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We present explicit and implicit large eddy simulations for fully developed turbulent pipe flows using a continuous-Galerkin spectral element solver. On the one hand, the explicit stretched-vortex model (by Misra & Pullin [45] and Chung & Pullin [14]), accounts for an explicit treatment of unresolved stresses and is adapted to the high-order solver. On the other hand, an implicit approach based on a spectral vanishing viscosity technique is implemented. The latter implicit technique is modified to incorporate Chung & Pullin virtual-wall model instead of relying on implicit dissipative mechanisms near walls. This near-wall model is derived by averaging in the wall-normal direction and relying in local inner scaling to treat the time-dependence of the filtered wall-parallel velocity. The model requires space-time varying Dirichlet and Neumann boundary conditions for velocity and pressure respectively. We provide results and comparisons for the explicit and implicit subgrid treatments and show that both provide favourable results for pipe flows at Re_τ = 2×10^3 and Re_τ = 1.8×10^5 in terms of turbulence statistics. Additionally, we conclude that implicit simulations are enhanced when including the wall model and provide the correct statistics near walls.

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... To the best of our knowledge, the highest turbulent Re used in LES of turbulent pipe flows were Re τ 2,000 and 180,000, by Berrouk et al. [43] and Ferrer et. al. [44]. The authors present explicit and implicit LES of fully-developed turbulent pipe flows using a continuous Galerkin spectral element solver, motivated by the work of Chung & Pullin [34]. ...

... The authors present explicit and implicit LES of fully-developed turbulent pipe flows using a continuous Galerkin spectral element solver, motivated by the work of Chung & Pullin [34]. For explicit LES they utilized the explicit stretched-vortex model of Chung & Pullin [34], more details can be found here [44]. On the other hand, From the above literature review, it is clear that only a few studies have been conducted with wall-modeled LES focusing on the moderate-to-high Re wall-bounded turbulent flow. ...

Despite the high relevance of wall-bounded turbulence for engineering and natural science applications, many aspects of the underlying physics are still unclear. In particular, at high Re close to many real-life scenarios, the true nature of the flow is partially masked by the inability of numerical simulations to resolve all turbulent scales adequately. To overcome this issue, we aim to numerically investigate fully-developed turbulent pipe flows at moderate-to-high Re ($361 \leq Re_\tau \leq 6,000$), employing LES. A grid convergence study, using the WALE subgrid stress model, is presented for $Re_\tau=361$. Additionally, the prediction accuracy of distinct subgrid-scale stress models, such as WALE, SMG, OEEVM, LDKM, and DSEM, is examined using a range of statistical measures. The results infer, as expected, that SMG and OEEVM are too dissipative, whereas WALE, LDKM, and, more surprisingly, DSEM perform rather well compared to experiments and results from DNS. Moreover, LES utilizing WALE are performed and investigated in detail for six different Reynolds numbers in the interval from $Re_\tau = 361$ to $6,000$ with gradually refined grids. These computations allow an insight into what turbulence information is retained when LES with a wall model is applied to such high Reynolds numbers in the limit of a relatively coarse grid. Second-order statistics for all values of $Re_\tau$ exhibited excellent agreement with the DNS data in the outer region. Surprisingly, results also revealed a dramatic deviation from the DNS data in the sub-viscous layer region irrespective of the $Re_\tau$, attributed to the considered scaling for mesh refinement. Overall, the WALE model enabled accurate numerical simulations of high-Reynolds-number wall-bounded flows at a fraction of the cost incurred if the inner layer was temporally and spatially resolved.

... The modification has been proposed in [29] using an increased penalty parameter (compared to the minimum required to ensure coercivity of the scheme) when discretising the viscous terms using an interior penalty formulation. When combined with high-order DG methods, ILES provide valuable results even for challenging problems such as the prediction of turbulent quantities, detached flows and laminar-turbulent transition on airfoils, see, for example, [89,29,90,83,74,91,92]. Finally, a promising approach is to combine robust energy-stable methods (e.g., skew symmetric variants) with ILES techniques (e.g., Spectral Vanishing Viscosity) to provide accurate simulation at high Reynolds numbers, see [85,74]. ...

We present the latest developments of our High-Order Spectral Element Solver (HORSES3D), an open source high-order discontinuous Galerkin framework, capable of solving a variety of flow applications, including compressible flows (with or without shocks), incompressible flows, various RANS and LES turbulence models, particle dynamics, multiphase flows, and aeroacoustics. We provide an overview of the high-order spatial discretisation (including energy/entropy stable schemes) and anisotropic p-adaptation capabilities. The solver is parallelised using MPI and OpenMP showing good scalability for up to 1000 processors. Temporal discretisations include explicit, implicit, multigrid, and dual time-stepping schemes with efficient preconditioners. Additionally, we facilitate meshing and simulating complex geometries through a mesh-free immersed boundary technique. We detail the available documentation and the test cases included in the GitHub repository.

... Other recent pipe DNSs include the studies by Boersma [139], Wu et al. [140] and Ahn et al. [141], the latter at Re τ = 3, 008. Wall modelling in turbulent pipe flows was discussed by Ferrer et al. [142] in the context of spectral elements. Furthermore, a DNS of the flow through a complex pipe, denoted as the Food and Drug Administration (FDA) nozzle (which has important implications in biomedical applications) was simulated by Sánchez Abad et al. [143] using Nek5000, and the flow can be observed in Figure 12. ...

The present review shows a summary of current trends in high-fidelity simulations of turbulent flows in moderately complex geometries. These trends are put in the historical context of numerical simulations, starting with early weather predictions and continuing with seminal direct-numerical-simulation work. Here we discuss high-fidelity simulations conducted in a number of complex geometries, including ducts, pipes, wings and obstacles, and describe the potential of the spectral-element method (SEM) to carry out such simulations. Finally, we provide a number of future directions where novel data-driven methods can exploit the great wealth of high-quality turbulence data in the literature.

