Critical Mach number — onset of local supersonic flow

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WHY do we care?

• WHY it matters: Above $M_{cr}$, a pocket of supersonic flow appears on the wing. This pocket usually ends in a shock wave → sudden rise in drag (drag divergence) → the infamous "sound barrier" buffeting at $M_\infty < 1$.
• WHAT it is NOT: It is not $M_\infty = 1$. The plane is still subsonic; only a local patch is sonic.
• The 80/20: Master one idea — "low surface pressure ⇒ high local speed ⇒ first point to go sonic" — and the whole derivation falls out.

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HOW: derive $M_{cr}$ from first principles

We need two physics facts and one geometry fact.

Fact 1 — Isentropic pressure–Mach relation

For steady, adiabatic, frictionless (isentropic) compressible flow, the stagnation pressure $p_0$ is constant along a streamline. From energy + the perfect-gas relations: $\frac{p_0}{p} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma-1}}$

Why this step? Energy conservation says total temperature is constant: $T_0/T = 1+\frac{\gamma-1}{2}M^2$. Isentropy links $p$ and $T$ by $p/T^{\gamma/(\gamma-1)}=\text{const}$. Combine → the formula above.

Fact 2 — Definition of pressure coefficient

$C_p \equiv \frac{p - p_\infty}{\tfrac12 \rho_\infty V_\infty^2}$ Using $\tfrac12\rho_\infty V_\infty^2 = \tfrac{\gamma}{2}p_\infty M_\infty^2$ (since $a^2=\gamma p/\rho$ and $V=Ma$): $C_p = \frac{2}{\gamma M_\infty^2}\left(\frac{p}{p_\infty}-1\right)$

Why this step? It rewrites the dynamic pressure purely in terms of $M_\infty$ — exactly what we want.

Putting it together — the critical pressure coefficient $C_{p,cr}$

Same stagnation pressure on the streamline gives:

