Dr. Sergey Boronin School of Computing, Engineering and Mathematics Sir Harry Ricardo Laboratories University of Brighton

Optimal spatial disturbances of axisymmetric viscous jets

2

Outline

Ø  Introduction •  Brief review of modal stability analysis •  Key ideas of algebraic instability and optimal disturbances

Ø Axisymmetric viscous jet in the air (main flow)

Ø Formulation of linear stability problem for spatially-growing disturbances

Ø Algorithm of finding optimal disturbances

Ø Evaluation of jet break-up length based on optimal disturbances

Ø Current issues/concerns

3

Linear stability analysis

Base plane-parallel shear flow: Small disturbances: Coefficients in governing equations depend on y only => Consider solutions in the form of travelling waves (equivalent to Fourier-Laplace transform)

}0,0),({ yU=Vx

y

U(y)

0

1

-1

z

pPp ʹ′+=ʹ′+= ,vVv

))(exp()(),( tzkxkiyt zx ω−+=ʹ′ qrq

⎩⎨⎧

=±=

=

0),1(,ty

Liq

qqω

q – vector of independent variables (normal velocity and normal vorticity for 3D disturbances), L – linear ordinary differential operator

4

Eigenfunctions and modal stability

Temporal stability analysis: ω - complex, kx, kz - real Eigenvalue problem:

⎩⎨⎧

=±

−=

0)1(qqq ωiL

System of eigenfunctions (normal modes): {qn(y), ωn(kx, kz)} (discrete part of the spectrum, wall-bounded flows) + {q(y), ω(kx, kz)} (continuous part of the spectrum, open flows)

Modal approach to the stability: Flow is stable ó for a given set of governing parameters, all modes decay (Im{ω(k)} < 0, ∀kx, kz)

5

Modal stability: pros and cons

ü Squire theorem (2D disturbances are the most unstable)

ü Modal theory predicts values of critical Reynolds numbers for several shear flows (plane channel, boundary layer)

ü Examples of failures: Poiseuille pipe flow (stable at any Re according to modal theory, unstable in experiments!)

ü Transition of shear flows is usually accompanied by 3D streamwise-alongated disturbances (”streaks”, see Fig.)

1 Alfredsson P.H., Bakchinov A.A., Kozlov V.V., Matsubara M. Laminar-Turbulent transition at a high level of a free stream turbulence. In: Nonlinear instability and transition in three-dimenasional boundary layers Eds. P.H. Duck, P. Hall. Dordrecht, Kluwer, 1996, P. 423-436. Fig. Visualization of streaks in boundary-layer flow1

6

Algebraic instability: mathematical aspect

Fig. Time-evolution of the difference of two decaying non-orthogonal vectors (P.J. Schmid. Nonmodal Stability Theory // Annu. Rev. Fluid Mech. 2007. V. 39. P. 129-162)

ü A necessity for linear “bypass transition” theories (non-modal growth)

ü Mathematical reason for non-modal instability: •  Linear differential operators involved are non-Hermitian

(eigenvectors are not orthogonal) •  Solution of initial-value problem is a linear combination of normal

modes, non-exponential growth is possible (see Fig.)

7

Algebraic instability: lift-up mechanism

2 M. T. Landahl. A note on the algebraic instability of inviscid parallel share flows // J. Fluid Mech. 1980. V. 98. P. 243-251

3 T. Ellingsen, E. Palm. Stability of linear flows // Phys. Fluids. 1975. V. 18. P. 487.

UtvuUvtu

ʹ′⇒=ʹ′+∂

∂ ~0

Inviscid shear flow U=U(y) Consider disturbances independent of x2:

y

x

U(y)

(linear growth, lift-up mechanism3)

Inviscid nature, but still holds for viscous flows at finite time intervals!

8

Optimal disturbances

zx

a b

kbka

dxdydzwvuab

tΕ

/2,/2

)}Real{}Real{}Real{(21),(

0 0

1

1

222

ππ

γ

==

++= ∫ ∫ ∫−

{ })(exp)exp()(),,,(1

zkxkitiytzyx zxn

N

nnn +⎟

⎠

⎞⎜⎝

⎛−= ∑

=

ωγ qq

Expanding the disturbance of wave numbers kx, kz into eigenfunction series:

(the set of coefficients {γn} is a spectral projection of a disturbance q)

Evaluation of the growth: density of the kinetic energy

Disturbances with maximum energy at a given time instant t: (optimal disturbances)

1),0(,max),(:? =→− γγγγ

EtΕ

9

Axisymmetric viscous jet in the air

• Axisymmetric stationary flow

• Both fluids (surrounding gas and jet liquid) are incompressible and viscous (Newtonian)

• Cylindrical coordinate system (z, r, θ)

• Parameters of fluids: (surrounding “gas” and jet liquid)

