Compressible Flow

Concept

Compressible and Supersonic Flow — Thermodynamic Foundations and Wave Phenomena

Scope: rigorous treatment of compressible flow, from first principles of conservation laws and thermodynamics. Covers isentropic relations, shocks, expansions, and one-dimensional flow models (Fanno, Rayleigh) with entropy and exergy analysis.

1. Fundamentals of Compressibility

Compressible flow occurs when density changes are non-negligible — typically when Mach number M>0.3M > 0.3.

1.1 Mach Number

M=ua,a=γRT.M = \frac{u}{a}, \quad a = \sqrt{γRT}.

Flow regimes:

RegimeMach numberDescription
IncompressibleM < 0.3Density ≈ constant
Subsonic0.3 < M < 1Compressibility effects
SonicM = 1Choked flow
Supersonic1 < M < 5Shock and expansion waves
HypersonicM > 5Strong shocks, dissociation

2. Conservation Equations for 1D Flow

For steady, adiabatic, inviscid flow:

2.1 Continuity

m˙=ρuA=constant.\dot{m} = ρuA = \text{constant}.

2.2 Momentum

dp+ρudu=0.dp + ρu du = 0.

2.3 Energy

h+u22=h0=constant.h + \frac{u^2}{2} = h_0 = \text{constant}.

For perfect gases: h=cpTh = c_pT, h0=cpT0h_0 = c_pT_0.

3. Isentropic Flow Relations

For reversible adiabatic (isentropic) processes: pρ−γ=constant,Tρ1−γ=constant.pρ^{-γ} = \text{constant}, \quad Tρ^{1-γ} = \text{constant}.

From energy and ideal gas laws: T0=T(1+γ−12M2).T_0 = T(1 + \frac{γ-1}{2}M^2). p0p=(1+γ−12M2)γ/(γ−1).\frac{p_0}{p} = (1 + \frac{γ-1}{2}M^2)^{γ/(γ-1)}. ρ0ρ=(1+γ−12M2)1/(γ−1).\frac{ρ_0}{ρ} = (1 + \frac{γ-1}{2}M^2)^{1/(γ-1)}.

4. Area–Mach Number Relation

Differentiating continuity and momentum equations yields: dAA=(M2−1)duu.\frac{dA}{A} = (M^2 - 1) \frac{du}{u}.

Result:

  • For M < 1: velocity increases → area decreases (convergent nozzle).
  • For M > 1: velocity increases → area increases (divergent nozzle).

Combined isentropic form: AA∗=1M[2γ+1(1+γ−12M2)](γ+1)/2(γ−1).\frac{A}{A^*} = \frac{1}{M}\left[\frac{2}{γ+1}(1 + \frac{γ-1}{2}M^2)\right]^{(γ+1)/2(γ-1)}.

A∗A^* corresponds to sonic throat (M = 1).

5. Nozzle Flow and Choking

For an isentropic converging–diverging (C–D) nozzle:

  • Subsonic inlet → throat (M=1) → supersonic exit.
  • Flow chokes when M=1M = 1 at throat — mass flow cannot increase further with downstream pressure decrease.

5.1 Mass Flow Rate

m˙=A∗p0γRT0(2γ+1)(γ+1)/2(γ−1).\dot{m} = A^* p_0 \sqrt{\frac{γ}{RT_0}} \left( \frac{2}{γ+1} \right)^{(γ+1)/2(γ-1)}.

Critical pressure ratio: p∗p0=(2γ+1)γ/(γ−1).\frac{p^*}{p_0} = \left(\frac{2}{γ+1}\right)^{γ/(γ-1)}.

6. Normal Shock Waves

A normal shock is a thin, nearly discontinuous compression wave that connects two steady states obeying conservation of mass, momentum, and energy.

6.1 Governing Relations

ρ1u1=ρ2u2,ρ_1u_1 = ρ_2u_2, p1+ρ1u12=p2+ρ2u22,p_1 + ρ_1u_1^2 = p_2 + ρ_2u_2^2, h1+u122=h2+u222.h_1 + \frac{u_1^2}{2} = h_2 + \frac{u_2^2}{2}.

