Thin Viscous Boundary Layer Research Articles

A model order reduction technique for the linearised compressible flow equations

AbstractViscous effects in acoustics are significant if the thickness of the oscillating viscous boundary layer (Stokes boundary layer) becomes comparable to characteristic problem dimensions. For air as an acoustic medium, this is frequently the case in small‐scale devices like micro‐electro‐mechanical systems or micro‐perforated panels for sound absorption. Accurate modelling of viscous effects can be done by the solution of the linearised compressible flow equations by the finite element method, which is computationally demanding due to the high number of unknowns arising in the problem: pressure, velocity and temperature. Furthermore, the discretisation must resolve the thin viscous boundary layers, creating a model with a high number of degrees of freedom. We suggest a projection‐based model order reduction procedure using the system eigenmodes as generalised coordinates. The procedure is tested based on a simple example problem in 2D. Here we show that the derived reduced order model is accurate and computationally highly efficient. The number of degrees of freedom can be reduced by several orders of magnitude, from around 30 000 to 100, thereby dramatically reducing the computation time for harmonic solutions.

Efficient modeling of sharp-edge acoustofluidics

Sharp-edge structures exposed to acoustic fields are known to produce a strong non-linear response, mainly in the form of acoustic streaming and acoustic radiation force. The two phenomena are useful for various processes at the microscale, such as fluid mixing, pumping, or trapping of microparticles and biological cells. Numerical simulations are essential in order to improve the performance of sharp-edge-based devices. However, simulation of sharp-edge structures in the scope of whole acoustofluidic devices is challenging due to the thin viscous boundary layer that needs to be resolved. Existing efficient modeling techniques that substitute the need for discretization of the thin viscous boundary layer through analytically derived limiting velocity fail due to large curvatures of sharp edges. Here, we combine the Fully Viscous modeling approach that accurately resolves the viscous boundary layer near sharp edges with an existing efficient modeling method in the rest of a device. We validate our Hybrid method on several 2D configurations, revealing its potential to significantly reduce the required degrees of freedom compared to using the Fully Viscous approach for the whole system, while retaining the relevant physics. Furthermore, we demonstrate the ability of the presented modeling approach to model high-frequency 3D acoustofluidic devices featuring sharp edges, which will hopefully facilitate a new generation of sharp-edge-based acoustofluidic devices.

Brachistochronous motion of a flat plate parallel to its surface immersed in a fluid

We determine the globally minimum time $T$ needed to translate a thin submerged flat plate a given distance parallel to its surface within a work budget. The Reynolds number for the flow is assumed to be large so that the drag on the plate arises from skin friction in a thin viscous boundary layer. The minimum is determined computationally using a steepest descent, where an adjoint formulation is used to compute the gradients. Because the equations governing fluid mechanics for this problem are nonlinear, multiple local minima could exist. Exploiting the quadratic nature of the objective function and the constraining differential equations, we derive and apply a ‘spectral condition’ to show the converged local optimum to be a global one. The condition states that the optimum is global if the Hessian of the Lagrangian in the state variables constructed using the converged adjoint field is positive semi-definite at every instance. The globally optimum kinematics of the plate starts from rest with speed $\propto t^{1/4}$ and comes to rest with speed $\propto (T-t)^{1/4}$ as a function of time $t$ . For distances much longer than the plate, the work-minimizing kinematics consists of an optimum startup, a constant-speed cruising, and an optimum stopping.

Classical 1/3 scaling of convection holds up to Ra = 1015

The global transport of heat and momentum in turbulent convection is constrained by thin thermal and viscous boundary layers at the heated and cooled boundaries of the system. This bottleneck is thought to be lifted once the boundary layers themselves become fully turbulent at very high values of the Rayleigh number [Formula: see text]-the dimensionless parameter that describes the vigor of convective turbulence. Laboratory experiments in cylindrical cells for [Formula: see text] have reported different outcomes on the putative heat transport law. Here we show, by direct numerical simulations of three-dimensional turbulent Rayleigh-Bénard convection flows in a slender cylindrical cell of aspect ratio [Formula: see text], that the Nusselt number-the dimensionless measure of heat transport-follows the classical power law of [Formula: see text] up to [Formula: see text] Intermittent fluctuations in the wall stress, a blueprint of turbulence in the vicinity of the boundaries, manifest at all [Formula: see text] considered here, increasing with increasing [Formula: see text], and suggest that an abrupt transition of the boundary layer to turbulence does not take place.

