Steady Transonic Flow Research Articles

MAGNETICALLY CONTROLLED ACCRETION FLOWS ONTO YOUNG STELLAR OBJECTS

(abridged) Accretion from disks onto young stars is thought to follow magnetic field lines from the inner disk edge to the stellar surface. The accretion flow thus depends on the geometry of the magnetic field. This paper extends previous work by constructing a collection of orthogonal coordinate systems, including the corresponding differential operators, where one coordinate traces the magnetic field lines. This formalism allows for an (essentially) analytic description of the geometry and the conditions required for the flow to pass through sonic points. Using this approach, we revisit the problem of magnetically controlled accretion flow in a dipole geometry, and then generalize the treatment to consider magnetic fields with multiple components, including dipole, octupole, and split monopole contributions. This approach can be generalized further to consider more complex magnetic field configurations. Observations indicate that accreting young stars have substantial dipole and octupole components, and that accretion flow is transonic. If the effective equation of state for the fluid is too stiff, the flow cannot pass smoothly through the sonic points in steady state. For a multipole field of order \ell, we derive a constraint on the polytropic index, n>\ell+3/2, required for steady transonic flow to reach free-fall velocities. For octupole fields, inferred on surfaces of T Tauri stars, n>9/2, so that the flow must be close to isothermal. The inclusion of octupole field components produces higher densities at the stellar surface and smaller hot spots, which occur at higher latitudes; the magnetic truncation radius is also modified. This contribution thus increases our understanding of magnetically controlled accretion for young stellar objects and can be applied to a variety of additional astrophysical problems.

HIGH RESOLUTION SOLUTIONS FOR THE SUPERSONIC FORMATION OF SHOCKS IN TRANSONIC FLOW

We present numerical solutions of two problems for the unsteady transonic small disturbance equations whose solutions contain shocks. The first problem is a two-dimensional Riemann problem with initial data corresponding to a slightly supersonic flow hitting the corner of an expanding duct at t = 0. The second problem is a boundary value problem that describes steady transonic flow over an airfoil. In both cases, the solutions contain regions of supersonic and subsonic flow, and an expansion wave interacts with a sonic line to produce a shock. We use high resolution methods, together with local grid refinement, to investigate the nature of the solution in the neighborhood of the point where the shock forms. We find that the shock originates in the supersonic region as originally proposed by Guderley, and very close to, but not at, the sonic line.

Triple-deck analysis of formation of supersonic and local separation regions in transonic steady flow over a roughness element on the surface of a body of revolution

Transonic axisymmetric flow over a body of rotation with a small roughness element located on its surface is considered. The body is manly cylindrical. The roughness height is assumed to be much smaller than the radius of the cylinder and such that a triple-deck flow is induced in its neighborhood. The goal of the work is to study the effect of the cylinder radius and the roughness shape on the triple-deck flow when the cylinder radius is of the same order as the transverse size of the triple-deck interaction region. In this case, the effect of three-dimensionality of the flow is exhibited even in the first approximation. Special attention is given to the structure of supersonic regions and closing shock waves arising in the outer potential region, as well as to local separation regions if they develop in the lower viscous boundary sublayer. Specifically, it is shown that, as the radius of the cylinder increases at a fixed roughness height, the shock intensity grows considerably, whereas the position of the main shock varies little.

A TVD Uncertainty Quantification Method with Bounded Error Applied to Transonic Airfoil Flutter

The Unsteady Adaptive Stochastic Finite Elements (UASFE) approach is a robust and efficient uncertainty quantification method for resolving the effect of random parameters in unsteady simulations. In this paper, it is shown that the underlying Adaptive Stochastic Finite Elements (ASFE) method for steady problems based on Newton-Cotes quadrature in simplex elements is extrema diminishing (ED). It is also shown that themethod is total variation diminishing (TVD) for one randomparameter and for multiple random parameters for first degree Newton-Cotes quadrature. It is proven that the interpolation of oscillatory samples at constant phase in the UASFE method for unsteady problems results in a bounded error as function of the phase for periodic responses and under certain conditions also in a bounded error in time. The twomethods are applied to a steady transonic airfoil flow and a transonic airfoil flutter problem.

