Wall-Modelled Large-Eddy Simulation of a hot Jet-In-Cross-Flow with turbulent inflow generation

Wall-Modelled Large-Eddy Simulation of a hot Jet-In-Cross-Flow with turbulent inflow generation

Accepted Manuscript Wall-Modelled Large-Eddy Simulation of a hot Jet-In-Cross-Flow with turbulent inflow generation S. Bocquet, J.-C. Jouhaud, H. Deni...

2MB Sizes 0 Downloads 11 Views

Accepted Manuscript Wall-Modelled Large-Eddy Simulation of a hot Jet-In-Cross-Flow with turbulent inflow generation S. Bocquet, J.-C. Jouhaud, H. Deniau, J.-F. Boussuge, M.J. Estève PII: DOI: Reference:

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

To appear in:

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

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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

40

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