Analytical solution of dam-break flood wave propagation in a dry sloped channel with an irregular-shaped cross-section

Accepted Manuscript Analytical solution of dam-break flood wave propagation in a dry sloped channel with an irregular-shaped cross-section Bo Wang, Yunliang Chen, Chao Wu, Yong Peng, Xiao Ma, Jiajun Song PII: DOI: Reference:

S1570-6443(16)30039-9 http://dx.doi.org/10.1016/j.jher.2016.11.003 JHER 382

To appear in:

Journal of Hydro-environment Research

Received Date: Revised Date: Accepted Date:

14 February 2016 2 November 2016 7 November 2016

Please cite this article as: B. Wang, Y. Chen, C. Wu, Y. Peng, X. Ma, J. Song, Analytical solution of dam-break flood wave propagation in a dry sloped channel with an irregular-shaped cross-section, Journal of Hydroenvironment Research (2016), doi: http://dx.doi.org/10.1016/j.jher.2016.11.003

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Analytical solution of dam-break flood wave propagation in a dry sloped channel with an irregular-shaped cross-section Bo Wanga, Yunliang Chenb, Chao Wu c, Yong Pengd*, Xiao Mae and Jiajun Songf a

Associate Professor, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan

University, Chengdu, Sichuan 610065, China. E-mail: [email protected] b

Associate Professor, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan

University, Chengdu, Sichuan 610065, China. E-mail: [email protected] c

Professor, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University,

Chengdu, Sichuan 610065, China. E-mail: [email protected] d

Associate Professor, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan

University, Chengdu, 610065, China. E-mail: [email protected] (Corresponding author) e

Master's Candidate, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan

University, Chengdu, Sichuan 610065, China. E-mail: [email protected] f

Master's Candidate, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan

University, Chengdu, Sichuan 610065, China. E-mail: [email protected] Abstract: Cross-sectional channel shape is a primary factor influencing dam-break floods. However, it is difficult to analytically understand the impact of the cross-sectional shape on flood wave propagation because most of the existing analytical solutions are only applicable to channels with specific cross-sections (e.g., rectangular, parabolic and triangular cross-sections). Here, a polyline cross-section representing a realistic scenario and derivation method were suggested for analyzing dam-break floods down a dry sloping channel. With the proposed model, the flow depth, average velocity and discharge profiles after an instantaneous dam break can be presented as three dimensionless curves. The effect of bed slope on the dam-break wave was evaluated. Both the flood acceleration on a downward-sloping channel and the flood retardation on an upward-sloping channel are illustrated clearly with the analytical solution in an example application. The effect of the bed

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slope on wave propagation becomes more prominent over time. The flow depth, average velocity and discharge at the dam site remain constant for a horizontal bed and vary with time for a sloping bed. Three-dimensional numerical simulations of the examples were performed using a large eddy simulation (LES) model along with the Volume of Fluid (VOF) approach for surface tracking. A comparison between the analytical and numerical solutions shows that the analytical model can provide the main features of dam-break floods and therefore can be used for the rapid prediction of dam-break flows. The model limitations are also presented. Author Keywords: Dam break; Flood propagation; Analytical solution; Irregular shape; Sloped channel; Numerical simulation. 1. Introduction Dam-break floods may cause major damage to lives and property in the path of the flood wave. A dam-break flood is strongly influenced by many factors (Katopodes and Strelkoff 1978; Sakkas and Strelkoff 1973; Su and Barnes 1970), among which the cross-sectional channel shape is the primary factor. Thus, analytical understanding of the relationship between the hydraulic characteristics and the cross-sectional shape of dam-break flow is of great importance. Although dam failures have motivated a large number of theoretical studies on the propagation of flood waves resulting from dam breaches, much of the existing analytical investigations were conducted using rectangular channels, including the well-known analytical solutions proposed by Ritter (1892), Dressler (1952), Whitham (1955) and Stoker (1957). Ritter (1892) derived an analytical solution approximately 120 years ago in a smooth channel with a rectangular cross-section. Dressler (1952) proposed a first-order correction for resistance using a perturbation procedure, and Whitham (1955) obtained an alternative solution by using a technique that was similar to the Pohlhausen method of boundary-layer theory. Stoker (1957) extended Ritter’s approach for initial water depths downstream the dam and proposed a theoretical solution consisting of three interrelated equations and three unknown variables for an infinitely long prismatic rectangular channel on a

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horizontal and frictionless channel bed. Subsequent achievements include the solutions by Hunt (1982, 1983, 1984), Mangeney et al. (2000), Fernandez-Feria (2006), Hogg (2006), Ancey et al. (2008), and Chanson (2009). Based on the solutions for a rectangular channel, analytical understanding of the relationship between the cross-sectional shape and the hydrodynamics of dam-break flow remain limited. Some investigators attempted to develop a dam-break wave model for channels with other cross-sectional shapes. In the analyses made by Sakkas and Strelkoff (1973), Thirriot (1973), and Lin et al. (1980), the cross-sectional area of flow is defined as A = khn+1 (or the top width of the flow as B = chn), where h is the flow depth, k and c are constants in the cross-section boundary equation, and n is a power-law index. This method is applicable only for rectangular, parabolic and triangular cross-sections in which n = 0, 0.5, and 1.0, respectively. Obviously, the aforementioned analytical solutions are only applicable to channels with a specific cross-section. Considering the complexity of the cross-sectional shape of natural rivers, herein, a polyline cross-section representing a realistic scenario is proposed and two shape parameters and one characteristic parameter are introduced into the polyline cross-section. Subsequently, the original integral equation of the dam-break flood is transformed into a model equation that can be treated as a purely mathematical problem by a mathematical transformation relating the flow depth to the shape parameters and the characteristic parameter. This approach was adopted by Wu et al. (1993) to develop a model for calculating the hydraulic characteristics of dam-break waves on horizontal channels and was later employed in solving the dam-break shock wave in a trapezoidal channel with an initial depth downstream of the dam (Wu et al., 1999). Although applying the models developed by Wu et al. (1993, 1999) into a rectangular channel with a dry and wet bed downstream of the dam yields the well-known solutions of both Ritter (1892) and Stoker (1957), the model applicability remains to be examined for general channel cross-sections and its limitations are yet unidentified. Moreover, the dam-break models established by Wu et al. (1993,

