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Current capabilities for modelling turbulence in industrial flows

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Abstract

This paper summarizes some of the more widely applied or promising schemes for computing heat and momentum transport in industrially relevant flows. Such flows typically involve complex flow domains, severe pressure gradients and regions of flow separation and reattachment. Models tuned by reference to equilibrium, simple shear flows cannot in general be relied upon to predict accurately the effects of these complexities on the transport processes. The main conclusions drawn are that second-moment closure offers a far more reliable basis for computing non-equilibrium turbulent flows than eddy-viscosity schemes, especially in flows with very complex strain fields or those substantially affected by external force fields. Moreover, where significant variations in shear stress occur across the near-wall viscosity-affected sublayer, the usual practice of employing wall functions needs to be replaced by at least a two-equation model in order to capture, even qualitatively, the consequent effects on wall heat transfer or skin friction coefficient.

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Abbreviations

A :

measure of ‘flatness’ of turbulent stress (equation (15))

A 2,A 3 :

second and third invariants ofa ij

a ij :

dimensionless anisotropic Reynolds stress\({{\left( {\overline {u_i u_j } - \tfrac{1}{3}\delta _{ij} \overline {u_k u_k } } \right)} \mathord{\left/ {\vphantom {{\left( {\overline {u_i u_j } - \tfrac{1}{3}\delta _{ij} \overline {u_k u_k } } \right)} k}} \right. \kern-\nulldelimiterspace} k}\)

C p :

specific heat at constant pressure

dϕ :

net diffusion rate of quantityφ

F ij :

rate of creation of\(\overline {u_i u_j } \) by Coriolis forces

G ij :

rate of creation of\(\overline {u_i u_j } \) by (other) body forces

G k :

rate of creation ofk by body forces

k :

turbulent kinetic energy

l :

turbulent length scalek 3/2/ε

l e :

equilibrium length scale

Nu:

Nusselt number

P ij :

rate of creation of\(\overline {u_i u_j } \) by shear

P k :

rate of creation ofk by shear

S ɛ :

source term inε equation

U :

streamwise velocity component

U B :

bulk velocity (pipe flow)

U i :

mean velocity inx i direction

U max :

maximum velocity

\(\overline {u_i u_j } \) :

kinematic Reynolds stress

\(\overline {u_i \theta } \) :

turbulent scalar flux in directionx i

x :

streamwise coordinate

x i :

Cartesian coordinate

y :

coordinate normal to stream flow with origin at wall

β i :

buoyancy vector (product of volumetric expansion coefficient times gravitational acceleration vector)

ΔΘ :

temperature above wall value

ε :

turbulence energy dissipation rate

ε ij :

viscous dissipation rate of\(\overline {u_i u_j } \)

Θ :

mean temperature

θ :

fluctuating temperature

ν t :

turbulent viscosity

ρ :

density

σ :

molecular Prandtl/Schmidt number

σϕ :

turbulent Prandtl number for diffusion ofφ

τ w :

wall shear stress

Φ ij :

pressure strain correlation

Φ ij1,Φ ij2,Φ ij3 :

turbulence, mean-strain and force-field parts ofφ ij

Φ ijw :

wall-reflection contributions toΦ ij

Ω k :

angular velocity vector in directionx k

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  1. Department of Mechanical Engineering, Thermodynamics and Fluid Mechanics Division, UMIST, P.O. Box 88, M60 1QD, Manchester, England

    B. E. Launder

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Launder, B.E. Current capabilities for modelling turbulence in industrial flows. Appl. Sci. Res. 48, 247–269 (1991). https://doi.org/10.1007/BF02008200

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