Analysing performance of Darrieus rotor

Streamtube mathematical model for Darrieus rotor

Two group of mathematical model for analysing and predicting Darrieus rotor performance are the simple momentum (streamtube) model and the complex vortex models. The momentum models are known to be unable to describe flow field around the turbine correctly. Strangely it seems to be the most widely used tool. The credit might be attributed to acceptable accuracy of the result, widely available literature, and code simplicity.
> Multiple streamtube model, [page2], [Matlab code]
> Single streamtube model
> Glauert empirical formula

Airfoil data for streamtube model

Darrieus wind turbine uses the old NACA symmetrical airfoil. The problems are NACA symmetry airfoil data at low Reynolds number and large range of angle of attack are scarce and seems inconsistent. Below are some comments on Strickland, Sheldahl, Jacobs, and Jesch low Reynolds data. If you need low Reynolds number airfoil data i.e. lower than 360e3, then why not try digitised Jacobs’s data for small angle of attack and then attach Sheldahl data to the remaining angle of attack. Complete airfoil data of 360° is not really important. Theoretically for tsr > 1, only data from 0 to not more than 90° is needed.
> Comparing Naca airfoil data

If the aspect ratio of the airfoil blade is low, modification need to be done to account for the finite aspect ratio. At angle of attack before stall, airfoil behaviour can be predicted accurately using Lanchester-Prandtl model. For angle of attack greater than stall, the airfoil behaviour is not clear. Anyway, the published Viterna Corrigan model extends the Prandtl model by adding some equations after stall angle to predict the lift and drag in these regions.
> Finite aspect ratio airfoil data modification [Matlab code]

Multiple streamtube model code in Matlab

Below are the streamtube model codes for simple STRAIGHT BLADED Darrieus rotor only and with the digitised Strickland Naca0012 airfoil data. A simple crude modification for the streamtube model when momentum equation fails was done using the Glauert empirical formula. This problem never seems to be addressed in research paper, any idea? The code accepts and uses variable Reynolds number data (but u can just throw in one single Re data like the sample run). No dynamic stall or pitch changing feature included. These codes were running fine under Matlab version 5.3.
> Strickland NACA0012 airfoil data at Re=300e3
> Viterna-Corrigan model code “makefinitedata.m”, sample run, & plotted output graph
> Airfoil data interpolation code “builddatatable.m”, sample run, & plotted output graph
> Multiple streamtubes code “predictcpcurve_sm.m”, sample run, & plotted output graph
> Additional scripts "batchsol.m" and "batchtablesplot.m" for batch processing

Power coefficient curve result from streamtube model

The first graphs (on the top of the page) shows result from the streamtube model with various solidity. Increasing solidity decreases tsr range, increases torque coefficients, sharper curve instead of the desired flat curve, and insignificant maximum Cp changes. Testing result also shows similar trend.

The graphs right above here shows expected result of Cp with decreasing aspect ratio if Viterna Corrigan model used to modified the airfoil data. Infinite airfoil data is not as smooth as the modified Viterna data so output curve is not as smooth also. The higher Cp values at low tsr range in infinite span airfoil blade is due to the Viterna Corrigan model predicting very high lift after stall and lower overall drag.

Dimensional analysis of Darrieus rotor performance

It seems that no dimensional analysis was mentioned at all for Darrieus rotor performance. Directly from the streamtube model momentum equation we found that Cp is a function of tsr and solidity. Further inspection into airfoil data we found airfoil type, aspect ratio and Re number. Are these 5 all the major significant dimensionless variables affecting the performance Cp? Here’s an ad hoc guessing type dimensional analysis which produced similar parameters plus one additional variable which shows an additional expected insight.
> Dimensionless analysis on Darrieus rotor efficiency

Links, research papers, and books on VAWT performance theory

J.H. Strickland (1975) "The Darrieus Turbine: A Performance Prediction Model Using Multiple Streamtubes" SAND75-0431 > A very neat organised complete paper on multiple streamtube and even with Fortran code included. However, you might want to normalised all the variable first if you intend to build program
G.F. Homicz (1991) "Numerical Simulation of VAWT Stochastic Aerodynamic Loads Produced by Atmospheric Turbulence: VAWT-SAL Code" SAND91-1124 > The newer double multiple streamtube concept is described here in sufficient detail including dynamics stall effect

Sheldahl, R. E., & Klimas, P. C. (1980). Aerodynamic characteristics of 7 symmetrical airfoil sections through 180-degree angle of attack for use in aerodynamics analysis of vertical axis wind turbine. Sandia National Laboratories: SAND80-2114. > The data looks good in the testing range from Reynolds number 0.35 to 0.70 million
Leo Lazauskus digitised Sheldahl airfoil data (the original Sheldahl pdf form were scanned poorly) and 'removed some abnormalities' of the half computer and half testing data.
Jacobs, Eastman N Sherman, Albert (1937). Airfoil section characteristics as affected by variations of the Reynolds number. NACA Report 586 > an old report but still in excellent gif image and pdf form. To avoid download, look directly at page 231 or 0005.gif for the start of NACA symmetry airfoil coordinate with data as low as Reynolds number 42000. Only small angle of attack are provided but the shape of the curves looks much promising than Sheldahl data.
Laurence K. Loftin, Jr. Hamilton A. Smith (1949). Aerodynamic characteristics of 15 NACA airfoil sections at seven Reynolds numbers from 0.7 x 10(exp 6) to 9.0 x 10(exp 6). NACA TN 1945 > Unfortunately, the report was scanned poorly with very dark blur graph
Abbott, Ira H Von Doenhoff, Albert E Stivers, Louis, Jr (1945). Summary of airfoil data. NACA Report 824 > This is the old popular book 'Theory of Wing Section'. Only NACA0009 available and at Re of millions

Wind Energy Explained book website at University of Massachusetts provide download for a set of programs for calculating performance (HAWT), wind data, dynamics (vibration), and generator too. This is probably the most complete code available online on such a wide topic.
TU-Berlin Klaus Kaiser commented a bit on HAWT and VAWT mathematical performance model. If you are about to use or build one, read first for some entertainment.

Spera, D. A. (Editor) (1994) [TOC], Wind turbine technology: Fundamental concepts of wind turbine engineering. New York, ASME Press. > Details on fixed wake method, blade element theory, and short overview on streamtube method, free vortex method, Glauert empirical formula, ….
Eggleston, D. M., & Stoddard, F. S. (1987). Wind turbine engineering design. New York: Van Nostrand Reinhold Company. > This textbook will cover the aerodynamics performance and structural design of HAWT.
Ion Paraschivoiu (2002), Wind Turbine Design - With Emphasis on Darrieus Concept, Polytechnic International Press. > Not yet seen new book. The author wrote research paper on streamtube model, so the book include this momentum theory, and other vortex theory. Streamtube model software, CARDAA included.