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S0045-7930(14)00254-0 http://dx.doi.org/10.1016/j.compfluid.2014.06.015 CAF 2597

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Computers & Fluids

Received Date: Revised Date: Accepted Date:

20 June 2013 24 April 2014 4 June 2014

Please cite this article as: Bocquet, S., Jouhaud, J.-C., Deniau, H., Boussuge, J.-F., Estève, M.J., Wall-Modelled Large-Eddy Simulation of a hot Jet-In-Cross-Flow with turbulent inflow generation, Computers & Fluids (2014), doi: http://dx.doi.org/10.1016/j.compfluid.2014.06.015

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Wall-Modelled Large-Eddy Simulation of a hot Jet-In-Cross-Flow with turbulent inﬂow generation S. Bocqueta,1,∗, J.-C. Jouhauda,2 , H. Deniaub,3 , J.-F. Boussugea,4 , M. J. Est`evec,5 a

Centre Europ´een de Recherche et de Formation Avanc´ee en Calcul Scientiﬁque, 42, Avenue Gaspard Coriolis, 31057 Toulouse Cedex, France b ONERA (French Aerospace Lab), Toulouse c Airbus Operations SAS, Rte de Narbonne, Toulouse

Abstract Hot jets-in-cross-ﬂow are frequently encountered in aeronautics and the accurate estimation of the wall temperature in the jet wake is crucial during the early design of a new aircraft. However, common two-equation RANS models fail at estimating the wall temperature in the jet wake. The use of Large-Eddy Simulation, which seems to be a promising solution at ﬁrst sight, is not applicable due to its prohibitive computational cost on such large Reynolds number wall-bounded ﬂows. For an aﬀordable cost, we propose a strategy which consists in : reducing the computational domain to a small region around the phenomenon of interest (RANS-LES embedded approach), perform a Wall-Modelled Large-Eddy Simulation (WMLES) in the reduced domain and generate a turbulent inﬂow at the reduced domain inlet. The test ∗

Corresponding author, email : [email protected], phone : +33 6 72 68 60 69 PhD 2 Senior Researcher 3 Senior Researcher 4 Aerodynamics Team Leader 5 Method and Tools Engineer 1

Preprint submitted to Computers and Fluids

June 17, 2014

case selected is a hot jet-in-cross-ﬂow experimentally studied by [Albugues, 2005]. We simulate the real geometry of the wind-tunnel model, which imposes strong constraints on the inﬂow generation and numerical method. It is shown that an advanced inﬂow generation, combining a stochastic velocity ﬂuctuation injection and a dynamic forcing term [Larauﬁe et al. , 2011], is mandatory to obtain a realistic turbulent ﬂow upstream of the jet. In the jet wake, the wall temperature estimated by the WMLES agrees well with the experimental measurements. Keywords: Jet-In-Cross-Flow, Large-Eddy Simulation, wall modelling, turbulent inﬂow generation Contents 1 Introduction

4

2 The Jet-In-Cross Flow conﬁguration

6

3 A strategy for the Large-Eddy Simulation of large Reynolds number wall-bounded ﬂows

8

3.1

The Wall-Modelled Large-Eddy Simulation . . . . . . . . . . .

8

3.2

The RANS-LES embedded approach . . . . . . . . . . . . . . 11

3.3

Turbulent inﬂow generation . . . . . . . . . . . . . . . . . . . 12 3.3.1

Stochastic inﬂow generation . . . . . . . . . . . . . . . 13

3.3.2

Dynamic forcing of turbulent ﬂuctuations . . . . . . . . 15

4 RANS computation of the entire wind tunnel test section

2

16

4.1

Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2

Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 18

4.3

Numerical method . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Large-Eddy Simulation modelling

21

5.1

Physical model . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.2

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 21

5.3

Numerical method . . . . . . . . . . . . . . . . . . . . . . . . 23

5.4

Meshing strategy . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Results and discussion

27

6.1

Inﬂow generation assessment upstream of the jet . . . . . . . . 27

6.2

Coherent structures of the Jet-In-Cross-Flow . . . . . . . . . . 31

6.3

Velocity ﬁeld in the jet wake . . . . . . . . . . . . . . . . . . . 32

6.4

Wall temperature in the jet wake . . . . . . . . . . . . . . . . 33

6.5

Spectral analysis in the jet wake and accuracy of the numerical method

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Conclusions

37

8 Acknowledgements

38

9 Appendix

43

9.1

Procedure used to inject the velocity ﬂuctuations . . . . . . . 43

9.2

Procedure used to apply the dynamic forcing method . . . . . 44

9.3

Choice of Sandham mode parameters . . . . . . . . . . . . . . 46

9.4

Computation of boundary layer thickness . . . . . . . . . . . . 48

3

1. Introduction Jet-In-Cross-Flow (JICF) are commonly encountered in aeronautics and their application range from turbine cooling in jet engines to ﬂow control and hot air exhaust in external aerodynamics, to state a few of them. JICF for hot air exhaust is the subject of this study and the application targeted here is the anti-icing system of aircraft engine nacelles (see Fig. 1 (a)). As shown in Fig. 1 (b), the anti-icing system consists in a circulation of hot air in the nacelle leading edge, which heats the leading edge and prevents ice accumulation. The hot air then enters in a plenum before exiting and mixing with the main ﬂow surrounding the aircraft. It appears that downstream of the jet, the hot air impacts the composite materials forming the engine nacelle. This composite material is thus submitted to repeated thermal stresses which can lead to abnormal fatigue and ﬁnally to structural damages. To prevent these damages, the composite materials are protected by thermal shields whose size should be minimized to avoid useless weight. As a result, an accurate description of the wall temperature ﬁeld in the jet wake is of special concern from the industrial point of view. During aicraft design, the Reynolds-Averaged Navier Stokes (RANS) approach is suitable for simulating most applications, with the advantage of a moderate computational cost. However, common one or two-equation RANS models have been shown to fail at predicting the wall temperature ﬁeld behind a hot JICF, as shown by Albugues [1], Jouhaud et al. [2] and Duda [3]. Duda also evaluated the suitability of Unsteady RANS (URANS) to simulate such JICF, without clear improvements of the results. The reason identiﬁed for the failure of (U)RANS is the presence of several large scale coherent 4

motions with broad spectral content, which determine the development of the jet wake and the wall temperature distribution downstream of the JICF. There is general agreement that LES is well suited for the simulation of JICF [2, 4]. However it is known that LES involves a prohibitive computational cost as soon as large Reynolds number wall-bounded ﬂows are concerned, such as the JICF described above. This computational cost is due to the presence of very small streaky turbulent structures in the inner layer of the boundary layer, which require a very ﬁne mesh to be properly captured. According to the estimates of Choi et al. [5], the LES of a turbulent wall-bounded ﬂow involves a number of computational cells proportional to Rex1.9 where Rex = ρ∞ U∞ x/μ∞ is the Reynolds number based on the distance x from the leading edge. To alleviate this expensive computational cost, hybrid RANS-LES methods [6] attempt to use RANS in the boundary layer and LES elsewhere. Among the hybrid RANS-LES methods, Wall-Modelled LES (WMLES) [7, 8] consists in resolving the turbulent structures in the outer layer of the boundary layer, while modelling the eﬀect of the smallest structures underneath. By modelling these very small structures, the number of cells needed becomes 6 proportional to Re0.4 x according to Chapman [9] , allowing a much smaller

number of cells than the one required by a wall-resolved LES. WMLES has been successfully applied to JICF by Jouhaud et al. [2] and Hallez et al. [4]. In addition to the WMLES approach, we will focus on two speciﬁc points : 1) further reducing the computational cost by limiting the 6

Choi et al. estimate the number of cells proportionnal to Rex for Rex > 106 .

5

computational domain to a small region around the JICF, which is called the ’RANS-LES embedded approach’ and 2) generating appropriate turbulent inﬂow at the inlet of the WMLES domain. Both the WMLES approach and the inﬂow generation are implemented in the elsA software [10], which is a multi-block structured compressible ﬂow solver, capable of massively parallel simulations and used by EADS and SAFRAN in their design process. The problem of inﬂow generation for LES has been the subject of several studies. However, inﬂow generation for WMLES has focused very minor attention and very few work exists on this subject. Thus, the investigation of inﬂow generation for WMLES constitutes the original part of this work. This study is structured as follows : 1) the JICF conﬁguration is described ; 2) The strategy chosen to tackle the LES of large Reynolds number wallbounded ﬂows, based on WMLES, the RANS-LES embedded approach and turbulent inﬂow generation, is presented ; 4) The RANS modelling ; 5) The LES modelling are then described ; 6) Results are discussed, starting with the eﬀect of inﬂow generation upstream of the JICF. Then, WMLES results in the jet wake are compared to experimental results. Finally velocity spectra are analyzed and the numerical method limitations are discussed. 2. The Jet-In-Cross Flow conﬁguration The conﬁguration studied is not the real anti-icing system but the representative wind tunnel model investigated experimentally by Albugues [1]. Fig. 2 (a) shows the wind tunnel test section, of dimension 5 × 1.4 × 1.8m respectively in the streamwise, spanwise and vertical directions. An airfoil of C = 0.7m chord is ﬁxed between the two lateral walls and contains internal 6

equipments able to generate the JICF (see Fig. 2 (c)). The use of an airfoil is motivated by the objective of reproducing the wall pressure distribution that takes place on a real aircraft engine nacelle. Hot air at a total temperature of 363K is supplied inside the airfoil through two symmetrical pipes at a given mass ﬂow rate qj , The hot air then mixes in a plenum located right below the ejector grid of thickness d = 2mm. The hot air ﬁnally exits through a square exhaust hole made in the ejector grid and interacts with the wind tunnel main ﬂow at ambient temperature, forming the JICF. It should be noted that the plenum walls are cooled by circulation of cold water in small pipes surrounding the plenum. Thus the total temperature of the hot air right below the exhaust hole has decreased to a value of about 353K [1]. The main ﬂow velocity is U∞ = 47.25m.s−1 at a static temperature T∞ = 295K, leading to a Mach number M∞ = 0.14 and a Reynolds number ReD =

