Based on the derivation of the continuity equation of water hammer, the modification coefficient used in the wave speed formula is discussed thoroughly. Mistakes presented in current formulas are analyzed. Finally in ...Based on the derivation of the continuity equation of water hammer, the modification coefficient used in the wave speed formula is discussed thoroughly. Mistakes presented in current formulas are analyzed. Finally in this paper a summarizing comment is given on the problem that whether or not the slope term Vsina should be included in the characteristic equations.展开更多
In engineering practice, single-phase water hammer models are still employed to analyze the water hammer of solid-liquid flow. According to the characteristics of solid-liquid flow, continuity equations and momentum e...In engineering practice, single-phase water hammer models are still employed to analyze the water hammer of solid-liquid flow. According to the characteristics of solid-liquid flow, continuity equations and momentum equations of pseudo-homogeneous flows are deduced, and a pseudo-homogeneous water hammer model is thus built and verified with experiment results. The characteristics of solid-liquid flow’s viscosity, resistance and wave velocity are considered in the model. Therefore, it has higher precision than a single-phase model.展开更多
The present work provides a novel method for calculating vertical velocity based on continuity equations in a pressure coordinate system.The method overcomes the disadvantage of accumulation of calculating errors of h...The present work provides a novel method for calculating vertical velocity based on continuity equations in a pressure coordinate system.The method overcomes the disadvantage of accumulation of calculating errors of horizontal divergence in current kinematics methods during the integration for calculating vertical velocity,and consequently avoids its subsequent correction.In addition,through modifications of the continuity equations,it shows that the vorticity of the vertical shear vector(VVSV) is proportional to-ω,the vertical velocity in p coordinates.Furthermore,if the change of ω in the horizontal direction is neglected,the vorticity of the horizontal vorticity vector is proportional to-ω.When ω is under a fluctuating state in the vertical direction,the updraft occurs when the vector of horizontal vorticity rotates counterclockwise;the downdraft occurs when rotating clockwise.The validation result indicates that the present method is generally better than the vertical velocity calculated by the ω equation using the wet Q-vector divergence as a forcing term,and the vertical velocity calculated by utilizing the kinematics method is followed by the O'Brien method for correction.The plus-minus sign of the vertical velocity obtained with this method is not correlated with the intensity of d BZ,but the absolute error increases when d BZ is >=40.This method demonstrates that it is a good reflection of the direction of the vertical velocity.展开更多
The differential equations of continuum mechanics are the basis of an uncountable variety of phenomena and technological processes in fluid-dynamics and related fields.These equations contain derivatives of the first ...The differential equations of continuum mechanics are the basis of an uncountable variety of phenomena and technological processes in fluid-dynamics and related fields.These equations contain derivatives of the first order with respect to time.The derivation of the equations of continuum mechanics uses the limit transitions of the tendency of the volume increment and the time increment to zero.Derivatives are used to derive the wave equation.The differential wave equation is second order in time.Therefore,increments of volume and increments of time in continuum mechanics should be considered as small but finite quantities for problems of wave formation.This is important for calculating the generation of sound waves and water hammer waves.Therefore,the Euler continuity equation with finite time increments is of interest.The finiteness of the time increment makes it possible to take into account the quadratic and cubic invariants of the strain rate tensor.This is a new branch in hydrodynamics.Quadratic and cubic invariants will be used in differential wave equations of the second and third order in time.展开更多
The authors analyze continuity equations with Stratonovich stochasticity,■ρ+divh[ρo(u(t,x)+∑_(i=1)^(N)a_(i)(x)w_(i)(t))]=0defined on a smooth closed Riemannian manifold M with metric h.The velocity field u is pert...The authors analyze continuity equations with Stratonovich stochasticity,■ρ+divh[ρo(u(t,x)+∑_(i=1)^(N)a_(i)(x)w_(i)(t))]=0defined on a smooth closed Riemannian manifold M with metric h.The velocity field u is perturbed by Gaussian noise terms Wi(t),:WN(t)driven by smooth spatially dependent vector fields a1(x),...,aN(x)on M.The velocity u belongs to L_(t)^(1)W_(x)^(1,2)with divh u bounded in Lf,for p>d+2,where d is the dimension of M(they do not assume div_(h) u∈L_(t,x)^(∞)).For carefully chosen noise vector fields ai(and the number N of them),they show that the initial-value problem is well-posed in the class of weak L^(2) solutions,although the problem can be ill-posed in the deterministic case because of concentration effects.The proof of this“regularization by noise”result is based on a L^(2) estimate,which is obtained by a duality method,and a weak compactness argument.展开更多
Based on Hillert's 3D grain growth rate equation, the grain growth continuity equation was solved. The results show that there are an infinite number of 3D quasi-stationary grain size distributions. This conclusio...Based on Hillert's 3D grain growth rate equation, the grain growth continuity equation was solved. The results show that there are an infinite number of 3D quasi-stationary grain size distributions. This conclusion has gained strong supports from results of different computer simulations reported in the literature.展开更多
文摘Based on the derivation of the continuity equation of water hammer, the modification coefficient used in the wave speed formula is discussed thoroughly. Mistakes presented in current formulas are analyzed. Finally in this paper a summarizing comment is given on the problem that whether or not the slope term Vsina should be included in the characteristic equations.
