Friction Reynolds Number Research Articles

An analysis of drag reduction using spanwise forcing on rough walls

Spanwise opposed wall-jet forcing has been shown to reduce the skin-friction drag of wall-bounded turbulent flows by suppressing the near-wall turbulent motion (Yao et al., 2018). In the present work, the response of this drag reduction mechanism to the presence of surface roughness is studied. To this end, direct numerical simulations of flow in smooth and rough plane channels at a matched friction Reynolds number (Reτ=180) are carried out. The roughness is generated by distributing discrete roughness elements at two different values of frontal solidity, namely 0.021 and 0.045. The roughness elements height is k+=18 (＋indicates viscous scaling). By varying the forcing amplitude A+, it is shown that, when other forcing parameters are fixed, maximum drag reduction in a rough channel is achieved at a considerably larger A+ than in a smooth channel. Notably, the strength of the wall-jet at which the maximum drag reduction is achieved, is comparable for the smooth and both rough cases. Additionally, it is observed that the maximum drag reduction values attained in rough channels are smaller compared to those observed in smooth channels, for both lower and higher frontal solidity cases — 2.4% and 2%, respectively. The reduced drag reduction potential originates from two observations; firstly, drag on roughness elements, which is dominated by pressure drag, is not reduced by the method in question. Secondly, the drag on channel floor is reduced to a lesser extent when roughness elements are present. The latter can be attributed to the observation that, while in all cases the forcing suppresses streamwise random turbulent fluctuations and shear stress, this suppression is less pronounced when roughness elements are present.

Direct Numerical Simulation of Turbulent Boundary Layer over Cubical Roughness Elements

The present study explores turbulence statistics in turbulent flow over urban-like terrain using direct numerical simulation (DNS). DNS is performed in a turbulent boundary layer (TBL) over 3D cubic roughness elements. The turbulence statistics at Reτ = 816 are compared with those of experimental and numerical studies for validation, where Reτ is the friction Reynolds number. The flow exhibits wake interference characteristics similar to k-type roughness. Logarithmic variations in streamwise and spanwise Reynolds stresses and a plateau in Reynolds shear stress are observed, reminiscent of Townsend’s attached-eddy hypothesis. The energy at long wavelengths near the top of elements extends to smaller scales, indicating a two-scale behavior and a potential link to amplitude modulation. The quadrant analysis of Reynolds shear stress is employed, revealing significant changes in the contributions of ejection and sweep events near the top of elements. The results of quadrant analysis in the outer region closely resemble those of a TBL over a smooth wall, aligning with Townsend’s outer-layer similarity. The analysis of the transport equation of turbulent kinetic energy highlights the role of the roughness elements in energy transfer, especially pressure transport. Streamwise energy is mainly reduced near upstream elements and redirected in other directions.

Effects of porous substrates on the structure of turbulent boundary layers

Three different porous substrates (with different pore sizes, $s$ , and permeabilities, $K$ ) are used to examine their effect on the structure of boundary layer flow over them. The flow is characterised with single-point hot-wire measurements as well as planar particle image velocimetry. In order to elucidate differences in shallow and deep flows past porous substrates, foams with two different thicknesses ( $h$ ) are used (for all three substrates). A wide range of friction Reynolds numbers ( $2000< Re_\tau <13\,500$ ) and permeability-based Reynolds numbers ( $1< Re_K< 50$ ) are attained. For substrates with $Re_K \sim 1$ , the flow behaviour remains similar to flow over impermeable smooth walls and as such Townsend's hypothesis remains valid. Very large-scale motions are observed over permeable foams even when the $Re_K > 1$ . In contrast, a substantial reduction in velocity disturbances and associated length scales is achieved for permeable foams with intermediate values of pore density and relative foam thickness ( $h/s$ ), which affects outer-layer similarity. As permeability is increased by increasing pore size, the foam becomes sparse relative to viscous scales at high Reynolds numbers. For such foams, the flow conforms to outer-layer similarity and is more akin to flow over rough surfaces. Permeability attenuates the wavelengths associated with the outer-layer peak.

