Karman Constant Research Articles

Numerical and analytical flow models in ecological channels with interaction of vegetation and freshwater

Aquatic vegetation interferes with river hydrodynamics, thus affecting the mass transport and energy transfer in an ecosystem. The flow over submerged vegetation is characterized by a complex velocity profile and multiple turbulence structures, which have been usually simulated using cylinders or strips in previous studies. Because the simplified vegetation configuration may hide or amplify some physical processes found in natural conditions, we investigate the velocity distribution and turbulence structure in foliaged vegetation flows using both analytical and numerical approaches. The main innovations and findings can be summarized as follows: 1) numerical and analytical models adopted in this paper accurately simulate the flow velocity profile in vegetated channel; 2) the Karman constant is found to be unsuitable for complex vegetation morphologies, so we proposed adjusted coefficient; 3) an image processing method is adopted to quantify the vegetation morphology accurately; 4) the existing mixing-layer thickness formula is found to be unsuitable for vegetation with leaves, an improved formula is proposed showing high correlation coefficient (0.9562) between measured and predicted data; 5) to ensure applicability to larger-scale hydrodynamic simulations, an analytical expression of Manning’s coefficient is proposed based on an analytical multi-layer flow velocity model. These research findings can provide theoretical support for the design of vegetated river and ecological restoration.

Dry Deposition Methods Based on Turbulence Kinetic Energy: Part 1. Evaluation of Various Resistances and Sensitivity Studies Using a Single-Point Model.

Different functions are used to account for turbulence strength in the atmospheric boundary layer for different stability regimes. These functions are one of the sources for differences among different atmospheric models' predictions and associated biases. Also, turbulence strength is underrepresented in some of the resistance formulations. To address these issues with dry deposition, firstly we take advantage of three-dimensional (3-D) turbulence information in estimating resistances by proposing and validating a 3-D turbulence velocity scale that is relevant for different stability regimes of boundary layer. Secondly, we hypothesize and validate that friction velocity measured by 3-D sonic anemometer can be effectively replaced by the new turbulence velocity scale multiplied by the von Karman constant. Finally, we (1) present a set of resistance formulations for ozone (O3) based on the 3-D turbulence velocity scale; (2) intercompare estimations of such resistances with those obtained using existing formulations; and, (3) evaluate simulated O3 fluxes using a single-point dry deposition model against long-term observations of O3 fluxes at the Harvard Forest (MA) site. Results indicate that the new resistance formulations work very well in simulating surface latent heat and O3 fluxes when compared to respective existing formulations and measurements at a decadal time scale. Findings from this research may help to improve the capability of dry deposition schemes for better estimation of dry deposition fluxes and create opportunities for the development of a community dry deposition model for use in regional/global air quality models.

Enhanced Turbulence in the Upper Mixed Layer Under Light Winds and Heating: Implications for Gas Fluxes

AbstractMeasurements of turbulence, as rate of dissipation of turbulent kinetic energy (ε), adjacent to the air‐water interface are rare but essential for understanding of gas transfer velocities (k) used to compute fluxes of greenhouse gases. Variability in ε is expected over diel cycles of stratification and mixing. Monin‐Obukhov similarity theory (MOST) predicts an enhancement in ε during heating (buoyancy flux, β+) relative to that for shear (u*w3/κz where u*w is water friction velocity, κ is von Karman constant, z is depth). To verify and expand predictions, we quantified ε in the upper 0.25 m and below from profiles of temperature‐gradient microstructure in combination with time series meteorology and temperature in a tropical reservoir for winds <4 m s−1. Maximum likelihood estimates of near‐surface ε during heating were independent of wind speed and high, ∼5 × 10−6 m2 s−3, up to three orders of magnitude higher than predictions from u*w3/κz, increased with heating, and were ∼10 times higher than during cooling. k, estimated using near‐surface ε, was ∼10 cm hr−1, validated with k obtained from chamber measurements, and 2–5 times higher than computed from wind‐based models. The flux Richardson number (Rf) varied from ∼0.4 to ∼0.001 with a median value of 0.04 in the upper 0.25 m, less than the critical value of 0.2. We extend MOST by incorporating the variability in Rf when scaling the influence of β+ relative to u*w3/κz in estimates of ε, and by extension, k, obtained from time series meteorological and temperature data.

