TECHNICAL PAPER Physical model study of Journal of the South African Institution of Civil Engineering bedrock scour downstream ISSN 1021-2019 Vol 62 No 3, September 2020, Pages 36–52, Paper 1000 of dams due to spillway ADÈLE BOSMAN PrEng is a lecturer in plunging jets Hydraulics at the University of Stellenbosch, South Africa. She is currently studying towards her PhD (Civil) degree in developing a numerical model for A Bosman, G R Basson predicting rock scour. She has more than 11 years’ experience mainly in river hydraulics and the design of large hydraulic structures. The erosive power of a free-falling high-velocity water jet, flowing from a dam spillway, could Contact details: create a scour hole downstream of the dam, endangering the foundation of the dam. Despite Civil Engineering Department Water Division extensive research since the 1950s, there is presently no universally agreed method to predict Stellenbosch University accurately the equilibrium scour depth caused by plunging jets at dams. These formulae yield Private Bag X1 a large range of equilibrium scour dimensions. The hydrodynamics of plunging jets and the Matieland 7602 subsequent scour of a rectangular, horizontal and vertical fissured rock bed were investigated T: +27 21 808 4356 E: [email protected] in this study by means of a physical model. Equilibrium scour hole geometries for different fissured dimensions (simulated with rectangular concrete blocks tightly prepacked in a regular PROF GERRIT BASSON PrEng MSAICE is a rectangular matrix), for a range of flow rates, plunge pool depths, and dam height scenarios professor in Hydraulic Engineering in the were experimentally established with 31 model tests. From the results, non-dimensional Civil Engineering Department at formulae for the scour hole geometry were developed using multi-linear regression analysis. Stellenbosch University. He obtained a PhD from Stellenbosch University in 1996 and The scour depth results from this study were compared to various analytical methods found has more than 30 years’ experience mainly in literature. The equilibrium scour hole depth established in this study best agrees with that in the fields of river hydraulics, fluvial predicted by the Critical Pressure method. morphology and the design of large hydraulic structures. He has worked on projects in 21 countries and is an Honorary Vice-President of ICOLD (International Commission on Large Dams). Introduction depth. New formulae were developed to Contact details: A free-falling high-velocity water jet, determine the scour hole geometry (depth, Civil Engineering Department flowing from a dam spillway, could scour length, width and volume) based on stream Water Division the bedrock downstream of a dam if the power and the rock’s movability number Stellenbosch University Private Bag X1 erosive capacity of the jet is large enough. and settling velocity. Matieland 7602 The foundation of the dam could be endan- T: +27 21 808 4355 gered by the scour hole. It is therefore E: [email protected] imperative to determine the equilibrium Rock scour geometry of the scour hole to incorporate Bedrock scour downstream of a dam is a it at the design stage to ensure the stability complex physical process and the complete of the dam. understanding of the air-water-rock phase Well documented cases, such as interaction in the scouring process is Kariba Dam on the Zambezi River (border required. The free-falling jet could be split between Zimbabwe and Zambia, Africa) into three different modules, namely the and Wivenhoe Dam in Australia are prime falling jet module, plunge pool module and examples where the plunge pool scouring rock mass module as defined by Bollaert had drastic consequences. The depth of and Lesleighter (2014) in Figure 1. the scour hole of Kariba Dam is approxi- Various researchers have studied the mately two-thirds of the dam height of hydrodynamics of a free-falling jet in the 128 m (Noret et al 2012), while that of the air and the plunge pool. It was found that

Wivenhoe Dam is one-third of the dam the fall height (Hw), issuance velocity (Vi), height of 33.87 m (full supply level to spill- discharge, air entrainment, outlet shape

way bucket lip) (Stratford et al 2013). and initial turbulence (Tu) influence the The purpose of this study was to behaviour of the jet as it falls through simulate rock scour due to plunging jets the air (Van Aswegen et al 2001). The jet in a physical model and to gain a greater trajectory and energy at impingement with understanding of the scouring process. The the plunge pool surface are dependent

physical model results were then compared on the velocity of the jet (Vj), whereas the Keywords: erodibility, impinging jets, physical model, plunge pool, to the predictions of different analytical breakup length (Lb: distance for the jet scour methods of the equilibrium rock scour core to dissipate in the air) is reliant on

Bosman A, Basson GR. Physical model study of bedrock scour downstream of dams due to spillway plunging jets. 36 J. S. Afr. Inst. Civ. Eng. 2020:62(3), Art. #1000, 17 pages. http://dx.doi.org/10.17159/2309-8775/2020/v62n3a4 the initial jet turbulence intensity (Manso et al 2008). Reservoir water level Manso et al (2008) investigated the D or B V i i behaviour of the jet as it travels through i the plunge pool. The jet velocity (Vj), impact angle (θ) and turbulence at enter- ing the plunge pool surface, as well as Falling jet Hw module the plunge pool depth (Y ) and geometry, δout determine the amount of energy that would be dissipated and how much of the jet core remains intact at impact with the bedrock. θj Tailwater level B , V According to Ervine and Falvey (1987), the j j inner core diffusion angle (αin) of highly yw Original bed level Plunge pool turbulent plunging round jets when travel- αout Y αin module ling through a plunge pool is approximately ys ° Rock 8 , whereas the outer expansion angle (αout) mass can increase up to 15°. module The plunge pool bed would scour if the erosive capacity of the jet exceeds the Xult ability of the rock to resist it. To deter- mine the conditions of the jet at impact Figure 1 Plunging jet modules involved in the rock scouring process (adapted from Bollaert 2002) with the rock bed, the velocity and pres- sure decay in the plunge pool need to be Table 1 Empirical formulae to predict plunge pool equilibrium rock scour depth defined. Ervine and Falvey (1987) defined Formula (empirical) Author(s) the maximum pressure of a circular jet at a certain distance below the plunge pool’s H 0.225q0.54 Veronese 1 (1937, as cited in y + y = 3.68 w w s 0.42 Whittaker & Schleiss 1984) free surface in terms of the velocity head d90 and pressure coefficients to account for Veronese 2 (1937, as cited in the fluctuating nature of the turbulent 0.225 0.54 yw + ys = 1.32 Hw q Wittler et al 1995) jet. Castillo et al (2014) determined the mean and fluctuating pressure coefficients Veronese 3 (1937, as cited in for rectangular jets. Factors affecting the 0.225 0.54 yw + ys = 1.9 Hw q Noret et al 2012) vulnerability of the erodibility of the rock are lithology, rock strength, joint spac- Yildiz & Uzucek (1994, as cited ing, joint orientation and joint condition 0.225 0.54 yw + ys = 1.9 Hw q sinθT in Alias et al 2008) (Annandale 2006). Extensive research has been conducted Damle (1966, as cited in Noret on the prediction of rock scour depth due 0.5 0.5 yw + ys = 0.362 Hw q et al 2012) to plunging jets since the 1950s. However, there is no universally agreed upon method Chee & Kung (1974, as cited in to accurately predict the equilibrium scour 0.2 0.6 yw + ys = 1.663 Hw q Noret et al 2012) depth. Rock scouring is a complex physi- cal process that is normally assessed in Martins (1975, as cited in Noret 0.1 0.6 practice by empirical methods combining yw + ys = 1.5 Hw q et al 2012) laboratory and field observations with some physics (Bollaert 2002). q0.6(1 + β)0.3y0.16 Mason-B (1989, as cited in Empirical or semi-empirical formulae, y + y = 3.39 s w s 0.3 0.06 Castillo & Carrillo 2014) developed from physical model tests and g dm prototype observations generally used to 0.6 0.05 0.15 3.27q Hw yw Mason & Arumugam (1985) predict the plunge pool scour depth, are yw + ys = g0.3d 0.1 summarised in Table 1. Most of the empiri- m cal formulae for rock scour rely on the fall H 0.35q0.7 Kotoulas (1967, as cited in y + y = 0.78 w height and unit discharge. w s 0.4 Whittaker & Schleiss 1984) d95 In Table 1 yw is the plunge pool 0.333 depth (m), ys is the scour hole depth (m), y Jaeger (1939, as cited in y + y = 0.6 q0.5H 0.25 ⎫ s ⎫ Hw is the effective head (m), q is the unit w s w ⎪ ⎪ Castillo and Carrillo 2014) 3 ⎭ dm⎭ discharge (m /s/m), d90 and d95 the rock diameter representing the 90% and 95% 0.97 1.35 q ∙ sin θ Mirskaulava (1967, as cited in y + y = ⎫ – ⎫ T + 0.25y percentile respectively, θ the impingement w s ⎪ ⎪ s Mason & Arumugam 1985) ⎭ √d90 √Hw ⎭1 – 0.175 cot θT angle with plunge pool surface, dm the

