Wave reflexion near a wall

1951 ◽  
Vol 47 (3) ◽  
pp. 528-544 ◽  
Author(s):  
A. Robinson
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Velocity Profile ◽  
Shear Layer ◽  
Continuous Wave ◽  
Rigid Wall ◽  
Main Flow ◽  
Supersonic Velocity ◽  
Main Stream ◽  
Numerical Examples

AbstractThe field of flow due to a shock wave or expansion wave undergoes a considerable modification in the neighbourhood of a rigid wall. It has been suggested that the resulting propagation of the disturbance upstream is largely due to the fact that the main flow in the boundary layer is subsonic. Simple models were produced by Howarth, and Tsien and Finston, to test this suggestion, assuming the co-existence of layers of uniform supersonic and subsonic main-stream velocities. The analysis developed in the present paper is designed to cope with any arbitrary continuous velocity profile which varies from zero at the wall to a constant supersonic velocity in the main stream. Numerical examples are calculated, and it is concluded that a simple inviscid theory is incapable of giving an adequate theoretical account of the phenomenon. The analysis includes a detailed discussion of the process of continuous wave reflexion in a supersonic shear layer.

The reflexion of a shock wave at a rigid wall in the presence of a boundary layer

1967 ◽  
Vol 30 (4) ◽  
pp. 699-722 ◽  
Author(s):  
L. F. Henderson
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Initial Conditions ◽  
Boundary Layer Separation ◽  
Rigid Wall ◽  
The Other ◽  
Main Stream ◽  
Wave System ◽  
Mach Stem ◽  
Mapping Procedure

The paper discusses the reflexion of a shock wave off a rigid wall in the presence of a boundary layer. The basic idea is to treat the problem not as a reflexion but as a refraction process. The structure of the wave system is deduced by a simple mapping procedure. It is found that a Mach stem is always present and that the bottom of this wave is bifurcated—called a lambda foot. The reflexion is said to be regular if the Mach stem and the lambda foot are confined to the boundary layer and irregular if either extends into the main stream. Two types of regular reflexion are found, one that has reflected compression waves and the other that has both reflected compression and expansion waves. Initial conditions are given that enable one to decide which type will appear. There are also two types of irregular reflexion, one that has a Mach stem present in the main stream and the other that is characterized by a four-wave confluence. Finally there are also two processes by which regular reflexions become irregular. One is due to the formation of a downstream shock wave that subsequently sweeps upstream to establish the irregular system and the other is due to boundary-layer separation which forces the lambda foot into the main stream.

On the generation of sound by turbulent boundary layer flow over a rough wall

1984 ◽  
Vol 395 (1809) ◽  
pp. 247-263 ◽  
Keyword(s):  
Boundary Layer ◽  
Near Field ◽  
Spectral Characteristics ◽  
Rigid Wall ◽  
Control Surface ◽  
Aerodynamic Noise ◽  
Stream Velocity ◽  
Main Stream ◽  
Wall Roughness ◽  
Boundary Layer Turbulence

The production of sound by scattering of the near field of low Mach number boundary-layer turbulence by a rough, rigid wall is examined on the basis of Lighthill’s theory ( Proc. R. Soc. Lond . A 211, 564 (1952)) of aerodynamic noise. The radiation is expressed in terms of the turbulence pressure spectrum on a control surface that is parallel to the mean plane of the wall and at a stand-off distance equal to the height of the wall roughness elements, the surface irregularities being modelled by a distribution of hemispherical bosses on an otherwise plane wall. The intensity of the sound produced by unit area of the wall varies as the sixth power of the main stream velocity and, for given wall roughness, increases as the boundary-layer thickness decreases. These conclusions are in accord with experimental observations reported by Hersh { AIAA paper no. 83-0786) of the generation of high frequency sound by turbulent flow from sand-roughened pipes, and it is shown how, for moderately rough pipes, the theory reproduces the spectral characteristics of Hersh’s data.

