Numerical simulation of complex\ud viscoelastic flows using discontinuous\ud galerkin spectral/hp element methods

Doctoral thesis English OPEN
Claus, Susanne
  • Subject: QA

Viscoelastic flows are characterised by fast spatial and temporal variations in the solution\ud featuring thin stress boundary near walls and stress concentrations in the vicinity of\ud geometrical singularities. Resolving these fast variations of the fields in space and time\ud is important for two reasons: (i) they affect the quantity of interest of the computation\ud (e.g. drag force); and (ii) they are commonly believed to be associated with the numerical\ud breakdown of the computation. Traditional discretisation methods such as finite differences\ud or low-order finite elements require a large number of degrees of freedom to resolve\ud these variations. Spectral methods enable this issue to be resolved by defining spatial expansions\ud that are able to represent such variations with a smaller number of degrees of\ud freedom. However, such methods are limited in terms of geometric flexibility. Recently,\ud the spectral/hp element method (Karniadakis and Sherwin, 2005) has been developed in\ud order to guarantee both spectral convergence, and geometric flexibility by allowing the\ud use of quadrilateral and triangular elements. Our work is the first attempt to apply this\ud method to viscoelastic free surface flows in arbitrary complex geometries.\ud The conservation equations are solved in combination with the Oldroyd-B or Giesekus\ud constitutive equation using the DEVSS-G/DG formulation. The combination of this formulation\ud with a spectral element method is novel. A continuous approximation is employed\ud for the velocity and discontinuous approximations for pressure, velocity gradient\ud and polymeric stress. The conservation equations are discretised using the Galerkin\ud method and the constitutive equation using a discontinuous Galerkin method to increase\ud the stability of the approximation. The viscoelastic free surface is traced using an arbitrary\ud Lagrangian Eulerian method.\ud The performance of our scheme is demonstrated on the time-dependent Poiseuille flow in\ud a channel, the flow around a cylinder and the die-swell problem.
  • References (39)
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    3.10. One dimensional advection of square wave on domain Ω = [0, 10] subdivided into 10 equally-sized elements. . . . . . . . . . . . . . . . . . . . .

    3.11. Comparison of Galerkin projection for the Gaussian function of ((a), (b)) spectral/hp element method and ((c), (d)) linear finite element method for DOF = 90 and the corresponding L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

    3.12. Comparison of Galerkin projection for the hat function of ((a), (b)) spectral/hp element method and ((c), (d)) linear finite element method for DOF = 90 and the corresponding L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

    3.13. Comparison of Galerkin projection for the square wave function of ((a), (b)) spectral/hp element method and ((c), (d)) linear finite element method for DOF = 90 and the corresponding L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

    3.14. Comparison of the numerical results for the advection equation after 30000 timesteps with Δt = 10−4 using the continuous Galerkin method and the discontinuous Galerkin method for the spectral/hp element method for P = 8, Nel = 10 ((a), (b)) and the linear finite element method, i.e. P = 1, for Nel = 45 ((c), (d)) for the smooth Gaussian function and the L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . .

    3.15. Comparison of the numerical results for the advection equation after 30000 timesteps with Δt = 10−4 using the continuous Galerkin method and the discontinuous Galerkin method for the spectral/hp element method for P = 8, Nel = 10 ((a), (b)) and the linear finite element method, i.e. P = 1, for Nel = 45 ((c), (d)) for the hat function and the L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . . . . . . . . .

    3.16. Comparison of the numerical results for the advection equation after 30000 timesteps with Δt = 10−4 using the continuous Galerkin method and the discontinuous Galerkin method for the spectral/hp element method for P = 8, Nel = 10 ((a), (b)) and the linear finite element method, i.e. P = 1, for Nel = 45 ((c), (d)) for the hat function and the L2 and L∞ error for increasing DOF ((e), (f)). . . . . . . . . . . . . . . . . . . . . .

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