Capturing transition and non-transition flows with a new shear stress transport model

Md Mizanur RAHMAN

School of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou 310018, China

KEYWORDS γ -Reθ model;Non-transition;Stress-intensity parameter;SST model;Transition

1.Introduction

Precise and dependable predictions of‘‘laminar-to-turbule nt”BL transitions are of vital importance in hydrodynamic,wind energy and aerospace industries.Modes of applications generate various mechanisms for transition.Flow instabilities due to the growth of Tollmien-Schlichting (T-S), cross-flow and inflectional instability waves in external flows convoke natural transition with Tu < 1%.5Turbomachinery applications associate the bypass transition mechanism with Tu>1%.6As the laminar BL breaks on account of a strong APG,separation-induced transition takes place and transition begins to develop within the separated shear-layer7.However,the laminarization of an existing turbulent BL influenced by a strong favorable-pressure gradient induces a reverse transition(i.e., turbulent-to-laminar)8.Transition is known to be sensitive to the pressure gradient and free-stream turbulence level Tu.All these transitional mechanisms are definitely cumbersome/challenging to be encapsulated in a single turbulence model.

Multiple transition models have been constructed to replicate the physics and mechanisms of transitional phenomena;the eNmethod8, LRN turbulence models9,10and empirical correlation-based models8,11belong to this group.Nevertheless,they are inappropriate for complex geometries since most of these transition models deal with non-local operations.A correlation-based γ - Reθtransition model has been developed by Menter, et al.12relying rigorously on local variables; the pattern remains susceptible to needs of the modern generalpurpose CFD code13.Using a‘‘flow-structure-adaptive”parameter(stemming from Bradshaw’s assumption)as a function of the eddy-to-laminar viscosity ratio, Xu, et al.14have developed the KDO (Kinetic Dependent Only) transition model which is incapable of differentiating an exact phase(e.g., transition state point in relation to relevant parameters)of the‘‘laminar-to-turbulent”transition.However, it can precisely predict onset locations of the transition and predicted results are reasonable for different types of transitions.The achievement is due to the Bradshaw flow-structure function and may be the length scale, taking the effects of free-stream length scale variations into account.Besides, the KDO model also retains the wall distance dependent Bradshaw parameter that signifies a wall constraint with a recourse to the correlation between wall-normal and stream-wise fluctuations and therefore, it can be regarded as an allowance for‘‘interaction-at-adistance”effects,promoting the transition process9.

The progress in predicting the‘‘laminar-to-turbulent”transition has been relatively limited due to flow intermittency in the transition region.RANS-based transition models have received much attention in the recent decades owing to their compromising accuracy and efficiency in simulating engineering problems.Liu, et al.15have developed a localcorrelation based transition model which avoids the‘‘trialand-error”inconsistency, associated with most correlationbased transition models for precise computations.This is undoubtedly a remarkable achievement in the local correlation-based transition modeling.Moreover, the latest state of art methodologies for integrating roughness and cross-flow effects in transition models has been detailed with relevant references by Zhang, et al.,16and Xu, et al.17Although available transition models are often overly complex, meaningful and inspiring numerical results have been obtained.Relying on the Structural Ensemble Dynamics(SED) theory18, Xiao and She19,20have proposed a new encouraging closure which models turbulence and transition phenomena in an integral manner.Using the SED concept,Rahman21has constructed an algebraic intermittency model which intrinsically preserves‘‘flow-structure-adaptive”characteristics in RANS simulations for various transition phenomena.The transition model is deemed to be a quite coherent choice owing to its simplicity and avoidance of the unwanted‘‘trial-and-error”operation,as mentioned earlier.In fact,with‘‘flow-structure-adaptive”features,the new transition model is potential for capturing bypass, natural, separation-induced and cross-flow transitions as the KDO model does; only wall roughness corrections (leading to an enhanced eddyviscosity)could make the turbulence model be capable of capturing the impact of roughness on the transition process22.

Model predictions are compared with several welldocumented test-cases, characterizing simple and nonequilibrium flows, and flows with bypass, natural and separation-induced transitions.Results demonstrate that the New SST (NSST) model has the potentiality to capture most features (e.g., produces almost identical predictions) on existing SST k -ω and γ -Reθmodels’perspective.

2.Model formulation

Eqs.(5)–(10) indicate the conversion of a k - ε model to a k - ω model.

