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DOE OSTI2024

A high-order finite difference method for moving immersed domain boundaries and material interfaces

Here, we present a high-order sharp treatment of immersed moving domain boundaries and material interfaces, and apply it to the advection-diffusion equation in two and three dimensions. The spatial discretization combines dimension-split finite difference schemes with an immersed boundary treatment based on a weighted least-squares reconstruction of the solution, providing stable discretizations with up to sixth order accuracy for diffusion terms and third order accuracy for advection terms. The temporal discretization relies on a novel strategy for maintaining high-order temporal accuracy in problems with moving boundaries that minimizes implementation complexity and allows arbitrary explicit or diagonally-implicit Runge-Kutta schemes. The approach is broadly compatible with popular PDE-specialized Runge-Kutta time integrators, including low-storage, strong stability preserving, and diagonally implicit schemes. Through numerical experiments we demonstrate that the full discretization maintains high-order spatial and temporal accuracy in the presence of complex 3D geometries and for a range of boundary conditions, including Dirichlet, Neumann, and flux conditions with large jumps in coefficients.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Direct Numerical Simulation of Transitional and Turbulent Flows Over Multi-Scale Surface Roughness—Part I: Methodology and Challenges

Abstract High-fidelity simulation of transitional and turbulent flows over multi-scale surface roughness presents several challenges. For instance, the complex and irregular geometrical nature of surface roughness makes it impractical to employ conforming structured grids, commonly adopted in large-scale numerical simulations due to their high computational efficiency. One possible solution to overcome this problem is offered by immersed boundary methods, which allow wall boundary conditions to be enforced on grids that do not conform to the geometry of the solid boundary. To this end, a three-dimensional, second-order accurate boundary data immersion method (BDIM) is adopted. A novel mapping algorithm that can be applied to general three-dimensional surfaces is presented, together with a newly developed data-capturing methodology to extract and analyze on-surface flow quantities of interest. A rigorous procedure to compute gradient quantities such as the wall shear stress and the heat flux on complex non-conforming geometries is also introduced. The new framework is validated by performing a direct numerical simulation (DNS) of fully developed turbulent channel flow over sinusoidal egg-carton roughness in a minimal-span domain. For this canonical case, the averaged streamwise velocity profiles are compared against results from the literature obtained with a body-fitted grid. General guidelines on the BDIM resolution requirements for multi-scale roughness simulation are given. Momentum and energy balance methods are used to validate the calculation of the overall skin friction and heat transfer at the wall. The BDIM is then employed to investigate the effect of irregular homogeneous surface roughness on the performance of an LS-89 high-pressure turbine blade at engine-relevant conditions using DNS. This is the first application of the BDIM to realize multi-scale roughness for transitional flow in transonic conditions in the context of high-pressure turbines. The methodology adopted to generate the desired roughness distribution and to apply it to the reference blade geometry is introduced. The results are compared to the case of an equivalent smooth blade.

Engineering↗

NASA NTRSDate not supplied

Wall-Modeled LES of a Swept Wing with Leading-Edge Ice Using LAVA Curvilinear, Unstructured, and Cartesian Solvers

Wall-modeled large-eddy simulation (LES) of a swept wing with leading-edge ice build up is performed using three mesh paradigms and associated flow solvers and compared to experimental results. The study focuses on an 8.9% scale model of the CRM65 swept wing featuring both high-fidelity and smooth leading-edge ice shapes. The assessment is conducted using the Launch, Ascent, and Vehicle Aerodynamics (LAVA) framework using the three actively supported meshing paradigms in LAVA: structured curvilinear overset, unstructured, andCartesian. For the iced configurations, these paradigms use hybrid body-fitted/source-term immersed-boundary, body-fitted, and ghost-cell immersed-boundary strategies, respectively.The unstructured and Cartesian mesh paradigms are particularly attractive for complex ice shapes as they avoid the manual mesh generation effort associated with the curvilinear approach. For the high-fidelity ice shape, good agreement with the experiment can be obtained with all three strategies; however, the curvilinear method is particularly sensitive to the source-term immersed-boundary timescale and span wise mesh resolution, and the Cartesian implementation is sensitive to the choice of numerical flux. For the smooth ice, larger discrepancies are observed across all methods. However, a mesh refinement study guided by flow visualizations leads to an improved comparison with the experiment that is particularly pronounced for the unstructured mesh paradigm.

