Osama A. Kandil

Unsteady transonic airfoil computation using implicit Euler scheme on body-fixed grid

The unsteady Euler equations have been derived for the flow relative motion with respect to a frame of reference that is rigidly attached to the moving airfoil. The resulting equations preserve the conservation form. The grid is generated once by an elliptic solver without any need for dynamic grid computation. An implicit approximately factored finite-volume scheme has been developed and implemented through a fully vectorized computer program. The scheme is based on the spatial approximate factorization of Beam and Warming. Implicit second-order and explicit second- and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The scheme is applied to steady and unsteady transonic rigid-airfoil flows. For forced harmonic airfoil motions, periodic solutions are achieved within the third cycle of oscillation. The results are in good agreement with the experimental data. sonic flows past a NACA 0015 airfoil at a constant pitch rate.18 The latter case is the same application considered in Ref. 16. The numerical solution has been obtained by using the flux-vector splitting and the flux-difference splitting meth- ods extended for dynamic meshes.19 In Ref. 18, it is noted that different turbulence models give different normal-force and pitching-moment coefficients. Numerical solutions of the unsteady Euler equations are less expensive than those of the unsteady Navier-Stokes equations. For the unsteady transonic flow, the unsteady Euler equations adequately model most of the real flow features, with the exception of viscous effects whenever they are substantial. The Euler equations model shock waves and their motion, entropy increase across shocks and entropy gradient, and vorticity production and convection behind shocks, as can be seen from Croccos theorem and the inviscid vorticity transport equa- tion. Recently, successful time-accurate solutions of the un- steady Euler equations have been presented for pitching air- foils19 and wings in transonic flows,20 and for the rolling oscillation of delta wings—a vortex-dominated flow problem in a locally conical supersonic flows.21 To the best of bur knowledge, Ref. 21 is the first work done for unsteady vortex- dominated flows with shock waves, which is directly applica- ble to maneuvering delta wings. In this paper, we present an implicit approximately-factored finite-volume scheme for the time-accurate numerical solution of the unsteady Euler equations of the flow relative motion with respect to an airfoil-fixed frame of reference. The scheme is applied to steady and unsteady transonic flows around a NACA 0012 airfoil. For the unsteady flows, the airfoil is in pitching oscillation about a small and moderate mean angle of attack at a large amplitude. The computational results are compared with the experimental data.

COMPUTATION OF VORTEX-DOMINATED FLOW FOR A DELTA WING UNDERGOING PITCHING OSCILLATION

The conservative, unsteady Euler equations for the flow relative to a moving frame of reference are used to solve for the three-dimensional steady and unsteady flows around a sharp-edged delta wing. The resulting equations are solved by using an implicit, approximately factored, finite-volume scheme. Implicit second-order and explicit second- and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The grid is generated by locally using a modified Joukowski transformation in crossflow planes at the grid-chord stations. The computational applications cover a steady flow around a delta wing, whose results serve as the initial conditions for the unsteady flow around a pitching delta wing at a large mean angle of attack. The steady results are compared with the experimental data, and the unsteady results are compared with results of a flux-difference splitting scheme.

Propagation of Waves in Cylindrical Hard-Walled Ducts with Generally Weak Undulations

