Supersonic Turbulent Boundary Layer Research Articles

Flow structure and cooling effect of seeping gas film in supersonic turbulent boundary layer

Seeping gas film cooling is an active cooling method that involves injecting a cooling gas film into the boundary layer through porous media. In this study, a sintered wire mesh porous material is used to generate the gas film through seepage flow. The flow of the gas film in the supersonic turbulent boundary layer is observed using nanometer particle plane laser scattering technology, while the temperature distribution of the porous wall surface is measured by infrared thermometry. It is observed that as the injection rate of the gas film exceeds 0.2 %, the supersonic turbulent boundary layer thickens significantly. The rate of thickness growth is promoted from 5.5 % to 17.1 % as the injection rate increases from 0.3 % to 0.8 %. Additionally, the injection of the gas film induces an oblique shock wave, and the shock angle increases with the enhancement of the injection rate, resulting in an uneven cooling effect along the injector length. The transient cooling efficiency of the seeping gas film under thermal non-equilibrium conditions is analyzed. The results indicate that the temperature amplitude on the porous wall surface decreases significantly when the injection rate is promoted from 0.2 % to 0.3 %. However, continuing to increase the coolant injection rate does not lead to a more significant improvement in cooling efficiency. When the injection rate reaches 0.5 %, the relative cooling efficiency reaches 50 %–60 %, which is much lower than the predicted values of the thin film theory.

Direct numerical simulations of supersonic flat-plate turbulent boundary layers with uniform blowing

The effect of uniform blowing on a spatially developing flat-plate turbulent boundary layer at Mach 2.25 is investigated using direct numerical simulations. Two values of the wall blowing ratio are considered, corresponding to low and high blowing rates. Uniform blowing is found to significantly reduce the near-wall turbulence anisotropy, although the turbulent kinetic energy still exhibits near-wall asymptotic behavior and the Reynolds analogy is relatively insensitive to changes in the blowing ratio. The pre-multiplied spectra of turbulent kinetic energy production demonstrate that increasing the blowing ratio significantly energizes the large-scale structures in the outer region, while suppressing the inner small-scale structures. An increase in the blowing ratio also has a strong influence on the behavior of the fluctuating wall pressure, amplifying the fluctuation intensity and reducing the dominant frequencies in the power spectrum. Two-point space–time correlations indicate that the characteristic length scale of the pressure fluctuations increases with increasing blowing ratio, whereas the convection velocity exhibits the opposite trend. Analysis of the reduced mean wall heat flux reveals that it is dominated by the relative balance between the work of the Reynolds stress and the turbulent transport of heat, but is insensitive to uniform blowing. Importantly, bidimensional empirical mode decomposition of the turbulent structures highlights the increasingly dominant contributions related to the significantly energized outer large-scale structures in the blowing region.

Aeroacoustic Loading of Impinging Supersonic Boundary-Layer Interaction on Statically Deformed Surfaces

This paper concerns the interaction of an impinging shock wave with a supersonic turbulent boundary layer over several distinct and permanently deformed surfaces, resulting in differences in the shock–boundary-layer interaction and the surface acoustic loading. High-order numerical simulations featuring two-dimensional surface deformations typically encountered in experiments are performed. The deformation amplitudes are up to half the incoming turbulent boundary-layer thickness. The results show that the high-pressure region about the shock impingement is significantly altered and can become narrower or wider depending on the local surface inclination of the deformed panel mode. The surface curvature is found to not significantly affect the separation and reattachment locations of the recirculation bubble. The power spectrum analysis of the pressure fluctuations along the panel’s midspan, where the surface attains the largest deformation amplitude, exhibits a rich and varied response. The pressure power spectrum is amplified in all of the surface deformation modes examined, with the magnitude of the amplification varying in the frequency domain, depending on the location and mode.