= \left(\frac{1+\frac{\gamma-1}{2}M_\infty^2}{1+\frac{\gamma-1}{2}M_{local}^2}\right)^{\frac{\gamma}{\gamma-1}}$$ Set the **local** point to sonic, $M_{local}=1$, and the free stream to $M_\infty=M_{cr}$: $$\boxed{\,C_{p,cr} = \frac{2}{\gamma M_{cr}^2}\left[\left(\frac{1+\frac{\gamma-1}{2}M_{cr}^2}{1+\frac{\gamma-1}{2}}\right)^{\frac{\gamma}{\gamma-1}} - 1\right]\,}$$ > [!formula] What this curve is > $C_{p,cr}$ vs $M_{cr}$ is a **universal** curve (depends only on $\gamma$). It says: *"to make local flow sonic at free-stream Mach $M_{cr}$, the minimum surface pressure must dip to exactly this $C_p$."* It is **independent of the airfoil**. ### The geometry fact — closing the loop with the airfoil The airfoil contributes its own **minimum-pressure** curve: how the lowest $C_p$ on the surface grows as $M_\infty$ rises. A simple model is the **Prandtl–Glauert** compressibility correction: $$C_{p} = \frac{C_{p,0}}{\sqrt{1-M_\infty^2}}$$ where $C_{p,0}$ is the *incompressible* minimum pressure coefficient (a fixed negative number from the wing's shape). > [!intuition] The intersection IS the answer > Plot two curves vs $M_\infty$: > 1. $C_{p,cr}(M_\infty)$ — the **gas-dynamics** requirement (universal). > 2. $C_{p,\min}(M_\infty)=C_{p,0}/\sqrt{1-M_\infty^2}$ — the **airfoil's** actual lowest pressure. > > Where they **cross**, the wing's minimum pressure point has *just* reached sonic. That crossing $M_\infty$ **is** $M_{cr}$. ![[3.1.24-Critical-Mach-number-—-onset-of-local-supersonic-flow.png]] --- ## Worked examples > [!example] Example 1 — Compute $C_{p,cr}$ for a given $M_{cr}$ > Find the critical pressure coefficient if $M_{cr}=0.70$, $\gamma=1.4$. > > **Step 1.** Inner ratio: > $\dfrac{1+0.2(0.49)}{1+0.2} = \dfrac{1.098}{1.2}=0.915$. *Why?* This is $\frac{p_\infty/p_0}{p^*/p_0}$ at sonic local point. > **Step 2.** Raise to $\gamma/(\gamma-1)=3.5$: $0.915^{3.5}=0.728$. *Why?* Isentropic $p$ ratio. > **Step 3.** $C_{p,cr}=\dfrac{2}{1.4(0.49)}(0.728-1)=\dfrac{2}{0.686}(-0.272)=-0.793$. > > **Answer:** $C_{p,cr}\approx -0.79$. A wing whose minimum $C_p$ reaches $-0.79$ at $M_\infty=0.70$ is at its critical Mach. > [!example] Example 2 — Find $M_{cr}$ for a real airfoil > An airfoil has incompressible minimum $C_{p,0}=-0.55$. Estimate $M_{cr}$. > > **Step 1.** Airfoil curve: $C_{p,\min}=\dfrac{-0.55}{\sqrt{1-M_\infty^2}}$. *Why?* Prandtl–Glauert scaling of suction peak. > **Step 2.** Solve $C_{p,\min}(M_\infty)=C_{p,cr}(M_\infty)$ numerically. > - Try $M=0.70$: airfoil $=-0.55/\sqrt{0.51}=-0.770$; required $C_{p,cr}=-0.793$. Airfoil not yet low enough. > - Try $M=0.71$: airfoil $=-0.55/\sqrt{0.496}=-0.781$; $C_{p,cr}(0.71)\approx-0.762$. Now airfoil is *below* the curve → already past sonic. > **Step 3.** Crossing lies $\approx M_{cr}\approx 0.705$. > > **Answer:** $M_{cr}\approx 0.70$. *Why this matters:* a **thinner** airfoil → less suction → smaller $|C_{p,0}|$ → **higher** $M_{cr}$. That's why fast jets have thin wings. > [!example] Example 3 — Qualitative forecast > **Forecast:** Sweep the wing back. Does $M_{cr}$ rise or fall? > **Verify:** Sweep reduces the velocity component *normal* to the leading edge that "sees" the airfoil, so the effective $M_\infty$ is $M_\infty\cos\Lambda$. The wing reaches sonic at a *higher* true $M_\infty$. → **$M_{cr}$ rises.** ✅ (Reason swept wings beat the sound barrier.) --- ## Common mistakes > [!mistake] "$M_{cr}=1$ — it's when the plane hits the speed of sound." > **Why it feels right:** "Sonic" naturally suggests the whole craft at Mach 1. > **The fix:** Air *accelerates* over curvature, so the **local** flow outruns the free stream. The plane is sonic *somewhere on its skin* while still flying at $M_\infty\approx 0.7$–$0.85$. $M_{cr}<1$ always. > [!mistake] "Lower (more negative) $C_{p,cr}$ ⇒ harder to reach sonic." > **Why it feels right:** Big numbers seem "more extreme." > **The fix:** $C_{p,cr}$ becomes *less* negative as $M_{cr}$ rises. The airfoil must merely *match* the universal curve. The relevant comparison is which curve the suction peak meets — read the **intersection**, not the magnitude alone. > [!mistake] Using Prandtl–Glauert right up to $M_{cr}$ as if exact. > **Why it feels right:** It's the simple closed form everyone quotes. > **The fix:** P–G is a *linearized* estimate; near $M_{cr}$ it under-predicts suction. Better corrections (Karman–Tsien, Laitone) give slightly *lower* $M_{cr}$. Use P–G for intuition, not final design. --- > [!recall]- Feynman: explain to a 12-year-old > Imagine squeezing your thumb over a hose — the water **shoots out faster** through the narrow gap. Air does the same over the **curved top of a wing**: it speeds up. If the plane flies fast enough, the air at the wing's fastest spot reaches the speed of sound *even though the whole plane is still slower than sound*. The exact plane-speed where this first happens has a name: the **critical Mach number**. After that point, a tiny "boom zone" forms on the wing and the plane suddenly feels lots of extra drag — that's the "sound barrier" that early pilots fought. > [!mnemonic] Remember it > **"CRITICAL = Curvature Rushes Inner-flow To 1 — Causes A Local boom."