Predicting head cut erosion and migration in concentrated flows typical of upland areas



[1] Soil erosion due to head cut development and migration can devastate agricultural lands, yet current prediction technology does not address this important erosion process. Here an analytical model of this erosional phenomenon is presented. Realistic, physically based approximations to the laws governing mass, momentum, and energy transfer in the neighborhood of the scour hole result in closed-form predictive algorithms for the magnitude of the plunge pool erosion and the rate of head cut migration. The model introduces a special treatment of nonventilated overfall conditions, is limited to homogeneous, unbounded soil layers, and is validated by available experimental measurements.

1. Introduction

[2] In upland concentrated flows such as rills, crop furrows, and ephemeral gullies, the occurrence and migration of head cuts, step changes in bed surface elevation where intense, localized erosion takes place, is commonly associated with significant increases in sediment load [Mosley, 1974; Bryan, 1990; Römkens et al., 1997]. Yet soil losses as a result of head cut migration are not explicitly addressed in soil erosion prediction technology such as the Revised Universal Soil Loss Equation [Renard et al., 1997] and the Water Erosion Prediction Project [Nearing et al., 1989]. The formation and development of head cuts and knickpoints in soils and their upstream migration have been linked to rill and gully erosion, concentration of overland flow, drainage development, and failure of reservoir embankments and spillway channels (see discussion by Bennett et al. [2000]).

[3] An experimental database has been compiled that describes systematic characteristics of actively migrating head cuts and their relation to overland flow rate, bed slope, and initial step height (Bennett [1999], Bennett et al. [2000], Bennett and Casalí [2001], collectively referred to herein as the Bennett data set). While these experiments were limited in both scope and scale, a number of key observations were made: (1) morphologic self-similarity of the migrating head cut, (2) conservation of scour hole geometry, and (3) commonality of plunge-pool erosion and scour hole hydrodynamics. This research was initiated to improve the understanding of soil erosion due to head cut migration and to enhance technology to predict soil losses in agricultural areas impacted by head cuts. These experimental observations now have provided the insight necessary to develop an analytical solution for predicting head cut scour depth, migration rate, and sediment flux.

[4] The objective of this study was to define predictive equations for the magnitude of head cut scour depth and the rate of head cut migration as well as other ancillary relations needed to complete the model. Direct numerical simulation of head cut erosion processes by describing soil and water as continuous, coupled media is possible but not practical primarily because of an inadequate understanding of the physics of soil erosivity and sediment movement within a head cut scour hole environment. Thus an approximate analytical procedure is adopted herein whereby the eroded soil mass can be readily computed. The model is applicable to relatively small upland concentrated flows with little or no upstream sediment supply, where the eroded soil depth can be treated as a homogeneous medium, and exfiltration erosion processes are neglected. Therefore the model is not suitable for settings impacted by shallow soil horizons and hard pans or where seepage forces and surface weathering can play a dominant role on soil erosion.

2. Phenomenological Basis of Model Construct

2.1. Self-Similar Behavior of Head Cut Migration

[5] Bennett et al. [2000] presented laboratory results on the growth and migration of head cuts in soils typical of upland concentrated flows. Their results showed that head cut propagation attains a steady state during which head cut geometry, migration rate, and sediment yield remain constant. These results clearly demonstrate that as a head cut migrates, it maintains a constant geometry, a behavior typical of self-similar propagating phenomena (Figure 1a). The term self-similar was coined many decades ago in reference to phenomena that display self or internal similitude [Ames, 1965], and it is used extensively in scaling and similarity theory practice [Barenblatt, 1996]. It is apparent from Figure 1a that the phenomenon of head cut migration in the controlled laboratory experiments that constitute the Bennett data set can be characterized as a uniform propagation, that is, a self-similar translation of a head cut profile invariant with respect to an observer moving with the head cut. Although it is not the intent of this paper to utilize similarity analysis, these experimental findings are used to formulate the modeling framework presented below.

Figure 1.

(a) Steady state head cut profile measured for qw = 8.32 × 10−3 m2s−1 in run 9 of Bennett et al. [2000]. The profile was constructed by time averaging instantaneous video images and the error bars depict the range of minimum and maximum bed elevations observed during head cut propagation. (b) Streamline pattern measured in a rigid bed model patterned after the averaged bed profile shown in Figure 1a, for qw = 7.07 × 10−3 m2 s−1, using laser Doppler anemometry (Bennett and Alonso, unpublished data set, 2001).

2.2. Flow Structure in the Plunge Pool Zone

[6] In Figure 1b a jet is seen emitting from the overfall brink and plunging into the pool. Flow entrainment on both sides of the diffusing jet creates recirculating eddies on each side, which behave as two wall jets after impinging on the pool bed. The wall jet directed toward the head cut face is responsible for most of the soil eroded, while the downstream wall jet erodes just the pool bottom and then becomes part of the sediment laden outflow. Flow reattachment at the free surface creates considerable upwelling at the transition from the pool to the downstream channel. Note also that the impinging jet reaches the bed very close to the deepest point in the pool. This flow structure was schematically presented and discussed by Bennett [1999] and Bennett et al. [2000]. Robinson et al. [2000] reported similar flow patterns for straight drop overfalls that were much larger in scale than those reported here.

[7] These phenomenological observations led Bennett [1999] and Bennett et al. [2000] to use the vertical and horizontal distances between the head cut brink and the point of maximum scour as normalizing scales for the head cut profile coordinates. It was found that such scaling results in a collapse of the profiles measured over all runs into a single head cut profile near the vicinity of the head cut face.

3. Model Description

[8] The foregoing observations and further experimental evidence presented in the Bennett data set yield the conceptual basis of the analysis. (1) Flow and erosional processes in the head cut region are treated as two-dimensional and as evolving in a homogeneous, infinitely wide, and vertically unbounded soil bed. (2) The sediment-laden flow and the soil matrix are characterized as separate continuous, homogeneous media. (3) The head cut shape and the associated flow structure do not change with time when viewed from a frame of reference fixed with respect to a head cut migrating at a constant rate. (4) The flow structure within the head cut is the basic flow mechanism that defines head cut formation and migration.

[9] The head cut region is divided into two connected domains: the overfall and the plunge pool. Normal flow upstream determines flow conditions at the brink, which in turn control the trajectory of the impinging jet. If the water level in the pool is well below the brink, the jet is considered operating under fully ventilated conditions; otherwise, the jet trajectory is corrected to account for reduced pressure under the nonventilated jet. When the jet plunges into the pool it becomes a submerged turbulent jet that diffuses along a rectilinear trajectory defined by the jet entry angle. After impinging on the soil bed the jet splits into two wall jets that provide the sole mechanism for soil erosion. These generalizations will be quantified below.

