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Computational results are presented for the transitional or turbulent flow about a prolate spheroid, at alpha = 10 deg or 30 deg, correspondingly, using an implicit, approximately factored, partially flux-split algorithm, based on the thin-layer equations. The computed flow field is in good agreement with available experimental data.
An efficient and accurate method for solving the viscous shock layer equations for hypersonic flows over long slender bodies is presented. The two first order equations, continuity and normal momentum, are solved simultaneously as a coupled set. The flow conditions included are from high Reynolds numbers at low altitudes to low Reynolds numbers at high altitudes. For high Reynolds number flows, both chemical nonequilibrium and perfect gas cases are analyzed with surface catalytic effects and different turbulence models, respectively. At low Reynolds number flow conditions, corrected slip models are implemented with perfect gas case. Detailed comparisons are included with other predictions and experimental data.
A solution procedure is presented which considerably improves the computational efficiency of the viscous-shock-layer technique, especially for long slender bodies. The 'predictor-corrector' procedure suggested for obtaining the shock shape beyond the nose region requires only a single global pass. The accuracy of the present method is demonstrated by comparison with globally iterated results over the entire body and with ground- and flight-test data. A good comparison of the results computed with the two methods is shown for different flowfield chemistry models and axisymmetric body shapes. The new procedure results in computer run times 1/3 to 1/2 of the times required for the full-body global iteration procedure. Further, the algebraic expressions used to specify the initial shock shape eliminate the need for a shock shape generated by external means and permit immediate introduction of the full viscous-shock-layer equations. Finally, the present method of solution for the VSL equations provides to the aerothermal designer a very efficient and accurate tool for detailed flowfield as well as future technology studies.
The computational equations governing the shock-layer flowfield over a blunt body dominated by radiation and ablation are derived. The flowfield considered is that expected around the vehicles entering the atmosphere of the planet Mars or the earth on return from Mars on a sprint-type manned mission. To determine the appropriate method of approach, the radiative transfer calculations are made first using the existing computer code RASLE (Nicolet et al., 1978), which uses the assumption of equilibrium and a coarse spectral band model for radiative transfer, and the code NONEQ which accounts for thermochemical nonequilibrium and carries out line-by-line calculation of radiation. By comparing the results, it is concluded that a thermochemical nonequilibrium model and a line-by-line spectral model must be used.
Continuum shock mixture models are reviewed and applied to determine the equations of state for five different compositions of Ni x Al y , as well as bulk Ni+Al reactive multilayers, by combining the fundamental property data for elemental nickel and aluminum. From the literature, we down-select and evaluate two analytical models for the mixture Hugoniot, i.e., the well-known method of kinetic energy averaging (KEA) and a recent model proposed by Jordan and Baer [J. Appl. Phys. 111, 083516 (2012)]. Fundamentally, the former method assumes pressure equilibrium, whereas the latter assumes a common particle velocity and mixture sound speed from compressible two-phase cavitating flows. Additionally, we construct thermodynamically complete equations of state by fitting Einstein oscillator series models for the specific heat at constant volume. Finally, the solid solution approximation is invoked for intermetallic compositions, which are not strictly physical mixtures. Overall, the KEA model provides a better fit to the available Ni x Al y and Ni+Al multilayer shock compression data; however, there are combinations of material properties where the performance of these two models is thought to be reversed. Moreover, the results of this work include the first analytical solution of Jordan–Baer that does not require numerical root finding, as well as proposed modifications to the Einstein oscillator series to incorporate some effects of local pressure–temperature equilibrium and reaction–diffusion. Future work is planned that will use these equations of state in mesoscale simulations to study shock-induced reaction in Ni+Al multilayers, and the intended application is illustrated with a brief 2D hydrocode example.
Results of analysis on damage distribution of a crack layer, in a model material, supported the self-similarity hypothesis of damage evolution which has been adopted by the crack layer theory. On the basis of measurements of discontinuity density and the double layer potential technique, a solution to the crack damage interaction problem has been developed. Evaluation of the stress intensity factor illustrated the methodology. Analysis of experimental results showed that Arrhenius type constitutive relationship described very well the expansion of the active zone of a crack layer.
Essential aspects are presented for computing flow past an oblique wing with the thin-layer Navier-Stokes equations. A new method is developed for generating a grid system around a realistic wing. This method utilizes a series of conformal transformations. The thin-shear-layer approximation and an algebraic eddy-viscosity turbulence model are used to simplify the Reynolds-averaged Navier-Stokes equations. An implicit, factored numerical scheme and the concept of pencil data structure are utilized. For the first time, some flow fields caused by the oblique wing in a supersonic free stream are discussed, emphasizing the separated vortex flows associated with such a wing.
A finite-volume scheme for numerical integration of the Euler equations was extended to allow solution of the thin-layer Navier-Stokes equations in two and three dimensions. The extended algorithm, which is based on a class of four-stage Runge-Kutta time-stepping schemes, was made numerically efficient through the following convergence acceleration technique: (1) local time stepping, (2) enthalpy damping, and (3) residual smoothing. Also, the high degree of vectorization possible with the algorithm has yielded an efficient program for vector processors. The scheme was evaluated by solving laminar and turbulent flows. Numerical results have compared well with either theoretical or other numerical solutions and/or experimental data.
Vortex flows caused by a swept wing are described in some detail. The thin-shear-layer Navier-Stokes equations with an algebraic eddy-viscosity turbulence model are solved. An implicit, factored numerical scheme is used. On the upper surface of the wing, flow is primarily along the free-stream and spanwise direction, respectively, at small and large angles of attack.
