Julian D. Cole

Problems in high speed flow prediction relevant to control

Three flow problems are discussed whose solutions suggest flow control schemes. These are 1) unsteady hypersonic flow over bodies in the Newtonian approximation, 2) a mechanism of hypersonic flow stabilization over acoustically semi-transparent walls and 3) store separation from cavities. Simplified systematic approximations based on asymptotic frameworks lead to compact computational models that elucidate the flow structure and opportunities for control. Besides generalizing the steady model of Cole, the Newtonian approximation in the unsteady context shows that unsteady body perturbations can lead to inflectional velocity profiles that can produce instabilities and boundary layer transition to enhance mixing in combustors and inlets. The absorbing wall illustrates a mechanism that can be exploited to damp 2 mode hypersonic instabilities. Simplified flow modeling based on systematic asymptotics for store separation from cavities shows the influence of the cavity shear layer on apparent mass effects that are important to damping in pitch and clearance from the parent body. Comparisons with free drop experiments are used for initial validations of the analytical models. * Senior Scientist, Fellow, AIAA f Principal Researcher, Member, AIAA * Margaret Damn Distinguished Professor, Mathematical Sciences, Fellow, AIAA § Professor ** Professor, Associate Fellow Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc. 1. Unsteady Newtonian thin shock layers and hypersonic flow stability 1.11ntroduction Although the stability of high speed flows has received much attention in the recent literature, major complicating aspects have not been treated in a unified way. These features include the combined effects of the finite shock displacement on the boundary layer, the nonparallelism of the flow and the vorticity introduced by the shock curvature. The relevant structure of the shock and boundary layers has been treated in [1][9]. In [6] and [7], the aforementioned stability issues were discussed within the Hypersonic Small Disturbance approximation for the inviscid deck strongly interacting with the hypersonic boundary layer. Equations of motion for the mean and fluctuating small amplitude flows were analyzed. Because of nonparallelism in this framework, the spatial part of the waves cannot be treated by the usual Fourier decomposition and an initial value rather than eigenproblem for spatial stability is obtained. The initial value problem leads to partial rather than ordinary differential equations that require a numerical marching method for their solution. Results indicate that the specific heat ratio 7 plays a major role in the stability of flow since it controls the reflection of waves from the shock and the radiation of energy in the shock layer whose thickness scales with 7 -1. Early experiments such as those described in [2] showed that for a practically interesting class of flows, the shock layer becomes very thin compared to the boundary layer near the nose of hypersonic flat plates. This feature and the desire to further understand the shock and boundary layer structure encourage the use of the Newtonian approximation 7 —> 1. The connection with flow stability motivates the study of this approximation in an unsteady context. In this chapter, limit process expansions will be discussed relevant to unsteady viscous interactions as a prelude to the analysis of hypersonic stability and transition. The application of these limits is an unsteady extension of the steady state analysis of [3]. Although the focus here is the treatment of viscous interaction, boundary layer stability, receptivity and transition, the results derived are useful in inviscid hypersonic unsteady aerodynamic methodology and load prediction as well. 1.2 Analysis Figure 1 schematically indicates strong interaction flow near the leading edge of a hypersonic body. The viscous boundary layer which is usually thin, occupies an appreciable fraction of the distance between the shock and body that will be considered without undue loss of generality a flat plate in what follows. Accordingly F(x,f} = Q, in the notation of Fig. 1. The results in this chapter will be expressed in terms of the boundary layer thickness function A(3c,r) = 0, which in the interpretation mentioned in the Introduction could be the body shape in an inviscid context. Copyright© 1998, American Institute of Aeronautics and Astronautics, Inc. The unsteady form of the Hypersonic Small Disturbance Theory (HSDT) equations [9] are applicable and are obtained as in [7] from limit process expansions of hatted variables defined as quantities normalized by their freestream counterparts, with p,T,u,v,fJL the density, temperature, horizontal, vertical components of the velocity vector, and viscosity respectively. If the freestream density, pressure and velocity are denoted as U,p^ and p^ respectively, then a pressure coefficient used in these expansions is defined as p = (P-PJ/P-U. Fig. 1 Schematic of hypersonic strong interaction flow. With these definitions and the coordinate system in Fig. 1 as well the normalization of the Cartesian dimensional coordinates x and y to the unit reference length L and the reference time scale L/U for the time t, unbarred dimensionless normalized counterparts of these independent variables are defined. If M^ and R^ are respectively the freestream Mach and Reynolds numbers, and 5 is a characteristic flow deflection angle, then the expansions are p=a(x,y,t;H,y)+--(1.1) T=T+— p = 8p+M = l+v =• §v+• • (1.2) (1.3) (1.4) (1.5) (1.6) where y = y/(L8}. These expansions are valid in the HSDT limit x, y, t, H = M o are fixed as 8 — > 0 ,

Modern Developments in Transonic Flow

A survey is given of transonic small disturbance theory. Basic equations, shock relations, similarity laves, lift and drag integrals are derived., The airfoil boundary value problem is formulated. Finite difference methods and computational algorithms are described. Results are compared with other calculation methods and experiments.

A Singular Perturbation Analysis of Induced Electric Fields in Nerve Cells

The electric field induced by a microelectrode inserted in a nerve cell is investigated in order to interpret the results obtained by the single and double probe techniques. The solution is obtained by means of a singular perturbation expansion in terms of the ratio of the membrane conductance to the cell conductivity. Both finite and infinite cells are considered and special attention is devoted to the spherical and cylindrical geometries.

Matched Asymptotic Expansions of the Green’s Function for the Electric Potential in an Infinite Cylindrical Cell

The potential is studied for a microelectrode current source inside a nerve fiber. The problem is represented by a point source in an infinitely long cylindrical conductor surrounded by a thin, low conductivity membrane bathed in a perfectly conducting medium. The potential satisfies Laplace’s equation with a mixed boundary condition containing a small parameter

Wave drag due to lift for transonic airplanes

\varepsilon

Limit process expansions and homogenization

. As

Excitation of convectively and absolutely unstable disturbances in three-dimensional boundary layers

\varepsilon \to 0

Note on the Axisymmetric Sonic Jet.

it approaches a homogeneous Neumann condition and the problem becomes singular. Asymptotic expansions are obtained in terms of

A blunt-nosed thin body in hypersonic flow

\varepsilon

On a blunt-body problem in hypersonic flow theory

by matching an inner expansion (valid at the source) to an outer expansion (valid away from the source). The inner expansion contains algebraic switchback terms whose orders are half-odd-integer powers of