The boundary layer near the stagnation point in hypersonic flow past a sphere

Abstract

When a sphere or other bluff body travels at supersonic speeds, a shock wave is formed close to the front surface. With increase of speed the air behind the shock is further compressed, and the shock wave moves closer to the surface. This paper considers the case where the region close to the stagnation point between the shock and the sphere can be taken to be a steady laminar boundary layer.The approximate solution of the equations of motion follows closely the classical work of Homann (1936), ideas similar to those of Lighthill (1957) being used to apply it to the problem in hand. It consists mainly in reducing the equations to ordinary differential form by assuming forms of the flow variables which satisfy the boundary conditions, notably at the shock wave. In addition, several transformations are employed in order to simplify the equations and to increase the range of solutions, and also to facilitate the use of the ‘Mercury’ electronic computer in solving them.The results give an insight into some aspects of hypersonic flows. Included in this paper are a selection of temperature and transverse velocity profiles across the boundary layer and several graphs relating such quantities as the shock stand-off distance and the skin-friction coefficient with Reynolds number. The last two mentioned are the most interesting. The first set gives the surprising result that the shock stand-off distance increases with increase in Reynolds number, whereas it is known that the skin-friction coefficient is inversely proportional to a decreasing power of the Reynolds number when it is lower than order 10³, but the indication is that it tends to the expected constant power of ½ when the Reynolds number is of order 10⁴.