International Conference on Methods of Aerophysical Research, ICMAR 008 LARGE EDDY SIMULATION OF TURBULENT ROUND IMPINGING JET B.B. Ilyushin, D.V. Krasinsky Kutateladze Institute of Thermophysics SB RAS 630090, Novosibirsk, Russia Large Eddy Simulation (LES) approach (see e.g. []) has for the past decade proven to be the "mainstream" line in research of turbulence modelling. In LES the major part of the turbulent kinetic energy is resolved "directly" whereas the effects of remaining scales smaller than the computational grid size (filterwidth) are accounted in a subgrid-scale (SGS) model. Compared to well-known RANS (Reynolds-Averaged Navier-Stokes equations) approach, universality of LES is higher because in LES the model assumptions are made only on subgrid, energy-negligible scales of the turbulent flow. Another advantage of LES approach compared to RANS is that it shows the dynamics of large-scale fluctuations as in LES inherently the time-dependent solution is obtained. To separate the "resolved" scales of motion from the "smaller" scales that are to be modelled, the spatial filtering operation is used in LES where typically the computational grid cell size is associated with the filter width (i.e. the cutoff length at which the scales are separated) in the physical space []. When filtering is applied to the Navier-Stokes equations written in isothermal incompressible case, they take the following form: u u u i + = + t x x x x u i x i = 0 i j ui τij p ν j ρ i j j x j In () the Cartesian tensor notation is used, any quantity in angular brackets i is assumed to be grid-filtered, and to close equations () the subgrid stress tensor τ ij = uu i j u i u j has to be modelled. The most widespread model for it is the Boussinesq eddy-viscosity formulation: sgs u u i j τij = νt + + δijτ kk () xj xi 3 The standard Smagorinsky model has been used in the work to provide the closure for the subgridscale eddy viscosity in (): sgs t CS Sij ij ν = S (3) where the filtered strain rate tensor is: S ij ( ui xj uj xi ) () + and the local filterwidth is determined according to Deardorff (see []) as a cubic root of the cell volume: = ( δ δ δ ) 3 x y z (hereinafter δ x, δ y, δ z denote the cell sizes in each coordinate direction). Smagorinsky model coefficient in LES computations has been taken as C S =0.48 according to an estimate given in []. The numerical algorithm used in the work to solve the grid-filtered 3-D unsteady Navier- Stokes equations has been earlier described and tested on DNS of jet flows in [], see also [3] where the algorithm details used in the present work are given. The spherical coordinate system is utilized in the numerical code, with this the computational domain is contained in a blunted cone with small B.B.Ilyushin, D.V.Krasinsky, 008
Section II opening angle. However, as a common practice in the study of axisymmetric flows, the results are presented hereinafter in the cylindrical coordinates (x, r, ϕ). The main features of the algorithm are as follows: finite-volume central differencing scheme on a staggered grid, with uniform cell size in circumferential direction and non-uniform grid in two other directions; time integration is done by explicit -nd order Adams-Bashforth method; solution of Poisson equation for pressure (obtained via the continuity equation) is efficiently done via the FFT in ϕ-direction reducing the problem to a set of -D elliptic equations solved by direct cyclic reduction method of P.Swarztrauber. The LES/DNS code has been parallelized using the Message Passing Interface (MPI) for data exchange between CPUs. In the previous works of authors [3 5] results of LES study on the dynamics and vortex structure of incompressible fluid flow of free round turbulent jet at Re=5000 have been reported. In the present paper the results of LES modelling of impinging turbulent round jet are shown. The impinging jet configuration (see Fig. ) is considered in case of the nozzle-to-wall distance H equal to three nozzle diameters, H/D jet =3, and Reynolds number taken is Re=8900 (based on the nozzle diameter D jet =5 mm and the mean flowrate U jet =0.5 m/s in water). LES computations were performed on 0 58 48 grid using supercomputer MVS-00k (Joint Supercomputer Center of RAS, Moscow), loading usually 4 CPUs. Comparison of LES-modelling results with data [6] of experimental investigation of impinging jet by Particle Image Velocimetry (PIV) method has been carried out, with this the same jet configuration and Re number were used in LES and in measurements. For the purpose of LES study of turbulent structure of impinging jet, several modern techniques of vortex identification (vorticity modulus, Q- and λ -criteria, see [7, 8]) have been employed in post-processing of LES results. A typical instantaneous flowfield of impinging jet is demonstrated in Fig. where a plane crosssection of the jet is shown in X,Y-coordinates (with Y denoting here the radial coordinate r) and Y=0 corresponds to jet axis. The jet issued from the nozzle exit (at X=0, Y<7.5 mm) develops downstreams until its impingement onto the wall (located at X=45 mm, see Fig. ) leading to the formation of near-wall jet flow in radial direction. The presence of vortical motion is identified by negative values of λ -criterion [5, 7] this can be seen as coloured spots (spots where λ <0) in Fig.. This λ -field, as well as velocity vector field, indicate the presence of locally-detached largescale vortex structures that evolve over the near-wall flow. From further visualization it is observed that some of these vortex structures are backward-moving and eventually exerting a reverse influence on the impinging jet dynamics. Typical structure of impinging jet flowfield is also Fig.. Instantaneous field of velocity vectors and contours of λ -criterion in impinging jet (zoomed domain, at time moment t=6.4 s).
