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At least 1,531 records · Page 85

The application of Green's theorem to the solution of boundary-value problems in linearized supersonic wing theory

Following the introduction of the linearized partial differential equation for nonsteady three-dimensional compressible flow, general methods of solution are given for the two and three-dimensional steady-state and two-dimensional unsteady-state equations. It is also pointed out that, in the absence of thickness effects, linear theory yields solutions consistent with the assumptions made when applied to lifting-surface problems for swept-back plan forms at sonic speeds. The solutions of the particular equations are determined in all cases by means of Green's theorem, and thus depend on the use of Green's equivalent layer of sources, sinks, and doublets. Improper integrals in the supersonic theory are treated by means of Hadamard's "finite part" technique.

Heaslet, Max A↗

Pretest Computational Assessment of Boundary Layer Transition in the NASA Juncture Flow Model with an NACA 0015-Based Wing

The first two phases of the NASA Juncture Flow experiment were carried out on a DLR-F6 swept-wing model and were designed to provide “CFD validation-quality” data toward the assessment and improvement of existing CFD turbulence models in predicting onset and extent of three-dimensional separated flow near the wing-juncture trailing-edge region. The next phase of experiments will involve an NACA 0015-based swept wing, as prior risk reduction experiments had indicated that this wing shape resulted in reduced separation near the juncture region than the DLR-F6 wing, thus providing a better option to evaluate the ability of CFD models to predict incipient turbulent separation. The NACA 0015 measurements will also include IR thermography to infer the variation of transition front with respect to an increasing angle of attack. The primary objective of this work is to computationally make a preliminary assessment of the transition front on both surfaces of the NACA 0015 wing at a crank-chord-based Reynolds number of 2.4 x 106 for four different angles of attack, (0°, 2.5°, 5°, and 7.5°) and to determine the dominant mechanisms responsible for transition. This assessment includes both RANS-based transition models from NASA’s OVERFLOW 2.3b flow solver and linear parabolized stability equations (PSE) stability analysis based on the Langley Stability and Transition Analysis code, LASTRAC. Linear PSE results indicate that the upper surface of the wing is dominated by Tollmien- Schlichting (TS) instabilities, and that the laminar flow region shrinks from about 50% chord to a very small region just downstream of the attachment line as the angle of attack is increased from 0° to 7.5°. Consequently, the transition fronts predicted by the Spalart- Allmaras-based amplification factor transport (AFT-2017b) equation model (which accounts for the TS instabilities alone) and the Menter’s shear-stress transport equation (SST2003)- based Langtry-Menter transition model with ability to account for both TS and crossflow effects (LM2015) compare well with those predicted using linear PSE. On the lower surface of the wing, stationary crossflow (CF) instabilities begin to appear on the inboard portion of the wing in addition to the TS-instabilities for the larger angles of attack (5° and 7.5°), further reducing the laminar flow extent within the inboard region. The LM2015 model that accounts for CF effects is able to replicate this trend but appears to predict a slightly earlier transition. The outcome of this effort will inform the experiment and, when the actual experimental data become available, provide further opportunity to assess and improve the various transition models.

CFD modeling↗

Observability of discretized partial differential equations

It is shown that complete observability of the discrete model used to assimilate data from a linear partial differential equation (PDE) system is necessary and sufficient for asymptotic stability of the data assimilation process. The observability theory for discrete systems is reviewed and applied to obtain simple observability tests for discretized constant-coefficient PDEs. Examples are used to show how numerical dispersion can result in discrete dynamics with multiple eigenvalues, thereby detracting from observability.

Cohn, Stephen E.↗

Efficient parallel solution of parabolic equations - Implicit methods on the Cedar multicluster

A class of implicit methods for the parallel solution of linear parabolic differential equations based on Pade and Chebyshev rational approximations to the matrix exponential are presented. It is pointed out that this approach incorporates both natural hierarchical parallelism, improved intrinsic efficiency, and fewer timesteps. These advantages lead to an extremely fast family of methods for the solution of certain time-dependent problems. These techniques are illustrated with numerical experiments on the University of Illinois Cedar multicluster architecture. The experiments indicate that implicit methods of very high degree offer great promise for the solution of certain parabolic problems when in computational environment with parallel resources. Hierarchically organized parallel computers, such as the Cedar multicluster, are found to be especially attractive for these schemes.

Gallopoulos, E.↗

Evaluation of dual-weighted residual and machine learning error estimation for projection-based reduced-order models of steady partial differential equations

Projection-based reduced-order models (pROMs) show great promise as a means to accelerate many-query applications such as forward error propagation, solving inverse problems, and design optimization. In order to deploy pROMs in the context of high-consequence decision making, accurate error estimates are required to determine the region(s) of applicability in the parameter space. The following paper considers the dual-weighted residual (DWR) error estimate for pROMs and compares it to another promising pROM error estimate, machine learned error models (MLEM). Here, we show how DWR can be applied to ROMs and then evaluate DWR on two partial differential equations (PDEs): a two-dimensional linear convection–reaction–diffusion equation, and a three-dimensional static hyper-elastic beam. It is shown that DWR is able to estimate errors for pROMs extrapolating outside of their training set while MLEM is best suited for pROMs used to interpolate within the pROM training set.

