Streamwise Domain Research Articles

A numerical study on the natural transition locations in the flat-plate boundary layers on superhydrophobic surfaces

In this paper, the natural transition locations in the flat-plate boundary layers on the superhydrophobic surfaces are studied by numerical methods. The laminar flow field in the whole stream-wise computational domain is obtained by solving the Blasius equation with the slip-velocity boundary condition on the wall. The boundary layer on the superhydrophobic surface becomes thinner than that on the ordinary surface. The linear instability analysis is performed on the laminar boundary layer, and the eN method is employed to predict the transition location. The two-dimensional (2D) Tollmien–Schlichting (T–S) waves are still more unstable than the three-dimensional (3D) ones on the superhydrophobic surfaces, so only the 2D waves are taken into consideration to predict transition. As the slip length becomes longer, the critical location of flow instability moves further downstream, and the unstable zone becomes smaller. Therefore, the superhydrophobic surfaces have the effect of delaying the natural transition and that the delay effect becomes stronger as the slip length becomes longer. The higher oncoming flow velocity leads to higher frequencies of the unstable T–S waves and the larger unstable zone. As the oncoming flow velocity rises, the transition location on the superhydrophobic surface moves once upstream and then downstream. Consequently, there is a “dangerous” oncoming flow velocity corresponding to the transition location, which is the closest to the lead edge. Furthermore, the transition delay effect of the superhydrophobic surface becomes stronger with the increase in the oncoming flow velocity.

The Influence of Computational Parameterization on Mean Flow and Turbulence Statistic in 2D Idealized Street Canyon: Computational Domain

The use of Computational Fluid Dynamics (CFD) as a research tool has been utilized in many ways to predict the turbulent flow nature and pollution dispersion in an urban environment such as street canyon. Apart from the advance advantages offered by this method, some problems can be arising on respect to accurate prediction of data and uncertainties of the assumption that are made in numerical modelling. The execution and evaluation of the CFD studies need to undergo validation and generic sensitivity analyses to minimize the computational error and have an accurate prediction of data. This study performs a series of large eddy simulation (LES) to investigate the flow fields within and above a two-dimensional idealized street canyon with a unity aspect ratio (ratio of street width, W to building height, H). The effect of domain size on turbulent statistics is evaluated through the implementation of three different domain sizes i.e. varied streamwise lengths with fixed spanwise length and vertical height for different grid resolution (coarse, medium and fine). Comparison with the available experimental results shown a good agreement for mean velocity profiles. However, current LES underestimates the higher-order turbulent statistics. For the mean flow, the LES predicts the streamwise flow in the upper part and reversed flow in the lower part of the canyon, indicating the well-known flow regime of skimming flow. The value of mean velocities for all cases of the run have almost matching profile inside and above the canyon. For all cases, the maximum values of standard deviation in the streamwise and vertical directions are located approximately near the roof-level windward wall represent the occurrence of velocities fluctuation. In contrast, the Reynold shear stress for all cases shows some discrepancy where the peaked of turbulence intensity near the roof height is not smoothly captured in the current LES. Study shows that the LES cannot properly simulate those statistics accurately when the streamwise domain is limited to 10H even with the increasing of grid resolution sizes. Nevertheless, these results are expected to provide additional information relevant to uncertainty estimation upon execution of computational fluid dynamics (CFD) simulation.

Comparison between temporal and spatial direct numerical simulations for bypass transition flows

ABSTRACT Temporally developing direct numerical simulations (T-DNS) are performed for free-stream turbulence induced bypass transition flow over a flat-plate under zero-pressure gradient, and results are validated using experimental data and spatially developing DNS (S-DNS) results. The temporal simulations predict the growth of near-wall Klebanoff modes in the pre-transition regime and their subsequent breakdown due to the sinuous-like instability, where the increase in free-stream turbulence intensity and length scale enhanced the instability. A formulation for the domain translation velocity is developed using the momentum integral boundary layer equation, to estimate the translation of the domain along the plate. The formulation was found to be robust for a range of free-stream turbulence intensities, and both the boundary layer growth and free-stream decay compared well with spatial simulations, confirming that the same translation velocity can be used for the entire domain. The optimal domain size for a temporal simulation is dictated by the length scale of the turbulent spots and errors due to streamwise periodicity, and is estimated to be to , where is the boundary layer thickness at the transition onset. The T-DNS prediction of the integral boundary layer parameters, mean flow, second and higher-order statistics, stress budgets, and free-stream decay were comparable to those of S-DNS, even though they were obtained on ten times smaller streamwise domain and numerical grid. However, T-DNS are limited for bypass transtion flows for which leading-edge effects are unimportant, and may have limitations for the accurate predictions of the near-wall Klebanoff modes that extend up to breakdown. Nonetheless, temporal approach is identified to be a viable inexpensive alternative to the spatial approach for canonical test cases for transition flow physics analysis.

