Calculation of the velocity profile in a nonlinear wave in a fluid layer flowing down a vertical wall

Abstract

CALCULATION OF THE VELOCITY PROFILE IN A NONLINEAR WAVE IN A FLUID LAYER FLOWING DOWN A VERTICAL WALL P. I. Geshev and Kh. Kh. Murtazaev UDC 532.62+532.594 A fluid flowing down a vertical surface due to gravity is an example of an active dissipative medium. The energy supply comes from the gravitational force while the dissipation comes from the viscous friction force. Studies of the linear stability of failing films of fluid [1, 2] have shown that smooth plane-parallel flow is unstable, no matter how small the Reynolds number. Usually motion in a film is considered laminar-wave flow up to Re - 300-400 [3]. Almost all theoretical research has been done in the long-wavelength approximation, which corresponds to experimental data. It has been shown [4] that the long-wavelength approximation can be used up to Re ~ 1000 for ordinary fluids. This approximation allows the complete system of the Navier-Stokes equations to be simplified to a boundary-layer system. An integral method [5] has been proposed which gives a semi-parabolic velocity profile. The assumption of self-similarity has been verified [6]. A system of two equations has been derived within the framework of the integral approach [7, 8]: one for the instantaneous thickness and one for the flow rate of the fluid for moderate Reynolds numbers. Stationary nonlinear running waves of the first kind, which are similar in form to sinusoidal waves, have been found from this approach [7, 8]. Highly nonlinear solutions of this system -- which correspond to waves with a smoothly sloping tail, a steep front, and a capillary ripple ahead of the wave -- can only be solved numerically [9, 10]. The development stages of both stimulated and naturally occurring waves, including two types of attractors, were examined [11] by extending an earlier theory [8, 12]. The theoretical results agreed quantitatively with experiment in the main part of the wave, but the capillary ripple, predicted by the integral approach, was much stronger and had a higher oscillation amplitude than in experiments. Nonlinear theory has been examined and the velocity profde has been determined for waves of the first kind in the long- wavelength approximation for stationary running waves [13]. Stationary nonlinear solutions, based on boundary-layer equations and integral equations for describing film flow, give good agreement with experiment [14]; a detailed comparison is in progress. A transient solution of the Navier-Stokes equations has been done in the long-wavelength approximation for the initial stage of wave development, up to the occurrence of reverse flow in the thinnest part of the t'tim [15]. Transient finite-element solutions for the complete system of Navier-Stokes equations have been found without any approximations [16]. A complete numerical solution using Galerkin's method has been presented for a stationary running wave in viscous fluid layers [17]. Calculations were done for various values of the dimensionless surface stress, including zero stress. Here a new pseudospectral method is presented for calculating the transient development of a wave within the framework of the long-wavelength approximation. It can compute the complete development of the wave until it becomes stationary. This stationary wave is compared with the solution to the stationary equations. The solution is extended parametrically to large Reynolds numbers in order to determine the effect of Reynolds number on the wave characteristics. Solutions found for the long- wavelength approximation are compared with solutions [17] of the complete system of equations. 1. PROBLEM FORMULATION A viscous incompressible fluid flows down a vertical plane by gravity. The flow is assumed to be two-dimensional and periodic with a wavelength X. The x-axis is directed downward along the gravity-acceleration vector g; the y-axis is perpendicular to the surface. We will assume that the wavelength is much larger than the film thickness. As Nusselt showed, a falling fluid film always has a trivial stationary solution with a smooth plane-parallel free surface: Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 1, pp. 53-61, January-February, 1994. Original article submitted November 17, 1992; revision submitted February 16, 1993. 0021-8944/94/3501-0053512.50 9 1994 Plenum Publishing Corporation 53