A Comprehensive Evaluation of Turbulence Models for Predicting Heat Transfer in Turbulent Channel Flow across Various Prandtl Number Regimes

Turbulent heat transfer in channel flows is an important area of research due to its simple geometry and diverse industrial applications. Reynolds-Averaged Navier–Stokes (RANS) models are the most-affordable simulation methodology and are often the only viable choice for investigating industrial flows. However, accurate modelling of wall-bounded flows is challenging in RANS, and the assessment of the performance of RANS models for heated turbulent channel flow has not been sufficiently investigated for a wide range of Reynolds and Prandtl numbers. In this study, five RANS models are assessed for their ability to predict heat transfer in channel flows across a wide range of Reynolds and Prandtl numbers (Pr) by comparing the RANS results with respect to the corresponding Direct Numerical Simulation data. The models include three Eddy Viscosity Models (EVMs): standard k−ϵ, low Reynolds number k−ϵLS, and k−ωSST, as well as two Reynolds Stress Models (RSMs): Launder–Reece–Rodi and Speziale–Sarkar–Gatski models. The study analyses the Reynolds number effects on turbulent heat transfer in a channel flow at a Pr of 0.71 for friction Reynolds number values of 180,395,640, and 1020. The results show that all models accurately predict velocity across all Reynolds numbers, but the accuracy of mean temperature prediction drops with increasing Reynolds number for all models, except for the k−ωSST model. The study also analyses the Pr effects on turbulent heat transfer in a channel flow with Pr values between 0.025 and 10.0. An error analysis is performed on the results obtained from different turbulence models, and it is shown that the k−ωSST model has the smallest error for the predictions of the mean temperature and Nusselt number for high-Prandtl-number flows, while the low Reynolds number k−ϵLS model shows the smallest errors for low-Prandtl-number flows at different Reynolds numbers. An analytical solution is utilised to identify Pr effects on forced convection in a channel flow into three different regimes: analytical region, transitional region, and turbulent diffusion-dominated region. These regimes are helpful to discuss the validity of the models in relation to the Pr. The findings of this paper provide insights into the performance of different RANS models for heat transfer predictions in a channel flow.

Introducing the Rankine vortex method for drag reduction in wall-bounded turbulent flows at low Reynolds number through streamwise vortex manipulation

It is well known that the near-wall streamwise vortices, together with the streaks, are the most important turbulent structures closely associated with drag reduction. Weakening or modifying the streamwise vortices are, thus, general approaches in near-wall turbulent control studies. In this study, a novel approach to manipulate the flow is introduced and applied to a turbulent channel flow in order to achieve drag reduction. The idea behind the “Rankine vortex method” is to generate a force based on the statistical and geometrical information of streamwise vortices. Direct numerical simulations of a turbulent channel flow at a frictional Reynolds number, Reτ, of 180 (based on the driving pressure gradient and channel half-width) are performed. The force is applied in the vicinity of the lower wall of the channel, and the results are comparatively analyzed for the cases with and without force implemented. A drag reduction of 10% is achieved. The theoretical flow control approach presented, along with the associated analysis, has the potential to enhance our current understanding of flow control mechanisms through the manipulation of near-wall turbulence.

Inflow turbulence generation for compressible turbulent boundary layers

It is still challenging to generate high-quality inflow turbulence for the direct numerical and large-eddy simulations of compressible turbulent boundary layers (CTBL). Recently, Wang et al. [“Inflow turbulence generation using an equivalent boundary layer model,” Phys. Fluids 35, 075110 (2023)] proposed a new inflow turbulence generation method based on an equivalent boundary layer model for incompressible turbulent boundary layers. In the present study, the compressible equivalent boundary layer (CEBL) model is proposed and applied to the direct numerical simulation of supersonic turbulent boundary layers. The streamwise equilibrious CEBL approximates the streamwise developing CTBL by adding source terms to the governing equations to recover the mean mass, momentum, and energy balances at a given Reynolds number. Direct numerical simulation is performed to CEBL at free-stream Mach number 5.86 and friction Reynolds number 380. Comparison with the CTBL statistics at the same parameters validates the fidelity and reliability of the CEBL model. Turbulence generated by CEBL as well as the digital filtering and recycling-rescaling methods is used, respectively, to construct the inflow conditions for the direct numerical simulation of supersonic turbulent boundary layers. Results show that the CEBL method has great superiority in reducing the adjustment length compared with the other two methods. In addition, a correction method designed for the high inflow Reynolds number is also introduced.