Study on Flow Fields of Boundary-Layer Separation and Hydraulic Jump during Rundown Motion of Shoaling Solitary Wave

2015 ◽  
Vol 09 (05) ◽  
pp. 1540002 ◽  
Author(s):  
Chang Lin ◽  
Ming-Jer Kao ◽  
Guang-Wei Tzeng ◽  
Wei-Ying Wong ◽  
James Yang ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Solitary Wave ◽  
Shear Layer ◽  
Flow Separation ◽  
Hydraulic Jump ◽  
High Speed ◽  
Flow Fields ◽  
Main Stream ◽  
Separated Shear Layer

The characteristics of flow fields for a complete evolution of the non-breaking solitary wave, having a wave-height to water-depth ratio of 0.363 and propagating over a 1:5 sloping bottom, are investigated experimentally. This study mainly focuses on the occurrences of both flow separation on the boundary layer under an adverse pressure gradient and subsequent hydraulic jump with the abrupt rising of free surface during rundown motion of the shoaling wave, together with emphasis on the evolution of vortex structures underlying the separated shear layer and hydraulic jump. A flow visualization technique with particle trajectory method and a high-speed particle image velocimetry (HSPIV) system with a high-speed digital camera were used. Based on the instantaneous flow images visualized and/or the ensemble-averaged velocity fields measured, the following interesting features, which are unknown up-to-date, are presented and discussed in this study: (1) Flow bifurcation occurring on both offshore and onshore sides of the explicit demarcation curve and the stagnation point during runup motion; (2) The dependence of the diffuser-like flow field, being changed from the supercritical flow in the shallower region to the subcritical flow in the deeper counterpart, on the Froude number during the early and middle stages of rundown motion; (3) The positions and times for the occurrences of the incipient flow separation and the sudden rising of free surface of the hydraulic jump; (4) The associated movement and evolution of vortex structures under the separated shear layer, the hydraulic jump and/or the high-speed external main stream of the retreated flow; and (5) The entrainment of air bubbles from the free surface into the external main stream of the retreated flow.

scholarly journals Instabilities in oblique shock wave/laminar boundary-layer interactions

2016 ◽  
Vol 789 ◽  
pp. 1-35 ◽  
Author(s):  
F. Guiho ◽  
F. Alizard ◽  
J.-Ch. Robinet
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Shear Layer ◽  
Laminar Boundary Layer ◽  
Oblique Shock ◽  
Oblique Shock Wave ◽  
Two Dimensional ◽  
Linear Framework ◽  
Displacement Thickness ◽  
The Impact

The interaction of an oblique shock wave and a laminar boundary layer developing over a flat plate is investigated by means of numerical simulation and global linear-stability analysis. Under the selected flow conditions (free-stream Mach numbers, Reynolds numbers and shock-wave angles), the incoming boundary layer undergoes separation due to the adverse pressure gradient. For a wide range of flow parameters, the oblique shock wave/boundary-layer interaction (OSWBLI) is seen to be globally stable. We show that the onset of two-dimensional large-scale structures is generated by selective noise amplification that is described for each frequency, in a linear framework, by wave-packet trains composed of several global modes. A detailed analysis of both the eigenspectrum and eigenfunctions gives some insight into the relationship between spatial scales (shape and localization) and frequencies. In particular, OSWBLI exhibits a universal behaviour. The lowest frequencies correspond to structures mainly located near the separated shock that emit radiation in the form of Mach waves and are scaled by the interaction length. The medium frequencies are associated with structures mainly localized in the shear layer and are scaled by the displacement thickness at the impact. The linear process by which OSWBLI selects frequencies is analysed by means of the global resolvent. It shows that unsteadiness are mainly associated with instabilities arising from the shear layer. For the lower frequency range, there is no particular selectivity in a linear framework. Two-dimensional numerical simulations show that the linear behaviour is modified for moderate forcing amplitudes by nonlinear mechanisms leading to a significant amplification of low frequencies. Finally, based on the present results, we draw some hypotheses concerning the onset of unsteadiness observed in shock wave/turbulent boundary-layer interactions.

scholarly journals Separated shear layer effect on shock-wave/turbulent-boundary-layer interaction unsteadiness

2018 ◽  
Vol 848 ◽  
pp. 154-192 ◽  
Author(s):  
David Estruch-Samper ◽  
Gaurav Chandola
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Turbulent Boundary Layer ◽  
Shear Layer ◽  
Low Frequency ◽  
Separated Flow ◽  
Layer Interaction ◽  
Separation Shock ◽  
Separated Shear Layer ◽  
Boundary Layer Interaction

