Law Of The Wall Research Articles

An Enhanced Depth-integrated Model for Flows over a Negative Step with Hydraulic Jump

Depth-integrated models play an important role of predicting flows for several practical problems. There are many contributions to develop an enhanced depth-integrated model to have the ability of evaluating vertical flow structures. However, it is still challenging for the depth integrated models to calculate flows over a negative step accompanying a hydraulic jump, because the separation zone behind the step and the pattern of the hydraulic jump depend on the several hydraulic conditions such as Froude number, step height and downstream water depth. In this study, assuming gravel bed rivers in which the bed is covered with relatively large roughness, an enhanced depth integrated model is developed to calculate various hydraulic jump patterns generated downstream from the negative step. The present method is based on General Bottom Velocity Computation method employing Dynamic Wall Law with 4th degree polynomial velocity distribution (GBVC4-DWL). The model with the virtual bed slope behind the negative step and the critical slope of wave breaking is validated through the comparisons with the experimental results on flows over a negative step for submerged jet. weak jump and undular jump conditions. Then the essential terms of equations which compose the present method to calculate the flow downstream of the negative step are investigated.

On the convergence and scaling of high-order statistical moments in turbulent pipe flow using direct numerical simulations

Direct numerical simulations of turbulent pipe flow in a flow domain of length L = 42R, friction Reynolds number in the range of 180 ≤ Reτ ≤ 1500, and two different wall-normal grid refinements were carried out and investigated in terms of high-order turbulence statistics. The phenomenology of large local wall-normal velocity fluctuations (velocity spikes) was discussed by means of time series and instantaneous flow-field realisations. Due to their rare appearance both in space and time, statistical high-order moments take a long time to converge. A convergence study was performed and for fully converged statistics the sensitivity of the grid resolution on the wall-normal kurtosis component value at the wall as well as the scaling behaviour of high-order statistics was investigated. The streamwise Reynolds stress as well as the streamwise skewness and the wall-normal flatness exhibited logarithmic Reynolds number dependencies in the vicinity of the wall and scaling laws were derived accordingly. In the bulk flow region, a sudden increase in magnitude in both the streamwise Reynolds stress and skewness was determined for the largest Reynolds number Reτ = 1500, while the profiles collapsed well in wall units for Reτ ≤ 720. Both Reynolds number dependencies in the near-wall and the bulk region could be related to large-scale outer-flow motions penetrating the buffer layer. While wavelengths related to larger-scale motions (λz ≈ 3R) were computed for Reynolds numbers up to Reτ = 720 by means of two-dimensional two-point velocity correlations, even larger wavelengths related to very-large-scale motions appeared for Reτ = 1500. They are probably the reason for the sudden increase in magnitude of streamwise Reynolds stress and skewness, respectively. With the aid of instantaneous flow-field realisations and conditional averaged statistics, the Reynolds dependency of the wall-normal flatness value at the wall was related to the scaling failure of the streamwise Reynolds stress peak. For the lowest Reynolds number (Reτ = 180), discrepancies between plane channel and pipe flow were found and discussed.

Laws of the wall for velocity and temperature in supersonic turbulent boundary layers

AbstractScaling laws for velocity and temperature profiles in the near‐wall region of sub‐ and supersonic turbulent boundary layers have been developed, which allow us to represent velocity and temperature profiles in compressible gas stream in terms of those in an incompressible boundary layer. They are obtained as asymptotic expansions of the solutions to the Reynolds equations in a small parameter — the Mach number based on the friction velocity and gas enthalpy on the wall. The leading term of the expansion for velocity corresponds to known Van Driest's formula. However, the obtained solution contains additional terms of order unity, which explains the contradiction between Van Driest's formula and experimental data. The law of the wall for temperature, which has been formulated for the first time, has an analogous structure. Besides the von Kármán constant and the turbulent Prandtl number in the logarithmic region, known for incompressible flow, the obtained relations contain three new universal constants, which do not depend on gas molecular properties and the specific heat ratio. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Observations of Tidal Straining Within Two Different Ocean Environments in the East China Sea: Stratification and Near‐Bottom Turbulence

