Spectral Element Methods For Incompressible Fluid Flow And Linear Stability Analysis Of Flow

The main research content of this thesis is numerical simulation of the problems influid mechanics by using spectral element method (SEM). The general idea is as follow.Firstly, primary algorithms of SEM, which is based on Cartesian and polar coordinatesystems respectively, are established and investigated. Some numerical examples withanalytical solutions are used for validating the precision of the methods and theeffectiveness of the program. Secondly, by combining SEM and time splitting method,unsteady incompressible Navier-Stokes equations are solved with different coordinatesystems. The accuracy of SEM is compared with other numerical methods. Lastly,linear stability analysis of lid-driven cavity flow is performed, where finitely longmodel is employed and the effect of various spanwise (i.e. z direction) length for theflow instability is studied. The instability mechanism is also studied via energy analysis.Based on the idea above, specific tasks have been accomplished and correspondingprograms have been developed as follows.(1) Chebyshev and Legendre SEM arereproduced, and Legendre and Fourier-Legendre SEM are proposed for solvingPoisson-type equations in polar coordinate. For SEM in polar coordinate, Legendreexpansion is adopted in radial direction, where Gauss-Radau points are chosen for theelement involving the origin and Gauss-Lobatto points are chosen for other elements.Fourier polynomial is applied in azimuthal direction. A number of Poisson-typeequations with analytical solution subject to the Dirichlet or Neumann condition arecomputed for validating accuracy and effectiveness of the method.(2) A number ofunsteady incompressible Navier-Stokes equations are solved by SEM and time splittingmethod, where time term is discretized by the time splitting technique to obtaincorresponding Poisson-types equations, which are solved by SEM. As a result, thelid-driven cavity flow, natural convection and free-surface flow are computed inCartesian coordinate, and the numerical results are compared with those in references orby finite volume method (FVM) to validate the accuracy and effectiveness of themethod. In polar coordinate, the wall driven flow on disk is calculated byFourier-Legendre SEM to investigate the effect of accelerated crucible rotationtechnique (ACRT) on the concentration homogenization of solution. Firstly, thenumerical isotope model is adopted to study the respective effect of ACRT and steadycrucible rotation on the concentration homogenization of isotope fluid, where the solution concentration is only determined by convection. The standard derivation ofsolution concentration is used to evaluate homogeneous level. Then numericalaccuracies of SEM and finite difference method (FDM) are studied. Secondly, the effectof ACRT on solution concentration homogenization is investigated to apply inhigh-temperature solution crystal growth method, where solution concentration isdetermined by both convection and diffusion. Six typical ACRT modes with differenttime periods are chosen for simulation to find optimum mode and time period byestimating the standard derivation of solution concentration. The results are expected toguide experiments.(3) The instability characteristic of lid-driven cavity flow is studiedbased on SEM and linear stability analysis theory, where the cavity (i.e. x-y plane)subjects to non-slip or slip boundary condition. The basic strategy is as follow. Thetwo-dimensional base flow is perturbed by a three-dimensional small disturbance, thenthe total velocity and pressure are substituted into the control equations and linearizing,and the disturbance is expressed as the normal mode. At last, the control equations forthe disturbance are obtained. Using the spectral element discretization, and then ageneralized eigenvalue problem is constituted from the weak forms of the disturbanceequations. The Arpack library is used to compute a few eigenvalues with largest realpart derived form the generalized eigenvalue problem. For different Reynolds number(Re) and wave number, the largest real part of eigenvalues determined the criticalparameter values of the flow instability. Then energy analysis is performed for thecritical flow to investigate the instability mechanism with different spanwise lengths.By computing various problems and analyzing their corresponding numericalresults, the results indicate following conclusions:①The present SEM is a high-precision numerical method with even relative fewernodes. For the SEM in polar coordinate, the1/r singularity at r=0can be avioidedeffectively by using the Gauss-Radau type quadrature points at the element includingthe pole. The clustering of collocation points near the pole can be prevented through thetechnique of domain decomposition to alleviate time step restriction for solvingtime-dependent problem with explicit time schemes.