 Hello, my name is Samin Grafava. I'm from the Institute of Continual Media Mechanics. I'm from Russia and today I want to tell you about helicity distribution in a convective vertical flows. So helicity is one of the most important characteristics for describing the structure of vortex flows. And helicity is a topological invariant which measures the degree of linkage of the vortex lines and defines like a dot product of velocity and vorticity vectors. And the volume integral calculated in specific domain gives a total value of helicity or a global helicity. So helicity is an invariant in order of question and existence of helicity can lead to reduction of turbulent viscosity and energy accumulation in the latch scale or play a significant role in the formation of latch scale vortex structure in the atmosphere of planet. But up to now the problem of helicity is unsolved because it's very complex to study. It requires a three-dimensional measurements of velocity in the volume or high resolution numerical simulation. And in our case, no, even realization of the flow with substantial value of helicity is quite complicated. In our work, we consider the helicity distribution in convective flow from localized heat source in rotating fluid layer. Our flow is not fully turbulent, but the structure of latch scale flow in our system is very promising for helicity formation. So differential characteristics on flow were studied by numerical simulation and experimental data were used for verification. So as the experimental model, we use cylindrical vessel. Sites and bottom was made from plexiglass, and the heater is a brass cylinder, which mounted in the center of the plush with the bottom. As the working fluid, we use silicon oil with kinematic viscosity 5-centistox, where depth of the fluid layer was always 30 millimeters and the surface of fluid was always free. Setup was placed on the rotating horizontal table. And the temperature of fluids was measured by thermocouples. And for getting the velocity fields, we used a 2D particle image velocity system. In the symmetry system, it includes dual pulsate laser digital camera, which placed in the rotating frame control unit at computer. So here you can see the scheme of experiment and some approaches of PAV system. Along with dimensional parameters, we use the system of non-dimensional parameter, which has flux Grashof number, Prandtl number, Reynolds number, and Eckman number. So the heating in the central area, sorry, I should... So heating in the central area initiates the intensive upward motion, and warm fluid cools on the upper layer and goes to the periphery, where cold fluid moves down. So after some time, the large-scale advective flow occupies the wall vessel. And in the presence of rotation, large-scale radial flow leads to the transport of angular momentum. So convergent flow takes the fluid with large value of angular momentum from the periphery to the center and produce cyclonic motion. And divergent flow takes the fluid with low value of angular momentum from the center to the periphery, resulting in anti-cyclonic motion. So this is...the structure of azimuthal flow is very typical for tropical cyclones. And here is the experimental video and some video from nature, and one can see many similarities. For studying the helicity distribution, we need to have three-dimensional velocity fields, but it's quite difficult. There are some experimental techniques. For example, tomographic particle image velocity symmetry, but they are very expensive and resource-demanding, so general results were received by using numerical CFD package flow vision. So the integrated domain was a cylindrical cavity. The sizes were the same as experimental, and the fluid is assumed to be incompressible and flow incompressible and laminar, and the fluid is a Newton, is compressible or is assumed to be Newton. So upper layer was stress-free. The bottom has a localized heat source and a tall boundary condition, and the additional condition was chosen close to experiment. There are some variable with the value of non-dimensional parameters for numerical simulation and experiment. For verification, we received the vertical profile of radial velocity, mean radial velocity, and fields of azimuthal velocity, averaging in azimuthal direction. So one can see a good agreement and quantitative, and let's go to the results. The large-scale objective flow in the lower part of the layer leads to formation of boundary layer with potentially unstable temperature stratification. You can see it here on the distribution of temperature at the vertical coordinate. It's a vertical profile under the heater, and in this boundary layer secondary structures appear under the heater. So there are visualizations of secondary flows by Schodograph method. Without rotation, in weak heating regime, the secondary flows have a form of ring-like rolls, and with increasing the heating, the structure of secondary flows became more complex. You can see the radial rolls and some transverse rolls. Transverse rolls move to the center of the vessel. In the presence of rotation, radial rolls have a spiral form because of intensive cyclonic motion, and simultaneously there is transverse rolls, which move to the center and float. They periodically float, and as a result, we have a system of convective plumps. These convective plumps are clearly seen on the instantaneous temperature filled in the vertical cross-section. These convective plumps move to the center and disturb the velocity field, the verticity field, so the existence of these secondary structures is very important for helicity formation. So the flow in our system is very complex and consists of different structures of different scales, and