Advances in Computation, Modeling and Control of Transitional and Turbulent Flows

The following sections are included:

• Preface
• Contents

The technique of large-eddy simulation (LES) continues to play an important role in the numerical simulation of fluid dynamic processes in engineering and scientific applications. This paper will review and discuss a few of the various schemes for LES applied to the incompressible Navier-Stokes equations and to the scalar advection-diffusion equation. In particular, subgrid models based on deconvolution will be discussed. An interesting connection between the tensor diffusivity model, a particular version of a deconvolution model, and Lagrangian particle methods for the vorticity transport equation and for the scalar convection-diffusion equation will be explored. In addition, the possibility of using super-resolution, i.e. recovering fine-scale information knowing only coarse-scale information, in LES will be investigated.

The convective envelope of the Sun exhibits many interesting fluid dynamical phenomena. Some of them are outlined here: differential rotation, weak poleward meridional circulation, thermal transport, helioseismic constraints on velocities, Reynolds stresses, momentum transport and the plume models for the large-scale motion. We consider these problems briefly from a fluid dynamics perspective and comment on the impact of the numerous observations of the Sun on our understanding of these properties.

Columnar vortices aligned with the rotation axis frequently dominate the large scales in rapidly-rotating turbulence, both in laboratory experiments and in numerical simulations. We argue that, often, these columnar vortices are simply low-frequency inertial wave packets propagating away from a localised disturbance (a turbulent eddy or buoyant blob). An important feature of these low-frequency wave packets is that they transport negative helicity upward and positive helicity downward (relative to the rotation vector). This generation and subsequent spatial segregation of positive and negative helicity is potentially important for planetary dynamos.

Recent numerical simulations suggest that magnetic field generation in planets occurs predominantly within columnar vortices located outside the tangent cylinder, and these dynamos may be classified (at least approximately) as α². The simulations, which operate in a regime very far from that of the planets, also suggest that the helicity outside the tangent cylinder is predominantly negative in the north and positive in the south. In the more viscous and weakly forced simulations this helicity is often attributed to Ekman pumping at the mantle. However, Ekman pumping is less evident in the less viscous and more strongly driven simulations, and almost certainly plays no significant dynamical role in planetary interiors. We argue that, in the absence of Ekman pumping, the observed helicity distribution is a consequence of low-frequency helical wave packets initiated near the equatorial plane by buoyant plumes floating out towards the mantle. This provides a mechanism for maintaining a dynamo in the Earth, as well as in the gas giants.

While the detailed dynamics of transitional and turbulent flows are complex and high-dimensional, many of the important characteristics can be captured by models of surprisingly low dimension. We review two approaches for constructing low-dimensional models from data: in particular, we discuss balanced proper orthogonal decomposition, which applies to linear systems and can give insight into transitional flows; and Koopman spectral analysis, which we use to extract coherent structures from snapshots of turbulent flows.

Here, we demonstrate by DNS of 3D zero-pressure gradient boundary layer that both K- and H- or N-types of transition, as described in Kachanov (Ann. Rev. Fluid Mech., 26, 1994) are consequences of low amplitude monochromatic deterministic excitation caused by the growth of spatio-temporal wave-front (STWF). In Bhaumik & Sengupta (Phys. Rev. E, 89, 043018, 2014), STWF has been established as the precursor of 3D routes of transition. One of the main feature of present DNS is the extreme accuracy of the used compact schemes over a significantly longer computational domain, including the leading edge of the plate. The velocity-vorticity ()-formulation used here helps achieving higher accuracy, maintain solenoidality of the vorticity vector and reduce aliasing error for solving Navier-Stokes equation (NSE). Results show that the H-type transition occurs for lower frequency of excitation, while K-type is seen to occur for higher frequency cases. This is in contrast to the theoretical view-point in the literature for H-type transition, which is claimed to occur via triad resonant interaction of spatial TS waves.

