No-slip Wall Research Articles

Coherent structure dynamics in near-wall turbulence

The regeneration and dynamics of near-wall longitudinal vortices – which dominate turbulence production, drag, and heat transfer – are analyzed using direct numerical simulation of turbulent channel flow. These dominant streamwise vortices are shown to result from nonlinear saturation of an instability of lifted low-speed streaks near a single wall, free from any initial vortex. The newly-found instability mechanism initiates streak waviness in the ( x, z) plane, generating streamwise vorticity sheets. Streak waviness in turn induces positive ∂u/ ∂x (i.e. positive VISA), which causes these sheets’ vorticity to then concentrate via stretching (rather than roll up) into new streamwise vortices. The instability requires sufficiently strong streaks ( y circulation per unit x>7.6 wall units) and is inviscid in nature, despite the close proximity of the no-slip wall. We find that self-annihilation of streaks due to viscous cross-diffusion of opposite-signed wall normal vorticity across each streak causes the instability amplification to scale in wall units, suggesting the relevance of our results to high Re as well. Significantly, the strongly 3D vortices generated by streak instability agree well with the CS educed from fully turbulent flow, suggesting the prevalence of this streak instability-based vortex formation mechanism. Simultaneous to vortex generation, an internal shear layer is generated across the streak from each streamwise vortex by stretching of spanwise vorticity. Such a shear layer eventually rolls up at the downstream end of a streamwise vortex, and the two link up and propagate outward to form a spanwise “arch” vortex. The newly generated streamwise vortices act to sustain/strengthen the preexisting streak which spawned them through localized lifting of near-wall fluid by induction. We develop a new spatiotemporal vortex generation mechanism, in which vortices leave behind “vortex-less” streaks, whose instability, due to the mechanism explained herein, initiates streamwise vortex formation.

Bénard–Marangoni convection at low Prandtl number

Surface-tension-driven Bénard convection in low-Prandtl-number fluids is studied by means of direct numerical simulation. The flow is computed in a three-dimensional rectangular domain with periodic boundary conditions in both horizontal directions and either a free-slip or no-slip bottom wall using a pseudospectral Fourier–Chebyshev discretization. Deformations of the free surface are neglected. The smallest possible domain compatible with the hexagonal flow structure at the linear stability threshold is selected. As the Marangoni number is increased from the critical value for instability of the quiescent state to approximately twice this value, the initially stationary hexagonal convection pattern becomes quickly time-dependent and eventually reaches a state of spatio-temporal chaos. No qualitative difference is observed between the zero-Prandtl-number limit and a finite Prandtl number corresponding to liquid sodium. This indicates that the zero-Prandtl-number limit provides a reasonable approximation for the prediction of low-Prandtl-number convection. For a free-slip bottom wall, the flow always remains three-dimensional. For the no-slip wall, two-dimensional solutions are observed in some interval of Marangoni numbers. Beyond the Marangoni number for onset of inertial convection in two-dimensional simulations, the convective flow becomes strongly intermittent because of the interplay of the flywheel effect and three-dimensional instabilities of the two-dimensional rolls. The velocity field in this intermittent regime is characterized by the occurrence of very small vortices at the free surface which form as a result of vortex stretching processes. Similar structures were found with the free-slip bottom at slightly smaller Marangoni number. These observations demonstrate that a high numerical resolution is necessary even at moderate Marangoni numbers in order to properly capture the small-scale dynamics of Marangoni convection at low Prandtl numbers.

Treatment of turbulent wall boundary conditions using linear-logarithmic elements

In the present paper special wall elements are introduced in order to implement near-wall boundary conditions accurately. These linear-logarithmic wall elements allow a no-slip wall condition with less resolution requirements than would else be the case. This approach is combined with a modified two-layer turbulence model, and applications to some two-dimensional cases are shown to illustrate the potential of the method.

Coupling Between a Viscoelastic Gas/Liquid Interface and a Swirling Vortex Flow

While the structure and dynamics of boundary layers on rigid no-slip walls in rotation dominated enclosed flows are still an area of active research, the interactions between rotating or swirling flows with a free surface have received comparatively less attention. For the most part, investigations in this area have been focused on clean free surfaces, which may be treated as stress-free. However, in most practical situations the surface is rarely clean, and even under laboratory conditions, it is quite difficult to achieve a clean free surface. Most impurities in liquids are surface active, and hence the name surface active agent or surfactant. These surfactants tend to establish an equilibrium surface concentration which alters the interfacial tension and interfacial viscoelastic properties of the gas/liquid interface. The coupling between the bulk swirling flow and the interface is provided via the tangential stress balances, and these stresses on the interface are dependent upon the surface concentration of surfactant, which in turn is altered by the interfacial flow. Forces acting on the interface include surface tension gradients (elastic) and the viscous resistance to shear and dilatation. These viscoelastic properties vary with the surfactant concentration on the surface. Here, we present numerical studies of flow in a cylinder driven by the constant rotation of the bottom endwall with the top free surface being contaminated by a Newtonian surfactant. Comparisons with a clean free surface and a no-slip stationary top endwall provide added insight into the altered dynamics that result from the presence of a small amount of surfactant.