We present an implicit Large Eddy Simulation (iLES) h/p high order (≥2) unstructured Discontinuous Galerkin–Fourier solver with sliding meshes. The solver extends the laminar version of Ferrer and Willden, 2012 [34], to enable the simulation of turbulent flows at moderately high Reynolds numbers in the incompressible regime. This solver allows accurate flow solutions of the laminar and turbulent 3D incompressible Navier–Stokes equations on moving and static regions coupled through a high order sliding interface. The spatial discretisation is provided by the Symmetric Interior Penalty Discontinuous Galerkin (IP-DG) method in the x–y plane coupled with a purely spectral method that uses Fourier series and allows efficient computation of spanwise periodic three-dimensional flows.
Since high order methods (e.g. discontinuous Galerkin and Fourier) are unable to provide enough numerical dissipation to enable under-resolved high Reynolds computations (i.e. as necessary in the iLES approach), we adapt the laminar version of the solver to increase (controllably) the dissipation and enhance the stability in under-resolved simulations. The novel stabilisation relies on increasing the penalty parameter included in the DG interior penalty (IP) formulation. The latter penalty term is included when discretising the linear viscous terms in the incompressible Navier–Stokes equations. These viscous penalty fluxes substitute the stabilising effect of non-linear fluxes, which has been the main trend in implicit LES discontinuous Galerkin approaches.
The IP-DG penalty term provides energy dissipation, which is controlled by the numerical jumps at element interfaces (e.g. large in under-resolved regions) such as to stabilise under-resolved high Reynolds number flows. This dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps, thus providing an efficient stabilization mechanism for iLES.
The IP-DG stabilisation is complemented with a Spectral Vanishing Viscosity (SVV) method, in the z-direction, to enhance stability in the continuous Fourier space. The coupling between the numerical viscosity in the DG plane and the SVV damping, provides an efficient approach to stabilise high order methods at moderately high Reynolds numbers.
We validate the formulation for three turbulent flow cases: a circular cylinder at Re=3900, a static and pitch oscillating NACA 0012 airfoil at Re=10000 and finally a rotating vertical-axis turbine at Re=40000, with Reynolds based on the circular diameter, airfoil chord and turbine diameter, respectively. All our results compare favourably with published direct numerical simulations, large eddy simulations or experimental data.
We conclude that the DG-Fourier high order solver, with IP-SVV stabilisation, proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery.

High Order DG methods with Riemann solver based interface numerical flux functions offer an interesting dispersion dissipation behaviour: dispersion errors are very low for a broad range of scales, while dissipation errors are very low for well resolved scales and are very high for scales close to the Nyquist cutoff. This observation motivates the trend that DG methods with Riemann solvers are used without an explicit LES model added. Due to under-resolution of vortical dominated structures typical for LES type setups, element based high order methods suffer from stability issues caused by aliasing errors of the non-linear flux terms. A very common strategy to fight these aliasing issues (and instabilities) is so-called polynomial de-aliasing, where interpolation is exchanged with projection based on an increased number of quadrature points. In this paper, we start with this common no-model or implicit LES (iLES) DG approach with polynomial de-aliasing and Riemann solver dissipation and review its capabilities and limitations. We find that the strategy gives excellent results, but only when the resolution is such, that about 40\% of the dissipation is resolved. For more realistic, coarser resolutions used in classical LES e.g. of industrial applications, the iLES DG strategy becomes quite in-accurate. The core of this work is a novel LES strategy based on split form DG methods that are kinetic energy preserving. Such discretisations offer excellent stability with full control over the amount and shape of the added artificial dissipation. This premise is the main idea of the work and we will assess the LES capabilities of the novel split form DG approach. We will demonstrate that the novel DG LES strategy offers similar accuracy as the iLES methodology for well resolved cases, but strongly increases fidelity in case of more realistic coarse resolutions.

The paper provides a brief introduction to the near-wall problem of LES and how it can be solved through modeling of the near-wall turbulence. The distinctions and key differences between different approaches are emphasized, both in terms of fidelity (LES, wall-modeled LES, and DES) and in terms of different wall-modeled LES approaches (hybrid LES/RANS and wall-stress-models). The focus is on approaches that model the wall-stress directly, i.e., methods for which the LES equations are formally solved all the way down to the wall. Progress over the last decade is reviewed, and the most important and promising directions for future research are discussed.

We present large-eddy simulations (LES) of separation and reattachment of a flat-plate turbulent boundary-layer flow. Instead of resolving the near wall region, we develop a two-dimensional virtual wall model which can calculate the time- and space-dependent skin-friction vector field at the wall, at the resolved scale. By combining the virtual-wall model with the stretched-vortex subgrid-scale (SGS) model, we construct a self-consistent framework for the LES of separating and reattaching turbulent wall-bounded flows at large Reynolds numbers. The present LES methodology is applied to two different experimental flows designed to produce separation/reattachment of a flat-plate turbulent boundary layer at medium Reynolds number
$Re_{{\it\theta}}$
based on the momentum boundary-layer thickness
${\it\theta}$
. Comparison with data from the first case at
$Re_{{\it\theta}}=2000$
demonstrates the present capability for accurate calculation of the variation, with the streamwise co-ordinate up to separation, of the skin friction coefficient,
$Re_{{\it\theta}}$
, the boundary-layer shape factor and a non-dimensional pressure-gradient parameter. Additionally the main large-scale features of the separation bubble, including the mean streamwise velocity profiles, show good agreement with experiment. At the larger
$Re_{{\it\theta}}=11\,000$
of the second case, the LES provides good postdiction of the measured skin-friction variation along the whole streamwise extent of the experiment, consisting of a very strong adverse pressure gradient leading to separation within the separation bubble itself, and in the recovering or reattachment region of strongly-favourable pressure gradient. Overall, the present two-dimensional wall model used in LES appears to be capable of capturing the quantitative features of a separation-reattachment turbulent boundary-layer flow at low to moderately large Reynolds numbers.

The measurements by M. V. Zagarola and A. J. Smits [ibid. 373, 33–79 (1998; Zbl 0941.76510)] of mean velocity profiles in fully developed turbulent pipe flow are repeated using a smaller Pitot probe to reduce the uncertainties due to velocity gradient corrections. A new static pressure correction [B. J. McKeon and A. J. Smits (2002)] is used in analysing all data and leads to significant differences from the Zagarola and Smits conclusions. The results confirm the presence of a power-law region near the wall and, for Reynolds numbers greater than 230×10 3 (R + >5×10 3 ), a logarithmic region further out, but the limits of these regions and some of the constants differ from those reported by Zagarola and Smits. In particular, the log law is found for 600<y + <0·12R + (instead of 600<y + <0·07R + ), and the von Kármán constant κ, the additive constant B for the log law using inner flow scaling, and the additive constant B * for the log law using outer scaling are found to be 0·421±0·002, 5·60±0·08 and 1·20±0·10, respectively, with 95% confidence level (compared with 0·436±0·002, 6·15±0·08, and 1·51±0·03 found by Zagarola and Smits). The data also confirm that the pipe flow data for Re D ≤13·6×10 6 (as a minimum) are not affected by surface roughness.

In this paper, we investigate the accuracy and efficiency of discontinuous Galerkin spectral method simulations of under-resolved transitional and turbulent flows at moderate Reynolds numbers, where the accurate prediction of closely coupled laminar regions, transition and developed turbulence presents a great challenge to large eddy simulation modelling. We take full advantage of the low numerical errors and associated superior scale resolving capabilities of high-order spectral methods by using high-order ansatz functions up to 12th order. We employ polynomial de-aliasing techniques to prevent instabilities arising from inexact quadrature of nonlinearities. Without the need for any additional filtering, explicit or implicit modelling, or artificial dissipation, our high-order schemes capture the turbulent flow at the considered Reynolds number range very well. Three classical large eddy simulation benchmark problems are considered: a circular cylinder flow at ReD=3900, a confined periodic hill flow at Reh=2800 and the transitional flow over a SD7003 airfoil at Rec=60,000. For all computations, the total number of degrees of freedom used for the discontinuous Galerkin spectral method simulations is chosen to be equal or considerably less than the reported data in literature. In all three cases, we achieve an equal or better match to direct numerical simulation results, compared with other schemes of lower order with explicitly or implicitly added subgrid scale models. Copyright © 2014 John Wiley & Sons, Ltd.

