Momentum Thickness Reynolds Number Research Articles

Transitional flow in a two-stage low-pressure axial turbine under different clocking of blades

Different modes of transitional flows are numerically analyzed in a two-stage low-pressure axial turbine, while being exposed to various clocking of second stator blades and results are presented in this paper. The numerical method utilizes coupled approach of SST k-ω turbulence andγ−Re˜θt transitional models. Kinematic of the wake flow in the blades passage and flow topology are presented and discussed. Different types of transitional flows including “separation induced transition” (i.e., “laminar separation bubble”), “reverse transition” and “bypass transition” are distinguished. Results of surface distribution of pressure and skin friction coefficients, turbulence intensity, intermittency and also transition momentum thickness Reynolds number are presented for the worst and optimum clocking cases, in detail. Results show that the onset points of separation induced transition and bypass transition move upstream by 6.64% and 5.47% in the optimum clocking in comparison to the worst case, respectively. In the optimum clocking the separation bubble length decreases by 1.57% and the onset point of the reverse transition moves downstream by 1.2%. All these beneficial effects happen basically as a result of impinging the first stage wake flow on the leading edge of the second stator blade suction side, which in turn, causes transition from laminar to turbulent to occur within the boundary layer.

Boundary-Layer Transition-Onset Variation Under Wide Range of Mach Numbers and Wall Temperature Ratios

Linear stability analysis is carried out for the boundary layer under a wide range of Mach numbers (4<Mae<16) and wall temperature ratios (0.1<Tw/Taw<1.0), and for the first time, the characteristics of the third mode are analyzed under different Mae and Tw/Taw. It is found that the unstable third mode appears at Mae=10 with an adiabatic wall, and its growth rate first increases and then decreases with increasing Mae. As the wall becomes colder, the unstable third mode is enhanced and appears at lower Mae. The transition-onset variations are investigated by the eN method. As Mae reaches 14, the third mode begins to promote the transition. The transition-onset Reynolds number of the third mode can be reduced by 11.4% compared to that of the second mode at Mae=16. The transition-onset momentum thickness Reynolds number Reθt varies significantly under different Mae and Tw/Taw. At 4<Mae<10, Reθt is around 800–2000 and increases with increasing Tw/Taw significantly. At 10<Mae<16, Reθt rises to 2000–3800 and is less sensitive to the change of Tw/Taw. Therefore, it is necessary to consider the effects of Mae and Tw/Taw on Reθt in transition prediction modeling for hypersonic boundary layers.

Local correlations for predicting the transition process in separated flows tuned with a large experimental database

This work provides new correlations based on local variables for characterizing the transition process developing in the case of separated flows. The goal is to improve the capability of correlation-based transition models through the use of local variables. This may indeed simplify the implementation of the correlations into modern numerical codes, especially dealing with parallel computation and unstructured meshes. A large experimental database considering about 90 different flow conditions has been used to tune the present correlations. The database accounts for the variation of the flow Reynolds number, the free-stream turbulence intensity and the adverse pressure gradient affecting the boundary layer developing over a flat plate. The parameter variation induces the formation of both short and long laminar separation bubbles. The vorticity Reynolds number, based on the second invariant of the vorticity tensor, has been used as the main local variable for the activation of the transition process. Since it is proportional to the momentum thickness Reynolds number at the separation position, the vorticity Reynolds number (local) can be used in place of its counterpart based on the momentum thickness (integral), which is the variable usually adopted for activation of transition. Then, correlations providing a critical threshold for the vorticity Reynolds number promoting transition are tuned, for both short and long bubbles. These may be used into transition models to directly provide the location where the turbulence production needs to be activated. The proposed correlations have been validated using a second database not adopted for tuning and finally tested with data concerning a separated flow case in a low-pressure turbine cascade.

