Normal Shock Relations

A normal shock wave is a fundamental yet extreme phenomenon in compressible fluid dynamics, representing the most abrupt way a supersonic flow can adjust to a downstream obstruction. Whether analyzing flow through a supersonic intake, over a transonic airfoil, or inside a high-speed propulsion system, understanding the precise, discontinuous property changes across a normal shock is critical for predicting performance, losses, and thermal loads. These relations, derived from conservation laws, allow you to quantify the sudden jumps in pressure, temperature, and density, and to calculate the irrecoverable loss in total pressure that defines the shock's inefficiency.

What is a Normal Shock Wave?

A normal shock wave is an extremely thin region, on the order of a few molecular mean free paths, across which flow properties change almost discontinuously. It is characterized as a stationary, irreversible compression wave that stands perpendicular to the flow direction. For a shock to exist, the upstream flow must be supersonic ($M_1>1$). The primary function of the shock is to decelerate this supersonic flow to a subsonic ($M_2<1$) state downstream, doing so through a process that is highly non-isentropic.

Think of it as a traffic jam propagating upstream: cars (air molecules) approach at high speed, are forced to slow down abruptly at the jam's front (the shock), and exit at a much lower speed and higher density. This process is not smooth—it generates entropy through viscous dissipation and thermal conduction within the shock structure, making it a fundamentally lossy mechanism. In engineering applications, shocks appear wherever a supersonic flow encounters a physical or thermodynamic blockage it cannot negotiate isentropically, such as in front of a blunt body or inside a converging-diverging nozzle operating at off-design conditions.

Governing Conservation Equations

The property changes across a normal shock are determined by applying the fundamental one-dimensional conservation equations to a control volume that straddles the shock. We assume steady, adiabatic flow of a calorically perfect gas (constant specific heats) with no shaft work or body forces. The shock is treated as a discontinuity, so we analyze the flow just upstream (state 1) and just downstream (state 2). The three governing equations are:

1. Conservation of Mass (Continuity): $${\rho }_1{u}_1={\rho }_2{u}_2$$ This states that the mass flow rate per unit area is constant. Here, $\rho$ is density and $u$ is velocity.
2. Conservation of Momentum: $$p_1+{\rho }_1{u}_1^2=p_2+{\rho }_2{u}_2^2$$ This equation balances the static pressure force with the momentum flux. The term $\rho u^2$ is often called the dynamic pressure in this context.
3. Conservation of Energy: $$h_1+\frac{u_1^2}{2}=h_2+\frac{u_2^2}{2}=h_0$$ This equation states that the total enthalpy $h_0$ remains constant across the shock, as the process is adiabatic. For a perfect gas with constant $c_p$, this becomes $c_p{T}_1+u_1^2/2=c_p{T}_2+u_2^2/2=c_p{T}_0$.

These three equations, combined with the perfect gas equation of state $p=\rho RT$, form a closed set that can be manipulated to derive all normal shock relations as functions of the upstream Mach number $M_1$ and the specific heat ratio $\gamma$.

Derived Property Ratios

By manipulating the conservation equations, we obtain algebraic relations for the ratios of key thermodynamic properties and the downstream Mach number. These are the primary working equations for shock calculations.

Mach Number Relation: The downstream Mach number is uniquely determined by the upstream Mach number: $$M_2^2=\frac{1+\frac{\gamma -1}{2}{M}_1^2}{\gamma M_1^2-\frac{\gamma -1}{2}}$$ This formula confirms two critical points: if $M_1>1$, then $M_2$ is always less than 1 (subsonic). In the limiting case of an infinitely weak shock ($M_1\to 1$), $M_2\to 1$.

Pressure, Density, and Temperature Ratios: The changes in static properties are given by:

• Pressure Ratio: $$\frac{p_2}{p_1}=1+\frac{2\gamma }{\gamma +1}(M_1^2-1)$$

This shows the static pressure always increases dramatically across the shock.

• Density Ratio (or Velocity Ratio): $$\frac{{\rho }_2}{{\rho }_1}=\frac{u_1}{u_2}=\frac{(\gamma +1)M_1^2}{2+(\gamma -1)M_1^2}$$

The density increases, but there is an upper limit: as $M_1\to \infty$, ${\rho }_2/{\rho }_1\to (\gamma +1)/(\gamma -1)$, which is 6 for air ($\gamma =1.4$).

