Transonic flutter analysis using time-linearization aerodynamics

The Transonic Small Disturbance (TSD) Equation is a common model equation for describing subsonic and supersonic flow close to the local speed of sound (transonic). In transonic flow there is an embedded region of locally supersonic flow inside an otherwise subsonic flow. The supersonic region is usually terminated by a shock discontinuity. The success of a numerical scheme for transonic flow prediction depends on its capability of capturing all the flow details and non-linearities including sharp shock profiles without oscillations near the shock. Most of the important phenomena in the TSD equation occur in the stream-wise direction. The nonlinearity and changes in the region of influence depend only on the stream-wise derivation. A suitable one-dimensional model equation derived from the TSD Equation is used. The one-dimensional equation is written in conservation law form. This nonlinear system of equations models the transition from supersonic to subsonic flow. In the numerical calculations the discretised problem is treated as a series of Riemann problems. We will investigate various techniques for solving these Riemann problems. It will be shown that the techniques do not allow non-physical expansion shocks to develop and that the techniques smooth out expansion shocks when these non-physical shocks are present in the initial velocity profile. A comparison will be made between the schemes based on the sharpness of the resulting shock profiles.

This paper investigates control surface reversal in the transonic flight regime. Linear and nonlinear rigid and aeroelastic analyses are performed while including the effects of flow viscosity with an interactive boundary layer in the prediction of this aeroelastic phenomenon. Transonic small disturbance theory is employed in the analysis of a typical fighter type wing to study the interactions among control surface deflections, structural flexibility, and embedded shocks in a flow field where a viscous boundary layer exists. Pressure distributions on the wing are examined, and control surface reversal calculations are presented. These results are discussed based on the predictions of the pressure coefficients generated by the solution of the transonic small-disturbance equation. Generalizations are offered concerning the effects of including viscosity in the prediction of steady aeroelastic phenomena in the transonic flight regime. Finally, the consequences of these findings on the preliminary design of aircraft structures are discussed.

The two-dimensional and steady transonic e ow of atmospheric moist air with equilibrium condensation around a thin airfoil is investigated. The study is based on an asymptotic analysis and numerical simulations. A smalldisturbance model is developed to explore the nonlinear interactions between the near-sonic speed of the e ow, the small thickness ratio and angle of attack of the airfoil, and the small amount of mass of water vapor in the air. The condensation process of water vapor in the air is assumed to be isentropic. The similarity parameters that govern the e ow problem are provided. The e owe eld may be described by a modie ed transonic small-disturbance (TSD)equation that includes parameters that are related to the condensation process. Murman and Cole’ smethod (Murman, E. M., and Cole, J. D., “ Calculation of Plane Study Transonic Flows,” AIAA Journal , Vol. 9, No. 1, 1971, pp. 114 ‐121.)is used for the numerical solution of the modie ed TSD problem. The results show that the e ow of moist air is similar to the e ow of dry air with an effective freestream Mach number that is greater than the freestream Mach number of moist air. The present approach is used to study the aerodynamic performances of airfoils in atmospheric transonic e ight with humidity.

The boundary-value problems describing the first-order corrections \[ O ( 1 A R , 1 A R ln ⁡ A R ) O\left ( {\frac {1}{{AR}},\frac {1}{{AR}}\ln AR} \right ) \] to two-dimensional flow about a lifting swept wing at transonic speeds ( M ∞ > 1 ) \left ( {{M_\infty } > 1} \right ) are derived. The corrections are found by the use of the method of matched asymptotic expansions on the transonic small disturbance equations. The wing is at a sweep angle of O ( ( 1 − M ∞ 2 ) 1 / 2 ) O\left ( {{{\left ( {1 - M_\infty ^2} \right )}^{1/2}}} \right ) in the physical plane, hence of O ( 1 ) O\left ( 1 \right ) in the transonic small disturbance plane. As has been noted for subsonic flow, the finite sweep angle necessitates the introduction of terms O ( ( A R ) − 1 ln ⁡ A R ) O\left ( {{{\left ( {AR} \right )}^{ - 1}}\ln AR} \right ) . These terms arise naturally in the matching process. Of particular interest is the derivation of the near field of a skewed lifting-line which is found by Mellin transform techniques. Also of interest is the fact that the influence of the nonzero sweep angle can be completely separated from the unswept solution.