②The incompressible fluid flow is successfully solved by SEM and time splittingmethod in Cartesian coordinate. Firstly, two-dimensional unsteady Burgers equationswith analytical solution are calculated to validate the feasibility and high accuracy of themethod. Secondly, lid-driven cavity flow, natural convection and free-surface flow arecomputed and compared with the results in references or solved by FVM, and the results show good agreement with them. So the present SEM can be used to solve fluidflow correctly, and builds the foundation of its applications to the fluid flow in polarcoordinate and stability analysis.③The comparison results between Fourier-Legendre SEM and FDM reveal thatfirst-order upwind difference scheme (FUD) has large errors due to the severe numericaldiffusion, and the numerical diffusion can only be alleviated slightly by increasingnodes. A little severe numerical diffusion is also existed in the second-order upwindscheme (SUS), but can be alleviated effectively by refining the mesh. However, there islittle numerical diffusion in the SEM, and the numerical solution of the steady cruciblerotation almost coincides with the theoretical analysis; moreover, the standard deviationchanges periodically in the accelerated rotation, and the amplitude of the standarddeviation does not shift along the time, which agrees well with the physical propertiesof the mathematical model. The results indicate the SEM is a high-precision numericalmethod with even fewer nodes with fine stability and convergence.④The effect of ACRT on solution concentration homogenization is investigatedby Fourier-Legendre SEM in high-temperature solution crystal growth method. Thenumerical results reveal that the optimum ACRT mode for concentrationhomogenization is a symmetrically trapezoidal mode with bidirectional rotation, and theoptimum dimensionless time period is T=0.1. Full mixing of the solution is determinedby both diffusion and convection in the present model, and convection can speed up theglobal mixing, and in the meantime, change the local concentration gradient, whichspeeds up the diffusion.⑤The instability characteristic of finitely long lid-driven cavity flow is studied,where the cavity (i.e. x-y plane) subjects to non-slip boundary condition. From theresults we can conclude that the critical Re and wave numer are818.43and3for thecubical cavity respectively. It is a stationary instability. The flow at critical state is morevisible near the cavity wall than in the center of the cavity. The most unstable area isnear the upstream rigid wall through energy analysis. This coincides well with theresults of the infinitely long model where period boundary conditions are adopted inspanwise direction. The instability mechanism of the flow is associated with centrifugalinstability related to a stationary TGL (Taylor-Goertler-like) mode. Then, the flowinstability for integer Λ is also studied. Two predictions are proposed based on thenumerical results to estimate critical parameters for different Λ. And the some examplesfor1<Λ<2are used to validate the proposed predictions. The results are compared with those by direct numerical simulation (DNS), and good agreement is obtained. Only alittle difference exists for Λ=1.8. Maybe the different accuracy of the method leads tothe error. All of the the flow are via stationary instabilities for different Λ, and instabilitymechanism does not alter with Λ. At last, the form of control equations for disturbancewith Λ=2π are same as those in the infinitely long model. Both critical values of the twomodels agree very well with each other. The flow lose its stability via oscillatinginstability for k≤11, i.e. Hopf bifurcation, but stationary instability for k>11. This issimilar to the infinitely long model with period boundary conditions and verifies theresults are right to explain flow instability mechanism in this paper.⑥The instability characteristic of finitely long lid-driven cavity flow is studied,where the cavity (i.e. x-y plane) subjects to slip boundary condition. We can concludethat the critical Re is337.02for cubrical cavity, and is less than half of the value infinitely long model with slip boundary condition. This indicates that non-slip boundaryconditions can stabilize the flow. The flow loses its stability by stationary instability.The critical wave number k=2is validated by DNS. The energy analysis show that thepositive total energy transfer rate peaks in the area near the cavity walls and leads to theinstability. This is the instability mechanism. The instability manner is different fromthe finitely long model with non-slip boundary conditions because the corner eddiesvanish in the flow in the present model. The spanwise length has little impact on thecritical Re, and all of the flows belong to stationary instability. In the meantime, theflow is more unstable than that with non-slip boundary conditions, and this furtherindicates that non-slip boundaries can stabilize the flow.