Abstract
The accurate treatment of finite-rate chemistry is possible by the application of stochastic turbulence models which generalize Reynolds-averaged Navier–Stokes equations. Usually, one considers linear stochastic equations. In this way, fluctuations are generated by uncorrelated forces and relax with a frequency that is independent of the actual fluctuation. It has been proved that such linear equations are well appropriate to simulate near-equilibrium flows. However, the inapplicability or unfeasibility of other methods also results in a need for stochastic methods for more complex flow simulations. Their construction requires an extension of the simple mechanism of linear stochastic equations. Two ways to perform this are investigated here. The first way is the construction of a stochastic model for velocities where the relaxation frequency depends on the actual fluctuation. This is a requirement to involve relevant mixing variations due to large-scale flow structures. The stochastic model developed is applied to the simulation of convective boundary layer turbulence. Comparisons with the results of measurements provide evidence for its good performance and the advantages compared to existing methods. The second way presented here is the construction of scalar equations which involve memory effects regarding to both the stochastic forcing and relaxation of fluctuations. This allows to overcome shortcomings of existing stochastic methods. The model predictions are shown to be in excellent agreement with the results of the direct numerical simulation of scalar mixing in stationary, homogeneous and isotropic turbulence. The consideration of memory effects is found to be essential to simulate correctly the evolution of scalar fields within the first stage of mixing.
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References
Baldyga, J. and Bourne, J.R., Turbulent Mixing and Chemical Reactions. John Wiley & Sons, Chichester (1999).
Canuto, V.M., Turbulent convection with overshooting: Reynolds stress approach. Astrophys. J. 392 (1992) 218–232.
Canuto, V.M., Minotti, F., Ronchi, C., Ypma, R.M. and Zeman, O., Second-order closure PBL model with new third-order moments: Comparison with LES data. J. Atmos. Sci. 51 (1994) 1605–1618.
Craft, T.J., Kidger, J.W. and Launder, B.E., Importance of third-moment modeling in horizontal, stably-stratified flows. In: Proceedings of the 11th Symposium on Turbulent Shear Flows, Grenoble, France (1997) pp. 2013–2018.
Craft, T.J., Developments in a low-Reynolds number second-moment closure and its application to separating and reattaching flows. Internat. J. Heat Fluid Flow 19 (1998) 541–548.
Dopazo, C., Recent developments in PDF methods. In: Libby, P.A. and Williams, F.A. (eds.), Turbulent Reacting Flows. Academic Press, San Diego, CA (1994) pp. 375–474.
Du, S., Wilson, J.D. and Yee, E., Probability density functions for velocity in the convective boundary layer, and implied trajectory models. Atmos. Environ. 28 (1994) 1211–1217.
Du, S., Wilson, J.D. and Yee, E., On the moments approximation method for constructing a Lagrangian stochastic model. Bound.-Layer Meteorol. 40 (1994) 273–292.
Du, S., Sawford, B.L., Wilson, J.D. and Wilson, D.J., Estimation of the Kolmogorov constant (C0) for the Lagrangian structure function, using a second-order Lagrangian model of grid turbulence. Phys. Fluids 7 (1995) 3083–3090.
Durbin, P.A. and Speziale, C.G., Realizability of second-moment closure via stochastic analysis. J. Fluid Mech. 280 (1994) 395–407.
Eswaran, V. and Pope, S.B., Direct numerical simulations of the turbulent mixing of a passive scalar. Phys. Fluids 31 (1988) 506–520.
Fox, R.O., The Fokker-Planck closure for turbulent molecular mixing: Passive scalars. Phys. Fluids A 4 (1992) 1230–1244.
Fox, R.O., Improved Fokker-Planck model for the joint scalar, scalar gradient PDF. Phys. Fluids 6 (1994) 334–348.
Fox, R.O., Computational methods for turbulent reacting flows in the chemical process industry. Rev. Inst. Français du Pétrole 51 (1996) 215–243.
Fox, R.O., On velocity-conditioned scalar mixing in homogeneous turbulence. Phys. Fluids 8 (1996) 2678–2691.
Gardiner, C.W., Handbook of Statistical Methods. Springer-Verlag, Berlin (1983).
Grabert, H., Projection Operator Technique in Nonequilibrium Statistical Mechanics. Springer-Verlag, Berlin (1982).
Heinz, S., Nonlinear Lagrangian equations for turbulent motion and buoyancy in inhomogeneous flows. Phys. Fluids 9 (1997) 703–716.
Heinz, S., Time scales of stratified turbulent flows and relations between second-order closure parameters and flow numbers. Phys. Fluids 10 (1998) 958–973.
Heinz, S., Connections between Lagrangian stochastic models and the closure theory of turbulence for stratified flows. Internat. J. Heat Fluid Flow 19 (1998) 193–200.
Heinz, S. and van Dop, H., Buoyant plume rise described by a Lagrangian turbulence model. Atmos. Environ. 33 (1999) 2031–2043.
Heinz, S., On Fokker-Planck equations for turbulent reacting flows. Part 2. Filter density function for large eddy simulation. Flow, Turb. Combust. 70 (2003) 153–181.
Heinz, S., Statistical Mechanics of Turbulent Flows. Springer-Verlag, Berlin (2003).
