Near-wall Velocity Profile Research Articles

Assimilating experimental data of a mean three-dimensional separated flow using physics-informed neural networks

In this article, we address the capabilities of physics-informed neural networks (PINNs) in assimilating the experimentally acquired mean flow of a turbulent separation bubble occurring in a diffuser test section. The training database contains discrete mean pressure and wall shear-stress fields measured on the diffuser surface as well as three-component velocity vectors obtained with particle image velocimetry throughout the volumetric flow domain. Imperfections arise from the measurement uncertainty and the inability to acquire velocity data in the near-wall region. We show that the PINN methodology is suited to handle both of these issues thanks to the incorporation of the underlying physics that, in the present study, are taken into account by minimizing residuals of the three-dimensional incompressible Reynolds-averaged Navier–Stokes equations. As a result, measurement errors are rectified and near-wall velocity profiles are predicted reliably. The latter benefits from the incorporation of wall shear-stress data into the PINN training, which has not been attempted so far to the best of our knowledge. In addition to demonstrating the influence of this novel loss term, we provide a three-dimensional, highly resolved, and differentiable model of a separating and reattaching flow that can be readily used in future studies.

Thermochemical non-equilibrium hypersonic flow over a rectangular cavity embedded on a compression ramp

This paper reports a systematic computational investigation that elucidates the fundamental thermochemical non-equilibrium physics that occurs when air at Mach number of 11 encounters a rectangular cavity of aspect ratio L/D = 2.0 embedded on a 25° compression ramp. The mechanistic details of this highly complex flow phenomenon are obtained by solving the compressible form of the Navier–Stokes equations in two dimensions using a finite-volume open-source library. Chemical and thermal non-equilibrium processes are treated using a five-species, 12-reaction chemical kinetics, and a two-temperature model, respectively. Following a detailed validation and grid sensitivity study, two simulations are conducted, one with isothermal boundary conditions and the other with conjugate heat transfer (CHT) to identify the effect of energy transmission to the material on surface heat flux. Fast Fourier transforms and near-wall velocity profiles inside and in the neighborhood of the cavity are used to identify primary oscillatory modes and shear layer dynamics. Two new descriptive states defined as “states I and II,” representative of the minimum and maximum deflection of the shear layer, are used to discuss the dynamical behaviors in the cavity, including the separation region before the cavity, trailing edge effects, frequency analysis of probe data collected at several key locations, and the effect of CHT on surface heat flux. It is found that the flow features at the cavity's center strongly influence the separation upstream of the cavity, and the transrotational temperature near the cavity's trailing edge is strongly correlated with the oscillations of the shear layer.

Data-driven prediction of the equivalent sand-grain roughness

Surface roughness affects the near-wall fluid velocity profile and surface drag, and is commonly quantified by the equivalent sand-grain roughness {k}_{s}. It is essential to estimate {k}_{s} for accurate fluid dynamic problem modeling. While numerous roughness correlation formulas have been proposed to predict {k}_{s} in the fully rough regime, most of them are restricted to certain roughness types, with various geometric parameters considered in each case, leading to ongoing disagreements regarding its parameterization and lack of universality. In this study, a Particle Swarm Optimized Backpropagation (PSO-BP) method is proposed to predict {k}_{s} based on the selected surface parameters from previous DNS, LES, and experimental results for flow behavior over various surface roughness. The PSO-BP model’s ability to predict {k}_{s} in the fully rough region is evaluated and compared with both the existing roughness correction formulas as well as the traditional BP model. An optimized polynomial function is also proposed to serve as a ‘white box’ for predicting {k}_{s}. It turns out that the PSO-BP method has better performance in the evaluation metrics compared to other methods, yielding a Mean Absolute Error (MAE) of 0.0390, a Mean Squared Error (MSE) of 0.0026 and a Mean Absolute Percentage Error (MAPE) of 28.12%. This novel approach for estimating {k}_{s} has practical applicability and holds promise for improving the precision and efficiency of calculations related to equivalent sand-grain roughness, and thus provides more accurate and effective solutions for CFD and other engineering applications.

