Logarithmic Layer Research Articles

Causality of energy-containing eddies in wall turbulence

Turbulent flows in the presence of walls may be apprehended as a collection of momentum- and energy-containing eddies (energy-eddies), whose sizes differ by many orders of magnitude. These eddies follow a self-sustaining cycle, i.e., existing eddies are seeds for the inception of new ones, and so forth. Understanding this process is critical for the modelling and control of geophysical and industrial flows, in which a non-negligible fraction of the energy is dissipated by turbulence in the immediate vicinity of walls. In this study, we examine the causal interactions of energy-eddies in wall-bounded turbulence by quantifying how the knowledge of the past states of eddies reduces the uncertainty of their future states. The analysis is performed via direct numerical simulation (DNS) of turbulent channel flows in which time-resolved energy-eddies are isolated at a prescribed scale. Our approach unveils, in a simple manner, that causality of energy-eddies in the buffer and logarithmic layers is similar and independent of the eddy size. We further show an example of how novel flow control and modelling strategies can take advantage of such self-similar causality.

Application of k-ε turbulence models with density corrections to variable density jets under subcritical/supercritical conditions

A density correction method is presented for improving the performance of turbulence models for variable-density jets under subcritical and supercritical conditions. The functional relationship between density and turbulence variables was obtained by considering the logarithmic layer and local equilibrium conditions and the basic form of the relationship was discussed with the large-eddy simulation results. The modified eddy viscosity and a source term of the -equation were introduced in four models, and the modified models were tested for helium, air, and carbon dioxide jets in the presence of co-flowing air. The velocity and density fields of the density-corrected models were in good agreement with the experiment. Cryogenic nitrogen jets under near-critical and supercritical pressures were selected for examining the applicability of the models to the variable density flow with heat transfer. The high velocity decay rates of the original models were found to be considerably reduced in the modified models. The effect of density correction was consistent among the four models with wall function and near-wall treatment.

Existence and properties of the logarithmic layer in oscillating flows

The existence and properties of the logarithmic layer in a turbulent streamwise oscillating flow are investigated through direct numerical simulations and wall-resolved large-eddy simulations. The phase dependence of the von Kármán constant and the logarithmic layer intercept is explored for different values of the Reynolds number and the depth-ratio between the water depth and the Stokes boundary layer thickness. The logarithmic layer exists for a longer fraction of the oscillating period and a larger fraction of the water depth with increasing values of the Reynolds number. However, the values of both the von Kármán and the intercept depend on the phase, the Reynolds number and depth-ratio. Additionally, the simulations characterized by a low value of the depth-ratio and Reynolds number show intermittent existence of the logarithmic layer. Finally, the Reynolds number based on the friction velocity does not support a previously mentioned analogy between oscillatory flows and steady wall-bounded flows.

Space–time formation of very-large-scale motions in turbulent pipe flow

We examine the origin of very-large-scale motions (VLSMs) in fully developed turbulent pipe flow at friction Reynolds number, $\mathit{Re}_{\unicode[STIX]{x1D70F}}=934$, using data from a direct numerical simulation. The VLSMs and the packet-like large-scale motions (LSMs) found in this study are very similar to those found in earlier studies. Three-dimensional time-evolving instantaneous fields show that one component of the process leading to the large streamwise length of VLSMs is the concatenation of adjacent streamwise LSMs caused by the continuous elongation of LSMs due to the strain component of the mean shear. Spatial organization patterns of the VLSMs and LSMs and their properties are studied by separating auto-correlation of the streamwise velocity fluctuations into the components of the VLSM and the LSM defined by low-pass/high-pass filtering in the streamwise direction. The structures of the two-point spatial correlations of the streamwise velocity component of the VLSMs and the LSMs in the streamwise-azimuthal plane are characterized by multiple maxima and complex patterns that beg explanation in terms of patterned coherent arrangements of the LSMs. Using proper orthogonal decomposition (POD), it is found that the X-shape correlation pattern of the VLSMs results from the superposition of very long helically inclined structures and streamwise-aligned structures. Further explanation of the patterns in the correlations of the VLSMs and LSMs is provided through the study of synthetically constructed arrangements of simple hairpin packet models of the LSM. Head-to-tail alignment of the model packets along streamwise and helical directions suggested by the eigenvalues of the POD creates a pair of long roll-cells centred above the logarithmic layer, and bracketing the LSMs. These roll-cells are pure kinematic consequences of the induction within the LSM packets, but they may also serve to organize smaller packets.