Journal of the South African Institution of Civil Engineering Volume 62 Number 3 September 2020 37 median diameter, and β the air-water rela- in which K is the erodibility index, MS is with the independent data set presented tionship (refer to Figure 1). the mass strength number, Kb the block by Van Schalkwyk et al (1995). Van Schalkwyk et al (1994; 1995) proved size number, Kd the discontinuity bond Bollaert (2002) also investigated that the hydraulic erodibility of rock can be shear strength number, and JS the relative rock scour due to falling jets. For this characterised by the Kirsten index (Kirsten ground structure number. Table 2 encapsu- investigation, near-prototype laboratory 1982) through studying rock scour forma- lates the formulae for the erodibility index experiments were carried out to reproduce tion of 42 spillways in South Africa and parameters. the dynamic pressure fluctuations at the United States of America. Annandale The scour threshold for the rock the plunge pool bottom and inside the (2006) improved on the rock scour concepts strength in terms of stream power per unit rock joints. Bollaert (2002) developed the developed by Van Schalkwyk et al (1994; area (kW/m2) is based on the erodibility Comprehensive Scour Model (CSM) to 1995) to develop a classification method index K (Annandale 2006): predict the equilibrium scour depth for called the Erodibility Index Method (EIM). both open-ended and closed joints. The 0.44 The method compares the erosive capacity Prock = 0.48K if K ≤ 0.1 (3) CSM consists of three methods describing of the water jet, to the erosive resistance of the failure of rock, namely Comprehensive 0.75 the rock. The erosive capacity of the jet is Prock = K if K > 0.1 (4) Fracture Mechanics (CFM), Dynamic defined by stream power per unit area and Impulsion method (DI), and Quasi-Steady varies with depth in the plunge pool calcu- According to the EIM, the maximum scour Impulsion method (QSI). The CFM lated with Equation 1. depth is reached when the available stream method assesses the instantaneous or time- power is equal to the resistive capacity of dependent joint propagation due to joint γQH w the rock (Ppool ≤ Prock). water pressures (fatigue failure and brittle Ppool = (1) Ai The EIM does, however, have fracturing) of closed-ended joints. some limitations, as the method only The DI method analyses sudden rock where Ai is the jet footprint (impact area determines the depth at the centreline of block ejection due to uplift pressures on of the jet) at different elevations below the jet and cannot determine the extent individual rock blocks from the completely the plunge pool water surface (m2), γQH of the scour hole. The EIM provides a jointed rock bed. The net impulse or maxi- is the stream power that stays the same at generalised evaluation for rock scour, mum dynamic impulsion underneath a impact with the plunge pool free surface, γ as it does not explicitly consider the rock block can be obtained by time integra- is the unit weight of water (N/m3), Q is the mechanisms of block removal, brittle tion of the net forces on the block (Bollaert 3 total discharge (m /s), Hw is the fall height fracture, or fatigue failure. Although et al 2015): (m), and P is the stream power per unit the geometric influence of rock joint pool ∆t,pulse 2 area in the pool at a certain depth (W/m ). orientation is considered, it is handled in I∆t,pulse = ∫0 (Fu – F0 – Gb – Fsh) ∙ dt The ability of the rock to resist scouring a two-dimensional context, and therefore = m ∙ V∆t,pulse (5) is defined by the erodibility index K (see does not capture the essence of block

Equation 2), which takes several geological removal in three-dimensional fractured in which Fu and F0 are the forces under rock characteristics into account, such as rock masses (Kieffer & Goodman 2012). and over the block (N) respectively, Gb is the mass strength of the rock, block size Pells (2016) argued that, to determine the immersed weight of the block (N), Fsh and shape, the interparticle friction, and the extent of the scour of unlined dam represents the shear and interlocking forces the joint orientation relative to the flow spillways in rock, a gradation indicating on the block (N), m is the mass of the block direction (Annandale 2006). erosion regions should rather be used (kg), V∆t,pulse is the average velocity experi- than a threshold line. The erosion regions enced by the rock block during time period

K = MS ∙ Kb ∙ Kd ∙ JS (2) presented by Pells (2016) compared well ∆t (m/s), ∆t,pulse is the time interval of

Table 2 Erodibility index parameters (adapted from Annandale 2006)

Rock characteristics Formulae Parameters

1.05 Ms = Cr ∙ (0.78) ∙ (UCS) when UCS ≤ 10 MPa UCS : Unconfined compressive strength : Coefficient of relative density Mass strength number ( ) Ms = Cr ∙ (UCS) when UCS > 10 MPa Cr Ms : Rock density ρr gρ 3 3 r γ : Unit weight of rock (27 ∙ 10 N/m ) Cr = r γr

RQD Kb = Jn RQD values range between 5 and 100 : Joint set number ranges between 1 and 5 Block size number ( ) Jn Kb RQD = (115 – 3.3J ) : Joint count number c Jc , , : Average spacing of joint sets 3 Jx Jy Jz J = ⎫ ⎫ + 3 c ⎪ 0.33⎪ ⎭ (Jx ∙ Jy ∙ Jz) ⎭

J : Joint roughness number Discontinuity bond shear strength ( ) r Jr Kd Kd = : Joint alteration number Ja Ja

38 Volume 62 Number 3 September 2020 Journal of the South African Institution of Civil Engineering bed. This scour prediction method is Qtotal called the Critical Pressure Method in this current study. A stable equilibrium scour hole is achieved when the condition B g Pactual < Pcritical is satisfied. The height a block should be lifted, generally hup/zb ≥ 1, for the particle to be displaced out of the bed matrix and become mobilised is quan- Wall jet region tified as follows (Annandale 2006):

2 Jet impingement L 1 region h = 2⎫ ⎫ ∙ ∙ up ⎪ ⎪ 4 2 2 ⎭ c ⎭ 2g ∙ xb ∙ zb ∙ ρr 2 2 [Pu ∙ xb – Gb – Fsh] (9) yup Qdown ydown Qup δ with the critical net uplift pressure (Pu) underneath a rock block with open-ended Rock bed joints being calculated with Equation 10 (Bollaert 2002):

2 Vj P = γC (10) Figure 2 Jet deflection on rock bed (adapted from Bollaert 2012) u I 2g

2 certain pressure pulse(s) and I∆t,pulse is the Gb = xb ∙ zb (γr – γw) (7) The QSI method (Bollaert 2012 calculates impulse on the rock block (Ns). the peeling of protruding rock blocks along The first step in the DI method is to Failure of a block is expressed by the thin layers in the wall jet region parallel define the maximum net impulsion (Imax) displacement it undergoes due to the net to the plunge pool floor as illustrated in as the product of a net force and a time impulsion. This is obtained by transfor- Figure 2. The QSI method is dependent on period. The corresponding pressure is mation of V∆t,pulse in Equation 5 into a the quasi-steady high-velocity wall jets par- made non-dimensional by the jet’s kinetic net uplift displacement, hup defined by allel to the pool floor and the subsequent V 2 Equation 8. As the block is ejected out of quasi-steady uplift forces on rock blocks, energy ⎫ ⎫. This results in a net uplift ⎪ ⎪ the rock bed, the kinetic energy (velocity) which are caused by the protrusions of the ⎭ 2g ⎭ pressure coefficient, Cup. The time period applied to the block is transformed into rock blocks. The rock blocks are said to is non-dimensionalised by the travel potential energy. be plucked or peeled out of the bed matrix period that is characteristic for pressure due to the wall jet, when the quasi-steady 2 L V∆t,pulse uplift force (Equation 11) can overcome waves inside rock joints, i.e. T = 2⎫ ⎫, hup = (8) up ⎪ ⎪ 2g the immersed weight (Equations 7) of the ⎭ c ⎭ with L the joint length (L = xb + 2zb), with protruding rock block. xb the width of the rock block and zb the in which hup is the height the rock block is 2 height of the rock block, and the mean lifted (m). Vxi, max FQSL = Cuplift ∙ ρw ∙ Aexp ∙ (11) pressure wave celerity c taken as 100 m/s. If the height that the block is lifted (hup) 2 Hence, the non-dimensional impulsion during the time period (∆t) is high enough, coefficient CI is defined by the product the particle or rock block would be ejected In Equation 11 Cuplift is the net uplift 2 Vj L out of the matrix and become mobilised. pressure coefficient, ρw is the density Cup ∙ Tup = [m ∙ s]. The maximum ‌g ∙ c Tightly jointed rock, for example, would of water, Aexp is the exposure area of a net impulsion (Imax) is obtained by multi­ require a vertical displacement close to its rock block, and Vxi, max (Equation 14) is 2 Vj L height (hup/zb ≥ 1) while rock which is not the wall jet velocity x-distance from the plication of CI by . Prototype-scaled g ∙ c as tightly jointed, would require a lower impingement point. analysis of uplift pressures resulted in the value of hup/zb. The deflection of the jet at the pool bot- following expression for CI (Bollaert et al The critical pressure for scour initiation tom occurs in both the up and downstream 2015): of a rock bed subjected to an impinging directions. The importance of each of these jet could be used in a similar manner as deflections directly depends on the jet 2 Y Y the DI method to determine when a rock impingement angle (δ) of the jet upon impact C = 0.0035 ∙ ⎫ ⎫ – 0.119 ⎫ ⎫ + 1.22 (6) I ⎪ ⎪ ⎪ ⎪ ⎭ Dj ⎭ ⎭ Dj ⎭ block would become dislodged from the with the rock bed, as set out in Table 3.

