Uncertainty

7.1. Pressure Uncertainty of Intensity-Based Methods

7.1.1. System Modeling

Uncertainty analysis is highly desirable in order to establish PSP as a quantitative measurement technique. Based on the Stern-Volmer equation, Sajben (1993) investigated error sources contributing to the uncertainty of PSP, and found that the uncertainty strongly depended on flow conditions and the surface temperature significantly affected the final measurement results. Oglesby et al. (1995a) presented an analysis of an intrinsic limit of the Stern-Volmer relation to the achievable sensitivity and accuracy. Mendoza (1997a, 1997b) studied CCD camera noise and its effect on PSP measurements and suggested the limiting Mach number for quantitative PSP measurements. From a standpoint of system modeling, Liu et al. (2001a) gave a general and comprehensive uncertainty analysis for PSP.

The following uncertainty analysis focuses on the intensity-ratio method widely used in PSP measurements. From Eq. (4.24), air pressure p can be generally expressed in terms of the system’s outputs and other variables

Vref(t, x) pref A(T)pref

V(t’, x’) B(T) B(T)

nc Пf h(x’) c(x’) q0( t’,X’)

Пcref ^f ref href ( x ) cref ( x ) q0ref(t, X )

where x = (x, y)T and x’ = (x’,y’ )T are the coordinates in the wind-off and wind – on images, respectively, X = (X, Y,Z)T and X’ = (X’,Y’,Z’ )T are the object space coordinates in the wind-off and wind-on cases, respectively, and t and t’ are the instants at which the wind-off and wind-on images are taken, respectively. Here, the paint thickness h and dye concentration c are expressed as a function of the image coordinate x rather than the object space coordinate X since the image

registration error is more easily treated in the image plane. In fact, x and X are related through the perspective transformation (the collinearity equations).

In order to separate complicated coupling between the temporal and spatial variations of these variables, some terms in Eq. (7.1) can be further decomposed when a small model deformation and a short time interval are considered. The wind-on image coordinates can be expressed as a superposition of the wind-off image coordinates and an image displacement vector Ax, i. e., x’ = x + Ax. Similarly, the temporal decomposition is t’ = t + At, where At is a time interval between the instants at which the wind-off and wind-on images are taken. If Ax and At are small, a ratio between the wind-off and wind-on images can be separated into two factors, Vref( t, x )/V( t’, x’) ~ Dt(At )Dx(Ax )Vref( t, x )/V( t, x),

where the factors Dt( At) = 1 – (dV / d t)(At)/V and

Dx(Ax) = 1 – (VV) • (Ax)/V represent the effects of the temporal and spatial changes of the luminescent intensity, respectively. The temporal change of the luminescent intensity is mainly caused by photodegradation and sedimentation of dusts and oil droplets on a surface. The spatial intensity change is due to model deformation generated by aerodynamic loads. In the same fashion, the excitation light flux can be decomposed into q0(t’,X’ )/q0ref(t, X) ~

Dq0(At )q0(t, X’ )/q0ref(t, X), where the factor

Dq0(dt) = 1 + (dq0/dt)(At)/q0ref represents the temporal variation in the excitation light flux.

The use of the above estimates yields the generalized Stern-Volmer relation

where

Without any model motion ( x’ = x and X’ = X ) and temporal illumination fluctuation, the factor U2 is unity and then Eq. (7.2) recovers the generic Stern – Volmer relation. Eq. (7.2) is a general relation that includes the effects of model deformation, spectral variability, and temporal variations in both illumination and luminescence, which allows a more complete uncertainty analysis and a clearer understanding of how these variables contribute the total uncertainty in PSP measurements.