Transonic Shock Research Articles

Multidimensional transonic shock waves and free boundary problems

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.

Analysis on the asymptotic behavior of transonic shocks for steady Euler flows with gravity in a flat nozzle as the gravity vanishes

This paper concerns the asymptotic behavior of transonic shocks for steady Euler flows with gravity in a flat nozzle as the gravity vanishes. One of the key points is to quantitatively determine the position of the shock front as the gravity is small. To this end, a free boundary problem for the linearized Euler system with small gravity will be proposed. Then one can obtain the information of the approximating position of the shock front. Once the initial approximation of the shock solution is obtained, a nonlinear iteration scheme can be constructed to show that the position of the shock front is a small perturbation of the initial approximation, which establishes a quantitative expression for the shock solution. Such an expression shows the asymptotic behavior as the gravity goes to zero.

Active control of a 3-D flow separation induced by a transonic shock

Active control of a 3-D flow separation induced by a transonic shock

Stability of Attached Transonic Shocks in Steady Potential Flow past Three-Dimensional Wedges

We develop a new approach and employ it to establish the global existence and nonlinear structural stability of attached weak transonic shocks in steady potential flow past three-dimensional wedges; in particular, the restriction that the perturbation is away from the wedge edge in the previous results is removed. One of the key ingredients is to identify a "good" direction of the boundary operator of a boundary condition of the shock along the wedge edge, based on the non-obliqueness of the boundary condition for the weak shock on the edge. With the identification of this direction, an additional boundary condition on the wedge edge can be assigned to make sure that the shock is attached on the edge and linearly stable under small perturbation. Based on the linear stability, we introduce an iteration scheme and prove that there exists a unique fixed point of the iteration scheme, which leads to the global existence and nonlinear structural stability of the attached weak transonic shock. This approach is based on neither the hodograph transformation nor the spectrum analysis, and should be useful for other problems with similar difficulties.

Multi-objective aerodynamic optimization of flying-wing configuration based on adjoint method

In order to solve the multi-objective multi-constraint design in aerodynamic design of flying wing, the aerodynamic optimization design based on the adjoint method is studied. In terms of the principle of the adjoint equation, the boundary conditions and the gradient equations are derived. The Navier-Stokes equations and adjoint aerodynamic optimization design method are adopted, the optimization design of the transonic drag reduction for the two different aspect ratio of the flying wing configurations is carried out. The results of the optimization design are as follows: Under the condition of satisfying the aerodynamic and geometric constraints, the transonic shock resistance of the flying wing is weakened to a great extent, which proves that the developed method has high optimization efficiency and good optimization effect in the multi-objective multi-constraint aerodynamic design of the flying wing.

Accretion-induced Black Hole Spin-up Revised by Numerical General Relativistic MHD

Abstract We investigate the accretion-induced spin-up of the black hole via numerical simulations. Our method is based on general relativistic magnetohydrodynamics of the slowly rotating flows in the Kerr metric, where possibly transonic shock fronts may form. We account for the changing black hole mass and spin during accretion that enforces dynamical evolution of the spacetime metric. We first study nonmagnetized flows with shocks, and we also include magnetic field endowed in the gas. The aim of this study is to verify whether the high-mass black holes may be produced with large spins, even though at birth the collapsars might have contained slowly or moderately spinning cores. In this way, we put constraints on the content of angular momentum in the collapsing massive stars. Our studies are also showing that shock fronts and magnetic fields may halt accretion and limit the black hole spin-up in the exploding supernovae.

On admissible positions of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle

This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution gives an initial approximating position of the shock front. Compared with the 2-D case, new difficulties arise due to the additional 0-order terms and singularities along the symmetric axis. New observations and careful analysis will be done to overcome these difficulties. Once the initial approximation is obtained, a nonlinear iteration scheme can be carried out, which converges to a transonic shock solution to the problem.

