is therefore also expected to apply to saltation b o u n d a r y layers where the saltating particles rise only to a fraction of the boundary-layer thickness. By comparison with plots of the defect law drawn for various values of the shear velocity V* and boundary-layer thickness 5, the value of V* could be deduced for each profile measured. This method of deducing V* was adopted because the more c o m m o n method, of comparing slopes of the logarithmic portions of the profile, was prone to excessive error in the present work, as it often is for rough-wall boundary layers [10]. Figure 4 shows some typical comparisons between the present velocity profile data and the velocity~lefect law for conventional turbulent boundary layers. It can be seen that the profiles for the saltating sand agree well with the usual defect law over its entire range. The data for polystyrene particles, on the other hand, agree only with the outer portion of the defect law. The skin-friction coefficient Cf has values near 0.005 for the sand and only ~0.0025 for the polystyrene. The values of Cf cannot be determined with great accuracy by the method used, especially for the polystyrene, but the error is no more than

96 0

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MATERIAL x(m) u"/U~h SAND 6.8 I.O SAND 6.8 1.5

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o ~ ~ ~ _ MLN IEEAFO NRCONVENTO I NAL TURBULENTBOUNDARY LAYERS

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Fig. 5. Velocity prof'des near saltating particle beds (the hatched band rep~eNnts previous data [1] for uniform sand of various grain sizes, non-uniform soil and blowing m o w , at various values of U*/U~). The key for these data is as in Fig. 4.

97 +25%. The measured boundary-layer thicknesses over the saltating polystyrene are ~ 50% of those over the saltating sand, a reflection of the difference in the Cf values. The logarithmic portions of the velocity profiles (i.e., the pox~ions which are linear in Fig. 4) are plotted in a different way in Fig. 5. This form of plot is based on the fact that the velocity profile near an aerodynamically rough surface can be described by the equation u / u * = 2.5 l n ( y / k ) + C

(1)

where C is a constant and k is an effective roughness height. Owen [8] has shown that this equation applies to flow over saltating sand and soil surfaces if u*2/2g is used as a measure of the effective roughness height, that is, k ~ u*:/2g

(2)

Kind [1] has shown t h a t field measurements of velocity profiles over saltating or drifting snow surfaces are also described well by eqns. (1) and (2). Kind [1] also showed that the constant of proportionality in eqn. (2) has a value near unity. Figure 5 shows that the velocity profiles over the saltating polystyrene particles do n o t agree with the others, indicating a significant difference between the flow over polystyrene on the one hand, and that over sand and snow on the other. If dynamic similarity prevails, the non
98 4. Discussion

The results of the preceding Section indicate that the saltation process for the sand particles was dynamically similar to snowdrifting in the field, b u t that this was not so for the polystyrene particles. Although the saltation processes were not dynamically similar, it does appear t h a t the initiation of saltation was dynamically similar for the t w o sets of particles; this is indicated b y the fact that the measured values of the threshold shear velocity U*th given in Table 1 agree well with those predicted using the methods of ref. 1. Many of the differences described in Section 3 can be explained in terms of the much greater non
99 distribution of u* that develops in a flow having given geometric boundaries of course depends significantly on the nature of the surface roughness. The characteristics of the roughness produced b y saltating polystyrene particles have been shown to be quite different from those produced b y saltating sand or b y drifting snow; the distributions of u* can therefore also be expected to differ, giving rise to differences in drift shapes. It remains to explain w h y the non-dimensional trajectory heights are much higher for the saltating polystyrene particles. In searching for an explanation two major differences between the sand and polystyrene particles come to mind: first, the polystyrene particles are spherical while the sand particles are irregular in shape, and second, the large difference between the mass densities of the two materials. Etkin [11] has shown that if drag is the only important aerodynamic force acting on particles, they should be completely characterized by their terminal falling velocity VTER; that is, particle shape or drag coeffic i e n t density and size are important only in combination as they affect VTER and are n o t each separately significant. Thus the shape and density differences cannot explain the observed non-similarity of behavior if drag is the only important aerodynamic force acting on the particles. If, on the other hand, aerodynamic lift is important for one set of particles, and not the other, nonsimilar behavior is to be expected. Spherical particles might conceivably be subject to larger lift forces than irregularly shaped ones in the highly sheared flow at the particle bed where the saltation trajectories are initiated. To investigate this, some of the experiments were repeated using spherical glass particles, which, except for their shape, had the same properties as the sand used earlier. Threshold shear velocity, particle transport rate, non
(3)

