Streamwise Vortices and Velocity Streaks in a Locally Drag-Reduced Turbulent Boundary Layer

This work aims to understand the changes associated with the near-wall streaky structures in a turbulent boundary layer (TBL) where the local skin-friction drag is substantially reduced. The Reynolds number is R e _?? = 1000 based on the momentum thickness or R e _τ = 440 based on the friction velocity of the uncontrolled flow. The TBL is perturbed via a local surface oscillation produced by an array of spanwise-aligned piezo-ceramic (PZT) actuators and measurements are made in two orthogonal planes using particle image velocimetry (PIV). Data analyses are conducted using the vortex detection, streaky structure identification, spatial correlation and proper orthogonal decomposition (POD) techniques. It is found that the streaky structures are greatly modified in the near-wall region. Firstly, the near-wall streamwise vortices are increased in number and swirling strength but decreased in size, and are associated with greatly altered velocity correlations. Secondly, the velocity streaks grow in number and strength but contract in width and spacing, exhibiting a regular spatial arrangement. Other aspects of the streaky structures are also characterized; they include the spanwise gradient of the longitudinal fluctuating velocity and both streamwise and spanwise integral length scales. The POD analysis indicates that the turbulent kinetic energy of the streaky structures is reduced. When possible, our results are compared with those obtained by other control techniques such as a spanwise-wall oscillation, a spanwise oscillatory Lorentz force and a transverse traveling wave.     

Pressure-Gradient Turbulent Boundary Layers Developing Around a Wing Section

A direct numerical simulation database of the flow around a NACA4412 wing section at R e _c = 400,000 and 5^° angle of attack (Hosseini et al. Int. J. Heat Fluid Flow 61, 117–128, 2016), obtained with the spectral-element code Nek5000, is analyzed. The Clauser pressure-gradient parameter β ranges from ? 0 and 85 on the suction side, and from 0 to ? 0.25 on the pressure side of the wing. The maximum R e _?? and R e _τ values are around 2,800 and 373 on the suction side, respectively, whereas on the pressure side these values are 818 and 346. Comparisons between the suction side with zero-pressure-gradient turbulent boundary layer data show larger values of the shape factor and a lower skin friction, both connected with the fact that the adverse pressure gradient present on the suction side of the wing increases the wall-normal convection. The adverse-pressure-gradient boundary layer also exhibits a more prominent wake region, the development of an outer peak in the Reynolds-stress tensor components, and increased production and dissipation across the boundary layer. All these effects are connected with the fact that the large-scale motions of the flow become relatively more intense due to the adverse pressure gradient, as apparent from spanwise premultiplied power-spectral density maps. The emergence of an outer spectral peak is observed at β values of around 4 for λ _z ? 0.65δ ₉₉, closer to the wall than the spectral outer peak observed in zero-pressure-gradient turbulent boundary layers at higher R e _?? . The effect of the slight favorable pressure gradient present on the pressure side of the wing is opposite the one of the adverse pressure gradient, leading to less energetic outer-layer structures.     

Improvement for expansion of parabolized stability equation method in boundary layer stability analysis

An improved expansion of the parabolized stability equation(iEPSE) method is proposed for the accurate linear instability prediction in boundary layers. It is a local eigenvalue problem, and the streamwise wavenumber α and its streamwise gradient dα/dx are unknown variables. This eigenvalue problem is solved for the eigenvalue dα/dx with an initial α, and the correction of α is performed with the conservation relation used in the PSE. The i EPSE is validated in several compressible and incompressible boundary layers. The computational results show that the prediction accuracy of the i EPSE is significantly higher than that of the ESPE, and it is in excellent agreement with the PSE which is regarded as the baseline for comparison. In addition, the unphysical multiple eigenmode problem in the EPSE is solved by using the i EPSE. As a local non-parallel stability analysis tool, the i EPSE has great potential application in the eNtransition prediction in general three-dimensional boundary layers.     

