A recently experiment about superfluid helium He II counterflow around a cylindrical obstacle was made by Zhang and Van Sciver using Particle Image Velocimetry technique (PIV). In this experiment was showed that apparently stationary normal fluid eddies are exist both downstream (at the rear) and upstream (in front) of the cylinder as shown in figure 1.1, a phenomenon that is not seen in classical fluids. In this interim report an introduction will be made of what is the project about and what work is done until now.
Figure 1.1: Normal fluid eddies both downstream and upstream of the cylinder.
Solid particles that are injected into fluids can be tracked by using the PIV technique which is the standard method that used for several years. In a normal fluid the solid particles are expected to follow the fluid flow but a recently experiment for helium II by using the PIV technique has shown some unexpected results. That surprising experimental results have shown as it mention before the existence of apparently stationary normal fluid eddies in the thermal counterflow past a cylinder.
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In order to interpret these observations two questions could be addressed. First, what do the particles in the fluid actually trace? One possible answer is that the particle traces the normal fluid because the Stokes drag of small particles in the experiment is much larger than the other forces exerted by the normal fluid. But from the other hand the solid particles interact with the quantized vortices which may reconnect to the particle surface and lead to the appearance of the additional force exerted on particles by the superfluid.
The other question is if the circulation cells of the particulate motion map the normal eddies or result from complex interactions of both normal fluid and quantized vortices of the superfluid component. One suggestion was that since the vortex tangle was relatively dilute was expected that the particle motion maps the normal flow. Another suggestion was that the observed large vortex structures were cause by the complex interaction between the two fluid components of He II.
In the present work it is believed that the existence of large - scale vortex structures is caused by the mutual friction between quantized vortices and the normal fluid and can be explained fully by using classical fluid dynamics without appeal to an interaction between the normal and the superfluid vortices.
The objectives of this project are first of all to show the existence of those stationary configurations of the vortex - antivortex pairs, both behind and in front of the disk. Another objective is to show that the vortices located sufficiently close to the corresponding stationary points will remain close to their initial locations and how long do they remain close to them. Finally, is to discuss a possible connection between the emergences of normal fluid eddies and the polarization of the vortex tangle in the superfluid component of helium II.
Finding the Stationary Positions of Point Vortices
One of the project's objectives which are related to the experiment was to find if there exist stationary locations of the vortex - antivortex pair, both downstream and upstream of the disk. The stationary points can only be complex conjugates on a complex plane, in the upper, and in the lower half - plane. The stationary points are shown in the figure 3.1 below:
Figure 3.1: Point vortices around the disk. At the left are is the vortex - antivortex pair downstream and at the right is the vortex - antivortex upstream of the disk.
To find the stationary point positions the following cubic equation was used:
λ: is the non - dimensional circulation
This equation was solved with 3 ways. The first method was Cardano's method using excel, also was solved using Matlab with root command and finally with fsolve command. The two m - files are shown in Appendix 1. The solution of this equation has given the values of and then these values where used in the following equation to find the values: (3.2) where
The range of λs that was used was from 0 to 5 with 0.1 intervals. So by the solution of the cubic equation the following graph was produced:
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Figure 3.2: Coordinates red dashed lines and blue solid line in the upper half - plane as functions of the non - dimensional circulation λ.
Analyse the motion of vortex - antivortex pairs of two vortices downstream of the cylinder
The next objective of the project was to see how the vortices move when particular stationary points chosen, the stationary points are depending on the non - dimensional circulation λ from the range that was mentioned before in section 3. To do this the Lagrangian Equations of Motion of Point Vortices in the Inviscid Flow around the Disk were used. The equations are shown below:
j = 1,2 so:
The equation 4.2 is for the upper half - plane vortex z1 and the equation 4.3 is for the lower half - plane vortex z2. Each of the vortices has two equations, one for real and one for imaginary solution, so the total equations are four. To analyse the motion, these four equations were solved with some initial conditions using Matlab.
First of all an m - file was made called Stream1down.m (Appendix 1) to set the ode equations to find how the vortices are move. Another m - file called Motion1down.m (Appendix 1) was made to set the required initial conditions to solve the ode equations. The stationary points in Motion1down.m are taken from the stationary1down.m (Appendix 1) and the λ value (q in m - files) that is required for Stream1down.m and in stationary1down.m is taken from the lambdainput.m (Appendix 1). One example of the graphs that were made is shown below, these three graphs are shown the trajectories for the vortex points of the vortex in the upper half-plane x1 vs y1 and the time - dependent trajectories x1(t) and y1(t) for λ = 1, 1.5, 1.8, 2 and 3.
Figure4.1: Trajectories of the vortex points x1 vs y1, time - dependent coordinates x1(t) and y1(t).