ρ∞ U∞ D μ∞

= 93000, with D = 30mm is the exhaust hole dimen-

sion and ρ∞ and μ∞ are respectively the main ﬂow density and molecular viscosity. Expressed using the exhaust hole dimension, the domain measures [−83D, 83D] × [−23D, 23D] × [−30D, 30D] respectively in the streamwise, spanwise and vertical directions. Transition is triggered upstream of the ejector grid so that the boundary layer is fully turbulent when it reaches the grid. The Reynolds number, Mach number, wall pressure distribution and the temperature diﬀerence between the hot and cold ﬂows, here equal to ΔT = Tj −T∞ = 57K, constitute a set of similarity parameters that characterize the JICF dynamics. In addition to these, two important similarity parameters can be further identiﬁed : the ratio of momentum between the hot and cold

7

ﬂow CR =

ρj Uj ρ∞ U∞

hole dimension

and the ratio of main ﬂow displacement thickness by exhaust δ1 . D

Here CR = 0.69 and

δ1 D

≈ 17 × 10−3 , the latter being

measured just upstream of the exhaust hole from the RANS computation described later. The X, Y , Z coordinates and U , V , W velocity components respectively stand for the streamwise, spanwise and vertical directions in the global frame of reference. The origin of this frame of reference is located at the middle of the exhaust hole downstream edge. x, y, z coordinates and u, v, w respectively denote streamwise, wall-normal and spanwise directions in the local boundary layer frame of reference. 3. A strategy for the Large-Eddy Simulation of large Reynolds number wall-bounded ﬂows 3.1. The Wall-Modelled Large-Eddy Simulation As seen in the introduction, the computational cost of a wall-resolved LES would be very expensive on this JICF. Indeed the Reynolds number based on the airfoil chord is Rec = 2.2 × 106 . To fulﬁll the criterion δx+ = 50, δy+ = 1 and δz+ = 15, we estimate that at least 300 × 106 cells would be necessary to discretize the small embedded domain shown in Fig. 3 with a conventional full-matching structured mesh . Here the superscript (.+ ) designates dimensions in wall units, that is : y+ = y

ρw u τ μw

with uτ being the friction velocity uτ =

(1)

τw , ρw

the subscript (.w ) denotes

values at the wall and the overlines stand for Reynolds-averaged quantities. 8

The number of cells would be much larger if the entire wind tunnel test section was considered. Compared to a wall-resolved LES, the WMLES approach consists in using coarse quasi-isotropic cells at the wall. The ﬁrst consequence is a drastic reduction in the number of cells, mostly obtained by relaxing the constraint + Δ+ z ≈ 15 to Δz = O(100). The second consequence is that the computation

advances in time much faster because, for an explicit time integration, the time step is proportional to the smallest cell dimension, which is generally the height of the wall-adjacent cells. Since the WMLES approach involves + heights of the order of Δ+ y = 0(100), while wall-resolved LES involves Δy = 1,

the time step of a WMLES is increased by about two orders of magnitude compared to LES. By combining a smaller number of cells and a larger time step, the WMLES approach leads to strong computational time savings. The use of very coarse wall-adjacent cells involves speciﬁc numerical treatments. Indeed, as shown in Fig. 4 (a), linear velocity and temperature proﬁles cannot be assumed in the wall-adjacent cells due to their large dimensions so that a ﬁnite diﬀerence approximation for the wall friction τw and wall heat w is not appropriate. Thus a wall model is introduced, which provides ﬂux φ w given the instantaneous wall-parallel an estimate for the wall ﬂuxes τw and φ LES velocity u1 and temperature T1 1 . This procedure is repeated in each wall-adjacent cells at each iteration in time. Another consequence of using very coarse wall-adjacent cells is that only the logarithmic layer turbulent structures are resolved, as shown in Fig. 4 (b). 1

In the elsA solver, conservative variables are stored at cell centers, thus the wall model

1 are considered at the ﬁrst cell center above the wall. input u 1 and T

9

In order to select an appropriate wall model, the thermal regime corresponding to this JICF needs to be investigated using the temperature ratio T∞ ΔT

and the Eckert number Ec =

2 U∞ Cp |ΔT |

2 T∞ = (γ − 1)M∞ . Here ΔT

T∞ ΔT

= 5.2

and Ec = 3.9 × 10−2 . Since Ec << P r−1 (P r being the molecular Prandtl number), Ec << 1 and

T∞ ΔT

>> 1, then the viscous heating, density variation

and molecular viscosity variation can be neglected and the temperature can be considered as a passive scalar. For this quasi-incompressible regime, the Reichardt law [11] is valid and gives a description of the velocity proﬁle over the entire inner layer of the boundary layer, from the top of the logarithmic region down to the wall. The Reichardt law reads: u1 1 1 y1+ + + exp(−0.33y1+ )) (2) = ln(1+κy1 )+(B− ln(κ))(1−exp(−y1 /11)− uτ κ κ 11 with κ = 0.41 and B = 5.25. In the region just downstream of the exhaust hole, where the ﬂow recirculates, Reichardt law is not strictly valid. However the discrepancies induced on the wall friction in this small recirculation region are not expected to induce signiﬁcant discrepancies on the quantity of interest, namely the wall temperature over the measurement plate. For more advanced wall modelling of recirculation regions, one can refer to [12, 13, 14]. The modelling of the temperature proﬁle depends on the thermal condiw = 0 and Tw = T1 for this tion imposed on the wall. On an adiabatic wall, φ quasi-isothermal regime. On an isothermal wall, the Kader law (Eq. 3) is used and allows to compute the wall heat ﬂux φw . This relation is appropriate for low Mach number ﬂow and can handle large variations of temperature 10

ratio T1 /Tw and Prandtl number, so that it is a fortiori valid here: ⎧ ⎪ T + =P ry1+ exp(−Γ) + [2.12 ln(1 + y1+ ) + β(P r)] exp(−1/Γ) ⎪ ⎪ ⎨ 1 10−2 (P ry + )4 Γ = 1+5P r3 y1+ and β(P r) = [(3.85P r(1/3) − 1.3]2 + 2.12 ln(P r) ⎪ 1 ⎪ ⎪ ⎩ T + = −(T1 −Tw )ρw Cp uτ 1 φ

(3)

w

More detailed information about the procedure used to solve this wall model and its validity can be found in Bocquet et al. [15]. 3.2. The RANS-LES embedded approach The Reynolds number Rec is not the only parameter determining the computational cost. It also depends on the size of the computational domain compared to the exhaust hole dimension D. On this wind tunnel model, the entire wind tunnel test section is already large compared to D, with dimensions equal to (166D; 46D; 60D). On the aircraft, if the entire nacelle was simulated, the ratio of computational domain size over D would be much larger than the one obtained on the model. Thus, even with the WMLES approach, the computational cost would be prohibitive and a reduction in the computational domain size is mandatory. To reduce the WMLES domain size, an embedded approach is used, which consists in two steps. First, a moderate computational cost RANS simulation is performed on the full wind tunnel test section. Secondly, as shown in Fig. 3, a smaller embedded domain around the area of interest is deﬁned and a WMLES is computed using the RANS ﬂow to specify the initial and boundary conditions. The embedded domain dimensions are about [−3.3D, 16D]×[−6.7D, 6.7D]×[0, 8D], leading to a volume of 0.056m3 , which is 23 times smaller than the full wind tunnel test section volume. 11

3.3. Turbulent inﬂow generation One consequence of the embedded approach is that a turbulent boundary layer must be injected at the domain inlet. Turbulent inﬂow generation for hybrid RANS-LES methods have mostly been studied in the context of Detached Eddy Simulation (DES), see for example Deck et al. [16] and Larauﬁe et al. [17]. Concerning WMLES, Wang and Moin [12] mention that they use a precursor method to generate inﬂow and Hallez et al. [4] indicate that they tested an inﬂow generation on a JICF without observing changes on the results. To our knowledge, no speciﬁc work has focused on the problem of inﬂow generation for WMLES. Consequently, the originality of the present study is the investigation of an inﬂow generation method in the context of WMLES. Several methods for inﬂow generation exist. First, instantaneous velocity ﬂuctuations can be generated by a precursor simulation, for example a bi-periodic turbulent plane channel and stored in a database. The work of Schl¨ uter et al. [18] provides an example of inﬂow generation for LES from a precursor database. Secondly, rescaling methods [19] use the instantaneous ﬂow ﬁeld on a plane parallel to the inlet but located at some distance downstream to deﬁne the inlet velocity ﬂuctuations. Due to the spatial boundarylayer development, these data need to be rescaled before they are injected at the inlet for turbulent generation. Rescaling methods allow to obtain a fully turbulent boundary layer at a small distance downstream of the inlet. However, depending on the initial ﬂow, a long time may be needed to evacuate the transient ﬂow and spurious frequencies may corrupt the results. Pami`es [20] proposes a review of solutions to alleviate these limitations. Finally,

12

stochastic methods use analytical functions to prescribe instantaneous velocity ﬂuctuations on the inlet plane [21, 20]. These functions are often a mix of deterministic and random functions. Due to the characteristics of WMLES, an inﬂow generation method for wall-resolved LES would not necessarily be appropriate for WMLES. The closest work to our case is certainly the study of Larauﬁe et al. , who successfully generate inﬂow turbulence in a Zonal Detached-Eddy Simulation (ZDES) on a much coarser mesh than the one used for wall-resolved LES. According to their choice, we use a stochastic inﬂow generation method combined with a volumic forcing source term acting on the wall normal velocity ﬂuctuations [22]. 3.3.1. Stochastic inﬂow generation The stochastic inﬂow generation method of Sandham et al. [21] is chosen and consists in prescribing several modes of velocity ﬂuctuations at diﬀerent distances above the wall. In the original method, distinction is made between inner and outer modes, respectively corresponding to buﬀer layer structures (streaks) and logarithmic layer structures. Here we only retain outer modes since inner modes are not discretized by the WMLES approach. The outer mode j for the streamwise and wall-normal ﬂuctuations are given by the following equations in the boundary layer frame of reference: c1j y exp(−y/yp,j )sin(ωj t)cos(βj z + φj ) 0.368 yp,j

