# Unsteady hypersonic boundary layers for slender axisymmetric bodies with large injection rates

TL;DR: A semi-similar solution of an unsteady hypersonic laminar compressible boundary layer flow over a slender axisymmetric body with massive blowing has been obtained when the free stream velocity varies arbitrarily with time.

Abstract: A semi-similar solution of an unsteady hypersonic laminar compressible boundary layer flow over a slender axisymmetric body with massive blowing has been obtained when the free stream velocity varies arbitrarily with time. The governing partial differential equations have been solved numerically by combining the implicit finite difference scheme with the quasi-linearization technique. The results have been obtained for (i) an accelerating/decelerating stream and (ii) a fluctuating stream. The skin friction responds to the fluctuation in the free stream more compared to the heat transfer. It is observed that the effect of large injection (blowing) rates is to move the viscous boundary layer away from the surface. The effect of the variation of the density-viscosity product across the boundary layer is found to be negligible for large blowing rates. Massive blowing reduces significantly the values of skin friction and heat transfer but the effect of the transverse curvature parameter is just reverse. Location of the dividing streamline increases as injection rate increases, but decreases with the increase of the transverse curvature parameter.

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01 Jan 2012

TL;DR: In this paper, the Hypersonic laminar compressible boundary layer heat transfer over a slender axisymmetric geometry with large injection rates and arbitrary free stream velocity variation with time, is studied using Differential Transform Method (DTM) combined with Pade approximants.

Abstract: The hypersonic laminar compressible boundary layer heat transfer over a slender axisymmetric geometry with large injection rates and arbitrary free stream velocity variation with time, is studied using Differential Transform Method (DTM) combined with Pade approximants. Homentropic external flow is assumed and ionization effects are neglected. The governing equations are transformed and an inviscid flow solution implemented simultaneously for the pressure gradient term. The key parameters dictating the unsteady dimensionless velocity and enthalpy fields are shown to be the Prandtl number (Pr), dissipation parameter (m), pressure gradient parameter (), transverse curvature parameter (A), density-viscosity product across the boundary layer (N), injection parameter (), temperature law exponent () and the length scale (R). The Differential Transform Method (DTM) with Pade approximants is required to solve the two-point boundary value problem, for the steady state case. DTM alone does not attain convergence due to the infinity boundary conditions. A number of important cases are considered. Excellent correlation is achieved with the DTM-Pade solutions and Runge-Kutta shooting quadrature. The importance of large injection rates (mass transfer at the wall) in actual hypersonic aerodynamics is also discussed.

10 citations

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TL;DR: In this paper, the effect of transverse curvature of the body on the perfect gas was investigated, and a method of applying the local-similarity approximation to obtain the approximate solution for nonsimilar cases was described.

Abstract: Axisymmetric viscous flow past unyawed very slender bodies of revolution is treated within the category of the perfect gas. Attention is paid especially to the effect of transverse curvature of the body. From the transformed equations, the similarity conditions are deduced, and the parameter characterizing the effect of transverse curvature is obtained. Several numerical solutions of similarity equations for hypersonic flows are presented, and upon the basis of these results, the effect of the transverse-curvature parameter is discussed. A method of applying the local-similarity approximation to obtain the approximate solution for nonsimilar cases is described, as are practical applications to incompressible flow past a long cylinder and to hypersonic flow past a very slender cone. Comparison with experimental results shows fair agreement with calculations using the local-similarity approximation in the present range of experimental flow conditions.

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01 Jan 1955

TL;DR: The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part, denoted as turbulence as discussed by the authors, and the actual flow is very different from that of the Poiseuille flow.

Abstract: The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part. These actual flows show a special characteristic, denoted as turbulence. The character of a turbulent flow is most easily understood the case of the pipe flow. Consider the flow through a straight pipe of circular cross section and with a smooth wall. For laminar flow each fluid particle moves with uniform velocity along a rectilinear path. Because of viscosity, the velocity of the particles near the wall is smaller than that of the particles at the center. i% order to maintain the motion, a pressure decrease is required which, for laminar flow, is proportional to the first power of the mean flow velocity. Actually, however, one ob~erves that, for larger Reynolds numbers, the pressure drop increases almost with the square of the velocity and is very much larger then that given by the Hagen Poiseuille law. One may conclude that the actual flow is very different from that of the Poiseuille flow.

17,321 citations

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TL;DR: The effect of large mass injection on the compressible, similar laminar boundary with favorable pressure gradient has been examined in this article, where the boundary layer may be divided into two regions: an inner region adjacent to the surface where the viscosity plays a minor role, and a viscous layer where the transition occurs from the inner layer to the inviscid flow outside the boundary surface.

Abstract: The effect of large mass injection on the compressible, similar laminar boundary with favorable pressure gradient has been examined. It is found that for high rates of injection, the boundary layer may be divided into two regions: 1) an inner region adjacent to the surface where the viscosity plays a minor role, 2) the viscous layer where the transition occurs from the inner layer to the inviscid flow outside the boundary layer. Matched asymptotic expansions appropriate for large injection rates have'been constructed for each layer and a uniformly valid solution has been obtained. In the case of the insulated wall, it turns out that the viscous outer layer contributes only small corrections to properties of the boundary layer. In the case of the highly cooled wall, on the other hand, the boundary layer is dominated by the viscous mixing between the inviscid outer flow and the high-density, low-speed gas adjacent to the wall. Simple expressions for heat-transfer rates, skin friction, and approximations for integral properties of the boundary layer have been derived, which are useful in future application in nonsimilar boundary-layer calculations.

50 citations

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TL;DR: In this paper, the strong interaction region in viscous hypersonic flow past a slender cone is considered and expressions for the pressure, shock, skin friction, and heat transfer are derived.

Abstract: The strong‐interaction region in viscous hypersonic flow past a slender cone is considered and expressions for the pressure, shock, skin friction, and heat transfer are derived. An approximate method by which the whole interaction region can be computed is also suggested.

45 citations