Sergey Boronin Research Fellow School of Computing, Engineering and Mathematics Sir Harry Ricardo Laboratories University of Brighton

Non-modal stability of round viscous jets

Centre for Automotive Engineering Research Workshop, 13th of January 2012

2

Outline

Ø Motivation, key ideas of modal and non-modal stability analisys

Ø Governing equations and boundary conditions

Ø Main flow velocity profiles

Ø Results and discussion

Ø Summary and conclusions

3

Motivation

Diesel jet break-up in internal combustion engines

Ø  Aim of the study: improvement of theoretical predictions of break-up lengths for round viscous jets

Ø  Stability of jets was analysed previously only within the framework of classical modal approach. Non-modal instability?

4

Key ideas of modal stability approach

Main plane-parallel shear flow: Small disturbances: Consider solutions in the form of travelling waves (equivalent to Fourier-Laplace transform):

}0,0),({ yU=V

x

y

U(y)

0

1

-1

z pPp ʹ′+=ʹ′+= ,vVv

))(exp()(),( tzkxkiyt zx ω−+=ʹ′ qrq

⎩⎨⎧

=±=

=

0),1(,ty

iLq

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

Solution is the system of eigenfunctions (normal modes): {qn(y), ωn(kx, kz)} The flow is stable if all normal modes decay (Im{ωn}<0, ∀n)

Eigenvalue problem:

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, pp. 423-436 (1996)

Fig. Visualization of streaks in boundary-layer flow1

6

Algebraic (non-modal) instability

Fig. Time-evolution of the difference of two decaying non-orthogonal vectors2

2 P.J. Schmid, Nonmodal Stability Theory, Annu.

Rev. Fluid Mech, V. 39, pp. 129-162 (2007)

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

Ø  Mathematical reason for non-modal instability: •  Differential operators are non-Hermitian (eigenvectors are not orthogonal) •  Solutions are linear combinations of normal modes, growth is possible even

if all modes decay (see Fig.)

7

Flow configuration

•  Surrounding gas and jet liquid are incompressible and viscous Newtonian fluids

• Cylindrical coordinate system (z, r, θ)

•  Parameters of fluids (gas and liquid, α = g, l)

ρα, µα are densities and viscosities vα, pα are velocities and pressures

r z

θ gas

liquid Interface

8

Non-dimensional governing equations

l

g

l

glg

l

llUL

rrr

rrzgl

µ

µζ

ρ

ρη

ζη

µρ

θα ====

∂

∂+⎥⎦

⎤⎢⎣

⎡∂

∂

∂

∂+

∂

∂=Δ= ,,ReRe,Re,11,, 2

2

22

2

011=

∂∂

+∂∂

+∂∂

θααα w

rrrv

rzu

αα

ααααα

αα

α

θu

zpu

rw

ruv

zuu

tu

Δ+∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

Re1

⎟⎠

⎞⎜⎝

⎛ −∂∂

−Δ+∂∂

−=−∂∂

+∂∂

+∂∂

+∂∂

22

2 2Re1

rvw

rv

rp

rwv

rw

rvv

zvu

tv αα

αα

αααααα

αα

α

θθ

⎟⎠

⎞⎜⎝

⎛ −∂∂

+Δ+∂∂

−=+∂∂

+∂∂

+∂∂

+∂∂

222

Re11

rwv

rwp

rrwvw

rw

rwv

zwu

tw αα

αα

ααααααα

αα

α

θθθ

Independent governing parameters: Rel – Reynolds number for jet liquid,

η, ζ – gas-to-liquid density and viscosity ratios

Flow scales: U – liquid velocity at the axis; L – jet radius; ρlU2 – liquid pressure pl; ρgU2 – gas pressure pg;

9

Boundary conditions

Interface Σ : H = r – h(z,θ, t) = 0

⎭⎬⎫

⎩⎨⎧

∂

∂−

∂

∂−=∇∇∇=

θh

rzhHHH 1,1,,/n

gas

liquid n

Σ Kinematic condition: 00 =−

∂

∂+

∂

∂⇔= v

zhu

th

dtdH

Continuity of velocity (no-slip): glglgl wwvvuu === ,,

Normal unit vector n:

Jump in stress due to capillary force R: Rss =− gl η( ) ;We,div

We1;

Re1 2

σααα

LUρnRnvvnp liij

ijjii ==∇+∇+−= ns

Ø Conditions at the interface:

