Theoretical Study On The Generation Of The Undular Bore Based On The KdV-Burgers Type Equation

Estuary regions tend to be densely populated and economically developed areas.After a tsunami occurs,rivers will become ideal high-speed passages for tsunami waves to invade upstream,and thus related disaster prevention and mitigation work is particularly important.Nevertheless,the related research on the problem of tsunami ascending into rivers,especially theoretical researches,are limited.Tsunami upstream propagation in rivers would generate the undular bore,i.e.,a leading near-solitary wave followed by a series of wave trains,but the underlying fluid mechanics influenced by the internal/external factors are still unclear and need further investigation.This study uses the theoretical analyzing method.Firstly,the uniformity of the first-order Korteweg-de Vries(KdV)-type equations and their solutions are investigated.Then,the influence of the bottom topography on wave profiles is discussed.Finally,based on the KdVBurgers equation,a theoretical model for undular bore generation in the circumstance of tsunami upstream propagation in rivers is established,which reveals the influence of viscosity and bottom slope on the formation of undular bore.KdV-type equations derived by different methods have different coefficients.This study introduces a unified dimensionless frame and re-derived the KdV equation in terms of seven existing types of methods.Eliminating the influence of different dimensionless methods,the nonlinear term coefficients of the dimensionless KdV-type equations remain the same,and thus derive the same expressions of the cnoidal wave and solitary wave.Nevertheless,different coefficients of the first-order partial derivative are presented due to the influence of second-order quantities related to the derivation process,which leads to different solutions of the wave celerities and water particle velocities.Compared with the second-order solutions,this study suggested suitable first-order KdV equation under different nonlinearity effects.The present study derives a new first-order steady KdV type equation over a bottom topography of small-scale disturbance with the stream function method.Considering the firstorder wave celerity over the bottom topography of large-scale steplike is a constant,being independent of the bottom topography,a spatial correspondence relation between the equilibrium wave profile and the bottom topography can be obtained by a uniform velocity equaling to the wave celerity but with opposite direction is added to the water body.The near-soliton solution indicates that the equilibrium waveforms are ascribed to its stability adjustments to the uneven bottom.The adaption of wave profiles to the bottom topography is piecewise,whose segmentation follows the variation of the uneven bottom topography.In addition,the variation of every single soliton of the oscillatory wave solution with bottom topography is the same as that of the nearsoliton solution.The theoretical equilibrium wave profiles over the bottom topography can be used as the initial wave profiles of different numerical calculation models.Based on the KdV-Burgers type equation,this study presents the influence of the viscosity and the bottom slope on the generation of the undular bore.Upon which,a theoretical model for the countercurrent undular bore propagation is established in the circumstance of tsunami upstream propagation in rivers.With a newly introduced parameter scaling the effects of the viscosity and the bottom slope,possible horizontal mean velocity distributions at the meeting area of the tsunami and river flow are specified,verified with the undular bore generation conditions,and finally confirmed to be the quasi-Poiseuille type with a convexly parabolic profile.The present model could also be applied to the horizontal bottom situation,showing broad applicability.The parametric analyses demonstrated that the competition between the dispersion(oscillation enhancement)and damping(oscillation attenuation)determines the steady undular bore profile.With an extra term from the newly introduced parameter,the present model describing the scenario of tsunami upstream propagation in rivers tends more to the KdV equation,whereas the classical KdV-Burgers equation describing the river flooding scenario biases more to the Burgers equation.Quantitative differences between the above two models are also revealed from the numerical simulation,indicating that even with the same basic conditions,the damping coefficient of the new model is significantly great than that of the classical KdV-Burgers model and derives an undular bore with less oscillation intensity and small wave amplitude.In addition,this study derives a modified steady KdV-Burgers equation by considering the influence of different nonlinearities and traveling speeds to the undular bores.It is confirmed that the fixed coefficient of the nonlinear term of the traditional steady KdV Burgers equation is selected to improve the accuracy of the approximation solution for small damping,while the resolution of the modified model shows that the error of the approximate solution caused by the variable nonlinear term coefficient is limited in the range of [-0.1,0.1].With a proper combination of the speed parameter and the viscous damping parameter,the modified model gives an adjustable theoretical undular bore profile,which breaks the limitation required in the traditional model,i.e.,the undular bore is solely determined by the viscous damping parameter.According to the phase-plane analysis,a new criterion for identifying two kinds of undular bores,i.e.,the strong bore and the weak bore,is proposed.Finally,the application of the modified model to fit with experiments validates the physical meaning of this newly introduced parameter.