In this paper, the authors consider that whether the equation of motion or the Continuity equation in the fundamental equation of water hammer they were derived under the assumption ……=0(……=0) and……=0(……=0) .

In accordance with the similarities between waves and hydraulic jumps , the expressions for estimating wave decay and energy dissipation in the surf zone are derived based on the fundamental equation of fluid mechanics .

According to the two-dimensional tidal wave fundamental equation,the tides and tidal current in the bight of Fengcheng harbour were computed by using finite difference method. The results basically reveal the characteristics of motion of tidal current in the bight of Fengcheng harbour in view of the comparison between the measured values and the calculated values.

Under the geomagnetic dipolar coordinate system, a partial differential equation of electric potential is deduced from the ionospheric dynamo theory and taken as the fundamental equation for the electric

Based on the theory of fluid mechanics, the fundamental equation of single stage liquid-gas jet pump and a simplified formula are deduced and then verified by the experimental results.

Based on the fundamental equation for unsteady flow in open channel the Preissman method is applied to establish the model for simulating the hydraulic transient in long distance conduit water transfer system with retaining weirs.

The temperature dependence of the investigated properties is analyzed on the basis of the activated complex theory and of the fundamental equation of A.I.

The fundamental equation is obtained that relates the critical radius of a nucleus and reaction parameters, such as temperature, metal particle oversaturation by carbon, the work of metal adhesion to graphite, and the metal-carbon bond energy.

The fundamental equation of a compressible discrete vortex method is derived.

For prediction of gradient retention times of analytes, the fundamental equation of gradient elution was numerically solved.

We consider the dissipative Landau-Lifshitz equation as the fundamental equation of motion and present a complete local bifurcation analysis in 1+1 dimensions.

The use of the method of charaeterictics to solve super-critical flow problems has been previously established by several authors.In most cases,frictional resistance and bottom slope have been neglected.Taking into consideration the above two factors(Fig.1),the equation of continuity and the equations of motion are respectively (hu)_x+(h_v)_y=0.(1) uux+vu_y=g sin i-((g/2h))(h~2osi)_x-τu/ρhq.(2) uv_x+vv_y=-((g/2h))(h~2cos i)_y-τv/ρhq.(3) Where the subscripts denote the variables with respect to which...

The use of the method of charaeterictics to solve super-critical flow problems has been previously established by several authors.In most cases,frictional resistance and bottom slope have been neglected.Taking into consideration the above two factors(Fig.1),the equation of continuity and the equations of motion are respectively (hu)_x+(h_v)_y=0.(1) uux+vu_y=g sin i-((g/2h))(h~2osi)_x-τu/ρhq.(2) uv_x+vv_y=-((g/2h))(h~2cos i)_y-τv/ρhq.(3) Where the subscripts denote the variables with respect to which partial differentiations are made. Making use of the condition of irrotational flow v_x-u_y=O,(4) the energy equation can be obtained d((q~2/2))+gcosidh=gsinidx+(1/2)gh sin i·i_xdx-(τqdz)/(ρhu) =(gusini+(1/2)guh sin i·i_(τq)/(ρh))(dx/u).(5) The above are the fundamental equations for the type of flow discussed. It can be shown that the two systems of characteristics in the physical plane and the u v plane(Fig.2)are represented by C~+:dy~+=ξ~+dx~+.(21a) Γ~+ξ~-(dv~+)/(du~+-Gdx~+)=-1.(21b) and C~-:dy~-=ξ~-dx~-.(22a) Γ~-:ξ~+.(dv~-)/(du~-)-Gdx~-=-1.(22b) Where ξ~±=(15) G=(gu sin i+(1/2)guh sin i·i_x+(τq/ρh))/(u~2-ghcosi).(23) The superscripts + and- refer to the pertinent system of characteristics. For flat-bottom,frictionless channels,G=0;(21b)and(22b)indicate that at cor- responding points the tangents of C~+ and C~- are perpendicular to those of Γ~- and Γ~+ respectively and the Γ characteristics are systems of epicycloids. Taking into consideration the varying bottom slope along the x-direction and the bottom friction τ,it can be shown that the velocity vector still bissets the C characteristics(Fig.2) and both A~+ and A~- are given by q~2sin~2A=ghcosi.(20) With G different from zero,(21b)(22b)can no longer be integrated to give analytic forms and the angles between the tangents,φ~+ and φ~-,no longer equal to π/2.How- ever,based on these two equations,a graphical method is proposed,as illustrated in Fig. 6.

