Pre-multiplied Energy Spectra Research Articles

Direct numerical simulation of turbulent, generalized Couette–Poiseuille flow

We present direct numerical simulation (DNS) and modelling of incompressible, turbulent, generalized Couette–Poiseuille flow. A particular example is specified by spherical coordinates $(Re,\theta,\phi )$ , where $Re = 6000$ is a global Reynolds number, $\phi$ denotes the angle between the moving plate, velocity-difference vector and the volume-flow vector and $\tan \theta$ specifies the ratio of the mean volume-flow speed to the plate speed. The limits $\phi \to 0^\circ$ and $\phi \to 90^\circ$ give alignment and orthogonality, respectively, while $\theta \to 0^\circ,\ \theta \to 90^\circ$ correspond respectively to pure Couette flow in the $x$ direction and pure Poiseuille flow at angle $\phi$ to the $x$ axis. Competition between the Couette-flow shear and the forced volume flow produces a mean-velocity profile with directional twist between the confining walls. Resultant mean-speed profiles relative to each wall generally show a log-like region. An empirical flow model is constructed based on component log and log-wake velocity profiles relative to the two walls. This gives predictions of four independent components of shear stress and also mean-velocity profiles as functions of $(Re,\theta,\phi )$ . The model captures DNS results including the mean-flow twist. Premultiplied energy spectra are obtained for symmetric flows with $\phi =90^\circ$ . With increasing $\theta$ , the energy peak gradually moves in the direction of increasing $k_x$ and decreasing $k_z$ . Rotation of the energy spectrum produced by the faster moving velocity near the wall is also observed. Rapid weakening of a spike maxima in the Couette-type flow regime indicates attenuation of large-scale roll structures, which is also shown in the $Q$ -criterion visualization of a three-dimensional time-averaged flow.

Flow Field Analysis of a Turbulent Channel Controlled by Scalloped Riblets

Riblets are small protruding surfaces along the direction of the flow, and are one of the most well-known passive turbulent drag reduction methods. We investigated a scalloped riblet, the shape of which was constructed by smoothly connecting two third-order polynomials and was not as sharp in the tip as corresponding triangular riblets with the same height-width ratio. Numerical simulations were performed for turbulent channel flow with and without riblet control at an estimated optimum width of W+=20 and a height-width ratio of 0.5. A drag reduction rate of 5.77% was obtained, which is generally larger than the reported drag reduction rates of corresponding triangular riblets from the literature. Mean flow fields and second-order statistics of velocity, vorticity, and Liutex, a quantity introduced to represent vortices, were reported. It was found that streamwise vortices just above the riblet tips, which have a length scale of 200−300 in wall units, have an important influence on those statistics and thus the turbulence generation cycle and the drag reduction mechanism. Pre-multiplied energy spectra of streamwise velocity and the Liutex component were reported to reveal the length scales in the flow field. Instantaneous vortical flow fields visualized by the Liutex method were provided with emphases on the streamwise vortices just above riblet tips. It should be noted that the class of scalloped riblets is suitable for investigations on the influences of curvatures at the tip and the valley of the riblet in future.

