Transonic Shock Research Articles

Stabilization Effect of Frictions for Transonic Shocks in Steady Compressible Euler Flows Passing Three-Dimensional Ducts

Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets. For the three-dimensional steady non-isentropic compressible Euler system with frictions, we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections. In this article, we are devoted to proving rigorously that a large class of these transonic shock solutions are stable, under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces. This manifests that frictions have a stabilization effect on transonic shocks in ducts, in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts. Except its implications to applications, because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system, we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.

On Stability of Transonic Shocks for Stationary Rayleigh Flows in Two-dimensional Ducts

Transonic shocks play a pivotal role in design of supersonic nozzles in gas dynamics. Previous studies have shown that for stationary compressible Euler flows in rectilinear ducts with constant cro...

Subsonic solutions to a shock diffraction problem by a convex cornered wedge for the pressure gradient system

We establish the global existence of subsonic solutions to a two dimensional Riemann problem governed by a self-similar pressure gradient system for shock diffraction by a convex cornered wedge. Since the boundary of the subsonic region consists of a transonic shock and a part of a sonic circle, the governing equation becomes a free boundary problem for nonlinear degenerate elliptic equation of second order with a degenerate oblique derivative boundary condition. We also obtain the optimal \begin{document}$ C^{0,1} $\end{document} -regularity of the solutions across the degenerate sonic boundary.

Analysis of a civil aircraft wing transonic shock buffet experiment

The physical mechanism governing the onset of transonic shock buffet on swept wings remains elusive, with no unequivocal description forthcoming despite over half a century of research. This paper elucidates the fundamental flow physics on a civil aircraft wing using an extensive experimental database from a transonic wind tunnel facility. The analysis covers a wide range of flow conditions at a Reynolds number of around . Data at pre-buffet conditions and beyond onset are assessed for Mach numbers between 0.70 and 0.84. Critically, unsteady surface pressure data of high spatial and temporal resolution acquired by dynamic pressure-sensitive paint is analysed, in addition to conventional data from pressure transducers and a root strain gauge. We identify two distinct phenomena in shock buffet conditions. First, we highlight a low-frequency shock unsteadiness for Strouhal numbers between 0.05 and 0.15, based on mean aerodynamic chord and reference free stream velocity. This has a characteristic wavelength of approximately 0.8 semi-span lengths (equivalent to three mean aerodynamic chords). Such shock unsteadiness is already observed at low-incidence conditions, below the buffet onset defined by traditional indicators. This has the effect of propagating disturbances predominantly in the inboard direction, depending on localised separation, with a dimensionless convection speed of approximately 0.26 for a Strouhal number of 0.09. Second, we describe a broadband higher-frequency behaviour for Strouhal numbers between 0.2 and 0.5 with a wavelength of 0.2 to 0.3 semi-span lengths (0.6 to 1.2 mean aerodynamic chords). This outboard propagation is confined to the tip region, similar to previously reported buffet cells believed to constitute the shock buffet instability on conventional swept wings. Interestingly, a dimensionless outboard convection speed of approximately 0.26, coinciding with the low-frequency shock unsteadiness, is found to be nearly independent of frequency. We characterise these coexisting phenomena by use of signal processing tools and modal analysis of the dynamic pressure-sensitive paint data, specifically proper orthogonal and dynamic mode decomposition. The results are scrutinised within the context of a broader research effort, including numerical simulation, and viewed alongside other experiments. We anticipate our findings will help to clarify experimental and numerical observations in edge-of-the-envelope conditions and to ultimately inform buffet-control strategies.

Non-isentropic multi-transonic solutions of Euler-Poisson system

In this paper, we consider the boundary value problem for the one-dimensional non-isentropic Euler-Poisson system. The physical subsonic inflow and outflow conditions are prescribed at boundaries. We prove the existence of multi-transonic solutions, i.e., the existence of subsonic-sonic-supersonic-shock-subsonic flows in fact nozzles. Furthermore, the monotonicity between the location of the transonic shock and the density at the exit of the nozzle will be elucidated.

Modal Analysis of Transonic Shock Buffet on 2D Airfoil

This paper presents a study of the shock buffet phenomenon on the RA16SC1 supercritical airfoil using the proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) modal approaches. The computational fluid dynamics simulation is validated, and the full buffet envelope of the RA16SC1 airfoil is mapped. POD and DMD techniques are then applied to compute a set of flow modes and corresponding dynamical data (for the DMD method). A dominant buffet mode is identified at the buffet frequency. The corresponding mode shape represents the interaction between the shock wave and boundary-layer. Higher modes, at multiples of the buffet frequency, show smaller coherent flow structures at the shock travel area. Further higher modes, of mixed frequencies, are associated with flow structures mainly in the boundary-layer area. Only a few modes are required for accurate reconstruction of the buffet flow field. Finally, it is shown that similar modes are present in all buffet angles of attack. The modal approach used in this study enables straightforward extraction of the underlying flow structures and dynamic properties of buffet flows.

