The Numerical Solution Of Finite-Volume Based Incompressible Navier-Stokes Equations And Fluid-Structure Interaction Problems By Runge-Kutta Methods

This thesis studies the numerical solution of the finite volume discretization of the in-compressible Navier-Stokes equations on non-staggered grids,as well as the fluid-structure interaction(FSI)problems that can be formulated as three-field problems concerning a fluid subsystem,its moving finite volume grid and a structure subsystem.The incompressible Navier-Stokes equations are the governing equations of the motion of viscous fluid substances with relatively low flow velocities(less than 0.3 Mach),and belong to non-linear partial-differential equations in general.The numerical solution of the Navier-Stokes equations can be done mainly in two steps.First,a spatial discretization method(e.g.,the finite difference method,the finite volume method or the finite element method)is employed to approximate the partial-differential form the Navier-Stokes equations by a system of differential and al-gebraic equations regarding the velocities and pressures at some discrete locations defined in the computational domain.The discrete locations at which the variables to be calculated are defined by the so-called grid.Then,the system of the spatially discretized Navier-Stokes equa-tions is solved in the time domain to obtain the values of the velocities and pressures at discrete time-instants.The incompressible Navier-Stokes equations,when discretized in space by a finite differ-ence method,a finite element method or a finite volume method on staggered grids,result in an index 2 differential-algebraic system.However,for the finite volume method on non-staggered grids,a spatial discretization method that enj oy s a great popularity in engineering communities,such a conclusion cannot be drawn directly.The finite volume method on non-staggered grids requires a unique discretization technique known as the momentum interpolation(or Rhie-Chow interpolation)which estimates the discrete velocity field defined at cell-faces through a custom interpolation of the fully-discrete momentum equation.The cell-face velocity field can be interpreted as a new variable,in addition to the velocity and pressure fields defined at cell centroids,that participates in the spatial-discretization of the incompressible Navier-Stokes equations.The conventional scheme of the momentum interpolation stemmed from the pio-neering work of Rhie and Chow.However,either in their study or the following studies on the refinement of the momentum interpolation scheme,the influence of momentum interpola-tions on the differential-algebraic nature of the spatially-discretized Navier-Stokes equations has never been addressed.Before the differential-algebraic nature regarding the finite volume formulation of the incompressible Navier-Stokes equations on non-staggered grids is clearly identified,the convergence analysis regarding any temporal discretization method applied to it can hardly proceed.Moreover,the computational effort required by momentum interpolations largely depends on the complexity of a particular temporal discretization method.For the rel-atively complicated Runge-Kutta methods,the momentum interpolation can be computational demanding.To address the issues stated above,a new momentum interpolation scheme is established based on which the semi-discrete Navier-Stokes equations can be strictly viewed as a system of differential-algebraic equations of index 2.The two distinct features of the proposed momen-tum interpolation scheme from the others are:(i)the interpolation is carried out with respect to the semi-discrete(i.e.,the spatially discretized)momentum equation rather than the fully-discrete one;(ii)the diffusion and convection terms of the semi-discrete momentum equation to be interpolated to each cell-face are partitioned by a specific scheme whose formulation relies on particular coefficients defined at the cell-faces.The convergence of the proposed momentum interpolation scheme,as well as its ability to preserve the state of a uniform flow on stationary or moving grids are examined in the thesisThe numerical method employed for the time-marching of the semi-discrete Navier-Stokes system,which may also be referred to as the temporal discretization method or time-integrator in the literature of computational fluid dynamics(CFD),is another main focus of the thesis The domain on which the Navier-Stokes equations are spatially discretized can be stationary or time-varying due to the movement of boundaries.In the latter case,if the moving boundaries correspond to the wet surfaces of some deformable or moveable structures with an internal or surrounding fluid flow,then the motion equations of the structure and the moving grid that accommodates the relative motion of the moving boundaries should also be introduced dur-ing the spatial discretization.This results in a system of differential and algebraic equations describing the FSI system mentioned earlier,the solution of which yields the response of the fluid and structure at different time-instantsMulti-step methods and Runge-Kutta methods are the two commonly used classes of methods for solving differential-algebraic problems.Compared to the multi-step methods,the Runge-Kutta methods combine high order with good stability,allow for adaptive time step-ping and are self-starting.It is also noteworthy that either multi-step methods or Runge-Kutta methods are originally conceived for the