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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. [ABSTRACT FROM AUTHOR]

We perform direct numerical simulations to investigate the characteristics of wall heat flux (WHF) in the interaction of an oblique shock wave at an angle of 33.2° and free-stream Mach number M∞ = 2.25 impinging on supersonic turbulent expansion corners with deflection angles of 0o (flat plate), 6o and 12o. The effect of the expansion on the WHF characteristics is analysed by comparing it to the interaction with the flat plate under the same flow conditions and a fixed shock impingement point. In the post-expansion region, the decreased mean WHF is found to collapse onto the flat plate case when scaled with the mean wall pressure. The statistical properties of the WHF fluctuations, including probability density function, frequency spectra, and space–time correlations, are comparatively analysed. The expansion causes an increase in the occurrence probability of negative extreme events, an enhancement of high-frequency energy, and an inhibition of intermediate-frequency energy. The increased expansion angle also results in a faster recovery of characteristic spanwise length scales and an increase in convection velocity. We use the mean WHF decomposition method in conjunction with bidimensional empirical mode decomposition to quantitatively analyse the impact of expansion on scale contributions. It is demonstrated that the presence of the expansion corner has no significant impact on the decomposed results, but it significantly reduces the contribution associated with outer large-scale structures. [ABSTRACT FROM AUTHOR]

Direct numerical simulation with up to 10 × 10 9 scale grid points based on graphics processing unit computation is carried out to investigate the bluntness effect on the hypersonic boundary-layer transition over a slender cone with zero angle of attack at Mach 6. Four cases with the nose radii of 1, 10, 20, and 40 mm are conducted, and the corresponding Reynolds number based on the nose radius varies from 1.0 × 10 4 to 4.0 × 10 5 . Random disturbances with a broad spectrum of frequencies and a wide range of azimuthal wavenumbers were applied to the wall to simulate disturbances caused by wall roughness. The numerical results show that as the nose tip radius increases, the transition position gradually moves downstream with increased transition region. For the case with a nose radius of 1 mm, the flow transition and entropy swallowing occur almost simultaneously, while for other cases, the transition takes place earlier than the entropy swallowing. In consequence, the disturbance amplitude upstream of the transition in the 1 mm case is much larger than that of other cases. To further study the mechanism of the transition, the frequency spectrum analysis is carried out. It is found that all cases exhibit two characteristic frequencies within the transition region, i.e., the high frequency and extremely low frequency. Owing to the influence of the entropy layer, the characteristic high frequency of the 1 mm case is significantly higher than that of other cases. With the increase in the nose radius, the characteristic frequency of the high frequency decreases gradually. [ABSTRACT FROM AUTHOR]

In this paper, direct numerical simulations in a Mach 6.0 hypersonic turbulent boundary layer over a 30 ° compression ramp are performed. The influence of shock wave/boundary layer interactions on the amplification of turbulent kinetic energy (TKE) and temperature fluctuation (TF) is explored, to provide an insight into the physical mechanism. In the initial part of the interaction region before the detachment of the shear layer, the amplification of the TKE and TF is found, via a frequency spectrum analysis, to be closely related to the low-frequency unsteadiness of the shock wave. Once the free shear layer is established, the shear component of the TKE production defined in the shear layer coordinate appears to act as the main contributor for the TKE amplification, owing to the mixing layer turbulence and the resultant Kelvin–Helmholtz instability. This is consistent with the result from the spectrum analysis that the TKE and TF amplification and their streamwise evolution are dominated by the spectral energy in the median-frequency range, arising from the mixing layer turbulence. As the flow moves downstream along the shock wave, the high-frequency spectral energy content of TF shows a decreasing trend, while the low-frequency spectral energy tends to increase gradually, implying that the shock wave low-frequency unsteadiness exists not only in the initial stage of the interaction region but also around the main shock wave. Under the combined influence of the shock wave intensity and interaction intensity, the median-frequency content appears to weaken first and then tends to increase before decreasing again. The variation amplitude appears to be small and generally dominates the distribution of the TF intensity. [ABSTRACT FROM AUTHOR]