Here, we have studied development of bypass transitional flow past a NACA 0012 aerofoil, with and without leading edge roughness elements. Calculations are performed for a Reynolds number of Re = 1.25 × 10⁶. We have used high accuracy, dispersion relation preserving (DRP) numerical schemes with appropriate numerical filters. We have discussed how bypass transition is triggered in an impulsively started flow past aerofoil. We have further analyzed the spectrum of temporal variation of vorticity at various locations on aerofoil surface along with variation of the RMS component of velocity fluctuations with the intermittency factor to characterize transitional flow.

Transition from steady state to intermittent chaos in the cubical lid-driven cavity flow is investigated numerically. Fully 3D global stability analysis has revealed that the flow experiences a Hopf bifurcation at Re_c = 1910. As for the 2D-periodic LDC flow, the unstable mode is related to a centrifugal instability of the primary vortex core. Once unstable, DNS shows that the flow is driven toward a limit cycle before eventually exhibiting intermittent chaotic dynamics.

Various physical mechanisms of transition to turbulence in zero pressure gradient (ZPG) flow past a flat plate are reported with the help of theoretical and computational studies. Spatio-temporal wave-fronts (STWF) have been reported in [Phys. Rev. Lett., 107, 154501 (2011); Phys. Rev. E., 89, 043018 (2014)], from the solution of two- (2D) and three-dimensional (3D) Navier-Stokes equation (NSE), as the precursor of transition to turbulence. This causes a shift in our understanding of transition showing centrality of STWF, as opposed to the view-point that the growth of so-called Tollmien-Schlichting waves under favourable condition eventually leads to turbulence. Having shown the commonality of STWF's role in causing transition by 2D and 3D routes, here we investigate the role of STWF for different routes of transition from the accurate direct numerical simulation of 2D NSE. Present work explains the computational requirements for accurately capturing and studying STWF.

The paper presents the results obtained using the two transport equation based transition models in the RANS framework, viz. k_T − k_L − ω model of Walters and Cokljat and γ−Re_θ SST model of Menter and Langtry. The relative performance of these two transition models for ERCOFTAC T3A flat plate test case at Re = 6.12 × 10⁵ and low Re SD7003 aerofoil at Re = 6 × 10⁴ are discussed in detail and compared with the available experimental/computational data. Finally, the two transitions models are applied to simulate three dimensional flow past a 6:1 prolate spheroid at Re = 4 × 10⁴.

Here DNS of incompressible flow through square duct and its receptivity to free stream turbulence (FST) is investigated for a sub-critical Reynolds number (Re) of 800 based on the side of the square duct and the oncoming velocity. An equilibrium flow is obtained first, whose receptivity to FST is studied, which is modelled by moving average time series model. This model corresponds to experimental results reported for channel flow.

The current work evolves a model for shear stress for a compressible mixing layer, with M_c being a parameter. This model is evolved with the data obtained from LES. It is seen that the shear stress to strain rate relation is remarkably linear in the self similar region of the mixing layer, albeit the proportionality constant depends on the prevailing convective Mach number. The model constants obtained from the simulations match with those which can be derived from experiments. By using this model, the growth rate obtained falls well within the range of experiments even at high M_c.

The inception and growth of centrifugally driven instabilities around an impulsively rotated cylinder is investigated through direct numerical simulations. Full three-dimensional simulation at a Reynolds number (Re) of 200, shows that the initial instability manifests as axisymmetric toroidal vortices stacked periodically along the cylinder axis. Based on this finding, two-dimensional axisymmetric simulations are performed for Reynolds numbers spanning seven orders of magnitude. A critical wavelength λ_c, defined as the wavelength of the first unstable mode is calculated over a range of Reynolds numbers based on the cylinder radius and wall velocity. A critical boundary layer thickness δ_c is defined based on vorticity and is found to scale as Re^−2/3 at large Reynolds numbers. The critical wavelength λ_c exhibits a similar scaling, so that λ_c ∼ δ_c for Re ≫ 1. A Taylor number based on the wavelength of the instability as the characteristic length scale is defined and its variation with the Reynolds number is discussed.

