Transonic Potential Flow Research Articles

Transonic potential flows in a convergent–divergent approximate nozzle

In this paper we prove existence, uniqueness and regularity of certain perturbed (subsonic–supersonic) transonic potential flows in a two-dimensional Riemannian manifold with “convergent–divergent” metric, which is an approximate model of the de Laval nozzle in aerodynamics. The result indicates that transonic flows obtained by quasi-one-dimensional flow model in fluid dynamics are stable with respect to the perturbation of the velocity potential function at the entry (i.e., tangential velocity along the entry) of the nozzle. The proof is based upon linear theory of elliptic–hyperbolic mixed type equations in physical space and a nonlinear iteration method.

Subsonic solutions to compressible transonic potential problems for isothermal self-similar flows and steady flows

We establish boundary and interior gradient estimates, and show that no supersonic bubble appears inside of a subsonic region for transonic potential flows for both self-similar isothermal and steady problems. We establish an existence result for the self-similar isothermal problem, and improve the Hopf maximum principle to show that the flow is strictly elliptic inside of the subsonic region for the steady problem.

Subsonic solutions for compressible transonic potential flows

We establish an existence theorem for transonic isentropic potential flows where the subsonic region is bounded by the sonic line and thus the governing equation may become degenerate on the boundary partly or entirely. It has been conjectured by experiments and numerical studies that the self-similar multidimensional flow changes its type, namely, hyperbolic far from the origin (supersonic region) and elliptic near the origin (subsonic region). Furthermore, the potential equation has a different nonlinearity compared to other transonic problems such as the unsteady transonic small disturbance equation, the nonlinear wave equation, and the pressure gradient equation. Namely, the coefficients of the potential equation depend on the gradients while others are independent of the gradients. We provide techniques to handle the gradients, establish interior and boundary gradient estimates for the potential flow in a convex region, and answer the conjecture, that is, the flow is strictly elliptic and the region is subsonic.

A variational approach to transonic potential flow problems

A variational approach to transonic potential flow problems

A variational approach to transonic potential flow problems

This paper continues considerations of transonic potential flow problems by variational methods. A functional which is associated with a boundary value problem for the (full) potential equation and which possesses a real physical meaning is minimized over a class of admissible functions. These functions have to satisfy a non-linear local entropy condition and a certain boundness constraint. Though this class is not a compact set of the underlying Hilbert space and though the functional need not be convex, the existence of a solution to the established variational problem can be proved by direct methods of the calculus of variations. Furthermore, some properties of minimizers concerning uniqueness, relation to the potential equation, and behaviour on supersonic regions are derived. Copyright © 2000 John Wiley & Sons, Ltd.

Transonic flow of dense gases around an airfoil with a parabolic nose

Transonic potential flow of dense gases of retrograde type around the leading edge of a thin airfoil with a parabolic nose is studied. The analysis follows the approach of Rusak (1993) for a perfect gas. Asymptotic expansions of the velocity potential function are constructed in terms of the airfoil thickness ratio in an outer region around the airfoil and in an inner region near the nose. The outer expansion consists of the transonic small-disturbance theory for dense gases, where a leading-edge singularity appears. Analytical expressions are given for this singularity by constructing similarity solutions of the governing nonlinear equation. The inner expansion accounts for the flow around the nose, where a stagnation point exists. A boundary value problem is formulated in the inner region for the solution of an oncoming uniform sonic flow with zero values of the fundamental derivative of gasdynamics (Γ=0) and the second nonlinearity parameter (Λ=0) around a parabola at zero angle of attack. The numerical solution of the inner problem results in a symmetric flow around the nose. The outer and inner expansions are matched asymptotically resulting in a uniformly valid solution on the entire airfoil surface. In the leading terms, the flow around the nose is symmetric and the stagnation point is located at the leading edge for every transonic Mach number, and small values of Γ and Λ of the oncoming flow and any shape and small angle of attack of the airfoil. Furthermore, analysis of the inner region in the immediate neighbourhood of the stagnation point reveals that the flow is purely subsonic, approaching critical conditions in the limit of large (scaled) distances, which excludes the formation of shock discontinuities in the nose region.

