Is there a universal log law for turbulent wall-bounded flows?

The history and theory supporting the idea of a universal log law for turbulent wall-bounded flows are briefly reviewed. The original idea of justifying a log law from a constant Reynolds stress layer argument is found to be deficient. By contrast, it is argued that the logarithmic friction law and velocity profiles derived from matching inner and outer profiles for a pipe or channel flow are well-founded and consistent with the data. But for a boundary layer developing along a flat plate it is not, and in fact it is a power law theory that seems logically consistent. Even so, there is evidence for at least an empirical logarithmic fit to the boundary-friction data, which is indistinguishable from the power law solution. The value of kappa approximately 0.38 obtained from a logarithmic curve fit to the boundary-layer velocity data, however, does not appear to be the same as for pipe flow for which 0.43 appears to be the best estimate. Thus, the idea of a universal log law for wall-bounded flows is not supported by either the theory or the data.     

Structure and dynamics of low Reynolds number turbulent pipe flow

Using large-scale numerical calculations, we explore the proper orthogonal decomposition of low Reynolds number turbulent pipe flow, using both the translational invariant (Fourier) method and the method of snapshots. Each method has benefits and drawbacks, making the 'best' choice dependent on the purpose of the analysis. Owing to its construction, the Fourier method includes all the flow fields that are translational invariants of the simulated flow fields. Thus, the Fourier method converges to an estimate of the dimension of the chaotic attractor in less total simulation time than the method of snapshots. The converse is that for a given simulation, the method of snapshots yields a basis set that is more optimal because it does not include all of the translational invariants that were not a part of the simulation. Using the Fourier method yields smooth structures with definable subclasses based upon Fourier wavenumber pairs, and results in a new dynamical systems insight into turbulent pipe flow. These subclasses include a set of modes that propagate with a nearly constant phase speed, act together as a wave packet and transfer energy from streamwise rolls. It is these interactions that are responsible for bursting events and Reynolds stress generation. These structures and dynamics are similar to those found in turbulent channel flow. A comparison of structures and dynamics in turbulent pipe and channel flows is reported to emphasize the similarities and differences.     

Some predictions of the attached eddy model for a high Reynolds number boundary layer

Many flows of practical interest occur at high Reynolds number, at which the flow in most of the boundary layer is turbulent, showing apparently random fluctuations in velocity across a wide range of scales. The range of scales over which these fluctuations occur increases with the Reynolds number and hence high Reynolds number flows are difficult to compute or predict. In this paper, we discuss the structure of these flows and describe a physical model, based on the attached eddy hypothesis, which makes predictions for the statistical properties of these flows and their variation with Reynolds number. The predictions are shown to compare well with the results from recent experiments in a new purpose-built high Reynolds number facility. The model is also shown to provide a clear physical explanation for the trends in the data. The limits of applicability of the model are also discussed.     

Behaviour of organised disturbances in fully developed turbulent channel flow

In our earlier work we have shown the relevance of stability theory in understanding the sustenance of turbulence in turbulent boundary layers. Here we adopt the same model to study the evolution of organised disturbances in turbulent channel flow. Since the dominant modes are wall modes we find that the stability characteristics in the two flows are nearly identical although the boundary conditions (at the edge of the boundary layer and at the centre of the channel) are different. Comparisons with the experiments of Hussain and Reynolds are also presented.     

The influence of nonparallelism in channel flow stability

Summary Considered below are the high Reynolds number flows of an incompressible fluid, and their nonparallel stability properties, in certain plane channels whose widths vary slowly in the streamwise direction. The first approximation to the steady-state flow is governed by the classical boundary layer equations but with the pressure unknown. Solutions, some including separation and reattachment, to this are obtained numerically for a range of the parameters involved, and the stability of the resulting flows is considered using fixed frequency disturbances and taking into account the nonparallel nature of the basic flow by use of a W.K.B. method. The calculations yield critical Reynolds numbers, below which all the disturbances decay downstream, and for various Reynolds numbers above the critical values the growths of the small disturbances are calculated. The results are specialized to a particular class of channels which are straight far upstream and far downstream but vary in between. So the predictions should be much more easily amenable to experimental investigation and comparisons than those of the more idealized, diverging channel, flow problem studied by Eagles & Weissman [1] and the only other basic flow seriously studied from the viewpoint of nonparallel flow stability theory, the Blasius boundary layer. The results also represent the first application of both quasiparallel and slightly non-parallel stability theory to channels involving slowly varying but finite changes in width.     