Local flow topology of a polymer-laden turbulent boundary layer

Fine-scale flow motions are measured in a Newtonian and polymer drag-reduced turbulent boundary layer (TBL) at a common momentum thickness Reynolds number $Re_{\theta }$ of 2300. Relative to the Newtonian TBL, the polymer-laden flow has a 33 % lower skin-friction coefficient. Three-dimensional (3-D) particle tracking velocimetry is used to measure the components of the velocity gradient tensor (VGT), rate of deformation tensor (RDT) and rate of rotation tensor (RRT). The invariants in these tensors are then used to distinguish the different types of fine-scale flow motions – a method called the $\varDelta$ -criterion. Joint probability density functions (j.p.d.f.s) of the VGT invariants, $Q$ and $R$ , for the Newtonian TBL produce the familiar tear-drop pattern, commonly seen in direct numerical simulations of Newtonian turbulence. Relative to the Newtonian TBL, the polymer-laden flow has significantly attenuated values of $R$ , implying an overall reduction in fluid stretching. The invariants in the RDT, $Q_D$ and $R_D$ , imply that straining motions of the polymeric flow are more two dimensional compared with the Newtonian flow. Moreover, j.p.d.f.s of $Q_D$ and the invariant in the RRT $Q_W$ , suggest that the flow consists of fewer biaxial extensional events and more shear-dominated flow. Few, if any, experimental investigations have measured the 3-D structure of fine-scale motions in a Newtonian and polymer drag-reduced TBL using the $\varDelta$ -criterion. We provide the first experimental evidence that supports the notion that an attenuation of fluid stretching, particularly biaxial straining motions, is central to the mechanism of polymer drag reduction.

Local correlation, compressibility, and crossflow corrections of γ-Reθ transition model for high-speed flows

Based on the original γ-Reθ transition model framework, in this work, an improved local correlation-based transition closure model is developed for high-speed flows. The local correlation between the vorticity Reynolds number and the momentum thickness Reynolds number obtained by compressible boundary-layer self-similar solutions, local compressibility correction including the pressure gradient parameter and momentum thickness Reynolds number, and local crossflow correlation are applied to improve the original γ-Reθ model for hypersonic transition predictions. The function Fonset1 used to control the transition onset and several relevant model parameters are also modified to make the improved model suitable for high-speed flow. The improved transition model is validated through several basic test cases under a wide range of flow conditions, including high-speed flat plates, sharp cones, double ramp, Hypersonic International Flight Research Experimentation, and complex hypersonic configuration X-33 vehicles. The numerical results show that the transition onset locations and the changes of heat transfer rate predicted by the present improved transition model are reasonably consistent with experimental results. The proposed model predicts the high-speed boundary layer transition behaviors induced by streamwise and crossflow instabilities with reasonable precision.

Response of a Turbulent Boundary Layer to an Imposed Synthetic Large-Scale Structure

The dynamic response of a zero-pressure gradient turbulent boundary layer (TBL) to an active flow control actuator was experimentally investigated. The experimental TBL had a sufficiently low momentum thickness Reynolds number, such that there were no energetic organized large-scale structures in the outer region, as evidenced from measurements of the premultiplied wavenumber–frequency spectra. The periodically pulsed plasma actuator, placed inside the outer region of TBL, introduced a synthetic large-scale structure, and the boundary-layer response to this synthetic structure downstream of the actuator at select wall-normal and streamwise locations was investigated. The turbulence amplitude modulating effect of the synthetic large-scale structure, within the inner and log-linear regions of the boundary layer, was isolated and analyzed using a phase-locked analysis. The dynamic interaction of the synthetic large-scale structure and smaller-scale turbulent motions was quantified using a modulation coefficient, and a strong positive correlation within the inner and log regions of the boundary layer was measured. The streamwise development of the synthetic large-scale structure and its modulating effect on the near-wall turbulence across several streamwise locations are described. Profiles of the phase speed at these streamwise locations were extracted and were found to be constant within the log region, further confirming a strong coupling between the near-wall fluctuations in turbulence intensity and the synthetic large-scale motions introduced in the outer region. Overall, the turbulence amplitude modulation effect induced by the synthetic large-scale structure was found to be dynamically similar to the large-scale modulation measured in canonical TBLs at higher Reynolds numbers.

