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NASA NTRS1982

Transonic potential flow and coordinate generation for bodies in a wind tunnel

An accurate method has been developed for computation of transonic potential flow about a 2-D lifting airfoil or an axisymmetric body in a wind tunnel. The computational mesh is nearly orthogonal everywhere and is generated by a sequence of Schwarz-Christoffel transformations and shearings to obtain an '0' grid near the body. A conservative finite-volume scheme for the full-potential equation and exact boundary conditions is used together with 'retarded' density to solve efficiently transonic flow with embedded shocks and large regions of supersonic flow. Supersonic free-stream flows are also solved with captured bow shocks and embedded subsonic regions.

Doria, M. L.↗

NASA NTRS1980

Numerical solution of the Navier-Stokes equations for high Reynolds number incompressible turbulent flow

The full Navier-Stokes equations for incompressible turbulent flow must be solved to accurately represent all flow phenomena which occur in a high Reynolds number incompressible flow. A two layer algebraic eddy viscosity turbulence model is used to represent the Reynolds stress in the primitive variable formulation. The development of the boundary-fitted coordinate systems makes the numerical solution of these equations feasible for arbitrarily shaped bodies. The nondimensional time averaged Navier-Stokes equations, including the turbulence mode, are represented by finite difference approximations in the transformed plane. The resulting coupled system of nonlinear algebraic equations is solved using a point successive over relaxation iteration. The test case considered was a NACA 64A010 airfoil section at an angle of attack of two degrees and a Reynolds number of 2,000,000.

Thompson, D. S.↗

NASA NTRS1980

Cycle II.5 aircraft aero-optical turbulent boundary-layer/shear-layer measurements

The aero-optical effects associated with propagating a laser beam through aircraft turbulent boundary layers and shear layers were examined. Observed laser optical performance levels were compared with those inferred from aerodynamic measurements of unsteady densities and correlation lengths within these random flows. Optical instrumentation included a fast shearing interferometer (FSI). A 9 cm diameter collimated helium neon laser beam made a double pass through the aircraft random flow via an airfoil mirror located one meter from the fuselage. Typical aircraft turbulent boundary layer thickness measured 0.3 meters. Averaging many FSI generated modulation transfer functions (MTFs) and Fourier transforming, this average yields the expected far field intensity degradation associated with an aircraft mounted laser system. Aerodynamic instrumentation included fine wire probes to measure unsteady temperature and mass flux. A laser doppler velocimeter measured unsteady velocity within the flows. An analysis of these data yielded point measurements of unsteady density and correlation length.

Gilbert, K.↗

DOE OSTI2023

Separable shape tensors for aerodynamic design

Airfoil shape design is a classical problem in engineering and manufacturing. In this work, we combine principled physics-based considerations for the shape design problem with modern computational techniques using a data-driven approach. Modern and traditional analyses of two-dimensional (2D) and three-dimensional (3D) aerodynamic shapes reveal a flow-based sensitivity to specific deformations that can be represented generally by affine transformations (rotation, scaling, shearing, and translation). We present a novel representation of shapes that decouples affine-style deformations over a submanifold and a product submanifold principally of the Grassmannian. As an analytic generative model, the separable representation, informed by a database of physically relevant airfoils, offers: (i) a rich set of novel 2D airfoil deformations not previously captured in the data, (ii) an improved low-dimensional parameter domain for inferential statistics informing design/manufacturing, and (iii) consistent 3D blade representation and perturbation over a sequence of nominal 2D shapes.

42 ENGINEERING↗

NASA NTRS1992

The semi-discrete Galerkin finite element modelling of compressible viscous flow past an airfoil

A method is developed to solve the two-dimensional, steady, compressible, turbulent boundary-layer equations and is coupled to an existing Euler solver for attached transonic airfoil analysis problems. The boundary-layer formulation utilizes the semi-discrete Galerkin (SDG) method to model the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby permitting the use of a uniform finite element grid which provides high resolution near the wall and automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes, through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past NACA 0012 and RAE 2822 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack. All results show good agreement with experiment, and the coupled code proved to be a computationally-efficient and accurate airfoil analysis tool.

Meade, Andrew J., Jr.↗

NASA NTRS1987

A comparison of two conformal mapping techniques applied to an aerobrake body

Conformal mapping is a classical technique which has been utilized for solving problems in aerodynamics and hydrodynamics. Conformal mapping has been successfully applied in the construction of grids around airfoils, engine inlets and other aircraft configurations. Conformal mapping techniques were applied to an aerobrake body having an axis of symmetry. Two different approaches were utilized: (1) Karman-Trefftz transformation; and (2) Point Wise Schwarz Christoffel transformation. In both cases, the aerobrake body was mapped onto a near circle, and a grid was generated in the mapped plane. The mapped body and grid were then mapped back into physical space and the properties of the associated grids were examined. Advantages and disadvantages of both approaches are discussed.