for better understanding, we divided the helicity into three parts, azimuthal helicity, radial helicity, and vertical helicity, and there are, in the table, mean values of each of components of helicity, and there are standard deviations for the case without rotation, and it is important to see that the value of mean helicity is much smaller than the value of RMS, and there are the fields of RMS of helicity, and RMS is concentrated in the area of secondary structures formation, so we can assume that secondary structures gives only a local value of verticity and only a large-scale flow of special structure can lead to the helicity formation. So let's go to rotating case in the presence of rotation. There are two main mechanisms that can lead to the existence of helicity as the first mechanism, we assume the strong correlation of vertical flow, upward flow in the center and cyclonic motion, and as a second mechanism, we can see the strong share of radial and azimuthal velocity on the periphery, but before describing the general results, there are some things that should be noted. It is common in fluid dynamics between theory of turbulence to divide the average and perturbation path. For the velocity, the composition will be like this as well as for verticity, and this one is steady-component and this one is perturbation path, and the time average of perturbation is equal to zero. For helicity, the composition will be the same, but the steady-component of helicity will include the time average product of velocity and verticity perturbation, and in our case, this path of steady-component of helicity is not zero, is approximately 20-12% of mean helicity, and it's concentrated here in the area of upward motion of cyclonic motion. For discussion, I will mean the global helicity, including this perturbation path. For studying helicity, it's very important to describe the integral values integrated in the azimuthal direction, so this quantity, because even a small local value of helicity on the periphery can lead to substantial value of helicity after integration, and here we present the field of this quantity, average in time, and you can see the asymmetrical distribution of integral helicity, and the global value of helicity is positive, and this is a very important result because it proves that the flow in our system is characterized by substantial value of helicity, and we have there in the central path to maximum. So let's consider the each component of helicity. There are fields of azimuthally integrated radial, azimuthal and vertical helicity. So radial helicity is smaller than azimuthal and vertical. Vertical helicity is concentrated as we expected in the area where we have a stronger relation between upward motion and cyclone, and here the large value of azimuthal helicity, negative value concentrated in the periphery where divergent flow replaced by convergent, and there are the maximum in the area of secondary structures formation. So as the next step, we want to estimate the level of helicity, and in the theory of turbulence, it's common to use this non-dimensional ratio. There are this one and this one in spectral density of helicity and kinetic energy, and this ratio is used to define the influence of helicity on spectral properties of turbulence, but in our case, we just multiply the integral of helicity on the characteristic size of our flow. It's the depth of the layer in order to have the same dimension as the integral kinetic energy, and this is the value in the table. So in our case, the relative level of helicity is approximately 20%. In comparison with non-dimensional rotating layer, this ratio was less than 1%. So it's very good, but there is one more important thing. The vortexes in our flow, this cyclone in the center, have a strong dependence on Reynolds. Up to some values, approximately 23, the cyclone exists in the center, so we call it a steady state. It exists on the center and it's quite robust, but when the Reynolds, more than 23, the vortex became unstable. It exists not in the center and moves around the vertical axis, and we call it unstable regime. So there is experimental video of unstable regime. The picture is not symmetrical. The cyclone is moved around the vertical axis. And here the same average helicity field for stable and unstable regime. And you can see that these fields very differ in an unstable regime. Helicity is concentrated in the center, but in unstable regime, helicity much smaller and spread near the lower boundary layer. There are some conclusions I have discussed during the presentation. I just want to add that here we focus only on the quantitative description of helicity field, but understanding of helicity distribution and dynamics requires analysis of helicity sources. And it will be done in future using a question for helicity balance. Thank you. Introduce helicity this way. Is it invariant? Or you measure it in a laboratory? Yes. So just in a co-rotating frame, can it be made larger or smaller, or is it invariant? We don't measure helicity because it requires a three-dimensional velocity field. We just use experimental data for verification. But we analyze the helicity already in steady regime. When our flow is already steady, so we don't describe the transient process when our flow is not steady, not in initial stage. But helicity is invariant in the order of question. And does it stay more or less constant in time when you measure it? So the global value of helicity is growth. And it has some fluctuation, yes, oscillation. And in non-rotational case, this oscillation has the same frequencies as the frequencies of secondary structure formation. But in rotating case, there are no special frequencies, just perturbation.