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.

Instrumentation Uncertainty quantification Experimental aerodynamics Pneumatic probing
Not sure where to start? 4 places to go

Start with your question

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?

You may also be asking

  • How does Reynolds number affect the result?
  • Where is the five-hole multi-facility project?
  • Is there a calculator implementation?

Where to go next

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 (200 K, 2000 K)

Sensor Architecture

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 modeled as second or third-order polynomials:

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

Uncertainty Propagation

The 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

Consider a yaw calibration with slope $m = 0.12\ \mathrm{rad}^{-1}$, 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.

A hard R&D problem is the kind of conversation I enjoy most, and most of my work has started as one.

Start a conversation

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).

This site documents peer-reviewed research, open curriculum, and premium STEM mentorship in one interconnected portfolio.

Full biography & services · Contact

Part of

This page sits within the broader knowledge structure on lucasrey.com:

  • Engineering - Central knowledge platform - tools, curriculum, notes, and research assets.

Related content from the same research and engineering work:

Part of Engineering

This page is part of the engineering knowledge platform on lucasrey.com.

Engineering knowledge platform

More tools, curriculum, notes, and research from the same body of work:

  • Engineering - Central knowledge platform - tools, curriculum, notes, research, and applied engineering work.

Where to go from here

Ways to take this further, in the order they usually happen.

I take a small number of advisory engagements alongside the doctorate.