Wavenumber-Frequency Spectra on a Launch Vehicle Model Measured via Unsteady Pressure-Sensitive Paint

Time histories of pressure fluctuations on a generic, “hammerhead” launch vehicle model were measured using unsteady pressure-sensitive paint over a Mach number range of . The calibration process, using a set of unsteady pressure sensors, was found to overcome some of the inherent problems of the fast-response paint, such as rapid photodegradation, nonlinearity in pressure response, and significant temperature sensitivity. The large data set exposed various critical transonic flow physics such as a coupling of the shock motion on the payload fairing (PF) with the separated flow region downstream, and an upstream convection of pressure fluctuations on the PF at certain Mach numbers. Area-weighted integration of the pressure fluctuation data provided a measure of the unsteady forces on various parts of the model. The unsteady force coefficients on the PF and the upper stage were the highest when transonic shocks were present and found to reduce significantly when supersonic flow set in. Time histories on a dense, regularly spaced, spatial grid points allowed for the calculation of wavenumber–frequency () spectra via Fourier transform. The spectra were dominated by the convected fluctuations; the acoustic domain was not discernible. These data, valuable for the buffet and vibroacoustics analysis of aerospace vehicles, are believed to be the first obtained for the transonic flight regime.

Graphene-Like Nanoflakes for Shock Absorption Applications

The effects of high-pressure shock waves generated by the detonation of explosives are of major interest to the strategic sector. We report interaction of transonic shock waves (1.1 Mach speed; pea...

Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations

We construct semi-hyperbolic patches of solutions, in which one family out of two families of wave characteristics start on sonic curves and end on transonic shock waves, to the two-dimensional (2D) compressible magnetohydrodynamic (MHD) equations. This type of flow patches appear frequently in transonic flow problems. In order to use the method of characteristic decomposition to construct such a flow patch, we also derive a group of characteristic decompositions for 2D self-similar MHD equations.

Experimental FSI study of adaptive shock control bumps

The shock stabilisation and wave drag reduction potential of a two-dimensional adaptive shock control bump has been studied in the Imperial College supersonic wind tunnel. The bump was modelled as a flexible aluminium alloy plate deformed through spanwise actuation, and several bump heights were tested beneath a Mach 1.4 transonic shock wave. Schlieren images and static pressure readings along the flexible plate allowed the study of the λ-shock structure generated by the bifurcation of the normal shock for a range of shock positions. All bumps tested were found to increase shock stability, but wave drag reduction was only observed for shocks close to the leading edge of the flexible plate. Positive deformations of the flexible plate for downstream shocks are believed to reduce supersonic flow reacceleration, and hence the strength of the rear leg of the λ-shock and wave drag, in comparison to a solid bump with the same shape. The position of the rear leg of the λ-shock was found to exhibit a bistable behaviour, and this is hypothesised to be caused by a complex coupling of aerodynamic and structural instabilities.

Pressure-based algorithm for compressible interfacial flows with acoustically-conservative interface discretisation

A pressure-based algorithm for the simulation of compressible interfacial flows is presented. The algorithm is based on a fully-coupled finite-volume framework for unstructured meshes with collocated variable arrangement, in which the governing conservation laws are discretised in conservative form and solved in a single linear system of equations for velocity, pressure and specific total enthalpy, with the density evaluated by an equation of state. The bulk phases are distinguished using the Volume-of-Fluid (VOF) method and the motion of the fluid interface is captured by a state-of-the-art compressive VOF method. A new interface discretisation method is proposed, derived from an analogy with a contact discontinuity, that performs local changes to the discrete values of density and total enthalpy based on the assumption of thermodynamic equilibrium, and does not require a Riemann solver. This interface discretisation method yields a consistent definition of the fluid properties in the interface region, including a unique definition of the speed of sound and the Rankine–Hugoniot relations, and conserves the acoustic features of the flow, i.e. compression and expansion waves. A variety of representative test cases of gas–gas and gas–liquid flows, ranging from acoustic waves and shock tubes to shock-interface interactions in one-, two- and three-dimensional domains, is used to demonstrate the capabilities and versatility of the presented algorithm in all Mach number regimes. The propagation, reflection and transmission of acoustic waves, shock waves and rarefaction fans in interfacial flows are predicted accurately, even for difficult cases that feature fluids with shock impedance matching, transonic shock tubes or strong shocks in gas–liquid flows, as well as on unstructured meshes.

