Neil D. Sandham

Three-dimensional simulations of large eddies in the compressible mixing layer

The effect of Mach number on the evolution of instabilities in the compressible mixing layer is investigated. The full time-dependent compressible Navier–Stokes equations are solved numerically for a temporally evolving mixing layer using a mixed spectral and high-order finite difference method. The convective Mach number Mc (the ratio of the velocity difference to the sum of the free-stream sound speeds) is used as the compressibility parameter. Simulations with random initial conditions confirm the prediction of linear stability theory that at high Mach numbers (Mc > 0.6) oblique waves grow more rapidly than two-dimensional waves. Simulations are then presented of the nonlinear temporal evolution of the most rapidly amplified linear instability waves. A change in the developed large-scale structure is observed as the Mach number is increased, with vortical regions oriented in a more oblique manner at the higher Mach numbers. At convective Mach numbers above unity the two-dimensional instability is found to have little effect on the flow development, which is dominated by the oblique instability waves. The nonlinear structure which develops from a pair of equal and opposite oblique instability waves is found to resemble a pair of inclined A-vortices which are staggered in the streamwise direction. A fully nonlinear computation with a random initial condition shows the development of large-scale structure similar to the simulations with forcing. It is concluded that there are strong compressibility effects on the structure of the mixing layer and that highly three-dimensional structures develop from the primary inflexional instability of the flow at high Mach numbers

Direct numerical simulation of 'short' laminar separation bubbles with turbulent reattachment

Direct numerical simulation of the incompressible Navier–Stokes equations is used to study flows where laminar boundary-layer separation is followed by turbulent reattachment forming a closed region known as a laminar separation bubble. In the simulations a laminar boundary layer is forced to separate by the action of a suction profile applied as the upper boundary condition. The separated shear layer undergoes transition via oblique modes and [Lambda]-vortex-induced breakdown and reattaches as turbulent flow, slowly recovering to an equilibrium turbulent boundary layer. Compared with classical experiments the computed bubbles may be classified as ‘short’, as the external potential flow is only affected in the immediate vicinity of the bubble. Near reattachment budgets of turbulence kinetic energy are dominated by turbulence events away from the wall. Characteristics of near-wall turbulence only develop several bubble lengths downstream of reattachment. Comparisons are made with two-dimensional simulations which fail to capture many of the detailed features of the full three-dimensional simulations. Stability characteristics of mean flow profiles are computed in the separated flow region for a family of velocity profiles generated using simulation data. Absolute instability is shown to require reverse flows of the order of 15–20%. The three-dimensional bubbles with turbulent reattachment have maximum reverse flows of less than 8% and it is concluded that for these bubbles the basic instability is convective in nature.

Compressible mixing layer growth rate and turbulence characteristics

Direct numerical simulation databases have been used to study the effect of compressibility on mixing layers. The simulations cover convective Mach numbers from 0.2 to 1.2 and all contain a fully resolved turbulent energy cascade to small spatial scales. Statistical information is extracted from the databases to determine reasons for the reduced growth rate that is observed as the convective Mach number is increased. It is found that the dilatational contribution to dissipation is negligible even when eddy shocklets are observed in the flow. Also pressure-dilatation is not found to be significant. Using an accurate relation between the momentum thickness growth rate and the production of turbulence kinetic energy together with integrated equations for the Reynolds stress tensor it is shown that reduced pressure fluctuations are responsible for the changes in growth rate via the pressure–strain term. A deterministic model for the required pressure fluctuations is given based on the structure of variable-density vortices and the assumption that the limiting eddies are sonic. Simple anisotropy considerations are used to close the averaged equations. Good agreement with turbulence statistics obtained from the simulations is found.