AbstractTidal straining describes the straining effect induced by the vertical shear of oscillatory tidal currents that act on horizontal density gradients. It tends to create tidal periodic stratification and modulate the turbulence in the bottom boundary layer (BBL). Here, we present observations of current, hydrology and turbulence obtained at two mooring stations that are characterized by two typical hydrological environments in the East China Sea (ECS). One is located adjacent to the Changjiang River's mouth, and the other is located over a sloping shelf which is far from the freshwater sources. Tidal straining induces a semidiurnal switching between stable and unstable stratification at both stations. Near‐bottom high‐frequency velocity measurements further reveal that the dissipation rate of turbulent kinetic energy (TKE) is highly elevated during periods when unstable stratification occurs. A comparison between the TKE dissipation rate ( ) and the shear production ( ) further reveals that the near‐bottom mixing is locally shear‐induced most of the time except during the unstable stratification period. Within this period, the magnitude of dissipation exceeds the expected value based on the law of the wall by an order of magnitude. The buoyancy flux that calculated by the balance method is too small to compensate for the existing discrepancy between the dissipation and shear production. Another plausible candidate is the advection of TKE, which may play an important role in the TKE budget during the unstable stratification period.

Near-Wall Stress Balance in Front of a Wall-Mounted Cylinder

The stress balance in the near-wall flow in front of a cylinder mounted on a flat plate at moderate Reynolds number is investigated by applying highly resolved Large-Eddy Simulation (LES). The flow around wall-mounted bluff bodies is subject of research due to its wide relevance for engineering applications. However, the structure of the vortex system in front of such a bluff body is complex, bears strong velocity and pressure gradients in each spatial direction and has rich dynamics. Furthermore, the vortex system is located close to the investigated flat bottom wall (Dargahi, Exp. Fluids 8(1-2):1–12, 1989; Devenport and Simpson, J. Fluid Mech. 210:23–55, 1990). Thus, classical models for the treatment of the near-wall flow based on the logarithmic law of the wall or a power law cannot be expected to suffice in such kind of flow (Pope 2011). This paper assesses which contributors to the stress balance have significant influence on the balances residual and thus have to be considered by an approach to model the investigated near-wall flow. To do so, the momentum equation in streamwise direction is integrated in wall-normal direction and applied to the results gained from the LES. The evaluation of the stress balance along four selected wall-normal profiles indicates that the significance of each single term depends on where the profile is located. Outside the viscous layer, no term except the viscous stresses can be neglected in general. The amplitude of the pressure gradient as well as horizontal gradients of mean and fluctuating velocity are multiples of the estimated wall shear stress. Wall models not including a spatial approach are therefore most likely to fail in such kind of flow.

Scaling of the streamwise turbulence intensity in the context of inner-outer interactions in wall turbulence

The classical view of wall-bounded turbulence considers a near-wall inner region where all velocity statistics are universally dependent on distance from the wall when scaled with friction velocity and the kinematic viscosity of the fluid. This is referred to as an inner scaling and leads to Prandtl's law of the wall. Data from numerical simulations and experiments over the past decade or so, however, have provided compelling evidence that statistics of the fluctuating streamwise velocity do not follow inner scaling in this near-wall region and an interaction of outer and logarithmic regions exists, resulting in a Reynolds number dependence. In this paper we briefly review some of these studies and discuss the Reynolds number dependence of the streamwise turbulence intensity near the wall in terms of an inner-outer interaction. An established model for such an interaction between near-wall and logarithmic region turbulence is considered that comprises two mechanisms: superposition and modulation. Here outer-region motions, of which a fraction is wall-attached, are superimposed onto the near-wall dynamics, and concurrently the near-wall motions are modulated by this superimposed signature. We discuss to what extent the superposition effect can relate changes in the inner-scaled near-wall peak value of streamwise turbulence intensity to logarithmic region turbulence resembling features of attached eddies.

