Numerous studies in the past 40 years have established that turbulent flow fields are populated by transient coherent structures that represent patches of fluids moving cohesively for significant distances before they are worn out by momentum exchange with the surrounding fluid. Two particular well-documented structures are the hairpin vortices that move longitudinally above the wall and ejections inclined with respect to the wall that bring the fluid from the transient viscous layers underneath these vortices into the outer region of the boundary layer.
It is proposed that the Karman universal constant in the logarithmic law the sine of the angle between the transient ejections and the direction normal to the wall. The edge of the buffer layer is represented by a combination of the Karman constant and the damping function in the wall layer.
Computation of this angle from experimental data of velocity distributions in turbulent shear flows matches published traces of fronts of turbulence obtained from the time shifts in the peak of the correlation function of the velocity.
Key works: Turbulence, coherent structures, Karman constant, mixing-length, shear layers
We evaluate the effect of two experimental parameters on the slow dynamics of a Von-Kármán swirling flow driven by two propellers in a closed cylinder. The first parameter is the inertia mo- mentum of the propellers, and the second parameter is the aspect ratio, i.e. the distance between the propellers $H$ divided by the diameter $D$. We use a cell with a fixed diameter $D$ but where the distance between the propellers can be turned continuously and where the inertia from the pro- pellers can also be changed using different gears. No change on the dynamics is observed when the momentum of iner…
The von Karman flow apparatus produces a highly turbulent flow inside a cylinder vessel driven by two counter-rotating impellers. Over more than two decades, this experiment has become a very classic turbulence tool, studied for a wide range of physical systems by many groups, with incompressible flow, compressible flow, for magnetohydrodynamics and dynamo studies inside liquid metal, for particle tracking purposes, and recently with turbulent super-fluid helium. We present a direct numerical simulation (DNS) version the von Karman flow, forced by two rotating impellers. The cylinder geometry …
We present a novel moving immersed boundary method (IBM) and employ it in direct numerical simulations (DNS) of the closed-vessel swirling von Karman flow in laminar and turbulent regimes. The IBM extends direct-forcing approaches by leveraging a time integration scheme, that embeds the immersed boundary forcing step within a semi-implicit iterative Crank-Nicolson scheme. The overall method is robust, stable, and yields excellent results in canonical cases with static and moving boundaries. The moving IBM allows us to reproduce the geometry and parameters of the swirling von Karman flow experi…
The velocity-vorticity correlation integral (Chkhetiani invariant) is an invariant of a viscous Karman-Howarth equation. In the tree-dimensional space this invariant can naturally determine a turbulent diffusivity (viscosity). It is shown, using results of direct numerical simulations, that distributed chaos dominated by this integral can provide an adequate description of the turbulent Rayleigh-Bénard convection, stably stratified turbulence and Rayleigh-Taylor mixing at Prandtl number $Pr \sim 1$.
Scrupulous measurements and detailed data analysis of the torque in a swirling turbulent flow driven by counter-rotating bladed disks reveals an apparent breaking of the law of similarity. Potentially, such breakdown could arise from several possible factors, including dependence on dimensionless numbers other that $Re$ or velocity coupling to other fields such as temperature. However, careful redesign and calibration of the experiment showed that this unexpected result was due to background errorscaused by minute misalignments which lead to a noisy and irreproducible torque signal at low rota…
All previous experiments in open turbulent flows (e.g. downstream of grids, jet and atmospheric boundary layer) have produced quantitatively consistent values for the scaling exponents of velocity structure functions. The only measurement in closed turbulent flow (von Kármán swirling flow) using Taylor-hypothesis, however, produced scaling exponents that are significantly smaller, suggesting that the universality of these exponents are broken with respect to change of large scale geometry of the flow. Here, we report measurements of longitudinal structure functions of velocity in a von Kármán …
Revisiting the mixing length scaling in pressure-gradient turbulent boundary layers via a symmetry approach
Department of Mechanics, School of Mechanics and Engineering Science（力学与工程科学学院力学系）；State Key Laboratory for Turbulence and Complex Systems, Peking University（北京大学湍流复杂系统全国重点实验室）
Turbulent flow evolution and energy cascades are significantly different in two-dimensional (2D) and three-dimensional (3D) flows. Studies have investigated these differences in obstacle-free turbulent flows, but solid boundaries have an important impact on the cross-over between 3D to 2D turbulence dynamics. In this work, we investigate the span effect on the turbulence nature of flow past a circular cylinder at Re=10000. It is found that even for highly anisotropic geometries, 3D small-scale structures detach from the walls. Additionally, the natural large-scale rotation of the Kármán vortic…
The classical scaling theory of turbulent parallel flow provides a framework for the description of the mean velocity profile through two functions of one variable, traditionally named law of the wall and law of the wake, and a universal logarithmic law characterized by von Kármán's constant. Despite its widespread adoption in research and in teaching, discrepancies between this theory and both experiments and numerical simulations have been repeatedly observed in the literature. Recently we have shown that in the logarithmic layer such discrepancies can be physically interpreted and analytica…
