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.     

Stability and nonlinear wavy regimes in downward film flows on a corrugated surface

The linear and nonlinear stability of downward viscous film flows on a corrugated surface to freesurface perturbations is analyzed theoretically. The study is performed with the use of an integral approach in ranges of parameters where the calculated results and the corresponding solutions of Navier-Stokes equations (downward wavy flow on a smooth wall and waveless flow along a corrugated surface) are in good agreement. It is demonstrated that, for moderate Reynolds numbers, there is a range of corrugation parameters (amplitude and period) where all linear perturbations of the free surface decay. For high Reynolds numbers, the waveless downward flow is unstable. Various nonlinear wavy regimes induced by varying the corrugation amplitude are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 110–120, January–February, 2007.     

Numerical investigations on vortical structures of viscous film flows along periodic rectangular corrugations

Two-dimensional gravity-driven film flows along a substrate with rectangular corrugations are studied numerically by using Finite Volume Method. The volume of fluid (VOF) method is utilized to capture the evolution of free surfaces. The film flows down an inclined plate are simulated to validate the numerical implementation of the present study. Results obtained indicate that the phase shift between the surface wave and the wall corrugation increases as the Reynolds number. The parametric studies on the interesting resonant phenomenon indicate that the peak Reynolds numbers increase as the raise of the wall depth or the decline of the inclination angle. The dependence of the flow fields is analyzed on the Reynolds numbers and wall depth in details. It is found that the vortical structures in the steady flows, either produced by the interaction between capillary wrinkling and inertia, or by the rectangular geometry, are closely related to the remarkable deformation of the free surfaces. This conclusion is also confirmed by the transient flow development of two typical simulations, i.e., flows in capillary–inertial regime and in inertial regime.     

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.     

饱和蒸发和热毛细力作用下低雷诺数液膜在波纹竖壁上的流动

考虑表面蒸发压力和热毛细力作用情况下,对饱和蒸发状态下低雷诺数自由降落液膜在小波幅正弦型波纹壁面上的流动进行理论分析。对控制微分方程及边界条件进行量纲一化并引入流函数,对微分方程及边界条件进行摄动展开,得到了这种情况下液膜流动的简化分析模型,求出了近似解析解。讨论了壁面波纹、表面张力、蒸发压力、热毛细力对液膜流动的影响。研究表明:液膜的波动幅度随蒸发强度和热毛细力的增大而增大;液膜波动与壁面波纹的相位差随蒸发强度增大而增大,随热毛细力增大而减小。     

Wave Regimes of Viscous Film Flow down a Vertical Cylinder

The flow of a viscous liquid film down a vertical cylinder in the gravity field is considered. In the case of small Reynolds numbers for long-wave perturbations on a cylinder of radius much greater than the film thickness, the problem can be reduced to a single nonlinear equation for the evolution of the film thickness perturbation. For axially symmetric solutions, this equation coincides with the well-known Sivashinsky-Kuramoto equation. The results of a numerical analysis of this equation for three-dimensional stationary traveling solutions of the problem are reported. The effect of the problem parameters on the solution behavior is demonstrated. Soliton type solutions are presented.     

Experimental study of the stochastic nature of wave phenomena on the surface of a liquid film

An experimental study was made of wave phenomena on the surface of a liquid film freely flowing down the walls of a vertical channel in the range of Reynolds numbers for film flow (Re=Γ/η= 50–2500, where Γ is the mass spray density and η is the dynamic viscosity) at various distances from the entrance. The working fluid (water) was fed into the operating section at a temperature of 15–30 °C. The dependence of typical wave parameters on mode parameters was obtained.     

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.     

Evaporation of Falling and Shear-Driven Thin Films on Smooth and Grooved Surfaces