Numerical experiments with several variant WENO schemes for the Euler equations

AbstractNumerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126:202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.

Transonic viscous inviscid interactions in narrow channels

AbstractUnsteady as well as steady transonic flows through channels which are so narrow that the classical boundary layer approach fails are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting interaction problem for laminar flows is formulated for both perfect and dense gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes one‐dimensional in the leading order approximation. The latter allows an interpretation of the flow in the core region by means of the theory of one‐dimensional transonic inviscid flow through a Laval nozzle while preserving the essential features of the interaction problem associated with the internal structure of pseudoshocks. The sensitivity of a separation bubble caused by a pseudoshock of sufficient strength to perturbations under the condition of choked flow will be demonstrated. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.

Dynamics of rotating flows in plasma accelerator channels with a longitudinal magnetic field

Results are presented from numerical simulations of axisymmetric plasma flows that occur in a coaxial accelerator with a longitudinal magnetic field. The simulations were carried out based on a two-dimensional MHD plasma dynamic model for the general case of a three-component magnetic field. The steady plasma flows are calculated in solving the time-dependent MHD problem by the relaxation method. The results of simulations of steady transonic flows are compared with the solutions that were obtained in the smooth accelerator channel approximation. The main regular features of plasmodynamic processes are revealed. It is found that current sheets arise in the plasma flow in a comparatively strong longitudinal magnetic field.

Viscous/inviscid interaction procedures for compressible aerodynamic flow simulations

Steady transonic flows over a wing are calculated based on a hierarchical formulation where the potential flows are corrected due to the entropy and vorticity effects of shock waves and/or laminar boundary layers. Preliminary results are in agreement with standard Euler and Navier–Stokes calculations. The merits of the present approach are briefly discussed.

Assessment of explicit algebraic Reynolds-stress turbulence models in aerodynamic computations

In the aerodynamic industrial design process, the use of numerical simulation, including viscous effects, is of ever increasing importance. As simple, standard Boussinesq-viscosity turbulence models have proven insufficient to correctly predict complex flow situations, attention is drawn to more reliable approaches towards the modelling of turbulence. This work aims at assessing the potential of Explicit Algebraic Reynolds Stress Models (EARSM) for application-oriented aerodynamic computations. To this end, two different EARSM are investigated on a variety of configurations in sub- and transonic steady flow, ranging from 2D aerofoils to 3D wing/body-configurations. Is is demonstrated that an increased over-all simulation quality is achieved. Thus, while their overhead with respect to standard linear approaches remains limited, EARSM constitute a valuable extension to the model range available to the aerodynamic design engineer.

Solution Method for Singular Initial Value Problems of One dimensional Steady Transonic Flow in a Dual-mode Scramjet

Singular initial value problems arise in solving one-dimensional steady transonic flow of dual-mode scramjet. The existing solution method has the problems of large initial value errors in principles. This paper puts forward an improved algorithm based on variable transformation, and constructs a non-singular one-dimensional steady transonic flow equation by defining a new variable. The improved algorithm can eliminate the singularity of the differential equation, and can solve the singular initial value problems of one-dimensional steady transonic flow of dual-mode scramjet.

An Implicit Fluctuation Splitting Scheme for Turbomachinery Flows

Abstract This paper describes an accurate, robust and efficient methodology for solving two-dimensional steady transonic turbomachinery flows. The Euler fluxes are discretized in space using a hybrid multidimensional upwind method, which, according to the local flow conditions, uses the most suitable fluctuation splitting (FS) scheme at each cell of the computational domain. The viscous terms are discretized using a standard Galerkin finite element scheme. The eddy viscosity is evaluated by means of the Spalart-Allmaras turbulence transport equation, which is discretized in space by means of a mixed FS-Galerkin approach. The equations are discretized in time using an implicit Euler scheme, the Jacobian being evaluated by two-point backward differences. The resulting large sparse linear systems are solved sequentially using a preconditioned GMRES strategy. The proposed methodology is employed to compute subsonic and transonic turbulent flows inside a high-turning turbine-rotor cascade.

