Accurately predicting hypersonic transitional flow on cone via a symmetry approach

Weito BI, Kexin ZHENG, Zhou WEI,b, Zhensu SHE,*

a State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China

b State Key Laboratory of Aerodynamics and Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China

KEYWORDS

Abstract A new algebraic transition model is proposed based on a Structural Ensemble Dynamics(SED) theory of wall turbulence, for accurately predicting the hypersonic flow heat transfer on cone.The model defines the eddy viscosity in terms of a two-dimensional multi-regime distribution of a Stress Length(SL)function,and hence is named as SED-SL.This paper presents clear evidence of precise predictions of transition onset location and peak heat flux of a wide range of hypersonic Transitional Boundary Layers(TrBL)around straight cone at zero incidence,to an unprecedented accuracy as validated by over 70 measurements for varying five crucial influential factors (Mach number, temperature ratio, cone half angle, nose Reynolds number and noise level).The results demonstrate the universality of the postulated multi-regime similarity structure, in characterizing not only the spatial non-uniform distribution of the eddy viscosity in hypersonic TrBL on cone,but also the dependence of the transition onset location on the five influential factors.The latter yields a novel correlation formula for transition center Reynolds number which takes similar functional form as the SL function within the symmetry approach.It is concluded that the SED-SL model simulates TrBL around cone with uniformly high accuracy, and then points out to an optimistic alternative way to construct hypersonic transition model.

1.Introduction

During the past decades,there have been tremendous interests in studying aerodynamics of high-speed flying vehicles, and a very challenging problem is the prediction of transition in Hypersonic Boundary Layer (HBL) for its first-order impact on the aerodynamic heating, drag, and control.1Compared to low-speed flows, HBL transition is subjected to more transition-influential factors and more instability mechanisms.2Anderson Jr3has concluded that the transition onset Reynolds number of HBL, in particular, depends on nearly twenty factors, whose accurate prediction is tremendously difficult.Formulating empirical correlation to quantify the effects of the many influential factors has been a longlasting research target, but few results are satisfactory, owing to the complexity from the interplay of multiple influential factors and from the wide variation in flow regimes in practical situations.2The latter becomes more intriguing in order to satisfy engineering demands.Specifically, at high speed and at varying attack angle,HBLs are governed by different instability mechanisms,4,5which may occur simultaneously on different surface areas,6resulting in great complexities in modeling and simulating HBL transitions.Note that nearly all transition models need to assume a correlation equation to define the transition onset,7and it is widely adopted that such correlation requires a correct description of the transition mechanism.8A consequence is that as more transition mechanisms (due to variation of many influential factors) come to interplay, transition models for HBL become increasingly more complicated,but with an accuracy either insufficient or even undetermined in practical situations.9To overcome this bottleneck, alternative approaches for transition prediction have been prospering during the past decade, towards unsteady simulations such as Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES),10,11and data-driven method such as machinelearning-aided modeling.12,13Effort is also made to improve the existing turbulence models via an uncertainty-sensitivity study.14Honestly speaking, all current approaches (DNS,LES, etc.) have made limited success for disentangling the complexities regarding variations of instability mechanisms;in particular, they reveal little physical understanding on the principle governing the HBL transition.

We propose a reflection that one should perhaps go beyond the conventional strategies focusing on detailed transition mechanism for constructing a transition model, but, instead,to discover a universal principle governing Transitional Boundary Layers (TrBL).A transition model under such a principle(once discovered)will have a significant improvement in both simplicity and accuracy.In this paper,we report a progress in this direction.First,we propose a notion of‘‘similarity structure”of TrBLs, which ensures a kind of similarity of the flows for varying Reynolds number(Re),Mach number(Ma),and other influential factors,to be defined below.Importantly,the similarity structure can only be‘‘seen”with proper similarity variable, which displays the right symmetry, which, for TrBLs, is the symmetry under dilation transformation from the wall in the wall-normal direction and that from the leading edge in the streamwise direction.The similarity variable is postulated to be a length being the characteristic size of eddies most relevant to turbulent transport of momentum.It has been identified in the Structural Ensemble Dynamics (SED) theory15,16to be the Stress Length (SL) function defined by the Reynolds stress and the mean shear,thus giving rise to an algebraic transition model called SED-SL.17,18