Turbulent flow over spanwise-varying roughness in a minimal streamwise channel

We report direct numerical simulations in a minimal streamwise domain of turbulent channel flow over spanwise-alternating patches of rough and smooth walls. Despite the minimal streamwise domain overpredicting streamwise-velocity fluctuations and inhibiting the meandering of long turbulent structures, it captures the rotational behaviour of mean secondary flows also observed in other studies with spanwise-varying roughness. To extend the study of spanwise-varying roughness, we prescribe a lateral velocity to the wall roughness to mimic flow over oblique patches of roughness. Far from the wall, long-lived turbulent structures are convected in the direction of the moving roughness, but their speeds are only weakly perturbed from a preferential value of around 40% of the friction velocity. The turbulence-driven secondary flows laterally convect at comparable speeds, but depend on the roughness patch width.

Analysis of the spanwise extent and time persistence of uniform momentum zones in zero pressure gradient and adverse pressure gradient turbulent boundary layers

Time-resolved velocity fields from direct numerical simulations (DNS) are used to investigate the spanwise extent and the time persistence of uniform momentum zones (UMZs) in a zero pressure gradient turbulent boundary layer (ZPG-TBL) and a self-similar adverse pressure gradient turbulent boundary layer (APG-TBL) at the verge of separation. The instantaneous detection methodology introduced by Adrian et al. [1] is used to detect the UMZs and is extended to take into account the spanwise domain length and the temporal evolution of the UMZs. The Reynolds number based on friction velocity (Reτ) ranges from 1176 to 1277 for the ZPG-TBL and from 1652 to 1745 for the self-similar APG-TBL within the domain of interest. For both the TBL cases, probability density functions (PDFs) of the number of UMZs are computed as a function of the streamwise extent, spanwise extent and time extent. For the ZPG-TBL, when the streamwise length of the domain is greater than or equal to 3 boundary layer thickness, the probability of finding 4 UMZs becomes almost negligible. For the APG-TBL, even when the streamwise domain length is taken as large as 1.3 boundary layer thickness, the probability of finding 4 UMZs is still significant. The spanwise extent of the UMZs is found to be shorter than their streamwise extent regardless of the pressure gradient in the flow. In the ZPG-TBL flow, the majority of the UMZs have a spanwise extent of the order of one-tenth of a boundary layer thickness while for the APG-TBL, it is found to be on the order of one-hundredth of a boundary layer thickness. In the ZPG-TBL, the probability of finding 2 UMZs that persist over a time period of 2 integral time scale is around 50%. Similarly, for the APG-TBL, the probability of finding 2 UMZs with a time persistence of 0.4 integral time scale is over 50%. In the case of the ZPG-TBL, it is observed that some of the UMZs with higher persistence in time have higher streamwise momentum and are found to be closer to the free-stream in general. This result is consistent with the previous observations by Laskari et al. [2]. In contrast, for the APG-TBL, UMZs with longer time persistence are found closer to the wall with lower streamwise momentum.

