Investigation of turbulence development in incompressible jets with zonal detached eddy simulation (ZDES) and synthetic turbulent inflow

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Investigation of turbulence development in incompressible jets with zonal detached eddy simulation (ZDES) and synthetic turbulent inflow F. Gand1,∗ ONERA – The French Aerospace Lab, F-92190 Meudon

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 25 February 2016 Revised 30 April 2016 Accepted 5 June 2016 Available online xxx

Hybrid RANS/LES simulations of two incompressible jets are performed with the Zonal Detached Eddy Simulation (ZDES). Two functioning modes of the ZDES for the selection of RANS and DES areas are evaluated, namely the user-defined mode (mode 1) and the global- or automatic- mode (mode 2). The RANS-to-LES transition occurs quickly downstream of the nozzle exit and is found to involve the same physics as a laminar to turbulent transition with vortex pairing near the nozzle exit. The effect of the delay in the RANS-to-LES transition on the jet flow development is analyzed. In particular, the delay in the formation of small-scale turbulent structures results in too high turbulence levels in the mixing layers. Furthermore, it is shown, for two cases, that the injection of synthetic turbulence at the nozzle inlet, originally targeted at reproducing the experimental turbulence level in the jet core, has a significant impact on the mixing layer as it accelerates the RANS-to-LES transition, reduces the spatial wavelength of the vortex pairing and promotes the production of fine-scale turbulence which leads to a better agreement with experiments. © 2016 Elsevier Inc. All rights reserved.

Keywords: CFD RANS/LES Jets Turbulence

1. Introduction Hybrid RANS/LES simulations of detached and free shear flows are increasingly used for industrial purposes as they present an acceptable trade-off between computational cost and physical accuracy (Deck et al., 2014, Spalart et al., 1997). Within the range of applications of interest, jet flows are very challenging since they involve several multidisciplinary aspects. Hybrid simulations, and more generally eddy-resolving ones, have therefore been widely used for instance for jets aerodynamics (Verrière et al., 2016, DeBonis, 2010, Mahak et al., 2014), thermal problems (Brunet, 2012, Zuckerman and Lior, 2006), and acoustics (Bogey et al., 2012, Xia et al., 2012, Eastwood et al., 2012, Shur et al., 2006, Tyacke et al., 2016). The major issue for hybrid simulations of such flows is that the most commonly used approaches rely on the natural flow instabilities to trigger the development of the resolved turbulence in shear layers (from RANS in attached areas to LES in detached ones). This is the basic principle of the Detached Eddy Simulation (DES) (Spalart et al., 1997) and its subsequent variations and improvements (Spalart et al., 2006, Shur et al., 2008, Shur et al., 2015, Deck, 2012), which have proved to be very efficient for massively separated flows where strong instabilities occur (Deck et al., 2014). ∗

1

Tel.: +33146734192. E-mail address: [email protected] Research scientist, Applied Aerodynamics Department.

However, jet flows are not governed by large-scale instabilities; therefore the RANS-to-LES transition can suffer from a delay which is sometimes referred to as the “grey area” issue. The most obvious technique to suppress the delay would be to resolve the attached boundary layer inside the nozzle, at least partly using Wall Modeled LES. This is done for instance for an axisymmetric jet in Brès et al. (2015) and shows very good results but does not seem applicable nor robust enough for complex geometries due to computational costs and implementation issues of turbulence generation methods for boundary layers on curvilinear grids. In the framework of global RANS/LES approaches (global in the sense that the model decides whether the local resolution should be RANS or LES, which is the most relevant approach for industrial applications) several proposals have been made to mitigate the RANS-to-LES transition length. Significant improvements have been achieved by optimizing the hybrid length scale so that the RANS eddy viscosity is removed in LES regions to avoid dampening the instabilities without compromising the safe treatment of the attached boundary layers (Shur et al., 2015, Deck, 2012, Kok and van der Ven, 2010, Kok and van der Ven, 2012). Another option is to improve the subgrid scale model in the LES areas, which has been tested in Mockett et al. (2015). The RANS-to-LES transition in jet flows simulations is all the more critical since the jet flow development and radiated noise are strongly influenced by the initial state of the mixing layer. This topic has been widely investigated, both experimentally and

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inflow conditions on the turbulence development in mixing layers within the framework of ZDES simulations of jets. The article is organized as follows. First, the numerical methods for turbulence modeling and freestream turbulence generation are presented. Two configurations of axisymmetric incompressible jets are then investigated: the first one with a thin initial boundary layer, the second one originating from a fully developed channel flow.

Nomenclature D

Nozzle exit diameter Boundary layer thickness Momentum thickness Length scale based on the vorticity Length scale based on the maximum grid spacing

δ θ ω max

2. Numerical methods

numerically. Initially laminar or transitional mixing layers are characterized by vortex parings at a reduced frequency based on the initial momentum thickness and jet velocity Stθ = 0.012 (“shear layer mode”) (Bogey and Bailly, 2010, Bogey et al., 2012, Kim and Choi, 2009, Zaman and Hussain, 1980, Zaman, 1985). These instabilities have significant consequences on the radiated noise and its spectral distribution (Bogey et al., 2012, Zaman, 1985). Besides, when the initial boundary layer is fully turbulent, the potential core is longer (Hussain and Zedan, 1978, Bogey et al., 2012). Another issue for jets simulations is the reproduction of inflow conditions at the nozzle inlet. In a steady framework, mass flow, total pressure and temperature have been shown to have a strong influence on the jet structure (Verrière et al., 2014) and are carefully controlled both experimentally and numerically. To go further into the accurate reproduction of flight and wind-tunnel conditions, the question of the turbulent rate in the jets core arises all the more since unsteady simulations are more and more used. To this effect, one can take advantage of turbulent inflow conditions which have been mostly developed in the framework on LES initialization for wall bounded flows, for instance (Smirnov et al., 2001, Jarrin et al., 2009). Besides, it seems that the residual turbulence in jets core is less challenging to generate than wall turbulence since it involves less anisotropy and turbulent scales variety. Some applications of the use of synthetic turbulent boundary conditions to reproduce jets core turbulent rate have shown promising results (Kim and Choi, 2009, Brunet, 2012) (Gand et al., 2015). In the same spirit, some work has been done to add upstream turbulence coming from the fan in jets simulations (Tyacke et al., 2016). Of interest, it is observed in these applications that the introduction of freestream turbulence in the jets core not only improves the overall realism of the computations but also has a significant impact on the RANS to LES transition in the mixing layer. The objective of this article is therefore to investigate the effect of adding freestream turbulence targeted at reproducing realistic

