Inviscid Transonic Flow Research Articles

Inverse Design of Transonic Airfoils Using Variable-Resolution Modeling and Pressure Distribution Alignment

Abstract The paper presents a computationally efficient and robust methodology for the inverse design of transonic airfoils. The approach replaces the direct optimization of an accurate, but computationally expensive, high-fidelity airfoil model by an iterative reoptimization of a corrected low-fidelity model. The low-fidelity model is based on the same governing fluid flow equations as the high-fidelity one, but uses coarser discretization and relaxed convergence criteria. The shape-preserving response prediction technique is utilized to align the pressure distribution of the low-fidelity model with that of the high-fidelity model. This alignment process is particularly suitable since a target pressure distribution is specified in the inverse design problem. The method is applied to constrained inverse airfoil design in inviscid transonic flow. The results show that the proposed method is able to match the target pressure distributions closely and requiring over 90 percent lower computational cost than when using only the high-fidelity model.

High effective WENO scheme for transonic inviscid flow computation on a helicopter rotor in hover

A high-order upwind scheme with high efficiency has been developed to compute the inviscid flow of a helicopter rotor in hover. For rotary-wing wake capturing, an improved fifth-order weighted essentially non-oscillatory (WENO) scheme is adopted to interpolate higher order left and right states across a cell interface with Roe Riemann solver updating inviscid flux. To improve the efficiency and convergence to steady state, three-level V-cycle multigrid relaxation scheme is used. The performance of the schemes is investigated in a transonic inviscid flow around hovering rotor. The results reveal that WENO has the great capabilities to capture shock with high resolution and has a lower numerical dissipation than MUSCL for capturing wake vorticity. Moreover, multigrid scheme can accelerate convergence to a great extent.

Robust optimization of dense gas flows under uncertain operating conditions

A robust optimization procedure based on a multi-objective genetic algorithm (MOGA) is used to generate airfoil profiles for transonic inviscid flows of dense gases, subject to uncertainties in the upstream thermodynamic conditions. The effect of the random variations on system response is evaluated using a non-intrusive polynomial chaos (PC) based method known as the probabilistic collocation method (PCM). After initial PCM simulations which showed that the dense gas system was highly sensitive to input parameter variation, a multi-objective genetic algorithm coupled to the PCM produced a Pareto front of optimized individual geometries which exhibited improvements in mean performance and/or stability over the baseline NACA0012 airfoil. This type of analysis is essential in improving the feasibility of organic Rankine cycle (ORC) turbines, which are typically designed to recover energy from variable sources such as waste heat from industrial processes.

Multi-fidelity design optimization of transonic airfoils using physics-based surrogate modeling and shape-preserving response prediction

A computationally efficient design methodology for transonic airfoil optimization has been developed. In the optimization process, a numerically cheap physics-based low-fidelity surrogate (the transonic small-disturbance equation) is used in lieu of an accurate, but computationally expensive, high-fidelity (the compressible Euler equations) simulation model. Correction of the low-fidelity model is achieved by aligning its corresponding airfoil surface pressure distribution with that of the high-fidelity model using a shape-preserving response prediction technique. The resulting method requires only a single high-fidelity simulation per iteration of the design process. The method is applied to airfoil lift maximization in two-dimensional inviscid transonic flow, subject to constraints on shock-induced pressure drag and airfoil cross-sectional area. The results showed that more than a 90% reduction in high-fidelity function calls was achieved when compared to direct high-fidelity model optimization using a pattern-search algorithm.

Multi-fidelity design optimization of transonic airfoils using shape-preserving response prediction

A computationally efficient methodology for transonic airfoil design optimization is presented. Our approach exploits a corrected physics-based low-fidelity surrogate that replaces, in the optimization process, an accurate but computationally expensive highfidelity airfoil model. Correction of the low-fidelity model is achieved by aligning its corresponding airfoil surface pressure distributions with that of the high-fidelity model using a shape-preserving response prediction technique. The presented method is applied to airfoil lift maximization in two-dimensional inviscid transonic flow, subject to constraints on shock induced pressure drag and airfoil cross-sectional area. More than a 90% reduction in high-fidelity function calls is achieved when compared to direct highfidelity model optimization.

