Wall-normal Gradient Research Articles

Sub-filter-scale shear stress analysis in hypersonic turbulent Couette flow

Direct numerical simulations of hypersonic turbulent Couette flows are performed for top-wall Mach numbers of 6, 7 and 8, inspired by non-reactive high-enthalpy wind tunnel free-stream conditions, with the goal of analysing the physical processes driving the sub-filter-scale (SFS) stresses to inform development of large-eddy simulation techniques for hypersonic wall-bounded flows. Semi-local scaling laws collapse mean profiles and second-order turbulent statistics well, in spite of the strong wall-normal gradients of temperature and density. On the other hand, the SFS shear stresses exhibit an unexpected profile characterized by a region of pronounced shear stress deficit, which becomes more pronounced for higher Mach numbers. Instantaneous visualizations suggest that the SFS shear stress deficit is induced by the counter-gradient resolved momentum transport driven by residual velocity motions at the interface between high-density low-speed streaks being ejected away from the wall, and low-density high-speed ones replacing the displaced fluid, qualifying this as a compressibility effect. It is shown that the SFS shear stresses are primarily driven by second-order interactions between residual velocities, in spite of their triply nonlinear nature. This, in turn, motivated a statistical quadrant analysis revealing the presence of a SFS shear stress deficit, that is, SFS processes driving momentum transport of the resolved field towards the top wall. Additionally, higher-Reynolds-number simulations reveal that there is an upper limit of spatial filter widths to resolve large-scale structures, and such deficit is observable at any Reynolds number when a reasonable spatial filter is applied.

Wall cooling effect on spectra and structures of thermodynamic variables in hypersonic turbulent boundary layers

The wall cooling effect on the spectra and structures of thermodynamic variables are investigated in hypersonic turbulent boundary layers. The density and temperature can be divided into the acoustic and entropic modes based on the Kovasznay decomposition. The intensities of the pressure and the acoustic modes of density and temperature attain the maximum values near the wall, while those of the entropy and the entropic modes of density and temperature achieve their primary peaks near the edge of the boundary layer. In the near-wall region, the pressure and the acoustic modes of density and temperature are significantly enhanced when the wall is strongly cooled, which can be attributed to the appearance of the travelling-wave-like alternating positive and negative structures. Moreover, the intensities of the entropy and the entropic modes of density and temperature become stronger near the wall as the wall temperature decreases, due to the appearance of the streaky entropic structures (SES). The SES are mainly caused by the advection effect of the strong positive wall-normal gradient of the mean temperature associated with ejection and sweep events. It is also found that the profiles of the intensities of the entropy, density and temperature are similar to each other far from the wall, which is mainly due to the reason that the entropic modes are dominant in the fluctuating density and temperature in the far-wall region. The acoustic modes of density and temperature only have significant contributions in the near-wall region.

Turbulent dispersion of a passive scalar in a smooth-wall turbulent boundary layer

The scalar dispersion of a ground-level point-source plume in a smooth-wall turbulent boundary layer is experimentally investigated using simultaneous particle image velocimetry and planar laser-induced fluorescence techniques. In the near-source region, the viscous sublayer is observed to trap dye, while in the far field, the half-width, vertical profiles and peak decay of the mean concentration and concentration variance exhibit self-similar behaviour and collapse with empirical relations. Full two-dimensional maps of the turbulent scalar fluxes show a net transport direction of upward and towards the incoming flow, with the vertical profiles collapsing well with Weibull-type exponential functions and the decay of peaks following power laws. Using the first-order gradient transport to model the turbulent scalar fluxes, maps of the anisotropic turbulent diffusivity tensor and an effective turbulent diffusivity coefficient are calculated. The streamwise and wall-normal turbulent scalar fluxes are driven dominantly by the wall-normal concentration gradient. The turbulent Schmidt number, relating the turbulent diffusivity and the turbulent (eddy) viscosity calculated using the Boussinesq hypothesis, varies with wall-normal position with values of the order of unity in the logarithmic layer.

Unsteady evolution of distributed roughness-induced vortices under re-entry conditions

AbstractThis study investigates a re-entry scenario of an Apollo-like space capsule with Direct Numerical Simulations (DNS). The simulation includes the chemical equilibrium gas model. Cross-flow-like vortices are induced through random distributed roughness patches on the capsule surface. Two different machine learning methods are used to predict the maximum vorticity magnitude downstream of a pseudo-random roughness patch, the wall-normal location of the vortex core and spanwise and wall-normal gradient maxima of u. A large DNS database is formed for training and testing of the neural networks. In order to understand the influence of the vorticity magnitude on the transition process, local one-dimensional inviscid (LODI) relations are used to describe perturbations at the inflow. The disturbance evolution in the streamwise direction is analysed with a two-dimensional Fourier transformation in time and space. We show how the vorticity magnitudes of the cross-flow-like vortices, spanwise and wall-normal derivatives of the streamwise velocity influence the transition location.

