New scaling laws for turbulent Poiseuille flow with wall transpiration

Abstract

Under consideration for publication in J. Fluid Mech. 1 New Scaling Laws for Turbulent PoiseuilleFlow with Wall Transpiration A fully developed, turbulent Poiseuille flow with wall transpiration, i.e. uniform blowing and suction on the lower and upper walls correspondingly, is investigated by both direct numerical simulation (DNS) of the three-dimensional, incompressible Navier-Stokes equations and Lie symmetry analysis. The latter is used to find symmetry transformations and in turn to derive invariant solutions of the set of two-and multi-point correlation equations. We show that the transpiration velocity is a symmetry breaking which implies a logarithmic scaling law in the core of the channel. DNS validates this result of Lie symmetry analysis and hence aids establishing a new logarithmic law of deficit-type. The region of validity of the new logarithmic law is very different from the usual near-wall log-law and the slope constant in the core region differs from the von Kármán constant and is equal to 0.3. Further, extended forms of the linear viscous sublayer law and the near-wall log-law are also derived, which, as a particular case, include these laws for the classical non-transpirating case. The viscous sublayer at the suction side has an asymptotic suction profile. The thickness of the sublayer increase at high Reynolds and transpiration numbers. For the near-wall log-law we see an indication that it appears at the moderate transpiration rates (0.05 < v 0 /u τ < 0.1) and only at the blowing wall. Finally, from the DNS data we establish a relation between the friction velocity u τ and the transpiration v 0 which turns out to be linear at moderate transpiration rates.

Symmetries and their importance for statistical turbulence theory

“…This content is based mainly on the PhD thesis Rosteck (2013) and papers where results of DNS calculations were compared with the theoretical findings i.e. Mehdizadeh and Oberlack (2010), Avsarkisov et al (2014), Oberlack and Rosteck (2010).…”

“…Therefore, the scaling symmetries, as stated in the previous sections can only be obtained in modified way. In the following, we will calculate different scaling laws and compare them to the recent data of Avsarkisov et al (2014). Older DNS data for this flow are available from Sumitani and Kasagi (1995), who studied this case involving heat transfer.…”

“…The mean velocity (366) and the Reynolds stresses (371-374) are compared to DNS data of Avsarkisov et al (2014) Re= 250, U T = 0.16 A list of parameters for the scaling laws (371-374) is given in 14,8 -111,8 -0,016 law as the latter represents the most general scaling law we derived from the governing symmetries.…”

“…which occur in the Reynolds stress tensor and can be generalized corresponding to the given formulas (356-360), so that finally the concise expressions R + 11 = −C 2 1 (A +x 2 ) 2γ + C 11 (A +x 2 ) γ−1 + E(A + 2x 2 )(A +x 2 ) γ−1 + B 11 + D 11x2 , In the following we compare both the algebraic and logarithmic approaches to the DNS data of Avsarkisov et al (2014) at the same Reynolds number and transpiration velocity. In Figure 14 the logarithm solution (375) and (378) (left-hand side graphs) and the algabraic solution (380) and (384) (right-hand side graphs) are fitted.…”

“…From these figures we observe that the algebraic approach can fit all quantities in a somewhat broader region than the logarithmic approach. Both approaches work similarly well also for different Reynolds numbers and transpiration velocities, see In order to distinguish which of the both approaches is to be preferred we have listed in table 9 and 10 the values of the respective formulas (375), (378) and (380), (384) fitted to the DNS data of Avsarkisov et al (2014).…”