Logarithmic Region Research Articles

Linear-model-based estimation in wall turbulence: improved stochastic forcing and eddy viscosity terms

We use Navier–Stokes-based linear models for wall-bounded turbulent flows to estimate large-scale fluctuations at different wall-normal locations from their measurements at a single wall-normal location. In these models, we replace the nonlinear term by a combination of a stochastic forcing term and an eddy dissipation term. The stochastic forcing term plays a role in energy production by the large scales, and the eddy dissipation term plays a role in energy dissipation by the small scales. Based on the results in channel flow, we find that the models can estimate large-scale fluctuations with reasonable accuracy only when the stochastic forcing and eddy dissipation terms vary with wall distance and with the length scale of the fluctuations to be estimated. The dependence on the wall distance ensures that energy production and energy dissipation are not concentrated close to the wall but are evenly distributed across the near-wall and logarithmic regions. The dependence on the length scale of the fluctuations ensures that lower wavelength fluctuations are not excessively damped by the eddy dissipation term and hence that the dominant scales shift towards lower wavelengths towards the wall. This highlights that, on the one hand, energy extraction in wall turbulence is predominantly linear and thus physics-based linear models give reasonably accurate results. On the other hand, the absence of linearly unstable modes in wall turbulence means that the nonlinear term still plays an essential role in energy extraction and thus the modelled terms should include the observed wall distance and length scale dependencies of the nonlinear term.

Velocity transformation for compressible wall-bounded turbulent flows with and without heat transfer.

In this work, a transformation, which maps the mean velocity profiles of compressible wall-bounded turbulent flows to the incompressible law of the wall, is proposed. Unlike existing approaches, the proposed transformation successfully collapses, without specific tuning, numerical simulation data from fully developed channel and pipe flows, and boundary layers with or without heat transfer. In all these cases, the transformation is successful across the entire inner layer of the boundary layer (including the viscous sublayer, buffer layer, and logarithmic layer), recovers the asymptotically exact near-wall behavior in the viscous sublayer, and is consistent with the near balance of turbulence production and dissipation in the logarithmic region of the boundary layer. The performance of the transformation is verified for compressible wall-bounded flows with edge Mach numbers ranging from 0 to 15 and friction Reynolds numbers ranging from 200 to 2,000. Based on physical arguments, we show that such a general transformation exists for compressible wall-bounded turbulence regardless of the wall thermal condition.

Adjoint Complement to the Universal Momentum Law of the Wall

The paper is devoted to an adjoint complement to the universal Law of the Wall (LoW) for fluid dynamic momentum boundary layers. The latter typically follows from a strongly simplified, unidirectional shear flow under a constant stress assumption. We first derive the adjoint companion of the simplified momentum equation, while distinguishing between two strategies. Using mixing-length arguments, we demonstrate that the frozen turbulence strategy and a LoW-consistent (differentiated) approach provide virtually the same adjoint momentum equations, that differ only in a single scalar coefficient controlling the inclination in the logarithmic region. Moreover, it is seen that an adjoint LoW can be derived which resembles its primal counterpart in many aspects. The strategy is also compatible with wall-function assumptions for prominent RANS-type two-equation turbulence models, which ground on the mixing-length hypothesis. As a direct consequence of the frequently employed assumption that all primal flow properties algebraically scale with the friction velocity, it is demonstrated that a simple algebraic expression provides a consistent closure of the adjoint momentum equation in the logarithmic layer. This algebraic adjoint closure might also serve as an approximation for more general adjoint flow optimization studies using standard one- or two-equation Boussinesq-viscosity models for the primal flow. Results obtained from the suggested algebraic closure are verified against the primal/adjoint LoW formulations for both, low- and high-Re settings. Applications included in this paper refer to two- and three-dimensional shape optimizations of internal and external engineering flows. Related results indicate that the proposed adjoint algebraic turbulence closure accelerates the optimization process and provides improved optima at no computational surplus in comparison to the frozen turbulence approach.