Large eddy simulations of shock wave/turbulent boundary layer interaction on a compression ramp at the Mach number M a ∞ = 5 and Reynolds number R e ∞ = 14 000 are performed to investigate the impact of the incipient and fully separated conditions on the development of the flow field. The quasi-dynamic subgrid-scale kinetic energy equation model, which combines the benefits of the gradient model with the eddy-viscosity model, has been applied. Compared with the previous experimental and numerical results, the simulation was validated. The flow structures, turbulence properties, vortex structures, and low-frequency unsteadiness are all investigated. The flow field of the incipient separation is attached and rarely impacted by shock. An evident separation bubble and localized high wall temperatures in fully separated flow are caused by the separation shock's significant reverse pressure gradient. The Reynolds stress components exhibit significant amplification in both cases, and the peak outward shifts from the near-wall region to the center of the free shear layer. Turbulent kinetic energy terms were analyzed, and the two scenarios show a significant difference. The power spectral density of the wall pressure fluctuations shows that the low-frequency motion of the incipient separation is not apparent relative to the fully separated flow. [ABSTRACT FROM AUTHOR]

In this paper, large-eddy simulation of the interaction between a shock wave and the hypersonic turbulent boundary layer in a compression corner with a fixed 34° deflection angle at Ma = 6 for different Reynolds number cases is conducted. For investigating the effects of the Reynolds number for hypersonic cases, three cases where the free-stream Reynolds numbers are 14000, 20000, and 30000/mm are selected. The averaged statistics, such as the mean velocity, the skin friction, the heat flux, and the wall pressure, are used in this paper. The flow structures in the compression ramp including the shock wave and interaction region are discussed. The decomposition of the mean skin-friction drag for the flat flow is extended to be used in the compression corner. In addition, the turbulent kinetic energy is studied through the decomposition of the mean skin-friction drag for the flat-plate region and the corner region. It is found that higher Reynolds numbers would increase the turbulent kinetic energy by turbulent dissipation at the interaction region, while higher Reynolds numbers would decrease the turbulent kinetic energy by turbulent dissipation after reattachment. In addition, it is also found that the turbulent kinetic energy is larger with a higher Reynolds number and higher turbulent kinetic energy inhibits the movement from the separation point to the inflection point (x = 0 mm), which deduces larger separation bubbles. [ABSTRACT FROM AUTHOR]

Direct numerical simulations of a supersonic turbulent boundary layer on a flat plate interacting with an impinging shock wave are carried out with two different incident shock angles at Mach 2.25. The effect of the interaction strength on the recovery process in the downstream region is systematically studied, including the turbulence evolution, the statistical and structural properties of wall pressure fluctuations, and the generation of mean skin friction and wall heat flux. The variations of the Reynolds stress components, the anisotropy tensor, and the turbulent kinetic energy budget in the two flow cases highlight a slow reversal tendency and an increasingly pronounced importance of the outer-layer large-scale structures in the relaxation region of the strong interaction. We find that the effect of increasing the interaction strength on the fluctuating wall pressure is reflected by a decrease in the characteristic frequencies, an increase in the spatial extent, and a decrease in the convection velocity. We decompose the mean skin friction and wall heat flux into different physically informed contributions and reveal that the mean wall heat flux generation is the same regardless of the interaction strength; in contrast, the generation mechanism of mean skin friction is found to be fundamentally changed. A novel scale-decomposition method is used to quantify the effect of the increased interaction strength on the leading components, and it is demonstrated that the energetic outer-layer large-scale structures are the dominant contributor in the recovery process as the interaction strength is increased. [ABSTRACT FROM AUTHOR]

It is well known that low- and high-speed velocity streaks are statistically asymmetric. However, it is unclear how different the low- and high-temperature structures (T-structures) are even though they are strongly coupled with the streamwise velocity. Therefore, this paper identifies three-dimensional wall-attached temperature structures in supersonic turbulent boundary layers over cooled and heated walls (coming from direct numerical simulations) and separates them into positive and negative families. Wall-attached T-structures are self-similar; especially, the length and width of the positive family are linear functions of the height. The superposed temperature variance in both positive and negative families exhibits a logarithmic decay with the wall distance, while the superposed intensity of the wall-normal heat flux in the negative family shows a logarithmic growth. The modified strong Reynolds analogy proposed by Huang, Coleman, and Bradshaw ["Compressible turbulent channel flows: DNS results and modelling," J. Fluid Mech. 305, 185–218 (1995)] is still valid in the negative family. The relative position between T-structures of opposite signs depends on the wall temperature and that in the cooled-wall case differs significantly from the relative position between low- and high-speed streaks, especially those tall ones. In the cooled-wall case, although positive temperature fluctuations below and above the maximum of the mean temperature can cluster to large-scale wall-attached structures, they are very likely dynamically unrelated. [ABSTRACT FROM AUTHOR]

