Diploma report Implementation and verification of a simple ... - LPAS

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4.5 Buoyant bubble In order to study the behaviour of numerical methods, a standard non-linear test problem was designed for the ’Workshop on Numerical Methods for Solving Nonlinear Flow Problems’ that was held at the NCSA in 1990 [8]. This test-problem consists in a density current in an otherwise homogeneous and isentropic two-dimentional fluid. It is initiated as a cold bubble of air that subsequently descends to the ground. Then, as the density current spreads out laterally at the bottom boundary, Kelvin-Helmholtz shear instability rotors form along the top of the cold air boundary. A grid-converged solution for this problem could be found in the literature and is shown in Figure 12. Figure 12: Plots of θ’ at 0, 300, 600 and 900 s for the 25 m resolution compressible reference solution [8]. Contour interval is 1 ◦ C and the contours are centered around 0 ◦ C. A background state that is in hydrostatic equilibrium is chosen but the potential temperature θ is set constant over the whole domain (N = 0). Then the Exner function π and therefore also the pressure are computed from (59) with θ constant. A value of 10 5 Pa is chosen for the surface pressure and θ is set equal to the surface temperature of T s = θ = 300 K. The lateral boundary conditions for the problem are the solid wall, and the vertical boundary conditions the solid wall with background, both are described in section 3.7. The size of the domain is of 6.4 km in the vertical direction and of 51.2 km in the lateral direction centered at x = 0 km. The temperature perturbation is specified by the following function ⎧ ⎨ 0 if L > 1, ∆T = ⎩−15 cos(πL) + 1 (75) if L 1, 2 where L = √ ( x − x c ) x 2 + ( z − z c ) r z 2 , (76) r 35
x c = 0 m, x r = 4000 m, z c = 3000 m and z r =2000 m. The thermal perturbation has a minimum temperature of -15 ◦ C and is centered at x = 0 m and z = 3000 m. These initial conditions provide the negative buoyancy necessary to initiate a density current. The simulations are initiated with a thermal perturbation as described above and our model is integrated using Leapfrog and centered-in-space differences. Asselin and spatial filters are used to prevent oscillations in solutions. The value µ = 0.1 is chosen for all simulations, while ν is adjusted according to cells size and time step in order to always obtain a diffusion coefficient K of 75 m 2 s −1 , as this particular value is chosen for the reference problem in [8]. Simulations are made using 80, 130 and 200 m resolution. According to [8], the flow features of the test problem should be adequately resolved for the first simulation and marginally resolved for the two others. The θ ′ solutions at 0, 300, 600 and 900 s from the three simulations are shown in Figure 13 and Figure 14. z[m] 4000 3000 2000 1000 0 80m t=0s z[m] 4000 3000 2000 1000 0 80m t=300s z[m] 4000 3000 2000 1000 0 80m t=600s z[m] 4000 80m 3000 t=900s 2000 1000 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 x[m] Figure 13: Plots of θ’ at 0, 300, 600 and 900 s for the 80 m resolution solution. interval is 1 ◦ C and the contours are centered around 0 ◦ C. Contour Our model seems to be capable of correctly reproducing the basic characteristics of the flow in the test problem, even though the propagation speed of the density current is found to be slightly bigger in our model and the front of the perturbation ends a little further than in the reference model. In the solution from the 80 m resolution simulation, the three rotors are correctely reproduced. In the solution from the 130 m resolution simulation, the formation of the first rotor at 600 s is well reproduced. However, the third rotor is missing in the solution at 900 s. With 200 m resolution, the rotors are even less developed and there are significant quantitatives and qualitative differences between our solutions and the 25 m reference solution. 36
Page 1: Diploma report Implementation and v
Page 4 and 5: 6 Acknowledgements 38 A Derivation
Page 6 and 7: In practice, however, many physical
Page 8 and 9: In developing the conservation laws
Page 10 and 11: and this expression can be used to
Page 12 and 13: and z j−1/2 respectively. They ca
Page 14 and 15: 3.3 Numerical fluxes There are diff
Page 16 and 17: 3.4.1 Forward Scheme The forward sc
Page 18 and 19: 3.6.1 Gas dynamics (c = 1) A simple
Page 20 and 21: Figure 3: The lower left corner of
Page 22 and 23: π. In order to fill the ghost cell
Page 24 and 25: 4 Results 4.1 Comparison of differe
Page 26 and 27: time filter cst = 0 time filter cst
Page 28 and 29: in the (ij) grid cell is then s ij
Page 30 and 31: 400 t=0s 250 t=9s 350 200 300 150 2
Page 32 and 33: 4.4 Advection Advection of a densit
Page 34 and 35: 1000 u’ ∆x=∆z=25m t=100s 800
Page 38 and 39: 4000 4000 z[m] 3000 2000 1000 0 130
Page 40 and 41: A Derivation of the differential fo

Diploma report Implementation and verification of a simple ... - LPAS

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