Second-order Boundary-layer Effects Research Articles

Influence of second-order boundary layer effects in hypersonic flow over slender blunt cones

The aim of this study is to determine the influence of second-order effects in the aggregate on supersonic axisymmetric flow over slender blunt cones and also to determine the region of applicability of approximate methods of taking into account the strongest of these second-order effects — entropy layer absorption. A system of complete viscous shock layer equations containing all the terms of the gas dynamic Euler equations and all the second-order terms of asymptotic boundary layer theory is chosen as the gas-dynamic model. Within the calculation domain the problem is solved in a unified manner.

Second-order boundary layer solution for a continuous moving surface in a non-Newtonian fluid

An analysis is presented to investigate the second-order boundary layer effects on the flow characteristics of a continuously moving flat surface through an otherwise quiescent Ostwaald-de Waele type power law non-Newtonian fluid. Numerical solutions are provided for the first two terms of the series representing the velocity field. Details are presented for the velocity distribution and skin friction factor dependence upon the generalized local Reynolds number. The viscosity index was allowed to vary from 0.2 to 3.0.

Semi-similar solutions of the unsteady compressible second-order boundary layer flow at the stagnation point

Semi-similar solutions of the unsteady compressible laminar boundary layer flow over two-dimensional and axisymmetric bodies at the stagnation point with mass transfer are studied for all the second-order boundary layer effects when the free stream velocity varies arbitrarily with time. The set of partial differential equations governing the unsteady compressible second-order boundary layers representing all the effects are derived for the first time. These partial differential equations are solved numerically using an implicit finite-difference scheme. The results are obtained for two particular unsteady free stream velocity distributions: (a) an accelerating stream and (b) a fluctuating stream. It is observed that the total skin friction and heat transfer are strongly affected by the surface mass transfer and wall temperature. However, their variation with time is significant only for large times. The second-order boundary layer effects are found to be more pronounced in the case of no mass transfer or injection as compared to that for suction.

Second-order boundary-layer effects for the unsteady laminar incompressible three-dimensional stagnation-point flow

All the second-order boundary-layer effects on the unsteady laminar incompressible flow at the stagnation-point of a three-dimensional body for both nodal and saddle point regions have been studied. It has been assumed that the free-stream velocity, wall temperature and mass transfer vary arbitrarily with time. The effect of the Prandtl number has been taken into account. The partial differential equations governing the flow have been derived for the first time and then solved numerically unsteady free-stream velocity distributions, the nature of the using an implicit finite-difference scheme. It is found that the stagnation point and the mass transfer strongly affect the skin friction and heat transfer whereas the effects of the Prandtl number and the variation of the wall temperature with time are only on the heat transfer. The skin friction due to the combined effects of first- and second-order boundary layers is less than the skin friction due to, the first-order boundary layers whereas the heat transfer has the opposite behaviour. Suction increases the skin friction and heat transfer but injection does the opposite.

Unsteady compressible second-order boundary layers at the stagnation point of two-dimensional and axisymmetric bodies

The unsteady laminar compressible boundary-layer flow over two-dimensional and axisymmetric bodies at the stagnation point with mass transfer has been studied for all second-order boundary layer effects when the basic potential flow admits self-similarity. The solutions for the governing equations are obtained by using an implicit finite-difference scheme. Computations have been carried out for different values of the parameters characterizing the unsteadiness in the free stream velocity, wall temperature, mass transfer rate and variable gas properties. The results are found to be strongly affected by the unsteadiness in the free stream velocity. For large injection rates the second-order boundary layer effects may prevail over the first-order boundary layer, but reverse is true for suction. The wall temperature and the variation of the density-viscosity product across the boundary layer appreciably change the skin-friction and heat-transfer rates due to second-order boundary-layer effects.

Second-order boundary layers for steady, incompressible, three-dimensional stagnation point flows

It is well known that the classical boundary-layer theory represents the asymptotic solution of the Navier-Stokes equations for large Reynolds numbers. The succeeding approximation is the second-order boundary layer one which includes the effects of surface curvature, vorticity interaction and boundary-layer displacement. The second-order boundary-layer effects become important when the boundarylayer thickness becomes comparable with the characteristic body length. Van Dyke [1] and Gersten and Gross [2] have given an excellent survey of higher-order boundary layers.