In-depth analyses of existing direct numerical simulations (DNS) data from various sources supported a logical and important classification of generic turbulent boundary layers (TBL), namely Type-A, -B and -C TBL, based on distribution patterns of time-averaged wall-shear stress. Among these types, Type-A TBL and its related law, as represented by the DNS data of turbulence on a zero-pressure-gradient semi-infinite flat-plate, was investigated in terms of analytical formulations of velocity independent on Reynolds ( ) number. With reference to the analysis from von Karman in developing the con…
A major challenge in computation of engineering flows is to derive and improve turbulence models built on turbulence physics. Here, we present a physics-based modified $k-ω$ equation for canonical wall bounded turbulent flows (boundary layer, channel and pipe), predicting both mean velocity profile (MVP) and streamwise mean kinetic energy profile (SMKP) with high accuracy over a wide range of Reynolds number ($Re$). The result builds on a multi-layer quantification of wall flows, which allows a significant modification of the $k-ω$ equation. Three innovations are introduced: First, an adjustme…
We propose a minimal spectral theory for boundary layer turbulence that captures very well the profile of the mean square velocity fluctuations in the stream-wise direction, and gives a quantitative prediction of the Townsend-Perry constants. The phenomenological model is based on connecting the statistics in the streamwise direction with the energy spectrum of the streamvise velocity fluctuations. The original spectral theory was proposed earlier to explain the friction factor and the von Kármán law. We generalized it by including fluctuations in the wall-shear stress and the streamwise veloc…
The water landing of an amphibious aircraft is a complicated problem that can lead to uncomfortable riding situation and structural damage due to large vertical accelerations and the consequent dynamic responses. The problem herein is investigated by solving unsteady incompressible Reynolds-averaged Navier-Stokes equations with a standard k-omega turbulence closure model. The theoretical solutions established by the von Karman's momentum theory are also employed. In order to validate the relationships between the initial vertical velocity and the peak value of vertical acceleration, free fall …
Wall-bounded turbulent flows are widely observed in natural and engineering systems, such as air flows near the Earth's surface, water flows in rivers, and flows around a car or a plane. The universal logarithmic velocity profile in wall-bounded turbulent flows proposed by von Kármán in 1930 is one of the few exact physical descriptions of turbulence. However, the mean velocity and temperature profiles cannot be adequately described by this universal log law when buoyancy effects are present. Monin-Obukhov similarity theory (MOST), proposed in 1954, has been the cornerstone theory to account f…
By analyzing the Karman-Howarth equation for filtered velocity fields in turbulent flows, we show that the two-point correlation between filtered strain-rate and subfilter stress tensors plays a central role in the evolution of filtered-velocity correlation functions. Two-point correlations-based {\it statistical priori tests} thus enable rigorous and physically meaningful studies of turbulence models. Using data from direct numerical simulations of isotropic and channel flow turbulence we show that local eddy viscosity models fail to exhibit the long tails observed in the real subfilter stres…
In turbulent wall sheared thermal convection, there are three different flow regimes, depending on the relative relevance of thermal forcing and wall shear. In this paper we report the results of direct numerical simulations of such sheared Rayleigh-Bénard convection, at fixed Rayleigh number $Ra=10^6$, varying the wall Reynolds number in the range $0 \leq Re_w \leq 4000$ and Prandtl number $0.22 \leq Pr \leq 4.6$, extending our prior work by Blass et al. (2020), where $Pr$ was kept constant at unity and the thermal forcing ($Ra$) varied. We cover a wide span of bulk Richardson numbers $0.014 …
The trinity of so-called "canonical" wall-bounded turbulent flows, comprising the zero pressure gradient turbulent boundary layer, abbreviated ZPG TBL, turbulent pipe flow and channel/duct flows has continued to receive intense attention as new and more reliable experimental data have become available. Nevertheless, the debate on whether the logarithmic part of the mean velocity profile, in particular the Kármán constant $κ$, is identical for these three canonical flows or flow-dependent is still ongoing. In this paper, which expands upon Monkewitz and Nagib (24th ICTAM Conf., Montreal, 2016),…
A series of direct numerical simulations of Taylor-Couette (TC) flow, the flow between two coaxial cylinders, with the outer cylinder rotating and the inner one fixed, were performed. Three cases, with outer cylinder Reynolds numbers $Re_o$ of $Re_o=5.5\cdot10^4$, $Re_o=1.1\cdot10^5$ and $Re_o=2.2\cdot10^5$ were considered. The radius ratio $η=r_i/r_o$ was fixed to $η=0.909$ to mitigate the effects of curvature. The vertical aspect ratio $Γ$ was fixed to $Γ=2.09$. Being linearly stable, outer cylinder rotation TC flow is known to have very different behavior than pure inner cylinder rotation T…
The role of instability in the growth of a 2D, temporally evolving, `turbulent' free shear layer is analyzed using vortex-gas simulations that condense all dynamics into the kinematics of the Biot-Savart relation. The initial evolution of perturbations in a constant-vorticity layer is found to be in accurate agreement with the linear stability theory of Rayleigh. There is then a stage of non-universal evolution of coherent structures that is closely approximated not by Rayleigh stability theory, but by the Karman-Rubach-Lamb linear instability of monopoles, until the neighboring coherent struc…
