Fourth-order Fully Nonlinear Boussinesq Equations And Their Simplified Models

Recently,the great efforts have been made on Boussinesq equations,and correspondingly,the Boussinesq-type wave models are widely used in coastal engineering. But there are still some limitations,e.g.,lower dispersion accuracy leads to model only applicable for intermediate or shallow water region and weak nonlinearity makes model failure when describing strong nonlinear wave motion.The present paper aims at higher order Boussinesq-type nonlinear wave models,the main contents of the paper are as following:(1)Based on Euler equations,a new set of fully nonlinear higher order Boussinesq equations(BouN4D4)is derived.The equations are accurate to O(μ~4,ε~5μ~4)(μ=kh andε=A/h are two parameters denoting dispersion and nonlinear properties,and k,h and A present wave number,characteristic water depth and characteristic wave amplitude respectively).The equations possess the following characteristics:the equations have dispersion accurate to Padé[4,4]and applicable for deep water cases;fully nonlinearity leads to the model applicable for strong nonlinear wave motion;adoption ofσ-transformation during derivation makes the equations are applicable for rapidly varying bottom problems. Two models,BouN2D4 and BouN2D2,which can be regarded as the simplified versions of BouN4D4,are also presented.Both of them are fully nonlinear up to O(μ~2)while different in dispersion accuracy,the former is approximated to Padé[4,4]and the latter to Padé[2,2]. Theoretical analyses are conducted for these three sets of Boussinesq equations to demonstrate their properties,including dispersion,shoaling and nonlinearity.Numerical models based on these three sets equations are built,and the processes of regular waves propagation over submerged bar and wave group evolution in deep water are simulated and the computational results are inter-compared to show the effects of dispersion and nonlinearity on numerical results.After extended to take wave breaking and moving shoreline into account,1D model of BouN4D4 and 2D model of BouN4D4 and BouN2D4 are used to simulate the wave propagation and breaking in the near-shore region and the numerical results are verified against several experimental data sets.Additionally,model BouN2D4 is enhanced to improve its ability for wave Bragg reflection problem and the numerical results show that the improvement is effective.(2)Though fully nonlinear up to O(μ~4)and in a compact form,model BouN4D4 is still lengthy and this brings us with difficulty in the processes of model building and application use.Compared to BouN4D4,models BouN2D4 and BouN2D2 are simpler in forms but still have higher-order derivatives in the governing equations,which will also be hard to deal with in numerical scheme.To overcome this problem,another three sets of higher-order Boussinesq equations are derived through introducing the mild slope assumption. They are referred to as BouN4P6-1,BouN2P6-1 and BouN2P2 and the order accuracy of dispersion and nonlinearity of them are the same as those for BouN4D4,BouN2D4 and BouN2D2.The ftrst set is derived by introducing nonlinear terms of order O(μ~4)to the six-parameter Boussinesq equations,and the latter two are obtained by improving the six-parameter model and two-parameter model to fully nonlinear up to O(μ~2). Theoretical analyses are conducted for the above equations and the 1D numerical models are built and the effects of mild slope on the numerical results are studied in detail;(3)Another two sets of Boussinesq equations are presented to discuss the effects of numbers of free parameters on models' properties.These two models have the same order accuracy in dispersion and nonlinearity as model BouN2P6-1 and BouN4P6-1,respectively but only contain four parameters;(4)Through introducing new nonlinear terms into the expression of computation velocity,a new set of higher order Boussinesq is derived,which can be regarded as the extension of six-parameter equations,the new equations possess better nonlinear properties, especially the third order nonlinearity than those in Zou.The above method is also used for models BouN2P4 and BouN4P4 to improve their nonlinear properties.The Theoretical analyses show that the method adopted is effective on improving models' nonlinear properties;(5)The best approximation of dispersion in the above equations is the Padé[4,4] expansion of the exact of linear dispersion relation,and this limit the application of models in much deeper water.Hence,two sets of higher-order Boussinesq models are derived by firstly taking the velocity at the sea bottom and by adopting mild slope assumption,and then by introducing the computational velocity.Resulting from the derivation,two sets equations with simpler expressions and Padé[6,6]and Padé[8,8]dispersions are obtained,respectively. Theoretical analyses are conducted for them;(6)By neglecting all dispersive terms and higher order nonlinear terms,fully nonlinear shallow water equations are obtained.The numerical scheme based on Finite Volume Method (FVM)is built and the corresponding numerical model is verified against analytical and experimental data;(7)The above equations,including Boussinesq-type equations and fully nonlinear shallow water equations,are not applicable when rotational motion exists.To deal with this problem,a new model is derived by firstly dividing the wave velocity into two pars:potential part and rotational part and then by repeating the derivation process of model BouN4D4.The properties of the model are analyzed and the method of determining vortex distribution is discussed.