Critical Standard Normalized Rapid Suppression Hydraulic Index and Its Estimation

Abstract

As a standard to by which characterize the ability of ships to ascend rapidly, the rapid suppression hydraulic index is the main indicator of the rapid regulatory effect and an important component of channel engineering research. This study reviewed the current determination methods and forms of expression of the rapid suppression hydraulic index and revealed that the existing comprehensive indices of unit energy have obvious disadvantages, such as critical standards not being unified. Based on the balance principle of a ship’s effective propulsion and navigation resistance, according to the unified idea of the critical rapid suppression index, the dimensionless expression of the critical standard normalized rapid suppression hydraulic index was derived. The concepts of flow velocity fraction, slope fraction and their combined rapid suppression index are proposed, and the assessment method of rapid forming and suppressing with the critical standard of 1 is provided. Using an example, we verified the feasibility and convenience of the application of the critical standard normalized rapid suppression hydraulic index in judging rapid formation reaches, navigation obstruction areas and the comparison of the ability of several ship types to ascend rapidly. Based on various ship slope flow rapid suppression indices tested by real ship tests and theoretical analysis through the dimensions, main influencing factors and regression analysis, we introduced the dimensionless power load ratio and established an empirical formula for estimating the flow velocity fraction and slope fraction by the main parameters of the ship type, which can provide a reference for determining the rapid suppression hydraulic index when lacking detailed ship type data. The application scope of the estimated formula is discussed. The methods and data for rapids regulation are provided, which is more convenient for rapids remediation projects, will reduces rapids hindering navigation and then will develop the shipping industry of mountainous rivers to promote the sustainable development of mountainous economic construction.

1. Introduction

The development of shipping is essential for a country’s improved transportation, and channel regulation is one of the main measures designed to improve the channel level. Even for continuous channelized rivers, most of the dehydration sections at the end of the reservoir need regulation to meet the corresponding standards [1]. Mountainous rivers are widely distributed in China’s Yangtze River, Yellow River, Pearl River, Lancang River and other water systems. According to the statistical bulletin on the development of the transportation industry in 2018, the navigation mileage of inland channels in China was 127,100 km, and most of them are mountainous channels. Almost all navigable rivers in China have undergone channel regulation. The channel regulation of the Chuanjiang River is arguably the epitome of channel regulation in mountainous areas in China. After three stages from the 1950s to the 1990s, 123 rapids were regulated, and the channel grade was greatly improved [2,3]. At present, research and construction designed to further improve the channel grade are being carried out in combination with river ecology. From the 1960s to the beginning of the 21st century, more than 100 rapids were regulated, and the Lancang River channel has been raised to the fifth grade inland channel standard through a number of channel regulation projects [4]. Currently, the construction of the fourth-grade channel project is being stepped up, and 500 t-grade ships will be able to pass through the channel from Lancang port to the China Myanmar 244 boundary pillar in the near future.

Rapids are common in mountainous rivers and are one of the key objects of channel regulation [5]. For geological and geomorphological reasons, mountainous rivers often present many rapids related risks. Of all the beach hazards in mountain rivers, those relating to rapids account for about 50% of all the beach hazards in Lancang River, those relating to rapids account for 80% [6]. The rapids’ characteristics are steep slope and flow; the velocity of the river is large and is steep. Thus, the sum of the flow resistance and slope resistance formed by the river reaching the ship is greater than the effective propulsion when the ship enters the rapids; the ship cannot ascend the rapids by itself, causing rapids hindering navigation. (It should be noted that “rapids” here does not refer to the rapids with Froude number Fr > 1 in general hydraulics, but is defined for the navigation flow conditions. The rapids on the channel usually belong to the slow flow with Fr < 1.) The rapid suppression index is the flow condition that characterizes this critical state. The rapid suppression index usually refers to the maximum surface flow velocity and water surface slope of a route that can maintain the minimum allowable speed and ascend the rapid by itself under the condition of rated power and rated load of a ship [7], which is expressed as a series of paired combinations of velocity and slope. It is not only a standard for the regulation of channel rapids in mountainous areas but also a reference basis for the effective analysis of regulation projects [8,9]. Therefore, the rapid suppression index is an important component of channel engineering [10].

1.1. Determination Method of the Rapid Suppression Index

Summarizing the current research literature [11,12,13,14,15], the main methods to determine the rapid suppression index are the empirical analysis method [16,17], real ship test method [16,18,19], ship model test method [19,20,21], numerical ship model method [22,23] and propulsion resistance balance calculation method [16,18,24,25].

1.2. Expression of the Rapid Suppression Index

As the standard by which to determine whether the rapids are formed and whether they reach the effect of rapid suppression after regulation, the expression of the rapid suppression index mainly comprises the slope flow index, comprehensive index of unit energy and power index.