Previously, we have shown that the SED-SL model can be applied to airfoil flows at different subsonic Ma,Re and angles of attack with an unprecedented accuracy19(up to a few counts in the drag prediction),indicating the validity of the model and also the concept of the similarity structure.This work extends the SED-SL to hypersonic TrBLs to assess its potential merit in realistic high-speed applications, by focusing on straight cones at zero incidence, which is a benchmark configuration for high-speed flows and possesses a large amount of experimental and flight data, under varying multiple influential factors.2Currently, we limit ourselves to simple transition mechanism at zero incidence, as comparing with those with non-zero incidence and non-canonical configuration,although preliminary study shows also encouraging results for flows at moderate incidence.20The flows have been studied for over seventy years and documented in numerous reports and papers, especially in a series of review articles.2,21,22However,uniformly accurate description of data by turbulence model is not yet available;in other words,no transition model is able to accurately predict the surface heat flux distribution uniformly well for a wide range of flow parameters.9,23

We present here clear evidence of a uniformly accurate prediction of transition onset location and peak heat flux for a wide range of hypersonic Transitional Boundary Layers(TrBL) on cone at zero incidence, to an unprecedented accuracy, validated by several dozens of measurements in varying five crucial influential factors (Mach number, temperature ratio,cone half angle,nose Reynolds number and noise level).A crucial finding is a novel symmetry-based correlation between the transition center Reynolds number(to define transition onset) and the five transition-influential factors.This correlation appears much simpler than the traditional correlation relations,because of the similarity nature of the transition center Reynolds number.These results demonstrate the universality of the postulated multi-regime similarity structure,which suggests an optimistic alternative way to construct hypersonic transition model.

The remainder of this paper is organized as follows.Section 2 explains the theoretical basis of the model and describes in detail the construction of the SED-SL model and the numerical method.Section 3 validates the SED-SL model with several sets of experimental data.Section 4 discusses the results and concludes the paper with future perspectives.

2.SED-SL model and numerical implementation

2.1.Symmetry principle for TrBL and its physical interpretation

A fundamental question in the theoretical study of wall turbulence is how to quantify the mean flow property as a function of flow condition, and a promising road is to study it through the concept of invariance or similarity.Dimensional analysis is the most well-known example of displaying a dilation invariance under the variation of physical units.In TrBL,two spatial dilation invariance groups exist, namely one away from the wall in the wall-normal direction and another away from the leading edge in the streamwise direction.As explained in Refs.15,16,18, when dilation group invariance encounters a symmetry-breaking, due to turbulent fluctuations which induce a random-dilation transformation,a generalized invariance principle can be formulated to yield, in the wall-normal direction, a four-layer structure (describing viscous sublayer,buffer layer, log-layer and wake region), and similarly in the streamwise direction, a three-layer structure to describe laminar-turbulence transition.This generalized dilation invariance reads:F(x)=cxα[1+(x/xc)p]γ/p=cxαΠ(x),which formulates a power-law jump from xαto xα+γat xc(with p called transition sharpness, often taken to be a big positive integer like 4).Here Π(x) = [1+(x/ xc)p]γ/pis called a universal dilation ansatz, or dilation-symmetry-breaking principle because it is quite universal.We will see that this principle will continue to hold when one formulates the law governing the variations of the SL function with major influential factors, such as Ma,etc.The results presented below demonstrate that discovering such an invariance principle helps significantly to find efficient RANS model for TrBL, definitively for hypersonic flows.

Note that choosing the SL function (instead of the mean velocity)is crucial to identify similarity structure of TrBL.This function is argued15,16to be the right variable to display the simplest dilation symmetry-breaking to form a multi-layer structure,which,as shown below,can be expressed with a product of multiple factors, each for one layer, to describe the whole profile.It is rather encouraging to see the similar multi-layer structures to work well in the wall-normal and the streamwise directions, which is a strong evidence that the generalized invariance principle is universal for TrBL.This is a signature of the self-organization principle for the ensemble of turbulent eddies in TrBL under the constraint of the wall,so that flow properties in different regions of TrBL are similarly linked.