Analytic derivation of Monin-Obukhov similarity function for open atmospheric surface layer

The Monin-Obukhov (MO) similarity function φm of the atmospheric surface layer (ASL) describing the deviation from the log law of the canonical turbulent boundary layer because of thermal stratification has been traditionally determined empirically. This study presents a unified analytic expression derived from a symmetry-based theory of wall turbulence, called structural ensemble dynamics (SED), which postulates a generalized dilation symmetry principle expressing the effect of the wall on turbulence, leading to an analytic multi-regimes expression for the mixing length. For ASL in unstable and stable conditions (i.e., UC and SC), a unified two-regime formula of the mixing length is proposed, leading to a φm, similar to the Businger-Dyer (BD) formula; with a simplified model energy balance equation, φm is completely specified with no free parameter. Furthermore, the theory allows the study of the open ASL’s underlying additional physical processes such as bottom-up or top-down flux due to pressure variations Tp. Assuming that Tp is decomposed into shear-like and buoyancy-like components, we propose new explanations for two important features of typical ASL: a significantly smaller Karman constant of 0.36 and a varying φm for SC mean speed profiles. The theory is validated by the data obtained at Kansas and also at Qingtu Lake Observation Array in Northern China for a variety of heat flux conditions. In conclusion, due to pressure variations, we assert that ASL is intrinsically open and that the current theory offers a new basis for its quantification.

The O’KEYPS Equation and 60 Years Beyond

Some 60 years ago, six researchers obtained a semi-empirical equation that describes how the stability correction function for the mean velocity profile ( $$\phi _\mathrm{m}$$ ) in the atmospheric surface layer varies with the stability parameter—the famous O’KEYPS equation. Their derivations are essentially based on interpolation of the turbulent eddy viscosity between neutral and convective conditions. Comparing the O’KEYPS equation with new theoretical developments—such as phenomenological and cospectral budget models—suggests that Heisenberg’s eddy viscosity provides a unifying framework for interpreting the behaviour of $$\phi _\mathrm{m}$$ . The empirical coefficient in the O’KEYPS equation, which is on the order of 10 based on data fitting to observations, is found to be primarily linked to the increase of the size of turbulent eddies as instability increases. The ratio of the sizes of turbulent eddies under convective and neutral conditions is on the order of $$1{/}\kappa $$ , where $$\kappa $$ is the von Karman constant, and is modulated by the turbulent Prandtl number.

Flow Resistance Equation in Sand-bed Rivers and Its Practical Application in the Yellow River

To fully reflect the effect of the flow characteristics, sidewall conditions and sediment concentrations on the bed roughness of sand-bed rivers, this study established a new flow resistance equation by introducing a comprehensive influence coefficient presented via a combination of power-function forms of the relative flow velocity, von Karman constant of sediment-laden flows and the ratio of particle size to viscous sublayer thickness. The comprehensive influence coefficient accordingly acts as a synthesized factor representing the combined effects of the flow intensity, bed material movement, energy consumption condition, and relative friction condition in the near-wall region of sediment-laden flows. Based on the field data from sediment-laden flows under scenarios of both high and low sediment concentrations, the performance of the proposed equation was validated to achieve the best accuracy in the calculation of Manning’s roughness coefficient compared with that of several previously presented flow resistance equations. Furthermore, the proposed flow resistance equation was adopted to quantify the stable channel width of the Lower Yellow River (LYR), i.e., the optimum main channel width for sediment transportation in the typical wandering reach of the LYR. The calculated stable channel width is consistent with the current river training width of the LYR, indicating that the proposed equation has great potential as a theoretical tool that can be used to support the determination of the river training strategy for the LYR.

