 I don't want there is a long just okay not there so you cannot escape and there is a long time motivation for the calculations we will show today they in my case they run as long as 20 years ago when I was in Japan working with professor Nishihara and starting to study the Rismar Meshkov instability. I will speak about vorticity which is an important quantity in these flows generated by corrugated shocks they will always appear and they will more in they are more important as higher is the compression level of the fluid. There are some works quite old right now 30 or 20 years ago or even more about Mayron and Saphman and by Micaelian in United States where in which they calculated or compute the kinetic energy in the compressed fluid after the corrugated shocks separated from the surface but with two fluids this is a very difficult problem with two fluids we have two shocks and then or a rarefaction here and then this complicates the calculations by the way all our results will be analytical so I will be restrict only to linear theory calculations so that the results so that the calculations can be kept analytical and written by hand even if they take several pages long this works of Saphman Mayron and Micaelian actually they are now a bit outdated in the sense that they did not consider compressibility of the fluids. Vorticity is one of the consequences of taking into account the compressibility of the fluid in the whole time perturbation history so they used impulsive model. I will not deal here because we have not started yet with two fluids but only with one fluid flows single shock moving into a homogeneous fluid the shock is corrugated and this is what is called Ritz-Mermeskov light flows. I will set the notation and the steps how to say the notation and the physical minimum basis to understand the development of the calculation that follow later so if we consider a rippled piston whichever piston it is rigid or free surface or ablation surface if it is corrugated the shock will be corrugated it will generate pressure waves, vorticity, entropy spots and the field itself will be strongly influenced by the boundary conditions at the piston and interaction between piston and shock. I must recall here that this is not necessary at all but the pressure perturbations emitted by the shock waves here are not running waves at least they are running along the shock surface but not into the back so they are pressure evanescent waves. This is something tricky and sometimes it is forgotten and the abuse of language may give two contradictions. Just very simple scenery so we can see that a rippled shock wave in front of it the fluid is homogeneous behind it creates perturbations in density velocity pressure will define the shock numbers upstream and downstream sorry even though I teach every week my mouse tends to get dry as I speak then it's a bother. A very important parameter in these flows is the slope of the ranking at the final state so the larger is alpha so the nearer to 90 degrees so the larger will be the perturbations in after vorticity so this is something to be taken into account and I define here this quantity this quantity will be very important for the later discussion it is the time at which the shock arrives to position X beta is is the shock speed I use the letter beta because it looks nice and simply that's all. The usual approach is writing the fluid perturbations the fluid equations perturbing them obtaining here the wave equation Laplacian pressure equal to the second derivative of pressure and for the sake of simplicity and that everyone can understand my language so I just comment that this is the scaling of the dimensionalizations of the normalizations of the different quantities so the fluid equations transforming this and the wave equation transforms into this along the history while several ways of dealing with this analytically Fraley was probably the first one to deal with this and solved this equation with Laplace transforms in Cartesian coordinates but even though that's quite interesting the system of equations that results at the end is really horrible and very difficult to understand the algebra or at least to follow it however there is a very nice reference here of the 1960s in which these people used this variable change kind of imaginary polar coordinate transformation if you like to call it like that in which this is the relation between Cartesian and the new coordinates in these new coordinates the wave equation is separable so the wave equation splits into these two quantities and to these two equations the solutions of which are this infinite series written here these infinite series are very nice to handle analytically and they are represented by some coefficients here with the index 2 and Bessel functions ordinary Bessel functions of order new here and here h is just a pressure gradient so it's the derivatives of the pressure along the normal direction if we go back to Cartesian coordinates and try to write the pressure perturbations in the whole space between shock and piston surface this will look like this this combination of time and normal position and here again we have this invariant quantity the how to say the interval so to say a natural question is and this is not trivial to answer how can we integrate this analytically to obtain information on the velocity fields at any time and at any position in space so this is actually difficult if we do it by brute force just asking Mathematica to integrate it in time and it will take a lot of time and we'll collapse