TTT↗

NASA NTRSDate not supplied

A Wall-Modeled LES Perspective for the High Lift Common Research Model Using LAVA

A new immersed boundary Wall-Modelled Large Eddy Simulation (WMLES) formulation is developed to study high-lift aerodynamics on the NASA High-Lift Common Research Model (HL-CRM). A sequence of Cartesian Octree grids with sizes ranging from 100 Million through 2.02 Billion grid points is utilized to systematically assess grid-sensitivity and convergence for the in-tunnel (QinetiQ) configuration of the model, and remarkable agreement between the immersed boundary and the curvilinear body-aligned WMLES formulations is reported on grids with comparable resolutions. In the free-air configuration of the model, consistent predictions between the Curvilinear Overset and the Cartesian Octree formulations are reported for angles of attack up to C(L,max) at a=19.57. However, some differences in the onset of stall are seen between the two methods for a>20°: while the curvilinear WMLES experiences wing-root separation with increasing angle of attack (Topology A), the Cartesian Octree formulation shows a different flow topology characterized by boundary layer weakness on the main element, emanating from the pylon-wing attachment (Topology B). In order to obtain further insight into the two-distinct topologies, carefully designed numerical experiments to isolate effects of the model standoff and the tunnel wall-boundary layers are conducted using the immersed boundary WMLES formulation. The increased incidence angle-of-attack on the inboard portion of the wing due to the standoff is shown to be sufficient for triggering a switch from Topology B to Topology A in Cartesian WMLES. The role of the floor boundary layer is further examined in detail by identification of additional corner-flow vorticity generated by the viscous juncture flow interactions between the floor boundary layer and the standoff leading to formation of a strong coherent and persistent vortex on the belly-side of the fuselage. The intensity of this vortex is shown to increase with the thickness of the floor boundary layer. A further increase in the incidence angle of attack near the leading-edge strake caused by the presence of this belly-side vortex is quantified for two-distinct floor boundary layers. Both of the floor boundary layers considered result in the onset of large scale wing-root separation at a=21.47in non-confined (free-air) configurations.

TTT↗

DOE OSTI2023

Virtual Flow Solver - Geophysics: A 3D Incompressible Navier-Stokes Solver

Virtual Flow Solver - Geophysics (VFS-Geophysics) is a three-dimensional (3D) incompressible Navier-Stokes solver based on the Curvilinear Immersed Boundary (CURVIB) method. The CURVIB is a sharp interface type of immersed boundary (IB) method that enables the simulation of fluid flows in the presence of geometrically complex moving bodies. The CURVIB method can be applied to wind/MHK turbine simulations and energy applications. VFS-Geophysics is the result of many years of research work by several graduate students, post-docs, and research associates that have been involved in the Computational Hydrodynamics and Biofluids Laboratory directed by Professor Fotis Sotiropoulos. The preparation of the present manual has been supported by the U.S. Department of Energy (DE-EE 0005482).

16 TIDAL AND WAVE POWER↗

DOE OSTI2025

Concurrent two-way coupling of global and local models across internal boundaries with non-matching discretizations

Coupling local and global models enables efficient simulation of multiscale systems, where global models capture large-scale behavior and local models, with enhanced physics, resolve finer details over a smaller region. Here, this paper presents a mathematically consistent method for coupling physics-based models of varying fidelity across adjacent, non-overlapping subdomains, even when discretizations do not match at the immersed interdomain interfaces. Incompressible Navier-Stokes equations (NSE) constitute the global model while residual-based turbulence model serves as the local high-fidelity model. In addition, a scalar advection-diffusion equation that models the convection of an active scalar field is appended to the turbulence model in the local domain. This scalar field does not have its complement in the global model, giving rise to unequal number of equations at the immersed boundary between local and global models. Interdomain coupling terms are derived via the Variational Multiscale Discontinuous Galerkin (VMDG) method with new developments in scale representation and efficient fine-scale estimation. While transient laminar flows modeled with NSE in the global domain can be resolved with relatively coarse mesh, turbulent flow calculations in the local model require much finer spatial discretizations as well as smaller time-step for appropriately resolving the turbulent flow physics. The proposed framework also accommodates non-matching meshes at the immersed boundaries. Test problems in 2D and 3D numerically showcase the concurrent two-way coupling of unknown fields across the immersed boundaries. The 3D test presents a case with an unequal number of equations, where the scalar field represents the convection of contaminant concentration. This provides more detailed physics in the local region and highlights its application in climate modeling and atmospheric sciences.