Inflow turbulence generation for compressible turbulent boundary layers

It is still challenging to generate high-quality inflow turbulence for the direct numerical and large-eddy simulations of compressible turbulent boundary layers (CTBL). Recently, Wang et al. [“Inflow turbulence generation using an equivalent boundary layer model,” Phys. Fluids 35, 075110 (2023)] proposed a new inflow turbulence generation method based on an equivalent boundary layer model for incompressible turbulent boundary layers. In the present study, the compressible equivalent boundary layer (CEBL) model is proposed and applied to the direct numerical simulation of supersonic turbulent boundary layers. The streamwise equilibrious CEBL approximates the streamwise developing CTBL by adding source terms to the governing equations to recover the mean mass, momentum, and energy balances at a given Reynolds number. Direct numerical simulation is performed to CEBL at free-stream Mach number 5.86 and friction Reynolds number 380. Comparison with the CTBL statistics at the same parameters validates the fidelity and reliability of the CEBL model. Turbulence generated by CEBL as well as the digital filtering and recycling-rescaling methods is used, respectively, to construct the inflow conditions for the direct numerical simulation of supersonic turbulent boundary layers. Results show that the CEBL method has great superiority in reducing the adjustment length compared with the other two methods. In addition, a correction method designed for the high inflow Reynolds number is also introduced.

Characterization of very-large-scale motions in supersonic and hypersonic turbulent boundary layers

Very-large-scale motions are commonly observed in moderate- and high-Reynolds-number wall turbulence, constituting a considerable portion of the Reynolds stress and skin friction. This study aims to investigate the behaviour of these motions in high-speed and high-Reynolds-number turbulent boundary layers at varying Mach numbers. With the aid of high-precision numerical simulations, numerical experiments and theoretical analysis, it is demonstrated that the very-large-scale motions are weakened in high-Mach-number turbulence at the same friction Reynolds numbers, leading to the reduction in turbulent kinetic energy in the outer region. Conversely, the lower wall temperature enhances the very-large-scale motions but shortens the scale separation between the structures in the near-wall and outer regions.

VISUALIZATION OF TURBULENT EVENTS VIA VIRTUAL/AUGMENTED REALITY

Mixed reality technology, i.e., virtual (VR) and augmented (AR) reality, has spread from research laboratories to enter the homes of many. Further, the widespread adoption of these technologies has caught the scientific community's attention, which is constantly researching potential applications. Backed by the continued enhancement of high-performance computing in hardware and software, we are applying mixed reality technologies as a scientific visualization tool for fluid dynamics purposes. In particular, we show a virtual wind tunnel (along with the simplified methodology to replicate it) that enables the user to visualize complex and intricate turbulent flow patterns within an immersive environment. Briefly, high spatial/temporal resolution numerical data over supersonic turbulent boundary layers subject to concave and convex wall curvature has been creatively "pipelined" for VR/AR visualization via several scripts, software, and apps, which are further explained and described along the manuscript. The intention is to present a technique of how to visualize fluid flows to be the most convenient for the user, especially if one is slightly unfamiliar with scientific visualization. Whereas VR/AR applications are principally discussed here for flow visualization, the lessons learned can be certainly extended to other disciplines involving three-dimensional time-dependent databases.

Wall shear stress and wall heat flux in a supersonic turbulent boundary layer subjected to concave surface curvature

Wall shear stress and wall heat flux in a supersonic turbulent boundary layer subjected to concave surface curvature

Assessment of Reynolds number effects in supersonic turbulent boundary layers

Wall-resolved large-eddy simulations (LES) are performed to investigate Reynolds number effects in supersonic turbulent boundary layers (TBLs) at Mach 2.0. The resulting database covers more than a decade of friction Reynolds number Reτ, from 242 to 5554, which considerably extends the parameter range of current high-fidelity numerical studies. Reynolds number trends are identified on a variety of statistics for skin-friction, velocity and thermodynamic variables. The efficacy of recent scaling laws as well as compressibility effects are also assessed. In particular, we observe the breakdown of Morkovin’s hypothesis for third-order velocity statistics, in agreement with previous observations for variable-property flows at low Mach number. Special attention is also placed on the size and topology of the turbulent structures populating the TBL, with an emphasis on the outer-layer motions at high Reynolds number. The corresponding streamwise spectra of streamwise velocity fluctuations show a clear separation between inner and outer scales, where energetic peaks are found at streamwise wavelengths of λx+≈700 and λx/δ0≈6. The spanwise spacing of the outer-layer structures, in turn, is found to be insensitive to the Reynolds number and equal to ∼0.7δ0. It is also found that the integral length-scales in spanwise direction for the temperature, streamwise and spanwise velocity fields appear to progressively collapse with increasing Reynolds number. The modulating influence that the outer-layer structures exert on the near-wall turbulence is also clearly visible in many of the metrics discussed. In addition, the present LES data is further exploited to assess the Reτ-sensitivity of uniform momentum regions in the flow. We find that the resulting probability density function of the number of zones as well as its evolution with Reτ agrees well with incompressible data. This suggests that uniform zones, which have been associated with outer-layer dynamics, are not strongly influenced by compressibility at the considered Mach number.