Direct Numerical Simulation and Theory of a Wall-Bounded Flow with Zero Skin Friction

We study turbulent plane Couette-Poiseuille (CP) flows in which the conditions (relative wall velocity ΔU _w ≡ 2U _w , pressure gradient dP/dx and viscosity ν) are adjusted to produce zero mean skin friction on one of the walls, denoted by APG for adverse pressure gradient. The other wall, FPG for favorable pressure gradient, provides the friction velocity u _τ , and h is the half-height of the channel. This leads to a one-parameter family of one-dimensional flows of varying Reynolds number Re ≡ U _w h/ν. We apply three codes, and cover three Reynolds numbers stepping by a factor of two each time. The agreement between codes is very good, and the Reynolds-number range is sizable. The theoretical questions revolve around Reynolds-number independence in both the core region (free of local viscous effects) and the two wall regions. The core region follows Townsend’s hypothesis of universal behavior for the velocity and shear stress, when they are normalized with u _τ and h; on the other hand universality is not observed for all the Reynolds stresses, any more than it is in Poiseuille flow or boundary layers. The FPG wall region obeys the classical law of the wall, again for velocity and shear stress. For the APG wall region, Stratford conjectured universal behavior when normalized with the pressure gradient, leading to a square-root law for the velocity. The literature, also covering other flows with zero skin friction, is ambiguous. Our results are very consistent with both of Stratford’s conjectures, suggesting that at least in this idealized flow turbulence theory is successful like it was for the classical logarithmic law of the wall. We appear to know the constants of the law within a 10% bracket. On the other hand, that again does not extend to Reynolds stresses other than the shear stress, but these stresses are passive in the momentum equation.     

Turbulent Drag Reduction by a Near Wall Surface Tension Active Interface

In this work we study the turbulence modulation in a viscosity-stratified two-phase flow using Direct Numerical Simulation (DNS) of turbulence and the Phase Field Method (PFM) to simulate the interfacial phenomena. Specifically we consider the case of two immiscible fluid layers driven in a closed rectangular channel by an imposed mean pressure gradient. The present problem, which may mimic the behaviour of an oil flowing under a thin layer of different oil, thickness ratio h₂/h₁ =?9, is described by three main flow parameters: the shear Reynolds number Re _τ (which quantifies the importance of inertia compared to viscous effects), the Weber number We (which quantifies surface tension effects) and the viscosity ratio λ = ν₁/ν₂ between the two fluids. For this first study, the density ratio of the two fluid layers is the same (ρ₂ = ρ₁), we keep Re _τ and We constant, but we consider three different values for the viscosity ratio: λ =?1, λ =?0.875 and λ =?0.75. Compared to a single phase flow at the same shear Reynolds number (Re _τ =?100), in the two phase flow case we observe a decrease of the wall-shear stress and a strong turbulence modulation in particular in the proximity of the interface. Interestingly, we observe that the modulation of turbulence by the liquid-liquid interface extends up to the top wall (i.e. the closest to the interface) and produces local shear stress inversions and flow recirculation regions. The observed results depend primarily on the interface deformability and on the viscosity ratio between the two fluids (λ).     

Evaluation of the Spectral Element Dynamic Model for Large-Eddy Simulation on Unstructured,Deformed Meshes

In this paper, the Spectral-Element Dynamic Model (SEDM), suited for Large-Eddy Simulation (LES) using Discontinuous Finite Element Methods (DFEM), is assessed using unstructured meshes. Five test cases of increasing complexity are considered, namely, the Taylor-Green vortex at Re =?5000, the turbulent channel flow at Re_τ =?587, the circular cylinder in cross-flow at Re_D =?3900, the square cylinder in cross-flow at Re_D =?22400 and the channel with periodic constrictions at Re_h =?10595. Various discretization parameters such as the grid spacing, polynomial degree and numerical flux are assessed and very accurate results are reported in all cases. This consistency in the results demonstrates the versatility of the SEDM approach and its ability to gage the actual resolution and quality of the mesh and, accordingly, to introduce an amount of sub-grid dissipation which is adapted to the spatial discretization considered.     

