Conversion to Pressure

Conversion to Pressure Conversion to Pressure Подпись: (5.31)

In PSP measurements, conversion of the luminescent intensity to pressure is complicated by the temperature effect of PSP especially when the surface temperature distribution is not known. Empirically, a priori calibration relation between air pressure and the relative luminescent intensity is expressed by a polynomial

The experimentally determined coefficients Cl, C2 and C3 in Eq. (5.31) can be expressed as a polynomial function of temperature. If a distribution of the surface temperature is not given and the thermal conditions in a priori laboratory calibrations are different from those in wind tunnel tests, a priori relation Eq.

(2.31) cannot be directly applied to conversion to pressure. To deal with this problem, a short-cut approach is in-situ PSP calibration that directly correlates the luminescent intensity to pressure data from taps distributed on a model surface. In this case, the constant coefficients Cl, C2 and C3 in Eq. (5.31) are determined using least-squares method to achieve the best fit to the pressure tap data over a certain range of pressures. Through in-situ calibration, the effect of a non-uniform surface temperature distribution is actually absorbed into a precision error of least – squares estimation. When the temperature effect of PSP overwhelms a change of the luminescent intensity produced by pressure, in-situ calibration has a large precision error. In addition, when the pressure tap data do not cover the full range of pressure on a surface, in-situ PSP conversion may lead to a large bias error in data extrapolation outside the calibration range of pressures.

Подпись: P Poff Conversion to Pressure Подпись: к Подпись: off Подпись: 2 Подпись: (5.32)

A hybrid method between in-situ and a priori methods is the so-called K-fit method originally suggested by M. Morris and recapitulated by Woodmansee and Dutton (1998). Eq. (5.31) is re-written as

where If = I( pojf, Tojf ) is the luminescent intensity in the wind-off conditions, and Kj = Iref / If and KP = pref / Pf are called the K-factors. The reference

conditions under which a priori calibration is made in a laboratory are generally different from the wind-off conditions in a wind tunnel. While the factor KP = pref / p0ff is known, the factor KI = Iref / If is generally not known and has to be determined since illumination conditions and photodetectors used in laboratory may be different from those in wind tunnel. Given the coefficients C1 , C2 and C3 at a known temperature on an isothermal surface, KI can be determined using a single data point from pressure taps. When the surface temperature data near a number of pressure taps are provided by other techniques like TSP and IR camera, a more accurate value of KI can be obtained using least – squares method with larger statistical redundancy. In the worst case where the surface temperature distribution is totally unknown, assuming an average temperature over the surface, we still able to estimate KI by fitting the pressure tap data. Similar to in-situ calibration, the effect of a non-uniform temperature distribution is absorbed into a precision error of least-squares estimation for KI.

Подпись: ref I Подпись: = g(Thf Подпись: (5.33)

Bencic (1999) used a similarity variable of the luminescent intensity to scale the temperature effect of certain PSP

corr

where g(T) was a function of temperature to be determined by a priori calibration. Under this similarity transformation, the calibration curves for the paint at different temperatures collapsed onto a single curve with the temperature – independent coefficients, i. e.,

(5.34)

In this case, instead of using a 2D calibration surface in the parametric space, only a single one-parameter relation Eq. (5.34) was used to convert the luminescent intensity ratio to pressure. Bencic (1999) found that this similarity was valid for a

Ruthenium-based PSP used at NASA Glenn. In fact, as pointed out in Section

3.6, this similarity is a property of the so-called ‘ideal’ PSP that obeys the following relations (Puklin et al. 1998; Coyle et al. 1999)

Подпись: I ref ( pref ’ Tref ) I(P,Tf) Conversion to Pressure Подпись: (5.35)

I(p, T)/I(p, Tref) = g(T),

Puklin et al. (1998) found that PtTFPP in FIB polymer was an ‘ideal’ PSP over a certain range of temperatures. Note that this similarity (or invariance) is not the universal property of a general PSP.

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