Direct numerical simulation of supersonic boundary layers over a microramp: effect of the Reynolds number

Microvortex generators are passive control devices smaller than the boundary layer thickness that energise the boundary layer to prevent flow separation with limited induced drag. In this work, we use direct numerical simulations (DNS) to investigate the effect of the Reynolds number in a supersonic turbulent boundary layer over a microramp vortex generator. Three friction Reynolds numbers are considered, up to $Re_\tau = 2000$ , for fixed free stream Mach number $M_\infty =2$ and fixed relative height of the ramp with respect to the boundary layer thickness. The high-fidelity data set sheds light on the instantaneous and highly three-dimensional organisation of both the wake and the shock waves induced by the microramp. The full access to the flow field provided by DNS allows us to develop a qualitative model of the near wake, explaining the internal convolution of the Kelvin–Helmholtz vortices around the low-momentum region behind the ramp. The overall analysis shows that numerical results agree excellently with recent experimental measurements in similar operating conditions and confirms that microramps effectively induce a significantly fuller boundary layer even far downstream of the ramp. Moreover, results highlight significant Reynolds number effects, which in general do not scale with the ramp height. Increasing Reynolds number leads to enhanced coherence of the typical vortical structures in the field, faster and stronger development of the momentum deficit region, increased upwash between the primary vortices from the sides of the ramp – and thus increased lift-up of the wake – and faster transfer of momentum towards the wall.

Supersonic turbulent boundary layer on a plate. I. Closure relations

The turbulent boundary layer on a flat plate in a supersonic flow is considered. The study consists of several parts. In this first one, we prove that the turbulent shear stress and turbulent enthalpy flux can be represented as functions of dimensionless parameters that are expressed in terms of the distance from the wall and mean velocity and enthalpy gradients. The existence of such functional relationships is a consequence of the fact that the flow on the plate is completely determined by a finite set of constant quantities: the free-stream parameters, the wall enthalpy, and the parameters specifying the dependence of viscosity on temperature and the equation of state for a perfect gas. The asymptotic structure of the relationships between the mean quantities is derived from the properties possessed by the solutions of the Navier–Stokes and energy equations. The established functional relations are closure relations of a general form, which together with the boundary-layer and energy equations give a well-posed problem for the velocity and enthalpy fields. For various characteristic flow regions in the boundary layer, this problem will be solved in subsequent parts of the study. This will make it possible to develop a theory of compressible boundary layers based on first principles, without involving particular hypotheses and approximate turbulence models.

Compressibility effects in supersonic and hypersonic turbulent boundary layers subject to wall disturbances

In the present study, we investigate the compressibility effects in supersonic and hypersonic turbulent boundary layers under the influence of wall disturbances by exploiting direct numerical simulation databases at Mach numbers up to 6. Such wall disturbances enforce extra Reynolds shear stress on the wall and induce mean streamline curvature in rough wall turbulence that leads to the intensification of turbulent motions in the outer region. The turbulent and fluctuating Mach numbers, the density and the velocity divergence fluctuation intensities suggest that the compressibility effects are enhanced by the increment of the free-stream Mach number and the implementation of the wall disturbances. The differences between the Reynolds and Favre average due to the density fluctuations constitute approximately $9\,\%$ of the mean velocity close to the wall and $30\,\%$ of the Reynolds stress near the edge of the boundary layer, indicating their non-negligibility in turbulent modelling strategies. The comparatively strong compressive events behaving as eddy shocklets are observed at the free-stream Mach number of $6$ only in the cases with wall disturbances. By further splitting the velocity into the solenoidal and dilatational components with the Helmholtz decomposition, we found that the dilatational motions are organized as travelling wave packets in the wall-parallel planes close to the wall and as forward inclined structures in the form of radiated waves in the vertical planes. Despite their increased magnitudes and higher portion in the Reynolds normal and shear stresses, the dilatational motions show no tendency of contributing significantly to the skin friction and the production of turbulent kinetic energy due to their mitigation by the cross-correlation between the solenoidal and dilatational velocity components.