Stochastic receptivity analysis of boundary layer flow

We utilize the externally forced linearized Navier-Stokes equations to study the receptivity of pre-transitional boundary layers to persistent sources of stochastic excitation. Stochastic forcing is used to model the effect of free-stream turbulence that enters at various wall-normal locations and the fluctuation dynamics are studied via linearized models that arise from locally parallel and global perspectives. In contrast to the widely used resolvent analysis that quantifies the amplification of deterministic disturbances at a given temporal frequency, our approach examines the steady-state response to stochastic excitation that is uncorrelated in time. In addition to stochastic forcing with identity covariance, we utilize the spatial spectrum of homogeneous isotropic turbulence to model the effect of free-stream turbulence. Even though locally parallel analysis does not account for the effect of the spatially evolving base flow, we demonstrate that it captures the essential mechanisms and the prevailing length-scales in stochastically forced boundary layer flows. On the other hand, global analysis, which accounts for the spatially evolving nature of the boundary layer flow, predicts the amplification of a cascade of streamwise scales throughout the streamwise domain. We show that the flow structures that can be extracted from a modal decomposition of the resulting velocity covariance matrix, can be closely captured by conducting locally parallel analysis at various streamwise locations and over different wall-parallel wavenumber pairs. Our approach does not rely on costly stochastic simulations and it provides insight into mechanisms for perturbation growth including the interaction of the slowly varying base flow with streaks and Tollmien-Schlichting waves.

On the detection of internal interfacial layers in turbulent flows

A novel approach to identify internal interfacial layers, or IILs, in wall-bounded turbulent flows is proposed. Using a fuzzy cluster method (FCM) on the streamwise velocity component, a unique and unambiguous grouping of the uniform momentum zones (UMZs) is achieved, thus allowing the identification of the IILs. The approach overcomes some of the key limitations of the histogram-based IIL identification methods. The method is insensitive to the streamwise domain length, can be used on inhomogeneous grids, uses all the available flow field data, is trivially extended to three dimensions and does not need user-defined parameters (e.g. number of bins) other than the number of zones. The number of zones for a given snapshot can be automatically determined by an a priori algorithm based on a kernel density estimation algorithm, or KDE. This automated approach is applied to compute the average number of UMZs as a function of Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ in turbulent channel flows in several numerical simulations. This systematic approach reveals a dependence of the Reynolds number on the average number of UMZs in the channel flow; this supports previously reported observations in the boundary layer. The fuzzy clustering approach is applied to the turbulent boundary layer (experimental, planar particle image velocimetry) and channel flow (numerical, direct numerical simulation) at varying Reynolds numbers. The interfacial layers are characterized by a strong concentration of spanwise vorticity, with the outer-most layer located at the upper edge of the log layer. The three-dimensional interface identification reveals a streak-like organization. The large-scale motion (LSM) at the outer region of the channel flow boundary layer modulates the outer IIL. The corrugations of the outer IIL are aligned with the LSM and the conditional correlation of the inner and outer IIL height shows that extreme near-wall events leave their mark on the outer IIL corrugations.

Azimuthal organization of large-scale motions in a turbulent minimal pipe flow

Direct numerical simulation data for turbulent minimal pipe flows with Reτ = 927, 1990, and 2916 are examined to explore the azimuthal (or spanwise) organization of their large-scale structures. We chose a streamwise-minimal unit with a streamwise domain length of Lx+≈1000, which is the characteristic streamwise length of near-wall streaks. The spanwise scales of most of the energetic motions and their contributions to the total energy are comparable with those of the streamwise long-domain simulation. In the azimuthal energy spectra of the streamwise velocity fluctuations (u), the large-scale energy increases with Reτ and three outer peaks (λθ = 0.7–0.8, π/2 and π) become evident when Reτ = 2916. The presence of the outer peaks at λθ = 0.7–0.8 and π/2 is consistent with the results of the long-domain simulation. The peak at λθ = 0.7–0.8 is associated with large-scale motions and the other two peaks are associated with very-large-scale motions (VLSMs). The maximum spanwise wavelength increases linearly with the wall-normal distance from the wall. A kz−1 region is evident in the range 0.3R < λz (=rλθ) < R, which indicates the presence of self-similar motions. The conditional two-point correlation with a cut-off wavelength of λz = 0.9R shows that there is a strong correlation between the enhanced energy in the outer region and the wall-attached structures, which were extracted from the time evolution of the streamwise-averaged u field (u2D). The spanwise sizes (lz) of the attached u2D structures scale with their height (ly) in the log region and their time scales (lt) follow ltuτ/lz = 2, which is consistent with the bursting time scale. Their spanwise sizes lie in the range R < lz < 3R, for which lt increases significantly, which indicates that these structures are associated with VLSMs and make the dominant contributions to the enhanced energy in the outer region. These structures penetrate to the wall region as a manifestation of the footprint and modulate the small-scale energy. The negative-u2D structures induce congregative motions in the near-wall region.