Elastic waves in porous media saturated with non-wetting fluid

Elastic waves have widely been used as a non-destructive probing method in oilfield exploration and development, and the most well-known applications are in seismic exploration and borehole sonic logging. For waves in porous media, it is popular to use the Biot theory, which incorporates the wave-induced global flow, accounting for the frictional attenuation. The Biot theory assumes that the fluid is wetting to the solid matrix. However, the fluid is not always wetting the rock in real reservoirs. It was previously revealed that a non-wetting fluid parcel tends to slip on the solid wall pore boundary where the intermolecular potential between the fluid and solid wall is weaker than in wetting fluid conditions. This particular slippage feature means that the coupling relationship between the fluid and solid frame and frictional dissipation is likely to be very different between non-wetting and wetting fluid situations. We characterise this wave-induced slippage using an apparent viscosity for the non-wetting fluid within the thin viscous boundary layer. This apparent viscosity is smaller than the viscosity of the bulk fluid. We demonstrate that the slip correction affects the dynamic permeability and dynamic tortuosity and results in slippage/wettability dependent phase velocities and attenuation of the fully fluid-saturated rock.

Microbubble dynamics in a viscous compressible liquid near a rigid boundary

Abstract This paper is concerned with microbubble dynamics in a viscous compressible liquid near a rigid boundary. The compressible effects are modelled using the weakly compressible theory of Wang & Blake (2010, Non-spherical bubble dynamics in a compressible liquid. Part 1. Travelling acoustic wave. J. Fluid Mech., 730, 245–272), since the Mach number associated is small. The viscous effects are approximated using the viscous potential flow theory of Joseph & Wang (2004, The dissipation approximation and viscous potential flow. J. Fluid Mech., 505, 365–377), because the flow field is characterized as being an irrotational flow in the bulk volume but with a thin viscous boundary layer at the bubble surface. Consequently, the phenomenon is modelled using the boundary integral method, in which the compressible and viscous effects are incorporated into the model through including corresponding additional terms in the far field condition and the dynamic boundary condition at the bubble surface, respectively. The numerical results are shown in good agreement with the Keller–Miksis equation, experiments and computations based on the Navier–Stokes equations. The bubble oscillation, topological transform, jet development and penetration through the bubble and the energy of the bubble system are simulated and analysed in terms of the compressible and viscous effects.

Precessing cylinders at the second and third resonance: turbulence controlled by geostrophic flow.

We investigate, via both asymptotic analysis and direct numerical simulation, precessionally driven flow of a homogeneous fluid confined in fluid-filled circular cylinders that rotate rapidly about their symmetry axis and precess about a different axis and that are marked by radius-height aspect ratios Γ=1.045945 and Γ=1.611089. At these radius-height aspect ratios, the Poincaré force resonates directly with the two special inertial modes that have the simplest vertical structure. An asymptotic analytical solution in closed form describing weakly precessing flow is derived in the mantle frame of reference for asymptotically small Ekman numbers, showing quantitative agreement with the result of direct nonlinear numerical simulation. Our numerical simulation makes use of a finite-element method with the three-dimensional tetrahedralization of a cylindrical cavity that allows the construction of dense nodes in the vicinity of the bounding surface of the cavity for resolving the thin viscous boundary layer. It is found that axisymmetric geostrophic flow in the alternating eastward and westward direction can be generated and maintained by nonlinear and viscous effects in the viscous boundary layer. It is also found that, when the precessing rate is moderate and, consequently, the geostrophic flow is weak, nonlinear interaction between the resonant inertial mode and the nonesonant inertial modes driven by the Poincaré force and the boundary-layer influx leads to strongly turbulent flow with irregular temporal-spatial fluctuation. When the cylinders are strongly precessing such that the geostrophic flow becomes predominant, however, the effect of the geostrophic flow controls/stabilizes its nonlinear dynamics, leading to weakly turbulent flow that can be largely described by a dominant quasisteady geostrophic component and a weak nonaxisymmetric component localized in the region where the geostrophic flow is weak.

