Stagnation and sonic reference conditions

Stagnation (total) and sonic (starred) reference states, which of them survive shocks, friction and heat addition, and the characteristic Mach number M*.

Drafted with Aria, reviewed by the AiCanCode.org team. Spotted an error? Use Give Feedback at the bottom of the page.

Why it matters

Stagnation (total) quantities are what probes and reservoirs actually measure, and they are the bookkeeping variables of high-speed flow: T₀ tracks energy, p₀ tracks losses. Sonic reference values (T*, p*, a*, A*) let one table describe every isentropic duct, shock and friction or heat-addition problem.

Key ideas

Stagnation state. Imagine slowing the flow at a point to zero velocity. If this is done adiabatically, the energy equation gives the stagnation enthalpy h₀ = h + V²/2 and, for a perfect gas, the stagnation temperature T₀ = T + V²/(2·cp). If it is also done reversibly (isentropically), you get the stagnation pressure p₀ and density ρ₀. The stagnation state is a property of the flow at that point, whether or not the flow is actually brought to rest there.

  • T₀ is constant in any adiabatic flow with no shaft work, including across shocks and in flow with friction (Fanno).
  • p₀ is constant only in isentropic flow. Any irreversibility — shock, friction, mixing — lowers it. The entropy rise is Δs = −R·ln(p₀₂/p₀₁) when T₀ is constant.
  • T₀ changes when heat is added (Rayleigh flow) or work is done (compressor, turbine).
  • A stationary thermometer in a fast stream reads close to T₀, not T (in practice the recovery temperature, slightly below T₀).

Static vs dynamic pressure. In compressible flow the stagnation pressure is not p + ½ρV². Expanding the isentropic relation gives p₀ − p = ½ρV²·(1 + M²/4 + M⁴/40 + …). The bracket is the compressibility correction; at M = 0.3 it adds about 2%.

Sonic (starred) reference state. The flow at a point could also be imagined to be accelerated or decelerated isentropically to M = 1. Its properties there are T*, p*, ρ*, a* and the area A* that would pass the same mass flow. For a given T₀ and p₀ they are fixed numbers: for air T*/T₀ = 0.8333, p*/p₀ = 0.5283, ρ*/ρ₀ = 0.6339.

Characteristic Mach number. M* = V/a* uses the fixed reference a* instead of the local a. Its useful properties: M* < 1 when M < 1, M* = 1 at M = 1, M* > 1 when M > 1, and M* stays finite as M → ∞ (limit √((γ+1)/(γ−1)) = 2.449 for air). It appears in Prandtl's normal-shock relation M₁*·M₂* = 1.

Maximum speed. If the gas expanded to T = 0, all enthalpy would become kinetic energy: V_max = √(2·cp·T₀).

Formulas

T₀ = T + V²/(2·cp) — adiabatic; cp = γR/(γ−1) = 1004.5 J/(kg·K) for air.

T₀/T = 1 + (γ−1)/2 · M²

p₀/p = (T₀/T)^(γ/(γ−1)), ρ₀/ρ = (T₀/T)^(1/(γ−1)) — isentropic.

T*/T₀ = 2/(γ+1); p*/p₀ = (2/(γ+1))^(γ/(γ−1)); ρ*/ρ₀ = (2/(γ+1))^(1/(γ−1)).

a* = √(2γ·R·T₀/(γ+1)) — speed of sound at the sonic state (m/s).

M*² = (γ+1)·M² / (2 + (γ−1)·M²) — characteristic Mach number.

V_max = √(2·cp·T₀) (m/s).

s₂ − s₁ = −R·ln(p₀₂/p₀₁) — entropy change between two states with equal T₀ (J/(kg·K)).

Symbols: T, p, ρ static values; subscript 0 stagnation; superscript * sonic reference; V speed (m/s); cp specific heat at constant pressure (J/(kg·K)); R gas constant (J/(kg·K)); γ ratio of specific heats.

Worked examples

Example 1 (standard). Air at T = 300 K, p = 101.325 kPa moves at M = 2. Find T₀ and p₀. γ = 1.4.

  1. T₀/T = 1 + 0.2 × 2² = 1.8, so T₀ = 300 × 1.8 = 540 K.
  2. p₀/p = 1.8^3.5 = 7.824.
  3. p₀ = 101.325 × 7.824 = 792.8 kPa.

Answer: T₀ = 540 K, p₀ ≈ 793 kPa.

Example 2 (GATE level). An aircraft flies at M = 2.2 at 11 km (T = 216.65 K, p = 22.63 kPa). Find T₀, p₀, the sonic reference speed a* and the characteristic Mach number M*. Air, γ = 1.4, R = 287 J/(kg·K).

  1. T₀ = 216.65 × (1 + 0.2 × 4.84) = 216.65 × 1.968 = 426.4 K.
  2. p₀ = 22.63 × 1.968^3.5 = 22.63 × 10.69 = 242.0 kPa.
  3. a* = √(2 × 1.4 × 287 × 426.4 / 2.4) = √142 773 = 377.8 m/s.
  4. Flight speed V = M·a = 2.2 × √(1.4 × 287 × 216.65) = 2.2 × 295.0 = 649.1 m/s.
  5. M* = V/a* = 649.1/377.8 = 1.718. Check with the formula: M*² = 2.4 × 4.84 / (2 + 0.4 × 4.84) = 11.616/3.936 = 2.951, M* = 1.718.

Answer: T₀ ≈ 426 K, p₀ ≈ 242 kPa, a ≈ 378 m/s, M ≈ 1.72**. (The skin and any probe see temperatures near 426 K even though the air is at −56 °C.)