Numerical studies of two-dimensional, transonic flows of dense gases of retrograde type, known as BZT gases, around thin airfoils are presented. The computations are guided by a recent asymptotic theory of Rusak & Wang (1997). It provides a uniformly valid solution of the flow around the entire airfoil surface which is composed of outer and inner solutions. A new transonic small-disturbance (TSD) equation solver is developed to compute the nonlinear BZT gas flow in the outer region around most of the airfoil. The flow in the inner region near the nose of the airfoil is computed by solving the problem of a sonic flow around a parabola. Numerical results of the composite solutions calculated from the asymptotic formula are compared with the solutions of the Euler equations. The comparison demonstrates that, in the leading order, the TSD solutions of BZT gas flows represent the essence of the flow character around the airfoil as computed from the Euler equations. Furthermore, guided by the asymptotic formula, the computational results demonstrate the similarity rules for transonic flows of BZT gases. There are differences between the self-similar cases that may be related to the error associated with the accuracy of the asymptotic solution. A discussion on the flow patterns around an airfoil at transonic speeds and at various upstream thermodynamic conditions is also presented. The paper provides important guidelines for future studies on this subject.

A new small-disturbance model for a steady transonic flow of moist air with non-equilibrium and homogeneous condensation around a thin airfoil is presented. The model explores the nonlinear interactions among the near-sonic speed of the flow, the small thickness ratio and angle of attack of the airfoil, and the small amount of water vapour in the air. The condensation rate is calculated according to classical nucleation and droplet growth models. The asymptotic analysis gives the similarity parameters that govern the flow problem. Also, the flow field can be described by a non-homogeneous (extended) transonic small-disturbance (TSD) equation coupled with a set of four ordinary differential equations for the calculation of the condensate (or sublimate) mass fraction. An iterative numerical scheme which combines Murman & Cole's (1971) method for the solution of the TSD equation with Simpson's integration rule for the estimation of the condensate mass production is developed. The results show good agreement with available numerical simulations using the inviscid fluid flow equations. The model is used to study the effects of humidity and of energy supply from condensation on the aerodynamic performance of airfoils.

A transonic unsteady aerodynamic and aeroelasticity code has been developed for application to realistic aircraft configurations. The new code is called CAP-TSD which is an acronym for Computational Aeroelasticity Program - Transonic Small Disturbance. The CAP-TSD code uses a time-accurate approximate factorization algorithm for solution of the unsteady transonic small-disturbance equation that is efficient for solution of steady and unsteady transonic flow problems including supersonic freestream flows. The new code can treat complete aircraft geometries with multiple lifting surfaces and bodies. Applications to wings in supersonic freestream flow are presented. Comparisons with selected exact solutions from linear theory are presented showing generally favorable results. Calculations for both steady and oscillatory cases for the F-5 and RAE tailplane models are compared with experimental data and also show good overall agreement. Selected steady calculations are further compared with a steady flow Euler code.

In order to get the details of flow fields in steam turbines, three-dimensional turbulent flow calculations are useful. However in a design procedure, three-dimensional flow calculations are only possible in the last design stage, because they need in-depth boundary conditions of both geometries and flows. At such a late time in the procedure, it is difficult to go back and change main design parameters, such as flow area and stage load. Both three-dimensional flow patterns and non-equilibrium condensation caused by rapid expansions of steam have important roles with respect to steam turbine performance particularly in low-pressure sections. The steam turbine internal efficiency can be improved by taking account of these effects in the early design stage, especially in flow pattern design. This paper describes a multi-stage through-flow calculation technique including both three-dimensional flow efffects and phase changes from vapour to small droplets. To compute the high-speed two phase steam flow, a flux-splitting procedure including non-equilibrium homogeneously condensation is introduced. Three-dimensional blade forces are calculated by using angles of both blade camber and radial lean. The blade camber lines can be decided without in-depth blade geometries. Therefore this computational technique is applicable in the flow pattern design. The calculation results agree well with fully three-dimensional flow calculation and the calculation can predict supersaturating states and Wilson lines which are defined as the maximum supercooling.

