Near-wall Region Of Turbulent Flow Research Articles

On the streamwise velocity variance in the near-wall region of turbulent flows

We study the behaviour of the streamwise velocity variance in turbulent wall-bounded flows using a direct numerical simulation (DNS) database of pipe flow up to friction Reynolds number${{Re}}_{\tau } \approx 12000$. The analysis of the spanwise spectra in the viscous near-wall region strongly hints to the presence of an overlap layer between the inner- and the outer-scaled spectral ranges, featuring a$k_{\theta }^{-1+\alpha }$decay (with$k_{\theta }$the wavenumber in the azimuthal direction, and$\alpha \approx 0.18$), hence shallower than suggested by the classical formulation of the attached-eddy model. The key implication is that the contribution to the streamwise velocity variance$(\langle{u}^2\rangle)$from the largest scales of motion (superstructures) slowly declines as${{Re}}_{\tau }^{-\alpha }$, and the integrated inner-scaled variance follows a defect power law of the type$\langle u^2 \rangle ^+ = A - B \, {{Re}}_{\tau }^{-\alpha }$, with constants$A$and$B$depending on$y^+$. The DNS data very well support this behaviour, which implies that strict wall scaling is restored in the infinite-Reynolds-number limit. The extrapolated limit distribution of the streamwise velocity variance features a buffer-layer peak value of$\langle u^2 \rangle ^+ \approx 12.1$, and an additional outer peak with larger magnitude. The analysis of the velocity spectra also suggests a similar behaviour of the dissipation rate of the streamwise velocity variance at the wall, which is expected to attain a limiting value of approximately$0.28$, hence slightly exceeding the value$0.25$which was assumed in previous analyses (Chen & Sreenivasan,J. Fluid Mech., vol. 908, 2021, R3). We have found evidence suggesting that the reduced near-wall influence of wall-attached eddies is likely linked to the formation of underlying turbulent Stokes layers.

Wall-shear-stress measurements using volumetric µPTV

Accurately determining the wall-shear-stress, tau _textrm{w}, experimentally is challenging due to small spatial scales and large velocity gradients present in the near-wall region of turbulent flows. To avoid these resolution requirements, several indirect iterative fitting methods, most notably the Clauser chart method, exist for determining overline{tau }_{w} by fitting the mean velocity profile further away from the near-wall region in the log-law layer. These methods often require proper selection of fitting constants, assumptions of a canonical flow state, and other empirical-based generalizations. To reduce the amount of ambiguity, determining the near-wall velocity gradient by assuming a linear relationship between the mean streamwise velocity and wall normal distance in the viscous sublayer can be used. However, this requires an accurate unbiased measurement of the near-wall velocity profile in the region below five viscous spatial units, which can be less than 50 µm for high Reynolds number flows. Therefore, in this study a method for a volumetric defocusing microparticle tracking velocimetry method is presented that is capable of resolving the flow in the viscous sublayer of a turbulent boundary layer up to U_{textrm{e}}=44.7,m/s (Re_{theta }=27250). This method allows for the measurement of the near-wall flow through a single optical access for illumination and imaging and serves as an excellent complement of larger scale measurements that require near-wall information. The overline{tau }_textrm{w} values determined from the defocusing approach were found to be in good agreement values obtained from a simultaneous parallax PTV measurement. Furthermore, analysis of the diagnostic plot and cumulative distribution of measured fluctuations in the near-wall region, showed that both methods are capable of accurately determining mean velocity and fluctuation profiles in the self-similar viscous sublayer region.

Coherent near-wall structures and drag reduction by spanwise forcing

We examine interaction mechanisms between the drag-reducing streamwise-traveling waves of spanwise wall velocity applied at the wall of a turbulent channel flow, and near-wall turbulent coherent structures. In particular, we focus our analysis on the drag-reduction-induced modifications of the quasi-streamwise vortices that populate the near-wall region of turbulent flows through a conditional averaging procedure. We observe that, while the streamwise-traveling waves share most of the drag-reducing mechanisms with the spanwise wall oscillations, some important differences related to the finite phase speed of the wave and the different properties of the generalized Stokes layer exist.