Direct numerical simulations are carried out to identify the effects of shock impingement on the behavior of bump flow at freestream Mach number of 2.25. Two cosine-shaped bump cases, with and without an impinging oblique shock at an angle of 33.2°, are compared. The shock impingement exhibits a remarkable influence on the pattern of the shock system and on the size of the separation region. A spectral analysis finds that low-frequency unsteadiness is significantly enhanced by the impingement interaction, and the proper orthogonal decomposition highlights the low-frequency breathing motion of the separation bubble, which is accurately reconstructed using only the first ten low-order modes. Downstream of the bump, both the Reynolds stress components and the turbulence kinetic energy exhibit a general amplification, with the peaks reoccurring at outer wall-normal locations. A turbulent kinetic energy budget analysis shows the greatly increased production in the outer layer which is balanced by turbulent transport and dissipation. An anisotropy-invariant map analysis identifies enhanced isotropic turbulence in the vicinity of the bump, which is qualitatively modified into a two-component axisymmetric state around the reattachment point. In addition, the mean skin friction decomposition suggests that the shock impingement has little influence on the predominant contribution of turbulence kinetic energy production, apart from the spatial growth dominance at the bump summit in the absence of the impinging shock. Interestingly, a scale-decomposed analysis quantitatively demonstrates that the contributions of small-scale structures are attenuated, but those of large-scale ones are relatively increased, with a contribution of more than 80% with shock impingement. [ABSTRACT FROM AUTHOR]

Direct numerical simulation of a spatially developing supersonic turbulent boundary layer at a Mach number of 2.25 and a friction Reynolds number of Reτ = 769 subjected to an expansion corner with a deflection angle of 12° is performed to investigate the effect of expansion on the characteristics of the wall heat flux (WHF). The effect of expansion on the statistical and structural properties of the fluctuating WHF is analyzed systematically in terms of probability density function, frequency spectra, and space-time correlations. Normalization using the local root mean square value yields good collapse of the probability density function curves. Unlike with wall pressure frequency spectra, it is found that expansion has little influence on the low-frequency components of the WHF spectrum. The correlation results show that the main effect of expansion is to increase the characteristic length scales and convection velocity of the WHF fluctuation in the post-expansion region. Furthermore, a direct comparison between the conditionally averaged flow fields and those presented in the authors' previous work [Tong et al., Phys. Fluids 34, 015127 (2022)] is performed to uncover the effect of expansion on the flow structures associated with extreme positive and negative WHF fluctuation events. We highlight that the extreme positive event emerges below a small hot spot under the action of a strong Q4 event, whereas the extreme negative event is relatively insensitive to expansion and still occurs between a pair of strong oblique vortices. In addition, we decompose the mean WHF into seven physics-informed contributions and quantify the effect of expansion on the dominating components with the aid of the bidimensional empirical mode decomposition method. The scale-decomposed results demonstrate quantitatively that expansion decreases the contribution of the large-scale structures in the outer region but the small-scale structures in the near-wall region contribute heavily to the mean WHF generation in the downstream region. [ABSTRACT FROM AUTHOR]

The negative and positive fluctuations of wall shear stress τ ′ x and wall heat flux q ′ w can be related to the wall-attached paired up large-scale velocity and temperature streaks. It is justifiable to infer the spatially paired-up coexistence of those wall flow quantities. The present study aims at testifying this hypothesis. We establish such relations between the negative and positive wall shear stress by exploiting a direct numerical simulation database over heated and cooled walls at the friction Reynolds number of 800 and the Mach number of 2.25. The clustering method is adopted for the search of the in-pair structures. It is found that the τx- and qw-structures are less self-similar for flows over cold walls. As they become wider, the τx-structures are increasingly more streamwise stretched, while the trend is reversed for qw-structures. τx-structures of opposite signs are paired up and aligned in the spanwise directions as the wall-attached streamwise velocity, and are left behind by streamwise rollers. The relative position between qw-structures of opposite signs, on the other hand, is sensitive to the wall temperature. Scrutinizing the statistical structures, we elucidate that such spatial coherence is determined by the meandering of velocity streaks that yields strong streamwise gradients of the streamwise velocity. [ABSTRACT FROM AUTHOR]