A Dual-Grid Hybrid RANS/LES Model for Under-Resolved Near-Wall Regions and its Application to Heated and Separating Flows

A hybrid RANS/LES model for high Reynolds number wall-bounded flows is presented, in which individual Reynolds-Averaged Navier-Stokes (RANS) and Large Eddy Simulations (LES) are computed in parallel on two fully overlapping grids. The instantaneous, fluctuating subgrid-scale stresses are blended with a statistical eddy viscosity model in regions where the LES grid is too coarse. In the present case, the hybrid model acts as a near-wall correction to the LES, while it retains the fluctuating nature of the flow field. The dual computation enables the LES to be run on isotropic grids with very low wall-normal and wall-parallel resolution, while the auxiliary RANS simulation is conducted on a wall-refined high-aspect ratio grid. Running distinct, progressively corrected simulations allows a clearer separation of the mean and instantaneous flow fields, compliant with the fundamentally dissimilar nature of RANS and LES. Even with the wall-nearest grid point lying far in the logarithmic layer, velocity and temperature predictions of a heated plane channel flow are corrected. For a periodic hill flow, the dual-grid system improves the boundary layer separation and velocity field prediction both for a constant-spaced and a wall-refined LES grid.

Wall turbulence response to surface cooling and formation of strongly stable stratified boundary layers

This paper investigates the processes by which stable boundary layers are formed through strong surface cooling imposed on neutrally stratified wall-bounded turbulence using high-resolution direct numerical simulation at a moderate Reynolds number. The adjustment of the flow to the imposed strong surface cooling is investigated. We further focus on a strongly stable case where turbulence partially collapses. We show that, due to a significant reduction in turbulence production, turbulence becomes patchy, with a band of turbulence coexisting with quiet regions. The nature of the quiet regions, which are often characterized as laminar, is investigated and shown to be consistent with viscously coupled stratified turbulence. The one-dimensional longitudinal streamwise velocity spectrum exhibits kx−5 and kx−3 behavior in the buffer and logarithmic layers, respectively, adjacent to an active region of three-dimensional turbulence with a kx−5/3 spectrum. Scenarios for turbulence recovery from such a patchy state are also discussed. We show that the presence of outer layer turbulence above z+ ≈ 300 is a key requirement for recovery. For higher values of stratification, it is shown that inner layer turbulence is damped entirely and outer layer turbulence is damped subsequently.

Mapping spheroid rotation modes in turbulent channel flow: effects of shear, turbulence and particle inertia

The rotational behaviour of non-spherical particles in turbulent channel flow is studied by Lagrangian tracking of spheroidal point particles in a directly simulated flow. The focus is on the complex rotation modes of the spheroidal particles, in which the back reaction on the flow field is ignored. This study is a sequel to the letter by Zhao et al. (Phys. Rev. Lett., vol. 115, 2015, 244501), in which only selected results in the near-wall buffer region and the almost-isotropic channel centre were presented. Now, particle dynamics all across the channel is explored to provide a complete picture of the orientational and rotational behaviour with consideration of the effects of particle aspect ratio ranging from 0.1 to 10 and particle Stokes number from 0 (inertialess) to 30. The rotational dynamics in the innermost part of the logarithmic wall layer is particularly complex and affected not only by modest mean shear, but also by particle inertia and turbulent vorticity. While inertial disks exhibit modest preferential orientation in either the wall-normal or cross-stream direction, inertial rods show neither preferential tumbling nor spinning. Examination of the co-variances between particle orientation, particle rotation and fluid rotation vectors explains the qualitatively different ‘wall mode’ rotation and ‘centre mode’ rotation. Inertialess spheroids transition between the two modes within a narrow zone ($15<z^{+}<35$) in the buffer region. If the spheroids have inertia, the transition zone between the two modes shifts to the inner part of the logarithmic layer, i.e. $z^{+}\geqslant 40$. We ascribe the transition of inertialess spheroids from the ‘wall mode’ to the ‘centre mode’ rotation to the changeover between the time scales associated with mean shear and small-scale turbulence. Inertial spheroids, however, transition between the two rotational modes when the Kolmogorov time scale becomes comparable to the time scale for particle rotation, i.e. the effective Stokes number is of order unity. The aforementioned findings reveal, in addition to the effects of particle shape and alignment, the importance of the characteristic local time scale of fluid flow for the rotation of both tracer and inertial spheroids in turbulent channel flows.

