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The Chebyshev matrix collocation method is applied to obtain the spatial modes of the Orr-Sommerfeld equation for Poiseuille flow and the Blausius boundary layer. The problem is linearized by the companion matrix technique for semi-infinite domain using a mapping transformation. The method can be easily adapted to problems with different boundary conditions requiring different transformations.
A vortex-lattice method for predicting the aerodynamics of wings having separation at the sharp edges in incompressible flows is extended to compressible subsonic flows using a modified Prandtl-Glauert transformation. Numerical results showing the effect of freestream Mach number on the aerodynamic coefficients are compared with available experimental data for several planforms. It is shown that the proposed method is suitable for predicting the aerodynamic loads on low-aspect wings at moderate angles of attack for high subsonic freestream Mach number. The method is limited to angles of attack up to 12 deg for high subsonic freestream Mach number and to angles of attack up to 20 deg for Mach number not exceeding 0.5.
The Sun lies at the center of space weather and is the source of its variability. The primary input to coronal and solar wind models is the activity of the magnetic field in the solar photosphere. Recent advancements in solar observations and numerical simulations provide a basis for developing physics-based models for the dynamics of the magnetic field from the deep convection zone of the Sun to the corona with the goal of providing robust near real-time boundary conditions at the base of space weather forecast models. The goal is to develop new strategic capabilities that enable characterization and prediction of the magnetic field structure and flow dynamics of the Sun by assimilating data from helioseismology and magnetic field observations into physics-based realistic magnetohydrodynamics (MHD) simulations. The integration of first-principle modeling of solar magnetism and flow dynamics with real-time observational data via advanced data assimilation methods is a new, transformative step in space weather research and prediction. This approach will substantially enhance an existing model of magnetic flux distribution and transport developed by the Air Force Research Lab. The development plan is to use the Space Weather Modeling Framework (SWMF) to develop Coupled Models for Emerging flux Simulations (CMES) that couples three existing models: (1) an MHD formulation with the anelastic approximation to simulate the deep convection zone (FSAM code), (2) an MHD formulation with full compressible Navier-Stokes equations and a detailed description of radiative transfer and thermodynamics to simulate near-surface convection and the photosphere (Stagger code), and (3) an MHD formulation with full, compressible Navier-Stokes equations and an approximate description of radiative transfer and heating to simulate the corona (Module in BATS-R-US). CMES will enable simulations of the emergence of magnetic structures from the deep convection zone to the corona. Finally, a plan will be summarized on the development of a Flux Emergence Prediction Tool (FEPT) in which helioseismology-derived data and vector magnetic maps are assimilated into CMES that couples the dynamics of magnetic flux from the deep interior to the corona.
The Sun lies at the center of space weather and is the source of its variability. The primary input to coronal and solar wind models is the activity of the magnetic field in the solar photosphere. What is needed is to develop physics-based models for the dynamics of the magnetic field from the deep convection zone of the Sun to the corona with the goal of providing robust near real-time boundary conditions at the base of space weather forecast models. NASA strategic capabilities efforts toward this goal will be summarized during the talk. These have the objective to enable characterizing and predicting the magnetic field structure and flow dynamics of the Sun by assimilating data from helioseismology and magnetic field observations into physics-based realistic magnetohydrodynamics (MHD) simulations. It is argued that an effort in integrating first-principle modeling of solar magnetism and flow dynamics with real-time observational data via advanced data assimilation methods is needed for transformative steps in space weather research and prediction.
The feasibility of the method of lines for solutions of physical problems requiring nonuniform grid distributions is investigated. To attain this, it is also necessary to investigate the stiffness characteristics of the pertinent equations. For specific applications, the governing equations considered are those for viscous, incompressible, two dimensional and axisymmetric flows. These equations are transformed from the physical domain having a variable mesh to a computational domain with a uniform mesh. The two governing partial differential equations are the vorticity and stream function equations. The method of lines is used to solve the vorticity equation and the successive over relaxation technique is used to solve the stream function equation. The method is applied to three laminar flow problems: the flow in ducts, curved-wall diffusers, and a driven cavity. Results obtained for different flow conditions are in good agreement with available analytical and numerical solutions. The viability and validity of the method of lines are demonstrated by its application to Navier-Stokes equations in the physical domain having a variable mesh.
