Mathematics of the Navier-Stokes-Equations for Engineers ... · Mathematical properties of NSE remains of interest and may be linked to the turbulence problem (however deﬁned)

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


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


Acknowledgements

Special thanks go toAndreas Nessel and Dr. Georg Richter


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!


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


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.


Generalize Riemannian to Lebesque integral

Important for existence of limits in function spaces


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


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.


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.


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.


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.


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.


Meaning

The idea behind generalizing functions stems mainly from the factthat non-classical functions (like δ-function) should not beexcluded as a solution.


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.


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 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.


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.


Some Results


2D NSE

Weak and strong solutions exist.

Physical reason may be seen in a more fixed vorticity (no vortexstretching).


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.


Further Selected Results

Leray/Hopf (1934, 1951

Birnir’s book (2013) will not be mentioned here

Otelbaev (2014)

Terence Tao (2016)


Leray/Hopf

Weak solutions in 3D exist.Uniqueness is an open question.


Otelbaev

Claimed (positive) proof in 2014 but it was found to be wrong veryshortly.


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.


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.


Many thanks for your patience!


Documents

Mathematics of the Navier-Stokes-Equations for Engineers ... · Mathematical properties of NSE remains of interest and may be linked to the turbulence problem (however deﬁned)