From 1e3867261baea5efa67ac9f850bd6246a5ad2aea Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Wed, 14 Nov 2007 03:56:20 +0000 Subject: [PATCH] More text git-svn-id: https://svn.dealii.org/trunk@15497 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-22/doc/intro.dox | 74 ++++++++++++++++++++++---- 1 file changed, 65 insertions(+), 9 deletions(-) diff --git a/deal.II/examples/step-22/doc/intro.dox b/deal.II/examples/step-22/doc/intro.dox index f9444b1735..b92744a94c 100644 --- a/deal.II/examples/step-22/doc/intro.dox +++ b/deal.II/examples/step-22/doc/intro.dox @@ -26,10 +26,12 @@ Boussinesq equations that read as follows: \nabla \cdot \kappa \nabla T &=& \gamma. @f} Here, ${\mathbf u}$ is the velocity field, $p$ the pressure, and $T$ -the temperature of the fluid. As can be seen, velocity and pressure +the temperature of the fluid. $\varepsilon ({\mathbf u}) = \frac 12 +[(\nabla{\mathbf u}) + (\nabla {\mathbf u})^T]$ is the symmetric +gradient of the velocity. As can be seen, velocity and pressure solve a Stokes equation describing the motion of an incompressible fluid. The forcing term of the fluid motion is the buoyancy of the -fluid, expressed as the product of the Raleigh number $\mathrm{Ra}$, +fluid, expressed as the product of the Rayleigh number $\mathrm{Ra}$, the temperature $T$ and the gravity vector ${\mathbf g}$. (A possibly more intuitive formulation would use $\mathrm{Ra} \; (T-\bar T) \mathbf{g}$ as right hand side where $\bar T$ is the average @@ -49,13 +51,67 @@ diffusion (heat conduction) term. In these equations, $\eta$ and $\kappa$ denote the viscosity and diffusivity coefficients. In the more general case and in many -physical applications, they may and $\eta$ often will depend -on the temperature, but we will neglect this dependence for the -purpose of this tutorial program. The term $\gamma$ on the right hand -side denotes the heat sources and may be a spatially and temporally -varying function. - - +physical applications, they may and $\eta$ often will depend on the +temperature, but we will neglect this dependence for the purpose of +this tutorial program. The term $\gamma$ on the right hand side +denotes the heat sources and may be a spatially and temporally varying +function. + +$\mathrm{Ra}$, called the Rayleigh +number is a dimensionless number that describes the ratio of heat +transport due to convection induced by buoyancy changes from +temperature differences, and of heat transport due to thermal +diffusion. A small Rayleigh number implies that buoyancy is not strong +relative to viscosity and fluid motion $\mathbf u$ is slow enough so +that heat diffusion $\kappa\Delta T$ is the dominant heat transport +term. On the other hand, a fluid with a high Rayleigh number will show +vigorous convection that dominates heat conduction. + +For most fluids for which we are interested in computing thermal +convection, the Rayleigh number is very large, often $10^6$ or +larger. From the structure of the equations, we see that this will +lead to large pressure differences and large velocities. Consequently, +the convection term in the convection-diffusion equation for $T$ will +also be very large and an accurate solution of this equation will +require us to choose small time steps. Problems with large Rayleigh +numbers are therefore hard to solve numerically for similar reasons +that make solving the Navier-Stokes +equations hard to solve when the Reynolds number +$\mathrm{Re}$ is large. + + +

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 +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 +constraint that has to hold at each time instant. The main difference +to @ref step_21 "step-21" is that the algebraic constraint there was a +mixed Laplace system of the form +@f{eqnarray*} + \mathbf u + {\mathbf K}\lambda \nabla p &=& 0, \\ + \nabla\cdot \mathbf u &=& f, +@f} +where now we have a Stokes system +@f{eqnarray*} + -\nabla \cdot \eta \varepsilon ({\mathbf u}) + \nabla p &=& f, \\ + \nabla\cdot \mathbf u &=& 0, +@f} +where $\nabla \cdot \varepsilon (\cdot)$ is an operator similar to the +Laplacian $\Delta$ applied to a vector field. + +Given the similarity to what we have done in @ref step_21 "step-21", +it may not come as a surprise that we choose a similar approach, +although we will have to make adjustments for the change in operator +in the top-left corner of the differential operator. As it will turn +out, it is actually a lot easier to solve the Stokes problem than it +is to solve the mixed Laplacian, and we will make full use of this +fact! -- 2.39.5