Periodic Roughness Research Articles

Wall laws for fluid flows at a boundary with random roughness

AbstractThe general concern of this paper is the effect of rough boundaries on fluids. We consider a stationary flow, governed by incompressible Navier‐Stokes equations, in an infinite domain bounded by two horizontal rough plates. The roughness is modeled by a spatially homogeneous random field, with characteristic size ε. A mathematical analysis of the flow for small ε is performed. The Navier's wall law is rigorously deduced from this analysis. This substantially extends former results obtained in the case of periodic roughness, notably in [16, 17]. © 2007 Wiley Periodicals, Inc.

Fabrication of Spatially Periodic Double Roughness Structures by Directional Viscous Fingering and Spinodal Dewetting for Water-Repellent Surfaces

Water-repellent and self-cleaning properties of lotus leaves are considered to be due to its double roughness structure, protrusion structure (approximately 20 microm) and hairy structure (0.2-1.0 microm). In this study, attempts to fabricate a spatially periodic double roughness structure by two far-from-equilibrium self-organization phenomena, a directional viscous fingering and a spinodal dewetting, were made. A mixture of an octylsilyl titanium dioxide particle having an average diameter of 35 nm suspended in volatile silicone, decamethyl cyclopentasiloxane, and octyl p-methoxycinnamate was spread on a glass plate by dragging an applicator across the top. Formation of a stripe pattern parallel to the direction of dragging, called directional viscous fingering, was sometimes observed. Influences of spreading conditions on the pattern formation were analyzed. In addition, attempts were made to apply the stripe pattern formation to the preparation of a water repellent surface. We have succeeded in preparing a highly water-repellent surface by immersing a glass plate, on which a spatially periodic stripe pattern having a characteristic wavelength of 200-700 microm was formed, in water, after the completion of evaporation of decamethyl cyclopentasiloxane. In this case, dewetting patterns having a characteristic wavelength at around 5 microm were formed at the bottom part of the stripe patterns. Neither the surface on which only the mesoscopic spatially periodic stripe pattern was formed nor the one on which only the microscopic dewetting pattern was formed showed high water-repellent properties, indicating that the coexistence of the two different scales of patterns increased the water-repellent properties of the hydrophobic surface.

Low Grazing Scattering from Sinusoidal Neumann Surface with Finite Extent: Total Scattering Cross Section

This paper deals with the scattering of a transverse magnetic (TM) plane wave from a perfectly conductive sinusoidal surface with finite extent. By use of the undersampling approximation and a rectangular pulse approximation, an asymptotic formula for the total scattering cross section at a low grazing limit of incident angle is obtained explicitly under conditions such that the surface is small in roughness and slope, and the corrugation width is sufficiently large. The formula shows that the total scattering cross section is proportional to the square root of the corrugation width but does not depend on the surface period and surface roughness. When the corrugation width is not large, however, the scattered wave can be obtained by a single scattering approximation, which gives the total scattering cross section proportional to the corrugation width and the Rayleigh slope parameter. From the asymptotic formula and the single scattering solution, a transition point is defined explicitly. By comparison with numerical results, it is concluded that the asymptotic formula is fairly accurate when the corrugation width is much larger than the transition point.

An experimental study and modelling of roughness effects on laminar flow in microchannels

Three different approaches were used in the present study to predict the influence of roughness on laminar flow in microchannels. Experimental investigations were conducted with rough microchannels 100 to 300μm in height (H). The pressure drop was measured in test-sections prepared with well-controlled wall roughness (periodically distributed blocks, relative roughness k* =k/0.5H≈0.15) and in test-sections with randomly distributed particles anchored on the channel walls (k* ≈0.04–0.13). Three-dimensional numerical simulations were conducted with the same geometry as in the test-section with periodical roughness (wavelength L). A one-dimensional model (RLM model) was also developed on the basis of a discrete-element approach and the volume-averaging technique. The numerical simulations, the rough layer model and the experiments agree to show that the Poiseuille number Po increases with the relative roughness and is independent of Re in the laminar regime (Re<2000). The increase in Po observed during the experiments is predicted well both by the three-dimensional simulations and the rough layer model. The RLM model shows that the roughness effect may be interpreted by using an effective roughness height keff. keff/k depends on two dimensionless local parameters: the porosity at the bottom wall; and the roughness height normalized with the distance between the rough elements. The RLM model shows that keff/k is independent of the relative roughness k* at given k/L and may be simply approximated by the law: keff/k = 1 − (c(ϵ)/2π)(L/k) for keff/k>0.2, where c decreases with the porosity ϵ.