This paper presents an experimental study on shock-wave/turbulent-boundary-layer interaction unsteadiness and delves specifically into the shear layer’s role. A range of axisymmetric step-induced interactions is investigated and the scale of separation is altered by over an order of magnitude – mass in the recirculation by two orders – while subjected to constant separation-shock strength. The effect of the separated shear layer on interaction unsteadiness is thus isolated and its kinematics are characterised. Results point at a mechanism whereby the depletion of separated flow is dictated by the state of the large eddy structures at their departure from the bubble. Low-frequency pulsations are found to adjust in response and sustain a reconciling view of an entrainment–recharge process, with both an inherent effect of the upstream boundary layer on shear layer inception and an increase in the mass locally acquired by eddies as they develop downstream.

Analysis of the two-dimensional dynamics of a Mach 1.6 shock wave/transitional boundary layer interaction using a RANS based resolvent approach

2019 ◽  
Vol 862 ◽  
pp. 1166-1202 ◽  
Author(s):  
N. Bonne ◽  
V. Brion ◽  
E. Garnier ◽  
R. Bur ◽  
P. Molton ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Shear Layer ◽  
Density Wave ◽  
Reflected Shock Wave ◽  
Oblique Shock Wave ◽  
Two Dimensional ◽  
Mean Flow ◽  
Transitional Boundary Layer ◽  
Resolvent Approach

A two-dimensional analysis of the resolvent spectrum of a Mach 1.6 transitional boundary layer impacted by an oblique shock wave is carried out. The investigation is based on a two-dimensional mean flow obtained by a RANS model that includes a transition criterion. The goal is to evaluate whether such a low cost RANS based resolvent approach is capable of describing the frequencies and physics involved in this transitional boundary layer/shock-wave interaction. Data from an experiment and a companion large eddy simulation (LES) are utilized as reference for the validation of the method. The flow is characterized by a laminar boundary layer upstream, a laminar separation bubble (LSB) in the interaction region and a turbulent boundary layer downstream. The flow exhibits low amplitude unsteadiness in the LSB and at the reflected shock wave with three particular oscillation frequencies, qualified as low, medium and high in reference to their range in Strouhal number, here based on free stream velocity and LSB length ($S_{t}=0.03{-}0.11$, 0.3–0.4 and 2–3 respectively). Through the resolvent analysis this dynamics is found to correspond to an amplifier behaviour of the flow. The resolvent responses match the averaged Fourier mode of the time dependent flow field, here described by the LES, with a close agreement in frequency and spatial distribution, thereby validating the resolvent approach. The low frequency dynamics relates to a pseudo-resonance process that sequentially implies the amplification in the separated shear layer of the LSB, an excitation of the shock foot and a backward travelling density wave. As this wave hits back the separation point the amplification in the shear layer starts again and loops. The medium and high frequency modes relate to the periodic expansion/reduction of the bubble and to the turbulent fluctuations at the reattachment point of the bubble, respectively.

scholarly journals Effects of nozzle-exit boundary-layer profile on the initial shear-layer instability, flow field and noise of subsonic jets

2019 ◽  
Vol 876 ◽  
pp. 288-325 ◽  
Author(s):  
Christophe Bogey ◽  
Roberto Sabatini
Keyword(s):  
Boundary Layer ◽  
Velocity Profile ◽  
Shear Layer ◽  
Nozzle Exit ◽  
Momentum Thickness ◽  
Shear Layer Instability ◽  
Instability Waves ◽  
Exit Velocity ◽  
Exit Boundary ◽  
Boundary Layer Profile

