Study on simplified calculation method for lateral natural frequency of water within U-shaped aqueducts

The water sloshing in aqueducts under seismic excitations is essentially a sloshing problem of water in liquid storage tanks. However, the present calculation method for the lateral natural frequency of water within U-shaped aqueducts is extremely complicated, inefficient, and only valid for the first lateral natural frequency calculation. Based on an existing calculation method derived from the Rayleigh quotient for the natural frequency of water in 2D (two-dimensional) containers with arbitrary shapes, a simplified calculation method for the lateral natural frequency of water within U-shaped aqueducts is proposed. The Taylor series, along with the β and Γ functions, is used to simplify the complex non-elementary integrals as algebraic expressions, making this calculation method convenient and efficient enough to calculate the first four lateral natural frequency of water in U-shaped aqueducts. Furthermore, a numerical model established by OpenFOAM-9 is employed to validate the accuracy of the proposed method. Validation shows the proposed calculation method is 9 times as precise as the method specified in Chinese code for the first lateral natural frequency calculation, and it can also calculate the second- to fourth-order lateral frequencies of the water in U-shaped aqueducts precisely. Results also show when water level inside the U-shaped aqueduct is beyond the semicircle part, aqueduct width significantly influences water natural frequency, whereas water depth has a minor impact.