On pressurerobustness, coherent structures and vortexdominated flows
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Title 
On pressurerobustness, coherent structures and vortexdominated flows

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CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Release Date 
2019

Language 
English

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Subject Area  
Abstract 
In vortexdominated 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 pressurerobust solvers are needed for longtime simulations of vortex dominated flows at high Reynolds numbers. Numerical benchmarks support the theory.

00:00
Goodness of fit
Fluid mechanics
Physicalism
Numerical analysis
Pressure
00:41
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>
01:15
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
01:40
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
04:38
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
05:07
Vortex
Order (biology)
Pressure
05:34
Vortex
Vortex
Degrees of freedom (physics and chemistry)
Multiplication sign
Letterpress printing
Numerical analysis
Object (grammar)
Mereology
Pressure
Pressure
Mach's principle
06:37
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
08:03
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
09:37
Sign (mathematics)
Multiplication sign
Metrologie
Point cloud
Mach's principle
10:18
Link (knot theory)
Multiplication sign
Algebraic structure
Object (grammar)
11:14
Reynolds number
Inclusion map
Vortex
Process (computing)
Water vapor
Maxima and minima
Mortality rate
Turbulence
Product (business)
12:16
Vortex
Geometry
Point (geometry)
Reynolds number
Vortex
Ring (mathematics)
Algebraic structure
Object (grammar)
String theory
Turbulence
12:56
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
13:46
Probability density function
Reynolds number
Maxwell's demon
Vortex
Velocity
Multiplication sign
Numerical analysis
Vertex (graph theory)
Turbulence
Curvature
Mach's principle
14:07
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
FinitePunktmengenMethode
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
17:23
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
17:53
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
18:15
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
19:00
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>
20:55
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
22:25
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
23:39
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
00:00
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
00:19
and so I will make and talk about physics their lives and about numerical mathematics of peonies I just want to mention coworkers 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
00:43
speak about Z
00:47
incompressible memories stoke situations and
00:50
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
01:16
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
01:31
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
01:42
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
03:41
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
04:43
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 coworker
05:08
for so it read the publicist together what do you say Mr. Yau Gration
05:16
watch X and what we see here
05:21
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
05:37
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
06:38
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
08:03
I don't use so 1st seeing our 1st
08:07
video to see what is happening here is a
08:10
very It's KelvinHelmholtz the very 1st year long times relation but Phillips showed
08:18
up and in the cabin handharvested 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 NavierStokes 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
09:45
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
10:21
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
11:14
ensuing we have similar settings and
11:20
on another 1st thing these waters appear what is what is is appear this is well
11:27
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
12:17
Sweeney and it's very similar what I
12:20
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 counterrotating 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
12:57
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
13:10
show you hear some pictures if
13:12
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
13:49
infotainment plant I sorry maybe I saw of the time there would be a city
14:02
and maybe a short in the and these
14:06
all these things they really they'd just
14:08
get turbulence here in this due to this process of class cascades that
14:14
everything will be destroyed in 3 D due to what extraction find number at a speed about
14:22
my topic OK I claim that the
14:29
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 patternbased 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 loworder people claim high is much better than you speak spp Presbytere be due about structure Preserving stings wellbalanced 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 redzone 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
17:27
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
17:52
gradient OK always a gradient
17:59
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
18:09
OK and so let's about shortly
18:13
about vector fields because all these things
18:16
are forces or in some sense and they
18:19
vector fields and importers again hamlets the hammer sparse decomposition can decompose every guy into a gradient hot and put into a divergencefree 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
19:03
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 NavierStokes because the dynamics of the was not sorry interest and now we go to
19:49
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
20:58
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
21:09
right hand side is the same if they
21:11
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 NavierStokes 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
22:27
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
23:04
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
23:41
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
24:28
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
24:55
the