A generalized framework for high order anisotropic mesh adaptation

We present a method for anisotropically adapting meshes for high order ( p > 2 ) finite volume methods. We accomplish this by assuming a polynomial error measure based on the local reconstruction. We then use Fourier series to choose a metric function that approximates our error measure. This approach is theoretically valid for any solution order, with any number of variables and any number of dimensions. Using both second and third order solvers, we present examples of two-dimensional subsonic viscous and inviscid flow around the NACA-0012 airfoil. These test cases demonstrate that mesh refinement based on our metric converges to an accurate solution much faster than with uniform refinement. For subsonic viscous flows, we also show that anisotropic refinement is more efficient than isotropic refinement. By limiting the magnitude of our error measure for a transonic inviscid flow, we demonstrate that our metric can be effective even in the presence of discontinuities. We also examined using second derivatives to estimate the error for third order solutions. While this is less theoretically sound, we found little difference in the resulting accuracy.

Numerical solution of transonic and subsonic flow of compressible inviscid and viscous fluid

AbstractThe work presents two numerical solutions of compressible flows problems with high and very low Mach numbers. Both problems are numerically solved by finite volume method and the explicit MacCormack scheme using a grid of quadrilateral cells. Moved grid of quadrilateral cells is considered in the form of conservation laws using Arbitrary Lagrangian–Eulerian method. In the first case, inviscid transonic flow through cascade DCA 8% is presented and the numerical results are compared to experimental data. The second case, numerical solution of unsteady viscous flow in the channel for upstream Mach number M∞=0.012 and frequency of the wall motions 100 Hz is presented. The unsteady case can represent a simplified model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds to the human vocal tract. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Turbine cascade design via multigrid-aided finite-difference progressive optimization

This paper proposes an efficient and robust procedure for the design optimization of turbomachinery cascades in inviscid and turbulent transonic flow conditions. It employs a progressive strategy, based on the simultaneous convergence of the design process and of all iterative solutions involved (flow analysis, gradient evaluation), also including the global refinement from a coarse to a sufficiently fine mesh. Cheap, flexible and easy-to-program Multigrid-Aided Finite Differences are employed for the computation of the sensitivity derivatives. The entire approach is combined with an upwind finite-volume method for the Euler and the Navier-Stokes equations on cell-vertex unstructured (triangular) grids, and validated versus the inverse design of a turbine cascade. The methodology turns out to be robust and highly efficient, the converged design optimization being obtained in a computational time equal to that required by 15 to 20 (depending on the application) multigrid flow analyses on the finest grid.

Continuous Adjoint Method for Unstructured Grids

Adjoint-based shape optimization methods have proven to be computationally efficient for aerodynamic problems. The majority of the studies on adjoint methods have used structured grids to discretize the computational domain. Because of the potential advantages of unstructured grids for complex configurations, in this study we have developed and validated a continuous adjoint formulation for unstructured grids. The hurdles posed in the computation of the gradient for unstructured grids are resolved by using a reduced gradient formulation. The methods to impose thickness constraints on unstructured grids are also discussed. The results for two- and three-dimensional simulations of airfoils and wings in inviscid transonic flow are used to validate the design procedure. Finally, the design procedure is applied to redesign the shape of a transonic business jet configuration; we were able to reduce the inviscid drag of the aircraft from 235 to 216 counts resulting in a shock-free wing. Although the Euler equations are the focus of the study in this paper of the adjoint-based approach, the solution of the adjoint system and gradient formulation can be conceptually extended to viscous flows. The approach presented in this study has been successfully used by the first and third authors for viscous flows using structured grids. However, particular aspects of the design process, such as the robustness of the mesh deformation process for unstructured grids, need more attention for viscous flows and are therefore the subject of ongoing research.

