The effect of boundaries on the velocity deficit and the limit of gaseous detonations
Boundary effects on gaseous detonation velocity deficit and limit using approximation into boundary layer equations, comparing to experimental data
Tsuge, S.↗
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Results for “Approximate boundary”
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Numerical solution to the glancing sidewall oblique shock wave/turbulent boundary layer interaction in three dimension
A supersonic three-dimensional viscous forward-marching computer design code called PEPSIS is used to obtain a numerical solution of the three-dimensional problem of the interaction of a glancing sidewall oblique shock wave and a turbulent boundary layer. Very good results are obtained for a test case that was run to investigate the use of the wall-function boundary-condition approximation for a highly complex three-dimensional shock-boundary layer interaction. Two additional test cases (coarse mesh and medium mesh) are run to examine the question of near-wall resolution when no-slip boundary conditions are applied. A comparison with experimental data shows that the PEPSIS code gives excellent results in general and is practical for three-dimensional supersonic inlet calculations.
Anderson, B. H.↗
Numerical solution to the glancing sidewall oblique shock wave/turbulent boundary layer interaction in three-dimension
A supersonic three-dimensional viscous forward-marching computer design code called PEPSIS is used to obtain a numerical solution of the three-dimensional problem of the interaction of a glancing sidewall oblique shock wave and a turbulent boundary layer. Very good results are obtained for a test case that was run to investigate the use of the wall-function boundary-condition approximation for a highly complex three-dimensional shock-boundary layer interaction. Two additional test cases (coarse mesh and medium mesh) are run to examine the question of near-wall resolution when no-slip boundary conditions are applied. A comparison with experimental data shows that the PEPSIS code gives excellent results in general and is practical for three-dimensional supersonic inlet calculations.
Anderson, B. H.↗
Euler procedure for three-dimensional transonic wall interference
Based on an optimization formulation, a procedure has been developed to evaluate Mach number and angle-of-attack corrections. The Euler equations are assumed to be the flow governing equations. To obtain efficient solutions for the optimization problem, the iterative solutions for the flow variables and the design parameters are simultaneously updated. In addition to the model lift and geometry, the procedure requires pressure measurements near the tunnel walls. The tunnel boundary conditions are based on the introduction of Reimann invariants for a one-dimensional flow normal to the boundary. Computations are performed to verify that the errors introduced by this approximate boundary-condition formulation are acceptably small. The correction scheme is applied to an aircraft configuration in an open jet. The results indicate that the optimization scheme is highly efficient with the rate of convergence of the flow solution nearly equal to the corresponding rate of a regular analysis problem.
Rizk, Magdi H.↗
Torsional stiffness of thin-walled shells having reinforcing cores and rectangular, triangular, or diamond cross section
A theoretical investigation has been made of the Saint-Venant torsion of certain composite bars. These bars are composed of two materials -- one material in the form of a thin-walled cylindrical shell and the other material in the form of a core which fills the interior of the shell and is bonded to it. An approximate boundary-value problem is formulated on assumptions similar to those of the theory of torsion of hollow thin-walled shells (Bredt theory). This boundary-value problem is solved exactly for a rectangular cross section and approximately for slender triangular and diamond cross sections. Results for the torsional stiffness constants are presented graphically.
Mccomb, Harvey G , Jr↗
Intermediate boundary conditions for LOD, ADI and approximate factorization methods
A general approach to determining the correct intermediate boundary conditions for dimensional splitting methods is presented. The intermediate solution U is viewed as a second order accurate approximation to a modified equation. Deriving the modified equation and using the relationship between this equation and the original equation allows us to determine the correct boundary conditions for U*. This technique is illustrated by applying it to locally one dimensional (LOD) and alternating direction implicit (ADI) methods for the heat equation in two and three space dimensions. The approximate factorization method is considered in slightly more generality.
Leveque, R. J.↗
Approximate solution of the shear flow boundary layer on a flat plate.
Nonsimilar solution of incompressible boundary layer over flat plate in presence of shear flow, using approximate method
SHEAR FLOW↗
Approximate method of integration of laminar boundary layer in incompressible fluid
A method is given for the approximate solution of the equations of the two-dimensional laminar boundary layer in an incompressible fluid. The method is based on the use of a system of equations of successive moments that is easily solved for simple supplementary assumptions. The solution obtained is given in closed form by simple formulas and is claimed to be no less accurate than the complicated solutions previously obtained, which were based on the use of special classes of flows.
BOUNDARY LAYER - WING SECTIONS↗
Parameter estimation in moving boundary problems
An approximation scheme which can be used to estimate unknown parameters in moving boundary problems is described. The model equations are fairly general nonlinear diffusion/reaction equations of one spatial variable. Conditions on the parameter sets and model equations are given. It is shown that the estimates obtained using the approximations will converge to best-fit parameters for the original model equations.