Table 3 Discharge distribution at impingement for different jet angles (Bollaert 2012) with Y the total plunge pool depth (yw + ys) as defined in Figure 1, and Dj the jet Jet angle (δ) with rock bed 10° 20° 30° 40° 90° diameter at impact with the plunge pool free surface. Qup 1.5% 6.0% 7.0% 12.0% 50.0% The immersed weight of the rock block Qdown 98.5% 94.0% 93.0% 88.0% 50.0% is defined by Equation 7.

Journal of the South African Institution of Civil Engineering Volume 62 Number 3 September 2020 39 Table 4 Physical model test conditions The wall jet thicknesses yup and ydown are determined with Equations 12 and 13 Prototype value for limiting Item Model value respectively, with initial discharge Qtotal model scale of 1:20 scale and jet thickness B impinging on a flatbed g 3 m 60 m (Hmin) (Bollaert 2012). Drop height, H 4 m 80 m (H ) (relative to bedrock surface) med Q y 1 5 m 100 m (Hmax) up = up = ∙ (1 – cos δ) (12) 0.224 m3/s 80 m3/s/m (Q ) Qtotal Bg 2 max Discharge, Q 0.13 m3/s 45 m3/s/m (Q ) ( = 0.25 m) med bmodel Qup ydown 1 3 3 = = ∙ (1 + cos δ) (13) 0.1 m /s 35 m /s/m (Qmin) Qtotal Bg 2 0.5 m 10 m (TWLmin) Plunge pool depth, TWL 1 m 20 m (TWL ) Once the jet deflects, the wall jets may be max characterised by their initial flow velocity Rock size 0.1 m × 0.1 m × 0.05 m 2.0 m × 2.0 m × 1.0 m (rock size 1) x × y × z 0.1 m × 0.1 m × 0.075 m 2.0 m × 2.0 m x 1.5 m (rock size 2) VZbottom (Equation 15) and their initial thickness yup or ydown at the point of deflec- vertical (90⁰) tion. Initiating from this impingement loca- Rock joint angle 45⁰ against the flow direction, illustrated in Figure 5 tion, the wall jets develop radially outwards 45⁰ in the flow direction (135⁰), illustrated in Figure 5 following self-preserving velocity profiles as given by Equation 14 (Beltaos & Rajaratnam

1973 as cited in Bollaert et al 2015). Vxi, max 1 000 300 2 700 5 469 expresses the decay of the maximum cross-sectional jet velocity with the relative distance from the start of the wall jet (lateral distance X divided by the initial thickness of 1 Issuance canal width the deflected jet yup or ydown): = 250 mm

Vxi, max 3.5 = (14) VZbottom X 4

6 000 6 2

ydown 000 5 TWL 4 000 4

The initial flow velocity at impingement 2 300

VZbottom (Equation 15) depends on the dif- fusion angle (α in Figure 1) of the imping- Circular basin 3 ing jet and on its development length 5 800 through the plunge pool (yw + ys). VZbottom continuously changes during scour forma- tion (Bollaert et al 2015). The velocity Figure 3 Model setup: (1) issuance canal, (2) plunge pool, (3) rectangular hand-packed decay through the plunge pool would concrete blocks to simulate a uniform rock bed mass with regular open-ended joints, increase for a greater jet diffusion angle (α) (4) scaffolding (adapted from a General Arrangement Drawing by PERI Formwork Scaffolding Engineering (Pty) Ltd) VZbottom Z = core (for rectangular jets) (15) Vj yw + ys with Zcore the distance required for the jet core to diffuse in the plunge pool depth (m), which is generally taken as four times the jet thickness (m) at impact (Bg).

Experimental Work

Model scale Bollaert and Schleiss (2005) determined that the pressures inside the rock joints are accurately reproduced for a physical model scale of approximately 1:10. Boushaba et al (2013) found that the scale effects on the plunging jet velocity, initial turbulence, plunge pool aeration and the dynamic Figure 4 Photograph of experimental setup for lowest spillway height

40 Volume 62 Number 3 September 2020 Journal of the South African Institution of Civil Engineering 45° 45°

Dipping angle of “horizontal” Horizontal and vertical joints Dipping angle of “horizontal” joints against flow direction joints in flow direction

Figure 5 Dipped joint structure orientation pressures at the bedrock surface are negli- canal to ensure that uniform, fully devel- respectively). A similar prototype rock gible for a physical model scale not exceed- oped flow would be reached. The issuance density of 2 650 kg/‌m3 should be used in ing 1:20. Model scales greater than 1:100 canal could be adjusted to three different the model since gravity cannot be scaled. could cause incorrect aeration results. fixed heights above the movable rock bed: The paver block densities used were Therefore, a model scale not exceeding 3 m, 4 m and 5 m (prototype heights for 2 355.4 kg/m‌ 3 and 2 388.1 kg/m3 respec- 1:20 is recommended for the experimental 1:20 model scale: 60 m, 80 m and 100 m tively. The test conditions are summarised results from this study. Additionally, with respectively). The tailwater depth at the in Table 4. the available laboratory height, a 1:20 plunge pool was also adjustable, namely A total of 31 experimental tests were model scale is able to replicate realistic 0.5 m and 1 m (prototype: 10 m and 20 m carried out. Ten per cent of the experi- prototype spillway heights (i.e. 60 m up respectively). Plunging jets were evalu- ments were repeated to ensure repeatability to 100 m). ated for unit discharges of 0.10 m3/s/m to of the results, and showed a maximum 0.224 m3/s/m (prototype: 35 m3/s/m to deviation of 15%. Figure 3 shows the Model layout 80 m3/s/m). The broken-up rock bed mass laboratory model setup (dimensions shown The experimental investigation was was modelled by using tightly hand-packed in Figure 3 are model dimensions in mm). conducted in the hydraulics laboratory concrete paver blocks (cobblestones) Figure 4 shows a photograph of the experi- of the Civil Engineering Department at emulating a uniform three-dimensional mental setup. The three joint structures Stellenbosch University, South Africa. open-ended horizontal and vertical rock investigated are illustrated in Figure 5. The three-dimensional physical model joint network. The two rock sizes tested During each of the experimental tests, (based on Froude scale laws) consisted of were rectangular concrete blocks with x, the scouring of rock by the free-falling a rectangular horizontal issuance canal, y, z dimensions of 0.1 m × 0.1 m × 0.05 m rectangular jet that exited the spillway was replicating an uncontrolled spillway. Flow and 0.1 m × 0.1 m × 0.075 m (prototype: monitored until scouring in the plunge straighteners were installed in the issuance 2 m × 2 m × 1 m and 2 m × 2m × 1.5 m pool stopped and equilibrium conditions

(a) (b) (c)

Figure 6 Bed deformation process (viewed from upstream)

Journal of the South African Institution of Civil Engineering Volume 62 Number 3 September 2020 41 were reached (after 6 to 8 hours of testing), Bed level called Case A. The bed profile of the equi- (m) librium scour hole was then surveyed. The –0.5 1.0 –0.6 same test was then repeated, but instead –0.7 0.5 –0.8 of leaving the scoured blocks from the –0.9 –1.0 pool, the deposited rocks downstream of Flow direction 0 –1.1 the scour hole were continuously removed

