THE HYDROELASTIC SIMULATION AND CALCULATION OF HIGH ARCH DAM VIBRATION INDUCED BY FLOOD DISCHARGE FLOW

Lian Jijian  Yang Min  Hu Minggang

Civil Engineering School of Tianjin University, Tianjin 300072, China

E-mail:ljj＠public.tjuc.com.cn

Tel. +86-22-27406390 

Abstract: High-velocity flow induced structure vibration and flood discharging energy dissipation in high arch dam are two main problems in hydropower station construction of China. This paper systematically introduces the research findings of authors from several facets such as hydrodynamic load character that can cause high arch dam to vibrate, the analytic method of flow-induced vibration and the hydroelastic simulation. The analytic method of vibration response of high arch dam by flood discharge flow was verified by tests of high arch dam hydroelastic models for Ertan, Xiaowan, Goupitan and Xiloudu hydropower station. It was also verified by prototype observation of flood discharge flow induced vibration of Ertan arch dam, both of which shows a good agreement.

Keywords: flow induced vibration, high arch dam, simulation of hydroelastic model, calculation model

1    INTRODUCTION

There is few established high arch dam with massive flood discharge capacity and discharge power in the world. The constructed and constructing high arch dam in China get to the world’s highest level in height of dam, discharge capacity and discharge power, e.g. Ertan, Xiaowan, Goupitan and Xiloudu, as shown in Tab.1. Among these dams, the discharge capacity of Xiloudu arch dam even reaches 52300m³/s, about 60% flood is discharged through dam body, and the flood discharge power reaches 100000MW, which can rank the first in the world. It shows that the flood discharge energy dissipation is a critical problem, which has already become one of the controlling factors in the project layout. These dams applied the flood discharge methods of jointing surface spillways and middle or bottom outlets and the water plunge pool to dissipate energy. Because there is no operation experience of flood discharge under massive discharge on the high arch dam in the world, the dam vibration induced by high velocity flow is an important problem and it is arousing people’s attention. A lot of experiments and theoretical analysis for the vibration were performed in Ertan, Xiaowan, Goupitan and Xiloudu arch dam in the past ten years. Recently, the prototype observation for the vibration of Ertan arch dam is conducted. This paper applies random analysis of flow fluctuating load to establish the theoretical analytic model of high arch dam vibration by flood discharge flow.The hydroelsatic model which meets the similitude of hydrodynamic condition and structural dynamic condition at the same time is use to simulate the arch dam vibration.

2    THE CHARACTERISTICS OF DYNAMIC LOADS

The characteristics of dynamic loads that causes high arch dam to vibrate is complicated and diversified, which includes: (1) the impact fluctuating load caused by discharge flow nappes impacting cushion pool. (2) the wave load directly acting on the downstream dam body in water plunge pool. (3) the orifice fluctuating load acting on the spillways and outlets. It is shown in the Fig.1.

Table 1    The flood discharge power of several arch dams

2.1    Impact Load

2.1.1    Amplitude of point fluctuating pressure

There are many influencing factors on the dynamic loads as discharge flow is acting on the surface of apron or riverbed. The factors mainly include the incidence velocity of nappe U₀, incidence angle θ, the thickness of incidence nappe d₀, downstream depth h and density of water ρ. The max root-mean-square of fluctuating pressure can be expressed as follows by dimensional analysis:.

                  (1)

After collating the experimental data and curve fitting, the following formula can be achieved:

                   (2)

Not only the maximum fluctuating pressure on the stagnation is required, but also the distribution of fluctuating pressure is necessary. The distribution of fluctuating pressure can be obtained by collating the ratio of the fluctuating pressure on every testing point to the maximum stagnation fluctuating pressure and the ratio of distance from corresponding point to stagnation x to the effective depth of water cushion h₀., shown as:

                               (3)

where                                   

x is positive value when the point is at downstream stagnation , R is the asymmetry factor of jet flow in water cushion R=0.9.

2.1.2    Amplitude of area fluctuating pressure

According to the correlation analysis, the amplitude of area fluctuating pressure in the rectangle with length L and width B can be calculated by the following formula:

       (4)

where , are constant, and =6.5, =1.05 in plunge pool, h is the depth of water cushion. The actual measurement amplitude of area fluctuating impact load of Xiloudu Arch Dam is shown in Table.2

Table 2    Impact and wave fluctuating load of xiluodu arch dam(rms,kpa)

2.2    Wave load and orifice load

According to the tests, the wave fluctuating load is about one fifth to one sixth of the impact fluctuating load, see Tab.2. Where as the mode and position of the fluctuating load acting on the orifice is very complicated, it is difficult to measure directly. In this paper, The equivalent fluctuating load on the discharge orifice can be achieved by inverse analysis of flow-induced vibration. The fluctuating pressure acting on the discharge orifice and the velocity head has such relation: , here  should vary with the flow condition. It can be considered that the global fluctuating load on orifice is proportionate to fluctuating pressure and its acting area . Let the acting area and discharge area A have the following relation: , then the global fluctuating load on orifice  can be expressed as follow:

