Caves and karst: Research in speleology

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Caves and karst: Research in speleology
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Caves and Karst: Research in Speleology
Cave Research Associates
Cave Research Associates
Tumbling Creek Cave Foundation
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Contents: Channel hydraulics of free-surface streams in caves / William B. White and Elizabeth L. White. Cave Notes(vols. 1-8) and Caves and Karst: Research in Speleology(vols. 9-15) were published by Cave Research Associates from 1959-1973. In 1975, the Tumbling Creek Cave Foundation compiled complete sets of the journals in three volumes. The Foundation sells hardbound copies of the material to support its activities.
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Tumbling Creek Cave Foundation Collection
Original Version:
Vol. 12, no. 6 (1970)
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CA YES AND KARST Research in Speleology Volume 12, No.6 Hydraulic Radius = R = dw 2d t w Wetted Perimeter = P = 2d T W Hydraulic Depth = D = d figure I. Sketch defining the hydraulic parameters for a rectangular open channel CHANNEL HYDRAULICS OF FREE-SURFACE STREAMS IN CAVES' by WILLIAM B. WHITE*'" and ELIZABETH 1. WHITE u Abstract Elementary fluid mechanics is applied to cave conduits in which the generating stream had a free air surface. These are mainly passages with canyon-like cross-sections. Examples are taken from the central Kentucky karst. Estimation of Reynolds and Froude numbers for typical water flows in caves predicts that most cave streams flow in a sub-critical turbulent regime. Channel widths vary systematically with both velocity and discharge. Slopes of channels calculated from the Manning equation agreewith measured values for small canyons. Large canyons have very flat gradients and ate interpreted as high points in an undulating conduit which lies near the ground-water level. Introduction The shapes of limestone caves are exceedingly complex. The passages themselves are crooked and intersect at a variety of angles and positions. Cross-sections may vary from elliptical to irregular. Wall and ceiling surfaces are usually sculptured with an immense variety of relief detail. The question which must be posed is whether or not the informa, don imprinted on this complex array of shapes can be deciphered inca meaningful knowledge about the evolution of the cave or abouc events in the drainage system of which the cave was a part. This paper is a portion of a project on the hydrology of the Central Kentucky Karst undertaken by the Cave Research Foundation as part of the United States' contribution to the International Hydtolcqic Decade. *" Materials Research Laboratory and Department of Geochemistry and Mineralogy, The Pennsylvania State University, University Park, PA 16802. *** Department of Civil Engineering, Ibid. 41


CAVES AND KARST CAVES AND KARST CAVES AND KARST is a publication of Cave Research Associates. Subscriptions are available for $2.50 per year (six issues) or $6.00 for Volumes 12 through 14. Mid-year subscriptions receive the earlier numbers of the volume. Correspondence, contributions, and subscriptions should be addressed to: CA~E RESEARCH ASSOCIATES, 3842 Brookdale Blvd., Castro Valley., Calif. 94546 Editor: Arthur l. lange Associate Editor: Alan D. Howard Editors: R. deSaussure, J. F. Quinlan, Sylvia-F. Graham, Mary l. Hege, Lee Christianson @ Copyright 1970, Cave RBI .. arch Aucclate' There are two competing sets of factors that control the shapes of cave passages. One set comprises the spatial variations in the rates of solution of the bedrock caused by the distribution and geometry of joints and bedding planes, by the variations in solubility of the limestone, and by the distribution of such lithologic features as shale beds, sandy layers and cherr nodules. The second set comprises the hydrodynamic force associated with moving water. If flow velocities are low, the passage tends to be etched out into a complex shape controlled by the vagaries of the rates of solution. At higher velocities the hydrodynamic forces become dominant and tend to generate smooth curvilinear shapes. Some predictions about flow behavior and thus cave patterns and sculpturing should be possible by the application of the laws of fluid mechanics. In applying these laws we distinguish conduit flow to mean water circulating in a cave passage, and subdivide it into pipe flow through passages completely filled with water, and channel flow of water having a free air surface. The ground plans, shapes, and sculpturing of the cave passages may then be tested against hydrodynamic predictions. One of the first attempts to apply fluid mechanics to problems of ground-water Bow in limestone was by Otto Lehmann (1932). The significance of Lehmann's approach was largely overlooked by later researchers, and it is only within the past decade that extensive application of the theory has been made. White and Longyear (1962) showed that Rows in conduits are turbulent and provide some restrictions on mechanisms of cavern development. Flow velocities are recorded in some passages by the scallop markings. Curl (1966) and Goodchild (1969) demonstrated a Reynolds number relation between scallop length and velocity and test the accuracy of the scallop length as a measure of velocity. Deike and White (1963) showed that the big caves of south-central Kentucky could be interpreted from a simple drainage-net model. Meander length and width of sinuous cave passages fit the same equation that describes surface rivers, and the numerical values of the constants are similar (DEIKE & WH1TE, 1969). Howard (1964) and Palmer (1969) both applied fluid mechanics to ground-water motion through small openings and used the results to interpret cave development. The purpose 'of the present paper is to explore the behavior of open channel Bows in caves. We first develop some of the concepts of open channel hydraulics and then apply these to predict behavior of such flows in caves. We then examine the patterns and solution sculpturing of real caves to see how well the theoretical model fits. The paper is another contribution to the continuing attempt to interpret cave systems as remnants of well developed underground drainage systems. Some characteristics of underground channels Channels are usually classified as rigid or erodable. Rigid channels have fixed walls that are not affected by the fluid flow. Concrete or metal flumes and spillways are examples. Erodable channels have walls of unconsolidated materials that can be moved and redeposited by moving fluid. Most natural channels, creeks, and rivers, that typically have beds cut in their own alluvium are of this type. Cave passages are natural examples of rigid channels, at least so far as short-term flood events are concerned. 42