Ø Kinematic condition at the axis3 (r → 0): θθθ

sin~,,0, llll wvpu→

∂

∂

∂

∂

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

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

10

Main flow: immersed jet

Two velocity profiles for immersed jet (single fluid, no interface):

Ø  “Top-hat” profile (close to the orifice)

Ø  “Smooth” profile (far downstream)4

( ){ }⎩⎨⎧

>−−

≤=

1,/1exp1,1

)( 221 rrr

rUδ

( )22211)(r

rU+

=

4 L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon, 1959.

Fig. Velocity profiles of the immersed jet considered

11

Main flow: jet in the air

{ }

,,:0

,,:1,0)(:

,0,0),(,,

∞<=

ʹ′=ʹ′==

→∞→

=

ll

glgl

g

glgl

PUr

UUUUrrUrrU

ζ

V

Ø  “Local” axisymmetric velocity profile (cylindrical jet of the fixed radius r = 1):

zr = 1

gas

liquid

Fig. Jet in the air at ζ=0.1, “model” velocity profiles 1 and 2 considered (liquid at r<1 and gas at r>1)

1 2

12

Linear stability problem

( )

,ReRe,11

2Re11

2Re1

Re1

011

2

2

22

2

22

22

ζη

θ

θθ

θ

θ

ααα

α

ααα

α

ααα

α

ααα

α

αα

ααα

αα

α

ααα

lgrrr

rrz

rwv

rwp

rzwU

tw

rvw

rv

rp

zvU

tv

uzpUv

zuU

tu

wrr

rvrz

u

=∂

∂+⎥⎦

⎤⎢⎣

⎡∂

∂

∂

∂+

∂

∂=Δ

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠

⎞⎜⎝

⎛ −∂

∂+Δ+

∂

∂−=

∂

∂+

∂

∂

⎟⎠

⎞⎜⎝

⎛ −∂

∂−Δ+

∂

∂−=

∂

∂+

∂

∂

Δ+∂

∂−=ʹ′+

∂

∂+

∂

∂

=∂

∂+

∂

∂+

∂

∂

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

13

Ø  Continuity

Ø  Kinematic condition

Ø  Force balance

ll vxhU

th

=∂

∂+

∂

∂glglggll wwvvUhuUhu ==ʹ′+=ʹ′+ ,,

Linearized boundary conditions Conditions at the interface at ( is the disturbance of the interface) 1=r

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂

∂+

∂

∂=+

∂

∂+

∂

∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂

∂+

∂

∂−

∂

∂−=

∂

∂−

⎟⎟⎠

⎞⎜⎜⎝

⎛ʹ′ʹ′+

∂

∂+

∂

∂=ʹ′ʹ′+

∂

∂+

∂

∂

ggg

lll

g

lg

l

ll

ggg

lll

ww

yw

wwyw

hzhh

Weyv

pyvp

Uhzv

yu

Uhzv

yu

θζ

θ

θζ

η

ζ

2

2

2

21Re2

Re2

,

Ø  Kinematic condition at the axis (r → 0): θθθ

sin~,,0, llll wvpu→

∂

∂

∂

∂

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

1<<h

14

Spatial normal modes and eigenvalue problem

Travelling waves (normal modes growing in z): { } },,,{,)(exp)(),,,( * pwvutmkzirtrz =−+= qqq ωθθ

(k – complex wave number, m – integer azimuthal number, ω - real frequency)

Ø  Eigenvalue problem:

0:0:10:0

0

=∞→

==

==

=

qqq

q

CrBrAr

L (L – second order linear differential operator 4x4)

Ø  System of eigenvalues and eigenfunctions: { } ,...2,1,,:,,We,,,Re =∀ nkm nnl Aωηζ (+ continuum part of the spectrum)

Ø  Modal stability analysis: flow is stable if , ∀n 0}Im{ >nk

homogeneous boundary conditions

15

Non-modal stability: optimal disturbances

( )

1,/2,/2

}Re{}Re{}Re{21),(

0 0 0

222

>>==

++= ∫ ∫ ∫Rbma

dtrdrdwvuab

tΕa b R

ωππ

θγ

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

tmizikrtrz nn

nn ωθγθ −⎟⎠

⎞⎜⎝

⎛= ∑

∞

=

qq

Ø  Expanding the disturbance into series of eigenfunctions at given m, ω:

