Analysis of main influencing factors of energy dissipation and flow diversion effects of rough-strip energy dissipators in curved spillways based on Plackett–Burman design

h_r

Relative height of R-SED

λ

Height ratio of R-SED

α

R-SED spacing

θ

R-SED angle

δ

R-SED thickness

R

Centerline radius of the bend

B

Bend width

β

Bend angle

i

Bottom slope of the bend

Average water depth of the inlet cross-section of the bend

h₁

Height of the R-SED at the concave bank

h₂

Height of the R-SED at the convex bank

M

Marking point of an enlarged view of the R-SED at the concave bank

N

Marking point of an enlarged view of the R-SED at the convex bank

L₁

Length of inlet straight section

L₂

Length of outlet straight section

h

Water depth at each measurement point

Q

Discharge flow rate of the spillway

C₀

Flow rate coefficient of the right triangular thin-walled weir

H

The head over the right triangular thin-walled weir

P₁

Height of the right triangular thin-walled weir

B₁

Width of the flow diversion canal at the upstream of the right triangular thin-walled weir

p_ij

Proportion of the ith evaluation object under the jth index

E_j

Entropy value of the jth index

W_j

Weight of the jth index

⁠,

Euclidean distance between each object and the ideal solution

Relative closeness of each object to the ideal solution

η

Energy dissipation rate of R-SEDs

E₁

Total mechanical energy per unit weight of water in the upstream flow cross-section

Z₁

Minimum elevation of the spillway bottom of the upstream flow cross-section

H₁

Average water depth of the upstream flow cross-section

v₁

Average flow velocity of the upstream flow cross-section

E₂

Total mechanical energy per unit weight of water in the downstream flow cross-section

Z₂

Minimum elevation of the spillway bottom of the downstream flow cross-section

H₂

Average water depth of the downstream flow cross-section

v₂

Average velocity of the downstream flow cross-section

φ

Kinetic energy correction factor

g

The acceleration of gravity

SCV

Coefficient of variation of superelevation

σ_i

Standard deviation of transverse water surface superelevation Δy for all calculated sections under the ith condition

μ_i

Average transverse water surface superelevation Δy of all the calculated sections under the ith condition

Δy_ij

The difference between the water level at the concave bank of the bend and the horizontal plane where the center of the water surface is located at the jth calculated cross-section under the ith condition

k

Superelevation coefficient

v_ij

Average velocity of the jth calculated section under the ith condition

B_i

Water surface width of the open channel based on the centerline water surface elevation under the ith condition

r_i

Centerline radius of the bend under the ith condition

Δy

Transverse water surface superelevation of the bend

W_(η)

Weight of the energy dissipation rate

W_(SCV)

Weight of the coefficient of variation of superelevation

Influenced by topographic and geological conditions, engineering characteristics, construction conditions and economic indicators (Seo & Shin 2018; Yang et al. 2019; Damarnegara et al. 2020), some spillways or natural river channels must have corners, thus forming bends. The water flowing through the bend is called bend flow, which is different from straight-section water flow. When water flows through a bend, uneven distribution of water flow (Zhang et al. 2015) and cross-sectional flow velocity (Pradhan et al. 2018) occurs on both banks of the bend due to centrifugal inertia forces. Thus, secondary flows are developed (Seyedashraf & Akhtari 2015). These unfavorable flow structures lead to sediment movement, riverbed evolution and river channel deformation (Olsen 2003).