The authors divide the effect of the shape of a valley cross-section on the earthquake hydrodynamic pressures into two parts: one due to the solidity ratio s=A/Bh, or the cross-sectional area A divided by the free-surface width B and the depth h, and the other due to the width-depth ratio w=B/H. As the ordinary cross-sections of valleys are essentially symmetric, three basic shapes, namely, rectangle, semi-circle, and isosceles right triangle, are chosen for analysis. Earthquakes in both the longitudinal and...

The authors divide the effect of the shape of a valley cross-section on the earthquake hydrodynamic pressures into two parts: one due to the solidity ratio s=A/Bh, or the cross-sectional area A divided by the free-surface width B and the depth h, and the other due to the width-depth ratio w=B/H. As the ordinary cross-sections of valleys are essentially symmetric, three basic shapes, namely, rectangle, semi-circle, and isosceles right triangle, are chosen for analysis. Earthquakes in both the longitudinal and the laterally transverse direction with respect to the valley axis are treated. The fundamental equations and hypotheses used follow those of Westergaard and Werner, and also borrowed from the two authors are the expressions for pressures on rectangular and semicircular dam surfaces due to longitudinal earthquake. Equation (8) gives a definition of the wave number c per unit length as related to the density ρand the bulk elastic modulus K of the water, the velocity ν_s, of sound in the water, and the circular frequency ω and the period T of the assumed simple harmonic seismic waves. In Eqs. (9) and (10) are introduced for the pressures and moments on dam surfaces the pressure coefficient C_p, the total pressure coefficient C_p, the coefficient of moment about the water line C_(MZ) due to longitudinal earthquake; and the corres ponding C'_p, C'_p, and C'_(MZ) and the coefficient of moment about the center line C'_MY due to transverse earthquake. In these equations, the symbol α denotes the acceleration coefficient; γ, the specific weight of water; A'=A/2, half the symmetric area; and b=B/2, the half width. Equations (11a) to (11c) and (12a) to (12d) are the derived expressions of the various coefficients for rectangular surface; Eqs. (13a) to (13c) and (14a) to (14d) are the ones for semi-circular surface, with the reduced Eqs. (14'a) to (14'd) in the condition c=0; and Eqs. (15a) to (15c) and (16a) to (16c), and also the reduced Eqs. (16'a) to (16'c) for c=0, are for isosceles right triangular surface. Figure 2 shows the effect of s on the conditions of resonances, for which the upper and the lower two curves correspond respectively to the case of transverse and longitudinal earthquake. Here the units of h and T refer respectively to meter and second.In Fig. 5 is shown the effect of s on the magnitudes of the various pressure and moment coefficients when the compressibility of water is ignored. Although this figure is for ω=2, it is considered that C_p and C_(MZ) for longitudinal earthquake depend on s only,and Fig. 5 alone is sufficient for their estimation whatever be the value w. Figure 6 shows the effect of ω on the various coefficients for transverse earthquakes when c=0 and s=1. Because C' is influenced by both s and w, it is suggested that when a C' is to be estimated, it is first obtained from Fig. 5 for the given s and then multiplied by the corresponding one obtained from Fig. 6 for the given w and again divided by C' for ω=2 from Fig. 6.

The design criteria for protecting a spillway surface from cavitation erosion are discussed. The spillway profile first of equal cavitation number and then of equal safety pressure are analyzed, and two corresponding curves for the protecting purpose obtained. Although the fundamental equations derived for their expressions come to be identical, the constants contained therein are different. A numerical integration method for solving the equations is given, which simple in form gives accurate results,...

The design criteria for protecting a spillway surface from cavitation erosion are discussed. The spillway profile first of equal cavitation number and then of equal safety pressure are analyzed, and two corresponding curves for the protecting purpose obtained. Although the fundamental equations derived for their expressions come to be identical, the constants contained therein are different. A numerical integration method for solving the equations is given, which simple in form gives accurate results, and is expected to be common in use for calculating spillway profiles.Finally, demonstrative calculations are made for two actual spillway profiles, both of high dams. Evidently, the profiles calculated are very effective even with large unit-width overflow discharges in avoiding erosion from cavitation, because the cavitation number of the overflow can be raised by 40~49% and the effective drop of head diminished by 24~40% by the profiles.