Linearised small-perturbation potential equation

Why it matters

The full equation for compressible potential flow is nonlinear and hard to solve. For thin wings and slender bodies at small angles, it reduces to one linear equation — the basis of the Prandtl–Glauert rule for subsonic flow, Ackeret theory for supersonic flow, and the panel and supersonic area-rule methods used in early design. Knowing where it comes from tells you exactly when those quick results can be trusted.

Key ideas

Starting point. For steady, inviscid, irrotational (isentropic) flow, the velocity is the gradient of a potential, V = ∇Φ. Combining continuity, Euler's equation and a² = (∂p/∂ρ)ₛ gives the full potential equation, in which the local speed of sound depends on the local velocity through the energy equation. It is nonlinear.

Small perturbations. Write the velocity as the free stream plus small disturbances: u = V∞ + u', v = v', w = w', with u'/V∞, v'/V∞, w'/V∞ ≪ 1, and define the perturbation potential φ with u' = ∂φ/∂x, v' = ∂φ/∂y, w' = ∂φ/∂z. Substituting and dropping products of small quantities gives the linearised small-perturbation equation (1 − M∞²)·φxx + φyy + φzz = 0.

When the linearisation is valid.

• Thin or slender bodies at small angle of attack, so the disturbances really are small.
• M∞ not close to 1. Near M∞ = 1 the coefficient (1 − M∞²) becomes small and the dropped term M∞²(γ+1)(u'/V∞)φxx is comparable to it — the transonic regime needs a nonlinear equation.
• M∞ not too large (hypersonic): the product M∞·τ (τ = thickness ratio or angle) must stay small; otherwise the perturbations are not small relative to the speed of sound.
• No strong shocks (they create entropy and vorticity); weak shocks are acceptable because the entropy jump is third order.

Character of the equation.

• M∞ < 1: (1 − M∞²) > 0, the equation is elliptic, like Laplace's equation in stretched coordinates. A disturbance is felt everywhere; the Prandtl–Glauert transformation x' = x, y' = β·y (with β = √(1 − M∞²)) maps it onto incompressible flow.
• M∞ > 1: (1 − M∞²) < 0, the equation is a hyperbolic wave equation. In 2-D its general solution is φ = f(x − λy) + g(x + λy), with λ = √(M∞² − 1). Disturbances travel only along Mach lines at the Mach angle μ = sin⁻¹(1/M∞), and only downstream.

Boundary condition. Flow tangency at the body surface y = y_s(x) becomes, after linearisation, v'(x, 0) = V∞·dy_s/dx applied on the axis y = 0 rather than on the actual surface (for 2-D bodies).

Pressure coefficient. Expanding the exact isentropic Cp and keeping first-order terms gives Cp = −2u'/V∞ for 2-D flow. For slender bodies of revolution the cross-flow term must be kept: Cp = −2u'/V∞ − (v'² + w'²)/V∞².

Formulas

(1 − M∞²)·φxx + φyy + φzz = 0 — linearised perturbation potential equation.

u' = ∂φ/∂x, v' = ∂φ/∂y, w' = ∂φ/∂z — perturbation velocities (m/s).

Cp = −2u'/V∞ — linearised pressure coefficient (2-D, thin bodies).

v'(x, 0) = V∞·(dy_s/dx) — linearised tangency condition.

β = √(1 − M∞²) (subsonic), λ = √(M∞² − 1) (supersonic).

Supersonic 2-D surface pressure (simple wave): Cp = ±2θ/√(M∞² − 1), where θ is the local surface slope relative to the free stream (+ for compression).

Symbols: φ perturbation potential (m²/s); M∞, V∞ free-stream Mach number and speed (m/s); u', v', w' perturbation velocity components (m/s); x along the stream (m); y_s(x) body surface shape (m); θ surface slope (rad); Cp pressure coefficient (dimensionless).

Worked examples

Example 1 (standard). In a stream at M∞ = 0.6, V∞ = 200 m/s, the axial perturbation velocity at a point on a thin airfoil is u' = 12 m/s. Find the linearised Cp and the coefficient of φxx. Compare with the exact isentropic Cp. γ = 1.4.

1. Cp = −2u'/V∞ = −2 × 12/200 = −0.120.
2. Coefficient 1 − M∞² = 1 − 0.36 = 0.64 (positive, so elliptic).
3. Exact isentropic: Cp = (2/(γM∞²))·[(1 + (γ−1)/2·M∞²·(1 − V²/V∞²))^(γ/(γ−1)) − 1] with V = 212 m/s: 1 − 1.1236 = −0.1236; 1 + 0.072 × (−0.1236) = 0.99110; 0.99110^3.5 = 0.96921; Cp = (2/0.504) × (−0.03079) = −0.1222.

Answer: Cp ≈ −0.120 (linear) vs −0.122 (exact); the 2% difference is the neglected second-order terms.

Example 2 (GATE level). A thin 2-D surface in a M∞ = 2 stream has a local slope of 0.05 rad into the flow. Using linearised theory find the perturbation velocities and Cp. Take V∞ = 600 m/s.

1. λ = √(4 − 1) = 1.732.
2. Tangency: v' = V∞ × 0.05 = 30 m/s.
3. For the upper surface, φ = f(x − λy), so u' = f' and v' = −λf': u' = −v'/λ = −30/1.732 = −17.3 m/s.
4. Cp = −2u'/V∞ = 2 × 17.3/600 = +0.0577. Check: 2θ/λ = 2 × 0.05/1.732 = 0.0577.

Answer: u' ≈ −17.3 m/s, v' = 30 m/s, Cp ≈ +0.058 (compression on a surface inclined into the flow).

Common mistakes

• Writing Laplace's equation ∇²φ = 0 for compressible flow; the (1 − M∞²) factor is the whole point.
• Using Cp = −2u' without dividing by V∞ — Cp must be dimensionless.
• Applying linear theory at M∞ ≈ 1 or for blunt bodies.
• Applying the tangency condition on the true surface instead of on y = 0.
• Dropping the (v'² + w'²) term for slender bodies of revolution.
• Forgetting that the supersonic equation is hyperbolic: there is no upstream influence.

For GATE AE

Expect: identifying the type of the equation for subsonic or supersonic flow, the coefficient 1 − M∞², the linearised Cp formula, the tangency condition, conditions of validity (thin body, M not near 1, not hypersonic), and short calculations linking slope, perturbation velocity and Cp. It underpins Prandtl–Glauert and Ackeret questions, so be fluent with β and λ.

Quick check

1. What type of PDE is the linearised equation at M∞ = 2?
2. Write the linearised 2-D pressure coefficient.
3. Why does the linear theory fail near M∞ = 1?
4. On which line is the linearised tangency condition applied?
5. What is 1 − M∞² at M∞ = 0.8?

Answers: 1. hyperbolic; 2. Cp = −2u'/V∞; 3. the coefficient of φxx tends to zero and the neglected nonlinear term dominates; 4. on the axis y = 0; 5. 0.36.