(4)

c2j y 2 exp(−(y/yp,j )2 )sin(ωj t)cos(βj z + φj ) 2 0.368 yp,j

(5)

uj =

vj =

13

The spanwise mode is obtained by imposing a divergence-free condition on the velocity ﬂuctuations. A white noise (W N )0.05U∞ of absolute maximum value equal to 0.05U∞ is also added to the streamwise ﬂuctuation. In order to conﬁne the white noise to the boundary layer, we use a damping function similar to that used in the mode deﬁnitions : uj = uj +

y −y/yp,j 1 e (W N )0.05U∞ 0.368 yp,j

(6)

In practice, the injection of these velocity ﬂuctuations in a compressible ﬂow solver requires further steps which depend on the type of inlet boundary condition. A detailed description of these steps can be found in appendix (see Sec. 9.1). Table 1: Parameters deﬁning the injected modes j

c1j

c2j

ωj (s−1 )

βj (m−1 )

φj (rad)

yp,j (m)

1

0.24U∞

−0.1U∞

0.67ω2

1.33β2

0.

1.33yp,2

2

0.24U∞

−0.1U∞

2π7850

2π180

0.1

8.5 × 10−4

3

0.24U∞

−0.1U∞

1.33ω2

0.67β2

0.15

0.67yp,2

As in the original work of Sandham et al. , we inject three outer modes. The parameters deﬁning the modes are given in Table. 1 and have been determined as explained in Appendix 9.3. WMLES computations of a ﬂat plate boundary layer, with the same ﬂow solver and numerical method as used here, have shown that these parameters allowed to generate a fully turbulent boundary layer after a distance of 25 inlet boundary layer thicknesses downstream of the inlet plane.

14

3.3.2. Dynamic forcing of turbulent ﬂuctuations In addition to the injection of velocity ﬂuctuations, the dynamic forcing term initially proposed by Spille-Kohoﬀ and Kaltenbach [22] and modiﬁed by Larauﬁe et al. [17] is used in a small volume adjacent to the inlet plane. This forcing term allows to dynamically adjust the amplitude of the wall-normal root mean squared (RMS) velocity toward a speciﬁed target. Indeed, following Larauﬁe et al. , the wall-normal RMS velocity appears in the turbulent shear stress production:

P−ρu v = v 2

∂u ∂y

(7)

As a result, it is expected that forcing an appropriate level of wall normal ﬂuctuations would lead to a correct level of turbulent shear stress and thus a realistic turbulent wall friction and velocity proﬁle. The dynamic forcing relies on a source term fy deﬁned as follows :

e = ρ(v 2 target − V 2 ) r = αe

2 2 fy = r(V − V ) if u2 < 0.36U∞ and V 2 < 0.16U∞

(8)

fy = 0 otherwise where . denotes temporal low-pass ﬁltering of characteristic time Δf as deﬁned in Appendix. 9.2. A target wall normal RMS velocity v 2 target is determined from the RANS ﬂow (see Appendix. 9.2). The source term f (y) is ﬁnally added to the wall normal LES momentum equation in each cell

15

contained within a forcing volume (see Fig. 6) which extends over a distance Δc from the inlet. This dynamic forcing requires to set three parameters: the forcing intensity α, the length Δc over which the forcing is applied and Δf . WMLES computations of a ﬂat plate boundary layer have shown that α = and Δc = 18 inlet boundary layer thicknesses is suﬃcient to generate a fully turbulent boundary layer over a distance smaller than 25 inlet boundary layer thicknesses downstream of the inlet plane. Δf is chosen equal to twice the longest period of the Sandham modes, which ensures that all injected modes are suﬃciently averaged in time. While the numerical schemes and turbulence models used in this work are already built-in methods of the elsA software, the wall model approach, the stochastic inﬂow generation and the dynamic forcing methods described above are new functionalities that have been implemented by the authors. 4. RANS computation of the entire wind tunnel test section 4.1. Physical model The ﬂuid considered is air, modelled as a perfect gas, of constant heat capacity Cv = 717.5J.kg −1 .K −1 and perfect gas constant γ = Cp /Cv = 1.4. We set P r = 0.72 and the molecular viscosity is assumed to follow Sutherland T law : μ(T ) = μref ( Tref )1.5

Tref +S , T +S

with Tref = 288.15K, μref = 1.789 ×

10−5 kg.m−1 .s−1 and S = 110.4K. The ﬂuid behaviour is given by the compressible Reynolds-Averaged NavierStokes equations :

16

∂ρ + div[ρU] = 0 ∂t ∂ρU (9) + div[ρU ⊗ U + pI − τ − τr ] = 0 ∂t ∂ρE + div[ρEU + pU − (τ + τr )U + Cp (φ + φt )] = 0 ∂t where U and E respectively refer to mass-weighted averaged velocity and total energy, while ρ, p, τ and φ refer to Reynolds-averaged quantities. Air is considered as a Newtonian ﬂuid with Stokes hypothesis, leading to τ = − 23 μ(divU)I + 2μD with D the deviatoric part of the stress tensor. The heat ﬂux vector is assumed to follow Fourier law : φ = −λgrad(T), with λ = − CPprμ being the molecular thermal diﬀusivity. The Boussinesq hypothesis is used to model the Reynolds tensor, which gives τr = − 23 μt (divU)I + 2μt D. To model the turbulent heat ﬂux, the turbulent Prandtl hypothesis leads to φt =

C p μt grad(T). P rt

The turbulent viscosity is modelled by the Spalart-

Allmaras model and the turbulent Prandtl number P rt is set to 0.9. The Spalart-Allmaras evolution equation reads: ∂ ν˜ + div[˜ ν U] = Sν˜ ∂t

(10)

The source term Sν˜ is given by: ν˜2 ˜ ν + Cb2 ∇ρ˜ ν · ∇˜ ν − Cw1 fw ρ 2 Sν˜ = Cb1 Sρ˜ σ d

(11)

ρ˜ ν ν˜ χ (12) S˜ = |rotU| + 2 2 fv2 , fv2 = 1 − , χ= κd 1 + χfv1 μ 1/6 6 1 + Cw3 ν˜ fw = g , g = r + Cw2 (r6 − r) , r = (13) 6 6 ˜ 2 d2 g + Cw3 Sκ ρ˜ ν χ3 (14) ν fv1 , fv1 = 3 , χ= μt = ρ˜ 3 χ + Cv1 μ 17

Cb1 = 0, 1355 , Cb2 = 0, 622 , σ =

2 Cb1 1 + Cb2 , κ = 0, 41 Cw1 = 2 + (15) 3 κ σ

Cw2 = 0, 3 , Cw3 = 2 , Cv1 = 7, 1

(16)

4.2. Boundary conditions Table 2: Boundary conditions for the RANS computation Inlet

Outlet

Airfoil

Upper face

Plenum

Injection pipes,

of the ejector grid

walls

upper, bottom and lateral wind tunnel walls

Main ﬂow :

pressure imposed

no-slip

no-slip

no-slip

NSCBC,

p∞ = 101325P a

adiabatic

isothermal

adiabatic

Ti = 296K,

with non-uniform

U∞ = 47.25m.s−1 ,

temperature

ν = 0.024

distribution

Hot air injection : mass ﬂow rate imposed, Tij = 353K, qj = 0.01771kg.s−1

Boundary conditions for the RANS computation are given in Table 2. A Navier-Stokes Characteristic Boundary Condition (NSCBC) [23], previously implemented in the elsA software by Fosso et al. [24], is prescribed at the inlet to impose total temperature and velocity. The adiabatic condition imposed on the airfoil is justiﬁed because the measurement plate used in the experiments can be considered as adiabatic (see [1], p. 61). The choice of an adiabatic condition on the walls inside the plenum and an isothermal condi18

wall slip

tion on the upper side of the ejector grid is less obvious and requires some justiﬁcation. First, since the plenum is cooled, an isothermal condition could be imposed on its walls. However there is no experimental data giving the distribution of temperature on the plenum walls. Even a ﬂuid-structure simulation resolving the temperature inside the plenum walls would be very diﬃcult to set up, because it would require to compute all the cooling system and all the internal airfoil equipments until simple thermal boundary conditions can be applied. As a result we follow the methodology used by Duda [3] : the plenum walls are considered as adiabatic and the total temperature right below the exhaust hole obtained in the experiments is imposed at the inlet of the injection pipes. Since the injection pipes and the plenum are adiabatic, the total temperature should be conserved until the exhaust hole. Secondly, the choice of an isothermal boundary condition on the upper side of the ejector grid is justiﬁed by the following arguments. It is known that air/metal interfaces can usually be considered as isothermal. However the plate is here thin (2mm) compared to its streamwise and spanwise extensions and the choice of an isothermal condition is not straightforward. Hallez et al. [4] propose to use two quantities for deciding if the isothermal condition is appropriate. First the ratio τs /τf of diﬀusive time into the solid (τs = d2 /αs ) by the convective time into the ﬂuid (τf = D/U∞ ), with αs =

λs ρs C p s

being the solid thermal diﬀusivity. Since τs /τf = 1580, a tem-

perature ﬂuctuation propagates much slower into the solid than in the ﬂuid. Secondly, they propose to use the ratio of solid eﬀusivity by ﬂuid eﬀusivity, which is estimated to 1450 for this air/copper interface. This ratio shows that