Investigation of turbulent boundary layer flows with adverse pressure gradient by means of 3D Lagrangian particle tracking with Shake-The-Box

A large-scale 3D Lagrangian particle tracking (LPT) investigation of a turbulent boundary layer (TBL) flow developing across different pressure gradient regions is presented in this study. Three high-speed multi-camera imaging systems, LED illumination and helium-filled soap bubbles (HFSB) tracers have been adopted to produce time-resolved sequences of particle images over a large volume encompassing approximately 3 m in the streamwise direction, 0:8 m in the spanwise direction and 0:25 m in the wall-normal direction. Individual tracers have been reconstructed and tracked within the imaged volume by means of the Shake-The-Box algorithm (STB, Schanz et al. (2016)); the FlowFit data assimilation algorithm (Gesemann et al. (2016)) has been used to evaluate the spatial velocity gradients and to interpolate the scattered LPT results onto a regular grid. Thanks to the large size of the investigated volume and to the time-resolved nature of the recorded images, the entire spatial extent of the large-scale coherent motions within the logarithmic region of the TBL (i.e. superstructures) could be captured and their dynamics investigated during their development over several boundary layer thickness in the streamwise direction, from the zero pressure gradient region (ZPG) to the adverse pressure gradient region (APG). Two free-stream velocities were investigated, namely 7 and 14m=s, corresponding to Ret ~ 3,000 and 5,000 respectively. The results confirm the location and scale of the elongated high- and low-momentum structures in the logarithmic region, as well as their meandering in the spanwise direction. Two-point correlation statistics show that the width and spacing of the superstructures are not affected by the transition from the ZPG to the APG region. The analysis of the instantaneous flow realizations from both a Lagrangian and Eulerian perspective indicates the presence of significant fluid particle elements exchange across the interfaces of the large-scale structures.

Direct numerical simulation of a turbulent plane Couette-Poiseuille flow with zero-mean shear

Direct numerical simulations of a turbulent Couette-Poiseuille flow with zero-mean-shear at the moving wall (SL-flow) is performed to examine flow features compared to those for a turbulent pure Poiseuille flow (P-flow). Profiles of the streamwise mean velocity, indicator function and ratio of production to dissipation show that the logarithmic region is significantly elongated for the SL-flow compared to that for the P-flow at a similar Reynolds number. In addition, the magnitudes of the Reynolds stresses are found to be larger in both inner and outer layers for the SL-flow than those for the P-flow. The spanwise spectra of the production term in the turbulent kinetic energy equation are examined to provide a structural basis for explaining the statistical behaviors. In addition, because the growth of the energy-containing motions extends to the outer layer further for the SL-flow due to the presence of a positive mean shear throughout the entire wall layer, the self-similar behavior of the energy balance between the production and transport terms with respect to the self-similar wavenumber is found far from the wall. We also find the increase in the number of uniform momentum zones in the SL-flow, revealing the hierarchical distribution of the energy-containing eddies which are composed of multiple uniform momentum zones. These coherent motions lead to the elongation of the logarithmic region for the SL-flow. Finally, investigation of the turbulent energy transfer process in a spectral domain for the SL-flow demonstrates importance of outer layer very-long structures, and these structures attribute to the energy transport process in an entire flow field.

Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region

We study the global, i.e., radially averaged, high Reynolds number (asymptotic) scaling of streamwise turbulence intensity squared defined as I2=u2¯/U2, where u and U are the fluctuating and mean velocities, respectively (overbar is time averaging). The investigation is based on the mathematical abstraction that the logarithmic region in wall turbulence extends across the entire inner and outer layers. Results are matched to spatially integrated Princeton Superpipe measurements [Hultmark et al., “Logarithmic scaling of turbulence in smooth- and rough-wall pipe flow,” J. Fluid Mech. 728, 376–395 (2013)]. Scaling expressions are derived both for log- and power-law functions of radius. A transition to asymptotic scaling is found at a friction Reynolds number Reτ∼11 000.