The bend flow is one of the topics in the field of hydraulics research. In 1876, Thomson (1876) first proposed the problem of bend circulating flow through experimental research. Currently, various research methods of bend flow have been developed, mainly including model tests, numerical simulations and theoretical studies. The research on bend flow is mainly divided into the study of basic water flow characteristics and the study of engineering measures to improve the flow pattern of bend flow. The study of the basic characteristics of the bend flow is focused on the water depth distribution (Seyedashraf & Akhtari 2015; Zhang et al. 2015; Qin et al. 2016; Zhou et al. 2017; Maatooq & Hameed 2020), flow velocity distribution (Han et al. 2011; Seyedashraf & Akhtari 2015; Vaghefi et al. 2015; Qin et al. 2016; Zhou et al. 2017; Moncho-Esteve et al. 2018; Pradhan et al. 2018; Schreiner et al. 2018; Seo & Shin 2018; Hu et al. 2019; Kim et al. 2020; Yan et al. 2020) and secondary flow evolution (Booij 2003; Huai et al. 2012; Ramamurthy et al. 2013; Seyedashraf & Akhtari 2015; Engel & Rhoads 2016; Gu et al. 2016; Zhou et al. 2017; Moncho-Esteve et al. 2018; Schreiner et al. 2018; Seo & Shin 2018; Hu et al. 2019; Shaheed et al. 2021). Based on the understanding of the basic bend flow characteristics, scholars have improved the bend flow pattern using engineering measures such as permeable spurs (Yang et al. 2019), guide walls (Zhang et al. 2015), vanes (Ranjan et al. 2006; Han et al. 2011) and riprap (Martín-Vide et al. 2010). However, compared with the research on the basic characteristics of bend flow, the research on related engineering measures is still rare. Thus, in this paper, a simpler rough-strip energy dissipator (R-SED) was added to the bend of curved spillways. The simple shape and convenient construction of R-SEDs can effectively solve the adverse hydraulic phenomena in the curved spillway, which will facilitate the safe and stable operation of the curved spillway and the continuous downstream water supply (such as industrial and irrigation water).

The research on the R-SED was first conducted based on the hydraulic model test (the geometric scale of the model was 1:50) of the curved spillway of Project 635 Reservoir in Xinjiang, China, and the research was mainly based on the laboratory test. The actual operation of the spillway reveals that the water depth and flow velocity of the concave and convex banks of the spillway differed greatly at a discharge flow rate of 800 m³/s. The water flow in the bend was splashing and turbulent. In order to solve the above adverse water flow problem, the R-SED was arranged at the bottom of the bend, exhibiting good energy dissipation and flow diversion effects in the bend (Li 2016). Due to the different scaling ratios (geometric ratios) of different curved spillway test models and the special and different geological conditions of each project, laboratory-scale model dimensions were adopted for the research and analysis in this study.

The R-SED has both energy dissipation and flow diversion effects on the bend flow, and the influencing factors affecting these effects are generally divided into R-SED layout parameters (such as relative height of R-SED, height ratio of R-SED, R-SED spacing, R-SED angle and R-SED thickness) and curved spillway engineering parameters (such as centerline radius of the bend, bend width, bend angle and bottom slope of the bend). In the existing R-SED study, Li (2016) mainly analyzed the results of the R-SED in the hydraulic model test of the curved spillway of Project 635 Reservoir in Xinjiang, China, using the single-factor test method. Li (2016) only considered the influence of three parameters of the R-SED arrangement (i.e., height ratio of R-SED, R-SED angle and R-SED thickness) on the energy dissipation and flow diversion effect without including the influence of the engineering parameters of the curved spillway. Li et al. (2020) mainly analyzed the influence law of six factors (height ratio of R-SED, R-SED spacing, R-SED angle, centerline radius of the bend, bend width and discharge flow rate) on the effect of energy dissipation and flow diversion of R-SEDs using the orthogonal test method (six factors and three levels), and the energy dissipation rate and the water flux dispersion coefficient for evaluation energy dissipation and flow diversion effects were adopted, respectively. However, the influence considered influencing factors are still not comprehensive and the calculation formula of the water flux dispersion coefficient is complicated for practical application.

The Plackett–Burman design is a two-level partial factorial experimental design method, which is mainly used for experimental designs with a large number of factors and where the significance of these factors relative to the response variable is not determined (Abdel-Fattah et al. 2005; Ghanem et al. 2017). The method mainly analyzed two levels of each factor and determined the significance of each factor by comparing the difference between the two levels of each factor with the overall difference. Two levels (i.e., high (1) and low (−1)) were selected for each influencing factor in the experimental design, with the high level two times larger than the low level.