1.2.1. Slope Flow Index

Slope flow index is the abbreviation of slope (water surface slope) and flow velocity index. It is the basic expression of the rapid suppression index and conforms to the basic definition of the rapid suppression index. The slope flow index is the most widely used expression [2,6,12,13,26], which is mainly reflected in a series of discrete slope and flow velocity paired combinations, as shown in

Table 1. Rapid suppression index of a typical slope flow.

Number	Ship Type	Slope Flow Index									Water Discharge $\nabla$/m3	Length of the Ship L/m
①	Chuanjiang 881 kw Top 1000 t Fleet	J/‰	1	2	3						1162	92.5
		U/m·s−1	3.9	3.5	3.1
②	Lancang River 510 kw 300 t cargo ship	J/‰	1	2	3	4	5	6	7	8	405	46.2
		U/m·s−1	4.2	4.0	3.8	3.6	3.3	3.0	2.7	2.4
③	Lancang River 1440 kw 500 t cargo ship	J/‰	1	2	3	4	5	6	7	8	731.3	54.4

(U—rapid entrance surface flow velocity; J—rapid reach water surface slope). It is also often expressed in the form of a curve, which is also a beach capacity curve. As shown in Figure 1, the left area is the rapid suppression area, and the right area is the rapid formation area. Any set of indices can be found on the index curve, which has the ad-vantage of continuous and non-discontinuous.

1.2.2. Comprehensive Index of Unit Energy

The comprehensive index of unit energy is the velocity gradient, a comprehensive index, and the assessment formula of rapid suppression is

$$E=\frac{U^2}{2g}+\Delta z$$

(1)

where E is the comprehensive rapids suppression index expressed in unit energy, with length dimensions (L); U is the rapid entrance surface flow velocity; Δz is the rapid reach water drop; g is gravitational acceleration, taken as $g$ = 9.81 m·s⁻².

Through the real ship test in Qingtan of the Chuanjiang River and empirical evidence [2,9], the critical rapid suppression index E_C = 1.4 is determined. The rapid suppression index e can be obtained by substituting the specific flow velocity U and adding Δz at the rapid entrance to the above formula. When E < E_C, it indicates that the rapid is suppressed and the ship can ascend rapidly by itself; when E > E_C, it indicates that the rapid is formed and the ship cannot ascend rapidly by itself; when E = E_C, it is in a critical state.

Xu et al. [16] believe that the calculated length and size of Δz in Formula (1) will change with different rapids, which may lead to different indicators for the same ship type, and the matching between Δz and slope J is poor, so the comprehensive index of unit energy expressed is proposed, as shown in the following assessment formula:

$$E=\frac{U^2}{2g}+\lambda LJ$$

(2)

where L is the ship length, λ is the multiple of the ship length and λLJ represents the water surface drop within λ times the length of the ship. Thus, it overcomes the disadvantage of the uncertainty of Δz and poor matching with the slope flow index in Formula (1).

The method of determining the critical index E_C is as follow: taking the slope flow indices U and J as a series of data groups, E and λ of each ship type are obtained by fitting through Equation (2), and E is E_C at this time.

1.2.3. Power Index

When Cao [26] applied the conceptual model to study the ascending rapids ability of the Wujiang 409 motor barge, he proposed the power method. The basic idea is that when the effective power provided by the main engine of the ship is greater than the power consumed by the flow resistance, the rapid power is suppressed—that is,

$$P_E>P_W$$

(3)

where P_E is the ship’s effective power, which can be calculated by the relevant empirical formula; P_W is the power consumed by navigation resistance, which can be calculated by the following formula:

$$P_W=\rho g\nabla \left(U+V_a\right)\left(J+J^{\prime }\right)$$

(4)

where $\rho$ is flow density, taken as $\rho$ = 1 t·m⁻³; $\nabla$ is the displacement calculated by volume; V_a is the opposite speed; $J^{\prime }$ is the additional water surface slope.

Considering that there are few studies on the propulsion characteristics of ships in mountainous channels, the calculation of effective power is cumbersome, and the accuracy is difficult to grasp. In addition, Equation (4) only considers the slope resistance part and does not reflect the flow resistance, so whether its expression is appropriate needs further discussion, making it not widely used.

1.3. Research Progress the of the Rapid Suppression Index

Cao proposed the power method from the perspective of the comparison between the effective power of the ship and the power of the current on the beach, and directly calculated the hydraulic index of the beach [26]. Tong and Xu put forward a series of combinations of velocity and specific decrease in the rapids, combining them with the calculation of crash landing resistance, and established the navigable hydraulic index of Lancang River with convenient application [18]. Xu and Zeng derived a reasonable expression of the hydraulic index of the shoal by analyzing the force balance between the effective thrust and navigation resistance of ships, and found that the calculated length of the shoal mouth section should be 1.5 to 2.0 times the length according to the research results of the existing hydraulic index of the shoal on the Chuanjiang River and Lancang River [27]. Based on the measured data of 3000 t ships sailing on the flood rapids at the end of the reservoir, and combined with the theoretical re-search method of Chuanjiang water conservancy index, Zhang Peng determined the hydraulic index of these 3000 t ships sailing on the flood rapids in the Three Gorges reservoir area [28]. Duan and Xu analyzed the convenience and feasibility of combining the index of breakaway slope flow to form a comprehensive index of breakaway slope, pointed out that the comprehensive index of breakaway slope and its theoretical assessment were complicated and difficult to apply in practice, and put forward the comprehensive index and its simplified basic assessment [29]. Xu and Filer obtained the preliminary shoal water index representing the ship type by using the shoal formation flow method based on the measured results and research results of several breakwaters of the Lancang River. The preliminary shoal water index was obtained by curve fitting with the shoal identification expression, and a reasonable series of shoal water indices was determined for convenient application [30].