THEORETICAL CALCULATION OF THE VALUE OF THE VON KARMAN CONSTANT

The article considers various approaches to assessing the physical essence of the von Karman constant characterizing the vertical distribution of the flow velocity. On the one hand, the von Karman constant is a proportionality coefficient between the length of the mixing length and the depth; on the other hand, it characterizes the tilt angle of the vertical velocity profile. It is considered that this parameter is universal, that means it is constant as long as the averaged velocity distribution is constant. However, depending on estimation way of this constant, its values under the same conditions may differ to 2 orders. Two methods for estimating the von Karman constant are considered. In the first case, determination of the von Karman constant is on the grounds of direct measurements of the maximum and average flow velocities, depth and slope of the water surface in a particular area. Propagation of the obtained value to other objects will lead to errors. In the second case, the parameter is defined as a function of the coefficient of hydraulic friction. The authors showed that the von Karman constant is a function of the coefficient of turbulent exchange (viscosity) and indirectly is a function of the depth. As a result of the calculations, it was shown that the maximum values of the von Karman constant observed on the bottom. Additionally, the authors propose a new version of the calculation of the von Karman constant through the tension shift for a turbulent flow. It is concluded that since for velocities <1 meter per second the changes in the von Karman constant values from 0.27 to 0.38 the maximum velocity with variation do not exceed 3%, which fits into the accuracy of the velocity measurements, it is practically impossible to estimate the value of von Karman constant by first method for flat rivers even with multiple measurements with standard hydrometric equipment. Keywords: the von Karman constant; coefficient of turbulent exchange (viscosity); Reynolds number; coefficient of hydraulic friction; velocity distribution diagram; turbulent mode

Velocity-defect laws, log law and logarithmic friction law in the convective atmospheric boundary layer

The mean velocity profile in the convective atmospheric boundary layer (CBL) is derived analytically. The shear-stress budget equations and the mean momentum equations are employed in the derivation. The multi-point Monin–Obukhov similarity (MMO) recently proposed and analytically derived by Tong & Nguyen (J. Atmos. Sci., vol. 72, 2015, pp. 4337–4348) and Tong & Ding (J. Fluid Mech., vol. 864, 2019, pp. 640–669) provides the scaling properties of the statistics in the shear-stress budget equations. Our previous and present studies have shown that the CBL is mathematically a singular perturbation problem. Therefore, we obtain the mean velocity profile using the method of matched asymptotic expansions. Three scaling layers are identified: the outer layer, which includes the mixed layer, the inner-outer layer and the inner-inner layer, which includes the roughness layer. There are two overlapping layers, the local-free-convection layer and the log layer, respectively. Two new velocity-defect laws are discovered: the mixed-layer velocity-defect law and the surface-layer velocity-defect law. The local-free-convection mean profile is obtained by asymptotically matching the expansions in the first two layers. The log law is obtained by matching the expansions in the last two layers. The von Karman constant is obtained using velocity and length scales, and therefore has a physical interpretation. A new friction law, the convective logarithmic friction law, is obtained. The present work provides an analytical derivation of the mean velocity profile hypothesized in the Monin–Obukhov similarity theory, and is part of a comprehensive derivation of the MMO scaling from first principles.

Spatial Variation of Statistical and Spectral Properties of the Stream Wise and Wall-Normal Velocity Fluctuations in the Near-Neutral Atmospheric Surface Layer