Large-scale structures in a turbulent channel flow with a minimal streamwise flow unit

Direct numerical simulations are used to examine large-scale motions with a streamwise length$2\sim 4h$($h$denotes the channel half-width) in the logarithmic and outer regions of a turbulent channel flow. We test a minimal ‘streamwise’ flow unit (Toh & Itano,J. Fluid Mech., vol. 524, 2005, pp. 249–262) (or MSU) for larger Kármán numbers ($h^{+}=395$and 1020) than in the original work. This flow unit consists of a sufficiently long (${L_{x}}^{+}\approx 400$) streamwise domain to maintain near-wall turbulence (Jiménez & Moin,J. Fluid Mech., vol. 225, 1991, pp. 213–240) and a spanwise domain which is large enough to represent the spanwise behaviour of inner and outer structures correctly; as$h^{+}$increases, the streamwise extent of the MSU domain decreases with respect to$h$. Particular attention is given to whether the spanwise organization of the large-scale structures may be represented properly in this simplified system at sufficiently large$h^{+}$and how these structures are associated with the mean streamwise velocity$\overline{U}$. It is shown that, in the MSU, the large-scale structures become approximately two-dimensional at$h^{+}=1020$. In this case, the streamwise velocity fluctuation$u$is energized, whereas the spanwise velocity fluctuation$w$is weakened significantly. Indeed, there is a reduced energy redistribution arising from the impaired global nature of the pressure, which is linked to the reduced linear–nonlinear interaction in the Poisson equation (i.e. the rapid pressure). The logarithmic dependence of$\overline{ww}$is also more evident due to the reduced large-scale spanwise meandering. On the other hand, the spanwise organization of the large-scale$u$structures is essentially identical for the MSU and large streamwise domain (LSD). One discernible difference, relative to the LSD, is that the large-scale structures in the MSU are more energized in the outer region due to a reduced turbulent diffusion. In this region, there is a tight coupling between neighbouring structures, which yields antisymmetric pairs (with respect to centreline) of large-scale structures with a spanwise spacing of approximately$3h$; this is intrinsically identical with the outer energetic mode in the optimal transient growth of perturbations (del Álamo & Jiménez,J. Fluid Mech., vol. 561, 2006, pp. 329–358).

Capturing Taylor–Görtler vortices in a streamwise-rotating channel at very high rotation numbers

In this paper, we study the scales and dynamics of Taylor–Görtler (TG) vortices in streamwise-rotating turbulent channel flows at moderate and high rotation numbers ($Ro_{\unicode[STIX]{x1D70F}}=7.5$, 15, 30, 75 and 150) with a fixed Reynolds number. In order to precisely capture TG vortices in the streamwise and spanwise directions, direct numerical simulations have been performed on 15 test cases of different domain sizes and rotation numbers. A two-layer pattern of TG vortices is identified, and the characteristic length scales of TG vortices are quantified using the premultiplied energy spectra. It is observed that as the rotation number increases, the spanwise scale of TG vortices remains stable but the streamwise scale increases rapidly. Three criteria have been used for judging a domain-size-independent solution in both physical and spectral spaces. The weakest criterion ensures accurate predictions of the first- and second-order statistical moments of the velocity, which requires a minimum streamwise domain size of $L_{1}=64\unicode[STIX]{x03C0}h$, where $h$ is one-half the channel height. However, the streamwise domain size needs to be stretched drastically to $L_{1}=512\unicode[STIX]{x03C0}h$ if the most stringent criterion is considered, which demands that all energetic eddies be fully captured based on a predefined threshold value (i.e. $12.5\,\%$ of the peak value) of the premultiplied two-dimensional energy spectrum. The effects of streamwise system rotation on the scales and dynamics of TG vortices are investigated by comparing the statistical results of rotating and non-rotating channel flows, and through the analysis of two-point correlations, premultiplied energy spectra, and budget balance of turbulent stresses.