The R-SEDs were installed at the bottom of the spillway bend. Each R-SED extended continuously from the concave bank to the convex bank, close to the bottom of the bend. The roughness of the bend bottom increased after the R-SEDs were added, and the water flow between adjacent R-SEDs collided and swirled, thus strengthening the energy dissipation effect of R-SEDs. Without the R-SED, the water flow in the spillway bend showed a phenomenon of increased flow depth at the concave bank and decreased flow depth at the convex bank; the discharge water flow continuously scoured the sidewall of the bend at the concave bank, which was not conducive to the structural safety and stability. Thus, in order to better improve the uneven distribution of water flow between the concave and convex banks of the bend and improve the flow diversion effect of R-SEDs, the height (h₁) of the R-SED at the concave bank was designed to be larger than the height (h₂) at the convex bank, i.e., h₁ > h₂. Therefore, the R-SED had a trapezoidal longitudinal section.

The relative height of R-SED (h_r) is the ratio between the height of the R-SED at the concave bank (h₁) to the average water depth at the inlet cross-section of the bend (⁠⁠), i.e., h_r=h₁/⁠. The height ratio of R-SED (λ) is the ratio of the height of the R-SED at the concave bank (h₁) to the height at the convex bank (h₂), i.e., λ=h₁/h₂. R-SED spacing (α) is the angle corresponding to the arc length between two adjacent R-SED centerlines along the bend centerline direction. R-SED angle (θ) is the angle between the centerline of the R-SED and the direction perpendicular to the bend centerline. R-SED thickness (δ) is the horizontal distance between the upstream and downstream faces of the R-SED, which is defined as the ratio of the horizontal distance to the bend width (B) for nondimensionalization. The centerline radius of the bend (R) is the distance between the bend centerline and the center of the curvature of the bend. Bend width (B) is the horizontal distance between the two banks of the spillway. Bend angle (β) is the angle corresponding to the bend flow path along the bend centerline. Bottom slope of the bend (i) is the slope along the spillway bottom. The schematic diagram of these parameters is shown in Figure 3.

Based on Table 1, the Plackett–Burman experimental design was developed. The specific test program is shown in Table 2.

• (a)

  Water depth measurement

The water level measurement probe was used to measure the water depth, with an accuracy of 0.1 mm. A total of 19 water depth measurement cross-sections were arranged along the spillway model, i.e., 0#–18#. Each cross-section was arranged with five measurement points (i.e., A–E). Considering the viscous resistance of the sidewall to the water flow, two near-bank measurement points (A and E) were located at 1 cm from the sidewalls of the concave and convex banks, respectively.

• (b)

  Flow velocity measurement

The hourly average flow velocity was measured using a Pitot tube. A total of 10 measurement cross-sections (i.e., 0#, 2#, 4#, 6#, 8#, 10#, 12#, 14#, 16# and 18#) were selected as flow velocity measurement sections and three measurement points (i.e., A, C and E) were selected for each section. The vertical measurement position of each measurement point was located at 2 h/3 from the bottom (h is the water depth at the measurement point).

The R-SED has a dual effect of energy dissipation and flow diversion on the bend flow. Evaluation indices need to be constructed to assess the energy dissipation effect and flow diversion effects of the R-SED under twelve test scenarios.

Cross-sections 4# and 12# were selected as the upstream and downstream sections of the bend, respectively. The horizontal plane where the bottom elevation of cross-section 12# was located was taken as the reference plane. The energy dissipation rate in the 12 test scenarios was calculated using Equations (3)–(5). The calculation results of the energy dissipation rate are shown in Table 3.

Table 3

Plackett–Burman test results

Test Number .	hr (cm) .	λ (–) .	α (°) .	θ (°) .	δ (cm) .	R (cm) .	B (cm) .	β (°) .	i (–) .	η (–) .	SCV (–) .
1	0.4	2	(1/6) β	15	(1/100)B	300	100	60	0.02	0.535	0.188
2	0.8	1	(1/6) β	15	(1/50)B	300	100	30	0.04	0.557	0.473
3	0.4	2	(1/3) β	15	(1/50)B	150	50	30	0.04	0.318	0.357
4	0.4	1	(1/6) β	15	(1/100)B	150	50	30	0.02	0.153	0.516
5	0.4	1	(1/3) β	30	(1/50)B	150	100	60	0.02	0.461	0.413
6	0.4	1	(1/6) β	30	(1/50)B	300	50	60	0.04	0.662	0.318
7	0.4	2	(1/3) β	30	(1/100)B	300	100	30	0.04	0.335	0.140
8	0.8	2	(1/3) β	15	(1/50)B	300	50	60	0.02	0.656	0.211
9	0.8	2	(1/6) β	30	(1/100)B	150	50	60	0.04	0.623	0.301
10	0.8	1	(1/3) β	15	(1/100)B	150	100	60	0.04	0.643	0.605
11	0.8	2	(1/6) β	30	(1/50)B	150	100	30	0.02	0.419	0.380
12	0.8	1	(1/3) β	30	(1/100)B	300	50	30	0.02	0.402	0.306

