A Study For Incompressible Fluid Flow Heat Transfer Simulation Algorithm

In order to improve the solving speed and numerical precision of incompressible flow and heat transfer on unstructured grid,the two-dimensional and the three-dimensional numerical programs were developed based on the analysis of SIMPLE algorithm,and some conclusions were acquired towards the acceleration convergence of algorithm and the appliance of unstructured grid.And the major contributions of the present work are as follows:Firstly,an algorithm amelioration for SIMPLE was introduced where the stability and steps of the present algorithm were almost same with SIMPLE while the convergence rate was faster than SIMPLE algorithm and with the smaller velocity sub-relaxation factor,the performance is better.With the velocity sub-relaxation factor from 0.5 to 0.85,the number of iteration cycles or CPU computing time consumed to obtain the convergent solution was about 60% to 100% of the one in SIMPLE.When the velocity relaxation factor increased from 0.85 to 1,it converged at a similar rate with SIMPLE.The thing mostly distinguished from SIMPLE was the assumption that the velocity satisfied with the mass conservation also met the discrete momentum conservation,not the discrete momentum conservation confined to the velocity sub-relaxation factor.And the satisfaction level of discrete momentum equation was taken into account in the process of solving the mass conservation equation.Thus the velocity from pressure-correction equation not only met the mass conservation equation,but also satisfied the conservation of momentum to some extent.Comparison among the amelioration,SIMPLE,SIMPLER,PISO and IDEAL showed that with the same velocity sub-relaxation factor,the amelioration can consume the least CPU time to achieve convergence.Secondly,toward the relatively less research report on the staggered technique in the unstructured grid,the dependency problem between the numerical results and the velocity sub-relaxation factor,the high memory expenditure matter and the problem of two velocity fields required to satisfy the momentum conservation velocity and the mass conservation velocity,a semi-staggered grid strategy was introduced where the pressure was stored on the vertex of cells and the velocity components and other scale variables were saved on the central of cells.The computer code verified the feasibility of this method and several classical cases proved that it can be used to solve flow problem with high Reynolds number and inclined grid.This semi-staggered mesh technique was independent of the mesh shape,but most economic on the two-dimensional triangular mesh and the three-dimensional tetrahedral mesh.The number of vertex in the triangular mesh was about half of the number of cells and the ratio on the tetrahedral mesh is about one in five or one in six,and the order of the pressure-correction equation also decreased with the same proportion.The disadvantages were that the pressure-correction equation had a large bandwidth and the number of non-zero elements in each line of each node was too large and different with each other,where the compression matrix storage technique was necessary in this model.In this paper,all variables in the discrete equations were stored by compression,and GMGRES with Saad incomplete ILU decomposition was used to solve all discrete equations,which unified all variables’ storage and the solution modules.Thirdly,the technique to apply the least square method on the cell interface on the discrete form of the diffusion term and the convection term,which avoided the interpolation problem of the gradient on element interface.It was also convenient to construct the convection term and diffusion term with high-order precision scheme in this model and this technique can perform as a reference for the construction of high-order scheme on the unstructured grid.The coefficient obtained by the least square method can be used to calculate the density,the diffusion coefficient and the velocity,where the calculation of those variables both needed the interface value.Numerical experiments including the pure diffusion case and the convective diffusion case both with analytical solution verified the least square method on the cell interface to solve the general convection-diffusion equation.Compared with the same order least-square method based on the center of cell commonly used in the literature,the L2-norm error of solving the diffusion equation on the orthogonal grid with this method was 1to 2 order of magnitude smaller,and the error was at least one order of magnitude less on the oblique or more oblique grid.When solving the convection-diffusion equation with the least-square method based on the center of cell and enhancing the intensity of convection,the error increased with the growth of the Peclet number,and rapidly came to the point with no reference value.Though there was a slight increase of the error of the least square method on the cell interface,this error can always keep at the same order of magnitude within the four order of magnitude change of the Peclet number,which indicated that the technique was not sensitive to the Peclet number.On the basis of the verified interface least squares interpolation method,3D SIMPLE code was developed with Date’s velocity-pressure coupling scheme on the collocated grid and the numerical results from the case of natural convection in square cavity and the case of lid-driven flow showed consistent with the ones in the literature.The biggest advantage of the least-square method on the interface of cell was that most of the interpolation calculation,including diffusion flux and many other computational items,can be completed in the grid preprocessing stage.There are only interpolation coefficients stored in the memory and the interpolation coefficient and the calculation formula of all variables are the same.Thus 2D and 3D code possessed the same operations in the discrete process,which greatly simplifies the programming complexity and facilitated the code maintenance and update.Finally,based on above algorithm researches,the calculation showed the evolution of temperature field in oil layer and pointed out the end time of the pre-heating stage in SAGD production with steam injection into the heavy oil.The coupling flow and heat transfer for the pre-heating stage was solved by the vorticity-flow function method and the least square method on the cell interface to analyze the influence of factors such as the distance among two wells and the viscosity of heavy oil on the pre-heating time.For the unsteady underground and surface flow in gas storage reservoir,the temperature change and the thermal equilibrium of the reservoir during the gas injection and the gas production period were analyzed with the least square method to process the convection term and diffusion term and with the deferred correction.The results showed that with the increase of the cycle time of gas storage operation,the temperature near the well decreased gradually,while the temperature in the area far away the well increased slightly.For the transient flow in gas injection of surface pipeline,a numerical method toward the one-dimensional unsteady flow and heat transfer equation based on finite volume method was developed,where the solving process reflected the influence of axial temperature gradient and the velocity gradient and the influence of kinetic energy of gas flow were also considered into the model.The algorithm and code were verified by the common cases in the literature and the results from the code were consistent with the ones in the literature.