Boundary-Layer Effects on Acoustic Transmission Through Narrow Slit Cavities.

We explore the slit-width dependence of the resonant transmission of sound in air through both a slit array formed of aluminum slats and a single open-ended slit cavity in an aluminum plate. Our experimental results accord well with Lord Rayleigh's theory concerning how thin viscous and thermal boundary layers at a slit's walls affect the acoustic wave across the whole slit cavity. By measuring accurately the frequencies of the Fabry-Perot-like cavity resonances, we find a significant 5% reduction in the effective speed of sound through the slits when an individual viscous boundary layer occupies only 5% of the total slit width. Importantly, this effect is true for any airborne slit cavity, with the reduction being achieved despite the slit width being on a far larger scale than an individual boundary layer's thickness. This work demonstrates that the recent prevalent loss-free treatment of narrow slit cavities within acoustic metamaterials is unrealistic.

Including fluid shear viscosity in a structural acoustic finite element model using a scalar fluid representation

An approximate boundary condition is developed in this paper to model fluid shear viscosity at boundaries of coupled fluid–structure system. The effect of shear viscosity is approximated by a correction term to the inviscid boundary condition, written in terms of second order in-plane derivatives of pressure. Both thin and thick viscous boundary layer approximations are formulated; the latter subsumes the former. These approximations are used to develop a variational formation, upon which a viscous finite element method (FEM) model is based, requiring only minor modifications to the boundary integral contributions of an existing inviscid FEM model. Since this FEM formulation has only one degree of freedom for pressure, it holds a great computational advantage over the conventional viscous FEM formulation which requires discretization of the full set of linearized Navier–Stokes equations. The results from thick viscous boundary layer approximation are found to be in good agreement with the prediction from a Navier–Stokes model. When applicable, thin viscous boundary layer approximation also gives accurate results with computational simplicity compared to the thick boundary layer formulation. Direct comparison of simulation results using the boundary layer approximations and a full, linearized Navier–Stokes model are made and used to evaluate the accuracy of the approximate technique. Guidelines are given for the parameter ranges over which the accurate application of the thick and thin boundary approximations can be used for a fluid–structure interaction problem.

Turbulence dynamics near a turbulent/non-turbulent interface

AbstractThe characteristics of the boundary layer separating a turbulence region from an irrotational (or non-turbulent) flow region are investigated using rapid distortion theory (RDT). The turbulence region is approximated as homogeneous and isotropic far away from the bounding turbulent/non-turbulent (T/NT) interface, which is assumed to remain approximately flat. Inviscid effects resulting from the continuity of the normal velocity and pressure at the interface, in addition to viscous effects resulting from the continuity of the tangential velocity and shear stress, are taken into account by considering a sudden insertion of the T/NT interface, in the absence of mean shear. Profiles of the velocity variances, turbulent kinetic energy (TKE), viscous dissipation rate ($\varepsilon $), turbulence length scales, and pressure statistics are derived, showing an excellent agreement with results from direct numerical simulations (DNS). Interestingly, the normalized inviscid flow statistics at the T/NT interface do not depend on the form of the assumed TKE spectrum. Outside the turbulent region, where the flow is irrotational (except inside a thin viscous boundary layer),$\varepsilon $decays as${z}^{\ensuremath{-} 6} $, where$z$is the distance from the T/NT interface. The mean pressure distribution is calculated using RDT, and exhibits a decrease towards the turbulence region due to the associated velocity fluctuations, consistent with the generation of a mean entrainment velocity. The vorticity variance and$\varepsilon $display large maxima at the T/NT interface due to the inviscid discontinuities of the tangential velocity variances existing there, and these maxima are quantitatively related to the thickness$\delta $of the viscous boundary layer (VBL). For an equilibrium VBL, the RDT analysis suggests that$\delta \ensuremath{\sim} \eta $(where$\eta $is the Kolmogorov microscale), which is consistent with the scaling law identified in a very recent DNS study for shear-free T/NT interfaces.