The hunt for the Kármán ‘constant’ revisited

The log law of the wall, joining the inner, near-wall mean velocity profile (MVP) in wall-bounded turbulent flows to the outer region, has been a permanent fixture of turbulence research for over hundred years, but there is still no general agreement on the value of the prefactor, the inverse of the Kármán ‘constant’ $\kappa$ , or on its universality. The choice diagnostic tool to locate logarithmic parts of the MVP is to look for regions where the indicator function $\varXi$ (equal to the wall-normal coordinate $y^+$ times the mean velocity derivative $\mathrm {d} U^+/\mathrm {d} y^+$ ) is constant. In pressure-driven flows, however, such as channel and pipe flows, $\varXi$ is significantly affected by a term proportional to the wall-normal coordinate, of order $O({Re}_{\tau }^{-1})$ in the inner expansion, but moving up across the overlap to the leading $O(1)$ in the outer expansion. Here we show that, due to this linear overlap term, ${Re}_{\tau }$ values well beyond $10^5$ are required to produce one decade of near constant $\varXi$ in channels and pipes. The problem is resolved by considering the common part of the inner asymptotic expansion carried to $O({Re}_{\tau }^{-1})$ , and the leading order of the outer expansion. This common part contains a superposition of the log law and a linear term $S_0 \,y^+{Re}_{\tau }^{-1}$ , and corresponds to the linear part of $\varXi$ , which, in channel and pipe, is concealed up to $y^+ \approx 500\unicode{x2013}1000$ by terms of the inner expansion. A new and robust method is devised to simultaneously determine $\kappa$ and $S_0$ in pressure-driven flows at currently accessible ${Re}_{\tau }$ values, yielding $\kappa$ values which are consistent with the $\kappa$ values deduced from the Reynolds number dependence of centreline velocities. A comparison with the zero-pressure-gradient turbulent boundary layer, further clarifies the issues and improves our understanding.

Wall-shear-stress measurements using volumetric µPTV

Accurately determining the wall-shear-stress, tau _textrm{w}, experimentally is challenging due to small spatial scales and large velocity gradients present in the near-wall region of turbulent flows. To avoid these resolution requirements, several indirect iterative fitting methods, most notably the Clauser chart method, exist for determining overline{tau }_{w} by fitting the mean velocity profile further away from the near-wall region in the log-law layer. These methods often require proper selection of fitting constants, assumptions of a canonical flow state, and other empirical-based generalizations. To reduce the amount of ambiguity, determining the near-wall velocity gradient by assuming a linear relationship between the mean streamwise velocity and wall normal distance in the viscous sublayer can be used. However, this requires an accurate unbiased measurement of the near-wall velocity profile in the region below five viscous spatial units, which can be less than 50 µm for high Reynolds number flows. Therefore, in this study a method for a volumetric defocusing microparticle tracking velocimetry method is presented that is capable of resolving the flow in the viscous sublayer of a turbulent boundary layer up to U_{textrm{e}}=44.7,m/s (Re_{theta }=27250). This method allows for the measurement of the near-wall flow through a single optical access for illumination and imaging and serves as an excellent complement of larger scale measurements that require near-wall information. The overline{tau }_textrm{w} values determined from the defocusing approach were found to be in good agreement values obtained from a simultaneous parallax PTV measurement. Furthermore, analysis of the diagnostic plot and cumulative distribution of measured fluctuations in the near-wall region, showed that both methods are capable of accurately determining mean velocity and fluctuation profiles in the self-similar viscous sublayer region.

Robust Eddy Viscosity Turbulence Modeling with Elliptic Relaxation and Compound Wall Treatment

This paper presents a holistic Reynolds-averaged Navier–Stokes (RANS) turbulence modeling framework for the computational fluid dynamics (CFD) simulations of complex wall-bounded turbulent flows. Based on the elliptic relaxation idea, the deployed eddy viscosity turbulence model reconstructs the near-wall stress anisotropy and nonviscous effects. The appropriate selection of the turbulent quantities that are being solved for, together with the zero value wall boundary condition for the related turbulent quantities, renders the model less sensitive to the near-wall grid nonuniformities and resolution. The unified near-wall velocity profile, obtained based on the boundary layer theory, is used to devise the compound near-wall treatment that ensures the robustness of the numerical simulation. The proposed turbulence modeling framework is implemented into the general-purpose open-source CFD code and validated against the generic test cases with satisfactory agreement.