2.1. Zonal detached eddy simulation The approach used in this work is the Zonal Detached Eddy Simulation (ZDES) (Deck, 2012) developed at ONERA. As mentioned in the introduction, this approach has been used with success to simulate a wide range of applications of industrial interest (Deck et al., 2014). One of the advantages of the ZDES is its flexibility illustrated in Fig. 1, which shows that ZDES covers several types of detached flows, which can be combined. In the present study, modes 1 and 2 of the ZDES are used. Mode 3, devoted to WMLES, is beyond the scope of this work. An example of the combined use of the three modes of ZDES within the same computation can be found in Deck and Laraufie (2013). The ZDES is based on the basic idea of the original Detached Eddy Simulation (Spalart et al., 1997) (DES97) which relies on the Spalart–Allmaras (SA) RANS model (Spalart and Allmaras, 1994). A brief description of the SA model is provided below, the reader is referred to the original paper (Spalart and Allmaras, 1994) for a full description. The SA model is based on the transport equation of a pseudo viscosity v˜ involving production, diffusion and destruction terms:



2

Dv˜ 1 v˜ 2 = cb1 S˜v˜ + [∇ .( (v + v˜ )∇ v˜ ) + cb2 (∇ v˜ ) ] −cw1 fw    σ Dt d w   production



di f f usion



 (1)



dest ruct ion

where, dw is the distance to the wall, cb1 , cb2 and cw1 are constants, S˜ is a modified vorticity magnitude involving a near wall function fv2 and fw is another near-wall function. The eddy viscosity entering the Boussinesq closure for the RANS equations is defined using a third near-wall function fv1 : νt = fv1 ν˜ . The three near-wall corrections fw , fv1 and fv2 were calibrated to ensure the correct behavior of ν˜ in the viscous, buffer, log-layer and outer parts of the boundary layer. The basic principle of DES97 is to

Fig. 1. Summary of the ZDES formulation (adapted from Deck et al., 2014).

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modify the destruction term of Eq. (1) so that the RANS model is reduced to a LES subgrid scale one in detached areas. To do so, the distance to the wall in Eq. (1) is replaced by dˇDES97 = min(dw , CDES ) with CDES = 0.65 and  = ࢞max is the maximum cell size. With this modification, away from the walls, when production and destruction terms are balanced, the eddy viscosity scales with the local mesh size and the local vorticity modulus: ν t ≈ S2 which is similar to Smagorinsky’s subgrid scale model. The ZDES mode 1, introduced to treat massive separations triggered by the geometry, relies on a user-defined zonal decomposition of the computational domain in RANS and DES areas. The I hybrid length scale d˜ZDES entering the SA model (Eq. (2)) model is equal to:



I d˜ZDES =

dw in RANS areas ˜I min(dw , CDES  ZDES )in DES areas

(3)

where dw is the wall distance; CDES is the original DES97 (Spalart 1/3 or  ˜I ˜I et al., 1997) constant and  ZDES = vol = (xyz ) ZDES =

ω =



Sω , where Sω is the average cross section of the cell normal to the vorticity vector ω. In the present study, the subgrid length scale based on the local vorticity vector direction ω (Chauvet et al., 2007) was chosen since there is a preferred mean vorticity direction in both applications. Therefore, a significant difference between ZDES mode 1 and DES97 is the definition of the length scale used in DES areas. Furthermore, the near-wall functions of the SA model are removed in LES areas for mode 1: fv1 = 1, fv2 = 0, fw = 0. The ZDES mode 2 was developed to deal with separations over smooth surfaces and works as a global approach where the model itself defines the RANS and DES areas, which makes the treatment of complex geometries straightforward for the user. To ensure that attached boundary layers are treated in RANS whatever the grid density, a shielding function fd is used. This function was defined in the framework of Delayed-DES (DDES) (Spalart et al., 2006) to be equal to 0 in the boundary layer and 1 elsewhere:

fd = 1 − tanh[(8rd ) ], rd = 3

ν + νt 2 Ui, jUi, j κ 2 dw



(4)

where ν is the kinematic viscosity, Ui,j is the velocity gradient and κ is the Karman constant. The hybrid length scale for ZDES mode 2 then reads: II ˜ II ) d˜ZDES = dw − fd max(0, dw − CDES  ZDES

(2)

II which ensures that d˜ZDES = dw in attached boundary layers where fd = 0, i.e., the SA model is forced in these areas. Conversely to DDES, the mode 2 of ZDES provides a specific definition of the subgrid length scale according to the flow resolution and a threshold ˜ II value fd0 = 0.8 determined in Deck (2012). When fd < fd0 ,  = ZDES max (max being the characteristic mesh length necessary to ensure the correct behavior of fd ). When fd > fd0 , the subgrid length ˜ II scale revolves to = ω or vol . As explained in Deck (2012), ZDES the near-wall functions of the SA model are not modified in mode 2 of ZDES.

