Formula & Calculator
Normal Shock Downstream Mach Number
Gives the (always subsonic) Mach number downstream of a normal shock as a function of the upstream Mach number.
Interpretation
Normal shock downstream Mach number: M₂² = (1 + ((γ−1)/2)·M₁²) / (γ·M₁² − (γ−1)/2). It gives the Mach number after a normal shock. Example: M₁=2, γ=1.4 → M₂² = (1+0.2×4)/(2.8−0.2) = 1.8/2.6 = 0.692 → M₂≈0.832.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| M2 | Downstream Mach number | |
| γ | Ratio of specific heats | |
| M1 | Upstream Mach number |
What it means
This relation gives the Mach number downstream of a normal shock, which is always subsonic. It is derived from the normal shock relations. The downstream Mach number depends only on the upstream Mach number and the specific heat ratio. For M₁ approaching infinity, M₂ tends to √((γ−1)/(2γ)), which for γ=1.4 is about 0.378. This relation is used in supersonic diffuser design and in evaluating shock losses. It helps in determining the recovery pressure and the flow conditions after a shock. Understanding this formula is essential for compressible flow analysis and for designing efficient intake systems.
Worked example
Normal Shock Downstream Mach – Two Examples
Real‑World| Parameter | Value |
|---|---|
| γ | 1.4 |
| M₁ | 2.0 |
| Parameter | Value |
|---|---|
| M₁ | 3.0 |
Common mistakes
- Normal shock downstream Mach number: M₂² = (1 + ((γ−1)/2)·M₁²) / (γ·M₁² − (γ−1)/2).
- M₁ > 1 ⇒ M₂ < 1 (shock decelerates flow).
- γ: Specific heat ratio.
- Valid for normal shock.
- Check: for M₁→∞, M₂→√((γ−1)/(2γ)).
Applications
The normal shock downstream Mach number, M₂² = (1 + ((γ−1)/2)M₁²)/(γM₁² − (γ−1)/2), gives the Mach number after a normal shock. This is crucial for designing supersonic inlets, as the downstream flow must be subsonic for the compressor. Engineers use it to ensure that the shock is positioned correctly and that the flow entering the engine is within acceptable limits. The shock also causes total pressure loss, which affects engine performance. By understanding the downstream Mach number, aerospace engineers can predict shock losses and design efficient supersonic propulsion systems.
- Supersonic inlet shock positioning and stability
- Engine performance impact of shock losses
- Design of shock‑trapping and bleed systems
- Supersonic combustion (scramjet) inlet design
- Supersonic vehicle aerodynamics (e.g., fighter aircraft)
Frequently Asked Questions
It gives the (always subsonic) Mach number downstream of a normal shock as a function of the upstream Mach number. It is used to compute post‑shock conditions.
M2 = downstream Mach number
M1 = upstream Mach number
γ = specific heat ratio
A normal shock always decelerates a supersonic flow to subsonic speeds. This is a fundamental property of shock waves.
For M1 = 2, M2 ≈ 0.577. For M1 = 5, M2 ≈ 0.415. As M1 → ∞, M2 → √((γ−1)/(2γ)) ≈ 0.378 for γ=1.4.
- Expecting M2 > 1 downstream of a normal shock; the flow is always subsonic.
- Using the formula for an oblique shock without resolving the normal component.
- Using the wrong value of γ.
M1 = 2.0, γ = 1.4. M2² = (1 + 0.2×4) / (1.4×4 − 0.2) = (1 + 0.8) / (5.6 − 0.2) = 1.8 / 5.4 = 0.3333. M2 = √0.3333 ≈ 0.577.
The downstream static pressure, temperature, and density are higher; the stagnation pressure is lower. The Mach number indicates the flow speed relative to the local speed of sound.
It determines the subsonic diffuser conditions after the shock, affecting the compressor inlet Mach number and overall pressure recovery.
They are linked; for a given M1, both M2 and p2/p1 are fixed by the shock relations.
M2 decreases asymptotically to the limit √((γ−1)/(2γ)) as M1 → ∞.