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Introduction

-The purpose of this program is to show how to effectively solve the incompressible time-dependent Navier-Stokes equations. -In previous tutorial programs (see for instance ****) we have seen how to solve the time-independent Stokes equations using a Schur-complement approach. However \ No newline at end of file + +

Motivation

+The purpose of this program is to show how to effectively solve the incompressible time-dependent +Navier-Stokes equations. These equations describe the flow of a viscous incompressible fluid and they read +@f{align*} + 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. + +In previous tutorial programs (see for instance step-20 and step-22) we have seen +how to solve the time-independent Stokes equations using a Schur-complement approach. For the +time-dependent case, after time discretization, we would arrive at a system like +@f{align*} + \frac1\tau u^k - \nu \Delta u^k + \nabla p^k = F^k, \\ + \nabla \cdot u^k = 0, +@f} +where $\tau$ is the time-step. Although the structure of this system is similar to the Stokes system and thus +it could be solved using a Schur-complement approach, it turns out that the condition number of the +Schur-complement is proportional to $\tau^{-2}$, which makes the system very difficult to solve. + + +

Projection Methods

+ +As we explained above, we need to come up with a different approach to solve the time-dependent Navier-Stokes +equations. The difficulty in their solution comes from the fact that the velocity and the pressure are coupled +through the constraint +@f[ + \nabla \cdot u = 0. +@f] +Projection methods aim at decoupling this constraint from the diffusion (Laplace) operator. + +Let us shortly describe how the projection methods look like in a semi-discrete setting. The objective is to +obtain a sequence of velocities $\{u^k\}$ and pressures $\{p^k\}$. We will also obtain a sequence $\{\phi^k\}.$ +Suppose that from the initial conditions, and an application of a first order method we have found +$(u^0,p^0,\phi^0=0)$ and $(u^1,p^1,\phi^1=p^1-p^0)$. Then the projection method consists of three steps: + +
    +
  • Step 0: Extrapolation. Define: + @f[ + u^\star = 2u^k - u^{k-1}, \quad p^\sharp = p^k + \frac43 \phi^k - \frac13 \phi^{k-1}. + @f] +
  • Step 1: Diffusion step. We find $u^{k+1}$ that solves + @f[ + \frac1{2\tau}\left( 3u^{k+1} - 4u^k + u^{k-1} \right) + + u^\star \cdot\nabla u^{k+1} + \frac12 \left( \nabla \cdot u^\star \right) u^{k+1} + -\nu \Delta u^{k+1} + \nabla p^\sharp + = f^{k+1}, + \quad + u|_{\partial\Omega} = 0. + @f] +
  • Step 2: Projection. Find $\phi^{k+1}$ that solves + @f[ + \Delta \phi^{k+1} = \frac3{2\tau} \nabla \cdot u^{k+1}, + \quad + \partial_n \phi^{k+1} = 0. + @f] +
  • Step 3: Pressure correction. Here we have two options: +
      +
    • Incremental Method in Standard Form. The pressure is updated by: + @f[ + p^{k+1} = p^k + \phi^{k+1}. + @f] +
    • Incremental Method in Rotational Form. In this case + @f[ + p^{k+1} = p^k + \phi^{k+1} - \nu \nabla \cdot u^{k+1}. + @f] +
    +
+ +Without going into details, which can be found in the corresponding literature, let us remark a few things about the +projection methods that we have just described: +
    +
  • The advection term $u\cdot\nabla u$ is replaced by its skew symmetric form + @f[ + u \cdot \nabla u + \frac12 \left( \nabla\cdot u \right) u. + @f] + This is consistent with the equation and it is needed to guarantee unconditional stability of the + time-stepping scheme. Moreover, to linearize the term we use the second order extrapolation $u^\star$ of + $u^{k+1}$. +
  • The projection step is a realization of the Helmholtz decomposition + @f[ + L^2(\Omega)^d = H \oplus \nabla H^1(\Omega), + @f] + where + @f[ + 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[ + u^{k+1} = v^{k+1} + \nabla \left( \frac{2\tau}{3} \phi^{k+1} \right), + @f] + with $v^{k+1}\in H$. Taking the divergence of this equation we arrive the projection equation. +
  • The most accurate form of the methods is the rotational one. We included the standard because it was + so easy to implement that it is worth having. Moreover, it is the one (in my experience) that should be + used if, for instance, the viscosity $\nu$ is variable. +
+ + +