Hommel, Mark J.↗

DOE OSTI2022

A comprehensive and fair comparison of two neural operators (with practical extensions) based on $\mathrm{FAIR}$ data

Neural operators can learn nonlinear mappings between function spaces and offer a new simulation paradigm for real-time prediction of complex dynamics for realistic diverse applications as well as for system identification in science and engineering. Herein, we investigate the performance of two neural operators, which have shown promising results so far, and we develop new practical extensions that will make them more accurate and robust and importantly more suitable for industrial-complexity applications. The first neural operator, DeepONet, was published in 2019 (Lu et al., 2019), and its original architecture was based on the universal approximation theorem of Chen & Chen (1995). The second one, named Fourier Neural Operator or FNO, was published in 2020, and it is based on parameterizing the integral kernel in the Fourier space. DeepONet is represented by a summation of products of neural networks (NNs), corresponding to the branch NN for the input function and the trunk NN for the output function; both NNs are general architectures, e.g., the branch NN can be replaced with a CNN or a ResNet. According to Kovachki et al. (2021), FNO in its continuous form can be viewed conceptually as a DeepONet with a specific architecture of the branch NN and a trunk NN represented by a trigonometric basis. In order to compare FNO with DeepONet computationally for realistic setups, we develop several extensions of FNO that can deal with complex geometric domains as well as mappings where the input and output function spaces are of different dimensions. We also develop an extended DeepONet with special features that provide inductive bias and accelerate training, and we present a faster implementation of DeepONet with cost comparable to the computational cost of FNO, which is based on the Fast Fourier Transform. Here we consider 16 different benchmarks to demonstrate the relative performance of the two neural operators, including instability wave analysis in hypersonic boundary layers, prediction of the vorticity field of a flapping airfoil, porous media simulations in complex-geometry domains, etc. We follow the guiding principles of FAIR (Findability, Accessibility, Interoperability, and Reusability) for scientific data management and stewardship. The performance of DeepONet and FNO is comparable for relatively simple settings, but for complex geometries the performance of FNO deteriorates greatly. We also compare theoretically the two neural operators and obtain similar error estimates for DeepONet and FNO under the same regularity assumptions.

42 ENGINEERING↗

NASA NTRS1981

Calculations of transonic potential flow over cascades

Transonic flow through a cascade was studied by using the full potential equation and the finite volume method of Jameson and Caughey. The C-type computational grid is generated by an electrostatic analogy and simple shearing transformation. The solution algorithm includes an option of using either an artificial density or an artificial viscosity formulation of the dissipative term. Using the developed code, flows through a cascade of NACA 0012 airfoils and flows through a cascade of shockless blades were computed. It is found that the designed flow through the shockless blade is accurately predicted, the artificial density formulation shows more tolerance to the mesh irregularity, and the C-type mesh does not extend very far upstream for a small pitch-cord ratio.

Jou, W. H.↗

NASA NTRS1994

Control theory based airfoil design for potential flow and a finite volume discretization

This paper describes the implementation of optimization techniques based on control theory for airfoil design. In previous studies it was shown that control theory could be used to devise an effective optimization procedure for two-dimensional profiles in which the shape is determined by a conformal transformation from a unit circle, and the control is the mapping function. The goal of our present work is to develop a method which does not depend on conformal mapping, so that it can be extended to treat three-dimensional problems. Therefore, we have developed a method which can address arbitrary geometric shapes through the use of a finite volume method to discretize the potential flow equation. Here the control law serves to provide computationally inexpensive gradient information to a standard numerical optimization method. Results are presented, where both target speed distributions and minimum drag are used as objective functions.

Reuther, J.↗

NASA NTRS1972

Numerical calculation of transonic flow about swept wings.

Description of a mixed elliptic-hyperbolic relaxation algorithm which calculates solutions to the three-dimensional, nonlinear transonic small disturbance potential equation for flows about thin swept lifting wings with free-stream Mach number less than 1. The algorithm is designed to treat supercritical flows, including cases with embedded shock waves. Nonrectangular planform shapes, including sweep and taper, are treated by a coordinate transformation which maps the wing planform into a rectangle. Computed results at angles of attack of 0 and 2 deg for a AR = 4, constant chord, 23.75 deg sweptback planform model with a Lockheed C141 airfoil section are compared with data obtained experimentally for both subcritical and supercritical flows. Subcritical results are also compared with those obtained by a subsonic 'panel' method.

