3.3.1
Demonstration of need for new calibration scheme
Using established techniques, such as those described by Zilliac (1989) and Galling- ton (1980), the calibration of multi-hole probes is dependent upon non-dimensional pressure coefficients derived from pressure differences. These coefficients are used to determine flow speed and direction. For reference, the numbering scheme of the holes on a seven-hole probe is as shown in figure 3.3.1. The probe is divided into seven sectors, each corresponding to a different hole and calibration region. For a full description on current seven hole probe calibration procedures, see appendix B.
Figure 3.3.1: Numbering scheme of a seven-hole probe, looking aft.
Most conventional calibration methods include using mutually exclusive and non-overlapping calibration regions in cone (θ) and roll (φ) space, as defined in 3.3.2. These regions can be discontinuous, and the algorithm used to select the appropriate calibration space could fail if the stagnation point falls in between
V
u v z y x w βα
θ φFigure 3.3.2: Seven hole probe co-ordinate system (adapted from Ericksen et al., 1995)
holes such that the pressures recorded at more than one hole are within the range of experimental uncertainty. Calibration of multi-hole probes can also be very time consuming to obtain a dense enough matrix of calibration data which yields reasonably low errors when using conventional interpolation schemes. Previous studies have proven a calibration matrix density of five degrees in pitch and yaw to be sufficient for seven hole probes, resulting in a 29×29 matrix for pitch and yaw angles ranging from -70 to 70 degrees (Zilliac, 1993). There have been a number of papers published on various methods of interpolation and data reduction for multi- hole probe calibration (Silva et al., 2003; Zilliac, 1993; Ericksen et al., 1995; Wenger and Devenport, 1999, among others). In fact, one outcome of this research is a method of data reduction using optimal design of experiments and D-Optimality. This method was presented in a conference paper and is described in detail in appendix C (a copy of the paper is also provided for reference in appendix F).
During the first stage of data gathering for this research, problems stemming from the conventional methods of seven hole probe calibration were overwhelmingly evident. Using accepted methods of probe calibration, as detailed in appendix B, figure 3.3.3 shows wake survey data from a seven-hole probe behind a smooth NACA0012 wing with a rounded tip at an incidence of 10◦
4 3.5 1.5 0. 5 0 -0.1 -0.05 0 0.1 0 -0.05 0.1 y/c (a) (b) 1 4 5 6 3 2 i = 7 0.05 -0.1 0.05 -0.1 -0.05 0.05 0.1 z/c y/c 7
Figure 3.3.3: (a) Contours of normalized vorticity, ζrc/v0, and (b) Contours of normalized vorticity showing calibration sector overlay using the conventional calibration methods
of the wing. The free-stream velocity was 10 m/s, and the probe was calibrated in-
situ before the test began. The figure shows contours of normalised vorticity, with
an overlay of the calibration sectors. Discontinuities such as these were not visible in raw pressure readings from the probe and are impossible in a vortex. Therefore, they must arise from the discontinuities in the calibration space (Shaw-Ward et al., 2014, and reproduced for convenience in G).
The conventional calibration of multi-hole probes requires the identification of pitch and yaw (or cone and roll) coefficients to determine flow angularity, reducing the number of independent variables from n to two (where n is the number of holes in the probe). For probes having more than five holes, this reduces the sensitivity of the probe as pressures have to be averaged at some point (see (B.0.1) for an example). Interpolation of flow velocities from calibration functions has also been shown to be a potential source of error (Shaw-Ward et al., 2014).
3.3.2
Description of calibration procedure
It was clear that, in order to proceed with this research, a new calibration procedure, which ideally eliminates the need for discontinuous functions, was necessary. Another motivation for this particular technique was to increase angular precision by keeping the number of independent variables at n, which also, consequently, means that the same approach can be used for any probe of n holes, with arbitrary geometry. The local stagnation pressure can still be approximated as the maximum pressure Pmax recorded, as in the conventional seven hole calibration. Without knowing the exact geometry or hole arrangement, the static pressure can be assumed as the minimum pressure Pmin recorded not subject to separated flow. The pressure coefficients of an n hole probe can then be defined as CP i = Pmax− Pi Pmax− Pmin (3.3.1) C0 = Pmax− P0 Pmax− Pmin (3.3.2) Cs = Pmax− Ps Pmax− Pmin , (3.3.3)
where Pi is the pressure recorded at the ith hole (i = 1, 2, ..., n for Pi 6= Pmax), and P0 and Ps are the reference total and static pressures, respectively. Given a set of calibration data collected at many angles (α, β), the functions fα and fβ can be defined such that
α = fα(CP 1, CP 2, ..., CP n) (3.3.4) β = fβ(CP 1, CP 2, ..., CP n) . (3.3.5)
where fα and fβ are empirical functions defined by calibration data collected in constant, uniform flow at a single velocity. The functions have the advantage of being continuous over the full range of the probe, eliminating the need to select between discrete functions. The arrangement and indexing of the holes
becomes arbitrary as well. Given experimental measurements of (CP 1, CP 2, ..., CP n), the flow angle can be interpolated from the calibration data set. The velocity magnitude can then be obtained using the conventional approach (see (B.0.11)). By removing the sectors in the calibration procedure and keeping the number of independent variables at n, the vorticity field from figure 3.3.3 becomes axisymmetric as shown in figure 3.3.4.
−0.1 −0.05 0 0.05 0.1 −0.1 −0.05 0 0.05 0.1 y/c z/c 3.25 2.5 1 0.25
Figure 3.3.4: Contours of normalized vorticity, ζrc/v0, using novel calibration method. Data is identical to that in 3.3.3.
For small n, it is also possible to fit polynomials to the continuous functions fα and fβ of order k with n variables. Related work (McParlin et al., 2013, included in F) suggested that a polynomial on the order of at least k = 6 is required. This results in 28 terms for a seven hole probe, but for example, if n = 19 (as for the probe described in Shaw-Ward et al., 2014) then 177,100 terms are required. This technique can also be applied to the calibration of triple-wire probes. However, the velocity response of triple-wire probes is nonlinear, so calibration is required in pitch, yaw, and speed, making a fine calibration grid impractical.