A locally implicit time-reversible sonic point processing algorithm for one-dimensional shallow-water equations

In this paper, we introduce a new locally implicit sonic point processing algorithm for one-dimensional shallow-water equations. The algorithm is based on transferring the shallow-water invariants along the characteristics around the sonic point, and it can be coupled with conservative-characteristic methods, which have problems modelling the transonic flows. For each sonic point, a system of two non-linear equations should be solved, which allows to use the proposed method with explicit methods for subsonic or supersonic flows without noticeable increase in calculation time. The important features of the algorithm include the second order of approximation in time and space and the property of time reversibility if monotonization procedures are disabled. Coupled with the CABARET scheme, the proposed method has been tested on several Riemann problems, including the Toro tests for transonic flows. The results have shown that the algorithm allows to obtain solutions with accurate transonic rarefaction and shock waves in both cases of wet and dry beds.

3-D Axisymmetric Transonic Shock Solutions of the Full Euler System in Divergent Nozzles

We establish the stability of 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles under small perturbations of an incoming radial supersonic flow and a constant pressure at the exit of the nozzles. To study 3-D axisymmetric transonic shock solutions of the full Euler system, we use a stream function formulation of the full Euler system for a 3-D axisymmetric flow. We resolve the singularity issue arising in stream function formulations of the full Euler system for a 3-D axisymmetric flow. We develop a new scheme to determine a shock location of a transonic shock solution of the steady full Euler system based on the stream function formulation.

Mitigation of transonic shock buffet on a supercritical airfoil through wavy leading edges

Mitigation of the shock buffet phenomenon over a supercritical airfoil by means of wavy leading edges (WLEs) is analyzed with implicit large eddy simulations. A Dassault Aviation's V2C airfoil is simulated in a transonic flow with Re∞=5.0×105 and M∞=0.7 at α=7.0°. This airfoil profile is designed for transonic flows, delaying the onset of wave drag and decreasing the skin friction drag. The results of this upstream flow condition on a straight airfoil are a large oscillation of lift and drag coefficients. In the first part, time-averaged aerodynamic characteristics over a straight leading edge (SLE) airfoil and three modified airfoils with different wavy amplitudes are compared. The results show that overall WLE airfoils are more efficient than SLE ones, and the airfoil with the lowest amplitude (h = 0.0075) is the most efficient, increasing the lift coefficient and decreasing the drag coefficient. Flow unsteadiness plays a key role for airfoils in transonic flows at moderate and high angles of attack. Hence, the second part of the paper is a detailed unsteady analysis of flow phenomena. The starting point is an investigation of unsteady aerodynamic performance. It is observed that WLE airfoils are capable of significantly decreasing low-frequency oscillations' amplitude, identified with shock buffet. The best performance is obtained with h = 0.0125 where a high-frequency oscillation becomes the dominant unsteady phenomenon. High-frequency oscillations are identified through the application of a frequency filtering method to the flow field. It is proved that the oscillation on the SLE airfoil is related to vortex shedding, while the one on WLE airfoils is related to laminar separation bubble (LSB) breathing. Overall, the unsteady analysis shows a connection between shock buffet and LSB breathing phenomena on wavy airfoils and identifies the WLE amplitude as a key parameter to control this relation and mitigate the shock buffet.

Leading Edge Shape Optimization of Whitcomb IL Airfoil in Transonic Flow Using a Multi-Objective Genetic Algorithm (MOGA)

Leading edge shape optimization of transonic airfoils requires creating an airfoil surface that reduces the drag divergence due to transonic shocks by either delaying them or reducing their strength at a given transonic cruise speed while maintaining the lift. Aircrafts like Boeing 757, Airbus A300, Boeing 777 and McDonnell Douglas AV-8B Harrier II use Whitcomb IL supercritical airfoil for efficient aerodynamic performance in transonic conditions. This study employs a multi-objective genetic algorithm for shape optimization of leading edge of the Whitcomb IL airfoil to achieve three objectives, namely dilation of shock, reduction in leading edge radius and increment of lift at a given transonic Mach number and at the given AoA. The commercially available software ICEM CFD and Fluent are employed for calculation of the flow field using the RANS (Reynolds-averaged Navier-Stokes equations) in conjunction with a two equation turbulence model. It is shown that the (MOGA) Multi-objective genetic algorithm can generate superior airfoil compared with Whitcomb IL airfoil by achieving three objectives. The optimized airfoil configuration is validated by wind tunnel testing facility.

