The localized differential quadrature (LDQ) method is a numerical technique with high accuracy for solving most kinds of nonlinear problems in engineering and can overcome the difficulties of other methods (such as di...The localized differential quadrature (LDQ) method is a numerical technique with high accuracy for solving most kinds of nonlinear problems in engineering and can overcome the difficulties of other methods (such as difference method) to numerically evaluate the derivatives of the functions.Its high efficiency and accuracy attract many engineers to apply the method to solve most of the numerical problems in engineering.However,difficulties can still be found in some particular problems.In the following study,the LDQ was applied to solve the Sod shock tube problem.This problem is a very particular kind of problem,which challenges many common numerical methods.Three different examples were given for testing the robustness and accuracy of the LDQ.In the first example,in which common initial conditions and solving methods were given,the numerical oscillations could be found dramatically;in the second example,the initial conditions were adjusted appropriately and the numerical oscillations were less dramatic than that in the first example;in the third example,the momentum equation of the Sod shock tube problem was corrected by adding artificial viscosity,causing the numerical oscillations to nearly disappear in the process of calculation.The numerical results presented demonstrate the detailed difficulties encountered in the calculations,which need to be improved in future work.However,in summary,the localized differential quadrature is shown to be a trustworthy method for solving most of the nonlinear problems in engineering.展开更多
Based on the second viscosity, the local differential quadrature （LDQ） method is applied to solve shock tube problems. It is shown that it is necessary to consider the second viscosity to calculate shocks and to sim...Based on the second viscosity, the local differential quadrature （LDQ） method is applied to solve shock tube problems. It is shown that it is necessary to consider the second viscosity to calculate shocks and to simulate shock tubes based on the viscosity model. The roles of the shear viscous stress and the second viscous stress are checked. The results show that the viscosity model combined with the LDQ method can capture the main characteristics of shocks, and this technique is objective and simple.展开更多
The boundary integral method（BIM） is used to simulate the 3-D gas bubble, generated within the two bubble pulsation periods in proximity to a free surface in an inviscid, incompressible and irrotational flow. The pr...The boundary integral method（BIM） is used to simulate the 3-D gas bubble, generated within the two bubble pulsation periods in proximity to a free surface in an inviscid, incompressible and irrotational flow. The present method is well validated by comparing the calculated shapes of the bubble and the free surface with both the experimental results and the numerical ones obtained by the Axisymmetric BIM code. The expansion, the collapse of the gas bubble and the further evolution of the rebounding non-spherical bubble are simulated. The various variation patterns of the free surface spike and the bubble centroid for different standoff distances, the buoyancy parameters and the strength parameters are obtained to reveal the nonlinear interaction between the bubble and the free surface. The amplitude of the second maximum bubble volume and the four typical patterns of the bubble jet and the free surface spike are examined in the context of the standoff distance. The large buoyancy is used to elevate the spray dome rather than the free surface spike.展开更多
文摘The localized differential quadrature (LDQ) method is a numerical technique with high accuracy for solving most kinds of nonlinear problems in engineering and can overcome the difficulties of other methods (such as difference method) to numerically evaluate the derivatives of the functions.Its high efficiency and accuracy attract many engineers to apply the method to solve most of the numerical problems in engineering.However,difficulties can still be found in some particular problems.In the following study,the LDQ was applied to solve the Sod shock tube problem.This problem is a very particular kind of problem,which challenges many common numerical methods.Three different examples were given for testing the robustness and accuracy of the LDQ.In the first example,in which common initial conditions and solving methods were given,the numerical oscillations could be found dramatically;in the second example,the initial conditions were adjusted appropriately and the numerical oscillations were less dramatic than that in the first example;in the third example,the momentum equation of the Sod shock tube problem was corrected by adding artificial viscosity,causing the numerical oscillations to nearly disappear in the process of calculation.The numerical results presented demonstrate the detailed difficulties encountered in the calculations,which need to be improved in future work.However,in summary,the localized differential quadrature is shown to be a trustworthy method for solving most of the nonlinear problems in engineering.
基金supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 50921001)the National Basic Research Program of China (973Program) (No. 2010CB832700)
文摘Based on the second viscosity, the local differential quadrature （LDQ） method is applied to solve shock tube problems. It is shown that it is necessary to consider the second viscosity to calculate shocks and to simulate shock tubes based on the viscosity model. The roles of the shear viscous stress and the second viscous stress are checked. The results show that the viscosity model combined with the LDQ method can capture the main characteristics of shocks, and this technique is objective and simple.
基金supported by the National Natural Science Foun-dation of China(Grant Nos.51221961,51279030)the National Key Basic Research Development Program of China(973 Pro-gram,Grant Nos.2013CB036101,2010CB832704)the Fundamental Research Funds for the Central Universities(Grant No.L2012016)
文摘The boundary integral method（BIM） is used to simulate the 3-D gas bubble, generated within the two bubble pulsation periods in proximity to a free surface in an inviscid, incompressible and irrotational flow. The present method is well validated by comparing the calculated shapes of the bubble and the free surface with both the experimental results and the numerical ones obtained by the Axisymmetric BIM code. The expansion, the collapse of the gas bubble and the further evolution of the rebounding non-spherical bubble are simulated. The various variation patterns of the free surface spike and the bubble centroid for different standoff distances, the buoyancy parameters and the strength parameters are obtained to reveal the nonlinear interaction between the bubble and the free surface. The amplitude of the second maximum bubble volume and the four typical patterns of the bubble jet and the free surface spike are examined in the context of the standoff distance. The large buoyancy is used to elevate the spray dome rather than the free surface spike.