The minimal-span channel for rough-wall turbulent flows

Roughness predominantly alters the near-wall region of turbulent flow while the outer layer remains similar with respect to the wall shear stress. This makes it a prime candidate for the minimal-span channel, which only captures the near-wall flow by restricting the spanwise channel width to be of the order of a few hundred viscous units. Recently, Chung et al. (J. Fluid Mech., vol. 773, 2015, pp. 418–431) showed that a minimal-span channel can accurately characterise the hydraulic behaviour of roughness. Following this, we aim to investigate the fundamental dynamics of the minimal-span channel framework with an eye towards further improving performance. The streamwise domain length of the channel is investigated with the minimum length found to be three times the spanwise width or 1000 viscous units, whichever is longer. The outer layer of the minimal channel is inherently unphysical and as such alterations to it can be performed so long as the near-wall flow, which is the same as in a full-span channel, remains unchanged. Firstly, a half-height (open) channel with slip wall is shown to reproduce the near-wall behaviour seen in a standard channel, but with half the number of grid points. Next, a forcing model is introduced into the outer layer of a half-height channel. This reduces the high streamwise velocity associated with the minimal channel and allows for a larger computational time step. Finally, an investigation is conducted to see if varying the roughness Reynolds number with time is a feasible method for obtaining the full hydraulic behaviour of a rough surface. Currently, multiple steady simulations at fixed roughness Reynolds numbers are needed to obtain this behaviour. The results indicate that the non-dimensional pressure gradient parameter must be kept below 0.03–0.07 to ensure that pressure gradient effects do not lead to an inaccurate roughness function. An empirical costing argument is developed to determine the cost in terms of CPU hours of minimal-span channel simulations a priori. This argument involves counting the number of eddy lifespans in the channel, which is then related to the statistical uncertainty of the streamwise velocity. For a given statistical uncertainty in the roughness function, this can then be used to determine the simulation run time. Following this, a finite-volume code with a body-fitted grid is used to determine the roughness function for square-based pyramids using the above insights. Comparisons to experimental studies for the same roughness geometry are made and good agreement is observed.

The minimal channel: a fast and direct method for characterising roughness

Roughness only alters the near-wall region of turbulent flow and leaves the outer-layer unaffected, making it a prime candidate for the minimal-span channel framework which only captures the near-wall flow. Recently, Chung et al. (J. Fluid Mech., vol. 773, 2015, pp. 418-431) showed that the minimal-span channel can accurately characterise the hydraulic behaviour of roughness. Following on from this, we aim to further optimise the minimal-span channel framework by primarily noting that the outer layer it produces is inherently incorrect, and as such modifications to this region can be made to improve performance. Firstly, a half-height channel with slip wall is shown to reproduce the near-wall behaviour seen in a standard channel, but with half the number of grid points. Next, a forcing model is introduced into the outer layer of a half-height channel. This reduces the high streamwise velocity associated with the minimal channel and allows for a larger computational time step. The streamwise length of the channel is also investigated independent of the previous improvements, and suggests the minimum length should be at least 3 times the spanwise width and also 1000 viscous-units long, whichever is longer. Finally, an investigation is conducted to see if varying the roughness Reynolds number with time is a feasible method for obtaining the full hydraulic behaviour of a rough surface, instead of running multiple simulations at fixed roughness Reynolds numbers.

