Theoretical Analysis of the Incompressible Laminar Flow in a Macro-Roughness Cell

2003 ◽  
Vol 125 (2) ◽  
pp. 309-318 ◽  
Author(s):  
Mihai Arghir ◽  
Nicolas Roucou ◽  
Mathieu Helene ◽  
Jean Frene
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Lubrication Theory ◽  
Textured Surface ◽  
Relative Importance ◽  
Two Dimensional ◽  
Textured Surfaces ◽  
Simplified Models ◽  
Inertia Effects ◽  
Laminar Shear

The present work deals with the analysis of the incompressible laminar shear driven flow in a channel of which one of the walls carries a macro roughness pattern while the opposite one has a parallel velocity. The problem is discussed from the standpoint of lubrication theory and it is shown that the usual simplified models as the Reynolds or the Stokes equations are not applicable. Numerical results are presented for three types of two dimensional macro-roughness and two versions of a three dimensional one. It is shown that a pressure generation effect occurs with increasing the relative importance of convective inertia. Previous analyses found in the literature discussed only the increase of the shear stress due to the presence of the macro roughness but the lift effect due to the pressure generation has never been enlightened up to now. It is further discussed that, extrapolated to a very large number of macro roughness characterizing a textured surface, this new effect could be added to the other lift generating mechanisms of the lubrication theory. It could thus bring a different light on inertia effects stemming from the use of textured surfaces.

scholarly journals A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Niklas Ericsson
Keyword(s):  
Fourier Expansion ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Countable Family ◽  
Angular Variable ◽  
Two Dimensional ◽  
Sobolev Norms ◽  
Well Posed ◽  
Variational Formulations

Abstract We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

Direct simulation of evolution and control of three-dimensional instabilities in attachment-line boundary layers

1995 ◽  
Vol 291 ◽  
pp. 369-392 ◽  
Author(s):  
Ronald D. Joslin
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Plate Boundary ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Computationally Efficient ◽  
Simulation Results ◽  
Dimensional Simulation ◽  
Attachment Line

The spatial evolution of three-dimensional disturbances in an attachment-line boundary layer is computed by direct numerical simulation of the unsteady, incompressible Navier–Stokes equations. Disturbances are introduced into the boundary layer by harmonic sources that involve unsteady suction and blowing through the wall. Various harmonic-source generators are implemented on or near the attachment line, and the disturbance evolutions are compared. Previous two-dimensional simulation results and nonparallel theory are compared with the present results. The three-dimensional simulation results for disturbances with quasi-two-dimensional features indicate growth rates of only a few percent larger than pure two-dimensional results; however, the results are close enough to enable the use of the more computationally efficient, two-dimensional approach. However, true three-dimensional disturbances are more likely in practice and are more stable than two-dimensional disturbances. Disturbances generated off (but near) the attachment line spread both away from and toward the attachment line as they evolve. The evolution pattern is comparable to wave packets in flat-plate boundary-layer flows. Suction stabilizes the quasi-two-dimensional attachment-line instabilities, and blowing destabilizes these instabilities; these results qualitatively agree with the theory. Furthermore, suction stabilizes the disturbances that develop off the attachment line. Clearly, disturbances that are generated near the attachment line can supply energy to attachment-line instabilities, but suction can be used to stabilize these instabilities.

Numerical simulation of boundary-layer transition and transition control

1989 ◽  
Vol 199 ◽  
pp. 403-440 ◽  
Author(s):  
E. Laurien ◽  
L. Kleiser
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Transition Process ◽  
Numerical Results ◽  
Boundary Layer Transition ◽  
Two Dimensional ◽  
Layer Transition ◽  
Longitudinal Vortices ◽  
Tollmien Schlichting

The laminar-turbulent transition process in a parallel boundary-layer with Blasius profile is simulated by numerical integration of the three-dimensional incompressible Navier-Stokes equations using a spectral method. The model of spatially periodic disturbances developing in time is used. Both the classical Klebanoff-type and the subharmonic type of transition are simulated. Maps of the three-dimensional velocity and vorticity fields and visualizations by integrated fluid markers are obtained. The numerical results are compared with experimental measurements and flow visualizations by other authors. Good qualitative and quantitative agreement is found at corresponding stages of development up to the one-spike stage. After the appearance of two-dimensional Tollmien-Schlichting waves of sufficiently large amplitude an increasing three-dimensionality is observed. In particular, a peak-valley structure of the velocity fluctuations, mean longitudinal vortices and sharp spike-like instantaneous velocity signals are formed. The flow field is dominated by a three-dimensional horseshoe vortex system connected with free high-shear layers. Visualizations by time-lines show the formation of A-structures. Our numerical results connect various observations obtained with different experimental techniques. The initial three-dimensional steps of the transition process are consistent with the linear theory of secondary instability. In the later stages nonlinear interactions of the disturbance modes and the production of higher harmonics are essential.We also study the control of transition by local two-dimensional suction and blowing at the wall. It is shown that transition can be delayed or accelerated by superposing disturbances which are out of phase or in phase with oncoming Tollmien-Schlichting instability waves, respectively. Control is only effective if applied at an early, two-dimensional stage of transition. Mean longitudinal vortices remain even after successful control of the fluctuations.