Shear stress-driven flow: the state space of near-wall turbulence as

An inner-scaled, shear stress-driven flow is considered as a model of independent near-wall turbulence as the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$. In this limit, the model is applicable to the near-wall region and the lower part of the logarithmic layer of various parallel shear flows, including turbulent Couette flow, Poiseuille flow and Hagen–Poiseuille flow. The model is validated against damped Couette flow and there is excellent agreement between the velocity statistics and spectra for the wall-normal height $y^{+}<40$. A near-wall flow domain of similar size to the minimal unit is analysed from a dynamical systems perspective. The edge and fifteen invariant solutions are computed, the first discovered for this flow configuration. Through continuation in the spanwise width $L_{z}^{+}$, the bifurcation behaviour of the solutions over the domain size is investigated. The physical properties of the solutions are explored through phase portraits, including the energy input and dissipation plane, and streak, roll and wave energy space. Finally, a Reynolds number is defined in outer units and the high-$Re$ asymptotic behaviour of the equilibria is studied. Three lower branch solutions are found to scale consistently with vortex–wave interaction (VWI) theory, with wave forcing localising around the critical layer.

Thin Layer Drying Kinetics of Indian Blackberry (Syzygium cumini L.) Pulp

Drying kinetics of fruit has importance in estimating optimum nutritional retainability, preservation requisites and process economy while processing into useful products. Drying kinetics of pulp of local variety of Indian blackberry was studied in a cabinet dryer. The drying experiment was conducted at 50 ºC, 60 ºC and 70 ºC, and moisture losses were recorded at 30-min intervals. Drying took place in two falling rate periods. Shift of first to second falling rate period was found to be irrespective of temperature, and started from 1.08 g water.g−1 dry matter. The drying data of Indian blackberry pulp was fitted into five commonly used Newton, Page, Peleg, Henderson and Pabis, and Logarithmic thin layer drying models. The logarithmic model adequately described the drying of Indian blackberry pulp. The activation energy values in first and second falling rate periods were 42.20 kJ.mol−1 and 61.62 kJ.mol−1, respectively.

Direct numerical simulations of Taylor–Couette turbulence: the effects of sand grain roughness

Progress in roughness research, mapping any given roughness geometry to its fluid dynamic behaviour, has been hampered by the lack of accurate and direct measurements of skin-friction drag, especially in open systems. The Taylor–Couette (TC) system has the benefit of being a closed system, but its potential for characterizing irregular, realistic, three-dimensional (3-D) roughness has not been previously considered in depth. Here, we present direct numerical simulations (DNSs) of TC turbulence with sand grain roughness mounted on the inner cylinder. The model proposed by Scotti (Phys. Fluids, vol. 18, 031701, 2006) has been modified to simulate a random rough surface of monodisperse sand grains. Taylor numbers range from $Ta=1.0\times 10^{7}$(corresponding to $Re_{\unicode[STIX]{x1D70F}}=82$) to $Ta=1.0\times 10^{9}$ ($Re_{\unicode[STIX]{x1D70F}}=635$). We focus on the influence of the roughness height $k_{s}^{+}$ in the transitionally rough regime, through simulations of TC with rough surfaces, ranging from $k_{s}^{+}=5$ up to $k_{s}^{+}=92$. We analyse the global response of the system, expressed both by the dimensionless angular velocity transport $Nu_{\unicode[STIX]{x1D714}}$ and by the friction factor $C_{f}$. An increase in friction with increasing roughness height is accompanied with enhanced plume ejection from the inner cylinder. Subsequently, we investigate the local response of the fluid flow over the rough surface. The equivalent sand grain roughness $k_{s}^{+}$ is calculated to be $1.33k$, where $k$ is the size of the sand grains. We find that the downwards shift of the logarithmic layer, due to transitionally rough sand grains exhibits remarkably similar behaviour to that of the Nikuradse (VDI-Forsch., vol. 361, 1933) data of sand grain roughness in pipe flow, regardless of the Taylor number dependent constants of the logarithmic layer. Furthermore, we find that the dynamical effects of the sand grains are contained to the roughness sublayer $h_{r}$ with $h_{r}=2.78k_{s}$.