[10] Four basic laws of physics must be satisfied for any continuous medium. These are conservation of mass (continuity), Newton's second law (momentum), conservation of energy (first law of thermodynamics), and constitutive relations (subsidiary laws). The second law of thermodynamics is automatically satisfied in the present analysis by assuming that isothermal conditions and friction without heat transfer are present. The remaining laws and the above conceptual framework are used herein to formulate analytical approximations to the governing equations that serve to quantify the most relevant features of head cut erosion processes. The notation and units used throughout this study are given in the notation section. The units are listed for completeness because all the equations developed below can be used with any consistent system of units.

3.1. Conservation of Mass During Head Cut Migration

[11] This section discusses the contribution of head cut erosion to the overall sediment balance at the head of a rill, crop furrow, or ephemeral gully. Consider the motion of water and sediment through a control volume of unit width surrounding a head cut migrating at the rate M (Figure 2). The inflow and outflow cross sections are assumed located well upstream and downstream, respectively, of the scour hole such that the flow can be regarded essentially uniform at those sections. The depth-averaged velocities measured by a stationary observer at the inflow and outflow sections are denoted as Vu and Vo, respectively, and I represents the point infiltration rate.

Figure 2.

Definition sketch of control volume used for mass conservation analysis.

[12] The motion is described with respect to a frame (x, y) fixed with respect to the migrating head cut and centered at the overfall mid-depth (Figure 2). Hence denoting the velocity fields with respect to the observer and the migrating frame of reference by equation image and equation image respectively, they are related by equation image At a time t0 the control volume contains a mass of sediment laden water m(t0). Over an infinitesimal time increment dt the head cut moves to a new location at which the mass contained in the control volume is that contained at t0 plus the net mass flux through the boundary Ω of the control volume. This condition can be expressed as

equation image

where ρ is the point density of the sediment laden water, and equation image is the unit normal vector taken positive in the outward direction. The boundary is divided into sections Ω1–2, Ω2–3, …, Ω6–1 for convenience. The infinitesimal mass flux in the neighborhood of any point on the surface Ω3–4 is (Figure 2):

equation image

where ρe is the soil bulk density, equation image and α is the angle between equation image Similar considerations apply at other points on Ω. Hence the integral on the right-hand-side of equation (1) yields

equation image

where ρw is the density of water, ρd is the bulk density of the deposited sediment, and C is the volumetric concentration of the suspended sediment load leaving the control volume. It should be noted that since y = f(x, t0) describes the shape of the head cut that remains invariant with time, the function f depends on x only. Hence integrating equation (2) and entering the result in equation (1) results in

equation image

where, q′w, 1–2 is the water discharge per unit width past the upstream section, SD is the scour depth, me = ρeMSD is the rate of soil erosion by head cut migration, md = ρdMdt is the rate of sediment past the surface Ω4–5 and deposited on the downstream side of the scour hole, dt is the thickness of the downstream deposit, qw,5–6 is the discharge of water per unit width through the downstream section, my = ρwC(Vo + M)do is the rate of sediment discharge where do denotes the downstream flow depth, and φ represents the infiltration losses along the soil bed. Variations in the bulk densities of eroded and deposited material are neglected.

[13] Invoking flow continuity and recognizing that the left hand-side of equation (3) vanishes because there cannot be sediment gains or losses within the control volume yields

equation image

Equation (3) shows that the rates of erosion and deposition are independent of the shape of the head cut, and depend only on the rate of head cut migration and the heights of the eroding and depositional faces. Equation (4) proves that, in the absence of upstream sediment supply, head cut erosion is the sole upstream source of sediment to be considered in models of developing rills and ephemeral gullies. A similar result was obtained by Begin et al. [1980] using a less rigorous analysis and an idealized head cut geometry.

3.2 Overfall Domain

[14] The flow in the neighborhood of the brink is treated as in a classical free overfall. This section discusses the characteristics of the flow past the brink and the nappe emitted from the brink and plunging into the pool (Figure 3). Here the nappe is regarded as an impinging jet and two types of jets, ventilated and nonventilated jets, are considered separately. In each case, the jet is considered as a stream tube with its axis coincident with the jet centerline. Although long jets are subjected to destabilizing shear by the surrounding medium [Goedde and Yuen, 1970], the short jets associated with head cuts justify ignoring this effect.

Figure 3.

Definition sketch of flow profile over a ventilated overfall.

3.2.1. Flow Characteristics at the Overfall Brink

[15] Hager [1983] developed the following relationship, valid for Fu ≥ 1, between the flow depth at the brink and upstream flow parameters:

equation image

where equation image is the Froude number of the upstream flow, g is the acceleration of gravity, V is average cross-sectional velocity, d is flow depth, and the subscripts b and u denote the brink and the far upstream flow sections, respectively. Flow continuity requires that

equation image

Equations (5) and (6) are used to characterize the flow conditions at the brink from known upstream flow parameters.

3.2.2. Ventilated Jet Profile

[16] Rouse [1957] analyzed the motion of a free jet under the action of no external forces other than gravitational attraction, and arrived at equations for the jet trajectory and velocity that have been used elsewhere [Fogle et al., 1993; Stein and Julien, 1994]. When Rouse's equations are related to the frame of reference (x, y) shown in Figure 3, they yield the following relationships used in the present analysis:

equation image
equation image
equation image
equation image

where d is the jet vertical thickness, Vx is the jet velocity component in the x-direction, and the absolute value of y is introduced because time is taken as not directional. It is apparent from equation (10) that x = Vbt, thus replacing this expression and equation (8) into equation (7) and introducing the convention tan θ = - dy/dx, one readily obtains the following expressions for the jet angle, θe, at the entry point:

equation image

where H is the jet plunging height given by

equation image

and h is the vertical distance from the brink to the pool surface (Figure 3). In addition, equations (9) and (10) lead to the following expressions for the jet thickness, be, and velocity, Ve, at the entry point:

equation image
equation image

[17] Equations (7)(10) are valid for noninertial flow conditions. However, when the approaching upstream flow becomes supercritical, the jet trajectory is also influenced by inertia. Hager [1983] developed a one-dimensional formulation of supercritical jet trajectory that involves a complex nonlinear function of the upstream Froude number and flow conditions at the brink. This notwithstanding, the relationships developed by Rouse lead to a less elaborate analysis, and hence it is desirable to examine the jet directional error that results from neglecting inertial effects. To this end, equation (7) and its first integral were normalized using equation (5) and the scaled coordinates ξ = x/du and η = y/du resulting in the relations:

equation image
equation image

[18] Equations (15) and (16) were used to compute values of θe = −tan−1 dη/dξ versus η = H/du for an arbitrary range ξ = 1.5, 2.0, 2.5, …, 10. The computed angles are plotted in Figure 4 as a function of the normalized plunging height and compared with entry angles measured by Robinson [1989] in a ventilated rectangular overfall. These angles were evaluated for Fu = 1.34, the average of Froude numbers reported by Robinson [1989] which differs little from the average value Fu = 1.43 of the Bennett data set (Table 1). Figure 4 shows that within the range of Froude numbers shown in Table 1, we can ignore inertia effects and proceed in our analysis using the equations presented by Rouse.

Figure 4.

