Analysing performance of Darrieus
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
Multiple streamtube model,
Single streamtube model
Glauert empirical formula
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°
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
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
Multiple streamtubes code
“predictcpcurve_sm.m”, sample run, & plotted output graph
Additional scripts "batchsol.m"
and "batchtablesplot.m" for batch processing
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.
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.
on Darrieus rotor efficiency
papers, and books on VAWT performance theory
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
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
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
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.
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.
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
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
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.
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.
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, ….
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.
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,