Dr. Sergey Boronin School of Computing, Engineering and Mathematics Sir Harry Ricardo Laboratories University of Brighton Optimal spatial disturbances of axisymmetric viscous jets
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!