Systematic study of the existing statistical data from direct numerical simulations (DNS) lead to a logical and important classification of generic turbulent boundary layers (TBL), namely Type A,B and C TBL, based on the distribution patterns of time averaged wall shear stress. Among these three types, Type A and B TBLs are investigated in terms of its universal statistical laws of velocity and temperature with independency on both Reynolds (Re) and Prandtl (Pr) numbers. With reference to the analysis by von Karman in developing the conventional law of the wall for Type A TBL, the current stud…
A large number of models which address the dynamics of particle-laden turbulent flows have been developed based on the assumption of local isotropy and use the Kolmogorov constant that correlates the spectral distribution of turbulent kinetic energy with the turbulent dissipation rate. Many turbulence models (Stochastic and LES models) use the Kolmogorov constant in the formulation. Compilation of a large number of experimental data for different flow configurations has revealed that the Kolmogorov constant is independent of Reynolds number in the limit of high Reynolds number (Sreenivasan, 19…
The two-dimensional flows around a cylinder between two parallel walls at Re=40 and Re=100 are simulated with computational fluid dynamics (CFD). The governing equations are Navier-Stokes equations. They are discretized with finite volume method (FVM) and the solution is iterated with PISO Algorithm. Then, the calculating results are compared with the numerical results in literature, and good agreements are obtained. After that, the mechanism of the formation of Karman vortex street is investigated and the instability of the entire flow field is analyzed with the energy gradient theory. It is …
The universal velocity log law first proposed by von Kármán in the near-wall region of turbulent shear flows is one of the cornerstones of turbulence theory. When buoyancy effects are important, the universal velocity log law is typically believed to break down according to Monin-Obukhov similarity theory (MOST), which has been used in almost all global weather and climate models to describe the dependence of the mean velocity profiles on buoyancy in the atmospheric boundary layer. In contrast to MOST, we propose new logarithmic profiles of near-wall mean velocity in the stably stratified atmo…
A series of Direct Numerical Simulations (DNS) of lean methane/air flames was conducted in order to investigate the enhancement of the turbulent flame speed and modifications to the reaction layer structure associated with the systematic increase of the integral scale of turbulence $l$ while the Karlovitz number and the Kolmogorov scale are kept constant. Four turbulent slot jet flames are simulated at increasing Reynolds number and up to $Re \approx 22000$, defined based on the bulk velocity, slot width, and the reactants' properties. The turbulent flame speed $S_T$ is evaluated locally at se…
In this study, we performed highly resolved large-eddy simulations (LES) to investigate the influence of variable properties on the forced turbulent convection in a channel. The constant heat flux boundary condition permits wall temperature fluctuations and thus induces variations of fluid properties. Since the effect of viscosity on the flow exhibits $Re_τ^{-1}$ scaling, we only considered $Re_τ= 180$ in the present study. Compared to the flow with constant properties, results indicate that the variable properties have trivial effects on the mean velocity and temperature profiles, Reynolds sh…
A composite representation of the turbulent boundary-layer velocity profile is proposed, which combines a recently determined accurate interpolation of the universal law of the wall with a simple analytical expression of the smooth transition of velocity to a constant value in the outer stream. Several examples are given of application of this representation to DNS and experimental data from the literature, and a conjecture is offered for the asymptotic approach of the velocity to its constant inviscid value.
Filamentary regions of high vorticity irregularly form and disappear in the turbulent flows of classical fluids. We report an experimental comparative study of these so-called " coherent structures " in a classical versus quantum fluid, using liquid helium with a superfluid fraction varied from 0% up to 83%. The low pressure core of the vorticity filaments is detected by pressure probes located on the sidewall of a 78-cm-diameter Von Kármán cell driven up to record turbulent intensity (R $λ$ $\sim$ $\sqrt$ Re 10000). The statistics of occurrence, magnitude and relative distribution of the fila…
The separating and reattaching turbulent flow past a rectangular cylinder is studied to describe how small and large scales contribute to the sustaining mechanism of the velocity fluctuations. The work is based on the Anisotropic Generalised Kolmogorov Equations (AGKE), exact budget equations for the second-order structure function tensor in the space of scales and in the physical space. Scale-space energy fluxes show that forward and reverse energy transfers simultaneously occur in the flow, with interesting modelling implications. Over the longitudinal cylinder side, the Kelvin-Helmholtz ins…
In this study, a new well-resolved large-eddy-simulation (LES) of an incompressible near-equilibrium adverse-pressure-gradient (APG) turbulent boundary layer (TBL) over a flat plate is presented. In this simulation, we have established a near-equilibrium APG over a wide Reynolds-number range. In this so-called region of interest (ROI), the Clauser--Rotta pressure-gradient parameter $β$ exhibits an approximately constant value of around 1.4, and the Reynolds number based on momentum thickness reaches $Re_θ=8700$. To the authors' knowledge, this is to date the highest $Re_θ$ achieved for a near-…