Turbulent boundary layers in thermal convection at moderately high Rayleigh numbers

In this work, we perform direct numerical simulations of Rayleigh–Bénard convection in a two-dimensional confined square cell for Rayleigh numbers (Ra) from 109 to 1013 and a Prandtl number (Pr) of 0.7. In contrast to a previous study in a periodic box conducted by Zhu et al. [Phys. Rev. Lett. 120, 144502 (2018)], our simulations apply two adiabatic sidewalls. In particular, boundary layer structures near the heating plates are examined using both mean velocity and temperature profiles in the impacting, shearing, and ejecting regions of the plumes. After an appropriate normalization using the wall units, the friction Reynolds numbers of our simulations exceed the critical value of 200 and follow Reτ∼Ra0.323, and we also observe the logarithmic mean velocity profiles (with the slope κv≈0.35) in the shearing regions and logarithmic mean temperature profiles (with the slope κθ≈2) in the ejecting regions. These logarithmic behaviors indicate that both the thermal and momentum boundary layers may have entered the fully developed turbulent state. However, for the Nusselt number (Nu), our data still follow the trend of classical 1/3 scaling, differing from the ultimate state reported before but agreeing with the three-dimensional results obtained by Iyer et al. [PNAS 117, 14 (2020)] for confined cells.

The statistical characteristics and auto-regeneration of backflow in non-Newtonian turbulent pipe flow

The backflow phenomenon in shear-thinning and shear-thickening fluids is investigated in pipe flows at friction Reynolds number Reτ=180 via direct numerical simulations. Conditional average results show that the extreme fluctuation of wall shear stress around the backflow regions is more abrupt under the shear-thinning effect. The statistical characteristics of the backflow at different flow indices from 0.5 to 1.5 show remarkable differences. The probability of the backflow events at the wall increases in both the shear-thinning and the shear-thickening fluids under different mechanisms. The backflow occurs more frequently and exists further away from the wall in the shear-thinning fluids owing to the suppressed near-wall turbulent structures and the laminarization at low flow indices. The increase in the probability of the backflow events in the shear-thickening fluids is caused by increased Q2 and Q4 events in the near-wall region. The variation in the size and the lifespan of the backflow regions with the flow index is very prominent which both increase with the shear-thinning effect and decrease as the flow becomes dilatant. In the weakly turbulent flow of shear-thinning fluid, large backflow regions appear near the leading edge of the turbulent spots where the off-axial turbulent fluctuations are significantly lowered. Observations show the linked evolution between the hairpin vortices and the backflow regions induced underneath the strong spanwise rotations. The backflow follows the auto-regeneration process of the hairpin vortices in a packet which results in coherent streamwise-aligned backflow regions under the hairpin packets confined closer to the wall.

Widest scales in turbulent channels

The widest spanwise scales in turbulent channel flows are studied through the use of three periodic channel-flow simulations at friction Reynolds number Reτ=550. The length and height of the channels are the same in all cases (Lx/h=8π and Ly/h=2, respectively), while the width is progressively doubled: Lz/h={4π,8π,16π}. The effects of increasing the domain width cannot be determined with statistical significance in our simulations, since the difference in the statistics between the simulations is of the same order as the errors of convergence. A channel flow similar to the smaller one [Del Álamo et al., “Scaling of the energy spectra of turbulent channels,” J. Fluid Mech. 500, 135–144 (2004)], which was averaged over a very long time, was used as a reference. The one-dimensional spanwise spectrum of the streamwise velocity is computed with the aim of assessing the domain-size effect on the widest scales. Our results indicate that 90% of the total streamwise energetic fluctuations is recovered without a significant influence of the size of the domain. The remaining 10% of the energy reflects that the widest scales in the outer layer are the ones most significantly affected by the spanwise length of the domain. The power-spectral density for kz = 0 remains constant even if the size of the domain in the spanwise direction is increased up to four times the standard spanwise length, indicating that wide, spanwise coherent structures are not an artifact of domain truncation.

Characterization and evolution of local streamline geometry in an incompressible turbulent channel flow

We investigated the statistical characterization and time evolution of local streamline geometry in typical regions of an incompressible turbulent channel flow at the friction Reynolds number Reτ∼1000. Local streamline structure is completely and uniquely determined by one magnitude factor—the magnitude of velocity gradient tensor (VGT) A—and four shape parameters—the second and third invariants of normalized VGT q and r, the intermediate eigenvalue of normalized strain-rate tensor a2, and the cosine of the angle between vorticity and the intermediate eigendirection of normalized strain-rate tensor | cos β|. As the distance to the wall decreases, the joint probability distribution function of q and r becomes more symmetrical and concentrated, while outside the viscous sublayer, the distribution of A in q–r plane gets dispersed. Interestingly, the inertial conditional mean trajectories (CMTs) exhibit a symmetrical picture only in the buffer layer, and outside the viscous sublayer, the pressure CMTs contributing to slow evolution from unstable focus compression geometry to stable focus stretching geometry tend to dominate the q–r plane as getting closer to the wall. Due to combined effects of inertia and pressure, the origin of the q–r plane (pure-shear geometry) acts as an attractor in the central region, the logarithmic region, and the upper part of the buffer layer while acts as a repeller in the lower part of the buffer layer and the viscous sublayer.