Measuring and Quantifying the Surface-Roughness Anisotropy of Modeled and Industrially Produced Surfaces

2005 ◽  
Vol 6-8 ◽  
pp. 573-582 ◽  
Author(s):  
C.M. Wichern ◽  
W. Rasp
Keyword(s):  
Surface Roughness ◽  
Rolling Direction ◽  
Three Dimensional ◽  
Roughness Parameter ◽  
Orientation Angle ◽  
Textured Surface ◽  
Angular Orientation ◽  
Textured Surfaces ◽  
Average Roughness ◽  
Dimensional Surface

‘Three-dimensional surface profilometry’ when used for analysis and product specification reports roughness parameters that provide an average surface description over a relatively large area. Many commercial sheet steels are produced with special textured surfaces for tribological benefits or appearance benefits. These surfaces, as well as others, may demonstrate high levels of roughness anisotropy that is not quantifiable by simple three dimensional surface parameters. This anisotropy can play an important role in the surface appearance of the finished product and in the tribological behaviour during forming. The current work presents a method for quantifying surface-roughness features as a function of angular orientation with respect to rolling direction. The measurement methodology was applied to several model surfaces and one industrially produced electron-beam textured-surface (EBT). This methodology extracts multiple surface-height profiles of the same angular orientation from a single surface and calculates an average roughness parameter for the orientation angle based on the multiple profiles. Particularly interesting results were the large number of profiles necessary to obtain repeatable values for the roughness variation with respect to direction and the strong influence of surface feature size on the repeatability of said results. These results indicate that care must be taken when using a single extracted profile to represent a ‘three-dimensional’ surface.

A Quasi-Generalized-Coordinate Approach for Numerical Simulation of Complex Flows

2006 ◽  
Vol 128 (6) ◽  
pp. 1394-1399 ◽  
Author(s):  
Donghyun You ◽  
Meng Wang ◽  
Rajat Mittal ◽  
Parviz Moin
Keyword(s):  
Coordinate System ◽  
Stokes Equations ◽  
Computational Cost ◽  
Three Dimensional ◽  
Cost Effective ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Curvilinear Coordinate ◽  
Generalized Coordinate

A novel structured grid approach which provides an efficient way of treating a class of complex geometries is proposed. The incompressible Navier-Stokes equations are formulated in a two-dimensional, generalized curvilinear coordinate system complemented by a third quasi-curvilinear coordinate. By keeping all two-dimensional planes defined by constant third coordinate values parallel to one another, the proposed approach significantly reduces the memory requirement in fully three-dimensional geometries, and makes the computation more cost effective. The formulation can be easily adapted to an existing flow solver based on a two-dimensional generalized coordinate system coupled with a Cartesian third direction, with only a small increase in computational cost. The feasibility and efficiency of the present method have been assessed in a simulation of flow over a tapered cylinder.

Three-dimensional impulsive vortex formation from slender orifices

2011 ◽  
Vol 666 ◽  
pp. 506-520 ◽  
Author(s):  
F. DOMENICHINI
Keyword(s):  
Stokes Equations ◽  
Intermediate Phase ◽  
Vortex Formation ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Vortex Ring Formation ◽  
Long Time ◽  
Definition Of

The vortex formation behind an orifice is a widely investigated phenomenon, which has been recently studied in several problems of biological relevance. In the case of a circular opening, several works in the literature have shown the existence of a limiting process for vortex ring formation that leads to the concept of critical formation time. In the different geometric arrangement of a planar flow, which corresponds to an opening with straight edges, it has been recently outlined that such a concept does not apply. This discrepancy opens the question about the presence of limiting conditions when apertures with irregular shape are considered. In this paper, the three-dimensional vortex formation due to the impulsively started flow through slender openings is studied with the numerical solution of the Navier–Stokes equations, at values of the Reynolds number that allow the comparison with previous two-dimensional findings. The analysis of the three-dimensional results reveals the two-dimensional nature of the early vortex formation phase. During an intermediate phase, the flow evolution appears to be driven by the local curvature of the orifice edge, and the time scale of the phenomena exhibits a surprisingly good agreement with those found in axisymmetric problems with the same curvature. The long-time evolution shows the complete development of the three-dimensional vorticity dynamics, which does not allow the definition of further unifying concepts.

scholarly journals Three-dimensional instabilities in compressible flow over open cavities

2008 ◽  
Vol 599 ◽  
pp. 309-339 ◽  
Author(s):  
GUILLAUME A. BRÈS ◽  
TIM COLONIUS
Keyword(s):  
Compressible Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Low Frequency ◽  
Vortical Flow ◽  
Cavity Flows ◽  
Two Dimensional ◽  
Mean Flow ◽  
Aspect Ratios ◽  
Open Cavities

