Dynamics of a reactive falling film at large Péclet numbers. II. Nonlinear waves far from criticality: Integral-boundary-layer approximation

2004 ◽  
Vol 16 (8) ◽  
pp. 3209-3226 ◽  
Author(s):  
Philip M. J. Trevelyan ◽  
Serafim Kalliadasis
Keyword(s):  
Boundary Layer ◽  
Nonlinear Waves ◽  
Falling Film ◽  
Integral Boundary ◽  
Boundary Layer Approximation

scholarly journals Nonlinear waves in counter-current gas–liquid film flow

2011 ◽  
Vol 673 ◽  
pp. 19-59 ◽  
Author(s):  
D. TSELUIKO ◽  
S. KALLIADASIS
Keyword(s):  
Boundary Layer ◽  
Liquid Film ◽  
Film Flow ◽  
Additional Term ◽  
Current Case ◽  
Integral Boundary ◽  
Boundary Layer Approximation ◽  
Long Wave ◽  
Liquid Film Flow ◽  
Counter Current

We investigate the dynamics of a thin laminar liquid film flowing under gravity down the lower wall of an inclined channel when turbulent gas flows above the film. The solution of the full system of equations describing the gas–liquid flow faces serious technical difficulties. However, a number of assumptions allow isolating the gas problem and solving it independently by treating the interface as a solid wall. This permits finding the perturbations to pressure and tangential stresses at the interface imposed by the turbulent gas in closed form. We then analyse the liquid film flow under the influence of these perturbations and derive a hierarchy of model equations describing the dynamics of the interface, i.e. boundary-layer equations, a long-wave model and a weakly nonlinear model, which turns out to be the Kuramoto–Sivashinsky equation with an additional term due to the presence of the turbulent gas. This additional term is dispersive and destabilising (for the counter-current case; stabilizing in the co-current case). We also combine the long-wave approximation with a weighted-residual technique to obtain an integral-boundary-layer approximation that is valid for moderately large values of the Reynolds number. This model is then used for a systematic investigation of the flooding phenomenon observed in various experiments: as the gas flow rate is increased, the initially downward-falling film starts to travel upwards while just before the wave reversal the amplitude of the waves grows rapidly. We confirm the existence of large-amplitude stationary waves by computing periodic travelling waves for the integral-boundary-layer approximation and we corroborate our travelling-wave results by time-dependent computations.

Long-Wave and Integral Boundary Layer Analysis of Falling Film Flow on Walls With Three-Dimensional Periodic Structures

2009 ◽  
Author(s):  
Tatiana Gambaryan-Roisman ◽  
Hongyi Yu ◽  
Karsten Lo¨ffler ◽  
Peter Stephan
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Inclination Angle ◽  
Film Flow ◽  
Three Dimensional ◽  
Falling Film ◽  
Integral Boundary ◽  
Long Wave ◽  
Surface Patterns ◽  
Periodic Microstructures

Falling films exhibit very complex wavy patterns, which depend on the properties of the liquid, the Reynolds number, the wall inclination angle, and the distance from the film inlet. The film hydrodynamics and the surface patterns have a high impact on heat and mass transfer. Our aim is to control and enhance heat and mass transport by using walls with specific micro topographies that influence the falling film flow, stability and wavy pattern. In the present work long-wave theory and integral boundary layer (IBL) approximation are used for modelling the falling film flow on walls with three-dimensional periodic microstructures. The wall topography is periodic both in the main flow direction and in the transverse direction. Examples of such microstructures are longitudinal grooves with sinusoidal path (or meandering grooves) and herringbone structures. The effects of the Reynolds number, the wall inclination angle and the longitudinal and transverse periods of the structure on the shape of liquid-gas interface are investigated. It is shown that, as opposed to straight grooves in longitudinal direction, grooves with meandering path may lead to significant interface deformations.

