3-hole probe uncertainty analysis

Full derivation of bias and precision contributions for three-hole pneumatic probes, why calibration polynomials and transducer metrology must be written down before loss coefficients or inlet profiles enter a validation statement.

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Instrumentation Uncertainty quantification Experimental aerodynamics Pneumatic probing
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Why visitors arrive: You are using or reviewing three-hole pneumatic probe data and need uncertainty quantification.

Your question: How do calibration coefficients propagate into flow angle and pressure uncertainty?

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Bias travels through polynomials

Three-hole probes recover flow angle and pressure from port readings through fitted calibration functions. A transducer bias that looks negligible in kilopascals can shift inferred yaw and total pressure by margins that RANS will absorb during tuning, unless partial derivatives are propagated at the actual operating point.

  • Focus: Triage of pressure-transducer bias and stochastic precision errors
  • Methodology: Taylor-series expansion for multi-variable propagation
  • Scope: Valid for subsonic and transonic regimes when the probe is calibrated at the test Mach and Reynolds numbers

Sensor Architecture

Plan view of a three-hole yaw probe: a head with a central port P C and two symmetrically placed side ports P L and P R, on a stem. Flow arrives at yaw angle alpha to the probe axis, which loads the side ports unequally.A YAW PROBE FROM ABOVE, SCHEMATICPLPCPRαFLOWa central port, and two side ports placed symmetrically
A YAW PROBE FROM ABOVEPLPCPRαFLOWa central port, and two symmetric side ports
Flow arriving at an angle loads the two side ports unequally; the angular coefficient below turns that difference into the yaw angle.

A three-hole (yaw) probe is an aerodynamic sensor used to recover local flow direction and pressures. The probe consists of three pressure ports: a central port ($P_C$) and two symmetrically placed side ports ($P_L$ and $P_R$).

Calibration workflow

  • Angular sweep through a known uniform flow field (typically ±40°)
  • Computation of non-dimensional coefficients for angle and pressure recovery
  • High-order polynomial fitting to relate coefficients to physical flow properties

The total uncertainty, $U_a$, for any measured variable $a$, is the root-sum-square of the bias ($B_a$) and precision ($S_a$) contributions:

\[ U_a = \sqrt{B_a^2 + S_a^2} \tag{1} \]

Coefficient definitions

\[ C_{\alpha} = \frac{P_L - P_R}{P_C - \frac{1}{2}(P_L + P_R)} \;,\quad C_{P0} = \frac{P_0 - P_C}{P_C - \frac{1}{2}(P_L + P_R)} \;,\quad C_{Ps} = \frac{P_0 - P_s}{P_C - \frac{1}{2}(P_L + P_R)} \tag{2} \]

The flow angle $\alpha$ is typically a linear function of $C_{\alpha}$, while pressure recovery coefficients are modelled as second or third-order polynomials (second order shown):

\[ C_{\alpha} = q + m\,\alpha \;,\quad \alpha = \frac{C_{\alpha}-q}{m} \tag{3} \]
\[ C_{P0} = a_0 + a_1\alpha + a_2\alpha^2 \;,\quad C_{Ps} = b_0 + b_1\alpha + b_2\alpha^2 \tag{4} \]

Uncertainty Propagation

The total uncertainty U a as the hypotenuse of a right triangle whose sides are the bias B a and the precision S a, so U a is the square root of B a squared plus S a squared.ROOT SUM SQUARE, DRAWNBabiasSaprecisionUatotal uncertainty is thehypotenuse, never the sum
ROOT SUM SQUARE, DRAWNBabiasSaprecisionUatotal uncertainty is the hypotenuse,never the sum
Equation 1 as a right triangle, schematic: bias and precision are independent, so they add in quadrature.