文摘In engineering practice, single-phase water hammer models are still employed to analyze the water hammer of solid-liquid flow. According to the characteristics of solid-liquid flow, continuity equations and momentum equations of pseudo-homogeneous flows are deduced, and a pseudo-homogeneous water hammer model is thus built and verified with experiment results. The characteristics of solid-liquid flow’s viscosity, resistance and wave velocity are considered in the model. Therefore, it has higher precision than a single-phase model.
基金National Key Basic Research Development Program"973"(2013CB430103,2009CB421503)National Natural Science Funding(41375058,41530427)State Key Laboratory of Severe Weather,Chinese Academy of Meteorological Sciences(2015LASW-A07)
文摘The present work provides a novel method for calculating vertical velocity based on continuity equations in a pressure coordinate system.The method overcomes the disadvantage of accumulation of calculating errors of horizontal divergence in current kinematics methods during the integration for calculating vertical velocity,and consequently avoids its subsequent correction.In addition,through modifications of the continuity equations,it shows that the vorticity of the vertical shear vector(VVSV) is proportional to-ω,the vertical velocity in p coordinates.Furthermore,if the change of ω in the horizontal direction is neglected,the vorticity of the horizontal vorticity vector is proportional to-ω.When ω is under a fluctuating state in the vertical direction,the updraft occurs when the vector of horizontal vorticity rotates counterclockwise;the downdraft occurs when rotating clockwise.The validation result indicates that the present method is generally better than the vertical velocity calculated by the ω equation using the wet Q-vector divergence as a forcing term,and the vertical velocity calculated by utilizing the kinematics method is followed by the O'Brien method for correction.The plus-minus sign of the vertical velocity obtained with this method is not correlated with the intensity of d BZ,but the absolute error increases when d BZ is >=40.This method demonstrates that it is a good reflection of the direction of the vertical velocity.
文摘The differential equations of continuum mechanics are the basis of an uncountable variety of phenomena and technological processes in fluid-dynamics and related fields.These equations contain derivatives of the first order with respect to time.The derivation of the equations of continuum mechanics uses the limit transitions of the tendency of the volume increment and the time increment to zero.Derivatives are used to derive the wave equation.The differential wave equation is second order in time.Therefore,increments of volume and increments of time in continuum mechanics should be considered as small but finite quantities for problems of wave formation.This is important for calculating the generation of sound waves and water hammer waves.Therefore,the Euler continuity equation with finite time increments is of interest.The finiteness of the time increment makes it possible to take into account the quadratic and cubic invariants of the strain rate tensor.This is a new branch in hydrodynamics.Quadratic and cubic invariants will be used in differential wave equations of the second and third order in time.
基金supported by the Research Council of Norway through the projects Stochastic Conservation Laws (No. 250674)(in part) Waves and Nonlinear Phenomena (No. 250070)
文摘The authors analyze continuity equations with Stratonovich stochasticity,■ρ+divh[ρo(u(t,x)+∑_(i=1)^(N)a_(i)(x)w_(i)(t))]=0defined on a smooth closed Riemannian manifold M with metric h.The velocity field u is perturbed by Gaussian noise terms Wi(t),:WN(t)driven by smooth spatially dependent vector fields a1(x),...,aN(x)on M.The velocity u belongs to L_(t)^(1)W_(x)^(1,2)with divh u bounded in Lf,for p>d+2,where d is the dimension of M(they do not assume div_(h) u∈L_(t,x)^(∞)).For carefully chosen noise vector fields ai(and the number N of them),they show that the initial-value problem is well-posed in the class of weak L^(2) solutions,although the problem can be ill-posed in the deterministic case because of concentration effects.The proof of this“regularization by noise”result is based on a L^(2) estimate,which is obtained by a duality method,and a weak compactness argument.
基金the National Natural Science Foundation of China (Grant No. 50171008).
文摘Based on Hillert's 3D grain growth rate equation, the grain growth continuity equation was solved. The results show that there are an infinite number of 3D quasi-stationary grain size distributions. This conclusion has gained strong supports from results of different computer simulations reported in the literature.