Mean Velocity Scaling of High-Speed Turbulent Flows Under Nonadiabatic Wall Conditions

The problem of scaling the near-wall mean velocity profiles of turbulent flows and collapsing them with the law of the wall has been traditionally studied using the conservation of momentum, by employing various levels of assumptions. In the van Driest transformation (“Turbulent Boundary Layer in Compressible Fluids,” Journal of the Aeronautical Sciences, Vol. 18, No. 3, 1951, pp. 145–216), the viscous stress was neglected, whereas in the Trettel and Larsson transformation (“Mean Velocity Scaling for Compressible Wall Turbulence with Heat Transfer,” Physics of Fluids, Vol. 28, No. 2, 2016, Paper 026102), the Reynolds stress was assumed to cancel out. Recent work by Griffin, Fu, and Moin (“Velocity Transformation for Compressible Wall-Bounded Turbulent Flows with and Without Heat Transfer,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 118, No. 34, 2021, Paper e2111144118) used a quasi-equilibrium assumption between turbulence production and dissipation in the log layer and demonstrated success for a wide variety of canonical flows. However, the extent of quasi-equilibrium is not verified, particularly for noncanonical flows. In this work, an alternate transformation is developed using the semilocal gradient of the van Driest transformed velocity, which is principally tied to the dynamics of vorticity transport in the boundary layer. In addition, a modified stress balance is utilized to directly account for and scale the buffer region. The transformation was benchmarked using a similar dataset as in Griffin et al., and comparable performance was obtained. Moreover, at supercritical pressures, the present transformation is found to produce significantly better collapse than existing works.

Experimental investigation of boundary layer flow near the cylinder wall in a direct-injection spark-ignition engine using particle image velocimetry

In a direct-injection spark-ignition (DISI) optical engine with a constant motored speed of 800 rpm, particle image velocimetry measurements on the in-cylinder bulk flow and the boundary layer flow near the cylinder wall were performed during the intake and compression strokes. Two different tumble flow levels were explored using a flap inserted in the intake port. The near-wall flow pattern under the effect of the bulk flow is more stable when the flap is fully closed with a higher intensity tumble. The serious optical distortion in the near-cylinder wall region under high magnification was addressed, and a high resolution of 76 μm × 76 μm within a field of view of 5 mm × 5 mm was achieved in six different measurement regions. Using inner scaling parameters (friction velocity uτ and viscous length-scale δυ), the dimensionless ensemble-averaged wall-tangential velocity profiles exhibit good consistency with the law-of-the-wall in the viscous sublayer but considerably departure in the logarithmic region. According to the low friction Reynolds number, the absence of the logarithmic layer was mainly attributed to the complete overlap between the viscosity-affected inner region and the outer region of the boundary layer. The near-wall velocity profiles were also normalized by three different outer scaling parameters. The collapse is significantly improved for y/ δ >0.07 when the velocity profiles are normalized by δ/ δ*(1− u/ u∞). The fluctuation RMS velocity in the wall-normal direction is greater, particularly for the high tumble flow level condition.

Accurate PIV measurement on slip boundary using single-pixel algorithm

To accurately measure the near-wall flow by particle image velocimetry (PIV) is a big challenge, especially for the slip boundary condition. Apart from high-precision measurements, an appropriate PIV algorithm is important to resolve the near-wall velocity profile. In our study, the single-pixel algorithm is employed to calculate the near-wall flow, which is demonstrated to be capable of accurately resolving the flow velocity near the slip boundary condition. Based on synthetic particle images, the advantages of the single-pixel algorithm are manifested in comparison with the conventional window-correlation algorithm. In particular, the single-pixel algorithm has higher spatial resolution and accuracy, and lower systematic error and random error for the case of the slip boundary condition. Furthermore, for experimental verification, micro-PIV measurements are conducted over a liquid–gas interface, and the single-pixel algorithm is successfully applied to the calculation of near-wall velocity under the slip boundary condition, especially negative slip velocity. The current work demonstrates the advantages of the single-pixel algorithm in analyzing complex flows under the slip boundary condition, such as in drag reduction, wall skin-friction evaluation, and near-wall vortex structure measurement.