Based on high-quality near-neutral atmospheric-surface-layer (ASL) data obtained from an observational site located in flat desert and existing experimental results for a friction Reynolds number Reτ 106, the change of the second-order statistics of the streamwise velocity component with height is consistent with the experimental results for Reτ < 106, while the second-order statistics of the wall-normal velocity component increase linearly with height. In combination with the experimental results for Reτ < 106, the variation of the slope of streamwise turbulence intensity with height and the variation of the slope and intercept of the wall-normal turbulence intensity with height are quantified to reveal that both the streamwise and wall-normal velocity variances follow a generalized logarithmic law, with the distribution of the streamwise and wall-normal velocity components satisfying sub- and super-Gaussian distributions. Spectral analysis shows that the variation of the wavenumber corresponding to the pre-multiplied streamwise and wall-normal spectra peaks with the ratio z/δ obeys the form $$ k_{x} \delta = a(\delta /z)^{b} $$ , where z is the height, kx is the streamwise wavenumber, and δ is the ASL thickness. For the pre-multiplied streamwise spectra, $$ b = 0.5 \pm 0.1 $$ and the value of a may have a Reynolds-number dependence, while for the pre-multiplied wall-normal spectra, $$ a = 2.2 \pm 0.09 $$ and $$ b = 1.0 \pm 0.1 $$ . The von Karman constant, the Townsend–Perry constant and other parameters also display a weak Reynolds-number dependence.

Properties of the scalar variance transport equation in turbulent channel flow

We study the total scalar variance equation for fully developed turbulent channel flow subject to uniform scalar generation. We find a total scalar variance leading balance four-layer structure similar to total kinetic energy balance. Differences in the two Karman constants, k and k_\ensuremath{\theta}, are clarified.

Sensitivity of surface drag predictions using Monin–Obukhov similarity theory in non-ideal flow conditions

Turbulent flux measurements in the atmospheric surface layer (ASL), acquired from a vertical tower of sonic anemometers, are used to investigate the transient behavior of the drag coefficient over a relatively flat, homogeneous, desert playa surface. The goal of the study is to quantify the extent to which Monin–Obukhov similarity theory (MOST) successfully predicts the empirical drag coefficient when the data are not necessarily restricted to idealized conditions. Here, “idealized” refers to the conditions under which MOST is assumed to be valid, namely that the ASL exhibits statistically stationary behavior and stresses are constant with height. In order to quantify the agreement between the empirical and theoretical drag coefficients, uncertainty bands as a function of time of day were calculated for each. These uncertainty bands account for random errors and nonstationarity effects in the momentum flux, horizontal wind velocity, and stability parameter, as well as uncertainties associated with the model parameters appearing in MOST, i.e., the von Karman constant and aerodynamic roughness length. Results show that the average uncertainties in the empirical and theoretical drag coefficients are about $$\pm \,30\%$$ and $$\pm \,20\%$$ , respectively. In the case of the former, the major contribution lies from uncertainty in the momentum flux measurements; while, the latter is dominated by uncertainty in the von Karman constant. Joint probability density functions reveal that large differences between the empirical and theoretical drag coefficient have a high probability of occurrence when the ASL is nonstationary and/or exhibits a non-constant stress layer.

Aerodynamic Parameters Over an Eroding Bare Surface: Reconciliation of the Law of the Wall and Eddy Covariance Determinations

AbstractAssessing accurately the surface friction velocity is a key issue for predicting and quantifying aeolian soil erosion. This is usually done either indirectly from the law of the wall (LoW) of the mean wind velocity profile or directly from eddy covariance (EC) of the streamwise and vertical wind velocity fluctuations. However, several recent experiments have reported inconsistency between friction velocities deduced from both methods. Here we reinvestigate the determination of aerodynamic parameters (friction velocity and surface roughness length) over an eroding bare surface and look at the possible reasons for observing differences on these parameters following the method. For that purpose a novel field experiment was performed in South Tunisia under the research program WIND‐O‐V (WIND erOsion in presence of sparse Vegetation). We find no significant difference between friction velocities obtained from both law of the wall and EC approaches when the friction velocity deduced from the EC method was extrapolated to the surface. Surface roughness lengths show a clear increase with wind erosion, with more scattered values when deduced from the EC friction velocity. Our measurements further suggest an average value of the von Karman constant of 0.407 ± 0.002, although individual wind events lead to different average values due probably to the definition of the ground level or to the stability correction. Importantly, the von Karman constant was found independent of the wind intensity and thus of the wind soil erosion intensity. Finally, our results lead to several recommendations for estimating the aerodynamic parameters over bare surface in order to evaluate saltation and dust fluxes.