Direct numerical simulations are performed to investigate the geometrical effects of riblets on the flow field in a fully-developed turbulent channel flow. The major focus of this paper is to compare the flow fields of the V-groove and the thin blade riblets. The objective is to determine the factors contributing to the more superior drag reduction performance of the thin blade riblets. In addition, the present work intends to elucidate flow phenomena discussed in the earlier studies, and to investigate the impact of riblet geometry on those phenomena.

Mechanism of flow separation control by the micro or small devices such as plasma actuators is discussed based on a series of computations of low speed flows over an NACA 0015 airfoil with devices attached near the leading edge. The DBD plasma actuator is shown to be very effective for controlling flow separation at a Reynolds number of 6.3 × 10⁴. Flow attachment phenomenon induced by these devices is strongly related to the turbulent transition but the study reveals that it is not the only factor for the flow separation control. Analysis of the phase-average flow structures reveals that there exist remarkable structures in the flow near the airfoil surface, which characterizes the actuator-oriented flows to reduce flow separation.

Here, a new capacitance based plasma model is developed using with frequency and pressure dependence incorporated. This model shows a good match for the induced body force with experimental results. High accuracy numerical schemes are used to solve time-accurate Navier-Stokes equation embeded with the plasma model.

Large-eddy simulations of spatially evolving turbulent boundary layers with uniform blowing or suction were performed at a moderate Reynolds number (based on free-stream velocity U_∞ and momentum thickness θ) of up to Re_θ ≈ 2500 in the uncontrolled case. Aiming at the control of external flows, the influence on the skin friction drag and turbulent statistics is investigated with the control amplitude of 0.1% of U_∞. While uniform blowing reduces the skin-friction coefficient and enhances the amplitude of all components of the turbulence intensities, uniform suction reverses the effect on the skin friction and scales turbulent intensities with wall units throughout the inner layer. A net-energy saving rate S of around 13% could be achieved by blowing over the controlled surface. Since the locally defined S develops in the streamwise direction, the global S increases as the streamwise length of the controlled region is expanded. The fact that the control gain has similar trend in the streamwise direction indicates that the control efficiency increases the streamwise length of the uniform blowing region. The 2D amplitude modulation map shows that the correlation between the energy of the small scale turbulence and the large scale modulation velocity is remarkably increased (decreased) around the diagonal and off-diagonal peaks by blowing (suction). These results are directly linked to the observed enhancement of turbulence intensities in the inner layer for the blowing and obtained inner-scaling for the suction case.

In the present study, an attempt is made to employ a novel technique to reduce the friction drag significantly while spending less actuating power. Regarding this, we have introduced periodically actuated weak sources of monopoles/dipoles in the shear layer region of turbulent channel flow at Re_τ = 180. The control is achieved by pressure perturbations using monopoles/dipoles as sources for generating weak pressure waves in shear layer region in both spanwise as well as in streamwise directions and this proposed scheme is practically achievable. A total of four simulation cases have been performed with different orientation of monopoles, dipoles and quadrupoles placed at the bottom wall of the turbulent channel. Results have been analysed based on spatio-temporal response, modification of turbulent structures and statistical analysis. Increased mass-flow rate (for fixed mean pressure-gradient) in the spatio-temporal response accompanied by reduction in magnitude of high-speed streaks as well as spatial density of near-wall coherent structures is a clear indication of reduction in turbulent drag. It is observed that upto 2-5% skin-friction turbulent drag-reduction is achieved with the current technique.

In a stably stratified turbulent flow, computation based on Boussinesq assumption (uniform fluid properties) for strongly temperature dependent fluid properties like viscosity and thermal conductivity results in an inaccurate estimation of momentum and heat transfer coefficients. Role of thermo-fluid properties adversely affect the turbulent structures and shows evidence of relaminarization at the hot wall of the channel. In this article the role of internal gravity wave, its behavior in Bousinesq and non-Boussinesq situation is elaborated via turbulent structure and energy spectra for turbulent channel flow of air with varying viscosity and thermal conductivity. Due to temperature dependent properties internal gravity wave shifted towards hot wall from the core of the channel.