Variation of the ventilated jet entry angle with normalized vertical drop distance computed with equations (15) and (16). The angles measured by Robinson [1989] are plotted for the average Froude number Fu = 1.34 of the Robinson data set.

Table 1. Summary of Experimental Data Sets Used in Model Validation
Data SourceRunqw,m2/sdu,mFuh, mh/duVentilated JetJet Entry Angle,degWater Tempearature,a °CSoil Bulk Density, kg/m3Deposit Bulk Density, kg/m3Deposit Thickness, mSediment Yield, kg/m/sScour Depth, mMigration Rate, m/s
  • a

    Average temperature for the data set.

  • b

    Estimated bulk density.

  • c

    Data not available.

  • d

    For the Bennett data set only.

Bennett and Casali [2001]30.00700.0141.350.0050.37no4222.0162114680.0470.00810.0670.00089
Bennett and Casali [2001]50.00710.0141.430.0010.09no4522.0153314230.0610.00900.0810.00083
Bennett and Casali [2001]60.00710.0141.330.0030.18no5222.015191447b0.0670.01510.0940.00109
Bennett and Casali [2001]70.00710.0161.110.0030.16no5022.014861447b0.0740.01830.0940.00142
Bennett and Casali [2001]80.00700.0141.390.0151.07no6222.0160213960.0530.02140.0890.00153
Bennett and Casali [2001]90.00710.0151.260.0070.50no6222.015451447b0.0830.01590.1060.00105
Bennett and Casali [2001]100.00710.0141.370.0120.89no6222.0151414510.0700.01990.1080.00114
Bennett and Casali [2001]110.00710.0141.410.0191.40no6922.0153715160.0620.02390.1030.00155
Bennett and Casali [2001]120.00710.0141.370.0221.55no7622.0155714560.0790.02700.1250.00123
Bennett and Casali [2001]130.00710.0141.440.0322.39no7222.0156213930.0810.02220.1320.00094
Bennett et al. [2000]10.00240.0081.080.0010.13no4222.01468N.A.c0.0160.00690.0250.00160
Bennett et al. [2000]30.00400.0081.800.0010.08no4722.01379N.A.0.0200.01090.0380.00170
Bennett et al. [2000]40.00420.0081.870.00020.03no4322.01369N.A.0.0250.01090.0400.00200
Bennett et al. [2000]60.00640.0131.380.0010.10no4422.01386N.A.0.0520.01310.0700.00160
Bennett et al. [2000]70.00690.0121.670.00050.04no4222.01369N.A.0.0400.01430.0590.00190
Bennett et al. [2000]2-A0.00710.0151.230.0231.52no6522.01558N.A.0.0700.02770.1190.00152
Bennett et al. [2000]2-B0.00720.0151.250.0322.16no6522.01564N.A.0.0560.02740.1140.00132
Bennett et al. [2000]2-C0.00720.0161.140.0301.84no6422.01583N.A.0.0620.02770.1120.00144
Bennett et al. [2000]2-D0.00720.0141.390.0261.86no6722.01622N.A.0.0580.03260.1090.00175
Bennett [1999]10.00530.0101.690.0030.33no5622.01464N.A.N.A.N.A.0.0800.00100
Bennett [1999]20.00530.0091.990.0090.96no4722.01382N.A.N.A.N.A.0.0730.00160
Robinson [1989]N.A.0.0620.0691.110.4706.85yes7324.0      
Robinson [1989]N.A.0.0620.0691.100.2794.07yes6924.0      
Robinson [1989]N.A.0.0310.0361.430.3058.40yes6824.0      
Robinson [1989]N.A.0.0310.0361.430.1524.20yes5824.0      
Robinson [1989]N.A.0.1230.1051.150.0510.48yes4124.0      
Robinson [1989]N.A.0.1240.1011.230.1911.88yes5624.0      
Robinson [1989]N.A.0.0620.0591.380.5729.66yes6924.0      
Robinson [1989]N.A.0.0620.0601.350.3816.38yes6724.0      
Robinson [1989]N.A.0.0620.0591.400.3055.21yes6524.0      
Robinson [1989]N.A.0.0620.0591.360.2674.49yes6024.0      
Robinson [1989]N.A.0.0990.0781.460.7139.18yes6724.0      
Robinson [1989]N.A.0.0990.0771.470.5336.89yes6624.0      
Robinson [1989]N.A.0.0990.0771.470.4575.91yes6724.0      
Robinson [1989]N.A.0.0990.0771.460.3814.92yes6724.0      
Robinson [1989]N.A.0.0990.0781.450.3053.92yes6424.0      
Robinson [1989]N.A.0.1230.1011.220.7016.91yes6724.0      
Robinson [1989]N.A.0.1230.1011.230.4574.55yes6524.0      
Geometric Meand 0.0060.0131.410.0050.414 55 150314440.0520.0170.0810.001

3.2.3 Jet Steepening in Nonventilated Overfalls

[19] Small head cuts typical of rills and ephemeral gullies are often not ventilated and the reduced pressure (suction) acting below the nappe may significantly affect the trajectory of the plunging jet. Christodoulou [1985] observed that the suction under the nappe is a function of the ratio between the critical depth of the approaching flow and the drop height. Similarly, Bormann and Julien [1991] and Robinson [1992] found that the steepening of the nonventilated nappe in submerged drop structures is related to drop height and tail water depth. While these findings relate jet entry conditions to measurable parameters, they cannot be used to compute the jet entry angle for a nonventilated head cut because here the drop height (i.e. the maximum scour depth SD) is an unknown parameter (Figure 2).

[20] Instead an alternate solution is obtained by examining the conservation of linear momentum within the jet section bounded by the brink section b and an arbitrary n−n section at a distance l from the brink measured along the jet axis (Figure 3). Considering the differential control volume centered about n−n (Figure 5), the space under the jet is assumed to remain void of water and at the pressure ps < 0 for any given h > 0, the inertial effects due to curvature of the jet are neglected, and pressure and weight are the only forces acting on the nonventilated jet. With these approximations, the horizontal and vertical components of the jet momentum change across the control volume are:

equation image
equation image

where f = − ∇(p + ρwg z), z represents vertical distance below an arbitrary datum, and d(l) represents the jet thickness at l. The videotaped records of the experiments in the Bennett data set show that, at the present head cut scale, momentum is conserved in the horizontal direction because the thickness of the nappe varies little in that direction. Integration of equation (18) requires an explicit characterization of the vertical pressure gradient, a lacking piece of experimental information. This obstacle is overcome by introducing, for simplicity, the linear approximation dp/dyps/[d(l)/ε] where ε > 1 is a calibrating pressure-gradient coefficient. This characterization assumes that the pressure varies linearly over the fraction d/ε of the jet thickness, and vanishes over the remaining portion (Figure 5). With these approximations, the horizontal and vertical components of the jet momentum change across the control volume reduce to:

equation image
equation image

Integration of equation (19) shows that equation (9) also holds for nonventilated jets.

Figure 5.

Definition sketch and control volume used in analysis of jet deflection under nonventilated conditions.

