Study On Gradient Smoothing Method And Its Coupling With Lattice Boltzmann Method Based On Unstructured Meshes

Fluid mechanics has a significant influence on both theoretical investigation and engineering application of naval architecture and ocean engineering.With the developments of numerical methods and hardware of computers,computational fluid dynamics has been a crucial tool in predicting the hydrodynamic characteristics of structures for ocean engineering.With respect to complex geometries in the area of naval architecture and ocean engineering,it typically requires high time consuming using the structured meshes,while the methods based on the unstructured meshes perform much better.However,using the unstructured meshes,it gives rise to a core question that how to compute the partial differential equations.In addition,in ocean engineering,many multi-scale phenomena exist.Unfortunately,to well reproduce these phenomena,it is difficult to achieve the aims for high computational accuracy and low computational costs depending on one numerical method based on one specific length-scale.For this point,by coupling the macro-scale methods and the micro-scale methods,it would be expected to well resolve the above challenge.In recent years,Gradient Smoothing Method(GSM)has been developed based on the unstructured meshes,which is designed for complex geometries.For the GSM,it is flexible,accurate and insensitive to the irregularity of meshes due to the gradient smoothing technique,which is very suitable for the numerical simulations of naval architecture and ocean engineering.Therefore,the study of GSM with unstructured meshes is carried out in terms of computational fluid dynamics.The main contents are as follows:(1)Based on the unstructured meshes,the convection equation is resolved using the GSM.In this work,we review and analyze the existing main schemes of convection for details,especially the Total Variation Diminishing(TVD)and Normalized Variable Diagram(NVD).Besides,the relations between the TVD and NVD are also studied.However,the above two schemes are based on the structured meshes.Here,we extend the TVD and NVD schemes to the unstructured meshes.Therefore,an algorithm is proposed for calculating the variables at upwind points based on the gradient smoothing technique,which has been verified in the framework of GSM.Firstly,the cell for each upwind point can be decided based on the coordinate of the point.And then based on different smoothing domains,three types of GSM are proposed,i.e.,node-based gradient smoothing method(nGSM),midpoint-based gradient smoothing method(mGSM)and centroid-based gradient smoothing method(cGSM).The results of the proposed methods are accurate compared with other numerical methods in a wide range of conditions,such as the discontinuous and continuous problems,steady and transient problems.(2)In order to simulate the free surface under the unstructured meshes,the GSM is employed based on the Volume of Fluid(VOF)model.The governing equation in VOF model is the convection equation.Under the structured meshes,the geometric reconstruction algorithm is conventionally used.However,this method is difficult to be applied to the unstructured meshes.To overcome this problem,high-resolution schemes based on the NVD are considered in this paper,such as CICSAM(Compressive Interface Capturing Scheme for Arbitrary Meshes),FBICS(Flux-Blending Interface-Capturing Scheme)and CUIBS(Cubic Upwind Interpolation based Blending Scheme).All needed upwind variables are computed using the above discretized methods in the framework of cGSM.The numerical results show that the GSM use high-resolution schemes can simulate the free surface accurately,for which the surface shape and its sharpness are well reproduced.(3)Numerical simulations for incompressible flows are the key issue in computational fluid dynamics.The Navier-Stokes governing equation is solved to accurately predict the hydrodynamic performances for the ocean structures.The study for incompressible flows is carried out based on the GSM using the unstructured meshes.On the one hand,the unstructured meshes are helpful to reduce the time cost of grid generation.On the other hand,it can improve computational efficiency through optimizing meshes.For coupling the velocity and pressure field,the artificial terms are introduced in the governing equation.The convection term and viscous term are discretized by gradient smoothing operation.The flexibility of meshes generation is displayed and the accuracy and efficiency of the GSM are verified through numerical examples.Meanwhile,the GSM is applied to a classical hydrodynamic problem,i.e.,the flow across a bluff cylinder.The effects of different bluff geometries on the wake flow are manifested in this paper.Also,with the changes of Reynolds number,the changes of drag coefficients,lift coefficients and Strouhal number are discussed and compared with each other based on the circular and triangular cylinders,with both steady flows and unsteady flows.The results demonstrate that the GSM can provide accurate and efficient numerical solutions in basic hydrodynamic problems of marine engineering.(4)Based on the numerical simulation for incompressible flows,the Lattice Boltzmann Method(LBM)is introduced and the coupling algorithm,i.e.,the GSM-LBM,is proposed.In this coupling algorithm,for one multi-scale computational domain,it can be divided into two parts,i.e.,the macro and micro computation domains.For each method,the GSM is responsible for the macro domain and LBM is used in the micro domain.The flow information transmits between two algorithms in the coupling region.Different from the previous FVM-LBM coupling algorithm,the unstructured meshes are used in the macro computation domain.In addition,the coupling method between the two algorithms is also improved.The GSM-LBM has been evaluated,which shows that this method is accurate and efficient.The numerical show that GSM-LBM is more efficient than LBM.Importantly,not only the flow information of the entire domain but also the mesoscale flow details can be captured using the GSM-LBM.Due to the unstructured meshes used in GSM,the computational efficiency for the coupling algorithm can be further improved if the grids are optimized through local refinements.In this way,it is easy to simulate complex geometries.