Transonic Flow Regime Research Articles

Organised modes and shock–vortex interaction in unsteady viscous transonic flows around an aerofoil: Part I: Mach number effect

The identification of successive stages in the transition of unsteady viscous transonic flow around an aerofoil is carried out by solving the time-dependent Navier–Stokes equations for a compressible fluid in two-dimensional approach. The numerical simulation is carried out at the Mach number range (0.2–0.98). At a fixed Reynolds number ( Re=10,000), it is found that this flow undergoes the following four transition steps: It remains steady up to the Mach number values (0.2–0.35) and afterwards it develops spontaneously, without any imposed artificial perturbation, an inherent unsteadiness corresponding to a near-wake von Kármán instability, in the Mach number range (0.35–0.9). It is found that there exists a critical Mach number between the values (0.90–0.95) for which the flow returns to a steady-state. Furthermore, the flow is found to be governed by two instability processes in the Mach number range (0.75–0.8), where, apart from the von Kármán mode (mode I), a lower frequency mode II appears, due to the formation of weakly supersonic alternating zones in the region upstream of the aerofoil, related to the buffeting phenomenon. A triple role played by the increasing compressibility effects to trigger the instability processes, to maintain and to inhibit them in the transonic flow regime is therefore analysed in detail.

Examples of non-uniqueness of almost-unidirectional gas flow

Modeling of non-linear phenomena always brings the question of uniqueness of the exact as well as numerical solution. The paper is devoted to the non-unique behavior of gases, which has been observed in special axisymmetric nozzles at transonic flow regimes. We describe the flow by means of the compressible Euler equations. Surprisingly, the non-uniqueness is recorded already by the simplest stationary quasi-one-dimensional model. In this case, we prove the non-uniqueness analytically. The non-unique behavior is present also in the three-dimensional model, which we illustrate numerically using an original mass-conserving axisymmetric finite volume scheme. Numerical example of non-unique solutions corresponding to an axisymmetric nozzle of a complicated geometry shows a good agreement between the quasi-one-dimensional analytical and axisymmetric three-dimensional numerical results.

Application of simulated annealing to inverse design of transonic turbomachinery cascades

In turbomachinery blade design, inverse methods and optimization techniques are often applied independently to produce high performance blade shapes. The idea of using an optimization algorithm to seek the optimal target distribution for an inverse design methodology has been explored. However, these efforts have been made mainly in the design of single aerofoils. In this paper, a new inverse design method is coupled with a simulated annealing algorithm to search for the optimum turbomachinery cascade shape. In order to speed up the algorithm, a database of generated designs is set up and the nearest match is selected to initialize subsequent calculation. The proposed computational procedure equips engineers with an automatic design tool with which the inverse method may be applied in isolation or combined with the optimization algorithm to produce the optimum. The inverse methodology is based on a cell vertex, finite volume time-marching flow solver that gives the viscous cascade flow solution in both the subsonic and the transonic flow regimes. The cascade shape is computed subject to an imposed distribution of the mass-averaged tangential velocity and a specified tangential thickness profile. The solver code is validated using experimental data and the accuracy of the inverse method is verified by regenerating a known cascade geometry starting from a different one using its mass-averaged tangential velocity distribution. In combining the inverse methodology with the optimization algorithm, the mass-averaged tangential velocity distribution is parametrized using a cubic B-spline curve and the proposed simulated annealing algorithm is applied to predict the optimal distribution by minimizing loss. The overall procedure is demonstrated to produce optimum shapes of a transonic axial turbine and an axial compressor rotor.

High Resolution Relaxed Upwind Schemes in Gas Dynamics

Numerical schemes based on relaxation are typically central difference schemes. In the case of supersonic flows, however, central differences are unphysical approximations. Introducing a shift in the relaxation approximation relaxed upwind schemes are constructed. Similar as central relaxed schemes, the new upwind versions also avoid the nonlinear Riemann problem and staggered grids. In addition, they simulate the physical domain of dependence correctly even in transonic flow regimes. The performance of the methods is illustrated by an acoustic shock interaction in gas dynamics.

A linearized method for the frequency analysis of three-dimensional fluid/structure interaction problems in all flow regimes

We present a computational fluid dynamics (CFD)-based linearized method for the frequency analysis of three-dimensional fluid/structure interaction problems. This method is valid in the subsonic, transonic, and supersonic flow regimes, and is insensitive to the frequency or damping level of the sought-after coupled eigenmodes. It is based on the solution by an orthogonal iteration procedure of a complex eigenvalue problem derived from the linearization of a three-field fluid/structure/moving mesh formulation. The key computational features of the proposed method include the reuse of existing unsteady flow solvers, a second-order approximation of the flux Jacobian matrix, and a parallel domain decomposition-based iterative solver for the solution of large-scale systems of discretized fluid/structure equations. While the frequency analysis method proposed here is primarily targeted at the extraction of the eigenpairs of a wet structure, we validate it with the flutter analysis of the AGARD Wing 445.6, for which experimental data is available.