One of the most important tasks in development of modern gas turbine combustors is the reduction of NO_x emissions. An effective way to reduce the NO_x emission is using the lean premixed prevaporization (LPP) concept. An important phenomenon taking place in LPP chambers is the evaporation of thin fuel films. To increase the fuel evaporation rate, the use of microstructured walls has been suggested. The wall microstructures make use of the capillary forces to evenly distribute the liquid fuel over the wall, so that the appearance of uncontrolled dry patches can be avoided. Moreover, the wall structures promote the thin film evaporation characterized by ultra-high evaporation rates. An experimental setup was built for the investigation of thin liquid films falling down on the outer surface of vertical tubes with either a smooth or structured surface. In the first testing phase water is used, fuel like liquids will be used later on. The thin film can be heated from both sides, by hot oil flowing inside the tube, and by hot compressed air flowing in co-current direction to the thin film. The film is partly evaporated along the flow. Results for the wavy film structure at different Reynolds numbers are reported. For theoretical investigations a model describing the hydrodynamics and heat transfer due to evaporation of the gravity- and shear-driven undisturbed liquid film on structured surfaces was developed. For low Reynolds numbers or low liquid mass fluxes the wall surface is only partly covered with liquid and the heat transfer is shown to be governed by the evaporation of the ultra-thin film in the vicinity of the three-phase contact line. A numerical model for the solution of a two-dimensional free-surface flow of a liquid film over a structured wall was also developed. The Navier–Stokes equations are solved using the Volume of Fluid (VOF) technique. The energy equation is included in the model. The model is verified by comparison with data from the literature showing favorable agreement. In particular, the proposed model predicts the formation of capillary waves observed in the experiments. The model is used to investigate the flow of liquid on a structured wall. This calculation is the first step towards the modeling of a three-dimensional wavy flow of a gravity- and shear-driven film along a wall with longitudinal grooves. It is found that due to the Marangoni effect, a circulating flow arises within the cavity, thereby leading to an enhancement in the evaporation rate.     

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.     

Flow stability of a conducting liquid flowing down an inclined plane in the presence of a magnetic field

The stability of a steady flow of incompressible, conducting liquid down an inclined plane in the presence of longitudinal and transverse magnetic fields is studied. Solutions of the linearized magnetohydrodynamic equations with corresponding boundary conditions are found on the assumption that the Reynolds number R_g and the wave number are small. It is shown that the longitudinal magnetic field plays a stabilizing role. It is known [1] that the flow of a viscous liquid over a vertical wall is always unstable. In this article it is shown that the instability effect at small wave numbers may be eliminated if the longitudinal magnetic field satisfies the conditions found. The case when the Alfvén number and the wave number are small and the Reynolds number is finite is also examined.     

Non-Adiabatic Surface Effects on Step-Induced Boundary-Layer Transition

The effect on step-induced boundary-layer transition of surface temperatures different from the adiabatic-wall temperature was investigated for a (quasi-) two-dimensional flow at large Reynolds numbers and at both low and high subsonic Mach numbers. Sharp forward-facing steps were mounted on a flat plate and transition was studied non-intrusively by means of the temperature-sensitive paint technique. The experiments were conducted in the Cryogenic Ludwieg-Tube Göttingen with various streamwise pressure gradients and temperature differences between flow and model surface. A reduction of the ratio between surface and adiabatic-wall temperatures had a favorable influence on step-induced transition up to moderate values of the step Reynolds number and of the step height relative to the boundary-layer displacement thickness, leading to larger transition Reynolds numbers. However, at larger values of the non-dimensional step parameters, the increase in transition Reynolds number for a given reduction in the wall temperature ratio became smaller. Transition was found to be insensitive to changes in the wall temperature ratio for step Reynolds numbers above a certain value. Up to this limiting value, the relation between the relative change in transition location (with respect to its value for a smooth surface) and the non-dimensional step parameter was essentially unaffected by variations in the wall temperature ratio. The present choice of non-dimensional parameters allows the effect of the steps on transition to be isolated from the influence of variations in the other factors, provided that both transition locations on the step and smooth configurations are measured at the same conditions.     

Stability Analysis of a Falling Film Flow Down a Plane with Sinusoidal Corrugations

The liquid viscous film falling down a vertical wall with sinusoidal relief is considered. The linear stability of steady-state flow with respect to time-periodic disturbances is studied using the Floquet theory. It is shown that in the case of applying corrugations the variation in the disturbance growth rate is proportional to the second power of their undulations. Depending on the relief parameters there exist two possibilities: the instability domain can expand or certain disturbances can be stabilized. The growth rates are obtained numerically and analytically in the approximation of low-amplitude corrugations. The development of waves from small disturbances is simulated within the framework of nonlinear equations and the formation of structures whose wavelength is significantly greater than the space relief period is found out.     

Surface roughness effects on squeeze film behavior in porous circular disks with couple stress fluid

The effect of surface roughness on squeeze film behavior between two circular disks with couple stress lubricant is analyzed when the upper disk has porous facing which approaches the lower disk with uniform velocity. The modified Stochastic Reynolds equation is derived on the basis of Stokes micro-continuum theory for couple stress fluid and Christensen Stochastic theory for the rough surface. Closed form solution of the Stochastic Reynolds equation is obtained in terms of Fourier–Bessel series. The importance of roughness and couple stress on bearing characteristics are presented in terms of load carrying capacity, squeeze time, and relative percentage of the load. It is observed that, effect of couple stress fluid, and surface roughness is more pronounced compared to classical case. These predictions enable design engineers to choose suitable parameters.     

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.     

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.     