Numerical investigations on transitional flows around forward and reversed hydrofoils

The laminar–turbulence transition phenomenon widely exists on the surface of many energy equipment, which is deserved to be studied because of the complex mechanics and some induced undesirable consequences. The goal of present work is to investigate the transitional flows around the forward and reversed hydrofoils at different incidences using the SST γ-Re˜θt transition model, with special emphasis on the dynamics of the transition. The effect of inflow turbulence condition is considered initially. Then, the difference between the original SST k-ω model and SST γ-Re˜θt transition model is analyzed, in terms of the near-wall velocity profiles and flow morphology. Afterwards, the change of the transition with the incidence for the reversed hydrofoil is clarified in detail. The primary results show that the flow separation near the sharp leading edge where the reverse dynamic vortex (RDV) appears makes the contribution to the transition. The size of RDV is much larger than laminar separation bubble (LSB) over the forward hydrofoil and it forms near the leading edge earlier. Moreover, the transition locations are mapped both for the forward and reversed hydrofoils. Finally, the effect of Reynolds number on the transition process for the reversed hydrofoil is presented. It is believed that this work can deep the understandings of the transition, especially for the reversed hydrofoils.

Characterization of the Velocity Field External to a Tube Bundle Using Spatial Filter Velocimetry Based on Variable Meshing Scheme

This paper presents results for the experimental vector velocity field obtained through spatial filter velocimetry in a triangular tube bundle with tubes of 20 mm O.D. and a transverse pitch of 25.2 mm. The experiments were conducted for single-phase water flows with Reynolds numbers ranging from 447 to 1842, and for air–water two-phase bubbly flow with liquid Reynolds number of 909 and gas Reynolds number of 42. The vector velocity field for each experimental condition was obtained using a variable meshing scheme to improve the spatial resolution in regions of low velocities. The results indicate that the variable meshing strategy is promising to increase the spatial resolution of the spatial filter velocimetry technique and to release from the constraint imposed by the uniform window sizes, especially for experimental data obtained with relatively low frame rates. Details of the flow structure of external flow across tube bundles, such as the near-wall velocity profiles, wake region and the effects of the gas buoyancy on vertical mean velocities are presented in this paper with high temporal and spatial resolution, which can be useful for validation of numerical models.

Effect of Particle Size and Surface Properties on the Sandbed Erosion with Water Flow in a Horizontal Pipe

Summary An experimental study was conducted to investigate the transport of sand particles over the sand bed deposited in a horizontal conduit by using turbulent flow of water. The main objectives were to determine the near-wall turbulence characteristics at the onset of bed erosion (i.e., near-wall velocity profile, Reynolds shear stresses, and axial-turbulent intensity); to determine critical velocity required for particle removal from the bed deposits; and more specifically, to determine how the sand-particle size and surface characteristics would influence the critical velocity required for the onset of bed erosion and the near-wall turbulence characteristics. A large-scale horizontal flow loop equipped with a nonintrusive laser-based particle-image velocimetry (PIV) system has been used for the experiments. The effect of sand-particle surface characteristics (i.e., wettability) on the critical velocity and the near-wall turbulence characteristics were investigated by using treated and untreated industrial sands of four different mesh sizes (i.e., 20/40, 30/50, 40/70, 100). The PIV technique was used to determine instantaneous local velocity distribution near the stationary sandbed fluid interface under subcritical and critical flow conditions. The near-wall velocity distribution measured directly at the sand bed/fluid interface together with the measured frictional pressure-loss values were then used for the evaluation of the Reynolds shear stresses and axial turbulent intensities acting at the bed/fluid interface. The results indicated that critical velocity for the onset of particle removal from sand beds increased with the increasing particle size. When sands with special surface treatment were used, it was observed that the critical velocity required for the onset of the bed erosion was significantly lower than that of required for the untreated sands. The degree of reduction in critical velocity varied between 14 and 40% depending on the particle size. In this study, by conducting experiments under controlled conditions, we provided much-needed fundamental data that can be used for the development of improved solid-transport design criteria and suitable mitigation technologies. In particular, we have shown the proof of concept that the surface-treated sand particles might have great potential for improving the transport efficiency of proppants used for hydraulic-fracturing operations.

