Wavy Bottom Research Articles

Mixed Convection Heat Transfer in a Lid Driven Cavity with Wavy Bottom Surface

The present numerical study is devoted to investigate the mixed convection flow and heat transfer in a lid-driven cavity with wavy bottom surface. The cavity upper wall is moving with a uniform velocity by unity and the other walls are no slip. The cavity vertical walls are insulated while the upper surface is maintained at a uniform temperature higher than the wavy bottom surface. The physical problem is represented mathematically by a set of governing equations and the developed mathematical model is solved by employing Galerkin weighted residual method of finite element formulation. The wide ranges of governing parameters, i. e., the Reynolds number (Re), the Grshof number (Gr) and the number of undulations (λ) on the flow structure and heat transfer characteristics are investigated in detail. It is found that these parameters have significant effect on the flow fields; temperature distributions and heat transfer in the cavity. Furthermore, the trend of skin friction and Nusselt number for different values of the aforementioned parameters are presented in this investigation.

Experimental Investigations on the Influence Factors of Paperboard Deep Drawing

Experiments were conducted to study which type of paperboard and process parameters are suitable for deep drawing of paperboard into trays. The tray is a shallow rectangle box with round corners, wavy bottom and indented walls. Paperboard used in this paper is anisotropic, and its mechanical properties include tensile strength and elongation. Results show that paperboard with the basis weight of 260g/m2 is the suitable type. Proper moisture can improve the tensile strength and ensure the surface quality. Improper blank shape and size can result in wrinkling and cracking.

Structural dissimilarity of large-scale structures in turbulent flows over wavy walls

Turbulent flows over a wavy wall are investigated in channels with a wavy bottom and a flat top with different channel heights. Flow structures are determined from proper orthogonal decompositions of velocity fields measured with particle image velocimetry. Three different channel heights are considered, which are characterized by blockage ratios β (half channel height to wave amplitude ratio) 3.3, 6.7, and 10. Measurements are evaluated at comparable Reynolds numbers (Re) around 10 000. Structural similarity of large-scale structures, which is valid at β = 6.7 and 10, no longer holds at β = 3.3. Furthermore, characteristic regions of flows over wavy walls exhibit different locations in the case of the smallest channel height.

Scaling weld or melt pool shape induced by thermocapillary convection

The molten pool shape and transport variables induced by thermocapillary force during welding (or melting) of workpieces, such as pure iron, titanium, high speed steel and stainless steel alloys, can be self-consistently predicted from scale analysis. Determination of the molten pool shape and transport variables is crucial due to their close relationship with the microstructures, strength and properties of the fusion zone. In this study, the pool excludes a strongly wavy bottom and the surface velocity profile has two peaks valid for Prandtl numbers lying between 0.3 and unity. The surface tension coefficient is negative and suitable for all pure liquid metals and alloys containing minor surface active solutes, giving rise to an outward surface flow. In view of high Marangoni number, the domain of scaling is divided into the hot and cold corner regions, boundary layers on the solid–liquid interface and ahead of the melting front. The results find that the width and depth of the pool, peak and secondary peak surface velocities, and maximum temperatures in the hot and cold corner regions can be explicitly and separately determined as functions of working variables, or Marangoni, Prandtl, Peclet, Stefan, and beam power numbers and solid-to-liquid thermal conductivity ratio. The scaled results agree with numerical data and available experimental data. This work has academic and practical importance. Successful scaling not only reveals physical mechanisms, but also provides quantitative predictions of the fusion zone shapes and transport variables prior to melting or welding.

Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline

We consider a thin film of a power-law liquid flowing down an inclined wall with sinusoidal topography. Based on the von Kármán–Pohlhausen method an integral boundary-layer model for the film thickness and the flow rate is derived. This allows us to study the influence of the non-Newtonian properties on the steady free surface deformation. For weakly undulated walls we solve the governing equation analytically by a perturbation approach and find a resonant interaction of the free surface with the wavy bottom. Furthermore, the analytical approximation is validated by numerical simulations. Increasing the steepness of the wall reveals that nonlinear effects like the resonance of higher harmonics grow in importance. We find that shear-thickening flows lead to a decrease while shear thinning flows lead to an amplification of the steady free surface. A linear stability analysis of the steady state shows that the bottom undulation has in most cases a stabilizing influence on the free surface. Shear thickening fluids enhance this effect. The open questions which occurred in the linear analysis are then clarified by a nonlinear stability analysis. Finally, we show the important role of capillarity and discuss its influence on the steady solution and on the stability.

