Presentation on theme: "Multi-Scale Behaviour in the Geo- Science I: The Onset of Convection and Interfacial Instabilities by Hans Mühlhaus The Australian Computational Earth."— Presentation transcript:
Multi-Scale Behaviour in the Geo- Science I: The Onset of Convection and Interfacial Instabilities by Hans Mühlhaus The Australian Computational Earth Systems Simulator (ACcESS)
Overview What is Geodynamics Mantle Convection, Spreading, Folding, Landscape Evolution, Earthquakes, Volcanoes The onset of Natural Convection: Linear instability analysis Numerical Simulations Nusselt Number Remarks on Weakly Non-Linear Analysis Galerkin methods
4 th talk: modeling of geological folds Director evolution n : the director of the anisotropy W, W n : spin and director spin D, D : stretching and its deviatoric part
Governing Equations Navier Stokes Equations: Heat Equation: v(t,x) is the velocity vector, T is the temperature, kg/m 2 ) is the density, Pas) is the viscosity, c p (10 -3 WK -1 s/kg) is the specific heat at constant pressure and k (4Wm -1 K -1 ) is the thermal conductivity
Governing Equations, cont. Temperature dependence of density: Simplified convection model: p ( 3 10 -5 K -1 ) is the thermal expansion coefficient, T 0 (288K) is the surface temperature x1x1 x2x2 T=T 1, v 2 =0 T=T 0, v 2 =0 T, 1, v 1 =0 H L
Governing Equations, cont. Nondimensionalisation: Raleigh Number: Prandtl Number: Relevant limit in Geophysics: Insertion and dropping tildes…..
Governing Equations, cont. Stream function and In this way we satisfy the incompressibility constraint div v=0 identically. Insertion into the velocity equations and the heat equation, assuming infinite Prandtl number gives: and Dropping nonlinear terms and insert the Ansatz, gives: andThus
Finite Element Approximations… Ra=10 4, mesh: 128X128
The Nusselt Number Definition: It can be shown that for zero normal velocity b.c.s and fixed top and bottom Temperature
The Nusselt Number Hint for derivation of Nu-Power relationship form And apply Gausss Theorem. For the given b.c.s it follows that
Finite Element Approximations… Ra=10 4, mesh: 128X128 Nusselt Number
Finite Element Approximations… Ra=10 5, mesh: 128X128
Finite Element Approximations… Ra=10 5, mesh: 128X128 Nusselt Number
Finite Element Approximations… Ra=10 6, mesh: 128X128
Finite Element Approximations… Ra=10 6, mesh: 128X128 Nusselt Number
Finite Element Approximations… Ra=10 7, mesh: 128X128
Finite Element Approximations… Ra=10 7, mesh: 128X128 Nusselt Number
Galerkin Method We consider the ansatz: Insert into: and We obtain: and
Rayleigh-Taylor Instabilities RT fingers evident in the Crab Nebula Consider two fluids of different densities, the heaviest above the lightest. An horizontal interface separates the two fluids. This situation is unstable because of gravity. Effectively, if the interfaces modified then a pressure want of balance grows. Equilibrium can be found again tanks to surface tension that's why there is a competition between surface tension and gravity. Surface tension is stabilizing instead gravity is destabilizing for this configuration.
Benchmark Problem Methodmesh γ0γ0 (v rms ) max reached at t= van Kekencoarse0.011300.003045212.14 van Kekenfine0.011640.003036209.12 Particle-in- cell 1024 el0.011020.003098222 Particle-in- cell 4096 el0.012440.003090215 Level set5175 el0.011350.003116215.06 Rayleigh-Taylor instability benchmark
Linear Instability Analysis Equilibrium to be satisfied in ground state at time t=t0 and at t0+dt: Ground state: n Continued Equilibrium: Stokes equation:
Linear Instability Analysis n Or, considering the incompressibility constraint :
Linear Instability Analysis: Boundary and interface conditions We assume that the velocities are zero on top and bottom; on the sides we assume that v1=0 and i.e. symmetry boundary. On the interface the velocities as well as the natural boundary terms have to be continuous. Natural b.c.s: replace n as well as its time derivative have to be continuous on the interface By. The vector in
Linear Instability Analysis: Boundary and interface conditions We assume that the velocities are zero on top and bottom; on the sides we assume that v1=0 and i.e. symmetry boundary. On the interface the velocities as well as the natural boundary terms have to be continuous. Natural b.c.s n Or: Result:
Exercises 1. The normal component of the surface velocity of a 2D half plane reads. The half plane is occupied by a Stokes fluid with the viscosity The normal stress is obtained as. Show that. 2. Consider a Rayleigh-Taylor instability problem involving 2 infinite half-planes; i.e.. The normal velocity of the Interface reads. Show that. Hint: Use the relationship for a gravity free Half-plane and note that.
Excercise 4: Solve convection equations and using the perturbation expansion up to terms of order. Hint: 1.Insert into pdes, collect coefficients of and 2.Individual coefficients must be equal to zero. Get etc and T 3 are a little bit harder to get since the pdes contain inhomogeneous terms which are proportional to (
Perturbation solution for the weakly nonlinear problem 4.We require a solubility condition (Fredholms alternative) In the present case this just means that the coefficient of the resonant inhomogeneous term must vanish. The coefficient has the form of an ode which can be written as: a Ra Ra crit
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