Quasi-3D two-phase model for dam-break flow over movable bed based on a non-hydrostatic depth-integrated model with a dynamic rough wall law

The assumptions of equilibrium flow conditions neglecting the flow acceleration for vertical velocity distributions and of the equilibrium wall law applied to the bottom boundary condition, in which the flow acceleration is neglected in the vicinity of the bed, are questionable when used to calculate sediment transport for flows that vary rapidly over time and space. Examples of such flows include dam-break flows and flows around structures. This study proposes a two-phase depth-integrated model for large-scale geophysical flow applications involving sediment transport phenomena and bed morphology. The model for the fluid phase is based on the non-hydrostatic quasi-3D method, and uses a dynamic rough wall law that employs continuity and momentum equations for the bottom boundary conditions.It was confirmed that the proposed model reduces to the previous bedload formulae for uniform flow conditions under the weak sediment transport condition. The model was applied in an experiment involving a dam-break flow on a movable bed channel with a suddenly enlarged section. The comparison between the experimental results and the results calculated with the proposed model and previous models demonstrates the validity of the proposed model. The comparison also highlights the advantages of introducing a quasi-3D two-phase model to evaluate vertical velocity distributions and non-equilibrium sediment motions.

Plate-spacing effect on unsteady flows and heat transfer characteristics in a channel with inclined plates

Flow and heat transfer characteristics in a channel with inclined plates are investigated numerically. To determine the effect of plate spacing, the ratios of plate pitch to channel height L/H = 1-9; plate angles a = 60°, 90° and 120°; and plate lengths d/H = 0.2-0.6 are adopted and unsteady simulations performed. As L/H increases, unsteady flows related to the supercritical Hopf bifurcation are developed and heat transfer characteristics severely changed. Hence, the peculiar relation between plate pitch and vortex evolution is observed. When the plate pitch increases, the observed vortices between the inclined plates can be classified into three patterns, namely, recirculation bubble, standing vortex, and traveling vortex. Unsteady features depending on the plate pitch are also examined by spectral analysis. As the plate pitch increases, the steady state of recirculation bubble is changed gradually to a quasi-periodic state of the traveling vortex through the periodic state of the standing vortex. From these results, the heat transfer enhancement is discussed with vortex evolution, flow oscillation, and law of the wall.

A Relatively Simple Integral Method for Turbulent Flow Over Rough Surfaces

The integral form of the equation for x momentum is solved for the skin friction coefficient, in external thin boundary layer flow, on surfaces whose technical roughness elements' size is given. This is done by using a “roughness depression function” in the law of the wall and wake which serves as the needed velocity profile. The method uses the equivalent sand grain size concept in its calculations. Predictions are made of the friction coefficient, Cf, as a function of momentum thickness Reynolds number and also, of Cf's dependence on the ratio of momentum thickness to the size of the technical (actual) roughness elements. In addition, boundary layer thicknesses and velocity profiles on rough surfaces are calculated and, when available, comparisons are made with the experimental data from a number of sources in the literature. Also, comparisons are made with the results of another major predictive scheme which does not use the equivalent sand grain concept.

Distribución vertical de sedimentos en suspension en la zona de desembocadura del Rio Magdalena, Colombia

Vertical measurements of velocity and suspended sediment concentration obtained in two climatic seasons at the mouth of the Magdalena River, Colombia, were used to estimate 1) bed shear velocity, bed shear stress and the bed roughness length by Wall law, 2) the settling velocity, by fitting distributions Rouse and Rouse equation modified and vertical profiles of suspended sediment concentration, 3) bed load sediment transport parameter estimates were determined with the formulations of Meyer-Peter & Muller (1948), Nielsen (1992), and Ribberink (1998). From the estimated parameters, a number of Rouse significantly less than one (0.01 to 0.18), a settling velocity (0.08-3.15 mm s -1 ) much less than bed shear velocity (15-70 mm s -1 ), and high values in shear stress (1.1-5.8 Pa) were obtained. It can be said therefore that in the River Magdalena predominates sediment transport in suspension. The values of bottom sediment transport calculated with the three formulations represent lower rates to 5% of total sediment transport. The formulation of Meyer-Peter & Muller (1948) underestimate the transport of bottom sediment due to the conditions under which it was originally calibrated, while Nielsen (1992) and Ribberink (1998) formulations can generate better estimates, because they were originally calibrated with fine sediment, similar to the sediment type reported in the study area.

