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Drag reduction
a b s t r a c t
The aim of this study is to investigate numerically the interaction between a dispersed phase composed of micro-bubbles and a **turbulent** **boundary** layer flow. We use the Euler–Lagrange approach based on Direct Numerical **Simulation** of the continuous phase flow equations and a Lagrangian tracking **for** the dispersed phase. The Synthetic Eddy **Method** (SEM) is used to generate the inlet **boundary** condition **for** the **simulation** of the **turbulent** **boundary** layer. Each bubble trajectory is calculated by integrating the force balance equation accounting **for** buoyancy, drag, added-mass, pressure gradient, and the lift forces. The numerical **method** accounts **for** the feedback effect of the dispersed bubbles on the carrying flow. Our approach is based on local volume average of the two-phase Navier–Stokes equations. Local and temporal variations of the bubble concentration and momentum source terms are accounted **for** in mass and momentum balance equations. To study the mechanisms implied in the modulation of the **turbulent** wall structures by the dispersed phase, we first consider simulations of the minimal flow unit laden with bubbles. We observe that the bubble effect in both mass and momentum equations plays a leading role in the modification of the flow structures in the near wall layer, which in return generates a significant increase of bubble volume fraction near the wall. Based on these findings, we discussed the influence of bubble injection methods on the modulation of the wall shear stress of a **turbulent** **boundary** layer on a flat plate. Even **for** a relatively small bubble volume fraction injected in the near wall region, we observed a modulation in the flow dynamics as well as a reduction of the skin friction.

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1. Introduction
Specifying inlet and outlet **boundary** conditions **for** compress- ible **simulation** still remains a key issue (Colonius [8] ) especially **for** unsteady ﬂows where wave reﬂections must be controlled. In this ﬁeld, characteristic **boundary** conditions have progressively become standard. Initially introduced by Thompson [48] , Euler Characteristic **Boundary** Conditions (ECBC) was then extended by Poinsot and Lele [34] to viscous ﬂows by proposing the Navier– Stokes Characteristic **Boundary** Conditions (NSCBC) approach. This **method** speciﬁes a given number of quantities –**for** example static pressure **for** **an** outlet, velocity and temperature **for** **an** inlet– on the **boundary** condition, and allowing the outgoing waves, com- puted by the numerical scheme, to leave the domain with min- imum reﬂection. The NSCBC strategy has been later extended to multi-species reacting ﬂows and to aeroacoustics (Baum et al. [3] , Okong’o and Bellan [31] , Moureau et al. [29] , Poinsot and Veynante

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Restrictions on the maximum allowable time step of explicit time integration methods can be very severe **for** direct and large eddy simulations of compressible **turbulent** flows at high Reynolds numbers, **for** which extremely small space steps have to be used close to solid walls in order to capture tiny and elongated **boundary** layer structures. A way of increasing stability limits is to use **implicit** time integration schemes. However, the price to pay is a higher computational cost per time step, higher discretization errors and lower parallel scalability. A successful **implicit** time scheme should provide the best possible compromise between these opposite requirements. In this paper, several **implicit** schemes assessed against two explicit time integration techniques, namely a standard four-stage and a six-stage optimized Runge–Kutta methods, in terms of computational cost required to achieve a threshold accuracy level **for** the **simulation** of compressible **turbulent** flows. Pre- cisely, a second-order backward scheme solved by means of matrix-free quasi-exact Newton subiterations is compared to time-accurate Runge–Kutta **implicit** residual smoothing (IRS) schemes. A new IRS scheme of fourth-order accuracy, based on a bilaplacian operator, is developed to improve the accuracy of the classical second-order approach. Numerical re- sults show that the proposed IRS scheme leads to reductions in computational time by about a factor 5 **for** **an** accuracy comparable to that of the corresponding explicit Runge- Kutta scheme.

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and to the mixing length model by Spalding (18)
reported in Galbraith and Head (19)
. The analytical and numerical approaches are then applied to model a **boundary** layer with and without **an** adverse streamwise pressure gradient. The remainder of the paper is structured in five sections. Section 2 presents the benchmark velocity profiles that are used in Sections 4 and 5 **for** the validation of the analytical and numerical predictions. Section 3.1 details the analytical **method** used to generate the composite velocity profile in a **turbulent** **boundary** layer. Section 3.2 details the numerical **method** based on the interactive **boundary** layer model. Section 4 validates both methods using zero pressure gradient velocity data over the Reynolds number range 422 ≤ Re θ ≤ 31,000 and presents a comparison of the new mixing length model with other mixing length schemes and experiment. Section 5 extends the validation to adverse pressure gradient **boundary** **layers** and summarises the limitations of the numerical methods. Concluding remarks are given in Section 6.

