Numerical simulations of a massively separated reactive flow using a DDES approach for turbulence modelling

Transcription

1 Numerical simulations of a massively separated reactive flow using a DDES approach for turbulence modelling Bruno Sainte-Rose, Nicolas Bertier, Sébastien Deck and Francis Dupoirieux Abstract Computations of a lean premixed methane - air flame in a lean stepped combustor are performed using a Delayed Detached Eddy Simulation approach to model turbulence. Two conditions for the outlet section are simulated and compared to an experimental database including mean velocity, mean temperature and instantaneous OH emission measurements. The main objective of this study is to assess of the efficiency of DDES for a massively separated reactive flow. Introduction Large Eddy Simulation is now of common use for reactive flows []. In fact, LES is well suited for massively separated flows as those found in swirled burners, ramjets and is effective to reproduce combustion instabilities. Indeed, LES is able to solve the most energetic structures of the turbulent flow while modelling the subgrid turbulent dissipation. On the other hand, Reynolds Averaged Navier Stokes approaches are progressively limited to parametric and optimisation studies in the design path of combustion devices. In certain cases where an accurate resolution of the wall flows Bruno Sainte-Rose Onera (Fundamental and Applied Energetics Department), 29 avenue de la Division Leclerc Châtillon. bruno.sainte-rose@onera.fr Nicolas Bertier Onera (Fundamental and Applied Energetics Department), 29 avenue de la Division Leclerc Châtillon. nicolas.bertier@onera.fr Sébastien Deck Onera (Applied Aerodynamics Department), 8 rue des Vertugadins 9290 Meudon. sebastien.deck@onera.fr Francis Dupoirieux Onera (Fundamental and Applied Energetics Department), Chemin de la Hunière 976 Palaiseau. francis.dupoirieux@onera.fr

2 2 B. Sainte-Rose et al. is required, solving the thin turbulent structures encountered at low wall units can become very costly with a LES, while the low Reynolds models make such computations far more affordable with RANS. As a consequence, hybrid RANS / LES methods such as DDES employed in this study are a solution to take into account simultaneously attached boundary layers thanks to RANS and massively separated flows thanks to the LES. Such an approach has been used and is presented here for the study of a lean premixed turbulent methane - air flame stabilized by a Backward Facing Step. An analysis of both the mixing layer and the turbulent flame is adressed in this paper. The influence of the acoustic conditionning of the outlet boundary is also scrutinized. 2 Delayed Detached Eddy Simulation for reactive flow The hybrid RANS / LES method chosen to model turbulence in this study is the DDES approach originally proposed by Spalart et al. [2]. DDES is applied to the k ω SST model [3] for which the destruction term of the equation of the turbulent kinetic energy k, i. e. ε, is thus modified [4], ε = β ωk = k3/2 l RANS k3/2 l DDES () and where the modified length scale l DDES is written as l DDES = l RANS f DDES max(0,l RANS C DES ) (2) where l RANS is the RANS length scale and is equal to k/β ω and is the mesh scale and equal to ( x y z ) /3. In equation 2, the f DDES function allows the model to yield the attached boundary layers to SST - RANS ( f DDES = 0) while switching to a Strelets DES approach [5] far from the wall ( f DDES = ). This modification of the original DES was motivated by some unphysical outcomes like artificial relaminarisation, also called Modelled Stress Depletion [6][7]. Concerning the modelling used for turbulent combustion, the source term of the species balance is obtained thanks to an Arrhenius law using a global reaction mechanism. The interaction with turbulence is modelled by a Dynamically Thickened Flame for LES introduced by Legier et al. [8]. 3 Presentation of the physical and numerical test case The test case presented in this study was used to validate the DES approach for reactive flows in the Onera CEDRE code [4]. The main features of the flow are presented in figure. The equivalence ratio of the methane air mixture is equal to 0.8. An experimental database covering both velocity [9] and temperature [] is