STEADY TWO-DIMENSIONAL TRANSONIC ANALYSIS USING A BOUNDARY INTEGRAL EQUATION METHOD

A boundary integral equation method for the simulation of two-dimensional steady transonic potential flows is presented. The method is based on a conservative differential full-potential formulation. The steady two-dimensional formulation is obtained as the limiting case of the unsteady three-dimensional one, as the iterative method used to obtain steady-state results is a pseudo-time-accurate three-dimensional technique. In the present formulation the full-potential equation appears in the form of a nonlinear wave equation for the velocity potential. All the nonlinear terms, which are expressed in conservative form, are moved to the right-hand side, and treated formally as nonhomogeneous terms. The paper includes a historical review on the development of integral formulations for transonic analysis. Numerical results are obtained for steady two-dimensional transonic flows. Comparisons with existing finite-difference, finite-element, and finite-volume results shows a good agreement. The convergence analysis for an increasing number of grid elements reveals a limit behaviour in good agreement with other numerical methods in both subcritical and supercritical problems. Finally, we present numerical results to demonstrate a remarkable feature of the formulation, that is that the transonic numerical results are quite insensitive to the geometry of the field volume elements; this makes the present formulation particularly appealing for optimal design applications.

Analysis of a FEM/BEM coupling method for transonic flow computations

A sensitive issue in numerical calculations for exterior flow problems, e.g. around airfoils, is the treatment of the far field boundary conditions on a computational domain which is bounded. In this paper we investigate this problem for two–dimensional transonic potential flows with subsonic far field flow around airfoil profiles. We take the artificial far field boundary in the subsonic flow region. In the far field we approximate the subsonic potential flow by the Prandtl–Glauert linearization. The latter leads via the Green representation theorem to a boundary integral equation on the far field boundary. This defines a nonlocal boundary condition for the interior ring domain. Our approach leads naturally to a coupled finite element/boundary element method for numerical calculations. It is compared with local boundary conditions. The error analysis for the method is given and we prove convergence provided the solution to the analytic transonic flow problem around the profile exists.

Transonic Flow Around Optimum Critical Airfoils

Optimum critical nonlifting airfoils that have the highest free stream Mach number $M_\infty $ for a given thickness ratio $\delta $ have recently been presented by Schwendeman, Kropinski, and Cole [Z. Angew. Math. Phys., 44 (1993), pp. 556–571]. The asymptotic shape of these airfoils as $M_\infty \to 1$ and $\delta \to 0$ consists of a sonic arc with a $( x^{2/ 5} )$ nose. In the present paper the transonic potential flow around these asymptotic shapes is analyzed. Asymptotic expansions of the velocity potential function are constructed in terms of $\delta $ in an outer region around the airfoil and in an inner region near the nose. The outer expansion consists of the transonic small disturbance theory, where a leading-edge singularity appears. The inner expansion accounts for the flow around the nose, where a stagnation point exists. The matching between the expansions results in a boundary value problem in the inner region for the solution of a uniform sonic flow about a two-dimensional $( x^{2/ 5} )$ surface. The numerical solution of the inner flow results in the pressure distribution on the nose. Finally, a uniformly valid pressure distribution over the entire airfoil surface is derived.

Transonic flow computations using streamfunction and potential function

AbstractA new formulation is proposed for two dimensional transonic potential flow calculations. The present approach takes advantage of both streamfunction and potential function formulations. A modified form of the partial differential equations and simplified boundary conditions are derived. Numerical examples show that a reliable method has been developed, capable of calculating highly transonic potential flows.

A second order finite difference error indicator for adaptive transonic flow computations

An adaptive finite element method for the calculation of transonic potential flows was developed. An error indicator based on first order finite differences of gradients is introduced as a local error estimator. It measures second order distributional derivatives. Estimates involving this error estimator, a residual and the error are given. The error estimator can be used as a criterion for mesh refinement. We also give some computational results.