The method of multiple scales is utilized to analyze the wave propagation in cylindrical hard-walled ducts having weak undulations which need not be periodic. Results are presented for two and three interacting modes. In the case of modes traveling in the same direction in a uniform duct, two interacting, spinning or nonspinning modes propagate unattenuated in an undulated duct. Moreover, neither of them can exist without strongly exciting the other. On the other hand, in the case of modes propagating in opposite directions, they may be cut off as a result of the interaction. OR two-dimension al ducts, straightforward expansions of the form (f>0 + e 7 were obtained by Isakovitch ] for the case of a waveguide with only one sinusoidally undulating wall, by Samuels 2 for the case of a waveguide with inphase wall undulations, and by Salant 3 for the general problem. Nayfeh4 showed that the above expansions are not uniform near the resonant frequencies because the correction term e 7 dominates the first term 0. He determined a uniform expansion for waves propagating in a two-dimensional hardwalled duct with sinusoidally perturbed walls by using the method multiple scales.5 In this paper, we extend the latter analysis to the case of linear waves propagating in a cylinderical hard-walled duct whose wall has weak undulations which need not be periodic. The gas is assumed to be inviscid, irrotational, and nonheat conducting. Dimensionless quantities are introduced by using the mean radius of the duct r0, the undisturbed speed of sound c, and the time r0/c as reference quantities. The dimensionless radius of the duct is assumed to have the form

Nonlinear forced vibration of orthotropic rectangular plates using the method of multiple scales

The method of multiple scales (MMS) in conjunction with the Galerkin method is used to analyze the nonlinear forced and damped response of a rectangular Orthotropic plate subjected to a uniformly distributed harmonic transverse loading. The effects of damping and in-plane loads are considered. The analysis considers simply supported as well as clamped plates. For each case, both movable and immovable edge conditions are considered. By using MMS, all possible resonances such as primary, subharmonic, and superharmonic reso- nances are studied. For the undamped response without in-plane loading, comparisons of the MMS results with those obtained by the finite-element method show excellent agreement. EVELOPMENT of composite materials comprising lam- inates of Orthotropic or multilayered anisotropic materi- als Recently has been receiving substantially growing research efforts. Due to the increasing demands for energy-efficient, high strength, minimum weight aircraft designs, many re- searchers believe that the use of composite materials offers promising alternatives for aircraft designs. Thin, laminated, composite plates subjected to transverse periodic loadings could encounter deflections of the order of plate thickness or even higher. Responses of this kind cannot be predicted by using the linear theory. Consequently, the need to study large- amplitude-deflection vibrations of composite structures is of paramount importance. The literature survey shows that the equations of motion for the large deflection analysis of heterogeneous anisotropic plates using the von Karman geometrical nonlinearity were first considered by Whitney and Leissa.1 Based on these equa- tions, different methods of analysis have been developed by several researchers. An excellent number of collections on nonlinear free and forced vibrations of composite plates cov- ering the work through 1979 can be found in the comprehen- sive book by Chia. 2 Bert3 has conducted a survey on the dynamics of composite plates for the period 1979-81. A re- view of the literature on nonlinear vibrations of plates can be found in the review paper by Sathyamoorthy4 and the book by Nayfeh and Mook.5 Large deflection analysis of symmetrically laminated rec- equations of motion are presented in terms of the lateral displacement and stress function. The equations are nondi- mensionalized following the transformation introduced by Brunelle and Oyibo.9 Though multimode analysis can be treated, the present study is focused on single-mode analysis. A deflection function representing the first mode and satisfy- ing the boundary conditions is assumed, and subsequently the stress function is found. Next, the Galerkin method is applied to obtain the modal equation, which is solved analytically by using the method of multiple scales (MMS).10 The MMS also provides solutions for subharmonic and superharmonic reso- nances. The effects of damping ratio, plate aspect ratio, and in-plane loading then are studied.

Computation of steady and unsteady vortex-dominated flows with shock waves

The unsteady Euler equations have been derived in the conservation form for the flow relative motion with respect to a rotating frame of reference. The resulting equations are solved by using a central-difference finite-volume scheme with four-state Runge-Kutta time stepping. For steady flow applications local time stepping is used, and for unsteady applications the minimum global time stepping is used. A three-dimensional fully vectorized computer program has been developed and applied to steady and unsteady maneuvering delta wings. The capability of the three-dimensional program has been demonstrated for a rigid sharp-edged delta wing undergoing uniform rolling in a conical flow and rolling oscillations in a locally conical flow.

