On pressure-robustness, coherent structures and vortex-dominated flows

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Video in TIB AV-Portal: On pressure-robustness, coherent structures and vortex-dominated flows

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On pressure-robustness, coherent structures and vortex-dominated flows
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CC Attribution 3.0 Germany:
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2019
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English

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Abstract
In vortex-dominated flows at high Reynolds numbers, vortex filaments interact with each other and reveal coherent geometrical structures, possibly persisting over long time scales. The force balance between the material velocity time derivative and the pressure gradient leads to complicated pressure gradients, which have to be handled accurately by the space discretization. It is argued that pressure-robust solvers are needed for long-time simulations of vortex dominated flows at high Reynolds numbers. Numerical benchmarks support the theory.
Goodness of fit Fluid mechanics Physicalism Numerical analysis Pressure
Classical physics Focus (optics) Variable (mathematics) Reynolds number Stokes' theorem Partial differential equation Fluid Vector space Conjugate gradient method Velocity Number theory Spacetime Iteration Object (grammar) Körper <Algebra>
Classical physics Dynamical system Process (computing) Multiplication sign Classical physics Focus (optics) Variable (mathematics) Reynolds number Fluid Term (mathematics) Number theory Numerical analysis Spacetime Figurate number
Point (geometry) Dynamical system Building Momentum Fluid mechanics Real number Algebraic structure Water vapor Mereology Mathematical structure Product (business) Vortex Reynolds number Fluid Conjugate gradient method Term (mathematics) Velocity Object (grammar) Boundary value problem Körper <Algebra> Turbulence Pressure Physical system Fundamental theorem of algebra Chemical equation Classical physics Algebraic structure Line (geometry) Fluid Conjugate gradient method Vortex Logic Numerical analysis Pressure Körper <Algebra> Fundamental theorem of algebra
Dynamical system Fluid mechanics Direction (geometry) Mathematical analysis Scherbeanspruchung Mach's principle Reynolds number Category of being Conjugate gradient method Vortex Fluid Pressure Fiber (mathematics) Pressure
Vortex Order (biology) Pressure
Vortex Vortex Degrees of freedom (physics and chemistry) Multiplication sign Letterpress printing Numerical analysis Object (grammar) Mereology Pressure Pressure Mach's principle
Dynamical system Group action Routing Line (geometry) Scherbeanspruchung Mach's principle Vortex Reynolds number Conjugate gradient method Mathematics Sign (mathematics) Vortex Friction Different (Kate Ryan album) Velocity Object (grammar) Turbulence
Point (geometry) Group action Theory of relativity Multiplication sign Weight Uniqueness quantification Physical law Point cloud Water vapor 3 (number) Inclusion map Reynolds number Vortex Friction Metrologie Right angle Turbulence Pressure Turbulence Set theory
Sign (mathematics) Multiplication sign Metrologie Point cloud Mach's principle
Link (knot theory) Multiplication sign Algebraic structure Object (grammar)
Reynolds number Inclusion map Vortex Process (computing) Water vapor Maxima and minima Mortality rate Turbulence Product (business)
Vortex Geometry Point (geometry) Reynolds number Vortex Ring (mathematics) Algebraic structure Object (grammar) String theory Turbulence
Zirkulation <Strömungsmechanik> Multiplication sign Chemical equation Numerical analysis Heat transfer Algebraic structure Insertion loss Water vapor Mach's principle Mathematical structure Vortex Event horizon Network topology Velocity Convex hull Directed set Pressure Condition number
Probability density function Reynolds number Maxwell's demon Vortex Velocity Multiplication sign Numerical analysis Vertex (graph theory) Turbulence Curvature Mach's principle
Point (geometry) Dissipation Consistency Multiplication sign Algebraic structure Open set Mereology Mathematical structure Mathematical structure Reynolds number Partial differential equation Conjugate gradient method Different (Kate Ryan album) Divergence Directed set Turbulence Fiber (mathematics) Social class Dissipation Covering space Scale (map) Process (computing) Theory of relativity Finite-Punktmengen-Methode Chemical equation Moment (mathematics) Numerical analysis Mathematical analysis Stress (mechanics) Commutator Heat transfer Algebraic structure Physicalism Maxima and minima Food energy Group action Flow separation Divergence Vortex Maxwell's demon Partial differential equation Order (biology) Numerical analysis Computational fluid dynamics Turbulence Pressure
Model theory Constraint (mathematics) Algebraic structure 1 (number) Equivalence relation Reynolds number Friction Conjugate gradient method Category of being Pressure Fundamental theorem of algebra Chemical equation Limit (category theory) Derivation (linguistics) Conjugate gradient method Divergence Vortex Normed vector space Number theory Numerical analysis Nichtlineares Gleichungssystem Stochastic kernel estimation Social class Körper <Algebra> Force
