Drag Divergence and Wave Shock A Path to Supersonic Flight Barriers
Mach Effects on Coefficient of Drag The Critical Mach Number is the velocity on the airfoil at which sonic flow is first acquired If the free-stream Mach number is increased slightly above the critical Mach number, a threshold is reached and supersonic flow occurs around the minimum pressure point on the airfoil; the coefficient of drag remains low If the free-stream Mach number is increased significantly towards M =1.0, there occurs a dramatic increase in the coefficient of drag
Minimum Pressure Location The area of maximum velocity on an airfoil occurs at the location of minimum pressure The location where the position on the airfoil divided by its chord length corresponds to the point of minimum pressure depends on the overall shape of the flow field around the airfoil The location of minimum pressure and maximum velocity is not at the point of maximum thickness in a local region of the airfoil; the entire flow field must be considered
Drag Divergence Mach Number Pressure-driven shock waves increase drag on the airfoil as the Mach number increases Flow separation occurs because of the adverse pressure gradient across these shock waves The dramatic increase in the coefficient of drag is a result of both the pressure-driven shock waves and flow separation away from the airfoil The free-stream Mach number at which the coefficient of drag begins to increase rapidly is called the Drag Divergence Mach Number M Critical <M Drag Divergence <1.0
Subsonic Wave Propagation Imagine a beacon that emits a sound pulse at regular intervals of time Consider the magnitude of one pulse from a beacon that is moving with velocity V < V Sound The sound from the beacon propagates in all directions at the prevailing speed of sound At time t, the beacon will have moved a distance V t Because V < V Sound the beacon will always be inside the envelope of the sound wave that it emits
Supersonic Wave Drag Again imagine a beacon that emits a sound pulse at regular intervals of time Consider the magnitude of one pulse from a beacon that is moving with velocity V > V Sound At time t, the sound wave will have travelled outward from the beacon by a distance V Sound t The beacon will have moved a distance V V > M 1.0 t New sound waves from the beacon will begin to form a stack inside the envelope of emitted sounds
Mach Wave Geometry The stack of sound waves inside the supersonic envelope align themselves along a tangent from the point where the beacon is located at time t to the circle drawn out by the initial sound wave This tangent line that intersects the outer edge of the sound circle is where the pressure disturbances begin to manifest themselves This geometric line is called the Mach Wave The vertex of the Mach wave originates at the moving beacon at time t
The Mach Angle The coalescing pressure waves from stacked sound waves create a specific wave geometry The Mach wave that is formed from the vertex of the moving beacon makes an angle Φ with respect to the direction of beacon movement This angle Φ is defined as the Mach Angle Φ Mach =sin 1 (1/ M )
Supersonic Shock Waves A large object moving at supersonic speeds will create a strong disturbance in the flow field This disturbance is called a Shock Wave, and is oblique with respect to the Mach wave angle As the airflow moves across this oblique shock wave, the pressure, density, and temperature increase, while the velocity and the Mach number decrease
Pressure on Shock Waves The pressure increases across the shock wave, and is greater than the free-stream pressure A net drag is produced on an airfoil with an inclined angle of attack because the pressure acts normal to the surface and the surface of the airfoil is inclined to the relative wind This net drag is caused by the inherent pressure increase across the oblique shock wave, and it is called Wave Drag
Wave Drag and Profile Optimization Supersonic profiles attempt to minimize the strength of the shock wave through thinning and the addition of sharp leading edges An expansion wave forms at the top of the leading edge of the airfoil, deflecting the flow field away from the free-stream A pressure decrease occurs across the fan-shaped region of the expansion wave emanating away from the airfoil
Wave Drag and Profile Optimization Along the top surface of the airfoil, the flow field is deflected away from the free-stream in the form of an expansion wave at the leading edge At the top side of the trailing edge, the flow field is reflected back towards the free-stream direction in the form of an oblique shock wave On the bottom surface of the airfoil, the flow field is deflected into the free-stream, causing a shock wave and an increase in pressure there At the bottom of the trailing edge, the flow field is reflected back towards the free-stream direction in the form of an expansion wave
Net Lift and Drag A net lift and net drag result from the pressure distribution from the oblique shock waves and expansion waves The expansion waves and shock waves at the leading edge of the airfoil result in an unequal top-bottom surface pressure distribution The pressure on the top surface of the airfoil is less than the free-stream pressure The pressure on the bottom surface of the airfoil is greater than the free-stream pressure
Aerodynamic Force from Pressure The net effect of the unequal top-bottom surface pressure distribution is an aerodynamic force that is normal to the airfoil Components of this aerodynamic force are Perpendicular to the relative wind: Lift (L) Parallel to the relative wind: Supersonic Wave Drag (D wave )
Mach Number and L+D Coefficients As the free-stream Mach number increases, a decrease occurs in both the coefficient of lift and the coefficient of drag The lift force (L) and drag force (D) tend to increase as the flight velocity increases because the dynamic pressure increases, particularly in the supersonic regimes P = 1 dynamic 2 ρ V 2
Lift and Drag Coefficients The lift and drag coefficients for thin airfoils at small to moderate angles of attack can be represented by the following approximations For α= Angle of Attack (radians) c l =Coefficient of Lift c l = (4 α) (M 2 1) c d,w =Coefficient of Wave Drag M =Free stream Mach Number c d,w = (4 α2 ) (M 2 1)