Numerical solution of boundary-layer flows with massive blowing

Abstract

MARCH 1971TECHNICAL NOTES 533 the technique used previously by Mitchell, Crocco, and Sirignano 1 and Crocco and Mitchell. 2 The relevant dependent variables u, p, r, C, cr and w are represented as power series in e. For example, p = p 0 + epi + e 2 p 2 + e 3 p 3 + . . . etc., where e = M 112 . The coefficients of e in these expansions are comprised of a steady-state and a time-dependent part. Thus, pi = pi + pi 1 , Pz = PZ + P2 1 etc. Both u\ and rj are taken to be of 0(e). Since, as will be discussed shortly, the analysis is carried out through 0 (e 5 ) , the combustion zone can actually occupy a fairly sizeable fraction of the chamber axial length, and is not of zero length as in the "concentrated combustion" model used by Mitchell, Crocco, and Sirignano. 1 It is necessary to use the simplest kind of coordinate stretching in order to ensure well-behaved periodic solutions even when waveforms are discontinuous. Therefore, the stretched time variable 5 is introduced and P = 2(1 + cPi + e 2 P 2 + ...). P is the nondimensional period of the oscillations and 2 is the wave travel time for an acoustic (zero amplitude) wave. The technique of multiple scales is applied with the introduction of a second axial variable y, which is defined as y = x/rj. Derivatives of u and w with respect to this variable are of 0(1) instead of 0(l/e) as they are with respect to x. All dependent variables are then considered to be functions of the 3 independent variables x, y, and 5. Thus, u = u(x,y,d), p = p(x,y,5), etc. The governing partial differential equations and the droplet vaporization equation are then rewritten in these variables, and the power series representations of the dependent variables are substituted into the equations which result.The system of equations is first solved for the steady state. To the order of approximation necessary for consistent solution of the time-dependent equations, the results are = w 2 = 1 -(1 -?/) 3 = 1, pz = yu 2 ui = Wi = Us = WS = pi = Pi = 00 = 0"! = (72 = 0 r 0 = (I-2/) 1 ' 2 = ui/2C Q Carrying out the analysis of the time-dependent equations through 0(e 3 ) and applying the appropriate boundary conditions leads to the following expressions: U2 1 = /(« -x) -f(d + x) p^1 -2rJ! f' J 8 -Ui l = pi 1 = = (71 1 = Wi l = = 0Here, $ = rji/ui (the steady-state droplet lifetime) and / is an arbitrary function periodic in 2. In order to determine the form of / and therefore of p 2 and u%, it is necessary to continue the analysis through 0 (e 5 ) . Doing this and applying the appropriate order boundary conditions, the following nonlinear integro-differential equation for / is finally derived:(87 -1 -where ^ = 2(7 -!)</),The developments that resulted in Eq. (2) are valid for either continuous or discontinuous pressure waves. In order to determine the waveform and amplitude of the pressure oscillations, this equation must be solved for either type of oscillations. Solution of Eq. (2) for small / is easily carried out by linearizing the equation and assuming / = sin?r6. A neutral stability relationship...

Stagnation Region Radiative Ablation of Asteroidal Meteoroids in the Rosseland Approximation

“…For this massive ablation [13], the thickness of the layer consisting of the ablation product becomes much larger than the δ given by Eq. (4).…”

“…The first term in Eq. (3) is nearly zero except in the interface region [13]. Numerical integration of Eq.…”

“…(12)(13)(14) in the stagnation region of interest because of the singularity occurring at the axis of symmetry r 0.…”

Three-dimensional laminar compressible boundary layers with large injection

“…t Present address : Aerodynamics Division, National Aeronautical Laboratory, Banga-C. S. Vimala and G. Nath t,he problem considered by Kubota & Fernandez (1968) directly from the original equations for the hard-blowing case. Nachtsheim & Green (1971) and Liu & Nachtsheim (1973) have proposed a purely numerical method through a conversion of the two-point boundary-value problem into a three-point boundary-value problem. The incompressible boundary-layer flow at a t,hree-dimensional stagnation point with strong blowing has been investigated by Walton (1973) by the method of matched asymptotic expansions.…”

“…The incompressible boundary-layer flow at a t,hree-dimensional stagnation point with strong blowing has been investigated by Walton (1973) by the method of matched asymptotic expansions. The methods of Kassoy (1971)) Nachtsheim & Green (1971) and Liu & Nachtsheim (1973) involve t,hree-point boundary-value problems which can be solved only by an inverse method. On the other hand, in the method due to Kubota & Fernandez (1968), one faces only a two-point boundary-value problem, which can be solved directly.…”