then that's not the way what we have found is that there is a extremely beautiful theorem called graph's theorem is a kind of addition theorem for Bessel functions in which each this expression here can be written as an infinite series this is some people may think what is nice in this change actually there is a lot of information we can extract from this transformation if we substitute these equivalences here and here into here we can integrate the equations very easily in time and very easily in space and get information on borticity kinetic energy production and follow the profiles of the perturbations as run goes from zero to infinity this graph theorem is already very well studied of course it is written in the in Watson's book a classical treatise on the theory of Bessel functions and if we continue I I arrived here very important point because this created a lot of confusion the last 20 years at least discussing with other people and still is creating problems so for example I will not show here how we get this expression but this is the tangential velocity at position x and time tau it's not at the shop not at the piston is that any position in the middle at any arbitrary time greater than tau zero it is composed of two terms this is the initial velocity the positive at that point by the shock then the shock goes far away and radiates pressure waves this pressure waves creates accelerations and negative accelerations and compress and right facts the fluid particle so given rise to delta y so this is the increment in velocity starting from this this quantity here if we plot it in space v y and vx the initial velocities created by the shock at different positions in the downstream this field is rotational all the vertices contained here and this is the action of the pressure waves so this is irrotational absolutely but it has an asymptotic in time which adds to this velocity here so this is important and not so easy to distinguish clearly which effect come from which zone after proceeding this is very brief vorticity is generated because of shock curvature but a clinic effect has nothing to do here so this is my conviction of course this is open to discussion so why vorticity is generated because simply tangential velocity must be conserved across and then with a velocity d in front of the shock and d minus u behind the projections are not enough and then the particles suffer a kind of twist when crossing the shock and this gives them vorticity and additional tangential velocity and this is all this effect was started in this year by Keblahan and he extended it to non-linear cases so the paper is actually difficult to read but it is more general than the results shown here just very briefly we need to speak in the complex plane language so we need to pass from the real-time domain to the domain of the Laplace transforms in order to get useful information this is just how to say how calculations go I start with the isolated shock boundary condition because it is the simplest and allows us to extract how to do the calculations for more complicated situations with two fluids or with rigid pistons or free surfaces I want just to mention I will not delve into the details of this here because it is it will be very long it will take more than half an hour this is the exact pressure Laplace pressure transform for the shock perturbations in the case of an isolated shock is this just this simple term it has singularities here is equal to plus minus I in the complex plane they give rise to the decay like t2 minus 3 helps so this this term is responsible for that behavior in ideal gases and if we see here we have here these coefficients if we want to plot any graph if you want to compare with experiments if we want to do any calculations we must have these coefficients pi 2 n plus 1 that accompany the vessel functions here so for this we must go to the boundary conditions at the shock front at the piston surface if it exists etc and we have found over the years four ways three of them are widely known one of them was a surprise for us because these coefficients can be written in finite terms there are no means of recurrences to solve for from P1 to P infinity so so it is this expression here this just to mention and the last method is if we are working with Laplace transform to get information in the real-time domain we need the inverse Laplace transform this is nothing that is so terrible it's just that you get used to it and well provide the integration contour take account of the branch points which are the only singularities here and this is an extremely nice and an extremely useful result for any analytical calculation that we will do later the pressure instead of writing it as an infinite series can be written as this integral this variable here is here so it is integrated in the interval from 0 to 1 where fp is this function here this function has no singularities for ideal gases and then a spontaneous acoustic emission is excluded from the problem it would be very interesting if it happened but fortunately not to complete things this phenomenon does not occur a typical solution of the equations that I have been showing up to now is this is the shock pressure perturbation here as a function of the distance traveled by the shock in units of lambda this is more convenient than k I hate kx in the in the legends this is gives more gives me directly here is the shock was here at one wavelength distance from the point it started to move what we show here is just the vorticity control left by the corrugated