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Niveau: Supérieur, Doctorat, Bac+8
No d'ordre : 2417 These presentee pour obtenir le titre de Docteur de l'Institut National Polytechnique de Toulouse Ecole Doctorale : Transfert, dYnamique des Fluides, Energetiques & Procedes Specialite : Energetique et Transfert par Adrien Toutant Modelisation physique des interactions entre interfaces et turbulence These soutenue le 4 decembre 2006 devant la commission d'examen : Dr. Alain Berlemont coria Rouen Rapporteur Dr. Marc Boucker edf Chatou Dr. Olivier Lebaigue cea Grenoble Encadrant Dr. Dominique Legendre imf-Toulouse Pr. Pierre Sagaut lmm Paris VI Rapporteur et President du jury Pr. Olivier Simonin imf-Toulouse Directeur de these

  • limit numerical

  • imf-toulouse pr

  • been validated

  • term has

  • phase equivalent

  • modelisation physique des interactions

  • interaction between


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Published 01 December 2006
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Language English
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oN d’ordre : 2417
These presentee pour obtenir le titre de
Docteur de l’Institut National Polytechnique de Toulouse
Ecole Doctorale : Transfert, dYnamique des Fluides, Energetiques & Procedes
Specialite : Energetique et Transfert
par
Adrien Toutant
Modelisation physique des interactions entre
interfaces et turbulence
These soutenue le 4 decembre 2006 devant la commission d’examen :
Dr. Alain Berlemont coria Rouen Rapporteur
Dr. Marc Boucker edf Chatou
Dr. Olivier Lebaigue cea Grenoble Encadrant
Dr. Dominique Legendre imf-Toulouse
Pr. Pierre Sagaut lmm Paris VI Rapporteur et President du jury
Pr. Olivier Simonin imf-Toulouse Directeur de theseAbstract
Turbulent two-phase ows abound in nature and engineering applications. The complex
interactions between interfaces and turbulence strongly impact the ow properties.
Consequently, there is both a scienti c and industrial interest to study this two-way coupling
phenomenon. Local measurements, e.g. deformations of interfaces, are very dicult and the only
tools able to provide information are numerical experiments. Unfortunately, Direct Numerical
Simulations (DNS) have to entail a number of degrees of freedom proportional to the third
power of the Reynolds number to correctly describe the ow behavior. This extremely hard
constraint makes it impossible to use DNS for industrial applications. In order to successfully
carry out industrial simulations their numerical cost has to be reduced.
Our strategy consists in using and improving DNS method in order to better understand
the interaction between interfaces and turbulence. Thanks to this reliable tool, we simulate
relevant test cases with the purpose of developing the Interfaces and Subgrid Scales concept.
ISS is a two-phase equivalent to the single-phase Large Eddy Simulation (LES) concept. In
this method, the geometry of interfaces is fully resolved. The challenge of ISS is to integrate
the two-way coupling phenomenon into subgrid models. First, we describe our DNS method.
Then, we explain our methodology to elaborate the ISS concept.
For DNS, we use a hybrid method front-tracking/Volume Of Fluid (VOF). This method is
based on a single uid formulation that allows writing only one momentum balance valid in
the entire domain. Indeed, each variable is expressed as a sum of the liquid contribution and
the gas contribution thanks to a phase indicator function.
– However, this formulation requires a closure technique for the dissipative term [88, 13].
Actually, this tensor which is constructed with the single uid velocity and dynamic
viscosity is not satisfactory. We have shown that the equivalent viscosity has to be
evaluated by a harmonic average in the normal direction and an arithmetic average in
thetangentialdirection.Twodi erentmethodsallowustoperformthisdemonstration:
one method originates in the discrete form of the dissipative tensor [29, 59]; the other
method consists in averaging the exact dissipative term on a control volume containing
the interface [6]. This closure technique has been validated thanks to two analytical
solutions : the two-phase Poiseuille ow, and the motion of a spherical bubble in a
creeping ow. The dissipative tensor is responsible of local momentum transfers across
the interfaces. That is the reason why this term has to be accurately evaluated for
studying the interaction between interfaces and turbulence.
– Another crucial point in order to study this interaction concerns the scheme for time
integration. Indeed, the position of the interfaces has to be very precise. For numerical
convenience, we choose a third order Runge Kutta scheme [102]. We verify the good
coupling between the momentum balance and the interface advection thanks to two
simpletestcases:thesolidrotationofaninterfaceandabubblesubmittedtoauniformiv Abstract