High-speed shock–boundary-layer interaction over deformed surfaces

This paper concerns the interaction of an oblique shock wave with a supersonic turbulent boundary layer over a thin panel surface, leading to shock–boundary layer interaction and panel buckling. We have performed high-order numerical simulations featuring various static two-dimensional surface deformations typically encountered in experiments. The deformation amplitudes we examined were at least half the height of the incoming turbulent boundary layer thickness. The results show that along the panel midspan, where the maximum deformation amplitude is located, the mean and root mean square pressure are affected by about 10%. Cases for which the pressure at the shock–boundary layer interaction was increased relative to the planar case showed to decrease downstream, and vice versa. Despite the weak response to the mean pressure amplitude, the mean pressure surface contour plots reveal that the streamwise, particularly the spanwise distribution, is affected more noticeably. For example, the surface deformation modes are shown to disrupt the spanwise constant mean pressure, forming higher (or lower) values at either the panel's midspan or edges, depending on the mode. Moreover, the surface curvature leads to a characteristic bending of the spanwise distribution, which can be concave or convex depending on the deformation mode. Analysis of the Reynolds stress anisotropy componentality at different heights from the buckled surface reveals a similar spanwise response of the turbulent velocity fluctuations. The results suggest that the deformation rate plays an important role alongside the deformation amplitude in the turbulent layer and shock–boundary layer interaction.

Response of a supersonic turbulent boundary layer to different streamwise adverse pressure gradients

An adverse pressure gradient (APG) has an impact on the boundary layer, increasing the turbulent intensity of the layer. The mean and turbulent properties of the turbulent boundary layer on a flat plate with different APGs were investigated at Mach 2.7 in the present work utilizing the particle image velocimetry and nanoparticle-based planar laser scattering techniques. According to analysis, the changing trends of boundary layer parameters are different depending on whether the local mainstream velocity or freestream velocity of the wind tunnel is used to normalize. Using the former might make the enhanced effect of the rising APG more visible. With the rise in APG, the principal strain rate, turbulent fluctuation, Reynolds stress, and turbulence production in the boundary layer all increased, while the turbulent boundary layer's thickness dropped. Furthermore, the heightened upward ejection and downward sweep events caused the streamwise turbulence intensity to reach its outer peak under the influence of strong APG. The characteristics of the spanwise vortex in the boundary layer are investigated in conjunction with the probability density function analysis. The growing APG, which primarily promote negative vorticity, can strengthen the rotational strength of spanwise vortices, which are a component of hairpin vortices. As APG rises, the number of small-scale vortices in the boundary layer increases and the fractal dimension grows. The increase in small-scale vortices tends to induce strong transportation and promotes turbulence intensity. Further investigation reveals that the increased volume change caused by the enhanced compression effect with increasing APG exacerbated the vorticity.

Decomposition of mean skin friction in incident shock wave/turbulent boundary layer interaction flows at Mach 2.25

The evolution characteristics of the mean skin friction beneath the supersonic turbulent boundary layer that interacts with incident shock waves at Mach 2.25 are analyzed using Direct Numerical Simulation (DNS). The separated and attached boundary layers in the interaction region that respectively correspond to 33.2° and 28° incident shock angles are considered. The mean skin friction recovery rate for the separated boundary layer is much gentler and distinctly less than that for the attached case where the skin friction completes its recovery within one boundary layer thickness. The novel mean skin friction decomposition method for compressible flows proposed by the recent research is applied in the interaction region to investigate the internal evolution characteristics quantitatively. The results reveal that the three decomposition components are distinctly unequal between the two cases. The contributions of the turbulent motions at different scales to the associated term are focused on using empirical mode decomposition technology. It indicates that the outer large-scale structures dominate separation and reattachment regions, while contributions from inner small-scale structures are limited. In contrast, contributions from the outer large-scale structures are dramatically reduced in the attached case, which results in the outer large-scale and inner small-scale motions being of equal importance.

Influence of optical aperture sizes on aero-optical effects induced by supersonic turbulent boundary layers.

