Drop characteristics of free-falling nappe for aerated straight-drop spillway

Abstract

A parallel study of regression and semi-theoretical equations is presented to forecast the drop force and the drop length of a free-falling nappe for an aerated straight-drop spillway. The ratio between the critical velocity and the exit velocity at the drop wall was introduced to establish semi-theoretical equations. Regression techniques were applied by using the dimensionless channel bed slope and the drop number to describe the drop characteristics. There were 64 groups of test data to calibrate the parameters of equations. Another 36 groups of test data were used to verify the developed model. The simulation results indicate that the semi-theoretical method provides reliable estimates for the drop force, and that the regression method can offer accurate predictions for the drop length.

Keywords:

• Drop force
• drop length
• drop number
• free-falling nappe
• regression
• semi-theoretical equation
• straight-drop spillway

1 Introduction

Check dams, drops, stepped spillways and cascades are common in stormwater systems, dam outlets and water treatment plants (Chanson and Toombes 1998). These structures may generate a free-falling nappe associated often with a scour hole (Chanson 1995, 1996, Chanson and Toombes 1998, Mossa et al. 2003, Chanson and Gualtieri 2008), yet the drop force is rarely discussed. The aim of this study is therefore to predict the drop force and the drop length of free-falling jets by semi-theoretical equations and regression techniques. A series of test data was used to calibrate the proposed equations, and further test data were used to verify these formulae.

2 Experimental apparatus and methods

The tests were conducted in a 12 m long, 0.3 m wide and 0.4 m deep channel, whose sidewalls were made of glass panels. The bottom slope was adjustable (Fig. 1). The discharge was accurately controlled by a constant head tank with a variable-speed electronic controller.

Figure 1 Experimental layout: (a) lateral view and (b) top view

The bottom slope upstream of the drop in a river is usually small because of silt accumulation, such that it was adopted to zero herein. Beyond the drop, a free-falling jet develops with an air cavity beneath the nappe (Chanson and Toombes 1998). The air cavity was ventilated by holes in the drop wall (Fig. 2). The flow depths in the drop vicinity and the bed pressures of the pool zone were measured by RPS-401^® powered ultrasonic sensors and eight KYOWA^® pressure transducers, respectively. The drop characteristics were investigated by varying the bottom slope S _o of the pool zone from 0 to 9%, the drop height H from 0.15 to 0.30 m and the unit discharge q from 0.0076 to 0.0402 m²/s (Table 1). Note that the Froude number is F = v/(gh)^1/2, where v is the water velocity and h the flow depth; the Reynolds number is R = vR _h /ν, where R _h is the hydraulic radius and ν the kinematic viscosity of water. Subscripts o and w relate to approach flow and tailwater conditions, respectively.

Figure 2 Illustration of free-falling nappe in aerated straight-drop spillway

Table 1 Experimental flow conditions

Sample	Parameter	Experimental conditions
Calibrated sample	S o	0%, 3%, 6%, 9%
	H	0.3, 0.25, 0.2, 0.15 m
	q	0.0076–0.0402 m^2/s
	F o	0.83–1.84
	R o	7642–34,594
	F w	3.53–20.40
	D	0.00026–0.04874
Simulated sample	S o	0%, 3%, 6%, 9%
	H	0.3, 0.25, 0.2, 0.15 m
	q	0.0138–0.0359 m^2/s
	F o	0.94–1.44
	R o	12,995–31,204
	F w	4.21–19.87
	D	0.00072–0.03845

Table 1 displays the test conditions. There were 64 groups of test data of Huang and Chen (2008) to calibrate the model. The authors conducted 36 groups of new test data to simulate the results. Most of approach flows to the drop were trans- and supercritical flows.

The drop number D = q ²/(gH ³) was excessively studied in the past, in which g is the gravitational acceleration and H the drop height. Rand (1955) found that the flow geometry of straight-drop spillways can be described by only D. Chinnarasri and Wongwises (2004) discovered that as D increases, the energy loss ratio decreases. The Federal Highway Administration (2006) classified the “low drop” structure if D > 1. Table 1 indicates that the present D values result in a “high drop”.

Figure 2 shows a free-falling nappe of an aerated straight-drop spillway, where u _e is the exit drop velocity. The distance measured (subscript m) from the drop to the maximum pressure position was defined as the drop length L _md (Fig. 2). The vertical unit drop force F _md was determined by the measured drop pressure P of the downstream channel as follows:

where L _i is the distance between the ith pressure transducers and the drop wall and ρ the water density.

3 Analysis of drop characteristics

3.1 Semi-theoretical equations

For critical (subscript c) flow,

where y _c is the critical depth and u _c the critical velocity (Fig. 2). A linear relationship between u _c and u _e is assumed as follows:

where α is a constant. Ferro (1992) suggested that α = 1.316. As the flow direction at the drop brink is nearly horizontal, the horizontal acceleration is close to zero, whereas the vertical acceleration equals to g (Chanson 1996). With t = time, the semi-theoretical (subscript t) drop length L _td is thus given as follows:

Solving Eqs (5) and (6) results in the following:

Using nonlinear regression, α = 1.312 was obtained, which is similar to Ferro's result.