Laminar Flow Heat Transfer Studies in a Twisted Square Duct for Constant Wall Temperature

Abstract: Three dimensional analysis of steady fully developed laminar flow inside twisted duct of square cross section flow area is studied for Reynolds number range of 100 – 2000 using a commercially available software. Twist ratio used are 2.5, 5, 10 and 20. Heat transfer results are generated for a uniform wall temperature case for Prandtl number range of 0.7 to 20. Maximum values for the product of friction factor and Reynolds number is observed for a twist ratio of 2.5 and Reynold number of 2000. Maximum Nusselt number is observed for the same values along with Prandtl number of 20. Correlations for friction factor and Nusselt number are developed involving swirl parameter. Local distribution of friction factor ratio and Nusselt number across a cross-section is presented. Heat transfer enhancement for Reynold number of 2000 and Prandtl number of 0.7 for twist ratio of 2.5, 5, 10, and 20 are 33 %, 18 %, 10 % and 7 % respectively. The results are significant because it will contribute to development of compact heat exchanger.

Characterization of very-large-scale motions in supersonic and hypersonic turbulent boundary layers

Very-large-scale motions are commonly observed in moderate- and high-Reynolds-number wall turbulence, constituting a considerable portion of the Reynolds stress and skin friction. This study aims to investigate the behaviour of these motions in high-speed and high-Reynolds-number turbulent boundary layers at varying Mach numbers. With the aid of high-precision numerical simulations, numerical experiments and theoretical analysis, it is demonstrated that the very-large-scale motions are weakened in high-Mach-number turbulence at the same friction Reynolds numbers, leading to the reduction in turbulent kinetic energy in the outer region. Conversely, the lower wall temperature enhances the very-large-scale motions but shortens the scale separation between the structures in the near-wall and outer regions.

An Air-Bearing Floating-Element Force Balance for Friction Drag Measurement

Abstract It is extremely difficult, if not impossible, for existing force balances to capture very small skin-friction drag (SFD) in a perturbed turbulent boundary layer (TBL), which is characterized by the unpredictable, nonuniform distribution of static surface pressure. A novel force balance is proposed, which combines the level principle, as deployed in Cheng et al.'s (2020, “A High-Resolution Floating-Element Force Balance for Friction Drag Measurement,” Meas. Sci. Technol., 32, p. 035301) force balance, with a single-degree-of-freedom air bearing mechanism. This mechanism acts to eliminate disturbances, such as nonuniform static pressure on the wall associated with high Reynolds number TBL or a TBL under control. As a result, the developed balance may be used to accurately measure SFD in the order of 10−3 N in a TBL with or without control. This balance has been successfully applied to measure the drag reduction (DR) of a TBL manipulated using one array of streamwise microjets, at friction Reynolds number Reτ = 3340 ∼ 5480.

The role of the areal parameters on turbulent flow over 2D Gaussian roughness

In the past decades, numerous efforts have been dedicated to establishing a direct correlation between a geometric parameter that represents wall roughness and the corresponding velocity reduction, known as the Roughness Function ΔU+. This reduction is influenced by various statistical measures of roughness height, including average roughness height, peak-to-valley roughness distance, roughness root mean square, and Effective Slope, among others. It has been demonstrated that a singular measure of roughness height cannot sufficiently predict the Roughness Function across all turbulent regimes. Consequently, many studies have concentrated on identifying a universal correlation between roughness geometry and the downward shift of the mean velocity profile. In this study, we investigated the correlation between various geometrical parameters and the Roughness Function using Large Eddy Simulation (LES) techniques in channel flows at a friction Reynolds number of Reτ=400. Given the complexity introduced by the random nature of irregular roughness, we explored specific aspects of the relationship between wall irregularities and the roughness function by studying 2D geometries. This approach allowed us to systematically investigate the impact of geometrical properties on the roughness function and isolate the effects of roughness density and coverage area. Several irregular rough surfaces, characterized by different average oscillations height, different distributions and different densities, were designed throughout 2D Gaussian functions. With the aim to find a universal correlation between the roughness geometry and the Roughness Function, new geometrical quantities were investigated, based on the global area occupied by the roughness A∗. The prediction of the roughness function can be thus obtained using apriori data. To predict the roughness function, we introduced a parameter called Effective Area (EA), which is derived from the correlation between the Effective Slope (ES) and the roughness area A∗. Our findings indicate that a single geometric parameter, whether ES or Area A, is insufficient to predict the roughness effect. Conversely, combining these two parameters enhances predictive accuracy, at least for the proposed roughness model. This improvement can be attributed to the ability of ES to interpret the roughness distribution and height, while the coverage area is effective in predicting roughness density over a flat plate.