** > And: *thin & swept ⇒ high $M_{cr}$* → **"Skinny swept wings sneak past sonic."** --- ## #flashcards/physics What is the critical Mach number $M_{cr}$? ::: The free-stream Mach number at which the flow somewhere on the body first reaches local Mach 1. Is $M_{cr}$ less than, equal to, or greater than 1? ::: Less than 1 — the plane is still subsonic while only a local patch is sonic. Why does flow accelerate over a wing's upper surface? ::: Curvature constricts the streamtube; by continuity the flow speeds up, dropping local pressure (suction peak). State the isentropic stagnation-to-static pressure ratio. ::: $p_0/p = (1+\frac{\gamma-1}{2}M^2)^{\gamma/(\gamma-1)}$. Give $C_p$ in terms of $p/p_\infty$ and $M_\infty$. ::: $C_p = \frac{2}{\gamma M_\infty^2}(p/p_\infty - 1)$. Write the critical pressure coefficient formula. ::: $C_{p,cr}=\frac{2}{\gamma M_{cr}^2}\big[\big(\frac{1+\frac{\gamma-1}{2}M_{cr}^2}{1+\frac{\gamma-1}{2}}\big)^{\gamma/(\gamma-1)}-1\big]$. Is the $C_{p,cr}$ curve airfoil-dependent? ::: No — it depends only on $\gamma$; it is a universal gas-dynamics curve. How do you find $M_{cr}$ graphically? ::: Intersect the universal $C_{p,cr}(M_\infty)$ curve with the airfoil's minimum-pressure curve $C_{p,0}/\sqrt{1-M_\infty^2}$. Effect of thinner airfoil on $M_{cr}$? ::: Less suction (smaller $|C_{p,0}|$) ⇒ higher $M_{cr}$. Effect of wing sweep on $M_{cr}$? ::: Raises it; only $M_\infty\cos\Lambda$ is "seen" by the airfoil, so sonic onset needs a higher true $M_\infty$. What happens just above $M_{cr}$? ::: A local supersonic pocket forms, usually terminated by a shock → drag divergence (sound barrier). For $M_{cr}=0.70$, $\gamma=1.4$, what is $C_{p,cr}$? ::: About $-0.79$. --- ## Connections - [[Isentropic Flow Relations]] — supplies $p_0/p$ and $T_0/T$. - [[Pressure Coefficient Cp]] — definition used to build $C_{p,cr}$. - [[Prandtl–Glauert Compressibility Correction]] — the airfoil's minimum-pressure curve. - [[Drag Divergence Mach Number]] — what happens *just above* $M_{cr}$. - [[Shock Waves & Normal Shocks]] — terminate the local supersonic pocket. - [[Swept Wings & Transonic Design]] — engineering response to $M_{cr}$. - [[Area Rule (Transonic)]] — minimizing wave drag past $M_{cr}$. ## 🖼️ Concept Map ```mermaid flowchart TD A[Subsonic flight] -->|air speeds over curved wing| B[Local flow accelerates] B -->|first reaches M_local=1| C[Critical Mach M_cr] C -->|above M_cr| D[Supersonic pocket on wing] D -->|ends in| E[Shock wave] E -->|causes| F[Drag divergence] F -->|felt as| G[Sound barrier buffeting] H[Isentropic p0-M relation] -->|combined with| J[Pressure coefficient Cp] J -->|set M_local=1| K[Critical Cp,cr formula] C -->|defines condition for| K K -->|depends only on gamma| L[Universal curve] K -->|intersect airfoil min-pressure| C ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, idea bahut simple hai. Jab plane curved wing ke upar se air ko guzaarta hai, toh us curvature ke kaaran air **tez** ho jaati hai — bilkul jaise hose ke mooh ko thumb se dabane par paani fast nikalta hai. Toh plane bhale hi subsonic ho (yaani $M_\infty < 1$), wing ke sabse fast point par local speed badhti rehti hai. Jis free-stream Mach number par yeh local flow **pehli baar** exactly sound ki speed ($M_{local}=1$) tak pahunchti hai, usko **critical Mach number** $M_{cr}$ kehte hain. Yaad rakho: $M_{cr}$ hamesha 1 se kam hota hai — sirf ek chhota patch sonic hota hai, poora plane nahi. > > Iska physics do cheezon se banta hai. Pehla — isentropic relation, jisme stagnation pressure $p_0$ ek streamline par constant rehta hai, isse milta hai $p_0/p = (1+\frac{\gamma-1}{2}M^2)^{\gamma/(\gamma-1)}$. Doosra — pressure coefficient ki definition $C_p$. Inko jod kar ek **universal curve** banti hai, $C_{p,cr}$ vs $M_{cr}$, jo sirf $\gamma$ par depend karti hai, kisi bhi airfoil par nahi. Phir airfoil apni khud ki minimum-pressure curve laata hai (Prandtl–Glauert: $C_{p,0}/\sqrt{1-M_\infty^2}$). Jahan dono curve **cross** karti hain, wahi $M_{cr}$ hai. > > Engineering ka asli mazaa yahan hai: **patli wing** ka suction kam hota hai (chhota $|C_{p,0}|$), isliye uska $M_{cr}$ zyada hota hai — yani woh sonic barrier ke paas tak fast ja sakti hai bina problem ke. Aur **swept wing** sirf $M_\infty\cos\Lambda$ "dekhti" hai, isliye uska bhi $M_{cr}$ badh jaata hai. Isi liye fighter jets ki wings patli aur peeche jhuki hui hoti hain. $M_{cr}$ ke thoda upar jaate hi wing par supersonic pocket banta hai, shock wave aata hai, aur drag suddenly badh jaata hai — yahi "sound barrier" wali dikkat thi. Toh $M_{cr}$ samajhna matlab transonic design ka 80% samajhna. ![[audio/3.1.24-Critical-Mach-number-—-onset-of-local-supersonic-flow.mp3]]