42 ENGINEERING↗

Reciprocal swimming at intermediate Reynolds number

In Stokes flow, Purcell's scallop theorem forbids objects with time-reversible (reciprocal) swimming strokes from moving. In the presence of inertia, this restriction is eased and reciprocally deforming bodies can swim. A number of recent works have investigated dimer models that swim reciprocally at intermediate Reynolds numbers Re≈1–1000. These show interesting results (e.g. switches of the swim direction as a function of inertia) but the results vary and seem to be case specific. Here, in this work, we introduce a general model and investigate the behaviour of an asymmetric spherical dimer of oscillating length for small-amplitude motion at intermediate Re. In our analysis we make the important distinction between particle and fluid inertia, both of which need to be considered separately. We asymptotically expand the Navier–Stokes equations in the small-amplitude limit to obtain a system of linear partial differential equations. Using a combination of numerical (finite element) and analytical (reciprocal theorem, method of reflections) methods we solve the system to obtain the dimer's swim speed and show that there are two mechanisms that give rise to motion: boundary conditions (an effective slip velocity) and Reynolds stresses. Each mechanism is driven by two classes of sphere–sphere interactions, between one sphere's motion and (1) the oscillating background flow induced by the other's motion, and (2) a geometric asymmetry induced by the other's presence. We can thus unify and explain behaviours observed in other works. Our results show how sensitive, counterintuitive and rich motility is in the parameter space of finite inertia of particles and fluid.

42 ENGINEERING↗

Verification of theory on weak turbulence relating to the sequence of diffuse plasma resonances in space.

An interpretation of the sequence of diffuse plasma resonances observed by space probes (Alouette 2 and ISIS-1 satellites) is developed in terms of wave-particle nonlinear interaction in a weakly turbulent plasma including the electrostatic electron cyclotron harmonic wave instability. The longest time duration of the center frequency of the diffuse plasma resonance is found to coincide with the most favorable condition for the electrostatic electron cyclotron harmonic wave instability which is obtained by solving the dispersion equation obtained for a linear approximation of the kinetic wave equation for the warm magnetoactive plasma. The electrostatic field due to the transmission of the intense rf pulse produces plasma turbulence involving nonlinear wave-wave interaction and temperature anisotropy which leads to instability. This instability supplies energy to the turbulence. The process can be thought of as a feedback system.

Oya, H.↗

Piecewise linear approximation for hereditary control problems

Finite dimensional approximations are presented for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems when a quadratic cost integral has to be minimized subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in case the cost integral ranges over a finite time interval as well as in the case it ranges over an infinite time interval. The arguments in the latter case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense. This feature is established using a vector-component stability criterion in the state space R(n) x L(2) and the favorable eigenvalue behavior of the piecewise linear approximations.

Propst, Georg↗

Solution of the two-dimensional Navier-Stokes equations using sparse matrix solvers

The use of direct sparse matrix solvers in the solution of the Navier-Stokes equations is investigated. The Yale Sparse Matrix Package and its implementation in the solution algorithm is described. The streamfunction-vorticity form of the Navier-Stokes equations are discretized and linearized and the resulting system of equations are solved using this package. Several viscous flow problems are investigated, including flow in a cavity and flow around a NACA0012 airfoil. Massively separated flow around a sine wave airfoil is investigated and high Reynolds number solutions are obtained. A solution of the unsteady flow around a Joukowski airfoil at high angle of attack is presented.

Bender, Erich E.↗

A finite element based method for solution of optimal control problems

A temporal finite element based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables that are expanded in terms of elemental values and simple shape functions. Unlike other variational approaches to optimal control problems, however, time derivatives of the states and costates do not appear in the governing variational equation. Instead, the only quantities whose time derivatives appear therein are virtual states and virtual costates. Also noteworthy among characteristics of the finite element formulation is the fact that in the algebraic equations which contain costates, they appear linearly. Thus, the remaining equations can be solved iteratively without initial guesses for the costates; this reduces the size of the problem by about a factor of two. Numerical results are presented herein for an elementary trajectory optimization problem which show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The goal is to evaluate the feasibility of this approach for real-time guidance applications. To this end, a simplified two-stage, four-state model for an advanced launch vehicle application is presented which is suitable for finite element solution.

Bless, Robert R.↗

Piecewise linear approximation for hereditary control problems

This paper presents finite-dimensional approximations for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems, when a quadratic cost integral must be minimized subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in the case where the cost integral ranges over a finite time interval, as well as in the case where it ranges over an infinite time interval. The arguments in the last case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense.