The minimal-span channel for rough-wall turbulent flows

Roughness predominantly alters the near-wall region of turbulent flow while the outer layer remains similar with respect to the wall shear stress. This makes it a prime candidate for the minimal-span channel, which only captures the near-wall flow by restricting the spanwise channel width to be of the order of a few hundred viscous units. Recently, Chung et al. (J. Fluid Mech., vol. 773, 2015, pp. 418–431) showed that a minimal-span channel can accurately characterise the hydraulic behaviour of roughness. Following this, we aim to investigate the fundamental dynamics of the minimal-span channel framework with an eye towards further improving performance. The streamwise domain length of the channel is investigated with the minimum length found to be three times the spanwise width or 1000 viscous units, whichever is longer. The outer layer of the minimal channel is inherently unphysical and as such alterations to it can be performed so long as the near-wall flow, which is the same as in a full-span channel, remains unchanged. Firstly, a half-height (open) channel with slip wall is shown to reproduce the near-wall behaviour seen in a standard channel, but with half the number of grid points. Next, a forcing model is introduced into the outer layer of a half-height channel. This reduces the high streamwise velocity associated with the minimal channel and allows for a larger computational time step. Finally, an investigation is conducted to see if varying the roughness Reynolds number with time is a feasible method for obtaining the full hydraulic behaviour of a rough surface. Currently, multiple steady simulations at fixed roughness Reynolds numbers are needed to obtain this behaviour. The results indicate that the non-dimensional pressure gradient parameter must be kept below 0.03–0.07 to ensure that pressure gradient effects do not lead to an inaccurate roughness function. An empirical costing argument is developed to determine the cost in terms of CPU hours of minimal-span channel simulations a priori. This argument involves counting the number of eddy lifespans in the channel, which is then related to the statistical uncertainty of the streamwise velocity. For a given statistical uncertainty in the roughness function, this can then be used to determine the simulation run time. Following this, a finite-volume code with a body-fitted grid is used to determine the roughness function for square-based pyramids using the above insights. Comparisons to experimental studies for the same roughness geometry are made and good agreement is observed.

Compressibility effects on the structural evolution of transitional high-speed planar wakes

The compressibility effects on the structural evolution of the transitional high-speed planar wake are studied. The relative Mach number ($Ma_{r}$) of the laminar base flow modifies two fundamental features of planar wake transition: (i) the characteristic length scale defined by the most unstable linear mode; and (ii) the domain of influence of the structures within the staggered two-dimensional vortex array. Linear stability results reveal a reduced growth (approximately 30 % reduction up to$Ma_{r}=2.0$) and a quasilinear increase of the wavelength of the most unstable, two-dimensional instability mode (approximately 20 % longer over the same$Ma_{r}$range) with increasing$Ma$. The primary wavelength defines the length scale imposed on the emerging transitional structures; naturally, a longer wavelength results in rollers with a greater streamwise separation and hence also larger circulation. A reduction of the growth rate and an increase of the principal wavelength results in a greater ellipticity of the emerging rollers. Compressibility effects also modify the domain of influence of the transitional structures through an increased cross-wake and inhibited streamwise communication as characteristic paths between rollers are deflected due to local$Ma$ gradients. The reduced streamwise domain of influence impedes roller pairing and, for a sufficiently large relative$Ma$, pairing is completely suppressed. Thus, we observe an increased two-dimensionality with increasing Mach number: directly contrasting the increasing three-dimensional effects in high-speed mixing layers. Temporally evolving direct numerical simulations conducted at$Ma=0.8$and 2.0, for Reynolds numbers up to 3000, support the physical insight gained from linear stability and vortex dynamics studies.

Shifted periodic boundary conditions for simulations of wall-bounded turbulent flows

In wall-bounded turbulent flow simulations, periodic boundary conditions combined with insufficiently long domains lead to persistent spanwise locking of large-scale turbulent structures. This leads to statistical inhomogeneities of 10%–15% that persist in time averages of 60 eddy turnover times and more. We propose a shifted periodic boundary condition that eliminates this effect without the need for excessive streamwise domain lengths. The method is tested based on a set of direct numerical simulations of a turbulent channel flow, and large-eddy simulations of a high Reynolds number rough-wall half-channel flow. The method is very useful for precursor simulations that generate inlet conditions for simulations that are spatially inhomogeneous, but require statistically homogeneous inlet boundary conditions in the spanwise direction. The method’s advantages are illustrated for the simulation of a developing wind-farm boundary layer.