Signatures of Air–Wave Interactions Over a Large Lake

The air–water exchange of momentum and scalars (temperature and water vapour) is investigated using the Lake-Atmosphere Turbulent EXchange (LATEX) dataset. The wind waves and swell are found to affect the coupling between the water surface and the air differently. The surface-stress vector aligns with the wind velocity in the presence of wind waves, but a wide range of stress–wind misalignment angles is observed during swell. The momentum transport efficiency decreases when significant stress–wind misalignment is present, suggesting a strong influence of surface wave properties on surface drag. Based on this improved understanding of the role of wave–wind misalignment, a new relative wind speed for surface-layer similarity formulations is proposed and tested using the data. The new expression yields a value of the von Karman constant ( $$\kappa $$ ) of 0.38, compared to 0.36 when using the absolute wind speed, as well as reduced data fitting errors. Finally, the ratios of aerodynamic to scalar roughness lengths are computed and various existing models in the literature are tested using least-square fitting to the observed ratios. The tests are able to discriminate between the performance of various models; however, they also indicate that more investigations are required to understand the physics of scalar exchanges over waves.

Dynamics of liquid–liquid flows in horizontal pipes using simultaneous two–line planar laser–induced fluorescence and particle velocimetry

Experimental investigations are reported of stratified and stratified–wavy oil–water flows in horizontal pipes, based on the development and application of a novel simultaneous two–line (two–colour) technique combining planar laser–induced fluorescence with particle image/tracking velocimetry. This approach allows the study of fluid combinations with properties similar to those encountered in industrial field–applications in terms of density, viscosity, and interfacial tension, even though their refractive indices are not matched, and represents the first attempt to obtain detailed, spatiotemporally–resolved, full 2–D planar–field phase and velocity information in such flows. The flow conditions studied span mixture velocities in the range 0.3–0.6 m/s and low water–cuts up to 20%, corresponding to in situ (local) Reynolds numbers of 1750–3350 in the oil phase and 2860–11,650 in the water phase, and covering the laminar/transitional and transitional/turbulent flow regimes for the oil and water phases, respectively. Detailed, spatiotemporally–resolved in situ phase and velocity data in a vertical plane aligned with the pipe centreline and extending across the entire height of the channel through both phases are analysed to provide statistical information on the interface heights, mean axial and radial (vertical) velocity components, (rms) velocity fluctuations, Reynolds stresses, and mixing lengths. The mean liquid–liquid interface height is mainly determined by the flow water cut and is relatively insensitive (up to 20% the highest water cut) to changes in the mixture velocity, although as the mixture velocity increases the interfacial profile transitions gradually from being relatively flat to containing higher amplitude waves. The mean velocity profiles show characteristics of both laminar and turbulent flow, and interesting interactions between the two co–flowing phases. In general, mean axial velocity profiles in the water phase collapse to some extent for a given water cut when normalised by the mixture velocity; conversely, profiles in the oil phase do not. Strong vertical velocity components can modify the shape of the axial velocity profiles. The axial turbulence intensity in the bulk of the water layer amounts to about 10% of the peak mean axial velocity in the studied flow conditions. In the oil phase, the axial turbulence intensity increases from low values to about 10% at the higher Reynolds numbers, perhaps due to transition from laminar to turbulent flow. The turbulence intensity showed peaks in regions of high shear, i.e., close to the pipe wall, and at the liquid–liquid interface. The development of the mixing length in the water phase, and also above the liquid–liquid interface in the oil phase, agrees reasonably well with predicted variations described by the von Karman constant. Finally, evidence of secondary flow structures both above and below the interface exists in the vertical velocity profiles, which is of interest to explore further.