numerical solution of ordinary differential equations(ODE's).However,the systems of DAE's have distinctive characteristics compared to the systems of ODE's,and are generally more difficult to solve.A numerical method may attain different orders of convergence in ordinary-differential problems and differential-algebraic problems.However,the differential-algebraic nature of the semi-discrete Navier-Stokes equations is not yet fully recognized in the CFD literature.Many researchers silently assume the order result of a numerical method for the temporal discretization of the Navier-Stokes equations is identical to that for ordinary-differential problems.At the present stage,the multi-step methods represented by the backward differentiation formula are popular in both commercial and open-source computational fluid dynamics packages,while,the use of Runge-Kutta methods for the solution of Navier-Stokes equations can only be found in academic fields,mainly due to the reason that the numerical difficulties arising from the implementation of Runge-Kutta methods can be a big obstacle in practice.Moreover,the consensuses on which subclasses of Runge-Kutta methods are more suitable for solving the Navier-Stokes equations and how to use them appropriately and more efficiently have not been reached in the literature yetDue to the reasons stated above,the pursuit of the low-storage,easy to implement and higher-order temporal discretization methods for the incompressible Navier-Stokes equations and fluid-structure interaction problems looks for either improvements of existing Runge-Kutta methods or developments of new ones.The detailed study is done mainly in three aspects:(1)For the time-marching of the semi-discrete incompressible Navier-Stokes equations on a stationary grid,which can be regarded as a quite special index 2 differential-algebraic problem,a new approach of applying the implicit Runge-Kutta(IRK)methods with a non-singular coefficient matrix is proposed.Compared to the standard direct approach,the pro-posed one significantly improves numerical efficiency and delivers higher-order pressures for cases concerning unsteady Dirichlet boundary conditions for velocities.Among all the IRK methods,the stiff-accurate diagonally implicit Runge-Kutta(DIRK)ones are appealing for their relatively low computational cost and some other favorable properties.The proposed approach allows them to attain the classical order(i.e.,the order of the local accuracy of a numerical method for ordinary differential problems)of convergence for both the velocity and pressure as long as the solutions of the considered Navier-Stokes systems exist and smoothly depend on time.Two classes of low-storage stiff-accurate DIRK methods based on the pro-posed approach are further developed to reduce the storage required by the implementation of the original schemes(2)For the solution of the semi-discrete incompressible Navier-Stokes equations on a moving grid and the three-field FSI problem,both of which can be regarded as a general non-linear index 2 differential algebraic problem,a new subclass of the partitioned Runge-Kutta method,named as the PEDIRK method in this thesis,is proposed.The PEDIRK method is developed mainly to improve the convergence property of the existing diagonally implicit type of Runge-Kutta methods for general non-linear index 2 problems.The distinct feature of the partitioned Runge-Kutta method,when compared to general Runge-Kutta methods,is that an additional set of Runge-Kutta coefficients and internal stage differential components is introduced which helps achieving higher orders of accuracy and convergence.The low storage and efficient implementation of the proposed PEDIRK method is also demonstrated to further strengthen its efficiency(3)The solution techniques for the fully-discrete incompressible Navier-Stokes equations obtained after temporal discretization are discussed.The discussion mainly focuses on the dif-ficulties encountered during the solution as a consequence of the non-linearity of the convec-tion term and the coupling between the velocity and pressure that lies in the incompressible Navier-Stokes equations.The goal of the discussion is to develop iterative solution algorithms that can attain a good balance between computational efficiency,numerical accuracy and soft-ware modularity.The effect of solution residual errors on the order of convergence of the temporal discretization is also investigatedIt is worth mentioning that the Runge-Kutta methods proposed in this thesis not only applies to the incompressible Navier-Stokes equations or the three-field FSI system considered in this thesis,but also to the more general differential-algebraic problems of mathematical interest.For the verification of the convergence of the proposed momentum interpolation schemes and Runge-Kutta methods,three groups of numerical experiments are conducted in this thesis The first group of the experiments concerns the simulation of the two-dimensional Taylor-Green Vortex with different boundary conditions and spatial discretization schemes.The sec-ond group of the experiments focuses on the flow past an oscillating circular which is either subjected to a forced transverse vibration or elastically mounted in both streamwise and trans-verse directions.The third group of the experiments concerns the identification of the flutter derivatives of an ideal plate.The numerical results obtained from these tests validate the ac-curacy and convergence of the proposed methods.