Inviscid Transonic Flow Research Articles

Anisotropic mesh adaptation for numerical solution of boundary value problems

AbstractWe present an efficient mesh adaptation algorithm that can be successfully applied to numerical solutions of a wide range of 2D problems of physics and engineering described by partial differential equations. We are interested in the numerical solution of a general boundary value problem discretized on triangular grids. We formulate a necessary condition for properties of the triangulation on which the discretization error is below the prescribed tolerance and control this necessary condition by the interpolation error. For a sufficiently smooth function, we recall the strategy how to construct the mesh on which the interpolation error is below the prescribed tolerance. Solving the boundary value problem we apply this strategy to the smoothed approximate solution. The novelty of the method lies in the smoothing procedure that, followed by the anisotropic mesh adaptation (AMA) algorithm, leads to the significant improvement of numerical results. We apply AMA to the numerical solution of an elliptic equation where the exact solution is known and demonstrate practical aspects of the adaptation procedure: how to control the ratio between the longest and the shortest edge of the triangulation and how to control the transition of the coarsest part of the mesh to the finest one if the two length scales of all the triangles are clearly different. An example of the use of AMA for the physically relevant numerical simulation of a geometrically challenging industrial problem (inviscid transonic flow around NACA0012 profile) is presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.

Parametric Study of Flutter for an Airfoil in Inviscid Transonic Flow

With the use of a state of the art inviscid computational fluid dynamic harmonic balance aerodynamic-Euler-based code, a systematic, parametric investigation is presented into how the several structural parameters and freestream Mach number of a transonic flow affect the flutter characteristics of a typical two-degree-of-freedom transonic airfoil configuration. The computational efficiency of the time-linearized option of the harmonic balance aerodynamic model allows a much more thorough exploration of the parameter range than has been possible previously.

Reduced-Order Modeling of an Elastic Panel in Transonic Flow

Reduced-order modeling is applied to a transonic aeroelastic panel that experiences oscillatory motions of a normal shock. Two-dimensional, transonic inviscid flow over an elastic panel produces transonic limit-cycle oscillations over a range of panel parameters. Proper orthogonal decomposition, in concert with domain decomposition, is shown to produce an accurate reduced-order model for the coupled aeroelastic system. Panel flutter in the transonic regime results in a large streamwise movement of a transonic normal shock across the panel surface. The accuracy and order reduction of the reduced-order model is quantified. In addition, the computational savings for this implementation is documented, and the robustness of the reduced-order model to changes in a panel parameter is explored.

Composite schemes used for 2D and 3D transonic flows computation

This work deals with numerical solution of 3D inviscid transonic flows in a channel using finite volume form of composite scheme. The combination of less dissipative Lax-Wendroffs cheme and more dissipative Lax-Friedrichs scheme is used in cell-vertex form and basic 3D five sides finite volume. Some 3D results of transonic flows in a channel that is an extension of 2D GAMM channel with a) a changed bump in z-direction b) not changed bump but with swept wing geometry are presented in the article.

Solvability of a problem for transonic flow with a local supersonic region

A nonlinear perturbation problem for steady two-dimensional inviscid transonic flow with a local supersonic region is considered. A damping condition is prescribed on a portion of the boundary in order to prevent the arising of shock waves in the flow. The existence of a smooth solution to the problem is established. A proof is based on the Fredholm alternative and on a technique for uniqueness analysis worked out by Morawetz (1957) and Cook (1978).

On laminar separation at a corner point in transonic flow

The separation of the laminar boundary layer from a convex corner on a rigid body contour in transonic flow is studied based on the asymptotic analysis of the Navier–Stokes equations at large values of the Reynolds number. It is shown that the flow in a small vicinity of the separation point is governed, as usual, by strong interaction between the boundary layer and the inviscid part of the flow. Outside the interaction region the Kármán–Guderley equation describing transonic inviscid flow admits a self-similar solution with the pressure on the body surface being proportional to the cubic root of the distance from the separation point. Analysis of the boundary layer driven by this pressure shows that as the interaction region is approached the boundary layer splits into two parts: the near-wall viscous sublayer and the main body of the boundary layer where the flow is locally inviscid. It is interesting that contrary to what happens in subsonic and supersonic flows, the displacement effect of the boundary layer is primarily due to the inviscid part. The contribution of the viscous sublayer proves to be negligible to the leading order. Consequently, the flow in the interaction region is governed by the inviscid–inviscid interaction. To describe this flow one needs to solve the Kármán–Guderley equation for the potential flow region outside the boundary layer; the solution in the main part of the boundary layer was found in an analytical form, thanks to which the interaction between the boundary layer and external flow can be expressed via the corresponding boundary condition for the Kármán–Guderley equation. Formulation of the interaction problem involves one similarity parameter which in essence is the Kármán–Guderley parameter suitably modified for the flow at hand. The solution of the interaction problem has been constructed numerically.

Solvability of a Boundary Value Problem for Transonic Flow in a Nozzle

A nonlinear perturbation problem for steady two-dimensional inviscid transonic flow in a nozzle is studied. The existence of a smooth solution to the problem is proved under the condition of positive acceleration of the given flow. The proof involves the method of singular perturbations for solving a linear problem associated with the nonlinear one. The technique for obtaining a priori estimates is simpler than that used in previous papers.

A study of non-oscillatory schemes based on LED principle for inviscid flow computation past airfoils

Several high resolution non-oscillatory discretization schemes have been formulated in a finite volume framework for the solution of Euler equations in order to compute inviscid transonic flow fields past airfoils. The theory is based on the local extremum diminishing (LED) principle, initially proposed by Jameson [12], [13] in 1993 for nonlinear scalar conservation laws. Higher order non-oscillatory schemes satisfying the LED criterion have been generated by using a suitable switching function and also by the use of flux limiters. For the implementation of these schemes to the solution of a system of conservation laws, a number of flux splitting techniques have been considered in the present work. Extensive numerical experiments indicate that the flux limited dissipation schemes, viz. the SLIP and USLIP schemes hold the promise of improving the accuracy of the results. The switched scheme with different types of flux splitting has also proved successful in the resolution of shock waves without oscillation.

An Euler correction method for computing two-dimensional unsteady transonic flows

AbstractAn Euler correction method is developed for unsteady, transonic inviscid flows. The strategy of this method is to treat the flow-field behind the shock as rotational flow and elsewhere as irrotational flow. The solution for the irrotational flow is obtained by solving the unsteady full-potential equation using Jameson's rotated time-marching finite-difference scheme. Clebsch's representation of velocity is followed for rotational flow. In this representation the velocities are decomposed into a potential part and a rotational part written in terms of scalar functions. The potential part is computed from the unsteady full potential equation with appropriate modification based on Clebsch's representation of velocity. The rotational part is obtained analytically from the unsteady momentum equation written in terms of Clebsch variables. This method is applied to compute the unsteady flow-field characteristics for an oscillating NACA 64A010 aerofoil. The results of the present calculation are found to be in good agreement with both Euler solution and experimental results.

Finite Volume Discretization Aspects for Viscous Flows on Mixed Unstructured Grids

A solution method for compressible turbulent flows on unstructured grids in two dimensions is described. The method can be used on grids consisting of triangular and/or quadrilateral cells. Control volumes are constructed from dual cells, and the solution variables are stored at the vertices of the grid. Grid-transparent algorithms are developed that do not require knowledge of cell types, leading to simple discretization schemes on mixed grids. The inviscid fluxes are computed from limited high-resolution schemes originally developed for unstructured triangular grids. They are easily applied to quadrilateral or mixed grids and are grid transparent. The discretization of the viscous fluxes is studied in detail. A positive, grid-transparent discretization of Laplace's equation is developed. The existence of tangential derivatives in the viscous terms prevents grid transparency. By neglecting tangential derivatives, an approximate form of the viscous fluxes is developed, which recovers grid transparency. The approximate form is shown to be similar to the thin-shear-layer approximation. Results are obtained for a transonic inviscid flow, a laminar separated flow, and a transonic turbulent flow

Three-dimensional simulation of nonstationary flow phenomena in “Last stage-exhaust hood” compartment

Three-dimensional nonstationary model of aerodynamical interaction of turbine stage and exhaust hood is realized, based on nonstationary 3D codes for calculation of inviscid transonic flow through stage[3] and exhaust hood[4] which consist of diffuser and space under casing. The codes are built with the use of the explicit Godunov’s 2nd order difference scheme. Some results of flow simulation through the compartments “stage-exhaust hood,” “stage-exhaust axial-radial diffuser” for wide range of volumetric flow rates are represented.

A NEW OPTIMAL SHAPE DESIGN PROCEDURE FOR INVISCID AND VISCOUS TURBULENT FLOWS

A new approach for optimal shape design is introduced. The main ingredients are an unstructured CAD-free framework for geometry deformation and automatic differentiation (AD) in reverse mode. Transonic inviscid and viscous turbulent flows are investigated. Both two- and three-dimensional configurations are considered. These cases involve up to several thousand control parameters.

圧縮性流れの陰的非構造解法における前処理付反復法の比較

Preconditioned iterative methods in an implicit unstructured grid solver of compressible flows are compared. The three-dimensional Euler and Navier-Stokes equations are discretized by the edge-based finite volume method. The artificial dissipation scheme for the inviscid numerical fluxes is developed for the unstructured grid from Jameson's scheme for the structured grid. The following methods are compared to the supersonic and transonic inviscid flows and subsonic viscous laminar and turbulent flows : the central difference methods with the scalar and matrix dissipation scheme and Roe's upwind method with the MUSCL sheme for the inviscid fluxes ; the scalar and matrix dissipation method for the left-hand side operator in the implicit scheme ; the ILU, DILU and SGS factorizations for the preconditioner ; and GMRES and Bi-CGSTAB methods for the linear system solver. The results show that a combination of the matrix dissipation in the inviscid fluxes and the scalar dissipation in the left-hand side works better than the other shemes. In the preconditioned iterative methods, a combination of DILU or SGS and the Bi-CGSTAB is recommended for the aspect of required memory.

Automatic grid generation and flow solution for complex geometries

The development of a new method for the automatic generation of computational grids around complex aerodynamic configurations is described. The grid-generation procedure uses a Cartesian cell, embedded, unstructured approach that enables the efficient placement of computational cells to resolve important flow regions close to the surface. The main difference between the present method and most existing grid-generation techniques is the avoidance of the need to explicitly generate a surface grid. This new approach is easily automated and is capable of producing grids around arbitrarily complex geometries. The capabilities of the method are demonstrated with a selection of inviscid transonic flows solutions.

The inviscid transonic flow about a cylinder

The two-dimensional inviscid transonic flow about a circular cylinder is investigated. To do this, the Euler equations are integrated numerically with a time-dependent technique. The integration is based on an high-resolution finite volume upwind method.Time scales are introduced and the flow at very short, short and large times is studied. Attention is focused on the behaviour of the numerical solution at large times, after the breakdown of symmetry and the onset of an oscillating solution have occurred. This solution is known to be periodic at Mach number between 0.5 and 0.6.At higher speed, however, a richer behaviour is observed. As the Mach number is increased from 0.6 to 0.98 the numerical solution undergoes two transitions. Through a first one the periodical, regular flow enters a chaotic, turbulent regime. Through the second transition the chaotic flow comes back to an almost stationary state. The flow in the chaotic and in the almost stationary regimes is investigated. A numerical conjecture for the behaviour of the solution at large times is advanced.

Compressible flowfield solutions with unstructured grids generated by Delaunay triangulation

A method is described that constructs three-dimensional unstructured tetrahedral meshes using the Delaunay triangulation criterion. The approach includes an automatic point creation technique and ensures that, given an initial surface triangulation which bounds a domain, a valid boundary conforming assembly of tetrahedra is produced. The efficiency of the proposed procedure reduces the computer time for the generation of realistic unstructured tetrahedral grids to the order of minutes on workstations of modest computational capabilities. The grids generated are used with two finite element algorithms to simulate inviscid transonic compressible flows. Flow solutions for aerospace configurations are presented and computed results compared with experimental data.

A PRESSURE-BASED EULER SCHEME FOR TRANSONIC INTERNAL AND EXTERNAL FLOW SIMULATION

SUMMARY The development of a pressure-based Euler scheme, based on a collocated grid arrangement and solving for Cartesian mass flux components is described. An implicit numerical dissipation model, which includes second and fourth difference terms expressed in pressure, is demonstrated. A smoothing function, which includes a two-level filter, is used to adjust the second order-dissipation. The successful calculations for the inviscid transonic aerodynamic flow around airfoil NACA0012 are presented. The transonic quasi-one-dimensional flow calculation in a converging-diverging nozzle is chosen as an example of an internal flow simulation.

Aerodynamic shape optimization using preconditioned conjugate gradient methods

In an effort to further improve upon the latest advancements made in aerodynamic shape optimization procedures, a systematic study is performed to examine several current solution methodologies as applied to various aspects of the optimization procedure. It is demonstrated that preconditioned conjugate gradient-like methodologies dramatically decrease the computational efforts required for such procedures. The design problem investigated is the shape optimization of the upper and lower surfaces of an initially symmetric (NACA-012) airfoil in inviscid transonic flow and at zero degree angle-of-attack. The complete surface shape is represented using a Bezier-Bernstein polynomial. The present optimization method then automatically obtains supercritical airfoil shapes over a variety of freestream Mach numbers. Furthermore, the best optimization strategy examined resulted in a factor of 8 decrease in computational time as well as a factor of 4 decrease in memory over the most efficient strategies in current use.

Multigrid convergence of an implicit symmetric relaxation scheme

The multigrid method has been applied to an existing three-dimensional compressible Euler solver to accelerate the convergence of the implicit symmetric relaxation scheme. This lower-upper symmetric Gauss-Seidel implicit scheme is shown to be an effective multigrid driver in three dimensions. A grid refinement study is performed including the effects of large cell aspect ratio meshes. Performance figures of the present multigrid code on Cray computers including the new C90 are presented. A reduction of three orders of magnitude in the residual for a three-dimensional transonic inviscid flow using 920 k grid points is obtained in less than 4 min on a Cray C90.

ANALYSIS/DESIGN CALCULATIONS OF TRANSONIC FLOWS

SUMMARY Analysis/design calculations of transonic flow are discussed and several improvements are made. The nonisentropic potential method is used to calculate the inviscid transonic flow analysis problem instead of the traditional potential method. An inverse integral 3D boundary layer method is used to calculate the boundary layer in the viscous transonic flow analysis problem. The viscous/inviscid interaction calculations are carried out by a semi-inverse coupling scheme. In design problem calculations, an improved residual-correction method is used. Three individual methods are combined in a global algorithm and computing code. The improvements speed up the convergence, increase applicability and computational efficiency. Some numerical results are given to illustrate that the present method provides an effective engineering tool of high accuracy and efficiency in three dimensional transonic analysis and design situations.