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Local stability analysis of a coaxial jetat low Reynolds number
Giacomo Gallino
supervisor: Alessandro Bottarosupervisor: Francois Gallaire
Genova, 20 Marzo 2013
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 1 / 20
Goal of the project: explain the transition between dripping and jetting
Dripping and jetting regime, experimental study by Guillot
Local stability analysis in the developing regionGiacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 2 / 20
Mathematical formulation of the problemRe = 0
No gravity terms
Axisymmetric domain
Axisymmetric flow
Stationary flow
∇ · u = 0,
∇p− µ∇2u = 0,
with no-slip conditions at the wall and symmetry at the axis.
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 3 / 20
Interface conditionsContinuity of velocities at the interface
u1(xint) = u2(xint).
Continuity of tangential stress and balance of normal stress
tT(σ2 − σ1)n = 0, nT(σ2 − σ1)n = κγ n,
κ being the curvature in the meridian plane.
Impermeability condition
ur cosα− uz sinα = U⊥ = 0 =⇒ ur − uz∂R0
∂z= 0.
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 4 / 20
Boundary elements method
Formulation of the Stokes equations in term of boundary integral equations(Pozrikidis 1992)
u(x0) = − 18πµ
∮lG(x, x0)f(x)dl(x) +
18π
∮lu(x)T(x, x0)n(x)dl(x),
G and T are the Green’s functionsf is a vector representing the stresses and u the velocities
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 5 / 20
Boundary integral equation of a domain with interface
8πµ2u(x0) = −∫
lG(x0, x)f(x)dl + µ1
∫l1
u(x) · T(x0, x) · n(x)dl+
µ2
∫l2
u(x) · T(x0, x) · n(x)dl−∫
intG(x0, x) ·∆f(x)dl+
(µ2 − µ1)
∫int
u(x) · T(x0, x) · n(x)dl.
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 6 / 20
Discretization
Converge when urint < 10−3 and u⊥(int) < 10−3.
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 7 / 20
FreeFem simulation
Pressure field and axial velocity fieldGiacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 10 / 20
Local stability analysis
Base flow taken at a given axial coordinate
Non dimensional number: Q = Q1Q2
, λ = µ1µ2
, Ka = dzpR2γ .
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 11 / 20
Linearize the equations adding small perturbations at the the base flow
P2P1Ur1
Uz1
Ur2
Uz2
R0
=
P2 + εp2P1 + εp10 + εur1
Uz1 + εuz1
0 + εur2
Uz2 + εuz2
R + εη
ε� 1
∇ · u = 0,
∇p− µ∇2u = 0.
Boundary conditionswall: ur2 = uz2 = 0,
axis: ur = 0,∂uz
∂r= 0,
∂p∂r
= 0.
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 12 / 20
Linearize the continuity of velocities condition at the interface
Uz1(R0 + εη) + εuz1(R0 + εη) = Uz2(R0 + εη) + εuz2(R0 + εη)
flattening hypothesis, Taylor expansion around R0
uz1(R0) +∂Uz1
∂r
∣∣∣∣R0
η = uz2(R0) +∂Uz2
∂r
∣∣∣∣R0
η
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 13 / 20
Modal expansion u = u(r)ei(kz−ωt), p = p(r)ei(kz−ωt), η = ηei(kz−ωt).
Aϕ = 0
where
A =
[domain1] [
0][
0] [
domain2]
interface conditions
ϕ =
(ur1 uz1 p1 ur2 uz2 p2 η
)T
Non trivial solution if det(A) = 0, this leads to an eigenvalue problem for ω.
ω = ωr(k) + iωi(k)
or in a non dimensional form
ω =16µ2R2
γ, k = kRout.
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 14 / 20
Local stability analysis performed in different sections z
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
k
ωr
z=0.1z=0.25z=0.5z=1z=2
0 0.2 0.4 0.6 0.8 1−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
kω
i
z=0.1z=0.25z=0.5z=1z=2
Real and imaginary part of ω for Q = 0.7, λ = 0.5, Ka ≈ 1 and R1 = 0.5The dispersion relation can be written as in Guillot study
ω = α(z)k + iA(z)((k/b(z))2 − (k/b(z))4).
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 15 / 20
Absolute instability criterion
Velocity of the back front of the perturbation
v−(z) = α(z)− A(z)(√
7 + 512b(z)2 −
√7 + 5
36b(z)4
)√24b(z)2√
7− 1
Labs > λmin, where λmin = 2π/kmax
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1
0
1
2
3
4
z
v −
Labs ≈ 0.125 < λminGiacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 16 / 20
Region of convective and absolute instability
0 0.2 0.4 0.6 0.8 110
−6
10−4
10−2
100
102
interface position
Ka
case analyzedthreshold
absolute regime
convective regime
Analytical solution and experimental data presented by Guillot
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 17 / 20
Results
0 0.5 1 1.5 2 2.5 31
2
3
4
5
6
z
v −
Numerical results for Q = 0.5636, λ = 0.2, Ka = 1, R1 = 0.2
Experimental study by GuillotGiacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 18 / 20
ConclusionsBetter agreement with the experimental results respect to previousstudies
Change in the behavior of the perturbations from the developing regionrespect to the fully developed flow
Future developmentsGlobal stability analysis
Giacomo Gallino (Universita di Genova-EPFL) Coaxial injectors 19 / 20