Direct numerical simulation of particle deposition onto a free-slip and no-slip surface

We consider here the direct numerical simulation (DNS) of channel flow with two different surfaces: a no-slip, fixed wall and on the opposite side a free-slip, free surface. The simulated velocity field agrees well with the experimental data for a free-surface flow obtained by Komori et al. [Int. J. Heat Mass Transf. 25, 513 (1982)]. The DNS is used to simulate particle trajectories, which are computed with a dynamic particle equation in which only the drag force given by the Stokes law is taken into account. For the particle time scale, nondimensionalized in terms of the fixed-wall friction velocity and the kinematic viscosity, we use the values τ+=5 and τ+=15. A statistically stationary condition is studied that is obtained by the introduction of a uniform distribution of particles at the beginning of the channel and by continuous removal through deposition at the two walls. The steady-state concentration distribution is nonuniform across the channel width, primarily due to the process whereby particles are trapped close to the surface. Moreover, we find that the wall–normal concentration profiles are self-similar. The deposition on both the no-slip and the free-slip wall can be described by a constant deposition coefficient, with for τ+=5 the larger value on the free-slip wall and for τ+=15 the opposite, i.e., the larger value over the no-slip wall. To study the deposition process in more detail we consider the cross channel particle fluxes and velocity statistics that are conditioned on deposition events. By means of instantaneous near-wall particle distributions we also consider the patterns of particles and their accumulation in certain areas of the flow. For a no-slip surface the well-known result that particles tend to collect in the low-speed streaks is confirmed. The patterns of particles near the free-slip surface are completely different, which can be explained in terms of the different types of coherent structures that are present near this surface.

Accuracy and performance of numerical wall boundary conditions for steady, 2D, incompressible streamfunction vorticity

Three recent papers have studied fourth-order compact discretizations of the streamfunction vorticity equations. They differed primarily in how the no-slip wall boundary conditions were handled. In this paper, these different formulas are compared to one another, as well as to three newly proposed formulas. Special consideration is paid to the truncation errors; in particular, it is shown that many well-known formulations are actually more accurate by O(h) than previously reported, where h is the mesh size. These new theoretical error rates are confirmed with an analytical model problem. The different formulas are then compared with published driven cavity results, both in terms of accuracy and performance, and the newly proposed high-order Jensen formula is judged to have the marginally best combination of these properties. © 1998 John Wiley & Sons, Ltd.

Finite element implementation of large eddy simulation for separated flows

In this paper, a finite element method is applied to the large eddy simulation of unsteady, vortex-shedding flows past a square cylinder. Galerkin FEM with linear, isoparametric elements is implemented. Unsteady computations are performed using the explicit (Euler) method, with a streamline upwind scheme for advection term. Subcycling strategy is incorporated into time integration. First, the results for two-dimensional, low Reynolds number flow with vortex shedding are presented to validate the computational methodology. Then, three-dimensional calculation of a flow past a square cylinder at Re=22 000 is discussed. The Smagorinsky model is used for parametrization of the subgrid scales. No-slip wall condition is enforced. The computed mean and root mean square of pressure as well as wake structure are compared with numerical and experimental results reported by other researchers.

Analytic Investigation of the Increase in Kinetic Energy at the Shear-Free Constant-Pressure Boundary of a Fluctuating Flow.