Ballhaus, W. F.↗

NASA NTRS1994

Optimum aerodynamic design via boundary control

These lectures describe the implementation of optimization techniques based on control theory for airfoil and wing design. In previous studies it was shown that control theory could be used to devise an effective optimization procedure for two-dimensional profiles in which the shape is determined by a conformal transformation from a unit circle, and the control is the mapping function. Recently the method has been implemented in an alternative formulation which does not depend on conformal mapping, so that it can more easily be extended to treat general configurations. The method has also been extended to treat the Euler equations, and results are presented for both two and three dimensional cases, including the optimization of a swept wing.

Jameson, Antony↗

NASA NTRS1978

Numerical solution of flow problems using body-fitted coordinate systems

The technique of boundary-fitted coordinate systems is based on a method of automatic numerical generation of a general curvilinear coordinate system having a coordinate line coincident with each boundary of a general multi-connected region containing any number of arbitrarily shaped bodies. Once the curvilinear coordinate system is generated, any partial differential system of interest can be solved on this coordinate system by transforming the equations and solving the resulting system in finite difference approximation on the rectangular transformed plane. This method of automatic body-fitted curvilinear coordinate generation is used to construct finite-difference solutions of the full, time dependent Navier-Stokes equations for the unsteady viscous flow about arbitrary two-dimensional airfoils, or any other two-dimensional bodies. Finally, initial results for three-dimensional applications are also presented.

Thompson, J. F.↗

NASA NTRS1983

Dimensional stability considerations for cryogenic metals

Work performed as part of an effort to identify, and where possible separate out, some of the factors that contribute to dimensional stability in cryogenic wind tunnel models is reported. Initial problems were encountered with two dimensional models made of 15-5 PH stainless steel, which warped significantly after being subjected to cryogenic testing in the 0.3 Meter Transonic Cryogenic Tunnel. Subsequently, an effort was undertaken to investigate the mechanisms that could cause model warpage during cryogenic testing. The two basic mechanisms that can lead to warpage are (1) metallurgical structural instability in which one phase transforms partially or fully into a second phase which has a different crystal structure and volume, and (2) deformation due to the creation, or relief, of unbalanced induced or residual stresses. In the case of the 15-5 PH airfoils, it is highly probable that metallurgical instability was responsible for most of the observed warpage. A particular specimen configuration was established for use in the systematic evaluation of the factors influencing warpage. Preliminary studies of a specimen made of VASCOMAX 200 suggest the possibility of manipulating the stresses in the surface layers by appropriate combinations of milling and grinding steps. This opens up the possibility of correcting or establishing the required surface profile of an airfoil.

Wigley, D. A.↗

NASA NTRS1982

Combined four-wall interference assessment in two-dimensional airfoil tests

Two different procedures are examined for combining the correction method developed by Barnwell and Sewall for the effects of the sidewall boundary layer in two-dimensional wind-tunnel tests with the assessment and correction method due to Kemp for the effects of the upper and lower tunnel walls. One procedure utilizes the similarity transformation defined by Sewall to eliminate velocity perturbations induced by the sidewall boundary layer from consideration in the assessment and correction method as sources of error. The other procedure combines perturbations from all four walls before assessment and correction. In general, the two procedures yield different corrections to Mach number and angle of attack, either of which can be considered valid. Pressure distributions corrected for higher order interference effects are provided by either procedure.

Kemp, W. B., Jr.↗

NASA NTRS1997

The Spectral and Statistical Properties of Turbulence Generated by a Vortex/Blade-Tip Interaction

The perpendicular interaction of a streamwise vortex with the tip of a lifting blade was studied in incompressible flow to provide information useful to the accurate prediction of helicopter rotor noise and the understanding of vortex dominated turbulent flows. The vortex passed 0.3 chord lengths to the suction side of the blade tip, providing a weak interaction. Single and two-point turbulence measurements were made using sub-miniature four sensor hot-wire probes 15 chord lengths downstream of the blade trailing edge; revealing the mean velocity and Reynolds stress tensor distributions of the turbulence, as well as its spanwise length scales as a function of frequency. The single point measurements show the flow downstream of the blade to be dominated by the interaction of the original tip vortex and the vortex shed by the blade. These vortices rotate about each other under their mutual induction, winding up the turbulent wakes of the blades. This interaction between the vortices appears to be the source of new turbulence in their cores and in the region between them. This turbulence appears to be responsible for some decay in the core of the original vortex, not seen when the blade is removed. The region between the vortices is not only a region of comparatively large stresses, but also one of intense turbulence production. Velocity autospectra measured near its center suggests the presence quasi-periodic large eddies with axes roughly parallel to a line joining the vortex cores. Detailed two-point measurements were made on a series of spanwise cuts through the flow so as to reveal the turbulence scales as they would be seen along the span of an intersecting airfoil. The measurements were made over a range of probe separations that enabled them to be analyzed not only in terms of coherence and phase spectra but also in terms of wave-number frequency (kappa-omega) spectra, computed by transforming the measured cross-spectra with respect to the spanwise separation of the probes. These data clearly show the influence of the coherent eddies in the spiral wake and the turbulent region between the cores. These eddies produce distinct peaks in the upwash velocity kappa-omega spectra, and strong anisotropy manifested both in the decay of the kappa-omega spectrum at larger wave-numbers and in differences between the kappa-omega spectra of different components. None of these features are represented in the von Karman spectrum for isotropic turbulence that is often used in broadband noise computations. Wave-number frequency spectra measured in the cores appear to show some evidence that the turbulence outside sets tip core waves, as has previously been hypothesized. These spectra also provide for the first time a truly objective method for distinguishing velocity fluctuations produced by core wandering from other motions.

