Numerical investigation of transonic flow over deformable airfoil with plunging motion

In this article, the transonic inviscid flow over a deformable airfoil with plunging motion is studied numerically. A finite volume method based on the Roe scheme developed in a generalized coordinate is used along with an arbitrary Lagrangian-Eulerian method and a dynamic mesh algorithm to track the instantaneous position of the airfoil. The effects of different governing parameters such as the phase angle, the deformation amplitude, the initial angle of attack, the flapping frequency, and the Mach number on the unsteady flow field and aerodynamic coefficients are investigated in detail. The results show that maneuverability of the airfoil under various flow conditions is improved by the deformation. In addition, as the oscillation frequency of the airfoil increases, its aerodynamic performance is significantly improved.

ANISOTROPIC MULTISTAGE FINITE ELEMENT METHOD FOR TWO DIMENSIONAL VISCOUS TRANSONIC FLOW IN TURBOMACHINERY

A new method based on the anisotropic tensor force finite element and Taylor-Galerkin finite element is presented in the present paper.Its application to two-dimensional viscous transonic flow in turbomachinery improves the conver- gence rate and stability of calculation,and the results obtained agree well with the experimental measurements.

HIGH ORDER ACCURACY SCHEME FOR 2-D TRANSONIC FLOWS

According to Taylor's expansion formula, a kind of derivatives computation method is presented, which can achieve arbitrary order of accuracy. Then it is combined with a flux difference splitting technique and a flux limiter to construct the desiredscheme that is suitable for non-oscillatory shock-capturing calculation. TVD typeRunge-Kutta method is used for temporal discretization. Several steady and unsteadynumerical experiments demonstrate that the scheme is a robust solver for transonicflows with high accuracy and high resolution of shock wave structures.

TIME MARCHING INTEGRAL EQUATION METHOD FOR UNSTEADY TRANSONIC FLOWS

In this paper,we have proposed a time marching intregral equation method which does not have the limitation of the time linearized integral equation method in that the latter method can not satisfac- torily simulate the shock-wave motions.Firstly,a model problem——one dimensional initial and boundary value wave problem is treated to clarify the basic idea of the new method.Then the method is implemented for 2-D and 3-D unsteady transonic flow problems.The introduction of the concept of a qua- si-velocity-potential simplifies the time marching integral equations and the treatment of trailing vortex sheet condition.The numerical calculations show that the method is reasonable and reliable.

NUMERICAL METHOD FOR SOLVING THE EULER EQUATION FOR UNSTEADY TRANSONIC FLOWS OVER OSCILLATING AIRFOILS

Through transformations, the time-dependent boundary condition on the airfoil contour and the boundary condition at infinity are brought fixed to the boundaries of a finite domain. The boundary conditions can thus be satisfied exactly without increasing the computational time. The novel scheme is useful for computing transonic, strong disturbance, unsteady flows with high reduced frequencies. The scheme makes use of curvefitted orthogonal meshes and the lattice control technique to obtain the optimal grid distribution. The numerical results are satisfactory.

TRANSONIC FLOW CALCULATION OF EULER EQUATIONS BY IMPLICIT ITERATING SCHEME WITH FLUX SPLITTING

Three dimensional Euler equations are solved in the finite volume form with van Leer's flux vector splitting technique. Block matrix is inverted by Gauss-Seidel iteration in two dimensional plane while strongly implicit alternating sweeping is implemented in the direction of the third dimension. Very rapid convergence rate is obtained with CFL number reaching the order of 100. The memory resources can be greatly saved too. It is verified that the reflection boundary condition can not be used with flux vector splitting since it will produce too large numerical dissipation. The computed flow fields agree well with experimental results. Only one or two grid points are there within the shock transition zone.

Numerical Analysis of Transonic Flow around Cones

A new solver is presented for transonic flow around cone-cylinder, axisymmetric bodies. Ground experiments almost always suffer from uncertainty due to operating in the presence of high levels of facility noise. Besides, experimental measurements of these mechanisms are not available at high-speed flows. Direct Numerical Simulations have made it possible to compute details of the transonic mechanisms but still a significant challenge due to the cost. This study aims to present a new solver to model transonic flows. To assess the new solver, the surface Mach number and the drag coefficient are investigated as the freestream Mach number varies. The results are in excellent agreement with experimental data, indicating the new model is capable of accurately predicting the aerodynamics coefficients at transonic flow regimes.

A LIMITING VISCOSITY APPROACH TO THE RIEMANN PROBLEM FOR TRANSONIC FLOW

In this paper we have obtained the existence of weak solutions of the small disturbance equations of steady two-dimension flow [GRAPHICS] with Riemann date [GRAPHICS] where v+ greater-than-or-equal-to 0, v- greater-than-or-equal-to 0 and u- less-than-or-equal-to u+ by introducing 'artificial' viscosity terms and employing Helley's theorem. The setting under our consideration is a nonstrictly hyperbolic system. our analysis in this article is quite fundamental.

