Stagnation (total) quantities — T₀, P₀, ρ₀ — derivations

1. What is "static" vs "total"?

WHAT we want: formulas linking $T_0, P_0, \rho_0$ to $T, P, \rho$ and $M$.

2. Deriving $T_0$ — from energy conservation

HOW — first principles. For an adiabatic, steady flow with no work and no elevation change, the enthalpy $h$ plus kinetic energy per unit mass is conserved:

$h + \frac{V^2}{2} = \text{const} = h_0$

Why this step? This is the First Law for an open system (control volume): $\dot Q - \dot W = \dot m\left[(h_2-h_1)+\tfrac12(V_2^2-V_1^2)\right]$. With $\dot Q=\dot W=0$, the bracket is zero, so $h+\tfrac12 V^2$ stays constant. At the stagnation point $V=0$, so the constant equals $h_0$ — the stagnation enthalpy.

For a calorically perfect gas, $h = c_p T$. Substitute:

$c_p T + \frac{V^2}{2} = c_p T_0 \quad\Rightarrow\quad T_0 = T + \frac{V^2}{2c_p}$

Why this step? $h=c_pT$ assumes constant specific heat — valid for ideal gases over moderate temperature ranges.

Now express in Mach number. Use $a^2 = \gamma R T$ (speed of sound) and the relation $c_p = \dfrac{\gamma R}{\gamma-1}$:

$\frac{T_0}{T} = 1 + \frac{V^2}{2c_p T} = 1 + \frac{V^2}{2 \cdot \frac{\gamma R}{\gamma-1}\cdot T} = 1 + \frac{(\gamma-1)}{2}\frac{V^2}{\gamma R T}$

Since $\gamma R T = a^2$ and $M = V/a$:

3. Deriving $P_0$ and $\rho_0$ — adding isentropy

HOW. For an isentropic process in an ideal gas, the temperature–pressure and temperature–density relations are:

$\frac{P_0}{P} = \left(\frac{T_0}{T}\right)^{\frac{\gamma}{\gamma-1}}, \qquad \frac{\rho_0}{\rho} = \left(\frac{T_0}{T}\right)^{\frac{1}{\gamma-1}}$

Why this step? From $Pv^\gamma=$const and ideal gas $Pv=RT$: eliminate $v$ to get $T P^{(1-\gamma)/\gamma}=$const, rearrange to the exponents above. Same trick gives the density relation.

Insert the $T_0/T$ result:

These three are the isentropic flow relations. Note they are consistent with the ideal gas law: $\dfrac{P_0}{P} = \dfrac{\rho_0}{\rho}\cdot\dfrac{T_0}{T}$. ✓

4. Low-speed limit — sanity check (Forecast-then-Verify)

Expand $\left(1+\tfrac{\gamma-1}{2}M^2\right)^{\gamma/(\gamma-1)}$ using the binomial series for small $M^2$:

$\frac{P_0}{P} \approx 1 + \frac{\gamma}{2}M^2 + \frac{\gamma}{8}M^4 + \dots$

So $P_0 - P \approx \tfrac{\gamma}{2}P M^2 = \tfrac{\gamma}{2}P \dfrac{V^2}{\gamma R T} = \tfrac12 \dfrac{P}{RT}V^2 = \tfrac12\rho V^2$.

Verified: incompressible Bernoulli is the $M\to 0$ limit. The $M^4$ term is the compressibility correction.

5. Worked Examples

6. Common Mistakes (Steel-man)

7. Active Recall

8. Flashcards

Connections

• Speed of sound a = √(γRT)
• Mach number and flow regimes
• Isentropic flow relations
• Normal shock waves — total pressure loss
• Bernoulli's equation (incompressible limit)
• Steady-flow energy equation (First Law for control volumes)
• Pitot tube measurement

Concept Map

Hinglish (regional understanding)