Reynolds number dependence of azimuthal and streamwise pipe flow structures

This paper revisits the study by Baileyet al.(J. Fluid Mech., vol. 615, 2008, pp. 121–138), adopting a higher-fidelity calibration approach to reveal subtle flow variations with Reynolds numbers that were not discernible previously. The paper aims therefore to provide insights into the characteristics of azimuthal and streamwise pipe flow structures adopting two-point joint statistics and spectral analysis for shear Reynolds numbers in the range$2\times {10^3}\le {Re_\tau }\le {16\times {10^3}}$, where${Re_\tau }$is based on the wall friction velocity$u_{\tau }$, the pipe radius$R$, and the fluid kinematic viscosity$\nu$. The streamwise velocity fluctuations were measured at four wall-normal locations,$0.1\le {x_{{2}}/R}\le {0.7}$, covering the logarithmic and core regions of fully developed turbulent pipe flow based on 35–41 azimuthal probe separations using, simultaneously, two single hot-wire probes. A uniquein situcalibration approach for both probes was adopted where a potential flow was insured, resulting in consistent and precise pipe flow data. The azimuthal velocity correlation, the cross-power spectral density and the coherence function of the streamwise velocity fluctuations are discussed, revealing a clear dependence of the azimuthal scales of the large and very large flow motions on the wall-normal location, the azimuthal separation, the streamwise wavenumber and the Reynolds number. Along the logarithmic region, a linear growth of the azimuthal scales of the large- and very-large-scale structures was observed; however, they do scale nonlinearly and reach their maximum sizes in the core region, i.e. near the centreline of the pipe. Additionally, the streamwise very-large- and large-scale motions were evaluated using the premultiplied energy spectra, showing wavelengths${\approx }{18R}$and${\approx }{3R}$for${Re_{\tau }}\approx {16\times {10^3}}$at half of the pipe radius, respectively.

A quantitative study on the turbulent kinetic energy redistribution in the near free-water-surface region of open channel flows

Open channel flows (OCFs) exhibit unique characteristics compared with other wall-bounded flows due to the presence of a free water surface. One of the well-known features is the turbulence kinetic energy (TKE) redistribution phenomenon in the near free-water-surface region. Unlike previous qualitative demonstrations, the present study aims providing a quantitative investigation of this phenomenon. Specifically designed super long domain direct numerical simulations (DNSs) of OCFs and closed channel flows (CCFs) are performed at three low-to-moderate friction Reynolds numbers (Reτ = 180–1000). The numerical configurations of the two flows (i.e., temporal and spatial resolution, domain size, Reynolds number, etc.) at identical Reτ are set to be the same, with the upper boundary being the only difference between them, to allow meaningful comparisons while super long domain sizes are adopted to facilitate fully spectral comparisons of the TKE features between the two flows. With such specifically designed DNS datasets, quantitative investigations of the TKE redistribution phenomenon in OCFs have been made. It is revealed that, as Reτ increases from 180 to 1000, the streamwise and spanwise TKE components of OCFs are higher than those of CCFs by 12%–23% and 28%–17%, respectively, while in the vertical direction OCFs are lower in TKE by 44%–31%. Overall, the TKE of OCFs is higher than that of CCFs by 3%–9% as Reτ increases from 180 to 1000. The comparison of OCFs and CCFs in pre-multiplied energy spectra reveals that VLSMs play a dominant role in the TKE redistribution phenomenon in OCFs.

Mechanistic study on modification of convective internal boundary layer by spanwise surface motion

Modification of a convective internal boundary layer (IBL) by spanwise motion of a warm surface is investigated by imposing different surface moving speeds in the present study. Our analysis shows that the spanwise surface motion reduces the Reynolds shear stress right after the increase in the surface temperature in the convective IBL. The maximum decreasing rate of the Reynolds shear stress is found to be approximately 75% at the largest moving speed of the warm surface considered in the manuscript. Due to the reduction of the Reynolds shear stress, the vertical momentum transport is fundamentally altered, and the mean flow accelerates immediately after the increase in the surface temperature. By scrutinizing the instantaneous and conditional averaged flow fields as well as the pre-multiplied energy spectra, we have attributed the reduction of the Reynolds shear stress to the suppression of the near-surface velocity streaks and quasi-streamwise vortices, and the delayed growth of the convective structures, such as thermal plumes. Our investigation suggests that the developments of the convective IBL can be influenced by a strong spanwise motion of the warm surface, which should be taken into consideration in the prediction model for practical applications.