shock I will not explain the equations because they are simple but we have not enough time and the main motivation of this talk is to determine the geometrical size of these spots these spots not only contain vorticity but also entropy so they are like patches of temperature or entropy if you want around which the fluid is revolving the the size is obviously here clearly seen in numbers so point 25 or so for this blue vortex here in the horizontal direction and one in the vertical direction is trivial so the if we pay attention this line here well my hand trembles a little so we'll coincide neatly with the first zero of the pressure perturbation function then if we know how to write the pressure function analytically time we could hopefully extract the zeros analytically too and this is what I will show next the former representations of the pressure functions are not the the unique it can be represented as an infinite productoria of this form where this here are j are just the zeros that I show it before x1 x2 x3 x4 this is this behavior is typical of Bessel functions and also of trigonometric functions if you look into regic and gratz time book and abramovic statement etc you will find this is standard knowledge the point is that to extract information on the roots of this pressure we need the initial derivatives the infinite in initial derivatives of the shock perturbation this is not a difficult task and the method we use was already developed by Leonhard Euler in the 18th century so it this person we are very grateful to him that he existed otherwise I cannot promise I would have arrived to this so this is the algebra the inherent algebra and I am afraid of the time so I will not explain here but these are one is the first root of the pressure perturbation and it is inside a nested chain of intervals so if we increase the index K we will get improved determinations of the first root and the same can be done for the following roots there is I will write it here for very weak shocks the the infinite series collapses into a unique term this one here this Bessel function decays like like R2 minus 1 half divided by 1 unit it is minus 3 halves well this possibility allows us to get information on the weak shock limit about the size of the vortices and this is what we show next we have here the zeros of the pressure function as a function of the shock Mach number so for very weak shocks they increase like this quantity which diverges if the shock becomes weaker the vortices stay attempted to become weaker but to separate from the contact surface when the shock become big become very strong and the level of compression is high the vortex size reduces to less than one wavelength it depends on the case and of course this is what we see here for this 5 with point 5 for this gamma for this air shock moving in air here we just plotted the differences between the different the different routes a very nice result it is that after the third route all the roots become equally spaced like the cell function zeros so there is a close connection between these functions that represent shock perturbations and vessel functions just this is to complete the picture when the shock run very far away it will have left velocity perturbations I show here the black line is the complete solution the blue line is the solution far from the piston and this exponential should be the solution near the piston this is for normal velocity and this is for tangential velocity we have here a shock Mach number equal to three it is not weak then an exclusively approximately exponential approximation for the velocity field is wrong so this can be seen here especially for the tangential velocities maybe you can match the normal velocity but you will be violating mass conservation because of that because you are not taking into account that actually the velocity field turns geometrically into circles right I need more word this is the basic question if the fluid is perturbed in velocity then it has a content of kinetic energy can we calculate it at least analytically for some cases well for isolated shock their answer is of course yes for other boundary conditions calculations will be more laborious but it is also true and in the future for the classical two fluids with marines configuration we hope to do so too so what we show next is just if we deal with the kinetic energy density in volume they find well this is the typical term multiplied by the density and integrate along a strip of length lambda over two on the tangential direction and up to infinity to have the whole content of energy in the space we get this quantity here this is a dimensionless integral this dimensionless integral compute this analytically from this expression I would say it is impossible then it must be transformed by with the fluid equations and we get this very nice result it can also be said this looks as horrible as this integral here but everything what we have developed before helps to get the analytical expression of this last integral and this is how to say our last observation what I plot to the left is the shock pressure time perturbation history at the shock front as it moves so here is distance travels by the shock divided by lambda these times here are the times of local maximum and local minimum and the times of zero crossing which coincide with the times of complete vortex structure generation and I plotted here the time evolution of these kinetic energy density as a function of time in space following the shock so tau a here corresponds to this first