We present below detailed evidence that a twodimensional multi-regime structure is valid to describe TrBL around a significant engineering model: cone.More strikingly, the universal dilation ansatz can be applied to the phase space of the influential factors (Ma, Tw, etc.), revealing a multi-state similarity of boundary layers under varying physical conditions.The latter yields a model to engineering applications.

2.2.SED-SL model

The HBL transition on cone can be described with the Favreaveraged form of RANS equations, which, under the thinlayer and Boussinesq approximations and assuming constant turbulent Prandtl number, are written as follows:

where overbar denotes Reynolds averaging and tilde denotes Favre averaging; ρ is fluid density, Ujis the j-th component of velocity, and xjrepresents the Cartesian coordinate.P=ρRT is the static pressure(R is the gas constant)and total entropy H-= ～E+P-/ρ-,where total energy E=cvT+K with cvbeing the specific heat at constant volume, T the static temperature and K the kinetic energy of the flow (K = UkUk/2).The molecular Prandtl number Pr is 0.72, and the turbulent Prandtl number Prtis set 0.9.The stress σ-ijis expressed by using the Boussinesq approximation:

Under the effects of varying influential factors (such as pressure gradient, Ma and Tw), the boundary layer becomes non-canonical, with a deformed four-layer structure.If the SL is the right similarity variable for non-canonical TBL, the deformation will be smooth, according to the SED theory, as long as the wall constraint remains dominant.Indeed, as one judges from the results below,TrBL appears to have only finite deformation being slow and continuous.Specifically,all deformation can be described by smoothly varying two multi-layer structure parameters: l0+and yb+uf, which determine the global size and location of the strongest eddies (which are vulnerable to environmental variation) and thus characterise the major variation of the boundary layer.This assertion has been validated as explained below.

To describe the TrBL,Xiao and She18postulated a streamwise multi-layer dilation with respect to the leading edge (i.e.x = 0) for l0+and yb+uf.In the current study, l0+and yb+ufare written as follows:

Function f(Rex)describes a streamwise three-layer development (Fig.1) for l0+and yb+uf.It includes two parameters:Rex*and β.Rex*= Re∞x* is the transition center Reynolds number - a newly-identified transition Reynolds number defined with x* being the transition center18.Rex*as a crucial engineering concern requires further modeling, which is analyzed in Section 2.3.β larger than unity physically defines a transition regime between the laminar and fully-turbulent regimes.It is the ratio of the location where the transitional boundary layer begins to relax to the fully-developed TBL to the location of the transition center.The larger the β,the wider the transition regime,and the stronger the transition overshoot(a phenomenon that the skin-friction coefficient and Stanton number at around the transition end considerably surpass their TBL values at the same Reynolds number).For most by-pass transitions, β is about 1.1.l0∞+in Eq.(10) denotes the nearwall eddy length of the fully-developed TBL and determines the magnitudes of skin friction and surface heat flux after transition.For most not-far-from-adiabatic compressible TBLs with less than moderate pressure gradient,we have found that l0∞+is about unity, i.e.the canonical TBL value, which is set in the calculations of Horvath et al.’s experiments in Section 3.1, and a slightly smaller value of 0.65–0.7 in the calculations of Grossir et al.’s experiments in Section 3.2 and 3.3; the latter takes into account the significant compression effect on the eddy size.Note that this adjustment of l0∞+is the only variation of our model parameter,and should be considered to be minor,in view of the cases covered.We leave the modeling of the (minor) variation of l0∞+for future study.

Fig.1 Function f(Rex)with Rex*=5.7× 106 and β=1.1(solid line) displays a streamwise three-layer development, compared to that with β=1.0(dash-dot line).The vertical dashed lines denote the locations of Rex* and βRex*.

2.3.A symmetry-based correlation for transition center Reynolds number

Rex*indicates the transition onset location and requires further modeling.In the study of the so-called canonical (or simplest) TrBL (incompressible, zero-pressure-gradient, smooth flat-plate TrBL with proper leading edge and subject to isotropic free-stream turbulent perturbations), Xiao and She18proposed that Rex*possesses a two-state similarity with the incoming turbulence intensity Tu: Rex*=3.3 × 106[1+(Tu/0.65%)4]-1.5/4, constructed with the universal dilation ansatz of the SED.15The new correlation is superior to the precious transition criteria (which often use Rexor Reθas the variable) in both prediction accuracy and physics (Rex*is a similarity variable of TrBLs, like yb+uf).18It is interesting to investigate if this one-variable correlation can be extended to describe the multiple influential factors in the hypersonic TrBLs over straight cones.