Width –1.2 –1.3 until equilibrium conditions were reached –0.5 –1.4 again (called Case B) after approximately –1.5 –1.6 16 to 20 hours, where after the so-formed –1.0 –1.7 –1.8 equilibrium bed profile was surveyed again. –1.9 Figure 6 (p 41) shows the bed deformation –3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0 2.5 –2.0 (a) Length –2.1 process for the horizontal and vertical –2.2 aligned joint structures. The topography of the bed profile was surveyed using a three- Bed level dimensional laser scanner, Z+F Imager (m) 1.0 –0.5 5006h, at a high resolution of 10 000 –0.6 pixels/360°. –0.7 0.5 –0.8 –0.9 Results of experimental work –1.0 0 Flow direction –1.1 –1.2 Although various flow rates, drop heights, Width –1.3 and tailwater levels were tested, only –0.5 –1.4 results of selected tests are presented in –1.5 –1.0 –1.6 this paper since all the tests followed simi- –1.7 –1.8 lar trends. –1.9 –3.0 –2.5 –2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0 2.5 Figure 7 shows the contour maps of –2.0 (b) Length –2.1 the surveyed scour holes of the model –2.2 for Cases A and B respectively for the 3 maximum discharge, highest drop height, Figure 7 Observed final bed level for Q = 80 m /s/m, H = 100 m, TWL = 20 m; (a) Case A (scour and and deepest plunge pool. It can be seen deposition) and (b) Case B (scour with deposited rock removed) from the contour maps that the scour hole depth, length, width and volume increase plunge pool water surface for all three to scour failure when the joint structure when the deposited rocks downstream of drop heights. is dipped 45° in the direction of the flow. the hole are removed, and the horizontal Figure 10 shows that the plunge pool The experimental results (Table 6) indicate distance from the spillway to the onset of acts as a water cushion, since the scouring that the rock with a joint structure dipped scour decreases. depth, width and length decrease with an 45° against the flow direction has a greater The longitudinal and lateral bed profiles increase in the plunge pool depth, in accor- scour resistance to the erosive capacity of for the model for the various flow rates, dance with Van Aswegen et al (2001) and the water. listed in Table 4 (35 m3/s/m, 45 m3/s/m Bollaert et al (2012). and 80 m3/s/m), for the highest fall height The bed profiles depicted in Figure 11 Development of FORMULAE (Hmax = 100 m) and the deepest plunge indicate that the scour depth, width and based on experimental pool depth (TWLmax = 20 m) are shown length decrease as the rock block size in Figure 8 for Case A (scour and deposi- increases. Larger rock blocks provide great- data OF THIS STUDY tion) and Case B (scour with deposited er erosive resistance than smaller blocks, A multi-linear regression analysis was rock removed). Figure 8 illustrates that the due to its submerged weight, in accordance done on the physical model results to find scour depth, width and length increase as with Annandale (2006). non-dimensional formulae that define the discharge increases. The scour results The geologic structure of the rock can the scour hole geometry (depth, length, for Case B presented in Figure 8 also indi- significantly influence the erodibility of width and volume). Three regression cate that greater scour could occur if the the rock. The effect of the rock joint angle models were analysed, namely Linear, deposited rocks downstream of the scour relative to the flow direction was therefore Logarithmic Transformed and Linear hole are removed by floodwaters. Test 2 in investigated by means of Figure 12. The Logarithmic, represented by Equations 16 Figure 8 is a repeated test of Test 1. three joint angles under investigation to 18 respectively. Figure 9 indicates that the scour depth, were 45° dipped against the direction of width and length for the three different the flow, vertical (90° to the horizontal Q0 = k + a ∙ x1 + b ∙ x2 + c ∙ x3 + … (16) drop heights (60 m, 80 m and 100 m) were axis) and 45° in the direction of the flow of similar magnitude. A possible reason (thus 135°). ln(Q0) = ln( k) + a ∙ ln(x1) + b ∙ ln(x2) + for the low sensitivity towards the drop Similar to the well-known rock scour c ∙ ln(x3) + … (17) height could be that the jet was already case of Ricobayo Dam in Spain (Annandale fully developed (the jet core is fully dis- 2006), the scour results shown in Figure 12 Q0 = k + a ∙ ln(x1) + b ∙ ln(x2) + sipated) at the impingement with the indicate that the rock is more conducive c ∙ ln(x3) + … (18)

42 Volume 62 Number 3 September 2020 Journal of the South African Institution of Civil Engineering 15 2 0 10 –2 5 –4 0 –6 –8

Elevation (m) Elevation –5 Elevation (m) Elevation –10 –10 –12 –15 –14 0 20 40 60 80 100 120 –30 –20 –10 0 10 20 30 Length (m) Width (m) 3 3 3 3 Qmin-Test 1 (35 m /s/m) Qmed (45 m /s/m) Qmin-Test 1 (35 m /s/m) Qmed (45 m /s/m) 3 3 3 3 Qmin-Test 2 (35 m /s/m) Qmax (80 m /s/m) Qmin-Test 2 (35 m /s/m) Qmax (80 m /s/m) Longitudinal cross-section: Case A – distance from spillway Lateral cross-section: Case A – distance from issuance centreline

15 2 0 10 –2 5 –4 0 –6 –8 –5 Elevation (m) Elevation (m) Elevation –10 –10 –12 –15 –14 0 20 40 60 80 100 120 –30 –20 –10 0 10 20 30 Length (m) Width (m) 3 3 3 3 Qmin-Test 1 (35 m /s/m) Qmed (45 m /s/m) Qmin-Test 1 (35 m /s/m) Qmed (45 m /s/m) 3 3 3 3 Qmin-Test 2 (35 m /s/m) Qmax (80 m /s/m) Qmin-Test 2 (35 m /s/m) Qmax (80 m /s/m) Longitudinal cross-section: Case B – distance from spillway Lateral cross-section: Case B – distance from issuance centreline

Figure 8 Observed prototype bed profile indicating the effect of discharge (H = 100 m, TWL = 20 m)

15 4 2 10 0 5 –2 –4 0 –6 –5 –8 Elevation (m) Elevation (m) Elevation –10 –10 –12 –15 –14 0 20 40 60 80 100 120 –30 –20 –10 0 10 20 30 Length (m) Width (m)

Hmin (60 m) Hmed (80 m) Hmax (100 m) Hmin (60 m) Hmed (80 m) Hmax (100 m) Longitudinal cross-section: Case A – distance from spillway Lateral cross-section: Case A – distance from issuance centreline

15 4 2 10 0 5 –2 –4 0 –6 –5 –8 Elevation (m) Elevation Elevation (m) Elevation –10 –10 –12 –15 –14 0 20 40 60 80 100 120 –30 –20 –10 0 10 20 30 Length (m) Width (m)

Hmin (60 m) Hmed (80 m) Hmax (100 m) Hmin (60 m) Hmed (80 m) Hmax (100 m) Longitudinal cross-section: Case B – distance from spillway Lateral cross-section: Case B – distance from issuance centreline

Figure 9 Observed bed profile indicating the effect of drop height (Q = 80 m3/s/m, TWL = 20 m)

Journal of the South African Institution of Civil Engineering Volume 62 Number 3 September 2020 43 25 25 20 20 15 15 10 10 5 5 0 0 –5 Elevation (m) Elevation (m) Elevation –10 –5 –15 –10 –20 –15 0 20 40 60 80 100 120 –30 –20 –10 0 10 20 30 Length (m) Width (m)

Bed level for TWLmax TWLmin (10 m) Bed level for TWLmax TWLmin (10 m) Bed level for TWLmin TWLmax (20 m) Bed level for TWLmin TWLmax (20 m) Longitudinal cross-section: Case A – distance from spillway Lateral cross-section: Case A – distance from issuance centreline

25 25 20 20 15 15 10 10 5 5 0 0 –5 –5 Elevation (m) Elevation (m) Elevation –10 –10 –15 –15 –20 –20 0 20 40 60 80 100 120 –30 –20 –10 0 10 20 30 Length (m) Width (m)

Bed level for TWLmax TWLmin (10 m) Bed level for TWLmax TWLmin (10 m) Bed level for TWLmin TWLmax (20 m) Bed level for TWLmin TWLmax (20 m) Longitudinal cross-section: Case B – dstance from spillway Lateral cross-section: Case B – distance from issuance centreline

Figure 10 Observed bed profile indicating effect of the plunge pool depth (Q = 80 m3/s/m, H = 100 m)

15 2 10 0 5 –2 –4 0 –6 –5 –8 Elevation (m) Elevation (m) Elevation –10 –10 –12 –15 –14 0 20 40 60 80 100 120 –30 –20 –10 0 10 20 30 Length (m) Width (m) Rock size 1 Rock size 2 – Test 1 Rock size 2 – Test 2 Rock size 1 Rock size 2 – Test 1 Rock size 2 – Test 2 Longitudinal cross-section: Case A – distance from spillway Lateral cross-section: Case A – distance from issuance centreline

15 2 0 10 –2 5 –4 0 –6 –8 –5 Elevation (m) Elevation (m) Elevation –10 –10 –12 –15 –14 0 20 40 60 80 100 120 –30 –20 –10 0 10 20 30 Length (m) Width (m) Rock size 1 Rock size 2 – Test 1 Rock size 2 – Test 2 Rock size 1 Rock size 2 – Test 1 Rock size 2 – Test 2 Longitudinal cross-section: Case B – dstance from spillway Lateral cross-section: Case B – distance from issuance centreline

Figure 11 Observed bed profile indicating effect of rock block size (Q = 80 m3/s/m, H = 100 m, TWL = 20 m)

44 Volume 62 Number 3 September 2020 Journal of the South African Institution of Civil Engineering 10 4 8 6 2 4 0 2 0 –2 –2 –4 Elevation (m) Elevation –4 (m) Elevation –6 –6 –8 –10 –8 0 20 40 60 80 100 120 –30 –20 –10 0 10 20 30 Length (m) Width (m) 90° joint angle 45° against flow direction 90° joint angle 45° against flow direction 45° in flow direction 45° in flow direction Longitudinal cross-section: Case A – distance from spillway Lateral cross-section: Case A – distance from issuance centreline