Propst, Georg↗

Unsteady flow computation of oscillating flexible wings

The problem of unsteady flow around flexible wings is solved using the unsteady, compressible, thin-layer Navier-Stokes equations in conjunction with the unsteady, linearized, Navier-displacement equations. Starting with the initial shape of the wing, the Navier-Stokes equations are solved on an initial structured grid to obtain the steady-flow solution which is used for the initial conditions. The forced deformation motion of the wing boundaries is then applied, and the problem is solved accurately in time. During the time-accurate stepping, the Navier-displacement equations are used to solve for the grid deformation and sequently, the Navier-Stokes equations are used to solve for the flowfield. Two applications are presented; the first is for a pulsating oscillation of a bending-mode airfoil in transonic flow, and the second is for a bending-mode oscillation of a sharp-edged delta wing in supersonic flow.

Kandil, Osama A.↗

Nonlinear effects in the two-dimensional adaptive-wall outer-flow problem

The importance of nonlinear effects to the solution of two-dimensional adaptive-wall outer-flow problems is presently evaluated by comparing outer-flow solutions computed on the basis of the transonic small perturbation equations with solutions based on the linear Prandtl-Glauert equation. Both methods are applied to simulated measurements of transonic flow past a two-dimensional airfoil in free air. Nonlinear effects are found to be important in the outer-flow solution only where the outer flow included supersonic flow.

Schairer, Edward T.↗

Unsteady vortex-dominated flow around wings with oscillating leading-edge flaps

Unsteady vortex-dominated flow around delta wings with oscillating leading-edge flaps represents an important classs of problems for supermaneuverability and flow control of advanced aircraft. The problem is solved using time accurate integration of the unsteady, compressible, thin-layer Navier-Stokes equations in conjunction with the unsteady, linearized, Navier-displacement equations. Starting with an initial configuration of the wing and its flaps, the Navier-Stokes equations are solved on an initial structured grid for the steady flow. The forced oscillation of the flaps is then applied, and the problem is solved accurately in time. The Navier-displacement equations are solved for the grid deformation and the Navier-Stokes equations are solved for the flowfield. Symmetric and anti-symmetric flaps oscillations are presented to study the effect of the flaps oscillation on the leading-edge vortical flow.

Kandil, Osama A.↗

A Particle Representation Model for the Deformation of Homogeneous Turbulence

In simple flows, where the mean deformation rates are mild and the turbulence has time to come to equilibrium with the mean flow, the Reynolds stresses are determined by the applied strain rate. Hence in these flows, it is often adequate to use an eddy-viscosity representation. The modern family of kappa-epsilon models has been very useful in predicting near equilibrium turbulent flows, where the rms deformation rate S is small compared to the reciprocal time scale of the turbulence (epsilon/kappa). In modern engineering applications, turbulence models are quite often required to predict flows with very rapid deformations (large S kappa/epsilon). In these flows, the structure takes some time to respond and eddy viscosity models are inadequate. The response of turbulence to rapid deformations is given by rapid distortion theory (RDT). Under RDT the nonlinear effects due to turbulence-turbulence interactions are neglected in the governing equations, but even when linearized in this fashion, the governing equations are unclosed at the one-point level due to the non-locality of the pressure fluctuations.

Kassinos, S. C.↗

Factorization of the Compressible Navier-Stokes Equations

The Navier-Stokes equations for a Newtonian ideal gas are examined to determine the factorizable form of the equations relevant to the construction of a factorizable relaxation scheme. The principal linearization of the equations is found by examining the relative magnitude of the terms for short-wavelength errors. The principal part of the operator is then found. Comparison of the factors of the Navier-Stokes and Euler equations differ qualitatively because of the coupling of entropy and pressure through thermal diffusion. Special cases of the factorization are considered.

Roberts, Thomas W.↗

Application of direct solvers to unstructured meshes for the Euler and Navier-Stokes equations using upwind schemes

The application of Newton iteration to inviscid and viscous airfoil calculations on unstructured meshes is examined. A cell-centered finite volume scheme is employed on an unstructured mesh consisting of triangles. Roe's flux difference splitting scheme is used to compute the inviscid fluxes. Higher order accuracy is achieved by an interpolation procedure that makes use of auxiliary gradients. The efficient solution of the sparse linear system of equations which arises upon linearization in time is addressed. Results are presented for inviscid and viscous test cases. The complications which arise due to the introduction of nonlinear limiters are addressed.

Venkatakrishnan, V.↗

Iterative solution of large, sparse linear systems on a static data flow architecture - Performance studies

The applicability of static data flow architectures to the iterative solution of sparse linear systems of equations is investigated. An analytic performance model of a static data flow computation is developed. This model includes both spatial parallelism, concurrent execution in multiple PE's, and pipelining, the streaming of data from array memories through the PE's. The performance model is used to analyze a row partitioned iterative algorithm for solving sparse linear systems of algebraic equations. Based on this analysis, design parameters for the static data flow architecture as a function of matrix sparsity and dimension are proposed.

Reed, D. A.↗