analysis of complex singularities in high-reynolds-number ...jftiago/pdeba... · workshop on...

Introduction Boundary Layer Theory - Navier Stokes Complex Singularity Tracking Work in progress

Analysis of complex singularities in high-Reynolds-numberNavier-Stokes solutions

Vincenzo Sciacca

joint work with

Marco Sammartino

Francesco Gargano

Kevin W. Cassel

Department of Mathematics and Computer Science

University of Palermo, Italy

Workshop on PDE’s and Biomedical ApplicationsDecember 4-6, 2014, Lisbon, Portugal


Introduction

The interaction of an incompressible fluid at high Reynolds number with a rigidboundary is one of the main interest phenomena in the classical fluid dynamics theory.

It is in fact known as such interaction is one of the possible mechanisms of thetransition from laminar to turbulent regimes.

Introduction to Boundary Layer Theory

Prandtl equation for the “impulsively started” disk.Singularity formation.

Comparison between Prandtl and Navier-Stokes solutions

Large-scale and small-scale phenomena.

Complex Singularity Tracking

Pade approximants and Polya method.

Work in progress


Navier-Stokes Equations

∂uNS

∂t+ uNS · 5uNS +5p =

1

Re∆uNS

5 · uNS = 0,

uNS(x, t = 0) = u0.

Euler Equations

∂uE

∂t+ uE · 5uE +5p = 0

5 · uE = 0,

uE(x, t = 0) = u0.

Without boundaries, ∥∥∥uNS − uE∥∥∥→ 0 for Re→ +∞,

Swann Trans AMS 1971 in R3,

Constantin & Wu Nonlinearity 1995 in R2 for initial data of “vortex patch” type.


Navier-Stokes equations

∂uNS

∂t+ uNS · 5uNS +5p =

1

Re∆uNS

5 · uNS = 0,

uNS |∂Ω = 0,

uNS(x, t = 0) = u0.

Euler Equations

∂uE

∂t+ uE · 5uE +5p = 0

5 · uE = 0,

ν · uE |∂Ω = 0,

uE(x, t = 0) = u0.

The different number of BC generates a Boundary Layer which expands to theinternal flow, due to non-linearity.


To study the Boundary Layer flux, Prandtl (1904) introduced the following scaling,valid near the boundary:

Y = y√Re with

∂u

∂Y= O(1).

This implies that:

∂u

∂x= O(1),

1

Re

∂2u

∂y2=

1

Re

∂2u

∂Y 2= O(1) = u

∂u

∂x. (1)

Prandtl Equations

∂tuP + uP ∂xu

P + vP ∂Y uP + ∂xp = ∂Y Y u

P

∂Y p = 0

∂xuP + ∂Y v

P = 0

uP (x, Y = 0) = vP (x, Y = 0) = 0

uP (x, Y →∞) −→ uE(x, y = 0)

uP (x, y, t = 0) = uin .

The procedure is the following: first solve Euler equations, to obtain the boundarydata uE(y = 0) and then solve Prandtl equation.

Conjecture : uNS = m(y√Re)uE +

(1−m

(y√Re))

uP +O(Re−

12

).


Kato (1984)

The following are equivalent:

limRe→+∞

∫ T

0

1

Re

∫d(x,∂Ω)< c

Re

∣∣∣5uNS∣∣∣2 dxdt = 0.

∥∥∥uNS − uE∥∥∥→ 0 per Re→ +∞ uniform in t ∈ [0, T ] .

If there is no energy anomalous dissipation at the boundary, in a layer of amplitude1/Re, and then in the entire domain, then the zero viscosity limit presents noturbulent behaviour.

If you want to solve the problem of the zero viscosity limit of the NS equations mustbe checked or improve (or disprove) Prandtl equations.

Well posedness:

Oleinik 1967 (∂Y uin > 0).

Xin and Zhang ’03 (favourable pressure gradient).

Sammartino and Caflisch 1998 (analytic initial data).

Lombardo, Cannone and Sammartino ’03, ’13 (analytic initial data in thestreamwise direction).