Assessment of the wall heat transfer in 3D-CFD in-cylinder simulations of high performance diesel engines

Heat transfer through the combustion chamber walls affects largely engine efficiency, emissions and engine component thermo-mechanical reliability. As a consequence, a correct estimation of the wall heat fluxes is mandatory. However, due to the complexity of the phenomena involved in the heat transfer mechanism, such an estimation is far from being trivial.The paper proposes a methodology aiming at evaluating wall heat transfer in 3D-CFD in cylinder combustion simulations of current production compression ignition engines. A high performance DICI engine is simulated, operated at part-load and medium revving speed. Particular care is devoted to both the evaluation of the auto-ignition delays and the selection of a heat transfer model able to predict correctly gas-to-wall heat transfer. As for the delays, they are evaluated adopting a validated mechanism for a Diesel surrogate composed by a blend of n–dodecane and m–xylene. As for the heat transfer, two different models are tested: a more “consolidated” one, widely available in commercial CFD codes, and a more recent one, based on an alternative formulation of the thermal law of the wall. Finally, in order to validate the proposed methodology, numerical heat transfer is compared to an experimental target.

Turbulent thermal boundary layer on a plate. Reynolds analogy and heat transfer law over the entire range of prandtl numbers

Arational asymptotic theory is proposed,which describes the turbulent dynamic and thermal boundary layer on a flat plate under zero pressure gradient. The fact that the flow depends on a finite number of governing parameters makes it possible to formulate algebraic closure conditions relating the turbulent shear stress and heat flux with the gradients of the averaged velocity and temperature. As a result of constructing an exact asymptotic solution of the boundary layer equations, the known laws of the wall for velocity and temperature, the velocity and temperature defect laws, and the expressions for the skin friction coefficient, Stanton number, and Reynolds analogy factor are obtained. The latter makes it possible to give two new formulations of the temperature defect law, one of which is identical to the velocity defect law and contains neither the Stanton number nor the turbulent Prandtl number, and the second formulation does not contain the skin friction coefficient. The heat transfer law is first obtained in the form of a universal functional relationship between three parameters: the Stanton number, the Reynolds number, and the molecular Prandtl number. The conclusions of the theory agree well with the known experimental data.

The contribution of active and inactive structures to the statistics of a turbulent pipe flow

The current study is a short follow-up to our most recent work (Stylianou et al., 2016), where among other things we have presented an effective criterion suitable for distinguishing large-scale coherent structures with low turbulent kinetic energy content (i.e. inactive structures) from their high energy counterparts (i.e. active structures). This criterion is based on the instantaneous values of the one-point turbulence Structure Tensors (Kassinos et al., 2001), which are effective in quantifying the different aspects of the coherent structures of turbulence. Same as in our previous work, here we perform Direct Numerical Simulation (DNS) of a fully developed turbulent pipe flow at bulk Reynolds number Reb=5300. Using simulation results, we accentuate the ability of this criterion to identify large-scale coherent structures and to differentiate between low energy and high energy content. Furthermore, based on this criterion we perform conditional averaging during the collection of statistics, which allows us to examine the contribution of the active and inactive structures to various large-scale and small-scale statistical quantities. Our results reveal that the characteristic trends in the turbulence statistics associated with the law of the wall, arise primarily due to contributions from the inactive structures. Moreover, in the log-law region both active and inactive structures contribute equally to the turbulence statistics.

Spectral derivation of the classic laws of wall-bounded turbulent flows.

We show that the classic laws of the mean-velocity profiles (MVPs) of wall-bounded turbulent flows—the ‘law of the wall,’ the ‘defect law’ and the ‘log law’—can be predicated on a sufficient condition with no manifest ties to the MVPs, namely that viscosity and finite turbulent domains have a depressive effect on the spectrum of turbulent energy. We also show that this sufficient condition is consistent with empirical data on the spectrum and may be deemed a general property of the energetics of wall turbulence. Our findings shed new light on the physical origin of the classic laws and their immediate offshoot, Prandtl’s theory of turbulent friction.

Direct Numerical Simulation and Theory of a Wall-Bounded Flow with Zero Skin Friction.

