A fourier-hermite pseudospectral method for penetrative convection
by Kwan-leung Tse
THESIS
1997
Ph.D. Mathematics
xii, 95 leaves : ill. ; 30 cm
Abstract
The first contribution of the thesis is the development of an efficient Fourier-Hermite pseudospectral method for solving the penetrative convection problem in an infinite domain. The advantage of this method over the use of a Fourier-Chebyshev pseudospectral method is that the collocation points are clustered in the central region rather than near the computational boundaries. Also, when solving the implicit equations, only one full matrix inversion is required compared to three for the Fourier-Chebyshev pseudospectral method. However, a disadvantage of the present method is the lack of a fast Hermite transform....[ Read more ]
The first contribution of the thesis is the development of an efficient Fourier-Hermite pseudospectral method for solving the penetrative convection problem in an infinite domain. The advantage of this method over the use of a Fourier-Chebyshev pseudospectral method is that the collocation points are clustered in the central region rather than near the computational boundaries. Also, when solving the implicit equations, only one full matrix inversion is required compared to three for the Fourier-Chebyshev pseudospectral method. However, a disadvantage of the present method is the lack of a fast Hermite transform.
In our problem, an S-shaped temperature profile in the absence of motion is prescribed in the vertical direction. All fluctuating variables are expanded in terms of Fourier-Hermite basis functions. The Hermite functions are scaled to adjust the length of the computational domain in the vertical. A time splitting method is used and the linear terms are marched with an implicit scheme. After discretization, the implicit part requires solving one full and two tridiagonal matrix equations for each horizontal wave number. The program is implemented in the Intel Paragon of HKUST. Numerical simulation results of resolution 643 are presented for low-to-moderate Rayleigh numbers with a Prandtl number of unity. The highly stable outer regions are seen to act as effective lids and all penetrative flow are contained within the computational box.
The second contribution of this thesis is the new statistical results obtained for turbulent penetrative convection. The first simulation result presented in this report is the temperature profiles. The temperature profile at highest R shows that the flow is well-mixed. The convective region extends for about 36.7% of the depth of the unstable layer on each side. Beyond the convection region, the flow is basically horizontal. The horizontally averaged variances and flux reach maximum values at a low R and then decrease as R is increased further. At the edges of the convective zone, the vertical velocity gradients are large. The stable regions there acts like walls. Velocity projection diagrams show that the flow structures are basically distorted rolls with plumes rising among them. The structures do not changed appreciably for the range of R studied in this report. There exist large vortex structures inside the stable regions. The integral scales of the structures increase as they move further away from the convective region. The penetration depth are also calculated by finding the zeros of mean vertical kinetic flux. The zero temperature isosurfaces are plotted. The middle isosurfaces are convoluted and broken when R = 8000. Energy and scaler-variance spectra indicate that higher R is needed for the inertial subrange to form. Fluid particles are then marked and their trajectories traced. The markers penetrate far into the stable regions. Equilibrium between depletion and entrainment is obtained. From the probability distribution functions of marker positions, we conclude the existence of another region beyond the convective region. Material from the unstable region overshoot to that region where they flow horizontally.
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