Linear and nonlinear optimal growth mechanisms for generating turbulent bands

Recently, many authors have investigated the origin and growth of turbulent bands in shear flows, highlighting the role of streaks and their inflectional instability in the process of band generation and sustainment. Recalling that streaks are created by an optimal transient growth mechanism, and motivated by the observation of a strong increase of the disturbance kinetic energy corresponding to the creation of turbulent bands, we use linear and nonlinear energy optimisations in a tilted domain to unveil the main mechanisms allowing the creation of a turbulent band in a channel flow. Linear transient growth analysis shows an optimal growth for wavenumbers having an angle of approximately$35^\circ$, close to the peak values of the premultiplied energy spectra of direct numerical simulations. This linear optimal perturbation generates oblique streaks, which, for a sufficiently large initial energy, induce turbulence in the whole domain, due to the lack of spatial localisation. However, spatially localised perturbations obtained by adding nonlinear effects to the optimisation or by artificially confining the linear optimal to a localised region in the transverse direction are characterised by a large-scale flow and lead to the generation of a localised turbulent band. These results suggest that two main elements are needed for inducing turbulent bands in a tilted domain: (i) a linear energy growth mechanism, such as the lift-up, for generating large-amplitude flow structures, which produce inflection points; (ii) spatial localisation, linked to the presence or generation of large-scale vortices. We show that these elements alone generate isolated turbulent bands also in large non-tilted domains.

Possibility for survival of macroscopic turbulence in porous media with high porosity

The direct numerical simulation (DNS) study by Jin et al. (J. Fluid Mech, vol. 766, 2015, pp. 76–103) shows that the turbulent structures are generally restricted in size to the pore scale, leading to the pore-scale prevalence hypothesis (PSPH). Although the PSPH has been validated under most conditions, it might become invalid as the porosity approaches unity. In order to investigate the valid domain of the PSPH, we have studied the turbulent flows in porous matrices which have one or two length scales using DNS and macroscopic simulation methods. The large porous elements are made of staggered arrays of square cylinders, which might stimulate strong macroscopic (large-scale) turbulence. The small porous elements are made of aligned arrays of spheres or cubes, which suppress the macroscopic turbulence. The analyses are performed for various values of the Reynolds number, Darcy number, pore-scale ratio and porosity. Turbulent two-point correlations, integral length scales and premultiplied energy spectra are calculated from the DNS and macroscopic simulation results to determine the length scale of the turbulent structures. Our numerical results show that the flow becomes turbulent when the Reynolds number is sufficiently large. However, the length scale of turbulence is not considerably affected by the Reynolds number, Darcy number and pore-scale geometry. The PSPH is valid when the porosity has small or medium values. At a sufficiently large Reynolds number, large-scale turbulence survives if the porosity is larger than a critical value. Our DNS and macroscopic simulation results show that this critical value is in the range 0.93–0.98 for porous matrices with large Darcy numbers (0.3–1.26 using the definition in this study). The dependence of the critical porosity on the pore-scale geometry still needs to be further investigated.

Investigation of the influence of miniature vortex generators on the large-scale motions of a turbulent boundary layer

Well resolved large-eddy simulation data are used to study the physical modulation effects of miniature vortex generators (MVGs) in a moderate Reynolds number zero pressure gradient turbulent boundary layer. Large-scale counter-rotating primary vortex pairs (PVPs) imposed by the MVG contribute to the formation of streamwise streaks by transporting high momentum fluids from the outer regions of the boundary layer towards the wall, giving rise to high-speed regions centred at the PVP. Consequently, low-speed regions are formed along the outer flank of the PVP, resulting in a pronounced alternating high- and low-speed flow pattern. The PVP also relates to regions with skin friction modification, where a local skin friction reduction of up to 15 % is obtained at the low-speed region, but the opposite situation is observed over the high-speed region. The MVG-induced flow feature is further investigated by spectral analysis of the triple decomposition velocity fluctuation. Pre-multiplied energy spectra of the streamwise MVG-induced velocity fluctuation reveal that the large-scale induced modes scale with the spanwise wavelength and the length of the MVG, but the energy peak is eventually repositioned to the size of the near-wall streaks in the streamwise direction. Analysis of the triple decomposition of the kinetic energy transport equations revealed the significance of the mean flow gradient in generating kinetic energy which sustains the secondary motion. There is also an energy transfer between the turbulent and MVG-induced kinetic energy independent of the mean flow.