[21] Using the equivalence cos θl = sin θl/tan θl in equation (19), combining this equation with equation (20), and integrating the result between θl = 0 and any value θl > 0 yields the following expression for the arc length along the jet axis:

equation image

where hs = ps/(ρwg) is the suction head below the nappe. The expression within brackets is closely approximated by the function tan2 θl in the range 0° ≤ θl ≤ 80°, thus:

equation image

It can be noted from this relationship that l → 0 as θl → 0 for a given flow discharge and suction, while θl increases with suction and decreases with qw for any given l > 0.

[22] Combining equations (7)(10) results in the following expression for the horizontal distance from the brink to the entry point (Figure 3):

equation image

and using equations (7) and (23) in the canonical definition of arc length yields

equation image

Carrying out the integration, the jet length between the brink and entry points can be expressed in terms of readily quantifiable parameters referred to the xy system of reference as follows:

equation image


equation image

is a monotonically increasing function of the tangent of the entry angle for gravitational, ventilated jets, Te, defined in equation (11). It should be noted that equation (25) approaches the correct limit as the tail water surface approaches the brink point from below because ΦH → 0 as Te → 0. Eliminating the arc length l from equations (22) and (25) yields the following relationship for the entry angle of nonventilated jets:

equation image

which exhibits the expected response, that is, the entry angle increases when the suction increases or the jet momentum decreases. Further, Christodoulou [1985] observed that the suction head at the brink of a rectangular overfall varies roughly in direct proportion to a distance that he compares to the drop h. Hence it is reasonable to assume that hs = −λh, where 0 < λ ≤ 1 is an empirical suction-head coefficient. Thus

equation image

There is no apparent reason to assume a priori that the coefficients (ε, λ) are mutually independent, and hence they could be integrated into a single calibrating parameter because they both multiply the same term of the preceding relationship. They are retained as separate parameters in the present analysis, however, to verify that calibration of the model yields the expected order of magnitude for the suction-head coefficient.

3.3. Plunge Pool Domain

[23] Figure 6 illustrates the flow pattern induced by the jet entering the pool. The jet's turbulent diffusion and the resulting circulations within the pool control the flow pattern and the concomitant boundary stresses. The jet enters the plunge pool at the entry point e and immediately begins to diffuse and entrain surrounding fluid as observed by Albertson et al. [1950] for two-dimensional neutrally-buoyant turbulent jets. Beyond the potential core limit (point p) the jet proceeds with decreasing centerline velocity, impinges on the soil bed at the stagnation point i, and bifurcates into two wall jets. Fluid entrainment and shear friction create recirculating zones on each side of the jet. Beltaos and Rajaratnam [1973] and Beltaos [1976a, 1976b] have conducted extensive experimental studies on the characteristics of turbulent two-dimensional jets impinging on flat, rigid walls. Stein et al. [1993] and Stein and Julien [1994] used an impinging jet analysis to estimate the vertical scour rate of erodible beds, and Stein and Nett [1997] tested the validity of using the jet analysis of Stein et al. [1993] with soil detachment parameters on several soil types.

Figure 6.

Definition sketch of flow pattern in the pool region showing impinging and wall jets, confined eddies, and control volume used in energy balance analysis. It should be noted that shear friction is neglected along the interface between the downstream wall jet and captive eddy.

3.3.1. Equilibrium Scour Depth

[24] One of the objectives of this study is to predict the maximum scour hole depth SD because this parameter rather than the pool shape determines the rate of soil erosion from head cuts (equation 3). The approach taken here is based on the conceptual model of equilibrium depth presented by Stein et al. [1993]. In this model the equilibrium scour depth is that obtained when the scour hole is so large that jet diffusion has established a balance between the eroding and resistive stresses and a steady state scour depth SD is reached.

[25] It is seen from Figure 6 that the following geometric function relates SD, the distance Ji between the jet entry point and the point of impingement, the jet entry angle θe, and the distance from the brink to the tail water surface, h [Stein et al., 1993]:

equation image

where the left-hand side characterizes the flow depth in the pool. Equation (29) implies that the maximum pool depth and the point of impingement coincide, and that the submerged jet trajectory does not exhibit any eccentricity. In actuality, when a jet plunges into the pool, the flow reattaches at a location further downstream, and a confined eddy is formed between the entry and reattachment points. When the presence of lateral walls result in nonventilated conditions, as is the case in the Bennett data set, the jet is pulled upstream by the reduction of pressure inside the captive eddy. This phenomenon is known as the Coanda effect and has been studied elsewhere [Sawyer, 1960; Newman, 1961; Rajaratnam and Subramanya, 1968]. Beltaos [1976a] shows that jet eccentricity is significant in the range 30° < θe < 60°, and this finding is confirmed by the streamline pattern shown in Figure 1b. However, it was decided in favor of neglecting jet eccentricity to retain the simple functional structure of equation (29).

[26] It is also important to note that equation (29) is based on the concept that when the scour hole becomes sufficiently large for a balance to exist between the eroding stresses induced by the diffused jet and the resistive soil stresses, both the scour depth and impingement distance reach their steady equilibrium limits [Blaisdell et al., 1981; Stein et al, 1993]. In actuality, the head cut is migrating continuously and the jet does not impinge on the bed at the same location for an extended period of time, thus preventing Ji from reaching its true equilibrium limit. In order to retain the steady state framework of the present analysis, we introduce the following ratio:

equation image

where Ji* represents a virtual equilibrium limit assumed to exist under migration conditions, SD and SD*are the corresponding true and virtual equilibrium scour depths, respectively, and β is treated as a calibrating coefficient.

[27] Points of maximum shear occur on both wall jet zones outside the impingement region, the latter being dominated by pressure gradients that are centered at the stagnation point and vanish rapidly outside this region [Beltaos, 1976a; Robinson, 1989]. Once the particles are detached, the flow carries them into the wall jet regions where shear stresses are dominant. However, little is known about soil entrainment by turbulent normal stresses and thus current measuring techniques focus exclusively on shear-stress characterizations [Hanson, 1990, 1991]. In order to take advantage of available data, and as proposed by Stein et al. [1993], scour by jet impingement is herein attributed to an effective maximum shear stress, τm, that is in turn related to the impinging velocity, Vi (Figure 6). It should be noted that this velocity is interpreted as the centerline velocity reached at the distance Ji in a jet diffusing in an unbounded fluid. This conceptualization results in:

equation image

where Cf is a bed shear coefficient. Beltaos [1976a] found that this coefficient can be approximated in the wall region as

equation image

for the jet Reynolds-number range 3 × 103 < Rj < 5 × 105 within the range 30° ≤ θe ≤ 90°. The coefficient δf is of order 1/20, Rj = qw/ν, and ν is the fluid kinematic viscosity. In the Bennett data set Rj varied between 2,400 and 8,300, and thus Cf can be expected to be of order 0.1 for small head cuts.