To study the interaction between the transient flow in pipe and the unsteady turbulent flow in turbine, a coupled model of the transient flow in the pipe and three-dimensional unsteady flow in the turbine is developed based on the method of characteristics and the fluid governing equation in the accelerated rotational relative coordinate. The load-rejection process under the closing of guide vanes of the hydraulic power plant is simulated by the coupled method, the traditional transient simulation method and traditional three-dimensional unsteady flow calculation method respectively and the results are compared. The pressure, unit flux and rotation speed calculated by three methods show a similar change trend. However, because the elastic water hammer in the pipe and the pressure fluctuation in the turbine have been considered in the coupled method, the increase of pressure at spiral inlet is higher and the pressure fluctuation in turbine is stronger.

Numerical simulation of 3D fluid–structure interaction flow using an immersed object method with overlapping grids

A three-dimensional Navier-Stokes solver is applied to the computation of unsteady nozzle flows resulting from fluctuating backpressure and validated by comparison with experimental measurements. The flow is modeled by the three-dimensional compressible Navier-Stokes equations using the Launder-Sharma near-wall k-e turbulence model. The mean-flow and turbulence-transport equations are integrated in time using a first-order implicit scheme with third-order MUSCL upwind-biased Van Leer flux-vector-splitting. The configuration studied is a three-dimensional nozzle, with thick sidewall boundary layers. The three-dimensional effect is produced by the shock wave-boundary layer interaction at the corners of the nozzle, where an important three-dimensional recirculating zone appears. The flow is unsteady because of backpressure fluctuation, produced experimentally by a rotating rod of elliptical cross section. The unsteady computations were run for a backpressure fluctuation frequency of 180 Hz.

The unsteady laminar three-dimensional boundary layer viscous flow and heat transfer over a permeable stretching/shrinking surface (sheet) is studied numerically. The system of nonlinear partial differential equations is transformed into ordinary differential equations via the similarity transformation. The resulting governing equations are then solved numerically by using the shooting method. Effects of the unsteadiness parameter, the stretching/shrinking parameter, the mass suction parameter and the Prandtl number on the skin friction coefficients, the local Nusselt number as well as the velocity and temperature profiles are presented and discussed in detail. The results also show that dual solutions exist for both cases of the stretching and shrinking surfaces.

To give insight into the influence of the clocking and the stator–rotor interaction, the unsteady three-dimensional (3D) flow through a two-stage turbine is simulated numerically, using a time marching Navier–Stokes computer code with a sliding mesh approach. A stator clocking is applied to the second stator vane over several circumferential positions. The numerical results are compared with the experimental one to check the availability of the code. Clocking effects on the turbine performance, wake trajectories, and outlet flow field are focused. A relative efficiency variation of about 0.52% is concluded among clocking positions. A link between the turbine efficiency and the wake trajectories on the midspan is shown based on the presented clocking analysis in the 3D unsteady flow field. The detailed illustration of the outlet flow field shows that the influence of the clocking at the outlet is focused on the temperature distribution.

Numerical simulations for transonic aerodynamic flows

Three-dimensional unsteady flow structure inside a deployed slat of high-lift configuration is simulated numerically to investigate the cause of slat noise. Because the Reynolds number of the flow is high and the geometry is complex, a zonal large eddy simulation/Reynolds-averaged Navier-Stokes hybrid method is used to reduce the overall computational cost. The power spectral density of the pressure coefficient is compared to the experiment, and some issues regarding grid resolution, order of numerical scheme, and number of subiterations for implicit time integration are discussed. Two different types of fluctuations (high-frequency narrowband and low-frequency broadband) are observed, and the computational results are consistent with the experiment. Careful observation of the unsteady computational data reveals longitudinal vortical structures under the lower surface of the slat, which is consistent with the previous studies. The low-frequency broadband fluctuation becomes large around this region. Therefore, it is assumed that these longitudinal vortical structures are causing the broadband noise from the slat.