Calculation Of Potential Flow Research Articles

Attached cavitation and the boundary layer: experimental investigation and numerical treatment

Attached cavitation on a wall with continuous curvature is investigated on the basis of experiments carried out on various bodies (circular and elliptic cylinders, NACA 16 012 foil). Visualization of the boundary layer by dye injection at the leading edge shows that a strong interaction exists between attached cavitation and the boundary layer. In particular, it is shown that the cavity does not detach from the body at the minimum pressure point, but behind a laminar separation, even in largely developed cavitating flow. A detachment criterion which takes into account this link between attached cavitation and boundary layer is proposed. It consists of connecting a cavitating potential-flow calculation and a boundary-layer calculation. Among all the theoretically possible detachment points, the actual detachment point is chosen to be the one for which the complete calculation predicts a laminar separation just upstream. This criterion, applied to the NACA foil, leads to a prediction which is in good agreement with experimental results.

Reply by Author to J. L. Hess

Miranda, L. R., Application of Computational Aerodynamics to Airplane Design, Journal of Aircraft, Vol. 21, June 1984, pp. 355-370. Hess, J. L., Calculation of Potential Flow about Arbitrary Lifting Bodies, McDonnell Douglas Rept. MDC J5679, Oct. 1972; (see also Computer Methods in Applied Mathematics and Engineering, Vol. 4, No. 3, Nov. 1974). Hess, J. L. and Friedman, D. M., An Higher Order Panel Method for Three-Dimensional Lifting Potential Flow, NADC-79277-60, Dec. 1981. Hess, J. L. and Friedman, D. M., Analysis of Complex Inlet Configurations Using a Higher-Order Panel Method, AIAA Paper 83-1828, July 1983. Hess, J. L., Improved Solution for Potential Flow About Arbitrary Axisymmetric Bodies by Use of a Higher-Order Surface Source Method, Computer Methods in Applied Mechanics and Engineering, Vol. 5, No. 3, May 1975. Duct Internal Pressure Distribution

Computation of Potential Flow on S2 Stream Surface for a Transonic Axial-Flow Compressor Rotor

A set of conservative full potential function equations governing the fluid flow along a given S2 streamsurface in a transonic axial compressor rotor was obtained. By the use of artificial density and a potential function/density iteration, this set of equations can be solved, and the passage shock on the S2 streamsurface can be captured. A computer program for this analysis problem has been developed and used to compute the flow field along a mean S2 streamsurface in the DFVLR transonic axial compressor rotor. A comparison of computed results with DFVLR L2F measurement at 100 percent design speed shows fairly good agreement.

Lift hysteresis of an oscillating slender ellipse

A theoretical investigation has been conducted to determine the timewise variation of lift on a slender elliptic cylinder moving at uniform speed but oscillating in pitch. The analysis couples a potential flow calculation, including the effect of a vortical wake, for the flow past the cylinder with a calculation of the unsteady, twodimensional, laminar boundary layer on the surfaces of the pitching ellipse. The coupling is achieved by matching the rate at which boundary-layer-developed vorticity is shed into the wake with the time rate of circulation about the ellipse. The unsteady lift is determined from an integration of the unsteady pressure distribution on the body. The effects of mean angle of attack and oscillation frequency on the lift hysteresis loops are determined. It is shown that the hysteresis loops change direction as the mean angle of attack is increased through the angle of attack corresponding to maximum steady lift.

Vortex dynamics of passive and active interfaces

Two classes of interface motion are exemplified. In the first example the advection of a line of passive marker particles by the unsteady flow due to a pair of vortex agitators is considered. The qualitative nature of the subsequent motion of the line depends sensitively on the parameters governing the unsteady flow. The advection of the particles determines either an integrable or a non-integrable dynamical system. The consequences of the transition from one regime to the other are explored by numerical experiments. In the second example a calculation of piecewise potential flow with a sharp interface is presented. The flow of two immiscible fluids of different but constant density and viscosity in a Hele-Shaw cell is considered. The interface separating the fluids is formally a tangential discontinuity of velocity and, thus, may be modelled as a vortex sheet. The evolution of fingers on the interface and the variability of the finger topology with the control parameters in the problem are accessible to numerical computation using a variant of the vortex-in-cell method. Results of numerical experiments employing such a code are presented and discussed. The paper concludes with some general speculations on the dynamics of interfaces.