3 DDES of a massively separated reactive flow 3 available along with OH emissions measurements [0]. The characteristics of the computational domain, meshes and boundary conditions of the computations can be found in [4] along with details on the numerical methods used in the CEDRE code. Fig. Focus on the particular regions of the reactive flow, height of the step h = 3.5 cm, U 0 = 50 m.s, Re h Streamwise acoustic modes are reproduced by applying Dirichlet conditions for the pressure at the outlet of the computational domain while the case without acoustic reflections at the outlet is obtained thanks to non reflecting NSCBC boundary conditions [2]. Three computations (2D-RANS, DDES with or without reflection at the outlet) are presented and discussed in the following sections. 4 Analysis of the recirculating region and of the mixing layer The time - dependent field is averaged during our unsteady computations. The first relevant analysis of the recirculating zone which can be made on our computations concerns the averaged reattachment length. The ratio between the mean reattachment length and the size of the step L r /h is compared to the experimental value obtained by Laser Doppler Velocimetry in table. One can observe that all the computations tend to overestimate this value, this is due in particular to the thermal conditions at the lower wall. Indeed, the lack of measurements compelled us to neglect the wall cooling by taking adabiatic walls. Moreover, on one side of the coin RANS results are of poor quality compared to the DDES computations, on the other, the imposed pressure at the outlet section of the DDES calculation, producing streamwise acoustic modes, results in a shorter recirculation length. Table Averaged size of the recirculating bubble LDV DDES w/o reflection DDES w/ reflection 2D-RANS L r /h 2.9 < < Another aspect was studied to appraise of the quality of the description of the reactive mixing layer. In figure 2, we plot the evolution of the vorticity thickness δ ω in the streamwise direction for our three computations and for the Laser Doppler Velocimetry measurements [9] in a range of 0 < x/h < 3.8. In this figure, the vor-

4 4 B. Sainte-Rose et al. ticity thickness is divided by the height of the step or by the momentum thickness θ respectively in the left and right sides of the figure. Let us be reminded of the expressions of δ ω (x) and θ(x): δ ω (x) = [U max[y] U min[y] ](x) U(x,y) max [y] y (3) ( ymax U(x,y) U min[y] (x) θ(x) = U(x,y) U ) min[y](x) dy y min U max[y] (x) U min[y] (x) U max[y] (x) U min[y] (x) (4) where U is the averaged streamwise velocity δ ω /h 0.6 DDES w/o reflexion 0.4 DDES w/ reflexion 2D - RANS 0.2 LDV LDV 0 0 X/h X/h δ ω /θ 2 0 Fig. 2 (left) Vorticity thickness and (right) ratio between vorticity and momentum thicknesses along the mixing layer It can be noticed in figure 2 that close to the step for 0 < x/h < 0.4, the evolution of the vorticity thickness is not exponential as for an inert shear layer but rather linear. This can be explained by the acceleration of the fluid located under the flame which tends to lower the maximum velocity gradient. We also observe a good agreement between the DDES computations w/o acoustic reflections and the experiments with a linear growth of about It is a consequence of the accuracy of the streamwise velocities discussed in [4]. Moreover, the comparison between the two DDES computations show that the shear layer is greatly impacted by the outlet conditions. The growth of the vorticity thickness is indeed doubled for 0 < x/h <.5 in the case of streamwise modes which highlights a coupling between acoustic and the size of the eddies growing from the corner of the step. 5 Analysis of the turbulent flame Let us now consider the results obtained for the averaged temperature flowfields. In figure 3, an iso-line of T = 500 K is displayed for our three computations and compared to the Coherent Anti- Stokes Raman Spectroscopy measurements []. This layout allows us to approximately locate the averaged computed flame. The results obtained close to the step for DDES are very satisfying since the flame angle

5 DDES of a massively separated reactive flow 5 is well reproduced whereas the RANS clearly underestimates the flame angle, the use of reflecting conditions at the outlet results in an increase of the flame angle. Fig K temperature iso - line for DDES w/o reflection (solid), DDES w/ reflection (dash-dot), RANS (dashed), CARS (circles), vertical dash lines: position of the CARS measurements T 2 T 2 is shown in the same figure. The difference If we now have a look at the temperature profiles in figure 4, the previous remarks concerning the location of the flame is therefore verified since we observe that the peak and the levels found for the fluctuating temperature are in very good agreement with the experiments except for the first profile for which the need to characterize of the termal conditions at the lower wall is evidenced. As for the temperature fluctuations, the value of T = between the two acoustic conditions discussed previsously is identified since the important wrinkling due to the pressure waves lead to an overestimation of the peak and of the width of the flame brush. - 0 Y/h T, K X/h=. X/h=4.3 X/h=6.0 X/h=9.7 X/h=3. X/h=20.3 T rms (K) Fig. 4 Mean and fluctuating temperature T and T DDES w/o reflection (solid), w/ reflection (dash-dot) and RANS (dashed only for T ), CARS measurements (diamonds) - 0 Y/h The acoustic modes appearing in the computations and their coupling with the dynamic of the flame are now scrutinized. For that purpose the temperature signal obtained at a point located at the corner of the step is analyzed. In figure 5 the temperature spectrum G T ( f ) for our two cases are plotted. The case with imposed pressure at the outlet clear displays quarter wave frequencies. Moreover, the analytic value of the quarter wave frequencies in a D wave duct of length L corresponds to f n = nc/4l, where c is the speed of sound in the burnt gases and n the number of the harmonics. This formula gives f 74 Hz, f Hz, f Hz etc., these frequencies are clearly evidenced in the right part of figure 5 where peaks at