Flow fields around asymmetrical micro vortex generators in supersonic flow

Motivated by the favorable effects of asymmetrical vortical structures on shock-wave boundary layer interactions, we performed implicit large-eddy simulations to analyze the ability of micro mechanical vortex generators (VGs) to generate such structures. For this purpose, we studied modified VG geometries, by skewing and varying the orientation of a microramp and introducing pairs of unevenly sized ramped-vanes. The modified VGs allow to study the effectiveness of these devices when operating in off-design conditions. The flow physics of all devices were investigated in a supersonic crossflow at Mach 2.5 and a momentum-thickness Reynolds number of 7000. The flow topology around each VG is very similar: the VG acts as an obstacle to the incoming flow and induces a complex system of shocks, recirculation regions, and vortical structures. We present a detailed study of the evolution of the dominant major counter-rotating vortex pair (CVP) to estimate the VGs' flow-control effectiveness. All modified configurations introduce asymmetry into the flow, yet to varying degree for different device types. The modifications made to the microramp have minimal impact on the boundary layer, while the uneven ramped-vane configurations have significant effect. Lateral spread of mechanical VG induced major CVP is not larger than the spanwise dimension of the VG. Consequently, the favorable effect of interactions between VGs arranged in arrays on flow-control effectiveness observed for air-jet VG arrays cannot be reproduced with mechanical VGs. Considering the large design offsets we introduced to the microramp and ramped-vane configurations, both device types are robust to off-design flight conditions.

Multi-scaling analysis of turbulent boundary layers over an isothermally heated flat plate with zero pressure gradient

A meticulous investigation into turbulent boundary layers over an isothermally heated flat plate with zero pressure gradient has been conducted. Eight distinct turbulence models, including algebraic yPlus, standard k-ω, standard k-ε, length-velocity, Spalart-Allmaras, low Reynolds number k-ε, shear stress transport, and v2-f turbulence models, are carefully chosen for numerical simulation alongside thermal energy and Reynolds-Averaged Navier-Stokes equations. A comparative analysis has determined that the Spalart-Allmaras model exhibits remarkable agreement with the results from direct numerical simulation, making it a reliable tool for predicting turbulent heat transfer and fluid flow, particularly at higher Prandtl and Reynolds numbers. Subsequently, a multi-scale investigation employs a comprehensive four-layer structure scheme and encompasses various momentum thickness Reynolds numbers of 1432, 2522, and 4000, and Prandtl numbers of 0.71, 2, and 5. The subsequent investigation reveals the governing non-dimensional numbers' substantial impact on the distribution and magnitude of mean thermal and flow characteristics. Notably, the scaling of mean thermal and momentum fields discloses the existence of a meso or intermediate layer characterized by a logarithmic nature unique to itself. The multi-scaling analysis of the flow field demonstrates greater conformity with the selected scaling variables primarily relying on the Reynolds number. Furthermore, the scaling of the energy field yields compelling outcomes within the inner and intermediate layers. However, according to the four-layer theory, minor discrepancies are observed in the outer layer when using the current scaling.

Forced Injection of a Spanwise-Inclined Jet in Supersonic Crossflow

Beneficial effects of forced air-jet vortex generators in controlling shock-induced flow separation motivated us to study forced jet injection in detail to understand the advantageous control effects. We performed implicit large-eddy simulations to examine the influence of sinusoidally forced injection of a spanwise-inclined jet in supersonic crossflow. Two forcing frequencies were considered (with and 0.0062) in a crossflow at Mach 2.5 and momentum-thickness Reynolds number of 7000. The underexpansion of the jet has a near-linear dependence on the injection pressure. Consequently, in the near field, the major vortices pulsate with the injection pressure. Statistical analysis using triple decomposition revealed that periodic oscillations of the major counter-rotating vortex pair (CVP) result in momentum transfer by major CVPs, also on their lateral sides, leading to enhanced mixing and lateral spreading, which is beneficial for separation control. Three-dimensional dynamic mode decomposition revealed that forced injection induces a large-scale clockwise rotation of the major vortex. These effects are weak at low forcing frequency. High-frequency jet forcing thus leads to more efficient control of flow separation. On the other hand, periodic oscillations cause these vortical structures to dissipate and breakdown faster in the midfield region than for steady jets. The placement of forced jets is therefore an important parameter when used for separation-control purposes.

Transport Equation Transition Modeling in Cerans For Hypersonic Flows

Scope of present work covers implementation to validation of the Langtry-Menter two equation γ-Reθt Local Correlation based Transition Model (LCTM) in CERANS code for modeling subsonic to hypersonic flow transition. The LCTM is coupled with SST k-ω turbulence model. The governing equations of γ-Reθt model is discretized in finite volume framework similar to the RANS model and implicitized using the point Jacobi method. Transition correlations based on freestream as well as local turbulence intensity and critical momentum thickness Reynolds number were integrated with the model, and the code is validated for several standard transition test cases involving low subsonic to hypersonic high enthalpy flows covering a wide range of turbulent intensities.