2.2. Synthetic eddy method To generate realistic turbulent inflow content, the Synthetic Eddy Method (SEM) introduced by Jarrin et al. (2009) is used in this work. The SEM is based on the generation of velocity fluctuations carried by synthetic eddies which are superimposed to a target mean flow to obtain a synthetic unsteady turbulent field used to feed an inlet boundary condition. The formulation relies on the prescription by the user of a target Reynolds stress tensor. In the present study, the Reynolds stress tensor is either inferred from a target turbulent rate (test case 1) or reconstructed from experimental data (test case 2). Please

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The length scale of turbulence σ has also to be prescribed. Since the SEM is used to generate freestream turbulence in this work, the length scale is explicitly given by the user (whereas for wall turbulence, the setup of the length scale is a key issue and should at least depend on the wall distance). In this work, the synthetic inflow condition is implemented in a python module which is coupled with the ONERA elsA software (Cambier et al., 2013) using the external coupling feature for boundary conditions. Preliminary tests and verifications of the implementation of the SEM in this module can be found in Gand et al. (2015). 3. Jet originating from a thin boundary layer 3.1. Test case and simulations parameters The round jet investigated is similar to one investigated experimentally in Davoust et al. (2012). The Reynolds number based on the nozzle exit velocity U0 = 20 m/s and diameter D = 0.15 m is equal to ReD = 2.1105 . During the experiments, the boundary layer inside the nozzle was tripped using sand paper so that the exit boundary layer is fully turbulent, of thickness δ 0 = 3 mm = 0.02 D and momentum thickness θ 0 = 0.36 mm = 0.0024 D. A sketch of the computational domain is presented in Fig. 2(a). In the inset of Fig. 2(a), the colors correspond to the boundary conditions being applied. The blue part of the inner wall of the nozzle corresponds to the location of the no-slip condition used to obtain in the simulations the boundary layer thickness measured at the nozzle exit. 40 grid points are used to discretize the nozzle boundary layer, 360 points are used in the azimuthal direction and the mesh contains 40.106 cells in total. The grid sizes along the lip line are given in Fig. 2(b). The simulations have been performed with the elsA software developed at ONERA (Cambier et al., 2013), which solves the compressible Navier–Stokes equations on structured multiblock meshes. The time integration is performed using an implicit LU-SSOR algorithm and a second-order accurate backward Gear scheme. The number of sub-iterations is set to reach a decrease of one order of magnitude of the inner iteration residuals to achieve second-order time accuracy. For the spatial integration, the diffusive fluxes are discretized using a second-order-accurate centered scheme. The convective terms are treated with the hybrid centered/upwind second-orderaccurate AUSM+P scheme (Mary and Sagaut, 2002) using MUSCL extrapolation of the third order. This version of the AUSM scheme involves a “wiggle” sensor to minimize numerical viscosity by applying some upwinding only in areas where the solution displays numerical oscillations, while the scheme is actually centered everywhere else (Mary and Sagaut, 2002). The AUSM+P scheme is well suited to the low-speed applications presented in this work since its dissipation is proportional to the fluid velocity. Six RANS/LES simulations are presented in this section as summarized in Table 1. Simulations ZDES1 and ZDES2 allow to compare the user-defined and global functioning modes of the ZDES, both use the hybrid length scale based on the vorticy ω . The results of a simulation using DDES (Spalart et al., 2006) are also presented in order to emphasize the critical impact of the length scale in LES areas. The use of the maximum grid spacing max as the filter width in DDES is actually known to lead to poor results in free shear flow cases (Deck, 2012, Shur et al., 2015, Mockett et al., 2015), which is further illustrated in the present paper. The fourth simulation, ZDES2-SEM, is identical to the ZDES2 case except for the inflow boundary condition which is carried out with the SEM, using a target turbulence rate of Tu = 7.5% at the inlet plane and a synthetic eddy length scale σ = 0.02 m. Due to the lack of

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Fig. 2. R4Ch test case computational setup. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 1 Simulations of the R4Ch jet test case. Pt and Tt are total pressure and temperature. Case

RANS/LES approach

Inflow condition

ZDES1 ZDES2 DDES ZDES2-SEM ZDES2-SEM2 ZDES2-SEM3

ZDES (nozzle: mode 0/jet: mode 1 ω ) ZDES (all domains: mode 2 ω ) DDES ZDES (all domains: mode 2 ω ) ZDES (all domains: mode 2 ω ) ZDES (all domains: mode 2 ω )

Steady uniform Pt, Tt Steady uniform Pt, Tt Steady uniform Pt, Tt SEM Jarrin σ = 0.02 Tu = 7.5% SEM Jarrin σ = 0.02 Tu = 35% SEM Jarrin σ = 0.05 Tu = 7.5%

experimental data for this latter parameter, this value corresponds to the typical size of a turbulence generating grid that could have been used in the experiments). The target Reynolds stress tensor required for the SEM method is computed by ui u j = T u2 × U02 × δi j (where U0 is the inlet mean velocity and δ ij is the Kronecker delta). These parameters were adjusted during the transient stage of the simulation to reproduce the turbulent rate measured in the experiment in the nozzle exit plane (SEM is applied at the nozzle inlet, see Fig. 2(a)). Actually, the effect of the SEM length scale and turbulence rate is studied with two additional simulations ZDES2SEM2 and ZDES2-SEM3 (see Table 1) whose results are reported in Section 3.3. For all simulations, the time step is 2 × 10−6 s so that the convective CFL number based on the maximal acoustic velocity is lower than 15 in the mixing layer, which is the value suggested in Daude et al. (2006) that has been used with success in several previous studies involving ZDES simulations (Deck et al., 2014, Deck and Laraufie, 2013, Brunet, 2012). The unsteady results are time averaged over 200 ms (∼25 D/U) after a transient stage estimated of around 120 ms (∼16 D/U). 3.2. Results Instantaneous flow visualizations are provided in Fig. 3. First, the expected failure of DDES to capture the jet instabilities is clearly illustrated. This comes from the definition of the subgrid viscosity in the DES areas based on the maximum grid spacing max which, for usual grids where the spanwise spacing is rather large compared to the other directions, leads to large values of subgrid viscosity that prevent the formation of resolved turbulence (see Fig. 4). Simulations ZDES1, ZDES2 and ZDES2-SEM appear qualitatively more satisfying in this regard. The different nozzle inlet boundary conditions are illustrated by the contours of velocity inside the nozzle in Fig. 3. The shape and size of the synthetic eddies generated by the SEM in simulation Please