In the cone flows, the influential factors with remarkable effects on Rex*include Ma,unit Re,the wall and total temperatures (Twand Tt), the incoming noise level, nose bluntness,cone half angle, roughness, angle of attack, etc.2Note that the current study focuses on zero angle of attack in order to significantly minimize the complexity, thus removing the transition mechanisms owing to cross-flow and streamwise vortex instability that occur at a non-zero angle of attack.This restriction has been widely used in the previous studies on cone flows.Also excluded is the surface roughness effect because of two reasons.First, roughness directly exerts inside the boundary layer and often dominates the transition and undermines the other effects.Second, roughness elements are of a wide variety in shape,size and distribution,and require more effort.Therefore, to cone flow with smooth surface, we propose the following symmetry-based correlation for Rex*:

where Tu is the free-stream noise level (Tu = prms/P, where prmsand P are the root-mean-square and mean static pressure of the incoming flow,respectively),Maeis Ma at the boundary layer edge,Tawis the adiabatic wall temperature,θ is the cone half angle, ReNis the nose Reynolds number (ReN= Re∞RNwhere RNis the nose-tip radius),and Recis a limiting Rex*for sharp cone at low intensity noise and moderate Mach number.Note that in principle the unit Re (i.e.Re∞) should not affect Rex*for a smooth,sharp cone,although it is not supported by most wind tunnel experiments.2Here we employ the viewpoint that the widely-observed Re∞-effect on transition is actually due to the varying intensity of pressure fluctuations radiating from the TBL on the wind tunnel wall as Re∞is changed.27Thus, we take Tu instead of Re∞as a key influential parameter.

Eq.(13) boldly assumes that Rex*independently correlates with five dimensionless parameters, which, though appearing to be questionable, may be a valuable first step, which was already used before;28in any way,it will be assessed by experimental data.A major difference between the present modeling and the previous ones is that the functions (Θ, Φ, etc.) in Eq.(13) all take similar form, as we apply the universal dilation ansatz of the SED; in other words, the multi-state similarity takes similar form in the physical space and the phase space of physical circumstances.TrBL is an excellent example for us to test the validity of this similarity principle.By incorporating available experimental data,these functions are determined as follows:

Here, we assume that Rex*is a product of five dimensionless functions,all of which are needed to describe experimental cone flows studied below, with each displaying a (generalized)power law.Somehow, the expression mimics the classical dimensional analysis expressing exact dilation invariance with changing physical units; however, Eq.(13) describes a more generalized similarity structure fully specifying the TrBLs over straight cones at zero incidence.With Eq.(13), the current RANS model is fully closed and the streamwise development of mean profiles are fully specified, including in particular the surface heat flux which is the target for comparison.Results presented below demonstrate the validity and the accuracy of this description,although future extension to cone flow with non-zero incidence may be non-trivial.Once fully established, it represents a major accomplishment in the study of TrBL for two reasons: one is the discovery of a universal feature in terms of the (generalized) power laws, and the other is the revelation of a series of critical parameters such as Maec≈5,θc≈5°,and Tuc≈0.65%,which separate different flow regimes.These two features differentiate the present model from previous correlation formula which yield few physics.Quite extensive validations of the power laws are presented in Section 3 for over 70 cases of cone flows.

We add a note to explain how the parameters in Eqs.(14)–(19) are determined.Note that once the functional form is fixed, the number of parameters is not many.For each factor,the dilation invariance ansatz includes just two empirical parameters: a threshold separating two regimes (e.g.0.65%in Eq.(19) separating natural from bypass transition), and a scaling exponent describing the second regime.So, the values can be determined by data of a few cases via a posterior test.Some difficulty may appear, since some parameters may change together in actual experimental conditions (e.g.changing Ma will change Tu in wind tunnels),but we have solved the problem by acquiring more data from the literature (which,due to space limitation, will be communicated elsewhere).