Transonic flows of dense gases over finite wings

Transonic inviscid flows of dense gases of the Bethe–Zel’dovich–Thompson (BZT) type over finite wings are numerically investigated. BZT gases are fluids of the retrograde type (i.e., that superheat when expanded), which exhibit a region of negative values of the fundamental derivative of gas dynamics Γ. As a consequence, they display, in the transonic and supersonic regime, nonclassical gas dynamic behaviors, such as rarefaction shock waves and mixed shock/fan waves. The peculiar properties of BZT fluids have received increased interest in recent years because of their possible application in energy-conversion cycles. The present research aims at providing insight about the transonic aerodynamics of BZT fluids past finite wings, roughly representative of isolated turbine blades with infinite tip leakage. This represents an important step toward the design of advanced turbine blades by using organic working fluids. An investigation of the flow patterns and aerodynamic performance for several choices of the...

On the use of orthogonal functions in fluid‐dynamic gradient‐based optimization

AbstractThis paper proposes a method to derive a set of symmetrical and antisymmetric orthogonal shape functions, which can be efficiently used in fluid‐dynamic design optimization. The inverse design of an airfoil in inviscid transonic flow conditions is proposed to demonstrate that a gradient‐based optimization using orthogonal profiles is 6–10 times faster than the same procedure using non‐orthogonal interpolation functions. Copyright © 2008 John Wiley & Sons, Ltd.

Vanishing Viscosity Method for Transonic Flow

A vanishing viscosity method is formulated for two-dimensional transonic steady irrotational compressible fluid flows with adiabatic constant \(\gamma \in [1,3)\) . This formulation allows a family of invariant regions in the phase plane for the corresponding viscous problem, which implies an upper bound uniformly away from cavitation for the viscous approximate velocity fields. Mathematical entropy pairs are constructed through the Loewner–Morawetz relation via entropy generators governed by a generalized Tricomi equation of mixed elliptic–hyperbolic type, and the corresponding entropy dissipation measures are analyzed so that the viscous approximate solutions satisfy the compensated compactness framework. Then the method of compensated compactness is applied to show that a sequence of solutions to the artificial viscous problem, staying uniformly away from stagnation with uniformly bounded velocity angles, converges to an entropy solution of the inviscid transonic flow problem.

Turbine cascade design via multigrid-aided finite-difference progressive optimization

This paper proposes an efficient and robust procedure for the design optimization of turbomachinery cascades in inviscid and turbulent transonic flow conditions. It employs a progressive strategy, based on the simultaneous convergence of the design process and of all iterative solutions involved (flow analysis, gradient evaluation), also including the global refinement from a coarse to a sufficiently fine mesh. Cheap, flexible and easy-to-program Multigrid-Aided Finite Differences are employed for the computation of the sensitivity derivatives. The entire approach is combined with an upwind finite-volume method for the Euler and the Navier-Stokes equations on cell-vertex unstructured (triangular) grids, and validated versus the inverse design of a turbine cascade. The methodology turns out to be robust and highly efficient, the converged design optimization being obtained in a computational time equal to that required by 15 to 20 (depending on the application) multigrid flow analyses on the finest grid.

Transonic viscous inviscid interactions in narrow channels

AbstractUnsteady as well as steady transonic flows through channels which are so narrow that the classical boundary layer approach fails are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting interaction problem for laminar flows is formulated for both perfect and dense gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes one‐dimensional in the leading order approximation. The latter allows an interpretation of the flow in the core region by means of the theory of one‐dimensional transonic inviscid flow through a Laval nozzle while preserving the essential features of the interaction problem associated with the internal structure of pseudoshocks. The sensitivity of a separation bubble caused by a pseudoshock of sufficient strength to perturbations under the condition of choked flow will be demonstrated. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Error Estimation and Grid Adaptation Using Euler Adjoint Method

An adjoint-based error estimation and grid adaptation study is conducted for three-dimensional inviscid flows with unstructured meshes. The error in an integral output functional of interest is estimated by a dot product of the residual error and adjoint variable vector. To suppress excessive mesh refinement in unnecessary regions due to high frequency noise contained in the residual vector, the flow residual is smoothed using a volume-weighted averaging process. Regions to be adapted are selected based on the error contribution of a local node to the global error. The adaptive regions are refined by the bi-section refinement algorithm. The present procedure is applied to threedimensional transonic inviscid flows around ONERA M6 wing and ONERA M5 airplane models. The same level of prediction accuracy for drag is achieved with much less mesh points than uniformly refined fine meshes. It is found that the residual smoothing strategy remarkably improves the accuracy of error estimation, and there exists an optimum number of smoothing for accurate error estimation.

