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Numerical solution of boundary value problem for nonlinear integro-differential equation system of hypersonic boundary layer
Numerical solutions for differences in ocean planetary boundary layer flow from that of steady-state spiral resulting from temporal wind variations
Numerical simulations are used to examine the roughness-induced transient growth in a laminar boundary-layer flow. Based on the spectral element method, these simulations model the stationary disturbance field associated with a nonsmooth roughness geometry, such as the spanwise periodic array of circular disks used by White and co-workers during a series of wind tunnel experiments at Case Western Reserve University. Besides capturing the major trends from the recent measurements by White and Ergin, the simulations provide additional information concerning the relative accuracy of the experimental findings derived from two separate wall-finding procedures. The paper also explores the dependence of transient growth on geometric characteristics of the roughness distribution, including the height and planform shape of the roughness element and the ratio of roughness due to spacing between an adjacent pair of elements. Results are used for a preliminary assessment of the differences between recently reported theoretical results of Tumin and Reshotko and the measurements by White and Ergin.
Numerical simulations demonstrate that laminar breakdown in a boundary layer induced by the secondary instability of two-dimensional Tollmien-Schlichting waves to three-dimensional subharmonic disturbances need not take the conventional lambda vortex/high-shear layer path.
Numerical simulations demonstrate that laminar breakdown in a boundary layer induced by the secondary instability of two-dimensional Tollmien-Schlichting waves to three-dimensional subharmonic disturbancews need not take the conventional lambda vortex/high-shear layer path.
Numerical integration of boundary layer ordinary differential equations
Numerical solutions of the instability wave pattern behind a harmonic point source in a Blasius boundary layer are used to form line sources by superposition. For infinite-length spanwise line sources of constant amplitude and phase, the result is just the two-dimensional normal mode of the same frequency; for a sinusoidal amplitude or linear phase distribution, the result is an oblique normal mode of the same spanwise wavenumber. A finite-length spanwise source simulates a vibrating ribbon. In a study of the influence of the source length on the downstream amplitude, the effective field of view is found to have a half-angle of about 16 deg. If the source tips are within this field, the amplitude may be either greater than or less than the comparable normal-mode amplitude, depending on the distance from the source and the spanwise location. For an oblique line source, the downstream wave development at each spanwise location is found to be close to, but not identical with, that of an oblique normal mode which originates at the source with the initial wave angle of the source and satisfies the irrotationality condition on the wavenumber vector.
Numerical and experimental studies have demonstrated that modal growth of planar Mack modes is responsible for laminar-turbulent transition on sharp cones at hypersonic speeds. However, the physical mechanisms that lead to transition onset upstream of the entropy-layer swallowing location over sufficiently blunt geometries are not well understood as yet. Modal amplification is too weak or nonexistent to initiate transition at moderate-to-large bluntness values. Nonmodal analysis shows that, with increasing nose bluntness, both planar and oblique traveling disturbances that peak within the entropy layer experience appreciable energy amplification. However, because of the relatively weak signature of the nonmodal traveling disturbances within the boundary-layer region, the route to transition onset subsequent to the nonmodal growth remains unclear. Thus, nonlinear parabolized stability equations (NPSE) and direct numerical simulations (DNS) have been used to investigate the potential transition mechanisms over a 7-degree blunt cone that was tested in the AFRL Mach-6 high-Reynoldsnumber facility. Computations are performed to separately follow the nonlinear development of two classes of inflow disturbances, namely, a pair of oblique traveling waves with equal but opposite angles with respect to the mean flow direction and a planar traveling wave. Results in both cases show an excellent agreement between the NPSE and DNS predictions, establishing that the NPSE is an accurate and efficient technique for predicting the nonlinear development for these particular nonmodal traveling disturbances. Computations reveal that the oblique mode interactions lead to the generation of stationary streaks inside the boundary layer that, in turn, facilitate the growth of a subharmonic sinuous disturbance. For relatively modest amplitudes of the inflow disturbance, the oblique-mode breakdown can lead to transition at the measured location of transition onset during the experiment. On the other hand, the nonlinear development of a planar traveling wave leads to the formation of inclined structures just above the boundary-layer edge and these structures are strongly reminiscent of the transitional events observed during blunt cone experiments by using schlieren flow visualizations.
Numerical solutions of the complete, time-averaged conservation equations using several eddy-viscosity models for the Reynolds shear stress to close the equations are compared with experimental measurements in a compressible, turbulent separated flow. An efficient time-splitting, explicit difference scheme was used to solve the two-dimensional conservation equations. The experiment used for comparison was a turbulent boundary layer that was separated by an incident shock wave in a Mach 2.93 flow with a unit Reynolds number of 5.7 x 10 to the seventh power m. Comparisons of predicted and experimental values of surface pressure, shear stress along the wall, and velocity profiles are shown. One of the tested eddy-viscosity models which allows the shear stress to be out of equilibrium with the mean flow produces substantially better agreement with the experimental measurements than the simpler models. A tool is thereby provided for inferring additional information about the flow, such as static pressures in the stream, which might not be directly obtainable from experiments.
