# Supersonic Laminar Flow in CFD Problems

### Key Takeaways

• Laminar flow is desirable in aerodynamics in order to keep drag low.

• Supersonic laminar flow corresponds to velocities above the speed of sound.

• The airfoil design will determine airflow behavior along the body and wings of an aircraft.

When the Concorde began its first supersonic flight in 1969, it was a landmark moment for the aeronautics industry. The promise of traversing transatlantic routes in approximately 4 hours stoked a near-obsession with supersonic flight. Ultimately, the Concorde retired in 2003, but research and development in supersonic flight still continue today.

CFD simulations are some of the primary tools used to design supersonic aircraft for commercial and military deployment, and understanding flow behavior is the main goal of these simulations. As aircraft speed increases beyond the critical Mach number, the supersonic laminar flow will dominate flow behavior as long as the aircraft’s wing is designed properly. The goal of these simulations is to minimize drag and ensure maximum airspeed within the laminar regime.

## The Transition to Supersonic Laminar Flow

The transition from subsonic to supersonic speed marks a progressive increase in drag due to increasing airspeed as well as a shockwave as the critical Mach number (Ma = 1) is passed. During supersonic flight, airflow across the wings and body of the aircraft should remain in the laminar flow regime, as the goal is to create the lowest possible drag. Eventually, the aircraft will move so fast that airflow across the body of the craft will become turbulent and will eventually limit the aircraft’s speed.

How do we ensure low drag flight with laminar flow across the wing and body? This requires designing the curvature of the body and wing such that airflow is always laminar, even at speeds beyond the speed of sound. We typically use the Mach number (Ma) to define when an aircraft enters the transonic flight regime as well as the Reynolds number to define a limit on airspeed that leads to turbulent flow along the aircraft and wing.

### Airfoils That Support Supersonic Laminar Flow

Airflow across an aircraft during flight depends on the shape of the airfoil and the body of the aircraft. Supersonic laminar flow places excessive drag on traditional airfoils because these create an earlier transition to flow separation and turbulence. Therefore, alternative airfoil styles are used to support supersonic laminar flow.

Typical airfoil design implements a higher radius of curvature (flatter) on the bottom surface and a shorter (cambered) radius of curvature on the top surface. The idea is that, at moderate airspeed, airflow will provide positive lift as described by Bernoulli’s equation. Here are a few common airfoil design types:

• Supercritical airfoil: This airfoil inverts the traditional design, where the top surface is flatter and the bottom has stronger curvature. This airfoil design compensates for negative lift with a higher attack angle so that the craft can climb and stay airborne.

• Biconvex airfoil: This airfoil style implements the same curvature on the top and bottom surfaces of the wing. The main parameter that determines flow separation and turbulence is the thickness to chord length ratio.

• Wedge airfoil: This airfoil resembles a diamond and is often designed to have a very shallow attack angle. It can experience a very high pressure coefficient on the front half of the wing, even at low  .

The goal in designing any of these airfoils is to determine the behavior of the airflow as the speed of sound is approached as well as to determine the Reynolds number where laminar flow ends and turbulence begins to develop.

### Approach to Ma = 1

Traditional airfoils will transition out of laminar flow without exceeding Ma = 1. On the approach to Ma = 1, the aircraft’s speed begins to reach the speed of sound and a pressure front begins to develop at the leading edge of the aircraft. As the craft approaches Ma = 1 and Ma > 1, the leading-edge wavefronts begin to compress on each other and will create a strong shockwave. This is shown conceptually below.

Wavefronts pile up along the front of the craft when Ma = 1 is approached. When Ma > 1, a shockwave occurs as the aircraft begins to exceed the speed of sound.

Well beyond Ma = 1, the aircraft should experience laminar airflow across the wings and body in order to be considered for supersonic flight. Very close to Ma = 1, the aircraft will experience mechanical oscillations (vibrations) known as buffeting. Therefore, another important task in supersonic airfoil design is to ensure that buffeting is not so extreme that the pilot could lose control or that anything on the aircraft could be damaged.

The alternative airfoil designs shown above can reduce the strength of the shockwave compared to a standard airfoil. This will further reduce flow separation, particularly in the supercritical airfoil. When studying airfoils to support supersonic laminar flow in a CFD simulation, the designer should determine the flow separation, maximum Reynolds number for laminar flow, and the buffeting/shockwave parameters to ensure steady laminar flow during flight.

Wakes can develop along the surface of a supersonic airfoil, such as in the approximate double-wedge design shown here.

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