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

Ø  Evaluation of the growth: density of the kinetic energy

Ø  Disturbance with maximum energy at a given position z (optimal disturbances): 1),0(,max),(:? =→− γγγ

γEtΕ

16

Ø  Variable mapping (non-uniform mesh, refinement with a decrease in r)

Ø  System of N spatial normal modes is found by orthonormalization method5

Ø  Optimal perturbations are found by Lagrange-multiplier technique (Resulting generalized eigenvalue problem for energy matrices is solved by QR-algorithm)

Algorithm of the numerical solution

{ } Nnkrm nnl ...1,),(:,,We,,,Re =∀ qωζη

5 S.K. Godunov, On the numerical solution of boundary-value problems for systems of linear differential equations [in Russian], Uspekhi mat. Nauk, V.16, Iss.3, pp.171-174 (1961).

17

Results: Immersed jet, “top-hat” velocity profile

Ø  “Top-hat” velocity profile is unstable for all values of azimuthal number m (first mode grows) Ø  Highest optimal-to-single mode energy ratio is 20, it corresponds to m=1 (Re=1000)

Fig. Normalized kinetic energy of the optimal disturbance for ‘top-hat’ velocity profile Vs. position downstream z for ω = 0.5, m=0 (a) and ω = 0.2, m=1 (b), Re = 1000. 1 – optimal energy, 2 – energy of the first mode only, 3 – ratio of optimal to single-mode energies

Non-modal stability of jet with “top-hat” velocity profile (close to the orifice)

Maximum optimal energy

increase for m=0 (~3 times)

Maximum optimal energy

increase for m=1 (~20 times)

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Ø  Smooth velocity profile is stable at m=0 and unstable at m>0 (P.J. Morris, 1976) Ø  Non-modal growth is significantly stronger for smooth profile and m>0 (optimal-to-single

mode energy ratio around 200 and corresponds to z~15)

Non-modal stability of jet with “smooth” velocity profile (far downstream)

Fig. Normalized kinetic energy of optimal disturbance for ‘smooth’ profile Vs. position downstream z for ω = 0.5, m=0 (a) and ω = 0.2, m=1 (b), Re = 1000. 1 – optimal energy, 2 – energy of the first mode only, 3 – ratio of the optimal to single-mode energies.

Results: Immersed jet, “smooth” velocity profile

19

Results: jet in the air, velocity profile 1

Ø  Non-modal growth decreases with an increase in frequency ω of the disturbance Ø  Highest optimal growth corresponds to non-axisymmetric disturbances (m>0) Ø  Highest optimal-to-single mode energy ratio is of order of 102 for ω=0.1

Fig. 1.  Optimal-to-single mode energy ratio for jet in the air with velocity profile 1 (Fig. 2) Vs. position downstream z for m=1, η=0.001, ζ=0.01, Re = 1000. Curves correspond to ω =0.1, 0.2, 0.5

Non-modal stability of jet in the air (Fig. 1), velocity profile is shown in Fig. 2.

Fig. 2.  Main flow velocity profile 1

20

Results: jet in the air, velocity profile 2

Fig. 1.  Optimal-to-single mode energy ratio for jet in the air with velocity profile 2 (Fig. 2) Vs. position downstream z for m=1, η=0.001, ζ=0.01, Re = 1000. Curves correspond to ω =0.1, 0.2, 0.5

Non-modal stability of jet in the air (Fig. 1), velocity profile is shown in Fig. 2.

Fig. 2.  Main flow velocity profile 2

21

Summary and conclusions

Ø  Linear stability problem for spatially-developing disturbances in round viscous jet is formulated. The effects of surface tension and ambient gas are considered

Ø  Numerical algorithm for modal and non-modal stability study is developed and validated

Ø  Parametric study of optimal spatial disturbances to both immersed jet and jet in the air is carried out

Ø  For the case of both jet flows, it is found that non-modal instability mechanism is strongest for non-axisymmetric disturbances (m>0) and it is damped with an increase in frequency of the disturbances. Maximum optimal-to-single mode energy ratio is of order of 102

Thank you for attention!

22

Ø Energy norm:

Energy norm and optimal spatial disturbances

( )WWVVUU

γγ***

****),(

++=

=++= ∫z

z

E

ErdrwwvvuuzE γ

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

tmizikrtzr n

N

nnn ωθγθ −⎟

⎠

⎞⎜⎝

⎛= ∑

=

qq

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)