An integral boundary layer equation for film flow over inclined wavy bottoms

We study the flow of an incompressible liquid film down a wavy incline. Applying a Galerkin method with only one ansatz function to the Navier–Stokes equations, we derive a second-order weighted residual integral boundary layer equation, which, in particular, may be used to describe eddies in the troughs of the wavy bottom. We present numerical results which show that our model is qualitatively and quantitatively accurate in wide ranges of parameters, and we use the model to study some new phenomena, for instance, the occurrence of a short wave instability (at least in a phenomenological sense) for laminar flows which does not exist over a flat bottom.

Effect of wall slip on the shear flow of a polymer in a channel with a wavy bottom

The effect of stick and wall slip boundary conditions on the specific features of the shear flow of viscous polymers in a confined two-dimensional channel with a wavy bottom is studied. The distribution of flow-rate disturbances across the transverse cross section of the channel is calculated by the numerical simulation of the Navier-Stokes equation for an incompressible fluid at arbitrary amplitudes and an arbitrary wave number of the wall. The wall slip is modeled by the introduction of a thin layer of a low-viscosity fluid at the bottom face. Slippage leads to a marked enhancement of flow rate disturbances including inertial advection. The results agree with the known analytical solutions for the low-amplitude wall wave.

Separation vortices and pattern formation

In this paper examples are given of the importance of flow separation for fluid patterns at moderate Reynolds numbers—both in the stationary and in the time-dependent domain. In the case of circular hydraulic jumps, it has been shown recently that it is possible to generalise the Prandtl–Karman–Pohlhausen approach to stationary boundary layers with free surfaces going through separation, and thus obtain a quantitative theory of the simplest type of hydraulic jump, where a single separation vortex is present outside the jump. A second type of jump, where an additional roller appears at the surface, cannot be captured by this approach and has not been given an adequate theoretical description. Such a model is needed to describe “polygonal” hydraulic jumps, which occur by spontaneous symmetry breaking of the latter state. Time-dependent separation is of importance in the formation of sand ripples under oscillatory flow, where the separation vortices become very strong. In this case no simple theory exists for the determination of the location and strengths of separation vortices over a wavy bottom of arbitrary profile. We have, however, recently suggested an amplitude equation describing the long-time evolution of the sand ripple pattern, which has the surprising features that it breaks the local sand conservation and has long-range interaction, features that can be underpinned by experiments. Very similar vortex dynamics takes place around oscillating structures such as wings and fins. Here, we present results for the vortex patterns behind a flapping foil in a flowing soap film, which shows the interaction and competition between the vortices shed from the round leading edge (like the von Karman vortex street) and those created at the sharp trailing edge.

Study of the sediment transport over flat and wavy bottom using large-eddy simulation

In the present paper the sensitivity of the flow and of sediment transport to bottom roughness is studied. First, a thorough numerical investigation of smooth-bottom channel flow at Re τ = 395 is performed using large-eddy simulations (LESs). A dynamic version of the wall-adapted local eddy-viscosity (WALE) model is used for this study, whereas the subgrid-scale (sgs) diffusion stress is based on a gradient hypothesis. The computed results compare well with direct numerical simulation (DNS) data and experiments. Next, rough-bottom cases, with the bottom having a sinusoidal, wavy shape are considered. It was found that the wavy bottom has a strong influence on the flow field and that the sediment transport is highly sensitive to the bottom waviness, in particular for larger wave heights. It is shown that for light and low-concentrated sediment the Rouse theory is also valid in the case of a wavy bottom, mainly in the outer zone. Finally, it is found that the turbulent-Schmidt-number profiles are not very sensitive to the wave height.

Modeling of flows over inclined wavy bottoms

AbstractWe study the flow of an incompressible liquid film down a wavy incline. Starting from the Navier–Stokes equations we derive an integral boundary layer equation (IBLe) by applying a Galerkin method with a single test and ansatz function. In comparison with older models this approach has several advantages. First, a linear stability analysis for stationary solutions yields a critical Reynolds number which corresponds in the limit of a flat incline with results directly obtained by the Navier–Stokes equations. Second, the IBLe is consistent with the pertinent Benney equation. Finally, the velocity profile is not assumed to be parabolic, so that also parameter regimes with recirculations can be modelled. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Granular flow transitions on sinusoidal surfaces

We report on a newly discovered bifurcation that occurs in dry grains flowing down a chute with a wavy bottom. We find that the bifurcation outwardly resembles the long-known fluid analogue of inviscid channel flow over a wavy bottom reported in 1886 by Lord Kelvin; however, in detail, the two situations differ significantly. We compare three distinct states seen in the granular system: a 'regular' flow in phase with the bottom wave; an 'antiregular' flow that is out of phase; and a 'flat' flow in which the surface slides nearly uniformly downhill. Additionally, we discuss evidence that sustained subsurface circulation in the granular bed, accompanied by strong fluctuations in flow velocities, can appear in granular flows over wavy surfaces.