1.4. Disadvantages of Existing Expressions

The deficiency of the slope flow is also obvious. First, it is composed of a discrete series of data, which expresses a large number of indicators. Second, it is not convenient to determine whether the rapid is formed or suppressed. When the slope and flow velocity are between the two groups of indicators, they need to be determined by numerical interpolation or with the help of the index curve (Figure 1). In addition, due to the large and discrete data groups, there are certain restrictions on the presentation of the results.

The comprehensive index [31,32] of the unit energy has obvious advantages in application, but it still has some disadvantages in practical application and further research, mainly reflected in the following:

• The critical index E_C is not unified, and E_C is different for different ship types, which inconveniences the application. As shown in Figure 2, the critical standard E_C is not unified, with multiple critical boundaries being required for comparing multiple ship types, which is chaotic, and the comparison of the ability to ascend rapids is not intuitive.
• The correlation with the slope flow index is not strong, and the corresponding relationship is inconsistent. Equation (2) specifically reflects the E_C and λ indicators. Although λ corresponds to the J indicator of the slope flow indicator, E_C has no corresponding item of the slope flow indicator, which is slightly less reasonable.
• The index E_C expressed by unit energy has length dimensions. If it can be changed into a dimensionless index, it has more universal significance.

The calculation of power index effective power is cumbersome, and the accuracy is difficult to achieve. In addition, Equation (4) only considers the slope resistance and does not reflect the flow resistance.

In summary, for the above deficiencies, the research on the navigation hydraulic index (rapid suppression hydraulic index) of 500 t rapids is carried out in the fourth-class waterway from the 244 boundary tablet of the Lancang River to Lincang Port. The proposed rapid suppression index can be applied to engineering design, considering the complex boundary and flow conditions of the Lancang River channel and the numerous influencing factors of the navigation process [31,33,34], the still water speed test, velocity test, and water level test is carried out, and the normalized rapid suppression index expression is proposed to provide feasibility and convenience for judging the application of rapid-forming reach, navigation-obstructing area, and ship-type beach-up capability comparison [32,35]. According to the slope flow rapid suppression index of various ship types tested by real ship test and theoretical analysis, the dimensionless power–load ratio is introduced through dimension and main influencing factors and regression analysis, and the empirical formula for estimating the flow velocity fraction and slope fraction through the main parameters of the ship type is established, which can provide a reference for determining the rapid suppression hydraulic index when there is a lack of detailed ship type data.

2. Methods

2.1. Test

2.1.1. Still Water Speed Test

According to the Feasibility Study Report of Lancang River 244-Boundary Monument to Lancang Port Four-level Waterway Construction Project, the main vessels of Lancang River level 4 waterway engineering design are 500 t cargo ships and 500 t container ships, but there are no real vessels on Lancang River so they could not be selected. Although the power-to-load ratio of the cargo ship “Baoshou 21” and the container ship “Ruifeng 9” is close to the design of a representative ship type, only the data do not match the ship, and the main scale is quite different, so the navigation resistance is difficult to match. Because, at present, the accuracy of calculating the ship thrust through experience and theory is high, and the error of calculating the navigation resistance is large, it is most reasonable to meet the requirement of similarity of navigation resistance, that is, the size of the ship body is similar. The Dragonair dump transportation 01 and 02 ship is designed for the Lancang River level 4 new waterway engineering development, the existing 500 t Lancang river shipping information is collected through the analysis of Dragonair dump transportation 01 and 02 ship, and is closer to the design type, so we chose, Dragonair dump ships carrying 02 ship for testing (Figure 3). The main ship type parameters are shown in

Table 2. Main types of parameters of the test ship.

Basic Parameter	Number	Category	Load W (t)	Displacement ${\nabla }_0$(m3)	Length L (m)	Wide B (m)	Draft T (m)	Power PS0 (kw)	Power Load Ratio PS0/${\nabla }_0$(kW/t)
	Test 1	500 t Self-unloader	434.45	675.267	52.32	8.6	1.95	2 × 530	1.57
Other parameter	Molded depth (m)	Light load displacement$\nabla$(m3)	Waterline length LW (m)	Waterplane area (m2)		Design speed V0 (km/h)	Rated rotation n0 (r/min)	Gearbox reduction ratio	Propeller shaft cross type
	2.80	240.817	51.103	422.6		21.0	1200	3:1	MAU4
	Pitch ratio II/D	Propeller disc area ratio	Propeller blade number	Propeller diameter D (m)		Block coefficient δ	Mid-section area coefficient β	Waterplane coefficient CW	Prismatic coefficient CP
	0.7765	0.50	4	1.70		0.788	0.973	0.962	0.809

.