Test Number .	hr (cm) .	λ (–) .	α (°) .	θ (°) .	δ (cm) .	R (cm) .	B (cm) .	β (°) .	i (–) .	η (–) .	SCV (–) .
1	0.4	2	(1/6) β	15	(1/100)B	300	100	60	0.02	0.535	0.188
2	0.8	1	(1/6) β	15	(1/50)B	300	100	30	0.04	0.557	0.473
3	0.4	2	(1/3) β	15	(1/50)B	150	50	30	0.04	0.318	0.357
4	0.4	1	(1/6) β	15	(1/100)B	150	50	30	0.02	0.153	0.516
5	0.4	1	(1/3) β	30	(1/50)B	150	100	60	0.02	0.461	0.413
6	0.4	1	(1/6) β	30	(1/50)B	300	50	60	0.04	0.662	0.318
7	0.4	2	(1/3) β	30	(1/100)B	300	100	30	0.04	0.335	0.140
8	0.8	2	(1/3) β	15	(1/50)B	300	50	60	0.02	0.656	0.211
9	0.8	2	(1/6) β	30	(1/100)B	150	50	60	0.04	0.623	0.301
10	0.8	1	(1/3) β	15	(1/100)B	150	100	60	0.04	0.643	0.605
11	0.8	2	(1/6) β	30	(1/50)B	150	100	30	0.02	0.419	0.380
12	0.8	1	(1/3) β	30	(1/100)B	300	50	30	0.02	0.402	0.306

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The addition of R-SEDs in the bend can induce the secondary distribution of water flow in the bend and the straight section downstream. To fully measure the flow diversion effect of R-SEDs, cross-section 4# at the bend and cross-section 18# at the straight section downstream were selected as the inlet and outlet cross-sections for the calculation of the coefficient of variation of superelevation. The calculation results are shown in Table 3.

• (a)

  Establishing a comprehensive performance evaluation system

Twelve groups of Plackett–Burman tests were used as feasibility study schemes, and the energy dissipation rate and the coefficient of variation of superelevation were used as target variables to construct the original matrix ⁠.

• (b)

  Determining the weight of each evaluation index using the entropy weight method

Minitab 21.1 software was used for statistical analysis of the data, including the energy dissipation rate, the coefficient of variation of superelevation and relative closeness.

Analysis of variance (ANOVA) was performed on the nine factors (predictor variables) using the energy dissipation rate as the response variable, as shown in Table 4. The significance level was selected as 0.05 and the factor was significant when P < 0.05. Table 4 shows that the probability P corresponding to the main effect was less than 0.05, rejecting the original hypothesis and indicating that the total effect of the regression was significant. Particularly, the relative height of R-SED and the bend angle were significant.

The coefficient of variation of superelevation was used as the response variable. The significance level was selected as 0.05. The ANOVA was performed on the nine factors, as shown in Table 5. Table 5 shows that the P-value corresponding to the main effect of the coefficient of variation of superelevation was less than 0.05, indicating that the original hypothesis was rejected and that the total effect of the regression was considered significant. The height ratio of R-SED and the centerline radius of the bend were significant.

The weights of each evaluation index were calculated using Equations (10)–(13) and the calculated parameters are shown in Table 6. The weight of the energy dissipation rate (W_(η)) and the coefficient of variation of superelevation (W_(SCV)) were 0.451 and 0.549, respectively, i.e., W_(SCV)> W_(η). This indicates that in these 12 Plackett–Burman tests, the coefficient of variation of superelevation carried more information than the energy dissipation rate and that the flow diversion effect of R-SEDs on the bend flow was larger than the energy dissipation effect.