19

a thermal ﬂuctuation in the ﬂuid induces a negligible thermal ﬂuctuation into the solid. Hallez et al. conclude that an isothermal condition can be imposed on the plate. It can be noted that Chatelain [25] uses a diﬀerent parameter based on the ratio of diﬀusive time into the solid by the diﬀusive time

in the ﬂuid called the non-dimensional solid thickness d++ = ρwμuwτ d αf /αs . Chatelain indicates that the isothermal condition is not justiﬁed if d++ → 0. Here d++ = 620, which conﬁrms that the isothermal condition is meaningful. Since the plate is heated by the hot air mixing in the plenum, it reaches an equilibrium temperature signiﬁcantly higher than the ambient temperature. For an accurate description of the boundary condition, a realistic temperature ﬁeld is imposed instead of a simple uniform temperature. This non-uniform temperature ﬁeld is provided by a thermally coupled ﬂuid-solid RANS simulation, resolving the temperature ﬁeld in the ejector grid [3]. A view of this temperature ﬁeld is shown in Fig. 20 where the ejector grid is visible. 4.3. Numerical method Convective ﬂuxes are reconstructed on cell interfaces by a third-order Roe scheme and diﬀusive ﬂuxes are computed by a centered second order 3 point stencil scheme. Time integration is implicit, the time derivative being discretized by a backward ﬁrst order approximation. Time advancement is performed using a local time step constrained to a maximum acoustic Courant-Friedrich-Lewy number (CFL) of 10. A larger CFL and/or a multigrid method may be chosen to accelerate the convergence but this was not further investigated because the RANS computation is very cheap compared to the WMLES one. 20

5. Large-Eddy Simulation modelling 5.1. Physical model In the WMLES, the ﬂuid behaviour is given by the compressible ﬁltered Navier-Stokes equations : ∂ ρ =0 + div[ ρU] ∂t ∂ ρU ⊗U + pI − τ − τr ] = 0 (17) + div[ ρU ∂t ∂ ρE U − ( + Cp (φ + φt )] = 0 + pU + div[ ρE τ + τr )U ∂t respectively refer to the mass-weighted ﬁltered velocity and E where U and total energy while ρ, p, τ and φ refer to Reynolds-ﬁltered quantities. Neglecting the cross term and Leonard term in the subgrid-scale tensor τr leads to the same formalism as for the RANS approach described above, except that the turbulent viscosity is replaced by the subgrid-scale viscosity μsgs and the turbulent Prandtl number replaced by a subgrid Prandtl number P rsgs . The Wall-Adapting Local Eddy-Viscosity (WALE) model developed by Nicoud and Ducros [26] is used to specify the subgrid-scale viscosity. The WALE model is able to produce the correct y 3 scaling for the subgrid scale viscosity near a wall without requiring a dynamical procedure. This eddyviscosity model is highly recommended for turbulent ﬂows involving walls. The subgrid Prandtl number P rsgs is set to 0.9 following Chatelain [25]. 5.2. Boundary Conditions Following the choices made in the RANS computation, the adiabatic version of the wall model is imposed on the airfoil while the isothermal version 21

Table 3: Boundary conditions for the WMLES computation Inlet

Outlet

Airfoil

Upper face

Top

of the ejector grid

Lateral faces

Main ﬂow :

imposed

adiabatic

isothermal

imposed

NSCBC

imposed Ti and pi ,

pressure,

wall-modelled

wall-modelled

pressure

velocity

from RANS

p∞ = 101325P a

with non-uniform

from RANS

and

Hot air injection :

temperature

temperature

distribution

from RANS

imposed mass ﬂow rate, Tij = 353K, qj = 0.01771kg.s−1

is imposed on the upper side of the ejector grid, using the same non-uniform temperature ﬁeld (see Sec. 4.2). The wall boundary condition has already been described and a description of the other boundary conditions is now provided. As remarked above, boundary conditions are often placed far away from the region of interest so that their inﬂuence can be neglected and simple conditions used, such as a uniform ﬂow ﬁeld. However, in the embedded domain, the boundary conditions are close to the JICF. Thus a proper deﬁnition of the boundary conditions becomes critical since they must impose the non-uniform ﬂow computed from the RANS simulation. The boundary conditions chosen are given in Table. 3. A boundary condition based on the characteristic equations imposes the total temperature, total pressure and velocity direction at the inlet. These quantities are computed from the RANS ﬂow. The inﬂow generation described in section 3.3 is 22

added to this boundary. The ﬁlter size Δf is set to

2(2π) ω1

(see Appendix 9.2),

in other words the ﬁlter size is twice the largest time period of the injected velocity ﬂuctuations. The parameter α is here set to 880, which will be shown to be high enough for obtaining a small adaptation distance. On both lateral boundaries, an NSCBC condition imposes the RANS velocity and temperature. The top boundary is modelled by an imposed pressure condition based on the characteristic equations, where the non-uniform static pressure ﬁeld is taken from the RANS solution. The outlet is well downstream of the jet wake and the pressure is found to be almost constant in the RANS solution. Thus we impose the same characteristic condition with a constant static pressure set to p∞ . The hot air injection and the injection pipes are modelled in the same way as in the RANS simulation. 5.3. Numerical method Convective ﬂuxes are reconstructed on cell interfaces by a skew-symmetric second order centered scheme operating on the primitive variables. This numerical scheme is stabilized by a Jameson-Schmidt-Turkel artiﬁcial viscosity [27, 28], where the second order and fourth order dissipation coeﬃcients are respectively set to k2 = 0 and k4 = 5 × 10−3 . A wiggle sensor [29, 30] applies the fourth order dissipation only where spurious oscillations are detected. Diﬀusive ﬂuxes are computed as in the RANS simulation. The time integration is explicit and the time derivative is discretized by a 4th order Runge-Kutta scheme. A time step of Δt = 2.56 × 10−7 s is used, leading to a maximum acoustic CFL number of 0.9.

23

5.4. Meshing strategy The meshing strategy for resolved LES of wall-bounded ﬂows essentially consists in prescribing the recommended cell dimensions at the wall, say + + (Δ+ x = 100, Δy = 1, Δz = 15). However, the meshing strategy for WMLES

is diﬀerent from the meshing strategy for wall-resolved LES. In addition, the behaviour of the wall-modelling approach and especially of the inﬂow generation method is strongly dependent on the meshing strategy used. These two reasons motivate the detailed description of the cell dimension choices given in this paragraph. After an overview of the WMLES mesh, we will focus on the inlet boundary. A global view of the WMLES mesh is shown in Fig. 5. The geometry is more complex than academic cases, which constraints the mesh generation and involves compromises on mesh quality. This non-ideal mesh quality explains why artiﬁcial viscosity is introduced to stabilize the computation. We now focus on the inlet boundary to explain the constraints on mesh generation involved by the inﬂow generation for WMLES. Two major constraints govern the mesh generation : • the WMLES mesh requirements, which imply the use of large quasiisotropic cells of dimension Δ = 0(100) (see Sec. 3.1) with about 20 cells to discretize the boundary layer thickness. To match these requirements, it is necessary to have Reτ > 2000 ; • the distance Xba − Xexhaust between the airfoil leading edge and the jet exhaust. As shown in Fig. 6, this distance is rather small on this JICF. As a consequence, the Reτ obtained at the inlet boundary is smaller 24

than 2000 and WMLES meshing requirements cannot be fulﬁlled. The short distance Xba −Xexhaust also requires an eﬃcient inﬂow generation, because we want the boundary layer to be fully turbulent at X = Xexhaust . Given these constraints, the compromise chosen is to place the inlet boundary so that Xin − Xexhaust = 27δin , which gives δin = 2, 1mm considering the RANS ﬂow 7 . We obtain Reτ = 420 at the inlet and given this small Reynolds number, we discretize the boundary layer thickness by ny = 15 cells, with a wall-adjacent cell height of Δy + 1 = 23. It can be noticed that ny = 15 is a small number of cells to discretize the outer part of the boundary layer and Chapman [9] recommends at least ny = 25. However, δ increases as the boundary layer develops so that ny increases accordingly. The small cell height Δ+ y 1 = 23 leads to another consequence in terms of computational cost. Indeed, because the WMLES imposes quasi-isotropic cells (say Δx = 5Δy and Δz = 2, 5Δy ), the cell dimensions along the wall+ parallel directions would be Δ+ x ≈ 100 and Δz ≈ 60, which is closer to

wall-resolved LES than WMLES. As a result, we impose here ﬂatter cells by choosing Δx = 8Δy and Δz = 4Δy , corresponding to Δ+ x = 190 and + Δ+ z = 94. This ﬂat cell shape combined with the small Δy 1 stress the

WMLES behaviour, which will be particularly interesting to analyze near the inlet. As we move from the inlet to the jet exhaust, Δy 1 is progressively increased to reach 0.21mm at the jet exhaust and 0.5mm in the jet wake at 7

The method used to compute the boundary layer thickness is given in Appendix 9.4

25

X/D = 8. Table 4 gives the cell dimensions at the inlet and at the middle of the dynamic forcing region. Table 4: Wall-adjacent cell size on the symmetry plane expressed in meter. Values in wall units are given in brackets. Xin +11δin corresponds to the middle of the dynamic forcing region. 103 Δx (Δ+ x)

inlet

Xin + 11δin

1 (190)

10 (170)

Δy (Δ+ y)

0.12 (23)

2 (33)

103 Δz (Δ+ z )

0.5 (94)

5 (82)

10

3

A comparison of the WMLES and RANS meshes is also provided in Fig. 7, which highlights the speciﬁc meshing imposed by the WMLES approach and the inﬂow generation. It should be noted that the cell dimensions described above concern the symmetry plane. Normally, we should apply these meshing requirements over the entire spanwise dimension of the WMLES domain. To limit the computational cost, we apply the previous meshing requirements in −2 < Y /D < 2, since the jet wake in entirely contained within this region. Outside this region, the spanwise dimension of the cells is progressively increased. We obtain a WMLES mesh containing 14×106 cells, which is a reasonable number considering that the real wind tunnel model geometry is reproduced and that a turbulent inﬂow generation is set over the whole width of the ejector grid.