Comparison and modification of turbulence models for active flow separation control over a flat surface

The present work studied various models for predicting turbulence in the problem of injecting a fluid microjet into the boundary layer of a turbulent flow. For this purpose, the one-equation Spalart–Allmaras (SA), two-equation k–ε and k–ω, multi-equation transition k-kL–ω, transition shear stress transport (SST), and Reynolds stress models were used for solving the steady microjet into the turbulent boundary layer, and their results are compared with experimental results. Comparing the results indicated that the steady solution methods performed sufficiently we for this problem. Furthermore, it was found that the four-equation transition SST model was the most accurate method for predicting turbulence in this problem. This model predicted the velocity along the x-axis in near- and far-jet locations with about 1% and 5% average errors, respectively. It also outperformed the other methods in predicting Reynolds stresses, especially at the center (nearly 5% error). Moreover, the modified four-equation transition SST model has improved the system's performance in predicting the studied parameters by utilizing Sørensen correlations in predicting Reθt (the transition momentum thickness Reynolds number), Flength (an empirical correlation that controls the length of the transition region), and Reθc (the critical Reynolds number where the intermittency first starts to increase in the boundary layer).

Spanwise-Inclined Jets in Supersonic Crossflow: Effects of Injection Pressure and Separation-Control Effectiveness

Motivated by the advantageous performance of air-jet vortex generators (AJVGs) in controlling shock-induced flow separation, we aim at understanding the effects of relevant injection parameters on the physical mechanisms involved, as well as on the control effectiveness. We performed implicit large-eddy simulations to examine the influence of injection pressure on a spanwise-inclined jet in supersonic crossflow at Mach 2.5 and with momentum-thickness Reynolds number 7000. Four injection pressures were considered. The jet and jet/crossflow interactions strengthen with injection pressure, which is reflected in an enhanced obliqueness and asymmetry of the flow topology, strengthening the induced major counter-rotating vortex pair (CVP). This strengthening of the major CVP stagnates at a certain threshold, above which the flow blockage and momentum influx caused by the jets become so strong that they counter the positive influence on the CVP. Subsequently, we installed these jets in an AJVG control setup for a 24 deg compression-ramp interaction with separation to analyze the control effectiveness and interplay of the injection pressure and jet spacing. The strongest effects on control effectiveness are not related to the injection pressure itself, but to interactions between the jet-induced shocks and vortical structures. Therefore, there is a strong link between jet spacing and injection pressure regarding the control effectiveness. Based on these observations, we provide recommendations for designing efficient AJVG control setups.

Compressible modification of boundary-layer momentum thickness localized method in transition model

In current transition models, Menter’s correlation between the maximum vorticity Reynolds number Revmax in the boundary layer and momentum thickness Reynolds number Reθ has been widely used in the localization of the boundary layer length scale. However, the existing correlations have problems in correctly reflecting compressibility effects and only relying on local variables. In this paper, the similarity solutions of laminar compressible boundary layer equations are adopted to study the relation between Revmax and Reθ. Through extracting the local Mach number Malocal and the ratio of wall temperature to the local temperature Tw/Tlocal right at the position where Revmax happens, the influences of the Mach number effect and wall heat transfer effect are characterized, meanwhile a new correlation only relying on local variables between Revmax and Reθ is proposed. For plat-plate boundary layers, the relative error is tiny within a large Mach number and wall-to-adiabatic temperature ratio range and the error caused by the streamwise pressure gradient is not of great concern for general pressure gradients. Furthermore, numerical results over common geometries, including wedges, sharp cones and blunt cones, demonstrate that the proposed correlation could give much better predictions of the relation between Revmax and Reθ than previous correlations. At last, the present correlation is applied to the γ-Reθ transition model as an application presentation.