                        (5)

where Q is the discharge quantity with unit m³/s, H is the acting water head on orifice with unit m, is the root-mean-square of global fluctuating load with unit ton. By isolating the impact load on plunge pool and the wave fluctuating load on downstream surface of arch dam, the dynamic response of arch dam by orifice fluctuating loads can be obtained by actual measurement, and then equivalent orifice fluctuating load is achieved by inverse analysis. So the coefficient of global fluctuating load on middle or bottom outlets C_f may be obtained, specified as 9×10^-3, and as to surface spillway, C_j = 7×10^-3.

2.3    Random analysis for fluctuating loads

Dynamic loads on two rectangles (length l, width b, L, B is the distance separately in x, y direction, shown as Fig.2.) are analyzed for correlation characteristics of point, area or global fluctuating loads, the relation between area fluctuating load and point fluctuating pressure is:

                     (6)

And correlation function of area fluctuating load  is as the following:

(7)

After integral transformation:

    (8)

where is the space-time correlation function of point fluctuating pressure, the correlation coefficient of point fluctuating pressure ρ_p(ζ,η,τ)= R_p(ζ,η,τ)/D_p, D_p is the variance of point fluctuating pressure, see Ref.(1). The test result shows that ρ_p(ζ,η) can be approximately described as product of longitude correlation coefficient ρ(ζ) and lateral correlation coefficient ρ(η), i.e. .As the value of L and B is zero, the converse coefficient of point fluctuating pressure C_p from Eq.(8) is:

                           (9)

where , .As L≠0 and B≠0, the correlation coefficient of area fluctuating load  is:

                            (10)

It shows that the correlation coefficient after point-area transformed among area fluctuating load and the correlation coefficient of point fluctuating pressure are in good consistence. For the fluctuating load in nappe impact area: ① as L=h/3,B=0, ρ_F=0.114; ② as L=0,B=h/3, ρ_F=0.031; ③ as L=B=h/3, ρ_F= 0.004.Apparently area fluctuating load can be regarded as independent random process as long as the distance is greater than a definite value (e.g. h/3) between them.

There is considerable difference between spectrum of point fluctuating pressure and that of area fluctuating load, as shown in Fig.3. Transform relationship of spectrum between the point and area fluctuating pressure can be achieved by correlation analysis, as taking L=B=0 in Eq. (8), we can obtain:

      (11)

And then using Taylor’s freezing assumption to transform the space-time correlation function of point fluctuating pressure to auto-correlative function, it can achieve the transformation between point and area frequency spectrum. It should be noted that vortex- transferring velocity V_e involved in Taylor’s assumption. The transform relationship between space-time correlation function and time correlation function is shown as:

                          (12)

After being transformed by Fourier transfer, the above equation adopted one-side spectrum, and made it dimensionless, the transform relationship of spectral density of point fluctuating pressure and area fluctuating pressure can be obtained as follow:

                          (13)

3    THE ANALYSIS OF FLOW-INDUCED VIBRATION

The kinematics differential equation of structure under the random load can be described as:

                   (14)

where [M] = mass matrix = the sum of structural mass matrix [M_s] and additional water mass matrix [M_w]; [K]=structural stiffness matrix;[C]=structural damping matrix; is acceleration, velocity and displacement vector respectively; [P]=random component of load. Eq.(14) can be solved by model superposition. Let {V}=[φ]{δ}, where [φ] = modal matrix; {δ} = generalized displacement vector of vibration. Because of low order mode is generally the main mode of vibration of l flow-induced vibration, Eq.(14) can be changed as follow:

                    (15)

where M_j, = generalized mass, ω_j = the jth order vibration frequency of structure; ξ_j = the jth order mode damping ratio; F_j(t) = the jth order structural component of random load, . The instant response of the jth order vibration mode can be achieved by Eq. (15), shown as follow:

                             (16)

where h_j(t) = the unit pulse response function .

                   (17)

where , from which displacement response of any node k can be illustrated as follow:

                (18)

The auto-correlation function of displacement response of any node k is:

                         (19)

After the Eq.(19) is transformed by Fourier transfer, the spectrum of displacement response can be obtained:

           (20)

where ;

H_j^*(ω) = conjugate complex numbers of H_j(ω). After transferring fluctuating pressure to area fluctuating load, the area fluctuating loads can be regarded as independent load processes as their distances are greater than h/3 , then the Eq.(20) can be rewritten as :

               (21)

The displacement response mean-square-root is:

                    (22)

After getting the displacement response vector of each node in element, the stress can be achieved by following equation:

                      (23)

where [D] = elastic matrix; [B] = strain matrix.