Ilyushin, B.B. and Kurbatski, A.F., Modeling of turbulent transport in PBL with third-order moments. In: Proceedings of the 11th Symposium on Turbulent Shear Flows, Grenoble, France (1997) pp. 2019–2024.
Juneja, A. and Pope, S.B., A DNS study of turbulent mixing of two passive scalars. Phys. Fluids 8 (1996) 2161–2184.
Kassinos, S., Reynolds, W.C. and Rogers, M.M., One-point turbulence structure tensors. J. Fluid Mech. 428 (2001) 213–248.
Kawamura, H., Sasaki, J. and Kobayashi, K., Budget and modeling of triple moment velocity correlations in a turbulent channel flow based on DNS. In: Proccedings of the 10th Symposium on Turbulent Shear Flows, Pennsylvania State University (1995) pp. 2613–2618.
Kerstein, A.R., Linear-eddy modeling of turbulent transport. II: Application to shear layer mixing. Combust. Flame 75 (1989) 397–413.
Kerstein, A.R., Linear-eddy modeling of turbulent transport. Part 3. Mixing and differential molecular diffusion in round jets. J. Fluid Mech. 216 (1990) 411–435.
Kerstein, A.R., One-dimensional turbulence: Model formulation and application to homogeneous turbulence, shear flows, and buoyant stratified flows. J. Fluid Mech. 392 (1999) 277–334.
Kuo, K.K., Principles of Combustion. Wiley Interscience, New York (1986).
Launder, B.E., Phenomenological modeling: Present and future. In: Lumley, J.L. (ed.), Wither Turbulence? Turbulence at the Crossroads. Springer-Verlag, Berlin (1990) pp. 439–485.
Lindenberg, K. and West, J.W., The Nonequilibrium Statistical Mechanics of Open and Closed Systems. VHC Publishers, New York (1990).
Luhar, A.K., Hibberd, M.F. and Hurley, P.J., Comparison of closure schemes used to specify the velocity PDF in Lagrangian stochastic dispersion models for convective conditions. Atmos. Environ. 30 (1996) 1407–1418.
Peters, N., Turbulent Combustion. Cambridge University Press, Cambridge (2001).
Pope, S.B., Probability distributions of scalars in turbulent shear flow. In: Bradbury, L.S.S., Durst, F., Launder, B.E., Schmidt, F.W. and Whitelaw, J.H. (eds.), Turbulent Shear Flows 2. Springer-Verlag, Berlin (1980) pp. 7–16.
Pope, S.B., PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11 (1985) 119–192.
Pope, S.B., On the relationship between stochastic Lagrangian models of turbulence and second-moment closures. Phys. Fluids 6 (1994) 973–985.
Pope, S.B., The vanishing effect of molecular diffusivity on turbulent dispersion: Implications for turbulent mixing and the scalar flux. J. Fluid Mech. 359 (1998) 299–312.
Pope, S.B., Turbulent Flows. Cambridge University Press, Cambridge (2000).
Risken, H., The Fokker-Planck Equation. Springer-Verlag, Berlin (1984).
Sawford, B.L., Recent developments in the Lagrangian stochastic theory of turbulent dispersion. Bound.-Layer Meteorol. 62 (1993) 197–215.
Sawford, B.L., Rotation of Lagrangian stochastic models of turbulent dispersion. Bound.-Layer Meteorol. 93 (1999) 411–424.
Speziale, C.G. and Xu, X., Towards the development of second-order closure models for nonequilibrium turbulent flows. Internat. J. Heat Fluid Flow 17 (1996) 238–244.
Thomson, D.J., Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J. Fluid Mech. 180 (1987) 529–556.
Thomson, D.J. and Montgomery, M.R., Reflection boundary conditions for random walk models of dispersion in non-Gaussian turbulence. Atmos. Environ. 28 (1994) 1981–1987.
Valiño, L. and Dopazo, C., A binomial Langevin model for turbulent mixing. Phys. Fluids A 3 (1991) 3034–3037.
Van Dop, H., Nieuwstadt, F.T.M. and Hunt, J.C.R., Random walk models for particle displacements in inhomogeneous unsteady turbulent flows. Phys. Fluids 28 (1985) 1639–1653.
Wilson, J.D. and Sawford, B.L., Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere. Bound.-Layer Meteorol. 78 (1996) 191–210.
Zubarev, D., Morozov, V. and Röpke, G., Statistical Mechanics of Nonequilibrium Processes. Vol. 1: Basic Concepts, Kinetic Theory. Akademie Verlag (VCH Publishers), Berlin (1996).
Zubarev, D., Morozov, V. and Röpke, G., Statistical Mechanics of Nonequilibrium Processes. Vol. 2: Relaxation and Hydrodynamic Processes. Akademie Verlag (VCH Publishers), Berlin (1997).
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Heinz, S. On Fokker–Planck Equations for Turbulent Reacting Flows. Part 1. Probability Density Function for Reynolds-Averaged Navier–Stokes Equations. Flow, Turbulence and Combustion 70, 115–152 (2003). https://doi.org/10.1023/B:APPL.0000004933.17800.46
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DOI: https://doi.org/10.1023/B:APPL.0000004933.17800.46