26

6. Results and discussion Time averaging is performed on the interval [t0 , tf ], where t0 = 4.6×10−2 s is chosen to evacuate the initial RANS ﬂow and tf = 0.017s. The change in RMS velocity proﬁles when choosing a larger t0 or tf is found to be negligible. For the velocity spectral analysis performed in Sec. 6.5, one computation is run until tf = 0.3s. We ﬁrst investigate the eﬀect of the inﬂow generation upstream of the jet before going to the results on the velocity and wall temperature ﬁelds in the jet wake. 6.1. Inﬂow generation assessment upstream of the jet We compare three WMLES simulations : • simulation inj c with the velocity ﬂuctuations of Sandham et al. and the dynamic forcing source term of Larauﬁe et al. ; • simulation inj with the velocity ﬂuctuations and without dynamic forcing source term; • simulation w/o inj without velocity ﬂuctuations nor forcing source term. The RANS simulation is taken as a reference data in the region upstream of the jet because of the lack of experimental data and because we are conﬁdent in the ability of the RANS approach to capture the aerodynamic ﬁeld as long as the ﬂow is attached. We focus on the symmetry plane and investigate the boundary layer development from the inlet to the jet exhaust hole. Considering the mean 27

wall pressure given in Fig. 8 (a), all simulations show the presence of an adverse pressure gradient due to the JICF, which can be seen as an obstacle to the cross-ﬂow. However, the behaviour of simulations inj and w/o inj is diﬀerent from the RANS and inj c cases, with a drop of pressure around X − Xin = 18δin . The non-dimensional pressure gradient p+ x =

μw ∂p w , ρw u3τ ∂x

shown in Fig. 8

(b), represents the eﬀect of the streamwise pressure gradient on the boundary layer. We observe that the inj and w/o inj simulations exhibit very large discrepancies compared to the inj c case, the latter remaining close to the RANS case. It is quite surprising to observe that even low order statistics like the mean pressure can be signiﬁcantly modiﬁed. As a result, we expect the boundary layer development to be strongly inﬂuenced depending on the inﬂow generation chosen. The mean wall friction is a good indicator to determine if the boundary layer is fully turbulent. Figure. 9 gives the streamwise component of the mean wall friction as a function of the distance from the inlet plane. It appears that the inj c simulation matches the RANS solution very well after an adaptation length of about 12δin , even if a drop of wall friction is observed close to the jet exhaust. This discrepancy is likely to be due to the WMLES approach rather than the inﬂow generation. Indeed, it is known that the WMLES approach does not provide good response to strong adverse pressure gradient, unless very speciﬁc methods are used, see for example Wang and Moin [12]. Looking at the inj and w/o inj cases, the wall friction is strongly underestimated and even becomes negative, which indicates a recirculation bubble. As a result these two simulations do not allow to obtain a turbulent boundary layer and

28

the very low wall friction tends to indicate the presence of a laminar ﬂow. Two reasons may explain the failure of using only a stochastic injection for WMLES: • stochastic injection cannot reproduce all the characteristics of wallbounded turbulence (whose dynamics remains largely unknown, see for example Jim´enez [31]); • in a WMLES computation, the wall normal RMS velocity is strongly dependent on numerical method and in general tends to be underestimated. Thus it seems diﬃcult to design a stochastic injection for WMLES capable of maintaining a suﬃcient level of wall normal RMS velocity. The success of the dynamic forcing method for WMLES is certainly due to the fact that this method is speciﬁcally designed to ensure a suﬃcient level of wall normal RMS velocity. Thus, the use of such a dynamic forcing seems particularly adapted to turbulent inﬂow generation in WMLES computations. A local analysis of the results is now provided through mean streamwise velocity proﬁles extracted at four stations (X − Xin )/δin = (11; 14; 17; 20) indicated by the dashed vertical lines in Fig. 9. These velocity proﬁles conﬁrm the ﬂow reversal at the third and fourth stations in simulations inj and w/o inj. The velocity proﬁle quality for the inj c case is investigated in more details

+ /uτ as a funcin Fig. 11 by plotting the non-dimensional velocity u = U tion of y + . Because the ﬂow reverses for the two other WMLES simulations, 29

it is not meaningful to add them to this plot. We observe that the velocity is underpredicted close to the wall. This is a consequence of the eﬀect of both the artiﬁcial dissipation and the adverse pressure gradient. WMLES computations performed on a boundary layer without pressure gradient (not shown here) have shown that the artiﬁcial dissipation eﬀect concerns the ﬁrst three points above the wall. The remaining underprediction can be attributed to the adverse pressure gradient, whose inﬂuence increases as the jet exhaust is approached. We now switch to ﬁrst-order statistics with the root-mean-squared (RMS) velocities shown in Fig. 12. The ﬂuctuation proﬁles of the inj c case appears similar at all stations even if the inﬂuence of the adverse pressure gradient is visible at the last station, where RMS velocities decrease. However, the inj and w/o inj simulations exhibit nearly zero velocity ﬂuctuation at the ﬁrst two stations, which conﬁrms the laminar behaviour of the boundary layer. At the third station, the velocity ﬂuctuations start increasing and much larger values than the inj c case can be observed at the last station. These results stress that without an appropriate inﬂow generation method, the boundary layer behaviour is unpredictible and certainly strongly dependent on the numerical method and mesh used. Since a turbulent boundary layer is obtained with the inj c simulation, whereas a laminar behaviour is observed when the inﬂow turbulence is not properly generated, diﬀerences on the wall-heat ﬂux on the ejector grid should φw be apparent. Figure 13 shows the Nusselt number N u = λ∞ (Tj −T∞ )/D on the ejector grid, which provides a global view of the inﬂow generation eﬀect on the whole domain width. The inj c case exhibits a wall heat ﬂux distribution

30

which is close to the RANS simulation. However, the inj and w/o inj cases underpredict the wall heat ﬂux for X/D < −1.8, which is coherent with the laminar boundary layer observed on these cases at the ﬁrst two stations. Then, for X/D > −1.8, the wall heat ﬂux is overpredicted around the exhaust hole, which is also coherent with the sudden burst observed on the velocity ﬂuctuations. This conﬁrms the complex and unpredictible boundary layer behaviour of cases inj and w/o inj. It also indicates that before assessing the inﬂuence of the ejector grid thermal boundary condition on the temperature ﬁeld in the jet wake, a realistic wall heat ﬂux distribution must ﬁrst be obtained on the ejector grid, which involves the use of an appropriate inﬂow generation method. These results show that the dynamic forcing method for inﬂow generation can be successfully applied to the WMLES approach on a real geometry wind tunnel model with an adverse pressure gradient at the inlet. The fact that the dynamic forcing method can be used for both the WMLES and ZDES approaches [17] reinforces its generality and usefulness. From now on, we will focus on the results of the inj c simulation in the jet wake. 6.2. Coherent structures of the Jet-In-Cross-Flow A ﬁrst insight into the WMLES results in the jet wake is provided by the coherent structures forming in a JICF. As shown in Fig. 14, several coherent structures develop in a JICF. The two structures used here for the assessment of the WMLES simulation are : • hairpin vortices generated in the shear layer at the jet-cross ﬂow interface. Andreopoulos [32] proposed experimental visualizations of these 31

vortices and indicates that their development depends on the turbulence level in the incoming cross-ﬂow boundary layer; • wake vortices, similar to the Von K´arm´an vortex street generated in the wake of a cylinder. As proposed by Duda [3], these vortices create a meandering of the jet wake. The two coherent structures described above are visualized by the instantaneous temperature, extracted from the inj c results. Figure 15 (a) shows the hairpin vortices on the symmetry plane. It appears that the hairpin vortices occupy a wide range of scales, which is coherent with Andreopoulos visualizations for a suﬃciently large Reynolds number ReD . Figure 15 (b) shows the jet meandering created by the wake vortices. It will be shown by the spectral analysis performed in Sec. 6.5 that the meandering frequency is correctly estimated by the inj c simulation. 6.3. Velocity ﬁeld in the jet wake We now focus on the velocity ﬁeld in the jet wake. Figure 16 and Fig. 17 show a global comparison of the mean streamwise velocity. In simulation inj c, the streamwise extent of the jet wake on the symmetry plane compares

/U∞ = 0.8 ending for fairly well with experimental data, the isoline U 5 < X/D < 6 in both cases. The RANS results show a much longer wake. Just downstream of the exhaust hole, the recirculation area predicted by simulation inj c shows a smaller extent and a lower backﬂow velocity than the experiment. A more local comparison of the mean streamwise and vertical velocity proﬁles is provided in Fig. 18 at X/D = 1 and X/D = 8. Despite a smaller 32

back ﬂow velocity in the recirculation region at X/D = 1, the inj c results compare fairly well with the experiments. The RANS case exhibit strong discrepancies with the experiments, especially on the vertical velocity component. A deeper comparison of the velocity ﬁeld is provided in Fig. 19, showing the three RMS velocity components on the symmetry plane at X/D = 1 and X/D = 8. The sharp maxima at X/D = 1 are not entirely recovered, which tends to indicate a lack of mesh reﬁnement in the recirculation region, explaining the discrepancies already observed on the velocity ﬁeld. At X/D = 8, the RMS velocity proﬁles are smooth and the inj c case shows very good agreement with experiments. 6.4. Wall temperature in the jet wake From the industrial point of view, the mean wall temperature prediction is of special concern to design thermal shields needed to protect the composite materials of the engine nacelle. The mean non-dimensional wall tempera T w −T∞ ture η = Tj −T∞ is shown in Fig. 20. The spanwise extent of the wall temperature distribution given by the inj c computation is in fair agreement with the experimental data. This is not the case for the RANS computation which exhibits a much narrower temperature distribution. It illustrates that the spanwise extent of the wall temperature distribution is controlled by the wake vortices. These structures are resolved by the WMLES simulation and not captured by the RANS approach, at least with the turbulence model used here. This is the main reason why usual one or two equation turbulence models such as the Spalart-Allmaras model cannot be trusted for the thermal shield design. 33