The effects of upstream perturbations on large-scale field and the proliferation of λ2 vortices

The effects of upstream perturbations on large-scale field are analyzed using direct numerical simulation data of a channel flow of frictional Reynolds number, Reτ, 394. A set of three and five spanwise blowing jets are used to create the upstream perturbations. The premultiplied energy spectra show that upstream perturbations led to the increase in range of large-scale wavenumbers in contrast to the unperturbed channel flow. The energy of wavenumbers lower than inertial sub-range (−5/3 power law) shows an increase in energy due to the perturbations and sustained downstream up to x = 10D from jets. Two-point correlation based Gaussian filter is used to extract large-scale features of correlation length greater than 2h (where h is channel half-height). Premultiplied energy spectra of the Gaussian filtered large-scales demonstrate the importance of jet spacing and diameter in exciting large-scale wavenumbers. The λ2 vortices of large-scale features formed a strong ring-type vortices at y+=60. Spectral turbulence kinetic energy (TKE) production depicts a secondary peak in TKE production at y+=60, which indicates the formation of a shear layer due to the upstream perturbations.

An investigation of particles effects on wall-normal velocity fluctuations in sand-laden atmospheric surface layer flows

Based on the high-quality observational data in the Qingtu Lake Observation Array (QLOA), the difference in the energy distribution, the scale of the coherent structures, and the amplitude modulation effect of the wall-normal velocity fluctuations between particle-free and particle-laden flow in the atmospheric surface layer are analyzed. The results show that the presence of particles enhanced the wall-normal turbulence intensity, especially the increase at the top of the logarithmic region is more significant though the particle mass loading decreases with the wall-normal distance. A further insight indicates that the increase in the length scale of the wall-normal fluctuating velocity coherent structure by particles is more significant further from the wall, which is supported by the premultiplied energy spectra and the two-point correlation. This leads to a drastic increase in kinetic energy of the large-scale coherent structures by the particle away from the wall and thus results in increased amplitude modulation effects of large-scale wall-normal velocity fluctuations onto small-scales.

Modulation of turbulence by saltating particles on erodible bed surface

Abstract

Effect of Reynolds number on turbulent boundary layer approaching separation

The effect of Reynolds number on turbulent boundary layer under adverse pressure gradient has been barely investigated in the available literature. So far, experimental or numerical research on that issue were performed either for low Re or for low range of Re to capture its clear effect on flow statistics. Hence, the purpose of the present paper is to examine whether the effect of Reynolds number can be observed when much higher Re values and its much wider range, than previously investigated, are considered. Herein, the momentum loss thickness based Reynolds number Reθ was changed from 4900 to 14600. Such a wide range of Reθ appeared to be sufficient to capture an evident footprint of Reθ on the flow. It was found that the separation point is shifted downstream with increasing Reynolds number due to increased convection velocity close to the wall. More detailed studies of the skewness factor and premultiplied energy spectra revealed that this phenomenon can be attributed to enhanced, with increasing Re, large- and small-scale interactions commonly known as the amplitude modulation.

Separating adverse-pressure-gradient and Reynolds-number effects in turbulent boundary layers

Wall turbulence is characterized by a near-wall cycle of streaks and quasistreamwise vortices apparent as an invariant inner peak in the premultiplied energy spectra. A second, outer peak is known to emerge in this spectral view and become energized with increasing Reynolds number (Re) as well as adverse pressure gradient (APG). An analysis of experimental data sets examines how this outer peak scales with Re and APG and whether their imprint on the near-wall small scales are different.