The still water speed and ‘Z’ -shaped test area were determined in the Jinghong Reservoir area. Through on-the-spot investigation, combining the straight-line length, river width, water depth of the water area and the convenience of loading and unloading, the test water area was determine to be in the reservoir area of the Xiangbi Mountain–Dagan River mouth, about 3 km downstream of Simao Port. Preliminary, 100% W₀, 75% W₀, 50% W₀ (W₀-rated load), and the other three load conditions, 50% n₀, 75% n₀, 90% n₀, 100% n₀ (n₀-rated speed) and the other four main engine speeds, and one upward and one downward heading condition, for no less than 24 voyages in the test.

The main test characteristics of calm water speed are track, speed, rudder angle, main engine speed and flow velocity, water level, wind speed and direction, water temperature, etc. The overall layout of the test is shown in Figure 4.

2.1.2. Rudder Angle and Engine Speed Test

Rudder angle and host speed were observed by video. A video camera was set up above the rudder angle indicator and the main engine speed dashboard vertically, and the network cable was connected to the laptop to monitor the whole process. The acquisition density set during the test was 25 frames/s. After the test was completed, image recognition technology was applied to digitize the rudder angle and the main engine speed. The accuracy of the rudder angle was 0.1°, and the accuracy of the main engine speed was 1 r/min. Figure 5 shows the change process of change of the main engine speed.

2.1.3. Track and Speed Test

One piece of RTK equipment (Figure 6) was set up on the central axis of the bow and stern of the ship, with a straight line spacing of 51.4 m. The X and Y coordinates of the bow and stern navigation process were measured in real time (2000 coordinate system, as below), and the allowable error of track positioning was ±1.5 mm on the map. After the track process was measured, the speed, drift angle and heading angle of the other side were calculated. During the test, one data is collected every 1 s, and the speed calculation time increment was also 1 s. Figure 7 shows the working conditions of the track and speed diagram.

2.1.4. Velocity and Water Level Test

The buoy method was used to measure the surface velocity and flow direction, and the RTK method was used to locate the measuring points. The buoy specifications used in each measurement are the same, and the layout range covers the track line of the section. The test section is generally arranged with three buoy flow directions, basically covering the track lines of all voyages.

We adopt the RTK measurement method for water level observation, and the elevation accuracy met the requirements of fourth-class leveling accuracy. At the beginning and end of RTK water level observation, compared with manual observation, the water level data collected dynamically by RTK removed gross errors such as elevation jump points caused by wind and waves, satellite loss of lock and data link interruption. During operation, real-time monitoring of PDOP values and RTK positioning status, record data were limited to RTK fixed solution. As the test water area is located in the reservoir area, the reservoir surface is almost horizontal, and the gradient is only 0.0041‰; it was only necessary to lay a water gauge at the head and tail of the test section.

2.1.5. Wind Speed, Wind Direction and Water Temperature Test

The wind speed and direction were measured by anemometer, and the water temperature was measured by thermometer. Through multiple observations during the test period, the wind speed, wind direction and water temperature did not change much, as shown in

Table 3. Test environment parameters.

Load Conditions	Test Date/Time	Water Temperature (°C)	Wind Velocity (m/s)	Wind Direction
100%W0	2021.10.30 p.m.	21.0	0.2	Due south
75%W0	2021.10.31 p.m.	20.5	0.4	Due south
50%W0	2021.11.01 a.m.	19.0	0.3	Southeast

.

2.2. The Theoretical Derivation of Critical Standard Normalized Rapid Suppression Index

The main engine power of the ship experiences many losses in deceleration devices, shafting, propeller propulsion and so on. The effective power actually propels the ship forward, which provides effective propulsion to overcome navigation resistance. According to the balance conditions of ship propulsion and resistance, the expression of the critical standard normalized rapid suppression index is deduced.

When a ship is ascending rapidly, the navigation resistance mainly includes flow resistance and slope resistance, where the calculation formula of flow resistance R_V is

$$R_V=C_D\rho A_s\frac{V_s^2}{2}$$

(5)

The calculation formula of the gradient resistance R_J is:

$$R_J={\alpha }_J\rho g\nabla J$$

(6)

where C_D is the comprehensive flow resistance coefficient considering the effects of friction resistance, viscous pressure resistance, and wave making resistance; A_s is the ship’s wet area; most inland river motor boats in China use $A_s=L_W\left(C_{1}T+\delta B\right)$; L_W is the ship’s waterline length; C₁ is the coefficient for motor boat C₁ = 1.8; δ is the block coefficient, defined as δ = $\nabla$/(B_WL_WT), where B_W, L_W, and T indicate the waterline width, waterline length, and draft corresponding to the displacement, respectively; $\nabla$, B_W = breadth B when designing draft; V_s is the ship relative speed of water around the ship; ${\alpha }_J$ is the correction coefficient considering the local increase of water surface slope when the ship ascends the rapids, usually taken as ${\alpha }_J=1.1~1.2$.