ρα, µα are densities and viscosities vα, pα are velocities and pressures (α = g, l)

r z

θ gas

liquid

10

Non-dimensional governing equations

( )

glULr

rrrz

rvv

rp

rvv

zvu

uzp

ruv

zuu

rrv

rzu

,,Re

1

Re1Re1

01

2

2

2

==

⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+∂∂

=Δ

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎠

⎞⎜⎝

⎛ −Δ+∂∂

−=∂∂

+∂∂

Δ+∂∂

−=∂∂

+∂∂

=∂∂

+∂∂

αµρ

α

αα

αα

α

ααα

αα

αα

ααα

αα

αα

(Reynolds numbers)

Axisymmetric stationary flow: vα = {uα , vα , 0}, ∂/ ∂θ = 0

11

Boundary conditions

At the infinity (r → ∞): ∞<→ gggg pwvu ,0,,

Interface Σ: H = r – h(z, t) = 0, n – normal unit vector:

1/0,1,/2

+⎟⎠

⎞⎜⎝

⎛∂∂

⎭⎬⎫

⎩⎨⎧

∂∂

−=∇∇=zh

zhHHn

gas liquid

n

Σ

Kinematic condition at the surface:

00 =−∂∂

+∂∂

⇔= vzhu

th

dtdH

Continuity of velocity (no-slip): [ ] [ ]( )gl fff −≡= 0v

12

Force balance at the interface

( ) ( )

l

gl

iizx

i

jigj

jgi

ggiigj

jlj

jli

liil

gl

LUρ

nnRR

R

nvvpnpnvvpnp

ρ

ρη

γ

ηη

==

−=⎟⎟⎠

⎞⎜⎜⎝

⎛+=

∇+∇+−=∇+∇+−=

=−

;We

,divWe111

We1

;Re1;

Re1

,

2

,,

n

RppDifference in stress at the surface is due to capillary force R:

- Weber number and density ratio

Kinematic condition at the axis r = 0 (all parameters should be finite)4:

0lim0

=∂

∂→ θ

l

r

v

4 G.K. Batchelor, A.E. Gill, Analysis of the stability of axisymmetric jets. J. Fluid. Mech., 1962, V.14, pp. 529-551

13

Axisymmetric jet flow, local velocity profile

{ }

0,,

,

,Re1

Re1,:

,0)()(,0,0),(

0

0

,,,,

=∞<

=−

ʹ′=ʹ′==

∞→→

==

rPUr

PP

UUUUrr

rrUzPPrU

ll

gl

gg

ll

gl

g

glglglgl

γη

η

V

Ø Assume that jet velocity profile varies slightly with z (on the scale of wave lengths λ considered) Ø For fixed z, consider “model ” axisymmetric solution: cylindrical jet of radius r0(z):

zr0

gas

liquid

Δz >> λ

14

Linear stability problem

( )

glULrr

rrrz

rwv

rwp

rzwU

tw

rvw

rv

rp

zvU

tv

uzpUv

zuU

tu

wrr

rvrz

u

,,Re,11

2Re11

2Re1

Re1

011

2

2

22

2

22

22

==∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

+∂∂

=Δ

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠

⎞⎜⎝

⎛ −∂∂

+Δ+∂∂

−=∂∂

+∂∂

⎟⎠

⎞⎜⎝

⎛ −∂∂

−Δ+∂∂

−=∂∂

+∂∂

Δ+∂∂

−=ʹ′+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

αµρ

θ

θθ

θ

θ

α

αα

ααα

α

ααα

α

ααα

α

ααα

α

αα

ααα

αα

α

ααα

Linearized Navier-Stokes equations for each fluid (α = l, g):

15

Normal modes

Normal modes: { })(exp)(),,,( * tmkzirtrz ωθθ −+=qq

( )

( )

2233

222

222

222

3244

1,1

,,,

Re2

Re2

Kdrd

rdrdrTK

drdr

drd

rS

rKmukrwirv

rmkK

rkUrKTkUimTT

rKUmkUiT

rKmS

−⎟⎠

⎞⎜⎝

⎛≡−⎟⎠

⎞⎜⎝

⎛≡

−≡Ω−≡+≡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ʹ′⎟⎠

⎞⎜⎝

⎛ ʹ′−−=Ω−

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ ʹ′−Ω−=⎟

⎠

⎞⎜⎝

⎛+Ω

φ

φφωαφ

φα

ωφα

Governing linear equations are reduced to analogues of Orr-Sommerfeld and Squire equations in cylindrical coordinates5:

+ zero b.c.

and conditions at the interface

5 D.M. Burridge and P.G. Drazin, Comments on ‘Stability of pipe Poiseuille flow’, Phys. Fluids, 1969, V.12, pp. 264–265

16

Solving eigenvalue problem

Condition for nontrivial solution is a dispersion relation: 0),WeRe,,,,( =ηωmkF

Temporal and spatial analysis: ( ) { }( ) { }),(),,(0,:

),(),,(0,:mkmkkkFikkkmkmkFi

iririr

iririr

ωω

ωωωωωωω

⇔=+=

⇔=+=

The goal is to find N normal modes with largest growth increments •  Methods for solving dispersion relation directly are not efficient

(e.g. orthonormalization method, result is a single mode, first guess is required!)