A direct numerical simulation of a fully developed turbulent pipe flow was performed to investigate the similarities and differences of very-large-scale motions (VLSMs) to those of turbulent boundary layer (TBL) flows. The Reynolds number was set to ReD = 35 000, and the computational domain was 30 pipe radii in length. Inspection of instantaneous fields, streamwise two-point correlations, and population trends of the momentum regions showed that the streamwise length of the structures in the pipe flow grew continuously beyond the log layer (y/δ < 0.3–0.4) with a large population of long structures (>3δ), and the maximum length of the VLSMs increased up to ∼30δ. Such differences between the TBL and pipe flows arose due to the entrainment of large plumes of the intermittent potential flow in the TBL, creating break-down of the streamwise coherence of the structures above the log layer with the strong swirling strength and Reynolds shear stress. The average streamwise length scale of the pipe flow was approximately 1.5–3.0 times larger than that of the TBL through the log and wake regions. The maximum contribution of the structures to the Reynolds shear stress was observed at approximately 6δ in length, whereas that of the TBL was at 1δ–2δ, indicating a higher contribution of the VLSMs to the Reynolds shear stress in the pipe flow than in the TBL flow.

Direct numerical simulations of fully developed turbulent pipe flow that span the Reynolds number range 90 ≲ δ+ ≲ 1000 are used to investigate the evolution of the mean momentum field in and beyond the transitional regime. It is estimated that the four layer regime for pipe flow is nominally established for δ+ ⩾ 180, which is also close to the value found for channel flow. Primary attention is paid to the magnitude ordering and scaling behaviors of the terms in the mean momentum equation. Once the ordering underlying the existence of four distinct balance layers is attained, this ordering is sustained for all subsequent increases in Reynolds number. Comparisons indicate that pipe flow develops toward the four layer regime in a manner similar to that for channel flow, but distinct from that for the boundary layer. Small but discernible differences are observed in the mean momentum field development in pipes and channels. These are tentatively attributed to variations in the manner by which the outer region mean vorticity field develops in these two flows.

Very large-scale motions in the form of long regions of streamwise velocity fluctuation are observed in the outer layer of fully developed turbulent pipe flow over a range of Reynolds numbers. The premultiplied, one-dimensional spectrum of the streamwise velocity measured by hot-film anemometry has a bimodal distribution whose components are associated with large-scale motion and a range of smaller scales corresponding to the main turbulent motion. The characteristic wavelength of the large-scale mode increases through the logarithmic layer, and reaches a maximum value that is approximately 12–14 times the pipe radius, one order of magnitude longer than the largest reported integral length scale, and more than four to five times longer than the length of a turbulent bulge. The wavelength decreases to approximately two pipe radii at the pipe centerline. It is conjectured that the very large-scale motions result from the coherent alignment of large-scale motions in the form of turbulent bulges or packets of hairpin vortices.

Detached-eddy simulation (DES) was first proposed in 1997 and first used in 1999, so its full history can be surveyed. A DES community has formed, with adepts and critics, as well as new branches. The initial motivation of high– Reynolds number, massively separated flows remains, for which DES is con-vincingly more capable presently than either unsteady Reynolds-averaged Navier-Stokes (RANS) or large-eddy simulation (LES). This review dis-cusses compelling examples, noting the visual and quantitative success of DES. Its principal weakness is its response to ambiguous grids, in which the wall-parallel grid spacing is of the order of the boundary-layer thickness. In some situations, DES on a given grid is then less accurate than RANS on the same grid or DES on a coarser grid. Partial remedies have been found, yet dealing with thickening boundary layers and shallow separation bubbles is a central challenge. The nonmonotonic response of DES to grid refinement is disturbing to most observers, as is the absence of a theoretical order of accuracy. These issues also affect LES in any nontrivial flow. This review also covers the numerical needs of DES, gridding practices, coupling with different RANS models, derivative uses such as wall modeling in LES, and extensions such as zonal DES and delayed DES.

We review wall-bounded turbulent flows, particularly high–Reynolds number, zero–pressure gradient boundary layers, and fully developed pipe and channel flows. It is apparent that the approach to an asymptotically high–Reynolds number state is slow, but at a sufficiently high Reynolds number the log law remains a fundamental part of the mean flow description. With regard to the coherent motions, very-large-scale motions or superstructures exist at all Reynolds numbers, but they become increasingly important with Reynolds number in terms of their energy content and their interaction with the smaller scales near the wall. There is accumulating evidence that certain features are flow specific, such as the constants in the log law and the behavior of the very large scales and their interaction with the large scales (consisting of vortex packets). Moreover, the refined attached-eddy hypothesis continues to provide an important theoretical framework for the structure of wall-bounded turbulent flows.

Aiming at the better prediction of inertial particle dispersion in LES of turbulent shear flow, a stochastic diffusion process of the Langevin type is adopted to model the time increments of the fluid velocity seen by inertial particles. This modelling is particularly crucial for dispersion and deposition of inertial particles with small relaxation times compared to the smallest LES-resolved turbulence time scales. Simple closures for drift and diffusion terms are described. The effects of inertia and external forces on particle trajectories are modelled. To numerically validate the proposed model, LES calculations are performed to track solid particles (glass beads in air with different Stokes' numbers) in a high Reynolds number, equilibrium turbulent pipe flow (Re = 50 000 based on the maximum velocity and the pipe diameter). LES predictions are compared to RANS results and experimental observations. Simulation findings demonstrate the superiority of LES compared to RANS in predicting particle dispersion statistics. More importantly, the use of a stochastic approach to model the subgrid scale fluctuations has proven crucial for results concerning the small-Stokes-number particles. The influence of the model on particle deposition needs to be assessed and additional validation for non-equilibrium turbulent flows is required.

Friction factor data from two recent pipe flow experiments are combined to provide a comprehensive picture of the friction factor variation for Reynolds numbers from 10 to 36,000,000.