2.1.Evaluation of model coefficients

The free-stream sensitivity due to ω is a notorious softness of the k - ω model.This occurrence particularly appears when simulating shear flows; almost arbitrary results are obtained through different (although reliable) free-stream values of ω.However,this problem can be avoided by incorporating a turbulent Cross-Diffusion (CD) term Cω(vT/k)?k/?xj?ω/?xjin the ω - equation as given in Eq.(10); such a term has been included in all previously proposed k - ω models2,25–28.A value of the CD coefficient Cω=0.25 is applied to the current model; however, Wilcox25has shown that Cω= 0.125 is adequate to prevent the model sensitivity from free-stream turbulence.Since k grows and ω declines in the viscous sublayer,it is essential to subdue the CD term in the wall proximity for wall-bounded flows;otherwise sublayer predictions will be deteriorated25,27.On the other hand, closure coefficients σk, σωand Cωcomply with constraints:

in order to have a smooth approach to free-stream values.In principle, the k -ω turbulence model needs to satisfy these criteria to achieve smooth solutions near a turbulent/nonturbulent interface in the immediate vicinity of BL edge,causing no difficult numerical issues to occur25.Consequently,σk=σω= 0.75 is set to be compatible with Eq.(11) and equal-diffusion assumption,as mentioned earlier.This is done so as to invite attention that Menter’s SST and γ - Reθmodels do not satisfy Eq.(11).

from which the value of von Karman constant κ ≈0.37 is procured; nevertheless, a range of κ ≈0.34–0.46 is obtainable from the literature31.

2.2.Formulations of Bradshaw parameter

Fig.1 Distributions of stress-intensity for flat-plate BL flow at Reθ = 4060.

Eq.(18) represents the interaction between non-transition and transition SST models.Regarding the transition onset location at Rμ=1.0 in principle distinguishes between laminar and turbulent states.Nevertheless, a pre-transitional/pseudolaminar allowance (an interpretation of the behavior for Rμ≤1.0) is assigned to the current approach since Rμ≤1.0 imitates characteristics of the intermittency involved in the γ -Reθtransition model.It is worth pointing out that an appropriate level of the eddy-viscosity evaluation in the transitional flow is‘‘recursive”(for instance, a‘‘trial-and-error”operation is needed by the γ -Reθtransition model for proper simulations); nonetheless, the damping effect of ～Rbon the eddyviscosity apparently disappears as the BL grows from its pretransitional form to the fully turbulent state.As mentioned earlier, Rb=Rbμpreserves the physical characteristic of a‘‘flow-structure-adaptive”function and the given allowance to Rμ≈1.0 can be interpreted as a reliable promoter for the existing transitional flow,perceivable from numerical examples presented afterward.In particular, the transition onset in the γ -Reθmodel is activated by the transport equation of Reθ.The Re effect has been implicitly invoked with the present approach since Rb=Rbμis calibrated with DNS data on a slice of Reθ=4060,focusing mainly on the regime before the transition onset.Ostensibly, the parent NSST model (given later)accounts for the evolution of turbulence (production of k as well as vTand sensibly transition length) after the transition onset.

It is worth mentioning that ～Rbhas been substantially modified from those of Xu,et al.4,14in order to make it compatible with the SST k - ω.Articulately, Eq.(21) represents a new Reynolds-stress constitutive formula.However, an arbitrary linear or non-linear Reynolds stress formulation may be utilized to calculate Reynolds stresses and the eddy-viscosity μT=ρvTis available from Eq.(18).The NSST model calculates Pkusing Eq.(20); therefore, the choice is arbitrary for the evaluation of Reynolds stresses.

2.3.NSST k - ω turbulence model

where vTand Pkare calculated from Eqs.(18)and(20),respectively.Associated constant coefficients are: Cμ= 0.09, σk=σω= 0.75, cω= 0.25, cω1= 0.5 and Cω2= 0.075.

To this end, it must be emphasized that Cμ= 0.068–0.083 in the log-layer as is observed from DNS data34–36and the Cμdistribution is seemingly quite irregular in other regions of the BL.In essence,the turbulence model may confront visible deficiencies to accurately predict complex flows.However,embedded SST characteristics of the k -ω turbulence model suppress generated shortcomings to a greater extent.The wall-boundary condition for ω given in Ref.2is applied which compromises with the one of Ref.37as long as y+≤3.