[28] The jet centerline velocity is given by [Albertson et al., 1950; Rajaratnam, 1976]:

equation image

where cd is a jet diffusion coefficient of order 5/2, and J is the distance from the entry point along the jet trajectory assumed greater than the length of the jet's potential core. When scour reaches the true-equilibrium limit, the condition τm = τc is assumed to hold, where τc is the soil's true critical-shear stress. Hence combining equations (31), (32), and (33) yields

equation image

Eliminating Ji between equations (30) and (34) and recalling equation (32) results in the virtual-equilibrium scour depth:

equation image


equation image

and τ*c = τc/β is the critical shear-stress at virtual equilibrium conditions. Equations (30) and (36) show that the true values of Ji and τc are modified simultaneously by adjusting the single coefficient β, and in this fashion, the observed scour depths are reached instantaneously in any given soil regardless of the migration rate. With this modification, the equilibrium model can be extended to head cuts migrating in a virtual soil whose erodibility properties are identified through empirical calibration of the coefficient β.

[29] It should be noted that the product cd2δf is of order 5/16 and independent of whether or not the jet is ventilated. Therefore the experimental coefficient σ depends essentially on fluid and soil properties. It is also noteworthy to observe that (1) SD* vanishes as H → 0 because θe → 0, and (2) SD* grows indefinitely as H → ∞ because θ → π/2 regardless of the degree of jet ventilation. However, the latter limit is physically impossible because eventually the jet becomes unstable and breaks down into droplets falling at terminal velocity that do not have the same erosive power of the jet [Goedde and Yuen, 1970].

3.3.2. Rate of Head Cut Migration

[30] As the head cut propagates upstream, scouring soil as it does so, energy must be consumed to overcome the resistance presented by the soil. Obviously, only the impinging jet that drives the flow in the plunge pool can supply that energy. A simple way to relate the supply and consumption of energy within the pool is to recall that the pool system must satisfy the first principle of thermodynamics, namely, that the heat added plus the total work done must equal the change in total energy of the system. To apply this law to the plunge pool domain, let us consider a control volume enclosing the flow in direct contact with the soil boundary (Figure 6). Noting that the control volume boundary does not change, the inflow and outflow control boundaries are positioned normal to the flow, and the process is steady, the control-volume form of the energy conservation law can be expressed as

equation image

where e = u + gζ + V2/2 is the stored specific energy of the fluid, u is the internal energy, gζ is the potential energy defined with respect to an arbitrary datum passing through the entry point, p denotes pressure, υ is the fluid specific volume, pυ is the work of the normal stresses, external heat sources are ignored, equation image are area and velocity vectors, and the subscripts e and d identify the inflow and outflow sections, respectively. The work term Wτ includes the work on the control volume boundaries by normal (pressure) and tangential (shear) stresses, which is largely expended in eroding the soil bed. Introducing the definition equation image and ignoring heat generated by viscous friction, equation (37) yields

equation image

[31] By further neglecting density variations (i.e., ρe = ρd ≈ ρw, ρdVdAd ≈ ρeVeAe = ρwqw) and recognizing that the pressure distribution outside the impingement region is not far from hydrostatic, the pressure at the outflow section can be approximated by pd = gρwe − ζd). Beltaos [1976a] found that the maximum velocity in the wall jet region, Vd,max, varies as

equation image

where α is the distance along the wall measured from the point of impingement. Moreover, studies of the classical wall jet [Rajaratnam, 1976] have established the similarity of velocity profiles along the jet cross-sections, and this similarity has been verified for both normal [Beltaos and Rajaratnam, 1973] and oblique [Beltaos, 1976b] impinging jets. Thus it is to be expected that the average velocity of the wall jet decays at the same rate as Vd,max, and by further assuming that Ad is far enough from the stagnation point such that α/be = 10 and rounding off the coefficient 5.5 to 5.0, equation (38) can be expressed as

equation image

The right-hand-side of equation (40) has the units (FLT−1)L−1 and therefore can be interpreted as the jet power consumed by head cut scour per unit width of soil bed. Obviously, any given amount of available jet power will result in varying amounts of scour depending upon the erodibility of the soil. It was shown in Section 3.1 that the rate of steady state head cut erosion depends only on the rate of migration and the height of the eroding face. The rate of migration can be interpreted as a rate of soil detachment per unit area uniformly distributed over the eroding face. Thus we can assume that

equation image

where kd represents the soil erodibility, τ is the point shear per unit area applied by the flow on the eroding face, and τc* = τc/β is the virtual critical stress introduced in equation (36). Equation (41) is commonly used to characterize the erosion of soil materials encountered in engineering applications [Stein et al, 1993; Stein and Nett, 1997]. Multiplying both sides of this equation by M and integrating over the portion of the head cut height directly exposed to erosive flow action, SD* − h, results in:

equation image

where r is a variable of integration. Using the units [kd] = F−1T−1L3, [τ] = FL−2, we note that the integral term on the R.H.S. of this equation has the dimensions (F LT−1L−1), i.e., jet power applied per unit width. Hence equation (42) can be expressed as

equation image

The energy rates (power) computed from equations (40) and (43) are expected to be equal under virtual-equilibrium conditions. Thus equating these expressions and solving for M we arrive at the final equation for the migration rate:

equation image


equation image

It should be noted that the parameter μ is a function of material properties and jet entry angle, and hence it will depend on jet ventilation conditions even when all other parameters remain constant.

4. Experimental Validation

[32] Equations (4), (11), (28), (35), and (44) for mass balance, jet entry angle, virtual-equilibrium scour depth, and rates of head cut migration were tested and validated with four data sets. The first data set comprises jet entry angles derived from water surface profiles measured by Robinson [1989] across ventilated, rectangular free overfalls. The remaining data sets comprise upstream bulk-flow parameters, jet entry angles, scour depths, and migration rates of head cuts freely developed in soils under nonventilated overfall conditions measured by Bennett [1999], Bennett et al. [2000], and Bennett and Casali [2001]. These data sets are summarized in Table 1 and do not incorporate all the experimental runs reported. As required by the present analysis, only those runs for which the nonventilated brink height h is greater than zero were used to evaluate equations (28), (35), and (44). In addition, data points with h/du < 0.1 were omitted from the entry angle validation because they are within the range of experimental errors. It is also important to note that in testing these equations only measured values were entered for the variables and coefficients appearing on the right-hand-side of the equations, the only exceptions being the calibrating parameters. For instance, the entry angle in equation (35) was obtained from Table 1 rather than being computed with equation (28). In this fashion, only experimental errors affected the results of the validations. It is also apparent from Table 1 that entry angles and scour depths in the Bennett data set are consistently greater in the range {R1} = {0.1 < h/du < 0.6} than those observed for the range {R2} = {0.6 < h/du < 2.4}, although such is not the case with the migration rates. Consequently, a distinction between these ranges is made in the following sections to further examine the influence of the tail water depth (i.e., h) on pool scour.