Fully developed incompressible turbulent pipe flow at bulk-velocity- and pipe-diameter-based Reynolds number ReD=44000 was simulated with second-order finite-difference methods on 630 million grid points. The corresponding Kármán number R+, based on pipe radius R, is 1142, and the computational domain length is 15R. The computed mean flow statistics agree well with Princeton Superpipe data at ReD=41727 and at ReD=74000. Second-order turbulence statistics show good agreement with experimental data at ReD=38000. Near the wall the gradient of with respect to ln(1−r)+ varies with radius except for a narrow region, 70 < (1−r)+ < 120, within which the gradient is approximately 0.149. The gradient of with respect to ln{(1−r)++a+} at the present relatively low Reynolds number of ReD=44000 is not consistent with the proposition that the mean axial velocity is logarithmic with respect to the sum of the wall distance (1−r)+ and an additive constant a+ within a mesolayer below 300 wall units. For the standard case of a+=0 within the narrow region from (1−r)+=50 to 90, the gradient of with respect to ln{(1−r)++a+} is approximately 2.35. Computational results at the lower Reynolds number ReD=5300 also agree well with existing data. The gradient of with respect to 1−r at ReD=44000 is approximately equal to that at ReD=5300 for the region of 1−r > 0.4. For 5300 < ReD < 44000, bulk-velocity-normalized mean velocity defect profiles from the present DNS and from previous experiments collapse within the same radial range of 1−r > 0.4. A rationale based on the curvature of mean velocity gradient profile is proposed to understand the perplexing existence of logarithmic mean velocity profile in very-low-Reynolds-number pipe flows. Beyond ReD=44000, axial turbulence intensity varies linearly with radius within the range of 0.15 < 1−r < 0.7. Flow visualizations and two-point correlations reveal large-scale structures with comparable near-wall azimuthal dimensions at ReD=44000 and 5300 when measured in wall units. When normalized in outer units, streamwise coherence and azimuthal dimension of the large-scale structures in the pipe core away from the wall are also comparable at these two Reynolds numbers.

In the outer region of fully developed turbulent pipe flow very large-scale motions reach wavelengths more than 8$R$–16$R$ long (where $R$ is the pipe radius), and large-scale motions with wavelengths of $2R$–$3R$ occur throughout the layer. The very-large-scale motions are energetic, typically containing half of the turbulent kinetic energy of the streamwise component, and they are unexpectedly active, typically containing more than half of the Reynolds shear stress. The spectra of the $y$-derivatives of the Reynolds shear stress show that the very-large-scale motions contribute about the same amount to the net Reynolds shear force, d$\overline{-u'v'}/{\rm d}y$, as the combination of all smaller motions, including the large-scale motions and the main turbulent motions. The main turbulent motions, defined as the motions small enough to be in a statistical equilibrium (and hence smaller than the large-scale motions) contribute relatively little to the Reynolds shear stress, but they constitute over half of the net Reynolds shear force.

A new pressure formulation for splitting methods is developed that results in high-order time-accurate schemes for the solution of the incompressible Navier-Stokes equations. In particular, improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods. A new family of stiffly stable schemes is employed in mixed explicit/implicit time-intgration rules. These schemes exhibit much broader stability regions as compared to Adams-family schemes, typically used in splitting methods. Their stability properties remain almost constant as the accuracy of the integration increases, so that robust third- or higher-order time-accurate schemes can readily be constructed that remain stable at relatively large CFL number. The new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation. A model Stokes problem is studied in detail, and several examples of Navier-Stokes solutions of flows in complex geometries are reported. Comparison is made with the previously used first-order in time spectral element splitting and non-splitting (e.g., Uzawa) schemes. High-order splitting/spectral element methods combine accuracy in space and time, and flexibility in geometry, and thus can be very efficient in direct simulations of turbulent flows in complex geometries.

A new simulation approach for high Reynolds number turbulent flows is developed, combining concepts of monotonicity in nonlinear conservation laws with concepts of large-eddy simulation. The spectral vanishing viscosity (SVV), first introduced by E. Tadmor [SIAM J. Numer. Anal.26, 30 (1989)], is incorporated into the Navier–Stokes equations for controlling high-wavenumber oscillations. Unlike hyperviscosity kernels, the SVV approach involves a second-order operator which can be readily implemented in standard finite element codes. In the work presented here, discretization is performed using hierarchical spectral/hp methods accommodating effectively an ab initio intrinsic scale separation. The key result is that monotonicity is enforced via SVV leading to stable discretizations without sacrificing the formal accuracy, i.e., exponential convergence, in the proposed discretization. Several examples are presented to demonstrate the effectiveness of the new approach including a comparison with eddy-viscosity spectral LES of turbulent channel flow. In its current implementation the SVV approach for controlling the small scales is decoupled from the large scales, but a procedure is proposed that will provide coupling similar to the classical LES formulation.

Large-eddy simulation (LES) has proven to be a computationally tractable approach to simulate unsteady turbulent flows. However, prohibitive resolution requirements induced by near-wall eddies in high-Reynolds number boundary layers necessitate the use of wall models or approximate wall boundary conditions. We review recent investigations in wall-modeled LES, including the development of novel approximate boundary conditions and the application of wall models to complex flows (e.g., boundary-layer separation, shock/boundary-layer interactions, transition). We also assess the validity of underlying assumptions in wall-model derivations to elucidate the accuracy of these investigations, and offer suggestions for future studies.

Turbulence closure models are central to a good deal of applied computational fluid dynamical analysis. Closure modeling endures as a productive area of research. This review covers recent developments in elliptic relaxation and elliptic blending models, unified rotation and curvature corrections, transition prediction, hybrid simulation, and data-driven methods. The focus is on closure models in which transport equations are solved for scalar variables, such as the turbulent kinetic energy, a timescale, or a measure of anisotropy. Algebraic constitutive representations are reviewed for their role in relating scalar closures to the Reynolds stress tensor. Seamless and nonzonal methods, which invoke a single closure model, are reviewed, especially detached eddy simulation (DES) and adaptive DES. Other topics surveyed include data-driven modeling and intermittency and laminar fluctuation models for transition prediction. The review concludes with an outlook.

We present wall-resolved large-eddy simulations (LES) of flow over a smooth-wall circular cylinder up to , where is Reynolds number based on the cylinder diameter and the free-stream speed . The stretched-vortex subgrid-scale (SGS) model is used in the entire simulation domain. For the sub-critical regime, six cases are implemented with . Results are compared with experimental data for both the wall-pressure-coefficient distribution on the cylinder surface, which dominates the drag coefficient, and the skin-friction coefficient, which clearly correlates with the separation behaviour. In the super-critical regime, LES for three values of are carried out at different resolutions. The drag-crisis phenomenon is well captured. For lower resolution, numerical discretization fluctuations are sufficient to stimulate transition, while for higher resolution, an applied boundary-layer perturbation is found to be necessary to stimulate transition. Large-eddy simulation results at , with a mesh of , agree well with the classic experimental measurements of Achenbach (J. Fluid Mech., vol. 34, 1968, pp. 625-639) especially for the skin-friction coefficient, where a spike is produced by the laminar-turbulent transition on the top of a prior separation bubble. We document the properties of the attached-flow boundary layer on the cylinder surface as these vary with . Within the separated portion of the flow, mean-flow separation-reattachment bubbles are observed at some values of , with separation characteristics that are consistent with experimental observations. Time sequences of instantaneous surface portraits of vector skin-friction trajectory fields indicate that the unsteady counterpart of a mean-flow separation-reattachment bubble corresponds to the formation of local flow-reattachment cells, visible as coherent bundles of diverging surface streamlines.