Direct numerical simulations of sink flow turbulent boundary layers are performed. These simulations have been performed at higher Reynolds numbers than the fairly recent experimental sink flow studies of Dixit and Ramesh (2008). The mean velocity profiles are found to be logarithmic, albeit with non-universal constants. The structure of the outer layer in a sink flow shows important differences with respect to a corresponding zero pressure gradient turbulent boundary layer.

Transport of miscible or immiscible admixtures in a compound vortex generated by a uniformly rotating disk located at the bottom of the cylindrical container is investigated. All admixtures form inside and on the surface of vortex flow well outlined coherent structures. Parameters of the structures depend on flow components and conditions of the experiment. All registered coherent structures are rather stable and reproduced. Their patterns change monotonously with variations of the flow parameters.

The single orifice hydraulic flat spray nozzle was evaluated with two global imaging techniques to characterize various aspects of the resulting spray. The two techniques were high resolution flow visualization and Particle Image Velocimetry (PIV). A CCD camera with 29 million pixels was used to capture shadowgraph images to realize ligament formation and collapse as well as droplet interaction. Quantitative analysis was performed to give the sizing information of the droplets and ligaments. This camera was then applied with a PIV system to evaluate the overall velocity field of the spray, from nozzle exit to droplet discharge. PIV images were further post-processed to determine the inclusion angle of the spray. The results from those investigations provided significant quantitative understanding of the spray structure. Based on the quantitative results, detailed understanding of the spray behavior was achieved which can be useful of computing validation. Further a comparative table determining the usage various non-intrusive techniques as per the development of the flows at various region of interest were concluded. This helps in selection of the most feasible technique required to be used in order to seek the detailed information of the certain flow characteristics.

In this paper we construct a comprehensive shell model for the buoyancy-driven turbulence, which is applicable to convective turbulence and stably-stratified turbulence. We simulate these models in the turbulent regime and show that the stably-stratified turbulence exhibit Bolgiano-Obukhbov scaling (E(k) ∼ k^−11/5), and the convective turbulence shows Kolmogorov spectrum (E(k) ∼ k^−5/3).

Direct numerical simulations in low–Prandtl number turbulent convection which are based on a spectral element method in a cylindrical cell are discussed. Here, we report results on the structure of the thermal boundary layer and on the statistics of the locally fluctuating Kolmogorov scales.

The effect of buoyancy on turbulent mixed convection flow through vertical and horizontal channels has been numerically studied. Turbulence is modeled by Reynolds averaged Navier-Stokes equations (RANS) with standard k − ϵ turbulence model. The present analysis is valid when the buoyancy force effects are small compared with the forced convection effects. The mixed convection flow problem is formulated by two dimensional unsteady incompressible flow with the buoyancy term modeled by Boussinesq approximation. The governing equations are solved by high accuracy compact finite difference schemes with four stage Runge-Kutta method for time integration. Results are reported for Reynolds number of 6000 with Richardson number varied from 0 to 0.5. The velocity, temperature profiles and average Nusselt number values are presented. The buoyancy shows significant effect on the flow and heat transfer characteristics in vertical channel in comparison with the horizontal channel. The effects of heated plate velocity relative to fluid velocity in a channel are reported. The present results are matching well with the experimental results available in the literature.

The slip velocity of the flow at the porous-fluid interface is studied using a 2D porous region, modeled as an array of circular cylinders in square and hexagonal arrangements to resemble the structure of real porous media. The Navier-Stokes and continuity equations are solved using a hybrid FEM-FVM solver by applying appropriate macroscopic boundary conditions. The existence of a slip velocity at the porous-fluid interface is demonstrated and it is observed that the slip velocity increases with increase in porosity and flow Reynolds number.

Details are provided on the development of a farfield boundary condition for the direct numerical simulation of an adverse pressure gradient (APG) turbulent boundary layer (TBL) at the verge of separation. The APG TBL achieves a region constant pressure velocity to freestream velocity ratio over a momentum thickness based Reynolds number range of Re_θ ≈ 2000 to 6000. Mean velocity deficit and Reynolds stress profiles are presented under both friction velocity and pressure velocity based scaling within this range. The pressure velocity scaling demonstrates a collapse of the statistical profiles.