Numerical calculation of unsteady transonic potential flow over three-dimensional wings with oscillating control surfaces

Numerical calculations of the unsteady transonic potential flow over three-dimensional wings with oscillating control surfaces are performed. For this purpose, a new grid system, which is appropriate for solving the control surface problems, is introduced into the computer code USTF3, which solves the unsteady three-dimensional full potential equation by a two-step semi-implicit time-marching technique. To validate the code, the unsteady pressure distributions on the NLR swept tapered wing with an inboard control surface and on the RAE swept tapered wing with a part-span control surface are calculated and compared with those of existing theories and experimental data. The qualitative behaviors of the unsteady pressure distributions due to the control surface oscillation are well predicted by the present method, but some quantitative discrepancies due to the viscous effects in the experiment are observed for supercritical Mach numbers.

Calculations of viscous transonic flow over airfoils

A simple and economic iterative scheme is presented for calculating compressible viscous boundary layers and wakes over airfoils and for matching the shear-layer calculations to the calculations of the transonic potential external flows. The iterative scheme is an extension and improvement of the scheme developed by Mahgoub and Bradshaw for calculating incompressible flow. A new iterative method in the shear-layer calculations has been designed and applied in the near-wake region. The computing time is only a little greater than in conventional displacement-surface calculations that ignore normal pressure gradients and consequently incur errors in the near wake. Some comparisons are made with full Navier-Stokes solutions and experimental data. ONSIDERABLE effort is currently being devoted to the solution of the full time-averaged Navier-Stokes equations, with appropriate turbulence model equation(s), using explicit time-dependent methods,1 implicit factorization methods,2 etc. However, the computing time is often too expensive for engineering purposes. It seems certain that methods involving separate calculations of the laminar and turbulent shear layers over airfoils and in the wake and of inviscid flow external to these layers, together with a matching process that enables the mutual interaction of these flows to be determined, will continue to have a practical advantage in both computing time and accuracy for a wide variety of engineering problems. In this paper we will present an iterative scheme of this kind, which can be regarded as an extension and improvement of a scheme developed for incompressible flow.3 The conventional displacement-surface methods may be inaccurate on highly curved surfaces or in the wake near the trailing edge where streamline curvature is extremely high, since they ignore the normal pressure gradient that may have a significant effect on the solution. With the new iterative method for the shear-layer calculations presented here, the effects of normal pressure gradient, even where they are relatively large and changing rapidly in the streamwise direction, can be included by iterative improvement of a marching calculation rather than by a fully elliptic calculation. Therefore, the whole iterative scheme remains simple and the cost of computing time is low. The pressure, but not the velocity, in the shear layer is stored from one iteration to the next: this means that only the pressure is allowed to transmit upstream influence, so that the method is nominally restricted to attached flow. Jameson's potential flow calculation method46 has been modified and applied in the present calculation of com- pressible flows external to the shear layers. It smears shock waves over at least two mesh widths, so discontinuiti es need not be included in the shear-layer calculation. The compressible boundary-layer calculation method described in Ref. 7 has been extended to an s, n coordinate system. A simple compressibility correction has been made to

Calculation of Potential Flow about Arbitrary Three Dimensional Wings using Internal Singularity Distributions

SummaryA first order panel method has been developed for calculating the incompressible potential flow about arbitrary three-dimensional wings. The method utilises a distribution of source and vorticity singularities on the mean camber surface of the wing and solves for the distribution by satisfying the boundary condition of zero normal flow at selected points on the surface of the wing. The method takes less computing time compared to other existing first order methods for the comparable numerical accuracy. This method can handle wings having cusped trailing edges and thin sections.