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ZDES2-SEM differ from the steady uniform injection used for the other simulations. In order to assess qualitatively the RANS-to-LES transition in the simulations, a close-up on the contours of Q criterion in the vicinity of the nozzle exit are also shown in Fig. 3. One can observe that azimuthal rings are created in the initial stages of the mixing layer for all ZDES simulation. The distance over which these vortex rings are destabilized seems to be slightly larger in ZDES mode 2 than in ZDES mode 1, but shorter when jet core disturbances are injected with ZDES mode 2. Furthermore, the spatial wavelength of these vortices (the distance between two consecutive vortices) appears to be shorter in the ZDES2-SEM case than in the other ones. This suggests a significant modification of the mixing layer development with turbulence injection in the jet core, which is further investigated in the following paragraphs. The switch from RANS to LES in the simulations is illustrated in Fig. 4. The abrupt cut of eddy viscosity in mode 1 is achieved thanks to the user-defined zones. One can see that the eddyviscosity drop is also very abrupt in ZDES mode 2 thanks to the use of ω in DES areas. Conversely, even though the same fd shielding function is used in DDES and ZDES mode 2, the eddy viscosity remains large in the jet with DDES, which is a known limitation of the length scale max used in this approach (Mockett et al., 2015, Shur et al., 2015). In order to validate the safe treatment of the nozzle boundary layer in RANS mode with the global approaches (DDES and ZDES mode 2), mean velocity and eddy viscosity boundary layer profiles are shown in Fig. 5. The results from a full RANS simulation with the Spalart–Allmaras model on the same grid are also reported as a reference. It is shown that DDES and ZDES mode 2 actually act as the SA model in this area where the flow is attached. Prior to analyzing the jet flow development, the nozzle exit boundary layer profiles from the simulations are compared to experimental data in Fig. 6. Integral quantities are provided in Table 2. The mean velocity profile is correctly reproduced by the

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Fig. 3. Visualizations of the R4Ch jet simulations (Q criterion isosurface and velocity contours in the nozzle, close-up: Q criterion contours at the nozzle exit).

Fig. 4. Eddy viscosity fields at the nozzle exit. Fig. 6. Mean velocity prifle and streamwise velocity fluctuations in the nozzle exit boundary layer. Table 2 Boundary layer quantities.

Fig. 5. Boundary layer velocity and eddy viscosity profiles at the nozzle exit.

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Case

δ (x 10−3 m)

δ /D

θ (x 10−3 m)

θ /D

Exp. ZDES1 ZDES2 DDES ZDES2-SEM

3.3 2.8 2.9 2.8 2.9

0.022 0.019 0.019 0.019 0.019

0.36 0.37 0.38 0.36 0.38

0.0024 0.0025 0.0025 0.0024 0.0025

simulations even though some discrepancies can be observed in the buffer zone. These are attributed to the fact that the development distance to achieve such a thin boundary layer is too short to allow the RANS model to generate a canonical boundary layer turbulence inflow,

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Fig. 8. Streamwise velocity and shear stresses profiles in the mixing layer. Fig. 7. Time averaged turbulent kinetic energy.

profile. However, this is not judged significant for the present study as long as the integral quantities of the boundary layer are correctly reproduced (see Table 2). The streamwise velocity fluctuations are also plotted in Fig. 6, which confirms that there is no resolved turbulence in simulations ZDES1, ZDES2 and DDES at the nozzle exit (except for the small peak near nozzle trailing edge due to the fact that the profiles are plotted at the experimental measurement location which is slightly downstream of the nozzle). Conversely, it appears that the disturbances introduced at the nozzle inlet in simulation ZDES2-SEM actually penetrate into the boundary layer, while it is still computed in RANS mode as depicted in Fig. 5. The distribution and intensity of these fluctuations are not realistic since the injected turbulence is not wall turbulence but this provides a different initial condition for the mixing layer development that may help the RANS to LES transition. Outside the boundary layer, the turbulent rate achieved in this case is around 1% which is close to the targeted experimental value estimated at 1.5% using hot-wire measurements. Overal views of the simulations are provided in Fig. 7. The delay in the turbulence development with DDES is critical and some discrepancies in the growth rate and turbulent intensity of the mixing layer between the three ZDES simulations are observed in the area comprised within 0 ≤ x/D ≤ 3. These discrepancies are emphasized in the profiles depicted in Fig. 8. First, the delay in the RANS-to-LES transition seems to affect simulations ZDES1 (mode 1) and ZDES2 (mode 2) at x/D = 0.3, where the experimental mean velocity profile is not recovered by these approaches and the turbulent shear stress level is underestimated. On the contrary, simulation ZDES2SEM appears to provide a quicker transition to fully developed turbulence in the mixing layer, which is further investigated in the next paragraphs. These differences in the initial stages of the mixing layer seem to affect its whole development in terms of turbulent shear stresses. At x/D = 2, the ZDES mode 2 shear stress profiles are slightly larger and the maximum is over-predicted compared to the ZDES1 and ZDES2-SEM simulations which are in better agreement with the experiments. Therefore it seems that the ZDES mode Please

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Fig. 9. Reynolds shear stresses along the lip line.