Wetting of Nanogrooved Polymer Surfaces

Molecular dynamics simulations were used to study the wetting of nanogrooved PE and PVC polymer surfaces. The contact angles, equilibrium states, and equilibrium shapes of two nanosized water droplets were analyzed on surfaces with 1D-arranged periodic roughness of various dimensions. The composite solid-liquid contact, which is preferred in practical applications and in which a droplet rests on top of the surface asperities, was observed on the roughest PE surfaces, whereas water filled the similar but slightly deeper grooves on PVC surfaces. The transition from the wetted to composite contact regime occurred when the contact angle with a flat surface reached the value at which the apparent Wenzel and Cassie contact angles are equal. Droplets on grooved PE surfaces with the composite contact exhibited contact angles in agreement with Cassie's equation, but the increase in hydrophobicity on smoother surfaces with the wetted contact was less than expected from Wenzel's equation. The difference between the simulated and theoretical values decreased as the dimensions of the surface grooves increased. Only a slight increase or even a slight decrease in the contact angles was observed on the grooved PVC surfaces, owing to the less hydrophobic nature of the flat PVC surface. On both polymers, the nanodroplet assumed a spherical shape in the composite contact. Only minor anisotropy was observed in the wetted contact on PE surfaces, whereas even a highly anisotropic shape was seen on the grooved PVC surfaces. The contact angle in the direction of the grooves was smaller than that in the perpendicular direction, and the difference between the two angles decreased with the increasing size of the water droplet.

Enhanced Hydrophobicity of Rough Polymer Surfaces

Molecular dynamics simulations were used to study the effect of periodic roughness of PE and PVC polymer surfaces on the hydrophobicity. Pillars of different lateral dimensions and heights were derived from flat crystalline surfaces, and the results of nanoscale simulations on the structured surfaces were compared with theoretical predictions of the Wenzel and Cassie equations. Hydrophobicity increased on all rough surfaces, but the increase was greater on the structured PE surfaces because of the larger water contact angle on the flat PE surface than the corresponding PVC surface. Equally sized pillar structures on the two polymers resulted in different equilibrium wetting geometries. Composite contacts were observed on rough PE surfaces, and the contact angle increased with decreasing contact area between the solid and the liquid. Opposite results were obtained for rough PVC surfaces; the contact angle increased with the solid-liquid contact area, in agreement with Wenzel's equation. However, the composite contact was observed if the energies of the wetted and composite contacts were almost equal. Good agreement was obtained between the simulated contact angles and equilibrium droplet shapes and the theories but there were also some limitations of the nanoscale simulations.

Effect of roughness on pressure fluctuations in a turbulent channel flow

Direct numerical simulation is used to investigate the nature of pressure fluctuations induced by surface roughness in a turbulent channel flow at Reτ=400 for three-dimensional periodic roughness elements, whose peaks overlap approximately 25% of the logarithmic layer. The three-dimensional roughness elements alter the pressure statistics significantly, compared to the corresponding smooth-wall flow, in both the inner and outer (core) regions of the channel. The direct consequence of roughness is an increased form drag, associated with more intense pressure fluctuations. However, it also alters the pressure fluctuations in the outer layer of the flow, and modifies the length scales defined by two-point correlations. We also find that the depth of the roughness sublayer defined by the pressure fluctuations is very different from that given by the large- and small-scale statistics from the velocity field.