Scale growth of structures in the turbulent boundary layer with a rod-roughened wall

Direct numerical simulation of a turbulent boundary layer over a rod-roughened wall is performed with a long streamwise domain to examine the streamwise-scale growth mechanism of streamwise velocity fluctuating structures in the presence of two-dimensional (2-D) surface roughness. An instantaneous analysis shows that there is a slightly larger population of long structures with a small helix angle (spanwise inclinations relative to streamwise) and a large spanwise width over the rough-wall compared to that over a smooth-wall. Further inspection of time-evolving instantaneous fields clearly exhibits that adjacent long structures combine to form a longer structure through a spanwise merging process over the rough-wall; moreover, spanwise merging for streamwise scale growth is expected to occur frequently over the rough-wall due to the large spanwise scales generated by the 2-D roughness. Finally, we examine the influence of a large width and a small helix angle of the structures over the rough-wall with regard to spatial two-point correlation. The results show that these factors can increase the streamwise coherence of the structures in a statistical sense.

Investigation of the influence of co-flow velocity ratio on a compressible plane jet exhausting into parallel streams

Two-dimensional numerical simulations of compressible plane jet exhausting into parallel streams is performed using sixth-order compact scheme with third-order low storage Runge–Kutta method. Characteristic based boundary conditions are utilized in all simulations. Sixth-order tri-diagonal spatial filter is applied to suppress any high wave-number oscillation created at the boundaries and due to unresolved scales and mesh non-uniformities. The study focuses on the influence of velocity difference between the parallel streams on fully developed plane jet. The two dimensional simulation is found capable of capturing correct jet spreading and decay. Time-averages over stream-wise domain are used to capture the mean field and turbulence structures. It is found that the potential core length increases as the velocity ratio increases. The predicted mean velocity profiles show self-similar behavior and collapse well with previous experimental results. Turbulent intensity and Reynolds shear stress change with change in velocity ratio. However, the mean streamwise excess velocity profiles are unaffected by the change in velocity ratio.

Role of Time Integration in Computing Transitional Flows Caused by Wall Excitation

Numerical investigation of receptivity and flow transition in spatio-temporal framework have shown the central role of spatio-temporal wave-front (STWF) created by wall excitation for transition of a two-dimensional (2D) zero pressure gradient boundary layer (ZPGBL) in Sengupta and Bhaumik (Phys Rev Lett 107:154501, 2011). Although the STWF is created by linear mechanism, it is the later nonlinear stage of evolution revealed by the solution of Navier---Stokes equation (NSE), which causes formation of turbulent spots merging together to create fully developed turbulent flow. Thus, computing STWF for ZPGBL from NSE is of prime importance, which has been reported by the present authors following earlier theoretical investigation. Similar computational efforts using NSE by other researchers do not report finding the STWF. In the present investigation we identify the main reason for other researchers to miss STWF, as due to taking a very short computational domain. Secondly, we show that even one takes a long enough domain and detect STWF, use of traditional low accuracy method will not produce the correct dynamics as reported by Sengupta and Bhaumik (2011). The role of time integration plays a very strong role in the dynamics of transitional flows. We have shown here that implicit methods are more error prone, as compared to explicit time integration methods during flow transition. For the present problem, it is noted that the classical Crank---Nicolson method is unstable for 2D NSE. Same error-prone nature will also be noted for hybrid implicit---explicit time integration methods (known as the IMEX methods). One of the main feature of present analysis is to highlight the accuracy of computations by compact schemes used by the present investigators over a significantly longer domain and over unlimited time, as opposed to those reported earlier in the literature for the wall excitation problem. A consequence of taking long streamwise domain enables one to detect special properties of STWF and its nonlinear growth. The main focus of the present research is to highlight the importance of STWF, which is a new class of spatio-temporal solution obtained from the linear receptivity by solving Orr---Sommerfeld equation and nonlinear analysis of NSE.