Effects of freestream turbulence on the secondary instability of the roughness-induced crossflow vortex in swept flat plate boundary layers

In a three-dimensional boundary layer with spanwise constant flow and chordwise acceleration, which is unique to the vicinity of the leading edge of a swept wing, an inviscid instability induces the crossflow vortex (CFV), around which a secondary instability further occurs to yield a turbulent transition. It is well known that surface roughness induces stationary CFVs and that CFVs are receptive to freestream turbulence (FST), but the interference of both effects remains unclear. To determine the transition mechanisms in mixed roughness and FST environments, we performed direct numerical simulations of a Falkner–Skan–Cooke boundary layer under various conditions considering the presence of cylindrical roughness and the peak FST wavelength. Without roughness, the FST with a long wavelength comparable to the most unstable mode promotes CFV induction, resulting in an earlier transition. Conversely, the short-wavelength FST interacts with a wake vortex induced by the presence of roughness, yielding a hairpin vortex as a secondary instability. We discuss the main production terms of the disturbance energy associated with each of the type-1 and -2 secondary instabilities, which are accompanied by finger and hairpin vortices, respectively. The rapid turbulent transition via the type-1 secondary instability is due to the spanwise velocity gradient of the developed CFV, whereas that via the type-2 is due to a wall-normal gradient and is similar to the fully developed wall turbulence. The wavelength of FST played a key role in inducing a high-frequency secondary instability on the CFV or the CFV itself, and the receptivity to FST differed between the CFV and wake vortex.

Immersed boundary based near-wall modeling for large eddy simulation of turbulent wall-bounded flow

This paper describes the coupling of the immersed boundary method and the near-wall modeling of large eddy simulation for high Reynolds number turbulent flows over complex geometries on Cartesian grids. To overcome the spurious oscillation problem arising from the imposition of boundary conditions on the stair-case off-wall boundaries, several key ingredients have been employed, such as interpolating the friction velocity instead of the flow velocity from near-wall fluid interior points, evaluating the gradients by the weighted least square method and correcting the wall-normal gradient of the tangential velocity with the law of the wall at the off-wall boundaries. Furthermore, a hybrid RANS-LES approach has been applied to the near-wall eddy viscosity, either through an empirical blending function or the Reynolds stress balance constraint. We systematically discuss the effects of the hybrid eddy viscosity in a turbulent channel flow and a high lift three-element airfoil. Enforcing the Reynolds stress balance constraint turns out to be very robust in the considered cases. For the high lift three-element airfoil flow, the overall wall pressure and skin friction are predicted reasonably well and smoothly. The flow details are in a good agreement with the experimental data as well.

Spatio-temporal dynamics of a two-layer pressure-driven flow subjected to a wall-normal temperature gradient

The present study investigates the linear spatio-temporal and weakly nonlinear stability of a pressure-driven two-layer channel flow subjected to a wall-normal temperature gradient commonly encountered in industrial applications. The liquid–liquid interface tension is assumed to be a linearly decreasing function of temperature. The study employs both numerical (pseudo-spectral method) and long-wave approaches. The general linear stability analysis (GLSA) predicts shear-flow and thermocapillary modes that arise due to the imposed pressure and temperature gradients, respectively. The previous stability analyses of the same problem predicted a negligible effect of the pressure-driven flow on the linear stability of the system. However, the GLSA reveals stabilising and destabilising effects of the pressure-driven flow depending on the viscosity ratio ( $\mu _r$ ), thermal conductivity ratio ( $\kappa _r$ ), interface position ( $H$ ) and the sign of the imposed temperature gradient ( $\beta _1$ ). The analysis predicts a range of $H$ for given $\mu _r$ and $\kappa _r$ , which can not be stabilised by the thermocapillarity. The numerically predicted long-wave instability is then captured using the long-wave asymptotic approach. The arguments based on the physical mechanism further successfully explain the role of $\mu _r$ , $\kappa _r$ , $H$ , the sign of $\beta _1$ and the interaction between the velocity and temperature perturbations in stabilising/destabilising the flow. The spatio-temporal analysis reveals the dominance of the spanwise mode in causing the absolutely unstable flow. The weakly nonlinear analysis reveals a subcritical pitchfork bifurcation without shear flow. However, with the shear flow, the streamwise mode undergoes a supercritical Hopf bifurcation.

