Mathematics of the Navier-Stokes-Equationsfor Engineers 2
A.P. Schaffarczyk, Kiel University of Applied Sciences
9. November 2016, small corrections 03 January 2018
2IAEwind task 29, MexNext3, 3rd Annual meeting ONERA, Meudon, 2016
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 3
Contents 1
Introduction
On the interrelationship of Maths, Physics and Engineering
Statement of the Millenium problem
What is a solution ?
Types of solutions and function spaces
Existence theorem example 1: ODE: Peano and Blasius’equation
Existence theorem example 2: linear PDE: CauchyKowalewskaya
Stability of solutions in terms of Mathematics
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 4
Contents 2
Connection to Turbulence
Onsager’s Observation or must turbulent flow be non-smooth?
2D vs 3D, Navier-Stokes vs Euler
Some recent results
Meaning for Engineers / Summary
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 5
Acknowledgements
Special thanks go toAndreas Nessel and Dr. Georg Richter
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 6
Physics ↔ Maths ↔ Engineering
Physicists: Condense oberservation or formulate theory
Mathematicians: Supports with language and (sometimes)with go or no-go theorems, generalize
Engineers: What can we do with it?In time and with a fixed budget!
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 7
Types of solution of a PDE:
Closed or analytical solutions: von Karman’s rotating disk
Power series (van Dyke’s matched asymptotic expansions)
Smooth (strong) solution
Generalized (weak) solution
Numerical Solution: Table of data
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 8
Physics in Euler/Navier-Stokes Equation
ρ (ut + u · ∇u) = −∇p + f + µ∆u (1)
Left side: pure kinematics (mathematics); ρ can be put on theother sideRight side: connection to forces/stresses emerging from materiallaws.Role of pressure will be discussed later.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 9
Generalize Riemannian to Lebesque integral
Important for existence of limits in function spaces
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 10
Basic function spaces (Set of possible solutions)
Finite Energy, Lebesque → Lp(=2)
E =1
V
∫
V∈R3
|u(x , t)|2d3x := ||u||22 (2)
Finite Dissipation, (Enstrophy), Sobolev → W k=1,p=2
ǫ =1
V
∫
V
|∇ × u(x , t)|2d3x (3)
Smoothness → Cn,C∞
Rapid decreasing, test function → S(chwartz space) which aredense (with some suitable Topology) in Lp
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 11
Application: Taylor’s microscale of turbulence
Doering (2009) page 123 set:
λ = ||u||2/||∇u||2 (4)
This is very remarkable because λ is local and independent of ReN.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 12
Statement of the Millenium Problem
Prove that there exist a smooth solutionu(x , t) ∈ C∞(Rd × [0,∞)) of the initial value problem:
ut + u · ∇u = ν∆u + 1ρ∇p + f (5)
u(x , 0) = u0(x) ∈ S (6)
either
in whole space R3, or
on torus T3 (periodic BC)
Remark:p may be eliminated by using:p = ∆−1trace(∇u)2,with ∆−1 being the inverse Laplacian.Usually this is completed by mass conservation ∇ · u = 0.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 13
Weak vs Strong Solutions
Leray’s (1934) idea about turbulence was that a certainbreak-down or blow-up of classical (=smooth) solutions mayindicate and thereby define turbulence mathematically.As he was not able to (dis)prove existence of smooth solutions heweakened the notation of functions to allow for more singularbehaviour.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 14
Example of a generalized function
Point force (δ ”function” for engineers):Let x ∈ R. Then:
δ(x) :=
{
0 if x 6= 0not defined if x = 0 .
(7)
Now, like ∞ · 0 = 1 it is demanded that:
∫
R
δ(x) dx = 1 . (8)
This is clearly (mathematical) nonsense.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 15
Mathematicians did better: Schwartz’ theory ofdistributions
Let S be the Schwartz set of rapid decaying (∼ e−|x |) functions.Then δ : S 7→ R may be well-defined via:
∀f ∈ S :
∫
R
δ(x) · f (x) dx = f (0) . (9)
f ist called a test function.Remark: By partial integration derivatives are no problem.But multiplication of distributions like δ · δ is NOT defined. Forthat reason meaningful (renormalizable) Relativistic Quantum FieldTheory was delayed by some 20 years.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 16
Meaning
The idea behind generalizing functions stems mainly from the factthat non-classical functions (like δ-function) should not beexcluded as a solution.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 17
1st Example of a succesful Existence theorem
Regarde Flat plate boundary layer with uniform inflow.Reduce Navier-Stokes to Blasius’ Equation to reach at:f ′′′ + f · f ′′ = 0 f (0) = f ′(0) = 0, f ′(∞) = 1.Simplify to a set of three first order ODEs
With: f = y1, f′ = y2, f
′′ = y3it follows: (10)
F1 = −y1 · y3, (11)
F2 = y1, (12)
F3 = y2 . (13)
Now apply Theorem of Peano:
Theorem
Let: F : G 7→ Rn ∈ C0.
