From d3076f7d52b27cbf0faaa8326ce354a962fd668f Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Sun, 16 Aug 2009 23:50:06 +0000 Subject: [PATCH] We were incorrectly referring to something as the 'Rayleigh' number, but it wasn't actually. Correct this. git-svn-id: https://svn.dealii.org/trunk@19279 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-31/doc/intro.dox | 17 +++++++++++------ deal.II/examples/step-31/step-31.cc | 11 +++++++---- 2 files changed, 18 insertions(+), 10 deletions(-) diff --git a/deal.II/examples/step-31/doc/intro.dox b/deal.II/examples/step-31/doc/intro.dox index f06b2d745f..abe5a04005 100644 --- a/deal.II/examples/step-31/doc/intro.dox +++ b/deal.II/examples/step-31/doc/intro.dox @@ -29,7 +29,7 @@ can be neglected, the equations that describe such behavior are the Boussinesq equations that read as follows: @f{eqnarray*} -\nabla \cdot (2 \eta \varepsilon ({\mathbf u})) + \nabla p &=& - -\mathrm{Ra} \; T \mathbf{g}, + -\rho \beta \; T \mathbf{g}, \\ \nabla \cdot {\mathbf u} &=& 0, \\ @@ -51,7 +51,8 @@ will draw extensively on the experience we have gained in that program, in particular with regard to efficient linear Stokes solvers. The forcing term of the fluid motion is the buoyancy of the -fluid, expressed as the product of the Rayleigh number $\mathrm{Ra}$, +fluid, expressed as the product of the density $\rho$, the thermal expansion +coefficient $\beta$, the temperature T and the gravity vector g pointing downward. (A possibly more intuitive formulation would use $-\mathrm{Ra} \; (T-\bar T) \mathbf{g}$ as right hand side where $\bar T$ is the average @@ -83,14 +84,18 @@ sometimes, as in the case of rock minerals at temperatures close to their melting point, $\eta$ may change by orders of magnitude over the typical range of temperatures. -$\mathrm{Ra}$, called the Rayleigh -number, is a dimensionless number that describes the ratio of heat +We note that the Stokes equation above could be non-dimensionalized by +introducing the Rayleigh +number $\mathrm{Ra}=\frac{\|g\| \beta \rho}{\eta \kappa} \delta T L^3$ using a +typical length scale $L$, typical temperature difference $\delta T$, density +$\rho$, thermal diffusivity $\kappa$, and thermal conductivity $\kappa$. +$\mathrm{Ra}$ 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 u is slow enough so -that heat diffusion $\kappa\Delta T$ is the dominant heat transport +that heat diffusion $\kappa\nabla T$ 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. diff --git a/deal.II/examples/step-31/step-31.cc b/deal.II/examples/step-31/step-31.cc index f8f5ace887..f48ee54347 100644 --- a/deal.II/examples/step-31/step-31.cc +++ b/deal.II/examples/step-31/step-31.cc @@ -130,13 +130,15 @@ using namespace dealii; // // Finally, or maybe firstly, at the top of // this namespace, we define the various - // material constants we need ($\eta,\kappa$ - // and the Rayleigh number $Ra$): + // material constants we need ($\eta,\kappa$, + // density $\rho$ and the thermal expansion + // coefficient $\beta$): namespace EquationData { const double eta = 1; const double kappa = 1e-6; - const double Rayleigh_number = 10; + const double beta = 10; + const double density = 1; template @@ -1773,7 +1775,8 @@ void BoussinesqFlowProblem::assemble_stokes_system () const Point gravity = -( (dim == 2) ? (Point (0,1)) : (Point (0,0,1)) ); for (unsigned int i=0; i