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1999) are applicable for the flat slope and the effect of bed slope cannot be considered by these model. Studies (Lauber and Hager 1998; Ancey et al. 2008) indicate that bed slope has great effect on propagation of dam-break wave. Both of the model experiment by Lauber and Hager (1998) and the theoretical analysis by Ancey et al. (2008) are carried out basing on vertical two-dimensional cases. Obviously, these results cannot be used for the irregular channel discussed in present paper. The goals of this investigation are to develop an analytical solution for dam-break floods in a sloped channel with irregular shapes, to evaluate the effect of bed slope on dam-break waves, to validate the proposed model against 3D numerical simulations, and to unravel its limitations. 2. Basic Equations and Solution for a Dry Sloping Channel The propagation of a dam-break flood resulting from a sudden release of a mass fluid in a channel is governed by the unsteady shallow water equations or a form of the Saint-Venant equations. For an infinitely long prismatic channel, the one-dimensional equations are as follows (Chow 1959): ∂h ∂h A ∂u +u + =0 ∂t ∂x B ∂x

(1a)

∂u ∂u ∂h +u + g = g ( S0 − S f ) ∂t ∂x ∂x

(1b)

where h is the flow depth, x is the distance, t is the time, g is the acceleration due to gravity, u is the average velocity of flow, B is the water surface width, A is the cross-sectional area, S0 is the bed slope, and Sf is the friction slope. The effects of frictions on the dam-break flow are neglected in this study, i.e., an ideal (frictionless) fluid is considered, because the Saint-Venant equations cannot be solved analytically usually due to the nonlinear terms (e.g., the friction slope) (Chanson 2009). Applying the characteristic method to the system of partial differential equations composed of Eqs. (1a)–(1b) without the friction slope, yields a characteristic system of equations: h d  B   u + ∫0 g dh  = gS0 dt  A 

along

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dx A =u+ g dt B

(2a)

h d  B   u − ∫0 g dh  = gS0 dt  A 

along

dx A =u− g dt B

(2b)

Along the forward characteristics, there is h hup t  B   B   u + ∫0 g dh  −  uup + ∫0 g dh  = ∫0 gS0dt A   A  

(3)

where the subscript "up" represents the undisturbed water upstream and hup and uup are the initial flow depth and velocity in the undisturbed reservoir upstream of the dam, respectively. The condition uup = 0 is usually used in the analysis of dam-break flood problems. Substituting

uup = 0 in Eq. (3) yields u=∫

hup

0

g

h t B B dh − ∫ g dh + ∫ gS0 dt 0 0 A A

(4)

Substituting Eq. (4) into Eq. (2b) yields

∫

h

0

hup B A dh + =∫ 0 A B

 1 dx  t B dh −  − ∫ gS0 dt  0 A  g dt 

(5)

Eq. (5) is the characteristic transform of the Saint-Venant equations, which is applicable for a prismatic channel with arbitrary cross-sections. 2.1. Cross-sectional Subdivision

The analytical solution of the governing Eq. (5) is associated with the channel cross-section. To separate the shape parameters of the cross-section from the integral function of Eq. (5), a cross-sectional description is proposed in which an arbitrary cross-section is represented by a polyline cross-section, as shown in Fig. 1. M is the slope coefficient, and d is the vertical distance from the bottom to the slope break-point. The subscripts l and r are used to denote the left and right slopes, respectively, and i and j denote the ith and jth broken line, respectively. b is the bottom width. Assuming the water surface is bounded by the ith broken line on the left and the jth broken line on the right, the cross-section is divided into N (= I+J−1) subsections by the horizontal lines (dashed lines) that pass through the break points. The water surface width B and cross-sectional area A

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summed up as follows: I −1

J −1

i =1

j =1

B = b + ∑ ( M li − M li +1 )d li + ∑ ( M rj − M rj +1 )d rj + ( M lI + M rJ ) h J −1  1  I −1 A = Bh −  ∑ ( M li − M li +1 )d li2 + ∑ ( M rj − M rj +1 )d rj2 + ( M lI + M rJ ) h 2  2  i =1 j =1 

(6)

(7)

Two combinative parameters of the cross-section are introduced as follows:

α=

1 M lI + M rJ

β 2 = 1+

I −1 J −1   b M M d + − + ( ) ( M rj − M rj+1 ) d rj  ∑  ∑ li li +1 li i =1 j =1  

J −1  I −1 1 2 2  ∑ ( M li − M li +1 ) d li + ∑ ( M rj − M rj +1 ) d rj  α ( M lI + M rJ )  i =1 j =1  2

(8)

(9)

It should be noted that α is not a number, but has dimensions of a length. All the other parameters are instead non-dimensional. Then, expressions for B and A can be rewritten as B = (M lI + M rJ )(h + α ) A=

[

1 (M lI + M rJ ) (h + α )2 − (αβ )2 2

(10)

]

(11)

2.2. Mathematical Transformation

The integral parts of Eq. (5) are distinctly related to the combinative parameters, i.e., α and β. To eliminate the connection between the integral parts and the combinative parameters, a transformation is introduced as follows:

h + α = αβ (2W 2 + 1)

(12)

dh = 4αβ WdW

(13)

where W is a characteristic parameter of the flow depth. 2.3. Model Development

Substituting Eq. (12) into Eqs. (10)–(11) yields B = (M lI + M rJ )αβ (2W 2 + 1)

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(14)

A = 2(M lI + M rJ )(αβ ) W 2 (W 2 + 1) 2

(15)

Substituting Eqs. (13)–(15) into Eq. (5) establishes the model for a dam-break flood as W = KWup

G (Wup ) + C − x*

 2W 2 + 1  G (W ) +  2   W +1 

−1 2

(16)

where K and C are two dam-break characteristic parameters, x* is an independent variable, and G(Wup) and G(W) are two functions of Wup and W, respectively. These variables are defined as follows: α up β up αβ

K=

C=

x* =

Wup

Wup

1 2α up β up

(17)

α up ( β up −1)

∫α ( β

−1)

B dh A

 1 dx  t 1 − ∫ gS0 dt   0 2α up β up  g dt 

2 G (Wup ) = Wup

G (W ) =

2 W

(18)

(19)

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∫

Wup

0

 2W 2 + 1   2  dW  W +1 

(20)

12

∫

W

0

 2W 2 + 1   2  dW  W +1 

(21)