Direct numerical simulations of forced and unforced separation bubbles on an airfoil at incidence

Direct numerical simulations (DNS) of laminar separation bubbles on a NACA-0012 airfoil at Re-c = 5 x 10(4) and incidence 5 degrees are presented. Initially volume forcing is introduced in order to promote transition to turbulence. After obtaining sufficient data from this forced case, the explicitly added disturbances are removed and the simulation run further. With no forcing the turbulence is observed to self-sustain, with increased turbulence intensity in the reattachment region. A comparison of the forced and unforced cases shows that the forcing improves the aerodynamic performance whilst requiring little energy input. Classical linear stability analysis is performed upon the time-averaged flow field; however no absolute instability is observed that could explain the presence of self-sustaining turbulence. Finally, a series of simplified DNS are presented that illustrate a three-dimensional absolute instability of the two-dimensional vortex shedding that occurs naturally. Three-dimensional perturbations are amplified in the braid region of developing vortices, and subsequently convected upstream by local regions of reverse flow, within which the upstream velocity magnitude greatly exceeds that of the time-average. The perturbations are convected into the braid region of the next developing vortex, where they are amplified further, hence the cycle repeats with increasing amplitude. The fact that this transition process is independent of upstream disturbances has implications for modelling separation bubbles.

Compressible mixing layer - Linear theory and direct simulation

Results from linear stability analysis are presented for a wide variety of mixing layers, including low-speed layers with variable density and high Mach number mixing layers. The linear amplification predicts correctly the experimentally observed trends in growth rate that are due to velocity ratio, density ratio, and Mach number, provided that the spatial theory is used and the mean flow is a computed solution of the compressible boundary-layer equations. It is found that three-dimensional modes are dominant in the high-speed mixing layer above a convective Mach number of 0.6, and a simple relationship is proposed that approximately describes the orientation of these waves. Direct numerical simulations of the compressible Navier-Stokes equations are used to show the reduced growth rate that is due to increasing Mach number. From consideration of the compressible vorticity equation, it is found that the dominant physics controlling the nonlinear roll-up of vortices in the high-speed mixing layer is contained in an elementary form in the linear eigenfunctions. It is concluded that the linear theory can be very useful for investigating the physics of free shear layers and predicting the growth rate of the developed plane mixing layer

Wall Pressure and Shear Stress Spectra from Direct Simulations of Channel Flow

Wall pressure and shear stress spectra from direct numerical simulations of turbulent plane channel flow are presented in this paper. Simulations have been carried out at a series of Reynolds numbers up to Re? = 1440, which corresponds to Re = 6:92 x 10(4) based on channel width and centerline velocity. Single-point and two-point statistics for velocity, pressure, and their derivatives have been collected, including velocity moments up to fourth order.§ The results have been used to study the Reynolds number dependence of wall pressure and shear stress spectra. It is found that the point spectrum of wall pressure collapses for Re? ? 360 under a mixed scaling for frequencies lower than the peak frequency of the frequency-weighted spectrum, and under viscous scaling for frequencies higher than the peak. Point spectra of wall shear stress components are found to collapse for Re? ? 360 under viscous scaling. The normalized mean square wall pressure increases linearly with the logarithm of Reynolds number. The rms wall shear stresses also increase with Reynolds number over the present range, but suggest some leveling off at high Reynolds number.

Closure strategies for turbulent and transitional flows

Turbulence modelling is a critically important area in any industry dealing with fluid flow, having many implications for computational fluid dynamics (CFD) codes. It also retains a huge interest for applied mathematicians since there are many unsolved problems. This book provides a comprehensive account of the state-of-the-art in predicting turbulent and transitional flows by some of the world’s leaders in these fields. It can serve as a graduate-level textbook and, equally, as a reference book for research workers in industry or academia. It is structured in three parts: Physical and Numerical Techniques; Flow Types and Processes; and Future Directions. As the only broad account of the subject, it will prove indispensable for all working in CFD, whether academics interested in turbulent flows, industrial researchers in CFD interested in understanding the models embedded in their software (or seeking more powerful models) or graduate students needing an introduction to this vital area.

