Compressible Potential Flow Research Articles

Exact Solution of a Compressible Potential Flow Past a Body of Revolution at Zero Incidence : Especially for High Subsonic Flows

Exact Solution of a Compressible Potential Flow Past a Body of Revolution at Zero Incidence : Especially for High Subsonic Flows

Exact Solution of Compressible Potential Flow Past a Body of Revolution at Zero Incidence : Especially for High Subsonic Flow

An entirely new solution for a compressible potential flow past a given body of revolution at zero incidence is predicted. A complex velocity for the flow is solved exactly through a complex integral equation with use of pseudo-analytic function theories in just the same way as in two-dimensional cases given by the authors. Owing to nonlinearity and the existence and uniqueness of the solution successive iteration is introduced. The present method has a wide range of applicabilities to flow fields such as those on meridional through flows in three-dimensional cases and so on. Moreover, for inverse problems only a few supplemental arguments may be needed. Numerical illustrations of our method, which fully depends on correspondence between two-dimensional incompressible flows and those in the present case, are given for incompressible and for high sub-sonic flows past ellipsoids. Obtained results show a good agreement with experimental values.

The solution of the transonic equation by optimal control and finite element methods

The transonic equation for compressible potential flow has previously been solved using the finite element method in conjunction with an optimal control algorithm. However, the method is complex and computation time is high. In this paper, by using a different performance index in the optimal control algorithm, the computation time can be significantly reduced. A numerical example is given in which the computation time is four times shorter than that using the previous method.

Rigorous Solution of Compressible Potential Flow Past Two-Dimensional Cascades of Airfoils : Especially for High Subsonic Flow

A rigorous solution of complex velocity of a compressible potential flow past two-dimensional cascades of airfoils is obtained by introducing a pseudoanalytical function with respect to complex coordinates. A basic differential equation about complex velocity is derived from a continuous relation and an irrotational flow condition and then reduced to an inhomogeneous complex singular integral equation. Using successive substitution which depends on theorems of quasi-conformal mappings a solution of the equation can be numerically given with the help of results of an incompressible potential flow, since the complex velocities of compressible and incompressible flows are in one-to-one correspondence with each other. Using this function-theoretic means the solution is applicable both to axial and to radial flows and also to inverse problems. Numerical illustrations of the method are given for high subsonic potential flows through A3K7 turbine blades in cascade.

Rigorous Solution of Compressible Potential Flow Past Two-Dimensional Cascades of Airfoils : Especially for High Subsonic Potential Flow

本研究は二次元翼列を過ぎる圧縮性ポテンシャル流れの満足する厳密解を具体的にかつ実用的に与えるものである. 圧縮性流れは非圧縮性ポテンシャル流れの解を用いて翼列無限上下流をも含めた流れの存在するすべての領域で求められ, また数値計算手法においても特別な仮定を必要としない. 流れの満足する擬等角写像についての性質を用いれば軸流機械, ふく流機械を問わず, また間接問題にも本解法を適用することができる.

Finite-Element Approach to Compressor Blade-to-Blade Cascade Analysis

A study of the application of the finite-element method to compressible potential flows in the context of axial turbomachinery is undertaken. Some novel finite-element approaches relating to the analysis as well as to the mesh generation are presented. The solutions use a pseudovariational integral, applicable to shockless transonic flows and possessing a physical basis for the iteration. Appropriate grids are generated automatically for all cascades at any solidity. The scheme is based upon an approximate mapping of a blade into a near circle around which a suitable grid, with layers reproducing the airfoil shape, is constructed. Such mapping homogenizes the gradients by geometrically condensing regions of low gradients on the main part of the blade while magnifying regions of steep gradients near the leading and trailing edges. The analysis, however, is carried out in the original physical plane. Successful comparisons are made with other results for incompressible and compressible flows.

Aerodynamics of Airfoils with Porous Trailing Edges

SummaryThe aerodynamics of a thin airfoil of arbitrary camber having a porous trailing edge in steady, subsonic, compressible potential flow is investigated. In the special case of a flat plate airfoil with a porous trailing edge, an exact, closed form solution is obtained using complex variable theory. The pressure loading on the airfoil, the lift and pitching moments are exhibited explicitly along with typical numerical results. The corresponding situation in supersonic flow is also considered.