Equation (3) is obtained b y applying an approximate modification to the exact solution for a circular cylinder in inviscid flow with uniform shear [12] ; it is supported b y the experimental data of Eichhorn and Small [13] over the parameter range of interest here. Since the initial portions of the trajectories are nearly vertical for solid particles saltating in air [14], it is possible to estimate VREL -- U -- 5U* The estimate in eqn. (4) comes from using the data of Fig. 5 at the lowest

(4)

100

values of y. Assuming, for the present purposes, that eqns. (1) and (2) are valid near the surface, results in du/dy ~- 2.5 u*/y

(5)

Then L / W = 25 pu*2/o gy

(6)

where W is the weight of the particle. For the aerodynamic drag D, it is possible to estimate D -~ 0 . 5

PVREL27rd2CD/4

(7)

Then D / W = 20 pu*2/o gd

(8)

using CD -~ 1 and VRELfrom eqn. (4). Comparison of eqns. (6) and (8) indicates that LID = d/y

(9)

This indicates that when a particle is just starting to rise from the surface (y -~d), lift forces are about as significant as drag forces. Other workers [15] have reached the same conclusion. However, for solid particles in gases, y quickly becomes much greater than~d and then L ~ D. Most previous workers have in fact assumed that lift forces are negligible over the majority of the saltation trajectory when o ~ p *. However, the ratio of lift to particle +weight is also important. As a rising particle approaches its maximum height, y ~- k, where k is given by eqn. (2) with the constant of proportionality about equal to unity [1]. Then from eqn. (6), L / W ~- 50 p/o

(10)

For sand in air (o/p = 2200), eqn. (10) indicates that the lift is indeed negligible in comparison with particle weight; however, for polystyrene in air (alp = 130), eqn. (10) indicates that the lift force may equal about half the particle weight when y -~ u*2/2g, and will be still larger at lower heights. Thus the vertical impuIse due to lift force appears to be much more significant for low
*White a n d S c h u l z [16] h a v e f o u n d e v i d e n c e o f s u b s t a n t i a l lift forces in t h e i n t e r m e d i a t e stages o f s a l t a t i o n trajectories, b u t t h e m a g n i t u d e s w e r e m u c h smaller t h a n t h o s e n e e d e d to explain the present e x p e r i m e n t a l results.

101

supports this approach if aerodynamic lift is mainly responsible for carrying particles up from the bed. If, on the other hand, the rising particles have acquired most of their vertical kinetic energy from the impact of other particles striking the bed, the analysis indicates that the density ratio o/p should not appear in the parameters. If the data of Fig. 5 were re-plotted with (egy/pu .2) as the abscissa, the disagreement between the sand and polystyrene results would be much greater; moreover, the snow and sand data would no longer agree. If the observed heights of the saltation trajectories were non-dimensionalized using eu*2/pg instead of u*2/2g, the non-dimensional heights for sand and polystyrene would still differ b y a factor of about four, the polystyrene values becoming t o o low instead of t o o high. A similar c o m m e n t applies to the nondimensional mass transport rates. Thus the present experimental results do not support the use of a densimetric roughness-height scale; they indicate that aerodynamic lift forces have a significant influence on saltation trajectories only if the density ratio o/p is relatively low. In summary, the present work indicates that aerodynamic lift becomes significant relative to particle weight in the initial stages of saltation trajectories if the density ratio o/p is relatively low. Consequently the saltation process with low (~/p is not dynamically similar to that for snow or sand in air; nondimensional trajectory heights are larger and this gives rise to other dissimilarities. Important discrepancies in snowdrift shapes and particle transport rates occur. On the basis of eqn. (10), a lower limit of 600 for alp is tentatively suggested if dynamic similarity with actual snowdrifting is desired. It thus appears that the preferred procedure for modeling of snowdrifting phenomena is to use a value of at least 600 for (~/p even though this implies abandoning the Froude-number similarity requirement. The present results show that correct snowdrift shapes and particle transport rates can be obtained using this approach. Efforts should be made to keep the saltation trajectory lengths small in comparison with the smallest snowdrift length that is of interest [1]. Water cannot be used as the working fluid if a high value of o/p is required. Iversen [2] arrived at similar conclusions using different reasoning. He obtained good comparisons with field snowdrift patterns by using dense glass spheres of diameter ~ 5 0 ~m in a wind tunnel. 5. Conclusions Experimental results have been presented which show that for relatively low values of the particle-to-fluid density ratio e/p the saltation process is not dynamically similar to that in snowdrifting. Important discrepancies in snowdrift shapes and particle transport rates consequently occur if relatively low e/p values are used when modeling snowdrifting phenomena. The discrepancies appear to stem from aerodynamic lift forces which are significant compared to particle weight when e/p is less than ~ 6 0 0 . It is r e c o m m e n d e d that particles having alp greater than 600 be ttsed for

