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Spatial Wave
 

 
 
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In this part we study Kelvin Helmholtz instabilities when both phases are flowing towards the same direction with a different velocity. The instability is moving with time building spatial waves. The results are here.
A) Mesh and Models
A-1 Meshes
All meshes were built with Gambit. The domain is 2 Dimensional. The length is 1m, the height 0,2m. 

For this study we mainly used 2 different kind of meshes:
one the variable step size and one with fixed step size 
 
 

  •  Variable step size mesh
The first mesh used had a fixed space step in length (5 mm - 200 elements) and a variable step in height (varying from 1 mm in the middle to 3 mm near the edge - 100      elements). The total element number is 20000. The idea was to have a very detailed grid at the interface while the wave is small.

Click here to see a picture of this mesh

The boundary conditions are: 2 Velocity Inlets on the left: lower inlet y=0 to 0.1m and upper inlet y= 0.1 to 0.2m. 1 Outflow on the right. The upper and lower edges are Walls.

This configuration proved to rather a bad choice. This mesh needs a long time to compute and the strong difference between element height and length deformed the waves. Besides by the nature of instabilities the interface soon occupies a large part of the domain and moves both upwards and forwards. That's why we used the fixed step size mesh with square elements for the results. The best best mesh geometry is probably one with smaller elements within the conical zone where the instabilities develop. It is consequently a good idea to refine meshes on when the first results are at hand. 

  • Fixed step size mesh
The first mesh used has a fixed space step in length (3 mm - 333 elements) and in height (3 mm - 67 elements). The total element number is 22311. 

Click here to see a picture of this mesh

The boundary conditions are: 2 Velocity Inlets on the left: lower inlet y=0 to 0.1m and upper inlet y= 0.1 to 0.2m. 1 Outflow on the right. The upper and lower edges are Walls.

This mesh gave good results. The biggest limitation is the size the element, that is not small enough to depict smaller structures: fully developed waves, smaller wavelength... 

A-2 Models
As with the mesh, this study was our first try to simulate the Kelvin Helmholtz instabilities, therefore most of our choices are not optimized or representative of a specific physical phenomenon. The objective was to get an insight on possibilities and limitations of Fluent for further investigations. In first contact we describe how the first simulations determined the setup used for results.
 
 
  • First contact


We used Fluent 5.0.2 with a VOF model as two phases model. Other settings are 2D, unsteady, laminar, segregated. The upper Velocity inlet delivers the heaviest fluid (with 3 m/s) and the lower Velocity inlet the light one (with 1 m/s) to ensure instability. We chose to operate with gravity (on the y-axis: -10 m/s2), and surface tension (Fluent's default value: 0,0375 N/m representative of an air-water interface). As initial state both fluids have the same speed (1 m/s) while the upper half is patched to be the heavy fluid. 

 At this time we started the computation with 0.02s to 0.04s time step and made three observations:

1. As the upper velocity inlet started bringing fluid with a greater velocity than the initial state a big instability formed. 
2. When the density difference is too big the instability did not appear as clearly. 
3. When the steady state was reached "tilting the box" by modifying the gravity's direction wouldn't create any instabilities. 
  • Setup used for results


Changes are in Bold

We used Fluent 5.0.2 with a VOF model as two phases model. Other settings are 2D, unsteady, laminar, segregated. The upper Velocity inlet delivers the heaviest fluid (with 3 m/s) and the lower Velocity inlet the light one (with 1 m/s) to ensure instability. We chose to operate with gravity (on the y-axis: -10 m/s2), and surface tension (Fluent's default value: 0,0735 N/m representative of an air-water interface). As initial state each fluid has the same speed as its respective inlet while the upper half is patched to be the heavy fluid. 

As result of our observations we decided to create a perturbation:  the upper Velocity inlet delivering V=V0+A*sin(wt) using an UDF (sinusoidal-v.c). Moreover we used fluids with a small density difference: liquid water (998 kg/m3) and liquid fuel (960 kg/m3). The perturbation on speed leads indirectly to a perturbation on the interface height. Increasing flow rate "pushes" the upper phase downwards.