Effects of Fuel Lewis Number on Localised Forced Ignition of Globally Stoichiometric Stratified Mixtures: a Numerical Investigation

The influences of fuel Lewis number Le_F on localised forced ignition of globally stoichiometric stratified mixtures have been analysed using three-dimensional compressible Direct Numerical Simulations (DNS) for cases with Le_F ranging from 0.8 to 1.2. The globally stoichiometric stratified mixtures with different values of root-mean-square (rms) equivalence ratio fluctuation (i.e. ?^′= 0.2, 0.4 and 0.6) and the Taylor micro-scale l_? of equivalence ratio ? variation (i.e. l_?/l_f= 2.1, 5.5 and 8.3 with l_f being the Zel’dovich flame thickness of the stoichiometric laminar premixed flame) have been considered for different initial rms values of turbulent velocity u^′. A pseudo-spectral method is used to initialise the equivalence ratio variation following a presumed bi-modal distribution for prescribed values of ?^′ and l_?/l_f for global mean equivalence ratio 〈?〉=1.0. The localised ignition is accounted for by a source term in the energy transport equation that deposits energy for a stipulated time interval. It has been observed that the maximum values of temperature and the fuel reaction rate magnitude increase with decreasing Le_F during the period of external energy deposition. The initial values of Le_F, u^′/S_b(?=1), ?^′ and l_?/l_f have been found to have significant effects on the extent of burning of the stratified mixtures following localised ignition. For a given value of u^′/S_b(?=1), the extent of burning decreases with increasing Le_F. An increase in u^′ leads to a monotonic reduction in the burned gas mass for all values of Le_F in all stratified mixture cases but an opposite trend is observed for the Le_F=0.8 homogeneous mixture. It has been found that an increase in ?^′ has adverse effects on the burned gas mass, whereas the effects of l_?/l_f on the extent of burning are non-monotonic and dependent on ?^′ and Le_F. Detailed physical explanations have been provided for the observed Le_F, u^′/S_b(?=1), ?^′ and l_?/l_f dependences.     

Perspectives on the Phenomenology and Modeling of Boundary Layer Transition

The characteristics of the coherent structures in a strongly decelerated large-velocity-defect boundary layer are analysed by direct numerical simulation. The simulated boundary layer starts as a zero-pressure-gradient boundary layer, decelerates under a strong adverse pressure gradient, and separates near the end of the domain, in the form of a very thin separation bubble. The Reynolds number at separation is R e _?? =3912 and the shape factor H=3.43. The three-dimensional spatial correlations of (u, u) and (u, v) are investigated and compared to those of a zero-pressure-gradient boundary layer and another strongly decelerated boundary layer. These velocity pairs lose coherence in the streamwise and spanwise directions as the velocity defect increases. In the outer region, the shape of the correlations suggest that large-scale u structures are less streamwise elongated and more inclined with respect to the wall in large-defect boundary layers. The three-dimensional properties of sweeps and ejections are characterized for the first time in both the zero-pressure-gradient and adverse-pressure-gradient boundary layers, following the method of Lozano-Durán et al. (J. Fluid Mech. 694, 100–130, [2012]). Although longer sweeps and ejections are found in the zero-pressure-gradient boundary layer, with ejections reaching streamwise lengths of 5 boundary layer thicknesses, the sweeps and ejections tend to be bigger in the adverse-pressure-gradient boundary layer. Moreover, small near-wall sweeps and ejections are much less numerous in the large-defect boundary layer. Large sweeps and ejections that reach the wall region (wall-attached) are also less numerous, less streamwise elongated and they occupy less space than in the zero-pressure-gradient boundary layer.     

Predicting Turbulent Spectra in Drag-reduced Flows

In the present work we describe how turbulent skin-friction drag reduction obtained through near-wall turbulence manipulation modifies the spectral content of turbulent fluctuations and Reynolds shear stress with focus on the largest scales. Direct Numerical Simulations (DNS) of turbulent channels up to Re _τ = 1000 are performed in which drag reduction is achieved either via artificially removing wall-normal turbulent fluctuations in the vicinity of the wall or via streamwise-travelling waves of spanwise wall velocity. This near-wall turbulence manipulation is shown to modify turbulent spectra in a broad range of scales throughout the whole channel. Above the buffer layer, the observed changes can be predicted, exploiting the vertical shift of the logarithmic portion of the mean streamwise velocity profile, which is a classic performance measure for wall roughness or drag-reducing riblets. A simple model is developed for predicting the large-scale contribution to turbulent fluctuation and Reynolds shear stress spectra in drag-reduced turbulent channels in which a flow control acts at the wall. Any drag-reducing control that successfully interacts with large scales should deviate from the predictions of the present model, making it a useful benchmark for assessing the capability of a control to affect large scales directly.     

Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 1: Linear Stability Analysis

The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β _T and a solutal mobility ratio β _C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β _T is seen to enhance the instability for fixed β _C , Le and λ. For fixed β _C and β _T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β _T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β _C , but β _T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β _C only.     

Effects of Lewis Number on Head on Quenching of Turbulent Premixed Flames: A Direct Numerical Simulation Analysis

The head on quenching of statistically planar turbulent premixed flames by an isothermal inert wall has been analysed using three-dimensional Direct Numerical Simulation (DNS) data for different values of global Lewis number Le(0.8, 1.0 and 1.2) and turbulent Reynolds number Re_t. The statistics of head on quenching have been analysed in terms of the wall Peclet number Pe (i.e. distance of the flame from the wall normalised by the Zel’dovich flame thickness) and the normalised wall heat flux Φ. It has been found that the maximum (minimum) value of Φ(Pe) for the turbulent Le=0.8 cases are greater (smaller) than the corresponding laminar value, whereas both Pe and Φ in turbulent cases remain comparable to the corresponding laminar values for Le=1.0 and 1.2. Detailed physical explanations are provided for the observed Le dependences of Pe and Φ. The existing closure of mean reaction rate \(\overline {\dot {\omega }}\) using the scalar dissipation rate (SDR) in the near wall region has been assessed based on a-priori analysis of DNS data and modifications to the existing closures of mean reaction rate and SDR have been suggested to account for the wall effects in such a manner that the modified closures perform well both near to and away from the wall.     

Experimental Control of Turbulent Boundary Layers with In-plane Travelling Waves

The experimental control of turbulent boundary layers using streamwise travelling waves of spanwise wall velocity, produced using a novel active surface, is outlined in this paper. The innovative surface comprises a pneumatically actuated compliant structure based on the kagome lattice geometry, supporting a pre-tensioned membrane skin. Careful design of the structure enables waves of variable length and speed to be produced in the flat surface in a robust and repeatable way, at frequencies and amplitudes known to have a favourable influence on the boundary layer. Two surfaces were developed, a preliminary module extending 152 mm in the streamwise direction, and a longer one with a fetch of 2.9 m so that the boundary layer can adjust to the new surface condition imposed by the forcing. With a shorter, 1.5 m portion of the surface actuated, generating an upstream-travelling wave, a drag reduction of 21.5% was recorded in the boundary layer with Re _τ =?1125. At the same flow conditions, a downstream-travelling produced a much smaller drag reduction of 2.6%, agreeing with the observed trends in current simulations. The drag reduction was determined with constant temperature hot-wire measurements of the mean velocity gradient in the viscous sublayer, while simultaneous laser Doppler vibrometer measurements of the surface recorded the wall motion. Despite the mechanics of the dynamic surface resulting in some out-of-plane motion (which is small in comparison to the in-plane streamwise movement), the positive drag reduction results are encouraging for future investigations at higher Reynolds numbers.     

Kovasznay Mode Decomposition of Velocity-Temperature Correlation in Canonical Shock-Turbulence Interaction

The correlation coefficient R_uT between the streamwise velocity and temperature is investigated for the case of canonical shock-turbulence interaction, motivated by the fact that this correlation is an important component in compressible turbulence models. The variation of R_uT with the Mach number, the turbulent Mach number, and the Reynolds number is predicted using linear inviscid theory and compared to data from DNS. The contributions from the individual Kovasznay modes are quantified. At low Mach numbers, the peak post-shock R_uT is determined by the acoustic mode, which is correctly predicted by the linear theory. At high Mach numbers, it is determined primarily by the vorticity and entropy modes, which are strongly affected by nonlinear and viscous effects, and thus less well predicted by the linear theory.     

Stress distributions in a semi-infinite plate due to a pin determined by interferometric method

The stress distributions in a semi-infinite plate due to a loaded pin of the same material as the plate are systematically investigated by an interferometric method which has been developed by the authors. For the experiments, a finite plate of diallylphthalate with a circular hole is used. It is supported at one side and loaded in the direction normal to the opposing straight edge by a pin which just fits the hole. The ratio of the distancee between the hole center and the straight edge to the diameterd of the hole is varied in steps from 4.0 to 1.0. At each step, the distributions of principal stresses σ₁ and σ₂ along the hole edge, line of symmetry and straight edge, which have not been fully investigated especially whene/d is small, are obtained separately from the isopachic and isochromatic fringes of the interfero-stress patterns. The relations between the maximum values of these stresses and the shape factore/d are determined.     