Stability and receptivity characteristics of a laminar separation bubble on an aerofoil

Stability characteristics of aerofoil flows are investigated by linear stability analysis of time-averaged velocity profiles and by direct numerical simulations with time-dependent forcing terms. First the wake behind an aerofoil is investigated, illustrating the feasibility of detecting absolute instability using these methods. The time-averaged flow around an NACA-0012 aerofoil at incidence is then investigated in terms of its response to very low-amplitude hydrodynamic and acoustic perturbations. Flow fields obtained from both two- and three-dimensional simulations are investigated, for which the aerofoil flow exhibits a laminar separation bubble. Convective stability characteristics are documented, and the separation bubble is found to exhibit no absolute instability in the classical sense; i.e. no growing disturbances with zero group velocity are observed. The flow is however found to be globally unstable via an acoustic-feedback loop involving the aerofoil trailing edge as a source of acoustic excitation and the aerofoil leading-edge region as a site of receptivity. Evidence suggests that the feedback loop may play an important role in frequency selection of the vortex shedding that occurs in two dimensions. Further simulations are presented to investigate the receptivity process by which acoustic waves generate hydrodynamic instabilities within the aerofoil boundary layer. The dependency of the receptivity process to both frequency and source location is quantified. It is found that the amplitude of trailing-edge noise in the fully developed simulation is sufficient to promote transition via leading-edge receptivity.

Low-order stochastic modelling of low-frequency motions in reflected shock-wave/boundary-layer interactions

A combined numerical and analytical approach is used to study the low-frequency shock motions observed in shock/turbulent-boundary-layer interactions in the particular case of a shock-reflection configuration. Starting from an exact form of the momentum integral equation and guided by data from large-eddy simulations, a stochastic ordinary differential equation for the reflected-shock-foot low-frequency motions is derived. During the derivation a similarity hypothesis is verified for the streamwise evolution of boundary-layer thickness measures in the interaction zone. In its simplest form, the derived governing equation is mathematically equivalent to that postulated without proof by Plotkin (AIAA J., vol. 13, 1975, p. 1036). In the present contribution, all the terms in the equation are modelled, leading to a closed form of the system, which is then applied to a wide range of input parameters. The resulting map of the most energetic low-frequency motions is presented. It is found that while the mean boundary-layer properties are important in controlling the interaction size, they do not contribute significantly to the dynamics. Moreover, the frequency of the most energetic fluctuations is shown to be a robust feature, in agreement with earlier experimental observations. The model is proved capable of reproducing available lowfrequency experimental and numerical wall-pressure spectra. The coupling between the shock and the boundary layer is found to be mathematically equivalent to a first-order low-pass filter. It is argued that the observed low-frequency unsteadiness in such interactions is not necessarily a property of the forcing, either from upstream or downstream of the shock, but an intrinsic property of the coupled system, whose response to white-noise forcing is in excellent agreement with actual spectra.

The Late Stages of Transition to Turbulence in Channel Flow

The late stages of transition, from the ?-vortex stage up to turbulence, are investigated by postprocessing data from a direct numerical simulation of the complete K-type transition process in plane channel flow at a Reynolds number of 5000 (based on channel half-width and laminar centreline velocity). The deterministic roll-up of the high-shear layer that forms around the ?-vortices is examined in detail. The new vortices arising from this process are visualized by plotting three-dimensional surfaces of constant pressure. Five vortices are observed, with one of these developing into a strong hairpin-shaped vortex. Interactions between the different vortices, and between the two channel halves, are found to be important. In the very last stage of transition second-generation shear layers are observed to form and roll up into new vortices. It is postulated that at this stage a sustainable mechanism of wall-bounded turbulence exists in an elementary form. The features which are locally present include high wall shear, sublayer streaks, ejections and sweeps. Large-scale energetic vortices are found to be an important part of the mechanism by which the turbulence spreads to other spanwise positions. The generality of the findings are discussed with reference to data from simulations of H-type and mixed-type transition