# Analytical and numerical study of normal shock response in a uniform duct

TL;DR: In this article, the response of an inviscid shock to external pressure perturbations in a constant area duct is analyzed in terms of fundamental processes like perturbation propagation and its interaction with shock.

Abstract: The response of an inviscid shock to external pressure perturbations in a constant area duct is analyzed in terms of fundamental processes like perturbation propagation and its interaction with shock. The results of these elementary processes are formulated analytically and with a Riemann wave tracking method to enable the prediction of shock movement for both upstream and downstream perturbations. The predictions thus obtained are compared with the finite-volume based numerical simulations of the Euler equations. This study shows that the shock responds nonlinearly to perturbations and the nonlinearity has a cumulative effect. Contact surfaces generated during the interaction of normal shock with perturbations, which was ignored in previous investigations, are shown to be important in order to capture this cumulative nonlinearity. The nonlinearity alters the positive and negative duty cycles, which results in a net displacement of shock after responding to one full cycle of sinusoidal perturbation. This drift in shock location is pronounced at low supersonic Mach numbers (1.2–3) but is also present at higher Mach numbers. Furthermore, it is demonstrated that the duty cycle variations are higher for perturbations originating downstream of shock than those originating upstream. The variations in frequency and amplitude are found to merely scale the response and do not introduce any new physics.The response of an inviscid shock to external pressure perturbations in a constant area duct is analyzed in terms of fundamental processes like perturbation propagation and its interaction with shock. The results of these elementary processes are formulated analytically and with a Riemann wave tracking method to enable the prediction of shock movement for both upstream and downstream perturbations. The predictions thus obtained are compared with the finite-volume based numerical simulations of the Euler equations. This study shows that the shock responds nonlinearly to perturbations and the nonlinearity has a cumulative effect. Contact surfaces generated during the interaction of normal shock with perturbations, which was ignored in previous investigations, are shown to be important in order to capture this cumulative nonlinearity. The nonlinearity alters the positive and negative duty cycles, which results in a net displacement of shock after responding to one full cycle of sinusoidal perturbation. This ...

##### Citations

More filters

••

[...]

TL;DR: In this article, high-speed schlieren imaging and high-resonance frequency pressure measurements were used to capture the flow features during the shock train movement, and the analysis was extended to complex situations with incident shocks.

Abstract: The oscillation characteristics of the shock train in an isolator have been investigated in a direct-connect wind tunnel at Mach 2.7. High-speed schlieren imaging and high-resonance frequency pressure measurements were used to capture the flow features during the shock train movement. The oscillation features without the effects of incident shocks were analyzed first. As the shock train moved upstream, the low-frequency part of the oscillation was found to develop. The analysis was then extended to complex situations with incident shocks. It was revealed that the shock wave-boundary layer interactions considerably influence the shock train behavior. The interactions were classified into three patterns: (I) single interaction, (II) multi-interactions on the same side, and (III) multi-interactions on different sides. Experimental results indicated that the oscillation could be affected in temporal scale by pattern II and enhanced in spatial scale by pattern III. The data also showed that the pressure rise induced upstream propagates to the exit, causing phase offsets in the wall pressure histories and making the pressure distributions diverge from their stable state. This phenomenon suggested a possible physical mechanism for the oscillation during shock train movement, which was verified by additional tests with large backpressure rising rate. It was found that there exists a critical frequency which is related to the pressure ramping rate during the oscillation. If the dominant frequency of the backpressure varies beyond this critical frequency, the pressure distribution could be forced into a steady state before the oscillation was induced. Otherwise the oscillation could not be suppressed.

36 citations

••

[...]

TL;DR: In this paper , two types of instabilities were observed in the unstart shock system: instability in the streamwise direction and instability in vertical direction with an asymmetrical effect on the pressures at the walls.

Abstract: Unstable movement of the unstart shock may pose a threat to the safety of a scramjet. The perturbation induced by the unstable movement can also influence the shock structure and the downstream flow, possibly causing a dynamic load on the wall or affecting downstream combustion. Without a thorough analysis of the isolator flow or by ignoring its properties, it is not possible to understand some of the phenomena prevalent in downstream combustion. In this study, two types of instabilities were observed in the unstart shock system. It is shown that if the flow distortion is not severe, the instability in the streamwise direction plays a dominant role. Sequential displacement of the downstream shock was observed in this mode. The time delay between sequential shock motions indicates their response to the movement of the first separation shock. With a highly distorted flow, a flapping mode that resulted in instability in the vertical direction with an asymmetrical effect on the pressures at the walls was observed. In this situation, the shock structure is successively attached to the wall from the head to the tail. By conducting a dynamic mode decomposition analysis, several oscillatory modes, characterized by low-frequency periodicity in the streamwise and vertical directions, were revealed in the shock system. Subsequently, the feasibility of considering the periodical deflection of the incoming flow induced by the significantly unequal amplitudes of shock movements at the two walls as the underlying mechanism for the flapping mode is explored.

11 citations

••

[...]

TL;DR: In this paper, a radiation model for the hydroxyl radical (OH) A-X band was developed, and it was validated using the benchmark data, and a curve fit parameters, such as a peak-to-peak ratio and the absolute peak intensity of the P-branch, were proposed to evaluate the rotational temperature and the number density from the measured emission spectra.