Computation of nonlinear supersonic potential flow over three-dimensional surfaces

A fully implicit three dimensional marching technique is presented for the computation of inviscid supersonic flows. The nonconservative full potential equation is solved on spherical cross flow planes using standard transonic SLOR techniques. The three dimensional bow shock is fitted as a boundary with isentropic jump conditions and embedded shocks are captured. Applications of these techniques are presented for three dimensional bodies, wings, and wing-body configurations and the computed solutions are compared to experimental pressure data. Zero lift wave drag calculations are also presented and compared with measurements to evaluate the accuracy of the present techniques.

Multi-grid calculation of three-dimensional transonic potential flows

A multi-grid algorithm has been developed to speed the iterative convergence of calculations for the transonic potential flow past swept wings and wing-fuselage combinations. The method is based upon a fully-conservative, finite-volume approximation to the steady potential equation which is second-order accurate everywhere in the flow field except near shock waves. The multi-grid scheme is incorporated within the framework of an alternating successive-line-overrelaxation (SLOR) solver of the difference equations. Computed results confirm the second-order accuracy of the scheme, and demonstrate the effectiveness of the multi-grid procedure.

Application of multigrid methods for integral equations to two problems from fluid dynamics

Multigrid methods are applied in order to solve efficiently the nonsparse systems of equations that occur in the numerical solution of the following problems from fluid dynamics: (1) calculation of potential flow around bodies and (2) calculation of oscillating disk flow. Problem (1) is reformulated as a boundary integral equation of the second kind that is approximated by a first-order panel method resulting in a full system of equations. This method is in widespread use for aerodynamic computations. The second problem is described by the Navier-Stokes and continuity equations. By means of the von Kármán similarity transformations these equations are reduced to a nonlinear system of parabolic equations which are approximated by implicit finite difference techniques. From the periodic conditions in time one obtains a nonsparse system of equations. For these two problems from fluid dynamics the fast convergence of multigrid methods for integral equations is established by numerical experiments.

Discharge Coefficient for Venturi Flowmeter With Short Laying Length

This paper presents a method for predicting the discharge coeffcient for a venturi flowmeter with a short laying length where the static pressure is not uniform at the throat due to streamline curvature. The discharge coefficient is determined by combining potential flow calculations and one-dimensional viscous flow considerations. For the potential flow, an accurate computational technique proposed by the author is used to calculate the pressure at the throat tap by assuming that the total pressure is equal to the average one at the throat. The average total pressure is related to the inlet pressure by use of a generalized empirical equation based on one-dimensional considerations. Validity of the method is verified by comparison with published experimental data for short venturi flowmeters.

A multigrid method for the transonic full potential eqaation discretized with finite elements on an arbitrary body fitted mesh

A multigrid method for the acceleration of transonic potential flow calculations based on a Galerkin finite element approach is described. In order to allow the use of arbitrary body fitted meshes, it is necessary to introduce nonuniform interpolation and residual weighting. Emphasis is put on the construction of these operators consistent with the finite element approximation, while standard successive line overrelaxation is used as a smoothing step. Substantial convergence acceleration is obtained and results are presented for different transonic flow configurations including shocks.

A multigrid algorithm for steady transonic potential flows around aerofoils using newton iteration

The application of multigrid relaxation to transonic potential-flow calculation was investigated. Fully conservative potential flows around aerofoils were taken as test problems. The solution algorithm was based on Newton iteration. In each Newton iteration step, multigrid relaxation was used to calculate correction potentials. It was found that the iteration to the circulation has to be kept outside the multigrid algorithm. In order to obtain meaningful norms of residuals (to be used in termination tests of loops), difference formulas with asymptotic scaling were introduced. Nonlinear instability problems were solved by upwind differencing using mass-flux-vector splitting instead of artificial viscosity or artificial density. It was also found that the multigrid method cannot efficiently update shock positions due to the (mainly) linear character of individual multigrid relaxation cycles. For subsonic flows, the algorithm is quite efficient. For transonic flows, the algorithm was found robust; it efficiency should be increased by improving the iteration on the shock positions; this is a highly nonlinear process.