Gap ratio effects on the coherent structures surrounding a near-wall square cylinder

Turbulent flow around a square cylinder offset at different heights from an upstream rough wall are investigated using time-resolved particle image velocimetry. The Reynolds number based on the freestream velocity and cylinder height was 12750 and the incoming turbulent boundary layer thickness was 7.2 times the cylinder height. The examined gaps beneath the cylinder were 0.0, 0.3, 0.5, 1.0, 2.0, 4.0, and 8.0 times the cylinder height. The results show that the mean recirculation length remains relatively constant for cases with a gap size greater than or equal to 1.0 times the cylinder height. Flow acceleration through the gap under the cylinder is strongest with a gap size of 2.0 times cylinder height. Downstream of the cylinder, the vertical velocity fluctuations contribute strongly to the production of turbulent kinetic energy by interacting with mean wall-normal velocity gradient in the vertical direction. The dominant frequency embedded in the incoming turbulent boundary layer persists in the wake of the cylinder when the gap size is less than 1.0 cylinder height. Kelvin-Helmholtz vortices originating from the top leading edge of cylinder occur at Strouhal numbers of 2.77-3.11 for the cases with gap ratio ranging from 0.3-8.0 cylinder heights, while spectral migration towards lower frequencies is observed for cases with gap sizes equal to 0.3 and 0.5 cylinder heights. The spatial trajectory of top Kelvin-Helmholtz vortices extends further away from the cylinder as the gap size increases. Spectral proper orthogonal decomposition analysis shows that the von Kármán vortex shedding structures possess the highest spectral energy for all offset cases despite regular vortex shedding being disrupted when the gap size is 0.3 times the cylinder height.

Evidence that uniform momentum zones originate from roughness sublayer structure interactions in fully rough channel turbulence

Fully rough wall-sheared turbulence is composed of an inner and outer layer, the former occupied by sinuous structures sustained by the well-known autonomous inner cycle, the latter occupied by inclined parcels of momentum deviations relative to the average. The outer-layer structures are also known as uniform momentum zones (UMZs), where the presence of successive UMZs manifests instantaneously with a distinct ‘staircase’ pattern in streamwise velocity. Direct numerical simulation (DNS) of fully rough channel turbulence was recently used to interpret UMZs as wakes originating from antecedent bluff-body-like interactions within the inner layer (Anderson & Salesky, J. Fluid Mech., vol. 906, 2020, A8). Wake-scaling arguments agreed precisely with instantaneous results from DNS. Foremost among these results was evidence that the instantaneous wall-normal gradient of streamwise velocity exhibited scaling of ${\sim }x_3^{-1/2}$ , where $x_3$ is wall-normal location; this is in contrast to logarithmic scaling, which requires the wall-normal gradient of streamwise velocity to scale as ${\sim }x_3^{-1}$ . Herein, wake-scaling arguments have been advanced and compared against results from wall-modelled large-eddy simulation (LES). New results from the ‘bottom up’ wake-scaling arguments agree with LES results, such that the shear associated with the instantaneous staircase-like streamwise velocity follows a clear trend. We have leveraged conditional sampling during LES – predicated upon instantaneous low-frequency, high-magnitude surface stress values known to correspond to large-scale inertial-layer coherence – to further assess the predictive value of the wake-scaling arguments. Model results compare favourably.

Investigation of vortical structures in the wake of pseudo-random roughness surfaces in hypersonic reacting boundary-layer flows

The influence of roughness on the laminar-turbulent transition of hypersonic flows is investigated with Direct Numerical Simulations (DNS) in a re-entry scenario of an Apollo-like space capsule. A Mach 20 flow regime is simulated for conditions at an altitude of 57.5 km. This study uses two different generating shapes (sinusoidal and triangular shaped waves) for random distributed patches. Stronger streamwise vorticity is observed for most configurations with the triangular base functions. The simulation includes the chemical equilibrium gas model. Vortical structures in the wake of both roughness patch types interact with an acoustic perturbation derived from local one-dimensional inviscid (LODI) relations. A breakdown of the cross-flow-like vortex in the wake of the roughness patch is observed for both roughness patches. The growth of modal instabilities is studied with a spatio-temporal Fourier analysis. Both y- and z-type instabilities, which are present in cross-flow vortices at low speeds, are observed for the sinusoidal patch. In the triangular configuration, the y-type instability is the dominating instability which is in line with the stronger wall-normal velocity gradients caused by the increase in spanwise slope of the patch.