•  Reduction of differential eigenvalue problem to algebraic one is the most reliable

•  Eigenvalue k enters the governing equations non-linearly, reformulation of governing equations is needed

(addition of new variables, but reduction the order of k)

17

Reduction of the differential eigenvalue problem to algebraic one

New variables: ( )

zwAwApAuA

zvAvA

tzr

∂∂

====∂∂

==

=

*6

*5

*4

*3

*2

*1

**

,,,,,

:,,, θAA

(L – 2nd-order linear differential operator in r)

***

AA Lz=

∂∂

{ })(exp)(),,,(* tmkzirtrz ωθθ −+= AA

Governing linear equations:

Normal modes:

AA Lik =

Eigenvalue problem:

18

Boundary conditions at r = 0, r → ∞

Gas disturbances decay at r → ∞: 6...1,0 =→ igA

Kinematic condition at the axis r = 0 (all parameters should be finite):

0lim0

=∂

∂→ θ

l

r

v

6...1,0:1;0,0,:1

;0:0

62512431

654321

==>

=+=+ʹ′===ʹ′=

===ʹ′=ʹ′===

iAmiAAiAAAAAAm

AAAAAAm

i

19

Boundary conditions are specified at perturbed interface (r = r0+h) and linearized to undisturbed interface r = r0: 1) Continuity

2) Kinematic condition

3) Force balance

( )hivU

ikh ω−=1

[ ] [ ] [ ] [ ] 0,0,0 513 ===ʹ′+ AAUhA

Linearized boundary conditions at the interface

Disturbed interface: ⎭⎬⎫

⎩⎨⎧

∂

∂−

∂

∂−=<<+=

θθ

hrz

hhtzhrr0

01,1,,1);,,( n

[ ] { } ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛ −−+=⎥⎦

⎤⎢⎣

⎡ ʹ′−−

=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−+ʹ′+=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+ʹ′ʹ′+ʹ′+

hrmhiA

UiA

UrhAA

BrUA

rAAimAimh

rUUhAA

20

2

1220

14

20

50

151

032

11We1

Re2

0Re1;0

Re1

ωω

[ ]( )gl fff η−≡

20

Ø Finite-difference method (non-uniform mesh!) reducing differential eigenvalue problem to algebraic eigenvalue

problem for matrix – discrete analogue of differential operator L Ø QR-algorithm for the solution of algebraic eigenvalue problem (factorization into unitary and upper-diagonal matrices)

Ø System of N normal modes (N is large enough)

Numerical solution of the eigenvalue problem

{ } Nnknnlg ...1,,:,We,,Re,Re =∀ Aωη

21

Ø Energy norm:

Energy norm and optimal spatial disturbances

( )WWVVUU

γγ*2**

**2**),(

rE

EdrwwrvvuuzE

z

z

++=

=++= ∫γ

{ })(exp)exp()(),,,(1

tmizikrtzr n

N

nnn ωθγθ −⎟

⎠

⎞⎜⎝

⎛= ∑

=

AA

Ø Maximization of energy functional: 1,max:? 0

** =→− γγγγγ EΕzγ

Euler-Lagrange equations: 00 =+ EEz σ

Optimal disturbances correspond to eigenvector with highest eigenvalue σ

(Ez is positive Hermitian quadratic form)

(generalized eigenvalue problem for energy matrix)

22

Possible application for break-up length evaluation

max)(,1)0(:, →= zEEpoptoptvOptimal disturbance growth is maximal in the spatial interval [0, z]

Example of optimal spatial growth (pipe flow)6

Ø Threshold energy for break-up should be specified (experiments?)

Ø Break-up of the jet with arbitrary disturbances occurs further upstream

Ø Optimal break-up lengths provide lower-bound estimate for real jet break-up lengths at a given ω, m

Ø Superposition of waves with different ω, m?

6 M.I. Gavarini, A. Bottaro, F.T.M. Nieuwstadt, Optimal and robust control of streaks in pipe flow, J. Fluid. Mech, 2005, V. 537. pp.187-219

23

Current issues/concerns

Ø  Problem is formulated in the most general way. Possible simplifications?

Ø  Choosing the appropriate “local” jet velocity profiles Ug(r), Ul (r)?

Ø  Range of governing parameters of interest?

Ø  Evaluation of the jet break-up based on optimal perturbations?

24

Thank you for attention!