Constraint (mathematics) Model theory Orthogonality Hodge theory Term (mathematics) Equivalence relation Derivation (linguistics) Reynolds number Divergence Finite element method Friction Conjugate gradient method Vector field Social class Nichtlineares Gleichungssystem Object (grammar) Force
Model theory Orthogonality Hodge theory Perturbation theory Mereology Derivation (linguistics) Category of being Friction Conjugate gradient method Sparse matrix Vector field Helmholtz decomposition Autoregressive conditional heteroskedasticity Nichtlineares Gleichungssystem Curve fitting Position operator
Point (geometry) Dynamical system Group action Insertion loss Water vapor Coefficient of determination Goodness of fit Conjugate gradient method Velocity Different (Kate Ryan album) Pressure Forcing (mathematics) Moment (mathematics) Numerical analysis 3 (number) Degree (graph theory) Category of being Conjugate gradient method Fluid statics Order (biology) Linearization Physics Gravitation Numerical analysis Object (grammar) Pressure Körper <Algebra> Randbedingung <Mathematik>
Point (geometry) Ocean current Fluid mechanics Equivalence relation Grothendieck topology Reynolds number Conjugate gradient method Term (mathematics) Velocity Convex set Pressure Social class Identical particles Forcing (mathematics) Numerical analysis Term (mathematics) Conjugate gradient method Velocity Fluid statics Partial differential equation Vector field Normed vector space Numerical analysis Normal (geometry) Social class Pressure Thermal conductivity Körper <Algebra> Resultant Force
Point (geometry) Constraint (mathematics) Multiplication sign Variance Mach's principle Category of being Divergence Finite element method Sign (mathematics) Vortex Invariant (mathematics) Conjugate gradient method Velocity Order (biology) Chain Iteration Mathematician Fundamental theorem of algebra Pressure
Point (geometry) Dynamical system Fluid mechanics Constraint (mathematics) Multiplication sign Algebraic structure Orthogonality Solid geometry Equivalence relation Reynolds number Vector space Category of being Curve fitting Pressure Social class Fundamental theorem of algebra Forcing (mathematics) Projective plane Stress (mechanics) Algebraic structure Perturbation theory Continuous function Equivalence relation Stokes' theorem Category of being Conjugate gradient method Divergence Vortex Normed vector space Number theory Numerical analysis Stochastic kernel estimation Social class Nichtlineares Gleichungssystem Körper <Algebra> Fundamental theorem of algebra Force
of the neurons good morning everybody and as you probably know I always speak about pressure was close us and I did this last year and but the news the news is now that the more you understand from what physical flows they are useful
and so I will make and talk about physics their lives and about numerical mathematics of peonies I just want to mention co-workers and friends and people will have worked with me also I would guess month from the letters in the rest this Parliament and the what I and I want to
speak about Z
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high when those numbers important for my talk is that we discretize this object in primer where workers that means as a vector PDE this velocity field which is fluoridated ends the gradient of the pressure of for the good do was a which is a DA tracing you could the wife to derive and other iterations but we want to focus on this primary object and we
look at hiring those numbers that means that C is constantly term the new here in front of the fridge and time is very small all in my job I would say it
wants to learn something about classic if you dynamics important figure of 1 saying that is hearts as that is so you look and it will be some kind of info
attainment and this is not only on the on science it's also on I would say communication and this is the part on that but this is related to my science of course because I worked on numerics of this year we how to build algorithms which simulated curently and presented these things which worked well was that so this big about high will some of flowers and what hot news flows has some features many people in my community speak always about urban decay that means big structures becomes smaller structures and that many other whizzy disease and in part because is of course what exterminated flows into 20 what takes shedding the production by Watters's by boundaries and important year in my come into the don't be too much about coherent structures that means interval everything is and some things the big so you think about their structures and some people in the this Applied Sciences say no at the picture of fluid dynamics and Numerical guys is not often so correct because they speak about all you were about to swoop wants the like laying as an improve understanding of numeric that we have improved understanding how to build numerics for longtime preservation of coherent structures and a message to it's much is not where we're spread is the dominant correct in fields in momentum balance I'm real initial think dominant legend fields which balances a pressure gradient of crops for many flow some of them and issue