International Conference on Methods of Aerophysical Research, ICMAR 008 illustrated in Figures 4 where the instantaneous snapshots of axial and radial velocity components and the pressure increment field are demonstrated. The time-averaged turbulent kinetic energy (TKE) field shown in Fig. 5 demonstrates a typical structure of the turbulent intensity field in impinging jet. This structure can be represented as Fig.. Instantaneous field of axial velocity component (at time moment t=7.3 s), m/s. Fig. 3. Instantaneous field of radial velocity component (at time moment t=7.3 s), m/s. Fig. 4. Instantaneous field of pressure (increment over P atm ), Pa. 3
Section II Fig. 5. Mean turbulent kinetic energy field in impinging jet, m /s. consisting of two zones: zone inside the jet shear layer at x/d jet >.5 where the stream turn begins and injection process combined with effect of backward-moving vortex structures (see the vector field in Fig. ) takes place; zone spreading over impingement wall (at 0.5 < r/d jet < 3) where the near-wall flow evolves in which the local instantaneous flow detachments can occur (see Fig. ). The process of evolution of vortex structures is illustrated in Fig. 6 where the development of vorticity modulus ( v ) field at three consecutive time moments is shown. Radial distributions of mean flow quantities obtained in LES computation on the statistical averaging period of s are plotted in Figs. 7 8 at cross-section of x/d jet =. These LES profiles are also compared there against the corresponding data of PIV -D measurements [6]. Profiles of mean velocity axial U and radial V components (non-dimensionalized on the mean axial centerline (r=0) velocity at x/d jet =) are presented in Fig. 7. Radial profiles of turbulent fluctuations are shown U 0 in Fig. 8 where dashed lines denote the mean-square components of velocity fluctuations ' u ', w related to. The solid black line in Fig. 8 denotes the distribution of mean turbulent kinetic energy (TKE), U 0 ( ) k t u v w v ', = ' + ' + ' (with overbar sign herein denoting the time-averaging), while the dots denote TKE values obtained from PIV experiment [6]. It should be noted that only u ' - and v' -components were used to calculate PIV TKE values, neglecting w' -component not measured in PIV. This assumption can be justified by consideration that w' -component obtained in LES computations is negligibly smaller than u ' - and v' -components as seen from Fig. 8. It can be noticed from Fig. 7 that PIV data exhibit somewhat higher rate of jet expansion, thus the jet is seen to be wider at x/d jet = section than it is predicted in LES. This discrepancy may be explained by the fact that the axial position of zone (see above), where turbulent fluctuations start to grow downstreams, is sensitive to the nozzle inlet conditions for which it is difficult to reproduce them in LES modelling with full correspondence to experimental conditions. However for TKE distribution good agreement between LES and PIV data can be noticed from Fig. 8 for the outer side of the jet shear layer (r/d jet >0.5) while some discrepancy is still seen for the inner side. Further improvement of LES predictions can be achieved by the use of more sophisticated SGS model such as the approximate localized dynamic procedure for Smagorinsky model [9]. 4
International Conference on Methods of Aerophysical Research, ICMAR 008 Fig. 6. Vorticity contours in impinging jet at three consecutive time moments (with time step T = 0.05 s). 5
Section II 0.8 0.6 U/U 0, LES U/U 0, PIV data [6] V/U 0, LES V/U 0, PIV data [6] 0.4 0. 0-0. 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r/djet Fig. 7. Radial profiles of dimensionless mean velocity components in impinging jet at x/d jet =. 0.05 0.04 0.03 u' /U 0, LES v' /U 0, LES w' /U 0, LES TKE/U 0, LES TKE/U 0, PIV data [6] 0.0 0.0 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r/djet Fig. 8. Radial profiles of dimensionless mean-square velocity fluctuations and turbulent kinetic energy in impinging jet at x/d jet =. Authors wish to thank their colleagues M.Yu. Hrebtov, D.F. Sikovsky and V.M. Dulin for fruitful discussions. The work has been supported by the Russian Foundation for Basic Research (Grant 06-0-0074-а) and by the Grant for Leading Research Schools ( НШ-6749.006.8). REFERENCES. Sagaut P. Large Eddy Simulation for Incompressible Flows: an Introduction. Springer-Verlag, 00. 39 pages.. Boersma B.J., Brethouwer G., Nieuwstadt F.T.M. A numerical investigation on the effect of the inflow conditions on the self-similar region of a round jet // Phys.Fluids 998, v.0, pp.899-909. 6
International Conference on Methods of Aerophysical Research, ICMAR 008 3. Ilyushin B.B., Krasinsky D.V. Large Eddy Simulation of the Turbulent Round Jet Dynamics // Thermophysics and Aeromechanics 006, Vol. 3, No., pp.43-54. 4. B.B. Ilyushin, D.V. Krasinsky, and M.Yu. Hrebtov LES Study of the Vortex Structure of Turbulent Round Jet // Proc. of 3-th Int. Conf. on the Methods of Aerophysical Research (ICMAR-007, Edt. V.M. Fomin), Novosibirsk: Publ. House Parallel, 007, Part III, pp.4 46. 5. Hrebtov M.Yu. Analysis of vortex identification criteria on the basis of Large Eddy Simulation of a turbulent jet // Abstracts of the IX-th All-Russian Conf. of Young Scientists Actual Problems in Thermophysics and Hydroaerodynamics, pp.9-30 (in Russian) / Novosibirsk, Russia, October 7-0, 006. 6. S.V. Alekseenko, A.V. Bilsky, V.M. Dulin, D.M. Markovich Experimental study of an impinging jet with different swirl rates // Int. J. Heat&Fluid Flow 007, vol. 8, pp. 340 359. 7. Jeong J., Hussain F. On the identification of a vortex // J. Fluid Mech. 995, v.85, pp.69-94. 8. Dubief Y., Delcayre F. On coherent-vortex identification in turbulence // Journal of Turbulence 000, v.(0). 9. Piomelli U., Liu J. Large-eddy simulation of rotating channel flows using a localized dynamic model // Phys.Fluids 995, v.7(4), pp.839-848. 7