Modeling of combined effects of surface roughness and blowing for Reynolds-averaged Navier–Stokes turbulence models

A new modeling strategy adapted to Reynolds-averaged Navier–Stokes turbulence models is proposed to predict combined effects of roughness and blowing boundary conditions. First, an analysis of experimental data is presented, leading to a specific description of the velocity profile in the logarithmic region of transpired turbulent boundary layers over rough walls. This analysis points out the deficiencies of existing roughness corrections to predict the effect of blowing in the presence of surface roughness. Indeed, these corrections tend to underestimate skin friction coefficients and Stanton numbers with the addition of blowing. The failure of existing models derives from an inaccurate estimation of the velocity shift of the logarithmic law given by roughness corrections. Concretely, roughness corrections underestimate the apparent velocity shift of the logarithmic law with blowing. To recover the expected law of the wall, an additional contribution on the velocity shift, characterizing blowing/roughness interactions, is integrated to standard roughness corrections. To that end, a modification of the equivalent sand grain height, adapted to k−ω based turbulence models, is proposed to take blowing effects into account. Furthermore, an extension of Aupoix's thermal correction [B. Aupoix, Int. J. Heat Fluid Flow 56, 160–171 (2015)] to blowing is presented to predict combined thermal effects of roughness and blowing. The assessment of the proposed corrections is performed using k−ω shear stress transport model on a large set of experimental data and proves the relevance of the strategy for incompressible and compressible turbulent boundary layers.

Experimental study on turbulent asymptotic suction boundary layers

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Heat transfer and skin-friction in a turbulent boundary layer under a non-equilibrium longitudinal adverse pressure gradient

The experimental data on the effect of weak and moderate non-equilibrium adverse pressure gradients (APG) on the parameters of dynamic and thermal boundary layers are presented. The Reynolds number based on the momentum thickness at the beginning of the APG region was Re** = 5500. The APG region was a slot channel with upper wall expansion angles from 0 to 14°. The profiles of the mean and fluctuation velocity components were measured using a single-component hot-wire anemometer. The friction coefficients were determined using two methods, namely, the indirect Clauser method and the direct method of weighting the lower wall region on a single-component strain-gage balance. The heat transfer coefficients were determined by a transient method using an IR camera. It is noticed that in the pressure gradient range realized the universal logarithmic region in the boundary layer profile is conserved. The values of the relative (divided by the parameters in zero gradient flow at the same value of Re**) friction and heat transfer coefficients, together with the Reynolds analogy factor, are determined as functions of the longitudinal pressure gradient. The values of the relative friction coefficient reduced to cf/cf0 = 0.7 and those of the heat transfer to St/St0 = 0.9. A maximum value of the Reynolds analogy factor (St/St0)/(cf/cf0) = 1.16 was reached for the pressure gradient parameter β = 2.9.

Interactions between scales in wall turbulence: phase relationships, amplitude modulation and the importance of critical layers

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Intermediate scaling and logarithmic invariance in turbulent pipe flow

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Temporal characteristics of the probability density function of velocity in wall-bounded turbulent flows

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Non--type behaviour of roughness when in-plane wavelength approaches the boundary layer thickness

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The late start of the mean velocity overlap log law at – a generic feature of turbulent wall layers in ducts

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LOGARITHMIC AMPLIFIERS FOR SOFTWARE HARDWARE MAGNETIC TRACKING SYSTEMS