Wind turbine wake intermittency dependence on turbulence intensity and pitch motion

Turbulence intermittency characteristics of the flow behind pitching and fixed wind turbines are assessed via hot-wire anemometry in a wind tunnel experiment. The pitching wind turbine model is free to oscillate in the streamwise direction to simulate pitch motion. Two inflow conditions are considered: 15% and 1.8% turbulent intensities. Empirical mode decomposition and Hilbert Huang transform are employed and validated by comparing the Hilbert energy spectrum with the Fourier energy spectrum. The extended self-similarity model indicates that pitching effects are more pronounced at locations where the flow is less turbulent due to its effect of being overshadowed by intermittency caused by tip vortex shedding. This agrees with arbitrary order Hilbert spectrum analysis (HSA) results. HSA is proven to be more accurate for scaling exponent estimation than structure functions as the latter results are significantly affected by the energetic scales. Premultiplied energy spectra show that pitch motion affects preferably large scales 0.1D−0.5D and the same amount of energy is contained on smaller scales compared to the fixed turbine, suggesting potential of higher power production. This work considers offshore wind turbine wakes by examining the pitch motion effects on the flow. Hence, results have direct implications on power production and quantification of fatigue loads due to pitch cyclic motion.

Secondary motion in turbulent pipe flow with three-dimensional roughness

The occurrence of secondary flows is investigated for three-dimensional sinusoidal roughness where the wavelength and height of the roughness elements are systematically altered. The flow spanned from the transitionally rough regime up to the fully rough regime and the solidity of the roughness ranged from a wavy, sparse roughness to a dense roughness. Analysing the time-averaged velocity, secondary flows are observed in all of the cases, reflected in the coherent stress profile which is dominant in the vicinity of the roughness elements. The roughness sublayer, defined as the region where the coherent stress is non-zero, scales with the roughness wavelength when the roughness is geometrically scaled (proportional increase in both roughness height and wavelength) and when the wavelength increases at fixed roughness height. Premultiplied energy spectra of the streamwise velocity turbulent fluctuations show that energy is reorganised from the largest streamwise wavelengths to the shorter streamwise wavelengths. The peaks in the premultiplied spectra at the streamwise and spanwise wavelengths are correlated with the roughness wavelength in the fully rough regime. Current simulations show that the spanwise scale of roughness determines the occurrence of large-scale secondary flows.

Mechanism of sweep event attenuation using micro-cavities in a turbulent boundary layer

Cavity arrays have been identified as a potential passive device to disrupt and capture sweep events, which are responsible for the excess Reynolds stresses in the boundary layer. In the present study, the mechanism of the attenuation of captured sweep events has been analyzed, as well as the non-linear relationship between the volume of the backing cavity and the reduction in sweep intensity. The influence of cavity array on the turbulent boundary layer has been analyzed, with a total of six different backing cavity arrangements with varying volumes. Three of the backing cavities have been used to determine the non-linear relationship between the effectiveness of the cavity array in reducing sweep intensity and the volume of the backing cavity. The other three have been used to determine the mechanism by which the arrays manipulate the captured sweep events. The pre-multiplied energy spectra of multiple velocity histories were significantly reduced, by up to 12.5%, in the low and mid-range wavelength values (λx+<104), which is associated with the coherent structures. The results show that the maximum reduction in sweep intensity of approximately 7% may be obtained when Reθ = 3771. It has been demonstrated that the non-linear relationship between sweep event intensity reduction and cavity volume has reached an upper limit in this investigation. Results from this study have revealed that the cavity array weakens the sweep intensity of the captured sweep events by damping the energy of the events through the friction losses in the cavity array and also in the large volume of the backing cavity.

Computational domain length and Reynolds number effects on large-scale coherent motions in turbulent pipe flow