10 4 8 6 2 4 0 2 0 –2 –2 –4 Elevation (m) Elevation –4 (m) Elevation –6 –6 –8 –10 –8 0 20 40 60 80 100 120 –30 –20 –10 0 10 20 30 Length (m) Width (m) 90° joint angle 45° against flow direction 90° joint angle 45° against flow direction 45° in flow direction 45° in flow direction Longitudinal cross-section: Case B – dstance from spillway Lateral cross-section: Case B – distance from issuance centreline

Figure 12 Observed equilibrium bed profile indicating effect of the rock joint angle (Q = 80 m3/s/m, H = 100 m, TWL = 20 m) where Q is the dependent variable, the with v the kinematic viscosity of water = 12 × R 0 Q = A × 18log ⎫ ⎫√R ∙ S (22) –6 down wall jet ⎪ ⎪ f independent variable, x1,2,3 … and k, a, b, 1.13 × 10 m²/s at 15˚C, and V*, the shear ⎭ d50 ⎭ and c are constants. velocity, is defined as:

Several parameters were incorporated with Qdown the respective discharge down- in the regression analysis, such as rock V* = √g ∙ ydown ∙ Sf (21) stream of the impingement region, based diameter, discharge, drop height, plunge on the total flow and jet impinging angle, δ pool depth, stream power, and uplift pres- with ydown denoting the jet thickness (Table 3), R the hydraulic radius, and Sf the sure. The main parameters influencing the downstream of the impingement region as energy slope. The wall jet flow area is defined flow and rock scour were selected based illustrated in Figure 2. The energy slope as Awall jet = ydown × Wj rock, with Wj rock being on the extensive literature study conducted Sf required for determining the shear the width of the jet at impingement with and experimental tests. velocity was determined from Chèzy the rock bed, taking the jet outer spreading Instead of incorporating bed shear (Equation 22) by assuming the hydraulic angle as 15° (Ervine & Falvey 1987) for highly stress explicitly, the movability number Awall jet turbulent jets. The median particle size (d ) radius R = (area of jet deflected 50 (Equation 19) based on the settling velo­city Wj rock is defined by Equation 23 (Pabst & Gregorova 3 (Vss) and the particle Reynolds number downstream of impingement/jet width at 2007) with Vblock the rock block volume (m ): (Equation 20) were used in analysing the impingement with the bedrock as illus- 1/3 rock scour. The movability number and trated in Figure 2). The surface roughness 6 d = ⎫ V ⎫ (23) 50 ⎪ block⎪ particle Reynolds number Rep (Armitage was approximated as ks = d50 median ⎭ π ⎭ & Rooseboom 2010) have the benefit that particle size (Equation 23). The settling they are based on the shear velocity V* velocity was elected as the representative The parameters describing the scour hole (Equation 21) the particles are experiencing of the transportability of large aggregate, geometry were calculated using ordinary due to the plunging jet, and the particle size. since it accounts for size, shape and least squares regressions, with due consid- density of the rock blocks. A settling tank eration given to the parsimony principle. V* was used in the laboratory to determine Due to the availability of many possible MN = (19) Vss that the blocks have a median prototype regressors, parsimony is emphasised to settling velocity (VSS) of 4.19 m/s and ensure avoiding over-fitting. The parameters V*d50 4.99 m/s for the small and large rock were made dimensionless by applying the Rep = (20) v blocks respectively. Buckingham theorem (Albrecht et al 2013).

Journal of the South African Institution of Civil Engineering Volume 62 Number 3 September 2020 45 Table 5 Regression formulae for equilibrium scour hole geometry for horizontal and vertical open joints*

Equation R2

Ds d V H V F y ln ⎫ CaseA⎫ = –468.532 ln ⎫ 50 ⎫ – 322.102ln ⎫ i ⎫ + 4.502ln ⎫ w⎫ – 483.385ln ⎫ SS ⎫ – 0.804ln ⎫ lift ⎫ – 0.817ln ⎫ w ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Hw ⎭ ⎭ Hw ⎭ ⎭ √gDi⎭ ⎭ Lb ⎭ ⎭ Vi ⎭ ⎭ Gb ⎭ ⎭ Hw ⎭ Depth V D q2 P P V* – 473.991ln ⎫ i i⎫ + 158.264ln ⎫ ⎫ – 0.419ln ⎫ SP ⎫ + 0.006ln ⎫ ⎫ – 473.536ln ⎫ ⎫ 0.998 Case A ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ v ⎭ ⎭ gHw ⎭ ⎭ Puplift ∙ VSS ⎭ ⎭ Pcrit⎭ ⎭ VSS⎭ V*d + 474.587ln ⎫ 50⎫ ⎪ ⎪ ⎭ v ⎭ Ds d V V F P y CaseB = –0.196 ln ⎫ 50 ⎫ – 0.578ln ⎫ i ⎫ – 0.942ln ⎫ SS ⎫ – 0.216ln ⎫ lift ⎫ – 0.129ln ⎫ SP ⎫ – 0.079ln ⎫ w ⎫ Depth ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hw ⎭ Hw ⎭ ⎭ √gDi⎭ ⎭ Vi ⎭ ⎭ Gb ⎭ ⎭ Puplift ∙ Vj ⎭ ⎭ Hw ⎭ 0.992 Case B V D q2 P V* V*d – 0.279ln ⎫ i i⎫ + 0.129ln ⎫ ⎫ + 0.006ln ⎫ ⎫ – 0.145ln ⎫ ⎫ + 0.315ln ⎫ 50⎫ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ v ⎭ ⎭ gHw ⎭ ⎭ Pcrit⎭ ⎭ VSS⎭ ⎭ v ⎭ Ls d V H V F y CaseA = –416.262 ln ⎫ 50 ⎫ – 281.275ln ⎫ i ⎫ + 2.804ln ⎫ w⎫ – 421.760ln ⎫ SS ⎫ – 0.666ln ⎫ lift ⎫ – 0.78ln ⎫ w ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hw ⎭ Hw ⎭ ⎭ √gDi⎭ ⎭ Lb ⎭ ⎭ Vi ⎭ ⎭ Gb ⎭ ⎭ Hw ⎭ Length V D q2 P P V* – 418.577ln ⎫ i i⎫ + 140.444ln ⎫ ⎫ – 0.157ln ⎫ SP ⎫ – 0.076ln ⎫ ⎫ – 417.815ln ⎫ ⎫ 0.987 Case A ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ v ⎭ ⎭ gHw ⎭ ⎭ Puplift ∙ VSS ⎭ ⎭ Pcrit⎭ ⎭ VSS⎭ V*d + 418.985ln ⎫ 50⎫ ⎪ ⎪ ⎭ v ⎭ Ls d 3.048 V –0.1878 H 1.427 V –3.052 F 0.485 y –0.081 V D 0.639 q2 0.002 CaseB =  ⎫ 50 ⎫ ∙ ⎫ i ⎫ ∙ ⎫ w⎫ ∙ ⎫ SS ⎫ ∙ ⎫ lift ⎫ ∙ ⎫ w ⎫ ∙ ⎫ i i⎫ ∙ ⎫ ⎫ Length ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ Hw ⎭ Hw ⎭ ⎭ √gDi⎭ ⎭ Lb ⎭ ⎭ Vi ⎭ ⎭ Gb ⎭ ⎭ Hw ⎭ ⎭ v ⎭ ⎭ gHw ⎭ 0.984 Case B P 0.019 P –0.001 V* 0.872 V*d –0.377 ∙ ⎫ SP ⎫ ∙ ⎫ ⎫ ∙ ⎫ ⎫ ∙ ⎫ 50⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Puplift ∙ VSS⎭ ⎭ Pcrit⎭ ⎭ VSS⎭ ⎭ v ⎭ Ws d V H V F y ln ⎫ CaseA⎫ = –35.453 ⎫ 50 ⎫ – 28.412ln ⎫ i ⎫ + 2.079ln ⎫ w⎫ – 43.005ln ⎫ SS ⎫ – 0.879ln ⎫ lift ⎫ + 0.102ln ⎫ w ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Hw ⎭ ⎭ Hw ⎭ ⎭ √gDi⎭ ⎭ Lb ⎭ ⎭ Vi ⎭ ⎭ Gb ⎭ ⎭ Hw ⎭ Width V D q2 P P V* – 37.850ln ⎫ i i⎫ + 12.705ln ⎫ ⎫ – 0.229ln ⎫ SP ⎫ + 0.079ln ⎫ ⎫ – 37.585ln ⎫ ⎫ 0.998 Case A ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ v ⎭ ⎭ gHw ⎭ ⎭ Puplift ∙ VSS ⎭ ⎭ Pcrit⎭ ⎭ VSS⎭ V*d + 38.229ln ⎫ 50⎫ ⎪ ⎪ ⎭ v ⎭ Ws d –2.228 V 4.390 H –1.844 V 6.531 F 0.853 y 0.048 V D 0.466 q2 0.614 CaseB =  ⎫ 50 ⎫ ∙ ⎫ i ⎫ ∙ ⎫ w⎫ ∙ ⎫ SS ⎫ ∙ ⎫ lift ⎫ ∙ ⎫ w ⎫ ∙ ⎫ i i⎫ ∙ ⎫ ⎫ Width ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ Hw ⎭ Hw ⎭ ⎭ √gDi⎭ ⎭ Lb ⎭ ⎭ Vi ⎭ ⎭ Gb ⎭ ⎭ Hw ⎭ ⎭ v ⎭ ⎭ gHw ⎭ 0.996 Case B P 0.379 P 0.021 V* 0.393 V*d –0.565 ∙ ⎫ SP ⎫ ∙ ⎫ ⎫ ∙ ⎫ ⎫ ∙ ⎫ 50⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Puplift ∙ VSS⎭ ⎭ Pcrit⎭ ⎭ VSS⎭ ⎭ v ⎭ Vol d V H V F y ln ⎫ CaseA⎫ = –131.752 ln ⎫ 50 ⎫ – 106.469ln ⎫ i ⎫ + 6.433ln ⎫ w⎫ – 159.640ln ⎫ SS ⎫ + 1.567ln ⎫ lift ⎫ – 0.724ln ⎫ w ⎫ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Hw ⎭ ⎭ Hw ⎭ ⎭ √gDi⎭ ⎭ Lb ⎭ ⎭ Vi ⎭ ⎭ Gb ⎭ ⎭ Hw ⎭ Scour V D q2 P P V* volume – 145.502ln ⎫ i i⎫ + 47.595ln ⎫ ⎫ – 0.012ln ⎫ SP ⎫ – 0134ln ⎫ ⎫ – 146.724ln ⎫ ⎫ 0.999 ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Case A ⎭ v ⎭ ⎭ gHw ⎭ ⎭ Puplift ∙ VSS ⎭ ⎭ Pcrit⎭ ⎭ VSS⎭ V*d + 146.313ln ⎫ 50⎫ ⎪ ⎪ ⎭ v ⎭ Vol d 3.061 V –5.062 H 2.519 V –8.050 F 1.330 y –0.470 V D –2.522 q2 1.238 CaseB =  ⎫ 50 ⎫ ∙ ⎫ i ⎫ ∙ ⎫ w⎫ ∙ ⎫ SS ⎫ ∙ ⎫ lift ⎫ ∙ ⎫ w ⎫ ∙ ⎫ i i⎫ ∙ ⎫ ⎫ Scour 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ volume Hw ⎭ Hw ⎭ ⎭ √gDi⎭ ⎭ Lb ⎭ ⎭ Vi ⎭ ⎭ Gb ⎭ ⎭ Hw ⎭ ⎭ v ⎭ ⎭ gHw ⎭ 0.998 Case B P 0.276 P 0.010 V* –2.566 V*d 2.966 ∙ ⎫ SP ⎫ ∙ ⎫ ⎫ ∙ ⎫ ⎫ ∙ ⎫ 50⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Puplift ∙ VSS⎭ ⎭ Pcrit⎭ ⎭ VSS⎭ ⎭ v ⎭ * Ranges: Discharge: 35 m3/s/m – 80 m3/s/m; Drop height: 60 m – 100 m; Tailwater depth: 10 m – 20 m; horizontal and vertical joint network