Aerodynamics of Stardust Sample Return Capsule

Successful return of interstellar dust and cometary material by the Stardust Sample Return Capsule requires an accurate description of the Earth entry vehicle''s aerodynamics. This desciption must span the hypersonic-rarefied, hypersonic-continuum, supersonic, transonic, and subsonic flow regimes. Data from numerous sources are compiled to accomplish this objective. These include Direct Simulation Monte Carlo analyses, thermochemical nonequilibrium computational fluid dynamics, transonic computational fluid dynamics, existing wind tunnel data, and new wind tunnel data. Four observations are highlighted: 1) a static instability is revealed in the free-molecular and early transitional-flow regime due to aft location of the vehicle''s center-of-gravity, 2) the aerodynamics across the hypersonic regime are compared with the Newtonian flow approximation and a correlation between the accuracy of the Newtonian flow assumption and the sonic line position is noted, 3) the primary effect of shape change due to ablation is shown to be a reduction in drag, and 4) a subsonic dynamic instability is revealed which will necessitate either a change in the vehicle''s center-of-gravity location or the use of a stabilizing drogue parachute.

Implicit and multigrid procedures for steady-state computations with upwind algorithms

A computational study for the convergence acceleration of Euler and Navier–Stokes computations with upwind schemes has been conducted in a unified framework. It involves the flux-vector splitting algorithms due to Steger–Warming and Van Leer, the flux-difference splitting algorithms due to Roe and Osher and the hybrid algorithms, AUSM (Advection Upstream Splitting Method) and HUS (Hybrid Upwind Splitting). Implicit time integration with line Gauss–Seidel relaxation and multigrid are among the procedures which have been systematically investigated on an individual as well as cumulative basis. The upwind schemes have been tested in various implicit–explicit operator combinations such that the optimal among them can be determined based on extensive computations for two-dimensional flows in subsonic, transonic, supersonic and hypersonic flow regimes. In this study, the performance of these implicit time-integration procedures has been systematically compared with those corresponding to a multigrid accelerated explicit Runge–Kutta method. It has been demonstrated that a multigrid method employed in conjunction with an implicit time-integration scheme yields distinctly superior convergence as compared to those associated with either of the acceleration procedures provided that effective smoothers, which have been identified in this investigation, are prescribed in the implicit operator.

External Boundary Conditions for Three-Dimensional Problems of Computational Aerodynamics

We consider an unbounded steady-state flow of viscous fluid over a three-dimensional finite body or configuration of bodies. For the purpose of solving this flow numerically, we discretize the governing equations (Navier--Stokes) on a finite-difference grid. Prior to the discretization, we obviously need to truncate the original unbounded domain by introducing an artificial computational boundary at a finite distance from the body; otherwise, the number of discrete variables will not be finite. This artificial boundary is typically the external boundary of the domain covered by the grid. The flow problem (both continuous and discretized) formulated on the finite computational domain is clearly subdefinite unless supplemented by some artificial boundary conditions (ABCs) at the external computational boundary. In this paper, we present an innovative approach to constructing highly accurate ABCs for three-dimensional flow computations. The approach extends our previous technique developed for the two-dimensional case; it employs the finite-difference counterparts to Calderón's pseudodifferential boundary projections calculated in the framework of the difference potentials method (DPM) of Ryaben'kii. The resulting ABCs appear spatially nonlocal but are particularly easy to implement along with the existing flow solvers. The new boundary conditions have been successfully combined with the NASA-developed production code TLNS3D and used for the analysis of wing-shaped configurations in subsonic and transonic flow regimes. As demonstrated by the computational experiments and comparison with the standard local methods, the DPM-based ABCs allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution. Moreover, they may provide for a noticeable speedup of multigrid convergence.

Improved Treatment of External Boundary Conditions for Three-Dimensional Flow Computations

An innovative numerical approach is presented for setting highly accurate nonlocal boundary conditions at the external computational boundaries when calculating three-dimensional compressible viscous flows over finite bodies. The approach is based on application of the special boundary operators analogous to Calderon's projections (Calderon, A. P.) and the difference potentials method by Ryaben'kii; it extends the previous technique developed for the two-dimensional case. The new boundary conditions methodology has been successfully combined with the NASA-developed code TLNS3D and used for the analysis of wing-shaped configurations in subsonic and transonic flow regimes. As demonstrated by the computational experiments, the improved external boundary conditions allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution. Moreover, they may provide for a noticeable speedup of convergence of the multigrid iterations