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In this study, a **boundary**-layer tripping **method** permitting to obtain **an** initially **turbulent** supersonic jet is studied. The influence of the tripped jet **boundary** **layers** on the flow and acoustic fields of the jet is analyzed. The impact of nozzle-exit turbulence levels on the noise radiation and notably on the acoustic components specific to supersonic jets (screech noise, broadband shock-associated noise, mixing noise and Mach wave radiation) is discussed.

transfer in the normal direction including phase change as well as source terms (previously described Galerkin **method**). It is worth noting that the dimension of U n+1 k depends on k, md and n since **layers** may appear or disappear during a time step.
There are two main difficulties in the construction of the **implicit** **method**. First, as all the cells of the mesh are coupled with their left and right neighbouring cells through the running liquid film, it is complicated to devise a global implicitation of the **method** (i.e. a **method** which would simultaneously yield U n+1 k in every cell). To overcome this problem, a fixed point algorithm is used. The general idea is to perform a local implicitation and iterate over the cells until convergence so as to construct a sequence

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144 En savoir plus

1. Introduction
Specifying inlet and outlet **boundary** conditions **for** compress- ible **simulation** still remains a key issue (Colonius [8] ) especially **for** unsteady ﬂows where wave reﬂections must be controlled. In this ﬁeld, characteristic **boundary** conditions have progressively become standard. Initially introduced by Thompson [48] , Euler Characteristic **Boundary** Conditions (ECBC) was then extended by Poinsot and Lele [34] to viscous ﬂows by proposing the Navier– Stokes Characteristic **Boundary** Conditions (NSCBC) approach. This **method** speciﬁes a given number of quantities –**for** example static pressure **for** **an** outlet, velocity and temperature **for** **an** inlet– on the **boundary** condition, and allowing the outgoing waves, com- puted by the numerical scheme, to leave the domain with min- imum reﬂection. The NSCBC strategy has been later extended to multi-species reacting ﬂows and to aeroacoustics (Baum et al. [3] , Okong’o and Bellan [31] , Moureau et al. [29] , Poinsot and Veynante

En savoir plus
Drag reduction
a b s t r a c t
The aim of this study is to investigate numerically the interaction between a dispersed phase composed of micro-bubbles and a **turbulent** **boundary** layer flow. We use the Euler–Lagrange approach based on Direct Numerical **Simulation** of the continuous phase flow equations and a Lagrangian tracking **for** the dispersed phase. The Synthetic Eddy **Method** (SEM) is used to generate the inlet **boundary** condition **for** the **simulation** of the **turbulent** **boundary** layer. Each bubble trajectory is calculated by integrating the force balance equation accounting **for** buoyancy, drag, added-mass, pressure gradient, and the lift forces. The numerical **method** accounts **for** the feedback effect of the dispersed bubbles on the carrying flow. Our approach is based on local volume average of the two-phase Navier–Stokes equations. Local and temporal variations of the bubble concentration and momentum source terms are accounted **for** in mass and momentum balance equations. To study the mechanisms implied in the modulation of the **turbulent** wall structures by the dispersed phase, we first consider simulations of the minimal flow unit laden with bubbles. We observe that the bubble effect in both mass and momentum equations plays a leading role in the modification of the flow structures in the near wall layer, which in return generates a significant increase of bubble volume fraction near the wall. Based on these findings, we discussed the influence of bubble injection methods on the modulation of the wall shear stress of a **turbulent** **boundary** layer on a flat plate. Even **for** a relatively small bubble volume fraction injected in the near wall region, we observed a modulation in the flow dynamics as well as a reduction of the skin friction.