Physical and numerical experiments and computations reported in the literature conclude that the kinetic energy of free-surface bounded turbulent flows appears to increase as the free surface is approached from below. There is in the previous literature no satisfactory explanation of the mechanism underlying this feature. In order to provide insight revealing the mechanism, an analytic investigation of an oscillating laminar flow bounded on the lower side by a horizontally moving no-slip wall and on the upper side by a shear-free constant-pressure boundary is presented herein. Although the motion is periodic, rather than turbulent, essential features including momentum diffusion are present and hence the flow is a valid candidate for indicating the mechanism. A mathematical solution for the velocity and the kinetic energy is analyzed to determine the physical mechanism leading to the increase in kinetic energy. The finding is that the relaxation of the shear stress at the free surface produces enhanced viscous acceleration of the fluid in a thin layer adjacent to the free surface. The enhanced acceleration produces a nearly constant kinetic energy in the layer

Inviscid dipole-vortex rebound from a wall or coast

A vortex approaching a no-slip wall ‘rebounds’ due to the creation of vorticity at the wall in a viscous boundary layer. Here it is demonstrated that a purely inviscid mechanism can also produce vortex rebound from a slip wall. In inviscid vortex rebound, vortex tube stretching generates the necessary vorticity to allow rebound, eliminating the need for viscous vorticity generation. This vortex stretching mechanism is demonstrated through numerical simulations and laboratory experiments on dipole-vortex rebound from a boundary. In an application to oceanography, numerical simulations of both quasi-geostrophic and shallow water dynamics are used to show that the β-effect at an eastern boundary can produce this inviscid rebound. Through a series of numerical experiments in which the strength of the β-effect is varied, a formula for predicting the point of separation of the vortices from the boundary in a dipole–coast collision is deduced. Through simulations, the flux of vorticity and fluid away from the boundary is measured as a function of β and initial angle of incidence. It is found that, in contrast to viscous vortex rebound, which typically does not produce a flux of material away from the boundary farther than a distance comparable to the initial vortex radius, the β-induced rebound does carry fluid far from the coast. Laboratory experiments in a rotating tank are used to show that a sloping bottom can also provide an inviscid mechanism for dipole-vortex rebound from the wall of the tank under certain conditions. A relation determining the conditions under which inviscid or viscous processes will dominate in the rebound of the dipole from a boundary is obtained.

Direct numerical simulation of turbulent flow over a wavy wall

The Navier–Stokes equations have been solved, by a pseudospectral method, for pressure-driven flows between a no-slip wavy wall and a slip flat wall. Periodic boundary conditions were used in the streamwise and spanwise directions. The physical domain is mapped into a computational domain that is a rectangular parallelepiped using a nonorthogonal transformation. The pseudospectral solution procedure employed in previous studies, for example, Lam and Banerjee [Phys. Fluids A 4, 306 (1992)], eliminated the pressure and solved for the wall–normal velocity and vorticity. The other velocity components were calculated using the definition of vorticity, and the continuity equation. This procedure leads to oscillations in the pressure field when solutions were attempted in the mapped computational domain. To overcome the problem, the procedure had to be modified and the pressure solved for directly using a fractional time step technique. For the cases examined here, these modifications resulted in spectral accuracy being maintained. Flow over sinusoidal wave trains has been simulated and the results compare well with available experiments. The simulations show significant effects of the wavy boundary on the mean flow and the turbulence statistics. The mean velocity profile differs substantially from the profile for the flat-wall case, particularly in the buffer region where the fluid is under the influence of both the wavy wall and the slip boundary. The velocity fluctuations in the streamwise direction decrease in the buffer region. This effect becomes more pronounced when the wave amplitude increases. Most of the redistribution of energy, from the streamwise direction to the spanwise and wall–normal directions, occurs in a thin layer close to the boundary, downstream of the wave troughs. The energy primarily redistributes into spanwise fluctuations. High shear stress regions form downstream of the wave troughs, and streaky structures and quasi-streamwise vortices are also seen to initiate in these regions. The length of the streaks, and the extent of the quasi-streamwise vortices, scale with wave length for the two cases investigated.

A finite element analysis of the squeeze flow of an elasto-viscoplastic paste material

A finite element analysis of squeeze flow has been implemented for a material that exhibits elasto-viscoplasticity. The formulation is based upon the assumption that linear elastic deformation occurs prior to yield and that the yield surface is strain rate hardening as defined by an associated viscoplastic flow rule. Both no-slip and lubricated wall boundary conditions are considered. The numerical simulation results are compared with experimental measurements involving a model elasto-viscoplastic material for which the material parameters were derived from tensile and ram extrusion measurements. Satisfactory agreement was obtained for the compressive forces as a function of displacement, the radial displacement fields and the wall normal and shear stress distributions.