A mixed-time-scale low-Reynolds-number one-equation turbulence model

In this paper, a new low-Reynolds-number (LRN) one-equation turbulence model for eddy viscosity is proposed. A mixed time scale, representing a combination of three time scales: two time scales made of strain-rate parameter S and vorticity parameter Ω and the turbulent time scale k/ϵ, is introduced into this model. The proposed model is derived from an LRN k−ϵ two-equation model where the mixed time scale has been proved to be very effective for predicting local flows over complex terrains. In the transport equation of the model, the mixed time scale is included in the production and the dissipation terms. The new model is evaluated in channel flows at various Reynolds numbers, boundary layer flows with or without pressure gradient and backward-facing step flows with different expansion ratios and Reynolds numbers. Then the grid convergence of the model is investigated. Finally, the model performance for different values of the weighting constant Cs in the mixed time scale is assessed. The results show that the proposed model reproduces the correct wall-limiting behaviour of turbulent quantities and performs well in the near-wall region of turbulent flows. The model could be expected to be adopted in hybrid Reynolds averaged Navier–Stokes/large eddy simulation methodology for complex wall-bounded flows at high Reynolds numbers.

Exact coherent structures in channel flow

Exact coherent states in no-slip plane Poiseuille flow are calculated by homotopy from free-slip to no-slip boundary conditions. These coherent states are unstable travelling waves. They consist of wavy low-speed streaks flanked by staggered streamwise vortices closely resembling the asymmetric coherent structures observed in the near-wall region of turbulent flows. The travelling waves arise from a saddle-node bifurcation at a sub-turbulent Reynolds number with wall-normal, spanwise and streamwise dimensions smaller than but comparable to 50+, 100+ and 250+, respectively. These coherent solutions come in pairs with distinct structure and instabilities. There is a three-dimensional continuum of such exact coherent states.

Regeneration mechanisms of near-wall turbulence structures

Direct numerical simulations of a highly constrained plane Couette flow are employed to study the dynamics of the structures found in the near-wall region of turbulent flows. Starting from a fully developed turbulent flow, the dimensions of the computational domain are reduced to near the minimum values which will sustain turbulence. A remarkably well-defined, quasi-cyclic and spatially organized process of regeneration of near-wall structures is observed. This process is composed of three distinct phases: formation of streaks by streamwise vortices, breakdown of the streaks, and regeneration of the streamwise vortices. Each phase sets the stage for the next, and these processes are analysed in detail. The most novel results concern vortex regeneration, which is found to be a direct result of the breakdown of streaks that were originally formed by the vortices, and particular emphasis is placed on this process. The spanwise width of the computational domain corresponds closely to the typically observed spanwise spacing of near-wall streaks. When the width of the domain is further reduced, turbulence is no longer sustained. It is suggested that the observed spacing arises because the time scales of streak formation, breakdown and vortex regeneration become mismatched when the streak spacing is too small, and the regeneration cycle at that scale is broken.

Subgrid-scale energy transfer in the near-wall region of turbulent flows

Direct numerical simulation databases of turbulent channel and pipe flow have been used in order to assess the energy transfer between resolved and unresolved motions in large-eddy simulations. To this end, the velocity fields are split into three parts: a statistically stationary mean flow, the resolved, and the unresolved turbulent fluctuations. The distinction between the resolved and unresolved motions is based on the application of a cutoff filter in spectral space. Within the buffer layer a backward transfer of averaged kinetic energy from subgrid to grid-scale turbulent motions has been found to exist, which is primarily caused by subgrid-scale stresses aligned with the mean rates of strain. Such reverse transfer generally cannot be described by the simple eddy-viscosity-type subgrid models usually applied in large-eddy simulations. The use of a conditional averaging technique revealed that the reverse transfer of energy within the near-wall flow is strongly enhanced by coherent motions, such as the well-known bursting events.

Large-Eddy Simulation of Turbulent Obstacle Flow Using a Dynamic Subgrid-Scale Model