Downstream marching iterative schemes for the solution of the Parabolized or Thin Layer (PNS or TL) Navier-Stokes equations are described. Modifications of the primitive equation global relaxation sweep procedure result in efficient second-order marching schemes. These schemes take full account of the reduced order of the approximate equations as they behave like the SLOR for a single elliptic equation. The improved smoothing properties permit the introduction of Multi-Grid acceleration. The proposed algorithm is essentially Reynolds number independent and therefore can be applied to the solution of the subsonic Euler equations. The convergence rates are similar to those obtained by the Multi-Grid solution of a single elliptic equation; the storage is also comparable as only the pressure has to be stored on all levels. Extensions to three-dimensional and compressible subsonic flows are discussed. Numerical results are presented.
Downstream marching iterative schemes for the solution of the Parabolized or Thin Layer (PNS or TL) Navier-Stokes equations are described. Modifications of the primitive equation global relaxation sweep procedure result in efficient second-order marching schemes. These schemes take full account of the reduced order of the approximate equations as they behave like the SLOR for a single elliptic equation. The improved smoothing properties permit the introduction of Multi-Grid acceleration. The proposed algorithm is essentially Reynolds number independent and therefore can be applied to the solution of the subsonic Euler equations. The convergence rates are similar to those obtained by the Multi-Grid solution of a single elliptic equation; the storage is also comparable as only the pressure has to be stored on all levels. Extensions to three-dimensional and compressible subsonic flows are discussed. Numerical results are presented.
The equation of motion was used to examine the problem of vibrations of a thin elastic plate covered with a thin viscous layer. Mathematical formulation and the solution of the problem are given.
The paper reports a numerical procedure developed for a two-dimensional azimuthal (or planar) invariant form of the thin-layer Navier-Stokes equations. Generalization of the governing equations is described, and the equations are solved with an implicit approximate factorization finite-difference scheme. Inviscid and viscous results are presented for both external and internal flows and for spinning and nonspinning bodies.
The viscous, Navier-Stokes solver for turbomachinery applications, MSUTC has been modified to include the rotating frame formulation. The three-dimensional thin-layer Navier-Stokes equations have been cast in a rotating Cartesian frame enabling the freezing of grid motion. This also allows the flow-field associated with an isolated rotor to be viewed as a steady-state problem. Consequently, local time stepping can be used to accelerate convergence. The formulation is validated by running NASA's Rotor 67 as the test case. results are compared between the rotating frame code and the absolute frame code. The use of the rotating frame approach greatly enhances the performance of the code with respect to savings in computing time, without degradation of the solution.
The flowfield about a NACA 0012 airfoil at a Mach number of 0.3 and Reynolds number of 1 million is computed through an angle of attack range, up to 18 deg, corresponding to conditions up to and beyond the maximum lift coefficient. Results obtained using the compressible thin-layer Navier-Stokes equations are presented as well as results from the compressible Euler equations with and without a viscous coupling procedure. The applicability of each code is assessed and many thin-layer Navier-Stokes benchmark solutions are obtained which can be used for comparison with other codes intended for use at high angles of attack. Reasonable agreement of the Navier-Stokes code with experiment and the viscous-inviscid interaction code is obtained at moderate angles of attack. An unsteady solution is obtained with the thin-layer Navier-Stokes code at the highest angle of attack considered. The maximum lift coefficient is overpredicted, however, in comparison to experimental data, which is attributed to the presence of a laminar separation bubble near the leading edge not modeled in the computations. Two comparisons with experimental data are also presented at a higher Mach number.
The eigenvalue spectrum of the Rayleigh equation is examined using three different solution techniques. In particular, a simple second-order finite difference scheme and two spectral methods, the Chebyshev tau and Chebyshev collocation methods, are used to discretize the equation. All of the approximation methods are shown to be capable of predicting the discrete spectrum as well as the continuous spectrum associated with the critical point singularity for the Rayleigh equation. The global eigenvalue methods considered here provide an efficient way of obtaining either an approximation to the complete eigenvalue spectrum or initial guesses for a local shooting procedure for the discrete part of the spectrum.
Observations and theoretical models of small-scale phenomena in the oceans are reviewed, with a focus on progress during the period 1983-1986. Topics examined include surface layers, equatorial turbulence, off-equator mixed layers, the scaling of mixing, turbulence concepts, laboratory results, internal waves and mixing, rings, the nature of the bottom layer, double diffusion and intrusions, salt fingers, and biological interactions. Also discussed are developments in instrumentation (fast sampling profilers with upward-profiling capability, deep profilers, ship-motion correction, horizontal samplers, small submersibles, submarines, towed packages, conductivity sensors, dissolved-oxygen sensors, and acoustic Doppler current profilers) and goals for future research.
We present a detailed analysis of a recently proposed perfectly matched layer (PML) method for the absorption of acoustic waves. The split set of equations is shown to be only weakly well-posed, and ill-posed under small low order perturbations. This analysis provides the explanation for the stability problems associated with the split field formulation and illustrates why applying a filter has a stabilizing effect. Utilizing recent results obtained within the context of electromagnetics, we develop strongly well-posed absorbing layers for the linearized Euler equations. The schemes are shown to be perfectly absorbing independent of frequency and angle of incidence of the wave in the case of a non-convecting mean flow. In the general case of a convecting mean flow, a number of techniques is combined to obtain a absorbing layers exhibiting PML-like behavior. The efficacy of the proposed absorbing layers is illustrated though computation of benchmark problems in aero-acoustics.