• Temperature Ratio: $$\frac{T_2}{T_1}=\frac{p_2}{p_1}\frac{{\rho }_1}{{\rho }_2}=\left[1+\frac{2\gamma }{\gamma +1}(M_1^2-1)\right]\frac{2+(\gamma -1)M_1^2}{(\gamma +1)M_1^2}$$

The temperature also increases significantly, as the kinetic energy of the supersonic flow is converted ("stagnated") into thermal energy.

Example Calculation: For air ($\gamma =1.4$) with an upstream Mach number $M_1=2.0$:

• $p_2/p_1=1+\frac{2.8}{2.4}(4-1)=4.5$
• ${\rho }_2/{\rho }_1=\frac{2.4\ast 4}{2+0.4\ast 4}=\frac{9.6}{3.6}\approx 2.667$
• $T_2/T_1=4.5/2.667\approx 1.687$
• $M_2^2=\frac{1+0.2\ast 4}{1.4\ast 4-0.2}=\frac{1.8}{5.4}\approx 0.333$, so $M_2\approx 0.577$

Total Pressure and Entropy Increase

While the total enthalpy ($T_0$) remains constant across the adiabatic shock, the total pressure $p_0$ does not. Total pressure is the pressure achieved if the flow were brought to rest isentropically. Since the shock process is irreversible, total pressure always decreases. This loss is a direct measure of the entropy generated.

The total pressure ratio across the shock is: $$\frac{p_{02}}{p_{01}}={\left[\frac{(\gamma +1)M_1^2}{2+(\gamma -1)M_1^2}\right]}^{\frac{\gamma }{\gamma -1}}{\left[\frac{\gamma +1}{2\gamma M_1^2-(\gamma -1)}\right]}^{\frac{1}{\gamma -1}}$$

From the Second Law of Thermodynamics, the entropy change per unit mass is: $$s_2-s_1=c_p\ln \left(\frac{T_2}{T_1}\right)-R\ln \left(\frac{p_2}{p_1}\right)$$ By substituting the property ratios, one can show that $s_2-s_1>0$ for $M_1>1$. A more convenient link is through total pressure: for a perfect gas, $$s_2-s_1=-R\ln \left(\frac{p_{02}}{p_{01}}\right)$$ Thus, a drop in total pressure ($p_{02}<p_{01}$) corresponds directly to an entropy increase. In propulsion systems, this loss in total pressure translates directly to a loss in thrust potential or efficiency.

Common Pitfalls

1. Confusing Static and Total Properties Across the Shock: A frequent error is assuming total temperature and total pressure are constant. Remember the cardinal rule: total enthalpy (and thus $T_0$ for a perfect gas) is constant across an adiabatic shock, but $p_0$ always decreases due to irreversibility. Using $p_{01}=p_{02}$ in a shock calculation will lead to grossly incorrect results.

1. Misapplying Isentropic Relations Through the Shock: The isentropic relations ($p/p_0=[1+(\gamma -1)M^2/2{]}^{-\gamma /(\gamma -1)}$, etc.) are only valid between two points on the same streamline in an isentropic process. The shock is non-isentropic, so you cannot use these relations to connect state 1 to state 2. You must use the specific normal shock relations derived from the conservation laws.

1. Forgetting the Supersonic Upstream Condition: The normal shock relations are only mathematically and physically valid for $M_1\ge 1$. Plugging a subsonic $M_1$ into the equations will often yield nonsensical results (e.g., a density ratio less than 1). If the upstream flow is subsonic, a standing normal shock cannot exist.

1. Ignoring the Perfect Gas Assumption: The classic relations assume constant $\gamma$ and $c_p$. At very high Mach numbers (e.g., hypersonic re-entry), real gas effects like vibrational excitation and dissociation become significant, and these simple relations become inaccurate. Always consider the validity of the perfect gas model for your application.

Summary

• A normal shock wave is a thin, irreversible discontinuity that decelerates a supersonic flow ($M_1>1$) to a subsonic state ($M_2<1$), causing abrupt increases in static pressure, temperature, and density.
• The property changes are governed by the conservation of mass, momentum, and energy for a steady, adiabatic, one-dimensional flow of a perfect gas, leading to the normal shock relations expressed as functions of $M_1$ and $\gamma$.
• While total temperature remains constant across the adiabatic shock, total pressure always decreases. This loss in $p_0$ is a direct and crucial measure of the irreversible entropy generation inherent to the shock process.
• In analysis, you must use the specific shock relations and not the isentropic formulas when connecting properties across the shock, and you must always verify the upstream flow is supersonic for the relations to be valid.