maximum or local minimum actually of the pressure perturbation tau b corresponds to the first zero crossing when the shock creates the first vertical structure and we see this picture here while the shock is near velocities are still changing because of the pressure waves they do not reach the asymptotic but when the shock is very far this kinetic energy density in space will resemble the red thick curve here and a nice surprise if we plot this asymptotic kinetic energy as a function of Mach number we see that for very weak shocks the contribution is a sudden exponential because in that limit impulsive model is okay and then with the exponential is more than enough but a shock strength increases I found this really was really tempting to continue studying if you look the local maximum the local minima as a function of the Mach number they tend to coincide in space and a sub structure is being created in the fluid in which kinetic energy is being how to say I would not say stratified but concentrated in spots of intense motion yes I don't remember the next time yes my time is my time is running so the natural question after we have seen this is could we predict analytically the position of this maximum and minimum the answer is yes it can be predicted and I will not explain all the calculations it consists just in taking the derivative of the kinetic energy making it equal to zero and we arrive after using incompressibility and vorticity generation etc etc etc that I will I will show here this first minimum corresponds with the zeros of the asymptotic tangential velocity it looks strange but it is that and the maximum correspond to the zeros of this function here this is normal velocity and this is proportional to the vorticity field so the vorticity fields always is behind and how to say influencing the state of affairs after the shock has disappeared from the center just here this is to complete the calculations the the integral that defines this kinetic energy can be analytically calculated so we do not need to wait for Mathematica to do these complicated integrals and this here the picture is as a function of gamma for different gammas as a function of the shock Mach number there are some asymptotic scale is scaling's there is an evolve it is we understand a curve that is an envelope so the red curve here to gamma equal to one is the envelope of all the curve so it will be the maximum kinetic energy that can be extracted from these corrugated shocks inside a single fluid just to mention the exact solutions are several pages long so it is no sense to write them by hand but the different physical limits can be extracted very nicely and this well weak shock limit it starts growing like the intensity squared and later we have strange powers here due to the vorticity field strong shock limit it grows logarithmically with no this is yes there should be maybe I put the wrong slide I'm sorry this is to show just that different limits can be extracted of course with some work but not so terrible we did the same for the free surface for the free surface results are even nicer so free surface means that at the surface where the shock started to move pressure perturbations are obliged to be zero like the surface of the sea this is a difficult approximation and then we see here velocity tangential normal velocity tangential velocity for different times following the shock and maximum and minima maximum and minima agree quiet neatly with the maximum minima and zero crossing of the shock pressure history so if we plot the kinetic energy density as a function of time we still preserve the same correlation between these times and the times at the at the shock time evolution just I am going to finish this has nothing to do with the kinetic energy but this is a nice result and if we plot at the free surface ripple the ripple amplitude as a function of time so its velocity will asymptotize in time quiet rapidly and its ripple amplitude will do the following that we have also observed in two fluids either in simulations or in experiments this is red line is the amplitude of the piston ripple piston corrugation as a function of time the dotted line is it the asymptotic the asymptotic here has obviously velocity multiplied by time but this ordinate here is not the initial post shock amplitude but where a lower value so the initial post shock amplitude should be the point at which the red cube starts but because of compressibility and you want to say if you would like because of what is it this is a matter of taste for the language actually because of the compressibility of the problem the at large times it seems as if the system started from a lower amplitude not size 0 asterisk or start and the nice point and this repeats at any almost at any experiment we have seen or simulation the time at which the asymptotic crosses for the first time the complete solution here coincides very neatly with the time when the shock reaches his first maximum so the shock has not yet formed the complete of artistic field and the interface knows what will be the value of the asymptotic velocity so this is sound strange but it lips open an interesting question if this is so and this is confirmed by many calculations this may mean that the asymptotic velocity to calculate it we need the whole time pressure history perhaps with much less information it would be calculated in the future and I think I have I am out of time I cannot just well it's not conclusions which are more or less the same things I have been saying up to now thank you very much