VOLUME 12, NO.6 Most man-made channels are approximated by some simple geometric figures such as (in cross-section) a rectangle, trapezoid, circle, or parabola. Each cross-section is characterized by a hydraulic radius and a hydraulic depth. Expressions for these are tabulated by Chow (1959) for many common geometries. Natural channels tend co have more complicated shapes and must be approximated to one of the simpler shapes. The defining parameters for a rectangular channel are shown in Figure 1. Many cave streams are underfitted to the conduit in which they flow, A large passage typically has a wide gravelled floor over which the stream flows. These streambeds are often armored with a layer of cobble-sized material, typically rounded sandstone cobbles and chert nodules. The channel banks are often poorly developed; depths may be shallow, in which case these channels approximate wide open channels with large width-to-depth ratios. These have various features in common with surface streams, such as the transport of clastic sediments by bedload, and alternating reaches of pools and riffles. Jones (in press) characterizes these as underground flood plains. Cave passages having thick sequences of clay and silt frequently contain streambeds incised into the sediment layers. These Streams have approximately trapezoidal crosssections and may be much smaller than the passage in which they occur. The channels often meander and have properties little different from surface streams on alluvial flood plains. The third type is of most interest here. These are the canyon-like passages cut into solid bedrock by incising underground streams The geometry is rectangular. Since rhe channel walls are in rock, bank slumpage does not take place, so that these canyons are possibly the most perfect rectangular channels occurring in nature. The total depth of the channels are vary variable. Canyons a meter or so wide and 15 to 30m deep are nor unknown. Since their banks do not erode, all previous stages of the channel are retained as the creating stream downcurs. The walls are nearly vertical in these canyon passages. In these channels there is no underground equivalent to rhe alluvial flood plain; therefore, one cannot easily define a discharge analogous to the bank-full stage of surface streams. Likewise the flood response is likely to be different because of extreme variations in stage. It is for this reason that an understanding of the channel hydraulics of such canyons is of great interest. Flow regimes in cave conduits Flow in pipes may be either laminar or turbulent. The regime is dependent on the balance between the momentum of the moving water and the viscous shear and is described by the dimensionless Reynolcls 1l1trflber. At low Reynolds numbers the viscous forces dominate the flow behavior and the flow is laminar. There is a transition region near N R = 2000 (in smooth circular pipes) ; at higher Reynolds numbers the momentum of the fluid is the dominant force and the flow is turbulent. When water is moving in a channel with a free air surface, the depth of the fluid need not be constant, and gravity forces become an important consideration. Gravity forces are compared wirh the fluid momentum by the dimensionless Proude number. Thus in channels, there are two characterizing numbers that yield four possible flow regimes. The Reynolds number for open channel flow can be defined by N .. VR R u (1 ) where V is the mean flow velocity; R, the hydraulic radius (defined for the rectangular channel of Figure 1) ; and v, the kinematic viscosity of the fluid. Note that the characteristic length in the dimensionless equation is defined differently than' it is, for pipe flow (where R is the pipe diameter). The onset of the turbulence in an open channel begins near N R = 500 because of this change in definition. The Froude number is defined by N V; ',/gD where g is the gravitational acceleration; and D, the hydraulic depth defined in Figure 1. (2) 43


