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Least-squaresfit.--As mentioned in the Test Procedure section, measurements obtained at four different fringe orienta- tions were used to calculate the axial, radial, and tangential n=l components of velocity. The measured ensemble-averaged 4 fringe normal velocity components were least-squares fit to the total ensemble-averaged velocity vector as follows: n=l 4 4 . :x: Z<:y. Z::.::o . 4 n=l 4 Z%:,. n=t 3V(j) 3V(j) -- ^ p 8n(j)= V(j).fn- Vf,(j) = -5(j)fx"+ V(j)fy,+ U(J)fz,- Vf_ (J) 4444 Y.SxSz. £:}. n=l n=l n=l n=l forj = 1,2 ..... NWN*NP, Here en(j) is the error between the pro- jection of the least-squares estimate of the total ensemble- averaged velocity vector _(j) = (h-(j), _(j), _(j)) in the fringe normal direction 3_n= (-fx., }Yn' }z. )and the measured ensemble- averaged velocity component in the fringe normal direction Vf. (j), where n denotes the nth fringe orientation, andj de- notes the jth window number. The least-squares estimate of the total ensemble-averaged velocity vector was then determined by minimizing the sum of the square of the errors between the total ensemble-averaged velocity component in the fringe nor- The ensemble-averaged velocity components in cylindrical coordinates were determined as follows: Vr(j) = _(j)* cos(0) + Y(j)* sin(0)' -Vo(j) = v(j) *cos(0) - g(j) *sin(0) (1I) Vz (j) = _(j) As a check of the least-squares method, the least-squares estimate of the calculated total ensemble-averaged velocity vector was projected in each of the four measured fringe nor- mal directions and compared to the measured components in each of the respective fringe normal directions. The difference was generally less than 2 percent of the measured component. Passage-averaging,--The circumferential distributions of Vr, VO, and Vzacross each rotor blade passage can be used to assess passage-to-passage variations in the ensemble-averaged flow field. However, our data set was quite large since there were NWN*NP = 200*20 = 4000 data points along the meas- urement line through each (r,z) survey point shown in figure 6. NASA Lewis will provide this unaveraged-passage data in ASCII format upon request. To arrive at a more compact data set that would meet the needs of most users, the measurements acquired in each of the NP blade passages surveyed were averaged across all the mea- sured blade passages to yield a passage-averaged circumferen- tial velocity distribution. The passage-average was formulated as follows: mal direction and the measured ensemble-averaged component in the fringe normal direction as follows: velocity 4 a£ :.(J) 4 aZ :.(:) n=l OS(j) 4 :o(J) n=l j=l,2 a_(j) 4 aE n=l a_(j) NP -- ' Z [k+(i-1).Nwq =0, ..... NWN* NP (9) This results in the following three equations, which can be solved simultaneously for the three unknowns (fffj),_(j),_(j)) that describe the least-squares estimate of the total ensemble- averaged velocity vector in window j: NP - NZ NP (8) Zfx fy" n=l n=l 4 4 £f;2 £fynfZ. n=l n=] _(j) Vr(k)= NP N_ j=l NP vo[k+(j-,),Nwr ] j=l (12) _z(k)= l NP-NzY--, +(j- 1),Nwq j=l n=l (10)PDF Image | Laser anemometer measurements of the three-dimensional rotor flow
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