In the present study, we investigate two-point statistics of fluctuating streamwise wall shear stress τ ′ x and wall heat flux q ′ w by exploiting a direct numerical simulation database of supersonic turbulent boundary layers over a heated wall and a cooled wall at the friction Reynolds number around 800. By separately investigating positive and negative families of τ x ′ and q w ′ with the aid of the conditional correlation analysis, we identify the asymmetrical deformation of τ ′ x and q ′ w , reminiscent of and ascribed to the asymmetrical deformations of sweeps and ejections events. The degree of such asymmetry is alleviated by the lower wall temperature. The spatial orientation of τ x ′ is insensitive to the wall temperature, whereas the spanwise elongated q ′ w that is closely related to the wall pressure is manifested merely in the cooled-wall case. The cross correlation between τ x ′ and the fluctuating streamwise velocity u′ reveals that low-speed streaks related to negative τ x ′ are more inclined to the wall than high-speed ones related to positive τ x ′ by 4 ° – 5 ° , and that the phase lag between negative τ x ′ and u′ is larger than that between positive τ x ′ and u′ except in the near-wall region. Such a difference is proportional to the wall distance and should be considered for models predicting near-wall and wall quantities using signals in the logarithmic layer. [ABSTRACT FROM AUTHOR]

The characteristics of wall heat flux (WHF) beneath a supersonic turbulent boundary layer interacting with an impinging shock wave with a 33.2° angle at Mach 2.25 are analyzed using direct numerical simulation. It is found that the QP85 scaling, defined as the ratio of the mean WHF and wall pressure, changes across the interaction. The probability density function of the WHF fluctuations normalized by the local root-mean-squared value is similar to that of wall shear stress. Comparing the WHF and wall pressure spectra shows that the low-frequency shock unsteadiness exhibits little influence on the spectrum. The space–time correlation of the fluctuating WHF reveals that both the streamwise correlation length scale and the convection velocity experience a sharp decrease in the separation region and subsequent recovery in the downstream region. Moreover, the mean WHF in an incident shock interaction is decomposed for the first time. An analysis of the velocity and temperature fluctuations based on bidimensional empirical mode decomposition is performed to evaluate the contribution of turbulent structures with specific spanwise length scales to the mean WHF generation. The decomposed results indicate that the contribution associated with the large-scale structures in the outer region is greatly amplified by the shock interaction and has the leading role in the generation downstream of the interaction. [ABSTRACT FROM AUTHOR]

We report the characteristics of wall shear stress (WSS) and wall heat flux (WHF) from direct numerical simulation (DNS) of a spatially developing zero-pressure-gradient supersonic turbulent boundary layer at a free-stream Mach number M∞ = 2.25 and a Reynolds number Reτ = 769 with a cold-wall thermal condition (a ratio of wall temperature to recovery temperature Tw/Tr = 0.75). A comparative analysis is performed on statistical data, including fluctuation intensity, probability density function, frequency spectra, and space–time correlation. The root mean square fluctuations of the WHF exhibit a logarithmic dependence on Reτ similar to that for the WSS, the main difference being a larger constant. Unlike the WSS, the probability density function of the WHF does not follow a lognormal distribution. The results suggest that the WHF contains more energy in the higher frequencies and propagates downstream faster than the WSS. A detailed conditional analysis comparing the flow structures responsible for extreme positive and negative fluctuation events of the WSS and WHF is performed for the first time, to the best of our knowledge. The conditioned results for the WSS exhibit closer structural similarities with the incompressible DNS analysis documented by Pan and Kwon ["Extremely high wall-shear stress events in a turbulent boundary layer," J. Phys.: Conf. Ser. 1001, 012004 (2018)] and Guerrero et al. ["Extreme wall shear stress events in turbulent pipe flows: Spatial characteristics of coherent motions," J. Fluid Mech. 904, A18 (2020)]. Importantly, the conditionally averaged flow fields of the WHF exhibit a different mechanism, where the extreme positive and negative events are generated by a characteristic two-layer structure of temperature fluctuations under the action of a strong Q4 event or a pair of strong oblique vortices. Nevertheless, we use the bi-dimensional empirical decomposition method to split the fluctuating velocity and temperature structures into four different modes with specific spanwise length scales, and we quantify their influence on the mean WSS and WHF generation. It is shown that the mean WSS is mainly related to small-scale structures in the near-wall region, whereas the mean WHF is associated with the combined action of near-wall small-scale structures and large-scale structures in the logarithmic and outer regions. [ABSTRACT FROM AUTHOR]