Table 6

Calculation of the weight of each evaluation index using the entropy weight method

Test number .	η .	SCV .	η* .	SCV* .	.	.	.	.	.	.
1	0.535	0.188	0.752	0.896	0.097	0.137	0.943	0.930	0.451	0.549
2	0.557	0.473	0.795	0.283	0.103	0.043
3	0.318	0.357	0.324	0.532	0.042	0.081
4	0.153	0.516	0.000	0.191	0.000	0.029
5	0.461	0.413	0.606	0.412	0.078	0.063
6	0.662	0.318	1.000	0.616	0.129	0.094
7	0.335	0.140	0.358	1.000	0.046	0.153
8	0.656	0.211	0.989	0.847	0.128	0.129
9	0.623	0.301	0.924	0.653	0.120	0.100
10	0.643	0.605	0.964	0.000	0.125	0.000
11	0.419	0.380	0.524	0.483	0.068	0.074
12	0.402	0.306	0.490	0.643	0.068	0.098

Test number .	η .	SCV .	η* .	SCV* .	.	.	.	.	.	.
1	0.535	0.188	0.752	0.896	0.097	0.137	0.943	0.930	0.451	0.549
2	0.557	0.473	0.795	0.283	0.103	0.043
3	0.318	0.357	0.324	0.532	0.042	0.081
4	0.153	0.516	0.000	0.191	0.000	0.029
5	0.461	0.413	0.606	0.412	0.078	0.063
6	0.662	0.318	1.000	0.616	0.129	0.094
7	0.335	0.140	0.358	1.000	0.046	0.153
8	0.656	0.211	0.989	0.847	0.128	0.129
9	0.623	0.301	0.924	0.653	0.120	0.100
10	0.643	0.605	0.964	0.000	0.125	0.000
11	0.419	0.380	0.524	0.483	0.068	0.074
12	0.402	0.306	0.490	0.643	0.068	0.098

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The relative closeness was a comprehensive evaluation index of the energy dissipation and flow diversion effects of R-SEDs. Based on the calculation results in Table 6, the relative closeness of each scenario to the ideal solution was calculated using Equations (14)–(18). Then, the overall ranking of the energy dissipation and flow diversion effect of R-SEDs on the bend flow under each test scenario was obtained, as shown in Table 7. Table 7 shows that, among the 12 test scenarios, Scenario 8 had the largest relative closeness and the highest overall rating.

Table 7

Calculation of the overall ranking of each test scenario using the TOPSIS method

Test number .	η .	SCV .	1-SCV .	.	.	.	.	.	.	.	Rank .
1	0.535	0.188	0.812	0.306	0.354	0.138	0.194	0.035	0.140	0.802	2
2	0.557	0.473	0.527	0.319	0.229	0.144	0.126	0.084	0.109	0.564	6
3	0.318	0.357	0.643	0.182	0.280	0.082	0.154	0.103	0.073	0.415	11
4	0.153	0.516	0.484	0.088	0.211	0.040	0.116	0.159	0.021	0.118	12
5	0.461	0.413	0.587	0.264	0.256	0.119	0.140	0.083	0.092	0.524	9
6	0.662	0.318	0.682	0.379	0.297	0.171	0.163	0.043	0.148	0.776	4
7	0.335	0.140	0.860	0.192	0.375	0.087	0.206	0.084	0.121	0.589	5
8	0.656	0.211	0.789	0.375	0.344	0.169	0.189	0.017	0.160	0.904	1
9	0.623	0.301	0.699	0.356	0.305	0.161	0.167	0.040	0.141	0.780	3
10	0.643	0.605	0.395	0.368	0.172	0.166	0.094	0.111	0.127	0.532	8
11	0.419	0.380	0.620	0.240	0.270	0.108	0.148	0.085	0.087	0.507	10
12	0.402	0.306	0.694	0.230	0.302	0.104	0.166	0.078	0.096	0.553	7