Estimating the Probability of Failure of a Nonlinear Aeroelastic System

A limit-cycle oscillation (LCO) can be characterized by a subcritical or supercritical bifurcation, and bifurcations are shown to be discontinuities in the stochastic domain. The traditional polynomial-chaos-expansion method, which is a stochastic projection method, is too inefficient for estimating the LCO response surface because of the discontinuities associated with bifurcations. The objective of this research is to extend the stochastic projection method to include the construction of B-spline surfaces in the stochastic domain. The multivariate B-spline problem is solved to estimate the LCO response surface. A Monte Carlo simulation (MCS) is performed on this response surface to estimate the probability density function (PDF) of the LCO response. The stochastic projection method via B-splines is applied to the problem of estimating the PDF of a subcritical LCO response of a nonlinear airfoil in inviscid transonic flow. A probability of failure based upon certain failure criteria can then be computed from the estimated PDF. The stochastic algorithm provides a conservative estimate of the probability of failure of this aeroelastic system two orders of magnitude more efficiently than performing an MCS on the governing equations.

The structural instability of transonic flow associated with amalgamation/splitting of supersonic regions

We study inviscid transonic flow in a channel with a bump modeling an airfoil at zero incidence. A few simple bump configurations are considered. Numerical simulations show the existence of singular free-stream Mach numbers, which trigger the splitting/amalgamation of local supersonic regions. An analysis of this phenomenon contributes to the understanding of the nonuniqueness of transonic solutions at certain free-stream conditions.

Structural Instability of Transonic Flow over an Airfoil

Inviscid transonic flow over the DSMA523a and Whitcomb airfoils and an airfoil described by a simple algebraic formula has been investigated numerically. The dependence of the flow structure and the lift coefficient on the freestream Mach number Minfin and the angle of attack α has been studied. The values of Minfin and α at which steady‐state flow becomes unstable have been revealed. The relationship between the flow nonuniqueness occurring in certain ranges of variation of Minfin and α and the instability has been analyzed.

Nonuniqueness of the Transonic Flow past an Airfoil

The inviscid transonic flow past a symmetric airfoil having a curvature minimum in the middle is numerically investigated. It is shown that at zero angle of attack α both symmetric and asymmetric steady-state flow patterns can exist on a certain freestream Mach number range Mmin <M∞ < Mmax. On this range, the asymmetric flows are stable against small perturbations, whereas the symmetric flows are stable only if M∞ does not coincide with a singular Mach number at which small variations in M∞ or α can result in flow restructuring.

An investigation of separation near corner points in transonic flow

The incipient separation from a corner in steady two-dimensional transonic flow is studied based on viscous-inviscid interaction at high Reynolds number. Of particular interest is the investigation of the dependence of the critical deflection angle (when a well-attached flow turns into a separated flow) on the Karman-Guderley parameter which characterizes the local flow field. In accordance with the procedure adopted, the analysis of the flow starts with the analysis of the boundary layer and then the solution of the Karman-Guderley equation describing the inviscid part of the flow near the corner point is investigated. The analysis of the inviscid transonic flow is performed based on the hodograph method and new solutions are obtained corresponding to the present flow topologies. In these solutions, the transonic flow appears to be subsonic everywhere except at the sonic corner point. Then, the interaction problem is formulated using the triple-deck model. Lastly, a procedure based on a semi-direct solution of the governing equations using Newton iterations is developed to obtain the numerical solution of the interaction problem