The Test Case

+ +The test case that we are going to carry out consist of the flow around a square obstacle. The geometry is +the following + +@image html geometry.gif + +with $H=4.1$. + +We impose no-slip boundary conditions on both the top and bottom walls and the obstacle. On the left side we +have the inflow boundary condition +@f[ + U = 4 U_m y (H-y)/H^2, \quad V = 0, +@f] +with $U_m = 1.5$, which corresponds to the Poiseuille flow for this configuration. Finally, on the right +vertical wall we impose the condition that the vertical component of the velocity should be zero. +The Reynolds number is $Re=100$ and the final time $T=10$. + + diff --git a/deal.II/examples/step-35/doc/results.dox b/deal.II/examples/step-35/doc/results.dox index f4c6feefb5..127518640b 100644 --- a/deal.II/examples/step-35/doc/results.dox +++ b/deal.II/examples/step-35/doc/results.dox @@ -1 +1,80 @@ +

Results

+ +We run the code with the following parameter-file.prm, which can be found in the +same directory as the source: +@verbatim + # First a global definition + # the type of method we want to use + set Method_Form = rotational + + subsection Physical data + # In this subsection we declare the physical data + # The initial and final time, and the Reynolds number + set initial_time = 0. + set final_time = 10. + set Reynolds = 100 + end + + subsection Time step data + # In this subsection we declare the data that is to be used for time discretization, + # i.e. the time step dt + set dt = 5e-3 + end + + subsection Space discretization + # In this subsection we declare the data that is relevant to the space discretization + # we set the number of global refines the triangulation must have + # and the degree k of the pair Q_(k+1)--Q_k of velocity--pressure finite element spaces + set n_of_refines = 3 + set pressure_fe_degree = 1 + end + + subsection Data solve velocity + # In this section we declare the parameters that are going to control the solution process + # for the velocity. + set max_iterations = 1000 # maximal number of iterations that GMRES must make + set eps = 1e-6 # stopping criterion + set Krylov_size = 30 # size of the Krylov subspace to be used in GMRES + set off_diagonals = 60 # number of off diagonals that ILU must compute + set diag_strength = 0.01 # diagonal strengthening value + set update_prec = 10 # this number indicates how often the preconditioner must be updated + end + + #The output frequency + set output = 5 + + #Finally we set the verbosity level + set verbose = false +@endverbatim + +Since the verbose parameter is set to false, we do not get any kind of output. +If we we were to set it to true we would get information on what the program is doing and +how many steps each iterative process had to make to converge, etc. + +A plot of the obtained results for $t=0,1,5,10$ is the following + +@image html visit0001.jpeg + +@image html visit0002.jpeg + +@image html visit0003.jpeg + +@image html visit0004.jpeg + +The contour lines correspond to the pressure, the color to the voriticity and the arrows to the velocity. +Here we only show a zoom into a region near the obstacle which is, after all, where the interesting things happen. + + +

Possible Extensions

+ +This program can be extended in the following directions: +
    +
  • Adaptive mesh refinement. As we have seen, we computed everything on a single fixed mesh. + Using adaptive mesh refinement can lead to increased accuracy while not significantly increasing the + computational time. +
  • Adaptive time-stepping. Although there is no theory about projection methods with variable time-step, + practice shows that they perform very well. This could be implemented too. +
  • Variable density incompressible flows. There are projection-like methods for the case of incompressible + flows with variable density. +
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