The Logarithmic model provided the Table 5 summarises the non-dimensional by a trial-and-error procedure using the best fit for the scour hole dimensions formulae for vertical rock joints obtained scour formula in Table 5, initially for a (scour depth, length, width and volume), from the regression analysis. To apply jet impinging onto the original flat bed. except the scour hole depth for Case B was these regression formulae, the equilibrium Iteration ceases when a reasonable per- predicted best by the Linear Log model. scour depth should first be determined centage difference is found. Thereafter the

46 Volume 62 Number 3 September 2020 Journal of the South African Institution of Civil Engineering scour parameters (i.e. movability number Table 6 Stability factors due to joint angle and dynamic pressure head) at the equi- Dipping angle of “horizontal” Dipping angle of “horizontal” Scour hole geometry librium scour depth can be used in order joints against flow direction joints in flow direction to obtain the other scour hole geometries Case A 0.93 1.29 (length, width and volume) by using the Depth formulae presented in Table 5. The regres- Case B 0.89 1.06 sion formulae should be applied inside the Case A 0.98 0.77 prototype ranges in which the formulae Length Q < 45 m3/s/m: Q > 45 m3/s/m: 3 Case B 0.82 were developed (discharge: 35 m /s/m – 0.42 0.89 80 m3/s/m; drop height: 60 m – 100 m; Case A 1.01 1.28 tailwater depth: 10 m – 20 m; horizontal Width and vertical joint network). Case B 0.9 1.13

The stabilising (or destabilising) fac- Q < 45 m3/s/m: Q > 45 m3/s/m: Case A 0.83 tors for joint structures orientated 45° in 1.22 0.48 and against the direction of the flow are Volume Q < 45 m3/s/m: Q > 45 m3/s/m: Case B 0.82 summarised in Table 6. The joint struc- 1.09 0.53 ture angle factors should be applied to the Ranges: Discharge: 35 m3/s/m – 80 m3/s/m; Drop height: 60 m – 100 m; predicted scour geometry results obtained Tailwater depth: 10 m – 20 m; 45° and 135° joints for the horizontal and vertical joint formulae in Table 5. The variations in the stabilising factors are small. However, results. The regression analysis results indi- the scour depth 14.3%, for the scour length additional experimental testing is required cate that a good correlation was achieved 22.5%, for the scour width 8.6%, and for the to ascertain whether linear interpolation between the predicted and observed values. scour volume 14.4%. is possible between the stabilising factors The regression formulae were confirmed for different degrees of joint structures inside the ranges in which the developed between 0° and 45° in and against the formulae were based by using experimental Comparison of physical flow direction. tests that did not form part of the regres- model results with scour The regression analysis results for sion analysis but with parameter values prediction methods the scour hole parameters (depth, length, inside the ranges of the tests. The devel- fROm literature width and volume) for the Case B sce- oped formulae have a good agreement with Eleven empirical formulae (Table 1), and the nario (deposited rocks removed) shown in the observed confirmation values with the EIM and CSM traditionally employed to Figure 13 with the 95% confidence band following percentage errors (i.e. percentage predict rock scour due to plunging jets were indicates the correlation between the for- differences between test values and those evaluated against the results from the physi- mulae predicted and the confirmation test predicted by the developed formulae) – for cal model study for a full-scale prototype.

(a) Scour depth (b) Scour hole length

0.25 R2 = 0.992 0.5 R2 = 0.984

0.20 0

0.15 predicted predicted

w –0.5 w Ls H Ds H 0.10

ln –1.0 0.05

0.05 0.10 0.15 0.20 0.25 –1.0 –0.5 0 0.5 Ds Ls ln Hw observed Hw observed

(c) Scour hole width (d) Scour hole volume –2 R2 = 0.996 R2 = 0.998 –0.4 –3 predicted

predicted –0.8 –4 3 w w H Ws Vol –1.2 H –5 ln ln –6 –1.6 –1.2 –0.8 –0.4 –6 –5 –4 –3 –2 Ws Vol ln ln 3 Hw observed Hw observed

Figure 13 Regression analysis outcome for Case B scenario with 95% confidence band