ABSTRACTWe present results from direct numerical simulations (DNS) of turbulent pipe flow at shear Reynolds numbers up to Reτ = 1500 using different computational domains with lengths up to . The objectives are to analyse the effect of the finite size of the periodic pipe domain on large flow structures in dependency of Reτ and to assess a minimum required for relevant turbulent scales to be captured and a minimum Reτ for very large-scale motions (VLSM) to be analysed. Analysing one-point statistics revealed that the mean velocity profile is invariant for . The wall-normal location at which deviations occur in shorter domains changes strongly with increasing Reτ from the near-wall region to the outer layer, where VLSM are believed to live. The root mean square velocity profiles exhibit domain length dependencies for pipes shorter than 14R and 7R depending on Reτ. For all Reτ, the higher-order statistical moments show only weak dependencies and only for the shortest domain considered here. However, the analysis of one- and two-dimensional pre-multiplied energy spectra revealed that even for larger , not all physically relevant scales are fully captured, even though the aforementioned statistics are in good agreement with the literature. We found to be sufficiently large to capture VLSM-relevant turbulent scales in the considered range of Reτ based on our definition of an integral energy threshold of 10%. The requirement to capture at least 1/10 of the global maximum energy level is justified by a 14% increase of the streamwise turbulence intensity in the outer region between Reτ = 720 and 1500, which can be related to VLSM-relevant length scales. Based on this scaling anomaly, we found Reτ⪆1500 to be a necessary minimum requirement to investigate VLSM-related effects in pipe flow, even though the streamwise energy spectra does not yet indicate sufficient scale separation between the most energetic and the very long motions.

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 spanwise spectra in wall-bounded turbulence

The pre-multiplied spanwise energy spectra of streamwise velocity fluctuations are investigated in this paper. Two distinct spectral peaks in the spanwise spectra are observed in low-Reynolds-number wall-bounded turbulence. The spectra are calculated from direct numerical simulation (DNS) of turbulent channel flows and zero-pressure-gradient boundary layer flows. These two peaks locate in the near-wall and outer regions and are referred to as the inner peak and the outer peak, respectively. This result implies that the streamwise velocity fluctuations can be separated into large and small scales in the spanwise direction even though the friction Reynolds number $$Re_\tau $$ can be as low as 1000. The properties of the inner and outer peaks in the spanwise spectra are analyzed. The locations of the inner peak are invariant over a range of Reynolds numbers. However, the locations of the outer peak are associated with the Reynolds number, which are much higher than those of the outer peak of the pre-multiplied streamwise energy spectra of the streamwise velocity.

Relationship between streamwise and azimuthal length scales in a turbulent pipe flow

The statistical relationships among the turbulence structures of the streamwise velocity fluctuations along the streamwise and azimuthal directions in a turbulent pipe flow were examined using direct numerical simulation data at Reτ = 3008. Two-point correlations of the streamwise velocity fluctuations showed a linear relationship between the streamwise and azimuthal length scales (lx and lθ), where lθ/lx = 0.07 along the wall-normal distance, indicating the long coherent structures called very-large-scale motions (VLSMs). The one-dimensional pre-multiplied energy spectra of the streamwise velocity fluctuations showed that the streamwise and the azimuthal wavelengths (λx and λθ) grew linearly along the wall-normal distance, λx/y = 20 and λθ/y = 7, respectively. The ratio between the two linear relationships was determined to be λθ/λx = 0.35, indicative of large-scale motions (LSMs). The energetic modes obtained from a proper orthogonal decomposition (POD) analysis using the translational invariance method showed that the averaged helical angles of the wall mode (ix < iθ; β < 0.1 rad, where ix and iθ are the streamwise and azimuthal mode numbers and β is the helical angle) and lift mode (ix ≥ iθ; β ≥ 0.1 rad) were related to lθ/lx = 0.07 (VLSMs) and λθ/λx ≈ 0.35 (LSMs), respectively. The superposition of the energetic POD modes showed the superimposed X-shaped patterns. The helical angle of the wall mode in the near-wall region was similar to that in the outer region, implying the existence of the VLSMs in the entire wall-normal distance. The LSMs showed more inclined X-shaped patterns. The LSMs were concatenated with the azimuthal offsets to form meandering VLSMs. Most of the VLSMs and LSMs in the near-wall region inclined smaller and larger than 10° (0.17 rad), respectively. In the core region, VLSMs were distributed more helically along the azimuthal direction due to the space limitations of the pipe geometry.