Field panel method with grid stretching technique for solving transonic potential flow around arbitrary airfoils

The Field Panel Method (FPM) with grid stretching technique, presented in this paper, was developed for solving transonic full potential flow around arbitrary airfoils at incidence. In this method, the total potential values are represented by boundary integrals together with a volume integral. The volume integral domain includes both inside and finite outside of the configuration and can be discretisized in a Cartesian grid which may penetrate into the configuration surface. Thus, we avoid the very difficult task of generating body-fitted grids around complex configurations. The boundary potential values are obtained by implementing a standard panel method (symmetrical singularity model), whereas the field potential values are estimated by solving the full potential equation (using AF3 scheme in a Cartesian grid) with approximate inner and proper outer boundary conditions. Furthermore, the grid stretching technique has been utilized that allows to capture the shock waves in a much better quality. It is also shown that both field grid and panel distribution have to be stretched at the same time. Results for transonic potential flows about NACA0012 and RAE2822 airfoils at different Mach numbers and incidences are obtained and compared with other numerical solutions. Great improvement in shock wave quality was achieved by using the present method.

Field panel method with grid stretching technique for solving transonic potential flow around arbitrary airfoils

Field panel method with grid stretching technique for solving transonic potential flow around arbitrary airfoils

Steady MHD flows with an ignorable co-ordinate and the potential transonic flow equation

The paper explores the interrelationship between the generalized Grad-Shafranov equation, or trans-field force balance equation, for steady MHD flows with an ingnorable co-ordinate, and work by Imai on field-aligned MHD flows. The development of Imai, assumes at the outset that the fluid velocity V is parallel to the magnetic field B, and exploits an analogy with steady compressible irrotational flow in ordinary fluid dynamics. In Imai's analysis the magnetic induction B is written in the form B = σb, where , and MA is the appropriate Alfvén Mach number. Gauss' law Δ. B = Δ. (σb) = 0 then plays a role analogous to the mass continuity equation in ordinary fluid dynamics, where σ corresponds to the density of the pseudo-fluid. Imai's analysis leads to a transonic equation for the field potential φ defined by b = Δφ. For a restricted class of flows the trans-field force balance equation formulation also leads to the transonic potential flow equation, but the assumption of an ignorable co-ordinate allows for the possibility of non-field-aligned flows with non-zero electric field potential ΦE The characteristics of the generalized Grad—Shafranov equation are related to the Mach cone and the group velocity surface for linear magnetosonic waves. The corresponding forms of the characteristics for the potential transonic flow equation in the (x, y) plane and in the (bx, by) hodograph plane are discussed. Sample solutions of the potential transonic flow equation for radial, helical and spiral flows are obtained by means of the hodograph transformation, and are used to illustrate the differences between hyperbolic and elliptic flows. The potential transonic flow equation is obtained for the case of an ignorable co-ordinate z of a rectangular Cartesian co-ordinate system (x, y, z), and also for the case of flows with an ignorable co-ordinate of a spherical polar co-ordinate system (r, θ, ω). Astrophysical applications are briefly discussed.

A hybrid integral equation finite volume scheme for transonic potential flow about complex configurations

AbstractA hybrid integral equation finite volume scheme has been developed for the calculation of transonic potential flow about complex configurations. A new technique has been used for evaluating the potential values in the field cells intersecting the body surface panels. These potential values then serve as Dirichlet boundary conditions for computing the potentials in the field by a finite volume Successive Line Over Relaxation (SLOR) scheme. In this approach there is no need to evaluate the potentials anywhere in the field by direct application of Green's third identity, thus significantly reducing computer processing time and storage requirement, while improving accuracy of surface pressure prediction and shock capture, as results indicate. The capability of tackling additional complex geometry with ease, the primary advantage of the integral equation approach, is demonstrated by using the same field grid for wing-alone and wing-body combination cases, while maintaining the solution accuracy.