2 induces a slightly larger RANS-to-LES transition which delays the production of fine scale dissipating structures in the mixing layer. Conversely, the shear stress profile at x/D = 2 for the ZDES2SEM simulation is in good agreement with the experimental data, which indicates that the injection of disturbances in the jet core have helped reduce the RANS-to-LES transition area and promote the production of small-scale turbulence in the mixing layer. This could be explained partly by the fact that some fluctuations penetrate into the nozzle boundary layer as shown in Fig. 6 (and even though the injected turbulence is isotropic). To further investigate the development of turbulence in the mixing layer and quantify the observations made in Fig. 3, the streamwise evolution of Reynolds shear stresses downstream of the nozzle lip are plotted in Fig. 9. The dual-peaked shape of the curve for the case ZDES2 is typical of the vortex pairing in an initially laminar or transitional mixing layer (Zaman and Hussain, 1980, Bogey et al., 2012, Bogey and Bailly, 2010). This is clearly attenuated in simulations ZDES1 and ZDES2-SEM which confirms the previous comments regarding the increased transition distance with ZDES mode 2 compared to ZDES mode 1 and its attenuation when introducing disturbances in the jet core. The effect of the use of SEM is similar to the one observed in Bogey et al. (2012) and Bogey and Bailly (2010) when laminar boundary layers are tripped.

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Fig. 10. Streamwise velocity spectra in the mixing layer.

Fig. 11. Streamwise mean velocity and fluctuations along the jet centreline.

The velocity spectra in the mixing layer presented in Fig. 10 further illustrate the activation of resolved turbulent energy and its frequency distribution in the simulations. At x/D = 0.3 the spectra exhibit a peak around the normalized frequency based on the initial momentum thickness of the mixing layer Stθ = 0.013. This is the frequency classically observed for the Kelvin–Helmholtz instabilities that trigger vortex pairing in transitional mixing layers (Zaman and Hussain, 1980, Bogey et al., 2012, Kim and Choi, 2009). This is not observed in the experiments since the nozzle exit boundary layer is fully turbulent. Further downstream, a fully developed turbulence is recovered in all simulations, although at x/D = 1 a low frequency bump is observed in the case ZDES2 and is attributed to the slightly longer transition for this simulation. A consequence of the increased delay in the RANS-to-LES transition in the ZDES2 simulation (and to a greater extent in the DDES simulation) is depicted in Fig. 11. As a matter of fact, the potential core length is underestimated in these cases and it is attributed to the increased jet mixing promoted by large structures which are not dissipated early enough in the ZDES2 simulation as commented previously. A similar observation is reported in Zaman and Hussain (1981) and Kim and Choi (2009), and corresponds to the effect of a laminar initial mixing layer on the jet flow development. In such situations, the shear-layer break-up process is more violent and induces higher turbulent intensities, which in turn leads to increased mixing and shorter potential core. Conversely, the potential core length is in very good agreement with the experimental data for simulations ZDES1 and ZDES2-SEM which shows that the early development of the shear layer is better captured in these simulations. The experimental evolution of the streamwise velocity fluctuations in Fig. 11 displays a change of slope corresponding to the Please

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end of the potential core around x/D = 5 depending on the dataset. It appears that using mode 2 of ZDES (with or without synthetic turbulence) gives a better prediction of the slope of the curve upstream of the end of the potential core compared to the ZDES1 simulation. Downstream of the end of the potential core, the simulations without synthetic turbulence exhibit a change of slope similar to the one observed in the experiments, whereas the curve for ZDES2-SEM simulation seems to be fairly monotonic. This behavior does not correspond to what is observed in the experiments and might be related to the parameters used for the SEM, further insights on this issue are given in Section 3.3. However, the spectral signature of the turbulent inflow used in ZDES2-SEM is visible in Fig. 12. Peaks appear at high frequencies, separated by a frequency band of f = 300 Hz. This corresponds to a wavelength λ = U/f (where U is the mean velocity at the nozzle inlet) equal to the length scale σ of the synthetic eddies. Nevertheless, the energy levels associated to these peaks are rather low (at least three orders of magnitude less that the most energetic levels) and are attenuated away from the nozzle exit. At x/D = 1 and x/D = 2, all ZDES simulations are in fairly good agreement with the experiments. In particular, the jet column mode (Zaman and Hussain, 1980, Gutmark and Ho, 1983) around StD = 0.6 based on the nozzle exit diameter is recovered in all cases. To summarize, it has been shown in this section that without disturbances, the RANS-to-LES transition in the simulations is similar to the laminar/turbulent transition occurring in transitional jets. Therefore, the delay in the transition to fully developed turbulence influences the mixing layer growth and jet development, which can have significant consequences for technical applications. Nevertheless, the addition of disturbances in the jet core accelerated this transition and improved the agreement with experimental data. Prior to evaluating the robustness of this result and its reproducibility for a jet emanating from a fully developed channel flow in part four, the effect of the SEM parameters (eddies length scale and turbulent rate) is assessed in the next section.

3.3. Effect of the SEM parameters Three simulations using SEM inflows are compared in this section (see Table 1). ZDES2-SEM is the one compared to the other ZDES simulations without turbulent inflow in the previous section. ZDES2-SEM2 uses the same eddies length scale as ZDES2-SEM but a higher inlet turbulent rate of 35% instead of 7.5%. ZDES2-SEM3 uses the same turbulent rate as ZDES2-SEM but a larger eddies length-scale: σ = 0.05 m instead of σ = 0.02 m. These differences in the SEM parameters are qualitatively illustrated in Fig. 13 which displays the streamwise velocity contours inside the nozzle.

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Fig. 12. Streamwise velocity spectra along the jet centerline.

Fig. 13. Snapshot of instantaneous streamwise velocity contours in the nozzle (the same color scale is used for all images). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 14. Streamwise evolution along the jet axis of normal stresses inside the nozzle. The inlet of the computational domain is at x/D = −4 and the nozzle exit is at x/D = 0.