A thermoacoustic heat pump driven by acoustic waves in a hypersonic boundary layer

Acoustic waves existing in hypersonic boundary layers act as a heat pump that transfers energy from the sonic line to the wall causing the wall temperature to rise, which explains the newly identified aerodynamic heating related to Mack's second mode from the perspective of thermoacoustic effects. The analysis of data from direct numerical simulations shows that Mack's second mode, belonging to the family of trapped acoustic waves, is highly amplified in a Mach 6 boundary layer and becomes sufficiently strong to affect the mean wall-normal temperature gradient, and the energy transport in the wall-normal direction due to the thermoacoustic effect balancing the thermal conduction brought by the additional temperature gradient.

Near-wall Taylor-series expansion solution for compressible Navier–Stokes–Fourier system

This paper presents the Taylor-series expansion solution of near-wall velocity and temperature for a compressible Navier–Stokes–Fourier system with a no-slip curved boundary surface. When the shear viscosity is a single-valued function of local fluid temperature, the near-wall velocity and temperature are explicitly expressed using the surface quantities including skin friction, surface pressure, surface dilatation, surface heat flux, surface temperature, surface curvature, and their relevant derivatives at the wall. In addition, the wall-normal pressure gradient at the wall is found to be contributed by three physical mechanisms including the skin friction divergence and surface dilatation effect as well as the coupled skin friction and surface heat flux with varying shear viscosity. Furthermore, without losing generality, we derive the near-wall Taylor-series expansion solution for the Lamb vector under the assumption of constant viscosities. Different physical mechanisms that are responsible for initial formation of the Lamb vector in the viscous sublayer are elucidated. The significance of the skin friction divergence and surface dilatation to the near-wall Lamb vector is highlighted.

Drag decomposition of laminar channel flows developing over convergent–divergent riblets

Convergent–divergent (C–D) riblets are known to induce large-scale secondary flow motion in laminar flow that may point to their potential for flow separation control and heat transfer enhancement. In this paper, a fully-developed pressure-driven laminar channel flow with C–D riblets is studied numerically. The effects of Reynolds number, riblet wavelength and riblet cross-sectional shape on the secondary flow and drag characteristics are investigated. An exact expression for the drag coefficient of this flow is derived based on the balance of kinetic energy principle for a rough-wall channel. Furthermore, the triple decomposition technique is used to identify the contribution to drag production from the mean velocity field, the small- and large-scale dispersive flow field. Our results show that the normalized drag increment starts to rise when the secondary flow begins to alter the velocity field across the span as the Reynolds number based on the channel height (ReH) increases above 100, a behavior which is very different from that of a laminar channel flow developing over homogeneous roughness in which the normalized drag increment is found to remain constant up to a much greater Reynolds number due to an absence of secondary flow. It is found that the drag produced by C–D riblets is predominantly caused by the wall-normal gradient of streamwise velocity. In addition, while the drag increment is predominantly caused by the mean and small-scale dispersive velocity at ReH≤100, the contribution from the large-scale dispersive velocity field increases rapidly with the Reynolds number and becomes dominant as ReH exceeds 400. It is also observed that the secondary flow scales with the riblet wavelength Λ, and its swirling strength peaks around Λ=1.25H, which is accompanied by a drag maxima at ReH≥400. Finally, among the three C–D riblet cross-sectional shape (rectangular, triangular and sinusoidal) examined here, the triangular riblet pattern is found to produce a secondary motion with a similar strength to the other two shapes but with less drag penalty.

Turbulence modulation in thermally expanding and contracting flows

We present direct numerical simulations of developing turbulent channel flows subjected to thermal expansion or contraction downstream of a heated or cooled wall. Using different constitutive relations for viscosity we analyse the response of variable property flows to streamwise acceleration/deceleration by separating the effect of streamwise acceleration/deceleration from the effect of wall-normal property variations. We demonstrate that, beyond a certain streamwise location, the flow can be considered in a state of ‘quasi-equilibrium’ regarding semilocally scaled variables. As such, we claim that the development of turbulent quantities due to streamwise acceleration/deceleration is localized to the region of impulsive heating/cooling, while changes in turbulence occurring farther downstream can be attributed solely to property variations. This finding allows us to study turbulence modulation in accelerating/decelerating flows using the semilocal scaling framework. By investigating the energy redistribution among the turbulent velocity fluctuations, we conclude that a change in bulk streamwise velocity has a non-local effect which originates from the change in mean shear and modifies the energy pathways through velocity-pressure-gradient correlations. On the other hand, the wall-normal property gradients have a local effect and act through the modification of the viscous dissipation. We show that it is possible to superimpose and compare the two different effects when using the semilocal scaling framework.