Velocity and spatial distribution of inertial particles in a turbulent channel flow

We present experimental observations of the velocity and spatial distribution of inertial particles dispersed in turbulent downward flow through a vertical channel at friction Reynolds numbers $\mathit{Re}_{\unicode[STIX]{x1D70F}}=235$ and 335. The working fluid is air laden with size-selected glass microspheres, having Stokes numbers $St=\mathit{O}(10)$ and $\mathit{O}(100)$ when based on the Kolmogorov and viscous time scales, respectively. Cases at solid volume fractions $\unicode[STIX]{x1D719}_{v}=3\times 10^{-6}$ and $5\times 10^{-5}$ are considered. In the more dilute regime, the particle concentration profile shows near-wall and centreline maxima compatible with a turbophoretic drift down the gradient of turbulence intensity; the particles travel at speed similar to that of the unladen flow except in the near-wall region; and their velocity fluctuations generally follow the unladen flow level over the channel core, exceeding it in the near-wall region. The denser regime presents substantial differences in all measured statistics: the near-wall concentration peak is much more pronounced, while the centreline maximum is absent; the mean particle velocity decreases over the logarithmic and buffer layers; and particle velocity fluctuations and deposition velocities are enhanced. An analysis of the spatial distributions of particle positions and velocities reveals different behaviours in the core and near-wall regions. In the channel core, dense clusters form which are somewhat elongated, tend to be preferentially aligned with the vertical/streamwise direction and travel faster than the less concentrated particles. In the near-wall region, the particles arrange in highly elongated streaks associated with negative streamwise velocity fluctuations, several channel heights in length and spaced by $\mathit{O}(100)$ wall units, supporting the view that these are coupled to fluid low-speed streaks typical of wall turbulence. The particle velocity fields contain a significant component of random uncorrelated motion, more prominent for higher $St$ and in the near-wall region.

Large-eddy simulation of an open-channel flow bounded by a semi-dense rigid filamentous canopy: Scaling and flow structure

We have carried out a large-eddy simulation of a turbulent open-channel flow over a marginally dense, fully submerged, rigid canopy. The canopy is made of a set of rigid, slender cylinders normally mounted on a solid wall. The flow in the canopy is resolved stem-by-stem by means of an immersed boundary method. It is found that the flow behavior can be categorized according to the velocity distribution into two separate spatial regions: the canopy itself and the outer region above it. Within the region occupied by the canopy elements, the velocity magnitude is found to be related to the local shear stress. Outside the canopy, a logarithmic velocity profile matching the canonical turbulent open-channel flow over rough walls is recovered albeit the strong manipulation exerted by the canopy on the buffer layer. In the innermost layer, the presence of the stems is responsible for redistributing the local momentum fluctuations from a streamwise to a spanwise leading component, inhibiting the survival of the wall streamwise velocity streaks. On the other hand, the outer region presents a structure very similar to the well-known logarithmic boundary layer with the presence of large and energetic streamwise velocity streaks generated by a system of quasistreamwise vortices. These vortices strongly contribute to the intracanopy fluctuations through vigorous sweep and ejection events that affect all the region occupied by the stems. Consistent with the results of previous investigations [H. Nepf, “Flow and transport in regions with aquatic vegetation,” Annu. Rev. Fluid Mech. 44, 123–142 (2012)], it is found that the inflection point in the mean velocity profile, produced by the drag discontinuity at the canopy tip, promotes the appearance of another system of spanwise oriented vorticity structures. However, different from previous results reported in the literature [J. Finnigan, “Turbulence in plant canopies,” Annu. Rev. Fluid Mech. 32, 519–571 (2000)], in our simulations, the presence of alternating head up–head down hairpin vortices generated by a mutual induction of the counter-rotating spanwise vortices is not observed. Instead, we advocate that the modulation of the spanwise coherent vorticity is due to the action of the external logarithmic layer structures (i.e., the outer streamwise vortices that penetrate the canopy) rather than by upwash and downwash motions induced by the mutual interaction of the spanwise rollers.

Turbulence, Sediment‐Induced Stratification, and Mixing Under Macrotidal Estuarine Conditions (Qiantang Estuary, China)

AbstractTime series of in situ measured velocity and suspended sediment concentration from Qiantang Estuary (China), and estimates of turbulence and sediment stratification parameters are presented. The data span a period of 9 days, and after phase averaged, they are used to explore spring‐neap tidal variations in flow, turbulence, and sediment stratification. A local balance between shear production, sediment‐induced buoyancy flux, and dissipation is found to hold during ebb for both neap and spring tides. During flood elevated turbulence dissipation rates are observed, attributed to nonlocal turbulence, most likely due to horizontal advection. Our results show that the effect of sediment stratification is successfully parameterized by adding the Monin‐Obukhov length scale to the classical logarithmic layer theory. Flood‐ebb asymmetry in both Rig and Rf is observed, with higher values attained during flood due to the higher sediment concentrations and the corresponding weaker velocity shear found at these times. Rf is found to increase with Rig when Rig < 0.25, and it attains a maximum value of ~0.2. Our estimates, consistently with those from previous studies, fall slightly below the commonly used theoretical predictions. Sediment stratification contributes to the decay of turbulence, and weak mixing still exists under high Rig numbers.