The computational procedure and numerical results are presented for a new method to solve Kuessner's integral equation in the case of subsonic compressible flow about harmonically oscillating planar surfaces with controls. Kuessner's equation is a linear transformation from pressure to normalwash. The unknown pressure is expanded in terms of prescribed basis functions and the unknown basis function coefficients are determined in the usual manner by satisfying the given normalwash distribution either collocationally or in the complex least squares sense. The present method of solution differs from previous ones in that the basis functions are defined in a continuous fashion over a relatively small portion of the aerodynamic surface and are zero elsewhere. This method, termed the local basis function method, combines the smoothness and accuracy of distribution methods with the simplicity and versatility of panel methods. Predictions by the local basis function method for unsteady flow are shown to be in excellent agreement with other methods. Also, potential improvements to the present method and extensions to more general classes of solutions are discussed.
The growth in computational power and algorithm development in the past few decades has granted the science and engineering community the ability to simulate flows over complex geometries, thus making Computational Fluid Dynamics (CFD) tools indispensable in analysis and design. Currently, one of the pacing items limiting the utility of CFD for general problems is the prediction of unsteady turbulent ows.1{3 Reynolds-averaged Navier-Stokes (RANS) methods, which predict a time-invariant mean flowfield, struggle to provide consistent predictions when encountering even mild separation, such as the side-of-body separation at a wing-body junction. NASA's Transformative Tools and Technologies project is developing both numerical methods and physical modeling approaches to improve the prediction of separated flows. A major focus of this e ort is efficient methods for resolving the unsteady fluctuations occurring in these flows to provide valuable engineering data of the time-accurate flow field for buffet analysis, vortex shedding, etc. This approach encompasses unsteady RANS (URANS), large-eddy simulations (LES), and hybrid LES-RANS approaches such as Detached Eddy Simulations (DES). These unsteady approaches are inherently more expensive than traditional engineering RANS approaches, hence every e ort to mitigate this cost must be leveraged. Arguably, the most cost-effective approach to improve the efficiency of unsteady methods is the optimal placement of the spatial and temporal degrees of freedom (DOF) using solution-adaptive methods.
A new method was developed to extract surface flow vectors from an oilflow visualization image that has oil streaklines. The method is analogous to the PIV processing where the image is divided into interrogation windows. A representative flow direction is obtained for each interrogation window using image processing with a line-detection algorithm. Repeating the process for the entire image, one can find the surface vector field. The Hough transformation, Radon transformation and Machine Learning were used as line-detection algorithms. The obtained vector field is then postprocessed to filter spurious vectors and to apply smoothing. The method was tested on 2D and 3D models with different flow complexities. The method was able to predict surface flow vectors for all cases tested. The predicted surface flow vectors were used in obtaining the surface skin friction. The successful implementation of the surface flow vectors enables interferometric skin friction measurements on surfaces beneath 3D complex flows. In addition, the surface flow vectors can be superimposed on the oilflow visualization images to better explain the surface flow topology.
A new method was developed to extract surface flow vectors from an oilflow visualization image that has oil streaklines. The method is analogous to the PIV processing where the image is divided into interrogation windows. A representative flow direction is obtained for each interrogation window using image processing with a line-detection algorithm. Repeating the process for the entire image, one can find the surface vector field. The Hough transformation, Radon transformation and Machine Learning were used as line-detection algorithms. The obtained vector field is then postprocessed to filter spurious vectors and to apply smoothing. The method was tested on 2D and 3D models with different flow complexities. The method was able to predict surface flow vectors for all cases tested. The predicted surface flow vectors were used in obtaining the surface skin friction. The successful implementation of the surface flow vectors enables interferometric skin friction measurements on surfaces beneath 3D complex flows. In addition, the surface flow vectors can be superimposed on the oilflow visualization images to better explain the surface flow topology.