Smooth transonic flow in an array of counter-rotating vortices

Numerical solutions to the steady two-dimensional compressible Euler equations corresponding to a compressible analogue of the Mallier & Maslowe (Phys. Fluids, vol. A 5, 1993, p. 1074) vortex are presented. The steady compressible Euler equations are derived for homentropic flow and solved using a spectral method. A solution branch is parameterized by the inverse of the sound speed at infinity, . Along these branches numerical evidence for the existence of extensive regions of smooth steady transonic flow, with local Mach numbers as large as 1.276, is presented.

Gridless Solution Method for Two-Dimensional Unsteady Flow

The main purpose of this paper is to develop a gridless method for unsteady flow simulation. A quadrantal point infilling strategy is developed to generate point and combine clouds of points automatically. A point-moving algorithm is introduced to ensure the clouds of points following the movements of body boundaries. A dual time method for solving the two-dimensional Euler equations in Arbitrary Lagrangian-Eulerian (ALE) formulation is presented. Dual time method allows the real-time step to be chosen on the basis of accuracy rather than stability. It also permits the acceleration techniques, which are commonly used to speed up steady flow calculations, to be used when marching the equations in pseudo time. The spatial derivatives, which are used to estimating the inviscid flux, are directly approximated by using local least-squares curve method. An explicit multistage Runge-Kutta algorithm is used to advance the flow equations in pseudo time. In order to accelerate the solution to convergence, local time stepping technique and residual averaging are employed. The results of NACA0012 airfoil in transonic steady flow are presented to verify the accuracy of the present spatial discretization method. Finally, two AGARD standard test cases in which NACA0012 airfoil and NACA64A010 airfoil oscillate in transonic flow are simulated. The computational results are compared with the experimental data to demonstrate the validity and practicality of the presented method.

An investigation of separation near corner points in transonic flow

The incipient separation from a corner in steady two-dimensional transonic flow is studied based on viscous-inviscid interaction at high Reynolds number. Of particular interest is the investigation of the dependence of the critical deflection angle (when a well-attached flow turns into a separated flow) on the Karman-Guderley parameter which characterizes the local flow field. In accordance with the procedure adopted, the analysis of the flow starts with the analysis of the boundary layer and then the solution of the Karman-Guderley equation describing the inviscid part of the flow near the corner point is investigated. The analysis of the inviscid transonic flow is performed based on the hodograph method and new solutions are obtained corresponding to the present flow topologies. In these solutions, the transonic flow appears to be subsonic everywhere except at the sonic corner point. Then, the interaction problem is formulated using the triple-deck model. Lastly, a procedure based on a semi-direct solution of the governing equations using Newton iterations is developed to obtain the numerical solution of the interaction problem

Experimental Investigation of the Periodic Unsteady Transonic Flow Field Around a Compressor Blade by Means of Particle Image Velocimetry (PIV)