accelerated eld of velocity. It is worth noting that this type of time scheme is required
to use a centered convective operator that is essential for the simulations of turbulence
in order to limit numerical dissipation.
In the following section, we explain how we use the above DNS method to develop the ISS
modeling.
– BeforetoelaboratecomplexLESmodelsthankstoDNSand a priori
tests,wehavestudied how under-resolution a ects the ow behavior. We have made several calculations
of the same physical case (one buoyant deformable bubble) with di erent resolutions.
We have observed that the fact that the zigzagging trajectory of the bubble is not found
is due to the bad description of its geometry. The simplest way to improve this
description consists in rening the Lagrangian mesh (low cost) but not the Eulerian one (high
cost). However, putting more than one Lagrangian point by Eulerian mesh creates
numerical instabilities. So, we have developed a method able to suppress these instabilities
by smoothing the interfaces. Our method does not modify the volume of bubbles and
cuts the high frequencies of interfaces deformation (i.e. cuto length equivalent to the
Eulerian grid size) without altering the low frequencies of the interface motion. It has
been validated with the analytical test case of bubble oscillations [51].
– Mathematically, under-resolution could be seen as a lter applied to the system of
equations [80]. Applying a lter, we have exhibited correlations or subgrid terms that
requireclosures.Wehaveshownthat,intwo-phaseows,thepresenceofadiscontinuity
leads to speci c subgrid terms. Indeed, interfaces create anisotropy and, as a result,
correlations between the velocity on the one hand, and the normal at the interfaces, the
viscosity or the density on the other hand, do exist. In order to sort out the subgrid
terms related to these correlations, we have made a priori tests thanks to several DNS
calculations. The most relevant of these DNS is the interaction of a strongly deformable
bubble and a homogeneous isotropic turbulence [96]. Comparing the maximum of the
norm of the subgrid terms with the maximum of the norm of the advection tensor,
we have found that subgrid terms related to interfacial forces and viscous e ect are
negligible. Consequently, in the momentum balance, only the subgrid terms related to
inertia have to be closed. The subgrid term related to advection is the equivalent of
the classical Reynolds tensor. So, the rst thing to do is to examine the eddy viscosity
assumption. Our results show that standard models based on this assumption do not
mimic the negative part of the equivalent turbulent viscosity. That is the reason why we
choose to use structural modeling based on the Leonard and Germano decomposition
[34]. Thanks to a priori tests performed on several DNS data, we demonstrate that this
type of decomposition reinterpreted near discontinuity, provide subgrid models that
integrate the two-way coupling phenomenon [46, 97].
– These models correspond to the rst step of our strategy. Indeed, in this step, interfaces
aresmoothand,interactionsbetweeninterfacesandturbulenceoccurinatransitionzone
whereeachphysicalvariablevariessharplybutcontinuously.Thenextchallengeconsists
in determining the jump conditions across the sharp equivalent interface corresponding
tothesubgridmodels ofthe transitionzone. We propose touse the matchedasymptotic
expansion method [12]. The rst results that we obtain for the velocity of the sharp
equivalent interface are very promising.
Inbrief,wehaveimprovedamixedfront-tracking/VOFmethodinordertosimulateaccurately
interactions between interfaces and turbulence. Thanks to reliable DNS, we study this
twoway coupling phenomenon in the purpose of developing the ISS concept, an equivalent toAbstract v
LES for two-phase ows. Further work concerns implementation and a posteriori tests of the
developed models. We hope that ISS will be able to simulate (at a low cost, around 2 or 3
times less of degrees of freedom by direction than DNS) turbulent two-phase ows with many
bubbles or droplets at high Reynolds numbers.Avant-propos
Enpremierlieu,jesuisextrˆemementreconnaissantal’ensembledesmembresdujurypour
l’interˆet qu’ils ont porte a mon travail. Merci a Pierre Sagaut d’avoir accepte de cumuler
avec
succesleslourdestˆachesderapporteuretpresident,merciaAlainBerlemontpoursescomplimentssurlefondetlaformedumanuscrit,merciaMarcBouckerd’avoirassurelacomposante
industrielle de ce jury, merci a Dominique Legendre d’avoir souligne les perspectives et les
approfondissements que necessiterait ce travail.
En deuxieme lieu, je tiens a remercier les personnes qui ont participe scienti quement et
de fa con tres signi cative a ce travail.
Merci a mon directeur de these et ancien ENSTA, Olivier Simonin, pour avoir accepte de
diriger cette these pourtant un peu eloignee de sa specialite premiere. Dans cette situation
delicate, il a joue, de fa con tres bene