Devenport, William J.↗

NASA NTRS1988

A numerical simulation of the full two-dimensional electrothermal de-icer pad

The ability to predict the time-temperature history of electrothermal de-icer pads is important in the subsequent design of improved and more efficient versions. These de-icer pads are installed near the surface of aircraft components, for the specific purpose of removing accreted ice. The proposed numerical model can incorporate the full 2-D geometry through a section of a region (i.e., section of an airfoil), that current 1-D numerical codes are unable to do. Thus, the effects of irregular layers, curvature, etc., can now be accounted for in the thermal transients. Each layer in the actual geometry is mapped via a body-fitted coordinate transformation into uniform, rectangular computational grids. The relevant heat transfer equations are transformed and discretized. To model the phase change that might occur in any accreted ice, in an enthalpy formulation the phase change equations are likewise transformed and discretized. The code developed was tested against numerous classical numerical solutions, as well as against experimental de-icing data on a UH1H rotor blade obtained from the NASA Lewis Research Center. The excellent comparisons obtained show that this code can be a useful tool in predicting the performance of current de-icer models, as well as in the designing of future models.

Masiulaniec, Konstanty C.↗

NASA NTRS1991

Compressible flows with periodic vortical disturbances around lifting airfoils

A numerical method is developed for solving periodic, three-dimensional, vortical flows around lifting airfoils in subsonic flow. The first-order method that is presented fully accounts for the distortion effects of the nonuniform mean flow on the convected upstream vortical disturbances. The unsteady velocity is split into a vortical component which is a known function of the upstream flow conditions and the Lagrangian coordinates of the mean flow, and an irrotational field whose potential satisfies a nonconstant-coefficient, inhomogeneous, convective wave equation. Using an elliptic coordinate transformation, the unsteady boundary value problem is solved in the frequency domain on grids which are determined as a function of the Mach number and reduced frequency. The numerical scheme is validated through extensive comparisons with known solutions to unsteady vortical flow problems. In general, it is seen that the agreement between the numerical and analytical results is very good for reduced frequencies ranging from 0 to 4, and for Mach numbers ranging from .1 to .8. Numerical results are also presented for a wide variety of flow configurations for the purpose of determining the effects of airfoil thickness, angle of attack, camber, and Mach number on the unsteady lift and moment of airfoils subjected to periodic vortical gusts. It is seen that each of these parameters can have a significant effect on the unsteady airfoil response to the incident disturbances, and that the effect depends strongly upon the reduced frequency and the dimensionality of the gust. For a one-dimensional (transverse) or two-dimensional (transverse and longitudinal) gust, the results indicate that airfoil thickness increases the unsteady lift and moment at the low reduced frequencies but decreases it at the high reduced frequencies. The results show that an increase in airfoil Mach number leads to a significant increase in the unsteady lift and moment for the low reduced frequencies, but a significant decrease for the high reduced frequencies.

Scott, James R.↗

NASA NTRS1937

Potential flow about arbitrary biplane wing sections

A rigorous treatment is given of the problem of determining the two-dimensional potential flow around arbitrary biplane cellules. The analysis involves the use of elliptic functions and is sufficiently general to include the effects of such elements as the section shapes, the chord ratio, gap, stagger, and decalage, which elements may be specified arbitrarily. The flow problem is resolved by making use of the methods of conformal representation. Thus the solution of the problem of transforming conformally two arbitrary contours into two circles is expressed by a pair of simultaneous integral equations, for which a method of numerical solution is outlined. As an example of the numerical process, the pressure distribution over certain arrangements of the NACA 4412 airfoil in biplane combinations is presented and compared with the monoplane pressure distribution.

Garrick, I E↗