Variational Multiscale Discontinuous Galerkin (VMD↗

DOE OSTI2022

The role of breakup and coalescence in fine-scale bubble-induced turbulence. I. Dynamics

We study the effect of bubble breakup and coalescence on fine-scale dynamics of bubbly turbulent flows using direct numerical simulations. We perform two different simulations of dilute bubbly flows of void fraction 0.5%: one with bubbles breaking up and coalescing and the other without these physical processes. The volume of the fluid method is used for simulating bubbles undergoing breakup and coalescence while the bubbles are treated as rigid spheres in the immersed boundary method simulation. The energy spectrum in both types of simulation, consistent with previous studies, exhibits a -3 slope. We follow a single infinitesimal fluid element as it evolves to understand velocity gradient dynamics using conditional mean trajectories. We note finite-time divergence when the fluid element evolves under the action of inertial and pressure dynamics. The inertial, pressure, and viscous velocity gradient dynamics, when considered individually, produce the same results for bubble-induced turbulence (BIT) as with the classical homogeneous isotropic turbulence (HIT). Yet when the overall velocity gradient dynamics is considered, BIT results in non-cyclic trajectories moving toward stable node and unstable saddle while classical HIT shows cyclic behavior in their trajectories that move toward the origin. Interestingly, both the volume of fluid and immersed boundary simulations produce similar results. Therefore, there are two main takeaways from this research. First, new velocity gradient models are needed for BIT as their velocity gradients behavior is entirely different from the HIT. Second, we can neglect the bubble topology, breakup, and coalescence while studying or modeling the fine-scale dynamics of BIT.

42 ENGINEERING↗

NASA NTRS1998

Studying Turbulence Using Numerical Simulation Databases: Proceedings of the Summer Program - No. 7

The Seventh Summer Program of the Center for Turbulence Research took place in the four-week period, July 5 to July 31, 1998. This was the largest CTR Summer Program to date, involving thirty-six participants from the U. S. and nine other countries. Thirty-one Stanford and NASA-Ames staff members facilitated and contributed to most of the Summer projects. A new feature, and perhaps a preview of the future programs, was that many of the projects were executed on non-NASA computers. These included supercomputers located in Europe as well as those operated by the Departments of Defense and Energy in the United States. In addition, several simulation programs developed by the visiting participants at their home institutions were used. Another new feature was the prevalence of lap-top personal computers which were used by several participants to carry out some of the work that in the past were performed on desk-top workstations. We expect these trends to continue as computing power is enhanced and as more researchers (many of whom CTR alumni) use numerical simulations to study turbulent flows. CTR's main role continues to be in providing a forum for the study of turbulence for engineering analysis and in facilitating intellectual exchange among the leading researchers in the field. Once again the combustion group was the largest. Turbulent combustion has enjoyed remarkable progress in using simulations to address increasingly complex and practically more relevant questions. The combustion group's studies included such challenging topics as fuel evaporation, soot chemistry, and thermonuclear reactions. The latter study was one of three projects related to the Department of Energy's ASCI Program (www.llnl.gov/asci); the other two (rocket propulsion and fire safety) were carried out in the turbulence modeling group. The flow control and acoustics group demonstrated a successful application of the so-called evolution algorithms which actually led to a previously unknown forcing strategy for jets yielding increased spreading rate. A very efficient algorithm for flow in complex geometries with moving boundaries based on the immersed boundary forcing technique was tested with very encouraging results. Also a new strategy for the destruction of aircraft trailing vortices was introduced and tested. The Reynolds Averaged Modeling (RANS) group demonstrated that the elliptic relaxation concept for RANS calculations is also applicable to transonic flows with shocks; however, prediction of laminar/turbulent transition remains an important pacing item. A large fraction of the LES effort was devoted to the development and testing of a new algorithmic procedure (as opposed to phenomenological model) for subgrid scale modeling based on regularized de-filtering of the flow variables. This appears to be a very promising approach, and a significant effort is currently underway to assess its robustness in high Reynolds number flows and in conjunction with numerical methods for complex flows. As part of the Summer Program two review tutorials were given on Turbulent structures in hydrocarbon pool fires (Sheldon Tieszen), and Turbulent combustion modeling: from RANS to LES via DNS (Luc Vervisch); and two seminars entitled Assessment of turbulence models for engineering applications (Paul Durbin) and Subgrid-scale modeling for non-premixed, turbulent reacting flows (James Riley) were presented. A number of colleagues from universities, government agencies, and industry attended the final presentations of the participants on July 31 and participated in the discussions. There are twenty-six papers in this volume grouped in five areas. Each group is preceded with an overview by its coordinator.