The wall-shear stress (WSS) fluctuations in the interaction of an oblique shock wave with a flat-plate turbulent boundary layer are investigated by means of direct numerical simulation (DNS) at Mach 2.25. The numerical results agree very well with previous experiments and DNS data in terms of turbulence statistics, wall pressure, and skin friction. The fluctuating WSS characteristics, including probability density function (PDF), frequency spectrum, space–time correlation, and convection velocity, are analysed systematically. It is found that the positively skewed PDF shape of the streamwise WSS fluctuations is significantly changed due to the presence of a separation bubble, while the PDF shape of the spanwise component is slightly affected, exhibiting a symmetric behaviour across the interaction. The weighted power-spectrum density map indicates that the low-frequency unsteadiness associated with the separated shock - exhibits little influence on the spectrum for either component, and no enhancement of the low-frequency energy is observed. A significant reduction in the spatial extent of the two-point correlation is observed, causing spanwise elongated coherence for the streamwise WSS fluctuations in the separation region. Moreover, the elliptic behaviour of the space–time correlations is essentially preserved throughout the interaction, and this is accompanied by a sudden reduction of the convection velocity in the separation bubble. [ABSTRACT FROM AUTHOR]

Direct numerical simulation (DNS) of transition over a hypersonic lifting body model HyTRV developed by China Aerodynamics Research and Development Center is performed. The free-stream parameters are: the free-stream Mach number is 6, the unit Reynolds number is 10000/mm, the free-stream temperature is 79 K, the angle of attack is 0, and the wall temperature is 300 K. Weak random blowing-and-suction perturbations in the leading range are used to trigger the transition. A high order finite-difference code OpenCFD developed by the authors is used for the simulation, and grid convergence test shows that the transition locations are grid-convergence. DNS results show that transition occurs in central area of the lower surface and the concaved region of the upper surface, and the transition regions are also the streamline convergence regions. The transition mechanisms in different regions are investigated by using the spectrum and POD analysis. [ABSTRACT FROM AUTHOR]

Direct numerical simulation (DNS) of transition over a hypersonic lifting body model HyTRV developed by China Aerodynamics Research and Development Center is performed. The free-stream parameters are: the free-stream Mach number is 6, the unit Reynolds number is 10000/mm, the free-stream temperature is 79 K, the angle of attack is 0, and the wall temperature is 300 K. Weak random blowing-and-suction perturbations in the leading range are used to trigger the transition. A high order finite-difference code OpenCFD developed by the authors is used for the simulation, and grid convergence test shows that the transition locations are grid-convergence. DNS results show that transition occurs in central area of the lower surface and the concaved region of the upper surface, and the transition regions are also the streamline convergence regions. The transition mechanisms in different regions are investigated by using the spectrum and POD analysis. [ABSTRACT FROM AUTHOR]