Agreement of Four Equations for Computing Dewfall in Northern Germany

The energy balance (EB), turbulent vapour transport (TVT), Penman-Monteith (PM) and Bowen ratio energy balance (BREB) equation were used to estimate dewfall based on meteorological data. Initially there were big disagreements between the estimates from these four equations. However, after multiplying the heat and vapour conductance terms by 0.33 the agreement was much better. This implies that the disagreements derived from improper conductance values. Initially we did not consider the effect of atmospheric stability on the conductances. With stability correction the conductances were on average 0.5 times the values without stability correction. To arrive at the aforementioned 0.33, the conductances with stability correction still need to be lower by a factor of 0.66. The value of the von Karman constant and the relationships for the zero plane displacement and the roughness length we used in our conductance computations are widely used, but not the only possible ones. With different values and relationships also suggested in the literature one can reach this factor. However, it is also possible that our wind speed data contributed to the fact that the conductances we computed were too high. Their computation for a given canopy—atmosphere system requires wind speeds from a wind profile in equilibrium with the vegetation. This in turn requires an adequate fetch around the investigated surface. The highly varied vegetation in and around the site where the study was conducted makes adequate fetch rather doubtful. To obtain valid conductance values the atmospheric stability conditions must be considered, the appropriate values for the von Karman constant, the zero plane displacement and roughness length must be used, and there must be adequate fetch. The BREB equation does not contain a conductance term and therefore does not suffer from the problems just stated. The other three equations do. However, the BREB, like the EB and TVT equations, need the surface temperature which is not routinely measured. This then leaves the PM equation from which this temperature has been eliminated as the only option. Hence, in a future study dewfall estimates from the PM equation should be compared with direct measurements with a high precision weighing lysimeter.

The study of droplet‐laden turbulent airflow over waved water surface by direct numerical simulation

AbstractThe objective of the present paper is to elucidate possible effects of sea spray on the momentum transfer in marine boundary layer under strong wind forcing conditions by performing direct numerical simulation (DNS) of turbulent, droplet‐laden airflow over a waved water surface. Three‐dimensional, turbulent Couette airflow is considered in DNS as a model of a constant‐flux layer in the atmospheric surface layer. Two‐dimensional stationary waves at the water surface are prescribed and assumed to be unaffected by the airflow and/or droplets. Droplets are considered as nondeformable spheres and tracked in a Lagrangian framework, and their impact on the carrier flow is modeled with the use of a point‐force approximation. The results show that drops dynamics and their impact on the carrier airflow is controlled by the drops velocity at injection, the ratio of drops gravitational settling velocity versus the product of air friction velocity and Karman constant ( ), and the wave slope, ka. Drops injected into the flow with the surrounding airflow velocity reduce the turbulent air‐stress and increase mean air velocity as compared to the droplet‐free case. On the other hand, the opposite effect is observed for drops injected with velocity equal to the water surface velocity, which increase the turbulent air stress and reduce the mean wind velocity. This modification of the airflow by drops is most pronounced for the ratio , increases with drops mass loading, and is reduced for steeper waves and smaller settling velocity.

Sensitivity of Turbine-Height Wind Speeds to Parameters in Planetary Boundary-Layer and Surface-Layer Schemes in the Weather Research and Forecasting Model

We evaluate the sensitivity of simulated turbine-height wind speeds to 26 parameters within the Mellor–Yamada–Nakanishi–Niino (MYNN) planetary boundary-layer scheme and MM5 surface-layer scheme of the Weather Research and Forecasting model over an area of complex terrain. An efficient sampling algorithm and generalized linear model are used to explore the multiple-dimensional parameter space and quantify the parametric sensitivity of simulated turbine-height wind speeds. The results indicate that most of the variability in the ensemble simulations is due to parameters related to the dissipation of turbulent kinetic energy (TKE), Prandtl number, turbulent length scales, surface roughness, and the von Karman constant. The parameter associated with the TKE dissipation rate is found to be most important, and a larger dissipation rate produces larger hub-height wind speeds. A larger Prandtl number results in smaller nighttime wind speeds. Increasing surface roughness reduces the frequencies of both extremely weak and strong airflows, implying a reduction in the variability of wind speed. All of the above parameters significantly affect the vertical profiles of wind speed and the magnitude of wind shear. The relative contributions of individual parameters are found to be dependent on both the terrain slope and atmospheric stability.