Contribution of Reynolds shear stress to near-wall turbulence in Rayleigh–Bénard convection

Direct numerical simulations of Rayleigh–Bénard convection flows are performed for Pr=0.7 with Ra=5×108 and 2×1010, and for Pr=0.021 with Ra=107 and 5×108, where Pr and Ra are the Prandtl number and the Rayleigh number, respectively. The velocity-gradient production based on the short-time averaging shows the near-wall intensive positive peak and negative region. The correlations between the Reynolds shear stresses and the wall-normal and transverse gradients of the horizontal mean velocities locally and temporally result in the positive and negative velocity-gradient production near the wall. The eigenvalues of the anisotropic Reynolds stress tensors on the barycentric map show that the short-lived anisotropy related to the one-directional stretching occurs along the wall-normal distance; this behavior is similar to that of a canonical turbulent channel flow. The results indicate that the local and temporal Reynolds shear stresses near the wall with the shear of the horizontal mean velocities play an important role in the near-wall velocity-gradient production, which leads to the turbulent regime in the Rayleigh–Bénard convection flow with an increase in Rayleigh number.

On the coupling of wall-model immersed boundary conditions and curvilinear body-fitted grids for the simulation of complex geometries

One of the main limitations in computational aerodynamics lies in the ability of CFD solvers to handle complex geometries while maintaining their accuracy. Among the currently available strategies, the Zonal Immersed Boundary Conditions (ZIBC) has shown its capability to introduce complex geometrical details for the simulations of high Reynolds turbulent flows. In this context, this work aims to extend the ZIBC framework, initially adapted to reproduce blockage effects, to take into account configurations where the influence of spatially developing boundary layers has to be accurately predicted. For this purpose, a compact sharp interface immersed boundary method has been developed for curvilinear grids. Moreover, a Thin Boundary Layer Equations (TBLE) wall model based on one-dimensional RANS Spalart-Allmaras equations is coupled with the IBC to model the inner part of compressible turbulent boundary layers. The coupling with the Zonal Detached Eddy Simulation (ZDES) approach is improved and permits to simulate turbulent flows capturing accurately the wall-normal gradients with a limited grid resolution. This strategy is applied to the simulation of a generic missile configuration (FG5) with an angle of attack of 10∘ using RANS and ZDES approaches. For the first time, wall quantities such as unsteady loads, including the viscous contribution, are accuratly reproduced on surfaces modelled with IBC using hybrid RANS/LES (ZDES) computations and for a realistic configuration at high Reynolds number. Such a reconstruction of physical quantities at the wall is achieved thanks to a precise reconstruction procedure using a TBLE wall model.

Growth mechanisms of second-mode instability in hypersonic boundary layers

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Uniform momentum zone scaling arguments from direct numerical simulation of inertia-dominated channel turbulence

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Transport of inertial particles in high-Reynolds-number turbulent boundary layers

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Imposing virtual origins on the velocity components in direct numerical simulations

The relative wall-normal displacement of the origin perceived by different components of near-wall turbulence is known to produce a change in drag. This effect is produced for instance by drag-reducing surfaces of small texture-size like riblets and superhydrophobic surfaces. To facilitate the research on how these displacements alter near-wall turbulence, this paper studies different strategies to model such displacement effect through manipulated boundary conditions. Previous research has considered the effect of offsetting the virtual origins perceived by the tangential components of the velocity from the reference, boundary plane, where the wall-normal velocity was set to zero. These virtual origins are typically characterised by slip-length coefficients in Robin, slip-like boundary conditions. In this paper, we extend this idea and explore several techniques to define and implement virtual origins for all three velocity components on direct numerical simulations (DNSs) of channel flows, with special emphasis on the wall-normal velocity. The aim of this work is to provide a suitable foundation to extend the existing understanding on how these virtual origins affect the near-wall turbulence, and ultimately aid in the formulation of simplified models that capture the effect of complex surfaces on the overlying flow and on drag, without the need to resolve fully the turbulence and the surface texture. From the techniques tested, Robin boundary conditions for all three velocities are found to be the most satisfactory method to impose virtual origins, relating the velocity components to their respective wall-normal gradients linearly. Our results suggest that the effect of virtual origins on the flow, and hence the change in drag that they produce, can be reduced to an offset between the virtual origin perceived by the mean flow and that perceived by the overlying turbulence, and that turbulence remains otherwise smooth-wall-like, as proposed by Luchini (1996). The origin for turbulence, however, would not be set by the spanwise virtual origin alone, but by a combination of the spanwise and wall-normal origins. These observations suggest the need for an extension of Luchini’s virtual-origin theory to predict the change in drag, accounting for the wall-normal transpiration when its effect is not negligible.