Viscous coupling of shear-free turbulence across nearly flat fluid interfaces

The interactions between shear-free turbulence in two regions (denoted as + and − on either side of a nearly flat horizontal interface are shown here to be controlled by several mechanisms, which depend on the magnitudes of the ratios of the densities, ρ+/ρ−, and kinematic viscosities of the fluids, μ+/μ−, and the root mean square (r.m.s.) velocities of the turbulence, u0+/u0−, above and below the interface. This study focuses on gas–liquid interfaces so that ρ+/ρ− ≪ 1 and also on where turbulence is generated either above or below the interface so that u0+/u0− is either very large or very small. It is assumed that vertical buoyancy forces across the interface are much larger than internal forces so that the interface is nearly flat, and coupling between turbulence on either side of the interface is determined by viscous stresses. A formal linearized rapid-distortion analysis with viscous effects is developed by extending the previous study by Hunt & Graham (J. Fluid Mech., vol. 84, 1978, pp. 209–235) of shear-free turbulence near rigid plane boundaries. The physical processes accounted for in our model include both the blocking effect of the interface on normal components of the turbulence and the viscous coupling of the horizontal field across thin interfacial viscous boundary layers. The horizontal divergence in the perturbation velocity field in the viscous layer drives weak inviscid irrotational velocity fluctuations outside the viscous boundary layers in a mechanism analogous to Ekman pumping. The analysis shows the following. (i) The blocking effects are similar to those near rigid boundaries on each side of the interface, but through the action of the thin viscous layers above and below the interface, the horizontal and vertical velocity components differ from those near a rigid surface and are correlated or anti-correlated respectively. (ii) Because of the growth of the viscous layers on either side of the interface, the ratio uI/u0, where uI is the r.m.s. of the interfacial velocity fluctuations and u0 the r.m.s. of the homogeneous turbulence far from the interface, does not vary with time. If the turbulence is driven in the lower layer with ρ+/ρ− ≪ 1 and u0+/u0− ≪ 1, then uI/u0− ~ 1 when Re (=u0−L−/ν−) ≫ 1 and R = (ρ−/ρ+)(v−/v+)1/2 ≫ 1. If the turbulence is driven in the upper layer with ρ+/ρ− ≪ 1 and u0+/u0− ≫ 1, then uI/u0+ ~ 1/(1 + R). (iii) Nonlinear effects become significant over periods greater than Lagrangian time scales. When turbulence is generated in the lower layer, and the Reynolds number is high enough, motions in the upper viscous layer are turbulent. The horizontal vorticity tends to decrease, and the vertical vorticity of the eddies dominates their asymptotic structure. When turbulence is generated in the upper layer, and the Reynolds number is less than about 106–107, the fluctuations in the viscous layer do not become turbulent. Nonlinear processes at the interface increase the ratio uI/u0+ for sheared or shear-free turbulence in the gas above its linear value of uI/u0+ ~ 1/(1 + R) to (ρ+/ρ−)1/2 ~ 1/30 for air–water interfaces. This estimate agrees with the direct numerical simulation results from Lombardi, De Angelis & Bannerjee (Phys. Fluids, vol. 8, no. 6, 1996, pp. 1643–1665). Because the linear viscous–inertial coupling mechanism is still significant, the eddy motions on either side of the interface have a similar horizontal structure, although their vertical structure differs.

Energy dissipation in microfluidic beam resonators

The fluid–structure interaction of resonating microcantilevers immersed in fluid has been widely studied and is a cornerstone in nanomechanical sensor development. In many applications, fluid damping imposes severe limitations by strongly degrading the signal-to-noise ratio of measurements. Recently, Burget al. (Nature, vol. 446, 2007, pp. 1066–1069) proposed an alternative type of microcantilever device whereby a microfluidic channel was embedded inside the cantilever with vacuum outside. Remarkably, it was observed that energy dissipation in these systems was almost identical when air or liquid was passed through the channel and was 4 orders of magnitude lower than that in conventional microcantilever systems. Here, we study the fluid dynamics of these devices and present a rigorous theoretical model corroborated by experimental measurements to explain these observations. In so doing, we elucidate the dominant physical mechanisms giving rise to the unique features of these devices. Significantly, it is found that energy dissipation is not a monotonic function of fluid viscosity, but exhibits oscillatory behaviour, as fluid viscosity is increased/decreased. In the regime of low viscosity, inertia dominates the fluid motion inside the cantilever, resulting in thin viscous boundary layers – this leads to an increase in energy dissipation with increasing viscosity. In the high-viscosity regime, the boundary layers on all surfaces merge, leading to a decrease in dissipation with increasing viscosity. Effects of fluid compressibility also become significant in this latter regime and lead to rich flow behaviour. A direct consequence of these findings is that miniaturization does not necessarily result in degradation in the quality factor, which may indeed be enhanced. This highly desirable feature is unprecedented in current nanomechanical devices and permits direct miniaturization to enhance sensitivity to environmental changes, such as mass variations, in liquid.

