Influence of Stefan flow on non spherical droplet vaporization:
DNS with a Level Set / Ghost Fluid Method
Sébastien Tanguy 1, Alain Berlemont 2
1
2
LEMTA CNRS UMR - 7563
2, avenue de la forêt de Haye
BP 160
54504 Vandoeuvre lès Nancy FRANCE
UMR6614-CORIA
Technopôle du Madrillet
BP 12 Avenue de l'Université
76801 Saint-Etienne-du-Rouvray Cedex France
Corresponding author: [email protected]
Tel: (33) 2 32 95 36 17
Fax: (33) 2 32 91 04 85
1. Introduction
The main objective of our work is to develop direct numerical simulation tools for droplet
vaporization without the assumption of spherical symmetry, both with interface tracking and precise
descriptions of velocity, temperature and mass fraction fields inside the droplet and in the
surrounding gas phase. The numerical method should describe the interface motion precisely,
handle jump conditions at the interface without artificial smoothing, take into account Stefan flow
and remain numerically robust for high density ratio or temperature gradients. We use both the
Level Set Method [1,2] and the Ghost Fluid Method [3,4,5,6] to accurately capture interface motion
and to handle suitable jump conditions.
2. Formalism
We consider 2D axi-symmetric incompressible flows and we assume that the fluid’s physical
properties are constant in space and time. The liquid phase (subscript “liq”) is mono-component and
the gas phase (subscript “gas”) differs from the liquid vapor. The jump operator [.] is defined as
follows:
[A]Γ = Aliq – Agas
(1)
Transport equations for temperature T and species mass fraction Y are:
(
)
(2)
(
)
(3)
r
∂T
∇ ⋅ (λ∇T)
+ V ⋅∇ T =
∂t
ρC p
r
∇ ⋅ ( ρDm ∇Y)
∂Y
+ V ⋅∇ Y =
ρ
∂t
r
where V is the velocity vector, λ is the thermal conductivity , ρ is the density , Cp is the specific
heat at constant pressure and Dm is the mass diffusion coefficient.
When evaporation is considered, jump conditions must be added to respect energy conservation and
mass conservation across the interface :
[
]
r
hlg m& − λ∇T ⋅ N Γ = 0
r
Γ
Γ
− m& Ygas
+ ρDm ∇Y ⋅ N Γ = − m& Yliq
[
]
(4)
(5)
where m& is the local vaporization mass flow rate per unit surface, hlg is the latent heat of
r
vaporization and N is the unit vector normal to interface. Noting that gradients of species mass
fraction are zero in a mono-component liquid, then Eq. (5) can be used to compute local
vaporization rate. On the interface, species mass fraction is 1 on the liquid side, and ClausiusClapeyron relation provides species mass fraction on the gas side. To respect mass conservation
across the interface, the following jump condition must be imposed :
[Vr ]Γ = m& ⎡⎢ 1ρ ⎤⎥
⎣ ⎦Γ
r
N
(6)
r
r
The interface velocity Vs is defined as the sum of the liquid phase velocity Vliq and the surface
r
regression velocity induced by vaporization Vvap which is defined by:
r
m& r
Vvap = −
N
ρliq
(7)
Level Set methods are based on the use of a continuous function φ to describe the interface between
two media. That function is defined as the signed distance between any point of the domain and the
interface, and the interface is described by the 0 level of that function. Solving a convection
equation determines the evolution of the interface:
∂φ r
m& r
+ (Vliq −
N).∇φ = 0
∂t
ρliq
(8)
The interface is defined by two different phases and all discontinuities must be carefully described.
Specific treatment is thus needed to describe the jump conditions numerically. In the GFM, ghost
cells are defined on each side of the interface and appropriate schemes are applied for jump
conditions. As defined above the distance function φ defines the interface, and jump conditions are
extrapolated on some nodes on each side of the interface. Following the jump conditions, the
discontinued functions are extended continuously and then derivatives are estimated. The method is
applied for any kind of discontinuities, with the assumption that the interface can be localized inside
a grid mesh and that the jump of the discontinuous variables is known. More details can be found
[3,4,5,6] on implementing the Ghost Fluid Method on various test-cases.
3. Results
In the far field, gas temperature is equal to 873 K and the vapor mass fraction is equal to 0.