Fourier time spectral method for subsonic and transonic flows

The time accuracy of the exponentially accurate Fourier time spectral method（TSM） is examined and compared with a conventional 2nd-order backward difference formula（BDF） method for periodic unsteady flows. In particular, detailed error analysis based on numerical computations is performed on the accuracy of resolving the local pressure coefficient and global integrated force coefficients for smooth subsonic and non-smooth transonic flows with moving shock waves on a pitching airfoil. For smooth subsonic flows, the Fourier TSM method offers a significant accuracy advantage over the BDF method for the prediction of both the local pressure coefficient and integrated force coefficients. For transonic flows where the motion of the discontinuous shock wave contributes significant higherorder harmonic contents to the local pressure fluctuations,a sufficient number of modes must be included before the Fourier TSM provides an advantage over the BDF method.The Fourier TSM, however, still offers better accuracy than the BDF method for integrated force coefficients even for transonic flows. A problem of non-symmetric solutions for symmetric periodic flows due to the use of odd numbers of intervals is uncovered and analyzed. A frequency-searching method is proposed for problems where the frequency is not known a priori. The method is tested on the vortex shedding problem of the flow over a circular cylinder.

Improved local amplification factor transport equation for stationary crossflow instability in subsonic and transonic flows

Transition prediction is a hot research topic of fluid mechanics.For subsonic and transonic aerodynamic flows,e^(N) method based on Linear Stability Theory(LST)is usually adopted reliably to predict transition.In 2013,Coder and Maughmer established a transport equation for Tollmien-Schlichting(T-S)instability so that the e^(N) method can be applied to general Reynolds-Average-Navier-Stokes(RANS)solvers conveniently.However,this equation focuses on T-S instability,and is invalid for crossflow instability induced transition which plays a crucial role in flow instability of three-dimensional boundary layers.Subsequently,a transport equation for crossflow instability was developed in 2016,which is restricted to wing-like geometries.Then,in 2019,this model was extended to arbitrarily shaped geometries based on local variables.However,there are too many tedious functions and parameters in this version,and it can only be used for incompressible flows.Hence,in this paper,after a large amount of LST analyses and parameter optimization,an improved version for subsonic and transonic boundary layers is built.The present improved model is more robust and more concise,and it can be applied widely in aeronautical flows,which has great engineering application value and significance.An extensive validation study for this improved transition model will be performed.

Liquid Phase Evaporation on the Normal Shock Wave in Moist Air Transonic Flows in Nozzles

This paper presents a numerical analysis of the atmospheric air transonic flow through de Laval nozzles. By nature, atmospheric air always contains a certain amount of water vapor. The calculations were made using a Laval nozzle with a high expansion rate and a convergent-divergent(CD) "half-nozzle", referred to as a transonic diffuser, with a much slower expansion rate. The calculations were performed using an in-house CFD code. The computational model made it possible to simulate the formation of the liquid phase due to spontaneous condensation of water vapor contained in moist air. The transonic flow calculations also take account of the presence of a normal shock wave in the nozzle supersonic part to analyze the effect of the liquid phase evaporation.

Computations of unsteady transonic flow over airfoil at low Reynolds number

Transonic flow over a thin airfoil at low Reynolds number was studied numerically by directly solving two-dimensional full Navier-Stokes equations through 5th order weighted essentially non-oscillatory(WENO) scheme without using any turbulence model.A series of distinguished unsteady phenomena for a thin 2-D transonic airfoil flow were presented.Due to continuous adverse pressure gradient in the subsonic flow downstream of the sonic line,the unsteady separated boundary layer with main vortex and secondary vortex was developed at the rear of the airfoil.At the trailing edge,the vortex-shedding was characterized by periodical connection of the main vortex and secondary vortex on the other side of the airfoil.The unsteady separation and vortex-shedding occurred with the same period.On the airfoil surface,the average pulse pressure related to the unsteady supersonic region was obviously smaller than that related to the vortex-shedding at the trailing edge.With the attack angle increasing from 0°to 2°,the frequency of vortex-shedding decreases about 4.2%.At last,the turbulence intensity and many second-order statistics in the wake region were investigated.

AN EFFICIENT METHOD FOR SOLVING THE MIXED DIRECT-INVERSE PROBLEM OF THE TRANSONIC ROTATIONAL FLOW IN PLANE CASCADES

The method in [1] has been extended to the case of rotational flow in this paper. A new method for dealing with the shock wave is presented. This method has the advantages of both the shock-fitting and the shock capturing methods. The direct problem and the mixed direct-inverse prob- lem of the rotational flow in a transonic plane cascade at both design and off design conditions are solved, and the results show that the present method has rapid convergence rate and high accuracy even for the flow with moderately strong shocks. The calculations have been carried out on the DPS-8 computer, and for the direct problem, only 50-80 iterations are needed, and 50-80 seconds of CPU time are required.

Influence of Air Humidity on Transonic Flows with Weak Shock Waves

The paper presents CFD results for the transonic flow of dry and moist air through a diffuser and a compressor rotor.In both test geometries,i.e.the Sajben transonic diffuser and the NASA Rotor 37,the air humidity impact on the structure of flows with weak shock waves was examined.The CFD simulations were performed by means of an in-house CFD code,which was the RANS-based modelling approach to compressible flow solutions.It is shown that at high values of relative humidity,above 70%,the modelling of the transonic flow field with weak shock waves by means of the dry air model may produce wrong results.