Source record↗

DOE OSTI2022

A particle resolved simulation approach for studying shock interactions with moving, colliding solid particles

This work applies a new combination of techniques for the fully resolved simulation of compressible, gas–particle multiphase flows. Here, the adaptive wavelet collocation method is used to dynamically, and efficiently, adapt the computational grid to localized flow features and the particles. A characteristic-based volume penalization method that imposes arbitrary Dirichlet, Neumann, or Robin-type immersed boundary conditions, is used to enforce the no-slip condition at particle surfaces. A hard-sphere collision model is applied to capture the particle–particle collisions. Proof of concept test cases are presented, showcasing the dynamic grid adaptation and fully resolved two-way coupling between the phases that is possible with this approach. Results for a shock-driven single cylinder under viscous and inviscid conditions are presented along with a demonstration of a shock interacting with a cloud of randomly distributed cylinders and spheres.

42 ENGINEERING↗

DOE OSTI2021

Pore-scale CFD simulations of clay mobilization in natural porous media due to fresh water injection

The present work investigates mechanisms of permeability impairment as a result of low-salinity fluid injection into brine-saturated porous media containing dispersible clays. Here, we present a computational fluid dynamics model at the pore-scale to simulate detachment, migration and straining of fine particles in porous media. The model uses an immersed boundary method to simulate the motion of clay fines in a fluid. In addition to the hydrodynamic forces, we model the Derjaguin-Landau-Verwey-Overbeek forces (DLVO) between clay fines and grains. Our simulations show the impact of the injected fluid's salinity and velocity on the concentration of clay fines retained on the grain surface. We see clay particles dislodging from the grain surface in clusters of up to 12 particles. Our simulation results also demonstrate clogging of narrow pore spaces by the detached particles from the grain surfaces.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

A high-order computational framework for particle-resolved simulations of disperse multiphase flows

This work presents a high-order numerical approach for particle-resolved simulations of disperse multiphase flows, where the Navier-Stokes equations for fluid flow are solved using a high-order spectral element method in the Eulerian framework, and the particle phase is directly simulated with a discrete element method. The coupling between particles and fluids is explicitly handled using an adapted direct-forcing immersed boundary method. Unlike the conventional schemes, a high-order barycentric Lagrange interpolation method and a Gaussian projection kernel are used to ensure accurate momentum exchange between local boundary points and surrounding fluid nodes in the framework of high-order fluid solver. Benchmark tests of increasing complexity are conducted to demonstrate the accuracy and efficiency of our method. Here, it is found that our approach exhibits an excellent convergence performance, as the fluid element/grid is refined and the number of boundary points increases. Compared to conventional low-order methods, the proposed high-order framework enables the use of substantially larger fluid elements while maintaining high accuracy in modeling fluid-particle interactions, owing to the enhanced resolution of high-order basis functions. Moreover, since the primary unknowns are stored at element or grid nodes, the high-order approach offers improved efficiency in both CPU memory usage and total computational cost.

42 ENGINEERING↗

DOE OSTI2021

A physics-based model for frost buildup under turbulent flow using direct numerical simulations

We present a new model for frost buildup under turbulent (and laminar) flow using direct numerical simulations. The physical model consists of two layers, the air and the frost. The air layer is fully resolved and consists of solving for the velocity, temperature, and vapor mass fraction fields. The frost layer thickness is resolved using conservation of mass and energy. Both phases are dynamically coupled using the immersed boundary method. Three-dimensional simulations are conducted in an open-channel configuration. A number of challenges need to be overcome to make these simulations feasible. First, to enforce far-field conditions of zero gradient and prescribed mean temperature and humidity, a source term is added to the energy and transport equations in the flow solver. Second, the mean frost thickness is subtracted after each time step to ensure a constant mean flow thickness and level of turbulence in the numerical domain. Third, a slow-time acceleration approach, which accelerates the frost buildup by a predetermined factor, is employed to bridge the gap between the fast turbulent and slow frost buildup time scales. Finally, a frost densification scheme is used to overcome the difficulties of vertically varying frost properties. The model is validated by comparing the frost thickness and frost thickness buildup rate over a period of one hour from a cooled flat plate experiment. As a result, both quantities compare favorably with experiments.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2023

