An algebraic non-equilibrium turbulence model of the high Reynolds number transition region

Abstract

We present a mixing length based algebraic turbulence model calibrated to pipe flow; the main purpose of the model is to capture the increasing turbulence production-to-dissipation ratio observed in connection with the high Reynolds number transition region. The model includes the mixing length description by Gersten and Herwig which takes the observed variation of the von Kármán number with Reynolds number into account. Pipe wall roughness effects are included in the model. Results are presented for area-averaged (integral) quantities which can be used both as a self-contained model and as initial inlet boundary conditions for computational fluid dynamics simulations.

An Algebraic Non-Equilibrium Turbulence Model of the High Reynolds Number Transition Region

“…In [11], we show that the global von Kármán number κ g transitions from a lower value to a higher value with increasing Re τ . Three mixing length definitions are considered and the decision is made to proceed with the Gersten-Herwig (subscript "G-H") expression.…”

“…where the subscript "g" indicates "global", A g,mean = 1.01 [4] and the global von Kármán number κ g is a function of Re τ [11]; z is the distance from the wall and z + = zu τ /ν kin is the normalised distance from the wall. Note that z/δ = z + /Re τ .…”

“…Here, we summarise the main components of our model; for details and derivations, we refer to the SI [11].…”

“…For equlibrium flows, we use the Prandtl (subscript "P") characteristic velocity, and for non-equilibrium flows, we use the Kolmogorov-Prandtl (subscript "K-P") characteristic velocity [11]. For the turbulent viscosity and the Reynolds (shear) stress of the streamwise fluctuating velocity u and the wall-normal fluctuating velocity v, we write:…”

“…The paper is organised as follows: In Section 2, we summarise previous relevant findings, followed by a model overview in Section 3. Model details can be found in the Supplementary Information (SI) document [11]. Results from our new model are provided in Section 4 and discussed in Section 5.…”