3.Computations

To show the accuracy and generality of NSST k -ω model,numerical computations of well documented test-cases are considered.Selected numerical experiments are the fullydeveloped channel flow, ERCOFTAC (‘‘European Research Community on Flow Turbulence and Combustion”) bypass transition of T3A and T3B BLs together with natural transition of T3A- BL38, separation-induced transition of airfoil-A39, Eppler-387 airfoil characterized by separation induced transition40,41, turbulent flow in asymmetric plane diffuser,transonic flow over RAE2822 airfoil, 3D axisymmetric hill and flow past 3D ONERA-M6 wing.Simulations from widely used SST k -ω model and γ -Reθtransition model are included for comparisons.In contrast to SST and γ -Reθmodels, the current NSST model has been sensitized to non-isotropic stress-intensity functions, parameterized with Reyand Rμ,respectively; both variables Reyand Rμcan be regarded as Reynolds numbers.A cell-centered finite-volume approach accompanied by a‘‘pseudo-compressibility”method is used to solve all flow equations; details of the numerical method can be found elsewhere42–45.

3.1.Turbulent channel flow

Turbulent channel flows at Reτ= (395, 640) with a fullydeveloped identity is simulated to substantiate the model competency in reproducing near-wall turbulence; DNS data are available from Refs.35,36.Using a one-dimensional (1D)RANS solver, computations are performed in a channel halfwidth.A non-uniform 1 × 64 grid resolution for Reτ= 395 and 1×128 grid resolution for Reτ=640 are presumably adequate to accurately delineate the flow features.The first computational cell-height remains at y+≈ 0.3 to resolve the viscous sublayer.Simulations from NSST and Menter’s SST in wall units are presented in Figs.2 and 3.As can be seen,predictive differences are almost indistinguishable as expected;nonetheless, the SST model under-predicts the mean velocity profile in the wake-deficit regime of BL at Reτ=640.Perhaps,the SST k -ω model has this complication.

3.2.Flat-plate BL flow with zero-pressure gradient

Fig.2 Developed channel flow at Reτ = 395.

Fig.3 Developed channel flow at Reτ = 640.

Table 1 Inlet conditions for flat-plate test-cases with NSST model.

Conventionally,Tu<1%is recognized as a natural transition state and after a critical Re, T-S waves appear and demand the presence of a transition state.Most LRN models are usually incapable of responding to such mechanisms9.The T3A- natural transition is familiar as one of the welldocumented benchmark test-cases for natural transition.Fig.4(c) shows Cfdistributions from different turbulence models for natural transition.As can be seen, the predictive performance of NSST is sensibly identical to that of the four-equation γ -Reθmodel.

3.3.Separation-induced flow over airfoil-A

The airfoil-A has been experimentally studied with ONERA39F1 and F2 wind tunnels in France;both turbulence models are applied to simulate this test-case in order to further substantiate their performances.It is well-known that experimental results of the F1 test-case are more susceptible to turbulence model validations.The measured angle of attack α = 13° is repaired to an α = 13.3°46for accounting wall effects in numerical simulations.The flow over the suction surface (upper surface) retains a small Laminar Separation Bubble(LSB) reattaching approximately at x/c = 0.12 (where c indicates the chord length); the growing BL separates at about x/c=0.85 and afterward,turbulent reattachment.Above all,the flow has complex features wherein the turbulent shear-layer thickness and length of separated regime are particularly tiny.The developed small recirculation region escalates toward the trailing edge46.

Experimental inlet conditions for the airfoil-A are given in Table 2; values for Tu and Rμare chosen to fit the transition with an LSB, appearing at x/c = 0.12.A non-uniform structured 384 × 128 C-type grid is used for the simulation of airfoil-A, ensuring the first near-wall cell location at y+< 1.0 A finer mesh is carefully arranged at the leadingedge regime to better forecast the LSB associated with the transition process.The computational domain extends for 40c in all directions and far-field boundary conditions are applied.

The suction-surface skin-friction coefficients of airfoil-A are plotted in Fig.5(a) together with experimental data.As can be observed,the NSST model captures the LSB well;however,the γ -Reθmodel presumably speculates this phenomenon partially.It appears that the NSST model under-predicts the peak in Cfrelative to those of the γ-Reθmodel and experimental data.Perhaps,the slow turbulence-generation inside the BL with the NSST model is responsible for this deficiency,although the model is activated properly by the transition artifact.Both transition models predict the turbulent separation on the tail, recognized by the negative Cfvalues.The SST model provides a fully turbulent Cf-profile over the airfoil surface.A good correlation with pressure-coefficient distributions is obtained among simulations and measurements as illustrated in Fig.5(b).