CAVES AND KARST 1.0 0.6 0.4 0.2 0.1 0.06 0.04 ~ ~ ~ 0.02 E I 0.01 .... 0W 0 0.002 0001 004 0.1 0.2 0.4 VELOCITY 4.0 10 2.0 (meters/sec) Figure 2. Flow regime for open channel flow, Assumptions in calculating this plot are that the channel is wide open (w > > d) and that the temperature of the water is 10 0 e. The grid contours the values of Reynolds number (N R ) and Proude number (N F ). Flows for which N F 1 are termed supercriticd. A convenient field test is to observe small gravity waves introduced in the flowing stream by a disturbance. These may propagate upstream in subcrirical but not supercritical flow. By using the model of the wide open channel (w d) containing water near cave temperature of lOoC, a plot of water depth versus flow velocity may be prepared (Figure 2) Showing the four possible Row regimes. Most cave streams have depths on the order of tenths of meters and move with velocities in the range of O.3m/ sec and less; thus placing these streams in the subcritical/ curbulenr regime of Figure 2. Indeed this is what is observed underground. Cave streams have a rough surface broken by irregularities in the bed. Smooth-surface, fast-moving water in supercritical flow is seldom observed. Hydraulic geometry of canyon passages Data for a few rectangular channels from Mammoth Cave and the Flint Ridge Cave 44 '0


VOLUME 12, NO.6 system, Kentucky are listed in Table 1 and plcered in Figure 3. The velocities were estimated from scallop measurements by means of Curl's (1966) equation N (5) VL R 'U' (3) NR(S) is a scallop Reynolds number and [ is the average scallop length. Goodchild's (1969) value of 11,500 was used as the scallop Reynolds number. Goodchild's careful analysis of scallop behavior shows that there is considerable variation in scallop lengths. Since the scallops that occur in canyon passages are small and the velocities are in the same range as those used in his model experiments, it seems probable that these estimates give at least an approximation to the actual flow velocities. The number of data points available is very small and any conclusion drawn from Figure 3 would be extremely tenuous. The curve fitted through the points can be described by a power function of the form, V -c.t w (4) Because of scatter, a two-thirds power re1a~ion could likely be the correct one. It seems likely that Equation 4 is/a subduced relation between velocity and channel width arising because both are related/to the discharge. A form of hydraulic geometry unique to underground conduits was discovered by Deike (1967) in Mammoth Cave. Robertson Avenue is a remarkable reach of passage, about lOOOm in length, which alternates between elliptical tube and canyon cross-section. Deike deduced from a measurement of the vertical profile that Robertson Avenue has a vertical undulation of 1.5 to 3m and that the tubular cross-sections correspond ro the low sections and the canyon cross-sections, to the high ones. Two typical cross-sections are shown in Figure 4. The scallops in the tubular section have a. mean dimension of 48crn, corresponding to a velocity of O.03mjsec. Since the scalloping is uniform on floor, walls, and ceil lpg, a pipe Ro'rv regime seems reasonable and requires a discharge of O.27m: l sec. In the canyon segments of the passage the discharge must be the same. The scallop length in the canyon is l Scm, yielding a .095m:~!sec flow velocity. The equation of continuity then predicts a flow depth of O.92m, a value that is not only reasonable bur in agreement with a rule-of-thumb estimate of a 1: 3 ratio of depth to width observed in many smaller canyons. Slopes of canyon passages The velocity of flow in Manning eqttation an open channel is usually calculated by means of the (5) Channel Width Scallop Estimated length velocity m It em in m/sec ft/se.c ~_. . .Dismal Valley 10.7 35. 51. 20. 0.01.9 0.096 Uppe r Salts 7.0 23. 56. 22. 0.027, 0.09 _~obertson Avenue 3.0 10. 16. 6.3 0.095 0.31 --Robertson Avenue 2.3 7.5 12. 4.75 0.12 0.4 Indian Avenue 0.9 3. 15. 6. 0.10 0.33 Be cky'e Alley 0.3 1.0 5. 2. 0.29 0.96 Table 1. Properties of rectangular channels of the Flint Ridge cave system arid Mammoth Cave, Kentucky, 46


