▪ Abstract We review the direct numerical simulation (DNS) of turbulent flows. We stress that DNS is a research tool, and not a brute-force solution to the Navier-Stokes equations for engineering problems. The wide range of scales in turbulent flows requires that care be taken in their numerical solution. We discuss related numerical issues such as boundary conditions and spatial and temporal discretization. Significant insight into turbulence physics has been gained from DNS of certain idealized flows that cannot be easily attained in the laboratory. We discuss some examples. Further, we illustrate the complementary nature of experiments and computations in turbulence research. Examples are provided where DNS data has been used to evaluate measurement accuracy. Finally, we consider how DNS has impacted turbulence modeling and provided further insight into the structure of turbulent boundary layers.

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DIRECT NUMERICAL SIMULATION: A Tool in Turbulence Research

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 5.24×107 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 Reθ = 80 to 1120. Mean and second-order statistics downstream of Reθ = 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.

Simulation of boundary layer transition induced by periodically passing wakes

The dynamics of the sublayer and buffer regions of wall-bounded turbulent flows are analysed using autonomous numerical simulations in which the outer flow, and on some occasions specific wavelengths, are masked. The results are compared with a turbulent channel flow at moderate Reynolds number. Special emphasis is put on the largest flow scales. It is argued that in this region there are two kinds of large structures: long and narrow ones which are endogenous to the wall, in the sense of being only slightly modified by the presence or absence of an outer flow, and long and wide structures which extend to the outer flow and which are very different in the two cases. The latter carry little Reynolds stress near the wall in full simulations, and are largely absent from the autonomous ones. The former carry a large fraction of the stresses in the two cases, but are shown to be quasi-linear passive wakes of smaller structures, and they can be damped without modifying the dynamics of other spectral ranges. They can be modelled fairly accurately as being infinitely long, and it is argued that this is why good statistics are obtained in short or even in minimal simulation boxes. It is shown that this organization implies that the scaling of the near-wall streamwise fluctuations is anomalous.

The large-scale dynamics of near-wall turbulence

Neuro-heuristics for nonlinear singular Thomas-Fermi systems

The present review article addresses the vibration behavior of bladed disks encountered e.g. in aircraft engines as well as industrial gas and steam turbines. The utilization of the dissipative effects of dry friction in mechanical joints is a common means of the passive mitigation of structural vibrations caused by aeroelastic excitation mechanisms. The prediction of the vibration behavior is a scientific challenge due to (a) the strongly nonlinear contact interactions involving local sticking, sliding and liftoff, (b) the model order required to accurately describe the dynamic behavior of the assembly, and (c) the multi-disciplinary character of the problem associated with the need to account for structural mechanical as well as fluid dynamical effects. The purpose of this article is the overview and discussion the current state of the art of vibration prediction approaches. The modeling approaches in this work embrace the description of the rotating bladed disk, the contact modeling, the consideration of aeroelastic effects, appropriate model order reduction techniques and the exploitation of the rotationally periodic nature of the problem. The simulation approaches cover the direct computation of periodic, steady-state externally forced and self-excited vibrations using the high-order harmonic balance method, the formulation of the contact problem in the frequency domain, methods for the solution of the governing algebraic equations and advanced simulation approaches, including the concept of nonlinear modes.

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Vibration Prediction of Bladed Disks Coupled by Friction Joints

The problem of determining the maximum forced response vibration amplification that can be produced just by the addition of a small mistuning to a perfectly cyclical bladed disk still remains not completely clear. In this paper we apply a recently introduced perturbation methodology, the Asymptotic Mistuning Model (AMM), to determine which are the key ingredients of this amplification process, and to evaluate the maximum mistuning amplification factor that a given modal family with a particular distribution of tuned frequencies can exhibit. A more accurate upper bound for the maximum forced response amplification of a mistuned bladed disk is obtained from this description, and the results of the AMM are validated numerically using a simple mass-spring model.Copyright © 2008 by ASME

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Asymptotic Description of Maximum Mistuning Amplification of Bladed Disk Forced Response

A new simple asymptotic mistuning model, which constitutes an extension of the well known fundamental mistuning model for groups of modes belonging to a modal family exhibiting a large variation of the tuned vibration characteristics, is used to analyze the effect of mistuning on the stability properties of aerodynamically unstable rotors. The model assumes that both the aerodynamics and the structural dynamics of the assembly are linear, and retains the first-order terms of a fully consistent asymptotic expansion of the tuned system where the small parameter is the blade mistuning. The simplicity of the model allows the optimization of the blade mistuning pattern to achieve maximum rotor stability. The results of the application of this technique to realistic welded-in-pair and interlock low-pressure-turbine rotors are also presented.

Stability Increase of Aerodynamically Unstable Rotors Using Intentional Mistuning

A method to address the rotor/stator interaction problem on an edge-based Reynolds-averaged Navier–Stokes solver is presented. The implementation of the phase-lagged boundary conditions and a new original conservative discretization of the sliding plane in an unstructured solver is discussed in detail. The numerical method is implicit in time, and convergence is accelerated using a multigrid technique where the coarser grids are obtained by agglomeration. Results of the present method are compared with the experimental results obtained at the National Aerospace Laboratory (The Netherlands) for a subsonic fan.

Efficient Edge-Based Rotor/Stator Interaction Method

Linear and Non-Validation of a Linear Multigrid Accelerated Unstructured Navier-Stokes Solver for the Computation of Turbine Blades on Hybrid Grids

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis in its impact on the scaling of the work per cycle curves using an asymptotic approach (Part I) and numerical simulations (Part II). A perturbation analysis of the linearized Navier-Stokes equations for real modes at low reduced frequency is presented and some conclusions are drawn. The first important result is that the loading of the airfoil plays an essential role in the trends of the phase and modulus of the unsteady pressure caused by the vibration of the airfoil. For lightly loaded airfoils the unsteady pressure and the influence coefficients scale linearly with the reduced frequency whereas the phase departs from π/2 and changes linearly with the reduced frequency. As a consequence the work-per-cycle scales linearly with the reduced frequency for any inter-blade phase angle and it is independent of its sign. For highly loaded airfoils the unsteady pressure modulus is fairly constant exhibiting only a small correction with the reduced frequency, while the phase departs from zero and varies linearly with it. In this case only the mean value of the work-per-cycle scales linearly with the reduced frequency. This behavior is independent of the geometry of the airfoil and the modeshape in first approximation. For symmetric cascades the work-per-cycle scales linearly with the reduced frequency irrespectively of whether the airfoil is loaded or not. Simulations using a frequency domain linearized Navier-Stokes solver have been carried out on a low-pressure turbine airfoil section, the NACA0012 and NACA65 profiles and a flat plate to show the generality and correctness of the analytical conclusions (Part II of the corresponding paper). Both, the traveling-wave and influence coefficient formulations of the problem are used in combination to increase the understanding and explore the nature of the unsteady pressure perturbations.Copyright © 2015 by ASME

Roque Corral

Papers

Asymptotic Description of Maximum Mistuning Amplification of Bladed Disk Forced Response

Stability Increase of Aerodynamically Unstable Rotors Using Intentional Mistuning

Efficient Edge-Based Rotor/Stator Interaction Method

Linear and Non-Validation of a Linear Multigrid Accelerated Unstructured Navier-Stokes Solver for the Computation of Turbine Blades on Hybrid Grids

The Low Reduced Frequency Limit of Vibrating Airfoils—Part I: Theoretical Analysis