Streamwise and spanwise slip over a superhydrophobic surface

The near-wall turbulent flow over a superhydrophobic surface (SHS) with random texture was studied using three-dimensional Lagrangian particle tracking velocimetry (3D-PTV). The channel was operated at a constant mass flow rate over the SHS and a smooth surface at a Reynolds number of 7000 based on the bulk velocity of $0.93~\text{m}~\text{s}^{-1}$ and the full channel height. The friction Reynolds number was 217, based on the friction velocity and half channel height. The 3D-PTV processing was based on the shake-the-box algorithm applied to images of fluorescent tracers recorded using four high-speed cameras. The SHS was obtained by spray coating, resulting in a root-mean-square roughness of $0.29\unicode[STIX]{x1D706}$ and an average texture width of $5.0\unicode[STIX]{x1D706}$, where $\unicode[STIX]{x1D706}=17~\unicode[STIX]{x03BC}\text{m}$ is the inner flow scale over the SHS. The 3D-PTV measurements confirmed an isotropic slip with a streamwise slip length of $5.9\unicode[STIX]{x1D706}$ and a spanwise slip length of $5.9\unicode[STIX]{x1D706}$. As a result, both the near-wall mean streamwise and spanwise velocity profiles over the SHS were higher than the smooth surface. The streamwise and spanwise slip velocities over the SHS were $0.27~\text{m}~\text{s}^{-1}$ and $0.018~\text{m}~\text{s}^{-1}$, respectively. The near-wall Reynolds stresses over the SHS were larger and shifted towards the wall when normalized by the corresponding inner scaling, despite the smaller friction Reynolds number of 180 over the SHS. The near-wall measurement of streamwise velocity showed that the shear-free pattern consists of streamwise-elongated regions with a length of $800\unicode[STIX]{x1D706}$ and a spanwise width of $300\unicode[STIX]{x1D706}$. The plastron dimensions correspond to the mean distance of the largest roughness peaks $(20~\unicode[STIX]{x03BC}\text{m})$ obtained from profilometry of the SHS. The drag reduction over the SHS was 30 %–38 % as estimated from pressure measurement and the flow field using the 3D-PTV.

Study of Conjugate Heat Transfer from Heated Plate by Turbulent Offset Jet in Presence of Freestream Motion using Low-Reynolds Number Modeling

The present study deals with conjugate heat transfer from a heated flat plate by a turbulent offset jet in presence of freestream motion. The turbulent convection in fluid and conduction in solid is solved in a coupled manner by simultaneously satisfying the equality of temperature and heat flux at the solid-fluid interface. The computations have been carried out using low-Reynolds number (LRN) k −ω SST model in the fluid region. The capability of LRN modeling have enabled to solve the entire boundary layer including the thin viscous sublayer due to which Moffatt vortices (secondary recirculation regions) have been captured near the corner of the wall where the turbulence Reynolds number is low. The bottom surface of solid plate is maintained at a constant temperature higher than the jet inlet temperature whereas the jet inlet temperature is same as that of the ambient. The present investigation reports the effects of offset ratio of jet (OR), Reynolds number of flow (Re), solid to fluid thermal conductivity ratio (K), solid slab thickness (S) and freestream velocity (U∞) on conjugate heat transfer arises due to solid and fluid interaction. The offset ratio is varied in the range OR = 3 − 11, Reynolds number in the range Re = 10000 − 25000, solid to fluid thermal conductivity ratio in the range K = 1 − 2000, solid slab thickness in the range S = 1−20 and freestream velocity in the range U∞ = 0.1 − 0.25. The effects of various parameters on the near-wall velocity profile, solid-fluid interface temperature, local Nusselt number variation along the plate, heat flux variation along the plate, etc. have been discussed in detail.