102

the modelling of snowdrifting even though this implies abandoning the Froude. number similarity requirement.

Acknowledgements Most of the experimental work was carried out in a wind tunnel provided by the Hydraulics Section of the National Research Council of Canada (NRC). Financial support was received through an NRC Operating Grant. During the early phases of the work one of the authors (R.J,K.) was attached to t h e Low Speed Aerodynamics Section of the NRC while on sabbatical leave from Carleton University. Support from all these sources is gratefully acknowledged.

References 1 R.J. Kind, A Critical examination of the requirements for model simulation of windinduced erosion/deposition phenomena such as snow drifting, Atmos. Environ.. 10 (1976) 219--227. 2 J.D. Iversen, Drifting snow similitude drift deposit zate correlation, Proc. 5th Int. Conf. o n Wind Engineering, Fort Collins, CO, July 1979. 3 G. Strom, G.R. Kelly, E.L. Keitz and R.F. Weiss, Scale model studies o n snow drifting, U.S. Army Snow, Ice and Permafrost Res. Establ., Res. Rep. 73 (1962). 4 F. Odar, Simulation of drifting snow, U.S. Army Cold Regions Res. Eng. Lab., Res. Rep. 174 (1965). 5 N. Isyumov, An approach to the prediction of snow loads, Ph.D. Thesis, University of Western Ontario, 1971. 6 K. Croce, in Snow Studies in Germany, National Research Council of Canada Tech. Memo. 20 (DBR) (May 1951). 7 D. Kobayashi, Studies of snow transport in low-level drifting snow, Inst. Low Temp. Sci., Sapporo, Japan, Rep. No. A31 (1973). 8 P.R. Owen, Saltation of uniform grams in air, J. Fluid Mech., 20 (1964) 225--242. 9 F.H. Clauser, The turbulent boundary layer, Adv. Appl. Mech., 4 (1956) 1-~51. 10 A.E. Perry, W.H. Schofield and P.N. Joubert. Rough wall turbulent boundary layers, J. Fluid Mech., 37 (1969) 383--413. 11 B. Etkin, Interaction of precipitation with complex flows, Proc. 3rd Int. Conf. on Wind Effects on Buildings and Structures, Tokyo, 1971, pp. 135--143. 12 G.K. Batchelor. An I n t r o d u c t i o n to Fluid Dynamics, Cambridge University Press. 1967, pp. 541--543. 13 R. Eichhorn and S. Small, Experiments on the lift and drag of small spheres suspended in a Poiseuille flow, J. Fluid Mech., 20 (1964) 513--527. 14 R.A. Bagnold, The Physics of Blown Sand and Desert Dunes, Methuen, London, 1941. 15 W.S. Chepil, Equilibrium of soil grains at the threshold of movement by wind, Proc. Soil Sci. Soc. Am., 23 (1959) 422--428. 16 B.R. White and J.C. Schulz, Magnus effect in saltation, J. Fluid Mech., 81 (1977) 497--512. 17 J.D. Ivereen, Drifting snow similitude, Proc. Am. Soc. Cir. Eng., J. Hydraul. Div., 105, HY6 (1979) 737--753.