A PRESSURE-BASED ALGORITHM FOR HIGH-SPEED TURBOMACHINERY FLOWS

The steady state Navier–Stokes equations are solved in transonic flows using an elliptic formulation. A segregated solution algorithm is established in which the pressure correction equation is utilized to enforce the divergence-free mass flux constraint. The momentum equations are solved in terms of the primitive variables, while the pressure correction field is used to update both the convecting mass flux components and the pressure itself. The velocity components are deduced from the corrected mass fluxes on the basis of an upwind-biased density, which is a mechanism capable of overcoming the ellipticity of the system of equations, in the transonic flow regime. An incomplete LU decomposition is used for the solution of the transport-type equations and a globally minimized residual method resolves the pressure correction equation. Turbulence is resolved through the k–e model. Dealing with turbomachinery applications, results are presented in two-dimensional compressor and turbine cascades under design and off-design conditions. © 1997 John Wiley & Sons, Ltd.

Dense Gas Flow in Minimum Length Nozzles

Recently, dense gases have been investigated for many engineering applications such as for turbomachinery and wind tunnels. Supersonic nozzle design can be complicated by nonclassical dense-gas behavior in the transonic flow regime. In this paper, a method of characteristics (MOC) is developed for two-dimensional (planar) and axisymmetric flow of a van der Waals gas. A minimum length nozzle design code is developed that employs the MOC procedure to generate an inviscid wall contour. The van der Waals results are compared to perfect gas results to show the real-gas effects on the flow properties and inviscid wall contours.

Partitioned procedures for the transient solution of coupled aroelastic problems Part I: Model problem, theory and two-dimensional application

In order to predict the dynamic response of a flexible structure in a fluid flow, the equations of motion of the structure and the fluid must be solved simultaneously. In this paper we present several partitioned procedures for time-integrating this focus coupled problem and discuss their merits in terms of accuracy, stability, heterogeneous computing, I/O transfers, subcycling and parallel processing. All theoretical results are derived for a one-dimensional piston model problem with a compressible flow, because the complete three-dimensional aeroelastic problem is difficult to analyze mathematically. However, the insight gained from the analysis of the coupled piston problem and the conclusions drawn from its numerical investigation are confirmed with the numerical simulation of the two-dimensional transient aeroelastic response of a flexible panel in a transonic non-linear Euler flow regime.

A Fully Three-Dimensional Inverse Method for Turbomachinery Blading in Transonic Flows

This paper presents a procedure to extend a recently developed fully three-dimensional inverse method for highly loaded turbomachine blades into the transonic-flow regime. In this inverse method, the required three-dimensional blade profile to produce a prescribed swirl schedule is determined iteratively using the blade boundary conditions. In the present implementation, the flow is assumed to be inviscid and the blades are assumed to be infinitely thin. The relevant equations are solved in the conservative forms and are discretized in all three directions using a finite-volume technique. Calculations are carried out for the design of high-pressure axial- and centrifugal-compressor rotors. These examples include prescribed swirl schedules corresponding to blade designs that are shock-free and blade designs that have rapid compression regions in the blade passage.

The instability of a liquid jet in a compressible airstream

The instability of a liquid jet immersed in a coflowing compressible airstream has been investigated on the basis of a linear, inviscid stability analysis, assuming uniform velocity profiles within the liquid and gas. Both two-dimensional and circular jets are investigated. Pronounced peaks in the temporal and spatial growth rates occur for values of the free-stream Mach number close to unity. A change from convective to absolute instability also occurs in this transonic flow regime of the gas.

Calculation of three-dimensional transonic turbine cascade flow

A computer code has been applied to inviscid and viscous flows through a linear turbine cascade operating in the transonic flow regime. The Euler and Navier-Stokes equations are solved by an explicit finite-volume time-stepping procedure based on the method of Jameson et al. for the integration of the Euler equations and extended to Navier-Stokes equations by Schwamborn. The inviscid and viscous results in two-dimensions represent the profile characteristics at midspan position and the three-dimensional, viscous results include sidewall effects. A particular consideration of the flow losses gives detailed information about secondary flow behavior. The computational results are compared to experimental data for the MTU-T5.1 profile at isentropic outflow Mach numbers of 0.7, 0.9, and 1.1.

Design of Turbomachinery Blading in Transonic Flows by the Circulation Method

The newly developed three-dimensional design technique for turbomachinery blading based on the circulation method has been successfully extended into the transonic flow regime. The main task involves replacing the classical Fourier series expansion technique by a finite-volume method. Calculations were carried out for the design of infinitely thin cascaded blades in transonic flows, including shocked and shock-free cascaded blades.