Variational methods (VM) sensitivity analysis, which is the continuous alternative to the discrete sensitivity analysis, is employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The determination of the sensitivity derivatives of the performance index or functional entails the coupled solutions of the state and costate equations. As the stable and converged numerical solution of the costate equations with their boundary conditions are a priori unknown, numerical stability analysis is performed on both the state and costate equations. Thereafter, based on the amplification factors obtained by solving the generalized eigenvalue equations, the stability behavior of the costate equations is discussed and compared with the state (Euler) equations. The stability analysis of the costate equations suggests that the converged and stable solution of the costate equation is possible only if the computational domain of the costate equations is transformed to take into account the reverse flow nature of the costate equations. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.
The development of an automated technique for the eduction of 3-D spatial patterns in vector or scalar diagnostics was completed. The method is based on an iterative convolution between a trial pattern and the data field. It was applied to the analysis of low Reynolds number turbulent channel flow and homogeneous shear flow. The results yielded new information on the dominant flow structures in these flows, particularly with respect to the spatial relationships between various forms of organized motion. A particular application of the pattern eduction method, which is tentatively referred to as an 'adaptive wavelet transformation', is proposed with the objective of investigating the way turbulence structure changes with scale. Preliminary results using data from homogeneous turbulent shear flow simulations are presented. At the low Reynolds numbers of the simulations, there is no evidence of scale similarity. The small scales appear to be associated with the edges of the larger scale vortical structures.
In the present paper, the motion of a gas in a plane-parallel Laval nozzle in the neighborhood of the transition from subsonic to supersonic velocities is studied. In a recently published paper, F. I. Frankl, applying the holograph method of Chaplygin, undertook a detailed investigation of the character of the flow near the line of transition from subsonic to supersonic velocities. From the results of Tricomi's investigation on the theory of differential equations of the mixed elliptic-hyperbolic type, Frankl introduced as one of the independent variables in place of the modulus of the velocity, a certain specially chosen function of this modulus. He thereby succeeded in explaining the character of the flow at the point of intersection of the transition line and the axis of symmetry (center of the nozzle) and in studying the behavior of the stream function in the neighborhood of this point by separating out the principal term having, together with its derivatives, the maximum value as compared with the corresponding corrections. This principal term is represented in Frankl's paper in the form of a linear combination of two hypergeometric functions. In order to find this linear combination, it is necessary to solve a number of boundary problems, which results in a complex analysis. In the investigation of the flow with which this paper is concerned, a second method is applied. This method is based on the transformation of the equations of motion to a form that may be called canonical for the system of differential equations of the mixed elliptic-hyperbolic type to which the system of equations of the motion of an ideal compressible fluid refers. By studying the behavior of the integrals of this system in the neighborhood of the parabolic line, the principal term of the solution is easily separated out in the form of a polynomial of the third degree. As a result, the computation of the transitional part of the nozzle is considerably simplified.
Recent experimental supersonic retropropulsion tests were conducted at the NASA Langley Research Center Unitary Plan Wind Tunnel Test Section 2 for a range of Mach numbers from 2.4 to 4.6. A 5-inch 70-degree sphere-cone forebody model with a 10-inch cylindrical aftbody experimental model was used which is capable of multiple retrorocket configurations. These configurations include a single central nozzle on the center point of the forebody, three nozzles at the forebody half-radius, and a combination of the first two configurations with no jets being plugged. A series of measurements were achieved through various instrumentation including forebody and aftbody pressure, internal pressures and temperatures, and high speed Schlieren visualization. Specifically, several high speed pressure transducers on the forebody and in the plenum were implemented to look at unsteady flow effects. The following work focuses on analyzing frequency traits due to the unsteady flow for a range of thrust coefficients for single, tri, and quad-nozzle test cases at freestream Mach 4.6 and angle of attack ranging from -8 degrees to +20 degrees. This analysis uses Matlab s fast Fourier transform, Welch's method (modified average of a periodogram), to create a power spectral density and analyze any high speed pressure transducer frequency traits due to the unsteady flow.
