Transonic Small Disturbance Potential Equation Research Articles

Analysis of compressible potential flow over aerofoils using the dual reciprocity method

AbstractThe use of the linearised potential model for the analysis of compressible flows is quite widespread, and provides good results for subsonic and supersonic flows. However, the calculation of aerofoils and wings subject to transonic flows requires a non-linear model, such as the transonic small-disturbance (TSD) potential equation. The solution of the problem by a singularity distribution requires singularities over the field, as well as panels on the boundary, characterising the procedure known as field panel method. The present work shows results of calculations of the transonic small-disturbance potential equation for flows without shock waves using the dual reciprocity method (DRM), which permits calculation of integrals only at the boundary of the problem, without the need of field distributions. This approach, compared to the field panel methods, takes considerably less computer time, and shows a significant improvement when compared to results of linear theory without much additional computer time, making this technique adequate to design phases of aircraft. Pressure distribution results show good agreement with other methods found in litterature. The low computational cost of the present method allows us to perform parametric tests and explore the effects of thickness and Mach number on the lift and pitching moment coefficients. A discussion of the physical effect of these parameters on the problem is presented, and the thickness of the aerofoil is shown to increase the lift and change the position of the aerodynamic centre. However, this non-linear effect depends on the precise shape of the thickness distribution.

Nonlinear Aeroelasticity: Continuum Theory, Flutter/Divergence Speed, and Plate Wing Model

This paper formulates flutter/divergence instability problems using continuum models for structure and air flow as coupled nonlinear partial differential equations. The structure model is a Kirchoff CFFF thin plate allowing for nonzero thickness and camber bending. The aerodynamics is modeled by the transonic small disturbance potential equation. The aeroelastic boundary conditions are derived for nonzero angle of attack. A central result is the time domain model as a nonlinear convolution/evolution equation in a Hilbert space. Flutter speed is characterized as a Hopf bifurcation point, completely determined by the linearized equations. The main tool in solving the linear equations is the Possio equation for nonzero angle of attack. Divergence speed is shown to be determined by an eigenvalue problem for linear operators. The corresponding stationary (steady state) solutions are more regular in the transonic range (as M goes to one) if the angle of attack is nonzero.

Transonic Small Disturbance Potential Equation

An analytical solution is presented to the transonic small disturbance potential equation with the Kutta‐ Joukowsky boundary conditions for an airfoil of zero thickness, generalizing it at the same time to nonzero angle of attack. In particular, an extension of the Possio integral equation is derived that is valid for nonzero angle of attack, as well as a formula for the divergence speed, showing explicitly a transonic dip, depending on the angle of attack.

Recent advances in transonic computational aeroelasticity

A transonic unsteady aerodynamic and aeroelasticity code called CAP-TSD has been developed for application to realistic aircraft configurations. The code permits the calculation of steady and unsteady flows about complete aircraft configurations for aeroelastic analysis in the flutter critical transonic speed range. The CAP-TSD code uses a time-accurate approximate factorization (AF) algorithm for solution of the unsteady transonic small-disturbance potential equation. The paper gives an overview of the CAP-TSD code development effort and presents results which demonstrate various capabilities of the code. Calculations are presented for several configurations including the General Dynamics one-ninth scale F-16C aircraft model and the ONERA M6 wing. Calculations are also presented from a flutter analysis of a 45° sweptback wing which agree well with the experimental data. The paper presents descriptions of the CAP-TSD code and algorithm details along with results and comparisons which demonstrate these recent developments in transonic computational aeroelasticity.

Efficient algorithm for unsteady transonic aerodynamics of low-aspect-ratio wings

An efficient coordinate transformation technique is presented for constructing grids for unsteady transonic aerodynamic computations for delta-type wings. The original shearing transformation yielded computations that were numerically unstable, and this paper discusses the sources of those instabilities. The new shearing transformation yields computations that are stable, fast, and accurate. Comparisons of those two methods are shown for the flow over the F5 wing that demonstrate the new stability. Also, comparisons are made with experimental data that demonstrate the accuracy of the new method. The computations were made by using a timeaccurate, finite-difference, alternating-direction-implicit (ADI) algorithm for the transonic small-disturbance potential equation.