6 6 B. Sainte-Rose et al. f = 73 Hz, f 2 = 352 Hz, f 3 = 525 Hz etc. can be observed corresponding to Strouhal numbers (based on an upstream velocity of 50 m.s and the height of the step) of St = 0.2, St 2 = 0.25, St 3 = 0.37 etc.. It is interesting to see that even with non reflecting boundary conditions the first harmonic can also be seen in the left part of the figure. However another peak at f = 433 Hz, St = 0.3 is evidenced, this frequency being close to the value obtained for the Kelvin - Helmholtz instability on the inert case. temperature spectrum, G T (f) f(hz) temperature spectrum, G T (f) f(hz) St h St h Fig. 5 Temperature spectrum G T ( f ) for DDES w/o reflection (left) and w/ reflection (right) A time - dependent sequence of heat release fields is displayed in figure 6 for DDES w/ acoustic reflection and allows us to compare our computations with instantaneous screenshots of Planar Laser Induced Fluorescence OH measurements obtained during a campaign aiming at reproducing combustion instabilities [0]. A strong wrinkling of the flame close to the step evidences a strong coupling between flame and eddies in response to streamwise acoustic modes, in our current case massive flash-back does not occur contrarily to the experiments.!"#$%!"&#$%!"'(#$%!"')#$%!"(*#$%!"+##$% Fig. 6 (top) Time dependent heat release during T 0 = 2.8 ms corresponding to a period of the first harmonic, (bottom) corresponding LIF screenshots (not time dependent) which evidences the reactive zone where OH is produced

7 DDES of a massively separated reactive flow 7 6 Conclusions In the present work, DDES computations were carried out on a BFS reactive flow. This study constitutes an extension of the work published in [4]. It indeed gives new insights by providing an investigation of the mixing layer and flame particularly to study the effect of acoustic modulations inside the chamber. DDES was demonstrated to provide a clear improvement to RANS by showing a good agreement with the experimental database for velocity and temperature. The behaviour of the flame and mixing layer with transverse acoustic waves was evidenced. To extend the use of DES for massively separated reactive flows, the study of a separated nozzle flow with combustion is now considered. Acknowledgements The authors want to acknowledge the Centre Nationale d Etudes Spatiales (CNES) for the funding of B. Sainte-Rose PhD studies. A. Laverdant and V. Sabelnikov are also gratefully acknowledged. References. T. Poinsot, D. Veynante, Theoretical and numerical combustion, R.T. Edwards Ed., P.R. Spalart, S. Deck, M.L. Shur, K.D. Squires, M. Kh. Strelets, A. Travin, A new version of detached-eddy simulation, resistant to ambiguous grid densities, Theor. Comput. Fluid Dyn., 20:8-95, F. R. Menter, Zonal two-equation k-omega turbulence models for aerodynamic flows, AIAA , B. Sainte-Rose, N. Bertier, S. Deck, F. Dupoirieux, A DES method applied to a Backward Facing Step reactive flow, accepted in C. R. Mécanique, M. Kh. Strelets, Detached-Eddy Simulation of Massively Separated Flows, AIAA , P. Sagaut, S. Deck, M. Terracol, Multiscale and multiresolution approaches in turbulence, Imperial College Press, P. R. Spalart, Detached Eddy Simulation, Annu. Rev. Fluid Mech. 4:8-202, J. P. Legier, T. Poinsot, D. Veynante, Dynamically thickened flame LES model for premixed and non-premixed turbulent combustion, Proceedings of the Center for Turbulence Research, P. Moreau, J. Labbe, F. Dupoirieux, R. Borghi, Experimental and numerical study of a turbulent recirculation zone with combustion, 5th Symposium on Turbulence and Shear Flow (985) 0. V. Sabelnikov, F. Grisch, M. Orain, Instabilities and structure of turbulent premixed flame in a lean stepped combustor, paper ISABE , P. Magre, G. Collin, P. Bouchardy, Application de la DRASC à l opération A3C (french), Onera Technical Report, T. Poinsot, S. Lele, Boundary conditions for direct simulations of compressible reacting flows, J. Comput. Phys., 0: 04-29, 992.