Abstract: Temperature determination in a shock tube is one of the most important factors to understand the relevant flow physics inside. In the present study, the reservoir temperature determination in a shock tube using the ultraviolet emission spectra of hydroxyl radical (OH) A-X band was carried out. A radiation model for the OH A-X transition was developed, and it was validated using the benchmark data. Curve fit parameters, such as a peak-to-peak ratio and the absolute peak intensity of the P-branch, are proposed to evaluate the rotational temperature and the OH number density from the measured emission spectra. In the shock tube experiments, humid air from the atmosphere was employed as a test gas, and a small amount of the ultraviolet OH emission was measured behind the reflected shock wave. The measured spectrum was converted to the rotational temperature and the number density using the present model of the OH A-X emission. Then, the evaluated rotational temperature was compared with the calculated values of the reservoir condition behind the reflected shock wave. A good agreement was detected between the measured and the calculated temperatures, which are 4020 ± 290 K and 4110 ± 220 K, respectively. It was recognized that the reservoir temperature behind the reflected shock wave is well described by the present model of the OH A-X ultraviolet emission.

6 citations

##### References

More filters

••

[...]

TL;DR: This work recognizes the need for additional dissipation in any higher-order Godunov method of this type, and introduces it in such a way so as not to degrade the quality of the results.

Abstract: We present the piecewise parabolic method, a higher-order extension of Godunov's method. There are several new features of this method which distinguish it from other higher-order Godunov-type methods. We use a higher-order spatial interpolation than previously used, which allows for a steeper representation of discontinuities, particularly contact discontinuities. We introduce a simpler and more robust algorithm for calculating the nonlinear wave interactions used to compute fluxes. Finally, we recognize the need for additional dissipation in any higher-order Godunov method of this type, and introduce it in such a way so as not to degrade the quality of the results.

3,698 citations

••

[...]

TL;DR: In this article, the development of a new flux-splitting approach for perfect-gas reacting-gas Navier-Stokes computations is presented, which is designed to capture a stationary contact discontinuity without excess numerical diffusion while providing a monotone resolution of strong normal shock waves.

Abstract: The development of a new flux-splitting approach for perfect-gas reacting-gas Navier-Stokes computations is presented in this work Three distinct variants are proposed, each of which is designed to capture a stationary contact discontinuity without excess numerical diffusion while providing a monotone resolution of strong normal shock waves The variants differ in their resolution of strong oblique shock waves and in their performance for unsteady flow situations A straightforward extension of the methods to general flows in thermo-chemical non-equilibrium is also proposed, and the construction of robust approximate linearizations of the schemes is discussed Comparisons of the new splittings with other upwinding techniques are presented for four steady-state test cases: Mach 8 viscous flow over a 15 ° wedge (perfect gas), Mach 6 viscous flow over a cone-flare configuration (perfect gas), Mach 16 viscous flow over a cylinder (five-species reacting-air), and a subsonic reacting shear layer (seven-species hydrogen-air chemistry) Shock tube simulations are also performed to ascertain the effectiveness of the schemes for unsteady flow situations It is shown that the new methods combine the desirable traits of more sophisticated Godunov-type schemes in the resolution of discontinuities with the robustness and simplicity of flux-vector splittings

476 citations

••

[...]

TL;DR: In this paper, the frequency response of a normal shock in a diverging channel is calculated for application to problems of pressure oscillations in ramjet engines, and two limits of a linearized analysis arc are discussed: one represents isentropic flow on both sides of a shock wave; the other may be a crude estimate to the influence of flow separation induced by the wave.

Abstract: The frequency response of a normal shock in a diverging channel is calculated for application to problems of
pressure oscillations in ramjet engines. Two limits of a linearized analysis arc discussed: one represents isentropic
flow on both sides of a shock wave; the other may be a crude appr'l'I;imation to the influence of flow separation
induced hy the wave. Numerical results arc given, and the influences of the shock wave on oscillations in the
engine are discus,ed.

130 citations

••

[...]

TL;DR: In this article, the behavior of shock capturing schemes which compute the numerical flux from a solution of Riemann's problem is investigated for a one-dimensional model problem consisting of a nearly stationary shock.

Abstract: An investigation of the behavior of shock capturing schemes which compute the numerical flux from a solution of Riemann's problem is performed. The schemes of Godunov, Roe, and Osher are examined for a one-dimensional model problem consisting of a nearly stationary shock. Both scalar and systems of equations are examined. It is found that for slow shocks there is a significant error generated when solving systems of equations, while the scalar results are well behaved. This error consists of a long wavelength noise in the downstream running wave families that is not effectively damped by the dissipation of the scheme. The source of this error is shown, and the implications for the performance of these schemes are considered. This error may contribute to the slow convergence to steady state reported by many researchers.

124 citations

••

[...]

TL;DR: In this paper, the response of choked nozzles and supersonic diffusers to one-dimensional flow perturbations is investigated and a set of boundary conditions is developed that extends the existing work to a nozzle of arbitrary geometry.

Abstract: The response of choked nozzles and supersonic diffusers to one-dimensional flow perturbations is investigated. Following previous arguments in the literature, small flow perturbations in a duct of spatially linear steady velocity distribution are determined by solution of a hyper-geometric differential equation. A set of boundary conditions is then developed that extends the existing work to a nozzle of arbitrary geometry. This analysis accommodates the motion of a plane shock wave and makes no assumption about the nozzle compactness. Numerical simulations of the unsteady, quasi-one-dimensional Euler equations are performed to validate this analysis and also to indicate the conditions under which the perturbations remain approximately linear. The nonlinear response of compact choked nozzles and supersonic diffusers is also investigated. Simple analyses are performed to determine the reflected and transmitted waveforms, as well as conditions for unchoke, 'over-choke' and unstart. This analysis is also supported with results from numerical simulations of the Euler equations. © Cambridge University Press 2007.

96 citations

##### Related Papers (5)

[...]

[...]