Ship waves on a viscous fluid of finite depth

In this paper we are concerned with the wave generation by a singular forcelet in a viscous fluid of finite depth, where the singularity is located far from the bottom and not very near the free surface. In the first part of this work, the image system of an Oseenlet bounded by a no-slip wall, is considered. It is found that the resultant velocity field can be described by a planar distribution of vertical Oseen doublets and a negative Oseenlet located at the mirror point of the singularity with respect to the plane wall. In the second part of the work we deal with the generation of waves by these solutions. By imposing the linearized free-surface conditions on the solutions obtained from the first part, the wave generated is shown to exhibit the Kelvin ship wave pattern that agrees with observation. The effects of water depth and of submergence on the wave amplitude are also investigated.

Three-dimensional vortex/wall interaction: Entrainment in numerical simulation and experiment

Three-dimensional interactions between an elliptic vortex ring and two no-slip parallel walls are visualized in a numerical simulation and an experiment. The vortex ring induces a surface vorticity layer on the wall which reconnects with the vortex ring. During the interaction core-area-varying axial waves are generated and carry the surface layer away from the wall. The vortex ring then becomes two tornado-like structures with strong upward helical flows near the wall surface, which provides entrainment of the surface fluid into the vortex structure. Similarities between the entrainment mechanisms of vorticity in the simulation and dye in the experiment from the surface layer are identified.

Inverse cascades of angular momentum

Most theoretical and computational studies of turbulence in Navier—Stokes fluids and/or guiding-centre plasmas have been carried out in the presence of spatially periodic boundary conditions. In view of the frequently reproduced result that two-dimensional and/or MHD decaying turbulence leads to structures comparable in length scale to a box dimension, it is natural to ask if periodic boundary conditions are an adequate representation of any physical situation. Here, we study, computationally, the decay of two-dimensional turbulence in a Navier—Stokes fluid or guiding-centre plasma in the presence of circular no-slip rigid walls. The method is wholly spectral, and relies on a Galerkin approximation by a set of functions that obey two boundary conditions at the wall radius (analogues of the Chandrasekhar— Reid functions). It is possible to explore Reynolds numbers up to the order of 1250, based on an RMS velocity and a box radius. It is found that decaying turbulence is altered significantly by the no-slip boundaries. First, strong boundary layers serve as sources of vorticity and enstrophy and enhance the early-time energy decay rate, for a given Reynolds number, well above the periodic boundary condition values. More importantly, angular momentum turns out to be an even more slowly decaying ideal invariant than energy, and to a considerable extent governs the dynamics of the decay. Angular momentum must be taken into account, for example, in order to achieve quantitative agreeement with the predicition of maximum entropy, or ‘most probable’, states. These are predicitions of conditions that are established after several eddy turnover times but before the energy has decayed away. Angular momentum will cascade to lower azimuthal mode numbers, even if absent there initially, and the angular momentum modal spectrum is eventually dominated by the lowest mode available. When no initial angular momentum is present, no behaviour that suggests the likelihood of inverse cascades is observed.

Numerical determination of turbulent fractal dimensions

This paper focuses on the fractal dimension of a line embedded in a homogeneous turbulent field. Based on an intuitive picture motivated by fractal theory, the introduction provides a physical approach for analysing the fractal dimension. The analysis is first applied to the case of an isotropic turbulent field which is suddenly subjected to steady rotation. It is shown that for a non-rotating field the fractal dimension D′ of the line increases with time. When expressed as a function of t/td, where td is the time of decay, the fractal dimension depends on both the Reynolds number and the rotation rate. However, using a suitable combination of linear (rotation rate Ω) and non-linear (td) mechanisms to define a characteristic time t*=td(1+αtdΩ), the dependence of the fractal dimension on rotation rate can be absorbed for Rossby numbers Ro≳1.54. The fractal dimension of a line immersed in periodic channel flow is also considered and the role of a no-slip wall is analysed. It shows that the fractal dimension increases rapidly from unity on the solid surface, and that this sharp and substantial increase occurs with 15 wall units. In the cases considered, this region corresponds to zones where the rms value of the velocity fluctuation is maximum.

An analytical and statistical examination of the low-wave-number region of the turbulent boundary layer wall pressure

The underlying physical mechanisms that generate the low-wave-number pressure components of the turbulent boundary layer wall pressure are examined based on a solution of linearized Navier-Stokes equations in the viscous sublayer with no-slip wall condition. The result indicates that the predominant low-wave-number pressure components are generated by viscous diffusion of shear stress fluctuations at the no-slip wall. These random pressure components have a wave-number spectrum spanning from zero wave number to well beyond the convective wave number. However, their contribution to the convective ridge of the wall pressure spectrum is negligible compared to those of other sources. In the low-wave-number limit, we reach the same conclusion reached by Chase [J. Fluid Mech. 225, 545–555 (1991)], i.e., the wave number spectral density does not vanish as the streamwise wave number approaches zero. The reasons for nonvanishing wave-number spectral density as k→0, for both shear stress and wall pressure, are established analytically. A statistical model is derived based on a probabilistic area-averaging on an assumed random process which is capable of producing the measured two-point correlation functions. The result is a space-time autocorrelation function that yields the experimental low-wave-number data. [Work supported by NSWC ILIR Program and ONR.]

