Stability of a viscous liquid film flowing down a periodic surface

The paper is devoted to a theoretical analysis of linear stability of the viscous liquid film flowing down a wavy surface. The study is based on the Navier–Stokes equations in their full statement. The developed numerical algorithm allows us to obtain pioneer results in the stability of the film flow down a corrugated surface without asymptotic approximations in a wide range over Reynolds and Kapitsa’s numbers. It is shown that in the case of moderate Reynolds numbers there is a region of the corrugation parameters (amplitude and period) where all disturbances decay in time and the wall corrugation demonstrates a stabilizing effect. At the same time, there exist corrugation parameters at which the steady-state solution is unstable with respect to perturbations of the same period as the period of corrugation. In this case the waveless solution cannot be observed in reality and the wall corrugation demonstrates a destabilizing effect.     

Viscous liquid film flow down an inclined corrugated surface. Calculation of the flow stability to arbitrary perturbations using an integral method

Viscous liquid film flow along an inclined corrugated (sinusoidal) surface has been studied. Calculations were performed using an integral model. The stability of nonlinear steady-state flows to arbitrary perturbations was examined using the Floquet theory. It has been shown that for each type of corrugation there is a critical Reynolds number for which unstable perturbations occur. It has been found that this value greatly depends on the physical properties of the liquid and geometric parameters of the flow. In particular, in the case of film flow down a smooth wall, the critical waveformation parameter depends only on the angle of inclination of the flow surface. The values of the corrugation parameters (amplitude and period) were obtained for which the film flow down a wavy wall is stable to arbitrary perturbations up to moderate Reynolds numbers. Such parameter values exist for all investigated angles of inclination of the flow surface.     

Experiments on the linear instability of flow in a wavy channel

The effects of wall corrugation on the stability of wall-bounded shear flows have been examined experimentally in plane channel flows. One of the channel walls has been modified by introduction of the wavy wall model with the amplitude of 4% of the channel half height and the wave number of 1.02. The experiment is focused on the two-dimensional travelling wave instability and the results are compared with the theory [J.M. Floryan, Two-dimensional instability of flow in a rough channel, Phys. Fluids 17 (2005) 044101 (also: Rept. ESFD-1/2003, Dept. of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, Canada, 2003)]. It is shown that the flow is destabilized by the wall corrugation at subcritical Reynolds numbers below 5772, as predicted by the theory. For the present corrugation geometry, the critical Reynolds number is decreased down to about 4000. The spatial growth rates, the disturbance wave numbers and the distribution of disturbance amplitude measured over such wavy wall also agree well with the theoretical results.     

Effect of Surface Corrugation on Convection in a Three-Dimensional Finite Box of Fluid Saturated Porous Material

The problem of finite amplitude thermal convection in a three-dimensional finite box of fluid saturated porous material is investigated, when the lower boundary of the fluid is corrugated. The nonlinear problem of three-dimensional convection in the box for the values of the Rayleigh number close to the classical critical value and for small values of the amplitude of the corrugations is solved by a perturbation technique. The preferred mode of convection is determined by stability analysis. In the absence of corrugation three-dimensional modes of convection can be either stable or unstable depending on the values of the aspect ratios of the box, while two-dimensional rolls are always stable, provided that the box aspect ratios allow the existence of such modes of convection. In the presence of boundary corrugation with the appropriate form, different three-dimensional or two-dimensional modes of corrugation can be stable or unstable. For a rough boundary with local roughness sites, the location, size, and number of the roughness elements plus the wave numbers of the convection modes and the box aspect ratios can all play a role leading to either stable or unstable particular three- or two-dimensional flow patterns. For a wavy boundary, resonant wave-vector excitation can lead to the preference of stable two- or three-dimensional flow patterns whose wave vectors are in a subset of those due to the wavy boundary, while nonresonant wave-vector excitation can lead to the preference of stable flow patterns whose wave vectors are not generally in a subset of those due to the wavy boundary. Heat transported by convection can either be enhanced or be reduced by certain proper forms of the corrugations and by appropriate values of the box aspect ratios. Due to the surface corrugation highly subcritical modes of convection are stable, while highly supercritical modes of convection are unstable. Received 24 July 1998 and accepted 11 April 1999     

An experimental study of a bio-inspired corrugated airfoil for micro air vehicle applications