Design and Analysis of a High Pitch to Chord Ratio Cascade Representative of Ducted Propfans

1991 ◽  
Author(s):  
A. Weber ◽  
W. Steinert ◽  
H. Starken
Keyword(s):  
Boundary Layer ◽  
Mach Number ◽  
High Efficiency ◽  
Pressure Ratio ◽  
Flow Analysis ◽  
High Pitch ◽  
Integral Boundary ◽  
Pressure Loss Coefficient ◽  
Distribution Boundary ◽  
Very High

Efforts to reduce the specific fuel consumption of a modern aero engine focus in particular on increasing the by-pass ratio beyond the current level of around 5. One concept is the counterrotating shrouded propfan operating at low overall pressure ratio and having only very few fan blades of extremely high pitch/chord ratios. The relative inlet Mach numbers cover a range from 0.7 at the hub to 1.1 at the tip section of the first rotor. A propfan cascade was designed by taking into account two characteristic features of a propfan blade-blade section: • a very high pitch/chord ratio of s/c = 2.25 • an inlet Mach number of M1 = 0.90 which leads to transonic flow conditions inside the blade passage In the design process a profile generator and a quasi-3D Euler solver were used iteratively to optimize the profile Mach number distribution. Boundary layer behavior was checked with an integral boundary layer code. The cascade design was verified experimentally in the transonic cascade wind tunnel of DLR at Cologne. The extensive experimental results confirm the design goal of roughly 5 degree flow turning. A total pressure loss coefficient of less than 1.5% was measured at design conditions. This validates the very high efficiency level the propfan concept is calling for. A 2D Navier-Stokes flow analysis code yields good results in comparison to the experimental ones.

The Effect of Real Turbine Roughness With Pressure Gradient on Heat Transfer

2003 ◽  
Author(s):  
Jeffrey P. Bons ◽  
Stephen T. McClain
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Pressure Gradient ◽  
Adverse Pressure Gradient ◽  
Combined Effects ◽  
Pressure Gradients ◽  
Smooth Wall ◽  
Reynolds Analogy ◽  
Integral Boundary ◽  
Favorable Pressure Gradient

Experimental measurements of heat transfer (St) are reported for low speed flow over scaled turbine roughness models at three different freestream pressure gradients: adverse, zero (nominally), and favorable. The roughness models were scaled from surface measurements taken on actual, in-service land-based turbine hardware and include samples of fuel deposits, TBC spallation, erosion, and pitting as well as a smooth control surface. All St measurements were made in a developing turbulent boundary layer at the same value of Reynolds number (Rex≅900,000). An integral boundary layer method used to estimate cf for the smooth wall cases allowed the calculation of the Reynolds analogy (2St/cf). Results indicate that for a smooth wall, Reynolds analogy varies appreciably with pressure gradient. Smooth surface heat transfer is considerably less sensitive to pressure gradients than skin friction. For the rough surfaces with adverse pressure gradient, St is less sensitive to roughness than with zero or favorable pressure gradient. Roughness-induced Stanton number increases at zero pressure gradient range from 16–44% (depending on roughness type), while increases with adverse pressure gradient are 7% less on average for the same roughness type. Hot-wire measurements show a corresponding drop in roughness-induced momentum deficit and streamwise turbulent kinetic energy generation in the adverse pressure gradient boundary layer compared with the other pressure gradient conditions. The combined effects of roughness and pressure gradient are different than their individual effects added together. Specifically, for adverse pressure gradient the combined effect on heat transfer is 9% less than that estimated by adding their separate effects. For favorable pressure gradient, the additive estimate is 6% lower than the result with combined effects. Identical measurements on a “simulated” roughness surface composed of cones in an ordered array show a behavior unlike that of the scaled “real” roughness models. St calculations made using a discrete-element roughness model show promising agreement with the experimental data. Predictions and data combine to underline the importance of accounting for pressure gradient and surface roughness effects simultaneously rather than independently for accurate performance calculations in turbines.

On the flow near the trailing edge of a flat plate

1968 ◽  
Vol 306 (1486) ◽  
pp. 275-290 ◽  
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Governing Equations ◽  
Trailing Edge ◽  
Navier Stokes Equations ◽  
Multiplicative Constant ◽  
Boundary Layer Approximation ◽  
Uniform Shear

It is shown that the boundary layer approximation to the flow of a viscous fluid past a flat plate of length l , generally valid near the plate when the Reynolds number Re is large, fails within a distance O( lRe -3/4 ) of the trailing edge. The appropriate governing equations in this neighbourhood are the full Navier- Stokes equations. On the basis of Imai (1966) these equations are linearized with respect to a uniform shear and are then completely solved by means of a Wiener-Hopf integral equation. The solution so obtained joins smoothly on to that of the boundary layer for a flat plate upstream of the trailing edge and for a wake downstream of the trailing edge. The contribution to the drag coefficient is found to be O ( Re -3/4 ) and the multiplicative constant is explicitly worked out for the linearized equations.

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