Time Wave

 In this part we study Kelvin Helmholtz instabilities when both phases are flowing in opposite direction with the same velocity. The instability is not moving with time building time 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.  The mesh has a 1 mm step size in height and 3 mm in length. The total element number is 200*333=66600 elements. This represents more or less the upper limit for the available computing power. This time the left and right edges are translationnal periodic boundaries. The upper and lower edges are still walls.  Click here to see a picture of this mesh. A-2 Models We used Fluent 5.0.2 with a VOF model as two phases model. Other settings are 2D, unsteady, laminar, segregated. We set gravity on the y-axis (-10 m2/s) and surface tension (0.0375n/m) as for the space wave study (except for the stability study) . We kept the same fluids liquid fuel and liquid water. Yet this time as we wanted to study stability the heavier fluid, water, is below. The default solution control setting was used. As the domain is now periodic, and since we are studying non-moving waves the perturbation must now be defined at the initial state only. To do this we are imposing a sinusoidal interface as initial condition with the UDF init.c as initialization function. This function defines a upper zone with velocity 0 and  lower zone with velocity 1. We then use the FluentAdapt>iso-value function to patch both zones with the right phase (fuel on top, water below) and velocity (-1 m/s for fuel and 1 m/s for water). We had to use this method because a direct initialization did not work (segmentation error). The periodic boundaries imposes the number of wavelength on the domain length to be a integer. Setting the perturbation at the initial state makes the time step size less critical, and makes the space step size even more important. A smaller step size will not only take thin structures into account, it will also be necessary for the correct dicretization of the sinusoidal interface. We have usually taken a 6mm amplitude as initial perturbation, enabling the sinus to be described over 6 elements.  As we mainly tried shorter wavelength, that a growing more rapidly (with the same speed gap between both phases), we mostly used a 0.005s time step to follow the evolution.  B) Results This time, the vortexes are almost stationary and developing uniformly everywhere on the interface. This enables us two see later stages of the Kelvin Helmholtz formation. Water is below this time. The wavelength is 0.1m. Speed are 1 m/s for water (left to right) and -1 m/s for fuel. In this part we studied the interface aspect, the evolution of the x-axis velocity, the velocity vectors, the instability growth rate, the stability of the flow. Interface evolution For example, this picture shows the vortexes short before the formation of the "cats eyes" at time 0.18s.
 Here an Animation of growing instabilities forming "cats eyes" with a 0.005s time step. This animation can be compared to the graphic below :
 X-axis velocity Looking at the the evolution of the velocity on the x-axis shows us that the vortexes form a "cushion zone" between the opposite flows. Here the x-velocity short after the star of the computation. We can see a very small transition zone with high gradient and some "hot spots" near the interface with high velocity magnitude
 The same picture when the vortexes are formed (0.22s later) show a large transition zone. The instabilities are located within that zone. Speed gradients are much smaller and are spread over that zone. The "hot spots" are disappearing to more homogeneous zones on the upper and lower parts.
 Here an Animation of the x-axis velocity with a 0.025s time step to observe the building of that transition zone.   Velocity vectors The velocity vectors clearly show the formation of the vortex as they are building are circular pattern. Here an image at time 0.15s showing the detail of two vortexes.
 Here an Animation showing velocity vectors with a 0.025s time step.   Growth rate In this part we will now try to compare the vortexes diameter growth rate to theoretical result. This is in fact the moment of truth for the computation since not only appearance and basic parameters are reviewed but a very significant value for an instability. Plotting the diameter evolution with the setting used for the animation gives the following result (in dark blue). Also plotted trend lines for this setting. The point are every 5 ms until 40 ms then every 10 ms. Besides the stabilization after 0.1s  we can see that: The starting growth follows rather well the exponential trendline (in red) . Yet the expected value in the exponential (33) is almost the half of the expected value. We assume that this value is linked of the amplitude of the initial perturbation (0.6 cm). This value is probably too big and dampers the initial growth. To support this idea we can compare with the result obtained with the spatial wave. To confirm this we made another computation starting with a amplitude of 0.2cm (in blue). This time the trendline (in green) has a bigger growth rate (57,6) that is close to the expected value. Yet the the exponential evolution seems to stop sooner.  The end of the exponential growth is corresponding to the peek x-axis velocity and to the moment when the vortexes start rolling-up. Between the start of the spiraling and the stabilized zone the vortexes get significantly bigger following a more or less linear evolution (trendline in red).
 Stability To study the stability of the flow we made several computation with varying wavelength (i.e. varying wave number). With the gravity and surface tension used so far the flow is unstable for all possible wavelength. That's the reason why for this study we took the following setting: gravity g=400 m2/s and surface tension 15n/m. This value are of course not representative of real flow but permit to keep the interesting zone to a small domain as it appears in the picture below. With this setting wave longer than 0.315m should be stabilized by the effect of gravity, and wave shorter then 0.117m should be stable because of the surface tension. Instabilities should be observable only between those values. We made computation with wavelength 0.5 m, 0.333 m, 0.2 m, 0.1m and 0.01m. Only the first case was stable. Whereas the instability for 0.333 m is acceptable because it is close to the limit, the phenomenon for 0.05 m is more problematic. In general any attempt to have short wavelength and stability failed.

 To investigate further we plotted the growth rate for these flow. The experimental value must be taken carefully as is it not computed very precisely for this study but the position of the point give a good idea of the numerical model's behavior. We can also regret the absence of a computation at 0.4m. Yet we can see that while the points 0.2 m and 0.333 m are not too bad, the accuracy decreases towards shorter wave. We can see that the growth rate is decreasing under 0.1m hinting at a stabilization. The reason why  it couldn't be obtain lies maybe at the minimal size of the perturbation 1mm, that may be too big to avoid the formation of vortexes. It is an important issue as the dicretization error are amplified to like in these pictures of a calculation with an 0.5 m wave.

 Generally the stability prediction is the weak point of this model as short wave are always unstable and long wavelength are not very precisely stabilized.    Comparison with the space wave model To conclude this study we would like to point out the advantages and downsides of both models: For the time wave model:  Advantages are: We can have clean and precise view of the flow evolution as all vortexes are non moving and from the size. We can use small speeds. The time step is not an important factor for the computation as the UDF is only active at the initial state. Downsides are: The number of vortexes must be an integer on the length of the domain. We must use either small space step or big perturbations to ensure a good dicretization of the interface. For the space wave model: Advantages are: We can freely choose the wavelength. We can have small perturbations of the interface's height. Downsides are: We must use high speeds. The evolution is not easy to view and study. We must use small time steps as the UDF is computed each time.

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