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

Calculation of Potential Flow About Three-Dimensional Bodies in Shallow Water with Particular Application to Ship Maneuvering

A combined two-and three-dimensional method is described for computing the potential flow about a three-dimensional body floating in the free surface in shallow water of infinite extent. The free surface is replaced by a rigid wall, and the problem then corresponds to the motion of a double body between two parallel walls. We utilize the fact that the flow is approximately two-dimensional in the middle part of a ship. The bow and stern region, however, are treated separately to take care of the end effects, which are very important in shallow water. Compared with a three-dimensional computation for the whole body, both the computer time and the necessary storage space are reduced. Numerical results of added-mass coefficients are given for circular and rectangular docks floating in the free surface and for prolate spheroids and a tanker-type ship (Tokyo Maru)for different water depths. Calculations of squat for Tokyo Maru are also presented.

An Improved Method of Turbulent Boundary Layer Theory to Solve Viscous Flow around Ship Stern

The authors have presented a new approach to solve the viscous flow around the stern end including the wake. The particular iterative procedure has been developed to satisfy precisely the condition on the boundary surface between the outer potential flow region and the inner viscous flow region, namely the boundary layer and wake (abbrv. BLW). The procedure makes the continuous solution of BLW to be obtainable avoiding the divergence of it at the stern end. The computational process is composed of two main parts, one of which is the potential flow calculation by the modified Hess & Smith method with the correction of the blow-out effect of BLW and the other is the boundary layer calculation by the integral method considering the cross flow of Okuno's reversed cross flow model. The numerical results for two examples, Series 60 (CB=0.60) and HSVA tanker (CB=0.85), have indicated the possibility of solution of the stern end flow continuing the wake without any divergence within the field of the boundary layer theory.

Calculation of Two-Dimensional Potential Flow through Cascade Using Conformal Transformation of Blades with Arbitrary Profiles in Cascade into a Row of Circular Cylinders

任意翼形によりなる直線翼列を横切る,二次元・非圧縮性流体のポテンシャル流れを,翼列を無限円柱列に等角写像することにより解いた.本解法によれば,物理面・写像面間の空間のひずみが極めて小さな写像が実現できるため,食違い角および弦節比の大きな翼列まわりのポテンシャル流れを精度良く解くことが出来る.さらに,本解法は,解の収束性が翼列配置により影響されないという,従来の写像法にない特色を持っている.

Alternating-Direction Implicit Algorithm for Unsteady Potential Flow

Art implicit finite-difference scheme is presented for the efficient computation of unsteady potential flow about airfoils. The formulation uses density and the velocity potential as dependent variables and is cast in conservation form to assure the theoretically correct determination of shockwave location and speed. To enable boundary conditions to be imposed directly on the airfoil surface, a time-varying sheared-rectilinear coordinate transformation is employed. Calculated time-history solutions on a pulsating airfoil are compared with the results of other unsteady transonic codes, including a previous method of the authors. The present method is demonstrated to be unconditionally stable and to give accurate solutions with sharply resolved shocks.

On the regularity of the principal value of the double-layer potential

In the present paper we discuss the regularity of the principal value of the potential due to a doublet distribution μ along the boundary S of a two-dimensional (2-D) open connected set. Assuming S to be a Lyapunov boundary and μ to be essentially bounded, we prove that the principal value in 2-D is more regular than the one in 3-D. This result is applied to the aerodynamics problem of calculation of potential flows around 2-D bodies.