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In this work we present **an** a posteriori high-order nite volume scheme **for** the computation of compressible **turbulent** ows. **An** automatic dissipation adjustment (ADA) **method** is combined with the a posteriori paradigm, in order to obtain **an** **implicit** subgrid scale model and preserve the stability of the numerical **method**. Thus, the numerical scheme is designed to increase the dissipation in the control volumes where the ow is under-resolved, and to decrease the dissipation in those cells where there is excessive dissipation. This is achieved by adding a multiplicative factor to the dissipative part of the numerical ux. In order to keep the stability of the numerical scheme, the a posteriori approach is used. It allows to increase the dissipation quickly in cells near shocks if required, ensuring the stability of the scheme. Some numerical tests are performed to highlight the accuracy and robustness of

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of Γ A (x; y) as **an** m × m matrix.
3) Assume 1 + n = 2. The first construction **for** complex coefficients is in [ AMcT ] **for** scalar operators (m = 1). **An** analogous estimate was obtained in [ DoK ], Theo- rem 2.21, **for** systems but was only carried out explicitly assuming strong ellipticity. See also [ CDoK ]. [ B , Chapter 4] used the construction in [ AMcT ] and showed uniqueness and also that it is possible to choose the constant of integration in such a way the symmetry relation holds. This construction extends mutatis mutandi to systems and does give the above estimates, with possible exception of uniqueness as the argument relies on properties of harmonic functions.

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In this paper, the above methodology (DNS combined with the- ory) is used to study the interaction between **an** evaporating liquid film and the **turbulent** **boundary** layer created in the vicinity of a wall in the generic configuration of the periodic **turbulent** channel flow. The liquid film flow is not solved but only its impact on the gaseous **boundary** layer is studied, through the **boundary** condition that reflects the film surface properties. The objective is to give a detailed understanding and build a model of the **boundary** layer structure above the film surface. In this two-way interaction, the liquid film evaporation is influenced by the near-wall gradients of species and the wall temperature, while the mass flux due to evaporation blows the **boundary** layer away from the wall, thereby changing the flow profiles and deviating significantly from the classical wall functions. It has been shown in previous studies that the **boundary** layer structure and more specifically the distance be- tween the wall and the laminar-**turbulent** transition depend on the

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Figure 10: S
attering of a plane wave by **an** air
raft. Time evolution of the E z
omponent at a sele
ted point.
7.2 Exposure of head tissues to a lo
alized sour
e radiation
We now
onsider a more realisti
problem whi
h
onsists in the **simulation** of the exposure of a geometri
al model of head tissues to **an** ele
tromagneti
wave emitted by a lo
alized sour
e. Starting from MR images of the Visible Human proje
t [RHGJ03℄, head tissues are segmented and the interfa
es of a sele
ted number of tissues (namely , the skin, the skull and the brain) are triangulated. Dierent strategies
**an** be used in order to obtain a smooth and a
urate segmentation of head tissues and asso
iated interfa
e triangulations. A rst strategy
onsists in using a mar
hing
ube algorithm [LC87℄ whi
h leads to huge triangulations of interfa
es between segmented subdomains. These triangulations
**an** then be regularized, rened and de
imated in order to obtain reasonable surfa
e meshes, **for** example using the YAMS [Fre03℄ re-meshing tool. Another strategy
onsists in using a variant of Chew's algorithm [Che93℄, based on Delaunay triangulation restri
ted to the interfa
e, whi
h allows to
ontrol the size and aspe
t ratio of interfa
e triangles [BO05℄. Surfa
e meshes of the skin, skull and brain resulting from su
h a pro
edure are presented on Fig. 11. Then, these triangulated surfa
es are used as inputs **for** the generation of volume meshes. In this study, the GHS3D tetrahedral mesh generator [GHS91℄ is used to mesh volume domains between the various interfa
es. Note that the exterior of the head must also be meshed, up to a
ertain distan
e from the skin. The
omputational domain is here arti
ially bounded by a sphere on whi
h the Silver-Müller
ondition is imposed. Moreover, a simplied mobile phone model is in
luded and pla
ed in verti
al position
lose to the right ear (see Fig. 12).

