Normal Shock Calculator

Normal Shock Wave Calculator

Note: For a shock wave to occur, M₁ must be greater than 1.0.

Standard value for air is 1.4.

Downstream Properties

Downstream Mach Number (M₂)	–
Static Pressure Ratio (P₂/P₁)	–
Static Temperature Ratio (T₂/T₁)	–
Density Ratio (ρ₂/ρ₁)	–
Stagnation Pressure Ratio (P₀₂/P₀₁)	–

Upstream Mach number must be greater than 1.0 for normal shock relations.

function calculateShock() { var m1 = parseFloat(document.getElementById('mach1').value); var gamma = parseFloat(document.getElementById('gamma').value); var resultsDiv = document.getElementById('results-area'); var errorDiv = document.getElementById('error-msg'); if (isNaN(m1) || isNaN(gamma) || m1 <= 1) { resultsDiv.style.display = 'none'; errorDiv.style.display = 'block'; return; } errorDiv.style.display = 'none'; resultsDiv.style.display = 'block'; var m1Sq = m1 * m1; var gp1 = gamma + 1; var gm1 = gamma – 1; // Downstream Mach Number M2 var m2 = Math.sqrt((1 + (gm1 / 2) * m1Sq) / (gamma * m1Sq – (gm1 / 2))); // Static Pressure Ratio P2/P1 var p2p1 = 1 + (2 * gamma / gp1) * (m1Sq – 1); // Density Ratio rho2/rho1 var rho2rho1 = (gp1 * m1Sq) / (2 + gm1 * m1Sq); // Temperature Ratio T2/T1 var t2t1 = p2p1 * (1 / rho2rho1); // Stagnation Pressure Ratio Po2/Po1 var term1 = ((gp1 / 2) * m1Sq) / (1 + (gm1 / 2) * m1Sq); var term2 = (gp1 / (2 * gamma * m1Sq – gm1)); var po2po1 = Math.pow(term1, (gamma / gm1)) * Math.pow(term2, (1 / gm1)); document.getElementById('res_m2').innerText = m2.toFixed(4); document.getElementById('res_p2p1').innerText = p2p1.toFixed(4); document.getElementById('res_t2t1').innerText = t2t1.toFixed(4); document.getElementById('res_rho2rho1').innerText = rho2rho1.toFixed(4); document.getElementById('res_po2po1').innerText = po2po1.toFixed(4); }

A normal shock wave is a thin region in a supersonic flow where physical properties such as pressure, temperature, and density change abruptly. These shocks occur perpendicular to the flow direction. They are a fundamental concept in high-speed aerodynamics, compressible fluid mechanics, and the design of jet engines and supersonic aircraft.

The Physics of Normal Shocks

When an object travels faster than the speed of sound (Mach 1), it creates pressure waves that cannot dissipate ahead of the object. These waves "pile up" to form a shock wave. In a normal shock:

• Flow Speed: Decreases from supersonic (M₁ > 1) to subsonic (M₂ < 1).
• Static Pressure: Increases significantly across the shock.
• Temperature: Increases, converting kinetic energy into internal thermal energy.
• Density: Increases as the gas is compressed.
• Stagnation Pressure: Decreases, representing an irreversible loss of total energy available for work (entropy increase).

Key Formulas Used

The calculations are based on the Rankine-Hugoniot relations for a calorically perfect gas. The most critical input is the Upstream Mach Number (M₁) and the Ratio of Specific Heats (γ).

Example Calculation:
If a flow is moving at Mach 2.0 (M₁ = 2.0) in air (γ = 1.4):

• The downstream Mach number (M₂) will be approximately 0.5774.
• The static pressure ratio (P₂/P₁) will be 4.500, meaning pressure increases by 4.5 times.
• The total pressure (Stagnation Pressure) will drop to about 72% of its original value.

Practical Applications

Aerospace engineers use these calculations for various critical tasks:

1. Engine Intake Design: Supersonic jet engines must slow down incoming air to subsonic speeds before it enters the compressor. This is often achieved through a series of shock waves.
2. Wind Tunnel Testing: Predicting the conditions behind shocks formed by models in supersonic test sections.
3. Re-entry Vehicles: Estimating the extreme heat and pressure loads on spacecraft entering planetary atmospheres at high Mach numbers.

Why Total Pressure (P₀) Drops

Unlike static pressure which rises, the total or stagnation pressure always decreases across a shock wave. This is a direct consequence of the Second Law of Thermodynamics. The shock process is adiabatic but highly irreversible, leading to a rise in entropy. In engineering, a higher stagnation pressure ratio is usually desired, as it indicates a more efficient process with fewer losses.