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

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4000 4000 z[m] 3000 2000 1000 0 130m t=0s z[m] 3000 2000 1000 0 200m t=0s 4000 4000 z[m] 3000 2000 1000 0 130m t=300s z[m] 3000 2000 1000 0 200m t=300s 4000 4000 z[m] 3000 2000 1000 0 130m t=600s z[m] 3000 2000 1000 0 200m t=600s z[m] 4000 3000 2000 1000 130m t=900s 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 x[m] z[m] 4000 200m 3000 t=900s 2000 1000 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 x[m] Figure 14: Plots of θ’ at 0, 300, 600 and 900 s for the 130 and 200 m resolution solutions. Contour interval is 1 ◦ C and the contours are centered around 0 ◦ C. As mentioned above, our simulations were made using the same diffusion constant (K = 75 m 2 s −1 ) as in the test problem in [8]. In consequence, our results are relatively noisy. In order to reduce this noise the value of the spatial filter coefficient ν should be increased. Results of a 130 m resolution simulation where ν is increased by a factor ten (ν=0.07) and the solution previousely obtained for the 130 m resolution simulation (ν=0.007, Figure 14) are shown togheter in Figure 15. z[m] 4000 3000 2000 1000 130m t=900s ν=0.007 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 x[m] z[m] 4000 3000 2000 1000 130m t=900s ν=0.07 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 x[m] Figure 15: Plots of θ’ at 900 s for the 130 m resolution solutions with ν=0.007 (left) and ν=0.07 (right). The solution with the stronger filter coefficient appears to be very smoothed. In this case it seems that part of the physical information is lost: the rotors form much slowlier. Moreover the propagation speed is reduced. These observations indicate that the results may be improved by adding more diffusion. The noise could be attenuated and the propagation speed reduced. So that our results would be even closer to the 25 m reference solution in Figure 12. 37
5 Conclusion A two-dimensional finite volume method modeling the Euler equations on a Cartesian grid was developped. Several types of numerical fluxes and various time integration methods were implemented. The model using Leapfrog time integration and central differences in space with initial hydrostatic equilibrium conditions was found stable for integration times up to ten hours if a numerical smoothing filter was applied. Sensitivity to numerical filtering was tested using a one-dimensional model and indicated that the oscillations in solutions that occur with Leapfrog scheme could be controlled using an Asselin time filter. Small-scale noise generated by numerical dispersion could be reduced by applying of a smoothing filter. Results also indicated that the source term formulation has a important effect on the velocities of the winds generated by numerical imprecisions. Grid-converged solutions could be obtained for various problems. In one space dimension the simulation of sound waves gave satisfying results for a Gaussian perturbation and the behaviour of the numerical solutions is in agreement with physical intuition. In two dimensions, advection of a density perturbation demonstrated the ability of the model to reproduce correct propagation speed. Finally, numerical simulations of a density-current test-problem showed that the model is capable of correctly reproducing the essence of all the important physics of the flow. Further improvements can be made by solving the instability problems arising with the splitting technique. To this end, changes could be done in the splitting method to include the numerical diffusion term. Besides this the order of the time integration can also be improved using a third-order Runge-Kutta scheme for example. These two modifications should permit to gain in accuracy. The numerical model that was developed here was found to work well and gave satisfying results. As next step, the implementation of a cut-cell boundary condition can be performed as this would permit to treat complex topography on the lower boundary. 6 Acknowledgements A would like to thank Oliver Fuhrer for his help and guidance during these four months. I would also like to thank Alain Clappier for offering me the possibility to do my diploma work in the LPAS group and the people in the group for their support. 38
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
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Page 36 and 37: 4.5 Buoyant bubble In order to stud
Page 40 and 41: A Derivation of the differential fo

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

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