Journal of the South African Institution of Civil Engineering Volume 62 Number 3 September 2020 47 and the stream power of the jet as it travels through the air and plunge pool. Figure 17 indicates that the EIM erosion threshold is compatible with the laboratory experimen- tal data for the Case B scenario when the scoured blocks were continually removed, since the data plots above the threshold line in the “scour” region (green markers). Mason and Arumugam (1985) proved to be the only identified empirical formula that incorporates the plunge pool depth with gravity acceleration. The implication of this is that models taking the plunge pool depth, and consequently the dis- Damle (1966): 2.1 m Martins (1975): 2.9 m sipation of the jet’s energy through the Physical model of this study (Case A): 4.2 m Uniform open-ended plunge pool, into consideration could joint rock bed Physical model of this study (Case B): 7.8 m offer better scour depth predictions. The Vernonese 2 (1984): 7.9 m; Critical Pressure: 8.0 m Mason (1985): 9.6 m experimental test results reiterate the EIM (2006): 10.5 m experimental finding that the scour depth Yildiz (1994): 16.9 m QSI (2012): 17.5 m greatly depends on the plunge pool depth. Chee (1974): 19.5 m The Jaeger equation (1939, as cited in Katoulas (1967); Veronese 3 (1984): 20.1 m Castillo & Carrillo 2014) also incorporated DI (2002): 30.0 m the plunge pool depth and had a median Veronese 1 (1984): 38.5 m percentage difference close to zero (25%); however, it had a wide percentage differ- Figure 14 Prototype equilibrium scour depth from physical model and literature scour prediction ence statistical spread. methods for Q = 45 m3/s/m, H = 100 m, TWL = 20 m Yildiz and Uzucek (1994, as cited in Alias et al 2008) and Mirskaulava (1967, as Only the DI and QSI methods of the CSM, The percentage difference results cited in Mason & Arumugam 1985) rely on and not the CFM method, were compared shown in Figure 16 indicate that the the impingement angle of the jet with the with the physical model test results, since methods are in weak agreement with plunge pool surface. Both equations have the CFM method analyses fatigue failure one another and generally overestimate a narrow statistical spread; however, the and brittle fracturing of closed-ended joints the observed scour depths with a mean Yildiz and Uzucek equation overestimates and the model’s rock mass comprised a fully difference of 99% (over-predict) for the the scour depth, whereas the Mirskaulava formed joint network. Case B scenario. The analytical method equation underestimates the scour depth. The prototype scour hole depth formed that agrees best with the experimental Also, the median percentage difference in a movable bed based on the physical results is the Critical Pressure method calculated using the Mirskaulava equation model study is compared in Figure 14 to (Annandale 2006), since the median per- (-78.6%) is closer to zero compared to the the scour prediction methods found in centage difference is close to zero (-25.2%). Yildiz and Uzucek equation (126.1%). A literature for the 45 m3/s/m flood, a drop However, the safest scour prediction possible reason is that the Mirskaulava height of 80 m and a plunge pool depth of method (accurate and conservative) would equation also incorporates the plunge pool 20 m. be the EIM (Annandale 2006), followed depth and 90% percentile rock dimension, The scour depth prediction methods by Mason and Arumugam’s (1985) empiri- whereas the Yildiz and Uzucek equation from literature were found to yield a wide cal formula, since the first quartile (25th does not. range of varying results for the same percentile) nears the zero percentage dif- The remainder of the empirical formu- input conditions as shown in Figure 14. ference (-12% and 15% respectively) with a lae are based on the drop height and unit Figure 15 is a correlation plot of scour narrow statistical spread. discharge with varying coefficients. A wide depths observed in the experimental tests The Critical Pressure method range in scour depths, both under- and versus the scour depths predicted by (Annandale 2006) proves to agree the clos- overpredicted, were calculated, indicating 15 selected formulae from literature by est with the laboratory data, presumably that the scour depth does not rely solely on applying prototype Case B physical model because it relies on the critical pressure for the discharge and drop height, and that the test conditions as input in the respective scour initiation of a rock bed subjected to scour depth is sensitive to the parameter formulae and methods. Figure 16 presents an impinging jet. The critical pressure is coefficients. Furthermore, these equations the statistical spread of the data in Figure dependent on the submerged weight of the do not incorporate the rock block size, thus 15. The percentage difference between the rock block, block dimensions, uplift pres- the same scour depth would be predicted observed prototype depth from the experi- sures inside the joints, diffusion of the jet if the discharge and drop height are kept mental tests and the depths obtained from through the air and plunge pool, as well as constant for various particle size scenarios. the respective literature methods is given the velocity of the jet and drop height. These empirical formulae are therefore predicted observed ds – ds The EIM presumably performed well considered less applicable for full-scale observed . by ds × 100 because it relies on rock material properties prototype cases.

48 Volume 62 Number 3 September 2020 Journal of the South African Institution of Civil Engineering 80 90 Over-predicted Over-predicted 70 80 60 70 60 50 50 40 40 30 30 20 20

Predicted scour depth (m) 10 Predicted scour depth (m) 10 Under-predicted Under-predicted 0 0 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 90 Observed scour depth (m) Observed scour depth (m) Veronese 1 Veronese 2 Veronese 3 Yildiz y = x Katoulas Jaeger Mirskaulava y = x

60 100 Over-predicted Over-predicted 90 50 80 40 70 60 30 50 40 20 30 10 20 Predicted scour depth (m) Predicted scour depth (m) 10 Under-predicted Under-predicted 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 80 90 100 Observed scour depth (m) Observed scour depth (m) Damle Chee Mason Martins y = x EIM Critical pressure DI QSI y = x

Figure 15 Comparison of equilibrium scour depths from the experimental work and the different prediction methods (Case B scenario)

The DI and QSI methods (Bollaert into account, whereas the QSI method is physical model scales, and that both meth- 2012) demonstrated the most significant the only identified scour prediction model ods were developed for vertical falling jets. spread of differences between all the meth- that takes wall jets into consideration. Both The developed regression formula ods. The DI method takes the uplift pulsat- methods over­estimated the scour depth, to predict the scour depth agreed well ing forces integrated over a pulse period presumably due to the challenges posed by with the experimental results. There is a

Under-predicted Over-predicted Method (%) (%) –200 –100 0 100 200 300 400 500 600 700 800 900 1 Veronese 1 (1937) 2 Veronese 2 (1937) 3 Veronese 3 (1937) 4 Yildiz and Uzucek (1994) 5 Damle (1966) 6 Chee and Kung (1974) 7 Martins (1975) 8 Mason and Arumugam (1985) 9 Katoulas (1967) 10 Jaeger (1939) 11 Mirskaulava (1967) 12 EIM (Annandale 2006) 13 Critical pressure (Annandale 2006) 14 DI (Bollaert 2002) 15 QSI (Bollaert 2012) 16 New Formula (Case B)

Figure 16 Scour depth distribution as a percentage difference for the different scour prediction methods from the experimental work (Case B)

Journal of the South African Institution of Civil Engineering Volume 62 Number 3 September 2020 49 ranked seventh and eighth respectively in 10 000.0 the least total SSR. Scour occurs

1 000.0 Conclusions

) This paper presents the results of equi- 2 librium plunge pool scour hole geometry 100.0 obtained from a physical model study. A model scale not exceeding 1:20 is recom- mended for the experimental results from 10.0 this study in order to limit scale effects.

Steam power (kW/m The physical model was used to simulate and test the following conditions: 1.0 QQ Rectangular plunging jets issuing horizontally. No scour Q 0.1 Q Jet issuance levels of 3 m, 4 m, and 5 m 0.01 0.1 1 10 100 1 000 10 000 (prototype: 60 m, 80 m and 100 m) Erodibility index above the rock bed level. Erosion No erosion Current study QQ Plunge pool tailwater depths of 0.5 m Annandale criteria 0% probability failure 100% probability failure and 1.0 m (prototype: 10 m and 20 m). QQ Unit discharges of 0.1 m3/s/m to Figure 17 EIM erosion potential threshold (Annandale et al 1996) overlaid with experimental data 0.224 m3/s/m (prototype: 35 m3/s/m to 80 m3/s/m). observed predicted 2 Q compact statistical range of differences SSR = ∑(ds – ds ) (24) Q Two rock sizes of 0.1 m × 0.1 m × with a median difference percentage close 0.075 m and 0.1 m × 0.1 m × 0.05 m to zero (0.8%), with the first quartile just The developed regression formula to (prototype: 2.0 m × 2.0 m × 1.5 m and below zero (-2%: overprediction) compared determine the scour hole depth ranked 2.0 m × 2.0 m × 1.0 m). to the literature scour prediction methods the highest with the least total SSR QQ Rock joint angles of 45°, 90° and 135° as seen in Figure 16. 26;13 (underpredicted; overpredicted), with the horizontal axis. The sum of squared residuals (SSR), followed by Martins (1975) and Damle Results from the physical model were used as defined by Equation 24, was compared (1966). However, Martins’ and Damle’s to develop non-dimensional multi-linear against the various scour prediction meth- empirical formulae have a high number of regression formulae to predict the scour ods in Figure 18 for the Case B scenario underpredictions. The EIM and Critical hole geometry (depth, length, width and (scour with deposited rocks removed). Pressure methods (Annandale 2006) volume). The proposed regression formulae

New formula 26;13 Veronese 1 (1937) 0;38111 Veronese 2 (1937) 69;884 Veronese 3 (1937) 0;8919 Yildiz and Uzucek (1994) 0;5828 Damle (1966) 657;61 Chee and Kung (1974) 0;9129 Martins (1975) 366;151 Mason and Arumugam (1985) 26;1161 Katoulas (1967) 0;10248 Jaeger (1939) 120;16476 Scour prediction methodsScour Mirskaulava (1967) 945;24 EIM (Annandale 2006) 184;1580 Critical pressure (Annandale 2006) 461;5055 DI (Bollaert 2002) 99;28317 QSI (Bollaert 2002) 340;20417 0 5 000 10 000 15 000 20 000 25 000 30 000 35 000 40 000 Sum of squared residuals Under-predicted Over-predicted