A shock indicator for adaptive transonic flow computations

An adaptive finite element method for the calculation of transonic potential flows was developed. A residual based error indicator is complemented by a shock indicator. For a good shock resolution mesh refinement as well as moving nodes were needed. An analysis of the method and computational results are given.

Analysis of the Finite Element Variational Crimes in the Numerical Approximation of Transonic Flow

The paper presents a detailed theory of the finite element approximations of two-dimensional transonic potential flow. We consider the boundary value problem for the full potential equation in a general bounded domain $\Omega$ with mixed Dirichlet-Neumann boundary conditions. In the discretization of the problem we proceed as usual in practice: the domain $\Omega$ is approximated by a polygonal domain, conforming piecewise linear triangular elements are used, and the integrals are evaluated by numerical quadratures. Using a new version of entropy compactification of transonic flow and the theory of finite element variational crimes for nonlinear elliptic problems, we prove the convergence of approximate solutions to the exact physical solution of the continuous problem, provided its existence can be shown.

Transonic flow around the leading edge of a thin airfoil with a parabolic nose

Transonic potential flow around the leading edge of a thin two-dimensional general airfoil with a parabolic nose is analysed. Asymptotic expansions of the velocity potential function are constructed at a fixed transonic similarity parameter (K) in terms of the thickness ratio of the airfoil in an outer region around the airfoil and in an inner region near the nose. These expansions are matched asymptotically. The outer expansion consists of the transonic small-disturbance theory and it second-order problem, where the leading-edge singularity appears. The inner expansion accounts for the flow around the nose, where a stagnation point exists. Analytical expressions are given for the first terms of the inner and outer asymptotic expansions. A boundary value problem is formulated in the inner region for the solution of a uniform sonic flow about an infinite two-dimensional parabola at zero angle of attack, with a symmetric far-field approximation, and with no circulation around it. The numerical solution of the flow in the inner region results in the symmetric pressure distribution on the parabolic nose. Using the outer small-disturbance solution and the nose solution a uniformly valid pressure distribution on the entire airfoil surface can be derived. In the leading terms, the flow around the nose is symmetric and the stagnation point is located at the leading edge for every transonic Mach number of the oncoming flow and shape and small angle of attack of the airfoil. The pressure distribution on the upper and lower surfaces of the airfoil is symmetric near the edge point, and asymmetric deviations increase and become significant only when the distance from the leading edge of the airfoil increases beyond the inner region. Good agreement is found in the leading-edge region between the present solution and numerical solutions of the full potential-flow equations and the Euler equations.

Analysis of the finite element variational crimes in the numerical approximation of transonic flow

The paper presents a detailed theory of the finite element approximations of two-dimensional transonic potential flow. We consider the boundary value problem for the full potential equation in a general bounded domain Ω \Omega with mixed Dirichlet-Neumann boundary conditions. In the discretization of the problem we proceed as usual in practice: the domain Ω \Omega is approximated by a polygonal domain, conforming piecewise linear triangular elements are used, and the integrals are evaluated by numerical quadratures. Using a new version of entropy compactification of transonic flow and the theory of finite element variational crimes for nonlinear elliptic problems, we prove the convergence of approximate solutions to the exact physical solution of the continuous problem, provided its existence can be shown.

Stability of the Numerical Solution of Transonic Potential Flow

AbstractThe influence of an artificial viscosity on the stability of numerical solution of transonic flows with shock waves is investigated. The variational principle with dissipation introduced through a physical viscosity is derived, and a simplified relation describing the stabilizing influence of the dissipative process is presented. The numerical solution of the potential equation by FEM is examined and compared with the formulation using the variational principle. The correspondence between physical viscosity μ and artificial viscosity μnum is shown both qualitatively and quantitatively. The necessary amount of artificial viscosity is investigated analytically and the results are compared with numerical experiments.