Singularities for Prandtl equation:

Van Dommelen and Shen ’80:

show numerically, with Lagrangian methods, that Prandtl solutions developed shocktype singularities at a finite time

These results are confirmed in Cowley and Van Dommelen ’90, o Hong e Hunter ’04.

E and Engquist ’97:

prove analytically, that for suitable initial data, different from VDS data, that Prandtlsolutions developed a shock type singularity.

Gargano, Sammartino and S. Physica D ’09:

using the complex singularity tracking method, characterize the Prandtl singularity andthey give the first numerical evidence that Prandtl equations are ill-posed in H1.


VDS datum

Consider the Prandtl and Navier-Stokes equations in the case of a disk in a uniformflux, impulsively started. The physical domain is [θ, r]=[0, π]×[a,∞], where a is theradius of the disk.

U

a

The inviscid irrotational solution of the Eulero equations, given in terms of thestreamfunction (ψx = −v, ψy = u), is the following:

ψ(θ, r) = U(r −a2

r) sin(θ),

where U is the streamwise component of the velocity at infinity.


VDS datum: Prandtl equation

In the case of the impulsively started disk, the equation of Prandtl are the following:

VDS Datum

∂tu+ u∂xu+ v∂Y u− U∂xU = ∂Y Y u [0, π]× [0,∞]

∂xu+ ∂Y v = 0

u(x, 0, t) = v(x, 0, t) = 0

u(x,∞, 0) = U

u(x, y, 0) = U

U = 2 sin(x)

The solution develops a singularity in a finite time

(Van Dommelen & Shen J.Comp.Phys. 80’).

To solve numerically the equation of Prandtl we used a fully-spectralFourier-Chebyshev (Pseudo–Spectral τ–method) numerical method and a RK2CN forthe advance in time.


VDS datum: Prandtl equation

t = 0.4

xY

0 1 2 3

1

2

t = 1

x

Y

0 1 2 30

3

6

x

Y

t = 1.4

0 1 2 3

10

20

t = 1.5

Y

x0 1 2 3

10

20

d)c)

a) b)

The formation of VDS singularity is due to the phenomenon of recirculation:Formation of a back-flow, and of a “stagnation point”, due to a adverse pressuregradient at time t ≈ 0.4;

At t ≈ 1 two “counter-rotating” vortex are visible;

The two vortex grow forming a “kink” at t ≈ 1.35 which evolves in a “sharpspike” at t ≈ 1.5;

The vorticity in the Boundary Layer is expelled to the external flow, and thenormal component of the velocity becomes infinite (in the BL scale) in thestreamwise location of the singularity: separation phenomenon.


VDS datum: NS equations

The Navier-Stokes equations in the vorticity-streamfunction form:

Vorticity-Streamfunction system

∂ω

∂t+u

r

∂ω

∂θ+ v

∂ω

∂r=

1

Re∆θ,rω, [0, π]× [1,∞]

∆θ,rψ = −ω,

u =∂ψ

∂r, v = −

1

r

∂ψ

∂θ,

ω(θ, r, t = 0) = 0,

ω(θ, r →∞, t) → 0,

u(θ, r = 1, t) = v(θ, r = 1, t) = 0.

where ∆θ,r is the Laplacian in cylinder coordinate, and the Reynolds number is

defined as Re = aUν

.

To solve numerically NS equations, for different Re (from Re = 103 at Re = 105), weused a Fourier-Chebyshev fully-spectral Galerkin-Collocation numerical method withan AD-BDI2 (with the influence matrix method to compute the BC).


The streamline for Prandtl and NS at different Re.


Large-Scale interaction

Prandtl Streamlines, t=1.1

0 0.5 1 1.5 2 2.5 30

5

NS Streamlines, Re=105, t=1.1

0 0.5 1 1.5 2 2.5 30

5

π

π

1.5 2 2.5 3−2

−1

0

1

2 t=1.1

PrandtlNS −−Re = 105

π

“Wall shear stresses” (the vorticity at the boundary):

τPw = ∂Y uP|Y=0

(Prandtl)

τNSw = Re−1/2∂yuNS|y=0(Navier-Stokes )

The LS interaction appears for all the Re:The evolution of the flow is similar to that predicted by the BL equation, it isvisible a single region of recirculation. However, one can notice some initialquantitative differences between Prandtl and NS solutions.