Assuming that the effective propulsion force of the ship ascending rapidly is F_e, when the ship keeps the minimum allowable speed and goes straight up at a constant speed, the propulsion and resistance are balanced, and the force balance equation is obtained:

$$F_e=R_V+R_J=C_D\rho A_s\frac{V_s^2}{2}+{\alpha }_J\rho g\nabla J$$

(7)

The above formula can explain the ship propulsion F_e in two parts. The first part is the corresponding propulsion required to overcome the flow resistance (Item 1 on the right), which can be called the corresponding flow propulsion, and the second part is the corresponding propulsion required to overcome the slope resistance (Item 2 on the right), which can be called the corresponding slope propulsion.

Divide each item of Equation (7) by C_DρgA_s, and let $E_C=\frac{F_e}{C_D\rho g{A}_s}$, $C_J^{\prime }=\frac{{\alpha }_J\nabla }{C_D{A}_s}$, which gives the following:

$$E_C=\frac{V_s^2}{2g}+C_J^{\prime }J$$

(8)

Considering

$$A_s=\left(C_{1}T+\delta B\right)L_W=\left(C_1\frac{T}{B}+\delta \right)B{L}_W$$

(9)

$$\nabla =\delta BT{L}_W$$

(10)

Hence:

$$C_J^{\prime }=\frac{{\alpha }_J\nabla }{C_D{A}_s}=\frac{{\alpha }_J}{C_D\left(\frac{C_{1}T}{\delta B}+1\right)}T=\lambda T$$

(11)

where $\lambda =\frac{{\alpha }_J}{C_D\left(\frac{C_{1}T}{\delta B}+1\right)}$. Considering that the physical concept of λTJ is not clear, we change it to λLJ, that is, the water surface drops in the range of λ times the length of the ship, and there is a linear relationship between the relative speed of water around the ship V_s and the flow velocity U. Therefore, Equation (8) becomes expression (2) of the existing unit energy index.

Considering that the main deficiency of Equation (2) is that the critical standard is not unified, the coefficient of the flow velocity term is not normalized, but the ship propulsion term is normalized, that is, dividing each item of Equation (7) by propulsion force F_e, and letting $C_U^{\prime }=\frac{C_D\rho A_s}{F_e}$, $C_J=\frac{{\alpha }_J\rho g\nabla }{F_e}$, is the following is obtained:

$$1=C_U^{\prime }\frac{V_s^2}{2}+C_{J}J$$

(12)

Since $C_U^{\prime }$ has an [L⁻²T²] dimension, the ship length L is introduced to make the coefficient dimensionless. Let $C_U=C_U^{\prime }gL$; then, the first item on the right of the above formula is $C_U^{\prime }\frac{V_s^2}{2}$ = $C_U^{\prime }gL\frac{V_s^2}{2gL}=C_U\frac{V_s^2}{2gL}$. Considering the linear relationship between the relative speed of water around the ship V_s and flow velocity U, the above formula becomes the dimensionless expression of the critical standard normalized rapid suppression index:

$$C_U\frac{U^2}{2gL}+C_{J}J=1$$

(13)

3. Results

3.1. Test Data

During the construction of the grade 4 channel from Lancang Port to the 244-boundary pillar on the Lancang River, the static water speed test of the 500 t cargo ship was carried out. The calculation parameters for reasonably determining the ship propulsion and navigation resistance were determined, and then, the rapid suppression indices of various ship types (see

Table 4. Main parameters of various ship types in Lancang River.

Ship Number	Load G/t	Water Discharge$\nabla$/m3	Length L/m	Wide B/m	Draft T/m	Power P/kW	Waterline Length LW/m	Block Coefficient δ	Remark
BT1	500	680	56	8.8	2.0	2 × 400	54.2	0.713	Design representative cargo ship
BT2	500	760	56	9.8	2.0	2 × 450	54.2	0.715	Design representative container ship
BT3	420	594.2	52.6	8.3	1.95	2 × 353	51.0	0.720	Actual cargo ship
BT4	420	678.8	53.4	8.6	1.9	2 × 426	51.2	0.812	Actual container ship
BT5	532	799	59.6	8.7	2.2	2 × 325	58.0	0.720	Long-term planning ship
BT6	434	675	52.32	8.6	1.95	2 × 530	51.1	0.788	Actual cargo ship (test ship)
BT7	320	445	46.2	7.6	1.75	2 × 255	44.0	0.760	300 t class actual cargo ship

) were proposed through a study on the calculation method of the ship propulsion and resistance balance [36]. The research results are listed in

Table 5. Slope flow beach index of each ship type.