Test number .	η .	SCV .	1-SCV .	.	.	.	.	.	.	.	Rank .
1	0.535	0.188	0.812	0.306	0.354	0.138	0.194	0.035	0.140	0.802	2
2	0.557	0.473	0.527	0.319	0.229	0.144	0.126	0.084	0.109	0.564	6
3	0.318	0.357	0.643	0.182	0.280	0.082	0.154	0.103	0.073	0.415	11
4	0.153	0.516	0.484	0.088	0.211	0.040	0.116	0.159	0.021	0.118	12
5	0.461	0.413	0.587	0.264	0.256	0.119	0.140	0.083	0.092	0.524	9
6	0.662	0.318	0.682	0.379	0.297	0.171	0.163	0.043	0.148	0.776	4
7	0.335	0.140	0.860	0.192	0.375	0.087	0.206	0.084	0.121	0.589	5
8	0.656	0.211	0.789	0.375	0.344	0.169	0.189	0.017	0.160	0.904	1
9	0.623	0.301	0.699	0.356	0.305	0.161	0.167	0.040	0.141	0.780	3
10	0.643	0.605	0.395	0.368	0.172	0.166	0.094	0.111	0.127	0.532	8
11	0.419	0.380	0.620	0.240	0.270	0.108	0.148	0.085	0.087	0.507	10
12	0.402	0.306	0.694	0.230	0.302	0.104	0.166	0.078	0.096	0.553	7

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The relative closeness was used as the response variable and a significance level of 0.05 was selected. An ANOVA was conducted to examine the importance of these nine influencing factors on the relative closeness, as shown in Table 8. Table 8 shows that the model was significant (P < 0.05). Particularly, the height ratio of R-SED, the centerline radius of the bend and the bend angle were significant for the relative closeness.

In this paper, based on 12 groups of Plackett–Burman tests, the R-SEDs set at the bottom of the curved spillway were studied using the entropy weight method and the TOPSIS method. Statistical analysis of the test results was performed using Minitab 21.1 software. The conclusions are drawn as follows:

• (1)

  The relative height of R-SED and the bend angle had significant effects on the energy dissipation effect of the R-SEDs. The height ratio and angle of R-SED had no significant effect on the energy dissipation effect of the R-SEDs. R-SED spacing was a negatively correlated factor, i.e., a smaller spacing of the R-SED can induce better the energy dissipation effects of the R-SEDs. The relative height of R-SED, R-SED thickness, the centerline radius of the bend, bend width, bend angle and bottom slope of the bend were positively correlated with the energy dissipation rate. The factors affecting the energy dissipation effect of the R-SED were in the following order: bend angle > relative height of R-SED > centerline radius of the bend > bottom slope of the bend > R-SED thickness > bend width > R-SED spacing > R-SED angle > height ratio of R-SED.

• (2)

  The height ratio of R-SED and the centerline radius of the bend had significant effects on the flow diversion effect of R-SEDs. The height ratio of R-SED, R-SED spacing, R-SED angle, the centerline radius of the bend and the bend angle had a negative correlation with the coefficient of variation of superelevation. The relative height of R-SED, R-SED thickness, the bend width and the bottom slope of the bend showed a positive correlation with the coefficient of variation of superelevation. The factors influencing the flow diversion effect of R-SEDs were in the following order: height ratio of R-SED > centerline radius of the bend > R-SED angle > relative height of R-SED > bend width > bottom slope of the bend > R-SED spacing > bend angle > R-SED thickness.

• (3)

  The weights of the energy dissipation rate and the coefficient of variation of superelevation were 0.451 and 0.549, respectively, indicating that the flow diversion effect of the R-SEDs was larger than their energy dissipation effect. The height ratio of R-SED, the centerline radius of the bend and the bend angle had significant effects on the overall energy dissipation and flow diversion effects of the R-SEDs. These three factors will be selected as the key factors for further steepest climbing tests and response surface test design. The factors influencing the overall energy dissipation and flow diversion effect of R-SEDs were in the following order: bend angle > centerline radius of the bend > height ratio of R-SED > relative height of R-SED > R-SED angle > R-SED thickness > bottom slope of the bend > R-SED spacing > bend width.

• (4)

  The R-SED performance achieved the highest overall rating with the following parameter selection: h_r = 0.8 cm, λ = 2, α = (1/3) β, θ = 15°, δ = (1/50) B, R = 300 cm, B = 50 cm, β = 60° and i = 0.02. This can be used as a reference for the R-SED design of similar curved spillway projects.

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 51769037) and the University Research Program Innovation Team Project of Xinjiang Uygur Autonomous Region (Grant No. XJEDU2017T004).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.