Small-Scale interaction

Prandtl Streamlines, t=1.35

0.5 1 1.5 2 2.5

NS Streamlines, Re=105, t=1.35

0.5 1 1.5 2 2.5

1.5 2 2.5 3−3

−2

−1

0

1

2 t=1.35

PrandtlNS −−Re = 105

π

“Wall shear stresses” (the vorticity at the boundary):

τPw = ∂Y uP|Y=0

(Prandtl)

τNSw = Re−1/2∂yuNS|y=0(Navier-Stokes )

For “moderate-high” Re (Re ≥ O(104)) the LS interaction evolves to a small-scaleinteraction, characterize:

Splitting of the recirculation region.

Formation of high gradients in the angular direction.



Let u(Z) = (Z − Z∗)α =∑∞k=−∞ uke

ikZ , be an analytic function with a singularityof algebraic type α in Z∗ = x∗ + iδ, then the asymptotic behaviour of its Fouriercoefficients is given by the following (Laplace formula):

uk ∼ C|k|−(1+α) exp (−δk) exp (ix∗k).

The exponential decay rate of the spectrum δ gives the width of the analyticity strip.

singularity ℑx

ℜx

strip of analyticity

δ

If the complex singularity reaches the real axis, the singularity is shown in the “realworld” as a blow up (of the solution or of its derivatives).The singularity time ts is the time when δ(ts) = 0. x∗ and α give, respectively, thereal location and the algebraic character of the singularity.(Sulem, Sulem & Frisch’83, Frish et al., Caflisch, Cowley, Pugh, Shelley, Tanveer)



Given a function u(z, w) =∑h,k ahke

ihzeikw, one defines the “shell–summed Fourier

amplitude” (Frisch et al. ’05) as

AK ≡∑

K≤|(h,k)|<K+1

ahk.

Using the asymptotic Laplace formula for AK , one can determines the distance δ ofthe complex singularity and its algebraic type α.

0.9 1 1.1 1.2 1.3 1.4 1.50

0.1

0.2

0.3

0.4

0.9 1 1.1 1.2 1.3 1.4 1.5

0.3

0.5

0.7

0.9

δSh

(t)

αSh

(t)

t

t

(a)

(b)

0 1600 20001200400 800

−16

−12

−8

−4

0

−2

−6

−10

−14

log|AK|

t=1.5

t=1.49

t=1.48

t=1.47

t=1.46K

t=1.25 t=1.40 t=1.45

Application to Prandtl equation in Della Rocca, Lombardo, Sammartino, S. J. App.Num. Math. ’05, Gargano, Sammartino, S. Physica D ’09.



Another possible method (Poincare 1899, Tsikh 1993), consists to evaluate theasymptotic Laplace formula to each direction of the bi-dimensional spectrum,

(h, k) = K(cos(θ), sin(θ)).

The distance δ is the minimum between all directions.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.05

0.1

0.15

0.2

0.25

θ/π

δ(θ)

0 100 200 300 400 500−80

−70

−60

−50

−40

−30

−20

−10

0

1.2 1.25 1.3 1.35 1.4 1.45 1.5

0

0.05

0.1

1.2 1.25 1.3 1.35 1.4 1.45 1.5

0.3

0.45

0.25

1.2 1.25 1.3 1.35 1.4 1.45 1.5

1.94

1.95

1.96

x∗(t)

δ(t)

α(t)

t

t

t

log|uk|

k

T=0.75

T=1

T=1.1

T=1.15

T=1.2

T=1.25

T=1.3

T=1.35

T=1.4

T=1.5

Application to Prandtl equation Della Rocca, Lombardo, Sammartino, S.J. App. Num.Math. ’05, Gargano, Sammartino, S.Physica D ’09



Application to NS Gargano, Sammartino, S., Cassel J. Fluid Mech. ’14

0 0.5 1

x 10−3

0

0.005

0.01

0.015

0.02

0.025

0.03

1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t

a) b)

1/Re

δNS(t)