An experimental study was conducted to investigate the aerodynamic characteristics of a bio-inspired corrugated airfoil compared with a smooth-surfaced airfoil and a flat plate at the chord Reynolds number of Re _C  = 58,000–125,000 to explore the potential applications of such bio-inspired corrugated airfoils for micro air vehicle designs. In addition to measuring the aerodynamic lift and drag forces acting on the tested airfoils, a digital particle image velocimetry system was used to conduct detailed flowfield measurements to quantify the transient behavior of vortex and turbulent flow structures around the airfoils. The measurement result revealed clearly that the corrugated airfoil has better performance over the smooth-surfaced airfoil and the flat plate in providing higher lift and preventing large-scale flow separation and airfoil stall at low Reynolds numbers (Re _C  < 100,000). While aerodynamic performance of the smooth-surfaced airfoil and the flat plate would vary considerably with the changing of the chord Reynolds numbers, the aerodynamic performance of the corrugated airfoil was found to be almost insensitive to the Reynolds numbers. The detailed flow field measurements were correlated with the aerodynamic force measurement data to elucidate underlying physics to improve our understanding about how and why the corrugation feature found in dragonfly wings holds aerodynamic advantages for low Reynolds number flight applications.     

Viscous Film Flow down Corrugated Surfaces

A theoretical analysis of a downward viscous film flow on corrugated surfaces is reported. The study is based on Navier–Stokes equations (for one and twodimensional surfaces) and on an integral model (for a threedimensional surface with double corrugation). The calculations were carried out in a wide range of Reynolds numbers and geometric characteristics of the surface with due allowance for the surfacetension force. The shape of the free surface of the liquid film and other characteristics of the flow are calculated. It is shown that, in the case of a onedimensional surface, there exists a range of parameters where the flow is predominantly governed by surfacetension forces; this flow can be adequately treated with the integral approach. In this range of parameters, on the surface with double corrugation, the average quantities of the downward flow in wide corrugation valleys are determined by the finetexture geometry.     

Dynamics of co-rotating vortices in a flow around a bio-inspired corrugated airfoil

Bio-inspired corrugated airfoils show favourable aerodynamic characteristics such as high coefficient of lift and delayed stall at low Reynolds numbers. Two-dimensional (2D) direct numerical simulation has been performed here on a corrugated airfoil at various angles of attack (0°, +5°, -5°) and Reynolds number of 280 to 6700. The objective is to analyse the pressure variation inside the corrugations and correlate it to the vortex movement across the corrugations and the overall aerodynamic characteristics of the corrugated airfoil. The flow characteristics have been examined based on the local Strouhal numbers in the corrugations of the airfoil. It is observed that the pressure variation in each corrugation is the result of vortex merging and separation in the corrugation which plays a major role in changing the flow characteristics. The Strouhal number of the flow is dictated by the most dominant local Strouhal number. The numerical results are further compared with experimental results obtained using particle image velocimetry, and the two set of results are found to match well. These results are significant because they elucidate the effect of corrugation, angle of attack, and Reynolds number on flow over a corrugated airfoil.     

Finite-Amplitude Equilibrium Solutions for Plane Poiseuille–Couette Flow

Two-dimensional nonlinear equilibrium solutions for the plane Poiseuille–Couette flow are computed by directly solving the full Navier–Stokes equations as a nonlinear eigenvalue problem. The equations are solved using the two-point fourth-order compact scheme and the Newton–Raphson iteration technique. The linear eigenvalue computations show that the combined Poiseuille–Couette flow is stable at all Reynolds numbers when the Couette velocity component σ₂ exceeds 0.34552. Starting with the neutral solution for the plane Poiseuille flow, the nonlinear neutral surfaces for the combined Poiseuille–Couette flow were mapped out by gradually increasing the velocity component σ₂. It is found that, for small σ₂, the neutral surfaces stay in the same family as that for the plane Poiseuille flow, and the nonlinear critical Reynolds number gradually increases with increasing σ₂. When the Couette velocity component is increased further, the neutral curve deviates from that for the Poiseuille flow with an appearance of a new loop at low wave numbers and at very low energy. By gradually increasing the σ₂ values at a constant Reynolds number, the nonlinear critical Reynolds numbers were determined as a function of σ₂. The results show that the nonlinear neutral curve is similar in shape to a linear case. The critical Reynolds number increases slowly up to σ₂∼ 0.2 and remains constant until σ₂∼ 0.58. Beyond σ₂ > 0.59, the critical Reynolds number increases sharply. From the computed results it is concluded that two-dimensional nonlinear equilibrium solutions do not exist beyond a critical σ₂ value of about 0.59. Received: 26 November 1996 and accepted 12 May 1997     

Flow and heat/mass transfer in a wavy duct with various corrugation angles in two dimensional flow regimes