A three-dimensional wave field over a bidirectionally periodic ripple bottom

For gravity wave trains propagating over an arbitrary wavy bottom, a perturbation expansion is developed to the second order so that the Bragg resonance effect of the ripple bottom on the free-surface wave can be analyzed. Both the resonant and non-resonant cases are treated and the singular behavior at resonance is avoided. This theory is successfully verified by reducing to simpler situations. Then, the analytical results for the special case of a unidirectional sinusoidal bottom are compared with experimental data for validation.

Creeping Newtonian Film Flow Down an Inclined Wavy Plane. Part II

AbstractWe consider the creeping film flow down an inclined wavy plane. For large wavelengths and thin films we consider two different approaches depending on the order of the amplitude of the wavy bottom profile and compare them to experimental results.

‘Phase diagram’ of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability

A unified view is given of the instabilities that may develop in two-layer Couette flows, as a ‘phase diagram’ in the parameter space. This view is obtained from a preliminary study of the single-fluid Couette flow over a wavy bottom, which reveals three flow regimes for the disturbances created at the bottom, each regime being characterized by a typical penetration depth of the vorticity disturbances and an effective Reynolds number. It appears that the two-layer flow exhibits the same flow regimes for the disturbances induced by the perturbed interface, and that each type of instability can be associated with a flow regime. Typical curves giving the growth rate versus wavenumber are deduced from this analysis, and favourably compared with the existing literature. In the second part of this study, we propose a mechanism for the long wavelength instability, and provide simple estimates of the wave velocity and growth rate, for channel flows and for semi-bounded flows. In particular, an explanation is given for the ‘thin-layer effect’, which is typical of multi-layer flows such as pressure driven flows or gravity driven flows, and according to which the flow is stable if the thinner layer is the less viscous, and unstable otherwise.

Currents induced by long waves propagating towards a beach over a wavy bed

For the understanding of longshore currents along a natural beach, the effects of bottom unevenness are considered to be important, especially for the flow in the swash zone. Currents in the swash zone are strongly influenced by the bed slope because the effect of gravity overwhelms the effect of the depth change. In the present paper, we investigate these effects and focus on waves propagating from offshore over a flat ocean basin of constant depth to a beach with a sloping wavy bottom. The waves are incident at a small angle to the beach normal, and the bed slope in the alongshore direction is varied slowly. To simplify the problem, only cnoidal waves and solitary waves are considered and the bed level is varied sinusoidally in the longshore direction.A perturbation method is applied to the two-dimensional nonlinear shallow water equation (two-dimensional NLSWE) for the wave motion in order to generate a more simplified model of wave dynamics consisting of a one-dimensional NLSWE for the direction normal to the beach and an equation for the alongshore direction. The first equation, the one-dimensional NLSWE, is solved by Carrier & Greenspan's transformation. The solution of the second one is found by extending Brocchini & Peregrine's solution for a flat beach. Two methods for the solution of the one- dimensional NLSWE are introduced in order to get a solution applicable to large-amplitude swash motions, where the amplitude is comparable to the beach length. One is the Maclaurin expansion of the solution around the moving shoreline, and the other is Riemann's representation of the solution, which exactly satisfies the one-dimensional NLSWE and the boundary conditions. After doing a consistency check by confirming that Riemann's method, a numerical solution, agrees with the exact solution for an infinitely long, sloping beach, we assumed that the Maclaurin series solution can also describe wave motion in the swash zone properly not only for this model but also for our ‘wavy’, finite beach model.The solution obtained from the Maclaurin series is then plugged into the equation for the alongshore direction to calculate the shore currents induced by wave run-up and back-wash motions, where a ‘weakly two-dimensional solution’ is derived from geometrical considerations. The results show that since the water depth near the shoreline is comparable to the bed level fluctuations, the flow is strongly affected by the bed unevenness, leading to recognizable changes in shoreline movement and the time-averaged velocity and the mass flux of the flow in the swash zone. More specifically, the inhomogeneity of the alongshore mass flux generates offshore currents because of the continuity condition for the fluid mass.

Non-temperate glacier flow over wavy sloping ground

AbstractThe mean, steady-state particle velocity in gravity-driven glacial flow over sinusoidal, sloping ground is computed using a Lagrangian description of motion. A Newtonian viscous fluid approximation is used for the ice. The glacier surface is free to move and is not subject to any stresses. At the bottom, the ice is frozen to the ground. The non-linear interaction between the basic downslope Poiseuille flow and the bottom corrugations yields a mean Lagrangian perturbation velocity that is always directed in the upslope direction near the ground. The requirement of mass balance imposes a mean negative surface slope in the corrugated region and an associated downslope perturbation flow in the upper part of the glacier. The no-slip condition at the wavy bottom induces a strong velocity shear in the ice, and particularly at the base. Analysis shows that the shear heating associated with shortwave perturbations could, in the case of a marginally frozen ground, lead to melting and subsequent sliding at wave crests along the bottom, while the ice stays frozen at the troughs. It is suggested that for glaciers the resulting high strain rates could lead to crevassing.