The Ekman Layer and Why Tea Leaves Go to the Center of the Cup

Consider a transparent, cylindrical container filled with water and sitting in the center of a record player turntable. When the turntable is started suddenly, the container rotates with the turntable, but the bulk of the fluid initially remains at rest. A thin (∼1 mm) viscous boundary layer (Ekman layer) forms almost immediately at the bottom and top (if there is a lid) of the fluid. Here we describe a laboratory or demonstration exercise in which we use dye crystals1 and fine particles2 as tracers to study the flow in the Ekman layer. We also give a general method for using fine particles to measure the time it takes for the bulk of the fluid to rotate with the turntable (spin-up time) for different rates of rotation and fluid depths.

An Analytical Model of the Dynamics of the Liquefied Vitreous Induced by Saccadic Eye Movements

An analytical model of the dynamics of the vitreous humour induced by saccadic movements within the eye globe is presented. The vitreous is treated as a weakly viscous Newtonian incompressible fluid, an assumption which is appropriate when the vitreous is liquefied or when it is replaced by aqueous humour after surgery. The thin viscous boundary layer generated during a saccadic movement on the side wall is neglected and the flow field is assumed to be irrotational. The vitreous chamber is described as a weakly deformed sphere and this assumption allows a linear treatment of the problem. An analytical solution is found in the form of an expansion of spherical harmonics. Results show that the non-spherical shape of the container generates a flow field characterised by significant velocities and strong three-dimensionality. The model allows the computation of the dynamic pressure on the wall, which may play a role in the generation of retinal detachments. Moreover, results suggest that the irregular shape of the globe may significantly modify tangential stresses on the boundary with respect to the case of motion within a sphere. A simplified analytical solution, for the case of two-dimensional flow within an impulsively rotated container, shows that boundary layer detachment is expected to occur for angles of rotation larger than a threshold value of 15° circa.

Multiple scattering of acoustic waves and porous absorbing media.

Porous media like air-saturated polymer foams with open cells, have a nontrivial frequency-dependent absorption that arises due to viscous and thermal effects at the scale of the rigid frame microstructure. In order to produce multiple scattering at ultrasonic frequencies, mesoscale scatterers are introduced in the porous medium host. The effective wave number of such a multiscale medium should take into account the peculiar absorption at the microscale and the multiple scattering at the mesoscale to describe precisely the propagation of a coherent acoustic wave. For this purpose, a simple model is developed. First, an equivalent fluid model, derived from a homogenization method, is used to describe the acoustic propagation in the host porous medium itself. Second, the scattering by the inclusions is described with a multiple scattering approximation (independent scattering approximation). This simple model allows to obtain the total effective wave number of the porous medium with mesoscale scatterers. After some validating results on the multiple scattering by an array of rigid cylinders in air, experiments on the multiple scattering by rigid cylinders embedded in a porous medium are presented and compared to the developed simple model. Incidentally, it appears that for the host medium itself, the equivalent fluid model is not capable to describe the high-frequency behavior whilst a multiple scattering approach with (thin) viscous and thermal boundary layers around the scatterers is accurate in the whole frequency range.

Large-scale velocity structures in turbulent thermal convection.

A systematic study of large-scale velocity structures in turbulent thermal convection is carried out in three different aspect-ratio cells filled with water. Laser Doppler velocimetry is used to measure the velocity profiles and statistics over varying Rayleigh numbers Ra and at various spatial positions across the whole convection cell. Large velocity fluctuations are found both in the central region and near the cell boundary. Despite the large velocity fluctuations, the flow field still maintains a large-scale quasi-two-dimensional structure, which rotates in a coherent manner. This coherent single-roll structure scales with Ra and can be divided into three regions in the rotation plane: (1) a thin viscous boundary layer, (2) a fully mixed central core region with a constant mean velocity gradient, and (3) an intermediate plume-dominated buffer region. The experiment reveals a unique driving mechanism for the large-scale coherent rotation in turbulent convection.

