Hypersonic viscous flow over slender bodies with sharp leading edges.

Abstract

The hypersonic viscous interaction is treated by the development of a set of equations valid throughout the boundary layer, shock-wave structure, and inviscid core. Although the primary interest is with the nature of the leading-edge continuum merged layer, where the hypersonic interaction parameter %« is of the order of the square of the Mach number, the theory is applicable for smaller values of %«> as well. Because of the parabolic nature of these equations, finite-difference solutions are attainable. The flow over a flat plate at zero incidence, as well as angle of attack, was considered. Calculated surface pressure, heat transfer, and shock-jump conditions in the merged layer were significantly below their strong-interaction values. These were approached downstream.specific heat at constant pressure and at constant volume, respectively .& = coefficient of thermal conductivity £ = length scale in tangential direction M = Mach number Ni = error terms in Eqs. (2-5) p = pressure R = gas constant Re m = Reynolds number based on freestream conditions T = temperature •u, v = tangential and normal velocity components, respectively x, y = tangential and normal body coordinates, respectively a = angle of attack 7 = ratio of specific heats y = C P /C V 6 = length scale in normal direction e = order of higher approximation in Eq. (5) X = mean free path JUL = coefficient of viscosity p = density a-= Prandtl number Xco = interaction parameter \l/ = stream function to = exponent in temperature-viscosity law Superscripts { ) = dimensional { ) = nondimensional Subscripts Jm = free molecular max = maximum value ref = local reference conditions RH = Rankine-Hugoniot .Y. t Associate Professor of Aerospace Engineering. Associate Member AIAA. s = schock wave si = shock layer stag = stagnation conditions 5 = boundary layer 0 = initial value oo = freestream quantities

Global pressure relaxation for laminar two‐dimensional internal flow

“…The subscript 'r' denotes reference value, h,, h,, h, are metric scale factors, u and u are the contravariant physical velocity components in the streamwise (5) and normal (q) co-ordinate directions, p is density, p is pressure, yr is the reference specific heat ratio, and the stagnation (or total) enthalpy H is defined (in non-dimensionai form) as H = 1 -2 T + -( u 2 + 0 ' ) .…”

“…where for convenience po (5) is identified with the value of pressure on the upper wall, P,,((); this portion is further split2' into a fully developed flow component pi,,(<) and a perturbation puw(5). Since only gradients of p occur in the equations, the value p f d , < can be added a priori in equation (3).…”

“…For compressible flow it is no longer numerically convenient to split the pressure, equation (14), since pressure is now related to density and velocity through the equation of state (5). Furthermore, for subsonic flow the downstream static pressure and upstream stagnation conditions completely determine the mass flow through the duct.…”

Interfacial instability of ferrofluid flow under the influence of a vacuum magnetic field

“…It is crucial to mathematically depict magnetics-related hydrodynamics. The parabolized stability equations (PSEs), which are first developed by Herbert [1] based on the parabolized Navier-Stokes (PNS) equations [2] , can be used to solve incompressible fluid problems. The nonlinearized PSE approach has been widely used to study the stability and transition.…”

Numerical computation of the hypersonic rarefied flow past the sharp leading edge of a flat plate

“…Initial values for each cell were specified and the solution was advanced in time for a small time increment by a two step finite difference process. In The area examined in this study extends from the freestream region ahead of the plate to the strong interaction 13 region. It can be argued that the Navier-Stokes equations are not valid in the near-free molecule and transition regions, but they can be solved in these regions and it is believed that this solution is at least a first-order approximation to the actual flow for moderate Mach numbers.…”