Rheology of concentrated fiber suspensions with a load-dependent friction coefficient

Here we investigate the effects of fiber aspect ratio, roughness, flexibility, and volume fraction on the rheology of concentrated suspensions in a steady shear flow using direct numerical simulations. We model the fibers as inextensible continuous flexible slender bodies with the Euler-Bernoulli beam equation governing their dynamics suspended in an incompressible Newtonian fluid. The fiber dynamics and fluid flow coupling is achieved using the immersed boundary method. In addition, the fiber surface roughness might lead to interfiber contacts, resulting in normal and tangential forces between the fibers, which follow Coulomb’s law of friction. The surface roughness is modeled as hemispherical protrusions on the fiber surfaces. Their deformation results in a normal load-dependent friction coefficient. Our simulations accurately predict the experimentally observed shear thinning in fiber suspensions. Furthermore, we find that the suspension viscosity η increases with increasing the volume fraction, roughness, fiber rigidity, and aspect ratio. The increase in η is the macroscopic manifestation of a similar increase in the microscopic contact contribution to the total stress with these parameters. In addition, we observe positive and negative first N 1 and second N 2 normal stress differences, respectively, with |N 2 | < |N 1 |, in agreement with previous experiments. Last, we propose a modified Maron-Pierce law to quantify the reduction in the jamming volume fraction by increasing the fiber aspect ratio and roughness. Our results and analysis establish the use of fiber surface tribology to tune the suspension flow behavior.

42 ENGINEERING↗

NASA NTRS2010

Instability of a Supersonic Boundary-Layer with Localized Roughness

A localized 3-D roughness causes boundary-layer separation and (weak) shocks. Most importantly, streamwise vortices occur which induce streamwise (low U, high T) streaks. Immersed boundary method (volume force) suitable to represent roughness element in DNS. Favorable comparison between bi-global stability theory and DNS for a "y-mode" Outlook: Understand the flow physics (investigate "z-modes" in DNS through sinuous spanwise forcing, study origin of the beat in DNS).

Marxen, Olaf↗

NASA NTRS1998

Annual Research Briefs, 1998

The topics contained in this progress report are direct numerical simulation of turbulent non-premixed combustion with realistic chemistry; LES of non-premixed turbulent reacting flows with conditional source term estimation; measurements of the three-dimensional scalar dissipation rate in gas-phase planar turbulent jets; direct simulation of a jet diffusion flame; on the use of interpolating wavelets in the direct numerical simulation of combustion; on the use of a dynamically adaptive wavelet collocation algorithm in DNS (direct numerical simulation) of non-premixed turbulent combustion; 2D simulations of Hall thrusters; computation of trailing-edge noise at low mach number using LES and acoustic analogy; weakly nonlinear modeling of the early stages of bypass transition; interactions between freestream turbulence and boundary layers; interfaces at the outer boundaries of turbulent motions; largest scales of turbulent wall flows; the instability of streaks in near-wall turbulence; an implementation of the v(sup 2) - f model with application to transonic flows; heat transfer predictions in cavities; a structure-based model with stropholysis effects; modeling a confined swirling coaxial jet; subgrid-scale models based on incremental unknowns for large eddy simulations; subgrid scale modeling taking the numerical error into consideration; towards a near-wall model for LES of a separated diffuser flow; on the feasibility of merging LES with RANS (Reynolds Averaging Numerical simulation) for the near-wall region of attached turbulent flows; large-eddy simulation of a separated boundary layer; numerical study of a channel flow with variable properties; on the construction of high order finite difference schemes on non-uniform meshes with good conservation properties; development of immersed boundary methods for complex geometries; and particle methods for micro and macroscale flow simulations.