We investigate the potential of linear dispersion–diffusion analysis in providing direct guidelines for turbulence simulations through the under-resolved DNS (sometimes called implicit LES) approach via spectral/hp methods. The discontinuous Galerkin (DG) formulation is assessed in particular as a representative of these methods. We revisit the eigen-solutions technique as applied to linear advection and suggest a new perspective to the role of multiple numerical modes, peculiar to spectral/hp methods. From this new perspective , "secondary" eigenmodes are seen to replicate the propagation behaviour of a "primary" mode, so that DG's propagation characteristics can be obtained directly from the dispersion–diffusion curves of the primary mode. Numerical dissipation is then appraised from these primary eigencurves and its effect over poorly-resolved scales is quantified. Within this scenario, a simple criterion is proposed to estimate DG's effective resolution in terms of the largest wavenumber it can accuratelyresolve in a given hp approximation space, also allowing us to present points per wavelength estimates typically used in spectral and finite difference methods. Although strictly valid for linear advection, the devised criterion is tested against (1D) Burgers turbulence and found to predict with good accuracy the beginning of the dissipation range on the energy spectra of under-resolved simulations. The analysis of these test cases through the proposed methodology clarifies why and how the DG formulation can be used for under-resolved turbulence simulations without explicit subgrid-scale modelling. In particular, when dealing with communication limited hardware which forces one to consider the performance for a fixed number of degrees of freedom, the use of higher polynomial orders along with moderately coarser meshes is shown to be the best way to translate available degrees of freedom into resolution power.

A direct numerical simulation of a turbulent pipe flow at a high Reynolds number of Reτ= 3008 over a long axial domain length (30R) was performed. The streamwise mean velocity followed the power law in the overlap region (y+ = 90–300; y/R = 0.03–0.1) based on the power law indicator function. The scale separation of the Reynolds shear stresses into two components of small- and large-scale motions (LSMs) revealed that the LSMs in the outer region played an important role in constructing the constant-stress layer and the mean velocity. In the pre-multiplied energy spectra of the streamwise velocity fluctuations, the bimodal distribution was observed at both short and long wavelengths. The kx−1 region associated with the attached eddies appeared in λx/R = 2–5 and λx/y = 18–160 at y+ = 90–300, where the power law was established in the same region. The kz −1 region also appeared in λz/R = 0.3–0.6 at y+ = 3 and 150. Linear growth of small-scale energy to large-scale energy induced the kx−1 region at high Reynolds numbers, resulting in a large population of the LSMs. This result supported the origin of very-large-scale motions in the pseudo-streamwise alignment of the LSMs. In the pre-multiplied energy spectra of the Reynolds shear stress, the bimodal distribution was observed without the kx−1 region.

We describe a high Reynolds number large-eddy-simulation (LES) study of turbulent flow in a long channel of length 128 channel half heights, delta, with the walls consisting of roughness strips where the long stream-wise extent invites a full relaxation of the mean velocities within each strip. The channel is stream-wise periodic and strips are oriented transverse to the flow resulting in repeated transitions between smooth and rough surfaces along the stream-wise direction. The present LES uses a wall model that contains Colebrook's empirical formula as a roughness correction to both the local and dynamic calculation of the friction velocity and also the LES wall boundary condition. This operates point-wise across wall surfaces, and hence changes in the outer flow can be viewed as a response to the temporally and/or spatially variant roughness distribution. At the wall surface, dynamically calculated levels of time- and span-wise-averaged friction velocity (u(tau)) over bar (x) over/undershoot and then fully recover towards their smooth or rough state over a stream-wise distance of order 10-30 delta depending on both roughness and Reynolds number. Also, the initial response rate in (u(tau)) over bar shows Reynolds number and roughness dependence over both transitions. The growth rate of the internal boundary layer (IBL), defined by the abrupt change in stream-wise turbulent intensity, is found to grow as x(0.70) on average over multiple simulation conditions for the case of a smooth-to-rough transition, which agrees with the experimental results of Antonia and Luxton (1971) [1] and Efros and Krogstad (2011)[2]. IBL profiles demonstrate a good collapse on delta/log(Re-tau*), where Re-tau* is the local Reynolds number based on (u(tau)) over bar at the point of full recovery.

Modal analysis of the flux reconstruction (FR) formulation is performed to obtain the semi-discrete and fully-discrete dispersion relations, using which, the wave properties of physical as well as spurious modes are characterized. The effect of polynomial order, correction function and solution points on the dispersion, dissipation and relative energies of the modes are investigated. Using this framework, a new set of linearly stable high-order FR schemes is proposed that minimizes wave propagation errors for the range of resolvable wavenumbers. These schemes provide considerably reduced error for advection in comparison to the Discontinuous Galerkin scheme and benefit from having an explicit differential update. The corresponding resolving efficiencies compare favorably to those of standard high-order compact finite difference schemes. These theoretical expectations are verified by a comparison of proposed and existing FR schemes in advecting a scalar quantity on uniform as well as non-uniform grids.

. In this paper we study the Legendre Spectral Viscosity (SV) method for the approximate solution of initialboundary value problems associated with nonlinear conservation laws. We prove that by adding a small amount of spectral viscosity, bounded solutions of the Legendre SV method converge to the exact scalar entropy solution. Our convergence proof is based on compensated compactness arguments, and therefore applies to certain 2 Theta 2 systems. Finally, we present numerical experiments for scalar as well as the one-dimensional system of gas dynamics equations, which confirm the convergence of the Legendre SV method. Moreover, these numerical experiments indicate that by post-processing the SV approximation, one can recover the entropy solution within spectral accuracy. Key Words. Conservation laws, Legendre polynomials, spectral viscosity, post-processing, compensated compactness, convergence, spectral accuracy. AMS(MOS) subject classification. 35L65, 65M10, 65M15. 1. Introduction...

We describe large-eddy simulations (LES) of the flat-plate turbulent boundary layer in the presence of an adverse pressure gradient. The stretched-vortex subgrid-scale model is used in the domain of the flow coupled to a wall model that explicitly accounts for the presence of a finite pressure gradient. The LES are designed to match recent experiments conducted at the University of Melbourne wind tunnel where a plate section with zero pressure gradient is followed by section with constant adverse pressure gradient. First, LES are described at Reynolds numbers based on the local free-stream velocity and the local momentum thickness in the range 6560–13,900 chosen to match the experimental conditions. This is followed by a discussion of further LES at Reynolds numbers at approximately 10 times and 100 times these values, which are well out of range of present day direct numerical simulation and wall-resolved LES. For the lower Reynolds number runs, mean velocity profiles, one-point turbulent statistics of the velocity fluctuations, skin friction and the Clauser and acceleration parameters along the streamwise, adverse pressure-gradient domain are compared to the experimental measurements. For the full range of LES, the relationship of the skin-friction coefficient, in the form of the ratio of the local free-stream velocity to the local friction velocity, to both Reynolds number and the Clauser parameter is explored. At large Reynolds numbers, a region of collapse is found that is well described by a simple log-like empirical relationship over two orders of magnitude. This is expected to be useful for constant adverse-pressure gradient flows. It is concluded that the present adverse pressure gradient boundary layers are far from an equilibrium state.