With the increase of flight speed, aero-optical effects induced by the turbulent boundary layer near the optical window become increasingly significant. The density field of the supersonic (Mach 3.0) turbulent boundary layer (SPTBL) was measured by nano-tracer-based planar laser scattering technique, and the optical path difference (OPD) was obtained through ray-tracing method. The influence of the optical aperture size on the aero-optical effects of SPTBL was studied in detail, and the underlying mechanisms were analyzed from the perspective of the turbulent structure scales. The influence of the optical aperture on the aero-optical effects is mainly due to turbulent structures with different scales. The beam center jitter (s x¯) and offset (x¯) are mainly caused by turbulent structures larger than the optical aperture size, while the beam spread about the center (x ' 2¯) is mainly caused by turbulent structures smaller than the optical aperture size. With the increase of optical aperture size, the proportion of turbulent structures larger than the optical aperture size decreases, which can suppress the beam jitter and the beam offset. Meanwhile, since the beam spread is primarily induced by small-scale turbulent structures with relatively strong density fluctuation intensity, the spread increases rapidly to its peak and then gradually stabilizes as the optical aperture size grows.

Post-shock turbulence recovery in oblique-shock/turbulent boundary layer interaction flows

The oblique shock impinging on the supersonic turbulent boundary layer leads to a mixing layer and the emergence of large-scale coherent structures within the interaction zone which leave significant velocity defect and turbulence amplification downstream. In the present study, we investigate the turbulence recovery in the post-shock region by exploiting direct numerical simulation data of the oblique-shock/turbulent boundary layer interaction flow at the incoming Mach number of $2.28$ and the shock angle of $33.2^\circ$ , with special attention paid to the contribution of the mixing layer and large-scale structures to flow dynamics. For that purpose, we propose to split the mean velocity, Reynolds stresses and spanwise spectra into a canonical portion that is constructed according to the statistics of canonical turbulent boundary layers, and a mixing-layer-induced portion. We found that the hidden mixing layer grows with the boundary layer thickness and that the induced mean shear and Reynolds stresses decay at different rates. The mean velocity recovers to the canonical profiles at a distance of 13 boundary layer thicknesses downstream where the mixing-layer-induced mean shear ceases to have strong impacts. The recovery of Reynolds stresses requires 10 boundary layer thicknesses in the near-wall region but a much longer streamwise extent in the outer region due to the slow decay of large-scale motions. These large-scale motions superpose on the near-wall turbulence, intensifying the turbulent fluctuations, yet having a trivial impact on the skin friction, for the contribution of the mixing-layer-induced mean shear and Reynolds shear stress are balanced by the advection term. We further establish a simple physical model capable of approximately predicting the streamwise evolution of mixing-layer-induced mean shear and turbulent kinetic energy. This model suggests that the complete recovery of turbulence in the outer region requires a streamwise extent of approximately 50 boundary layer thicknesses.

Resolvent-based analysis of hypersonic turbulent boundary layers with/without wall cooling

The ability of the low-rank approximation of hypersonic turbulent boundary layers with/without wall cooling is examined with the linear resolvent operator in a compressible form. The freestream Mach number of the base flow is 5.86, and the friction Reynolds number is 420. The wall-to-recovery temperature ratio is set as 1.0 and 0.25, respectively, corresponding to an adiabatic wall condition and a cold-wall condition. Different from the resolvent analysis of incompressible turbulent boundary layers, the optimal response mode in the wave-parameter space exhibits a relatively subsonic and a relatively supersonic region [Bae et al., “Resolvent-based study of compressibility effects on supersonic turbulent boundary layers,” J. Fluid Mech. 883, A29 (2020)], divided by the freestream relative Mach number of unity. The features of energy distribution of the optimal response mode in space and scales are examined, and the energy spectra of streamwise velocity and temperature fluctuations, carried by the optimal response mode, are discussed with typical subsets of streamwise and spanwise wavelengths. This reveals the dynamics of the near-wall small-scale and outer larger-scale motions and the distinction in the relatively subsonic/supersonic region. Moreover, the coherent structures, including the velocity and temperature streaks, quasi-streamwise vortices, and large-scale/very-large-scale motions, are identified in the optimal response mode. Special attention is paid to the effects of wall cooling.