Nonlinear Stability Analysis of a Two-Dimensional Model of an Elastic Tube Conveying a Compressible Flow

This paper examines the dynamic behavior of a two-dimensional channel whose upper and lower walls deform symmetrically with respect to the center line of the channel. Unsteady fluid dynamic forces acting on the internal wall are analytically evaluated on the basis of a linearized compressible potential flow theory. The effects of distributed springs outside the channel and an internal pressure on the stability characteristics are studied by considering small disturbances about flat and buckled equilibrium configurations of the wall. The analytic methods indicate that no flutter of the flat or buckled wall is predicted when the Mach number is small and the viscous damping coefficient is positive. Numerical results by the Runge-Kutta-Gill method suggest that nonlinear effect of flow should be taken into account to fully examine the dynamic characteristics of the channel conveying a flow.

A finite element approach to subsonic aerodynamics

AbstractA study of the application of the Finite Element Method to compressible potential flows, typified by the airfoil problem, is undertaken. Some novel approaches, believed to simplify solution techniques, are presented.The solutions use two pseudo‐variational integrals, appropriate to subsonic flows, and possessing a physical iterative basis. With constant‐derivatives triangular elements formulated for cylindrical co‐ordinates, accurate solutions are easily obtained for the flow over a circular cylinder. For arbitrary airfoils a simple mapping is used to transform them into near circles. An appropriate mesh is then constructed in the mapped plane. The paper then presents two solution approaches by which this non‐linear problem is solved in both the near circle plane and the airfoil plane.

Certain properties of numerical schemes used to solve the blade‐to‐blade flow problem

AbstractThe compressible potential flow on a blade‐to‐blade surface of a turbomachine is analysed using the Thomas algorithm. The numerical properties of resulting non‐linear simultaneous equations are studied with respect to grid aspect ratios and convergence. The heuristic approach has established which are the important factors that affect the flow solution in typical blade‐to‐blade configurations.

Cartesian Mesh Solution for Axisymmetric Transonic Potential Flow Around Inlets

A procedure has been developed for computing transonic solutions of the exact axisymmetric compressible potential flow equation for flow about inlets. A cartesian mesh is used in physical coordinates to eliminate restrictions on the geometries that can be treated. The method is shown to be fast and stable; and agreement between analysis and data is excellent. The solution procedure is valid for any geometry or any number of surfaces. The method was developed as a first step toward a general three-dimensional analysis and all parts of the procedure apply directly to three-dimensional flow computation with a minimum of further development.

On the Flow in a Centrifugal Impeller : 1st Report, Subsonic Compressible Flow

Subsonic compressible potential flow in a centrifugal impeller can be analyzed by using a pseudo-analytic function. In this paper, the foregoing method has been applied to the calculations of nonviscous and compressible fluid flows in a two-dimensional centrifugal impeller with thin logarithmic spiral vanes, and influences of compressibility on the velocity, pressure and density distributions along the vane surfaces under the condition of shokless entry have been calculated. Furthermore, heads and flow rates of the impeller in case of shockless entry have been shown charts.

Three dimensional analysis of compressible potential flows with the finite element method

AbstractSolutions of elliptic boundary value problems are obtained with variational formulations by minimization of functionals.Conventional tools of the finite element strategy are emphasized; linear or quadratic approximations on triangular elements, curved iso‐parametric elements, Ni shape funtions, Li homogeneous co‐ordinates. High speed of computation of the final assembled matrices require the use of the Formal–FORMAC calculus. Illusrations of the finite element method are chosen in the class of the 2d–3d compressible subsonic irrotational flows in a nozzle and around one or several lifting bodies of arbitary shape.

A Study of Compressible Potential and Asymptotic Viscous Flows for Comer Region

A new and direct approach for calculating the first higher order potential flow along an axial corner is presented. For the incompressible potential flow, the present approach demonstrates that the displacement effects in the corner may be visualized as the superposition of the displacement effects for the two intersecting semi-infinite plates forming the corner. The compressible subsonic potential flow is then obtained by the Prandtl-Glauert rule. Linearized airfoil theory is used to determine the potential flow for the supersonic case. The asymptotic viscous flow, to lowest order, for the corner problem has been calculated for general compressible flow. The analysis presented here recovers all the previously obtained lowest order asymptotic solutions. Cross flow velocity profiles have been given for M f between 0.001 to 4 for an adiabatic wall as well as for a prescribed temperature at the wall. The behavior of the cross flow skin-friction coefficient is shown to be quite different from that of the skin friction coefficient due to classical axial flow.