Development and convergence analysis of an effective and robust implicit Euler solver for 3D unstructured grids

This paper reports the development and convergence analysis in steady-state of an effective and robust implicit finite-volume solver for compressible Euler equations on three-dimensional unstructured grids. A second-order upwind scheme (MUSCL) was employed based on Roe's approximate Flux-Difference Scheme (FDS) by using Venkatrakishnan flux limiters. The construction of the linear system for the implicit scheme was performed by applying the backward Euler on the left-hand side of the conservation equation and Newton-type linearization on the right-hand side. The Jacobian matrix that resulted from the linearization process was computed analytically using Roe flux terms. In this phase, the defect-correction technique was employed allowing effective time-dependent computations by an implicit time-integration scheme. In this approach, the flux integral on the right-hand side is computed based on a high-order of accuracy whilst the left-hand side the Jacobian is performed based on the low-order. The resulting sparse and large system of linear equations is solved by a sequential Gauss–Seidel iterative method. Simulations were performed and the developed implicit defect-correction solver was validated and verified. In addition, convergence analysis comparing the implicit solver and the explicit Runge–Kutta of 5-steps using Implicit Residual Smoothing (IRS) were performed showing the significant speed-up of the implicit solver over the explicit one. Simulations were performed for case studies to demonstrate the robustness of the developed implicit defect-correction solver in solving typical problems of aerodynamic involving transonic condition and shock wave captures for internal and external flows. Finally, the main particularities of the implicit scheme were investigated and discussed considering the simulation results, showing also its capacity to serve as an effective preconditioner (start-up method) to other implicit techniques.

Influence of Mach number and angle of attack on the two-dimensional transonic buffet phenomenon

Within a narrow band of flight conditions in the transonic regime, self-sustained shock oscillations that involve the interaction between shock-waves and intermittently separated shear layers may develop. This phenomenon, known as transonic shock buffet, limits the flight envelope and is detrimental to both aircraft handling quality and structural integrity. In this investigation, numerical simulation of transonic shock buffet over the OAT15A aerofoil is performed to explore the buffet envelope. Unsteady Reynolds-Averaged Navier–Stokes simulations are validated against available experimental data to ascertain the most effective combination of simulation parameters to reproduce autonomous shock oscillations. From the baseline test case, the influence of Mach number and angle of attack on the nature of the buffet response is investigated. Radial Basis Function surrogate models are developed to represent the variation of buffet amplitude and frequency with flight condition. While the frequency is found to increase monotonically with both parameters, variation in buffet amplitude through the region of shock unsteadiness is more complex, particularly at high angles of attack. This is related to a bifurcation in the behaviour of the shock. As incidence increases from onset, the shock dynamics transition from periodic oscillations over the suction surface to quasi-periodic motions, whereby the shock is propelled forward into the oncoming flow during its upstream excursion.

Unsteady pressure fluctuations measured on a hammerhead space vehicle and comparison with Coe and Nute's 1962 data

A detailed survey of pressure fluctuations was conducted on a replica of the generic, “hammerhead” shaped space vehicle tested more than half century ago by Coe and Nute (1962). A large number of current and past space vehicles follow this general hammerhead configuration. The present test was conducted in the 11-by-11 ft Transonic Wind Tunnel of NASA Ames Research Center over the critical transonic Mach number range of 0.6 ≤ M ≤ 1.2 where typical vehicles encounter very high levels of surface pressure fluctuations during ascent through the atmosphere. Out of the many different types of instrumentations used for this test, data from the dynamic pressure sensors are presented in this paper. In addition, shadowgraph flow visualizations, and a qualitative measure of turbulent fluctuations, estimated from high-speed photography, are also presented. The data set provided a clear insight into the local high level of fluctuations from the formation of the transonic shocks on the payload fairing and also from the impingement of the separated shear layer on the second stage. Space–time correlation showed interesting physics of upstream propagation of pressure fluctuations on the payload fairing when local shock waves were present. A comparison of the magnitude of pressure fluctuations between the present test and that done earlier by Coe and Nute showed good agreement when the limited frequency bandwidth of equipment used in the latter was accounted for. Similarly, when accounting for the differences in the free-stream dynamic pressure, the spectra of surface pressure fluctuations were in reasonable agreement between the present and Coe–Nute tests. The present test provides a far more complete picture of pressure fluctuations due to the usage of a large number of pressure transducers (216 versus 30), wider frequency bandwidth (50 kHz versus 1 kHz), and unsteady pressure-sensitive paint.