A high-order compact finite-difference lattice Boltzmann method for simulation of natural convection

In the present work, compact finite-difference lattice Boltzmann method (CFDLBM) is extended to simulate natural convection in square and concentric-circular-annulus cavities, in two-dimensions, for Rayleigh numbers in the range of 103 to 109. Owing to the finite-difference discretization of the method, non-uniform and curvilinear grids with a finer near-wall grid resolution could be used in the present simulations. The use of high-order methods further enabled the simulations to be run on relatively coarse meshes when compared with simulations that used classical lattice Boltzmann method. The simulation results are analyzed with the help of streamlines, isothermal contours and temperature, velocity, Nusselt number variations along the walls. The average Nusselt number on the hot wall shows less than 1% error for natural convection in the square cavity study. The circular-annular-cavity is found to have laminar flow for Rayleigh numbers in the range of 103–106 and it becomes unsteady and chaotic from 107 onwards. The instantaneous and time-average contours of isotherms and streamlines were used to study the unsteady behavior of the flow. A universal near-wall velocity profile is found to exist in all the unsteady natural convection studies in the annular cavity case. The location of the maximum velocities occurring in the annulus is analyzed for all the Rayleigh numbers. The spectra of velocity magnitude fluctuations and temperature fluctuations indicate that the unsteadiness in the cavity is dominant around 0.001 Hz. Natural convection in sine-walled and concentric star shaped cavities are simulated to demonstrate the ability of the present solver to handle complex geometries. The present results indicate that CFDLBM can be successfully used to simulate steady and unsteady chaotic natural convection flows in both rectangular and curvilinear geometries accurately.

Numerical assessment of wall modelling approaches in scale-resolving in-cylinder simulations

Wall modelling in internal combustion engines (ICEs) is a challenging task due to highly specific boundary layers and a dynamically changing flow environment. Recent experimental (Jainski et al., 2013, Renaud et al., 2018) and direct numerical simulation (DNS, Schmitt et al., 2015a) studies demonstrate that scaled near-wall velocity and temperature profiles in ICEs deviate considerably from the law of the wall. Utilising the DNS data, the present paper focusses on benchmarking a scale-resolving approach with a 1-D non-equilibrium wall model (HLR-WT, Keskinen et al., 2017) in ICE-like flows. Specific emphasis is put on the compression stroke using different grids and two additional wall-modelled large eddy simulation (WMLES) reference approaches. The standard wall law based WMLES-1 produces highly grid-dependent underprediction of wall fluxes, to which WMLES-2 (Plensgaard and Rutland, 2013) and HLR-WT, employing engine-targeted wall treatments, yield considerable improvement. Differences between the improved methods are noted in detailed metrics. Throughout the compression stroke, HLR-WT provides a good match to the DNS in scaled mean boundary layer profiles for both velocity and temperature. With relevance to local heat flux distribution, the characteristic impingement-ejection process observed in the DNS is qualitatively replicated with WMLES-2 and HLR-WT. The non-equilibrium formulation of the latter allows for slight improvements in terms of local heat transfer fluctuation predictions. In contrast, coarse near-wall grids appear to be detrimental for such predictions with all approaches. The study provides evidence on the potential of the HLR-WT and WMLES-2 approaches in ICE near-wall flow prediction, advocating further investigations in more realistic engine configurations.

