scholarly journals Computation of the effective slip of rough hydrophobic surfaces via homogenization

2014 ◽  
Vol 24 (11) ◽  
pp. 2259-2285 ◽  
Author(s):  
Matthieu Bonnivard ◽  
Anne-Laure Dalibard ◽  
David Gérard-Varet
Keyword(s):  
Scaling Laws ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Volume Distribution ◽  
Hydrophobic Surfaces ◽  
Surface Distribution ◽  
Navier Stokes Equations ◽  
Small Areas ◽  
Newtonian Flows ◽  
Effective Slip

We present a quantitative analysis of the effect of rough hydrophobic surfaces on viscous Newtonian flows. We use a model introduced by Ybert and coauthors in [Achieving large with superhydrophobic surfaces: Scaling laws for generic geometries, Phys. Fluids 19 (2007) 123601], in which the rough surface is replaced by a flat plane with alternating small areas of slip and no-slip. We investigate the averaged slip generated at the boundary, depending on the ratio between these areas. This problem reduces to the homogenization of a nonlocal system, involving the Dirichlet to Neumann map of the Stokes operator, in a domain with small holes. Pondering on the works of Allaire [Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal. 113 (1990) 209–259; Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes, Arch. Rational Mech. Anal. 113 (1990) 261–298]. We compute accurate scaling laws of the averaged slip for various types of roughness (riblets, patches). Numerical computations complete and confirm the analysis.

Reconsideration of the Fan Scaling Laws: Part I — Theory

2003 ◽  
Author(s):  
Asad M. Sardar ◽  
William K. George
Keyword(s):  
Scaling Laws ◽  
Stokes Equations ◽  
Performance Testing ◽  
Navier Stokes ◽  
Speed Ratio ◽  
Navier Stokes Equations ◽  
Fan Performance ◽  
Dynamic Similarity ◽  
Scaling Parameters ◽  
Fan Speed

Generalized Fan Scaling Laws (GSFL) are derived for the scaling of fan performance. These follow from first principles using the Navier-Stokes equations appropriate to rotating and swirling flows. Not surprisingly, both Strouhal and Reynolds number similarity must be maintained. Thus for a geometrically similar family of fans, dynamic similarity is only possible if ΩD/U = constandUD/ν = const. If the second relation is solved for U and substituted into the first, it follows that full dynamic similarity is possible only if ΩD2/ν = const. This can be contrasted with the classical fan laws (CFSL) which for the same flow rate coefficient would imply that Q/ΩD3 (or U/ΩD) = const, implying that both fan size ratio and fan speed ratio are independent fan scaling parameters. Clearly for dynamic similarity to be maintained, the velocity and fan speed can not be varied independently (i.e. fan size and fan speed are not independent scaling parameters), contrary to the implications of the classical fan scaling laws. Further implications of the differences between the classical and generalized scaling laws for fan performance testing and design will be explored. Also several examples will be given in Part II as to how the generalized scaling laws can be applied in design practice.

Self-similar behaviour of a rotor wake vortex core

2014 ◽  
Vol 740 ◽  
Author(s):  
Mohamed Ali ◽  
Malek Abid
Keyword(s):  
Vortex Core ◽  
Scaling Laws ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Azimuthal Velocity ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Rotating Blade ◽  
Self Similar

AbstractWe report a self-similar behaviour of solutions (obtained numerically) of the Navier–Stokes equations behind a single-blade rotor. That is, the helical vortex core in the wake of a rotating blade is self-similar as a function of its age. Profiles of vorticity and azimuthal velocity in the vortex core are characterized, their similarity variables are identified and scaling laws of these variables are given. Solutions of incompressible three-dimensional Navier–Stokes equations for Reynolds numbers up to $Re= 2000$ are considered.

scholarly journals Well-posed and ill-posed behaviour of the -rheology for granular flow

2015 ◽  
Vol 779 ◽  
pp. 794-818 ◽  
Author(s):  
T. Barker ◽  
D. G. Schaeffer ◽  
P. Bohorquez ◽  
J. M. N. T. Gray
Keyword(s):  
Friction Coefficient ◽  
Blow Up ◽  
Stokes Equations ◽  
Momentum Balance ◽  
Navier Stokes ◽  
Major Step ◽  
Navier Stokes Equations ◽  
Newtonian Flows ◽  
Ill Posed ◽  
Well Posed

In light of the successes of the Navier–Stokes equations in the study of fluid flows, similar continuum treatment of granular materials is a long-standing ambition. This is due to their wide-ranging applications in the pharmaceutical and engineering industries as well as to geophysical phenomena such as avalanches and landslides. Historically this has been attempted through modification of the dissipation terms in the momentum balance equations, effectively introducing pressure and strain-rate dependence into the viscosity. Originally, a popular model for this granular viscosity, the Coulomb rheology, proposed rate-independent plastic behaviour scaled by a constant friction coefficient ${\it\mu}$. Unfortunately, the resultant equations are always ill-posed. Mathematically ill-posed problems suffer from unbounded growth of short-wavelength perturbations, which necessarily leads to grid-dependent numerical results that do not converge as the spatial resolution is enhanced. This is unrealistic as all physical systems are subject to noise and do not blow up catastrophically. It is therefore vital to seek well-posed equations to make realistic predictions. The recent ${\it\mu}(I)$-rheology is a major step forward, which allows granular flows in chutes and shear cells to be predicted. This is achieved by introducing a dependence on the non-dimensional inertial number $I$ in the friction coefficient ${\it\mu}$. In this paper it is shown that the ${\it\mu}(I)$-rheology is well-posed for intermediate values of $I$, but that it is ill-posed for both high and low inertial numbers. This result is not obvious from casual inspection of the equations, and suggests that additional physics, such as enduring force chains and binary collisions, becomes important in these limits. The theoretical results are validated numerically using two implicit schemes for non-Newtonian flows. In particular, it is shown explicitly that at a given resolution a standard numerical scheme used to compute steady-uniform Bagnold flow is stable in the well-posed region of parameter space, but is unstable to small perturbations, which grow exponentially quickly, in the ill-posed domain.