Direct numerical simulation databases of turbulent channel and pipe flow have been used in order to assess the energy transfer between resolved and unresolved motions in large‐eddy simulations. To this end, the velocity fields are split into three parts: a statistically stationary mean flow, the resolved, and the unresolved turbulent fluctuations. The distinction between the resolved and unresolved motions is based on the application of a cutoff filter in spectral space. Within the buffer layer a backward transfer of averaged kinetic energy from subgrid to grid‐scale turbulent motions has been found to exist, which is primarily caused by subgrid‐scale stresses aligned with the mean rates of strain. Such reverse transfer generally cannot be described by the simple eddy‐viscosity‐type subgrid models usually applied in large‐eddy simulations. The use of a conditional averaging technique revealed that the reverse transfer of energy within the near‐wall flow is strongly enhanced by coherent motions, such as the well‐known bursting events.

Large eddy simulation (LES) is reported for both smooth and rough-wall channel flows at resolutions for which the roughness is subgrid. The stretched vortex, subgrid-scale model is combined with an existing wall-model that calculates the local friction velocity dynamically while providing a Dirichlet-like slip velocity at a slightly raised wall. This wall model is presently extended to include the effects of subgrid wall roughness by the incorporation of the Hama's roughness function ΔU+(ks∞+) that depends on some geometric roughness height ks∞ scaled in inner variables. Presently Colebrook's empirical roughness function is used but the model can utilize any given function of an arbitrary number of inner-scaled, roughness length parameters. This approach requires no change to the interior LES and can handle both smooth and rough walls. The LES is applied to fully turbulent, smooth, and rough-wall channel flow in both the transitional and fully rough regimes. Both roughness and Reynolds number effects are captured for Reynolds numbers Reb based on the bulk flow speed in the range 104–1010 with the equivalent Reτ, based on the wall-drag velocity uτ varying from 650 to 108. Results include a Moody-like diagram for the friction factor f = f(Reb, ε), ε = ks∞/δ, mean velocity profiles, and turbulence statistics. In the fully rough regime, at sufficiently large Reb, the mean velocity profiles show collapse in outer variables onto a roughness modified, universal, velocity-deficit profile. Outer-flow stream-wise turbulence intensities scale well with uτ for both smooth and rough-wall flow, showing a log-like profile. The infinite Reynolds number limits of both smooth and rough-wall flows are explored. An assumption that, for smooth-wall flow, the turbulence intensities scaled on uτ are bounded above by the sum of a logarithmic profile plus a finite function across the whole channel suggests that the infinite Reb limit is inviscid slip flow without turbulence. The asymptote, however, is extremely slow. Turbulent rough-wall flow that conforms to the Hama model shows a finite limit containing turbulence intensities that scale on the friction factor for any small but finite roughness.

Time series velocity signals obtained from large-eddy simulations (LES) within the logarithmic region of the zero-pressure gradient turbulent boundary layer over a smooth wall are used in combination with an empirical, predictive inner-outer wall model [I. Marusic, R. Mathis, and N. Hutchins, “Predictive model for wall-bounded turbulent flow,” Science 329, 193 (2010)10.1126/science.1188765] to calculate the statistics of the fluctuating streamwise velocity in the inner region. Results, including spectra and moments up to fourth order, are compared with equivalent predictions using experimental time series, as well as with direct experimental measurements at Reynolds numbers Reτ = 7300, 13 600, and 19 000. The LES combined with the wall model are then used to extend the inner-layer predictions to Reynolds numbers Reτ = 62 000, 100 000, and 200 000 that lie within a gap in log (Reτ) space between laboratory measurements and surface-layer, atmospheric experiments. The present results support a loglike increase in the near-wall peak of the streamwise turbulence intensities with Reτ and also provide a means of extending LES results at large Reynolds numbers to the near-wall region of wall-bounded turbulent flows.

In this paper, direct numerical simulation of fully developed turbulent pipe flow is carried out at Reτ ≈ 170 and 500 to investigate the effect of the streamwise periodic length on the convergence of turbulence statistics. Mean flow, turbulence intensities, correlations, and energy spectra were computed. The findings show that in the near-wall region (below the buffer region, r+ ≤ 30), the required pipe length for all turbulence statistics to converge needs to be at least a viscous length of O(6300) wall units and should not be scaled with the pipe radius (δ). It was also found for convergence of turbulence statistics at the outer region that the pipe length has to be scaled with pipe radius and a proposed pipe length of 8πδ seems sufficient for the Reynolds numbers considered in this study.

We examine the dispersion and dissipation properties of the P
N
P
M
schemes for linear wave propagation problems. P
N
P
M
scheme are based on P
N
discontinuous Galerkin base approximations augmented with a cell centered polynomial least-squares reconstruction from degree N up to the design polynomial degree M. This methodology can be seen as a generalized discretization framework, as cell centered high order finite volume schemes (N=0) and discontinuous Galerkin schemes (N=M) are included as special cases.
We show that with respect to the dispersion error, the pure discontinuous Galerkin variant gives typically the best accuracy for a defined number of points per wavelength. Regarding the dissipation behavior, combinations of N and M exist that result in slightly lower errors for a given resolution. An investigation of the influence of the stencil size on the accuracy of the scheme shows that the errors are smaller the smaller the stencil size for the reconstruction.

Fully developed, statistically steady turbulent flow in straight, curved and helically coiled pipes is studied by means of direct numerical simulation to show the influence of curvature and torsion on turbulent flow. The incompressible Navier–Stokes equations, expressed in an orthogonal helical coordinate system, are integrated numerically by using second order central schemes in space and time. Besides the mean flow quantities and the rms values, vector plots and the stream function are shown to give an impression of the induced mean secondary flow.

During the last decade the spectral vanishing viscosity (SVV) method has been adopted successfully for large-eddy-type simulations (LES) with high-order discretizations in both Cartesian and cylindrical coordinate systems. For the latter case, however, previous studies were confined to annular domains. In the present work, we examine the applicability of SVV in cylindrical coordinates to flows in which the axis region is included, within the setting of an exponentially convergent spectral element–Fourier discretization. In addition to the ‘standard’ SVV viscosity kernel, two modified kernels with enhanced stabilization in the axis region are considered. Three fluid flow examples are considered, including turbulent pipe flow. The results, on the one hand, show a surprisingly small influence of the SVV kernel, while on the other, they reveal the importance of spatial resolution in the axis region.