4.1. Uncertainty Analysis

[33] Since observational errors were present in the measured variables, an analysis was performed to ascertain the degree of uncertainty in the experimental validation of the relations presented above. The analysis consisted of (1) determination of the data uncertainty, which means the most likely error committed during the measurement of the recorded variables, and (2) the propagation of the experimental errors, which means the way the data uncertainty affects the uncertainty of calculations performed with the recorded values of the variables. This analysis is restricted to the Bennett data set because the Robinson data set does not include the measured variables used to validate the mass balance, scour and migration rate algorithms.

[34] The measurements reported in the Bennett data set are essentially single-sample experiments because the uncertainty of the observations was not statistically determined by repeating the measurements a sufficiently large number of times. Therefore the uncertainty of each recorded variable is represented by an acceptable estimate of the true value as well as an acceptable measure of the error in the estimate. The estimate, equation image, of the true value of any measured variable X is given as either a single reading or the average of a few readings (Table 1), and the error in that estimate is specified as an uncertainty interval, ±ux, which is not a variable but a value suggested by the experience of the observers. Hence the uncertainty range of the variable is expressed as equation image ± ux, and the associated relative error as εX = uX/equation image. The relative errors estimated in this manner for the measured variables in the Bennett data set are shown in Table 2. The uncertainty in the estimation of water density and viscosity was considered negligible in relation to all other data uncertainty.

Table 2. Summary of Relative Error Estimates for the Measured and Computed Variables
Data SourceRunErrors due to Measurement Uncertainties in the Bennett Data SetPropagated Errors
Bennett and Casali [2001]30.0090.0790.1940.0720.1430.6250.0370.0060.0320.0460.0510.1980.0890.0900.1170.1010.1170.1380.0500.1800.357
Bennett and Casali [2001]50.0100.0590.8330.1990.1430.6250.0410.0020.0540.0490.0530.1670.0660.0660.1580.1720.1890.2090.0680.2300.519
Bennett and Casali [2001]60.0070.0560.3880.1540.1430.6250.0420.0060.0460.0490.0520.3050.0640.0650.1330.1910.2020.2310.0630.2420.465
Bennett and Casali [2001]70.0100.0680.3880.1200.1430.6250.0690.0020.0300.0500.0520.2130.0850.0860.1390.1520.1670.1940.0560.2130.415
Bennett and Casali [2001]80.0090.0510.0680.0810.1430.6250.0300.0130.0360.0470.0540.2290.0570.0580.0600.1720.1740.2200.0440.2220.386
Bennett and Casali [2001]90.0090.0540.1370.0800.1430.6250.0290.0070.0430.0490.0520.2080.0640.0640.0900.1770.1830.2220.0430.2290.388
Bennett and Casali [2001]100.0080.0650.0810.0640.1430.6250.0530.0070.0390.0500.0520.1010.0730.0730.0730.1510.1560.1940.0380.2080.367
Bennett and Casali [2001]110.0070.0580.0520.0730.1430.6250.0290.0060.0320.0490.0490.1260.0650.0660.0540.2340.2360.3020.0430.2710.411
Bennett and Casali [2001]120.0080.0720.0460.0660.1430.6250.0330.0050.0350.0480.0520.1700.0810.0820.0550.3470.3480.4490.0420.3700.481
Bennett and Casali [2001]130.0070.0590.0310.0690.1430.6250.0260.0020.0510.0480.0540.1350.0660.0660.0380.2840.2840.3740.0430.3090.439
Bennett et al. [2000]10.1370.1250.9740.1440.1430.6250.1720.005    0.1730.2200.2960.2390.2810.3250.1160.3160.493
Bennett et al. [2000]30.1240.1251.6190.1070.1430.6250.0740.011    0.1340.1830.2720.2050.2460.2580.0830.2780.431
Bennett et al. [2000]40.0790.1254.5830.3490.1430.6250.0600.009    0.1320.1530.2930.2880.3230.3290.1130.3410.791
Bennett et al. [2000]60.1030.0770.8100.1130.1430.6250.0510.013    0.0940.1390.1750.1630.1850.2060.0690.2390.417
Bennett et al. [2000]70.0250.0832.0000.1200.1430.6250.0930.020    0.0890.0930.1950.1210.1550.1630.0590.1950.408
Bennett et al. [2000]2-A0.0140.0530.0440.0760.1430.6250.0350.006    0.0630.0650.0470.2010.2030.2630.0440.2440.396
Bennett et al. [2000]2-B0.0140.0330.0310.0770.1430.6250.0850.014    0.0390.0420.0310.1890.1900.2510.0430.2290.393
Bennett et al. [2000]2-C0.0140.0250.0340.0780.1430.6250.0170.006    0.0310.0340.0310.1850.1860.2450.0430.2270.389
Bennett et al. [2000]2-D0.0140.0860.0380.0740.1430.6250.0240.008    0.0970.0980.0550.2300.2320.3020.0470.2650.410
Bennett [1999]10.0620.1200.3040.1080.1430.6250.0370.006    0.1290.1430.1690.2090.2250.2560.0630.2610.427
Bennett [1999]20.0620.1110.1150.0850.1430.6250.0270.016    0.1160.1310.1090.1510.1610.1910.0630.2180.384
Geometric mean 0.0190.0700.1930.0990.1430.6250.0430.0070.0390.0480.0520.1770.0800.0860.0970.1910.2050.2440.0560.2480.430

[35] Now, let Y be the result of a calculation involving n measured variables X1, X2, X3, …, XN. Then, the relation between the experimental uncertainty for these variables and the uncertainty interval for the result can be expressed as [Rabinovich, 2000]:

equation image

where the partial derivatives are evaluated at the point equation image, equation image1, equation image2, …, equation imageN so that the variations of the variables are small. In general, equation (46) can be reduced to the dimensionless form:

equation image

which yields the relative error of the estimate equation image in terms of the propagation of relative errors in the measured variables. Equation (47) was used to derive the relative errors associated with equations (5), (6), (11), (12), (14), (26), (28), (35), (36), (44), and (45) due to the propagation of measuring errors, and the resulting expressions are given in Appendix A. It is noteworthy to observe that these relationships are independent of the system of units and they yield the weighting factors associated with each propagated error. The experimental and computed propagated errors are shown in Table 2. The data columns listed in Tables 1 and 2 exhibit little variations among their arithmetic, geometric, and harmonic means with the exception of h, H, SD*, and qw. In these instances, the geometric mean gave a better approximation of the central tendency of the data as gleamed from inspection of the data. Therefore the last row in each of these tables gives the mean value of each experimental and propagated error in terms of the geometric mean. These mean values are used in the following sections.

4.2. Conservation of Mass

[36] Equation (4) states that regardless of the shape of the head cut and depositional front, the mass of soil eroded per unit width and unit time by head cut migration under steady state conditions must equal the mass of sediment deposited per unit width and unit time plus the sediment discharge, i.e. the sediment mass exiting the head cut area per unit width and unit time. To validate this statement the rates me = ρeMSD* and md = ρ Mdt were computed using measured data, and the rate my was determined from sediment samples collected at the flume outlet (Table 1). The sum of the md and my rates is plotted against the me rate on Figure 7 for all the runs in the Bennett data set listed on Table 1. Figure 7 shows that the sum of sediment deposition and discharge rates is indeed linearly related to the rate of head cut erosion as expected. However, the latter is exceeded in all cases by the added rates of deposition and discharge by about 22%. The cause for this systematic departure is not known, but the determination of bulk density may be questionable. Similarly, the experimental uncertainty of these results could not be examined because me and the variables entering in the calculation of my were not recorded.