Anomalous dissipation and kinetic-energy distribution in pipes at very high Reynolds numbers

A symmetry-based theory is developed for the description of (streamwise) kinetic energy K in turbulent pipes at extremely high Reynolds numbers (Re's). The theory assumes a mesolayer with continual deformation of wall-attached eddies which introduce an anomalous dissipation, breaking the exact balance between production and dissipation. An outer peak of K is predicted above a critical Re of 10^{4}, in good agreement with experimental data. The theory offers an alternative explanation for the recently discovered logarithmic distribution of K. The concept of anomalous dissipation is further supported by a significant modification of the k-ω equation, yielding an accurate prediction of the entire K profile.

高レイノルズ数円管流れにおける管摩擦係数と流速分布

The mean velocity profile and the friction factor was measured at high Reynolds number using high Reynolds number actual flow facility (Hi-Reff) at NMIJ, AIST. The Reynolds number range of this experiment is from ReD=1.2×104 to 1.8×107. The Karman and additive constants of the mean velocity profile have the Reynolds number dependency for ReD<5.0×105 and they are constant for ReD>5.0×105. The Karman constants obtained from the velocity profile measurement and the friction factor measurement are completely consistent, which is 0.383, for ReD>5.0×105.

After transition in a soft-walled microchannel

In comparison to the flow in a rigid channel, there is a multifold reduction in the transition Reynolds number for the flow in a microchannel when one of the walls is made sufficiently soft, due to a dynamical instability induced by the fluid–wall coupling, as shown by Verma &amp; Kumaran (J. Fluid Mech., vol. 727, 2013, pp. 407–455). The flow after transition is characterised using particle image velocimetry in the $x{-}y$ plane, where $x$ is the streamwise direction and $y$ is the cross-stream coordinate along the small dimension of the channel of height 0.2–0.3 mm. The flow after transition is characterised by a mean velocity profile that is flatter at the centre and steeper at the walls in comparison to that for a laminar flow. The root mean square of the streamwise fluctuating velocity shows a characteristic sharp increase away from the wall and a maximum close to the wall, as observed in turbulent flows in rigid-walled channels. However, the profile is asymmetric, with a significantly higher maximum close to the soft wall in comparison to that close to the hard wall, and the Reynolds stress is found to be non-zero at the soft wall, indicating that there is a stress exerted by fluid velocity fluctuations on the wall. The maximum of the root mean square of the velocity fluctuations and the Reynolds stress (divided by the fluid density) in the soft-walled microchannel for Reynolds numbers in the range 250–400, when scaled by suitable powers of the maximum velocity, are comparable to those in a rigid channel at Reynolds numbers in the range 5000–20 000. The near-wall velocity profile shows no evidence of a viscous sublayer for $(yv_{\ast }/{\it\nu})$ as low as two, but there is a logarithmic layer for $(yv_{\ast }/{\it\nu})$ up to approximately 30, where the von Karman constants are very different from those for a rigid-walled channel. Here, $v_{\ast }$ is the friction velocity, ${\it\nu}$ is the kinematic viscosity and $y$ is the distance from the soft surface. The surface of the soft wall in contact with the fluid is marked with dye spots to monitor the deformation and motion along the fluid–wall interface. Low-frequency oscillations in the displacement of the surface are observed after transition in both the streamwise and spanwise directions, indicating that the velocity fluctuations are dynamically coupled to motion in the solid.