Reynolds-stress and triple-product models applied to flows with rotation and curvature

Abstract

Turbulence models, with increasing complexity, up to triple product terms, are applied to the flow in a rotating pipe. The rotating pipe is a challenging case for turbulence models as it contains significant rotational and curvature effects. The flowfield starts with the classic fully developed pipe flow, with a stationary pipe wall. This well defined condition is then subjected to a section of pipe with a rotating wall. The rotating wall introduces a second velocity scale, and creates Reynolds shear stresses in the radial-circumferential and circumferential-axial planes. Furthermore, the wall rotation introduces a flow stabilization, and actually reduces the turbulent kinetic energy as the flow moves along the rotating wall section. It is shown in the present work that the Reynolds stress models are capable of predicting significant reduction in the turbulent kinetic energy, but triple product improves the predictions of the centerline turbulent kinetic energy, which is governed by convection, dissipation and transport terms, as the production terms vanish on the pipe axis. NomenclatureM ∞ free stream Mach number R pipe radius (reference length) R ij Reynolds-stress tensor = u i u j Re τ wall shear Reynolds number [= u τ R/ν w ] S ij strain rate tensor = 1 2 ∂u i ∂x j + ∂u j ∂x i

Turbulent Axial Odometer Model

“…The remainder of the model definition follows the previous TTR and Reynolds stress development closely [10,12,[19][20][21] The TAO turbulence model including turbulent transport (T i jk ) terms is:…”

“…The numbers associated with each curve in the legend are the value of Re θ for each curve, along with a leading initial denoting the experimental team. Again, as in the precursor [12], the region of interest is above the viscous sublayer and buffer layer, z + > 60. Numerous experimental results are shown, parameterized by Re θ .…”

“…The equation is designed to provide a non-dimensional quantity that is a Reynolds number formed by the length of streamlines continuously inside turbulent flow, the square root of the turbulent kinetic energy, and the molecular velocity ( √ kl p /ν). The equation is Galilean invariant, and when coupled with a modified version of the triple product model [12] provides a reasonable estimate of the axial Reynolds stress variations seen in experiment [4,[6][7][8]-something that no other current turbulence model is capable of.…”

Using Adaptive Mesh Refinement to Study Grid Resolution Effects for Shock-Boundary Layer Interactions

“…With judicious grid creation, a solution that would be considered grid converged can easily be obtained with 358 × 513 grid points. This grid system was devised as a baseline for the TTR [3,9,10] turbulence model, a model which can require more grid to obtain truly grid converged solutions, and was believed to be essentially grid converged for eddy viscosity models. A comparison of solutions for surface pressure, skin friction and a velocity profile just upstream of the shock/boundarylayer interaction region is shown in Fig.…”

Revised Reynolds-Stress and Triple Product Models

“…These are an attempt to fulfill the need for turbulent transport predictions in regions of separation, where their relative importance is larger than it is for attached flows.The TTR model was modified slightly when it was applied to a rotating pipe flow. 8 The modifications that improved separated flow prediction on the Bachalo-Johnson bump also improved predictions in the similar region of the pipe axis, where turbulent production vanishes and turbulent transport is balanced by convection and dissipation. This gives evidence that the separation prediction improvements were the result of improved physical modelling and not simply the addition of an additional model degree of freedom.One of the remaining issues with the higher order Lag models is that they are tuned to a outdated model of the flat plate Reynolds-stress distribution.…”