Figure 7.

Validation of equation (4) based on a comparison of head cut erosion rates computed from me = ρeMSD against the sum of deposition rates evaluated with md = ρdMdt and sediment discharge rates my measured at the flume outlet. All parameters entering in this evaluation are from the Bennett and Casali [2001] data set listed in Table 1. The dash-dotted line represents a linear fit of the data points forced through the origin.

4.3. Jet Entry Angle

[37] Equations (35) and (44) are dependent on the jet entry angle, and hence the ability to accurately predict this angle is critical in those instances where measured angles are not available. Figure 8 compares the entry angles listed in Table 1 with the angles predicted by equation (11) for ventilated jets and by equation (28) for nonventilated jets. The input variables db and Vb entering in these equations were computed from equations (5) and (6) using the measured upstream flow parameters. The slope of a linear fit for the Robinson data, forced through the origin of Figure 8, differs only 2% from the slope of the line of perfect agreement. Evaluation of equation (28) required identification of two empirical coefficients. The pressure-gradient coefficient was set equal to 2 for all runs, implying that the pressure within the nonventilated jets varies from 0 at the centerline of the jet to ps at the nonventilated boundary of the nappe. This value of ε serves as an educated guess until new experimental data becomes available. The suction-head coefficient was varied between 0.2 and 1.0 and independently adjusted for each range {Ri}, i =1,2, until a match was obtained between a linear fit for all the Bennett data, forced through the origin of Figure 8, and the line of perfect agreement. This calibration resulted in the values λ1 = 0.80 and λ2 = 0.30, which compare well with the observations reported by Christodoulou [1985]. The variation of the suction-head coefficient with the h/du range as well as the functional structure of equation (28) demonstrates the jet entry angle dependence on both upstream flow and tail water level.

Figure 8.

Comparison of jet entry angles computed with equations (11) and (28) versus the measured angles shown in Table 1. Equation (28) was evaluated with ε = 2, λ1{0.1 < h/du < 0.6} = 0.48 and λ2{0.6 < h/du < 2.4} = 0.30. The dashed lines delimit the mean data uncertainty range of the Bennett data set.

[38] The mean relative error of the measured entry angles in the Bennett data set is about 10% and the mean propagated error associated with equation (28) is about 6% (Table 2). Combining these errors yields the mean range of data uncertainty delimited by the dashed lines plotted in Figure 8. Although the Bennett data set exhibits considerably more scatter than the Robinson data set most of the computed angles lie within the uncertainty range. The contribution of individual measurement errors to uncertainty in the results was examined by substituting equations (A1), (A2), (A5), (A6), and (A8) into equation (A9), grouping terms by individual measurement error, evaluating the coefficients of the resulting equation using the mean measured parameters (Table 1), and finally substituting the mean values of the measurement errors (Table 2). This procedure leads to the following approximation of the propagated error for equation (28):

equation image

Equation (48) shows that for the Bennett data set, measuring errors in du, h, and the entry angle constitute the largest contributions to the propagated error in the predicted angles. For this reason, the scatter in the Bennett data set is attributed to the limited resolution afforded by the smaller physical scale of the experiments.

4.4. Equilibrium Scour Depth

[39] Equation (35) and its complementary equation (36) were evaluated with cd δf = 5/16, the measured water discharges, drop heights h, and jet entry angles listed in Table 1, and with the flow depth at the brink computed from equation (5). The same true critical-stress value τc = 1.75 Pa was assigned to each range {Ri} because the same soil was used in all the experimental runs. Although this parameter was not measured, the selected value is consistent with critical stresses specified for moderately resistant soils of the type used in the Bennett data set [Hanson and Simon, 2001]. The predicted scour depths at virtual equilibrium are compared in Figure 9 with the measured depths listed in Table 1. The virtual equilibrium parameters βi {Ri, i = 1,2} were evaluated by adjusting their values independently of each other until the slope of a linear relation fit for all the data, forced through the origin of Figure 9, matched the line of perfect agreement. This procedure resulted in the values β1 = 0.70 and β2 = 0.90 that reflect the expected trend because, as seen from equations (35) and (44), the virtual critical stress τc* = τc/β should decrease for the scour depth to increase as the rate of migration decreases. The data presented in Table 1 show that such response is confirmed by the experiments. The adequate fit achieved by treating β as the only degree of freedom in the calibration of equations (35)(36), is taken as experimental confirmation of the virtual-equilibrium conceptualization leading to these equations. An immediate corollary of the variation of the scour depth with {Ri} is its dependence on upstream flow parameters and tail water level. The mean measurement and propagated errors for the scour depth are approximately 4% and 25%, respectively (Table 2). These errors are combined in Figure 9 to demarcate the mean range of data uncertainty bounded by the dashed lines. It is apparent that the scatter of the predicted scour depths is largely the result of propagated errors. In order to characterize the distribution of these errors, equations (A1), (A2), and (A5) are substituted in equation (A10) and the result evaluated following the same procedure leading to equation (48). This yields the approximation:

equation image

which shows that in the present case the propagated error is mostly attributable to uncertainties in the measured jet entry angle and the selected true critical shear stress.

Figure 9.

Comparison of measured scour depths with scour depths at virtual equilibrium predicted by equations (35)(36) for cd2δf = 5/16, τc {0.1 < h/du < 2.4} = 1.75 Pa, β1 {0.1 < h/du < 0.6} = 0.90, and β2 {0.6 < h/du < 2.4} = 0.70. The dashed lines delimit the mean data uncertainty range.

4.5. Rate of Migration

[40] The rates of migration for the Bennett data set were predicted with equations (44) and (45). Measured values from Table 1 were substituted for all the variables in the right-hand-side of these equations. Figure 10 compares the migration rates presented in Table 1 with the predicted rates. The same soil-erodibility coefficient was used for all runs and it was adjusted until the data points clustered around the line of perfect agreement. This calibration resulted in the value kd {R1} = 8 × 10−8 m3 N−1s−1 that matches the average erodibility coefficient measured by Hanson and Simon [2001] for the threshold stress τc = 1.75 Pa introduced in the preceding section. The geometric mean of the measurement and propagated errors for the migration rate data are roughly 0.1% and 43%, respectively, (Table 2) and thus the scatter of the computed migration rates is entirely due to the propagation of measurement errors. The predicted rates match the order of magnitude of the measured rates but the data scatter is too large to draw any conclusions from this test without further examining the impact of propagated errors on the uncertainty of the computed data. To this end, equations (A1), (A2), and (A5) are substituted in equation (A11) and the result evaluated as explained above resulting in the approximation:

equation image

Thus the observed scatter is largely the result of (1) using the same kd value for all the experimental runs, which does not account for variations in soil erodibility due to variations in bulk density, aggregate stability, and moisture content in different runs, and (2) the limited resolution of the jet entry angle measurements afforded by the smaller physical scale of the experiments. As a matter of fact, by simply cutting in half the uncertainty of these parameters would decrease the mean propagated error of the predicted migration rates to 27%.