Eqs. (20)–(21) involve elliptic integrals and cannot be transformed into finite forms, except when Wup → 0 (W → 0) and Wup → ∞ (W → ∞). For convenience sake, when W has any value between 0 and ∞, the integrand is evaluated by expansion in the power series as follows: 1/2

 2W 2 + 1   2   W +1 

1/2

 1  1  = 2 1 −  2    2  W + 1 

2 3   1 1  1 1  1 1  = 2 1 − 2  2  − 5  2  − 7  2  −
  2  W + 1  2  W + 1  2  W + 1  

(22)

Substituting the first four terms into Eqs. (20)–(21) and integrating them, G(Wup) and G(W) can be approximated as

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2  275 arctan W 19  1  1  1   up  Gappr (Wup ) = 2 2 1 − 10 − 10  2 −    2 2  Wup + 1  29  Wup2 + 1   Wup   

 275 arctan W 19  1  1  1  2  Gappr (W ) = 2 2 1 − 10 − 10  2 −    2 W 2  W + 1  29  W 2 + 1   

(23)

(24)

To evaluate the error between Gappr(Wup) and G(Wup), two limits of both Gappr(Wup) and G(Wup), under the conditions Wup → 0 and Wup → ∞ are calculated as follows: lim G (Wup ) =

Wup → 0

lim Gappr (Wup ) =

Wup → 0

2 Wup

∫

Wup

0

2 Wup

∫

Wup

0

dW = 2

 275 19 1  dW = 2 2  1 − 10 − 10 − 9  2 2 2  

lim G (Wup ) =

Wup →∞

2 Wup

∫

Wup

0

2 dW = 2 2

lim Gappr (Wup ) = 2 2

Wup →∞

(25)

(26)

(27)

(28)

The relative error between Gappr(Wup) and G(Wup) is quantified to 0.54% when Wup approaches zero; and the value of Gappr(Wup) is equal to that of G(Wup) under Wup → ∞. The maximum relative error between Gappr(Wup) and G(Wup) is 0.54%. It is implied that the approximation of Gappr(Wup) for G(Wup) is reasonable and has a high degree. Of course, if more terms in the series of Eq. (22) are retained in Eq. (23), a more precise solution can be obtained. The same situation occurs for the relative error between Gappr(W) and G(W). Substituting Gappr(Wup) and Gappr(W) for G(Wu) and G(W) in Eq. (16), respectively, results as W = KWup

Gappr (Wup ) + C − x *

 2W 2 + 1  Gappr (W ) +  2   W +1 

−1 2

Eq. (29), as an approximation of Eq. (16), is used in the subsequent calculation. After rearrangement, Eq. (19) can be written in the following form:

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(29)

t dx = x*Wup 2 gαup β up + ∫ gS0dt 0 dt

(30)

Substituting Eqs. (14, 15 and 30) into Eq. (2b) yields the flow velocity as 12

*

u = x Wup

 W 2 +1  2 gαup β up + 2 gαβ W   2  2W + 1 

t

+ ∫ gS0dt 0

(31)

Combining Eqs. (15 and 31), the flow discharge can be obtained: Q = 2 ( M lI + M rJ )(αβ ) W 2 (W 2 + 1) 2

12  *  t  W 2 +1  gS dt •  x Wup 2 gαup β up + 2 gαβ W  +  2 ∫0 0    2W + 1  

(32)

3. Solution Procedure

A possible five-step strategy, primarily consisting of two parts, for the application of the present model is shown as follows: Part I: Determination of subsections and values of parameters, i.e., α, β, K, C in every subsection and x* corresponding to dli and drj. First, the combinative parameters of the channel cross-section, α and β, and the dam-break characteristic parameter K in all subsections are calculated from Eqs. (8, 9 and 17), respectively. Note that the values of I and J in Eqs. (6)–(9) are not fixed, as they depend on the subsection where the water surface is located. Using Channel I as an example, in the first subsection, α= 0.080 and β= 1.000 under the conditions of I = 1 and J = 1 are obtained; and in the second subsection, the conditions are I = 1 and J = 2. Note that the values of αup and βup are equal to the values of α and β, respectively, in the top subsection. Second, substituting the vertical distance from the bottom to the break points, dli and drj, into Eq. (12), the characteristic parameter of flow depth W is obtained in all subsections. Note that W has two values under a given dli or drj because two different sets of α and β from the adjacent subsections are used in Eq. (12). The function Gappr(W) is calculated from Eq. (24). Note that the values of Wup and Gappr(Wup) are equal to the values of W and Gappr(W) in the top subsection.

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Third, in the top subsection, the dam-break characteristic parameter C = 0 is obtained from Eq. (18) and the independent variable x* is calculated with Eq. (29) under h = hup. The value of x* in the top subsection is the minimum of x*. The maximum of x* is calculated by Eq. (29) under h = 0. Note that the value of x*, corresponding to a given dli or drj, is obtained from Eq. (29) by using C in the upper adjacent subsection. The value of C in other subsections is obtained from Eq. (29) by using x* corresponding to the upper distance from the bottom to the break points. Part II: Computation of hydraulic characteristics including the water depth h, flow velocity u and discharge Q. First, according to a given x*, which may be selected at random from the range of x* in Part I or be obtained from Eq. (19) under known x and t, the subsection that the given x* belongs to is identified. The corresponding values of α, β, K and C are determined. Second, the value of W is obtained from Eq. (29) by iterative computations. Then, the flow depth h, flow velocity u and discharge Q are predicted from Eqs. (12, 31 and 32), respectively. A flow chart representing the solution procedure is reported in Fig. 2. 4. Discussion 4.1. General remarks

To demonstrate the model application, two 10 m-long prismatic channels with different cross-sections (as shown in Fig. 3) were used. The initial upstream flow depths of 0.35 m and 0.40 m were specified for Channels I and II, respectively. Three bottom slopes, i.e., S0 = −0.02, 0, and 0.02, were considered for Channel I, and S0 = −0.01, 0, and 0.01 were considered for Channel II. The cross-sections of both channels were divided into six subsections. The values of α, β, K, C and x* of both Channels I and II are shown in Tables 1 and 2. Dimensionless variables were introduced into the analysis of the similarity of the water depth profile of dam-break floods on a horizontal, frictionless channel with a trapezoidal cross-section (Wu et al., 1999; Chen et al., 2011). LaRocque et al. (2013a) found that the upstream velocity profiles in the