A singular perturbation problem of non-newtonian flow between porous disks

The non-Newtonian flow between parallel porous stationary disks due to uniform suction at the disks is considered for both small and large suction Reynolds numbers. In the case of small suction Reynolds number the Navier-Stokes equations have been solved by a regular perturbation technique. The solution obtained is valid for both suction and injection Reynolds numbers. The velocity, pressure and shear distributions have been obtained and are compared with those of the Newtonian flow. We find, in the case of injection, that the combined effect of cross viscosity and visco-elastic co-efficients is to increase the maximum velocity at the centre of the channel and to decrease the magnitude of the velocity gradient at the disks. Whereas in the case of suction, velocity profile is flatter with higher magnitude for the velocity gradients at the disks.In the case of large suction, the Navier-Stokes equations have been solved by the method of matched asymptotic expansions. We find that the effect of large suction at the disks is to flatten the velocity profiles considerably and thereby to push the boundary layer towards the disks. The combined effect of visco-elastic and cross-viscosity terms is to decrease the radial velocity and to increase the axial velocity distributions.     

A physical model of the turbulent boundary layer consonant with mean momentum balance structure

Recent studies by the present authors have empirically and analytically explored the properties and scaling behaviours of the Reynolds averaged momentum equation as applied to wall-bounded flows. The results from these efforts have yielded new perspectives regarding mean flow structure and dynamics, and thus provide a context for describing flow physics. A physical model of the turbulent boundary layer is constructed such that it is consonant with the dynamical structure of the mean momentum balance, while embracing independent experimental results relating, for example, to the statistical properties of the vorticity field and the coherent motions known to exist. For comparison, the prevalent, well-established, physical model of the boundary layer is briefly reviewed. The differences and similarities between the present and the established models are clarified and their implications discussed.     

Hydrodynamic interaction of two neutrally-buoyant smooth spheres suspended in plane Poiseuille flow: the BEM simulations versus the MoR approximations

The hydrodynamic interaction between two particles suspended in shear flows is fundamental to the macroscopic characterization of suspension flows. Although such interaction in quiescent or linear shear flow is well understood, studies on that in a nonlinear shear field are rare. In this study, the hydrodynamic interaction between two neutrally-buoyant smooth spheres moving at negligible Reynolds numbers in an unbounded plane Poiseuille flow has been calculated by three-dimensional boundary element method (BEM) simulations. The BEM results have been compared with the analytical results obtained with the method-of-reflection (MoR) approximations. The BEM simulations have been found to provide satisfactory predictions if the number of elements on the spheres are more than 200, whereas the MoR approximations provide satisfactory predictions only when the minimum separation between the spheres is relatively large although this MoR method has the advantage to easily calculate the hydrodynamic interaction between two spheres freely moving at negligible Reynolds numbers in unbounded quadratic flow by solving ordinary differential equations. Furthermore, it is found that there is a preferential cross-streamline migration of the center-of-gravity of the sphere-pair in the plane of shear in plane Poiseuille flow which does not arise in simple shear flow. This migration is always directed towards low shear regions when the sphere having larger translational velocity approaches the other sphere, and reverses towards high shear regions when the faster sphere leads the other sphere in the plane of shear. There is also a cross-streamline migration of the center-of-gravity of the sphere-pair in the plane of vorticity, but this migration does not have a preferential direction. These migrations are symmetric about the point where the spheres are at the minimum separation, and are only significant when the hydrodynamic interaction of the spheres is strong. These results show that the migration of the center-of-gravity of the sphere-pair can be attributed to the nonlinearity of the shear field, which agrees with the MoR approximations. The hydrodynamic interaction between the two spheres has been quantified under various conditions by the BEM simulations for both identical and disparate spheres.     

