Paul A. Durbin

A Reynolds stress model for near-wall turbulence

A tensorially consistent near-wall second-order closure model is formulated. Redistributive terms in the Reynolds stress equations are modelled by an elliptic relaxation equation in order to represent strongly non-homogeneous effects produced by the presence of walls; this replaces the quasi-homogeneous algebraic models that are usually employed, and avoids the need for ad hoc damping functions. A quasi-homogeneous model appears as the source term in the elliptic relaxation equation-here we use the simple Rotta return to isotropy and isotropization of production formulae. The formulation of the model equations enables appropriate boundary conditions to be satisfied. The model is solved for channel flow and boundary layers with zero and adverse pressure gradients. Good predictions of Reynolds stress components, mean flow, skin friction and displacement thickness are obtained in various comparisons to experimental and direct numerical simulation data. The model is also applied to a boundary layer flowing along a wall with a 90°, constant-radius, convex bend. Because the model is of a general, tensorially invariant form, special modifications for curvature effects are not needed; the equations are simply transformed to curvilinear coordinates. The model predicts many important features of this flow. These include: the abrupt drop of skin friction and Stanton number at the start of the curve, and their more gradual recovery after the bend; the suppression of turbulent intensity in the outer part of the boundary layer; a region of negative (counter-gradient) Reynolds shear stress; and recovery from curvature in the form of a Reynolds stress ‘bore’ propagating out from the surface. A shortcoming of the present model is that it overpredicts the rate of this recovery. A heat flux model is developed. It is shown that curvature effects on heat transfer can also be accounted for automatically by a tensorially invariant formulation.

Simulations of bypass transition

Bypass transition in an initially laminar boundary layer beneath free-stream turbulence is simulated numerically. New perspectives on this phenomenon are obtained from the numerical flow fields. Transition precursors consist of long backward jets contained in the fluctuating u -velocity field; they flow backwards relative to the local mean velocity. The jets extend into the upper portion of the boundary layer, where they interact with free-stream eddies. In some locations a free-stream perturbation to the jet shear layer develops into a patch of irregular motion – a sort of turbulent spot. The spot spreads longitudinally and laterally, and ultimately merges into the downstream turbulent boundary layer. Merging spots maintain the upstream edge of the turbulent region. The jets, themselves, are produced by low-frequency components of the free-stream turbulence that penetrate into the laminar boundary layer. Backward jets are a component of laminar region streaks. A method to construct turbulent inflow from Orr–Sommerfeld continuous modes is described. The free-stream turbulent intensity was chosen to correspond with the experiment by Roach & Brierly (1990). Ensemble-averaged numerical data are shown to be in good agreement with laboratory measurements.

Numerical study of turbulent heat transfer in confined and unconfined impinging jets

An elliptic relaxation turbulence model (v2−f model) has been used to simulate the flow and heat transfer in circular confined and unconfined impinging jet configurations. The model has been validated against available experimental data sets. Results have been obtained for a range of jet Reynolds numbers and jet-to-target distances. The effects of confinement on the local heat transfer behavior has been determined. It has been shown that confinement leads to a decrease in the average heat transfer rates, but the local stagnation heat transfer coefficient is unchanged. The effect of confinement is only significant in very low nozzle-to-plate distances (H/D<0.25). In contrast, the flow characteristics in the nozzle strongly affects the heat transfer rate, especially in the stagnation region. Quantitative (up to 30% difference) and qualitative differences have been obtained when different nozzle velocity profiles were used.

Reynolds averaged simulation of unsteady separated flow

The accuracy of Reynolds averaged Navier–Stokes (RANS) turbulence models in predicting complex flows with separation is examined. The unsteady flow around square cylinder and over a wall-mounted cube are simulated and compared with experimental data. For the cube case, none of the previously published numerical predictions obtained by steady-state RANS produced a good match with experimental data. However, evidence exists that coherent vortex shedding occurs in this flow. Its presence demands unsteady RANS computation because the flow is not statistically stationary. The present study demonstrates that unsteady RANS does indeed predict periodic shedding, and leads to much better concurrence with available experimental data than has been achieved with steady computation.