CAVES AND KARST 03 0.2 RECTAN GULAR CHANNELS u ~ m m ~ E 0.1 008 >f0.06 u o .J0.0 4 w >003 0.02 CHANNEL WIDTH (m et er s) Figure 3. Plot of channel width versus flow velocity (estimated from scallop measurements) for some rectangular channels. (Numerical values given in Table 1.) where R is the hydraulic radius of the channel (in meters) and S is the slope. The constant n is determined by the roughness of the channel. Many values for Manning's n are tabulated by Chow (1959) and a value of 0.02 seems most appropriate for a rough bedrock-walled channel. It is of interest to compute the slopes (usually a difficult quantity to measure) for a variety of canyons. Three examples-a large canyon, an intermediate-sized canyon, and a small, typical shaft-drain canyon-were chosen. The various parameters are tabulated in Table 2. The large canyon was Upper Salts. Most of this passage has been highly modified by breakdown and the original solution outlines have been lost. In one short teach, near the Pike Chapman entrance, tbe solution walls are intact. Here the walls are vertical and parallel. The passage is 7m wide and at least 18m high. The walls are uniformly scalloped from floor to ceiling. The discharge through this passage is not known but was estimated to be in the range of O.6m'l/ sec by a variery of arguments relating the Salrs'trunk to paleolocations of Pike Spring. The intermediate-sized canyon is Robertson Avenue, sketched in Figure 4. Discharge and depth of flow were given above. Becky's Alley is typical of many small canyons that occur in rhe big central Kentucky caves. The narrow width remains nearly constant, although the passage reaches heights of 10m and more. At the bottoms of these narrow canyons one can often find an active level occupied by a small stream. The flow depth is typical for many observed canyons. The results are surprising. The slopes calculated for Upper Salts and Robertson Avenue are extremely small compared with known slopes for the big drainage trunks which, are on the order of a.Gm/ken. The steeper slope of Becky's Alley, somewhat larger than 46


VOLUME 12, NO.6 Upper Salts Robertson Becky's Avenue Avenue Alley V 1m) 0.027 0.094 0.29 w tm) 7.0 3.0 0.3 dim) 2.9 0.92 1.0 R (m) 1.60 0.57 0.061 ,--. -S (m/m) 1.6x10 7 7.56xI06 1,43xI0 3 (ft/mi) 8. 4x 10. 4 0.04 7.5 Table 2. Hvdreulic parameters and calculated slopes of three rectangular channels of the Flint Ridge cave system and Mammoth Cave, Kentucky. that of the major trunks is about as expected. These results cannot be written off to inaccuracies in the data. All parameters entering the Manning equation are reasonably precise except for the flow depth. Except for the narrow -myon, all passages are abandoned, so that the depth must be estimated by some method. The estimates are not without limits. The flow depth cannot be greater than the height of the passage nor smalJer than the height of a single scallop. The maximum range of error is from 1 to 20m in the case of Upper Salts, and either of these extreme values still gives channel slopes much smaller than the slope of the conduits. These very flat slopes are understandable in terms of the undulating conduit illustrated in Figure 5. The shorr reaches of canyon are, in effect, underground lakes, fed at one end and drained at the. other by flow from pipes. The very flat slopes represent the differences in elevation between the upstream and downstream ends. Conclusions The objectives of this short study were to examine some of the possibilities for interpreting the shapes of cave canyon passages in terms of the hydraulics of channel flow. It appears that J. considerable amount of information is imprinted on these channels and that it may be possible to determine the origin of particular canyons by their measurable properties. Much work is needed, however, to work out empirical relations analogous to equations that have already been developed for alluvial channels. In particular, two questions urgently need answering: Is rhere some simpler power-function law of the form of Equation 4 that relates width (the most accurately measurable parameter) to discharge? Is there a reliable relation between width and depth (i.e. is the 3: 1 ratio really valid)? A second class of questions not even touched upon in this paper deals with the seasonal variations in channel flows. Parameters such as velocity and channel width measured from an abandoned channel perhaps can be used to calculate other hydrological properties. But we do not know whether these properties refer to the mean flow, the flood flow, the most probable flow" the flow at seasonal low discharge, or some other flow. Somehow in these calculations the seasonal variations of che discharge must be taken into consideration. I C_~ 6.6ml 1.8 m Figure 4. Typical cross-section for the tubular and canyon reaches of Robertson Avenue, Mammouth Cave, Kentucky. 47