Direct numerical simulation of laminar, transitional and turbulent radially inward flow between closely spaced corotating disks

This study describes direct numerical simulation (DNS) of radially inward spiralling corotating disk flow with a narrow disk spacing, using the open source solver Nek5000 and the supercomputer SuperMUC-NG at Leibniz Supercomputing Centre. Knowledge about laminar and turbulent regime boundaries in this flow scenario is important for modelling and performance prediction of friction turbines. Simulations are performed in differently sized sections of the flat annulus that is formed by two opposing corotating disk surfaces. Three sets of operating conditions are covered, from the laminar, transitional and turbulent region of a previously determined stability chart respectively. Directly downstream of the inlet boundary, the flow is artificially perturbed with a random body force acting normal to the disk surfaces. Fourier analysis of the DNS flow field reveals that the artificial perturbation is dampened across all wavenumbers for the laminar conditions, while at the transitional conditions a small range of modes is weakly amplified towards the outlet. The identified unstable modes were previously correctly predicted by linear stability analysis. Comparison to experimental velocity profile measurements from a previous study at the same transitional operating conditions suggests strongly perturbed flow during the experiment. For inflow conditions leading to turbulent flow, average velocity profiles from DNS coincide with those from experiment and from commercial fluid simulation software with turbulence modelling (ANSYS CFX). Close to the walls, turbulent dissipation and turbulent kinetic energy distributions do not agree with the ANSYS CFX results. Friction Reynolds number settles at about 118 after turbulent flow has developed from the initial perturbation. Two point correlations and corresponding energy spectra are presented.

Transducer Resolution Effect on Pressure Fluctuations Beneath Hypersonic Turbulent Boundary Layers

The size of a pressure transducer is known to affect the accuracy of measurements of wall-pressure fluctuations beneath a turbulent boundary layer because of spatial averaging over the sensing area of the transducer. In this paper, the effect of finite transducer size is investigated by applying spatial averaging or wavenumber filters to a database of hypersonic wall pressure generated from a direct numerical simulation (DNS) that simulates the turbulent portion of the boundary layer over a sharp 7° half-angle cone at nominally Mach 8. A good comparison between the DNS and the experiment in the Sandia Hypersonic Wind Tunnel at Mach 8 is achieved after spatial averaging is applied to the DNS data over an area similar to the sensing area of the transducer. The study shows that a finite sensor size similar to that of the PCB132 transducer can cause significant attenuation in the root-mean-square and power spectral density (PSD) of wall-pressure fluctuations, and the attenuation effect is identical between cone and flat plate configurations at the same friction Reynolds number. The Corcos theory is found to successfully compensate for the attenuated high-frequency components of the wall-pressure PSD.

Reynolds number effects on a velocity–vorticity correlation-based skin-friction drag decomposition in incompressible turbulent channel flows

A form of skin-friction drag decomposition is given based on the velocity–vorticity correlations, $\langle v\omega _z\rangle$ and $\langle -w\omega _y\rangle$ , which represent the advective vorticity transport and vortex stretching, respectively. This identity provides a perspective to understand the mechanism of skin-friction drag generation from vortical motions and it has better physical interpretability compared with some previous studies. The skin-friction coefficients in incompressible turbulent channel flows at friction Reynolds numbers from 186 to 2003 are divided with this velocity–vorticity correlation-based identity. We mainly focus on the Reynolds number effects on the contributing terms, their scale-dependence and quadrant characteristics. Results show that the contributing terms and their proportions exhibit similarities and the same peak locations across the wall layer. For the first time, we find that the positive and negative regions in the spanwise pre-multiplied spectra of the turbulent inertia ( $\langle v'\omega _z'\rangle +\langle -w'\omega _y'\rangle$ ) can be separated with a universal linear relationship of $\lambda _z^+=3.75y^+$ . The linear relationship is adopted as the criterion to investigate the scale dependence of the velocity–vorticity coupling structures. It reveals that the negative and positive structures dominate the generation of friction drag associated with the advective vorticity transport and vortex stretching, respectively. Moreover, quadrant analyses of the velocity–vorticity correlations are performed to further examine the friction drag generation related to different quadrant motions.