THE INFLUENCE OF THE VISCOUS BOUNDARY LAYER ON THE CRITICAL STOKES NUMBER FOR PARTICLE IMPACTION NEAR A STAGNATION POINT

The existence of a critical value of the Stokes number, St c, below which no particle impaction takes place, has been demonstrated for inviscid stagnation flows in the absence of sedimentation. In the inviscid approximation, St c is inversely proportional to the normal gradient of the inward velocity at the stagnation point. In practice, the effect of viscosity is to increase the value of St c, because the convective flow velocity towards the surface is reduced. Using asymptotic matching techniques, we calculate approximately the increase in St c for large-Reynolds-number stagnation flows, either two-dimensional or axisymmetric, for which there is a thin viscous boundary layer near the surface. Numerical calculations for unseparated flow around spheres and cylinders show that the increase owing to viscosity is likely to be significant under all laminar flow conditions, even though the viscous boundary layer may be very thin. The sensitivity to the shape of the body on which the particles are deposited, and the likely effects of flow separation, are briefly discussed.

Approximate Dynamical Equations for Fronts Modified by the Planetary Boundary Layer

Abstract The planetary boundary layer (PBL) strongly modifies the structure of surface fronts; yet standard theories of frontogenesis ignore the PBL. As a first step toward understanding the effects of the PBL on frontogenesis, dominant terms are estimated through a scale analysis in an attempt to identify approximations to the primitive equations that are useful near fronts and within the planetary boundary layer. The scaling reveals the absence of small parameters in such flows and emphasizes the importance of turning of the wind with height in determining the frontal dynamics. Although without small parameters the scale analysis is at most suggestive, it indicates that the geostrophic momentum and Ekman momentum approximations will not be accurate, and that the cross-front acceleration may in practice be negligible, as proposed by Cullen. Analysis of a thin viscous boundary layer beneath a front supports these conclusions, as do diagnostics applied to numerical simulations. The scale analysis and diagn...

Compound difference schemes for time-dependent equations on non-uniform nets

A new approach to constructing monotonous second-order-accurate schemes for time-dependent Navier-Stokes equations in the compressible case is suggested. Flows with shock waves and thin viscous boundary layers are considered. The approach joins two different schemes while each of them is appropriate for its own type of flow. The resulting scheme is constructed as a splitting one, and the approach is applied at the stage governed by the Euler equations in Lagrangian variables. The approach turns out to be fruitful also for parabolic equations with ‘viscous’ terms. The method is illustrated by the problem on compression of deuterium in a conical solid-body target with viscous heating of deuterium.

Attenuated leaky Rayleigh waves

The propagation of leaky Rayleigh waves under the influence of viscous damping and heat conduction in boundary layers is studied using the matched asymptotic method. Viscosity of the fluid is considered unimportant except in a thin viscous boundary layer at the interface. A new characteristic equation is obtained in which the effect of boundary layer is shown by terms associated with R−1/2=(ων)1/2/ct, where R is the Reynolds number, ω is the frequency, ν is the kinematic viscosity of the fluid, and ct is the shear velocity of the solid substrate. One of the numerically obtained solutions gives the leaky Rayleigh wave speed and the attenuation coefficient. It is shown that, together with radiation, viscosity and heat conduction in the boundary layer also affect the attenuation of the leaky Rayleigh waves. Furthermore, it is shown that, because of the effect of the viscous boundary layer, the attenuated leaky Rayleigh wave speed can be smaller than the Rayleigh wave speed at the interface of a vacuum and a solid substrate. A critical Reynolds number of about 2500 is found beyond which a viscous boundary layer stops influencing leaky Rayleigh wave propagation. Finally, a new wave mode sustained by the viscous boundary layer alone is found in the limit of a small fluid–solid density ratio. This mode exists for appropriate frequency and layer thickness combinations. For air, the corresponding propagation speed is shown to be higher than the sound speed and the corresponding attenuation is significant. These results may be used to improve our interpretation of acoustic signature of materials.