not fall so I said we speak about what a system that's being about what is its 1st thing about what is is this is a truly vortex into question what text the important point is that users tend to forgo force balances the gradient of the pressure it the localized structure and the for the false is is a new what you through a quadratic term in the velocity balances suppression this makes pressure very difficult in some sense you have this jaunty but if you have a vortex and Big slim you you get sees how can structures of what it's for them and what its line what it's and you can have these things also curved as you see water to rings and like if quitting OK can have many different sex but this is a fundamental logic of rudiments this what it's filaments what is what OK this
has much to do with some pressure robust that's I speak about property of flow fiber switches pressure wasn't as new on in 2016 me the introduce this notion into the fluid dynamics is p for dynamics community and analysis so our 1st video by a friend and co-worker
for so it read the publicist together what do you say Mr. Yau Gration
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actually 6 autumn efforts 6 orders quite high this is a view gene message here and it's because people's purpose the bars Briskin has lacking here this is also the gene effort fix all the they have nearly the
same number of degrees of freedom of this is pressure robust and this is not this dust is what text refusing multicity that means here in this part of the food it is going you mathematical positive saying and here it's negative for and here it's well it is a compact support this object this interesting and Oceania policies I was a wise print about these other should be just translated to act but this is destroyed in really short time and us to this is pressure was this is not and we have a vortex so long let's go back I'm very slow in these things in my talk
OK as being a bit louder read about is what is a lot more text but since the 19th century you know a lot about how What is is inject had want to give you a very small inside and that so we have we can have 2 what is his next to each other this money is mathematical positive and this 1 is not enough to make a negative that then they both have a half you effect this guy here has been velocity effect on this guy going up and down routes and this guy has luminosity if it if it downwards so these tools what takes on the Baltics pass say we're powerless they don't attract and they just go down what's this is the other thing wanted to or what its path to sustain mathematics high OK of very starts to go around themselves and due to friction mail go together this is the race typical object so vorticity Miss different sign they repel have some straight men motion and if they have a difference the percent they will merge well it is is things and Maxim through dynamics videos again to look at Montes Connes
I don't use so 1st seeing our 1st
video to see what is happening here is a
very It's Kelvin-Helmholtz the very 1st year long times relation but Phillips showed
up and in the cabin hand-harvested saying we have small waters beginning from OK and due to friction the real right that become slower and the real says to begin metrology in the clouds to continue is settings and these to what is is now they stand then the alleged aggressor law texts if you're not pressure robust this thing will destroy after some time and you have to have made a long time these things become unstable and we have the 2nd merging this is now here and this is very difficult to have the exact time Ponders's it toward a problem here and toward a pine problem we have a unique solution Navier-Stokes we haven't distinct time commended what happened but this is very difficult for a long time so the the 3rd time point archive is as you saw this this what is it always the same what is it here OK therefore they much finally go to 2 new turbulence I report the effect is the important thing is dead
this actually flows spot in metrology they have these things appear that let's say as a sysadmin clouds I can see that and now we have thirtytwo time started to what a thousand 24 this is 20 and we look what is happening foot and I sent what is this was a hands and the March like here may be here on this going to be you have different sign the they're not not much It's that's quite fun isn't it there are some we
proudly they repel and somebody on the merchandise receiving the link exactly the basic facts and you can see generate this for a long long time this is well some attendees 6 this is 10 to the 4 this is this'll poker by the important thing in 20 Germans is small structures become bigger and bigger OK and this is not an sweetie the case their big structures goes smaller thank just to give an idea this Watters's out where we import object and you want to preserve them and the rats so but OK but to my talk the