Numerical simulations of the gas-seeding strategies required for planar laser-induced fluorescence in a Mach 10 (approximately Mach 8.2 postshock) airflow were performed. The work was performed to understand and quantify the adverse effects associated with gas seeding and to assess various types of seed gas that could potentially be used in future experiments. In prior experiments, NO and NO2 were injected through a slot near the leading edge of a flatplate wedge model used in NASA Langley Research Center's 31 in. Mach 10 air tunnel facility. In this paper, nitric oxide, krypton, and iodine gases were simulated at various injection rates. Simulations showing the deflection of the velocity boundary layer for each of the cases are presented. Streamwise distributions of velocity and concentration boundary-layer thicknesses, as well as vertical distributions of velocity, temperature, and mass distributions, are presented for each of the cases. A comparison between simulated streamwise velocity profiles and experimentally obtained molecular tagging velocimetry profiles using a nitric oxide seeding strategy is performed to verify the influence of such a strategy on the boundary layer. The relative merits of the different seeding strategies are discussed. The results from a custom solver based on OpenFOAM version 2.2.1 are compared against results obtained from ANSYS® Fluent version 6.3.
Numerical results support predictions of simplified theories. Report discusses aspects of algebraic and numerical modeling of flow in infinite half space on one side of infinitely-large, flat plate, with sinusoidally oscillating free-stream velocity along one axis of plate. Flow has rich variety of behaviors, including strong gradients of pressure, points of inflection, and reversal.
Numerical solution of boundary layer differential equations
Numerical solution of the three-dimensional incompressible Navier-Stokes equations is used to study the instability of a flat-plate boundary layer in a manner analogous to the vibrating-ribbon experiments. Flow-field structures are observed which are very similar to those found in the vibrating-ribbon experiment to which computational initial conditions have been matched. Streamwise periodicity is assumed in the simulation so that the evolution occurs in time, but the events which constitute the instability are so similar to the spatially occurring ones of the laboratory that it seems clear the physical processes involved are the same. A spectral and finite difference numerical algorithm is employed in the simulation.
Direct numerical simulations of sink-flow boundary layers, with acceleration parameters K between 1.5 x 10 to the -6th and 3.0 x 10 to the -6th, are presented. The three-dimensional, time-dependent Navier-Stokes equations are solved numerically, using a spectral method, with about one million degrees of freedom. The flow is assumed to be statistically steady, and self-similar. A multiple-scale approximation and periodic conditions are applied to the fluctuations. The turbulence is studied using instantaneous and statistical results. Good agreement with the experiments of Jones and Launder (1972) is observed. The two effects of the favorable pressure gradient are to extend the logarithmic layer, and to alter the energy balance of the turbulence near the edge of the boundary layer. At low Reynolds number the logarithmic layer is shortened and slightly displaced, but wall-layer streaks are present even at the lowest values of R(theta) for which turbulence can be sustained. Large quiescent patches appear in the flow. Relaminarization occurs at K = 3.0 x 10 to the -6th, corresponding to a Reynolds number R(theta) of about 330.
Steady boundary layer equations numerical solution method for planar and laminar problems, noting extension to axisymmetric and turbulent flow
A numerical method designed to solve the time-dependent, three-dimensional, incompressible Navier-Stokes equations in boundary layers is presented. The fluid domain is the half-space over a flat plate, and periodic conditions are applied in the horizontal directions. The discretization is spectral. The basis functions are divergence-free and a weak formulation of the momentum equation is used, which eliminates the pressure term. An exponential mapping and Jacobi polynomials are used in the semi-infinite direction, with the irrotational component receiving special treatment. Issues related to the accuracy, stability and efficiency of the method are discussed. Very fast convergence is demonstrated on some model problems with smooth solutions. The method has also been shown to accurately resolve the fine scales of transitional and turbulent boundary layers.
Recent numerical simulations of accretion disk boundary layers have shown qualitatively the importance tance of advected energy in the inner region of the disk. In this short paper we present quantitative results of advective boundary layers in the optically thick regime. Numerical results are obtained for various systems, by means of a one-dimensional time-dependent numerical code. At high accretion mass rates, dot-M approx. = 10(exp -4) solar mass/yr, or low values of the viscosity parameter, alpha approx. = 0.001-0.01 (characteristic of FU Orionis systems and some symbiotic stars), the optical thickness in the inner part of the disk becomes very large (tau much greater than 1). The disk, unable to cool efficiently, becomes geometrically thick (H/r approx. = 0.5). The energy dissipated in the dynamical boundary layer is radiated outward to larger radii and advected into the star. The boundary-layer luminosity is only a fraction of its expected value; the rest of the energy is advected into the star. The fraction of the advected energy is zeta = L(sub adv)/L(sub acc) approx. = 0.1 in symbiotic stars (accretion onto a low-mass main-sequence star) and zeta approx. = 0.2 in FU Ori systems (accretion onto a pre-main sequence star).
A finite difference turbulent boundary layer computer program has been developed. The program is primarily oriented towards the calculation of boundary layer performance losses in rocket engines; however, the solution is general, and has much broader applicability. The effects of transpiration and film cooling as well as the effect of equilibrium chemical reactions (currently restricted to the H2-O2 system) can be calculated. The turbulent transport terms are evaluated using the phenomenological mixing length - eddy viscosity concept. The equations of motion are solved using the Crank-Nicolson implicit finite difference technique. The analysis and computer program have been checked out by solving a series of both laminar and turbulent test cases and comparing the results to data or other solutions. These comparisons have shown that the program is capable of producing very satisfactory results for a wide range of flows. Further refinements to the analysis and program, especially as applied to film cooling solutions, would be aided by the acquisition of a firm data base.