Scale Model of a Glider

Scale Model of a Glider

You are asked to evaluate a model of a glider aircraft in a wind or water tunnel. The aircraft has a wingspan of 20 meters, a chord of 2 meters, and typical speeds for flight of 30 meters per second. There is a wind tunnel available which limits the wingspan to 4 meters. Perform a dimensional analysis using the Buckingham theorem assuming the following variables are critical: (The subscripts FS and M denote full scale and model)

Speed VFS (L/T)

Span SFS ( L )

Chord CFS ( L )

Density rFS ( M/L3)

Molecular Viscosity mFS (ML-1T-1)

VM

SM

CM

rM

mM

Perform the analysis for the full scale set of variables and than by symmetry apply the results to the model variables (obtaining a similar set of groups by inspection). Determine the conditions under which you can obtain similarity between full scale and the model, defining the needed wind tunnel characteristics.

F( V, S, c, r, m) =0

There are 5 variables and 3 dimensions (M, L, T )

\ N – K = 2 and there will be two dimensionless groups

For the k-set choose S, V, r -This does not form a dimensionless group and all dimensions are represented.

p1 = c SAVBrC

(L)(L)A(LT-1)B(ML-3)C

For L: 1+A+B-3C = 0

For Mass: C=0

For Time: -B=0 \A=-1

p1 = c/S

p2 = m SAVBrC

(ML-1T-1)(L)A(LT-1)B(ML-3)C

For L: -1+A+B-3C = 0 \A = -1

For Mass: 1+C = 0 C=-1

For Time: -1 –B = 0 B=-1

p2 = m/(SVr)

This is the form of the inverse Reynolds number – The two dimensionless groups can be expressed as:

p1 = c/S and p2 = (SVr)/m

To obtain similarity between full scale and the model, the ratios of chord to span must be matched in the full scale and the model (geometric similarity). Also the Reynolds numbers must be matched (Ratios of forces made equal). To do this, given a limit on span for the tunnel, either the density or speed must be changed.

For air: r= 1.225 Kilograms/M3

m=1.7894x10-5 Kilograms/(meter second)

CFS/SFS=2/20=.1

\ CM=.1SM For geometric similarity

CM= 40 centimeters

ReFS=(SFS VFS rFS)/mFS = ReM For dynamic similarity

And rFS=rM , mFS= mM for normal atmospheric conditions

VM = (SFC / SM) VFS

For a 4 cm wind tunnel span VM = 5 VFS

or 150 meters/second