On the departure of near-wall turbulence from the quasi-steady state

An examination is undertaken of the validity and limitations of the quasi-steady hypothesis of near-wall turbulence. This hypothesis is based on the supposition that the statistics of the turbulent fluctuations are universal if scaled by the local, instantaneous, wall shear when its variations are determined from footprints of large-scale, energetic, structures that reside in the outer part of the logarithmic layer. The examination is performed with the aid of direct numerical simulation data for a single Reynolds number, which are processed in a manner that brings out the variability of locally scaled statistics when conditioned on the local value of the wall friction. The key question is to what extent this variability is insignificant, thus reflecting universality. It is shown that the validity of the quasi-steady hypothesis is confined, at best, to a thin layer above the viscous sublayer. Beyond this layer, substantial variations in the conditioned shear-induced production rate of large-scale turbulence cause substantial departures from the hypothesis. Even within the wall-proximate layer, moderate departures are provoked by large-scale distortions in the conditioned strain rate that result in variations in small-scale production of turbulence down to the viscous sublayer.

Study on Flow Characteristics of a Turbulent Boundary Layer and Vortex Structure of High Pressure Guide Vanes in SCO2 Turbines

In order to further understand the aerothermodynamic performance and flow loss mechanism of SCO2 turbines, RANS equations and an SST Turbulence Model were chosen for a numerical study on the secondary flow and vortex structure of cascades using the commercial software CFX. The dimensionless vorticity analysis method was used to study the flow characteristics of the logarithmic layer and viscous sublayer in high pressure guide vane cascades. The new vortex structure and formation mechanism of the vortices were given and analyzed. Simulation results indicated that during the motion of the boundary layer in the cascades, the logarithmic layer and viscous sublayer obtain the different rotational direction vorticity, respectively. The endwall logarithmic layer and pressure side leg of the horseshoe votex gradually develop into the passage vortex, with the endwall viscous sublayer gradually developing into the corner viscous sublayer vortex II and the endwall viscous sublayer vortex I. The endwall viscous sublayer that rolled by the passage vortex is encountered with the upstream-side and radial boundary layer of the vane at the suction separation line, forming the suction separation line vortex beside the passage vortex. A pair of radial transition vortices are formed between the wake and the main stream.

Experimental evaluation of the mean momentum and kinetic energy balance equations in turbulent pipe flows at high Reynolds number

ABSTRACT In light of recent data from hot-wire anemometry and laser Doppler velocimetry, this article explores experimentally the momentum balance and kinetic energy production in fully developed turbulent pipe flow for shear Reynolds numbers in the range from two pipe facilities. It has become common practice to indirectly deduce the Reynolds shear stress via the mean flow data and the mean-momentum balance whenever the simultaneous measurements of the streamwise and wall-normal velocity fluctuations can not be performed precisely. The current assessment underlines, however, the importance of measuring the Reynolds shear stress directly, and the friction velocity independently from the mean-velocity profile to ascertain the quality of the data when utilising the momentum balance. The present analysis also reinforces the universality of the viscous stress gradient to the Reynolds shear stress gradient in the wall vicinity up to the inner limit of the logarithmic layer. The new set of the experimental data shows that Panton's stress function reproduces the measured Reynolds shear stress and kinetic energy production in turbulent pipe flows over a wide Reynolds number range to a high degree.