Second-order boundary-layer flow past sharp cones

Second-order boundary-layer effects of displacement thickness, transverse curvature, wall slip and temperature jump have been investigated in detail for flow past sharp cones of 5°, 10° and 15° semi-vertex angles. Similar solutions of the non-dimensional first- and second-order boundary-layer equations exist and numerical solutions of the resulting systems of ordinary-differential equations have been obtained using the finite-difference technique with variable step size. The second-order effects have been computed for different values of Mach number, wall temperature, and injection parameter. Some of the first-order results for wall heat transfer have been compared with the available experimental results at DFVLR.

Compressible second-order boundary layers for three-dimensional stagnation point flows with mass transfer

All the second-order boundary-layer effects have been studied for the steady laminar compressible 3-dimensional stagnation-point flows with variable properties and mass transfer for both saddle and nodal point regions. The governing equations have been solved numerically using an implicit finite-difference scheme. Results for the heat transfer and skin friction have been obtained for several values of the mass-transfer rate, wall temperature, and also for several values of parameters characterizing the nature of stagnation point and variable gas properties. The second-order effects on the heat transfer and skin friction at the wall are found to be significant and at large injection rates, they dominate over the results of the first-order boundary layer, but the effect of large suction is just the opposite. In general, the second-order effects are more pronounced in the saddle-point region than in the nodal-point region. The overall heat-transfer rate for the 3-dimensional flows is found to be more than that of the 2-dimensional flows.

Second-order boundary-layer flow with hard suction

The two-dimensional and axisymmetric compressible laminar boundary-layer flow over an arbitrary body with a porous wall is studied for the case in which suction at the wall is very high. Particular attention is paid to the second-order boundary-layer effects for the limiting case of hard suction, which leads to a singular perturbation problem. An important result is that curvature effects disappear and only displacement effects play a role in determining second-order behavior. It is shown that the second-order equations for the inner solution are greatly simplified, and analytical closed-form solutions are obtained for the shear stress and heat transfer at the wall. The shear stress is proportional to the suction-mass flux and the velocity at the outer edge of the boundary layer. The heat transfer is independent of second-order effects. In the example of the supersonic flow along a flat plate with homogeneous suction, the second-order boundary-layer effects lead to an increase of the drag.

Heat transfer in nonequilibrium dissociated hypersonic flow with surface catalysis and second-order effects.

An experimental investigation of heat transfer to hemispheres with both catalytic and noncatalytic surfaces in nonequilibrium dissociated hypersonic nitrogen flow has been conducted. The wind-tunnel flow conditions provided low Reynolds numbers, allowing the assumption of completely frozen shock and boundary layers while introducing the influences of second-order boundary-layer effects such as vorticity interaction. The results for copper surfaces, when correlated using the stagnation-point heat-transfer equation of Rosner together with Lees' heat-transfer distribution, are in good agreement with the second-order theory of Cheng under the assumption of complete surface recombination of atoms. Reductions in heat-transfer rate up to 40% were obtained with the use of noncatalytic coatings applied to the probes. Typically, values of the recombination coefficient range between 0.01 and 0.1 for the noncatalytic surfaces, compared with 1.0 for the catalytic surfaces.

Second-order boundary-layer effects in hypersonic flow past axisymmetric blunt bodies

First- and second-order boundary-layer theory are examined in detail for some specific flow cases of practical interest. These cases are for flows over blunt axisymmetric bodies in hypersonic high-altitude (or low density) flow where second-order boundary-layer quantities may become important. These cases consist of flow over a hyperboloid and a paraboloid both with free-stream Mach number infinity and flow over a sphere at free-stream Mach number 10. The method employed in finding the solutions is an implicit finite-difference scheme. It is found to exhibit both stability and accuracy in the examples computed. The method consists of starting near the stagnation-point of a blunt body and marching downstream along the body surface. Several interesting properties of the boundary layer are pointed out, such as the nature of some second-order boundary-layer quantities far downstream in the flow past a sphere and the effect of strong vorticity interaction on the second-order boundary layer in the flow past a hyperboloid. In several of the flow cases, results are compared with other theories and experiments.