From: Abner J. Salgado Date: Tue, 15 Sep 2009 18:52:10 +0000 (+0000) Subject: Added the implementation section X-Git-Tag: v8.0.0~7113 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=a24cf4697260aa26dacb40ac0b8d142b82816f06;p=dealii.git Added the implementation section git-svn-id: https://svn.dealii.org/trunk@19460 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-35/doc/intro.dox b/deal.II/examples/step-35/doc/intro.dox index 1578b37346..58c209f946 100644 --- a/deal.II/examples/step-35/doc/intro.dox +++ b/deal.II/examples/step-35/doc/intro.dox @@ -16,7 +16,22 @@ Navier-Stokes equations. These equations describe the flow of a viscous incompre u_t + u \cdot \nabla u - \nu \Delta u + \nabla p = f, \\ \nabla \cdot u = 0, @f} -supplemented by the boundary condition $u=0$ Here $u$ represents the velocity of the flow and $p$ the pressure. +where $u$ represents the velocity of the flow and $p$ the pressure. This system of equations is supplemented by +the initial condition +@f[ + u |_{t=0} = u_0, +@f] +with $u_0$ sufficiently smooth and solenoidal, and suitable boundary conditions. For instance, an admissible boundary +condition, is +@f[ + u|_{\partial\Omega} = u_b. +@f] +It is possible to prescribe other boundary conditions as well. In the test case that we solve here the boundary +is partitioned into two disjoint subsets $\partial\Omega = \Gamma_1 \cup \Gamma_2$ and we have +@f[ + u|_{\Gamma_1} = u_b, \qquad u\times n|_{\Gamma_2} = 0. +@f] +where $n$ is the outer unit normal. The boundary condition on $\Gamma_2$ is used to model outflow conditions. In previous tutorial programs (see for instance @ref step_20 "step-20" and @ref step_22 "step-22") we have seen @@ -106,7 +121,7 @@ projection methods that we have just described: @f] where @f[ - H = \left\{ v \in L^2(\Omega)^d:\ \nabla\cdot v =0 \ v\cdot n|_{\partial\Omega} = 0 \right\}. + H = \left\{ v \in L^2(\Omega)^d:\ \nabla\cdot v =0, \ v\cdot n|_{\partial\Omega} = 0 \right\}. @f] Indeed, if we use this decomposition on $u^{k+1}$ we obtain @f[ @@ -114,13 +129,26 @@ projection methods that we have just described: @f] with $v^{k+1}\in H$. Taking the divergence of this equation we arrive at the projection equation.
  • The more accurate of the two variants outlined above is the rotational - one. That said, in the program below we use the standard form because it is - much easier to implement. Moreover, in the author's experience, it is the one that should be - used if, for instance, the viscosity $\nu$ is variable. + one. However, the program below implements both variants. Moreover, in the author's experience, + the standard form is the one that should be used if, for instance, the viscosity $\nu$ is variable. + +

    Implementation

    + +Our implementation of the projection methods follows verbatim the description given above. We must note, +however, that as opposed to most other problems that have several solution components, we do not use +vector-valued finite elements. Instead, we use separate finite elements for each component of the velocity +and the pressure, respectively, and use different DoFHandler's for those as well. The main +reason for doing this is that, as we see from the description of the scheme, the dim components +of the velocity and the pressure are decoupled. As a consequence, the equations for all the velocity components +look all the same, have the same system matrix, and can be solved in parallel. Obviously, this approach +has also its disadvantages. For instance, we need to keep several DoFHandlers and iterators +synchronized when assembling matrices and right hand sides; obtaining quantities that are inherent to +vector-valued functions (i.e. divergences) becomes a little awkward, and others. + -

    The testcase

    +

    The Testcase

    The testcase that we use for this program consists of the flow around a square obstacle. The geometry is as follows: