Euler Equations For Transonic Flow Research Articles

Vanishing Viscosity Approach to the Compressible Euler Equations for Transonic Nozzle and Spherically Symmetric Flows

We are concerned with globally defined entropy solutions to the Euler equations for compressible fluid flows in transonic nozzles with general cross-sectional areas. Such nozzles include the de Laval nozzles and other more general nozzles whose cross-sectional area functions are allowed at the nozzle ends to be either zero (closed ends) or infinity (unbounded ends). To achieve this, in this paper, we develop a vanishing viscosity method to construct globally defined approximate solutions and then establish essential uniform estimates in weighted L p norms for the whole range of physical adiabatic exponents $${\gamma\in (1, \infty)}$$ , so that the viscosity approximate solutions satisfy the general L p compensated compactness framework. The viscosity method is designed to incorporate artificial viscosity terms with the natural Dirichlet boundary conditions to ensure the uniform estimates. Then such estimates lead to both the convergence of the approximate solutions and the existence theory of globally defined finite-energy entropy solutions to the Euler equations for transonic flows that may have different end-states in the class of nozzles with general cross-sectional areas for all $${\gamma\in (1, \infty)}$$ . The approach and techniques developed here apply to other problems with similar difficulties. In particular, we successfully apply them to construct globally defined spherically symmetric entropy solutions to the Euler equations for all $${\gamma\in (1, \infty)}$$ .

Solution adaptive direct variational grids for fluid flow calculations

In the present study, the discrete variational (DV) grid generation method is used to produce suitable two dimensional grids for solving the inviscid Euler equations for transonic flow past an airfoil. Through the use of a reference grid, the DV method provides excellent control over grid line clustering in the interior of the grid while maintaining smoothly varying grid line distributions. A solution adaptive grid procedure has been developed using the DV approach which re-distributes the existing grid points in order to resolve regions of high flow (solution) gradients. The grid point re-distribution is governed by the principle of equidistribution. The adaption procedure is done by creating an adapted reference grid which is used to drive the grid point re-distribution. Results for transonic flow past an airfoil are presented to demonstrate the feasibility of the DV-reference grid approach.

Knowledge-Based Nonlinear Boundary Integral Models of Compressible Viscous Flows Over Arbitrary Bodies: Taking CFD Back to Basics

The paper starts with a discussion of a knowledge-based CFD methodology. A new incompressible formulation known as SAVER is first introduced, employing a novel relaxation approach. This is then generalized through a modification of the boundary conditions to the GENESIS methodology, for analysis or design in compressible, rotational flow. A discussion is presented of how the basic causal nature of integral methods offers new insights into certain flow phenomena, such as shocks and separations, and facilitates aerodynamic sensitivity analysis. The paper presents a new class of vector fields approximating the Euler equations for transonic flows, and shows how GENESIS can be used to construct first an exact solution of these approximate fields, then a numerical solution of the residual error fields. The explicit representation of a shock discontinuity on the body boundary exploits its causal link with conditions at the sonic point to suppress, non-dissipatively, the mathematically-valid but physically-impossible formation of an expansion shock.

Block-structured solution of Euler equations for transonic flows

A block-structured solution scheme is developed for the solution of Euler equations for steady three-dimensional transonic flows*. The Euler equations are expressed in terms of Clebsch variables w = v + Svn. The resulting set of equations for conservation of mass, entropy (5) and the Clebsch variable j are solved using a block-based relaxation scheme. The flowfield is divided into a series of individual blocks representing local regions of the flowfield. The employed grid generation scheme provides local refinement of each block yet ensures one-to-one matching of grid points on the surfaces of adjoining blocks. For each of the blocks, either potential or Euler equations can be solved. For every block, the conservation of mass equation is solved to determine . For blocks with rotationality, two additional equations for determining S and T are solved. In this paper, the basic steps of the developed solution scheme are presented. The computational considerations involved in utilizing the present scheme are demonstrated for several two-dimensional sample problems.

Solution of the euler equations for transonic flow over a lifting aerofoil—the bernoulli formulation (Roe/Lytton Method)

The method of Lytton, updated from that of Sells, for solving the Euler equations governing steady inviscid supercritical flow past an aerofoil, is described. Roe's “Superbee” algorithm is incorporated into an explicit, flux-difference-splitting, non-time-split sequence of one-dimensional spatial operators in local-timestep mode. Results are presented for a parametric study of one supercritical case, as a check on the consistency of the method. The ability of the method to compute the flow over a bump on a channel wall is demonstrated. Results from the computation of a supercritical flow with recirculation are also shown.

Numerical solutions of the Euler equations governing axisymmetric and three-dimensional transonic flow

SummaryA computational method is presented for solving numerically the three dimensional Euler equations for transonic flow around practical aircraft forebodies. The Euler solver is pseudo-time dependent, split and cast in finite volume form. Shock waves are captured crisply without the need for additional smoothing by means of an operator-switching facility which more accurately reflects the direction of propagation of signals.The method is illustrated by examples of computed external, axisymmetric flows and some, simulated, realistic aircraft fore-bodies. The computational meshes employed in the three dimensional cases are essentially of cylindrical polar, flow conforming, type and relatively coarse. Closer attention to the mesh generation is expected to refine the results presented here. The method is versatile, robust, and holds promise for treating complex three dimensional geometries within economically viable run times.

Automatic adaptive grid refinement for the Euler equations

A method of adaptive grid refinement for the solution of the steady Euler equations for transonic flow is presented. Algorithm automatically decides where the coarse grid accuracy is insufficient, and creates locally uniform refined grids in these regions. This typically occurs at the leading and trailing edges. The solution is then integrated to steady state using the same integrator (FLO52) in the interior of each grid. The boundary conditions needed on the fine grids are examined and the importance of treating the fine/coarse grid inerface conservatively is discussed. Numerical results are presented.

Computation of flow around wings based on the Euler equations

Inviscid transonic flows containing either strong shock waves or complex vortex structure call for the Euler equations as a realistic model. We present here a computational procedure, termed WINGA2, for solving the Euler equations for transonic flow around aircraft upon a 0–0 mesh generated by transfinite interpolation. An explicit time-marching finite-volume technique solves the flow equations and features a non-reflecting far-field boundary condition and an internal mechanism for temporal damping together with a model for artificial viscosity. The method's convergence to a steady state is studied, and results computed on the CYBER 205 vector processor are presented. The Euler equation model is found to predict the existence of a tip vortex created by flow separating from the downstream region of the tip of the ONERA M6 wing where the radius of curvature approaches zero.

Recent developments in transonic Euler flow over a circular cylinder

Numerical solutions to the Euler equations for transonic flow over a circular cylinder indicate that the inviscid flow separates ahead of the rear stagnation point. Our understanding of this phenomenon and various solutions presented at a workshop on this subject are discussed.

Damped Euler-equation method to compute transonic flow around wing-body combinations

Inviscid transonic flows containing either strong shock waves or complex vortex structure call for the Euler equations as a realistic model. Presented here is a computational procedure for solving the Euler equations for transonic flow around a wing and fuselage upon an O-O mesh generated by transfinite interpolation. An explicit time-marching finite-volume procedure solves the flow equations and features a nonreflecting far field boundary condition, an internal mechanism for temporal damping, and use of the local time step, all of which improve the convergence of the computation. Converged after several hundred iterations, results computed on the CYBER 203 vector processor are compared with experimental data and potential-flow computations. The Euler-equation model is found to predict the existence of a tip vortex created by inviscid flow separation in the downstream region of the tip of the M6 wing where the radius of curvature approaches zero. AST year a workshop 1 was held in order to assess the currently used computational procedures, one against the other, in several carefully specified two-dimensional transonic flow problems. While reasonable agreement among the results given by the various full potential and Euler-equation methods was obtained in the subcritical cases, a disparity between the results of these two models was found to grow with cases of increasing shock strength. For the NACA 0012 airfoil and conditions M- 0.80 and a. = 1.25 deg, for example, the lift coefficients CL given by all the Euler methods ranged between 0.30 and 0.38, whereas the range of CL for the potential method was 0.28-1.1. This discrepancy has led to a reconsideration of the validity of the potential model when strong shocks are present in the flow, not only locally in terms of the isentropic shock jumps but also, and perhaps even more importantly, with regard to the correct modeling of vortex phenomena throughout the entire flowfield. In the potential representation the vorticity that is bound to the airfoil is accounted for by a jump in the potential across a line originating at the trailing edge and lying a priori along some chosen coordinate direction downstream, the so-called Kutta condition. The Euler equations, in contrast, admit vorticity in the solution and the equivalent to the Kutta condition evidently does not need to be enforced explicitly. This situation is currently under study by researchers using a variety of numerical methods. In three dimensions the flow past a finite wing is even more complex and less is known. Vortices, for example, are shed continuously from the wing tips and the entire trailing edge. Whether a Kutta condition or its equivalent is necessary in this case has not been in- vestigated, but because vorticity is so crucial to the realism achieved by inviscid flow models, interest has been aroused in questions like this and in numerical methods that solve the Euler equations. A number of methods 2'4 exist to solve the Euler equations for three-dimensional flow, but they have been developed exclusively for and applied only to internal flows. Apparently, the solution of the Euler equations for