In the present article, a study of the time‐dependent transonic flow field around an isolated airfoil in a Laval nozzle is presented. In order to obtain instantaneous flow field information all over the flow around the blade, the Particle Image Velocimetry (PIV) method has been employed allowing one to obtain velocity data in an illuminated light sheet with a high data acquisition rate and high spatial resolution. From these instantaneous PIV measurements the mean velocity field and turbulence quantities of the flow easily can be obtained by statistical means.A single compressor blade with a NACA 3506 profile and a chord length of 78 mm is centrally mounted in the test section. A vibration system driven by a hydraulic motor is externally attached to the blade. The rotational motion of the motor is translated to a plunging blade vibration perpendicular to the chord via an eccentric shaft and connecting arm. The vibration frequency can be varied between 0 and 100 Hz, and the average amplitude is approximately 0.4 mm. The entire blade‐motor assembly is mounted to the tunnel sidewall via a circular disc to allow easy adjustment of the inlet flow incidence angle. The exit pressure of the nozzle can be varied by a rotating flat plate (15 mm high, 39 mm wide, and 2 mm thick) that is mounted approximately 5 times the chord length downstream of the blade mid‐chord. The rotational frequency of this assembly can be varied between 0 and 100 Hz, producing upstream‐running waves at the doubled frequency (i.e., 0 to 200 Hz). The above excitation systems can be precisely synchronized and the phase lag between them can be freely varied. This allows for unsteady measurements to be conducted in the presence of only the downstream perturbation, only the blade vibration, or a combination of the two for different phase angles.For the first test series the existing PIV measurement system at Turbomachinery Institute (LTT) has been applied to measure the transonic steady flow field around the compressor blade. In another test series, measurements of the time‐dependent periodic flow field were conducted by means of PIV. This measurement series consisted of three subtasks: the measurement of the unsteady flow field with only back pressure variation, with only blade vibration, and finally with a combination of both back pressure variation and blade vibration. These results quantify not only the unsteady motion of the normal shock for each case on the suction side of the blade, but also give a detailed insight into the instantaneous transonic flow field around the compressor blade. Furthermore, the results of the set of unsteady measurements with both back pressure variation and blade vibration (3) is compared with the combination of the results of the sets of unsteady measurements of only back pressure variation (1) and only blade vibration (2) to validate the superposition principle.Finally, phase averaging of the instantaneous flowquantities produced a large database for statistical treatment (e.g., turbulence) and thus the ability to compare the averaged results with those from traditional measurement techniques.

Nonlinear Hyperbolic Waves in Multi-Dimensions. Monographs and Surveys in Pure and Applied Mathematics, Vol. 121.

7R10. Nonlinear Hyperbolic Waves in Multi-Dimensions.Monographs and Surveys in Pure and Applied Mathematics, Vol 121.- P Prasad (Dept of Math, Indian Inst of Sci, Bangalore, India). Chapman and Hall/CRC, Boca Raton FL. 2001. 338 pp. ISBN 1-58488-072-4. $89.95.Reviewed by Shi Tsan Wu (Dept of Mech and Aerospace Eng, Univ of Alabama, Sparkman Dr, Huntsville AL 35899).In this book, the authors have discussed the propagation of a curved nonlinear wavefront for the ordinary fluid dynamics (ie without considering magneto-fluids) according to two fundamental physical processes: i) at different points of the wavefront, it travels with different speeds depending on the local amplitude leading to a longitudinal stretching of rays; and ii) a lateral deviation of rays is produced due to non-uniform distribution of the amplitude of the wave front. This book consists of ten chapters. The first three chapters introduce all necessary mathematical concepts and some basic results. For example, the fundamentals of nonlinear hyperbolic waves are given in Chapter 1. In Chapter 2, the basic results of hyperbolic systems are presented. These results include i) discussion of hyperbolic system of first order equations in two independent variables with a canonical form of a system of linear and semi-linear equations, and ii) the hyperbolic system in more than two variables together with propagation of discontinuities of first order derivatives along rays is given. In Chapter 3, the results for simple wave high frequency approximation and ray theory are given. Specifically, the Huygen’s method of wavefront construction, Huygen’s method and ray theory, and Fermat’s principle are discussed in detail. The next seven chapters deal with specifics of the wave propagation of nonlinear hyperbolic systems; the derivation of weakly nonlinear ray theory (WNLRT) is given in Chapter 4. The stability of solutions for one-dimensional weakly nonlinear wave propagation as well as waves in a multi-dimensional steady transonic flow near a sonic singularity are discussed in Chapter 5. In Chapter 6, a special case of WNLRT for polytropic gas was presented in detail for two spatial dimensions. In Chapter 7, the authors presented a single conservation law approach to derive the compatibility conditions on a shock which is claimed as a New Theory of Shock Dynamics (NTSC). To establish this NTSC, the authors present a proof of the existence and uniqueness of the solution and made comparison of the numerical results deduced from NTSD with the exact solution which shows good agreement. Then, this NTSD is applied to a one-dimensional piston problem to illustrate its capabilities in Chapter 8. In Chapter 9, the authors presented a derivation of the compatibility conditions on a shock in multi-dimensions using shock ray theory. The final Chapter 10 includes the application of NTSD to the propagation of a curved weak shock. Overall, the book has three parts: i) The first three chapters have given all the necessary fundamental mathematical concepts, models and methods to calculate complete evolutionary history of a curved nonlinear wave front consisting of shock front. ii) The detailed discussions of WNLRT and NTSD are presented in Chapters 4-8; and iii) the specific applications of shock ray theory and NTSD are given in Chapters 9 and 10, respectively. In summary, Nonlinear Hyperbolic Waves in Multi-Dimensions includes numerical examples to illustrate theory, the quality of the graphic is good and an adequate subject index is provided. The author has achieved his goal as a textbook for advanced undergraduate applied mathematics. However, it could be used as a textbook for first year graduate students in engineering and physics if the author had presented some problems at the end of each chapter. This should be able to be remedied by the instructor. This reviewer strongly recommends that scientists and engineers who are interested in these topics read this book; it would be a useful reference book.