Fabrication of Flexible Gold Films with Periodic Sub-Micrometer Roughness and Their Wettability Control by Modification of SAM

Flexible honeycomb gold films supported by polymer sheets are fabricated by using polystyrene particle monolayers. The surfaces of the flexible gold films are covered with self-assembled monolayers (SAMs) of hydrophobic or hydrophilic thiol compounds, and the wettability of the modified surface is evaluated by measurements of the contact angles of water droplets. The contact angle of the film covered with hydrophobic SAM is ca. 150 degrees, which is greater than the value of 112 degrees for a flat gold surface, while the values for hydrophilic SAM are below 10 degrees.

Micro-roughness effects in (elasto)hydrodynamic lubrication including a mass-flow preserving cavitation model

An average Reynolds equation is proposed for predicting the effects of deterministic periodic roughness, taking JFO mass flow preserving cavitation model and elastohydrodynamic effects into account. For this, the asymptotic model is based upon a double scale analysis approach. The average Reynolds equation can be used both for the description of cavitation on a macroscopic scale in widening gap regions as well as for cavitation on the microscopic interasperity scale. Results of numerical simulations based on the model are presented for the case of a hydrodynamically lubricated journal bearings and an elasto-hydrodynamically lubricated point contact.

Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular-scale simulations versus continuum predictions

The influence of surface roughness on the slip behaviour of a Newtonian liquid in steady planar shear is investigated using three different approaches, namely Stokes flow calculations, molecular dynamics (MD) simulations and a statistical mechanical model for the friction coefficient between a corrugated wall and the first liquid layer. These approaches are used to probe the behaviour of the slip length as a function of the slope parameter ka = 2πa/λ, where a and λ represent the amplitude and wavelength characterizing the periodic corrugation of the bounding surface. The molecular and continuum approaches both confirm a monotonic decay in the slip length with increasing ka but the rate of decay as well as the magnitude of the slip length obtained from the Stokes flow solutions exceed the MD predictions as the wall feature sizes approach the liquid molecular dimensions. In the limit of molecular-scale wall corrugation, a Green–Kubo analysis based on the fluctuation–dissipation theorem accurately reproduces the MD results for the behaviour of the slip length as a function of a. In combination, these three approaches provide a detailed picture of the influence of periodic roughness on the slip length which spans multiple length scales ranging from molecular to macroscopic dimensions.

Influence of small imperfections on the stability of plane Poiseuille flow and the limitation of Squire’s theorem

In a recent paper [Luo and Wu, Phys. Fluids 16, 2852 (2004)], a theoretical model was proposed to illustrate the key role played by small imperfections in the subcritical transition of plane Poiseuille flow. In that model, the imperfections are represented by a steady but spatially periodic two-dimensional surface roughness, and a small oscillatory pressure gradient representing the inevitable background fluctuations. The present paper extends the previous work by considering an improved model, where the wall roughness is three-dimensional. A steady perturbation in the form of spatially periodic suction is also considered. In both cases, the resulting steady and unsteady components interact to produce a forcing that is in near resonance with a pair of oblique Tollmien-Schlichting (T-S) waves. The latter are generated as a result, and grow in proportion to the time during the initial stage. This excitation process and the subsequent nonlinear development of the T-S waves are studied by a weakly nonlinear analysis. A forced nonlinear (and nonlocal) amplitude equation is derived to provide a simple mathematical framework to link the critical Reynolds number for nonlinear instability to the level of imperfections. The theoretical result demonstrates that the onset critical Reynolds number is lower for three-dimensional than for two-dimensional disturbances if the imperfection exceeds a critical level, which turns out to be remarkably small. This offers a possible explanation as to why Squire’s theorem is applicable only when external perturbations are suppressed to an extremely low level. Further analysis of the nonlinear interactions indicates that the much reduced critical Reynolds number for three-dimensional external perturbations can be attributed to the fact that the excited pair of oblique T-S waves interact to generate extremely strong streaks, which then result in a very strong nonlinear destabilizing effect.