2.4.Numerical implementation

Two sets of grid have been applied for the computation.The first one is of a 7° half-angle sharp cone, which has a uniform mesh with grid number 31 in the circumferential direction(from 0°to 180°,i.e.a half-model simulation), an increasingly coarsening mesh with grid number 173 in the axial direction,and a stretching mesh with grid number 131 in the wallnormal direction.The physical dimension of the grid is:1275 mm of cone length, and 1325 mm of radius in the end plane of the cone.It has been verified that the first wallnormal mesh has a dimension of y+<0.8 over the whole cone surface for all simulations.Grid convergence is tested by doubling the grid number in both the circumferential and axial directions without observing noticeable differences on the surface heat flux distribution.The second grid is of a 5°half-angle sharp cone and similar to the first one.

The RANS equations have been solved by using CFL3D on a workstation with 16 CPU cores.Free-stream boundary condition is used for the inflow.Isothermal, no-slip condition is employed on the cone surface.Symmetry condition is applied on the symmetry plane(at circumferential angles 0°and 180°).Extrapolation condition is used for the outflow.As to the numerical algorithms, an implicit approximate-factorization method is applied for time advancing.The viscous fluxes are computed with the second-order central difference, and the inviscid fluxes are computed with the upwind flux-differencesplitting method.

Table 1 Flow conditions of Horvath et al.29(Ma = 6, θ = 5°, and RN = 0.00254 mm).

Fig.2 Normalized surface heat flux distributions predicted by SED-SL compared with experimental results and some of the most recent transition models for Cases H6a-H6h.

3.Validation and assessment

We first present the predictions of three groups of well-known experimental measurements, and then further establish an empirical relation between the presently introduced transition Reynolds number and traditionally used onset Reynolds number; this relation allows us to directly validate the proposed correlation formula.

3.1.Prediction of experiment of Ref.29

Here,we present the validation of the SED-SL model by comparing the numerical prediction of the surface heat flux with the experimental data of Horvath et al.29, which is a benchmark data set widely employed to validate hypersonic transition models.30–32The eight test cases possess the same temperature ratio Tw/T∞but different Re∞(Table 1).Case H6f has a smaller stagnation temperature compared with the others in order to study the effect of the absolute magnitude of temperature.The surface heat flux data are presented by using h/href, where h = q/(Haw- Hiw), q is the heat transfer rate, Hawand Hiwrepresent the adiabatic and isothermal wall enthalpy respectively,and hrefis the reference heat transfer rate based on the Fay-Riddell theory.

Note that Horvath et al.only provided a reference tunnel noise level of 2.8%-3.4% in their original paper.To predict Rex*for each flow case, the tunnel noise levels have been estimated by using the current correlation equations (Eqs.(13)–(19)), and listed in Table 1, which are quite close to the reference level of Horvath et al.29In the computation, l0∞+is set unity and β is 1.1, as the recommendation.Fig.2 shows the comparisons between the SED-SL and the experimental data,and the agreements are excellent for all test cases,remarkably.The differences between the numerical predictions and measurements with respect to RexT(xTis the location with minimum surface heat flux) are below 10% and mostly below 5%,with Stpeak(the peak dimensionless heat flux)being within 4%.As displayed in Fig.2,this is an outstanding record,compared with the most recent transition models in the literature,including Qin et al.32, Guo et al.33, Qiao et al.34and Liu et al.35

Owing to experimental uncertainties with respect to the incoming flow conditions,the sensitivity of the model is tested by assuming 5% deviations from the claimed values for the total temperature and noise level, respectively.As shown in Fig.3, when either the total temperature or the noise level is slightly increased (or decreased), the predicted transition location mildly moves upstream (or downstream), consistent with the conventional understanding of the wall-cooling and noise level effects on the hypersonic transition.