We study turbulent plane Couette-Poiseuille (CP) flows in which the conditions (relative wall velocity ΔU w ≡ 2U w , pressure gradient dP/dx and viscosity ν) are adjusted to produce zero mean skin friction on one of the walls, denoted by APG for adverse pressure gradient. The other wall, FPG for favorable pressure gradient, provides the friction velocity uτ , and h is the half-height of the channel. This leads to a one-dimensional family of flows of varying Reynolds number Re ≡ U w h/ν. We apply three codes, and cover three Reynolds numbers stepping by a factor of 2 each time. The agreement between codes is very good, and the Reynolds-number range is sizable. The theoretical questions revolve around Reynolds-number independence in both the core region (free of local viscous effects) and the two wall regions. The core region follows Townsend's hypothesis of universal behavior for the velocity and shear stress, when they are normalized with uτ and h; universality is not observed for all the Reynolds stresses, any more than it is in Poiseuille flow or boundary layers. The behavior at very high Re is unknown. The FPG wall region obeys the classical law of the wall, again for velocity and shear stress, but could suggest a low value for the Karman constant κ, possibly near 0.37. For the APG wall region, Stratford conjectured universal behavior when normalized with the pressure gradient, leading to a square-root law for the velocity. The literature, also covering other flows with zero skin friction, is ambiguous. Our results are very consistent with both of Stratford's conjectures, suggesting that at least in this idealized flow geometry the theory is successful like it was for the classical law of the wall. We appear to know the constants of the law within a 10% bracket. On the other hand, again that does not extend to Reynolds stresses other than the shear stress, but these stresses are passive in the momentum equation.

Changes in the boundary-layer structure at the edge of the ultimate regime in vertical natural convection

In thermal convection for very large Rayleigh numbers ($Ra$), the thermal and viscous boundary layers are expected to undergo a transition from a classical state to an ultimate state. In the former state, the boundary-layer thicknesses follow a laminar-like Prandtl–Blasius–Polhausen scaling, whereas in the latter, the boundary layers are turbulent with logarithmic corrections in the sense of Prandtl and von Kármán. Here, we report evidence of this transition via changes in the boundary-layer structure of vertical natural convection (VC), which is a buoyancy-driven flow between differentially heated vertical walls. The numerical dataset spans $Ra$ values from $10^{5}$ to $10^{9}$ and a constant Prandtl number value of $0.709$. For this $Ra$ range, the VC flow has been previously found to exhibit classical state behaviour in a global sense. Yet, with increasing $Ra$, we observe that near-wall higher-shear patches occupy increasingly larger fractions of the wall areas, which suggest that the boundary layers are undergoing a transition from the classical state to the ultimate shear-dominated state. The presence of streaky structures – reminiscent of the near-wall streaks in canonical wall-bounded turbulence – further supports the notion of this transition. Within the higher-shear patches, conditionally averaged statistics yield a logarithmic variation in the local mean temperature profiles, in agreement with the log law of the wall for mean temperature, and an $Ra^{0.37}$ effective power-law scaling of the local Nusselt number. The scaling of the latter is consistent with the logarithmically corrected $1/2$ power-law scaling predicted for ultimate thermal convection for very large $Ra$. Collectively, the results from this study indicate that turbulent and laminar-like boundary layer coexist in VC at moderate to high $Ra$ and this transition from the classical state to the ultimate state manifests as increasingly larger shear-dominated patches, consistent with the findings reported for Rayleigh–Bénard convection and Taylor–Couette flows.

Near-wall behaviors of oblique-shock-wave/turbulent-boundary-layer interactions

A direct numerical simulation (DNS) on an oblique shock wave with an incident angle of 33.2° impinging on a Mach 2.25 supersonic turbulent boundary layer is performed. The numerical results are confirmed to be of high accuracy by comparison with the reference data. Particular efforts have been made on the investigation of the near-wall behaviors in the interaction region, where the pressure gradient is so significant that a certain separation zone emerges. It is found that, the traditional linear and logarithmic laws, which describe the mean-velocity profiles in the viscous and meso sublayers, respectively, cease to be valid in the neighborhood of the interaction region, and two new laws of the wall are proposed by elevating the pressure gradient to the leading order. The new laws are inspired by the analysis on the incompressible separation flows, while the compressibility is additionally taken into account. It is verified by the DNS results that the new laws are adequate to reproduce the mean-velocity profiles both inside and outside the interaction region. Moreover, the normalization adopted in the new laws is able to regularize the Reynolds stress into an almost universal distribution even with a salient adverse pressure gradient (APG).

