Stabilization methods for FEM on moving meshes
4.5 Stabilization of the Navier–Stokes equations
4.5.3 Flow past an oscillating cylinder
To numerically validate derived stabilization scheme for the Navier–Stokes equations, a flow past oscillating and rotating elliptic cylinder is considered. The rotation is added to the standard oscillating cylinder problem in order to emphasize the change in triangle area (the main source of SCL problematics) during the mesh motion (see Figure 4.9).
Linearized problem (4.26) is being solved where outflow part of the boundary Γout is the
doi:10.6342/NTU202003676 artificial boundary. On Γoutthe Neumann no–stress condition is prescribed,
[ε∇ v −pI] n = 0 on Γout.
Inflow part of the boundary (left wall) Γin, rigid horizontal walls Γw and boundary of the cylinder Σ are Dirichlet boundaries where velocity is prescribed,
v = vD on ΓD(t), t ∈ (0, T ).
Parabolic velocity profile is prescribed on Γinwhose magnitude increases from 0 to 1,
v = (1 − 4y2)1
2(1 − χ{t<1/2}cos 2πt + χ{t≥1/2}) on Γin,
where χ denotes the characteristic function. The no–slip boundary condition is prescribed on the rigid walls and oscillating cylinder,
v = 0 on Γw
v =
2
5sin(π4t) sin(πt) +65y sin(2πt)sgn{sin(πt) sin(π4t)}sgn{1 − χ{t>4}} 0
on Σ, t ∈ (0, T ).
The tests have been ran for various Reynolds numbers up to 105 and all three stabilized methods produced smooth solutions without non–physical oscillations. Final simulation time was taken as T = 8 while the time step in all tests is ∆t = 0.01. The convergence criterion for the Newton method (difference between previous and current iteration) was set as
k(vn,k+1h , pn,k+1h ) − (vn,kh , pn,kh )kL∞ ≤ 10−8,
where k denotes the iteration. Maximal number of iteration was set to 30; if solution at current time step did not converge in 30 iterations, it is declared that method did not con-verge. Note that the stabilization parameter τKdefined in (4.30) is an increasing function of 1/ε, so the amount of artificial diffusion increases as ε decreases. Although it works
in favor of stabilization even on (relatively) very coarse meshes, this might influence the accuracy of the solution since the amount of artificial diffusion added is significant. Test have been performed for various finite element spaces (both satisfying and violating the LBB condition) and for implicit Euler and Crank–Nicolson temporal discretizations. In all the cases the method converged. Some of the finite element spaces tested for include
• [Pb1]2 × P1: piecewise linear polynomials enriched with the bubble function for velocity and piecewise linear polynomials for pressure, a space known to satisfy the LBB condition;
• [P2]2× P1: piecewise quadratic polynomials for velocity and piecewise linear poly-nomials for pressure, a space known to satisfy the LBB condition;
• [P1]2× P1: piecewise linear polynomials for velocity and piecewise linear polyno-mials for pressure; this space does not satisfy the LBB condition but the resulting scheme is inf–sup stable due to stabilization consequence discussed in 4.5.2;
• [P2]2 × P2: piecewise quadratic polynomials for velocity and piecewise quadratic polynomials for pressure; this space does not satisfy the LBB condition but the resulting scheme is inf–sup stable due to stabilization consequence discussed in 4.5.2.
Furthermore, a test without any stabilization was performed on a very fine mesh (h = 0.004) to investigate the influence of stabilization. It has been noticed that when increasing the mesh density, solution obtained by stabilized methods indeed converge towards the solution obtained on the very dense mesh (h = 0.004). In Figure 4.10, neces-sary number of iterations at each time instant is shown for the space [Pb1]2× P1and on two different meshes. For these meshes, method without stabilization does not converge. In Figure 4.11, the pressure field obtained by standard Galerkin and stabilized GLS method is shown at the last time instant standard Galerkin FEM converges. Finite element space employed was [Pb1]2× P1 and for the temporal discretization implicit Euler method was used. Sharp oscillations can be observed for standard Galerkin method. The same holds
doi:10.6342/NTU202003676
(a) (b)
Figure 4.9: Computational mesh for the flow past oscillating cylinder problem. Cylinder oscillates along {y = 0} line and rotates around its center. Initial shape and position of the cylinder are given by parametric description: (x(t), y(t)) = (12 + 0.08 cos(t), 0.1 sin(t)), t ∈ [0, 2π).
(a) Number of Newton’s iteration on mesh with characteristic size h = 0.05.
(b) Number of Newton’s iteration on mesh with characteristic size h = 0.03.
Figure 4.10: Number of Newton’s iterations on two different meshes where finite element space is chosen as [Pb1]2for velocity and P1 for pressure. ”Standard” denotes the method without stabilization but on the very fine mesh (h = 0.004).
for the velocity field; we show in Figure 4.12 the x–component of the velocity fields pro-duced by standard Galerkin and GLS stabilized schemes. All of the stabilized method produce smooth solution and converge at each time step. Finally, the L2 energy of the velocity produced by stabilized methods on coarse meshes was compared with the energy produced by standard Galerkin method on dense mesh (the ”referential solution”). It has been noticed that the energies produced by various stabilized methods converge towards the energy of the referential solution. This is illustrated in Figure 4.13. The results seem satisfactory.
(a) Pressure field obtained by standard Galerkin method on mesh with characteristic size h = 0.03.
(b) Pressure field obtained by GLS method on mesh with characteristic size h = 0.03.
Figure 4.11: Pressure field in area near the cylinder obtained by standard Galerkin and GLS method on the same mesh with characteristic size h = 0.03 at time t = 0.15. Newton method does not converge for standard Galerkin method for later times.
(a) x–component of the velocity field obtained by standard Galerkin method on mesh with character-istic size h = 0.03.
(b) x–component of the velocity field obtained by GLS method on mesh with characteristic size h = 0.03.
Figure 4.12: x–component of the velocity field in area near the cylinder obtained by standard Galerkin and GLS method on the same mesh with characteristic size h = 0.03 at time t = 0.15. Newton method does not converge for standard Galerkin method for later times.
(a) Energies of velocity field produced by stabilized methods on mesh with characteristic size h = 0.05.
(b) Energies of velocity field produced by stabilized methods on mesh with characteristic size h = 0.05.
Figure 4.13: Energies of velocity field produced by stabilized methods on two different meshes. ”Standard” denotes the method without stabilization but on very fine mesh (h = 0.004), i.e. our reference solution.
doi:10.6342/NTU202003676