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reservoir are self-similar using dimensionless variables. In the present work, the flow depth, average velocity and discharge were non-dimensionalized by the following scale: h hup

h* =

u ghup

u* =

Q* =

Q Aup ghup

(33a)

(33b)

(33c)

Ritter (1892) has presented a theory to the dam-break wave in a wide, smooth and horizontal channel with a dry tailwater portion and a reservoir extending infinitely upstream. Ritter's solution reads h* =

2 1 ( 2 − x* ) 9

(34a)

2 (1 + x* ) 3

(34b)

u* =

Q* =

2 2 (1 + x* )( 2 − x* ) 27

(34c)

As shown in Fig. 4, the dimensionless flow depth, average velocity and discharge are plotted against the independent variable x*. Different cross-sections of a channel have their own h*–x*, u*–x* and Q*–x* curves. These three curves for irregular channels move right compared with Ritter's solutions. For a prismatic channel with a given cross-section, the flow depth profile is self-similar by introducing the compound variable x*, in which the effects of distance, time, and bed slope on the dam-break flood wave are considered. Ritter's solution provides −1 and 2 for the values of x* corresponding to the leading edge of the negative wave and the wave front, respectively. The range of x* becomes larger for both Channels I and II. It is interesting that h* is equal to 0.554 and 0.562 for Channels I and II, respectively, when x* = 0. The values of h* are in the range of 4/9 and 16/25, and the borders correspond to a rectangular and triangular channel, respectively (Wu et al., 1999). The dimensionless discharge reaches the maximum when x* = 0. The description of the dam-break

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hydrograph is greatly simplified. The same situation occurs for the average velocity and discharge profiles; that is, the flow depth, average velocity and discharge of a dam-break flood can be represented by only three dimensionless curves. The flow depth, average velocity and discharge profiles along the length of the channel are analyzed in detail only for Channel I because of the similar profiles in Channel II. 4.2. Flow Depth, Average Velocity, Discharge

The flow depth, average velocity, discharge profiles along the length of the channel at t = 0.5 s, 1.0 s, 1.5 s and 2.0 s for Channel I are shown in Fig. 5. The top profiles at a given time correspond to the downward slope (S0 = 0.02), the bottom profiles are related to the upward slope (S0 = −0.02), and the middle profiles are related to the horizontal channel. The effect of the bed slope on the flow depth becomes more prominent as time increases. It is interesting that the flow depth, average velocity, discharge at the dam site (x = 0) for different times is fixed (h* = 0.554, u* = 0.624, and Q* = 0.481) for the horizontal channel (S0 = 0) and varies with time for the sloping channel (S0 ≠ 0). The flow depth at the dam site is larger than 0.554 for the downward slope and is smaller than 0.554 for the upward slope. Fig. 5(b) implies that the dam-break flood is promoted on the downward-sloping channel and is retarded on the upward-sloping channel. The velocity profiles become mild over time. The average velocity at the dam site is greater than 0.624 for the downward slope and less than 0.624 for the upward slope. It can be seen from Fig. 5(c) that discharge on the downward-sloping channel is increased due to the promotion of a dam-break flood, whereas the discharge on the upward-sloping channel is decreased due to the flood retardation. The discharge at the dam site is greater than 0.481 for the downward slope and is less than 0.481 for the upward slope. Besides, Fig. 5(c) shows that the location of the maximum discharge along the downward-sloping channel (S0 > 0) is on the upstream side of the dam and is farther from the dam with increasing time. The opposite case occurs for the upward-sloping channel (S0 < 0). The maximum discharge along the horizontal channel (S0 = 0) occurs exactly at the dam site.

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Moreover, the profiles of the flow depth, average velocity and discharge at different times will be compared with numerical simulations to examine the applicability of the present model in the next section. 5. Comparison between the Analytical Solution and Numerical Simulation

Commercial computational fluid dynamics packages, i.e., Fluent (Fluent Inc., 2006) and FLOW-3D (Flow Science Inc., 2007), have been widely used to simulate dam-break flow (Biscarini et al., 2010; Wang et al., 2010; LaRocque et al., 2013a, 2013b; Kocaman and Cagatay, 2015). Kikkert et al. (2015) simulated the dam break flow by standard k – ε model (SKE) and large eddy simulation (LES), then compared the simulated results with corresponding experimental data and found that LES produced the most accurate prediction for the dam break flow while prediction by SKE was poor. LaRocque et al. (2013a) studied the effect of turbulence modeling for dam-break flow in detail. In their paper, the performance of LES model, Reynolds-averaged Navier–Stokes modeling with the k − ε model and by turning k − ε model off has been compared. The quantitative comparisons have been carried out between experimental data and above three models. For example: relative root mean square errors of flow velocities in the upstream and downstream of the dam for above three models have been shown in the paper. All in all, they showed that the predictions by LES agreed well with the experimental data, but both of k − ε and no-turbulence models obviously underestimated the flow velocity. Herein, the LES turbulence model was adopted in Fluent (Fluent, 2006) to simulate the dam-break flow. The VOF method developed by Hirt and Nichols (1981) was used to track the free surface. Gambit, a mesh generation package, was used to create six sets of grids according to the dimensions of the six examples involving two channels (three bed slopes for each channel). The initial condition and the boundaries were similar to those adopted by LaRocque et al. (2013a, 2013b). The computational domain, having an upstream reservoir to a specified level and a dry floodable area, was initialized. The bank slopes and bottom were specified as walls and the top and downstream boundaries as pressure outlets. To reduce the grid size to save computational time, the grid sizes of

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both Channels I and II were uniform in both the longitudinal and vertical directions (∆x = 0.1 m and ∆z = 0.0052 m). In the transverse direction, the grid size was taken as ∆y = 0.005–0.024 m for