Evaluation of RANS and LES turbulence models for simulating a steady 2-D plane wall jet

PurposeThis paper examines various turbulence models for numerical simulation of a steady, two-dimensional (2-D) plane wall jet without co-flow using the commercial CFD software (ANSYS FLUENT 14.5). The purpose of this paper is to decide the most suitable and most economical method for steady, 2-D plane wall jet simulation.Design/methodology/approachSeven Reynolds-averaged Navier–Stokes (RANS) turbulence models were evaluated with respect to typical jet scaling parameters such as the jet half-height and the decay of maximum jet velocity, as well as coefficients from the law of the wall and for skin friction. Then, a plane wall jet generating from a rectangular slot of 1:6 aspect ratio located adjacent to the wall was investigated in a three-dimensional (3-D) model using large eddy simulation (LES) and the Stress-omega Reynolds stress model (SWRSM), with the results compared to experimental measurements.FindingsThe comparisons of these simulated flow characteristics indicated that the SWRSM was the best of the seven RANS models for simulating the turbulent wall jet. When scaled with outer variables, LES and SWRSM gave generally indistinguishable mean velocity profiles. However, SWRSM performed better for near-wall mean velocity profiles when scaled with inner variables. In general, the results show that LES performed reasonably well when predicting the Reynolds stresses.Originality/valueThe main contribution of this article is in determining the capabilities of different RANS turbulence closures and LES for the prediction of the 2-D steady wall jet flow to identify the best modelling approach.

Near-Wall Modification of Spalart–Allmaras Turbulence Model for Immersed Boundary Method

An improved immersed boundary method for turbulent flow simulation on Cartesian grids is proposed. To determine the appropriate boundary conditions, near-wall approximation of the mean-flow equation and the Spalart–Allmaras turbulence model is investigated. A modified near-wall velocity profile that is linear with respect to the distance from the wall and that can be resolved by a numerical scheme with second-order spatial accuracy is derived. As with the velocity profile, the eddy-viscosity profile is modified to maintain the balance of the shear stress. To obtain the modified profile as a numerical solution, the boundary condition for the tangential velocity and the damping function in the Spalart–Allmaras turbulence model is modified. Verification studies defined in the NASA Turbulence Modeling Resource are conducted. In all cases, the results confirm that the modified immersed boundary method produces a more accurate prediction of skin friction than the original immersed boundary method and show a consistent trend of grid convergence.

Turbulent Taylor–Couette flow with stationary inner cylinder

A series of direct numerical simulations were performed of Taylor–Couette (TC) flow, the flow between two coaxial cylinders, with the outer cylinder rotating and the inner one fixed. Three cases were considered, where the Reynolds number of the outer cylinder was $Re_{o}=5.5\times 10^{4}$, $Re_{o}=1.1\times 10^{5}$ and $Re_{o}=2.2\times 10^{5}$. The ratio of radii ${\it\eta}=r_{i}/r_{o}$ was fixed to ${\it\eta}=0.909$ to mitigate the effects of curvature. Axially periodic boundary conditions were used, with the aspect ratio of vertical periodicity ${\it\Gamma}$ fixed to ${\it\Gamma}=2.09$. Being linearly stable, TC flow with outer cylinder rotation is known to have very different behaviour than TC flow with pure inner cylinder rotation. Here, we find that the flow nonetheless becomes turbulent, but the torque required to drive the cylinders and level of velocity fluctuations was found to be smaller than those for pure inner cylinder rotation at comparable Reynolds numbers. The mean angular momentum profiles showed a large gradient in the bulk, instead of the constant angular momentum profiles of pure inner cylinder rotation. The near-wall mean and fluctuation velocity profiles were found to coincide only very close to the wall, showing large deviations from both pure inner cylinder rotation profiles and the classic von Karman law of the wall elsewhere. Finally, transport of angular velocity was found to occur mainly through intermittent bursts, and not through wall-attached large-scale structures as is the case for pure inner cylinder rotation.

Near-wall velocity profile measurement for nanofluids

We perform near-wall velocity measurements of a SiO2–water nanofluid inside a microchannel. Nanoparticle image velocimetry measurements at three visible depths within 500 nm of the wall are conducted. We evaluate the optical properties of the nanofluid and their effect on the measurement technique. The results indicate that the small effect of the nanoparticles on the optical properties of the suspension have a negligible effect on the measurement technique. Our measurements show an increase in nanofluid velocity gradients near the walls, with no measurable slip, relative to the equivalent basefluid flow. We conjecture that particle migration induced by shear may have caused this increase. The effect of this increase in the measured near wall velocity gradient has implications on the viscosity measurement for these fluids.