From the time - depended figures 4.1.2 and 4.1.3, it is observed that for vortices initially located at the rear of the cylinder, the period of time during the vortex points remain close to their initial locations increases with the non - dimensional circulation λ. For example for λ = 1 the vortex point stays close to its initial location for approximately 5 non - dimensional units of time and for λ = 2 for approximately 35. When λ = 3 the vortex point remain close to its initial location for more than 70 no - dimensional units of time.
Analyse the motion of vortex - antivortex pairs of four vortices downstream of the cylinder
The next task of the project was to see how the vortices move when there are two vortex - antivortex pairs this time, that means two vortices at the upper half plane and two vortices at the lower half -plane as shown in figure 5.1 below:
Figure 5.1: Point vortices in the potential flow around the disk, two vortex - antivortex pairs downstream (at rear) of the disk.
At this case z11 and z12 vortices have negative circulation (clockwise) and z21 and z22 vortices have positive circulation (anticlockwise) and now the equations are eight in total (two for each vortex). The stationary points are the same as in the case in section 3 and the m - files in Matlab were made with the same idea as before and can be seen in Appendix 1. One example of the graphs when λ = 1 is shown below in figure 5.2
Figure 5.2: 1: Trajectories of the vortex points for vortices z11, z12, z21 and z22. 2: Trajectories of z11 and z12. 3: Trajectories of z21 and z22.
Again at this case the from the time - depended figures that are not shown here but will be shown in the final report it was observed that for vortices initially located at the rear of the cylinder, the period of time during the vortex points remain close to their initial locations increases with the non - dimensional circulation λ as in the case with two vortices. Another observation from figures 5.2 is that the trajectories now are not just a line and moving but are making a spiral motion, rotating and moving simultaneously.
Analyse the motion of vortex - antivortex pairs of four vortices downstream of the cylinder when there is no other flow
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This objective was to see the motion of vortices when other flows are removed and leave just one pair of vortices of the same polarity. This can be done by removing some terms from the Lagrangian Motion Equations. The first attempt was to remove the term from equation 4.2 for example. To do this this term was removed from the Strem2down.m for all the equations and so one example of the graphs for λ = 1 that was made is shown below in figure 6.1.
Figure 6.1: NO FLOW case 1. Trajectories of the vortex points for vortices z11, z12, z21 and z22. 2: Trajectories of z11 and z12. 3: Trajectories of z21 and z22.
In this case it is expected to see the trajectories of the vortices to propagate to a straight line because the vortices in the upper half - pane are negative and the vortices in the lower half - plane are positive and they do not oppose each other. In figure 6.1 this thing is observed but not at the beginning. To see the behaviour of the vortices when there is no flow correct another term was removed from the Lagrangian equations, this term was of the equation 4.2 for example. The Stream2down.m was modified and both terms were removed and renamed to Stream2downnf2.m (Appendix 1). Motion m - file and stationarypoints m - file are the same as before with small changes. Below an example graph is presented about the no flow case for λ = 1 in figure 6.2.
Figure 6.1: NO FLOW case 2. Trajectories of the vortex points for vortices z11, z12, z21 and z22. 2: Trajectories of z11 and z12. 3: Trajectories of z21 and z22.
Now from figure 6.1 it can be seen clearly what was expected. The vortices motion propagates to straight line (spiral motion on a straight line) because the upper vortices are negative and the lower vortices are positive and do not oppose each other. Since the vortices on the upper half - plane have the same signs (negative) they rotate around each other and on the lower half - plane the vortices also have the same signs (positive) they rotate around each other.
Find how long does the vortices stay close to the initial locations
The task that now is in process is to find how long that the vortices stay close to their initial locations. The main idea is to make the time - dependent graphs for each case and then see where the line on the graph goes away from the initial location. To do this the data cursor can be used in Matlab but this will take too much time and won't be enough accurate. The basic idea is to set a range of tolerance on the graph and when the line goes out of these limits (upper and lower limit) the program will stop and give the time at this point. This time will be the time that the vortex point is close to its initial location. In figure 7.1 below is an example of this idea where are showing the upper and lower limit.
Figure 7.1: Time - dependent trajectories of the two vortices case x1(t).
The m - file that was made to find this time point is ready and is like the Motion2downp.m (Appendix 1). This program sets the upper limit, for example 30% of the initial location and lower limit the same. Then at each time point it compares the location and if it is smaller than upper limit and bigger than lower limit the program continuous to the next time point, if it exceeds the limits the program stops and gives the time at this point. This is the non - dimensional (t1) time which the vortex point was close to its initial location. To find the dimensional time (t) the following formula will be used:
(7.1) where a is the disk radius and U is undisturbed flow velocity.
Now the project is going with respect to Gantt chart on page 3, the matlab code is working correct and the next task is to interpret the solution and compare the results with Zhang and Van Sciver experiment. Then to make the final report, the poster and prepare for the presentation at the end of the semester in the time as is shown in the Gantt chart.