Review of Adaptive Shock Control Systems

Drag reduction plays a major role in future aircraft design in order to lower emissions in aviation. In transonic flight, the transonic shock induces wave drag and thus increases the overall aircraft drag and hence emissions. In the past decades, shock control has been investigated intensively from an aerodynamic point of view and has proven its efficacy in terms of reducing wave drag. Furthermore, a number of concepts for shock control bumps (SCBs) that can adapt their position and height have been introduced. The implementation of adaptive SCBs requires a trade-off between aerodynamic benefits, system complexity and overall robustness. The challenge is to find a system with low complexity which still generates sufficient aerodynamic improvement to attain an overall system benefit. The objectives of this paper are to summarize adaptive concepts for shock control, and to evaluate and compare them in terms of their advantages and challenges of their system integrity so as to offer a basis for robust comparisons. The investigated concepts include different actuation systems as conventional spoiler actuators, shape memory alloys (SMAs) or pressurized elements. Near-term applications are seen for spoiler actuator concepts while highest controllability is identified for concepts several with smaller actuators such as SMAs.

Non-uniqueness of transonic shock solutions to non-isentropic Euler–Poisson system

Non-uniqueness of transonic shock solutions to non-isentropic Euler–Poisson system

Structural Stability of the Transonic Shock Problem in a Divergent Three-Dimensional Axisymmetric Perturbed Nozzle

In this paper, we prove the structural stability of the transonic shocks for three-dimensional axisymmetric Euler system with swirl velocity under the perturbations for the incoming supersonic flow, the nozzle boundary, and the exit pressure. Compared with the known results on the stability of transonic shocks, one of the major difficulties for the axisymmetric flows with swirls is that corner singularities near the intersection point of the shock surface and nozzle boundary and the artificial singularity near the axis appear simultaneously. One of the key points in the analysis for this paper is the introduction of an invertible modified Lagrangian transformation which can straighten the streamlines in the whole nozzle and help to represent the solutions of transport equations explicitly. Furthermore, the simple but useful modified Lagrangian transformation makes the treatment for the singularities near the axis easy and clean. Such a technique may be helpful in studies of other problems for axisymmetric flows.

Global solutions of a two-dimensional Riemann problem for the pressure gradient system

<p style="text-indent:20px;">We are concerned with a two-dimensional Riemann problem for the pressure gradient system that is a hyperbolic system of conservation laws. The Riemann initial data consist of four constant states in four sectorial regions such that two shocks and two vortex sheets are generated between the adjacent states. The solutions keep the four constant states and four planar waves outside the outer sonic circle in the self-similar coordinates, while the two shocks keep planar until meeting the outer sonic circle at two different points and then generate a diffracted shock to connect these points, whose location is <em>apriori</em> unknown. Then the problem can be formulated as a free boundary problem, in which the diffracted transonic shock is the one-phase free boundary to connect the two points, while the other part of the sonic circle forms a fixed boundary. We establish the global existence of a solution and the optimal Lipschitz regularity of both the diffracted shock across the two points and the solution across the outer sonic boundary. Then this Riemann problem is solved globally, whose solution contains two vortex sheets and one global shock containing the two originally separated shocks generated by the Riemann data.</p>

On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure

AbstractThis paper concerns the existence of transonic shock solutions to the 2‐D steady compressible Euler system in an almost flat finite nozzle (in the sense that it is a generic small perturbation of a flat one), under physical boundary conditions proposed by Courant‐Friedrichs in [14], in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock front is some of the most desirable information one would like to determine. However, the location of the normal shock front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. Once an initial approximation is obtained, a further nonlinear iteration can be constructed and proved to lead to a transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived that yields the existence of the solutions to the proposed free boundary problem. By this condition, it will be shown that, for the proposed linear free boundary problem, if the nozzle is either strictly expanding or contracting, there exists only one solution as long as the given pressure at the exit lies within a certain interval; while if the nozzle is generic with both expanding and contracting portions, there may exist more than one solution for the same given receiver pressure, which implies the existence of more than one initial approximating shock front. This nonuniqueness of the initial approximations will lead to the nonuniqueness of the transonic shock solutions for generic nozzles with the same given receiver pressure, which shows the instability of the unperturbed normal shock solution under general perturbations of the nozzle boundary and how this instability may behave. © 2020 Wiley Periodicals LLC

