Optimal spatial disturbances of axisymmetric viscous jets · Coefficients in governing equations depend on y only => Consider solutions in the form of travelling waves (equivalent
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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!
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