Similarity in non-rotating and rotating turbulent pipe flows

1999 ◽  
Vol 379 ◽  
pp. 1-22 ◽  
Author(s):  
MARTIN OBERLACK
Keyword(s):  
Reynolds Number ◽  
Scaling Laws ◽  
Stokes Equations ◽  
Scaling Law ◽  
Navier Stokes ◽  
Invariant Solutions ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Pipe Flows ◽  
The Mean

The Lie group approach developed by Oberlack (1997) is used to derive new scaling laws for high-Reynolds-number turbulent pipe flows. The scaling laws, or, in the methodology of Lie groups, the invariant solutions, are based on the mean and fluctuation momentum equations. For their derivation no assumptions other than similarity of the Navier–Stokes equations have been introduced where the Reynolds decomposition into the mean and fluctuation quantities has been implemented. The set of solutions for the axial mean velocity includes a logarithmic scaling law, which is distinct from the usual law of the wall, and an algebraic scaling law. Furthermore, an algebraic scaling law for the azimuthal mean velocity is obtained. In all scaling laws the origin of the independent coordinate is located on the pipe axis, which is in contrast to the usual wall-based scaling laws. The present scaling laws show good agreement with both experimental and DNS data. As observed in experiments, it is shown that the axial mean velocity normalized with the mean bulk velocity um has a fixed point where the mean velocity equals the bulk velocity independent of the Reynolds number. An approximate location for the fixed point on the pipe radius is also given. All invariant solutions are consistent with all higher-order correlation equations. A large-Reynolds-number asymptotic expansion of the Navier–Stokes equations on the curved wall has been utilized to show that the near-wall scaling laws for at surfaces also apply to the near-wall regions of the turbulent pipe flow.

Reconsideration of the Fan Scaling Laws: Part II — Applications

2003 ◽  
Author(s):  
Asad M. Sardar ◽  
William K. George
Keyword(s):  
Scaling Laws ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Pressure Coefficient ◽  
Navier Stokes ◽  
Flow Measurements ◽  
Navier Stokes Equations ◽  
Fan Performance ◽  
Flow Rate Coefficient ◽  
Scale Models

The entire approach to the scaling of fan performance was reconsidered in Part I from first principles using the Navier-Stokes equations appropriate to rotating and swirling (incompressible) flows. Generalized fan scaling laws (GFSL) were derived, consistent with the fact that both Strouhal and Reynolds number must be maintained constant for full dynamical similarity to be possible. As a consequence, dynamic similarity is only possible if ΩD2/ν = const. (in addition to U/ΩD = constant, or equivalently, Q/ΩD3 = constant). This can be contrasted with the classical fan laws (CFSL) which for the same flow rate coefficient would imply that Q/ΩD3 = const. The differences in fan scaling laws between GFSL and CFSL lead to very different fan pressure and fan power scaling criteria. For example, using the GFSL, the pressure coefficient (or Euler number), similarity between the model and prototype fan results in ΔPm/ΔPp = 1/(Lm/Lp)2 (for constant kinematic viscosity), whereas the CFSL pressure scaling criteria lead to ΔPm/ΔPp = (Ωm/Ωp)2 (Lm/Lp)2. Clearly these are very different scaling results and affect the scaling of all the relevant fan performance parameters. In this paper, several applications will be described of experimental programs which utilized these GFSL to design dynamically similar scale models of automotive fans, and these designs are contrasted with what might have been done had the classical laws been used. Three of the facilities were built, and each allowed detailed flow measurements, which would not otherwise have been possible. In addition, the implications and advantages of making tests in a high-pressure facility are explored. Some suggestions will also be made as to how the generalized scaling laws can be applied in design practice.

Effects of stator splitter blades on aerodynamic performance of a single-stage transonic axial compressor

2020 ◽  
Vol 14 (4) ◽  
pp. 7369-7378
Author(s):  
Ky-Quang Pham ◽  
Xuan-Truong Le ◽  
Cong-Truong Dinh
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Aerodynamic Performance ◽  
Numerical Results ◽  
Single Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Operating Stability ◽  
Length Coverage

Splitter blades located between stator blades in a single-stage axial compressor were proposed and investigated in this work to find their effects on aerodynamic performance and operating stability. Aerodynamic performance of the compressor was evaluated using three-dimensional Reynolds-averaged Navier-Stokes equations using the k-e turbulence model with a scalable wall function. The numerical results for the typical performance parameters without stator splitter blades were validated in comparison with experimental data. The numerical results of a parametric study using four geometric parameters (chord length, coverage angle, height and position) of the stator splitter blades showed that the operational stability of the single-stage axial compressor enhances remarkably using the stator splitter blades. The splitters were effective in suppressing flow separation in the stator domain of the compressor at near-stall condition which affects considerably the aerodynamic performance of the compressor.

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