Gradient J/‰	Beach Flow Velocity U/m·s−1
	BT1	BT2	BT3	BT4	BT5	BT6	BT7
0	4.94	4.85	4.82	4.63	4.25	4.96	4.41
1	4.75	4.68	4.64	4.48	3.99	4.83	4.24
2	4.55	4.48	4.45	4.31	3.70	4.69	4.06
3	4.33	4.26	4.24	4.12	3.38	4.53	3.86
4	4.09	4.03	4.01	3.92	3.02	4.37	3.63
5	3.83	3.77	3.76	3.70	2.62	4.19	3.39
6	3.55	3.49	3.49	3.47	2.15	3.99	3.11
7	3.25	3.19	3.20	3.20	1.60	3.78	2.81
8	2.93	2.88	2.88	2.93	1.00	3.55	2.47

.

3.2. The Determination of Critical Standard Normalized Rapid Suppression Hydraulic Index

The physical (13) concept of the above formula is very clear, which means that the ship propulsion is regarded as 1, and the first and second items are the proportion of corresponding current propulsion and corresponding slope propulsion in the total propulsion, respectively. To facilitate the description and distinguish it from previous studies and considering that all items in the above formula are dimensionless parameters, C_U and C_J are called the velocity fraction and slope fraction, respectively, which means the proportion or component of propulsion. Different ship types have different C_U and C_J values, which are obtained by the slope flow index through regression analysis.

Table 6. The velocity fraction, slope fraction and critical rapid suppression index of the representative ship type.

Ship Type	CU	CJ	L(m)	Critical Rapid Suppression Index XuC (Fitting Values)
				J = 1‰	2‰	3‰	4‰	5‰	6‰	7‰	8‰
①	101.07	155.93	92.5	1.0030	0.9941	1.0030
②	46.43	88.49	46.2	0.9920	0.9965	1.0051	1.0178	1.0003	0.9919	0.9928	1.0030
③	41.44	48.08	54.4	0.9999	0.9999	0.9999	0.9999	0.9999	1.0000	1.0000	1.0000

shows the regression analysis results of the ship types listed in Table 1. The critical rapid suppression index corresponding to each water surface slope index is very close to 1, with a high correlation.

Define the rapid suppression index X_u:

$$X_u=C_U\frac{U^2}{2gL}+C_{J}J$$

(14)

By substituting the actual U and J of the rapid reach into the above formula, the rapid suppression index X_u can be cancelled, and the assessment conditions for forming and suppressing rapids are:

$$\left\{\begin{array}{l}{X}_u<1,\hspace{1em}rapid suppression\\ X_u=1,critical state\\ X_u>1,rapids\end{array}\right.$$

(15)

Therefore, the rapid suppression critical standard of various ship types is unified as 1, which is no longer changed due to ship types, and which is also the largest feature of the expression of this index. In addition, the slope flow index also corresponds to it; that is, the flow velocity index corresponds to the velocity fraction, and the slope index corresponds to the slope fraction. To distinguish it from the current index expression, X_u is called the rapid suppression index.

4. Discussion

4.1. Application of Critical Standard Normalized Rapid Suppression Index

Taking the flow conditions of the unnamed rapids group of the Lancang River (including watermelon rapid, unnamed upper rapid, unnamed rapid, embroidered rapid and Mengsong Rapid) as an example, the application of the critical standard normalized rapids suppression index was analyzed. The route, cross-section flow velocity and water surface slope distribution data were derived from the river engineering model test [37].

4.1.1. Analysis and Assessment of Navigation Obstruction Rapids Reach

Since the discrete series of slope flow indicators are converted into two indicators, flow velocity fraction and slope fraction, and combined into a rapid suppression index through Equation (14), it is more intuitive and convenient to assess rapid formation and suppression and to display the results. A considerable amount of information can be seen in Figure 8. First, it clearly distinguishes the reach that forms rapids and obstructs navigation (X_umax > 1) and the reach that suppresses rapids and does not obstruct navigation (X_umax < 1). Second, the navigation obstruction situation in each water period can be analyzed. For example, a watermelon rapid is a flood rapid, a nameless upper rapid, and nameless rapids are perennial rapids; embroidery beach is not a rapid, and Mengsong Beach is a medium low-flow period rapid. It can also be seen that Mengsong Rapid has the most serious low-flow obstruction to navigation, and a watermelon rapid has less obstruction to navigation. Slope flow indicators have difficulty analyzing a lot of information in a simple graph.