Stability of a laminar film flow

The problem of the wave motion of a liquid layer was first investigated by Kapitsa [1, 2], who gave an approximate analysis of the free flow and flow in contact with gas stream, and evaluated the influence of the heat transfer processes on the flow. The problem of the stability of such a flow was studied in detail by Benjamin [3] and Yih [4, 5], These authors proposed seeking the solution of the resulting Orr-Sommerfeld equation in the form of a series in a small parameter and developed a corresponding method of successive approximations. As the small parameter [3–5], they made use of the product of the disturbance wave number and the Reynolds number. In these studies, the tangential stress on the free surface was taken equal to zero, and the fluid film was always considered essentially plane. At the same time, there are certain types of problems of considerable interest in which neither of these assumptions is satisfied. A good example might be the problem on the stability of the annular regime of two-phase flow in pipes and capillaries, when the basic stream of one fluid is separated from the pipe walls by an annular layer of another fluid. In this case, the interface has a finite radius of curvature and the tangential stress on the interface may be significantly different from zero.In the present paper, the problem of the flow stability of a fluid layer with respect to small disturbances of the boundary surface is considered with account for both the finite radius of curvature of the boundary surface and the nonzero hydrodynamic friction at the boundary. The film is assumed to be quite thin. This enables us, firstly, to consider the Reynolds number small, to use the general method of [5], and, second ly, to consider the film thickness sufficiently small in comparison with the radius of curvature of the substrate on which the film lies. Furthermore, for evaluating the stability of the laminar flow of the curved film we can use the results obtained for a plane film with account for the terms which depend on the curvature of the substrate.As a rule, previous studies have considered only one-dimensional disturbances of the boundary surface. In the present paper, in the first approximation, the stability is examined in relation to two-dimensional disturbances of this surface, corresponding to three-dimensional flow disturbances.As an example, the results obtained are applied to the investigation of the stability of the free flow of a layer of fluid over an inclined plane under the sole influence of gravity.     

Experimental Study of the Linear Stability of a Falling Liquid Film in the Presence of a Turbulent Gas Stream

Using the local electrical conductivity method, the parameters of the linear waves generated on the surface of a falling liquid film in the presence of a co- or counter-current gas stream are measured. The Reynolds numbers of the fluid and the gas were varied from 24 to 125 and from 0 to 8000, respectively. The results are presented in the form of dispersion relations. In the case of the absence of a gas stream, the results are compared with calculations based on a linear integral theory. It is shown that a gas stream increases the instability of the film and a counter-current gas flow has a greater effect on the wave phase velocity than a co-current flow.     

Numerical study of the turbulent channel flow under space-dependent electromagnetic force control at different Reynolds numbers

In this paper, the control of turbulent channel flow by space-dependent electromagnetic force and the mechanism of drag reduction are investigated with the direct numerical simulation(DNS) methods for different Reynolds numbers. A formulation is derived to express the relation between the drag and the Reynolds shear stress. With the application of optimal electromagnetic force, the in-depth relations among characteristic structures in the flow field, mean Reynolds shear stress, and the effect of drag reduction for different Reynolds numbers are discussed. The results indicate that the maximum drag reductions can be obtained with an optimal combination of parameters for each case of different Reynolds numbers. The regular quasi-streamwise vortex structures, which appear in the flow field, have the same period with that of the electromagnetic force.These structures suppress the random velocity fluctuations, which leads to the absolute value of mean Reynolds shear stress decreasing and the distribution of that moving away from the wall. Moreover, the wave number of optimal electromagnetic force increases,and the scale of the regular quasi-streamwise vortex structures decreases as the Reynolds number increases. Therefore, the rate of drag reduction decreases with the increase in the Reynolds number since the scale of the regular quasi-streamwise vortex structures decreases.     

Effect of Heat Supply on the Stability of the Supersonic Swept Atiachment-Line Boundary Layer

The stability of a boundary layer with volume heat supply on the attachment line of a swept wing is investigated within the framework of the linear theory at supersonic inviscid-free-stream Mach numbers. The results of numerical calculations of the flow stability and neutral curves are presented for the flow on the leading edge of a swept wing with a swept angle χ=60° at various free-stream Mach numbers. The effect of volume heat supply on the characteristics of boundary layer stability on the attachment line is studied at a surface temperature equal to the temperature of the external inviscid flow. It is shown that in the case of a supersonic external inviscid flow volume heat supply may result in an increase in the critical Reynolds number and stabilization of disturbances corresponding to large wave numbers. For certain energy supply parameters the situation is reversed, the unstable disturbances corresponding to the main flow-instability zone are stabilized but another zone of flow-instability with small wave numbers and a significantly lower critical Reynolds number appears.