Theoretical and Numerical Studies of Transonic Flow of Moist Air Around a Thin Airfoil

Numerical studies of a two-dimensional and steady transonic flow of moist air around a thin airfoil with condensation are presented. The computations are guided by a recent transonic small-disturbance (TSD) theory of Rusak and Lee (2000) on this topic. The asymptotic model provides a simplified framework to investigate the changes in the flow field caused by the heat addition from a nonequilibrium process of condensation of water vapor in the air by homogeneous nucleation. An iterative method which is based on a type-sensitive difference scheme is applied to solve the governing equations. The results demonstrate the similarity rules for transonic flow of moist air and the effects of energy supply by condensation on the flow behavior. They provide a method to formulate various cases with different flow properties that have a sufficiently close behavior and that can be used in future computations, experiments, and design of flow systems operating with moist air. Also, the computations show that the TSD solutions of moist air flows represent the essence of the flow character computed from the inviscid fluid flow equations.

Variational Barrier Method of Adaptive Grid Generation in Hyperbolic Problems of Gas Dynamics

Application of the harmonic mapping using a variational approach to generate moving adaptive grids in the hyperbolic problems of gas dynamics is considered. Using the example of a three-point model of adaptation, the possibility to generate an unfolded mesh with strong grid lines condensing in the vicinity of discontinuities of the control/(monitor) function is demonstrated. The algorithm of redistributing the boundary nodes is suggested and consists of using constrained minimization of the discrete harmonic functional when constraints define the boundary of the domain. In real computations due to mesh adaptation it is possible to reduce the errors, caused by shock waves smearing over the cells, by many factors of ten. Modeling of the two-dimensional (2-D) supersonic gas flow in the channel has shown that the same accuracy on the adaptive grid with the same structure as the quasi-uniform mesh can be acheived while requiring less CPU memory by a factor of 25 and less running time by a factor of 50 to 60. Computational tests of the steady transonic and supersonic flow over an airfoil demonstrate the ability of the method to control mesh sizes across shocks.

Toward the Ultimate Conservative Scheme: Following the Quest

The aim of this paper is to develop a class of numerical schemes that work on triangular finite element type meshes, and which are devoted to the computation of steady transonic flows. The schemes are extensions of the positive streamwise invariant scheme of Struijs and are built directly on the system of the Euler equation for fluid mechanics. They are a blending between a first-order and a second-order scheme, which is realized from entropy considerations. It is formally second-order accurate at steady state. Several numerical examples are shown to demonstrate the stability and accuracy of these schemes.