Complete course materials for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Compressible Fluid Dynamics Academic Year 2025/2026 Contents 1 Conservation Laws and Thermodynamics of Motion in Compressible Flows 6 1.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.1 Mass Conservation . . . . . . . . . . . . .
Complete course materials for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Compressible Fluid Dynamics Academic Year 2025/2026 Contents 1 Conservation Laws and Thermodynamics of Motion in Compressible Flows 6 1.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.1 Mass Conservation . . . . . . . . . . . . .
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Contents 1 Conservation Laws and Thermodynamics of Motion in Compressible Flows 6 1.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.2 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.3 Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Thermodynamics of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Local Thermodynamic Equilibrium (LTE) . . . . . . . . . . . . . . . . . . . 8 1.2.2 Fundamental Relation and Postulates . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Thermodynamic Potentials and Maxwell Relations . . . . . . . . . . . . . 8 1.3 Thermodynamic Stability and Speed of Sound . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Thermodynamic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Reciprocity Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Ideal Gases and Polytropic Ideal Gases (PIG) . . . . . . . . . . . . . . . . . . . . . 10 1.4.1 Ideal Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.2 Equipartition Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.3 Polytropic Ideal Gases (PIG) . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Van der Waals Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5.1 Speed of Sound in VDW Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Similarity Theory and Adimensionalization 12 2.1 Similarity Theory and Buckingham Pi Theorem .…
For a Perfect Ideal Gas (PIG), where P = ρRT and P/ργ = const: (︃ )︃ ∂P = γRT =⇒ c2 = γRT (4.22) ∂ρ s 4.1.3 Isentropic Nature of Small Perturbations Consider pressure expressed as P = P(s, ρ). Its partial time derivative is: ∂s′ ∂ρ′ ∂s′ ∂ρ′ (︃ )︃ (︃ )︃ (︃ )︃ ∂P ∂P ∂P ∂P = + = + c̄2 (4.23) ∂t ∂s ρ ∂t ∂ρ s ∂t ∂s ρ ∂t ∂t ∂p′ ∂ρ′ (︂ )︂ Comparing this with the linearized relation ∂t = c̄2 ∂t , and noting that ∂P ∂s ̸= 0, we con- ρ clude: ∂s′ =0 (4.24) ∂t Physical Meaning: Small acoustic perturbations do not cause entropy to change over time. The acoustic wave propagation process is inherently isentropic (though it does not imply that baseline entropy is spatially uniform). 4.2 Derivation of Wave Equations and Decoupling 4.2.1 Wave Equation for Pressure and Density Taking the time derivative of the linearized mass conservation (4.11): ∂2 ρ ′ (︃ ′ )︃ ∂u 2 + ρ̄∇ · =0 (4.25) ∂t ∂t ′ ∇ p′ Substituting the momentum equation ∂u ∂t = − ρ̄ : ∂2 ρ ′ − ∇2 p ′ = 0 (4.26) ∂t2 Since p′ = c̄2 ρ′ , both pressure and density perturbations satisfy the classic homogeneous d’Alembert wave equation: ∂2 p ′ − c̄2 ∇2 p′ = 0 (4.27) ∂t2 ∂2 ρ ′ − c̄2 ∇2 ρ′ = 0 (4.28) ∂t2 4.2.2 Velocity Potential Assuming an irrotational perturbation field (∇ × u′ = 0), we introduce a velocity potential Φ such that u′ = ∇Φ. Sostituting into the linearized momentum equation (4.12): p′ p′ (︃ )︃ ∂Φ ∂Φ ∇ + = 0 =⇒ + = R(t) = 0 (4.29) ∂t ρ̄ ∂t ρ̄ where R(t) is an arbitrary spatial constant set to zero without loss of generality. Thus: ∂Φ p′ = −ρ̄ (4.30) ∂t 21
7.5.1 Isentropic Relations for Polytropic Ideal Gases (PIG) For a PIG, evaluating the integral for f ( p, s̄) yields an explicit algebraic expression: 2 f = c (7.39) γ−1 Consequently, the Riemann invariants for a PIG become: 2 J± = u ± c = const (7.40) γ−1 And the isentropic pressure-sound speed relations satisfy: (︃ )︃ γ2γ−1 (︃ )︃ γ−2 1 p c ρ c = , = (7.41) p0 c0 ρ0 c0 7.6 Simple Waves Example A Simple Wave Region in unsteady one-dimensional flow is defined as a domain adjacent to a uniform state where one family of Riemann invariants is globally constant throughout the entire flow field (J − = const or J + = const). 