Aspects of linear and nonlinear instabilities leading to transition in pipe and channel flows

The failure of normal-mode linear stability analysis to predict a transition Reynolds number (Retr) in pipe flow and subcritical transition in plane Poiseuille flow (PPF) has led to the search of other scenarios to explain transition to turbulence in both flows. In this work, various results associated with linear and nonlinear mechanisms of both flows are presented. The results that combine analytical and experimental approaches indicate the strong link between the mechanisms governing the transition of both flows. It is demonstrated that the linear transient growth mechanism is based on the existence of a pair of least stable nearly parallel modes (having opposite phases and almost identical amplitude distributions). The analysis that has been applied previously to pipe flow is extended here to a fully developed channel flow predicting the shape of the optimized initial disturbance (a pair of counter-rotating vortices, CVP), time for maximum energy amplification and the dependence of the latter on Re. The results agree with previous predictions based on many modes. Furthermore, the shape of the optimized initial disturbance is similar in both flows and has been visualized experimentally. The analysis reveals that in pipe flow, the transient growth is a consequence of two opposite running modes decaying with an equal decay rate whereas in PPF it is due to two stationary modes decaying with different decay rates. In the first nonlinear scenario, the breakdown of the CVPs (produced by the linear transient growth mechanism) into hairpin vortices is followed experimentally. The associated scaling laws, relating the minimal disturbance amplitude required for the initiation of hairpins and the Re, are found experimentally for both PPF and pipe flow. The scaling law associated with PPF agrees well with the previous predictions of Chapman, whereas the scaling of the pipe flow is the same as the one previously obtained by Hof et al. indicating transition to a turbulent state. In the second nonlinear scenario, the base flow of pipe when it is mildly deviated from the Poiseuille profile by an axisymmetric distortion is examined. The nonlinear features reveal a Retr of approximately 2000 associated with the bifurcation between two deviation solutions.     

Boundary layer growth on two circular cylinders

This paper deals with the problem of boundary layer growth on two circular cylinders in a flow with large Reynolds and Strouhal numbers. Analytic solutions for the stream function of the inner and outer flow field are obtained to the second order by using the method of matched asymptotic expansions. The dependence of the movement of the detachment points and the drag coefficients of the two cylinders on (i) the distance between them, (ii) the ratio of their radii, (iii) the Reynolds number and (iv) the acceleration parameter of the flow is investigated. The results obtained indicate that the mutual hydrodynamic interaction between two cylinders leads to some new relations and findings.     

Second-order boundary-layer theory of laminar jet flows

Summary Laminar jet flows are investigated by asymptotic expansions in terms of large Reynolds numbers and large distances from an orifice in a wall. For two-dimensional flow, previously known results are modified and extended. For axisymmetric flow, the expansion breaks down as the distance from the orifice tends to infinity. An error analysis indicates that the range of applicability of classical boundary layer theory is severely limited.With 4 Figures     

A particle–particle Reynolds stress transportation model of swirling particle-laden-mixtures turbulent flows

On the basis of the gas–particle Euler–Euler two-fluid approach, a new particle–particle Reynolds stress transportation model is proposed for closing the constitution equations of particle-laden-mixtures turbulent flows. In this model, binary particle-particle interaction originating from large-scale particle turbulent diffusions are fully considered in view of an extension closure idea of second-order-moment disperse gas–particle turbulent flows. The binary-particles turbulent flows with different density and same diameter are numerically simulated. The number density, the time-averaged velocity, the fluctuation velocity, the multiphase fluctuation velocity correlations, the normal and the shear Reynolds stress are obtained. Simulated results are in good agreement with experimental data. Binary mixture system has a unique transportation behavior with a stronger anisotropy due to particle inertia and multiphase turbulence diffusions. Fluctuation velocity correlation of axial–axial gas–particle is about twice larger than those of axial–axial particle–particle interaction. Moreover, both normal and shear Reynolds stress are redistributed.     