5 · 103

104

5 · 104

105

103δmNS

0 1 2 3 4 5 6−20

−15

−10

−5

0

5

Re = 105

log(K)

log(AK)

t=1.4

t=1.6

y =-log(K)

0.5 1 1.5 20.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55

t

105

104

1.5 · 103

αSh




0 1 2 3 4 5 6 7 8 9 10x 104

0.8

0.85

0.9

0.95

1

Re

0 1 2 3 4 5 6 7 8 9 10x 104

1.5

2

2.5

3

Re

T (!MNS)

Large-Scale beginning

t

T (!mNS)

a)

b)




0 1 2 3 4 5 6 7 8 9 10

x 104

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

T=1.4

T=1.5

T=1.6

T=1.7

Re

θ∗most singular direction θ∗ inthe spectrum vs Re numberat various time



The “singularity tracking method” gives the singularity closest to the real plane. Weanalyze the full set of complex singularities considering the wall shear for Prandtl andNS.

Pade approximants

This methodology allows to determine the position of the complex singularities.

Does not provide clear information regarding the characterization of complexsingularities.

Polya method

This methodology allows to capture the position and the characterization of “branches ” or poles of a power series.

When two or more singularities are close together, there are numerical problems.

This method was introduced by Pauls & Frish J. Stat. Phys. 07’.



Pade approximants: given a complex function as a Taylor (Fourier) series

u(z) =∞∑k=0

ukzk,

the Pade approximants PL/M is the rational function which approximates u:

PL/M : =

∑Li=0 aiz

i

1 +∑Mj=1 bjz

j= u(z) +O

(zL+M+1

),

The ai, bj are determinate from the following linear system (ill− posed):

min(α,M)∑i=0

bicα−i = aα α = 0, . . . , L;M∑i=0

bicL+β−i = 0, β = 1, . . . ,M.

This method only provides information on the location of the various complexsingularity, even those outside the radius of convergence, but does not provideinformation on their characterization.



Polya method: Let f be an analytic function:

f(z) =N∑k=0

ak/zk+1

︸︷︷︸Taylor(Fourier)

.

Let H be the radius of convergence, andK the the smallest convex and compactset that contains the singularities. Then:

F define an entire function ofexponential type,

F (ζ) =N∑k=0

akζk/n!

︸︷︷︸Borel−Laplace Transform

the following relation holds k(ϕ) = h(−ϕ) with

h(ϕ) = lim supr→∞

ln(F (reiϕ) indicatrix function

k(ϕ) = supZ∈K

<(Ze−iϕ) supporting function,

H = supϕ h(ϕ).



Consider f(z) ∼∑nj=1(z − cj)αj−1, with n complex singularities in cj = |cj |e−iγj ,

then the asymptotic behaviour along direction reiϕ, with ϕj−1 < −ϕ < ϕj :∣∣F (reiϕ)∣∣ = Crαj eh(ϕ)r [1 + ε(r)] ,

h(ϕ) = |cj | cos(ϕ− γj).

From the indicatrix function h(ϕ) (which is a piecewise-cosine function) oneobtains informations of the complex singularity location.

Form αj(ϕ) one obtains the algebraic character of the complex singularity.

High-precision numerical computation is used if two or more singularities are closetogether.

Both the method of Pade that Polya, if applied individually, do not give a completepicture of the complex singularity of a given function, therefore they should be usedtogether to provide a solid framework for analysing complex singularity.


Prandtl Singularity

0 500 1000 1500 2000

−160

−120

−80

−40

0

k

t = 1

t = 1.4

t = 1.5

t = 0.5

ln |τPk|

(d) The Fourier spectrum of τPw .

1.4 1.6 1.8 2 2.2 2.40.85

0.9

0.95

1

x

h(x)

(e) The indicatrix function h(x) at ts =

1.5 which is a cosine centred in xs ≈1.94.

0.5 1.5 2.5 3.5 4.5−14

−10

−6

−2

0

ln(k)

ln |τPk |y = −(1 + 7/6) ln(k)

(f) The algebraic type of singularity of

τPw .

