Flow through a Venturi pipe at low up to medium Reynolds number

The objective of the following test calculations is to examine the resulting accuracy of the employed flow solver if it is applied to nonsteady flows with complex behaviour. In contrast to the `flow around a cylinder' benchmark simulations which "only" showed periodical vortex shedding behind a circle, we present now results for a flow which does not seem to exhibit such an easy flow pattern in space and time.

The interesting flow quantity in this simulation is the flux through the upper small channel (see the following typical coarse mesh for a sketch of the geometry).

In "real life", this Venturi pipe represents a small device used to drain sailing boats. If the flow speed is sufficiently high, then -- due to the Bernoulli principle -- the narrowing section enforces a low pressure which creates a flux through this small device, out of the boat. Therefore, we are mainly interested in controlling the flux through this device, as a function of time.

In this simulation (until T=20), the inflow is steady parabolic, and we start from the corresponding Stokes solution. The aim of the following calculations is to demonstrate `graphically' the described results and problems via performing such types of CFD simulations. More explanations and remarks can be found in our Paper Archive, particularly in http:/www.featflow.de/ture/paper/habil.ps.gz and in Stefan Turek's CFD-book, Springer.

We aim to show how hard it is - even for this low up to medium Reynolds number range - to calculate the `grid independent' solution by simply refining the mesh in space and time only, if we ask for a quantitatively exact representation of the flux (through the upper inlet) or the integral mean value of the pressure (on the lower wall)! We employed the following two types of meshes `a' and `b' which are shown for level 3, i.e., after two regular refinements.

Grid a:

Grid b:

The following diagrams show the resulting flux and integral pressure coefficients in time. As explained more carefully in the Computational details , we prescribed a fixed velocity profile only at the left edge, and let the others as free as possible ('zero mean pressure'!). The resulting flux through the upper small device depends on the geometry and Reynolds number; and in this case, the result is that `flow is sucked from the boat'!!!

We applied an adaptive time step control with very small error tolerances so that the shown results are (more or less) `exact in time', that means that only the spatial error should be visible! For more information about different types of time stepping techniques (what happens for time steps too large, which time step size is needed to guarantee sufficient accuracy, how does the time step depend on the used `Navier-Stokes solver'???) and the comparison of various `Navier-Stokes solvers' (projection-like, fractional step, pressure correction, fully coupled, linearization techniques for the nonlinearity, ...) can be found in the previously described Paper Archive, and particularly in Stefan Turek's CFD-book, Springer.

The following plots show the results on different mesh levels (up to level 9!), on meshes `a' and `b', and a zoomed version with the results on the finest meshes to demonstrate that the result is (almost) mesh size- and grid type-independent!

While mesh levels 5 - 7 are too coarse, we can see that, on grid `a', the results on level 8 and 9 are almost identical, concerning flux as well as pressure values (note: the pressure values to be controlled are positioned directly on the boundary!!!). Additionally, the solutions on mesh type `b', and here particularly on level 8, strongly converge to the same solution (level 9) as on mesh `a' which seems to be the `grid -independent' reference solution!

Some conclusions for flow simulations in this Reynolds number regime can be stated:

• Even for this highly nonsteady flow without simple periodical flow patterns a precise numerical simulation seems to be possible, hereby achieving a grid-independent structure in space as well as in time.
• However, level 8 (about 200,000 cells) is needed for a `precise' simulation! As further simulations show, the number of mesh points and time steps increase for the higher Reynolds number case.
• Nevertheless, a qualitative simulation of the dynamic behaviour of the complete flow structure in space and time is possible, with much less requirements concerning space and time discretizations. But, the corresponding `exact' prediction of certain flow quantities in a pointwise sense may fail!
• For corresponding 3D calculations, or for higher Reynolds numbers, a reduction of mesh points and time steps while maintaining the same accuracy and same efficiency of the solvers is an absolutely necessary task!!! Possible techniques to achieve these improvements are:
  • `Optimal' meshes and `large' time steps (by hand or via error control, second order time stepping)
  • `Better' discretizations of the convective term (by adaptive upwinding or streamline-diffusion techniques)
  • `Faster' and `more robust' solvers (multigrid, better adapted to hardware)
  • `Improved' Navier-Stokes schemes (MPSC techniques)

For the mathematical background concerning these observations and suggestions for improvements, and with respect to general CFD for incompressible flow problems, look at our Paper Archive, and there especially at http://www.featflow.de/ture/paper/habil.ps.gz and in Stefan Turek's CFD-book, Springer.

• Distribution of temperature/concentration via Boussinesq model
  
  

  Visualization via tracking of concentration/temperature, starting from the left inflow (size 1) and the upper small device (size -1). The first row shows the results 'exact in time and space', that means calculated with the described adaptive time step control and on level 9. We performed the visualization for two different color maps: the first with `red for size 1' and `blue for size -1', while the second one uses `red for absolute size 1' while here blue stands for `blue for size 0' (where both are mixed!). The following rows are for level 8 - which looks more or less identical - followed by levels 7 and 6, all calculated on grid `a'. All movies occupy about 1 MB at the most. Playing all videos at the same time, a comparison shows that even level 7 is able to capture qualitatively the dynamic behaviour.
  

  level 9 (a)
  

  level 8 (a)
  

  level 7 (a)
  

  level 6 (a)
  
  

• Pressure
  
  

  Corresponding visualization via shaded pressure plots (less than 1 MB). First row for level 9 and 8, second row for level 7 and 6, all on grid `a'.
  

  
  

  
  
  

• Streamfunction
  
  

  Corresponding visualization via shaded streamline plots (less than 1 MB). First row for level 9 and 8, second row for level 7 and 6, all on grid `a'.
  

  
  

  
  
  

• Velocity
  
  

  Some representative vector plots on level 9 at T=20 (about 4 and 7 MB).