Simulation of boundary layer transition induced by periodically passing wakes

The interaction between an initially laminar boundary layer developing spatially on a flat plate and wakes traversing the inlet periodically has been simulated numerically. The three-dimensional, time-dependent Navier–Stokes equations were solved with 5.24×10 7 grid points using a message passing interface on a scalable parallel computer. The flow bears a close resemblance to the transitional boundary layer on turbomachinery blades and was designed following, in outline, the experiments by Liu & Rodi (1991). The momentum thickness Reynolds number evolves from Re θ = 80 to 1120. Mean and second-order statistics downstream of Re θ = 800 are of canonical flat-plate turbulent boundary layers and are in good agreement with Spalart (1988). In many important aspects the mechanism leading to the inception of turbulence is in agreement with previous fundamental studies on boundary layer bypass transition, as summarized in Alfredsson & Matsubara (1996). Inlet wake disturbances inside the boundary layer evolve rapidly into longitudinal puffs during an initial receptivity phase. In the absence of strong forcing from free-stream vortices, these structures exhibit streamwise elongation with gradual decay in amplitude. Selective intensification of the puffs occurs when certain types of turbulent eddies from the free-stream wake interact with the boundary layer flow through a localized instability. Breakdown of the puffs into young turbulent spots is preceded by a wavy motion in the velocity field in the outer part of the boundary layer. Properties and streamwise evolution of the turbulent spots following breakdown, as well as the process of completion of transition to turbulence, are in agreement with previous engineering turbomachinery flow studies. The overall geometrical characteristics of the matured turbulent spot are in good agreement with those observed in the experiments of Zhong et al . (1998). When breakdown occurs in the outer layer, where local convection speed is large, as in the present case, the spots broaden downstream, having the vague appearance of an arrowhead pointing upstream. The flow has also been studied statistically. Phase-averaged velocity fields and skin-friction coefficients in the transitional region show similar features to previous cascade experiments. Selected results from additional thought experiments and simulations are also presented to illustrate the effects of streamwise pressure gradient and free-stream turbulence.

Mode interaction and the bypass route to transition

The manner by which external vortical disturbances penetrate the laminar boundary layer and induce transition is explored. Linear theory suggests that the well-known Klebanoff mode precursor to transition can be understood as a superposition of Squire continuous modes. Shear sheltering influences the ability of free-stream disturbances to generate a packet of Squire modes. A coupling coefficient between continuous spectrum spectrum Orr–Sommerfeld and Squire modes is used to characterize the interaction. Full numerical simulations with prescribed modes at the inlet substantiate this approach. With two weakly coupled modes at the inlet, the boundary layer is little perturbed; with two strongly coupled modes, Klebanoff modes are produced; with one strongly coupled and one weakly coupled high-frequency mode, the complete transition process is simulated.

Perturbed vortical layers and shear sheltering

Abstract New theoretical results and physical interpretations are presented concerning the interactions between different types of velocity fields that are separated by thin interfacial layers, where there are dynamically significant variations of vorticity across the layers and, in some cases within them. It is shown how, in different types of complex engineering and environmental flow, the strengths of these interactions vary from the weakest kind of superposition to those where they determine the flow structure, for example by mutual exclusion of velocity fields from the other region across the interface, or by local resonance near the interface. We focus here on the excluding kinds of interactions between, on the one hand, elongated and compact regions containing vortical flows and large variations in velocity, and on the other hand various kinds of weak perturbation in the surrounding external flow region: rotational, irrotational; time-varying, steady; large, small; coplanar, non-coplanar; non-diffusive, diffusive. It is shown how all these kinds of external disturbances can be wholly, or partially, ‘blocked’ at the interface with the vortical region, so that beyond a certain sheltering distance into the interior of this region the fluctuations can be very small. For the special case of quasi-parallel co-planar external straining motions outside non-directional shear flows, weak sheltering occurs if the mean velocity of the shear flow increases – otherwise the perturbations are amplified. For non-parallel flows, the sheltering effect can be greater when the vorticity is distributed in thin vortex sheets. The mechanism whereby the vortical flow induces ‘blocking’ and ‘shear-sheltering’ effects can be quantitatively explained in terms of the small adjustments of the vorticity in the vortical layers, and in some cases by the change in impulse of these layers. If the vorticity in the outer part of the vortical region is weak, it can be ‘stripped away’ by the external disturbances until the remaining vorticity is strong enough to ‘block’ the disturbances and shelter the inner flow of the vortical region. The mechanisms presented here appear to explain on the one hand some aspects of the observed robustness of vortical structures and jet or plume like shear flows in turbulent and geophysical flows, and on the other hand the levels of external perturbation needed to erode or breakdown turbulent shear flows.

Toward Improved Prediction of Heat Transfer on Turbine Blades

Reynolds averaged computations of turbulent flow in a transonic turbine passage are presented to illustrate a manner in which widely used turbulence models sometimes provide poor heat transfer predictions. It is shown that simple, physically and mathematically based constraints can substantially improve those predictions.

Shear sheltering and the continuous spectrum of the Orr–Sommerfeld equation

The expansion into eigenfunctions of a general disturbance in a viscous flow is possible only when both the discrete and continuous modes of the Orr–Sommerfeld equation are employed. Proper implementation of the boundary conditions and a method for computation of the continuous modes are developed. The unique phenomenon known as shear sheltering is discussed and illustrated. It is shown that the penetration depth of disturbances into the boundary layer has a dependence on frequency and Reynolds number similar to that of a Stokes layer. A simple model that captures this dependence is developed.

Application of a near-wall turbulence model to boundary layers and heat transfer

Abstract A near-wall turbulence model based on k , ϵ, and v 2 equations is described. It is used to predict flow and heat transfer in a two-dimensional channel and in boundary layers. Good agreement with data on skin friction, Stanton number, mean velocity, and turbulent intensities is obtained. Solutions to the model show the correct Reynolds number dependence without building it into any of the coefficients. Zero and adverse pressure gradient boundary layers are calculated; in both cases, the results agree well with experiment.