Stability and asymptotic behavior of transonic flows past wedges for the full Euler equations

The existence, uniqueness, and asymptotic behavior of steady transonic flows past a curved wedge, involving transonic shocks, governed by the two-dimensional full Euler equations are established. The stability of both weak and strong transonic shocks under the perturbation of the upstream supersonic flow and the wedge boundary is proved. The problem is formulated as a one-phase free boundary problem, in which the transonic shock is treated as a free boundary. The full Euler equations are decomposed into two algebraic equations and a first-order elliptic system of two equations in Lagrangian coordinates. With careful elliptic estimates by using appropriate weighted Hölder norms, the iteration map is defined and analyzed, and the existence of its fixed point is established by performing the Schauder fixed point argument. The careful analysis of the asymptotic behavior of the solutions reveals particular characters of the full Euler equations.

A Variational Inequality Formulation for Transonic Compressible Steady Potential Flows: Radially Symmetric Transonic Shock

We establish a variational inequality formulation that captures the transonic shock for a steady compressible potential flow. Its critical point satisfies the transonic equation; moreover the associated jump conditions across its free boundary match the usual Rankine--Hugoniot jump conditions for a shock. By means of example we validate our formulation and establish the necessary and sufficient condition for the existence of a transonic shock. Numerical results are also discussed.

Stability of transonic shocks in steady supersonic flow past multidimensional wedges

We are concerned with the stability of multidimensional (M-D) transonic shocks in steady supersonic flow past multidimensional wedges. One of our motivations is that the global stability issue for the M-D case is much more sensitive than that for the 2-D case, which requires more careful rigorous mathematical analysis. In this paper, we develop a nonlinear approach and employ it to establish the stability of weak shock solutions containing a transonic shock-front for potential flow with respect to the M-D perturbation of the wedge boundary in appropriate function spaces. To achieve this, we first formulate the stability problem as a free boundary problem for nonlinear elliptic equations. Then we introduce the partial hodograph transformation to reduce the free boundary problem into a fixed boundary value problem near a background solution with fully nonlinear boundary conditions for second-order nonlinear elliptic equations in an unbounded domain. To solve this reduced problem, we linearize the nonlinear problem on the background shock solution and then, after solving this linearized elliptic problem, develop a nonlinear iteration scheme that is proved to be contractive.

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge, governed by the Euler equations, there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle—the steady weak shock with supersonic or subsonic downstream flow (determined by the wedge angle that is less than or greater than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which satisfy the entropy condition. The fundamental issue—whether one or both of the steady weak and strong shocks are physically admissible solutions—has been vigorously debated over the past eight decades. In this paper, we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes. For the static stability, we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small, and we finally present some recent results on the static stability of the steady supersonic and transonic shocks. For the dynamic stability for potential flow, we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem, and we finally survey some recent developments in solving this free boundary problem for the existence of the Prandtl-Meyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity. Some further developments and mathematical challenges in this direction are also discussed.

Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions

We analyze rarefaction wave interactions of self-similar transonic irrotational flow in gas dynamics for the two dimensional Riemann problems. We establish the existence result of the supersonic solution to the prototype nonlinear wave system for the sectorial Riemann data, and study the formation of the sonic boundary and the transonic shock. The transition from the sonic boundary to the shock boundary inherits at least two types of degeneracies (1) the system is sonic, and in addition (2) the angular derivative of the solution becomes zero where the sonic and shock boundaries meet.

Delayed Detached–Eddy Simulation of Shock Buffet on Half Wing–Body Configuration

This paper presents a numerical study of the transonic flow over a half wing–body configuration representative of a large civil aircraft. The Mach number is close to cruise conditions, whereas the high angle of attack causes strong separation on the suction side of the wing. Results indicate the presence of shock-wave oscillations inducing unsteady loads that can cause serious damage to the aircraft. Transonic shock buffet is found. Based on exploratory simulations using a baseline grid, the region relevant to the phenomenon is identified and mesh adaptation is applied to significantly refine the grid locally. Time-accurate Reynolds-averaged Navier–Stokes and delayed detached–eddy simulations are then performed on the adapted grid. Both types of simulation reproduce the unsteady flow physics, and much information can be extracted from the results when investigating frequency content, the location of unsteadiness, and its amplitude. Differences and similarities in the computational results are discussed in detail and are also analyzed phenomenally with respect to recent experimental data.