A procedure for computing the spot production rate in transitional boundary layers

The present work describes a method for the computation of the nucleation rate of turbulent spots in transitional boundary layers from particle image velocimetry (PIV) measurements. Different detection functions for turbulent events recognition were first tested and validated using data from direct numerical simulation, and this latter describes a flat-plate boundary layer under zero pressure gradient. The comparison with a previously defined function adopted in the literature, which is based on the local spanwise wall-shear stress, clearly highlights the possibility of accurately predicting the statistical evolution of transition even when the near-wall velocity field is not directly available from the measurements. The present procedure was systematically applied to PIV data collected in a wall-parallel measuring plane located inside a flat plate boundary layer evolving under variable Reynolds number, adverse pressure gradient (APG) and free-stream turbulence. The results presented in this work show that the present method allows capturing the statistical response of the transition process to the modification of the inlet flow conditions. The location of the maximum spot nucleation is shown to move upstream when increasing all the main flow parameters. Additionally, the transition region becomes shorter for higher Re and APG, whereas the turbulence level variation gives the opposite trend. The effects of the main flow parameters on the coefficients defining the analytic distribution of the nucleation rate and their link to the momentum thickness Reynolds number at the point of transition are discussed in the paper.Graphical abstract

Hypersonic shock wave and turbulent boundary layer interaction in a sharp cone/flare model

A direct numerical simulation of hypersonic Shock wave and Turbulent Boundary Layer Interaction(STBLI) at Mach 6.0 on a sharp 7° half-angle circular cone/flare configuration at zero angle of attack is performed. The flare angle is 34° and the momentum thickness Reynolds number based on the incoming turbulent boundary layer on the sharp circular cone is Reθ = 2506. It is found that the mean flow is separated and the separation bubble occurring near the corner exhibits unsteadiness. The Reynolds analogy factor changes dramatically across the interaction, and varies between 1.06 and 1.27 in the downstream region, while the QP85 scaling factor has a nearly constant value of 0.5 across the interaction. The evolution of the reattached boundary layer is characterized in terms of the mean profiles, the Reynolds stress components, the anisotropy tensor and the turbulence kinetic energy. It is argued that the recovery is incomplete and the near-wall asymptotic behavior does not occur for the hypersonic interaction. In addition, mean skin friction decomposition in an axisymmetric turbulent boundary layer is carried out for the first time. Downstream of the interaction, the contributions of transverse curvature and body divergence are negligible, whereas the positive contribution associated with the turbulence kinetic energy production and the negative spatial-growth contribution are dominant. Based on scale decomposition, the positive contribution is further divided into terms with different spanwise length scales. The negative contribution is analyzed by comparing the convective term, the streamwise-heterogeneity term and the pressure gradient term.

Capturing transition around low-Reynolds number hydrofoil with zero-equation transition model

Compared with the local correlation-based shear stress transport (SST) γ−Reθ transition model (where SST k−ω transport equations are coupled with intermittency γ and transitional momentum-thickness Reynolds number Reθ transport equations), relatively simple and convenient modifications are applied to the parent SST k−ω model for computing natural and separation-induced transitions over the hydrofoil at a low-Reynolds number (LRN). The curiosity toward hydrofoil performance at an LRN has been enhanced by increasing attention to autonomous marine systems, deserving numerical simulations for transitional flow using computational fluid dynamics. With the newly devised transitional SST (T-SST) model, the viscous sublayer blending function F2 is slightly modified, and a stress-intensity parameter as a function of eddy-to-laminar viscosity ratio RT is introduced; intended formulations are plausible and have significant impacts on the transition prediction. Owing to the inherent potential for predicting bypass transition, two anisotropic versions of the v¯2−f(V2F) turbulence model are selected to evaluate their competencies in capturing separation-induced and natural transitions. Results demonstrate that natural transition prediction is more challenging than separation-induced transition for the V2F model. Nonetheless, the T-SST model performs consistently well in replicating both transitional phenomena.

Flow fields around spanwise-inclined elliptical jets in supersonic crossflow

Motivated by favorable effects in separation control with air-jet vortex generators in aerospace applications, we performed implicit large-eddy simulations to analyze the flow physics of spanwise-inclined elliptical jets in supersonic crossflow and reveal the influence of aspect ratio. Orifices with similar hydraulic diameters and aspect ratios of 0.5, 1.0, and 2.0 were studied at Mach number M=2.5 and momentum-thickness Reynolds number Reθ=7000. The jet injection results in the formation of asymmetric flow features (bow and barrel shocks, upstream and downstream separation regions, and vortical structures), which are qualitatively similar for all cases. Compared with the more commonly-studied wall-normal injection, the axis-switching phenomenon and its effects are very mild for spanwise-inclined injection. For the low-aspect-ratio orifice, the clockwise-rotating major vortex strengthens and vice versa for the high-aspect-ratio orifice. Flow penetration increases with aspect ratio; however, the influence of the injection is limited within the boundary layer and the outer layer is undisturbed. In the far field, the enhanced momentum transfer makes the boundary layer more resistant to separation for all orifices. Combined with the amplified vorticity for elliptical cases, elliptical jet orifices become more effective for separation control purposes than circular orifices.