4.1.2. Analysis and Assessment of Navigation Obstruction Area

After converting the discrete slope flow index into a rapid suppression index, the analysis and assessment of the navigation obstruction area become simple. Figure 9 shows the contour line of the water rapid suppression index in the unknown rapid reach, which indicates that, in the area with X_u > 1, the ship cannot ascend rapidly by itself, and the area is a navigation obstruction area, which is difficult to achieve by using the slope flow index. The navigation obstruction area of the nameless rapid almost fills the whole cross section, so it is difficult for ships to ascend rapidly. At the exit section of the unnamed rapid, the navigation obstruction area accounts for approximately half of the navigation width, but the width of the navigable area on the right side is insufficient, so it is difficult for the ship to ascend rapidly. Although there is a navigation obstruction area at the inlet section of the unnamed rapid, it has a small range. There is a non-navigation obstruction area of nearly one navigation width on the left, so ships can ascend rapidly by slow flow.

4.1.3. Comparative Analysis of Multiple Ship Types

After the critical standards are unified, the comparison of the ascending rapids ability among ship types becomes more convenient and intuitive. According to the comparison of the maximum rapid suppression index X_umax of each ship type throughout the journey presented in Figure 10, only in ship type ③ is X_umax slightly greater than 1 in the small reach of the Mengsong Rapids, but the whole beach section basically does not form rapids, and the ship has the strongest ability to ascend them. For ship type ②, watermelon rapids do not form a rapid, but nameless upper rapids, nameless rapids and Mengsong Rapids all form a beach. For the ship type ①, except for embroidered rapids, they are all rapids, and their rapid potential is significantly stronger than ship type ②, which is the ship type with the weakest ability to ascend rapids. Comparing Figure 2 and Figure 10, the critical standard normalized rapid suppression index and the unit energy rapid suppression index, the comparison between ship types is obviously more convenient, intuitive and simple.

By normalizing the rapid suppression index, the traditional research on the rapid suppression index is innovated; we can judge the beach elimination section and the navigation obstruction area more simply and intuitively and solve the problem that the rapid suppression index is limited by the ship type. The normalized rapid suppression index makes the ship drivers better grasp the actual situation of the channel during the navigation of the inner river beach section and timely carry out risk warnings to avoid accidents.

4.2. Estimate of the Flow Velocity Fraction and Water Surface Slope Fraction

The determination process of a more reasonable slope flow rapid suppression index is very complex. It requires not only detailed ship type and propeller data but also real ship or ship model tests to provide parameters or coefficients for the reasonable calculations of ship propulsion and resistance [25,38]. In the planning or preliminary research stage of waterway engineering, there are often only the main ship type parameters, such as the main engine power, deadweight tonnage and ship type scale, and there are basically no detailed propeller data. It is often difficult to obtain more accurate rapid suppression indicators. Therefore, this study makes a tentative investigation of whether it is possible to establish the law through the rapid suppression indices of various ship types, determine the relationship between the rapid suppression indices and the main parameters of each ship type and then estimate the rapid suppression indices to provide a design basis for waterway engineering.

4.2.1. Estimated Formula of the Flow Velocity Fraction and Slope Fraction

First, the flow velocity fraction C_U and slope fraction C_J are fitted according to Formula (13) through the ship type parameters in Table 4 and the slope flow rapid suppression index in Table 5. The fitting results are shown in

Table 7. Velocity fraction C_U and slope fraction C_J of each ship type.

Parameter	BT1	BT2	BT3	BT4	BT5	BT6	BT7
L/m	56	56	52.6	53.4	59.6	52.32	46.2
CU	44.67	46.17	44.02	48.27	64.53	41.28	45.95
CJ	81.12	81.03	80.06	74.84	123.55	60.35	85.04
fitting deviation/%	−0.86~0.60	−0.99~0.68	−0.94~0.70	−1.15~0.77	−0.43~0.56	−1.07~0.76	−1.50~0.96

. Then, the estimated formulas of the flow velocity fraction and slope fraction are obtained through dimensional and correlation analyses.

The key parameters affecting the rapid suppression index are the machine power P and the displacement $\nabla$, followed by the square coefficient δ and the ship width draft ratio B/T, and then, considering the flow density ρ and the gravity acceleration g, there are:

$$C_U=f_1\left(P,\nabla ,\delta ,\rho ,g,B/T\right)$$

(16)

$$C_J=f_2\left(P,\nabla ,\delta ,\rho ,g,B/T\right)$$

(17)

Taking $\nabla$, ρ and g as the basic physical quantities, according to the dimensional analysis, we can obtain:

$$C_U=f_1^{\prime }\left(\frac{P}{\rho g\nabla \sqrt{g{\nabla }^{1/3}}},\delta ,\frac{B}{T}\right)$$

(18)

$$C_J=f_2^{\prime }\left(\frac{P}{\rho g\nabla \sqrt{g{\nabla }^{1/3}}},\delta ,\frac{B}{T}\right)$$

(19)

Define the dimensionless power load ratio $\Gamma =\frac{P}{\rho g\nabla \sqrt{g{\nabla }^{1/3}}}$. According to the ship type parameters in Table 4 and the C_U and C_J values in Table 7, it is found that B/T basically has no effect on C_U through the regression analysis. The regression formula is:

$$C_U=18.6exp\left(0.03663{\Gamma }^{-0.815}{\delta }^{1.06}\right)$$

(20)

$$C_J=1.395{\Gamma }^{-0.994}{\delta }^{-0.136}{\left(\frac{B}{T}\right)}^{-0.201}$$

