Convective discretization schemes for the turbulence transport equations in flow predictions through sharp u‐bends

Abstract

This paper presents finite volume computations of turbulent flow through a square cross‐sectioned U‐bend of curvature strong enough (Rc/D =0.65) to cause separation. A zonal turbulence modelling approach is adopted, in which the high‐Re k‐ε model is used over most of the flow domain with the low‐Re, I‐equation model of k‐transport employed within the near‐wall regions. Computations with grids of different sizes and also with different discretization schemes, demonstrate that for this flow the solution of the k and ε equations is more sensitive to the scheme employed in their convective discretization than the solution of the mean flow equations. To avoid the use of extremely fine 3‐Dimensional grids, bounded high order schemes need to be used in the discretization of the turbulence transport equations. The predictions, while encouraging, displayed some deficiencies in the downstream region due to deficiencies in the turbulence model. Evidently, further refinements in the turbulence model are necessary. Initial computations of flow and heat transfer through a rotating U‐bend, indicate that at rotational numbers (Ro = ΩD/Wb) relevant to blade cooling passages, the Coriolis force can substantially modify the hydrodynamic and thermal behaviour.

Numerical study of the thermal development in a rotating cooling passage

“…-1 1 2.5 -10 -5 5 -10 5 -5 Table 3 Constants appearing in turbulence model [11,16] supported this conclusion. Song et al [12] developed a higher order bounded numerical scheme named WACEB (weighted average coefficients ensuring boundedness) for interpolating the fluxes at the cell faces.…”

“…They employed three turbulence models k À e eddy viscosity model with one equation in the near wall region, a low-Re k À e eddy viscosity model, and a low-Re algebraic stress model. However, there are very limited numerical studies on the secondary flows inside rotating ducts oriented some angles with rotation axis [2,3,5,[9][10][11][12]. More recently, Hu et al [13] performed flow and heat transfer in the tip-turn region of a U-duct under rotating and non-rotating conditions with the passage with dimpled surface.…”

The computation of turbulent flow through stationary and rotating U-bends with rib-roughened surfaces

“…In the case of the DSM model, use of the apparent viscosity concept prevents numerical oscillations arising from the explicit presence of the Reynolds stress gradients in the momentum transport equations. For the discretisation of convective transport, earlier attempts to compute flows through U-bends of curvature strong enough to cause flow separation [7] revealed that it was necessary to employ a high-order discretisation scheme for all transport equations. The scheme previously employed, a bounded form of the QUICK [14] scheme proposed by Zhu and Leschziner [15] called LODA, while effective in minimising numerical errors, led to problems of numerical stability.…”

“…Moreover, as work in the author's group revealed [3,6], the introduction of second-moment closures, instead of using effective viscosity models, further improves flow and heat transfer predictions, for both curved and rotating flows. For flows in smooth U-bends of strong curvature, strong enough to cause flow separation, the author's earlier work [7], provided evidence that it also becomes necessary to employ high-order schemes for the discretisation of convective transport, in all transport equations for both mean and turbulent flow variables. The recent emergence of detailed LDA measurements [8] for flow through tight U-bends, stationary and rotating, motivated further numerical investigations [9].…”

The computation of flow and heat transfer through an orthogonally rotating square‐ended U‐bend, using low‐Reynolds‐number models

“…In the authors' group, previous numerical investigations and most earlier experimental studies focused on round-ended U-bends, shown in Figure 1b, see Bo et al [3] and Iacovides et al [4]. These studies employed low-Reynolds-number models at both effective-viscosity and second-moment closure levels.…”