However, it appears that both the RANS and the inj c simulations overestimate the wall temperature in the region X/D < 4. From Fig. 20, it can be inferred that the brutal switch from the isothermal condition on the ejector grid to the adiabatic condition at X/D ≈ 0.4 is problematic. We propose the following explanations: • the overestimation of wall temperature in the RANS computation is due to the fact that the mesh is not reﬁned along the streamwise direction at the switch of boundary condition. Thus thermal diﬀusion is too small in these coarse cells; • the overestimation of wall temperature in the inj c simulation is even larger than in the RANS case. Indeed, not only the mesh in not reﬁned in the streamwise direction, but the WMLES method assumes a boundary layer at equilibrium, which is not the case at the switch of boundary condition. We arrive here at the limits of the WMLES approach and for accurately capturing this change in thermal boundary condition, wall-resolved LES with very small streamwise cell size would be mandatory near X/D = 0.4, that is a quasi-DNS mesh. We now focus on a more local investigation of the wall temperature shown in Fig. 21. It can be noted that error bars of ±3% are added to the experimental temperature due to an uncertainty in the post-processing. Fortunately, this uncertainty is too small to prevent from drawing conclusions on the numerical results accuracy. On the symmetry plane, the wall temperature overestimation noted above for the inj c simulation can be observed for 34

X/D < 4. For X/D > 4, the inj c case is close to the experiments despite a remaining slight overestimation, while the RANS temperature shows larger discrepancies with an almost horizontal asymptote. Considering the spanwise distribution of wall temperature at X/D = 1, X/D = 2.3 and X/D = 8, the RANS data show a strong underprediction of the wall temperature spanwise extent. On the contrary, the wall temperature spanwise distribution for the inj c case agrees very well with the experiments, conﬁrming that the WMLES approach is well suited for the simulation of JICF. 6.5. Spectral analysis in the jet wake and accuracy of the numerical method A deeper level of validation of the WMLES is provided by a spectral analysis of two velocity signals extracted in the inj c computation. The location of the two probes is given in Table 5 and corresponds to the recirculation region forming just downstream of the jet exhaust. X/D

Y /D

Z/D

y/D

Probe 1

1

0.8

0.5

0.438

Probe 2

0.5

0

0.134

0.1

Table 5: Probe location

After removing the ﬁrst 4.6 × 10−2 s, the velocity signals contain 810, 000 time steps for a duration of about 0.2s with a sampling frequency of

1 Δt

=

3.9M Hz. Welch’s method [33] is used to compute the pseudo spectral density (PSD) G(f ), splitting the signal in 12 blocks using the Hanning window function with 50% overlap.

35

Before computing the PSD spectra, we recall that the numerical scheme used is second order accurate in space. Thus, we estimate that about 25 cells are required to accurately capture a wavelength (see for example Fosso [34]). To take into account this property of the numerical scheme, we apply a ﬁrst order low-pass ﬁlter whose cutoﬀ is computed as fc = U /25Δ, where U is the mean local velocity magnitude and Δ = (ΔX ΔY ΔZ )1/3 is the characteristic cell dimension. Figure 22 shows the raw and the low-pass ﬁltered spectra of the streamwise velocity at probe 1 and of the spanwise velocity at probe 2. We observe that the amplitude of the main peak at St = 0.14 is systematically underestimated. This is not surprising because we saw in Fig. 19 that the RMS velocities are underestimated at X/D = 1, which roughly corresponds to the location of the probes. The secondary peaks present at St > 0.14 in the simulation are not observed in the experiments. We conclude that their presence is not physical and may be explained by numerical instabilities at high frequencies. To conﬁrm this hypothesis, we now consider the ﬁltered PSD. We recall that above the ﬁlter cutoﬀ, wavelengths are considered to be under-resolved by the numerical scheme. We observe that the ﬁltering alters a signiﬁcant range of frequencies and that the ﬁltered PSD better agrees with the experiments. This result indicates that the artiﬁcial dissipation used here does not damp all the numerical wiggles generated by the second order centered scheme. It advocates the use of a higher order scheme or at least of a better treatment of the badly resolved frequencies, for example by ﬁltering them out during the simulation. Surprisingly, despite the lack of mesh reﬁnement

36

in the recirculation region, it does not prevent the WMLES from accurately predicting the peak frequency at St = 0.14. This frequency corresponds to the meandering of the jet wake [3] shown in Fig. 15 (b). 7. Conclusions In this study, we use the WMLES approach to investigate a complex geometry large Reynolds number hot JICF over cold walls. To further reduce the computational cost, the WMLES domain is limited to a small region around the JICF thanks to a RANS-LES embedded strategy. At the inlet of the WMLES domain, a turbulent inﬂow generation combining the stochastic method of Sandham et al. and a dynamic forcing term (Spille-Kohoﬀ and Kaltenbach, Larauﬁe et al. ) is used. Despite the geometrical constraints and the presence of an adverse pressure gradient at the inlet, the inﬂow generation method is able to generate a fully turbulent boundary layer at a small distance of 12 inlet boundary layer thicknesses downstream of the inlet. The key result is that the wall-heat ﬂux distribution is accurately estimated on the ejector grid upstream of the jet. Without the dynamic forcing term, the inﬂow generation fails at injecting a turbulent boundary layer. In the jet wake, the WMLES approach provides an accurate description of the velocity and temperature ﬁelds. In particular, the wall temperature distribution is largely improved compared to a Spalart-Allmaras RANS simulation. A spectral analysis of two velocity signals shows that the jet meandering frequency is accurately estimated. However, the recirculation region forming downstream of the jet exhaust is not well captured due to a lack 37

of mesh reﬁnement and/or a too small order of accuracy of the numerical method. As a result, this study provides a method for accurate simulations of large Reynolds number JICF at a manageable computational cost. The main diﬃculty and original aspect of this work is to use a turbulent inﬂow generation in the framework of the WMLES approach. Inﬂow generation for WMLES is expected to widen the range of application of WMLES to a larger number of large-Reynolds number wall-bounded ﬂows. 8. Acknowledgements The authors gratefully acknowledge AIRBUS Operations SAS and Foundation for research in aeronautics and space (FNRAE) for their support. We are grateful to Dr. B. Duda for the mesh generation and his comments on a draft version of this work. We would like to thank ONERA for providing the experimental results used to assess the WMLES approach in the jet wake. We also thank Dr. B. Aupoix for fruitful discussions about the wall modelling approach and Pr. P. Sagaut for proposing the use of the dynamic forcing method.

38

References [1] L. Albugues, Analyse exp´erimentale et num´erique d’un jet d´ebouchant dans un ´ecoulement transverse, Ph.D. thesis, Ecole Nationale Sup´erieure de l’a´eronautique et de l’espace (2005). [2] J.-C. Jouhaud, L. Y. M. Gicquel, B. Enaux, Large-Eddy Simulation Modeling for Aerothermal Predictions Behind a Jet in Crossﬂow, AIAA journal 45 (10) (2007) 2438–2447. [3] B. Duda, Numerical Investigations on a Hot Jet in Cross Flow Using Scale-Resolving Simulations, Ph.D. thesis (2012). [4] Y. Hallez, J.-C. Jouhaud, T. Poinsot, On the relative impact of subgridscale modelling and conjugate heat transfer in LES of hot jets in crossﬂow over cold plates, International Journal For Numerical Methods In Fluids 00 (2010) 1–24. [5] H. Choi, P. Moin, Grid-point requirements for large eddy simulation: Chapman’s estimates revisited, Tech. rep., Center for Turbulence Research : Annual Research Briefs (2011). [6] J. Fr¨ohlich, D. von Terzi, Hybrid LES/RANS methods for the simulation of turbulent ﬂows, Progress in Aerospace Sciences 44 (2008) 349–377. [7] W. Cabot, P. Moin, Approximate Wall Boundary Conditions in the Large-Eddy Simulation of High Reynolds Number Flow, Flow, Turbulence and Combustion 63 (1-4) (2000) 269–291.