Simulations are presented in a 2D-axisymetric configuration. The domain lengths are lx=4RD in the
x-direction and ly=2lx in the y-direction. The grid size is 64x128 nodes. A gas flow is imposed at
the top of the domain and a body force is computed and imposed at every time step to keep the drop
in the computational field. To estimate the body force, the acceleration of the drop due to the gas
flow is calculated from the liquid velocity field. Then an opposite acceleration source term is
introduced in the momentum Navier Stokes equation. The drop initial temperature is equal to 293
K. Both computations have been carried out for a quite low Weber number, as large droplet
deformations are not presented here. In the first case, the droplet diameter is equal to 10 μm and the
gas velocity is equal to 15 m.s-1 (We=0.06). In the second case, the droplet diameter is equal to 200
μm and the gas velocity is equal to 2 m.s-1 (We=0.044). We present on Fig.1 typical temperature
fields and vapor mass fraction fields around the droplet in both cases. It clearly appears that thermal
and vapor mass fraction boundary layers around the drop exhibit different behaviors. In the first
case ((a) and (c) on Fig.1 , the boundary layer is much thicker than in the second case ((b) and (d)
on Fig.1.That behavior is linked to the influence of Stefan flow around the drop. For small drops,
the local vaporization rate is larger, leading to higher Stefan flow velocities and we observe a
blowing of the boundary layer around the drop. It is also clearly observed on the Fig. 2 and Fig.3
where streamlines around the drop are presented. Streamlines starting from the droplet interface are
generated by the Stefan flow and on Fig. 2 they push the gas velocity streamlines away from the
drop leading to a larger liquid vapor film around the drop. Moreover, another very interesting
phenomenon arises when the Stefan flow is important. The boundary layer blowing leads to a
decrease in the viscous friction at the interface. Therefore, internal recirculation decreases, and the
transient heating of the drop is modified. On Fig. 3 we present the temperature field inside the drop
for the second case, and it is clear that recirculation zones inside the drop are important to describe
the dynamics of droplet heating. Hot points can be observed at the bottom and at the top of the drop,
whereas cool points are observed on the droplet sides. On Fig.3 we observe that the dynamic of
droplet heating is very different, convective transport is much lower and droplet heating is mainly
driven by thermal diffusion inside the drop.
4. Conclusions
An innovative numerical method for coupling interface capturing method with heat and mass
transfer and phase changes has been set up. A local vaporization mass flow rate is defined on the
droplet interface and no spherical symmetry assumption is applied. We focused on the ability of the
method to respect physical jump conditions across the interface without any smearing of the droplet
surface. The numerical method is robust and it allows fine description of the physical mechanisms
in droplet vaporization. We observe that the vapor film around the droplet is strongly dependent on
the Stefan Flow. The droplet heating mechanism is dominated by convective transport with internal
recirculation for low Stefan flow, and by diffusion when the Stefan is high enough to enlarge the
boundary layer around the droplet which leads to a high decrease of viscous friction at the interface.
ACKNOWLEDGEMENTS
This research is developed in ASTRA program (CNRS-ONERA) on "Joined experimental
and simulation methods for multi-component sprays”
.
(a)
(b)
(c)
(d)
Fig. 1: Temperature fields Tp: (a) Case 1 t=20 s, (b), case 2, t=0.024 s
Vapor mass fraction YM: (c) Case 1, t=20 s (d), case 2 t=0.024 s
Black line is the interface.
Fig. 2: Case 1: velocity streamlines and droplet temperature t=0, 10, 20
s
Fig. 3. Case 2 velocity streamlines and droplet temperature t= 0, 0.012, 0.024 s
REFERENCES
[1] S. Osher, J. A. Sethian , J. Comp. Phys. 79 (1988), 12-49
[2] M. Sussman , P. Smereka , S. Osher , J. Comput. Phys. 114 (1994) 146-159.
[3] R. Fedkiw, T. Aslam, B. Merriman, S. Osher, J. Comp. Phys. 152, 457-492, 1999
[4] M. Kang , R. Fedkiw , X.-D. Liu , J. Sci. Comput. 15 (2000) 323-360
[5] F. Gibou , R. Fedkiw , L.T. Chieng , M. Kang , J. Comput. Phys. 176 (2002) 205-27
[6] S. Tanguy, A. Berlemont , submitted J. Comp. Phys.