Figure 2 also shows the drop area, onto which the nappe impacts the channel generating flow recirculation and thus increasing the discharge q _b into the nappe, changing the nappe velocity near the channel. According to momentum conservation in the vertical direction, the semi-theoretical vertical unit drop force F _td is given as follows:

where u _n is the nappe velocity in a thick stream at the side of the pool and θ the jet impact inclination at the channel bottom. White (1943) proposed for the relationship between q and q _b

where y _a is the effective nappe width and y _b the below width, respectively. Therefore,

White (1943) neglected the pool presence in the downstream channel and deduced the impact jet velocity u into the pool for S _o  = 0 as follows:

This study considers the effect of the downward slope, and the height of the drop is therefore H + L _td S _o . Substituting Eq. (2) and the height of the drop into Eq. (11a) results in the following:

White (1943) assumed that u _w  = u _n and obtained the following:

Based on the momentum balance in the horizontal direction near the channel bottom gives (Fig. 2)

Substituting Eqs (2) and (3) into Eq. (11a) results in the following:

Therefore,

Substituting Eqs (10), (12) and (13c) into Eq. (8), the semi-theoretical vertical unit drop force F _td is given by

3.2 Regression equations

From Eq. (7), the regression (subscript r) drop length L _rd is given by

From the Buckingham π-theorem, the three terms π ₁  = q/(gH ³)^0.5, π ₂  = L _rd /H and π ₃  = S _o are relevant. The dimensionless parameter L _rd /H therefore is given as follows:

Using the same method for the vertical unit drop force gives π ₁  = q/(gH ³)^0.5, π ₂  = F _rd /(ρgH ²) and π ₃  = S _o , such that

Based on the dimensional analysis and the empirical equations developed by Chanson (1995), this study used calibrated data to determine the regression equations as follows:

Equations (18) and (19) can be used to calculate the drop length and the force for a given drop number and bottom slope. The prediction accuracy R ² is high enough to prove that the drop characteristics can be attained from D and S _o .

4 Discussion of drop characteristics

4.1 Accuracy of drop characteristics

Figure 3 compares the semi-theoretical and regression equations using dimensionless parameters. The root-mean-square error (RMSE) was also determined to estimate the difference between measured and predicted values.

Figure 3 Comparison between measured/predicted drop characteristics: (a) dimensionless drop length and (b) vertical unit drop force

4.1.1 Drop length prediction

Except for two values, the predictions using the semi-theoretical equation are within ±0.05. The RMSE (= 0.0328) of the semi-theoretical method is larger than that of the regression method (= 0.0194). Therefore, the regression approach can be applied to predict the drop length.

4.1.2 Unit drop force prediction

The semi-theoretical equation can be used to predict the unit drop force to be within ±0.02, except for two values within the range of 0.05. The RMSE of the semi-theoretical method (= 0.0069) is smaller than that of the regression method (= 0.0270). Therefore, semi-theoretical method provides a better estimate.

Scale effects may exist if dimensionless terms have different values between model and prototype (Chanson and Gualtieri 2008). Henderson (1966) stated that the drag coefficients of the model and prototype are identical for Reynolds number R > 1000. Herein, R > 10,000 as shown in Table 1. The dimensional analysis indicates that the drop characteristics are D, F and S _o . Therefore, the present method can be used to simulate free-falling jets of high drops (0.00026 ≤ D ≤ 0.04874), mild tailwater slopes (0% ≤ S _o  ≤ 9%) and transcritical approach flow (0.83 ≤ D ≤ 1.48).

4.2. Drop length comparison with literature

Rand (1955) (subscript r) estimated the drop length L _rd by the regression method for aerated nappes as follows:

Chanson (1995) assumed that the flow upstream of the drop is subcritical, and suggested for non-aerated nappes

Equations (20) and (21) indicate that the relative drop length is a function of y _c /H only. Substituting q = (y _c ³ g)^0.5 into D = (y _c /H)³, Eq. (18) can be rewritten as follows:

Equation (7) can be divided by H to express the dimensionless drop length as follows:

Substituting S _o  = 0 into Eq. (23) gives the following:

The FHA (2006) suggested for aerated nappes is given by

Equation (25) can be transformed into the following:

Equations (21), (22) and (24) have similar exponents of (0.525, 0.489 and 0.5), respectively, whereas Eqs (20) and (26) have the identical exponent 0.81.

Figure 4 shows the relationship between the dimensionless drop length and the dimensionless critical depth. Rand (1955) and Federal Highway Administration (2006) obviously overestimated the drop length. Although the equation of Chanson (1995) overestimates the drop length, his trend is closer to the present test data than the others. The discrepancy between Chanson's and the present data may be attributed to the downstream slope, resulting in a longer drop length.

Figure 4 Comparison of drop length L/H versus relative drop height y _c /H

5 Conclusions

The drop characteristics of a free-falling nappe were determined by regression techniques and a semi-theoretical approach. The semi-theoretical method provides an accurate estimation for the unit drop force, and the regression method accurately predicts the drop length. This study includes the effect of tailwater channel slope on the drop length. The results apply to relatively high drops, mild tailwater slopes and transcritical approach flow. Further research is required to generalize the present experimental limitations.

Notation

D	=	drop number (-)
F md	=	vertical unit drop force (N/m)
F	=	Froude number (-)
H	=	drop height (m)
L d	=	drop length (m)
P	=	drop pressure (N/m2)
q	=	unit discharge (m2/s)
R	=	Reynolds number (-)
S o	=	bottom slope (-)
u	=	velocity (m/s)
y c	=	critical flow depth (m)
α	=	velocity ratio (-)

Subscripts

a	=	effective
b	=	below
c	=	critical
d	=	drop
e	=	exit
m	=	measured
n	=	nappe
o	=	approach flow
r	=	regression
t	=	semi-theoretical
w	=	tailwater