Direct numerical simulations are performed to investigate the three-dimensional stability of compressible flow over open cavities. A linear stability analysis is conducted to search for three-dimensional global instabilities of the two-dimensional mean flow for cavities that are homogeneous in the spanwise direction. The presence of such instabilities is reported for a range of flow conditions and cavity aspect ratios. For cavities of aspect ratio (length to depth) of 2 and 4, the three-dimensional mode has a spanwise wavelength of approximately one cavity depth and oscillates with a frequency about one order of magnitude lower than two-dimensional Rossiter (flow/acoustics) instabilities. A steady mode of smaller spanwise wavelength is also identified for square cavities. The linear results indicate that the instability is hydrodynamic (rather than acoustic) in nature and arises from a generic centrifugal instability mechanism associated with the mean recirculating vortical flow in the downstream part of the cavity. These three-dimensional instabilities are related to centrifugal instabilities previously reported in flows over backward-facing steps, lid-driven cavity flows and Couette flows. Results from three-dimensional simulations of the nonlinear compressible Navier–Stokes equations are also reported. The formation of oscillating (and, in some cases, steady) spanwise structures is observed inside the cavity. The spanwise wavelength and oscillation frequency of these structures agree with the linear analysis predictions. When present, the shear-layer (Rossiter) oscillations experience a low-frequency modulation that arises from nonlinear interactions with the three-dimensional mode. The results are consistent with observations of low-frequency modulations and spanwise structures in previous experimental and numerical studies on open cavity flows.

Modelling of rotors with three-dimensional solid finite elements

2001 ◽  
Vol 36 (4) ◽  
pp. 359-371 ◽  
Author(s):  
A Nandi ◽  
S Neogy
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Cross Sections ◽  
Three Dimensional ◽  
Element Model ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Two Dimensional ◽  
Inertia Effects ◽  
Solid Finite Elements

A shaft is modelled using three-dimensional solid finite elements. The shear-deformation and rotary inertia effects are automatically included through the three-dimensional elasticity formulation. The formulation allows warping of plane cross-sections and takes care of gyroscopic effect. Unlike a beam element model, the present model allows the actual rotor geometry to be modelled. Shafts with complicated geometry can be modelled provided that the shaft cross-section has two axes of symmetry with equal or unequal second moment of areas. The acceleration of a point on the shaft is determined in inertial and rotating frames. It is found that the finite element formulation becomes much simpler in a rotating frame of reference that rotates about the centre-line of the bearings with an angular velocity equal to the shafts spin speed. The finite element formulation in the above frame is ideally suited to non-circular shafts with solid or hollow, prismatic or tapered sections and continuous or abrupt change in cross-sections. The shaft and the disc can be modelled using the same types of element and this makes it possible to take into account the flexibility of the disc. The formulation also allows edge cracks to be modelled. A two-dimensional model of shaft disc systems executing synchronous whirl on isotropic bearings is presented. The application of the two-dimensional formulation is limited but it reduces the number of degrees of freedom. The three-dimensional solid and two-dimensional plane stress finite element models are extensively validated using standard available results.

Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations

2002 ◽  
Vol 451 ◽  
pp. 261-282 ◽  
Author(s):  
F. GRAF ◽  
E. MEIBURG ◽  
C. HÄRTEL
Keyword(s):  
Experimental Data ◽  
Growth Rate ◽  
Linear Stability ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Interface Thickness ◽  
Gap Width ◽  
Stability Results ◽  
Hele Shaw Cell

We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

Drag force on a liquid domain moving inside a membrane sheet surrounded by aqueous medium

2015 ◽  
Vol 779 ◽  
pp. 468-482 ◽  
Author(s):  
V. Laxminarsimha Rao ◽  
Sovan Lal Das
Keyword(s):  
Drag Force ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Domain Size ◽  
Bilayer Membrane ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Liquid Domain ◽  
Viscosity Contrast ◽  
Surrounding Membrane

We compute the drag on a circular and liquid microdomain diffusing in a two-dimensional fluid lipid bilayer membrane surrounded by a fluid above and below. Under the assumptions that the liquids are incompressible and the flow is of low Reynolds number, Stokes’ equations describe the flow in the two-dimensional membrane as well as in the surrounding three-dimensional fluid. The expression for the drag force on the liquid domain involves Fredholm integral equations of the second kind, which we numerically solve using discrete collocation method based on Chebyshev polynomials. We observe that when the domain is more viscous than the surrounding membrane (including the rigid domain case), the drag force is almost independent of the viscosity contrast between the domain and the surrounding membrane, as also observed earlier in experiments by other researchers. The mobility also varies logarithmically with Boussinesq number${\it\beta}$for large${\it\beta}$. On the other hand, for a less viscous domain the dimensionless drag force reduces with increasing viscosity contrast, and a significant change in the drag force, from that when there is no viscosity contrast or when the domain is rigid, has been observed. Further, the logarithmic behaviour of the mobility no longer holds for less viscous domains. Our method of computing the drag force and diffusion coefficient is valid for arbitrary viscosity contrast between the domain and membrane and any domain size (subject to${\it\beta}\geqslant 5$).

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