Reynolds equations for turbulent flow Turbulence Jet in

Reynolds equations for turbulent flow

Turbulence Jet in cross-flow Vizualization Instantaneous

Turbulence • Statistics – time mean quantities – Mean value – Variance

Reynolds decomposition • Dec. – Mean value – Fluctuation • Properties • NS eq.

Ensemble average • • Quantity Ensemble average Ergodic process Estimator Choice of T, n

EA – choice of n and T • Statistically independent n > 200 • Pseudo-periodical sig. T >> Tsmax spectrum Tsmax

Properties of averaging • Linearity • Commutativity – with der. – with int. F of time const • Double averaging • Averaging of product • ATTENTION!!! Any nonlinear function!!!

Continuity equation instantaneous velocity field Is solenoidal Reynolds: Averaging Both mean and fluktuation velocity fields are solenoidal

Substantial derivative NONLINEAR ! Averaging ? Integration per-partés: = 0 (continuity)

Substantial derivative Extra term !

Reynolds equations Reynolds 1894 N-S eq. Averaging Reynolds eq. 0 Extra term !

Solvability of RANS • Equations: 3 + 1 • Unknowns: 3 1 Total 10 PROBLEM!!! Extra eqs are needed! 6 (symm)

Extra eqs for correlations Eq for fluctuations (subtract RANS from NS): Multiply by than averaging

Eqs for Reynolds stress • I the time change of the local correlation tensor, • II advection of Reynolds stress through a mean flow (not total!), • III interaction between the mean and fluctuation components of the flow and is related to the production of Reynolds stress, • IV advection in relation to the fluctuation part of the flow, • V effect of pressure, • VI diffusion and dissipation resulting from viscosity, which is expressed mainly in small scale perturbations

Eqs for Reynolds stress • • 4 + 6 = 10 eqs Reynolds stress Unknown tensors of 2 nd order Unknown tensor of 3 rd order Overall 75 unknowns !!!

Averaged Poisson eq. • Averaging

Reynolds stress Rearrangement Stress 1 Mean pressure Stress 2 Mean viscous Stress 3 Velocity fluctuations Reynolds stress

Reynolds stress • Reynolds stress tensor • 2 nd order tensor – Symmetric – Semidefinite

Reynolds stress • Physical meaning • Mean momentum flux in i direction induced by fluctuations in j direction • Mean momentum flux in j direction induced by fluctuations in i direction • Always 3 D!

Reynolds stress • Covariance tensor properties • Symmetric • Decomp. (isotropy): – Isotropic part IRROTATIONAL – Anisotropic part ROTATIONAL Kinetic energy

Reynolds stress x 2 x 1 • A: • B: A B For any transversal movement within BL the POSITIVE Reynolds stress is generated

Reynolds stress x 2 x 1 Isotropic fluctuations R 12 = 0 ISOTROPIC TURBULENCE x 1 Anisotropic fluctuations R 12 > 0 BOUNDARY LAYER

Quantities • Kinetic energy • Intensity of fluctuations Intensity of turbulence Isotropic turbulence

Dissipation rate • rate of dissipation Specific dissipation rate Izotropic turbulence

Reynolds equations Anisotropic part Rij Joseph Valentin Boussinesq 1877 Hypothesis of turbulent viscosity

Turbulent viscosity • Molecular viscosity – material „constant“, property of fluid • Turbulent viscosity – property of flow – Function of space – Function of time (for nonstationary case) – Is solution – Is input

Prandtl mixing length • Estimate x 2 u 1 Boundary layer x 1 x 2 Border of the BL ttot tt x 1

Transport equation • Advection • Production • Transport • Dissipation