Source-linked AI summary
Assessment of Numerical Lift Coefficient Data for a Circular Cylinder with Application to Bladeless Turbines
L. Ridgway Scott, Henry von Wahl
TL;DR
The paper addresses sparse and contradictory evidence on cylinder lift coefficients and their relevance to bladeless energy generation. It reviews two-dimensional finite element computations over Reynolds numbers 120–8000, compares them with published numerical and experimental results, and finds agreement in some cases but significant disagreement in others. The results support preliminary bladeless-turbine analysis while highlighting measurement, three-dimensionality, and wind-speed-dependent design limitations.
Problem
Lift data for cylinder flow are sparser and more contradictory than drag data, despite oscillatory lift's potential relevance to bladeless energy generation.
Method
The paper reviews two-dimensional finite element computations of incompressible Navier–Stokes flow using two high-accuracy methods and compares computed lift data with published computational and experimental results.
Results
Lift data for Reynolds numbers 120 to 8000 show good agreement with some published results but significant disagreement with others.
Takeaways & Limitations
The data provide input for preliminary bladeless-wind-turbine analysis and indicate the importance of measuring lift and simulating oscillating cylinders.
Takeaways & Limitations
Wind-induced vibration force varies by 4 orders of magnitude across the considered cases, so different energy-capture approaches may be needed at different wind speeds.
Abstract
from arXiv · showhide
We assess computed lift coefficient data for flow past a circular cylinder to evaluate their suitability for practical applications. Specifically, we consider lift coefficient data for a circular cylinder over Reynolds numbers from 120 to 8000. The results are obtained from two-dimensional finite element simulations of the incompressible Navier-Stokes equations using pressure robust discretizations. We compare the computed lift coefficients with published experimental and numerical results, finding good agreement in some cases but significant disagreement in others. Because lift fluctuations are central to vortex-induced vibration concepts, these data therefore provide input for the analysis and preliminary design of bladeless turbines.
1 Introduction
The paper shifts attention from extensively studied cylinder drag to sparser, contradictory lift data because oscillatory lift may support bladeless energy generation. It reviews prior computations, compares lift results across studies, and considers implications for vibrating-cylinder applications.
- Drag has received considerably more study than lift in flow around a cylinder.
- Oscillatory lift may contribute to industrial applications such as bladeless energy generation.For fixed cylinders, lift fluctuations are an order of magnitude greater than drag fluctuations.
- Existing lift data are sparse and contradictory, motivating emphasis on computational and experimental lift measurements.The authors are designing experiments to measure lift.
- The paper reviews prior finite element computations, compares reliable lift results with computational and experimental studies, and reports both agreement and discrepancies.These discrepancies suggest considering vibrating cylinders as contributors to applications.
- The paper applies the reviewed results to bladeless wind-turbine design after presenting the governing equations, numerical setup, literature comparison, and vibration-related research.
2 Setting the problem and model equations
The model is the incompressible, time-dependent Navier–Stokes problem around an obstacle with prescribed boundary conditions, represented in weak form for finite element discretization. Drag and lift are evaluated from the resulting boundary traction, with volume residual evaluation available as a more accurate and stable alternative.
- The incompressible Navier–Stokes equations are posed in a domain containing an obstacle with boundary Γ.
- The model imposes u = g on the outer boundary and u = 0 on the obstacle boundary Γ.
- The equations are rewritten in weak form to support finite element discretization.The formulation seeks velocity and pressure functions satisfying temporal, viscous, convective, and incompressibility terms.
- The bilinear forms encode viscous and incompressibility contributions, while the convective term represents nonlinear advection.The strain-rate operator D(v) and Frobenius inner product define the viscous form.
- Drag and lift are computed from a boundary traction functional using test directions (1, 0) and (0, 1), respectively.A non-conforming residual evaluation is described as more accurate and stable than direct surface integration.
3 Problem Set-up
The simulations use two pressure-robust finite element methods to compute lift and drag for flow past a circular cylinder across a prescribed domain and time interval. The MCS method’s numerical dissipation supports reliable mesh-converged results at higher Reynolds numbers, while SV requires full resolution of all scales.
- Geometry and boundary conditions: The computational domain is Ω = {(x, y) : −30 < x < 300, |y| < 30, x2 + y2 > 1}, with uniform inflow and no-slip cylinder boundary conditions.The cylinder diameter is the reference length, giving Re = 2/ν, and simulations run over [0, 500].
- Computational methods: The Scott–Vogelius (SV) method uses conforming piecewise quartic velocities, discontinuous cubic pressures, Crank–Nicolson time stepping, and the volume formulation for lift and drag.The finest mesh contains 551434 degrees of freedom and uses Δt = 0.01.
- Computational methods: The MCS method uses an H(div)-conforming formulation with nonconforming piecewise quartic velocities, upwind convection, penalty treatment, and boundary-based lift and drag evaluation.Its finest mesh contains 604047 degrees of freedom and uses Δt = 0.00025 because of a CFL condition.
- Computational methods: Both methods produce pointwise divergence-free velocities, conserve mass exactly, and are pressure robust.Pressure robustness means the velocity error is independent of the pressure error.
- Computational methods: Upwind convection is critical for MCS because its numerical dissipation captures periodic results on under-resolved meshes, whereas SV achieves comparable reliability only when all scales are resolved.The distinction is especially relevant at higher Reynolds numbers.
4 Quantities of interest
The lift data are examined through drag–lift phase diagrams and comparisons with experimental and numerical studies. Agreement is strong for some reference data, but substantial discrepancies and three-dimensional limitations remain across Reynolds-number ranges.