Numerical method for sound propagation with acoustic shock waves in transonic flow ducts

The fourth order MacCormack scheme with fourth viscous term is used to improve the shocked solutions for sound propagation in varying cross area and hard-wall ducts with transonic flow. The artificial viscous coefficient is given out by an empirical formula. It is shown from three calculation examples of acoustic shock waves that the new method is much better than the second order MacCormack method which is the best one of second order schemes. Moreover, CPU times of both methods are almost the same.

AN IMPLICIT ALGORITHM OF THIN LAYER EQUATIONS IN VISCOUS，TRANSONIC，TWO－PHASE NOZZLE FLOW

Omitting viscosity along flow direction, we have simplified the dimensionless N-Sequations in arbitrary curved coordinate system as the thin layer equations. Using theimplicit approximate-factorization algorithm to solve the gas-phase governing equ-ations and the characteristic method to follow the tracks of particles, we then obtainedthe full coupled numerical method of two-phase.transonic, turbulent flow. Here, par- ticle size may be grouped, the subsonic boundary condition at entry of nozzle is ireatedby quasi-characteristic method in reference plane and the algebraic model is used forturbulent flow. These methods are applied in viscous two-phase flow. calculation of ro-cket nozzle and in the prediciton of thrust and specific impulse for solid propellant ro-cket motor. The calculation results are in good agreement with the measurerment va-lues. Moreover, the influences of different particle radius, different particle mass frac-tion and particle size grouped on flow field have been discussed, and the influences of particle two-dimensional radial velosity component and viscosity on specific impulse ofrocket motor have been analysed.The method of this paper possesses the advantage of saving computer time. More important, the effect is more obvious for the calculation of particle size being grouped.

AN EFFICIENT FINITE-DIFFERENCE ALGORITHM FOR COMPUTING AXISYMMETRIC TRANSONIC NACELLE FLOW FIELDS

A finite difference method for computing the axisymmetric, transonic flows over a nacelle is presented in this paper. By use of the conservative full-potential equation, body-fitted grid, and the exact boundary conditions, a new AF scheme is constructed according to the criterion of optimum convergence. The proposed scheme has been applied to transonic nacelle flow problems. Computation for several nacelles shows the rapid convergence of this scheme and excellent agreement with the experimental results.

Transonic Rudder Buzz on Tailless Flying Wing UAV

Transonic rudder buzz responses based on the computational fluid dynamics or computational structural dynamics(CFD/CSD)loosely method are analyzed for a tailless flying wing unmanned aerial vehicle(UAV).The Reynolds-averaged Navier-Stokes(RANS)equations and finite element methods based on the detailed aerodynamic and structural model are established,in which the aerodynamic dynamic meshes adopt the unstructured dynamic meshes based on the combination of spring-based smoothing and local remeshing methods,and the lower-upper symmetric-Gauss-Seidel(LU-SGS)iteration and Harten-Lax-van Leer-Einfeldt-Wada(HLLEW)space discrete methods based on the shear stress transport(SST)turbulence model are used to calculate the aerodynamic force.The constraints of the rudder motions are fixed at the end of structural model of the flying wing UAV,and the structural geometric nonlinearities are also considered in the flying wing UAV with a high aspect ratio.The interfaces between structural and aerodynamic models are built with an exact match surface where load transferring is performed based on 3Dinterpolation.The flying wing UAV transonic buzz responses based on the aerodynamic structural coupling method are studied,and the rudder buzz responses and aileron,elevator and flap vibration responses caused by rudder motion are also investigated.The effects of attack,height,rotating angular frequency and Mach number under transonic conditions on the flying wing UAV rudder buzz responses are discussed.The results can be regarded as a reference for the flying wing UAV engineering vibration analysis.

The performance of proper orthogonal decomposition in discontinuous flows

In this paper, flow reconstruction accuracy and flow prediction capability of discontinuous transonic flow field by means of proper orthogonal decomposition （POD） method is studied. Although linear superposition of ＂high frequency waves＂ in different POD modes can achieve the reconstruction of the shock wave, the smoothness of the solution near the shock wave cannot be guaranteed. The modal coefficients are interpolated or extrapolated and different modal components are superposed to realize the prediction of the flow field beyond the snapshot sets. Results show that compared with the subsonic flow, the transonic flow with shock wave requires more POD modes to reach a comparative reconstruction accuracy. When a shock wave exists, the interpolation prediction ability is acceptable. However, large errors exist in extrapolation, and increasing the number of POD modes cannot effectively improve the prediction accuracy of the flow field.

REVIEW OF THE GREEN＇S FUNCTION METHOD FOR STEADY ＆UNSTEADY，SUBSONIC，TRANSONIC ＆SUPERSONIC AERODYNAMICS AROUND COMPLEX CONFIGURATIONS

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