1.7 2 2.31

1.1

1.2

x

αP (x)

(g) The algebraic decay rate is αP ≈7/6.


Prandtl Singularity

x0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0 0.5 1 1.5 2 2.5 3−2

0

2

4

x

τPw

∂xxτPw /100

a)

b)

xim

The Pade approximant P200/200 of τPw at t = 1.5.

τPw at t = 1.5 shows a blow up in its second derivative.


NS Singularity

1.8 2 2.2 2.4 2.6−20

0

20

!

!1.8 2 2.2 2.4 2.60

0.1

0.2

0.3a)

!im

"NSw

b)

The Pade approximat P300/300 of τNSw for Re = 105 at t = 1.58. Aredistinguishable three distinct groups of singularities.


NS Singularity

Prandtl syngularity (sP )

1.8 2 2.2 2.4 2.6−20

0

20

!

!1.8 2 2.2 2.4 2.60

0.1

0.2

0.3a)

!im

"NSw

b)



NS Singularity

Prandtl singularity (sP )

Large–scale singularity (sls)-related tothe formation of the Large–Scaleinteraction.

1.8 2 2.2 2.4 2.6−20

0

20

!

!1.8 2 2.2 2.4 2.60

0.1

0.2

0.3a)

!im

"NSw

b)



NS Singularity

Prandtl singularity (sP )

Large–scale singularity (sls)-related tothe formation of the Large–Scaleinteraction.

Small–scale singularity (sss)-related tothe formation of the Small–Scaleinteraction.

1.8 2 2.2 2.4 2.6−20

0

20

!

!1.8 2 2.2 2.4 2.60

0.1

0.2

0.3a)

!im

"NSw

b)



NS: the sP singularity

1.9 2 2.1 2.20

1

2

3

θ

7 8 9 10 11 12−2

−1.5

−1

−0.5

Re

yP

3.20 · Re−0.25

Re 103

Re 104

Re 5 · 104

Re 105

Prandtl

b)a)

θim yP

t = 0.1

t = 1.5

The evolution in time of sP and its minimum distance from the real axis yP respectto Re.

1.92 1.93 1.94 1.95 1.96 1.97 1.980

0.4

0.8

1.2

θ

Re = 103

Re = 104

Re = 105

αPNS(θ)

The algebraic characterization of sP at t = 1.5.


NS: the sls singularity

θ0 1 2 3

0

0.5

1

0 1 2 3−2

0

2

4

θ

θ0 1 2 3

0

0.5

1

0 1 2 3−2

0

2

4

θ

Re = 103

Prandtl

Re = 103

Prandtlτwτw

a) b)

d)c)

θimθimθim θim

The comparison between τw of NS (with Re = 103) and Prandtl, and the Padeapproximant P200/200 at t = 1 (a, b) and t = 1.45 (b, d). At t = 1.45, sls is related tothe formation of a gradient in τw close to its minimum.


NS: the sls singularity

2.2 2.4 2.6

0

1

2

3

!

8 10 12−1.3

−1

−0.7

−0.4

Re

Re ! 103

Re ! 104

Re ! 5 · 104

Re ! 105

yls

0.44 · Re!0.138

b)a)

!imyls

t = 3t = 1.5

t = 0.1

The evolution in time in the complex plane (θ, θim) of sls. After the large-scaleinteraction, the real part of sls moves “upstream” along the cylinder forRe = 104 − 105, while the opposite occurs for Re = 103.

At TLS the time when the large-scale interaction starts, the singularity remains ata distance yls from the real axis, which depends on Re as 0.44 ·Re−0.138

(yls → 0 for Re→∞).

The singularities of τNSw have αslsNS = 1/2 (t = 1).


NS: the sss singularity

9 10 11 12−4

−3.5

−3

Re

2.3 2.4 2.5 2.650

0.5

1

1.5

2

θ

Re − 104

Re − 5 · 104

Re − 105

yss

0.41 · Re−0.25

a) b)

yssθim

The evolution in time in the complex plane (θ, θim) of sss. When the small-scaleinteraction starts the singularity is a distance yss from the real axis, whichdepends on Re as 0.41 ·Re−0.25 (yss → 0 as Re→∞).