In this study, two dimensional heat/mass transfer characteristics and flow features were investigated in a rectangular wavy duct with various corrugation angles. The test duct had a width of 7.3 mm and a large aspect ratio of 7.3 to simulate two dimensional characteristics. The corrugation angles used were 100°, 115°, 130°, and 145°. Numerical analysis using the commercial code FLUENT, was used to analyze the flow features. In addition, the oil-lamp black method was used for flow visualization. Local heat/mass transfer coefficients on the corrugated walls were measured using a naphthalene sublimation technique. The Reynolds number, based on the duct hydraulic diameter, was varied from 700 to 5,000. The experimental results and numerical analysis showed interesting and detailed features in the wavy duct. Main flow impinged on upstream of a pressure wall, and the flow greatly enhanced heat/mass transfer. On a suction wall, however, flow separation and reattachment dominantly affected the heat/mass transfer characteristics on the wall. As the corrugation angle decreased (it means the duct has more sharp turn), the region of flow stagnation at the front part of the pressure wall became wider. Also, the position of flow reattachment on the suction wall moved upstream as the corrugation angle decreased. A high heat transfer rate appeared at the front part of the pressure wall due to main-flow impingement, and at the front part of the suction wall due to flow reattachment. The high heat/mass transfer region by the main-flow impingement and the circulation flow induced at a valley between the pressure and suction walls changed with the corrugation angle and the Reynolds number. As the corrugation angle decreased, the flow in the wavy duct changed to transition to turbulent flow earlier.     

Linear stability of viscous incompressible flow in a circular viscoelastic tube

Flow stability in rigid tubes has been the subject of much research [1]. The overwhelming majority of authors of both theoretical and experimental studies now conclude that Poiseuille flow in a circular rigid tube is linearly stable. However, real tubes all possess elastic properties, the influence of which has not been investigated in such detail. For certain selected values of the parameters characterizing an elastic tube it has been shown that with respect to infinitesimal axisymmetric perturbations Poiseuille flow in the tube can be unstable [2]. In this case boundary conditions that did not take into account the fairly large velocity gradient of the undisturbed flow near the tube wall were used. The present paper reports the results of a numerical investigation of the linear stability of Poiseuille flow in a circular elastic tube with respect to three-dimensional perturbations in the form of traveling waves propagated along the system (azimuthal perturbation modes with numbers 0, 1, 2, 3, 4, and 5 are considered). It is shown that the elastic properties of the tube can have an important influence on the linear stability spectrum. In the case of axisymmetric perturbations it is possible to detect an instability which, at Reynolds numbers of more than 200, exists only for tubes whose modulus of elasticity is substantially less than that of materials in common use. The instability to perturbations of the second azimuthal mode is different in character, inasmuch as at Reynolds numbers greater than unity it occurs in stiffer tubes. Moreover, as the Reynolds number increases it can also occur in tubes of greater stiffness. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 126–134, November–December, 1986.     

Numerical simulation of fluid flow and heat transfer characteristics in channel with V corrugated plates

A detailed numerical study is carried out to investigate fluid flow and heat transfer characteristics in a channel with heated V corrugated upper and lower plates. The parameters studied include the Reynolds number (Re = 2,000–5,500), angles of V corrugated plates (θ = 20°, 40°, 60°), and constant heat fluxs (q″ = 580, 830, 1,090 W/m²). Numerical results have been validated using the experimented data reported by Naphon, and a good agreement has been found. The angles of V corrugated plates (θ) and the Reynolds number are demonstrated to significantly affect the fluid flow and the heat transfer rate. Increasing the angles of V corrugated plates can make the heat transfer performance become better. The increasing Reynolds number leads to a more complex fluid flow and heat transfer rate. The numerical calculations with a non-equilibrium wall function have a better accuracy than with a standard wall function for solving high Reynolds numbers or complex flow problems.     

Stability of Poiseuille flow in the presence of a longitudinal magnetic field

The stability of the plane flow of an electrically conducting fluid with respect to small perturbations was studied at large Reynolds numbers in the presence of a longitudinal magnetic field. The dependence of the critical Reynolds number on the electrical conductivity is investigated. At large Reynolds numbers, a new branch of instability and a sudden change in the critical Reynolds numbers is found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 45–53, May–June, 2008.     

Laminar heat transfer characteristics of internally finned tube with sinusoidal wavy fin

Comparative numerical study of laminar heat transfer characteristics of annular tubes with sinusoidal wavy fins has been conducted both experimentally and numerically with Re = 299–1,475. The uniform heat flux is imposed on the tube outside wall surface. Two tube materials (copper and stainless steel) are considered. It is found that the fluid temperature profile is not linear but convex along the flow direction due to the axial heat conduction in tube wall, and the effects of axial heat conduction on the heat transfer decreases with an increase in Reynolds number or decrease in tube wall thermal conductivity. The axial distributions of local Nusselt number could reach periodically fully developed after 3–5 cycles. The convectional data reduction method based on the traditional method should be improved for tube with high thermal conductivity or low Reynolds numbers, Otherwise, the heat transfer performance of internally finned tube may be underestimated.     