4    HYDROELASTIC MODEL SIMULATION

Hydroelastic model tests for flow induced vibration is the simulation of fluid-solid coupled vibration system, which is composed of dam-water of reservoir-foundation-dynamic load. The dynamic load input system and the dynamic response system of structural are required to be similar at the same time.

4.1    Hydraulic condition similitude

The essence of simulation of dynamic load input system is the similitude issue of fluctuating pressure in a hydraulic model designed by the gravity similitude law. Generally, as R_e is considerable large , the gravity similitude law can meet main vortex similitude, and the fluctuating pressure is issue from larger scale vortex motion. Therefore, adopting gravity similitude law can make the main part of fluctuating pressure to be similar. The amplitude of fluctuating pressure usually accord with gravity similitude law, where as the frequency of fluctuating pressure may deviate from it to some extent.

4.2    Structural dynamic condition similitude

The dynamic condition similitude includes: the geometrical condition similitude, the physical condition similitude, the kinematics condition similitude and the boundary condition similitude. The flow condition similitude should be considered at the same time in order to meet the similitude of flow induced vibration. After analyzing, the structural material should meet such condition as: unit weight scale , elastic modulus scale , damping ratio scale , Poisson ratio scale . If we select the range of foundation and reservoir water reasonably, the similarity of hydraulic condition and structural dynamic condition will be met at the same time.

The similitude range of foundation of hydroelastic model affects the accuracy and reliability of experimental results notably. In principle, the range should include the area affected by loads. In the simulation experiments, on the condition of meeting the accuracy and reliability of result, the area adopted should be as small as possible for saving model material and fund. To ensure the accuracy of dam dynamic response under the dynamic load and the modal characteristic of dam, the study of reasonable foundation range is necessary. The conclusion can be achieved by finite element analysis, which is illustrated as follow: ① the similitude range of foundation has little effect to the dam modal characteristic, as the similitude range is larger than 1/2H (H=the maximum height of dam) in depth and 1/4H in up-downstream and 1/7H in dam abutment. Comparing the first fifth order frequencies of arch dam with deepest foundation (2H), the error is within 4%. ② the similitude range of foundation has more obvious effect to the vibration response of dam , especially the influence of the depth. As the similitude range is greater than or equal to 0.75H in depth, 0.5H in up-downstream and 0.25H in dam abutment, the dynamic responses of arch dam can be well simulated (see Ref.2,3).

5    COMPARISONS OF MEASURED RESULTS WITH CALCULATED RESULTS

5.1    Comparison of model measured results with calculted results

Based on the measured dynamic loads, the vibration responses of Xiaowan and Goupitan arch dams in the case of joint discharge of surface spillways and middle outlets is calculated. The comparison of the measured results with the calculated results is shown as Fig.4 to Fig.5. It shows that calculated results and experimental results are in a good agreement, whose absolute error is not more than 30%.

5.2    Comparison of calculated and prototype observation results

Taking account of the condition of prototype flood discharge, i.e. the upstream water level is 1175m and the discharge flow rate is 4500m³/s, the calculation is performed. The global fluctuating load of orifices can be computed by Eq.(5). The wave and impact fluctuating load may be roughly obtained based on the experimental results of hydraulic model of Ertan arch dam. Fig.6 and Fig.7 gives the comparison of the calculated and prototype observation results of vibration displacement in mean-square-root and in the largest amplitude. It is shown that the results of prototype observation is consistent with calculated results well. 

6    CONCLUSIONS

Based on the gravity similitude of hydraulics and elastic similitude of structural dynamics , hydroelastic model of high arch dam was established to simulate the fluid-solid coupled vibration system, which is compose of dam- water of reservoir- foundation- dynamic load. Theory analysis and test indicate that it is a reasonable method for studying the flow-induced vibration. Based on the calculation model proposed by this paper, the calculated results of the vibration response of arch dam by flood discharge flow has good consistence with those of model experiments and prototype observation, the error is within 30%. The proposed calculated method can be applied to predict the discharge flow induced vibration of arch dam primarily only by measuring the fluctuating loads in a convention hydraulic model.

References

[1]    Lin Jiyong, Lian Jijian. The calculation of amplitude value of fluctuating wall pressure under binary jet flow. JOURNAL OF HYDRAULIC ENGINEERING(in Chinese), 1988 (1)

[2]    Cui Guangtao. the research of discharging vibrant hydroelastic model of Ertan arch dam. JOURNAL OF TIANJIN UNIVERSITY, (in Chinese)1990 (1)

[3]    Cui Guangtao, Peng Xinmin. Discharging vibrant hydroelastic model of high arch dam. JOURNAL OF HYDRAULIC ENGINEERING, (in Chinese)1996 (4)