Common mistakes

  • Assuming p₀ is conserved across a shock or with friction; only T₀ is (for adiabatic flow).
  • Using p₀ = p + ½ρV² at high Mach number — it underestimates p₀.
  • Confusing M* (based on a*) with M (based on local a), or with the critical Mach number of an airfoil.
  • Using T₀ in place of T when computing the local speed of sound.
  • Forgetting that T*, p* depend only on T₀, p₀ — they do not change along an isentropic duct, but p* drops across a shock.

For GATE AE

Typical questions: T₀ and p₀ from M and static values (or the reverse), the fraction p*/p₀ = 0.528 for choking, T₀ from speed using cp, entropy change from a stagnation-pressure loss, and statements about which quantities stay constant across shocks, friction and heat addition. Memorise the air values 0.8333, 0.5283 and 0.6339.

Quick check

  1. Air at 300 K moves at 250 m/s. Find T₀ (cp = 1005 J/(kg·K)).
  2. What is p*/p₀ for air?
  3. Across a normal shock, which of T₀ and p₀ stays constant?
  4. What is the limiting value of M* as M → ∞ for γ = 1.4?
  5. If p₀ drops by 10% at constant T₀, what is the entropy rise for air?

Answers: 1. 331.1 K; 2. 0.528; 3. T₀; 4. 2.449; 5. −287 × ln 0.9 = 30.2 J/(kg·K).

Interview questions

All Compressible Aerodynamics interview questions

Try answering each one aloud before you open it.

  1. 1.What is the definition of stagnation conditions in compressible aerodynamics?Concept

    Stagnation conditions refer to the state of a fluid when it is brought to rest isentropically. This means that the fluid's kinetic energy is converted into internal energy without any losses due to friction or heat transfer. The stagnation pressure, temperature, and density are higher than the static conditions and are used as reference points in compressible flow analysis.

  2. 2.Explain the concept of sonic reference conditions.Concept

    The sonic (starred) state is the state the flow would reach if it were accelerated or decelerated isentropically to M = 1. Its temperature, pressure, density, speed of sound and area (T*, p*, ρ*, a*, A*) are fixed by the stagnation conditions: for air T*/T₀ = 0.833, p*/p₀ = 0.528 and ρ*/ρ₀ = 0.634. These are the reference values in isentropic, Fanno and Rayleigh tables, and A* is the throat area that would just choke the given mass flow.

  3. 3.How are stagnation temperature and static temperature related in a compressible flow?Concept

    In compressible flow, the stagnation temperature is always greater than or equal to the static temperature. The relationship is given by the equation T₀ = T + (V² / (2·cₚ)), where T₀ is the stagnation temperature, T is the static temperature, V is the velocity of the fluid, and cₚ is the specific heat at constant pressure. This equation shows that as the velocity increases, the difference between stagnation and static temperature increases.

  4. 4.Why are stagnation conditions important in the design of jet engines?Application

    Engine components are characterised by stagnation quantities because these include the kinetic energy of the stream: the work of a compressor or turbine per unit mass equals cp times the change in T₀, and the losses in intakes, combustors and nozzles show up as drops in p₀. Intake performance is quoted as stagnation-pressure recovery, compressors by p₀ ratio, and the nozzle exit velocity depends on the available p₀ and T₀. Working in static quantities would mix up energy changes with mere velocity changes.

  5. 5.What happens to the stagnation pressure if a shock wave passes through a flow?Application

    When a shock wave passes through a flow, the stagnation pressure decreases. This is because shock waves are non-isentropic processes, meaning they involve entropy increase and energy losses. As a result, the total pressure, or stagnation pressure, is reduced across the shock wave.

  6. 6.How does the Mach number affect the relationship between stagnation and static conditions?Application

    The Mach number significantly affects the relationship between stagnation and static conditions. As the Mach number increases, the difference between stagnation and static pressure, temperature, and density becomes more pronounced. This is because higher Mach numbers indicate higher kinetic energy, which translates into greater differences when the flow is brought to rest isentropically.

  7. 7.Calculate the stagnation temperature for an airflow with a static temperature of 300 K and a velocity of 250 m/s. Assume cₚ = 1005 J/(kg·K).Numerical

    To calculate the stagnation temperature (T₀), use the formula: T₀ = T + (V² / (2·cₚ)). Here, T = 300 K, V = 250 m/s, and cₚ = 1005 J/(kg·K). T₀ = 300 + (250² / (2·1005)) = 300 + (62500 / 2010) = 300 + 31.09 = 331.09 K.

  8. 8.If the static pressure of a flow is 101325 Pa and the Mach number is 0.8, calculate the stagnation pressure. Assume γ = 1.4.Numerical

    Use p₀ = p·(1 + (γ−1)/2·M²)^(γ/(γ−1)). With M = 0.8: 1 + 0.2 × 0.64 = 1.128, and 1.128^3.5 = 1.524. So p₀ = 101 325 × 1.524 ≈ 154.5 kPa. The incompressible estimate p + ½ρV² would give about 147 kPa, showing why the compressible relation is needed at this speed.

  9. 9.Explain why sonic conditions are used as reference points in supersonic flow analysis.Application

    For a given T₀ and p₀ the sonic values are constants, so every other state can be written as a ratio to them as a function of Mach number alone — that is how one table covers all isentropic ducts. The sonic area A* ties area to Mach number through A/A*, and a* defines the characteristic Mach number M* = V/a*, which stays finite as M → ∞ and gives the neat normal-shock result M₁*·M₂* = 1. Fanno and Rayleigh flows likewise use the sonic state as the end point that friction or heating drives the flow towards.

Finished this topic? Mark it so your progress, study plan and readiness keep up.

Stuck on something here?

PreviousOne-dimensional isentropic flow relationsNext Normal shock relations