Analysis of Mode Characteristics for Deformed Square Resonators by FDTD Technique

The mode frequencies and quality factors (Q-factors) in two-dimensional (2-D) deformed square resonators are analyzed by finite-difference time-domain (FDTD) technique. The results show that the deformed square cavities with circular and cut corners have larger Q-factors than the perfect ones at certain conditions. For a square cavity with side length of 2 /spl mu/m and refractive index of 3.2, the mode Q-factor can increase 13 times as the perfect corners are replaced by a quarter of circle with radius of 0.3 /spl mu/m. Furthermore the blue shift with the increasing deformations is found as a result of the reduction in effective resonator area. In square cavities with periodic roughness at sidewalls which maintains the symmetry of the square, the Q-factors of the whispering gallery (WG)-like modes are still one order of magnitude larger that those of non-WG-like modes. However, the Q-factors of these two types of modes are of the same order in the square cavity with random roughness. We also find that the rectangular and rhombic deformation largely reduce the Q -factors with the increasing offset and cause the splitting of the doubly degenerate modes due to the breaking of certain symmetry properties.

An Average Flow Model of the Reynolds Roughness Including a Mass-Flow Preserving Cavitation Model

Abstract An average Reynolds equation for predicting the effects of deterministic periodic roughness, taking Jakobsson, Floberg, and Olsson mass flow preserving cavitation model into account, is introduced based upon the double scale analysis approach. This average Reynolds equation can be used both for a microscopic interasperity cavitation and a macroscopic one. The validity of such a model is verified by numerical experiments both for one-dimensional and two-dimensional roughness patterns.

Towards transition modelling for supersonic laminar flow control based on spanwise periodic roughness elements

Laminar flow control (LFC) is one of the key enabling technologies for quiet and efficient supersonic aircraft. Recent work at Arizona State University (ASU) has led to a novel concept for passive LFC, which employs distributed leading edge roughness to limit the growth of naturally dominant crossflow instabilities in a swept-wing boundary layer. Predicated on nonlinear modification of the mean boundary-layer flow via controlled receptivity, the ASU concept requires a holistic prediction approach that accounts for all major stages within transition in an integrated manner. As a first step in developing an engineering methodology for the design and optimization of roughness-based supersonic LFC, this paper reports on canonical findings related to receptivity plus linear and nonlinear development of stationary crossflow instabilities on a Mach 2.4, 73 degrees swept airfoil with a chord Reynolds number of 16.3 million.

Heterogeneous Multiscale FEM for Diffusion Problems on Rough Surfaces

We present a finite element method for the numerical solution of diffusion problems on rough surfaces. The problem is transformed to an elliptic homogenization problem in a two dimensional parameter domain with a rapidly oscillating diffusion tensor and source term. The finite element method is based on the heterogeneous multiscale methods of E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87--132]. For periodic surface roughness of scale $\varepsilon$ and amplitude $O(\varepsilon)$, the method converges at a robust rate, i.e., independent of $\varepsilon$, to the homogenized solution.

Resonance Scattering by a Small Particle above a Perfectly Reflecting Periodic Rough Surface

Resonance Scattering by a Small Particle above a Perfectly Reflecting Periodic Rough Surface

Solving Diffusion Problems on Rough Surfaces with a Hierarchical Multiscale FEM

AbstractDiffusion on rough surfaces is a basic problem for many applications in engineering and the sciences. Solving these problems with a standard finite element method is often difficult or even impossible, due to the computational work and the amount of memory needed to triangulate the whole surface with a mesh which resolves its oscillations. We discuss in this paper a hierarchical Finite Element Method of “heterogeneous multiscale” type, which only needs to resolve the surface's fine scale on small sampling domains within a macro triangulation of the underlying smooth surface. This method converges, for periodic surface roughness and sufficiently small amplitude, at a robust (i.e. scale independent) rate, to the homogenized solution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Spectral shift and Q change of circular and square-shaped optical microcavity modes due to periodic sidewall surface roughness