Experimental Study on Three-Dimensional Deformation Field of Portevin–Le Chatelier Effect Using Digital Image Correlation

In view of its high precision and high efficiency, three-dimensional digital image correlation (3D-DIC) is widely used to accurately measure full-field deformation. A spatiotemporal experimental study using 3D-DIC to explore the Portevin–Le Chatelier (PLC) deformational behavior, provides a new insight into the whole 3D deformation field, including the out-of-plane displacement, and in particular the relationship between the serrations and the strain field in the deformation bands corresponding to individual serrations. Specimens 1, 2 and 3 mm thick of 5456 Al-based alloy were tested in uniaxial tension at room temperature at strain rates from 1.8 × 10^?4 to 9.1 × 10^?3s^?1. The spatial and temporal characteristics of the strain localization were quantitatively analyzed. The out-of-plane displacement increment field (w) of the localized bands was observed by 3D-DIC, and found to be related to the specimen thickness and the in-plane strain increment. The largest displacement increments were respectively 15, 10 and 5 μm for 3, 2 and 1 mm specimens at maximum strain increment of about 12000 με. The elastic shrinkage outside the deformation bands was found to be an essential characteristic of the PLC effect. The width of the PLC band (w_band) increased with increasing thickness; the angle of the PLC band (??_band) was not affected by either specimen thickness or serration amplitude. Temporally, the serrations in the plots both of in-plane strain and out-of-plane displacement vs. time coincided throughout the entire loading procedure. When PLC banding occurred, the serration amplitude within the bands was found to be proportional to the maximum strain increment in the direction of the applied tensile force (??_max).     

On the orders of the best approximations of integrals of functions by integrals of rank <Emphasis Type="Italic">σ</Emphasis>

We study the values e _σ(f) of the best approximation of integrals of functions from the spaces L _p (A, dμ) by integrals of rank σ. We determine the orders of the least upper bounds of these values as σ → ∞ in the case where the function ? is the product of two nonnegative functions one of which is fixed and the other varies on the unit ball U _p (A) of the space L _p (A, dμ). We consider applications of the obtained results to approximation problems in the spaces S _p ^? .     

Morse Index and Linear Stability of the Lagrangian Circular Orbit in a Three-Body-Type Problem Via Index Theory

It is well known that the linear stability of the Lagrangian elliptic solutions in the classical planar three-body problem depends on a mass parameter β and on the eccentricity e of the orbit. We consider only the circular case (e = 0) but under the action of a broader family of singular potentials: α-homogeneous potentials, for \(\alpha \in (0, 2)\), and the logarithmic one. It turns out indeed that the Lagrangian circular orbit persists also in this more general setting. We discover a region of linear stability expressed in terms of the homogeneity parameter α and the mass parameter β, then we compute the Morse index of this orbit and of its iterates and we find that the boundary of the stability region is the envelope of a family of curves on which the Morse indices of the iterates jump. In order to conduct our analysis we rely on a Maslov-type index theory devised and developed by Y. Long, X. Hu and S. Sun; a key role is played by an appropriate index theorem and by some precise computations of suitable Maslov-type indices.     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

Investigation of Particle Deposition Characteristics in Vicinity of Laidback Fan-shaped Film Cooling Holes

To design efficient film cooling systems and mitigate particulate deposition, it is very important to know the influences of the design parameters of film cooling holes on particulate deposition. However, most previous research focused on round film cooling holes. Particle deposition characteristics near shaped film cooling holes need to be studied further. In the present study, numerical computations were carried out to simulate the particle deposition behavior on gas turbine disk samples with laidback fan-shaped film cooling holes by using CFD-DPM (Computational fluid dynamics-discrete particle method). The critical velocities for particle sticking and detachment were determined by EI-Batsh model. Compared with round holes, shaped holes mitigate the particle-wall collision for small particles (d_p≤2μm, ρ_p=990kg/m ³), but promote particle-wall collision for large particles (d_p≥4μm, ρ_p=990kg/m ³). Adding the lateral and forward expansion angle can both cause the decrease of particle deposition efficiency, however, the effect of lateral expansion angle on particle deposition is more active.