This study conducted a direct numerical simulation of an incident shock wave impinging at an angle of 33.2° on a 12° supersonic turbulent expansion corner at Mach 2.25 to determine the influence of expansion on the physics of interaction. The point of nominal impingement was located at the tip of the expansion corner. This scenario was compared in detail with interactions in the case of a flat plate under the same inflow conditions. The expansion led to a significant reduction in the wall pressure and the size of the separation bubble. The pre-multiplied spectra of the fluctuating wall pressure indicated that the motion induced by the low-frequency shock was strongly inhibited by the presence of an expansion corner. The root mean-squared wall pressure declined rapidly downstream of the corner, and relaxed at a nearly constant level very close to its upstream value. The evolution of the reattached boundary layer was analyzed in terms of its velocity profile, map of anisotropic invariance, and turbulent kinetic energy, and a quick recovery process was clearly identified. Moreover, the analysis of decomposition of the mean skin friction revealed the dominant contribution of the turbulent kinetic energy regardless of the effect of expansion. Unlike in the case of flat-plate interaction, the component of negative spatial growth became positive owing to large positive streamwise heterogeneity, and could be neglected. Bidimensional empirical-mode decomposition was used to decompose the fluctuations into four modes with specific spanwise length scales, and the primary mechanism for the generation of skin friction was linked to small-scale structures in the near-wall region. [ABSTRACT FROM AUTHOR]

The effect of wall expansion on the structural and statistical characteristics of wall-shear stress (WSS) fluctuations was investigated by direct numerical simulations of a supersonic turbulent boundary layer over a sharp expansion corner with various deflection angles (β = 00, 20, 50 and 100). It is found that the two-dimensional fields of WSS are characterised as streamwise-elongated streaky structures being aligned in the spanwise direction, resembling low- and high-speed streaks in the buffer region of the flow. Due to the relaminarization effect, these WSS steaks experience a sudden weakening shortly after the expansion corner, but present gradual regrowth with their length scales even exceeding those of the flat-plate case in the far downstream. A strong streamwise-alignment of the instantaneous WSS vector is evident in the case of the largest deflection angle, suggesting a distinct reduction of the intermittency in the relaminarization process. Furthermore, the characteristic time scale of the spanwise component of WSS is quasi-invariant to the expansion effect, while the peak frequency of the streamwise component increases with the increase of the deflection angle. [ABSTRACT FROM AUTHOR]

Uncertainty-based design optimization (UBDO) is becoming increasingly important with great attention to critical system reliability and robustness. However, most traditional UBDO methods only consider low-order response moments and a limited number of aleatory uncertainties, which desiderates to be resolved. In this study, an efficient UBDO approach based on cumulative distribution matching is presented to address all response moments under mixed aleatory and epistemic uncertainties. The mathematical formulation of cumulative distribution function (CDF) matching is systematically derived including the distance metric definition and numerical computation process. Uncertainty analysis within the CDF matching optimization loop is also investigated, including the identification of active subspaces and the propagation of constructed surrogate model in reduced dimensions. Extended active subspaces are furthered discussed for mixed uncertainty analysis to obtain the bounds of CDF efficiently, instead of the time-consuming full-dimensional simulation. The proposed approach is illustrated by three typical optimization examples, i.e., response function matching, standard NASA test problem, and satellite system design in aerospace engineering. The UBDO results based on the CDF matching are discussed and compared with the probability density function matching, which exhibits higher efficiency and better convergence in tackling constrained optimization design problems subject to mixed uncertainties. [ABSTRACT FROM AUTHOR]

Direct numerical simulations of shock wave and supersonic turbulent boundary layer interaction in a 24° compression ramp with adiabatic and cold-wall temperatures are conducted. The wall temperature effects on turbulence structures and shock motions are investigated. The results are validated against previous experimental and numerical data. The effects of wall cooling on boundary layer characteristics are analysed. Statistical data show that wall cooling has a significant effect on the logarithmic region of mean velocity profile downstream the interaction region. Moreover, the influence of wall temperature on Reynolds stress anisotropy is mainly limited in the near-wall region and has little change on the outer layer. As the wall temperature decreases, the streamwise coherency of streaks increases. Based on the analysis of instantaneous Lamb vector divergence, the momentum transport between small-scale vortices on cold-wall condition is significantly enhanced. In addition, spectral analysis of wall pressure signals indicates that the location of peak of low-frequency energy shifts toward higher frequencies in cold case. Furthermore, the dynamic mode decomposition results reveal two characteristic modes, namely a low-frequency mode exhibiting the breathing motion of separation bubble and a high-frequency mode associated with the propagation of instability waves above separation bubble. The shape of dynamic modes is not sensitive to wall temperature. [ABSTRACT FROM PUBLISHER]