The singularities have α = 1/2.


0 0.5 1 1.50.15

0.35

0.55

0.75

t

0 0.5 1 1.50.3

0.5

0.7

0.9

1

t

Re 103

Re 104

Re 5 · 104

Re 105

Re 104

Re 5 · 104

Re 105

dss

a) b)

dls

The evolution in time of the distance dls in the complex plane between sP andsls for different Re, and the evolution in time of the distance dss between sPand sss.

Conclusions:

In both cases, the distance decreases with the increase of Re, and this supports theconjecture that asymptotically all singularities are reduced to the singularity of PrandtlsP .


Navier BC:

Navier Boundary Condition:

uNS · ν = 0,1

Re

(∂uNS

∂ν+ CuNS

)· τ = −β(Re)uNS · τ .

If

limRe→+∞

1

Re

(∂uNS

∂ν

)= 0,

then each weak limit uNS is a dissipative solution of the Euler equations.

Dirichlet BC: β = ∞

In recent years several of the classical fluid problems of this type have been recast tomodel flows on a nanoscale or microscale and the Navier-slip conditions becomerelevant in certain applications related to hemodynamics and high-altitude flows aswell.In the present study Navier boundary conditions allows for slip on the disk, and weconsider β(Re) = Re−α .


Navier BC:

0 0.1 0.2 0.3 0.4 0.50.9

0.95

1LS time formation

t

Re = 103

Re = 104

Re = 105

0 0.05 0.1 0.15 0.2 0.25 0.3 0.351.3

1.4

1.5

t

SS time formation

Asymptotic no−slip value

Asymptotic no−slip value

β(Re)

β(Re)The value β = Re−1 is somewhat critical, as for β = Re−α, α < 1 both large–scale and small–scaleinteractions forms as in the no-slip case.

For β = Re−1 no reciruclation region forms and viscous-inviscid no interactions are present.

The time formations of both large and small scale interaction are delayed, and as α→ 0 this time tends tothe time determined by the no-slip condition.


Complex singularity analysis for Navier BC

0 0.1 0.2 0.3 0.4 0.5−8

−7

−6

−5

−4

−3

β(Re)

log(δ

m)

Re = 103

Re = 104

Re = 105

Asymptotic no− slip value



For β = Re−1 no complex singularities are detected, while for β = Re−α, α < 1, NS solutions have complexsingularities. As α→ 0 The width of analicity of NS solution tends to that predicted by the slip-case.


Bibliography:

F. Gargano, M. Sammartino, V. Sciacca,K. Cassel, Analysis of ComplexSingularity in High-Reynolds-numbers Navier-Stokes solutions, J. Fluid Mech,,vol.747 (2014).

F. Gargano, M. Sammartino, V. Sciacca,K. Cassel, Viscous-InviscidInteractions in a Boundary-Layer Flow Induced by a Vortex Array, Acta Applic.Math., vol. 132 (2014).

F. Gargano, M. Sammartino, V. Sciacca,High Reynolds numberNavier-Stokes solutions and boundary layer separation induced by a rectilinearvortex, Computer and Fluids, vol.52 (2011).

F. Gargano, A.M. Greco, M. Sammartino, V. Sciacca, UnsteadySeparation for High Reynolds numbers flow, Proc. WASCOM 2009, (2010).

F. Gargano, M. Sammartino, V. Sciacca, Singularity Formation andseparation Phenomena in Boundary Layer Theory, Physica D, vol.238(19)(2009).

F. Gargano, M.C Lombardo, M. Sammartino, V. Sciacca, Singularity andseparation in Boundary Layer, London Math. Society Lecture Notes Series,(2009).

G. Della Rocca, F. Gargano, M. Sammartino, V. Sciacca, High Reynoldsnumber Navier-Stokes solutions and boundary layer separation induced by arectilinear vortex array, Proc. WASCOM 2007, (2008).

G. Della Rocca, M.C. Lomabrdo, M. Sammartino, V. Sciacca,Singularity Tracking for Camassa-holm and Prandtl’s equations,App.Numer.Maths, vol.56(8) (2006).

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