High-speed flow measurement and visualisation

Why it matters

Every number in compressible aerodynamics — Mach number, pressure distributions, shock positions — ends up being checked in a wind tunnel or in flight. A pitot probe that is correct at low speed gives the wrong Mach number at M = 0.8 if read with Bernoulli, and in supersonic flow it sits behind its own shock. Knowing how probes and optical methods behave in compressible flow is basic test-engineering competence.

Key ideas

Pitot probes in subsonic compressible flow. A pitot tube brings the flow to rest nearly isentropically, so it reads the stagnation pressure p₀. With a static tap reading p, the Mach number follows from the isentropic relation. Bernoulli's p₀ − p = ½ρV² underestimates the Mach number once compressibility matters (above about M = 0.3); the error grows with M.

Pitot probes in supersonic flow. A detached bow shock forms ahead of the probe. On the probe axis this shock is locally normal, so the probe reads p₀₂, the stagnation pressure behind a normal shock — not the free-stream p₀₁. The Rayleigh pitot formula relates p₀₂/p₁ to M₁, where p₁ is the free-stream static pressure (measured by a wall tap or a carefully aligned static probe). It must be solved by iteration or a table. Combining the pitot reading with the reservoir pressure p₀₁ (from the settling chamber of a tunnel) gives p₀₂/p₀₁ and hence M₁ from the normal-shock table.

Static pressure is measured with flush wall taps or slender cone-cylinder probes placed so that their own shocks do not reach the orifices. In supersonic tunnels wall taps in the test section are most reliable.

Mach number from waves. In a uniform supersonic stream a small disturbance produces Mach waves at μ = sin⁻¹(1/M); measuring the angle on a schlieren image gives M. A wedge or cone probe with measured shock angle gives M via the θ–β–M relation or cone charts.

Temperature. A stationary probe recovers most, but not all, of the stagnation temperature because heat is conducted away within its boundary layer: T_r = T + r·(T₀ − T), with recovery factor r ≈ √Pr (about 0.85) for laminar and ∛Pr (about 0.89) for turbulent boundary layers. Stagnation (shielded) probes are designed to approach r = 1. Take r for a given probe from its calibration.

Optical density-based methods (no probe in the flow):

• Shadowgraph responds to the second derivative of density (∂²ρ/∂x²); simple and good for sharp features such as shocks.
• Schlieren uses a knife edge at the focus to respond to the first derivative of density (∂ρ/∂x normal to the edge); more sensitive, shows expansions, shear layers and weak waves; colour schlieren encodes gradient direction.
• Interferometry (Mach–Zehnder, holographic) responds to density itself, giving quantitative density fields. All rely on the refractive index of a gas varying linearly with density (Gladstone–Dale: n − 1 = K·ρ, K about 2.3 × 10⁻⁴ m³/kg for air — a data-book constant).

Other techniques. Hot-wire anemometry (sensitive to mass flux and total temperature in compressible flow — needs careful calibration), laser Doppler velocimetry and particle image velocimetry (particles must be small enough to follow flow through shocks), pressure-sensitive and temperature-sensitive paints, oil-flow and surface visualisation for separation lines, and thin-film or infrared thermography for heat transfer.

Formulas

p₀/p = (1 + (γ−1)/2·M²)^(γ/(γ−1)) — subsonic pitot-static (isentropic).

p₀₂/p₁ = [ (γ+1)²M₁² / (4γM₁² − 2(γ−1)) ]^(γ/(γ−1)) · (1 − γ + 2γM₁²)/(γ+1) — Rayleigh pitot formula (M₁ > 1).

p₀₂/p₀₁ — normal-shock stagnation-pressure ratio (table), for use with reservoir pressure.

μ = sin⁻¹(1/M) — Mach angle.

T_r = T + r·(T₀ − T) — recovery temperature; r from probe calibration.

n − 1 = K·ρ — Gladstone–Dale relation (K from data book).

Symbols: p, p₁ free-stream static pressure (Pa); p₀, p₀₁ free-stream stagnation pressure (Pa); p₀₂ pitot pressure behind a normal shock (Pa); M, M₁ free-stream Mach number; γ ratio of specific heats; T, T₀, T_r static, stagnation and recovery temperature (K); r recovery factor; n refractive index; ρ density (kg/m³).

Worked examples

Example 1 (standard). A pitot-static probe in a subsonic stream reads p₀ = 150 kPa and p = 100 kPa. Find M correctly and with the incompressible formula. γ = 1.4.

1. Isentropic: (1 + 0.2M²)^3.5 = 1.5, so 1 + 0.2M² = 1.5^(1/3.5) = 1.1228, M² = 0.6141, M = 0.784.
2. Incompressible: p₀ − p = ½ρV² = (γ/2)·p·M², so M² = 50/(0.7 × 100) = 0.714, M = 0.845.

Answer: M = 0.784 (Bernoulli would give 0.845, an 8% error).

Example 2 (GATE level). In a supersonic tunnel a pitot probe reads 5.00 times the test-section static pressure. Find the free-stream Mach number. γ = 1.4.

1. Rayleigh pitot formula: p₀₂/p₁ = 5.00. As a first check, at M = 2 the formula gives 5.640, so M < 2.
2. At M = 1.85: p₀₂/p₁ = 4.90; at M = 1.90: 5.14.
3. Interpolating: M₁ ≈ 1.85 + 0.05 × (5.00 − 4.90)/(5.14 − 4.90) = 1.87.

Answer: M₁ ≈ 1.87. Using the isentropic relation instead (p₀/p = 5) would give 1.71 — badly wrong.

Common mistakes

• Using the isentropic p₀/p relation with a pitot reading in supersonic flow; the probe reads p₀₂ behind a normal shock.
• Using Bernoulli for pitot data at M > 0.3.
• Placing a static probe where its own shock or the model's shock reaches the orifice.
• Assuming a thermocouple probe reads T₀ exactly; it reads the recovery temperature.
• Mixing up what schlieren (density gradient) and shadowgraph (second derivative) show.

For GATE AE

Expect: Mach number from subsonic pitot-static readings, Rayleigh pitot calculations (both directions), pitot-to-reservoir ratios with the normal-shock table, Mach angle from a photograph, and which optical method responds to ρ, ∂ρ/∂x or ∂²ρ/∂x². Practise the Rayleigh pitot formula at a couple of trial Mach numbers.

Quick check

1. What pressure does a pitot tube read in supersonic flow?
2. Which optical technique responds to the first derivative of density?
3. A schlieren image shows Mach waves at 30°. What is M?
4. Find p₀₂/p₁ at M = 2 for air.
5. Why does a stagnation-temperature probe read slightly below T₀?

Answers: 1. the stagnation pressure behind the normal part of its bow shock, p₀₂; 2. schlieren; 3. 2.0; 4. 5.64; 5. heat conduction in its boundary layer gives a recovery factor below 1.