39

[8] U. Piomelli, E. Balaras, Wall-Layer Models for Large-Eddy Simulations, Annual Review of Fluid Mechanics 34 (2002) 349–374. [9] D. R. Chapman, Computational Aerodynamics Development and Outlook, AIAA journal 17 (12) (1979) 1293–1313. [10] L. Cambier, M. Gazaix, elsA: an eﬃcient object-oriented solution to CFD complexity, in: 40th AIAA Aerospace Science Meeting and Exhibit, Reno, AIAA 2002-0108, 2002, pp. 14–17. [11] H. Reichardt, Complete representation of the turbulent velocity distribution in smooth pipes, Z. Angew. Math. Mech 31 (1951) 208. [12] M. Wang, P. Moin, Dynamic wall modeling for large-eddy simulation of complex turbulent ﬂows, Physics of Fluids 14 (2002) 2043–2051. [13] M. Breuer, B. Kniazev, M. Abel, Development of wall models for LES of separated ﬂows using statistical evaluations, Computers and Fluids 36 (2007) 817–837. [14] C. Duprat, G. Balarac, O. M´etais, P. M. Congedo, O. Brugi`ere, A walllayer model for large-eddy simulations of turbulent ﬂows with/out pressure gradient, Physics of Fluids 23 (2011) 015101. [15] S. Bocquet, P. Sagaut, J.-C. Jouhaud, A compressible wall model for large-eddy simulation with application to prediction of aerothermal quantities, Physics of Fluids 24 (2012) 065103. [16] S. Deck, P. E. Weiss, M. Pami`es, E. Garnier, Zonal Detached Eddy

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Simulation of a spatially developing ﬂat plate turbulent boundary layer, Computers & Fluids 48 (1) (2011) 1–15. [17] R. Larauﬁe, S. Deck, P. Sagaut, A dynamic forcing method for unsteady turbulent inﬂow conditions, Journal of Computational Physics 230 (2011) 8647–8663. [18] J. Schl¨ uter, H. Pitsch, P. Moin, S. Shankaran, S. Kim, J. Alonso, Towards Multi-component Analysis of Gas Turbines by CFD : Integration of RANS and LES Flow Solvers, in: ASME TURBO EXPO, 2003. [19] T. S. Lund, X. Wu, S. K.D., Generation of turbulent inﬂow data for spatially-developing boundary layer simulations, Journal of Computational Physics 140 (1998) 233–258. [20] M. Pami`es, P.-E. Weiss, E. Garnier, S. Deck, P. Sagaut, Generation of synthetic turbulent inﬂow data for large eddy simulation of spatially evolving wall-bounded ﬂows, Physics of Fluids 21 (2009) 045103. [21] N. D. Sandham, Y. F. Yao, A. A. Lawal, Generation of synthetic turbulent inﬂow data for large eddy simulation of spatially evolving wallbounded ﬂows, International Journal of Heat and Fluid Flow 24 (2003) 584–595. [22] A. Spille-Kohoﬀ, H.-J. Kaltenbach, Generation of turbulent inﬂow data with a prescribed shear-stress proﬁle, in: Third AFSOR Conference on DNS and LES, 2001. [23] T. J. Poinsot, S. K. Lele, Boundary Conditions for Direct Simulations of 41

Compressible Viscous Flows, Journal of computational physics 101 (1) (1992) 104–129. [24] A. Fosso Pouange, H. Deniau, N. Lamarque, T. Poinsot, Comparison of outﬂow boundary conditions for subsonic aeroacoustic simulations, International Journal for Numerical Methods in Fluids 68, 10 (2012) 1207–1233. [25] A. Chatelain, Simulation des Grandes Echelles d’´ecoulements turbulents avec transferts de chaleur, Ph.D. thesis, CEA Grenoble (2004). [26] F. Nicoud, F. Ducros, Subgrid-scale stress modelling based on the square of the velocity gradient, Flow, Turbulence and Combustion 62(3) (1999) 183–200. [27] A. Jameson, Analysis and design of numerical schemes for gas dynamics, 1: Artiﬁcial diﬀusion, Upwind biasing, Limiters and their eﬀect on accuracy and multigrid convergence, Computational Fluid Dynamics 4 (1995) 171–218. [28] A. Jameson, Analysis and design of numerical schemes for gas dynamics, 2:Artiﬁcial diﬀusion and Discrete shock structure, Computational Fluid Dynamics 5 (1995) 1–38. [29] I. Mary, P. Sagaut, Large Eddy Simulation of Flow around an Airfoil Near Stall, AIAA journal 40 (6) (2002) 1139–1145. [30] B. Raverdy, I. Mary, P. Sagaut, High-Resolution Large-Eddy Simulation of Flow Around Low-Pressure Turbine Blade, AIAA 41, 3 (2003) 390– 397. 42

[31] J. Jim´enez, Cascades in Wall-Bounded Turbulence, Annual Review of Fluid Mechanics 44 (2011) 27–45. [32] J. Andreopoulos, On the structure of jets in a crossﬂow, Journal of Fluid Mechanics 157 (1985) 163–197. [33] P. D. Welch, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging over Short, Modiﬁed Periodograms, IEEE Transactions on Audio and Electroacoutics Au-15 (1967) 70–73. [34] A. Fosso Pouange, Sch´emas Volumes Finis pr´ecis :

application a`

l’a´eroacoustique num´erique de jets subsoniques, Ph.D. thesis, Sciences m´ecaniques, acoustique et ´electronique de Paris (2011). [35] C. Wilcox, D., Turbulence Modeling for CFD, 2nd Edition, DCW Industries, Inc., La Ca˜ nada, CA, 2006.

9. Appendix 9.1. Procedure used to inject the velocity ﬂuctuations These inﬂow velocity ﬂuctuations are injected according to the following steps. At each solver iteration : 1. compute velocity ﬂuctuations in boundary layer frame according to Eq. 6, Eq. 4 and Eq. 5 ; 2. change of frame of reference to compute velocity ﬂuctuations in the global frame (X, Y, Z). They are noted Uj , Vj and Wj ; 43

3. compute the ﬂuctuating conservative state from the velocity ﬂuctuations. This step depends on the type of inlet boundary condition chosen. In our case, it is a characteristic boundary condition imposing the total pressure, total temperature and velocity direction. We choose to add the velocity ﬂuctuations so that the total temperature is unchanged :

(ρU ) = ρ (ρV ) = ρ

Uj

j

Vj

(ρW ) = ρ (ρE) =

j

j

Wj

1 γ−1 {[(ρU 2 γρ

+ (ρU ) )2 + (ρV + (ρV ) )2 + (ρW + (ρW ) )2 ]

− [(ρU )2 + (ρV )2 + (ρW )2 ]} (18) 4. compute the perturbed conservative state ()p on the inlet by adding the ﬂuctuating conservative state to the inlet conservative state, ρ being unchanged : (ρU )p = ρU + (ρU ) (ρV )p = ρV + (ρV ) (ρW )p = ρW + (ρW )

(19)

(ρE)p = ρE + (ρE) 9.2. Procedure used to apply the dynamic forcing method The procedure used by Larauﬁe et al. to compute the source term is given in the following :

44

1. interpolate on the LES mesh the turbulent viscosity μt and the modulus of vorticity ||ω|| computed from the RANS solution ; 2. compute the target wall-normal RMS with the following equations, which involve assumptions that are reasonable for a zero pressure gradient boundary layer [35] : u v = μt ∗ ||ω|| k = u v /0.3

(20)

v 2 target = 49 k where k designates the turbulent kinetic energy, 3. compute a low-pass ﬁltered value of the streamwise (u) and wallnormal (v) LES velocity in the boundary layer frame. . is deﬁned as an exponential time ﬁlter : 1 f = Δf

t

exp( −∞

τ −t )f (τ )dτ Δf

(21)

In practise, it is convenient to use the following recurcive relation, obtained by considering the time derivative of Eq. 21 and approximating the time derivative of f by a ﬁrst order ﬁnite diﬀerence. We obtain : f (t + Δt ) =

Δt Δt f (t) + (1 − ) f (t) Δf Δf

(22)

where Δt is the simulation time step, 4. compute the streamwise and wall-normal mean squared LES velocity : u2 = (u − u)2 v 2 = (v − v)2 45

(23)

5. compute the source term fy from the error between the targeted and the LES wall-normal mean squared velocity :

e = ρ(v 2 target − v 2 ) r = αe 2 2 fy = r(v − v) if u2 < 0.36U∞ and v 2 < 0.16U∞

(24)

fy = 0 otherwise It can be remarked that fy is non-zero only if the streamwise and wallnormal mean squared LES velocity take realistic values, 6. compute the source term fy in the global frame (X, Y, Z) to obtain (fX , fY , fZ ), 7. add (fX , fY , fZ ) to the right hand side of the respective momentum equations. 9.3. Choice of Sandham mode parameters The procedure used to choose the parameters deﬁning the three outer Sandham modes is described. We start from the observation that turbulent structures in a WMLES are not physical in the ﬁrst few cell layers above the wall. Indeed artiﬁcial streaks develop in the inner boundary layer region (see for example the review by Piomelli [8]) and are strongly dependent on the numerical method. Thus, we consider that optimal parameters for inﬂow turbulent generation in a WMLES should be based on numerical arguments.

To facilitate the choice of parameters for the Sandham modes, and as done in the original paper by Sandham et al. , the ﬁrst and third modes are 46

deﬁned from the second mode according to Table. 1. Thus we only have to determine the spatial and temporal periodicities of the second mode β2 and ω2 , the wall distance at which the second mode is injected yp,2 and the mode amplitudes c1j and c2j , which are chosen equal for the three modes. These parameters are determined as follows: • the spatial wavenumber β2 along the spanwise direction z is obtained considering that, with the numerical method used in the WMLES, at least 8 consecutive cells are necessary to resolve a turbulent scale. Thus the smallest injected wavelength should be equal to 8Δz . According to Table. 1, the smallest injected wavelength (or larger wavenumber βj ) is the one of mode one: β1 = 1.33β2 . Thus we set 1.33β2 =

2π ; 8Δz

• the mode frequency ω2 is determined from a longitudinal wavelength and a convection velocity: ω2 =

2π . λx /Uc

Following the previous reason-

ing, λx is set to 8Δx , and Uc is set to U∞ , corresponding to the most critical case (largest mode frequency); • the mode amplitudes c1j and c2j and wall distance injection yp,2 are determined from RMS velocity proﬁles obtained from a WMLES of a periodic turbulent channel (not presented here) performed with the same ﬂow solver and numerical method used here. The mode as amplitudes √ √ 2 2 are thus deﬁned as c1j = 2 and c2j = 2 , u v max max

2 is maximum. and yp,2 is deﬁned as the wall distance at which u Note that the phase shift between modes is equal to the original values by Sandham et al. .