Radiation loss and resonant frequency shift due to sidewall surface roughness of circular and square high-contrast microcavities are estimated and compared by using a boundary integral equations method. An effect of various harmonic components of the contour perturbation on the Whispering-Gallery (WG) modes in the circular microdisk and WG-like modes in the square microcavity is demonstrated. In both cases, contour deformations that are matched to the mode field pattern cause the most significant frequency detuning and Q-factor change. Favorably mode-matched deformations have been found, enabling one to manipulate the Q-factors of the microcavity modes.

Wave period and flow asymmetry effects on the transition to turbulence in relation to sediment dynamics

The presence or absence of turbulence in near‐bed flows is known to affect patterns of sediment transport. Therefore the accurate prediction of turbulence is important when considering the initiation of motion of medium to coarse sands as such grains may be eroded by conditions similar to those causing turbulence. An experimental study is described to investigate boundary layer stability under nonlinear (asymmetric) oscillatory flows. In addition, the results are combined with data collected elsewhere to better explain the dependence of initial transition to turbulence on bed roughness and oscillatory period. An oscillating trolley system simulating such oscillatory flows at prototype scale is described. The transition to turbulence under asymmetric (nonlinear) oscillatory flow is investigated utilizing visual techniques and the high‐resolution measurement of near‐bed velocity. These data are used in conjunction with the extensive data sets of Li [1954] and Manohar [1955] for linear oscillatory flows, collected using similar equipment and methods. Relationships are presented to describe the transition over smooth and fixed (granular) roughness beds under linear and nonlinear flows. The transition to turbulence is shown to have a strong positive linear correlation with both oscillatory period and grain roughness up to a maximum Reynolds number for linear flows. An additional positive nonlinear relationship with asymmetry is observed; this was proportionately greater with increasing bed roughness (from smooth beds to a uniform bed roughness of 550 μm). These relationships suggest that transition is regulated by a balance between stabilizing and destabilizing forces or conditions, namely, rates of fluid acceleration, the time frame for development of turbulence, and sources of the initial flow perturbation. It is argued that (wave) period dependence observed in threshold of motion data may be explained by a combination of (a) period dependency in the transition to turbulence or of (b) translation of the proposed regulatory mechanisms for the transition to turbulence.

Influence of small imperfections on the stability of plane Poiseuille flow: A theoretical model and direct numerical simulation

It is well known that the stability of plane Poiseuille flow is extremely sensitive to small imperfections that are inevitably present. In this paper, a simple model is proposed, in which the imperfections are represented by a steady but spatially periodic surface roughness and a small oscillatory pressure gradient. A steady perturbation in the form of spatially periodic suction is also considered. For both cases, the resulting steady and unsteady components interact to produce a forcing that is in resonance with the Tollmien–Schlichting (TS) wave. The latter is excited as a result, and grows in proportion to time during the initial stage. The subsequent nonlinear development of the TS wave is shown to be governed by a forced (nonlocal) amplitude equation, which provides a simple framework to link the subcritical nonlinear instability explicitly to external forcing. The validity of the amplitude equation was checked against the direct numerical simulation (DNS), carried out in the case of wall suction. The results indicate that for a fixed level of external disturbance, there exist two distinctive regimes of response: A large-amplitude regime for the Reynolds numbers above a critical value, and a small-amplitude regime below it. The large-amplitude response regime in DNS was found to correspond to the occurrence of a finite-time singularity in the solution to the amplitude equation. This observation allows the critical Reynolds number for the large-amplitude response regime to be defined as one that divides the singular and bounded solutions of the amplitude equation. Estimate based on this equation shows that presence of small imperfections may reduce the critical Reynolds number for subcritical instability to values well below 5772.22.