Spinks, Debra↗

DOE OSTI2025

Explicit block encodings of boundary value problems for many-body elliptic operators

Simulation of physical systems is one of the most promising use cases of future digital quantum computers. In this work we systematically analyze the quantum circuit complexities of block encoding the discretized elliptic operators that arise extensively in numerical simulations for partial differential equations, including high-dimensional instances for many-body simulations. When restricted to rectangular domains with separable boundary conditions, we provide explicit circuits to block encode the many-body Laplacian with separable periodic, Dirichlet, Neumann, and Robin boundary conditions, using standard discretization techniques from low-order finite difference methods. To obtain high-precision, we introduce a scheme based on periodic extensions to solve Dirichlet and Neumann boundary value problems using a high-order finite difference method, with only a constant increase in total circuit depth and subnormalization factor. We then present a scheme to implement block encodings of differential operators acting on more arbitrary domains, inspired by Cartesian immersed boundary methods. We then block encode the many-body convective operator, which describes interacting particles experiencing a force generated by a pair-wise potential given as an inverse power law of the interparticle distance. This work provides concrete recipes that are readily translated into quantum circuits, with depth logarithmic in the total Hilbert space dimension, that block encode operators arising broadly in applications involving the quantum simulation of quantum and classical many-body mechanics.

Kharazi, Tyler [University of California, Berkeley↗

NASA NTRS2004

Frequency Domain Calculations Of Acoustic Propagation

Two complex geometry problems are solved using the linearized Euler equations. The impedance mismatch method1 is used to impose the solid surfaces without the need to use a body-fitted grid. The problem is solved in the frequency domain to avoid long run times. Although the harmonic assumption eliminates all time dependence, a pseudo-time term is added to allow conventional iterative methods to be employed. A Jameson type, Runge-Kutta scheme is used to advance the solution in pseudo time. The spatial operator is based on a seven-point, sixth-order finite difference. Constant coefficient, sixth-derivative artificial dissipation is used throughout the domain. A buffer zone technique employing a complex frequency to damp all waves near the boundaries is used to minimize reflections. The results show that the method is capable of capturing the salient features of the scattering, but an excessive number of grid points are required to resolve the phenomena in the vicinity of the solid bodies because the wavelength of the acoustics is relatively short compared with the size of the bodies. Smoothly transitioning into the immersed boundary condition alleviates the difficulties, but a fine mesh is still required.

Lockard, David P.↗

NASA NTRS2004

Interaction of a Boundary Layer with a Turbulent Wake

The objective of this grant was to study the transition mechanisms on a flat-plate boundary layer interacting with the wake of a bluff body. This is a simplified configuration presented and designed to exemplify the phenomena that occur in multi-element airfoils, in which the wake of an upstream element impinges on a downstream one. Some experimental data is available for this configuration at various Reynolds numbers. The first task carried out was the implementation and validation of the immersed-boundary method. This was achieved by performing calculations of the flow over a cylinder at low and moderate Reynolds numbers. The low-Reynolds number results are discussed, which is enclosed as Appendix A. The high-Reynolds number results are presented in a paper in preparation for the Journal of Fluid Mechanics. We performed calculations of the wake-boundary-layer interaction at two Reynolds numbers, Re approximately equal to 385 and 1155. The first case is discussed and a comparison of the two calculations is reported. The simulations indicate that at the lower Reynolds number the boundary layer is buffeted by the unsteady Karman vortex street shed by the cylinder. This is shown: long streaky structures appear in the boundary layer in correspondence of the three-dimensionalities in the rollers. The fluctuations, however, cannot be self-sustained due to the low Reynolds-number, and the flow does not reach a turbulent state within the computational domain. In contrast, in the higher Reynolds-number case, boundary-layer fluctuations persist after the wake has decayed (due, in part, to the higher values of the local Reynolds number Re achieved in this case); some evidence could be observed that a self-sustaining turbulence generation cycle was beginning to be established. A third simulation was subsequently carried out at a higher Reynolds number, Re=3900. This calculation gave results similar to those of the Re=l155 case. Turbulence was established at fairly low Reynolds number, as a consequence of the high level of the free-stream perturbation. An instantaneous flow visualization for that case is shown. A detailed examination of flow statistics in the transitional and turbulent regions, including the evolution of the turbulent kinetic energy (TKE) budget and frequency spectra showed the formation and evolution of turbulent spots characteristic of the bypass transition mechanism. It was also observed that the turbulent eddies achieved an equilibrium, fully developed turbulent states first, as evidenced by the early agreement achieved by the terms in the TKE budget with those observed in turbulent flows. Once a turbulent Reynolds stress profile had been established, the velocity profile began to resemble a turbulent one, first in the inner region and later in the outer region of the wall layer. An extensive comparison of the three cases, including budgets, mean velocity and Reynolds stress profiles and flow visualization, is included. The results obtained are also presented.

Piomelli, Ugo↗