Large Eddy Simulation (LES) is an approach to compute turbulent flows based on resolving the unsteady large-scale motion of the fluid while the impact of the small-scale turbulence on the large scales is accounted for by a sub-grid scale model. This model distinguishes LES from any other method and reduces the computational demands compared with a Direct Numerical Simulation. On the other hand, the cost typically is still at least an order of magnitude larger than for steady Reynolds-averaged computations. The LES approach is attractive when statistical turbulence models fail, when insight into the vortical dynamics or unsteady forces on a body is desired, or when additional features are involved such as large-scale mixing, particle transport, sound generation etc. In recent years the rapid increase of computer power has made LES accessible to a broader scientific community, and this is reflected in an abundance of papers on the method and its applications. Still, however, some fundamental aspects of LES are not conclusively settled, a fact residing in the intricate coupling between mathematical, physical, numerical and algorithmic issues. In this situation it is of great importance to gain an overview of the available approaches and techniques.
Pierre Sagaut, in the style of a French encyclopedist, gives a very complete and exhaustive treatment of the different kinds of sub-grid scale models which have been developed so far. After discussing the separation into resolved and unresolved scales and its application to the Navier-Stokes equations, more than 140 pages are directly devoted to the description of sub-grid scale models. They are classified according to different criteria, which helps the reader to find his or her way through the arsenal of reasonings. The theoretical framework for which these models have mostly been developed is isotropic turbulence. The required notions from classical turbulence theory are summarized together with notions from EDQNM theory in two concise and helpful appendices. Further sections deal with numerical and implementational issues, boundary conditions and validation practice. A final section assembles a few key applications, cumulating in a condensed list of some general experiences gained so far.
The book very wisely concentrates on issues particular to LES, which to a large extent is sub-grid scale modelling. Classical issues of CFD, such as numerical discretization schemes, solution procedures etc, or post-processing are not addressed. Limiting himself to incompressible, non-reactive flows, the author succeeds in describing the fundamental issues in great detail, thus laying the foundations for the understanding of more complex situations. The presentation is essentially theoretical and the reader should have some prior knowledge of turbulence theory and Fourier transforms. The text itself is well written and generally very clear. A pedagogical effort is made in several places, e.g. when an overview over a group of models is given before these are described in detail. A few typing errors and technical details should be amended in a second edition, though, such as the statement that a filter which is not a projector is invertible (p 12), but this is not detrimental to the quality of the text.
Overall the book is a very relevant contribution to the field of LES and I read it with pleasure and benefit. It constitutes a worthy reference book for scientists and engineers interested in or practising LES and may serve as a textbook for a postgraduate course on the subject.
Jochen Fr?hlich

A primitive-variable formulation for simulation of time-dependent incompressible o ws in cylindrical coordinates is developed. Spectral elements are used to discretise the meridional semi-plane, coupled with Fourier expansions in azimuth. Unlike pre- vious formulations where special distributions of nodal points have been used in the radial direction, the current work adopts standard Gauss{Lobatto{Legendre nodal- based expansions in both the radial and axial directions. Using a Galerkin projection of the symmetrised cylindrical Navier{Stokes equations all geometric singularities are removed as a consequence of either the Fourier-mode dependence of axial bound- ary conditions or the shape of the weight function applied in the Galerkin projection. This observation implies that in a numerical implementation, geometrically singular terms can be naively treated by explicitly zeroing their contributions on the axis in integral expressions without recourse to special treatments such as l'Hopital's rule. Exponential convergence of the method both in the meridional semi-plane and in azimuth is demonstrated through application to a three-dimensional analytical solution of the Navier{Stokes equations in which o w crosses the axis.

Where spectral methods are concerned, the spectral vanishing viscosity (SVV) method offers an inter- esting way of computing high Reynolds number flows since it allows stabilization of the calculations whilst preserving the exponential rate of convergence of the spectral approximation. Here we first show how to implement the SVV method in an existing Navier-Stokes solver and then investigate the sensitivity of the numerical results to its main characteristic parameters, namely the SVV amplitude and the SVV activation mode, by focusing on the computation of a turbulent wake in a cylinder, embedded in a channel-like domain, at Reynolds number Re = 3900.

We present a new implementation of the spectral vanishing viscosity method appropriate for alternative formulations of large-eddy simulations. We first review the method and subsequently present results for turbulent incompressible channel flow.

We extend the splitting scheme of G. E. Karniadakis, M. Israeli and S. A. Orszag [J. Comput. Phys. 97, No. 2, 414-443 (1991; Zbl 0738.76050)] to temporally and spatially varying viscosity, while retaining the decoupling of viscous term. The derivation of the algorithm and a simplified von Neumann stability analysis for the one-dimensional diffusion equation are presented, demonstrating that for a linear diffusion equation, the scheme is unconditionally stable if the implicit part of viscosity is larger than the explicit part.

We present in this paper high resolution, two-dimensional LDV measurements in a turbulent pipe flow of water over the Reynolds number range 5000–25000. Results for the turbulence statistics up to the fourth moment are presented, as well as power spectra in the near-wall region. These results clearly show that the turbulence statistics scaled on inner variables are Reynolds-number dependent in the aforementioned range of Reynolds numbers. For example, the constants in the dimensionless logarithmic mean-velocity profile are shown to vary with Reynolds number. Our conclusion that turbulence statistics depend on the Reynolds number is consistent with results found in other flow configurations, e.g., a channel flow. Our results for the pipe flow, however, lead nevertheless to quite different tendencies.

Measurements of the mean velocity profile and pressure drop were performed in a fully developed, smooth pipe flow for Reynolds numbers from 31×10 ³ to 35×10 ⁶ . Analysis of the mean velocity profiles indicates two overlap regions: a power law for 60< y ⁺ <500 or y ⁺ <0.15 R ⁺ , the outer limit depending on whether the Kármán number R ⁺ is greater or less than 9×10 ³ ; and a log law for 600< y ⁺ <0.07 R ⁺ . The log law is only evident if the Reynolds number is greater than approximately 400×10 ³ ( R ⁺ >9×10 ³ ). Von Kármán's constant was shown to be 0.436 which is consistent with the friction factor data and the mean velocity profiles for 600< y ⁺ <0.07 R ⁺ , and the additive constant was shown to be 6.15 when the log law is expressed in inner scaling variables.
A new theory is developed to explain the scaling in both overlap regions. This theory requires a velocity scale for the outer region such that the ratio of the outer velocity scale to the inner velocity scale (the friction velocity) is a function of Reynolds number at low Reynolds numbers, and approaches a constant value at high Reynolds numbers. A reasonable candidate for the outer velocity scale is the velocity deficit in the pipe, U CL − Ū , which is a true outer velocity scale, in contrast to the friction velocity which is a velocity scale associated with the near-wall region which is ‘impressed’ on the outer region. The proposed velocity scale was used to normalize the velocity profiles in the outer region and was found to give significantly better agreement between different Reynolds numbers than the friction velocity.
The friction factor data at high Reynolds numbers were found to be significantly larger (>5%) than those predicted by Prandtl's relation. A new friction factor relation is proposed which is within ±1.2% of the data for Reynolds numbers between 10×10 ³ and 35×10 ⁶ , and includes a term to account for the near-wall velocity profile.