Origins of atypical shock buffet motions on a supercritical aerofoil

To expand the allowable flight regime of contemporary aircraft, a fundamental understanding of edge-of-the-envelope flow phenomena, such as transonic shock buffet, is necessary. In this study, Unsteady Reynolds-Averaged Navier-Stokes simulations are employed to probe the evolution of shock motions throughout the buffet envelope. At deep-buffet freestream conditions, atypical shock motions are shown to develop, in which an alternating strong/weak shock formation is evident with significant upstream pressure wave propagation. The transition between typical, single-frequency shock oscillations and these high-periodicity aerodynamic responses with increasing incidence is found to exhibit characteristics of a period-doubling route to chaos. Analysis of these flows through Higher-Order Dynamic Mode Decomposition reveals the exotic buffet dynamics emerge from the interplay between a dominant oscillatory buffet mode and its related even sub- and superharmonics, which appear with equivalent modal energies at high-incidence conditions. Correspondingly, the tendency for upstream pressure wave propagation is found to be correlated to a reversal in the suction surface leading edge phase gradient of the dominant oscillatory modes, which also develops with increasing angle of attack.

Surface Interactions of Transonic Shock Waves with Graphene-Like Nanoribbons

Graphene-like nanoribbons (GLNRs) were fabricated (length—20 μm; width—2 μm) and subjected to blast-like pulsed pressure >1.5 GPa (pulse speed ≈1 Mach, impulse duration, ≈µs) to examine the amount of absorption. GLNRs prepared by the chemical vapor deposition technique via controlled biomass combustion were subjected to investigate the structure–property characteristics using microspectroscopic techniques. Following this, GLNRs were employed to high strain rate (HSR) studies with the help of the technique known as split Hopkinson pressure bar (SHPB) to evaluate numerous dynamic parameters. The parameters were extracted from variations in the stress and strain rates. Their analysis provided insight into the damping response of blast energy within GLNRs. By and large, the impact generated modified the microstructure, exhibiting modifications in the number of layers, conjugated loops, and dynamic disorder. Signal processing analysis carried out for incident and transmitted impulse pressure revealed an interaction mechanism of shock wave with GLNR. Details are presented.

Numerical Investigation for Separation Characteristics of Transonic Shock Oscillations Based on Parallel Computing

Shock-induced oscillation is transonic flow instability phenomenon, which brings additional vibration to the aircraft. In this study, the RANS-LES methods with parallel computing were applied to investigate the transonic shock oscillations over a supercritical airfoil. The third-order MUSCL and fifth-order WENO based Roe scheme with adjustable dissipation were applied to study the effects of numerical dissipation. The results shown the numerical dissipation of the spatial schemes has a great influence on the calculation, and the small scale flow structure are suppressed by excessive numerical dissipation. The numerical results present the separation evolution on the upper surface. And periodic variation characteristics of the separation region downstream of the shock wave was discussed. The ratio of the time taken by the shock moves downstream and upstream is 1:1.

Convexity of Self-Similar Transonic Shocks and Free Boundaries for the Euler Equations for Potential Flow

We are concerned with geometric properties of transonic shocks as free boundaries in two-dimensional self-similar coordinates for compressible fluid flows, which are not only important for the understanding of geometric structure and stability of fluid motions in continuum mechanics but also fundamental in the mathematical theory of multidimensional conservation laws. A transonic shock for the Euler equations for self-similar potential flow separates elliptic (subsonic) and hyperbolic (supersonic) phases of the self-similar solution of the corresponding nonlinear partial differential equation in a domain under consideration, in which the location of the transonic shock is apriori unknown. We first develop a general framework under which self-similar transonic shocks, as free boundaries, are proved to be uniformly convex, and then apply this framework to prove the uniform convexity of transonic shocks in the two longstanding fundamental shock problems -- the shock reflection-diffraction by wedges and the Prandtl-Meyer reflection for supersonic flows past solid ramps. To achieve this, our approach is to exploit underlying nonlocal properties of the solution and the free boundary for the potential flow equation.