The features of flow past a cylinder with a vertical slot at Reynolds number of 3500 based on high-speed visualizations of smoke-flow and computations are presented. The pressure differential across the slot causes natural periodic suction and blowing across the cylinder. The ‘dead-water’ region that forms behind the basic circular cylinder is replaced by a highly dynamic and complex vortex dominated periodic base flow. The separated shear layer on one side rolls up into a large vortex and reattaches at the base. The reattaching vortex entrains the opposite shear layer and promotes extended regions of attached flow featuring multiple separations and reattachments. Computations show qualitative agreement with experiments and clearly bring out the occurrence of periodic wall jets transverse to the cylinder, caused by oscillatory flow across the slot excited by the external shedding. These wall jets lead to the formation of multi-pole vortices in the shear layers and periodic ejection of vorticity leading to ‘aerodynamic tripping’ of the shear layers. It seems likely that the separated shear layers undergo bypass transition to turbulence and reattach onto the base, resulting in formation of periodic separation bubbles around the base of the slotted cylinder. Variation of instantaneous static pressures around the slotted cylinder shows oscillations with increasing amplitude from the stagnation point, which tend to coalesce towards the beginning of the slot. Soon after the slot, large amplification of pressure occurs due to the wall-jet. Time-averaged pressure distributions indicate increased pressures in the entire separated flow region of the slotted cylinder as compared to that of the basic cylinder. The slotted cylinder shows dramatic reductions of more than 80% in the mean and fluctuating drag and lift forces. The present studies show that the vortex-induced vibrations of a circular cylinder could be passively controlled using natural ventilation through a slot across the cylinder.

Direct numerical simulation (DNS) studies of compressible flow past low pressure (LP) turbine blades have been carried out using a new compressible code ANUROOP. The code has been developed at JNCASR and validated for both free-shear and wall-bounded flows. The mid-section of the Pratt and Whitney LP turbine blade T106A has been chosen for detailed analysis. The set-up enables direct comparison with both experimental and computational data available on the blade. The Reynolds number (Re) of the flow is 51,831 based on inlet velocity and axial chord of the blade.

The grid size has been varied from 25 to 160 million, and shows that the flow near transitional Re depends crucially on resolution. The separation bubble at the leading edge of the suction side present at low resolutions disappears at the highest resolution, with consequent changes in blade fluid dynamics that are striking. Around the leading edge, the high curvature demands a correspondingly near wall resolution, which plays a major role in determining the pressure distribution.

Impulsive transient jet originating from the open end of a short driver section shock tube has been simulated numerically by solving the unsteady Navier-Stokes equation in axisymmetric form using AUSM+ scheme for a driver section pressure ratio of 10. The simulations are performed using second order central and fifth order upwind schemes with different spatial resolutions. The effect of higher order schemes and spacial resolution on the accuracy of the solution is studied in detail. It has been observed that the results obtained from the fifth order scheme with lower spacial resolution and denser grids along the free shear layer matched close with experimental smoke flow visualization.

The following sections are included:

• Introduction
• Results
• Conclusions
• References

We study internal wall-bounded flows by means of direct numerical simulations of canonical Poiseuille and Couette flow at computationally high Reynolds number. The two flows exhibit a number of similarities, in that they both feature a near log layer in the mean velocity profile and in the wall-parallel velocity variances, and exhibit excess energy production over dissipation in the outer layer. When closely inspected, the mean velocity profiles exhibit systematic deviations from the alleged log law, which are more severe in Couette flow. The latter also exhibits clearer outer-layer activity associated with the formation of quasi-streamwise rollers, and which implies the emergence of a secondary outer-layer peak of the streamwise turbulence intensity.

A new methodology for finding hybrid RANS/LES damping functions is proposed. The Flow Simulation Methodology framework is modified from Speziale's original formulation [AIAA 36,173 1998] to allow for fewer grid points away from solid boundaries. The proposed functional form is regressed using an evolutionary algorithm. The solution is trained by considering DNS data of a separated flow. This sophisticated approach to building the damping function allows for the development of a hybrid methodology adaptable to any level of required modelling. The regressed model is compared to highly resolved reference LES on the classical two dimensional periodic hills case, for which the agreement is very good.

