Which law states that for an adiabatic and steady flow, the Mach number can not increase across a normal shock wave?

Explanation:

Mach Number Across a Normal Shock Wave

• A normal shock wave is a sudden and nearly discontinuous change in the flow properties of a compressible fluid (usually a gas) that occurs when the flow transitions from supersonic to subsonic speeds. The Mach number (M) is a crucial parameter in this context, defined as the ratio of the fluid velocity to the speed of sound in the medium.
• In an adiabatic and steady flow situation, the Mach number cannot increase across a normal shock wave. This behavior is governed by the fundamental principles of gas dynamics and is mathematically expressed using the Rankine-Hugoniot relation.

Rankine-Hugoniot Relation

• The Rankine-Hugoniot relation is a set of equations derived from the conservation laws of mass, momentum, and energy across a shock wave. These equations describe the relationship between the pre-shock and post-shock states of the fluid. Specifically, they ensure the continuity of mass, momentum, and energy across the shock front. Here’s how the Rankine-Hugoniot relation explains why the Mach number cannot increase across a normal shock wave:

1. Conservation of Mass:

The mass flow rate is conserved across the shock wave. Mathematically, this is expressed as:

ρ₁U₁ = ρ₂U₂

where:

• ρ₁, ρ₂: Densities of the fluid before and after the shock
• U₁, U₂: Velocities of the fluid before and after the shock

2. Conservation of Momentum:

The momentum equation across the shock wave is given by:

P₁ + ρ₁U₁² = P₂ + ρ₂U₂²

where:

• P₁, P₂: Static pressures before and after the shock

3. Conservation of Energy:

The total energy (including internal energy and kinetic energy) is also conserved across the shock wave, expressed as:

h₁ + (U₁² / 2) = h₂ + (U₂² / 2)

where:

• h₁, h₂: Specific enthalpies before and after the shock

4. Implications for the Mach Number:

Combining these conservation equations reveals that the post-shock Mach number (M₂) is always less than 1 (subsonic), while the pre-shock Mach number (M₁) is greater than 1 (supersonic). This means the flow transitions from supersonic to subsonic across the normal shock wave.

In essence, the Rankine-Hugoniot relation establishes the fundamental physics that prevents the Mach number from increasing across a normal shock wave. Instead, the Mach number decreases as a result of the increase in pressure, temperature, and density, and the decrease in velocity