Murphy, K. A.↗
Boundary conditions for exterior acoustic problems
Problems of acoustics are generally posed in unbounded regions. Numerical computations require that the infinite regions be truncated with artificial finite boundaries. Boundary conditions must be developed at these artificial boundaries which approximate the conditions for outgoing waves. These conditions should be asymptotic in the distance of the boundaries from the source terms and should give rise to well posed problems interior to fairly arbitrary regions, which can be chosen on the basis of computational efficiency rather than for mathematical (i.e., well posedness) reasons. A family of such conditions is presented and their properties are discussed. These conditions can be thought of as extensions of the Sommerfeld radiation condition for outgoing waves.
Bayliss, A.↗
Spinning mode sound propagation in ducts with acoustic treatment and sheared flow
The propagation of spinning mode sound was considered for a cylindrical duct with sheared steady flow. Calculations concentrated on the determination of the wall optimum acoustic impedance and the maximum possible attenuation. Both the least attenuated and higher radial modes for spinning lobe patterns were considered. A parametric study was conducted over a wide range of Mach numbers, spinning lobe numbers, sound frequency, and boundary layer thickness. A correlation equation was developed from theoretical considerations starting with the thin boundary layer approximation of Eversman. This correlation agrees well with the more exact calculations for inlets and provides a single boundary layer refraction parameter which determines the change in optimum wall impedance due to refraction effects.
Rice, E. J.↗
Spinning mode sound propagation in ducts with acoustic treatment and sheared flow
The propagation of spinning mode sound was considered for a cylindrical duct with sheared steady flow. The calculations concentrated on the determination of the wall optimum acoustic impedance and the maximum possible attenuation. Both the least attenuated and higher radial modes for spinning lobe patterns were considered. A parametric study was conducted over a wide range of Mach numbers, spinning lobe numbers, sound frequency, and boundary layer thickness. A correlation equation was developed from theoretical considerations starting with the thin boundary layer approximation of Eversman. This correlation agrees well with the more exact calculations for inlets and provides a single boundary layer refraction parameter which determines the change in optimum wall impedance due to refraction effects.
Rice, E. J.↗
Flame Shapes of Nonbuoyant Laminar Jet Diffusion Flames
The shapes (flame-sheet and luminous-flame boundaries) of steady nonbuoyant round hydrocarbon-fueled laminar-jet diffusion flames in still and coflowing air were studied both experimentally and theoretically. Flame-sheet shapes were measured from photographs using a CH optical filter to distinguish flame-sheet boundaries in the presence of blue CO2 and OH emissions and yellow continuum radiation from soot. Present experimental conditions included acetylene-, methane-, propane-, and ethylene-fueled flames having initial reactant temperatures of 300 K, ambient pressures of 4-50 kPa, jet exit Reynolds number of 3-54, initial air/fuel velocity ratios of 0-9 and luminous flame lengths of 5-55 mm; earlier measurements for propylene- and 1,3-butadiene-fueled flames for similar conditions were considered as well. Nonbuoyant flames in still air were observed at micro-gravity conditions; essentially nonbuoyant flames in coflowing air were observed at small pressures to control effects of buoyancy. Predictions of luminous flame boundaries from soot luminosity were limited to laminar smokepoint conditions, whereas predictions of flame-sheet boundaries ranged from soot-free to smokepoint conditions. Flame-shape predictions were based on simplified analyses using the boundary layer approximations along with empirical parameters to distinguish flame-sheet and luminous flame (at the laminar smoke point) boundaries. The comparison between measurements and predictions was remarkably good and showed that both flame-sheet and luminous-flame lengths are primarily controlled by fuel flow rates with lengths in coflowing air approaching 2/3 lengths in still air as coflowing air velocities are increased. Finally, luminous flame lengths at laminar smoke-point conditions were roughly twice as long as flame-sheet lengths at comparable conditions due to the presence of luminous soot particles in the fuel-lean region of the flames.
Xu, F.↗
Unsteady-flow-field predictions for oscillating cascades
The unsteady flow field around an oscillating cascade of flat plates with zero stagger was studied by using a time marching Euler code. This case had an exact solution based on linear theory and served as a model problem for studying pressure wave propagation in the numerical solution. The importance of using proper unsteady boundary conditions, grid resolution, and time step size was shown for a moderate reduced frequency. Results show that an approximate nonreflecting boundary condition based on linear theory does a good job of minimizing reflections from the inflow and outflow boundaries and allows the placement of the boundaries to be closer to the airfoils than when reflective boundaries are used. Stretching the boundary to dampen the unsteady waves is another way to minimize reflections. Grid clustering near the plates captures the unsteady flow field better than when uniform grids are used as long as the 'Courant Friedrichs Levy' (CFL) number is less than 1 for a sufficient portion of the grid. Finally, a solution based on an optimization of grid, CFL number, and boundary conditions shows good agreement with linear theory.