We apply the dynamic subgrid-scale model to a large-eddy simulation of turbulent channel flow with a square rib mounted on one wall. The Reynolds number Re is 3.21 x 10 3 based on the mean velocity above the obstacle and the obstacle's height. Near-wall structures are resolved with the no-slip boundary condition. The results show better agreement with direct numerical simulation than large-eddy simulation with a fixed model constant, verifying the value of the dynamic subgrid-scale model for simulating complex turbulent flows. ARGE-EDDY simulation (LES) is an accurate method of simulating complex turbulent flows in which the large flow structures are computed while small scales are modeled. The rationale behind this method is based on two observations: most of the turbulent energy is in the large structures, and the small scales are more isotropic and universal. Therefore, LES may be more general and less geometry dependent than Reynolds-averaged modeling, although it comes at higher cost. Even though LES has been used by many investigators, most research has been limited to flows with simple geometry. In engineering applications, however, one encounters more complicated geometries. Here we shall consider a rectangular parallelepiped mounted on a flat surface. Related flows are those over surfaces protruding from submarines (conning towers or control fins), wind flows around buildings, and airflows over computer chips, among others. The most distinctive features associated with these flows are three dimensionality, flow separation due to protruding surfaces, and large-scale unsteadiness. As a model flow, we consider a plane channel flow in which a two-dimensional obstacle is mounted on one surface (see Fig. 1). This relatively simple geometry contains flow separation and reattachment. Flow in this geometry has been studied by Tropea and Gackstatter1 for low Re and Werner and Wengle2 and Dimaczek et al.3 for high Re, among others. Recently, Germano et al. 4 suggested a dynamic subgridscale model (DSGSM) in which the model coefficient is dynamically computed as computation progresses rather than specified a priori. This approach is based on an algebraic identity between the subgrid-scale stresses at two different filter levels and the resolved turbulent stresses. They applied the model to transitional and fully turbulent channel flows and showed that the model contributes nothing in laminar flow and exhibits the correct asymptotic behavior in the nearwall region of turbulent flows without an ad hoc damping function. This is a significant improvement over conventional subgrid-scale modeling.

Vortices interacting with nonslip walls: Stirring and mixing of primary and secondary vorticity

The interaction of vortices with fixed boundaries is an important problem in many areas of practical interest. During the approach of the vortices to the wall, vorticity is created at the wall, and then mixing and stirring of primary and secondary vorticity lead to different scenarios that depend on the Re number. This problem is relevant to bursting in turbulent boundary layers, which is responsible for turbulence production. In the near-wall region of turbulent flows, vortex stretching plays a fundamental role. As a prelude to attempting the solution by direct simulation of 3-D Navier–Stokes equations, it is convenient to describe the mechanism associated with 2-D or axisymmetric flows. In the present study, axisymmetric vortex rings and vortex dipoles approaching a nonslip flat wall have been considered. In both cases, experimental results based on flow visualization are available.1,2 The purpose of the present paper is to do a numerical simulation with the same conditions used in the real experiments and follow the flow evolution for a much longer time than has been possible in such experiments. Both for axisymmetric and 2-D cases the numerical simulation shows, as revealed by the experiments, that, at the initial stage, very thin layers of vorticity of sign opposite to that of the primary vortex are generated at the wall. This mechanism is almost independent of Re. This thin and intense layer immediately becomes unstable and rolls up creating a secondary vortex. This vortex interacts with the primary vortex and different flow structures are generated, depending on Re. The location where these structures form depends, in large measure, on whether vortex stretching is present or not. Particularly relevant to the bursting phenomenon is the case at intermediate Re (e.g., Re=1600). Here, a new dipole is created near the wall after multiple rebounding and pairing of secondary vortices. This new structure has sufficient strength to move itself away from the wall. We conclude that mixing and stirring of vorticity are the relevant mechanisms influencing the whole process in a two-dimensional simulation. This ejection of a new structure was previously noticed in the interaction of vortex rings with a wall. From flow visualization, Walker et al., at Re=Γ/ν>3500, observed the occurrence of azimuthal instabilities and, immediately after that, they noticed a new vortex ring was rapidly ejected away from the wall.2 From these observations, those authors argued that the ejection was due to the azimuthal instabilities. In the present paper, axisymmetric calculations have been done in the same range of numbers used in the experiment of Walker et al.2 From the time evolution of the vorticity distribution, we argue that vortex pairing is the main mechanism that contributes to the generation of the new ring rather than azimuthal instabilities. The numerical simulations clearly show that this new ring is created above the primary ring and not at its interior.