From 589706c2b03488606297f16fac90b298d143b43d Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Wed, 14 Nov 2007 17:56:15 +0000 Subject: [PATCH] A bit more text git-svn-id: https://svn.dealii.org/trunk@15498 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-22/doc/intro.dox | 72 +++++++++++++++++++++++++- 1 file changed, 71 insertions(+), 1 deletion(-) diff --git a/deal.II/examples/step-22/doc/intro.dox b/deal.II/examples/step-22/doc/intro.dox index b92744a94c..c8f58b01e0 100644 --- a/deal.II/examples/step-22/doc/intro.dox +++ b/deal.II/examples/step-22/doc/intro.dox @@ -82,11 +82,44 @@ equations hard to solve when the Reynolds number $\mathrm{Re}$ is large. +Note that a large Rayleigh number does not necessarily involve large +velocities in absolute terms. For example, the Rayleigh number in the +earth mantle has a Rayleigh number larger than $10^6$. Yet the +velocities are small: the material is in fact solid rock but it is so +hot and under pressure that it can flow very slowly, on the order of +at most a few centimeters per year. Nevertheless, the can lead to +mixing over time scales of many million years, a time scale much +shorter than for the same amount of heat to be distributed by thermal +conductivity. + + +

%Boundary and initial conditions

+ +Since the Boussinesq equations are derived under the assumption that +inertia of the fluid's motion does not play a role, the flow field is +at each time entirely determined by buoyancy difference at that time, +not by the flow field at previous times. This is reflected by the fact +that the first two equations above are the steady state Stokes +equation. Consequently, we do not need initial conditions for either +velocities or pressure. On the other hand, the temperature field does +satisfy an equation with a time derivative, so we need initial +conditions for $T$. + +As for boundary conditions: if $\kappa>0$ then the temperature +satisfies a second order differential equation that requires +temperature data all around the boundary for all times. Similarly, the +velocity field requires us to pose boundary conditions. These may be +no-slip no-flux conditions $\mathbf u=0$ on $\partial\Omega$ if the +fluid sticks to the boundary, or no normal flux conditions $\mathbf n +\cdot \mathbf u = 0$ if the fluid can flow along but not across the +boundary, or any number of other conditions that are physically +reasonable. In this program, we will use no normal flux conditions. +

Solution approach

Like the equations solved in @ref step_21 "step-21", we here have a -system of differential-algebraic equations: with respect to the time +system of differential-algebraic equations (DAE): with respect to the time variable, only the temperature equation is a differential equation whereas the Stokes system for $\mathbf u$ and $p$ has no time-derivatives and is therefore of the sort of an algebraic @@ -114,6 +147,43 @@ is to solve the mixed Laplacian, and we will make full use of this fact! +

Time stepping

+ +The structure of the problem as a DAE allows us to use the same +strategy as we have already used in @ref step_21 "step-21", i.e. we +use time lag scheme: first solve the Stokes equations for velocity and +pressure using the temperature field from the previous time step, then +with the new velocities update the temperature field for the current +time step. In other words, in time step $n$ we first solve the Stokes +system +@f{eqnarray*} + -\nabla \cdot \eta \varepsilon ({\mathbf u}^n) + \nabla p^n &=& + \mathrm{Ra} \; T^{n-1} \mathbf{g}, + \\ + \nabla \cdot {\mathbf u}^n &=& 0, +@f} +and then +@f{eqnarray*} + T^n + - + k_n\nabla \cdot \kappa \nabla T^n + &=& + T^{n-1} + + + k_n\gamma + - + k_n{\mathbf u}^n \cdot \nabla T^{n-1}, +@f} +where $k_n$ is the time step size. Note how the temperature equation is +solved semi-explicitly: diffusion is treated implicitly whereas +advection is treated explicitly using the just-computed velocity +field. + + +

Linear solvers

+ + + -- 2.39.5