Huff, Dennis L.↗
Flame Shapes of Nonbuoyant Laminar Jet Diffusion Flames
The shapes (flame-sheet and luminous-flame boundaries) of steady nonbuoyant round hydrocarbon-fueled laminar-jet diffusion flames in still and coflowing air were studied both experimentally and theoretically. Flame-sheet shapes were measured from photographs using a CH optical filter to distinguish flame-sheet boundaries in the presence of blue C02 and OH emissions and yellow continuum radiation from soot. Present experimental conditions included acetylene-, methane-, propane-, and ethylene-fueled flames having initial reactant temperatures of 300 K, ambient pressures of 4-50 kPa, jet exit Reynolds number of 3-54, initial air/fuel velocity ratios of 0-9 and luminous flame lengths of 5-55 mm; earlier measurements for propylene- and 1,3-butadiene-fueled flames for similar conditions were considered as well. Nonbuoyant flames in still air were observed at micro-gravity conditions; essentially nonbuoyant flames in coflowing air were observed at small pressures to control effects of buoyancy. Predictions of luminous flame boundaries from soot luminosity were limited to laminar smoke-point conditions, whereas predictions of flame-sheet boundaries ranged from soot-free to smoke-point conditions. Flame-shape predictions were based on simplified analyses using the boundary layer approximations along with empirical parameters to distinguish flame-sheet and luminous-flame (at the laminar smoke point) boundaries. The comparison between measurements and predictions was remarkably good and showed that both flame-sheet and luminous-flame lengths are primarily controlled by fuel flow rates with lengths in coflowing air approaching 2/3 lengths in still air as coflowing air velocities are increased. Finally, luminous flame lengths at laminar smoke-point conditions were roughly twice as long as flame-sheet lengths at comparable conditions due to the presence of luminous soot particles in the fuel-lean region of the flames.
Xu, F.↗
Flame and Soot Boundaries of Laminar Jet Diffusion Flames
The shapes (flame-sheet and luminous-flame boundaries) or steady weakly buoyant round hydrocarbon-fueled laminar-jet diffusion flames in still and coflowing air were studied both experimentally and theoretically. Flame-sheet shapes were measured from photographs using a CH optical filter to distinguish flame-sheet boundaries in the presence of blue CO2 and OH emissions and yellow continuum radiation from soot. Present experimental conditions included acetylene-, methane-, propane-, and ethylene-fueled flames having initial reactant temperatures of 300 K. ambient pressures of 4-50 kPa, jet-exit Reynolds numbers of 3-54, initial air/fuel velocity ratios of 0-9, and luminous flame lengths of 5-55 mm; earlier measurements for propylene- and 1,3-butadiene-fueled flames for similar conditions were considered as well. Nonbuoyant flames in still air were observed at microgravity conditions; essentially nonbuoyant flames in coflowing air were observed at small pressures to control effects of buoyancy. Predictions of luminous flame boundaries from soot luminosity were limited to laminar smoke-point conditions, whereas predictions of flame-sheet boundaries ranged from soot-free to smoke-point conditions. Flame-shape predictions were based on simplified analyses using the boundary-layer approximations along with empirical parameters to distinguish flame-sheet and luminous-flame (at the laminar smoke point) boundaries. The comparison between measurements and predictions was remarkably good and showed that both flame-sheet and luminous-flame lengths are primarily controlled by fuel flow rates with lengths in coflowing air approaching 2/3 of the lengths in still air as coflowing air velocities are increased. Finally, luminous flame lengths at laminar smoke-point conditions were roughly twice as long as flame-sheet lengths at comparable conditions because of the presence of luminous soot particles in the fuel-lean region of the flames.
Xu, F.↗
AN AEROTHERMOCHENMCAL ANALYSIS OF EROSIVE BURNING OF SOLID PROPELLANT
In a solid propellant rocket of a side burning type, the flow rate of the hot combustion gases parallel to the burning surface is approximately proportional to the distance from the leading edge of the propellant. In the present paper, therefore, the erosive burning of the solid propellant is analyzed by tile boundary-layer approximation in aerothermochemistry for the case where the velocity, u ∞ , of the hot combustion gas stream outside the boundary layer increases linearly with the distance, x, from the leading edge of the propellant, i.e., u ∞ = Gx, and the effects of the hot gas stream on the burning rate of the propellant are examined.
GAS STREAM↗
Stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations
Implicit, noniterative, finite difference schemes were recently developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations. When applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable (A-stable). As applied in practice the algorithms often face a severe time step restriction. A major source of the difficulty is the treatment of the numerical boundary conditions. One conjecture was that unconditional stability requires implicit numerical boundary conditions. An apparent counter example was the space time extrapolation considered by Gustafsson, Kreiss, and Sunstrom. Spatial (implicit) and space time (explicit) extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals are examined. The results indicate that for unconditional stability with a finite number of spatial mesh intervals, the numerical boundary conditions must be implicit.
Beam, R. M.↗