The work deals with the problem of signal conversion in magnetic tracking devices. Magnetic tracking technology is based on computing the spatial position of an object being tracked upon measuring reference magnetic fields in low-frequency electromagnetic radiation spectrum. Magnetic tracking devices are key components of navigation sensors for virtual and augmented reality. It has been shown that the main problem one faces when developing sensory devices for magnetic tracking is the fact that signals should be measured in a wide measurement range. We have analyzed possible ways to solve the stated problem by digital and combined methods. The latter have proven to be more efficient. They consist in signal amplification due to analog compression, which is performed by logarithmic amplifiers whose negative feedback circuits contain components with non-linear volt-ampere characteristics (typically, diodes or bipo- lar transistors are used). It has been shown that the parameters of logarithmic signal compression can be controlled by modified circuits with auxiliary resistance dividers. The resistance dividers scale the logarithmic volt-ampere characteristics of emitter p-n junctions of bipolar n-p-n and p-n-p transistors. A substantial advantage of circuits with resistance dividers is that they provide the possibility to expand the range of the output voltage of logarithmic amplifiers and optimize the transition between the linear and logarithmic amplification regions. The work presents the results of simulation and experimental investigations into a logarithmic amplifier for a magnetic tracking system. Simulation was carried out using SPICE (Simulation Program with Integrated Circuit Emphasis) models. We applied an integrated approach,which provides collections of transient characteristics of logarithmic amplifiers at different sets of the parameters of resistance dividers. The simulation results have been verified using our own software- firmware magnetic tracking tools – Magnetic Tracking System Integrated Development Environment. The signal converter was built upon a programmable system-on-chip PSoC 5LP by Cypress Semiconductor.

Hot-wire experimental investigation on turbulent Prandtl number in a rotating non-isothermal turbulent boundary layer

This experiment used a parallel array of hot wire probes to simultaneously measure the temperature and velocity fields in the non-isothermal turbulent boundary layer of a rotating straight channel. The Reynolds numbers are 15,000 and 25,000, respectively. The rotation numbers are 0, 0.07, 0.14, 0.21 and 0.28, respectively. The purpose of this study is to calculate the turbulent Prandtl number in a rotating non-isothermal turbulent boundary layer. Due to the difficulty in measuring local turbulent Prandtl numbers, this study focuses on the average turbulent Prandtl numbers in the logarithmic region instead. Under static conditions, this value is taken as 0.9 normally. This research finds that rotation conditions can affect the turbulent Prandtl number by affecting the properties of velocity and temperature boundary layers. The change range of the turbulent Prandtl number is roughly 0.6–1.1. The influence of the leading side is greater than that of the trailing side, especially at high rotation numbers. This can provide validation and guidance for numerical simulation. Other information within the turbulent boundary layer is also discussed. It is hoped that this study would enhance our understanding of the mechanism of turbulent flow in the turbulent layer at rotating conditions.

Statistical behaviour of self-similar structures in canonical wall turbulence

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Spectral-scaling-based extension to the attached eddy model of wall turbulence

Two-dimensional (2-D) spectra of the streamwise velocity component, measured at friction Reynolds numbers ranging from 2400 to 26000, are used to refine a model for the logarithmic region of turbulent boundary layers. Here, we focus on the attached eddy model (AEM). The conventional AEM assumes the boundary layer to be populated with hierarchies of self-similar wall-attached ($Type\,A$) eddies alone. While $Type\,A$ eddies represent the dominant energetic large-scale motions at high Reynolds numbers, the scales that are not represented by such eddies are observed to carry a significant proportion of the total kinetic energy. Therefore, in the present study, we propose an extended AEM that incorporates two additional representative eddies. These eddies, named $Type\,C_A$ and $Type\,SS$, represent the self-similar but wall-incoherent low-Reynolds number features, and the non-self-similar wall-coherent superstructures, respectively. The extended AEM is shown to better predict a greater range of energetic length scales and capture the low- and high-Reynolds number scaling trends in the 2-D spectra of all three velocity components. A discussion on spectral self-similarity and the associated $k^{-1}$ scaling law is also presented.

On the structure of streamwise wall-shear stress fluctuations in turbulent channel flows

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

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