Time-delayed characteristics of turbulence in pulsatile pipe flow

Direct numerical simulations are performed to study temporal variations of the wall shear stresses and flow dynamics in the turbulent pulsatile pipe flow. The mechanisms, responsible for the paradoxical phenomenon for which the amplitude of the oscillating wall shear stress in the turbulent flow is smaller than that in the laminar flow for the same pulsation conditions, are investigated. It is shown that the delayed response of turbulence in the buffer layer generates a large magnitude of the radial gradient of the Reynolds shear stress near the wall, which counteracts the effect of the oscillating pressure gradient on the change of the streamwise velocity and hence reduces the amplitude of the wall shear stress. Such a delayed response consists of two processes: the delayed development of near-wall streaks and the subsequent energy redistribution from the streamwise velocity fluctuation to the other two co-existing components. This is a dynamical manifestation of the viscoelasticity of turbulent eddies. As the frequency is reduced, the variation of the friction Reynolds number results in a phase-wise variation of the time scale and intensity of the turbulence response, causing the hysteresis of the wall shear stress. Such a phase asymmetry is amplified by the increase of the pulsation amplitude. An examination of the energy spectra reveals that the near-wall streaks are stretched in the streamwise direction during the acceleration phase, and then break up into small-scale structures in the deceleration phase, accompanied by the enhanced dissipation that transforms the turbulent kinetic energy into heat.

Direct numerical simulations of turbulent channel flow with wall transpiration using lattice Boltzmann method

Abstract Direct numerical simulations of turbulent channel flow with wall transpiration are conducted using the multi-relaxation time lattice Boltzmann method for the first time. Flows with two friction Reynolds numbers, $Re_\tau$ = 150 and 250, are investigated, and the explored transpiration rates based on the friction velocity range from 0.05 to 0.26. A successive block mesh-refinement strategy is adopted to resolve the near-wall structure, where the grid sizes are $\Delta ^+ \sim$1, 2, and 4. Under such resolution, the size of the maximum grid density employed is $775 \times 10^6$. At the low transpiration rate ($V^+=0.05$), the turbulence level at the suction side is damped relative to the less altered values along the injection side. However, at the high level of wall transpiration, i.e. $V^+=0.26$, turbulence levels are severely retarded on both the injection and suction sides. The predicted mean and turbulence quantities show good agreements with the available direct numerical simulation data, indicating the effectiveness of lattice Boltzmann method combined with mesh refinement in reproducing the correct physics of turbulent channel flows with different levels of wall transpiration.

Total and static temperature statistics in compressible turbulent plane channel flow

The paper studies the statistics of total and static temperature (total $h_t$ and static $h$ enthalpy) in compressible turbulent plane channel flow using direct numerical simulations (DNS) data covering the range of centreline Mach numbers $0.3\lessapprox \bar {M}_{{{CL}}_x} \lessapprox 2.5$ and Huang–Coleman–Bradshaw friction Reynolds numbers $100\lessapprox Re_{\tau ^\star }\lessapprox 1000$ . For this class of very-cold-wall flows, the DNS data for correlation coefficients and joint probability density functions (p.d.f.s) show that $h_t'$ is invariably very strongly correlated with the streamwise velocity fluctuation $u'$ , in contrast to static temperature (static enthalpy $h'$ ) whose correlation with $u'$ weakens rapidly with increasing wall distance. We study various correlations and joint p.d.f.s of $h_t'$ and $h'$ in relation to the fluctuating velocity field, including the turbulent Prandtl number $Pr_{{T}}$ , and discuss the predictions of Reynolds analogy. The scaling of the mean enthalpy and the fluctuating enthalpy variance and fluxes with respect to inner and outer velocity scales is investigated. The complex behaviour and scaling of different terms in the transport equations for the enthalpy variance and fluxes are discussed.