Fluent recommends using the VOF model with the following setting intheControls>Solution menu: Presto, Quick, Body weighted. We used both these and the default settings obtaining very similar results in most cases. The main difference is that the default setting is more stabilizing: instabilities are only obtained when a perturbation is added, whereas the Presto... setting leads sometimes to "spontaneous" instabilities. Even if this sounds like a good effect, it is most of the time a bad one. The perturbation is not a clean harmonic because of the finite time and space steps. Other perturbation frequencies appear, create instabilities and as these are not controlled the flow gets often very unstructured (that problem was also encountered in the time wave study). A more stabilizing setting proved to be more convenient to get the expected result. the counter part is the need for a bigger perturbation. Another difference is that the interface seems to be thinner in the recommended setting, smaller structures remains longer clearly visible. Yet the best way to have small waves is to have a tight mesh. 

Go there for some pictures on that subject.
 
 
 

    B) Results
     
     
  • Computation results
This first case shows the development of the Kelvin Helmholtz instabilities as they move away from the inlet. The growing spiraling vortices can be clearly seen.

 
 

 


Here an Animation of growing instabilities with a 0.01s time step.

Different parameters of the flow can be read out of these pictures: Animation.

  1. The measured wave celerity is  2.2 m/s. We expected 2 m/s (+/-0.2 m/s for the perturbation). The value is within the expected range. 
  2. The measured wavelength is 0.35m. (the expected wave length is 0.31m, +/- 0.03m). This value is a bit small and can be explained by the difficulty of comparing two vortex that are not at the same size. It is also possible that the speed perturbation distorts the wave more than expected. 
  • Comparison with an experimental result.


This picture can be compared to he result of a similar experience:


 
 

 


While the vortexes are following the same evolution confirming the validity of the result, some differences can be pointed out.

On the experimental picture the formation of "cat eyes" on the right side is visible. This is not the case on the computation for several reasons. 

Firstly, the chosen domain length (1m) is too short, and further evolution does not appear within the displayed part. There are more than one way to avoid this but we have each time to face limitations by the modelling. 

  1. We could decrease the flow speed to allow more time for the vortex formation. The problem is that the perturbation must have a minimal value around 0.2 m/s to have the expected effect on the flow. As the perturbation must remain little compared to the flow speed, it isn't possible to decrease the speed very much under the selected value of 3 m/s. 
  2. Another possible change was to increase the perturbation frequency, leading to shorter wavelength. This would also help seeing the further evolution of the wave, but by decreasing the wavelength the spatial step size (3 mm) gets too little to simulate correctly the instabilities. Moreover higher frequencies need small time step (like 0.001s) to discretize correctly the sinusoidal perturbation.  
  3. We could also try to use a longer mesh. Yet again this would be needing more computation time.
In fact each solution is probably a feasible way to obtain the "cats eyes". Yet we preferred the more radical way of rethinking our approach of the problem to avoid most of the limitations of the model. The results of this work is developed in Time Wave.

Besides the experimental structures are much thinner. This can be explained by Fluent capabilities, and the used mesh., and by the perturbation amplitude that is probably larger for the computed result. This was another point for us to search for a better modelling of these instabilities.
 
 

  • Growth rate
Study of the vortex growth rate is an important step to validate the result as it is the important characteristic of an instability. 

Remark: this part of the study was complete when the time wave result were at hand.

 On his plot the evolution of he vortex size with time hints, in spite of the availability of only a few points and a not perfect match, towards a starting exponential growth that would confirm the theory. Furthermore the expected growth rate is an expected 22 in the exponential. Our value, 26, is close to this result. Besides this chart gives another interesting result: the initial perturbation seems to have a 0.17 cm amplitude. This is an important figure when compared to the one used for the time wave study







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