Channel I and ∆y = 0.005–0.021 m for Channel II. The minimum and maximum ∆y were specified for the bottom and top of the computation domain, respectively. The selected time step of 0.001 s was based on the Courant-Friedrichs-Lewy (CFL) condition. For a horizontal bed, the water is at rest initially in the partial region of the reservoir and is not affected by the upward flood wave. On the other hand, for a sloping bed, the fluid in the unaffected region of the reservoir will start to move from the beginning of dam breaking because of the gravity effect. So, both of its velocity and discharge are nonzero, and the resulting flow depth varies with increasing time. These phenomena were predicted by the LES model and the details are presented in Figs. 6–9 along with the analytical solution. The dam-break flood down a channel with a polyline cross-section exhibits three-dimensional characteristics. The transversal water surface is not horizontal because the flow is backed by both the left and right slopes. This phenomenon is apparent downstream of the dam. Herein, the flow depth along the channel bottom centerline at t = 0.5 s, 1.0 s, 1.5 s and 2.0 s for Channels I and II obtained from the LES model (symbols) is compared with the analytical solution (lines) in Fig. 6. The variation characteristics of flow depth profiles with increasing bed slope are captured by the LES model. The upstream flow depth in the numerical model is larger than that of the analytical solution. The opposite case occurs downstream of the dam. This phenomenon also occurred in the study conducted by LaRocque et al. (2013a), in which the scatter plots of measurement were mainly in the interval of the numerical and analytical solutions. The average velocity along the length of the channel at t = 0.5 s, 1.0 s, 1.5 s and 2.0 s for different locations from the dam for Channels I and II are shown in Fig. 7. The average velocities with both the analytical solution (lines) and the numerical simulation (symbols) are comparable with each other. It is an exception that the analytical model in which the dam-break flow is analyzed as an

14

ideal-fluid flow overpredicts the velocity in the wave tip region where the flow resistance dominates (Chanson, 2009). The flow velocity does not increase rapidly in the wave tip zone. The analytical and numerical flow velocities are compared more clearly in Fig. 8. It can be seen that the analytical solutions are larger generally than those by the LES model and the relative errors are less than 20%. Moreover, the relative large errors occur in the zone of low velocity (below 1m/s). Overall, the agreements between analytical and numerical solutions are good. Comparisons of the computed (lines) and simulated (symbols) discharge along the length of the channel at t = 0.5 s, 1.0 s, 1.5 s and 2.0 s for different locations for Channels I and II are presented in Fig. 9. The analytical solution shows a better match with the numerical simulation in the reservoir. In particular, the analytical solution can provide a reasonable approximation of the discharge at the dam site, which is a critical parameter used for hazard classifications and emergency action plans. However, the discharge predicted by the analytical model is greater than that obtained by the LES model on the downstream side of the dam. The discharge in the unaffected region could not be estimated by the analytical model because the analytical solution assumes that the rest of the water is initially stored in the reservoir. 6. Model Limitations

The cross-sectional description method is a key factor of the present model. However, not all channel cross-sectional shapes can be described by this approach, and therefore, the analytical solution developed in this study is not applicable to a channel with an arbitrary cross-sectional shape. It should be noted that the effect of friction is neglected during the derivation of analytical solution, so the present model cannot be used for the channel considering the effect of friction. The other limitations of the present model are discussed in this section. 6.1. Disappearance of the Horizontal Stages

As shown in Fig. 10, a cross-sectional shape, consisting of several horizontal stages, is described by a polyline composed of fifteen line segments. Three horizontal line segments on each

15

slope represent the horizontal stages, i.e., dl1 = dl2, dl3 = dl4, dl5 = dl6, dr1 = dr2, dr3 = dr4, and dr5 = dr6. According to Eqs. (6)–(7), the water surface width B and cross-sectional area A are expressed as follows: 6

6

i =1

j =1

B = b + ∑ ( M li − M li +1 )d li + ∑ ( M rj − M rj +1 )d rj + ( M l 7 + M r 7 ) h 6  1 6 A = Bh −  ∑ ( M li − M li +1 )d li2 + ∑ ( M rj − M rj +1 )d rj2 + ( M l 7 + M r 7 ) h 2  2  i =1 j =1 

(35)

(36)

Inserting dl1 = dl2, dl3 = dl4, dl5 = dl6, dr1 = dr2, dr3 = dr4, and dr5 = dr6 into the above two expressions results in the following B = b + M l1d l 1 + M l 3 ( d l 3 − d l 2 ) + M l 5 ( dl 5 − d l 4 ) + M l 7 ( h − d l 6 ) + M r1d r1 + M r 3 ( d r 3 − d r 2 ) + M r 5 ( d r 5 − d r 4 ) + M r 7 ( h − d r 6 ) 1 A = Bh −  M l 1d l21 + M l 3 ( d l23 − d l22 )+ M l 5 ( d l25 − d l24 ) + M l 7 ( h 2 − d l26 ) 2 + M r1d r21 + M r 3 ( d r23 − d r22 )+ M r 5 ( hr25 − d r24 ) + M r 7 ( h 2 − d r26 )

(37)

(38)

The width of the horizontal stages is not considered in Eq. (37), i.e., the value of B is less than the true value. Furthermore, the area of all rectangular regions (indicated by the shadow) from the bottom to all horizontal stages is not deducted from the area obtained by Eq. (38), i.e., the value of A is greater than the true value. In other words, the horizontal stages seem to disappear from the channel cross-section described by Eqs. (6)–(7). This result indicates that the present model is not suitable for channels with horizontal stages. However, this limitation is not significant due to following two reasons: the first is the channel with cross-section including horizontal stage are few in real cases; the second is the elevation of horizontal stage can be adjusted slightly if this case with horizontal stage is found, and this approximate treatment only cause small errors. 6.2. Second Combinative Parameter

The introduction of the second combinative parameter of the cross-section, β, simplifies the expression of the cross-sectional area A. However, the definition of β leads to another limitation.