3.4.Flow over LRN Eppler-387 airfoil

To further validate model competencies, the turbulence flow over Eppler-387 airfoil at Re = 3 × 105is simulated which induces a separation-induced transition at an α = 2°.This test-case corresponds to wind tunnel experiments carried out in the‘‘University of Illinois at Urbana-Champaign (UIUC)low-turbulence subsonic wind tunnel”40and at the‘‘NASA Langley Low-Turbulence Pressure Tunnel”(LTPT)41.Table 3 represents inlet parameters for the Eppler-387 airfoil; according to an average turbulence level in the series of LTPT measurements, the turbulence intensity is selected as Tu = 0.1%.A non-uniform C-type 384 × 128 grid resolution is employed for simulations.The grid is heavily clustered near the wall and the first near-wall y+values on the upper surface range from 0.03 to 0.3.Nevertheless, guidelines of typical LRN RANS models suggest that the magnitude of near-wall y+remains in the range 0.1

To maintain a consistent eddy-viscosity level in the vicinity of separation-induced transition due to an LRN flow is a challenging task.In accordance with both LTPT and UIUC experiments, laminar flow extends from the leading edge to approximately 0.40c.It is found that the NSST model underestimates the near-wall eddy-viscosity, convoking a phenomenon known as‘‘modeled stress depletion”(e.g., insufficient eddy-viscosity level generated in near-wall regions).To avoid this problem with the NSST closure, a limited viscous production Plimk is added to the k - equation (e.g., Eq.(22))to ascertain proper generation of k at the wall-proximity forarbitrary low Tu levels.The viscous production term is modeled resorting to the evidence that both strain-rate and vorticity invariants have significant impacts on transitional physics 12,48:

Table 2 Inlet conditions for airfoil-A with NSST model.

Comparisons of drag coefficient CDand lift coefficient CLfrom both turbulence models are illustrated in Table 4.As can be seen, the discrepancy of computed coefficient from the NSST model is minor when compared with the fourequation γ -Reθand measured data.Lift coefficients adhering to both transition models have fairly good agreement with experimental evidence.The RANS NSST model exhibits reasonable convergence acceleration although the test-case is likely to suffer from flow unsteadiness due to the low-flow speed and the appearance of an LSB.Remarkably, all turbulence closures agree well with the experiment for CL.Asexpected, the SST model overpredicts the CDcompared to the measure data due to the overly turbulence generation.

Table 3 Inlet conditions for Eppler-387.

Cpdistributions around the airfoil surface from three turbulence models are contrasted in Fig.6(a).As can be observed,small fluctuations appear at the Cpprofile due to a slight unsteadiness in the flow field.Nonetheless, the solution converges which is evaluated by monitoring drag and lift coefficients individually.Iteration counts are terminated as both coefficients do not change more than 0.01%.Except the SST model, both transition closures capture pressure bumps on the upper surface of airfoil,confirming the existences of LSBs.All turbulence models perform equally well outside the LSB region; differences are almost negligible when compared with measurements.

Fig.6(b) illustrates Cfdistributions on the back surface of airfoil from two turbulence models.As can be noticed, details of separation-induced transitional features are better manifested.Above the airfoil surface, transition takes place in the free shear layer with the appearance of an LSB, which can be detected by a negative Cfregime with a reverse flow on the airfoil back surface.It is worthy of attention that the LSB is very small relative to the dimensions of airfoil and therefore, a finer mesh is needed to predict this occurrence.Remarkably, the NSST turbulence model reasonably traces both separation and reattachment points.In contrast, the γ -Reθmodel exhibits a large negative Cf-spike implying a large reverse flow; this feature is conceptually confirmed but not experimentally with UIUC data40.The γ -Reθmodel provides the largest separation length.Separated and reattaching points are shown in Table 5;separation length can be calculated from the table.Note-worthily, numerical simulations are also carried out with Rμ= 0.1–1.0 using Tu = 0.1%; however, the transition location remains unchanged.This is due to the fact that downstream of the separation point, the reattachment takes place in a way that is familiar to be unsteady as periodically generated vortices impinge on the solid surface50,51.