Process Intensification with Coaxially Placed Entry Region Vanes Assembly Turbulence Promoter in Homogeneous Flow

The entry region vane turbulence promoters are inserted at the downstream of a circular electrolytic cell and limiting current data are measured at the copper micro electrodes fixed on an electrode support. The effect of flow rate of the electrolyte, effect of velocity on mass transfer at the wall, effects of geometric parameters diameter of the vane (dv) from 0.02 m to 0.04 m, the angle of the vane (γ) from 15° to 60°, sectorial angle (α)/number of the vanes (N) from 4 to 8, diameter of the annular rod of mass transfer is investigated. The electrolyte was equimolal potassium ferricyanide, potassium ferrocyanide and excess sodium hydroxide. Limiting current data had been obtained for the reduction of potassium ferricyanide ion. The mass transfer correlation is based on the law of the wall similarity. The effect of each parameter was studied in terms of friction factor. A model was developed for mass transfer. The correlation may be extended to a wider range of parameters by virtue of the law of the wall. The experimental data on mass transfer was modelled in terms of mass transfer function and Reynolds number geometric parameters.

Critical bed shear stress and threshold of motion of maerl biogenic gravel

A determination of the critical bed shear stress of maerl is a prerequisite for quantifying its mobility, rate of erosion and deposition in conservation management. The critical bed shear stress for incipient motion has been determined for the first time for samples from biogenic free-living maerl beds in three contrasting environments (open marine, intertidal and beach) in Galway Bay, west of Ireland. The bed shear stress was determined using two methods, Law of the Wall and Turbulent Kinetic Energy, in a rotating annular flume and in a linear flume. The velocity profile of flowing water above a bed of natural maerl grains was measured in four runs of progressively increasing flow velocity until the flow exceeded the critical shear stress of grains on the bed. The critical Shields parameter and the mobility number are estimated and compared with the equivalent curves for natural quartz sand. The critical Shields parameters for the maerl particles from all three environments fall below the Shields curve. Along with a previously reported correlation between maerl grain shape and settling velocity, these results suggest that the highly irregular shapes also allow maerl grains to be mobilised more easily than quartz grains with the same sieve diameter. The intertidal beds with the roughest particles exhibit the greatest critical shear stress because the particle thalli interlock and resist entrainment. In samples with a high percentage of maerl and low percentage of siliciclastic sand, the lower density, lower settling velocity and lower critical bed shear stress of maerl results in its preferential transport over the siliciclastic sediment. At velocities ∼10 cm s−1 higher than the threshold velocity of grain motion, rarely-documented subaqueous maerl dunes formed in the annular flume.

The influence of shear-dependent rheology on turbulent pipe flow

Direct numerical simulations of turbulent pipe flow of power-law fluids at $Re_{\unicode[STIX]{x1D70F}}=323$ are analysed in order to understand the way in which shear thinning or thickening affects first- and second-order flow statistics including turbulent kinetic energy production, transport and dissipation in such flows. The results show that with shear thinning, near-wall streaks become weaker and the axial and azimuthal correlation lengths of axial velocity fluctuations increase. Viscosity fluctuations give rise to an additional shear stress term in the mean momentum equation which is negative for shear-thinning fluids and which increases in magnitude as the fluid becomes more shear thinning: for an equal mean wall shear stress, this term increases the mean velocity gradient in shear-thinning fluids when compared to a Newtonian fluid. Consequently, the mean velocity profile in power-law fluids deviates from the law of the wall $U_{z}^{+}=y^{+}$ in the viscous sublayer when traditional near-wall scaling is used. Consideration is briefly given to an alternative scaling that allows the law of wall to be recovered but which results in loss of a common mean stress profile. With shear thinning, the mean viscosity increases slightly at the wall and its profile appears to be approximately logarithmic in the velocity log layer. Through analysis of the turbulent kinetic energy budget, undertaken here for the first time for generalised Newtonian fluids, it is shown that shear thinning decreases the overall turbulent kinetic energy production but widens the wall-normal region where it is generated. Additional dissipation terms in the mean flow and turbulent kinetic energy budget equations arise from viscosity fluctuations; with shear thinning, these result in a net decrease in the total viscous dissipation. The overall effect of shear thinning on the turbulent kinetic energy budget is found to be largely confined to the inner layers, $y^{+}\lesssim 60$.