The bias on the angular coefficient $C_{\alpha}$ is propagated from the individual pressure channel biases $B_{P_L}$, $B_{P_R}$, and $B_{P_C}$:

\[ B_{C_{\alpha}} = \sqrt{ \dfrac{(P_C - P_R)^2 B_{P_L}^2 + (P_L - P_C)^2 B_{P_R}^2 + (P_R - P_L)^2 B_{P_C}^2} {\left[P_C - \frac{1}{2}(P_L + P_R)\right]^4} } \tag{5} \]

The sensitivity of the pressure recovery is determined via partial derivatives of the calibration polynomials:

\[ \frac{\partial C_{P0}}{\partial C_{\alpha}} = \left(a_1 + 2a_2\alpha \right)\frac{1}{m} \;,\quad \frac{\partial C_{Ps}}{\partial C_{\alpha}} = \left(b_1 + 2b_2\alpha \right)\frac{1}{m} \tag{6} \]
\[ B_{C_{P0}} = \left|\frac{\partial C_{P0}}{\partial C_{\alpha}}\right| B_{C_{\alpha}} \;,\quad B_{C_{Ps}} = \left|\frac{\partial C_{Ps}}{\partial C_{\alpha}}\right| B_{C_{\alpha}} \tag{7} \]

Worked uncertainty mindset

Equation 5 evaluated at the worked example: port pressures P L 102, P R 98 and P C 105 kilopascals, with a bias of 50 pascals on each transducer. The share of the squared bias on C alpha from each port: left port 66 percent; central port 22 percent; right port 12 percent. The example yields C alpha 0.80, alpha 6.6 degrees, a bias on C alpha of 0.017 and a bias on alpha of 0.14 degrees.EQUATION 5 AT THE WORKED EXAMPLEOPERATING POINTPL102 kPaPR98 kPaPC105 kPabias on each transducerBP50 PaEach port's share of BCα²PL(PC − PR)² = 49 kPa²66%PC(PR − PL)² = 16 kPa²22%PR(PL − PC)² = 9 kPa²12%WHAT THE EXAMPLE YIELDSCα = 0.80equation 2α = 6.6°equation 3BCα = 0.017equation 5Bα = 0.14°BCα / m
EQUATION 5 AT THE WORKED EXAMPLEOPERATING POINTPL102 kPaPR98 kPaPC105 kPaBP50 PaEach port's share of BCα²PL(PC − PR)² = 49 kPa²66%PC(PR − PL)² = 16 kPa²22%PR(PL − PC)² = 9 kPa²12%WHAT THE EXAMPLE YIELDSCα = 0.80equation 2α = 6.6°equation 3BCα = 0.017equation 5Bα = 0.14°BCα / m
Computed from the example's own numbers. With the same bias on every transducer, the left port supplies about two thirds of the squared bias on Cα, because its weight in equation 5 is the largest at these pressures. The weights are port pressure differences, so another operating point moves the shares.

Consider a yaw calibration with slope $m = 0.12\ \mathrm{deg}^{-1}$, intercept $q = 0.01$, and port pressures at a representative operating point $P_L = 102\ \mathrm{kPa}$, $P_R = 98\ \mathrm{kPa}$, $P_C = 105\ \mathrm{kPa}$. If each transducer bias is $B_P = 50\ \mathrm{Pa}$, equation (5) in the derivation above yields a bias on $C_\alpha$ that propagates through equations (6) to (7) into $C_{P0}$ and $C_{Ps}$, often comparable to the facility repeatability you would otherwise treat as noise.

The point is not the specific numbers, it is that bias propagation must be evaluated at the actual port pressures and angles used in the campaign, not at a generic “±1%” rule of thumb. Pair this note with the experimental aerothermal measurement article for traceability context and the CFD-experiment validation article for how pneumatic inputs constrain simulation comparisons.

This note supports applied probe work documented across the engineering platform, calibration campaigns, peer-reviewed publication, and future calculator modules.

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About this work

Lucas Rey, aerothermal systems engineer and academic tutor.

  • University of Oxford: DPhil Researcher, Thermofluids Institute.
  • University of Cambridge: Alumnus.
  • Rolls-Royce: Sponsored researcher (High-pressure turbine programme).

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