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.

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.

Performance analysis of a finite noncircular floating ring bearing in turbulent flow regime

2016 ◽  
Vol 231 (7) ◽  
pp. 869-888 ◽  
Author(s):  
Sandeep Soni ◽  
DP Vakharia
Keyword(s):  
Steady State ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Speed Ratio ◽  
Turbulent Regime ◽  
New Variety ◽  
Navier Stokes Equations ◽  
Turbulent Flow Regime ◽  
Oil Flow ◽  
Steady State Performance

The present paper investigates the turbulence effect on the steady-state performance of a new variety of journal bearing, i.e. the noncircular floating ring bearing. This particular bearing consists of the journal, floating ring, as well as lower and upper lobes. The shaft and the floating ring are cylindrical while surfaces of the bearing are noncircular. The classical Navier–Stokes equations and continuity equation in cylindrical coordinates are being satisfactorily adapted with the linearized turbulent lubrication model of Ng and Pan. These improved equations are being solved by the finite element method using Galerkin’s technique and an appropriate iteration strategy. The proposed bearing has a length-to-diameter ratio of 1 and operates over different values of the ratio of clearances (i.e. 0.70 and 1.30). The steady-state performance parameters computed are presented in terms of an inner and outer film eccentricity ratios, load-carrying capacity, attitude angle, speed ratio, friction coefficient variable, oil flow, and temperature rise variable for the Reynolds number up to 9000. The present analysis predicts better performance in the turbulent regime as compared to the laminar regime for the noncircular floating ring bearing.

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.

Numerical Investigation of an Optimised Horizontal Axis Tidal Stream Turbine

2019 ◽  
Author(s):  
Hassan El Sheshtawy ◽  
Ould el Moctar ◽  
Thomas E. Schellin ◽  
Satish Natarajan
Keyword(s):  
Output Power ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Speed Ratio ◽  
Navier Stokes Equations ◽  
Tidal Turbines ◽  
Tidal Turbine ◽  
Sst Turbulence Model ◽  
Tidal Stream ◽  
Tidal Stream Turbine

Abstract A tidal stream turbine was designed using one of the optimised hydrofoils, whose lift-to-drag ratio at an angle of attack of 5.2 degrees was 4.5% higher than that of the reference hydrofoil. The incompressible Reynolds-averaged Navier Stokes equations in steady state were solved using k-ω (SST) turbulence model for the reference and optimised tidal stream turbines. The discretisation errors and the effect of different y+ values on the solution were analysed. Thrust and power coefficients of the modelled reference turbine were validated against experimental measurements. Output power and thrust of the reference and the optimised tidal turbines were compared. For a tip speed ratio of 3.0, the output power of the optimised tidal turbine was 8.27% higher than that of the reference turbine of the same thrust.

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.

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.

Parametric Study on Aerodynamic Performance of a Centrifugal Fan With Additionally Installed Splitter Blades

2013 ◽  
Author(s):  
Man-Woong Heo ◽  
Jin-Hyuk Kim ◽  
Kyung-Hun Cha ◽  
Kwang-Yong Kim
Keyword(s):  
Flow Separation ◽  
Stokes Equations ◽  
Flow Analysis ◽  
Three Dimensional ◽  
Aerodynamic Performance ◽  
Navier Stokes ◽  
Centrifugal Fan ◽  
Height Ratio ◽  
Navier Stokes Equations ◽  
Fan Performance

Aerodynamic Performance of a centrifugal fan with additionally installed splitter blades in the impeller has been investigated numerically using three-dimensional Reynolds-averaged Navier-Stokes equations. The shear stress transport turbulence model and hexahedral grids system were used to analyze the flow in the centrifugal fan. From results of the flow analysis, considerable energy loss by flow separation was observed in the blade passages. Splitter blades were applied between the main blades to reduce the loss and enhance fan performance. The chord length ratio of splitter to main blade, the angle between splitter and main blade, and the height ratio of outlet and inlet of impeller were selected as the geometric parameters, and their effects on the aerodynamic performance of the centrifugal fan have been investigated. The performance of the centrifugal fan with added splitter blades was improved conspicuously compared to the centrifugal fan without splitter blades. It was found that the installation of splitter blades in the impeller is effective to improve the aerodynamic performance of a centrifugal fan by reducing the flow separation generated between main blades in the impeller.

Influence of rotation on the near-wake development behind an impulsively started circular cylinder

1985 ◽  
Vol 158 ◽  
pp. 399-446 ◽  
Author(s):  
Madeleine Coutanceau ◽  
Christian Ménard
Keyword(s):  
Circular Cylinder ◽  
Stokes Equations ◽  
Flow Characteristics ◽  
Navier Stokes ◽  
Speed Ratio ◽  
Flow Topology ◽  
Near Wake ◽  
Navier Stokes Equations ◽  
Magnus Effect ◽  
Point Trajectories

The early phase of the establishment of the flow past a circular cylinder started impulsively into rotation and translation is investigated by visualizing the flow patterns with solid tracers and by analysing qualitatively (flow topology) and quantitatively (velocity distributions and singular-point trajectories) the corresponding photographs. The range considered corresponds to moderate Reynolds numbers (Re [les ] 1000). The rotating-to-translating-speed ratio α increases from 0 to 3.25 and the motion covers a period during which the cylinder translates 4.5 or even 7 times its diameter. The details of the mechanisms of the near-wake formation are considered in particular and the increase of the flow asymmetry with increase in rotation is pointed out. Thus the existence of two regimes has been confirmed with the creation or non-creation of alternate eddies after an initial one E1 Furthermore, the new phenomena of saddle-point transposition and intermediate-eddy coalescence have been identified in the formation or shedding of respectively the odd and even subsequent eddies Ei (i = 2,3,…) when they exist. The very good agreement between these experimental data and the numerical results of Badr & Dennis (1985), obtained by solving the Navier-Stokes equations and presented in a parallel paper, confirms their respective validity and permits the determination of the flow characteristics not accessible, or accessible only with difficulty, to the present experiments. These flow properties such as drag and vorticity are capable of providing information on the Magnus effect for the former property and on unsteady separated flows for the latter.

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