One of the problems for the hypersonic flights is the lack of accurate methods for predicting the boundary layer parameters at high Mach numbers and temperatures. At high Mach numbers and temperatures, the real gas effects should be included in calculations of the compressible laminar and turbulent boundary layers. By introducing the Mangler-Levy-Lees transformation, the compressible laminar boundary layer equations were transformed into two ordinary differential equations. To solve these ordinary differential equations for the flow over a flat plate, the Runge-Kutta method of order 5 and 6 was used with a secant method of solving the simultaneous nonlinear equations for the boundary layer conditions. Instead of using the perfect gas assumptions, the properties of air in the temperature range of 100 to 2500 K at M(e) = 8 were used for the calculations. The calculations were carried out for free flight conditions. The real gas effects were significant for the conditions of high freestream Mach numbers, and the boundary layer parameters from the real gas solutions were quite different from the existing formulas and solutions.
This paper describes modeling methods for the three core components of a Metabolic heat regenerated Temperature Swing Adsorption (MTSA) subassembly: the sorbent bed, a sublimation (cooling) heat exchanger (SHX), and a condensing icing (warming) heat exchanger (CIHX). The primary function of the MTSA, removing carbon dioxide from a ventilation loop, is performed via the sorbent bed. The CIHX is used to heat the sorbent bed for desorption and to remove moisture from the ventilation loop while the SHX is alternately employed to cool the sorbent bed via sublimation of a spray of water at low pressure to prepare the reconditioned bed for the next cycle. This paper describes a system level model of the MTSA as developed in Thermal Desktop and SINDA/FLUINT including assumptions on geometry and physical phenomena, modeling methodology and relevant pa ra mete rizatio ns. Several areas of particular modeling interest are discussed. In the sorbent bed, capture of the translating CO2 saturation front and associated local energy and mass balance in both adsorbing and desorbing modes is covered. The CIHX poses particular challenges for modeling in SINDA/FLUINT as accounting for solids states in fluid submodels are not a native capability. Methods for capturing phase change and latent heat of ice as well as the transport properties across a layer of low density accreted frost are developed. This extended modeling capacity is applicable to temperatures greater than 258 K. To extend applicability to the minimum device temperature of 235 K, a method for a mapped transformation of temperatures from below the limit temperatures to some value above is given along with descriptions for associated material property transformations and the resulting impacts to total heat and mass transfer. Similar considerations are shown for the SHX along with assumptions for flow mechanics and resulting model methods for sublimation in a flow.
The present report concerns a method of computing the velocity and pressure distributions on bodies of revolution in axially symmetrical flow in the subsonic range. The differential equation for the velocity potential Phi of a compressible fluid motion is linearized tn the conventional manner, and then put in the form Delta(Phi) = 0 by affine transformation. The quantity Phi represents the velocity potential of a fictitious incompressible flow, for which a constant superposition of sources by sections is secured by a method patterned after von Karman which must comply with the boundary condition delta(phi)/delta(n) = 0 at the originally specified contour. This requirement yields for the "pseudo-stream function" psi a differential equation which must be fulfilled for as many points on the contour as source lengths are assumed. In this manner, the problem of defining the still unknown source intensities is reduced to the solution of an inhomogeneous equation system. The pressure distribution is then determined with the aid of Bernoulli's equation and adiabatic equation of state. Lastly, the pressure distributions in compressible and incompressible medium are compared on a model problem.
In this paper we present numerical solutions to several optimal control problems for an unsteady viscous flow. The main thrust of this work is devoted to simulation and control of an unsteady flow generated by a circular cylinder undergoing rotary motion. By treating the rotation rate as a control variable, we can formulate two optimal control problems and use a central difference/pseudospectral transform method to numerically compute the optimal control rates. Several types of rotations are considered as potential controls, and we show that a proper synchronization of forcing frequency with the natural vortex shedding frequency can greatly influence the flow. The results here indicate that using moving boundary controls for such systems may provide a feasible mechanism for flow control.
An alternative approach which employs multigrid concepts in the iterative solution of spectral equations was examined. Spectral multigrid methods are described for self adjoint elliptic equations with either periodic or Dirichlet boundary conditions. For realistic fluid calculations the relevant boundary conditions are periodic in at least one (angular) coordinate and Dirichlet (or Neumann) in the remaining coordinates. Spectral methods are always effective for flows in strictly rectangular geometries since corners generally introduce singularities into the solution. If the boundary is smooth, then mapping techniques are used to transform the problem into one with a combination of periodic and Dirichlet boundary conditions. It is suggested that spectral multigrid methods in these geometries can be devised by combining the techniques.