Subgrid-scale (SGS) turbulence models for large eddy simulations (LES) are quantitatively assessed for the strongly-stratified helical transition-to-turbulence, Arnold-Beltrami-Childress (ABC) problem through comparisons to in-house direct numerical simulations (DNS). LES predictions using the classical non-dynamic and dynamic Smagorinsky models (SSM and DSM, respectively) are compared to a modified Smagorinsky model (MSM) designed specifically for natural convection. DSM predicts the best results when compared to DNS, although none of the SGS models are able to predict the peak of nonlinearities.

Here an attempt has been made to demonstrate relative strengths and weaknesses of various methods of simulating transitional flows. Flow simulations have been done using compressible RANS and DNS for compressible and incompressible formulations. Flow over the WTEA-TE1 airfoil is simulated for free-stream Mach no. M_∞ = 0.1815 and angle of attack α = −0.0328°. In particular, the pressure and skin friction distributions are compared. In additions RMS signals for the DNS calculations are made, including Fourier Transforms of pressure signals over specific experimental measurement stations.

This paper examines the lower bound of the sound that a circular turbulent cold jet (Mach number 0.88) radiates to the far field, for the given jet exit velocity and nozzle size. By applying discrete wavelet transform (DWT) and Fourier transform in azimuth, the jet flow is decomposed into wave packets (WP), coherent eddies (CE), and incoherent structures (IS). With the time-averaged component of the hydrodynamic near field unchanged, WP are filtered out using wavelets. It is shown that the remaining IS, which has a Gaussian PDF by definition, is de-correlated. Using the same approach, the acoustic far field is filtered in the low azimuthal modes, leading to 9-15dB lower overall far-field sound compared with the original jet at different axial angles. Sound spectra show that the high-frequency component dominates and remains unaffected by the filtering process. After IS are removed, the values of the correlations between the axisymmetric modes of the near and far field are significantly higher than those reported for flow-acoustic correlations of the axisymmetric mode. It is suggested that the sound reduction after all modes are filtered is the best that can be achieved in practice, given the proposed constraints.

We employ a new technique comprised of multiple simultaneous Large-Eddy Simulations (LES) to analyze the dynamics of turbulence, with emphasis on jet noise. Unlike methods which seek to use correlation of signals at different locations, here we isolate and tag perturbations in different regions of interest and examine the manner in which they are modulated and filtered by the turbulence. The technique essentially uses two (or more) simultaneous, synchronized Large Eddy Simulations (LES). One, denoted baseline, provides the natural perturbations existing in the jet. These are processed and injected into a twin to provide a perturbation boost, which can then be tracked in the twin. Several simulations on a Mach 1.3 cold jet demonstrate the effectiveness of the approach in determining the self and mutual interactions in core turbulence that ultimately yield the near acoustic field. Detailed 1-D and 2-D statistical analyses yield unique insight into the nature of intermittency, spatio-temporal correlations, wave-packet generation and directivity.

A series of direct numerical simulations (DNS) were conducted of turbulent jets issuing from acoustically lined pipe. The inclusion of the pipe in the simulations with a fully turbulent flow inside ensures that all possible noise generation mechanisms are represented. Earlier results from a similar pipe/jet configuration, albeit without acoustic liner treatment, showed contamination of the far field noise field by interior noise from the pipe. Therefore, here two key modifications were made. Firstly, the interior pipe walls were acoustically lined using an impedance condition. Secondly, the turbulent pipe inflow boundary condition was modified to reduce spurious noise introduced into the axisymmetric mode. The sound radiation from the pipe/jet configuration was analysed using a phased array source breakdown technique. It is demonstrated that the modification of the inflow boundary results in a strongly reduced contribution of the interior noise component to the farfield noise in the axisymmetric mode, while the acoustic liner is effective in reducing the interior noise contribution to farfield noise in the higher azimuthal modes. This enables the source breakdown analysis to extract the jet mixing noise contributions to the farfield much more clearly.