(21)

Considering that the indices of δ and B/T in C_J are small and have no obvious impact on C_J, Equation (21) can be roughly expressed as:

$$C_J=1.055{\Gamma }^{-1}$$

(22)

The C_U and C_J obtained by the slope flow index fitting shown in Table 7 are called the estimated values, and the calculated values of Equations (20)–(22) are called the simulated values. Figure 11 shows a comparison between them. The simulated value of C_U is in good agreement with the estimated value. In Formula (21), except for one measuring point, the simulated value of C_J has a high degree of coincidence with the estimated value. The simulated value of Equation (22) C_J is slightly less consistent, but it also has a high degree of simulation. Equations (20)–(22) are appropriate for the estimation of C_U and C_J and can be used to estimate the flow velocity fraction and slope fraction of the appropriate ship type.

After obtaining the flow velocity fraction C_U and slope fraction C_J from the main parameters of the ship type, we can not only assess the rapid formation, rapid suppression and navigation obstruction area of the river reach through Equations (14) and (15) but also conveniently calculate a series of slope flow indicators through Equation (13).

4.2.2. Application Scope of Estimate Formula

Limited by the data, the above research only selected the data of the Lancang River motor cargo ship, and the application scope was thus limited to a certain extent. The range of ship type parameters was $\nabla =$ 445~799 m³, L = 46.2~59.6 m, T = 1.75~2.0 m, P = 510~900 kW and δ = 0.713~0.812. According to the statistics of the real ship test and analysis data, the applicable conditions were as follows: ship Froude number F_r = $\frac{V_s}{\sqrt{gL}}$ 0.15~0.3 and block coefficient δ = 0.70~0.82.

Through the estimated formula put forward for the lack of a rapid suppression index of the beach section calculations, a quite accurate rapid suppression index provides a design basis for waterway engineering, saving project costs, shortening the project time and ensuring the safety of ship navigation on the beach.

5. Conclusions

In this paper, the review and analysis of the research status of the rapid suppression hydraulic index, compared with previous research, established that the traditional inland river rapid suppression hydraulic index is not uniform and has an inconvenient application, weak applicability and other shortcomings. However, based on the principle of effective ship thrust and sailing resistance balance, this study also put forward the critical normalized rapid suppression index, so that the critical index will not be affected by ship type. The feasibility and convenience of its application in relation to the beach forming area, navigable area and ship’s beaching ability were verified by experiments. The critical standard normalized beach dissipation hydraulic index and its prediction formula were studied. This study provides methods and data for rapids regulation, which is more convenient for rapids remediation projects, will reduce rapids hindering navigation and then will develops the shipping industry of mountainous rivers to promote the sustainable development of mountainous economic construction. This paper was found to be rich in content, real in terms of data and innovative and applicable. The main conclusions are as follows:

(1)

The deficiency of the existing unit energy index is that not only is the critical index E_C not unified but also different ship types have different E_C values, which makes it inconvenient to use. Additionally, the matching relationship of the slope flow index is not strong.

(2)

The dimensionless expression of the critical standard normalized rapid suppression hydraulic index was derived, which reflects the distribution of the corresponding flow propulsion and corresponding slope propulsion the total propulsion. In particular, the critical standard is unified as 1, which is no longer affected by the change in ship type, making its application convenient. The flow velocity fraction C_U and slope fraction C_J are fitted through the known slope flow index and then combined into the rapid suppression index X_u, which can easily assess the rapid-forming reach and navigation obstruction area.

(3)

The empirical formulas for estimating the flow velocity fraction and slope fraction were established by using the main parameters of each ship type. Then, a series of slope flow indices were calculated through the expression of the critical standard normalized rapid suppression hydraulic index, which provides a reference for determining the rapid suppression hydraulic index in the absence of detailed ship type data.

Although this paper has made a breakthrough in traditional research and put forward a new type of normalized rapid suppression index, which has contributed to the research of rapid suppression, it still has several shortcomings. Due to the limited data, this study only selected the data of the Lancang River mobile cargo ship, and the scope of application was thus limited. Next, in future real ship test or ship model tests, the principle of effective thrust and navigation resistance balance of the ship could be applied. Based on the idea of normalizing the critical beach dissipation index, the amount of test data could be further studied, and then, the navigation principle of other mountain channel ships could be studied and summarized, and the mathematical model calculations could be verified against each other. The ship motion response test and resistance measurement test could be added to analyze the three degrees of freedom (roll, pitch and heave) motion response and resistance during the process of the ship going up the beach, so as to understand this, which is more intuitive and conducive to the force analysis of the ship and reveals the relationship between the motion of the ship going up the beach and the force. This will lay the foundation for further study of the characteristics of ships running aground on the beach.