3.2.Prediction of high Ma experiment of Ref.36

Grossir et al.conducted a high Ma experiment at the Longshot hypersonic facility of the VKI in Belgium.36Maeis over 8, so the TrBLs are truly hypersonic.Tw/Tawis about 0.2, so the cone surface is significantly cooled.The facility is based on the principle of a gun tunnel with a piston used to compress the test gas.During the compression, the experimental conditions in the test section vary with time, so that the freestream Re,Ma,Ttand noise level are not constant.In the current simulation, the flow conditions (listed in Table 2) are set for the test cases according to the decaying properties and the noise level properties of the tunnel provided by Grossir et al.36,37

Grossir et al.employed a Stanton number with reference to the local flow conditions, which is expressed as Stlocal=q/[ρeUe(rHt- Hw)], where q is the heat flux, r is the recovery factor, Htis the stagnation enthalpy, and subscript e denotes boundary layer edge.Fig.4 shows the Stlocaldistributions predicted by the SED-SL for Cases G10a-G10d, compared with the experimental data.The numerical simulations accurately predict the surface heat flux distributions for all the test cases.As shown in Table 2, the differences between the numerical simulations and measurements with respect to RexTand Stpeakare very small, considering the measurement uncertainty.

A remarkable feature of Grossir et al.’s cases is l0∞+～0.65–0.7 < 1, as shown in Table 2; this decrease of l0∞+for Cases G10a-G10d is interpreted as the result of high compressibility and strong wall-cooling,both of which have a squeezing effect on the near-wall eddy size.A further analysis and modeling of l0∞+on Ma and Tw/Tawshould be conducted in the future with diagnostics of more experimental data in a wider parameter range.

Fig.3 Surface heat fluxes predicted by SED-SL for Case H6f with perturbations of total temperature, and noise level.

At larger Ma (Ma∞>12), the ionized gas effects play roles and the current setting of the model with respect to the fluid properties, turbulent Prandtl number, etc., needs to be modified accordingly.But the self-organization principle would still apply,so that the additional effect can be described with moderate variations of the model parameters.In other words, we believe that the multi-regime structure of the TrBL should preserve as the wall-constraint;this assertion needs to be tested in the future.

3.3.Prediction of blunt cone cases of Ref.36

Fig.5 Stanton number distributions predicted by SED-SL compared with experimental results for Cases G10b, G10e, and G10f.

Table 2 Simulation conditions of Grossir et al.36′s experiments(θ = 7° and β = 1.1 in the computation).

Fig.4 Stanton number distributions predicted by SED-SL compared with experimental results for Cases G10a-G10d.

Fig.6 Predicted RexT and Rexp by SED-SL compared with measured ones for different test cases(Errors are all bounded within 10%).

Fig.7 Validation of Rex* vs Mae relationship(Symbols denote flight and wind tunnel data and lines denote models).Note that stars of different color represent different experiments collected by SFW.

Grossir et al.also studied the nose bluntness effect in their experiments.36As shown in Table 2, Cases G10b, G10e and G10f are measured under the same tunnel condition but of different nosetip radius.ReNof Cases G10b, G10e and G10f are 2200,1.925×104and 5.7×104,respectively.So,according to Eq.(15), Case G10b is sharp, the other two cases are of small bluntness,and Rex*of Cases G10e and G10f are 1.43 and 3.12 times that of Case G10b respectively.Fig.5 compares the measured and the SED-SL predicted Stanton number distributions of the three cases.The simulation results agree with the experimental ones very well, showing that the current model correctly predicts the nose bluntness effect on the hypersonic transition of straight cone.

3.4.Validation of the correlation equation of Rex*

Now,let us present a direct validation of the correlation equation of Rex*by using experimental data.Note that Rex*is a new concept and has not been studied in the literature.One uses mostly the so-called transition onset Reynolds number RexT(defined with the location with minimum surface heat flux) and Rexp(defined with the location with maximum surface heat flux), to quantify the transition onset.Since our RANS calculation predicts the entire surface heat flux profile very accurately(as shown from the results of the last three subsections), it is expected that our predicted RexTand Rexpare also close to experimental measurement.Fig.6 shows the comparison between predicted RexT(and Rexp) by our RANS calculation and the measurements, which is indeed the case:relative errors are all bounded within 10%.This forms a basis for us to establish an empirical relation between Rex*and RexT(and Rexp).

