Nathan Gerber

Characteristic calculation of flowfields with chemical reactions.

The accuracy and the relative simplicity of the construction of the method discussed have been amply demonstrated by example problems solved in Refs. 1 and 2 and in this paper. Geometrical constructions have been used in these examples, i.e., distances measured directly on the drawing board and circles drawn by compass. These geometrical constructions and numerical calculations can be easily programed on a digital computer, and greater accuracy can be expected. Further refinement of the method by considering the change of radius of curvature according to the Gauss equation is also feasible. Currently, this method is being applied to nozzle design problems and problems involving viscous fluids.

Shock curvature and gradients at the tip of pointed axisymmetric bodies in non-equilibrium flow

Abstract : The shock curvature and flow variable gradients at the tip of a pointed body caused by nonequilibrium effects are considered. Coordinates introduced by Chester are used since they offer a convenient way of treating the boundary conditions. The desired functions are obtained by solving numerically a system of linear ordinary differential equations. These equations have a singularity; the nature of the singularity is found analytically, and its numerical treatment is discussed. The specific nonequilibrium effect considered is vibrational relaxation in a pure diatomic gas. Representative results are given for flow of N2 over a cone for a comprehensive range of Mach numbers and cone angles. There is a point analagous to the Crocco point. These results compare favorably with those obtained by South and Newman using an approximate method. Another check is made by comparison with characteristic calculations extrapolated to the origin.

Dynamics of the fluid in a spinning coning cylinder

The fluid motion inside a cylinder that simultaneously spins and cones is determined according to linear theory for small coning angles. The Navier-Stokes equations are solved by expansions in spatial eigenfunctions. This form of spectral method gives an efficient solver over a range of Reynolds numbers; cases for Re ≤ 2500 have been computed. The results are validated by comparing computed pressure and moment coefficients with experimental observations and numerical calculations

Decay of streamwise vorticity downstream of a three-dimensional protuberance

Streamwise vortices generated in boundary layers are known to persist for hundreds of boundary-laye r thicknesses. This remarkable property is the main topic of this paper. The particular case of the flowfield generated by a three-dimensional protuberance immersed in a flat plate boundary layer is studied using numerical modeling. Experimental measurements in a downstream crossflow plane are used to construct initial data for a marching calculation using the boundary-layer and boundary-region approximations; the latter includes crossflow diffusion terms. The former fails to predict the flowfield, but results from the latter agree qualitatively with aspects of the flowfield determined from the measurements of Tani et al. and are also in fair quantitative agreement.

Note on shock reflection coefficient.

Abstract : The shock reflection coefficient, R sub w, is defined here as the ratio of pressure drop across a reflected disturbance from a shock wave to that across the incident disturbance wave (which is generated downstream of the shock). The treatment of shock reflection coefficients in the work of previous investigators is reviewed briefly. A derivation is given of a general expression for R sub w in terms of shock curvature for nonequilibrium axisymmetric flow. Sample computations show the effects of nonequilibrium flow and three-dimensionality on the behavior of R sub w. It is shown that R sub w can become infinite in these situations, although for two-dimensional ideal gas flow its magnitude is always finite and usually very small.

High Reynolds Number Flows in Rotating and Nutating Cylinders: Spatial Eigenvalue Approach

The spatial eigenvalue expansion procedure that we have used previously to study viscous flow in rotating and nutating cylinders is studied in the high Reynolds number limit. It is shown that in this limit there is a significant simplification in the procedure employed to determine the amplitudes of the eigenfunctions used to describe the forced velocity field in the cylinder. More precisely, it is shown that these amplitudes can, to the order of Reynolds number used in this paper, be simply expressed in terms of Bessel functions. The method is used to calculate the pressure coefficient at a point on the end wall. Detailed comparisons with experimental results are made. It is found that the approach provides an accurate and extremely rapid way of determining this pressure coefficient. I. Introduction and Formulation I T is well known that there can be a significant difference in behavior in flight between liquid-filled and solid-filled projectiles. The difference is caused by the motion of the liquid inside the spinning projectile. This motion causes forces to act on the projectile that can ultimately cause the flight of the projectile to be prematurely terminated by instability. The initial motion of the projectile necessarily causes the fluid motion in the cylinder to be time dependent; later it can be assumed that the flow is steady. It is the latter situation that is investigated here. Our concern then is with the motion of an incompressible viscous fluid of kinematic viscosity v in a cylinder of length 2A a and radius a. The cylinder is spinning at a constant rate about its axis, which, in turn, is nutating at a constant rate and yaw angle about an axis directed along its trajectory. Throughout this calculation, a is used as an appropriate length scale and the magnitude of a typical velocity is taken to be (ft + T cos#0)fl, where ft is the magnitude of the angular velocity of the projectile as observed in the aeroballistic reference frame, T is the dimensional coning frequency, and K0 is the coning angle. (In the aeroballistic frame, one Cartesian coordinate lies along the cylinder axis and a second coordinate lies in the plane of the cylinder axis and trajectory; see, e.g., Ref. 1.) KQ is assumed small throughout this work so that it is possible to determine the properties of the forced motion using a simple perturbation expansion in powers of K0. We nondimensionalize the coning frequency as follows,