Effects of Nose Bluntness on Hypersonic Boundary-Layer Receptivity and Stability over Cones

The receptivity to freestream acoustic disturbances and the stability properties of hypersonic boundary layers are numerically investigated for boundary-layer flows over a 5 straight cone at a freestream Mach number of 6.0. To compute the shock and the interaction of the shock with the instability waves, the Navier-Stokes equations in axisymmetric coordinates were solved. In the governing equations, inviscid and viscous flux vectors are discretized using a fifth-order accurate weighted-essentially-non-oscillatory scheme. A third-order accurate total-variation-diminishing Runge-Kutta scheme is employed for time integration. After the mean flow field is computed, disturbances are introduced at the upstream end of the computational domain. The appearance of instability waves near the nose region and the receptivity of the boundary layer with respect to slow mode acoustic waves are investigated. Computations confirm the stabilizing effect of nose bluntness and the role of the entropy layer in the delay of boundary-layer transition. The current solutions, compared with experimental observations and other computational results, exhibit good agreement.

Transonic vortex flows past delta wings: integral equation approach

The steady full-potential equation is written in the form of Poissons equation, and the solution of the velocity field is expressed in terms of an integral equation. The solution consists of a surface integral of vorticity distribution on the wing and its free-vortex sheets and a volume integral of source distribution within a volume around the wing and its free-vortex sheets. The solution is obtained through successive iteration cycles. The density gradient in the source distribution is computed by using a type-differencing scheme. The method is applied to delta wings, and the numerical examples show that a curved shock is captured on the suction side of the wing. It is attached to the lower surface of the leading-edge vortex but does not necessarily reach the wing surface. The present solution does not suffer from the numerical diffusion problem usually encountered with the finite-difference solutions of Euler equations.

Two-mode nonlinear vibration of orthotropic plates using method of multiple scales

Nonlinear forced oscillation of a rectangular orthotropic plate subjected to uniform harmonic excitation is solved using the method of multiple scales. The governing equations are based on the von Karman type geometrical nonlinearity, and the effect of damping is included. The general multimode solution is developed for simply supported boundary conditions, and the solution is specialized for two-symmetric modes analysis. The primary resonances and the subharmonic and superharmonic secondary resonances are studied in detail. HIN laminated composite panels subjected to transverse periodic loadings can encounter deflections of the order of panel thickness or even higher. The effect of these periodic excitations on the panel can be very severe. Responses of this kind cannot be predicted by linear theory. Consequently, the need to study large deflections using nonlinear methods of analysis is of paramount importance. The formulation of the equations governing the fundamen- tal kinematic behavior of the laminated composite plates in the presence of the von Karman geometrical nonlinearity is attributed to Whitney and Leissa.1 Based on these equations, various methods have been developed to solve nonlinear free and forced vibrations of composite panels. A good survey on mainly nonlinear free and forced vibrations of isotropic plates is given in a book by Nayfeh and Mook.2 The most compre- hensive work on geometrically nonlinear analysis of both static and dynamic behavior of the laminated panels through 1972 is collected in a book by Chia. 3 Bert4 has conducted a survey on the dynamics of composite panels for the period of 1979-81. A review of literature on linear vibrations of plates can be found in a review paper by Sathyamoorthy.5 Relatively few investigations have been reported on the nonlinear forced vibration of isotropic or composite panels under harmonic excitations. Yamaki6 presented a one-term solution for free and forced vibrations of the rectangular plates, using Galerkins method. Lin7 studied the response of a nonlinear flat panel to periodic and randomly varying load- ings. Nonlinear forced vibrations of beams and rectangular plates were studied by Eisely8 using a single-mode Galerkins method in conjunction with the Linstedt-Duffing perturbation technique. Free and forced response of beams and plates undergoing large-amplitude oscillations using the Ritz averag- ing method were studied by Srinivasan.9 Bennett10 studied the nonlinear vibration of simply supported angle-ply laminated plates by considering the instability regions of the response of such plates subjected to harmonic excitations. Nonlinear free and forced vibration of a circular plate with clamped bound-