An axisymmetric nacelle and turboprop inlet analysis including powersimulation

An axisymmetric nacelle and turboprop inlet analysis is presented. The transonic small disturbance potential equation is solved in nonconservaiive form by means of column relaxation on a Cartesian mesh. Exact surface boundary conditions are implemented to account for the influence caused by the strong interatcion of neighboring bodies. The Kutta condition is included so that flow-through as well as sting mounted configurations can be simulated. The analysis includes an actuator disk model to simulate power input for either ducted fans or propellers. The actuator disk formulation specifies the necessary jump conditions both at the disk and along the jet plume or propeller wake slipstream. Comparisons are made with exact full potential solutions and with wind tunnel data.

Finite difference computation of the aerodynamic interference of wing—pylon—store combinations at transonic speeds

In this paper a computational method is described for calculating aerodynamic interference of a wing—pylon—store combination. This method is based on the mixed finite difference method presented first by Murman and Cole in 1971 [1]. The governing equation is a steady flow transonic small disturbance potential equation. It differs from that of [1] in that the transonic small disturbance potential equation is derived from the exact potential equations without the assumption of small disturbance in the x (longitudinal) direction. The flow field around the wing is described by Cartesian coordinates and around the store by cylindrical coordinates. The effect of the pylon is taken into account together with its vortex sheet. The boundary condition on the surface of the wing—pylon—store combination is satisfied. On the vortex sheets the Kutta condition is satisfied. The potential value in the far field is approximated by integrating the linearized potential equation. The program is written for the TQ-6 computer. It can calculate all the flow field requirements including the perturbation potential ϑ, the down wash speed ϑy, the side wash speed ϑz, the pressure distributions and the aerodynamic coefficients. Three examples of computation are presented in the paper. They indicate the convergence and level of agreement with wind tunnel test values.

Unsteady transonic flow over wings including inviscid/viscous interaction

A numerical procedure is presented for computing the unsteady transonic flowfield about swept wings undergoing time-dependent motions. The outer inviscid portion of the flow is assumed to be governed by the modified unsteady transonic small disturbance potential equation which is integrated in the time domain by means of an alternating direction implicit algorithm. Dominant effects of the shock/boundary-layer interaction are accdunted for by a simple empirically defined model. Viscous flow regions adjacent to the wing surface and in the trailing wake are described by a set of integral equations appropriate for compressible turbulent shear layers. The two-dimensional boundary-layer equations are applied quasistatically stripwise across the span. Coupling with the outer inviscid flow is implemented through use of the displacement thickness concept. Validity of the assumptions underlying the method is established by comparison with experimental data for the flow about a high aspect ratio transport wing having an advanced airfoil section.

Numerical Solution of Three-Dimensional Unsteady Transonic Flow over Swept Wings

An algorithm for the calculation of unsteady transonic flow over three-dimensional swept wings undergoing general unsteady motion has been developed. The equation considered is the transonic small-disturbance potential equation with second-order terms to insure proper swept shock jump conditions. An alternating direction-implicit approximate factorization difference scheme is employed. A shearing transformation of the coordinates is applied in order to map a swept-tapered planform onto a rectangle in the computational domain. Use of a rotated type-dependent differencing scheme provides computational stability as well as proper shock capture. Computed results for an infinite-yawed wing are found to agree with two-dimensional solutions when simple sweep theory is applied for both the steady and unsteady cases. Steady three-dimensional results compare favorably with existing steady methods. Unsteady results are presented for a swept wing undergoing sinusoidal pitching motion at M0.9. It is found that the shock position oscillates over about 16% of chord at the wing tip.