For ideal gases: p2p1=1+2γγ+1(M12−1),\frac{p_2}{p_1} = 1 + \frac{2γ}{γ+1}(M_1^2 - 1), ρ2ρ1=(γ+1)M12(γ−1)M12+2,\frac{ρ_2}{ρ_1} = \frac{(γ+1)M_1^2}{(γ-1)M_1^2 + 2}, T2T1=p2/p1ρ2/ρ1.\frac{T_2}{T_1} = \frac{p_2/p_1}{ρ_2/ρ_1}.

6.2 Downstream Mach Number

M22=1+(γ−1)2M12γM12−(γ−1)2.M_2^2 = \frac{1 + \frac{(γ-1)}{2}M_1^2}{γM_1^2 - \frac{(γ-1)}{2}}.

6.3 Entropy Increase

Δs=cpln⁡T2T1−Rln⁡p2p1>0.Δs = c_p \ln\frac{T_2}{T_1} - R \ln\frac{p_2}{p_1} > 0.

7. Oblique Shocks and Shock Polars

For supersonic flow deflected by a wedge or compression corner:

tan⁡θ=2cot⁡βM12sin⁡2β−1M12(γ+cos⁡2β)+2.\tan θ = 2 \cot β \frac{M_1^2 \sin^2 β - 1}{M_1^2(γ + \cos 2β) + 2}.

Where:

  • θ: flow deflection angle
  • β: shock angle

Weak and strong shock solutions exist; the weak branch is physically stable for attached flow.

7.1 Shock Polar

Graphical representation of pressure ratio vs. flow deflection, used for shock reflection and interaction analysis.

8. Expansion Waves — Prandtl–Meyer Flow

Isentropic expansion turning flow around a convex corner produces continuous rarefaction waves.

Prandtl–Meyer function: ν(M)=γ+1γ−1tan⁡−1[γ−1γ+1(M2−1)]−tan⁡−1M2−1.ν(M) = \sqrt{\frac{γ+1}{γ-1}} \tan^{-1}\left[\sqrt{\frac{γ-1}{γ+1}(M^2 - 1)}\right] - \tan^{-1}\sqrt{M^2 - 1}.

Deflection angle: θ=ν(M2)−ν(M1).θ = ν(M_2) - ν(M_1).

Static pressure ratio: p2p1=(1+γ−12M121+γ−12M22)γ/(γ−1).\frac{p_2}{p_1} = \left(\frac{1 + \frac{γ-1}{2}M_1^2}{1 + \frac{γ-1}{2}M_2^2}\right)^{γ/(γ-1)}.

9. Fanno Flow (Adiabatic, Frictional)

Flow in constant area duct with wall friction and no heat transfer.

9.1 Governing Equations

4fLD=1−M2γM2+γ+12γln⁡(γ+1)M22+(γ−1)M2.\frac{4fL}{D} = \frac{1 - M^2}{γM^2} + \frac{γ+1}{2γ} \ln\frac{(γ+1)M^2}{2 + (γ-1)M^2}.

9.2 Properties Along Fanno Line

  • As L increases, M → 1.
  • Entropy increases monotonically.

9.3 Maximum Flow Condition

At M=1, choked frictional flow — analogous to sonic choking in nozzles.

10. Rayleigh Flow (Constant Area with Heat Transfer)

For heat addition/removal in constant area ducts (e.g., combustion, heat exchangers).

10.1 Governing Relations

T0T=(1+γ−12M2),\frac{T_0}{T} = (1 + \frac{γ-1}{2}M^2), p0p=(1+γ−12M2)γ/(γ−1).\frac{p_0}{p} = (1 + \frac{γ-1}{2}M^2)^{γ/(γ-1)}.