Turbulent forced convection of nanofluid in a wavy channel using two phase model

Two phase mixture model is used to numerically simulate the turbulent forced convection of Al₂O₃-Water nanofluid in a channel with corrugated wall under constant heat flux. Both mixture and single phase models are implemented to study the nanofluid flow in such a geometry and the results have been compared. The effects of the volume fraction of nanoparticles, Reynolds number and amplitude of the wavy wall on the rate of heat transfer are investigated. The results showed that with increasing the volume fraction of nanoparticles, Reynolds number and amplitude of wall waves, the rate of heat transfer increases. Also the results showed that the mixture model yields to higher Nusselt numbers than the single phase model in a similar case.     

Ligament formation in sheared liquid–gas layers

We perform numerical simulations of two-phase liquid–gas sheared layers, with the objective of studying atomization. The Navier–Stokes equations for two-dimensional incompressible flow are solved in a periodic domain. A volume-of-fluid method is used to track the interface. The density ratio is kept around 10. The calculations show good agreement with a fully viscous Orr–Sommerfeld linear theory over several orders of magnitude of interface growth. The nonlinear development shows the growth of finger-like structures, or ligaments, and the detachment of droplets. The effect of the Weber and Reynolds numbers, the boundary layer width and the initial perturbation amplitude are discussed through a number of typical cases. Inversion of the liquid boundary layer is shown to yield more readily ligaments bending upwards and is thus more likely to produce droplets.     

An experimental investigation of the flow structure over a corrugated waveform in a transpired air collector

An experimental investigation of the flow dynamics in a channel with a corrugated surface is presented. Particle image velocimetry was used to obtain two-dimensional velocity fields at three different locations along the channel length, over a range of Reynolds numbers. The results show a significant impact of the corrugation waveform on the mean and turbulent flow structure inside the channel. Strong bursting flow originating from the trough, sweeping flow from the bulk region and the vortex shedding off the crest were observed. Their interactions created a complex three-dimensional flow structure extended over almost the entire channel. The mean velocity profiles indicate a strong diffusion of shear. The profiles of various turbulent properties show the enhancement of turbulence in the vicinity of the waveform. It was found that the turbulence in the channel was almost entirely produced in this region above the corrugation trough. Significant momentum transfer from the corrugation wall by the turbulent velocity field was also observed. The mean and turbulent flow behaviour was found to be periodic with respect to the waveform over most of the channel length. The results show the presence of strong turbulence even at the Reynolds number that falls within the conventional laminar range.     

Nonlinear evolution of Görtler vortices in the near-wall part of a boundary layer

The evolution of Görtler vortices with wavelength smaller than the thickness of the boundary layer on a concave surface is modelled asymptotically at high Reynolds and Görtler numbers. It is known that in the initial linear stage of their evolution such vortices have the largest increment of amplitude growth. Numerical results demonstrate that taking the nonlinear interaction of the flow parameters into account considerably reduces the growth rate and leads to the forming of a perturbed vortex region core; profiles of the flow characteristics in the different stages of vortex evolution are presented.     

Vibration analysis of corrugated beams: The effects of temperature and corrugation shape

This paper deals with the vibration analysis of corrugated beams under large amplitude displacements and temperature variations. The dynamic response depends on the shape, the period and the amplitude of the corrugations as well as on the temperature variations. These different parameters were taken into account using a homogenization process. We coupled the harmonic balance method and the Galerkin technique to derive a frequency amplitude equation, which includes the corrugation shape and temperature. Finally, the influence of different shapes of corrugation (sinusoidal, triangular and square) on the nonlinear vibration response of a corrugated beam was compared to results obtained for a planar beam.     

Active Control of Wave Instabilities in Three-Dimensional Compressible Flows

In many fluid flows of practical importance transition is caused by the linear growth of wave instabilities, such as Tollmien–Schlichting waves, which eventually grow to a finite size at which stage secondary instabilities come into play. If transition is to be delayed or even avoided in such flows, then the linear growth of the disturbances must be prevented since control in the nonlinear regime would be a considerably more difficult task. Here a strategy for active control of two-dimensional incompressible and compressible Tollmien–Schlichting waves and its use in controlling the more practically relevant problem of crossflow instability which arises in swept-wing flows is discussed. The control is through an active suction/blowing distribution at the wall though the same result could be achieved by variable wall heating. In order to control the instability it is assumed that the wall shear stress and pressure are known from measurements. It is shown that, certainly at finite Reynolds numbers, it is sufficient to know the flow properties at a finite number of points along the wall. The cases of high and finite Reynolds numbers are discussed using asymptotic and numerical methods respectively. It is shown that a control strategy can be developed to stop the growth of all two-dimensional Tollmien–Schlichting waves at finite and large Reynolds numbers. Some discussion of nonlinear effects in the presence of active control is given and the possible control of other instability mechanisms investigated. Received 1 May 1998 and accepted 24 September 1998