16

Here, Channel I is taken as an example. If one only changes the value of Mr4, i.e., Mr4 = 2.0 instead of Mr4 = 0.8, and keeps the other parameters of the channel cross-section constant, an unexpected result occurs. The calculated results are shown in Table 3. In the sixth subsection, β2 is negative. As a consequence, the subsequent computations cannot be completed. To avoid the aforementioned unreasonable results, β2 in all subsections must satisfy

β2 ≥ 0

(39)

Substituting Eq. (10) into Eq. (39) yields  I −1

J −1

 i =1

j =1



( M lI + M rJ )  ∑ ( M li − M li +1 ) d li2 + ∑ ( M rj − M rj +1 ) d rj2  I −1 J −1   ≥ b + ∑ ( M li − M li +1 ) d li + ∑ ( M rj − M rj +1 ) d rj  i =1 j =1  



2

(40)

Thus, the present model can only work when the condition described by Eq. (40) is satisfied in all subsections. Eq. (40) is an applicable condition of the analytical solution from the pure mathematical point of view. For the typical cross-sections (such as rectangular, triangular, trapezoidal, triangular, trapezoidal cross-sections), Eq.40 are satisfied automatically. Namely, the analytical solution is applicable for all of above typical cross-sections. Generally speaking, the Eq.40 cannot be satisfied for cross-section with narrow top and larger bottom which is very few in practical rives. 7. Conclusions

By introducing a cross-sectional description and mathematical transformation into the Saint-Venant equations, an analytical solution of dam-break flow over a dry sloping channel with a polyline cross-section was derived. The solution procedure was tested by two prismatic channels involving different cross-sections and considering variation of the bed slope. The results show that all of flow depth, average velocity and discharge are reasonably determined by the proposed approach. Moreover, the effect of slope is studied by the proposed model. For example, it has been demonstrated that the dam-break flood was accelerated on a downward slope and was retarded on an

17

upward slope. The effects of the bed slope on the flow depth, average velocity and discharge become more prominent over time. The flow depth at the dam site remains constant at different times for the horizontal channel and varies with time for a sloping channel. The same situation occurs for the average velocity and discharge. The location of the maximum discharge along the downward-sloping channel is on the upstream side of the dam and is farther from the dam with increasing time. The opposite case occurs for the upward-sloping channel. The maximum discharge along the horizontal channel occurs exactly at the dam site. In order to verify the proposed model further, the flow depth, average velocity and discharge with the analytical solution were compared with the results obtained from the LES model. The comparison showed that the analytical model could provide the main features of dam-break floods down a channel with a polyline cross-section. However, there are some shortcomings of the analytical solution. In the wave tip region where the flow resistance dominates, the present model overestimates the velocity. The discharge is also overpredicted on the downstream side of the dam. Overall, this study shows that the analytical solution can be used for the rapid prediction of dam-break flows. It needs to be noted that the analytical solution developed in this study is not applicable to channels with horizontal stages. Acknowledgements

The authors are grateful to Associate Professor Hong Xiao (Sichuan University) for his constructive comments. This study was funded by the National Natural Science Foundation of China (Grant Nos: 51209155, 51409183, 51579166 and 5151101425). Notation

A

= cross-sectional area of flow

α, β

= combinative parameters of the cross-section

B

= water surface width

b

= bottom width

18

c, k

= constants in the cross-section boundary equation

d

= vertical distance from the bottom to the slope break-point

G

= function of W

g

= acceleration due to gravity

h

= flow depth

hup

= initial flow depth in the undisturbed reservoir upstream of the dam

K,C

= dam-break characteristic parameters

l,r

= subscripts to denote the left and right slopes, respectively

i ,j

= the ith and jth broken line respectively

M

= slope coefficient

N

= number of total subsections

n

= a power-law index

Q

=discharge

S0

= bed slope

Sf

= friction slope

t

= time

u

= average velocity of flow

uup

= velocity in the undisturbed reservoir upstream of the dam

x

= the distance

x*

=an independent variable

W

= a characteristic parameter of the flow depth

∆x

= space step in x direction

∆y

= space step in y direction

∆z

= space step in z direction

19

References

Ancey C., Iverson R. M., Rentschler M., and Denlinger R. P. An exact solution for ideal dam-break floods on steep slopes. Water Resources Research, 2008, 44(1):567-568. Biscarini, C., Di Francesco, S., Manciola, P., 2010. CFD modelling approach for dam break flow studies, Hydrol. Earth Syst. Sci. 14(4), 705–718. Chanson, H., 2009. Application of the method of characteristics to the dam break wave problem, J. Hydraul. Res. 47(1), 41–49. Chow, V.T., 1959. Open channel hydraulics, McGraw-Hill, New York. Dressler, R.F., 1952. Hydraulic resistance effect upon the dam-break functions, J. Res. Natl. Bur. Stand. 49(3), 217–225. Fernandez-Feria, R., 2006. Dam-break flow for arbitrary slopes of the bottom, J. Eng. Math. 54, 319–331. Flow Science Inc. FLOW-3D user manual, 2007, Santa Fe NM. Fluent Inc. Fluent 6.3 user manual, 2006, Lebanon, NH. Hogg, A.J., 2006. Lock-release gravity currents and dam-break flows, J. Fluid Mech. 569, 61–87. Hunt, B., 1982. Asymptotic solution for dam-break problems, J. Hydraul. Div.-Asce 108(1), 115–126. Hunt, B., 1983. Asymptotic solution for dam break on sloping channel, J. Hydraul. Eng.-Asce 109(12), 1698–1706. Hunt, B., 1984. Perturbation solution for dam-break floods, J. Hydraul. Eng.-Asce 110(8), 1058–1071. Kikkert GA, Liyanage T, Shang C. 2015. Dam-break generated flow from an infinite reservoir into a positively inclined channel of limited width. Journal of Hydro-environment Research, 9:519–531. Kocaman, S., Cagatay, H., 2015. Investigation of dam-break induced shock waves impact on a

20

vertical wall, J. Hydrol. 525, 1–12. LaRocque, L., Imran, J., Chaudhry, M., 2013a. Experimental and numerical investigations of two-dimensional dam-break flows, J. Hydraul. Eng.-Asce 139(6), 569–579. LaRocque, L., Imran, J., Chaudhry, M., 2013b. 3D numerical simulation of partial breach dam-break flow using the LES and k–ε turbulence models, J. Hydraul. Res. 51(2), 145–157. Lauber G., Hager W. H. Experiments to dam break wave: Sloping channel. Journal of Hydraulic Research, 1998, 36(5): 761-773. Lin, B.N., Gong, Z.Y., Wang, L.X., 1980. Dam site hydrographs due to sudden release, Journal of Tsinghua University, 20(1), 17–31 (in Chinese). Mangeney, A., Heinrich, P., Roche, R., 2000. Analytical solution for testing debris avalanche numerical models, Pure Appl. Geophys. 157, 1081–1096. Oertel, M., Bung, D.B., 2012. Initial stage of two-dimensional dam-break waves: laboratory versus VOF, J. Hydraul. Res. 50(1), 89–97. Ritter, A., 1892. Die fortpflanzung der wasserwellen, Z. Ver. Dtsch. Ing., 36(33), 947–954 (in German). Stoker, J.J., 1957. Water waves, Interscience/Wiley, New York. Wang, B., Wu, C., Chen, Y.L., 2010. Discussion of “Dam-Break Flows: Acquisition of Experimental Data through an Imaging Technique and 2D Numerical Modeling”, Journal of Hydraulic Engineering, 125(7), 457–458. Whitham, G.B., 1955. The effects of hydraulic resistance in the dam-break problem, Proc. R. Soc. London, 227(1170), 399–407. Wu, C., Dai, G.Q., Wu, C.G., 1993. Model of dam-break floods for channels of arbitrary cross section, J. Hydraul. Div.-Asce 119(8), 911–923. Wu, C., Huang, G.F., Zheng, Y.H., 1999. Theoretical solution of dam-break shock wave, Journal of Hydraulic Engineering, 125(11), 1210–1215.