ensuing we have similar settings and
on another 1st thing these waters appear what is what is is appear this is well
some of 50 thousand and as you want extending his of and they are produced in the common what extreme that and he and the rake they're very slow and the doing exactly the same thing they're Murch and we will and after that if the rates they were just from supported and they make converging processes as a song that will be wrong and here we see what heading west of us had the production of what right just to get to the idea this is by some guys from the wrong on the assumptions and this is on Hardrick at that OK now we go to the to
Sweeney and it's very similar what I
mean this what is is Michael him structures is my point point OK these are coherent structures defined objects geometric objects and and string we have that is lead forming vortex ring sentiment years because we knew vortex rings and they have counter-rotating and this was the predicted 1858 by what's but this can happen so it's quite a very old things can't this is what experiments and active and the the see coherent structures you can make this very
long time they were not dialed this is interesting but now they go to the case of the we haven't this to turban and the Kay and I will just
show you hear some pictures if
filaments they can have various tracing to you do what extraction conditions instinctual wrote experiments they made no strange 6 of these are coherent structures I mean you can happen Waters's and what you can see it's really nice and I would say pressure loss is interesting that all Mason no we did a good precious because they're overtaking and think great reviews a pressure balance as tender for the it's making that more complicated than the velocity will try this was my
infotainment plant I sorry maybe I saw of the time there would be a city
and maybe a short in the and these
all these things they really they'd just
get turbulence here in this due to this process of class cascades that
everything will be destroyed in 3 D due to what extraction find number at a speed about
my topic OK I claim that the
newest thing which people have no observed and raise oppression how can this be any food enemies 450 it's OK science to 1965 by our cover for example but there are many open questions for example we have not only PDE based solvers you have of particle based on for example pattern-based solvers were used by this problem of 50 thousand from the PM CET December so why do we have a very different kind of fibers if everything is perfect as most of its 1st question then our people speak in rising in torrid this says to me you have in my community of depict always think about adding viscosity but we have to call him structures and how do we preserve stand in and the relation is the issue which is now own then we have a big zoo of realizations artificial dissipation the originals as this hyper diffusion and so on and I'm not sure that you like that there's lots of things it's people tried to go from door to stabilize the 6 then we have high order versus low-order people claim high is much better than you speak spp Presbytere be due about structure Preserving stings well-balanced schemes nematics and I wants to save to use this common contest things are in fact also pressure whole past this is important point to the majority people have invented this at the same time I would say and so on so they invented I mean it's just they about that and they understood that it works well and they have some moment a of your order that you know what due to the gradient feets that in just over a balance and there's a max sky next in red-zone Yongqiu LPs and on and to find a different scheme of 101 in whatever you call but it works really only on Cuba's on stress and given my commute to be adaptive I don't use my head after methods is some petitioners statements several months or likes to the next and the next thing is pressure will bust very old also I think from 65 about that and parts people wanted to be exact but it is difficult to beat and then there was a discussion by babushka breath boards and these guys on this said divergency message so I mean there they're not really report in my community and had to be don't need that but if 50 they are the world's these are the pressure will bastard of wanted to explain that their message were there but nobody wanted to move because people just make analysis and not related to physics I would say OK so my my main point
is if you go to Highway those numbers and we have no forcing you can say in the incompressible Allah cases and limit in some sense the have the mature under that durative unity past you what you and this balances minus grid and pay so the important observation off on my talk is the material developed world FIFA is always a
gradient OK always a gradient
and I want to say great influence very difficult he had 4 of my father's the mind of the main objects is just a grid and the
OK and so let's about shortly
about vector fields because all these things
are forces or in some sense and they