Fluorescent particle image velocimetry using atomized liquid particles

Aerosolized fluorescent particles of Kiton Red 620 dye in a water/glycol fluid are generated using a Venturi-type atomizer and shown to provide effective flow seeding for fluorescent particle image velocimetry (PIV), which can mitigate the detrimental effects of laser reflections from surfaces. Ninety two percent of particles by number concentration were found to be <1 μm in diameter, an acceptable size threshold for gas-flow PIV purposes. A PIV application was conducted in a wind tunnel (freestream velocity U ∞ = 27 m s−1), using the particles for measurement of the boundary layer flow approaching a forward-facing step (approach boundary layer momentum thickness Reynolds number of . Particles were generated from solutions with dye molar concentrations of and mol l−1, and PIV images were obtained for both elastic Mie scattering and filtered, Stokes-shifted fluorescent light. Raw images indicate that the fluorescence yield of the mol l−1 solution provides PIV images with high contrast, even in the near-surface regions where Mie scattering image contrast is highly affected by surface reflections. Boundary layer profiles are processed in the region of adverse pressure gradient leading up to the forward-facing step, where the fluorescent PIV was found to perform comparably to the most optimized Mie scattering PIV; both approaches obtained data as near to the wall as 30 μm, or two viscous wall units in our flow of interest. These results indicate that the new seeding method holds promise for near-surface measurement applications with more complicated three-dimensional geometries, where it is impossible to arrange PIV cameras to reject surface-scattered light.

Fluorescent Particle Image Velocimetry using Atomized Liquid Particles

Abstract Aerosolized fluorescent particles of Kiton Red 620 dye in a water/glycol fluid are generated using a Venturi-type atomizer and shown to provide effective flow seeding for fluorescent PIV, which can mitigate the detrimental effects of laser reflections from surfaces. 92% of particles by number concentration were found to be &lt; 1 μm in diameter, an acceptable size threshold for gas-flow PIV purposes. A PIV application was conducted in a wind tunnel (freestream velocity U∞ = 27 m/s), using the particles for measurement of the boundary layer flow approaching a forward-facing step (approach boundary layer momentum thickness Reynolds number of Re_θ=5930). Particles were generated from solutions with dye molar concentrations of 2.5×10^(-3) and 1.0×10^(-2) mol/L, and PIV images were obtained for both elastic Mie scattering and filtered, Stokes-shifted fluorescent light. Raw images indicate that the fluorescence yield of the 1.0×10^(-2) mol/L solution provides PIV images with high contrast, even in the near-surface regions where Mie scattering image contrast is highly affected by surface reflections. Boundary layer profiles are processed in the region of adverse pressure gradient leading up to the forward-facing step, where the fluorescent PIV was found to perform comparably to the most optimized Mie scattering PIV; both approaches obtained data as near to the wall as 30 μm, or 2 viscous wall units in our flow of interest. These results indicate that the new seeding method holds promise for near-surface measurement applications with more complicated three-dimensional geometries, where it is impossible to arrange PIV cameras to reject surface-scattered light.

Influence of the gas model on the boundary layer characteristic parameters and transition prediction for the Mars entry capsule

<p indent="0mm">Turbulent transition is an important factor affecting the thermal environment of spacecraft. A boundary layer transition prediction method was developed for both smooth and rough thermal protection system surfaces of a large-scale entry capsule in the process of entering the Martian atmosphere. The effects of the gas model and wall temperature constraints on the boundary layer characteristic parameters and transition features were analyzed. The results indicate that the dissociation reaction of carbon dioxide and its thermodynamic non-equilibrium effect will lead to increases in the momentum thickness Reynolds number and the roughness-height-based Reynolds number of the boundary layer. The radiative-equilibrium wall temperature constraint also results in a higher momentum thickness Reynolds number and roughness-height-based Reynolds number compared to those of the cold wall conditions. Therefore, the thermochemical non-equilibrium and radiative-equilibrium wall temperature boundary condition should be included in the analysis model to obtain the actual transition features.