Pipe-Wall Friction in Vertical Sand-Slurry Flows

This article describes the results of extensive laboratory tests of vertical flows of three sand fractions (0.12, 0.37, and 1.84 mm sands) in a 150 mm pipe. The tests revealed an interesting phenomenon of a surprisingly low contribution of the medium sand to the total friction of the mixture flow in the vertical pipe. The frictional pressure drop in highly concentrated flows at high velocities was lower for slurries of the medium sand than for slurries of both the fine sand and the coarse sand. The solids friction at the pipe wall is analyzed taking into account effects of particle-particle interactions and particle-liquid interactions in the boundary layer of a vertical flow of settling slurry. The analysis suggests that the observed phenomenon is associated with the hydrodynamic liquid lift force acting on solid particles traveling near a pipe wall. This off-wall force seems to be more effective for the medium sand particles than for the fine sand particles and coarse sand particles interacting with liquid in the boundary layer of the mixture flow. The excessive frictional pressure drop due to the presence of solids in vertical flows seems to be primarily produced by the combined effect of the Bagnold collisional force (the force that colliding particles exert against the pipe wall) and the liquid lift force acting on solid particles in the near-wall zone of the slurry flow.     

逆压梯度下近壁湍流的喷射和扫掠

采用Fourier谱展开和紧致有限差分格式，选用两组共振三波为相干结构的初值，计算了其在零压和逆压梯度作用下的演化。对演化后期流场的2，4象限的运动进行了详细的分析。结果发现，在逆压梯度下，扫掠对雷诺应力的贡献要强于喷射。无论是在零压梯度还是逆压梯度下，uv和u2在法向的输运主要是靠Q2和Q4这两种运动来完成的。零压梯度下喷射部分对输运的贡献大于扫掠的部分。而在逆压梯度下喷射部分对输运的贡献明显减少，扫掠的作用要强于喷射。     

Analytical solution of time periodic electroosmotic flows: analogies to Stokes' second problem.

Analytical solutions of time periodic electroosmotic flows in two-dimensional straight channels are obtained as a function of a nondimensional parameter kappa, which is based on the electric double-layer (EDL) thickness, kinematic viscosity, and frequency of the externally applied electric field. A parametric study as a function of kappa reveals interesting physics, ranging from oscillatory "pluglike" flows to cases analogous to the oscillating flat plate in a semi-infinite flow domain (Stokes' second problem). The latter case differs from the Stokes' second solution within the EDL, since the flow is driven with an oscillatory electric field rather than an oscillating plate. The analogous case of plate oscillating with the Helmholtz-Smoluchowski velocity matches our analytical solution in the bulk flow region. This indicates that the instantaneous Helmholtz-Smoluchowski velocity is the appropriate electroosmotic slip condition even for high-frequency excitations. The velocity profiles for large kappa values show inflection points very near the walls with localized vorticity extrema that are stronger than the Stokes layers. This have the potential to result in low Reynolds number flow instabilities. It is also shown that, unlike the steady pure electroosmotic flows, the bulk flow region of time periodic electroosmotic flows are rotational when the diffusion length scales are comparable to and less than the half channel height.     

Numerical simulation of the separated fluid flows at large Reynolds numbers

The variety of flow regimes (steady separated, periodically separated-‘Karman vortex street’, unsteady turbulent) and their characteristic peculiarities (separation and reattachment points, secondary separation, boundary layer, instability of the shear mixing layer, etc.) require the construction of effective numerical methods, which will be able to simulate adequately the considered flows. MERANGE ? SMIF–a splitting method for physical factors of incompressible fluids¹-is used for calculations of the steady and unsteady fluid flows past a circular cylinder in a wide range of Reynolds numbers (10° < Re < lo6). The finite-difference scheme for this method is of second order accuracy in the space variables, has minimal numerical viscosity and is also monotonic. Use of the Navier-Stokes equations with the corresponding transformation of Cartesian co-ordinates allows the calculations to be made by one algorithm both in a boundary layer and out of it. The method allows calculations at Re = ∞ cc and simulation of d‘Alembert’s paradox. Some results on the classical problem of the flow around a circular cylinder for a wide range of Reynolds numbers are discussed. The crisis of the total drag coefficient and the sharp rise of the Strouhal number are simulated numerically (without any turbulence models) for the critical Reynolds numbers (Re ≈ 4 × 10⁵), and are in a good agreement with experimental data.     