- Periodic and chaotic regimes: For Reynolds numbers between 50 and 1000, the flow develops a periodic von Kármán vortex street with periodic drag–lift profiles.Figure 1 presents phase diagrams for Re = 120 and Re = 1000 after the initial start-up phase.
- Periodic and chaotic regimes: At Re = 2000, the drag–lift phase diagram is chaotic, and SV and MCS produce slightly different phase trajectories but similar lift-coefficient standard deviations.Small trajectory differences are expected in chaotic flow, while the lift statistics remain comparable.
- Comparison with prior data: Figure 3 compares MCS and SV lift data with TCA numerical results and Norberg data as functions of Reynolds number.The plotted markers distinguish the computational methods and published sources.
- Comparison with prior data: The computed lift data show significant agreement with the TCA numerical simulations, whose results are similar across the turbulence models considered.The cited comparison also reports agreement between wind-tunnel experiments and numerical simulations for other flow quantities.
- Model scope and discrepancies: For Reynolds numbers up to a few thousand, the two-dimensional model is likely to agree substantially with three-dimensional data for sufficiently long cylinders, but three-dimensional effects become important near Re = 10000.Spanwise lift variation can reduce effective lift.
- Model scope and discrepancies: Around Re = 1000, reported r.m.s. lift coefficients range from 0.05 to 0.7, indicating substantial variance among prior data and possible under-resolution in some simulations.The drag–lift profile remains periodic at Re = 1000, although some simulations could not verify this because of inadequate accuracy and numerical dissipation.
- Model scope and discrepancies: Additional data report r.m.s. sectional lift coefficients from about 0.2 to 0.6 for Re between 4 · 10^4 and 10^5, while other comparisons span 0.03 to 0.6.These studies disagree with Figure 3 and may be affected by cylinder vibration in experiments.
- Experimental reference data: Fixed-cylinder experiments report a mean lift coefficient of 0.3842 with standard deviation 0.0873, while individual experiments range approximately from 0.2 to 0.55.The spread is attributed mainly to three-dimensional effects and unavoidable relative cylinder motion.
5 Industrial implications
The paper uses computed lift data to make a preliminary assessment of bladeless wind-generator parameters, including Reynolds numbers, Strouhal periods, and oscillation frequencies. It emphasizes that practical energy extraction remains unassessed and that forces vary strongly with wind speed and may change for vibrating cylinders.
- Industrial design assessment: Computed lift data support a superficial assessment of Strouhal periods and vortex-flow frequencies for bladeless wind-generator designs.The assessment is not intended as an in-depth evaluation of turbine-design challenges.
- Industrial design assessment: Cylinders from 1 millimeter to 10 centimeters in diameter were selected as commercially obtainable examples for the design calculations.The largest example corresponds to about 4 inches and common plastic pipe dimensions.
- Industrial design assessment: Table 1 reports Reynolds numbers, physical time, interpolated Strouhal periods, and oscillation frequencies for untethered cylinders across lengths and wind speeds.Wind speeds are also expressed in miles per hour for convenience.
- Industrial design assessment: Many practical-size cylinders could operate in commonly occurring winds, but their oscillations could be fast enough to appear as a blur.The paper highlights these observations from Table 1 without claiming a complete turbine-design assessment.
- Scope and limitations: The study does not assess how wind-induced vibrations could generate electricity, and the associated force varies by 4 orders of magnitude across Table 1.Because lifting force scales quadratically with wind speed, different wind speeds may require different energy-capture approaches.
- Scope and limitations: The Figure 3 and Table 1 data will likely change when cylinders are vibrating rather than fixed.
6 Studies of vortex-induced vibrations
Prior work on vortex-induced vibrations includes moving-boundary simulations, turbulence-model studies, and several wind-energy-harvesting concepts. The cited literature also raises concerns about the accuracy of some OpenFoam results at low Reynolds numbers.
- Vibrating-cylinder simulations: Vibrating-cylinder studies require more complex simulations with moving boundaries, including implementations using OpenFoam at Re=100.
- Vibrating-cylinder simulations: OpenFoam-computed Strouhal numbers and frequencies were reported to err by more than a factor of two for Re ∈[55, 161].
- Vibrating-cylinder simulations: A turbulence-model study at Re=150 used the k −ω SST model with ten listed turbulence coefficients and OpenFoam implementations.
- Bladeless energy harvesting: Related energy-harvesting work covers electromagnetic harvesters, triboelectric extraction, piezoelectric arrays, and magnetorheological elastomers for broadening resonance ranges.
7 Conclusions
The study presents lift data over Reynolds numbers 120–8000, compares them with experimental and numerical results, and applies them to preliminary bladeless-turbine considerations. Agreement is good in some cases but significant disagreement remains, motivating further lift measurements and vibrating-cylinder simulations.
- Conclusions: Lift data were presented for flow past a cylinder over Reynolds numbers ranging from 120 to 8000.
- Conclusions: Computed lift data showed good agreement with experimental data in some cases but significant disagreement in others.
- Conclusions: Lift fluctuations are typically larger than drag fluctuations, as indicated by the differing vertical and horizontal scales in Figures 1 and 2.
- Conclusions: The data were used to estimate Strouhal periods and frequencies for practical cylinder sizes and realistic wind speeds, showing potential for bladeless wind turbines.
- Conclusions: The findings suggest that more attention should be given to measuring lift and that flow past oscillating cylinders should be simulated.