Figure 10.

Migration rates predicted by equations (44)(45) with kd {0.1 < h/du < 2.4} = 8 × 10−8 m3 N−1s−1 versus measured rates from Table 1. The dashed lines delimit the mean data uncertainty range.

5. Conclusions

[41] An analytical model is developed to predict scour and migration of head cuts at scales typical of rills on hillslopes, crop furrows on agricultural fields, and ephemeral gullies in upland areas. The conceptual framework is limited to two-dimensional head cuts retreating at a steady rate in homogeneous cohesive soil layers, with no upstream sediment supply, and the head cut migration process is characterized as a self-similar propagation phenomenon. Closed form algorithms are developed to describe the jet trajectory at the overfall, and the resulting virtual-equilibrium scour depth and rate of head cut retreat. An enhancement to standard jet theory is introduced to address nonventilated overfalls; however, the model is not applicable to fully submerged overfalls. The algorithms depend uniquely on readily measurable physical soil properties, upstream bulk flow parameters, and the tail water stage, and all the model parameters are physically based. The mass balance test was inconclusive due to limited data. Calibration of the model parameters yielded realistic parameter values falling within the expected order of magnitude, and resulted in the model predicting almost exactly the linear trend of the measured jet entry angles, scour depths, and head cut migration rates. The mean propagated error for the predicted entry angles is about equal to the mean experimental error for the measured angles. The mean propagated error for the scour rate is four times as large as the mean measurement error, and mostly due to uncertainties in the input entry angles and critical shear stress. The head cut migration rates are very sensitive to uncertainties in the soil erodibility and jet entry angle data. Three useful corollaries of this study are: (1) head cut erosion is the sole upstream source of sediment in growing rills and ephemeral gullies, (2) characterizations of hydrodynamic processes governing head cut erosion depend on accurate evaluation of upstream flow parameters and tail water stage, and (3) equilibrium-scour concepts developed for stationary plunge pool analysis cannot be extended to migrating head cuts without relaxing the soil properties. This model provides efficient algorithms that can be readily incorporated in available soil erosion prediction technology.

Appendix A:: Propagation of Experimental Errors

[42] The following equations give the relative errors associated with equations (5), (6), (11), (12), (14), (26), (28), (35), (36), (44), and (45) due to the propagation of measuring errors.

equation image
equation image
equation image
equation image
equation image
equation image
equation image

for ventilated overfalls

equation image
equation image

for nonventilated overfalls

equation image
equation image

exchange areas of control volumes, m2.


outflow area of the pool control volume, m2.


inflow area of the pool control volume, m2.


volumetric concentration of suspended sediment load.


bed friction coefficient.


upstream Froude number.


vertical distance from mid-depth at the brink to the tail water surface, m.


infiltration rate per unit area of bed, m s−1.


distance from the entry point along the axis of an unbounded jet, m.


distance from jet entry to impingement under true equilibrium, m.


distance from jet entry to impingement under virtual equilibrium, m.


horizontal distance from brink to jet entry point, m.


rate of head cut migration, m s−1.


range of h-to-du ratio.


wall jet Reynolds number.


scour depth under true-equilibrium conditions, m.


scour depth under virtual-equilibrium conditions, m.


arctangent of entry angle for ventilated jets, radians.

equation image

flow velocity field with respect to a stationary observer, m s−1.

equation image

flow velocity field with respect to the migrating control volume, m s−1.


average jet velocity at the brink point, m s−1.


average velocity of wall jet exiting the pool region, m s−1.


average jet velocity at point of entry to the pool, m s−1.


centerline velocity of diffusing jet at the distance Ji, m s−1.


average jet velocity at the distance l from the brink, m s−1.


average flow velocity at the downstream channel outlet, m s−1.


average velocity of the upstream channel inflow, m s−1.


mechanical work on control surface by normal and shear stresses, Joules.


measured variable, variable.


result of calculation with measured variables, variable.


turbulent diffusion coefficient of submerged jet.


thickness of jet, m.


thickness of jet at the entry point, m.


vertical thickness of jet, m.


flow depth at the brink point, m.


depth of flow at the downstream channel outlet, m.


depth of constructed bed downstream of pool, m.


depth of the upstream channel flow, m.


stored specific energy of flowing water, m2 s−2.


gravitational acceleration, m s−2.


vertical distance from brink to tail water surface, m.


ambient suction head under nonventilated nappe, m.


soil erodibility coefficient, m3 N−1 s−1.


distance from the brink along the jet axis, m.


mass of sediment laden water contained in control volume, kg.


rate of soil erosion per unit width, kg m−1 s−1.


rate of sediment deposition per unit width, kg m−1 s−1.


rate of sediment discharge per unit width, kg m−1 s−1.

equation image

outward unit vector normal to the control surface.


hydrodynamic pressure within flowing water, Pa.


ambient suction under nonventilated nappe, Pa.


water discharge per unit width, m2 s−1.


variable of integration.


time, s.


experimental uncertainty associated with measured variable X, variable.


horizontal coordinates of reference frame migrating with head cut , m.


vertical coordinates of reference frame migrating with head cut, m.


vertical distance below an arbitrary datum, m.


function defined in equation (26).


boundary of control volume.


distance along pool bed from the impingement point, m.


calibrating equilibrium-scour coefficient.


empirical constant entering in equation (32).


calibrating pressure-gradient coefficient.


relative error associated with the variable X.


vertical distance to datum passing through the jet entry point, m.

equation image

vertical distance of flow area centroid to preceding datum, m.


dimensionless vertical coordinate of points on jet centerline.


jet entry angle, radians.


jet angle at the distance l from the brink along the jet axis, radians.


calibrating suction-head coefficient.


parameter defined in equation (45) .


kinematic viscosity of water, m2 s−1.


dimensionless horizontal coordinate of points on jet centerline


mass density of water, kg m−3.


parameter defined in equation (36).


critical shear stress for erosion initiation, Pa.


virtual critical shear stress for erosion initiation, Pa.


maximum shear stress applied by wall jet, Pa.


infiltration losses per unit width of soil bed, m2 s−1.


fluid specific volume, m3 kg−1.


[43] This research builds on the work of previous investigators that made critical contributions to the fields of fluid-jet and soil-erosion mechanics; our respectful indebtedness goes to all of them. We thank K.M. Robinson for kindly making available his data for our validation work. E.J. Langendoen and two anonymous reviewers provided helpful comments. This article is listed in the Journal Series no. 2002-19 of the Montana Agricultural Experiment Station, Montana State University.