CAVES AND KARST Tube Canyon with _l?-I Tube Remnont~' Tube -G-y-" -==--,, ~ = ----~~~ ~. -= -: --L LONG ITU DINAL PRO FI LE L Figure 5. Model for Robertson Avenue, Mammoth Cave, Kentucky. The amplitude of the undulations are on the order of 3m. The "wavelength" is on the order of 300m. References CHOW, VEN TE (1959). Open-channel Hydrau/i~J. McGraw-Hill, New York. 680p. CURL, RANE L. (1966). Scallops and flutes. CaIle Research Group of Great Britahl Tramactions 7: 121160, I DEIKE, GEORGE H. & WILLIAM B. WHIT.E (1963). Paleohydrology of Mammoth Cave and the Flint Ridge cave system. Geological Societ')' of America, Proceedings (abstract-manuscript in preparation) DEIKE, GEORGE H, 111 (1967). The Deoeloprnent of Caverns in the Mam,moth Cave Region. PhD Thesis, The Pennsylvania State University. 235p. DEIKE, GEORGE H. & WILLIAM B. WHITE (1%9). Sinuosity in limestone solution conduits. American Journal of Science 267, 230-241. GOODCHILD MICHAEL F. (1969). The Generation of Small-scale Relief Peeuees of Eroded Limestone: A Study of Brosionol Scallops. PhD Thesis, MacMaster Univers.ity. 168p. HOWARD, ALAN D. (1964). Processes of limestone cavern development. lrtternational Journal of Speleology L 47-60. JONES, WILLIAM K. (in press). Characteristics of the underground flood plain. National Speleological Society BuLletill. LEHMANN, OTTO (1932). Die Hydrographic des Karstes. Enzyklopadie der Erdku1tde, Franz Deuticke, Leipzig. 212p. PALMER, ARTHUR N. (1969). A Hydrologic Study of the Tndiana Kerst. PhD Thesis, Indiana University. 181p WHITE, WILLIAM B. & JUDITH LONGYEAR (L962). Some limitations on spelecgeneric speculation imposed by the hydraulics of ground-water flow through Iimesrone.. National Speleological Society, Nittany Grotto Newsletter. 10: 155-167, The velocities of dry channels used in this paper were calculated from measured scallop lengths using Goodchild's value of 11,500 for the scallop Reynolds number. New results by P. Blumberg (R. L CURL, personal communication) indicate that the scallop Reynolds number of 22,000 originally proposed by Curl is more nearly correct. The discrepancy arises because of the definition of velocity. The channel velocity is approximated by vmflxwhich occurs near the center of the channel (for a circular pipe). The velocity profile is logarithmic and becomes zero at the wall. Goodchild measures his velocities at an arbirary one inch (2.54cm) above the bed. His scallop Reynolds number relates to a velocity less than mean or maximum channel velocity (which, of course, is the correct velocity to use for calculating discharge). The uncertainty in scallop Reynolds number affects the numbers listed in this paper but does not change the conclusions in any significant way. 48

Contents: Channel hydraulics of free-surface streams in
caves / William B. White and Elizabeth L. White.
Cave Notes(vols. 1-8) and
Caves and Karst: Research in Speleology(vols. 9-15)
were published by Cave Research Associates from 1959-1973. In
1975, the Tumbling Creek Cave Foundation compiled complete
sets of the journals in three volumes. The Foundation sells
hardbound copies of the material to support its