Investigation of the Boundary Layer Flow Under Engine-Like Conditions Using Particle Image Velocimetry

The turbulent boundary layer flow in internal combustion (IC) engines has a significant effect on the in-cylinder flow and the wall heat transfer. A detailed analysis of the in-cylinder near-wall flow was carried out on an optical steady flow test bench by using high-resolution particle image velocimetry (PIV) in order to characterize the in-cylinder boundary layer flow in this study. The difference between the in-cylinder boundary layer and the canonical turbulent boundary layer was analyzed. The experimental results show that small-scale vortices with a length scale of about 1–2 mm in the instantaneous flow fields appeared in the wall jet region due to the entrainment of the free jet in the outer region of the wall jet. The viscous sublayer thickness decreased from 0.5 mm to 0.3 mm as the valve lift increased from 2.32 mm to 7.975 mm and the pressure drop from 0.5 kPa to 1 kPa. The dimensionless velocity profile is in good agreement with the law of the wall in the viscous sublayer. However, no obvious logarithmic law distribution region was observed in the logarithmic layer. The distribution of the Reynolds stress and the turbulent kinetic energy is similar to that of the canonical turbulent boundary layer. But the Reynolds stress had a much larger magnitude because the turbulent velocity measured in this boundary layer included not only the turbulence generated by wall shear but also the large-scale turbulent vortices caused by the wall jet.

Numerical study of plane Couette flow: turbulence statistics and the structure of pressure–strain correlations

AbstractThe paper presents the results of direct numerical simulation of turbulent plane Couette flow. The calculations were performed for Reynolds numbers Re =U0H/ν(His the height of the channel, ±U0/2 is the motion velocities of the lower and upper walls, respectively,νis the kinematic viscosity) from 5200 (where viscous effects significantly affect the flow structure) to 80000 (where a logarithmic layer is clearly observed). Estimates of terms of the equation for the balance of turbulent Reynolds stresses are obtained, which indicate the importance of the kinetic energy transport by velocity fluctuations. The vertical transport of the turbulent momentum flux is less important and partly compensated by the transport of pressure fluctuations. It is shown that in the logarithmic layer the normal components of the ‘pressure–strain rate’ correlation tensor are described in the framework of the ‘isotropization of production’ model, and in the central part of the channel they are described by the linear Rotta model [29]. The additive model considering both the interaction of the velocity field fluctuation and the influence of the mean velocity gradient is a good approximation only for the off-diagonal component of the tensor entering the balance equation for the turbulent momentum flux.

Characteristic scales of Townsend's wall-attached eddies.

Townsend (The Structure of Turbulent Shear Flow, 1976, Cambridge University Press) proposed a structural model for the logarithmic layer (log layer) of wall turbulence at high Reynolds numbers, where the dominant momentum-carrying motions are organised into a multiscale population of eddies attached to the wall. In the attached-eddy framework, the relevant length and velocity scales of the wall-attached eddies are the friction velocity and the distance to the wall. In the present work, we hypothesise that the momentum-carrying eddies are controlled by the mean momentum flux and mean shear with no explicit reference to the distance to the wall and propose new characteristic velocity, length and time scales consistent with this argument. Our hypothesis is supported by direct numerical simulation of turbulent channel flows driven by non-uniform body forces and modified mean velocity profiles, where the resulting outer-layer flow structures are substantially altered to accommodate the new mean momentum transfer. The proposed scaling is further corroborated by simulations where the no-slip wall is replaced by a Robin boundary condition for the three velocity components, allowing for substantial wall-normal transpiration at all length scales. We show that the outer-layer one-point statistics and spectra of this channel with transpiration agree quantitatively with those of its wall-bounded counterpart. The results reveal that the wall-parallel no-slip condition is not required to recover classic wall-bounded turbulence far from the wall and, more importantly, neither is the impermeability condition at the wall.

Flow structure within a vegetation patch in a gravel-bed river

Abstract Investigation of the interactions between submerged vegetation patch and flow structure is of crucial importance for river engineering. Most of hydraulic models have been presented for fully developed flows over uniform vegetation in the laboratory conditions; however, the mentioned interactions are complex in river flows where the flow is not developed along small patch. This reveals a gap between developed and non-developed flow along the vegetation patch. This study was conducted in a gravel-bed river in the central Iran. The results reveal that the flow structure in evolving flow (non-developed flow) along the patch resembles that in shallow mixing layer. Accordingly, a shallow mixing layer model and modified equations are combined to quantify evolving area along the patch. The evolving shallow mixing layer equations for the flow along a non-uniform vegetation patch reach a reasonable agreement with field data. However, the spreading coefficient of this model less than one was reported in literature, 0.06 and 0.12. In addition, the flow immediately downstream the vegetation patch behaves similar to a jet and is parameterized by two conventional models, conventional logarithmic law and mixing layer theory. These models present a reasonable agreement with the measured velocity profiles immediately downstream the patch.