Fig.5 Coefficient distributions on suction surface of Airfoil-A.

Table 4 Comparisons of force coefficients with experiments for Eppler-387 airfoil.

The inherent unstable feature of vortices collides with the airfoil surface, leading to a relatively high shear-stress, which makes the transition location be independent of Rμvalues in a certain range.Naturally, the NSST behaves like a‘‘flow-str ucture-adaptive”transition model.The SST model produces excessive turbulence generation far ahead of the LSB,showing no separation at all.The flow remains attached and most turbulent on the suction (back) surface.

3.5.Turbulent flow in asymmetric plane diffuser

The opening angle and expansion ratio of this configuration are 10° and 4.7, respectively.The set-up is capable of producing flow separation and reattachment on the deflected wall with an Re = 2 × 104founded on an inlet channel height h and a reference (center-line) velocity Ur.In essence, the flow characteristics can be recognized as an APG driven dominated flow.A plane channel has been attached to the diffuser entrance in order to create fully-developed inlet boundary conditions.The plane asymmetric diffuser flow is simulated to further evaluate the model performance; experimental data areachievable from Buice and Eaton52.The computational domain length is set to 76h.

Table 5 Separation and reattachment points from turbulence models for Eppler-387 airfoil.

Computations involving a 120×72 non-uniform grid resolution is found to be satisfactory to bring about a gridindependent solution.The thickness of the first near-wall cell remains at y+<1 on both flat and deflected walls.Predicted Cfdistributions for both models are illustrated in Figs.7(a)-Fig.7(b); they replicate almost identical results.Figs.8–9 represent stream-wise mean velocity and turbulent kinetic energy profiles,respectively.NSST predictions reasonably match SST and experimental data.As is evidenced, almost equivalent imprints are produced by both NSST and SST models.

3.6.Transonic flow past RAE2822 airfoil

Due to the appearance of shock waves around an airfoil surface in a transonic flow, a degraded performance is obtained.Nevertheless, the lost proficiency could be recovered to a greater extent with supercritical airfoils (for instance,RAE2822) which ascertains a‘‘roof-top type pressure”allotment.The airfoil with a critical Ma can initiate the cruise Ma of transonic aircraft.Generally, numerical experiments of this test-case are performed to evaluate the model performance in replicating the suction peak and the onset of the shock wave.Two types of measured data are achievable53:α = 2.8°, Re= 6.5 × 106, and Ma = 0.73; and α = 2.72°,Re= 6.5 × 106, and Ma = 0.75.Numerical simulations are conducted on a C-type non-uniform 384 × 128 grids; meshes are densely compressed near the wall and the first near-wall cell thickness remains below one in y+unit.

Fig.6 Coefficient distributions on back surface of Eppler-387 airfoil.

Fig.7 Diffuser flow: Cf distributions with bottom wall and top wall.

Fig.8 Diffuser flow: Stream-wise velocity profiles at selected locations.

Fig.10(a)-Fig.10(b) demonstrate pressure distributions with experimental data, evincing an encouraging agreement between simulations and measured data.Predictions of NSST are almost identical to those of the SST model.Note-worthily,the SST model with the first working condition predicts the shock-location slightly upstream of the experimental location.The reasoning could be that the SST model is insensitive to turbulence transition, whereas the measurement accompanies a transition trip for the‘‘laminar-to-turbulent”flow near the leading edge.As can be observed, Cpis nearly constant up to 50% of the chord length, authenticating the existence of a‘‘roof-top type pressure”distribution at design conditions.The Cpdistributions also dictate an aspect of APG,prevailing over 50% of the chord length.

3.7.3D axisymmetric hill

Fig.9 Diffuser flow: Kinetic energy profiles at selected positions.

Fig.10 Cp over RAE2822 airfoil.

Fig.11 3D hill flow:Computational mesh in plane of symmetry at z/h = 0.

The existence of a 3D separation on the leeside is acknowledged by the flow over an axisymmetric 3D hill.Simpson,et al.54have conducted the experiment at‘‘Virginia Polytechnic Institute and State University”.This test-case is rather challenging for RANS turbulence models as ascertained at Gothenburg in Sweden on the 11th‘‘ERCOFTAC Workshop on Refined Turbulence Modeling”.Generally,RANS closures provide erroneous results over the 3D-hill leeside regime and downstream thereafter.