The influence of the nozzle-exit boundary-layer profile on high-subsonic jets is investigated by performing compressible large-eddy simulations (LES) for three isothermal jets at a Mach number of 0.9 and a diameter-based Reynolds number of $5\times 10^{4}$, and by conducting linear stability analyses from the mean-flow fields. At the exit section of a pipe nozzle, the jets exhibit boundary layers of momentum thickness of approximately 2.8 % of the nozzle radius and a peak value of turbulence intensity of 6 %. The boundary-layer shape factors, however, vary and are equal to 2.29, 1.96 and 1.71. The LES flow and sound fields differ significantly between the first jet with a laminar mean exit velocity profile and the two others with transitional profiles. They are close to each other in these two cases, suggesting that similar results would also be obtained for a jet with a turbulent profile. For the two jets with non-laminar profiles, the instability waves in the near-nozzle region emerge at higher frequencies, the mixing layers spread more slowly and contain weaker low-frequency velocity fluctuations and the noise levels in the acoustic field are lower by 2–3 dB compared to the laminar case. These trends can be explained by the linear stability analyses. For the laminar boundary-layer profile, the initial shear-layer instability waves are most strongly amplified at a momentum-thickness-based Strouhal number $St_{\unicode[STIX]{x1D703}}=0.018$, which is very similar to the value obtained downstream in the mixing-layer velocity profiles. For the transitional profiles, on the contrary, they predominantly grow at higher Strouhal numbers, around $St_{\unicode[STIX]{x1D703}}=0.026$ and 0.032, respectively. As a consequence, the instability waves rapidly vanish during the boundary-layer/shear-layer transition in the latter cases, but continue to grow over a large distance from the nozzle in the former case, leading to persistent large-scale coherent structures in the mixing layers for the jet with a laminar exit velocity profile.

Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory

1967 ◽  
Vol 299 (1459) ◽  
pp. 491-507 ◽  
Keyword(s):  
Boundary Layer ◽  
Velocity Profile ◽  
Pressure Gradient ◽  
Asymptotic Behaviour ◽  
Power Law ◽  
Boundary Layer Equation ◽  
Rigid Wall ◽  
Velocity Profiles ◽  
Boundary Layer Theory ◽  
Prandtl Boundary Layer

Consider the Prandtl boundary layer equation for the steady two-dimensional laminar flow of an incompressible viscous fluid past a rigid wall. On the basis of an arbitrary velocity profile at some initial position on the wall, the analysis presented shows that, for a power law streaming speed U ( x ) ═ C ( x + d ) m ( m ≽ 0), the velocity profile which develops downstream is asymptotically given by the well known Falkner-Skan similarity solution. Moreover, for a streaming speed satisfying (6), the velocity profile which develops downstream is asymptotically unique, though of course the particular form of the resulting profile depends on the precise nature of the exterior stream. The rate of convergence for this asymptotic behaviour is estimated, as well as corresponding rates for the convergence of the skin friction coefficient. This result verifies the tacit assumption of a number of writers that the downstream velocity profile is essentially independent of the initial profile, and also supplies a theoretical justification for the role of similar solutions in boundary layer theory. We also prove the existence of concave velocity profiles whenever the pressure gradient is favourable. It follows that, for streaming speeds which correspond to a favourable pressure gradient, concave velocity profiles play somewhat the same role as similarity profiles do for a power law streaming speed.

scholarly journals On Shock Propagation through Double-Bend Ducts by Entropy-Generation-Based Artificial Viscosity Method

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 837 ◽  
Author(s):  
Arnab Chaudhuri
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Reynolds Number ◽  
Shear Layer ◽  
Entropy Generation ◽  
Compressible Flows ◽  
Artificial Viscosity ◽  
Shock Propagation ◽  
Viscosity Method ◽  
Shock Dynamics

Shock-wave propagation through obstacles or internal ducts involves complex shock dynamics, shock-wave shear layer interactions and shock-wave boundary layer interactions arising from the associated diffraction phenomenon. This work addresses the applicability and effectiveness of the high-order numerical scheme for such complex viscous compressible flows. An explicit Discontinuous Spectral Element Method (DSEM) equipped with entropy-generation-based artificial viscosity method was used to solve compressible Navier–Stokes system of equations for this purpose. The shock-dynamics and viscous interactions associated with a planar moving shock-wave through a double-bend duct were resolved by two-dimensional numerical simulations. The shock-wave diffraction patterns, the large-scale structures of the shock-wave-turbulence interactions, agree very well with previous experimental findings. For shock-wave Mach number M s = 1.3466 and reference Reynolds number Re f = 10 6 , the predicted pressure signal at the exit section of the duct is in accordance with the literature. The attenuation in terms of overpressure for M s = 1.53 is found to be ≈0.51. Furthermore, the effect of reference Reynolds number is studied to address the importance of viscous interactions. The shock-shear layer and shock-boundary layer dynamics strongly depend on the Re f while the principal shock-wave patterns are generally independent of Re f .

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