On the linear stability of Stokes layers

Oscillatory flows occur naturally, with applications ranging across many disciplines from engineering to physiology. Transition to turbulence in such flows is a topic of practical interest and this article discusses some recent work that has furthered our understanding of the stability of a class of time-periodic fluid motions. Our study starts with an examination of the linear stability of a classical flat Stokes layer. Although experiments conducted over many years have demonstrated conclusively that this layer is unstable at a sufficiently large Reynolds number, it has only been relatively recently that rigorous theoretical confirmation of this behaviour has been obtained. The analysis and numerical calculations for the planar Stokes layer were subsequently extended to flows in channels and pipes and for the flow within a torsionally oscillating circular cylinder. We discuss why our predictions for the onset of instability in these geometries are in disappointingly poor agreement with experimental results. Finally, some suggestions for future experimental work are given and some areas for future theoretical analysis outlined.     

Turbulence simulation and multiple scale subgrid models

 The problem of modelling turbulence in CFD is due to the wide range of length scales present in a turbulent flow. The physics of these scales is examined, and the need for models of the small scale motions is made clear. A review is given of methods of turbulence modelling. These methods can be divided into two classes: Reynolds averaged Navier–Stokes (RANS) models, and large eddy simulation (LES). The Reproducing Kernel Particle method (RKPM) is then presented and proposed as a class of filters for LES of inhomogeneous turbulent flows. Important properties of the method are discussed, including the effectiveness of the RKPM reproduction as a low-pass filter. The commutation of the filtering operation with differentiation is demonstrated, showing that the commutation error can be made arbitrarily small. A one-dimensional non-linear example problem is solved using a Galerkin method in which a bi-scale constitutive model is used for the subgrid scale terms. The extension of the method to the three-dimensional equations of fluid dynamics is then outlined, where the method is used as a filter in a dynamic subgrid stress model. Emphasis is placed on the multi-scale properties of RKPM, which allow the reproduction of different scales of the solution using the same set of nodal parameters.     

Interception of two spherical particles with arbitrary radii in simple shear flow

Summary The interception of two force-free and torque-free spherical particles with arbitrary radii freely suspended in simple shear flow is investigated in the limit of vanishing Reynolds number. At any instant, the flow is computed in a frame of reference with origin at the center of one particle using a cylindrical polar coordinate system whose axis of revolution passes through the center of the second particle. The problem is formulated as a decoupled system of integral equations for the zeroth, first, and second Fourier coefficients of the boundary traction with respect to the meridional angle. The derived integral equations are solved with high accuracy using a boundary element method that features adaptive discretization and automatic time-step adjustment according to the inter-particle gap. The results illustrate particle trajectories and describe the particle rotation and evolution of the stress tensor during the interception. The particle interaction is found to always cause a positive shift in the rotational phase angle due to the rolling motion at close contact. As the gap between two particles tends to zero, the shear stress diverges even though the net force and torque exerted on each particle remain zero, independent of the particle relative radius. A frictional force for rough surfaces and small gaps eliminates the slip velocity and promotes the rolling motion.     

Self similar unsteady boundary layers

Summary This paper deals with unsteady laminar boundary layers contiguous to self-similar flows. The following general result is obtained: when the self-similar structure of the external flow is apparent in the boundary layer system of coordinates, and when there is no heat transfer through the wall, then the boundary layer is self-similar too. This property can be applied to a lot of known self-similar motions. Some examples are considered for which the final form of the boundary layer equations is formulated.