vector fields and importers again hamlets the hammer sparse decomposition can decompose every guy into a gradient hot and put into a divergence-free part and they are orthogonal things out for the visit to scan political with check this guy if the whole forced to puts divergent 3 parts support for my talk the fundamental property here is not for the hamlet projector that's the afforded of every feud is you will like Codd Gladys is the world because if you have here gradient field we don't need a divergent free pass to get to the position OK the riot weighted fit
important 1 of his dogs though is this thing if a glass of water there's no velocity but there's a false there's the gravity it's a gradient and this gradient is completely absorbed by the gradient of the pressure so it means for the velocity there's no if this is a really big no loss of tea archives in Valiant under any great influence how large it could be this is important and important property that means the tool that guy cannot be the white object for the moment on a dynamics of Navier-Stokes because the dynamics of the was not sorry interest and now we go to
numerics credited as special is my point here we have a 1st order method we have different kind of of gradient is higher and higher polynomial order the and to get in my head how does a decays my best what of the velocity future busy world but if might not put on the good degrees more and more I did this 1st order methods more more hours it is method by not pressure will cost if I go to a 2nd order methods I can make this guy here to the world no pop here my my does of orders find this effect with 3rd order method I can do more that's here now I have the pleasure of us message 1st order vigilant obeys the great influence of phone Marek's initial many people use 2nd ordered hood that's good for on linear pressures but if the grading for the force is more complicated you have a problem this pressure was meant and not on independent could see how they treat gradient knowledge so my
point is these hands hearts projector would I called about you induces an experience classes of forces true forces like that might hear my falls on the
right hand side is the same if they
produce the same velocity features that means a credit they only differ by gradient that is absorbed by the pressure gradient of that means the White Hmong for Navier-Stokes America's not the to of f but most objective and since emigrating year of that guy would be 0 it's Osanai norms and a seminar on use it removed dozens of forces OK and people have not explored of that within a numeric is very strange result very basic it has it in a men's conductances font high removes number flows because of the nonlinear convex term of force in a true on the site assumptions and they could be a clue into a gradient field could be no force in the wide occurrence thus means our high 1 summer of flowers that are not at all difficult because effectively there's no false this is the exact using when Russia what text for agression what X guy differed standing it is balanced by gradient so if you're high wears number there no falls in the wide the current task forces OK I don't want to
show the video again but this is my point here the material derivative my question what 6 It's a gradient and this this guy here is this a steady solution of the or the iteration this compact support and this moving thing is just him in variance to that and people have they the solvers have problems resist any solution of the order fresh people didn't know that for a long time I would say if you want to read that I don't
want to show we can have this sign is saying it did not understand the point of the crew stars evolve accorded fundamental invariance property so we have a I shies preprint from 2018 from August to where we have now the right language to communicate more to mathematicians I called this strange property that chains of the force by grand and and the velocity doesn't feel chains accord as fundamental invent poverty than the other way let's just assume I'm on and occurrence of fossils now with the new language components and much more much easier messages the
divergent constraints and in who was a reversal in use is has a seminar on how much I have what what's projector degrading fits as a connoisseur of hot project this induces increments stars of forces personal busses and rail assures that in equivalence classes around merits as my important point it the best possible way and this is a fundamental property for flaws was high when those numbers and we need that for a long time preservation of his Korean structures the others are in preparation we have a concept for a compressible flow services I'm adjustment method because this can be used as a new kind of rabbit but in scheme for compressible solids as well I don't want to speak about relevance is not so
many things 2 people from that is a master at work would really wanted to do this thing seriously food dynamics so it is I want many things for the organization I want it was also out of class and deal for not pick the term interested in these things and this is my talk again that it
the
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