The evolution of the normal stresses inside the nozzle (between the inlet of the computational domain in x/D = −4 and the nozzle exit in x/D = 0) is presented in Fig. 14. The effect of the higher turbulent rate in simulation ZDES2-SEM2 is obvious, even though it seems that a large amount of the injected turbulence is quickly dissipated downstream of the inlet. A similar drop of the turbulent quantities is also observed for the ZDES2-SEM simulation and is strongly attenuated in ZDES2-SEM3. This is attributed to the ratio between the eddies length-scale and the axial grid size near the SEM inlet, which seems more appropriate in the latter case since it corresponds to twice the axial spacing near the inlet. Of interest, the dissipation of the injected turbulence is more important for the axial component of the normal stresses, and isotropic turbulence is recovered at the nozzle exit (x = 0) only in the ZDES2-SEM3 case. Therefore the SEM parameters used influence the actual turbulent rate and isotropy achieved at the nozzle exit, which may be altered by the grid dissipation. The boundary layer profiles at the nozzle exit plotted in Fig. 15 show that the mean velocity profile is not significantly altered by the different SEM parameters in terms of integral quantities. As observed in the previous section, some resolved turbulence penetrate into the boundary layer treated in RANS, and the level of resolved Please

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Fig. 15. Mean and fluctuating streamwise velocity profiles in the nozzle exit boundary layer.

turbulence obtained in the boundary layer scales with the turbulent rate achieved in the jet core. The differences observed in the nozzle boundary layer regarding the level of resolved turbulent kinetic energy are likely to

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Fig. 16. Streamwise evolution of shear stresses downstream of the nozzle lip (left) and streamwise velocity spectra close to the nozzle lip (right).

Table 3 Impinging jet experimental conditions.

Fig. 17. Streamwise evolution of the mean and fluctuating axial velocity along the jet axis.

impact the early development of the mixing layer, which is depicted in Fig. 16. The streamwise evolution of the shear stresses close to the nozzle lip (Fig. 16(a)) indicates that using a higher turbulent rate seems to improve the slope of the curve upstream of the maximum, which is consistent with the higher level of resolved turbulence in the boundary layer for this simulation. The streamwise velocity spectra close to the nozzle lip (Fig. 16(b)) is therefore closer to a fully turbulent one. It is noteworthy that the peak of the spectra around Stθ = 0.013 observed for simulations ZDES2-SEM and ZDES2-SEM2 is almost completely suppressed in the case with a higher turbulent rate, which is consistent with the fact that the RANS boundary layer actually conveys a large amount of resolved turbulence in this case. The global jet development is also altered by the turbulent rate achieved in the simulations. Fig. 17 shows that the potential core length is reduced with a larger turbulent rate – and one may even question the notion of potential core in this case with high turbulent intensity in the jet core. The velocity spectra along the jet axis in Fig. 18 at the nozzle exit illustrate the state of the jet core turbulence achieved with the different SEM parameters. The effect of the eddies length scale is visible at high frequencies where the bumps for ZDES2-SEM are roughly twice as large as the ones observed for ZDES2-SEM3. This supports the assumption made in the previous section regarding the wavelength of these bumps being equal to the eddies length scale. Conversely, the spectra for the ZDES2-SEM2 simulation with a higher turbulent rate are more broadband and closer to fully developed turbulence. In this case, it seems that the turbulent intenPlease

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Quantity

Notation

Value

Bulk velocity Jet temperature Freestream temperature Jet exhaust diameter Reynolds number Nozzle-to-plate distance

Vj Tj Ta D ReD H/D

25,6 m/s 130°C 25°C 60 mm 59 0 0 0 3

sity is such that the spectral signature of the SEM is no longer visible at high frequencies. Downstream of the nozzle exit, at x/D = 1 and x/D = 2, the jet column mode is no longer observable in the ZDES2-SEM2 case which is attributed to the strong mixing in the jet core induced by the high turbulent level. The physical effect of the turbulent rate on the overall jet development is beyond the scope of the present article and would require more experimental data for an in-depth analysis. As a conclusion, the study conducted in the present section provides preliminary insights on the choice of the SEM parameters and their effect on the simulation. First, the length scale used for the eddies has a significant impact on the turbulent rate actually achieved at the nozzle exit due to the possible dissipation by the grid. This leads to a constraint on the grid sizes near the SEM inlet if the targeted length scale is known from the experiments. No significant effect of the length scale on the jet development has been observed in this study. On the contrary, a high value for the SEM turbulent rate leads to an increase of the resolved turbulence penetrating into the boundary layer, thus to a reduction of the RANSto-LES transition. This also impacts the jet flow development as the increased mixing in the jet core reduces the potential core length and attenuates the jet column mode.

4. Jet originating from a fully developed channel flow 4.1. Test case and simulations performed The configuration investigated in this section is a heated jet impinging on a flat plate, taken from the experiments analyzed in Grenson et al. (2016) (illustrated in Fig. 19) devoted to the study of heat transfer and flow dynamics on the plate. The flow conditions are presented in Table 3. In this work, the focus is put only on the jet flow development area, not on its interaction with the plate. The main difference with the case investigated in the

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Fig. 18. Streamwise velocity spectra along the jet centerline. Table 4 Simulations of the impinging test case. Case

RANS/LES approach

Inflow condition

ZDES ZDES-SEM

ZDES mode 2, ω ZDES mode 2, ω

Steady, fully developed channel flow SEM Jarrin, σ = 0 0 02 m, target Re tensor from exp.

in Section 3 to assess the efficiency of the method without taking into account the anisotropy near the wall. The value of the length scale is equal to five times the axial grid size at the pipe exhaust to avoid the dissipation of the eddies in the first cells according to the results presented in Section 3.3. ZDES mode 1 is not tested in this case because mode 2 is required to capture the reattachment on the impinged plate, the location of which is not known a priori. 4.2. Results

Fig. 19. Basics of impinging jets configurations (from Carlomagno and Ianiro, 2014).