Decomposition of radiating and non-radiating fluctuating components are important to the study of sound generation and acoustic feedback mechanisms present in a variety of free shear flows. Prior work primarily focused on the development of a decomposition operator for subsonic flows based on the acoustic dispersion relationship which is invariant to the position in time and space, such as found in Goldstein and Sinayoko et al. This paper presents the development of an operator that approximately decomposes linear fluctuating flow variables into their radiating and non-radiating components locally in time and space using the wavelet and curvelet transforms as a basis. The operator has been applied to a large-eddy simulation of a highly compressible under-expanded impinging jet to study its radiating and non-radiating fields.

We aim to demonstrate a proof-of-concept of thermodynamics-acoustics-based strategies for controlling the growth of instability in high-speed compressible shear flows. While compressible flows pose many challenges, they also offer novel opportunities for flow control. We show that the equipartition of energy between the potential energy of pressure field and dilatational kinetic energy can be exploited for design of novel thermodynamics-based flow control strategies.

A Gibbs phenomenon detector that is useful in damping numerical oscillations in hybrid solvers for compressible turbulence is proposed and tested. It is designed to function in regions away from discontinuities where commonly used discontinuity sensors are ineffective. Preliminary tests indicate that using it alongside a traditional discontinuity sensor in a hybrid code leads to more effective damping of numerical oscillations while leaving the physical oscillations intact.

The wavepacket is a model that bears close resemblance to naturally-occurring laminar turbulent transition processes in a boundary layer, but its large number of constituent spectral modes presents a challenge during analysis of the results. In this work, we leverage the extensive flow data available from direct numerical simulations (DNS) to perform a three-dimensional (3D) discrete Fourier transform of a wavepacket in order to investigate the most energetic modes in the flow. Through windowing, it is possible to isolate the wavepacket at different stages of its development, and build up an overall picture of its transition. Furthermore, an inverse transform is applied to subsets of the full spectrum, and this provides insight into the physical structure of its key components.

The following sections are included:

• Introduction
• Instability Modes and SLE Equation
• Results and Discussion
• Conclusions and Future Work
• References

We performed a comparative study of extraction of large-scale flow structures in Rayleigh Bénard convection using proper orthogonal decomposition (POD) and Fourier analysis. We show that the free-slip basis functions capture the flow profiles successfully for the no-slip boundary conditions. We observe that the large-scale POD modes capture a larger fraction of total energy than the Fourier modes. However, the Fourier modes capture the rarer flow structures like flow reversals better. The flow profiles of the dominant POD and Fourier modes are quite similar. Our results show that the Fourier analysis provides an attractive alternative to POD analysis for capturing large-scale flow structures.

Using direct numerical simulation we demonstrate that for isotropic stably-stratified flows follow Bolgiano-Obukhov scaling, i.e., the kinetic energy spectrum E_u(k) ∼ k^−11/5, the entropy spectrum E_θ(k) ∼ k^−7/5, and kinetic energy flux Π_u(k) ∼ k^−4/5. This is due to the conversion of kinetic energy to potential energy because of buoyancy. For Rayleigh Bénard convection, we argue that Π_u(k) is a nondecreasing function of k due to the positive energy supply rate by buoyancy. Our numerical simulations show that the convection turbulence follows Kolmogorov-like scaling (E_u(k) ∼ k^−5/3) with Kolmogorov's constant approximately 1.4.

We conduct the DNS of a buoyant turbulent cloud under rapid rotation in a 512³ periodic box at low Rossby numbers (Ro). The initial condition chosen is relevant to the equatorial plane of the Earth's inhomogeneous outer core. A random but uniform distribution of buoyant blobs of different sizes is centrally placed in the box. It is observed that columnar structures form and grow towards the box boundary. Using helicity as a diagnostic, we confirm that these structures are formed by inertial waves which arise from the buoyant cloud. A near perfect helicity segregation of negative in the north (+z) and positive in the south (−z) exists at Ro = 0.01 but not for Ro = 0.1. Moreover, at Ro = 0.1, there is a significant distortion in the buoyancy field which vanishes at Ro = 0.01. These observations are attributed to a wave-induced mean flow at higher Ro. The kinetic energy distribution shows oscillations in the buoyant cloud region. A POD analysis of horizontal velocity field indicates that these are stationary inertial waves of frequency 2Ω.