Eddy viscosity transport model for turbulent flow

An eddy viscosity transport model that is integrable to no-slip walls is described. The transport equation accommodates significant departure of the turbulence from equilibrium. The model uses an elliptic relaxation equation to avoid a need for damping functions; nonlocal wall effects are produced by solving the elliptic equation. Length scales are determined locally, without reference to distance to a surface. The model bifurcates from a laminar to a turbulent solution at a reasonable critical Reynolds number in plane channel flow. An assessment of the model is provided via boundary layer computations: these show that it behaves well in zero and adverse pressure gradients, although premature separation occurs in a strong adverse pressure gradient. Good results are obtained for the trailing edge separation bubble of an aerofoil at large angle of attack.

An experimental and theoretical study of the squeeze-film deformation and flow of elastoplastic fluids

Lubrication theory is commonly employed to analyse the squeeze-film flow of plastic fluids under no-slip wall boundary conditions. Solutions exist for both Bingham and Herschel-Bulkley fluids but they infer that there exists a rigid or unyielded core and flow zones adjacent to the platens; it has been recognised previously that such a velocity field is kinematically inconsistent. Furthermore, the pressure boundary condition at the edge of the platens is conventionally set to zero which is inconsistent with experimental data presented here for a model Herschel-Bulkley fluid (Plasticine). An attempt has been made to examine the lubrication theory in more detail by a comparison with equilibrium stress analysis for rigid-plastic solids. Squeeze-film measurements were carried out using a model Herschel-Bulkley fluid and the results were consistent with the theory, suggesting that it is a useful first approximation. Nevertheless, the approach does not resolve the kinematic inconsistency resulting in lubrication theory.

Vortex rings impinging on walls: axisymmetric and three-dimensional simulations

Accurate numerical simulations of vortex rings impinging on flat boundaries revealed the same features observed in experiments. The results for the impact with a free-slip wall compared very well with previous numerical simulations that used spectral methods, and were also in qualitative agreement with experiments. The present simulation is mainly devoted to studying the more realistic case of rings interacting with a no-slip wall, experimentally studied by Walker et al. (1987). All the Reynolds numbers studied showed a very good agreement between experiments and simulations, and, at Rev > 1000 the ejection of a new ring from the wall was seen. Axisymmetric simulations demonstrated that vortex pairing is the physical mechanism producing the ejection of the new ring. Three-dimensional simulations were also performed to investigate the effects of azimuthal instabilities. These simulations have confirmed that high-wavenumber instabilities originate in the compression phase of the secondary ring within the primary one. The large instability of the secondary ring has been explained by analysis of the rate-of-strain tensor and vorticity alignment. The differences between passive scalars and the vorticity field have been also investigated.

Length scales and the energy balance for turbulence near a free surface

The structure of turbulence near a free surface is examined by using results obtained from a direct simulation of flow between a no-slip wall and a shear free boundary, which serves as a model of a waveless free surface. An energy balance analysis shows that the pressure-strain term is the dominant producing term for the spanwise component of the turbulent kinetic energy. In addition, the dissipation rates for the horizontal components of the turbulence are reduced near the free surface, whereas the dissipation rate for the vertical component remains approximately constant. Two-point correlations, energy spectra, and length scales reveal important free surface induced effects. The length scales near the free surface are compared with the scales near the centerline of normal turbulent channel flow. This comparison reveals an increase by a factor of three in the streamwise length scales associated with the spanwise velocity fluctuations and an increase by a factor of two in the spanwise length scales for the streamwise velocity fluctuations. The length scales normal to the free surface are decreased for all velocity components. This indicates a more pancake-like eddy structure near the free surface compared with the structure near the centerline of a normal channel. The energy spectra show qualitative agreement with the Hunt-Graham model, though higher resolution calculations will be required to make more quantitative comparisons. The streaky structure in free surface bounded turbulent channel flow is noticeably more persistent than in normal turbulent channel flow. This is principally due to the attachment of streamwise oriented wall eddies to the free surface.