previous section is that the jet originates from a fully developed channel flow. A global view of the numerical setup is depicted in Fig. 20(a). Note that the channel is not included in the computational domain. The inlet boundary condition corresponds to the velocity profile measured at the channel exit. The grid sizes at r = 0.5 D are plotted in Fig. 20(b). The mesh comprises around 17.106 nodes. The numerical parameters are similar to the ones detailed in Section 3. The only difference is that a 5th order MUSCL extrapolation is used for the AUSM+P scheme, which has been shown to improve the accuracy of the simulations in a ZDES and LES framework (Gand et al., 2015, Marty et al., 2015). In this case, the time step is 1.10−6 s and the unsteady results are time averaged over 95 ms (∼40 D/U) after a transient stage estimated at around 24 ms (∼10 D/U). Two simulations are compared with experimental data as shown in Table 4, the first one (named ZDES in this section) involves the ZDES mode 2 with a steady inlet. In the second one (ZDES-SEM), synthetic turbulence was added in the inlet plane. The target Reynolds stress tensor for SEM was taken from the LDV measurements, and the length scale was chosen uniform as Please

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The mean velocity and velocity fluctuations just downstream of the channel exit (i.e., the inlet boundary condition) are plotted in Fig. 21. The mean velocity profile is correctly reproduced in both simulations, and the use of the SEM allows to recover satisfying turbulent statistics in simulation ZDES-SEM while there is no resolved turbulence in the simulation ZDES due to the steady inlet, which is the same as if the flow was solved with a RANS model inside the channel. Fig. 22 illustrates the coherent structures resolved in the simulations, with a focus on the initial stage of the mixing layer. These visualizations are similar to the ones in Fig. 3: vortex pairing occurs in the ZDES simulation while the mixing layer is fully turbulent right downstream of the inlet in the ZDES-SEM simulation. In this regard, the reduction of the spatial wavelength of the vortex pairing process with turbulent inflow observed in the previous case (see Fig. 3) is also found here. These observations are confirmed by the turbulent kinetic energy fields in Fig. 23. High levels of resolved turbulence are reached close to the nozzle exit in the ZDES-SEM simulation, whereas the RANS-to-LES transition can be estimated at 0.4 D in the ZDES simulation. This difference seems to affect the whole development of the mixing layer and also affects the impingement conditions, thus the heat flux on the plate, which will be the topic of future work. The profiles in Fig. 24 confirm the overall better agreement of the ZDES-SEM simulation with the experimental data. Indeed, the mean velocity profile from the ZDES simulation does not agree with the experimental one at x/D = 0.5 due to the delay in the RANS-to-LES transition, whereas the agreement is very good for simulation ZDES-SEM at all stations. The late development of small scale structures in the ZDES case results in the overprediction of the turbulent fluctuations as illustrated in the visualizations

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Fig. 20. Numerical setup for the impinging jet simulation.

Fig. 21. Exit profiles.

Fig. 22. Instantaneous flow visualizations. Top: iso-Q criterion colored by streamwise velocity and contours of temperature. Bottom: contours of Q criterion in the vicinity of the pipe exhaust. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 23. Time-averaged turbulent kinetic energy.

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Fig. 25. Streamwise velocity spectra.

same result was found for the first test case (Fig. 12). The spectra along the centerline illustrate the need for turbulence injection in the jet core for this case as it dramatically modifies the state of the jet before the impingement of the plate, which may have a significant impact of the heat transfer. 5. Conclusions and perspectives

Fig. 24. Mean velocity and radial fluctuations profiles.

in Fig. 23, which was also observed in the previous case (see Figs. 7 and 8). Nevertheless, the ZDES-SEM simulation underpredicts the levels of turbulent kinetic energy downstream from x/D = 1.5 which could be attributed to the underprediction of turbulence levels in the jet core, despite the synthetic inflow. Indeed, it seems that the injected turbulence is not correctly convected in the jet core in simulation ZDES-SEM and this seems to affect the development of the shear layer downstream of the transition area. This unexpected dissipation of the turbulence in the jet core might be due to the grid stretching in the streamwise direction. To improve this result, the grid should be refined and the SEM length scale should be modified to take into account the turbulence anisotropy at the pipe exhaust (this is a fully developed channel flow). However, the ZDES-SEM results show that even with a partial description of the actual turbulence at the pipe exhaust, the turbulence development in the shear layer is improved. The frequency distribution of the turbulent energy in the mixing layer and along the centerline is presented in Fig. 25. The experimental spectra in the mixing layer are recovered at x = 1D with the ZDES simulation due to the delay in the turbulence development, contrary to the ZDES-SEM simulation which is again in very good agreement with the wind-tunnel data close to the nozzle exit at x/D = 0.3. Along the centerline, the absence of turbulence in the jet core in simulation ZDES results in a large bump at StD = 0.6 which corresponds to the jet column mode. This frequency is also found in the experimental spectra but it is superimposed on broadband turbulence. This is correctly reproduced by the ZDES-SEM simulation but the overall level of the spectra is underestimated which could be due to improper tuning of the SEM in this case (especially the use of a uniform length scale for the eddies). Besides, the spectral signature of the SEM is visible at high frequency with low-energy bumps separated by a frequency band of f = 14 kHz which corresponds to a wavelength equal to the synthetic eddies size. The Please