47

9.4. Computation of boundary layer thickness The common deﬁnition of boundary layer thickness is the wall distance δ at which u = 0.99U∞ . This criterion is found to lack robustness especially for complex conﬁguration such as this JICF, where the boundary layer upstream of the jet develops over a curved wall and experience an adverse pressure gradient due to the jet exhaust. As a result we look for a more robust criterion for determining boundary layer thickness. We choose the following : τ (δ) = τmax

(25)

where τ is the total shear, τmax is the maximum value of the total shear and a small number chosen here as = 0.01. In other words, with this criterion, the boundary layer thickness is the wall distance at which the total shear becomes small enough compared to the maximum value of the total shear. U . For a WMLES computation, For a RANS computation, τ = (μ + μt ) ∂∂y U − ρu v . or more generally for an LES, τ = (μ + μsgs ) ∂∂y

48

List of Figures 1

(a) : hot air exhaust from the anti-icing system on a A380 engine. (b) : the diﬀerent components of the anti-icing system. 52

2

(a) : global view of the wind-tunnel model. (b) : side-view of the model. The origin O of the global frame of reference (X, Y, Z) is located at the middle of the exhaust hole downstream edge. (c) : exploded view of the model. The measurement plate is used for infrared wall temperature measurements. 53

3

Two-step RANS-LES embedded approach : a RANS computation is ﬁrst performed in the whole domain. Then a WMLES is performed in the embedded domain, using the RANS solution as initial and boundary conditions. . . . . . . . . . . . . . 54

4

Description of the wall modelling approach: (a) Velocity proﬁle discretization by the WMLES approach compared to a wall-resolved LES. The dotted arrow is the slope that would be obtained without a wall model. (b) Resolved turbulent structures : WMLES does not resolve the structures in the viscous sublayer and buﬀer layer. . . . . . . . . . . . . . . . . 54

5

Overview of the embedded WMLES mesh. The geometry of the injection pipes and plenum is faithfully reproduced. . . . . 55

6

Geometrical constraints on inﬂow generation : the small distance Xexhaust −Xba involves a thin boundary layer at the inlet boundary Xin , whose meshing is diﬃcult to conciliate with the WMLES meshing requirements. . . . . . . . . . . . . . . . . . 56

49

7

Comparison of the WMLES and RANS meshes near the inlet boundary. The vertical plane corresponds to the symmetry plane and the horizontal surface corresponds to the airfoil. The boundary layer thickness δ at the inlet is shown for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8

p w −p∞ 2 (a) and non-dimensional pressure 1/2ρ∞ U∞ μw ∂p w (b) on the symmetry plane. The ρw u3τ ∂x

Wall pressure Cp = gradient p+ x =

dashed horizontal line on (b) corresponds to p+ x = 0.005. The shaded zone indicates the region where the dynamic forcing term is used. Hot jet exhaust starts at X − Xin = 27δin (end of the x-axis n the plot). . . . . . . . . . . . . . . . . . . . . . 58 9

Streamwise component of the mean wall friction CfX = τw · X on the symmetry plane. The vertical dotted lines indicate the locations of the four stations (X −Xin )/δin = (11; 14; 17; 20). 59

10

Mean streamwise velocity proﬁles on the symmetry plane.

. . 59

11

Non-dimensional velocity proﬁles in wall units on the symmetry plane, shifted by 10 units for clarity. . . . . . . . . . . . . 60

12

streamwise, wall-normal and spanwise RMS velocity, respec-

13

tively shifted by 0.2, 0.05 and 0.1 units for clarity. . . . . . . . 61 w φ Nusselt number N u = on the ejector grid. . . 62 λ∞ (Tj − T∞ )/D Some of the coherent structures of a JICF. The core of the

14

wake is composed of two counter-rotating vortices. Hairpin vortices are supposed to be generated in the shear layer at the jet-cross ﬂow interface. Periodic wake vortices are shed as the cross ﬂow passes around the jet. . . . . . . . . . . . . . . . . . 63 50

15

(a) : instantaneous temperature η = (T − T∞ )/(Tj − T∞ ) on the symmetry plane. Hairpin vortices are visible. (b) : instantaneous wall temperature, showing the jet meandering

17

due to the wake vortices. . . . . . . . . . . . . . . . . . . . . . 64

on the symmetry plane. . . . . 65 Mean streamwise velocity U

at X/D = 1 and X/D = 8. . . 66 Mean streamwise velocity U

18

Mean streamwise and spanwise velocity at X/D = 1 and

16

X/D = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 19

RMS velocity at X/D = 1 et X/D = 8. Error bars correspond to the uncertainty in experimental measurement estimated at

20 21

±5% (see Albugues [1] p.96). . . . . . . . . . . . . . Tw − T∞ Non-dimensional wall temperature η = . T j − T∞ Tw − T∞ Non-dimensional wall temperature η = Tj − T∞ 0 (symmetry plane), X/D = 1, X/D = 2, 3 and

. . . . . . 68 . . . . . . 69 at Y = X/D =

8. Error bars correspond to the uncertainty in experimental measurement estimated at ±3%. . . . . . . . . . . . . . . . . . 70 22

Raw and low-pass ﬁltered velocity spectra. Vertical lines correspond to the ﬁlter cutoﬀ frequency. . . . . . . . . . . . . . . 71

51

(a)

(b)

Fig. 1: (a) : hot air exhaust from the anti-icing system on a A380 engine. (b) : the diﬀerent components of the anti-icing system.

52

Fig. 2: (a) : global view of the wind-tunnel model. (b) : side-view of the model. The origin O of the global frame of reference (X, Y, Z) is located at the middle of the exhaust hole downstream edge. (c) : exploded view of the model. The measurement plate is used for infrared wall temperature measurements.

53

Fig. 3: Two-step RANS-LES embedded approach : a RANS computation is ﬁrst performed in the whole domain. Then a WMLES is performed in the embedded domain, using the RANS solution as initial and boundary conditions.

Fig. 4: Description of the wall modelling approach: (a) Velocity proﬁle discretization by the WMLES approach compared to a wall-resolved LES. The dotted arrow is the slope that would be obtained without a wall model. (b) Resolved turbulent structures : WMLES does not resolve the structures in the viscous sublayer and buﬀer layer.

54

Fig. 5: Overview of the embedded WMLES mesh. The geometry of the injection pipes and plenum is faithfully reproduced.

55

Fig. 6: Geometrical constraints on inﬂow generation : the small distance Xexhaust − Xba involves a thin boundary layer at the inlet boundary Xin , whose meshing is diﬃcult to conciliate with the WMLES meshing requirements.

56

WMLES

RANS

Fig. 7: Comparison of the WMLES and RANS meshes near the inlet boundary. The vertical plane corresponds to the symmetry plane and the horizontal surface corresponds to the airfoil. The boundary layer thickness δ at the inlet is shown for comparison.

57

(a)

Fig. 8: Wall pressure Cp = p+ x =

μw ∂p w ρw u3τ ∂x

(b)

p w −p∞ 2 1/2ρ∞ U∞

(a) and non-dimensional pressure gradient

(b) on the symmetry plane. The dashed horizontal line on (b)

corresponds to p+ x = 0.005. The shaded zone indicates the region where the dynamic forcing term is used. Hot jet exhaust starts at X − Xin = 27δin (end of the x-axis n the plot).

58

Fig. 9: Streamwise component of the mean wall friction CfX = τw · X on the symmetry plane. The vertical dotted lines indicate the locations of the four stations (X − Xin )/δin = (11; 14; 17; 20).

Fig. 10: Mean streamwise velocity proﬁles on the symmetry plane.

59

Fig. 11: Non-dimensional velocity proﬁles in wall units on the symmetry plane, shifted by 10 units for clarity.

60

Fig. 12: streamwise, wall-normal and spanwise RMS velocity, respectively shifted by 0.2, 0.05 and 0.1 units for clarity.

61

RANS

inj c

inj

w/o inj

Fig. 13: Nusselt number N u =

φ w

λ∞ (Tj − T∞ )/D

62

on the ejector grid.

Fig. 14: Some of the coherent structures of a JICF. The core of the wake is composed of two counter-rotating vortices. Hairpin vortices are supposed to be generated in the shear layer at the jet-cross ﬂow interface. Periodic wake vortices are shed as the cross ﬂow passes around the jet.

63

(a)

(b)

Fig. 15: (a) : instantaneous temperature η = (T−T∞ )/(Tj −T∞ ) on the symmetry plane. Hairpin vortices are visible. (b) : instantaneous wall temperature, showing the jet meandering due to the wake vortices.

64

inj c

expe

RANS

on the symmetry plane. Fig. 16: Mean streamwise velocity U

65

X/D = 1 : inj c

X/D = 8 : inj c

X/D = 1 : expe

X/D = 8 : expe

X/D = 1 : RANS

X/D = 8 : RANS

at X/D = 1 and X/D = 8. Fig. 17: Mean streamwise velocity U

66

X/D = 1

X/D = 8

Fig. 18: Mean streamwise and spanwise velocity at X/D = 1 and X/D = 8.

67

X/D = 1

X/D = 8

Fig. 19: RMS velocity at X/D = 1 et X/D = 8. Error bars correspond to the uncertainty in experimental measurement estimated at ±5% (see Albugues [1] p.96).

68

inj c

expe

RANS

Fig. 20: Non-dimensional wall temperature η =

69

Tw − T∞ . Tj − T∞

Y =0

X/D = 1

X/D = 2, 3

X/D = 8

Fig. 21: Non-dimensional wall temperature η = plane), X/D = 1, X/D = 2, 3 and X/D = 8.

Tw − T∞

at Y = 0 (symmetry Tj − T∞ Error bars correspond to the

uncertainty in experimental measurement estimated at ±3%.

70

Probe 1

Probe 2

Fig. 22: Raw and low-pass ﬁltered velocity spectra. Vertical lines correspond to the ﬁlter cutoﬀ frequency.

71

(IGHLIGHTS

x x x x x

Wall temperature distribution downstream a hot jet-in-cross-flow is crucial in aeronautics We simulate the real geometry of a wind tunnel model of hot-jet-in-cross-flow Wall-Modelled Large-Eddy Simulation with turbulent inflow generation is used Only an inflow generation with a dynamic forcing term leads to a realistic flow upstream of the jet The wall temperature in the jet wake compares well with experimental measurements