We compute the evolution of a helium bubble in ambient air after a shock of Mach 1.22 passes over it. A thirteenth order upwind scheme is employed for this purpose, in the framework of the AUSM⁺ algorithm. The basic features of this class of problems have been captured, and the present numerical results match well with earlier experimental and numerical findings. In addition, the high spectral accuracy and low numerical dissipation of the explicit Euler solver reveals features of the deformed bubble – particularly after the formation of small-scale rolled-up vortices due to the Kelvin-Helmholtz instability – that are difficult to detect with low order upwind schemes.

In the present study, a novel methodology of 2D Direct Numerical Simulation (DNS), incorporating non-Boussinesq, inhomogeneous formulation with non-periodic boundary conditions is used to study the Rayleigh-Taylor Instability (RTI) of miscible fluid at the interface. The evolution of RTI is shown with density contours and mixing length. The system considered here is completely isolated and hence entropy evolution of the system reported here is relevant from non-equilibrium thermodynamics point of view.

The applicability of a high resolution low-pass differential spatial filter for unstructured grids was investigated for typical high-order one-dimensional Finite Element methods. The goal was to investigate the accuracy and stability of the filter when applied on high-order elements. The one-dimensional differential filter was discretized based on Lagrange and Hierarchical Legendre base functions. Manufactured solutions were used to demonstrate the performance of the filter on two one-dimensional test cases for elements of order up to six. The results suggest that the proposed filter is effective for explicit filtering in Large Eddy Simulations using high-order methods.

In the symposium we have seen a very wide range of simulations for transitional and turbulent flows. In writing this epilogue, the author takes the responsibility of identifying some common elements which add to success of desired DNS and LES. Of course, the single common element of any such activities is to be aware of numerical dispersion relation and how it relates to physical dispersion relation. We also investigate suitability of a specific combined implicit-explicit (IMEX) time integration method in performing DNS, which has been used to explore the so-called bypass transition. While IMEX methods have often been used, yet its properties and potentials are not assessed. Specifically, error analysis by solving a model equation that mimics some of the physical processes in DNS is not performed. A model equation with exact solution allows tracking of error with time for IMEX time integration methods, with either explicit or implicit spatial discretization. This analysis is motivated by recent DNS of flow transition over a flat plate reported in Bhaumik and Sengupta (Phys. Rev. E, 89, 043018, 2014) and Sayadi et al. (J. Fluid Mech., 724, 480-509, 2013), both trying to mimic the classical experimental results on flow transition in Schubauer and Skramstad (J. Aero. Sci., 14, 69, 1947). These two simulations show distinctly different computed solutions, yet the average flow quantities show remarkable similarity. In Bhaumik and Sengupta (2014), explicit dispersion relation preserving time integration method has been used everywhere in the domain. In contrast, in Sayadi et al. (2013), implicit time integration method is used very close to the plate, while explicit time integration method is used in rest of the domain. Such splitting of domain into explicit and implicit regions causes spurious internal reflection of the signal at the interface between these regions, creating unphysical, spurious and upstream-propagating q-waves which considerably contaminates solution. This is investigated in detail along with other issues of high resolution computing here to show suitable methods for DNS.

A panel discussion was organized on the final day of the symposium to take stock of the proceedings and to take views of experts on status and future of HPC related to transition and turbulence. The panel consisted of Prof. T. K. Sengupta (IIT Kanpur), Dr. P. Spalart (The Boeing Company), Prof. D. Gaitonde (Ohio State University), Prof. J. Jimenez (Politecnica de Madrid, Spain), Prof. M. Deville (EPFL, Switzerland), Prof. S. K. Lele (Stanford University), Prof. T. W. H. Sheu (NTU, Taiwan) and Prof. P. A. Davidson (Cambridge University) and was moderated by Dr. Vidyadhar Mudkavi (CSIR NAL)…