The origin of coordinate system is set at x0=-4.11h upstream from the hill-center, where the inflow section is located.The 3D-hill computational grid in the plane of symmetry at z/h = 0 is shown in Fig.11.Inlet flow variables at x0=-4.11h are interpolated from experimental data at x=0 with removing the hill;symmetric conditions are applied to the span-wise side boundaries.A non-uniform 160 × 80 × 64 grid resolution is seemingly adequate to accurately simulate the characteristic flow; the grid distributions ensure the first near-wall cell thickness to be remained at y+< 1.

Fig.12 Coefficient distributions over 3D hill surface at z/h=0 (Plane of symmetry).

Fig.13 3D Onera-M6 wing: Coefficients of wall-pressure at selected cross section.

At the center-plane, Cfdistributions along the 3D-hill surface are plotted in Fig.12(a); the formation of recirculation lengths by two models can be calculated from Cfdistributions.The predicted recirculation regions by the NSST and SST models are 0.34≤x/h ≤2.72 and 0.3≤x/h ≤2.7, respectively;differences are almost negligible.However,the experiment evidences that after crossing the hill crest,the flow in the plane of symmetry separates at x/h = 0.96.The flow reattaches to the hill-foot at x/h = 2, having a very shallow separation zone.Discrepancies between experiments and predictions rectify much stronger reverse flow, as speculated by both turbulence models at the leeward-central section with the absence of above-mentioned down-wash.These characteristics are also congruent with the quick recovery of measured pressure behind the 3D-hill, relative to the plateaus in simulations as reflected in Fig.12(b) with the feature of massive separation.Apparently, both models produce almost analogous solutions for Cpcurves.

3.8.3D ONERA-M6 wing

The Onera-M6 wing is an extensively used 3D test-case to validate both numerical and turbulence models for transonic flows.This configuration has a free-stream Re = 11.71 × 106and a free-stream Ma = 0.8395 with an α = 3.06°.A structured grid having four blocks with 1572864 cells is utilized in the simulation.The non-uniform grid resolution guarantees y+<1 , translating the minimum normalized near-wall cellthickness as 2 × 10-5.This test-case is characterized by a shock-induced BL separation due to the interactions between BL and shock-wave.However, the numerical study validates two independent turbulence models, focusing exclusively on available measured data for pressure coefficients55.

Six locations(y/b=20%,44%,65%,80%,90%and 95%,b is the span-wise length)in the span-wise direction are considered to measure wall-pressure coefficients.Computed pressure coefficients with measured data are compared in Fig.13.As can be observed,both NSST and SST turbulence models exhibit good agreement with measurements.Remarkably,adjacent to the upper wing-surface leading-edge, minor variations (i.e.,slight over-predictions) are visible.Nevertheless, both models capture the overall Cpprofiles, especially at the trailing edge and on the lower side of wing.Indistinctive differences in pressure distributions exist between NSST and SST models.

4.Conclusions

Several well-documented test-cases are computed; comparisons demonstrate that NSST predictions are almost equivalent to those of SST and γ -Reθmodels.The NSST model has several attractive attributes over SST and four-equation γ -Reθmodels:it avoids blending functions and stress–strain misalignment usually involved in a linear eddy-viscosity model; it requires no transport equations for γ and Reθto be solved; it is Galilean invariant;transitional flow can be‘‘fine-tuned”with a single parameter Rμfor an arbitrary Tu and finally,it reduces the computational effort considerably relative to the γ -Reθtransition model.The NSST model has a benign property in speculating Reynolds-stress transport effects since Reynolds stresses are related to k and can handle various types of flow phenomena.The secret is that modeling constituents of the RANS closure are‘‘flow-structure-adaptive”.Presumably,different transitional/non-transitional mechanisms naturally appear at the evolution of turbulent transport equations.The influence of cross-flow transition will be introduced with the NSST model in the near future.Conclusively,transition simulations with the NSST model consider the free-stream turbulence intensity Tu in a range of 0.1%–6% and the corresponding range of Rμis 0.1–2.0.This information approves that the NSST model can avoid the‘‘trial-and error”inconsistency (associated with the correlation-based γ -Reθtransition model for precise computations)to a greater extent.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by Hangzhou Dianzi University Research Supporting Fund of Zhejiang Province, China (No.GK218803299037).