21

Figure captions list

Fig. 1. Notations of an irregular-shaped cross-section. Fig. 2. Flow chart of the solution procedure for the proposed model. Fig. 3. Cross-sectional shapes of Channels I and II. Fig. 4. Dimensionless flow depth, average velocity and discharge profiles along the independent variable. Fig. 5. Profiles of (a) flow depth, (b) average velocity and (c) discharge for Channel I obtained from the analytical solution. Fig. 6. Flow depth predicted by both the analytical and numerical models. Fig. 7. Average velocity predicted by both the analytical and numerical models. Fig. 8. Comparison of flow velocities predicted by the analytical and numerical models. Fig. 9. Discharge predicted by both the analytical and numerical models. Fig. 10. Cross-sectional shape and notation of a channel with horizontal stages.

22

Fig. 1. Notations of an irregular-shaped cross-section.

23

Begin Input Mli , Mrj, dli , drj, hup, b, S0, Sf

Calculate α, β and K with Eqs. (8, 9 and 17), respectively, in all subsections. Note that the values of αup and βup are equal to the values of α and β in the top subsection.

Part I

Calculate W and Gappr(W) with Eqs. (12) and (24), respectively, in all subsections under every dli or drj. Note that the values of Wup and Gappr(Wup ) are equal to the values of W and Gappr(W) in the top

Calculate C and x* with Eq. (29) in all subsections except that C = 0 is obtained from Eq. (18) in the top subsection.

For a given x*, find the subsection that the given x* belongs to and determine the corresponding values of α, β, K and C.

Part II Calculate the value of W using Eq. (29) by iterative computations and predict h, u and Q from Eqs. (12, 31 and 32), respectively.

Stop

Fig. 2. Flow chart of the solution procedure for the proposed model.

24

6 5 4

1.0 0.8

0.35

0.8 1.0

3

0.26 1.2

0.30

0.6

2 0.16

0.22

1

0.10

1.3

0.20

Channel I

6 0.5

0.7

5 0.85

4 0.40 0.7

0.34

1.0

3 0.28

2

0.9 1.0

1

0.20 0.15 0.10

0.20

Channel II Fig. 3. Cross-sectional shapes of Channels I and II.

25

1.0

Channel I Channel II Ritter (1892) * h u* Q*

0.8

*

*

h,u,Q

*

0.6

0.4

0.2

0.0 -1

0

1

2

3

x*

Fig. 4. Dimensionless flow depth, average velocity and discharge profiles along the independent variable.

26

1.0

A

t = 0.5 s S0 = 0.02 0 -0.02

0.8

t = 1.0 s

Enlarged view

0.6

0.02 0 -0.02

0.65

h

*

0.61

t = 1.5 s 0.4 0.57

0.02 0 -0.02

0.53

t = 2.0 s

0.2 0.49 0.45 -0.3

0.0 -15

0.0

-9

0.02 0 -0.02

0.3

-3 x / h up

3

9

15

B 2.5

t = 0.5 s 0.8

S0= 0.02

Enlarged view

2.0

0 -0.02

1.5

0.02 0 -0.02

t = 1.0 s

u

*

0.6

t = 1.5 s 1.0 0.4 -0.5

0.0

0.02 0 -0.02

0.5

t = 2.0 s

0.5

0.0 -15

C

0.02 0 -0.02 -9

-3 x / h up

3

9

15

0.7

t = 0.5 s S0 = 0.02

0.6

0 -0.02

0.5

t = 1.0 s 0.02 0 -0.02

Q

*

0.4

t = 1.5 s

0.3

0.02 0 -0.02

0.2

t = 2.0 s 0.1 0.0 -15

0.02 0 -0.02 -9

-3

x / hup

3

9

15

Fig. 5. Profiles of (a) flow depth, (b) average velocity and (c) discharge for Channel I obtained from the analytical solution.

27

0.35 0.30 0.25

0.25

0.20 0.15

0.20 0.15

0.10

0.10

0.05

0.05

0.00 -5

-3

-1 x (m) 1

3

0.00 -5

5

0.30

0.30

0.25

0.25

0.20

0.20

0.15

0.10

0.05

0.05 -3

-1 x (m) 1

3

-1 x (m) 1

3

t = 2.0 s

0.00 -5

5

5

0.15

0.10

0.00 -5

-3

0.35

t = 1.5 s

h (m)

h (m)

0.35

t = 1.0 s

0.30

h (m)

h (m)

0.35

t = 0.5 s S0 = Analytical LES 0.02 0 -0.02

-3

-1 x (m) 1

3

5

(a) Channel I 0.4

0.01 0 -0.01

0.2

0.1

0.0 -5

-3

-1 x (m) 1

3

0.0 -5

5

-3

-1 x (m) 1

3

0.4

t = 1.5 s

0.3

5 t = 2.0 s

0.3 h (m)

h (m)

0.2

0.1

0.4

0.2

0.1

0.0 -5

t = 1.0 s

0.3 h (m)

0.3 h (m)

0.4

t = 0.5 s S 0 = Analytical LES

0.2

0.1

-3

-1 x (m) 1

3

5

0.0 -5

-3

-1 x (m) 1

(b) Channel II Fig. 6. Flow depth predicted by both the analytical and numerical models.