A near-wall subgrid-scale (SGS) model is used to perform large-eddy simulation (LES) of the developing, smooth-wall, zero-pressure-gradient flat-plate turbulent boundary layer. In this model, the stretched-vortex, SGS closure is utilized in conjunction with a tailored, near-wall model designed to incorporate anisotropic vorticity scales in the presence of the wall. Large-eddy simulations of the turbulent boundary layer are reported at Reynolds numbers based on the free-stream velocity and the momentum thickness in the range . Results include the inverse square-root skin-friction coefficient, , velocity profiles, the shape factor , the von Kármán ‘constant’ and the Coles wake factor as functions of . Comparisons with some direct numerical simulation (DNS) and experiment are made including turbulent intensity data from atmospheric-layer measurements at . At extremely large , the empirical Coles–Fernholz relation for skin-friction coefficient provides a reasonable representation of the LES predictions. While the present LES methodology cannot probe the structure of the near-wall region, the present results show turbulence intensities that scale on the wall-friction velocity and on the Clauser length scale over almost all of the outer boundary layer. It is argued that LES is suggestive of the asymptotic, infinite Reynolds number limit for the smooth-wall turbulent boundary layer and different ways in which this limit can be approached are discussed. The maximum of the present simulations appears to be limited by machine precision and it is speculated, but not demonstrated, that even larger could be achieved with quad- or higher-precision arithmetic.

In recent years there has been significant progress made towards understanding the large-scale structure of wall-bounded shear flows. Most of this work has been conducted with turbulent boundary layers, leaving scope for further work in pipes and channels. In this article the structure of fully developed turbulent pipe and channel flow has been studied using custom-made arrays of hot-wire probes. Results reveal long meandering structures of length up to 25 pipe radii or channel half-heights. These appear to be qualitatively similar to those reported in the log region of a turbulent boundary layer. However, for the channel case, large-scale coherence persists further from the wall than in boundary layers. This is expected since these large-scale features are a property of the logarithmic region of the mean velocity profile in boundary layers and it is well-known that the mean velocity in a channel remains very close to the log law much further from the wall. Further comparison of the three turbulent flows shows that the characteristic structure width in the logarithmic region of a boundary layer is at least 1.6 times smaller than that in a pipe or channel.

Streamwise and wall-normal turbulence components are obtained in fully developed turbulent pipe over a Reynolds number range from 1.1 × 105 to 9.8 × 106. The streamwise intensity data are consistent with previous measurements in the same facility. For the wall-normal turbulence intensity, a constant region in v'r.m.s. is found for the region 200 ≤ y+ ≤ 0.1R+ for Reynolds numbers up to 1.0 × 106. An increase in v'r.m.s. is observed below about y+ 100, although additional measurements will be required to establish its generality. The wall-normal spectra collapse in the energy-containing region with inner scaling, but for the low-wavenumber region a y/R dependence is observed, which also indicates a continuing influence from the outer flow on the near-wall motions.

The possibility of using the spectral vanishing viscosity method for the spectral element computation of high Reynolds number incompressible flows is investigated. An exponentially accurate stabilized formulation is proposed and then applied to the computation of the 2D wake of a cylinder. Such a formulation can be easily implemented in existing spectral element solvers, since only modifying the computation of the viscous term while preserving the symmetry of the corresponding bilinear form.

Single normal hot-wire measurements of the streamwise component of velocity were taken in fully developed turbulent channel and pipe flows for matched friction Reynolds numbers ranging from 1,000 ≤ Re
τ ≤ 3,000. A total of 27 velocity profile measurements were taken with a systematic variation in the inner-scaled hot-wire sensor length l
+ and the hot-wire length-to-diameter ratio (l/d). It was observed that for constant l
+ = 22 and \(l/d \gtrsim 200\), the near-wall peak in turbulence intensity rises with Reynolds number in both channels and pipes. This is in contrast to Hultmark et al. in J Fluid Mech 649:103–113, (2010), who report no growth in the near-wall peak turbulence intensity for pipe flow with l
+ = 20. Further, it was found that channel and pipe flows have very similar streamwise velocity statistics and energy spectra over this range of Reynolds numbers, with the only difference observed in the outer region of the mean velocity profile. Measurements where l
+ and l/d were systematically varied reveal that l
+ effects are akin to spatial filtering and that increasing sensor size will lead to attenuation of an increasingly large range of small scales. In contrast, when l/d was insufficient, the measured energy is attenuated over a very broad range of scales. These findings are in agreement with similar studies in boundary layer flows and highlight the need to carefully consider sensor and anemometry parameters when comparing flows across different geometries and when drawing conclusions regarding the Reynolds number dependency of measured turbulence statistics. With an emphasis on accuracy, measurement resolution and wall proximity, these measurements are taken at comparable Reynolds numbers to currently available DNS data sets of turbulent channel/pipe flows and are intended to serve as a database for comparison between physical and numerical experiments.

1. Objects and results of the investigation .—The results of this investigation have both a practical and a philosophical aspect. In their practical aspect they relate to the law of resistance to the motion of water in pipes , which appears in a new form, the law for all velocities and all diameters being represented by an equation of two terms. In their philosophical aspect these results relate to the fundamental principles of fluid motion; inasmuch as they afford for the case of pipes a definite verification of two principles, which are— that the general character of the motion of fluids in contact with solid surfaces depends on the relation between a physical constant of the fluid and the product of the linear dimensions of the space occupied by the fluid and the velocity .

Both the inherent intractability and complex beauty of turbulence reside in its large range of physical and temporal scales. This range of scales is captured by the Reynolds number, which in nature and in many engineering applications can be as large as 10(5)-10(6). Here, we report turbulence measurements over an unprecedented range of Reynolds numbers using a unique combination of a high-pressure air facility and a new nanoscale anemometry probe. The results reveal previously unknown universal scaling behavior for the turbulent velocity fluctuations, which is remarkably similar to the well-known scaling behavior of the mean velocity distribution.

In this paper we present a formulation of spectral vanishing viscosity (SVV) for the stabilisation of spectral/hp element methods applied to the solution of the incompressible Navier–Stokes equations. We construct the SVV around a filter with respect to an orthogonal expansions, and prove that this methodology provides a symmetric semi-positive definite SVV operator. After providing a few simple one- and two-dimensional examples to demonstrate the utility of the SVV, we examine how it can be applied to a spectral/hp element discretisation of the Navier–Stokes equations using a velocity correction splitting scheme. We provide three fluid flow examples to help illustrate the pros and cons of this approach on stability and accuracy.