On generalized Bragg scattering of surface waves by bottom ripples

We study the generalized Bragg scattering of surface waves over a wavy bottom. We consider the problem in the general context of nonlinear wave–wave interactions, and write down and provide geometric constructions for the Bragg resonance conditions for second-order triad (class I) and third-order quartet (class II and class III) wave– bottom interactions. Class I resonance involving one bottom and two surface wave components is classical. Class II resonance manifests bottom nonlinearity (it involves two bottom and two surface wave components), and has been studied in the laboratory. Class III Bragg resonance is new and is a result of free-surface nonlinearity involving resonant interaction among one bottom and three surface wave components. The amplitude of the resonant wave is quadratic in the surface wave slope and linear in the bottom steepness, and, unlike the former two cases, the resonant wave may be either reflected or transmitted (relative to the incident waves) depending on the wave–bottom geometry. To predict the initial spatial/temporal growth of the Bragg resonant wave for these resonances, we also provide the regular perturbation solution up to third order. To confirm these predictions and to obtain an efficient computational tool for general wave–bottom problems with resonant interactions, we extend and develop a powerful high-order spectral method originally developed for nonlinear wave–wave and wave–body interactions. The efficacy of the method is illustrated in high-order Bragg resonance computations in two and three dimensions. These results compare well with existing experiments and perturbation theory for the known class I and class II Bragg resonance cases, and obtain and elucidate the new class III resonance. It is shown that under realistic conditions with moderate to small surface and bottom steepnesses, the amplitudes of third-order class II and class III Bragg resonant waves can be comparable in magnitude to those resulting from class I interactions and appreciable relative to the incident wave.

The stability of capillary-gravity waves on flow over a wavy bottom

The stability of free surface waves on a fluid flowing over a wavy bottom is considered. The mechanism for instability is the three-wave resonance conditions of Phillips. Previous studies not including surface tension have shown that there are two sets of disturbance waves which exist. It has now been found that with the inclusion of surface tension, two types of disturbances still exist, but may be multiply-valued, resulting in as many as four disturbance wave sets. The regions in parameter space with different numbers of solutions are identified. The zero surface tension case can be proven to be always unstable, but not the nonzero case. Numerical evaluation of the growth rates shows that the nonzero case is always unstable, and the instability may be a result of either type of disturbance wave. Increasing the value of surface tension from a non-zero value has been found to lead to a stronger instability. Increasing the surface tension from zero, however, may weaken the instability.

Sand ripples under sea waves Part 1. Ripple formation

In the present paper we formulate a predictive theory of the formation of sand ripples under sea waves. The theory is based on a linear stability analysis of a flat sandy bottom subject to a viscous oscillatory flow. The conditions for decay or amplification of a bottom perturbation are determined along with the wavelength of the most unstable component as a function of the Reynolds number of the flow and of the Froude and Reynolds numbers of the sediments. A comparison between theoretical findings and experimental data supports the validity of the present theory. An analytical solution for viscous oscillatory flow over a small-amplitude wavy bottom is determined for arbitrary values of the ratio r between the amplitude of fluid displacement and the wavelength of bottom waviness. Previous works by Lyne (1971) and Sleath (1976), who considered small or large values of r, are thus extended.

Baroclinic instability over wavy topography

The occurrence of new unstable modes of quasigeostrophic baroclinic oscillation of rotating stratified shear flow over a wavy bottom is examined. To obtain a tractable mathematical problem, the bottom topography is considered as a perturbation modifying the oscillations. It is found that combinations of a top-intensified and a bottom-intensified Eady mode, each stable without topography, can be destabilized by topography if certain resonant conditions are met. These are that (i) the two modes possess the same wavefrequency, and (ii) topography possesses a wavenumber c bridging the gap between the wavenumbers of the modes a, b, i.e. c = a – b. Growth rate of this instability (called type A) is proportional to the amplitude of the topographic component. There are two special cases: (i) when one of the basic modes is a marginally neutral mode – according to the classical analysis without topography – the instability is stronger (type M) with growth rate proportional to the 2/3-power of topographic amplitude; (ii) when both modes are marginally neutral the instability is even stronger (type M2) with growth rate proportional to the square root of topographic amplitude.These topographic instabilities, like classical baroclinic instability, draw their energy from the available potential by transporting buoyancy down the mean gradient associated with the geostrophic shear flow.