Subsonic and Transonic Potential Flow over Helicopter Rotor Blades

Compressible potential flows over nonlifting, hovering helicopter blades are described by suitable linear and nonlinear equations of motion for subsonic and transonic cases, respectively. Analytical and numerical results are presented for the linearized subsonic three-dimensional flow in the tip region. When the tip Mach number is transonic, the flowfield is calculated using a computational method that is a formal extension to three dimensions of recently developed nonlinear two-dimensional relaxation schemes. Calculations are presented for rectangular blades with 6% thick biconvex sections. Calculations show the relative importance of tip Mach number and aspect ratio on the growth and extent of shock waves in the tip region, and indicate a significant reduction in shock strength with decreasing aspect ratio. ELICOPTER rotor blades that can operate at high tip Mach numbers without the penalties usually associated with super-critical flow would permit high-forward speeds at relatively low-advance ratios. As the advance ratio decreases, the portion of the retreating blade that is stalled would be diminished, the portion of the retreating blade that is unstalled could operate at smaller angles of attack, and the advancing blade could generate lift without also generating an excessive rolling moment. Vibratory loads would be diminished, and severe unsteady pitching moments on the lifting portion of the retreating blade would decrease if that portion could usefully operate at higher tip speeds and smaller angles of attack. The design of rotor blade geometries that can support a transonic flowfield that is nearly shock-free requires a method for computing the detailed pressure distribution on the blade for a specified rotor geometry and advance ratio. However, the unsteady aerodynamic environment of a lifting rotor that is operating at or near the critical Mach number is extremely complicated. It is easy to appreciate the difficulty of describing such flowfields when one recalls that we do not yet have an adequate model for describing the details of the flow about a lifting, hovering rotor operating in an essentially incompressible flow. Our purpose here is therefore to focus on a portion of the high-speed rotor problem that is sufficiently limited to allow for some analytical and numerical treatment, but which retains important linear and nonlinear features of the flow. We consider a two-bladed nonlifting and hovering rotor operating in the range of tip Mach numbers where compressibility effects are important. The pressure distribution near the rotor tip is studied. In this region, the flowfield is fully three-dimensional, and blade element (strip) theory is in no sense a reasonable approximation to the rotor aerodynamics. It is assumed that the flow is inviscid. The flow is steady relative to a coordinate system attached to the blades. Analytical and numerical results presented here are based on suitable approximations to the full three-dimensional potential equation for compressible flow. When the tip Mach number is subcritical, the potential equation is the usual acoustic

On vacuum points and the possibility of a Liouville theorem for compressible potential flows. I

Solutions of Chaplygin's equation with γ = 3 for the initial value problem with analytic initial data on the vacuum line are used as a basis for a classification of vacuum points. Three types of vacuum points are distinguished: (i) points of the generation type, (ii) points of the reflection type, (iii) vacuum limit points. A vacuum limit point may either lie on a supersonic limit line or be isolated. It is conjectured that a Liouville theorem for compressible potential flows exists, stating that the uniform flow is the only compressible potential flow free of limit lines and vacuum reflection lines throughout the entire flow plane.

On vacuum points and the possibility of a Liouville theorem for compressible potential flows. II

Abstract Solutions of Chaplygin's equation with γ = 3 for the initial value problem with analytic initial data on the vacuum line are used as a basis for a classification of vacuum points. Three types of vacuum points are distinguished: (i) points of the generation type, (ii) points of the reflection type, (iii) vacuum limit points. A vacuum limit point may either lie on a supersonic limit line or be isolated. It is conjectured that a Liouville theorem for compressible potential flows exists, stating that the uniform flow is the only compressible potential flow free of limit lines and vacuum reflection lines throughout the entire flow plane.

Three-dimensional compressible potential flow in a curved duct

The three-dimensional problem of compressible potential flow in a bend of a circular duct is studied by means of a perturbation method. As perturbation parameter is used the quotient between the radius of the duct and the radius of curvature of the duct, and the solution includes terms of the first and second order of this parameter. Numerical calculations have been performed for the values 0.2 and 0.1 of the perturbation parameter, and the plotted result shows how the axial velocity distribution changes from the straight to the curved region of the duct.

Flügelschwingungsformen in ebener kompressibler Potentialströmung

On the basis of energy considerations a survey is given of the possible forms of oscillations for flutter with two degrees of freedom in a plane compressible potential flow. It is found that the possible forms do not much depend on Mach Number in the whole range from 0 to ∞, with some exceptions in the neighbourhood ofM=1, where forms with only one degree of freedom (pure torsion) may occur. Conclusions are drawn for methods of preventing flutter. Limits of the reduced frequency (depending onM) are given for the possibility of flutter in two and one degree of freedom.