Then there exist (at least) one solution of y ′ = F (x , y); y(a) = y0.
READY! (first done by famous Hermann Weyl in 1942)A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 18
2nd Example of an Existence theorem
NSE reduce to a linear PDE if inertia terms are discarded:This is called Stokes or creeping flow.
µ∆u = ∇p (14)
This is a linear non homogenous Laplace equation, sometimescalled a Poisson equation.Here, a well established existence theorem byCauchy-Kowalewskaya exists.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 19
Stability - existence of solution for short times
The Millenium Problem demands for proofs for 0 < t < T → ∞.We know from experience that - under some conditions - smooth(laminar) flow can be become instable. More precicely this meansthat arbitrary small disturbances may grow exponentially in time.In the language of Mathematicians (see Doering, Annu Rev FluidMech, (2009) Eq (45) loc. cit.) a regularity time scale t⋆ can beintroduced as:
t⋆ = Cν3
||∇u0||42, (15)
so that for 0 < t < t⋆ u(t) is regular. u0 is the initial condition.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 20
A Glimpse of turbulence:Connecting Kolmogorov scaling and weak solutions
According to Kolmogorov and Obukhov
Sn(∆x) := 〈(u(x +∆x)− u(x))n〉 = Cn · (∆x)(n/3) with Cn ∈ R.(16)
Therefore n = 1 does NOT lead to an ordinary derivative if∆x → 0.Only some kind of Holder continuity holds instead:
|f (x)− f (y)| ≤ C · ‖x − y‖α, α 6∈ N . (17)
This may be interpreted as turbulent velocities being not smooth
in general. This is sometimes called Onsager’s Observation.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 21
Deterministic vs Statistical Approach to Turbulence
Averaging ” < • > ” in the sheet before was not defined.One big challenge is to investigate how the deterministic systemdescribed by NSE may be become statistical.There are two approaches:
Regard NSE as a Dynamical System or
Postulate an invariant measure = probability density function.
In engieering the second approach is well known as ReynoldsAveraging.Additive or multiplicative noise (via f) must be intoduced ifstationarity is demanded.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 22
Some Results
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 23
2D NSE
Weak and strong solutions exist.
Physical reason may be seen in a more fixed vorticity (no vortexstretching).
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 24
Euler Equations
Limit ν → 0 highly non-trivial (changes order of PDE form 2 to 1)
ρ (ut + u · ∇u) = −∇p + f . (18)
The following differences to NSE are obvious:
No dissipation
Conservation of Energy, Momentum and Helicity
Theorems of Kelvin (Circulation) and Helmholtz (Preservationof vortex streamlines)
T. Tao (see below) conjectures about blow-up solutions for Euler’sequations as well very recently.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 25
Further Selected Results
Leray/Hopf (1934, 1951
Birnir’s book (2013) will not be mentioned here
Otelbaev (2014)
Terence Tao (2016)
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 26
Leray/Hopf
Weak solutions in 3D exist.Uniqueness is an open question.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 27
Otelbaev
Claimed (positive) proof in 2014 but it was found to be wrong veryshortly.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 28
Terence Tao (*1975, Fields medal winner in 2006)
Claimed (2014/2016) blow-up solutions for an averaged version ofNSE and Euler’s equations and proposed a program towardsextension to full NSE.
This would correspond to disprove this Millenium Problem.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 29
Summary
Mathematical properties of NSE remains of interest and maybe linked to the turbulence problem (however defined).
If defined via the closure problem of engineering models aclear relationship not easy to see.
Theoretical Physicists (as proposed on Feynman’s lastblackbord) should re-enter the scene.
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 30
Many thanks for your patience!
A.P. Schaffarczyk, Kiel University of Applied Sciences Mathematics of the Navier-Stokes-Equations for Engineers 31