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Hybrid RANS/LES simulations of two incompressible jets at low speed are presented in this article. The focus is put on the RANSto-LES transition in the early stages of the mixing layer as it is a key area for the global jet flow development and radiated noise in more complex configurations. The comparison between ZDES and DDES illustrates the importance of the choice of the subgrid length scale in free shear layers where no global flow instability drives the generation of resolved LES content. As shown in previous work (Chauvet et al., 2007, Deck, 2012, Gand et al., 2015), the use of a length scale based on the local flow characteristics ω quickly reduces the eddy viscosity level in separated areas which in turn allows instabilities to grow. The area downstream of the nozzle lip still displays a RANSto-LES transition, even if its length is significantly reduced compared to DDES. It is noteworthy that the RANS-to-LES transition in the present ZDES simulations displays the same physical properties as a laminar/turbulent transition, with vortex pairing occurring at Stθ ∼0.012. This transition delays the formation of fine scale turbulence and therefore too high turbulence intensities are observed in the mixing layers. Depending on the length of the transition area, this can induce large errors in the growth rate of the mixing layers and overall jet flow development which is critical for applications. In order to assess the efficiency of using turbulent inflow as a mitigation tool to further reduce the RANS-to-LES transition, the Synthetic Eddy Method of Jarrin et al. (2009) was used to try and reproduce the freestream turbulence in jet core. The reason of the choice of this method is that it seems straightforward even for technical configurations since it aims at reproducing somewhat broadband isotropic turbulence via an inflow boundary condition – and not wall turbulence as a Wall Resolved LES inside the nozzle would require, which is not an easy task especially in curvilinear grids. In both cases investigated, the injection of synthetic turbulence in the jets core not only improved the realism of the simulations by taking into account an additional experimental parameter but accelerated the RANS-to-LES transition and reduced the spatial wavelength of the vortex pairing. In the first case with a thin boundary layer, this is most likely due to the penetration of the freestream turbulence into the boundary layer which is still treated in RANS. The more realistic turbulence development in the mixing

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layers resulted in a better match between simulations and experiments regarding the growth rate and turbulent intensities. Some insights on the effect of the SEM parameters have been gained in Section 3.3 but need to be assessed on different cases and compared to experimental results. Some limitations of the approach have been identified. Some vortex pairing still occurs in the first case with the addition of jet core turbulence (Figs. 3 and 10), which does not have consequences for the jet flow development but might produce tonal noise, especially in technical configurations at higher Reynolds and Mach numbers. In order to completely suppress vortex pairing, one might try to increase the injected turbulence rate, which should be evaluated in future work. Besides, the spectral signature of the synthetic turbulence could be observed at the nozzle exit, which might be an issue for acoustic prediction in compressible jets cases. This will be the topic of a following work. As mentioned in the introduction, the most radical solution to suppress the transition area would be to resolve the nozzle boundary layer, or at least its outer region. This type of simulation requires the injection of plausible wall turbulence and a grid refinement which might still be prohibitive for technical configurations. It has also been seen that the turbulent inflow is not transparent in a spectral point of view, which could be counter-productive for jet noise prediction. This needs to be assessed and will be the topic of future work. Acknowledgments The author would like to thank R. Courtier from ONERA for providing the experimental data for the R4Ch test case. P. Grenson and P. Reulet from ONERA are also thanked for providing the experimental data for the impinging jet. The author is grateful for the comments of S. Deck from ONERA. The simulations of the R4Ch jet were carried out using HPC resources from GENCI-TGCC (Grant 2014-t20142a7215). References Bogey, C., Bailly, C., 2010. Influence of nozzle-exit boundary-layer conditions on the flow and acoustic fields of initially laminar jets. J. Fluid Mech. 663, 507–538. doi:10.1017/S0022112010003605. Bogey, C., Marsden, O., Bailly, C., 2012. Influence of initial turbulence level on the flow and sound fields of a subsonic jet at a diameter-based Reynolds number of 105 . J. Fluid Mech. 701, 352–385. doi:10.1017/jfm.2012.162. Brès, G., Jaunet, V., Le Rallic, M., Jordan, P., Colonius, T., Lele, S., 2015. Large eddy simulation for jet noise: the importance of getting the boundary layer right. 21st AIAA/CEAS Aeroacoustics Conference, 26 June 2015, Dallas, TX AIAA Paper 2015--2535. Brunet, V., 2012. Random turbulent flow generation coupled with ZDES for civil aircraft jet configurations. 30th AIAA Applied Aerodynamics Conference, 25–28 June 2012, New Orleans, Louisiana AIAA Paper 2012--2896. Cambier, L., Heib, S., Plot, S., 2013. The Onera elsA CFD software: input from research and feedback from industry. Mech. Indus. 14 (3), 159–174. doi:10.1051/ meca/2013056. Carlomagno, G.M., Ianiro, A., 2014. Thermo-fluid-dynamics of submerged jets impinging at short nozzle-to-plate distance: a review. Exp. Therm Fluid Sci. 58, 15–35. doi:10.1016/j.expthermflusci.2014.06.010. Chauvet, N., Deck, S., Jacquin, L., 2007. Zonal detached eddy simulation of a controlled propulsive jet. AIAA J. 45 (10), 2458–2473. doi:10.2514/1.28562. Daude, F., Mary, I., Comte, P., 2006. Local optimization of the convergence rates of implicit time advancements for LES of complex flows. 36th AIAA Fluid Dynamics Conference and Exhibit AIAA Paper 2006--3545. Davoust, S., Jacquin, L., Leclaire, B., 2012. Dynamics of m = 0 and m = 1 modes and of streamwise vortices in a turbulent axisymmetric mixing layer. J. Fluids Mech. 709, 408–444. doi:10.1017/jfm.2012.342. DeBonis, J., 2010. A high-resolution capability for large-eddy simulation of jet flows. 40th Fluid Dynamics Conference and Exhibit AIAA Paper 2010--5023. Deck, S., 2012. Recent improvements of the Zonal Detached Eddy Simulation (ZDES) formulation. Theor. Comput. Fluid Dyn. 26, 523–550. doi:10.1007/ s00162-011-0240-z. Deck, S., Gand, F., Brunet, V., Ben Khelil, S., 2014. High-fidelity simulations of unsteady civil aircraft aerodynamics: stakes and perspectives. application of zonal detached eddy simulation. Phil. Trans. R. Soc. A doi:10.1098/rsta.2013.0325. Deck, S., Laraufie, R., 2013. Numerical investigation of the flow dynamics past a three-element aerofoil. J. Fluid Mech. 732, 401–444. doi:10.1017/jfm.2013.363.

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