Transonic & Shock-Wave CFD

What Is It?

Transonic flow is flow in the Mach number range where both subsonic and supersonic regions coexist — typically around M = 0.8 to 1.2. In this regime, the flow accelerates to supersonic speed over parts of the body (the upper surface of a wing, for example) and then decelerates back to subsonic through a shock wave. The shock is a thin discontinuity where the pressure, density and temperature jump and the velocity drops. Transonic flow is the operating regime for most commercial aircraft, and the shock waves it produces are a primary source of drag (wave drag) and a driver of stability and buffet behaviour.

Why It Matters

Transonic flow is one of the most challenging regimes for CFD. The coexistence of subsonic and supersonic flow, the presence of shock waves and the interaction between shocks and boundary layers create physical and numerical complexities that require careful treatment. The shock location, the shock strength and the shock–boundary-layer interaction determine the drag, the lift and the stability. Getting these wrong — predicting the shock in the wrong place or at the wrong strength — produces incorrect aerodynamic forces and incorrect performance predictions. Understanding the physics and the numerical requirements of transonic CFD is essential for credible high-speed aerodynamic analysis.

A shock is a physical discontinuity that the numerical scheme must capture without destroying the rest of the flow field. The mesh must be fine enough to resolve the shock, the numerical scheme must be able to represent the discontinuity, and the shock–boundary-layer interaction must be captured for correct drag prediction.

Shock Formation

A shock wave forms when supersonic flow is decelerated — typically when the supersonic region over a body meets the downstream subsonic flow. The shock is a thin region (a few mean free paths in reality) where the pressure, density and temperature increase sharply and the velocity decreases. The shock is a compression discontinuity — the flow crosses it from supersonic to subsonic. The shock strength depends on the upstream Mach number — a higher supersonic Mach number produces a stronger shock with a larger pressure jump. In transonic flow over a wing, a shock typically forms on the upper surface where the supersonic region ends.

Shock Properties

Normal and Oblique Shocks

A normal shock is perpendicular to the flow direction — the flow crosses it directly from supersonic to subsonic. A normal shock produces the strongest pressure jump for a given upstream Mach number. An oblique shock is at an angle to the flow — the flow is deflected as it crosses the shock, and the downstream flow may remain supersonic (if the deflection angle is below the maximum). Oblique shocks occur on wedges, cones and at the leading edges of supersonic bodies. In transonic flow over a wing, the shock is typically near-normal on the upper surface.

Shock–Boundary-Layer Interaction

When a shock wave interacts with the boundary layer on a surface, the result is one of the most complex phenomena in transonic aerodynamics. The shock imposes a strong adverse pressure gradient on the boundary layer. The boundary layer may thicken in response, and if the shock is strong enough, the boundary layer may separate. The separated flow creates a lambda shock structure — the shock bifurcates near the wall as the separated boundary layer creates a subsonic region. The shock–boundary-layer interaction increases drag, can cause buffet (unsteady shock oscillation) and can reduce lift. Capturing this interaction in CFD requires both shock-capturing capability and boundary-layer resolution.

Shock–boundary-layer interaction:

Shock wave hits boundary layer → strong adverse pressure gradient
→ boundary layer thickens or separates
→ lambda shock structure near the wall
→ possible buffet (unsteady shock oscillation)
→ increased drag; reduced lift; unsteady loading

Transonic Aerofoil Behaviour

A transonic aerofoil has a typical flow structure: subsonic flow at the leading edge, acceleration to supersonic over the upper surface, a shock wave where the supersonic region ends, and subsonic flow downstream of the shock. As the free-stream Mach number increases, the supersonic region grows, the shock moves aft and strengthens, and the wave drag increases. At some point, the shock becomes strong enough to cause boundary-layer separation — the drag diverges (drag rise) and the aerofoil performance degrades. The Mach number at which this occurs is the drag divergence Mach number, a key design parameter for transonic aircraft.

Numerical Requirements

Transonic CFD has specific numerical requirements beyond those of subsonic or incompressible flow. The shock must be captured by the numerical scheme — the mesh must be fine enough in the shock region to resolve the discontinuity, and the numerical scheme must handle the sharp gradients without producing oscillations or excessive smearing. The solver must be a compressible solver (density-based or pressure-based with compressible formulation). The turbulence model must handle the shock–boundary-layer interaction — SST k–ω is commonly used. Convergence can be difficult — the shock location may oscillate during convergence, and the shock–boundary-layer interaction may produce unsteady behaviour.

• Compressible solver formulation (density-based or pressure-based compressible)
• Mesh refinement in the shock region — the shock must be resolved
• Shock-capturing numerical scheme — second-order or higher with appropriate flux limiter
• Turbulence model that handles shock–boundary-layer interaction (SST k–ω)
• Convergence monitoring — shock location and force coefficients must stabilise

Shock-Capturing Schemes and Numerical Dissipation

Shock-capturing schemes are numerical methods that represent shock waves as sharp gradients in the mesh rather than as explicit discontinuities. The scheme must balance accuracy (sharp shock representation) with stability (no oscillations). First-order schemes are stable but smear the shock over several cells. Second-order schemes provide sharper shocks but can produce oscillations near the discontinuity. Flux limiters (Roe, AUSM, HLLC) switch between high-order and low-order schemes based on the local gradient — high-order in smooth regions, low-order near shocks. The choice of scheme and limiter affects the shock sharpness and the overall solution accuracy.

MESH CONSIDERATION: The mesh in the shock region must be fine enough to resolve the shock. If the shock is smeared over too many cells, the pressure jump, the wave drag and the shock–boundary-layer interaction will be incorrectly predicted. Refine the mesh in the expected shock location.

Wave Drag

Wave drag is the drag associated with shock wave formation. It is distinct from skin-friction drag and pressure (form) drag. Wave drag arises from the entropy increase across the shock — the total pressure decreases, representing an irreversible loss of energy. Wave drag increases rapidly as the Mach number exceeds the drag divergence Mach number. In transonic aircraft design, reducing wave drag is a primary objective — it drives the development of supercritical aerofoils, area ruling and swept wings. CFD predicts wave drag by integrating the pressure distribution, but the prediction depends on accurate shock location and strength.

Key Takeaways

• Transonic flow has coexisting subsonic and supersonic regions with shock waves
• Shocks are thin discontinuities — pressure, density and temperature jump; velocity drops
• Shock–boundary-layer interaction can cause separation, buffet and lambda shock structures
• Transonic CFD requires compressible solver, shock-capturing schemes and mesh refinement at the shock
• Wave drag arises from the entropy increase across the shock and increases rapidly above the drag divergence Mach number