Figure 18 Comparison of the sum of squared residuals for the various scour prediction methods (Case B)

50 Volume 62 Number 3 September 2020 Journal of the South African Institution of Civil Engineering were confirmed to be inside the ranges Notation in which the formulae were developed by using experimental test data that did not The following symbols are used in this paper: form part of the regression analysis. These B Jet issuance thickness proposed regression formulae agree reason- i ably well with the experimental data on q Core thickness at impingement (Castillo et al 2014) which they are based. Bg = √2gH The scour depth results from the physi- w cal model were compared to various ana- q Impingement jet thickness for horizontal B = + 4 × (1.14T )√B (√2H – 2√B) lytical methods found in literature. It was j u i w i discharge channel (Castillo et al 2014) √2gHw found that, regarding the more frequently Thickness of jet impinging on the rock bed, with used formulae from literature, the Critical jet’s outer spreading angle as 15° Bj rock Pressure method agrees the closest with (Ervine & Falvey 1987) the experimental data and regression for- 2 Y Y mulae developed in this study, whereas the C = 0.035 ⎫ ⎫ – 0.119 ⎫ ⎫ + 1.22 Dynamic impulsion coefficient (Bollaert et al 2015) I ⎪ ⎪ ⎪ ⎪ EIM method (Annandale 2006) followed by ⎭ Bg⎭ ⎭ Bg⎭ Mason and Arumugam’s (1985) empirical V 2 j Lift force underneath block (Bollaert 2002) formula were more conservative than the Fuplift = CI γw(xb ∙ yb) formulae developed in this study. 2g g Gravitational acceleration constant

Submerged weight of rock block (Bollaert 2002) Acknowledgements Gb = Vb(ρr – ρw)g

The assistance toward this research of Fall height Hw PERI Formwork Scaffolding, Continental 0.32 Jet breakup length (Horeni 1956) Cobbles, Kaytech Engineered Fabrics, Lb = 6q and Horts Geo-Solutions, as well as 1% Exceedance dynamic pressure head Mr NF Katzke and Mr DE Bosman, is P acknowledged with gratitude. The opinions 1 Critical pressure head expressed, and conclusions arrived at, are Pcrit = Puplift ∙ γ those of the authors.

γQwHw Stream power per unit area at rock bed PSP = (Annandale 2006) B ∙ W REFERENCES j rock j rock C γV 2 Albrecht, M, Nachtsheim, C, Albrecht, T & Cook, R I j Uplift pressure (Annandale 2006) Puplift = 2013. Experimental design for engineering 2g dimensional analysis. Technometrics, 55(3): 257–270. q Unit discharge DOI: 10.1080/00401706.2012.746207. Alias, N, Mohamed, T, Ghazali, A & Mohd, M 2008. Rock volume Vb Impact of takeoff angle of bucket type energy Issuance velocity dissipater on scour hole. American Journal of Vi Applied Sciences, 5(2): 117–121. 2 Velocity at impingement with plunge pool Annandale, G, Smith, S, Nairns, R & Jones, J 1996. Vj = √Vi + 2gHw (Annandale 2006) Scour Power. Civil Engineering, 66(7): 58–60. V Rock block settling velocity Annandale, G W 2006. Scour Technology: Mechanics SS and Engineering Practice. New York: McGraw-Hill. Shear velocity V* = √9.81 ∙ γw ∙ Sf Armitage, N & Rooseboom, A 2010. The link between Flow width at issuance Movability Number and Incipient Motion in river Wi sediments. Water SA, 36(1): 89–96. Thickness of jet impinging on the rock bed, with jet’s outer spreading angle as 15° Bollaert, E 2002. Transient water pressures in joints Wj rock (Ervine & Falvey 1987) and formation of rock scour due to high-velocity jet impact. PhD Thesis, Lausanne, Switzerland: Ecole Rock block size xb, yb, zb, d50 Polytechnique Fédérale de Lausanne. Scour depth Bollaert, E 2012. Wall jet rock scour in plunge pools: A ys quasi-3D prediction model. Hydropower and Dams, Initial plunge pool depth yw XX: 1–9. Bollaert, E & Lesleighter, E 2014. Spillway rock scour Rock density ρr experience and analysis – The Australian scene over Water density the past four decades. Proceedings, 5th International ρw Symposium on Hydraulic Structures, Brisbane, v Kinematic viscosity of water at 15°C = 1.13×10–6 m2/s Australia.

Journal of the South African Institution of Civil Engineering Volume 62 Number 3 September 2020 51 Bollaert, E & Schleiss, A2005. Physically based Ervine, D & Falvey, H 1987. Behaviour of turbulent Pells, S 2016. Erosion of rock in spillways. PhD Thesis. model for evaluation of rock scour due to water jets in the atmosphere and in plunge pools. Kensington, Australia: University of New South high-velocity jet impact. Journal of Hydraulic Proceedings of the Institution of Civil Engineers, Wales. Engineering, 131(3): 153–165. DOI: 10.1061/ Part 2, 83(1): 295–314. Stratford, C, Bollaert, E & Lesleighter, E 2013. Plunge (ASCE)0733-9429(2005)131:3(153). Horeni, P 1956. Disintegration of a free jet of water pool rock scour analysis techniques: Wivenhoe Bollaert, E, Munodawafa, M & Mazvidza, D 2012. in air. Byzkumny ustav vodohospodarsky prace a Dam spillway, Australia. Proceedings, Hydro 2013 Kariba Dam plunge pool scour: Quasi-3D numerical studie, Sesit 93. Conference, Innsbruck, Austria. predictions. Proceedings, ICSE 6-265, Paris, Kieffer, D & Goodman, R 2012. Assessing scour Van Aswegen, W J, Dunkley, E, Blake, K R K 2001. pp 627–634. potential of unlined rock spillways with the Block Plunge pool scour reproduction in physical hydraulic Bollaert, E, Stratford, C & Lesleighter, E 2015. Scour Spectrum. Geomechanics and Tunnelling, models. Stellenbosch: Water Research Commission. Numerical Modelling of Rock Scour: Case Study 5: 527–536. Van Schalkwyk, A, Dooge, N & Pitsiou, S 1995. Rock of Wivenhoe Dam (Australia). London: Taylor & Kirsten, H 1982. Classification system for excavation mass characteristics for evaluation of erodability. Francis. in natural materials. The Civil Engineer in South Proceedings, 11th European Conference on Soil Boushaba, F, Manso, P, Schleiss, A, Yachouti, A & Africa, 24: 293–308. Mechanics and Foundation Engineering, Volume 3, Daoudi, S 2013. Numerical and experimental high- Manso, P, Bollaert, E & Schleiss, A 2008. Evaluation of Copenhagen, Denmark. Reynolds jet diffusion and impact pressures in flat high-velocity plunging jet-issuing characteristics as Van Schalkwyk, A, Jordaan, J & Dooge, N 1994. Die and laterally confined aerated pools. International a basis for plunge pool analysis. Journal of Hydraulic erodeerbaarheid van verskillende rotsformasies Journal of Hydraulic Engineering, 2(6): 133–141. Research, 46(2): 147–157. onder variërende vloeitoestande. Report WRC DOI: 10.5923/j.ijhe.20130206.02. Mason, P & Arumugam, K 1985. Free jet scour 302/1/95. Pretoria: Water Research Commission. Castillo, L & Carrillo, J 2014. Scour analysis below dams and flip buckets. Journal of Hydraulic Whittaker, J & Schleiss, A 1984. Scour related to downstream of Paute-Cardenillo Dam. Proceedings, Engineering, 111(2): 220–235. energy dissipators for high head structures. Zürich: 3rd IAHR Europe Congress, Porto, Portugal, Noret, C, Girird, J, Munodawafa, M & Mazvidza, D Eidgenössischen Technischen Hochschule. pp 1–10. 2012. Kariba Dam on Zambezi River: Stabilizing the Wittler, R, Mefford, B, Abt, S, Ruff, J & Annandale, Castillo, L, Carillo, J & Blàzquez, A 2014. Plunge pool natural plunge pool. Proceedings, ICSE 6-265, Paris. G 1995. Spillway and dam foundation erosion: dynamic pressures: A temporal analysis in the Pabst, W & Gregorova, E 2007. Characterization of Predicting progressive erosion extents. Proceedings, nappe flow case. Journal of Hydraulic Research, particles and particle systems. Proceedings, ICT, International Conference on Water Resources 53(1): 101–118. DOI: 10.1080/00221686.2014.968226. Prague, p 122. Engineering, San Antionio, TX, pp 1–5.

52 Volume 62 Number 3 September 2020 Journal of the South African Institution of Civil Engineering