Next, Maehas important influences on the stability and transition of compressible boundary layers.When Mae> 4,the Mack second mode dominates and Maehas a stabilizing effect on the second mode, and thus the transition onset is postponed with increasing Mae.Eq.(16) is a modeling of the two-state influence of Maeon Rex*.A difficulty with respect to wind tunnel experiment is that Maecannot be varied under a fixed tunnel condition for a fixed cone27,so that the Tu-effect can hardly be removed.One method is to utilize flight data that have been measured with nearly zero incoming noise.Fig.7 shows the variation of Rex*with Maefor several sets of flight data (listed in Table 3), compared with Eq.(16) and Beckwith’s wind tunnel data correlation.39Eq.(16) depicts the data with a reasonable accuracy.Fig.7 also includes five wind tunnel cases collected by Stainback,Fischer and Wagner(SFW) for sharp cones measured at different tunnels but with the same noise level and the same temperature ratio.40The cone half-angles are different, which does not affect the relationship between Rex*and Maeaccording to Schneider.2So the data set is qualified.As shown in Fig.7, the SFW data are close to the current model, especially for the slope in the hypersonic regime.For the current data sets,Beckwith’s correlation displays more deviations.Note that Beckwith’s correlation was attained by a sixth order polynomial data fitting,28,39which cannot describe the Ma-similarity of the TrBLs, in principle.

Note that Figs.2-7 contain more than 70 cases reported from both wind tunnel and flight data,which constitute a solid basis for supporting the validity of the correlation formula(Eqs.(13)–(19)).Furthermore, the validation has been extended to more cases (e.g., with different RN, different surface temperature, etc.) with very encouraging results, which,due to space limitation, will be reported elsewhere in the near future.

To conclude, the symmetry-based correlation Eqs.(13)–(19)are validated with uniformly high accuracy beyond similar work recently published in leading journals.Nevertheless, we do not consider the present setup as a mature model for engineering applications; more data need to be studied in thefuture to investigate the applicable scope of the new correlation formula.Also more influential factors will be explored,such as surface roughness, different spectrum of incoming noise, absolute value of total temperature, etc.

Table 3 Flow conditions of straight cone test cases used for validating the correlation formula (Eqs.(13)–(19)).

4.Conclusions

(1) The predictions of the SED-SL model are compared with the measurements of the hypersonic heat transfer on zero-incident straight cones, in more than about 70 cases covering a significant range of five important influential factors (including Ma, temperature ratio, noise level,cone half-angle and nose bluntness),which demonstrate an unprecedented uniform accuracy ever achieved before.The success represents a significant advancement in RANS modeling of engineering wall flows, in two aspects.First, it is a sign that an essence (i.e.true similarity structure) of wall turbulence is captured, namely,the universal multi-regime description of the SL function defines the invariant distribution of the eddy viscosity.Second, it offers a more optimistic view for future RANS modelling of the transition: accurate prediction is possible when fundamental understanding of wall turbulence is taken into consideration.

(2) A crucial revelation is that three physical parameters(i.e.Rex*, l0∞+, and β) of the SED-SL are able to parameterize a variety of TrBLs.The most important one is Rex*that possesses a symmetry-based correlation with five crucial transition-influential factors, depicting the multi-state similarity of the TrBLs over zeroincident straight cones.l0∞+is a coefficient quantifying globally the eddy size for after-transition TBL, and it decreases from unity when strong wall-cooling or very high Mach numbers take place.This corrects surface heat flux in after-transition region that is usually overestimated by most RANS models.β models the transition overshoot strength by inserting a ‘‘transition”zone,which is very physical,since in most cases the transition is not so abrupt.

Future study will explore the boundary of the validity of the concept of universal multi-regime similarity structure in complex TrBLs.We are confident that as more cases are studied, a well-documented and widely-adaptive transition model for industrial application will appear, which we will pursue in the future.Additional merits of the present algebraic model are noteworthy.First,it may produce an alternative and better(more smooth and accurate) wall-function constraint for unsteady simulations such as DES or LES.Second, it paves a way to a deeper understanding of the transition physics,which have been the main bottlenecks in the current transition model studies.

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 study was supported by the National Numerical WindTunnel Project, China (No.NNW2019ZT1-A03), and the National Natural Science Foundation of China (Nos.91952201, 11372008 and 11452002).