Direct numerical simulations of fully developed turbulent pipe flows for Reτ = 180, 544 and 934

Direct numerical simulations (DNSs) of fully developed turbulent pipe flows with a 30R streamwise domain for Reτ=180, 544 and 934 were performed to see the turbulence statistics and the interactions between the near-wall and outer regions. There are some slight discrepancies between previous DNS data for turbulent pipe flows with low Reynolds numbers Reτ≈180 and the present data. These discrepancies arise because DNS results are sensitive to numerical conditions such as grid resolution and distribution. Our DNS data for various Reynolds numbers show that the near-wall peak in the streamwise turbulence intensity increases with increasing the Reynolds number. Evidence to support this finding was found in pre-multiplied energy spectra and the two-point amplitude modulation (AM) covariance. It was determined that both the long wavelength energy near the wall and the non-linear effects of AM contribute to the growth of the near-wall peak in the streamwise Reynolds stress.

Numerical study of fully developed flow and heat transfer in a wavy passage

A two-dimensional numerical study of fully developed fluid flow and heat transfer through a horizontal wavy surface is presented. Time dependent Navier–Stokes and energy equation have been solved using finite volume method. Numerical calculations are performed for a wavy surface described by sine function y = 2a × sin2 (πx/L) with ratios Hmin/Hmax and L/a fixed to 0.4 and 8 respectively. Attempts have been made to foresee the effect of streamwise domain length, an integer multiple of periodic domain length, on the flow and heat transfer characteristics. It is established that the flow and heat transfer characteristics do not show any dependence on the length of the periodic domain thereby confirming that geometric and flow periodicity are identical. The effect of Reynolds numbers, for Re in the range 25–1000, on the flow field and heat transfer has been presented. For steady flow the heat transfer rates are found to be very low and for unsteady flow with increased mixing between core and near wall fluids enhanced heat transfer rates are obtained.

Very-large-scale motions in a turbulent boundary layer

Direct numerical simulation of a turbulent boundary layer was performed to investigate the spatially coherent structures associated with very-large-scale motions (VLSMs). The Reynolds number was varied in the range Reθ = 570–2560. The main simulation was conducted by using a computational box greater than 50δo in the streamwise domain, where δo is the boundary layer thickness at the inlet, and inflow data was obtained from a separate inflow simulation based on Lund's method. Inspection of the three-dimensional instantaneous fields showed that groups of hairpin vortices are coherently arranged in the streamwise direction and that these groups create significantly elongated low- and high-momentum regions with large amounts of Reynolds shear stress. Adjacent packet-type structures combine to form the VLSMs; this formation process is attributed to continuous stretching of the hairpins coupled with lifting-up and backward curling of the vortices. The growth of the spanwise scale of the hairpin packets occurs continuously, so it increases rapidly to double that of the original width of the packets. We employed the modified feature extraction algorithm developed by Ganapathisubramani, Longmire & Marusic (J. Fluid Mech., vol. 478, 2003, p. 35) to identify the properties of the VLSMs of hairpin vortices. In the log layer, patches with the length greater than 3δ–4δ account for more than 40% of all the patches and these VLSMs contribute approximately 45% of the total Reynolds shear stress included in all the patches. The VLSMs have a statistical streamwise coherence of the order of ~6δ; the spatial organization and coherence decrease away from the wall, but the spanwise width increases monotonically with the wall-normal distance. Finally, the application of linear stochastic estimation demonstrated the presence of packet organization in the form of a train of packets in the log layer.

Direct numerical simulation of a turbulent boundary layer up to Re θ = 2500

Direct numerical simulations (DNSs) of a turbulent boundary layer (TBL) with Re θ = 570–2560 were performed to investigate the spatial development of its turbulence characteristics. The inflow simulation was conducted in the range Re θ = 570–1600 by using Lund’s method. To resolve the numerical periodicity induced by the recycling method, we adopted a sufficiently long streamwise domain of x/ θ in,i = 1000 (=125 δ 0 ,i ), where θ in,i is the inlet momentum thickness and δ 0 ,i is the inlet boundary layer thickness in the inflow simulation. Furthermore, the main simulation with a length greater than 50 δ 0 was carried out independently by using the inflow data, where δ 0 is the inlet boundary layer thickness of the main simulation. The integral quantities and the first-, second- and higher-order turbulence statistics were compared with those of previous data, and good agreement was found. The present study provides a useful database for the turbulence statistics of TBLs. In addition, instantaneous field and two-point correlation of the streamwise velocity fluctuations displayed the existence of the very large-scale motions (VLSMs) with the characteristic widths of 0.1–0.2 δ and that the flow structure for a length of approximately ∼6 δ fully occupies the streamwise domain statistically.