Artificial-neural-network-based nonlinear algebraic models for large-eddy simulation of compressible wall-bounded turbulence

In this paper, we propose artificial-neural-network-based (ANN-based) nonlinear algebraic models for the large-eddy simulation (LES) of compressible wall-bounded turbulence. An innovative modification is applied to the invariants and the tensor bases of the nonlinear algebraic models through using the local grid widths along each direction to normalise the corresponding gradients of the flow variables. Furthermore, the dimensionless model coefficients are determined by the ANN method. The modified ANN-based nonlinear algebraic model (MANA model) has much higher correlation coefficients and much lower relative errors than the dynamic Smagorinsky model (DSM), Vreman model and wall-adapting local eddy-viscosity model in the a priori test. The significantly more accurate estimations of the mean subgrid-scale (SGS) fluxes of the kinetic energy and temperature variance are also obtained by the MANA models in the a priori test. Furthermore, in the a posteriori test, the MANA model can give much more accurate predictions of the flow statistics and the mean SGS fluxes of the kinetic energy and the temperature variance than other traditional eddy-viscosity models in compressible turbulent channel flows with untrained Reynolds numbers, Mach numbers and grid resolutions. The MANA model has a better performance in predicting the flow statistics in supersonic turbulent boundary layer. The MANA model can well predict both direct and inverse transfer of the kinetic energy and temperature variance, which overcomes the inherent shortcoming that the traditional eddy-viscosity models cannot predict the inverse energy transfer. Moreover, the MANA model is computationally more efficient than the DSM.

One-dimensional turbulence modeling of compressible flows. I. Conservative Eulerian formulation and application to supersonic channel flow

Accurate but economical modeling of supersonic turbulent boundary layers is a standing challenge due to the intricate entanglement of temperature, density, and velocity fluctuations on top of the mean-field variation. Application of the van Driest transformation may describe well the mean state but cannot provide detailed flow information. This lack-in modeling coarse and fine-scale variability is addressed by the present study using a stochastic one-dimensional turbulence (ODT) model. ODT is a simulation methodology that represents the evolution of turbulent flow in a low-dimensional stochastic way. In this study, ODT is extended to fully compressible flows. An Eulerian framework and a conservative form of the governing equations serve as the basis of the compressible ODT model. Computational methods for statistical properties based on ODT realizations are also extended to compressible flows, and a comprehensive way of turbulent kinetic energy budget calculation based on compressible ODT is put forward for the first time. Two canonical direct numerical simulation cases of supersonic isothermal-wall channel flow at Mach numbers 1.5 and 3.0 with bulk Reynolds numbers 3000 and 4880, respectively, are used to validate the extended model. A rigorous numerical validation is presented, including the first-order mean statistics, the second-order root mean square statistics, and higher-order turbulent fluctuation statistics. In ODT results, both mean and root mean square profiles are accurately captured in the near-wall region. Near-wall temperature spectra reveal that temperature fluctuations are amplified at all turbulent scales as the effects of compressibility increase. This phenomenon is caused by intensified viscous heating at a higher Mach number, which is indicated by the steeper profiles of viscous turbulent kinetic energy budget terms in the very near-wall region. The low computational cost and predictive capabilities of ODT suggest that it is a promising approach for detailed modeling of highly turbulent compressible boundary layers. Furthermore, it is found that the ODT model requires a Mach-number-dependent increase in a viscous penalty parameter Z in wall-bounded turbulent flows to enable accurate capture of the buffer layer.

Effects of wall disturbances on the statistics of supersonic turbulent boundary layers

In the present study, we perform direct numerical simulations to investigate the spatial development and basic flow statistics in the supersonic turbulent boundary layers at the free-stream Mach number of 2.0 over smooth and disturbed walls, the latter of which enforces extra Reynolds shear stress in the streamwise direction to emulate the drag increment and mean streamline curvature effects of rough walls. Such disturbances escalate the growth rate of turbulent boundary layer thickness and the shape factor. It is found that under the rescaled global coordinate, the mean velocity, Reynolds stress, and pressure fluctuation variance manifest outer-layer similarity, whereas the average and fluctuation variances of temperature and density do not share such a property. Compressibility effects are enhanced by the wall disturbances, yet not sufficiently strong to directly impact the turbulent kinetic energy transport under the presently considered flow parameters. The generalized Reynolds analogy that relates the mean velocity and temperature can be satisfied by incorporating the refinement in modifying the generalized recovery coefficient, and that associates the fluctuating velocity and temperature work reasonably well, indicating the passive transport of temperature fluctuations. The dispersive motions are dominant and decay exponentially below the equivalent sand grain roughness height ks, above which the wall disturbances are distorted to form unsteady motions responsible for the intensified density and pressure fluctuations in the free-stream traveling isentropically as the acoustic radiations.