Locally implicit total-variation-diminishing schemes on unstructuredtriangular meshes

A numerical solution procedure that includes locally implicit total-variation-diminishing schemes and adaptive mesh generation techniques has been developed in this work. In a Cartesian coordinate system, the Euler equations are solved by using a cell-centered finite volume algorithm. A new construction of symmetric total-variation-diminishing schemes on unstructured triangular meshes is presented. The validation of the present solution-adaptive methods is confirmed by comparison with related numerical results for inviscid flows around an isolated NACA 0012 airfoil and passing through a channel with a circular arc bump in transonic and supersonic flow regimes. To further prove the feasibility of this approach, a two-element airfoil flow is also investigated. Furthermore, one unsteady transonic channel flow is studied to demonstrate the reliability and capability of the present solution procedure for a time accurate calculation.

Self-excited compressible flow in a pipe–collar nozzle

The turbulent flow in a duct with an abrupt increase in its cross-sectional area is very unstable and exhibits strong oscillations. Such a flow is investigated experimentally using a simple device — a pipe–collar nozzle, composed of a convergent nozzle and a pipe lengthened by a collar with an enlarged cross-section. The oscillating flow characteristics are considered over a wide range of flow velocities including both subsonic and transonic flow regimes. Two modes of symmetric and one of asymmetric oscillations are distinguished. For the first mode of symmetric oscillation the jet is separated in each phase of the oscillation cycle, whereas for the second the jet alternately separates and reattaches to the collar wall. Both modes of symmetric oscillations are controlled by shear-layer instability of the collar flow. For low velocities and short collars the oscillation is also affected by the organ-pipe resonance in the preceding pipe. The asymmetric oscillation appears at relatively large flow Mach numbers. This mode exhibits a radial resonance of the jet.

Inertial impaction of fine particles at moderate reynolds numbers and in the transonic regime with a thin-plate orifice nozzle

Because the smallest size of very fine particles that can be separated in an impactor is proportional to the ratio Re/M 2, there is an interest in studying these instruments at moderate Reynolds numbers, Re, and at Mach numbers, M, up to sonic conditions. Here an inertial impactor of fixed geometry is tested at Reynolds numbers between 40 and 840, and in the transonic flow regime, at downstream to upstream pressure ratios between 0.42 and 0.79. Earlier theoretical and experimental work has shown that conventional impactors, with nozzles having relatively large length-to-diameter ratios, loose their resolution at Re smaller than several hundreds due to boundary layer growth at the nozzle walls. This effect is minimized here by means of nozzles having rapidly converging walls down to their exit, embodied by a thin-plate orifice with thickness to diameter ratio of 0.014. The nozzle to plate distance is equal to the orifice diameter. Test dioctyl sebacate (DOS) aerosols, with diameters ranging from 0.05 to 0.24 μm, were generated using a Differential Mobility Analyzer (DMA). Collection efficiencies were measured on-line at constant Reynolds and variable Mach numbers using two electrometers; one to monitor the current of charged particles impacting on the collecting plate, the other for the current of uncollected particles. A new method based on sweeping over the Mach number for a given test aerosol, rather than on varying the aerosol size at a fixed Mach number, provides an inversion technique which eliminates completely the effects of multiply charged particles from the DMA. Measurements show excellent resolution even at Re = 100, as well as through the transonic region, so that neither compressibility nor viscous phenomena limit the size of the smallest particle that can be inertially separated with high resolution. The limiting factor is Brownian diffusion. Its negative effects on particle collection efficiency curves are measured here for the first time using NaCl particles with diameters between 0.008 and 0.016 μm.

Self-excited oscillations in internal transonic flows

Compressible internal flows in a velocity regime close to Mach-number one are known to exhibit an oscillatory character and produce a very high intensity noise. Such flows appear in equipment and installations, e.g. engines, valves, pipe systems and so on. Depending on the particular geometries the shock waves that appear interact with the existing boundary layers or free shear layers leading very often to flow instabilities. The different types of oscillations in internal transonic flows are systematically described in this paper. The mechanisms leading to the oscillatory self-excited flow character are elucidated and substantiated by experimental, and in some cases theoretical, data. As the flows are strongly geometry dependent, the following four most often encountered configurations are considered and their properties and peculiarities are pointed out: the curved duct, the convergent-divergent nozzle, the sudden enlargement of duct cross-section and the jet-cavity interaction. In the cases of the curved duct and the convergent-divergent nozzle the supersonic flow region is terminated by a shock wave which causes flow separation. At some values of the supplied to ambient pressure ratio, strong oscillations appear. During these oscillations strong acoustic disturbances (shock waves) are periodically transmitted upstream. Similarly, in a duct with a sudden enlargement of its cross-section, the downstream and upstream part of the flow are often affected by severe oscillations. In this flow geometry several mechanisms of instability have been detected. The jet and the upstream-facing cavity give a very wide variety of possible oscillatory configurations. Particularly in the transonic and supersonic flow regime, oscillations are very strong and lead to noise of high intensity and high gas temperatures at the base of the cavity.