28

3

5

5

0.02 4 0 -0.02

4

3

3

2

2

1

1

0 -5

-3

-1

x (m)

1

3

0 -5

5

4

4

3

3

2

2

1

1

-3

-1

x (m)

1

3

-3

-1

x (m)

1

3

5

t = 1.5 s

u (m/s)

u (m/s)

5

0 -5

t = 1.0 s

t = 0.5 s

S0= Analytical LES

u (m/s)

u (m/s)

5

0 -5

5

5

t = 2.0 s

-3

-1

x (m)

1

3

5

(a) Channel I 0.01 4 0 -0.01

4

3

3

2

2

1

1

0 -5

-3

-1

x (m)

1

3

5

0 -5

5

4

4

3

3

2

2

1

1

0 -5

-3

-1

x (m)

1

3

5

t = 1.0 s

-3

-1

x (m)

1

3

5

t = 1.5 s

u (m/s)

u (m/s)

5

t = 0.5 s

S0= Analytical LES

u (m/s)

u (m/s)

5

0 -5

5

t = 2.0 s

-3

-1

x (m)

1

3

(b) Channel II Fig. 7. Average velocity predicted by both the analytical and numerical models.

29

5

3.0

h

Analytical solution (m/s)

2.5 2.0 1.5 1.0 0.5 0.0 0.0

t

S0

0.35m 0.5s 0.02 0.35m 0.5s 0 0.35m 0.5s -0.02 0.35m 1.0s 0.02 0.35m 1.0s 0 0.35m 1.0s -0.02 0.35m 1.5s 0.02 0.35m 1.5s 0 0.35m 1.5s -0.02 0.35m 2.0s 0.02 0.35m 2.0s 0 0.35m 2.0s -0.02 perfect line 20% deviation 0.5

1.0 1.5 2.0 LES prediction (m/s)

2.5

3.0

(a) Channel I 3.0

h

Analytical solution (m/s)

2.5 2.0 1.5 1.0 0.5 0.0 0.0

0.5

1.0 1.5 2.0 LES prediction (m/s)

2.5

3.0

t

S0

0.4m 0.5s 0.01 0.4m 0.5s 0 0.4m 0.5s -0.01 0.4m 1.0s 0.01 0.4m 1.0s 0 0.4m 1.0s -0.01 0.4m 1.5s 0.01 0.4m 1.5s 0 0.4m 1.5s -0.01 0.4m 2.0s 0.01 0.4m 2.0s 0 0.4m 2.0s -0.01 perfect line 20% deviation

(b) Channel II Fig. 8. Comparison of flow velocities predicted by the analytical and numerical models.

30

0.15

t = 0.5 s

0.15

S0 = Analytical LES 0.02 0 -0.02

0.12

t = 1.0 s

0.12

0.09

3

3

Q (m /s)

Q (m /s)

0.09

0.06

0.06

0.03

0.03

0.00 -5 0.15

-3

-1

x (m)

1

3

0.00 -5

5

0.15

t = 1.5 s

0.12

0.09

0.09

-1 x (m)

1

3

5

-3

-1

1

3

5

-3

-1

x (m)

1

3

5

-3

-1

x (m)

1

3

5

t = 2.0 s

3

3

Q (m /s)

Q (m /s)

0.12

-3

0.06

0.06

0.03

0.03

0.00 -5

-3

-1

x (m)

1

3

0.00 -5

5

x (m)

(a) Channel I 0.15

t = 0.5 s

0.15

S0 = Analytical LES 0.01 0 -0.01

0.12

0.12

0.09

3

3

Q (m /s)

Q (m /s)

0.09

0.06

0.06

0.03

0.03

0.00 -5

0.15

t = 1.0 s

-3

-1

x (m)

1

3

0.00 -5

5

0.15

t = 1.5 s

0.12

0.09

0.09

3

3

Q (m /s)

Q (m /s)

0.12

0.06

0.06

0.03

0.00 -5

t = 2.0 s

0.03

-3

-1

x (m)

1

3

5

0.00 -5

(b) Channel II Fig.9. Discharge predicted by both the analytical and numerical models.

31

B M r7 M r6

M l7 M r5 M r4

M l6 M l5 M l4

h d l6

dl5 dl4

h

M r3 M r2

M l3 dl3 dl2

M l2 d l1

M r1

M l1

d r1

dr3 d r2

d r5 dr4

d r6

b

Fig. 10. Cross-sectional shape and notation of a channel with horizontal stages.

32

Table 1. Relevant parameters and dimensionless distance in every subsection for Example Channel I. Subsection

1

2

3

4

5

6

α

0.080

0.150

0.239

0.137

0.097

0.141

β

1.000

1.083

1.103

0.968

0.411

1.032

K

1.349

0.947

0.744

1.049

1.912

1.000

C*

0.061

−0.047

−0.131

0.046

0.414

0.000

W

0.000

0.791

0.519

0.674

0.507

0.609

0.921

1.000

1.994

2.117

1.007

1.089

G appr(W)

2.011

2.151

2.083

2.121

2.080

2.105

2.185

2.204

2.396

2.413

2.206

2.226

h

0.00

0.10

0.16

0.22

0.26

0.30

0.35

x*

2.287

0.672

0.221

−0.156

−0.371

−0.569

−0.805

Table 2. Relevant parameters and dimensionless distance in every subsection for Example Channel II. Subsection

1

2

3

4

5

6

α

0.118

0.131

0.115

0.145

0.208

0.277

β

1.000

1.018

0.972

1.070

1.162

1.188

K

1.671

1.568

1.717

1.454

1.166

1.000

C*

0.293

0.273

0.314

0.225

0.093

0.000

W

0.000

0.652

0.604

0.743

0.829

0.955

0.782

0.932

0.714

0.796

0.662

0.728

Gappr(W)

2.011

2.116

2.104

2.139

2.161

2.193

2.149

2.187

2.132

2.153

2.118

2.135

h

0.00

0.10

0.15

0.20

0.28

0.34

0.40

x*

2.428

0.824

0.456

0.144

−0.293

−0.588

−0.862

Table 3. First and second combinative parameters in every subsection of the modified cross-section for Example

Channel I. Subsection

1

2

3

4

5

6

α

0.080

0.150

0.239

0.137

0.097

−0.0353

β2

1.000

1.173

1.216

0.937

0.168

−27.206

p

1.000

1.083

1.103

0.968

0.411

—

35

We derive an analytical model of dam-break flow on polyline cross-sectional channels. Slope impacts on the flood propagation are presented with the analytical solution. The analytical solution provides the main features of dam-break floods.

36