Dimensionless form: TT∗=(γ+1)M2(1+γM2)2,pp∗=1Mγ+12(1+γM2).\frac{T}{T^*} = \frac{(γ+1)M^2}{(1 + γM^2)^2}, \quad \frac{p}{p^*} = \frac{1}{M}\sqrt{\frac{γ+1}{2(1 + γM^2)}}.

10.2 Heat Addition Effects

  • Subsonic flow: heat addition increases M (toward choking).
  • Supersonic flow: heat addition decreases M (toward M=1).

11. Entropy and Exergy in Compressible Flow

Entropy change per unit mass: ds=cpdln⁡T−Rdln⁡p.ds = c_p d\ln T - R d\ln p.

11.1 Across Shock

Δs=cpln⁡T2T1−Rln⁡p2p1>0.Δs = c_p \ln\frac{T_2}{T_1} - R \ln\frac{p_2}{p_1} > 0. Irreversibility leads to exergy destruction: E˙D=m˙T0Δs.\dot{E}_D = \dot{m} T_0 Δs.

11.2 In Isentropic Flow

No exergy loss; total pressure p0p_0 constant.

In shocks and Fanno/Rayleigh flows, p0p_0 decreases due to viscous or heat transfer irreversibility.

12. Combined Shock–Expansion and Nozzle Applications

Supersonic nozzles and diffusers often feature alternating shocks and expansions:

  • Overexpanded nozzle → internal shock → exit pressure rise.
  • Underexpanded → expansion fan at nozzle exit.

Shock-expansion analysis used in design of supersonic inlets and rocket nozzles.

13. Example Calculations

CaseEquationNotes
Mach from pressure ratioM=2γ−1[(p0/p)(γ−1)/γ−1]M = \sqrt{\frac{2}{γ-1}\left[(p_0/p)^{(γ-1)/γ} - 1\right]}Inverse isentropic relation
Normal shock p-ratiop2/p1=1+2γ/(γ+1)(M12−1)p_2/p_1 = 1 + 2γ/(γ+1)(M_1^2-1)Normal shock relation
Choked flow mass fluxG∗=p0γRT0(2/(γ+1))(γ+1)/2(γ−1)G^* = p_0\sqrt{\frac{γ}{RT_0}}(2/(γ+1))^{(γ+1)/2(γ-1)}Maximum flow rate

14. Summary of Key Relations

PhenomenonEquationRemarks
Isentropic T–M relationT0/T=1+(γ−1)/2M2T_0/T = 1 + (γ-1)/2 M^2Energy conservation
Area–Mach relationA/A∗=(1/M)[(2/(γ+1))(1+(γ−1)/2M2)](γ+1)/2(γ−1)A/A^* = (1/M)[(2/(γ+1))(1 + (γ-1)/2 M^2)]^{(γ+1)/2(γ-1)}Flow acceleration
Normal shock entropyΔs=cpln⁡(T2/T1)−Rln⁡(p2/p1)Δs = c_p\ln(T_2/T_1) - R\ln(p_2/p_1)Irreversible jump
Fanno flow friction4fL/D=(1−M2)/(γM2)+(γ+1)/2γln⁡[(γ+1)M2/(2+(γ−1)M2)]4fL/D = (1 - M^2)/(γM^2) + (γ+1)/2γ \ln[(γ+1)M^2/(2 + (γ-1)M^2)]Adiabatic, frictional
Rayleigh flow heatT/T∗=(γ+1)M2/(1+γM2)2T/T^* = (γ+1)M^2/(1+γM^2)^2Heat transfer, choking
  • Thermodynamics/02_First_and_Second_Laws.md — foundation for energy and entropy balances.
  • Thermodynamics/10_NonEquilibrium_Thermodynamics.md — entropy generation and transport coupling.
  • Fluid_Dynamics/08_TwoPhase_Heat_Transfer_and_Critical_Flow.md — compressibility in flashing and critical flow.
  • Aero_Thermodynamics/HighSpeed_Design.md — applications in propulsion and aerospace systems.