7.6.1 Mathematical Proof of Straight Characteristics in Simple Waves Consider a right-running simple wave region where the J − Riemann invariant is constant ev- erywhere across the domain: J − = u − f ( p, s̄) = J0− = const everywhere (7.42) Along any specific Ck+ characteristic line (identified by index k), the J + invariant is also constant by definition: Jk+ = uk + f k = const along Ck+ (7.43) Combining equations (7.42) and (7.43) allows us to solve explicitly for local velocity uk and state function f k along Ck+ : Jk+ + J0− uk = = const along Ck+ (7.44) 2 Jk+ − J0− fk = = const along Ck+ (7.45) 2 Because f k is constant along Ck+ , all thermodynamic state variables—including pressure pk , density ρk , and sound speed ck = c( f k )—remain strictly constant along that specific Ck+ line. dX + Consequently, the characteristic propagation slope dtk is spatially and temporally uniform along Ck+ : dXk+ = uk + ck = const along Ck+ (7.46) dt 7.6.2 Limit of Uniform Flow If both Riemann invariants are globally constant everywhere (J + = J0+ and J − = J0− ), the flow field is uniform. Velocity u0 and sound speed c0 are constant throughout space and time,…
10.5.1 Characteristic Equation and Slopes ⃓ The derivative ∂u becomes indeterminate along directions where the quadratic coeffi- ⃓ ∂y ⃓ ( x,Y ( x )) cient vanishes, defining the Characteristic Equation: (︃ )︃2 (︁ 2 dY dY (︁ 2 v − c2 + u − c2 = 0 )︁ )︁ − 2uv (10.26) dx dx ± Solving for the characteristic slopes dY dx via the quadratic formula: √ dY ± √︁ uv ± u2 v2 − ( u2 − c2 ) ( v2 − c2 ) uv ± c2 M2 − 1 = = (10.27) dx u2 − c2 u2 − c2 2 2 where M2 = u c+2 v is the local Mach number squared. 10.5.2 Mathematical Classification of the PDE The discriminant ∆ = c4 ( M2 − 1) dictates the mathematical nature of steady 2D compressible flow: • Subsonic Flow (M < 1): ∆ < 0 =⇒ No real characteristic directions exist. The PDE is Elliptic. • Sonic Flow (M = 1): ∆ = 0 =⇒ One unique real characteristic direction exists. The PDE is Parabolic. • Supersonic Flow (M > 1): ∆ > 0 =⇒ Two distinct real characteristic families C + and C − exist. The PDE is Hyperbolic. Fundamental Remark: In steady flow, the PDE classification depends implicitly on the solution itself (M). In contrast, the unsteady Euler equations are always hyperbolic in time regardless of the Mach number. 10.6 Compatibility Equations and the Prandtl-Meyer Function √ (M > 1), expressing velocity components in polar flow coordi- In steady 2D supersonic flow nates (U, θ )—where U = u2 + v2 is speed and θ = arctan(v/u) is streamline inclination—the compatibility equation along characteristics C ± simplifies to: √︁ dU dθ ± = ± M2 − 1 (10.28) U Using the isentropic relation for a Perfect Ideal Gas, dU U = 1 dM : 1+ γ − 1 2 M 2 M √ ± M2 − 1 dM dθ = ± (10.29) 1 + γ −1 M 2 M 2 Integrating yields: θ ± − θ P± = ± [ν( M ) − ν( MP )] (10.30) where ν( M ) is the Prandtl-Meyer Function. 61
Figure 14.2: Distribution of −C p over an airfoil chord x/c for increasing free-stream Mach numbers (M1 < M2 ), illustrating local exceedance of sonic pressure C ∗p (M = 1). Figure 14.3: Internal wave pattern of a local supersonic bubble enclosed by a sonic line (M = 1), showing Prandtl-Meyer expansion fans reflected as compression waves terminating in a normal shock wave. 1. Prandtl-Meyer Expansion Waves (C + ): As supersonic flow passes over the convex up- per surface contour, continuous expansion waves (C + characteristics) radiate outward toward the sonic boundary M = 1. 2. Reflection as Compression Waves (C − ): The sonic line (M = 1) acts as a constant- pressure isobar boundary. To satisfy pressure equilibrium at the sonic boundary, incident C + expansion waves reflect back toward the airfoil surface as C − compression waves. 3. Wave Coalescence and Terminating Normal Shock Wave: The reflected C − compres- sion waves travel downward toward the wall, decelerating the supersonic flow. Because compression waves naturally converge, they coalesce near the downstream end of the bubble to form a terminating Normal Shock Wave. The shock wave abruptly decelerates the flow from supersonic (M > 1) back to subsonic (M < 1). 14.2.2 Wave Drag Divergence As free-stream Mach number M∞ increases beyond Mcr , the supersonic bubble expands and the terminating normal shock strengthens and shifts downstream. Total pressure losses across the shock wave produce a sharp rise in drag known as the Drag Divergence Mach Number (Mdd ). 81
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