Full potential integral solution for transonic flows with and without embedded Euler domains

Two methods are presented to solve transonic airfoil flow problems. The first method is based on the integral equation solution of the full-potential equation in terms of the velocity field. A shock capturing-shock fitting scheme has been developed. In the shock-fitting part of the scheme, shock panels are introduced at the shock location. The shock panels are fitted by using the Rankine-Hugoniot relations. The second method is a coupling of the integral solution of the full-potential equation with the pseudotime integration of Euler equations, which are used in a small embedded region around the shock. This scheme is caUed the integral equation-embedded Euler scheme. The two methods are applied to NACA 0012 and 64A010A airfoils over a wide range of Mach numbers, and the results are in good agreement with the experimental data and other computational results. The schemes converge within a number of iterations that is one order of magnitude less than the finite-difference schemes. ment methods for solving transonic airfoil flow problems. The IE solution has several advantages in comparison with the finite-difference solution. With the IE formulation (IEF), the far-field boundary conditions are automatically satisfied, and only a small computational domain is needed around the source of disturbance. Moreover, the accuracy of the method depends on the evaluation of integrals rather than derivatives, and hence coarse grids can be used within the small computa- tional domain. Because of the obvious advantages of the meth- ods that are based on the IEF, it is highly desirable to develop these methods fully and extend them to treat transonic flows over a wide range of Mach numbers. Computational schemes that are based on the IE formula- tion have recently been developed by several investigators. Piers and Sloof10 and Tseng and Morino11 have developed IE schemes for the TSP equation, while Kandil and Yates,12 Os- kam,13 Erickson and Strande,14 Sinclair,15 and Kandil and Hu16 have developed IE schemes for the FP equation. Since the potential equation cannot accurately treat flows with strong shocks, the IE schemes for the FP equation must be modified to accurately treat flows with strong shocks. In this paper we present two methods to solve for the tran- sonic airfoil flow problems that are based on the IE solution only or on the IE solution with small embedded Euler-domain solution. The latter method can efficiently treat flows with strong shocks. For the former method, a shock capturing- shock fitting (SCSF) scheme is developed, whereas for the lat- ter method an integral equation-embedded Euler (IEEE) scheme is developed. The SCSF scheme takes about 20-30 iterations for convergence, whereas the IEE scheme takes about 10 IE iterations and a few hundred of Euler equations iterations (over a very small computational domain) for con- vergence. The methods are applied to NACA 0012 and 64A010A airfoils over a wide range of Mach numbers. The results are in good agreement with the experimental data and other results that were obtained by using finite-difference and finite-volume methods with TSP, FP, and Euler equations.

Receptivity of Hypersonic Boundary Layers Due to Acoustic Disturbances Over Blunt Cone

The transition process induced by the interaction of acoustic disturbances in the freestream with boundary layers over a 5-degree straight cone and a wedge with blunt tips is numerically investigated at a free-stream Mach number of 6.0. To compute the shock and the interaction of shock with the instability waves the Navier-Stokes equations are solved in axisymmetric coordinates. The governing equations are solved using the 5 –order accurate weighted essentially non-oscillatory (WENO) scheme for space discretization and using third-order total-variation-diminishing (TVD) Runge-Kutta scheme for time integration. After the mean flow field is computed, acoustic disturbances are introduced at the outer boundary of the computational domain and unsteady simulations are performed. Generation and evolution of instability waves and the receptivity of boundary layer to slow and fast acoustic waves are investigated. The mean flow data are compared with the experimental results. The results show that the instability waves are generated near the leading edge and the non-parallel effects are stronger near the nose region for the flow over the cone than that over a wedge. It is also found that the boundary layer is much more receptive to slow acoustic wave (by almost a factor of 67) as compared to the fast wave.