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The other approach goes the other way around. Indeed, in finite dimension, it is known that if ˙
y = F (y) + Bu where F is quadratic is controllable, then ˙ y = F (y) + Ay + Bu is controllable too (see [57 , Theorem 3.8]). Likewise, **for** fluid systems, trying to get a small time controllability result implies to work at high Reynolds number (ie. with big fluid velocities, or low viscosity) inside the domain. Therefore, inertial forces prevail and the fluid system behaves like its null viscosity hyperbolic limit system. In our case, we expect to deduce results **for** Navier-Stokes from the Euler sytem. **For** Euler, global controllability has been shown in [ 51 ] by Coron **for** the 2D case (see also [ 53 ]) and by Glass **for** the 3D case in [ 94 ]. Their proofs rely on the return **method** introduced by Coron in [ 50 ] (see also [ 55 , Chapter 6]). **For** Navier-Stokes, things get harder. In [ 59 ], Coron and Fursikov show a global controllability result in the case of a 2D manifold without **boundary**. In [ 86 ], Fursikov and Imanuvilov show a global exact controllability result **for** 3D Navier-Stokes with a control acting on the whole **boundary** (ie. Γ = ∂Ω).

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Numerical simulation of resin transfer molding using linear boundary element methodF. Fabrice Schmidt, P Lafleur, Florentin Berthet, Pierre Devos.[r]

6
that rotation causes a suppression of the enstrophy pro- duction similar to the effects of confinement, favoring the two-dimensionalization of the flow and the development of the inverse energy cascade. Nonetheless, this effect is not accompanied by the presence of a range of scales in which the enstrophy is conserved by the large-scale dy- namics. This is likely to affect the interactions between 2D and 3D modes. In the case of stably stratified fluid **layers**, it has been shown that the conversion of kinetic energy into potential energy, which is promptly trans- ferred toward the small diffusive scales, provides a fast dissipative mechanism which suppresses the large scale energy transfer [45]. Investigating the interactions be- tween 2D vortical modes and 3D potential modes will improve the understanding of this process.

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We are interested in this work with a precise asymptotic description of the solution y ε when ε is small.
As a first motivation, we mention that system (1) can be seen as a simple example of complex models where the diffusion coefficient is small compared to the others. Actually, as discussed in [5], the model problem (1) is **an** embedded system of the Navier-Stokes system with non-characteristic **boundary** condition and viscosity coefficient equals to ε. A second motivation comes from the numerical approximation of (1) that may be not straightforward **for** small values of ε (we refer to [8], [23]). A third motivation comes from the asymptotic controllability property of (1) studied in [7] and which exhibits surprising behaviors, leaving many open questions.

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The **simulation** presented in figures 6 and 7 were first repeated **for** two specific density ratios r p / r
= 8 and 16 (St and Re varying in the abovementioned range) and one specific Reynolds number Re = 1, without any lubrication model (19). We plot in figure 8 the restitution coefficient e /e n = –V R /V T (see figure 7 **for** definitions) as a function of Stokes number (23). **For** comparison, we included available experiment data of the rebound of a spherical inclusion with a wall or another particle. While the numerical results are in good agreement with experimental data **for** St 200, the restitution coefficient is clearly overestimated at lower St. This can be attributed to the low resolution of the flow field when the gap between the particle and the wall is of the order of the grid size. As a consequence, the film pressure stemming from the drainage of the liquid in the gap is underestimated so the particle rebound is artificially enhanced. This issue is overcome when one adds a lubrication force (19) in (3). Figure 9 shows the results obtained with the coupled IBM- DEM **method** with the lubrication model (19) **for** the case r p / r = 8. The numerical results fall in the range of the experimental data.

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The interaction between a **boundary** layer and wall-impedance is a classic problem in aeroacoustics. Numerous theoretical investigations by Rienstra and co-workers 2,8–10 , to-
gether with some companion experimental efforts 11 , have looked at the stability properties
of **boundary** **layers** over homogeneous IBCs. In particular, the presence of hydro-acoustic in- stabilities was predicted under specific conditions, which were deemed to be rarely found in aeronautical practice. Such instability occurs when wall-normal acoustic wave propagation (controlled by the IBCs) becomes hydrodynamically significant. This type of instability has been reproduced in the present work, in a fully developed compressible **turbulent** flow, by tuning the characteristic resonant frequency of a mass-spring-damper model **for** the IBCs (a damped Helmholtz oscillator) to the characteristic hydrodynamic time scale of the flow. While the present results are purely numerical, experimental proof of concept of the pro- posed flow control strategy has already been successful obtained in the context of laminar flow separation control over **an** airfoil by Yang and Spedding 12 .

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