Chinese Journal of Aeronautics, (2016), 29(1): 66–75

Chinese Society of Aeronautics and Astronautics& Beihang University

Chinese Journal of Aeronautics

cja@buaa.edu.cnwww.sciencedirect.com

Boundary-layer transition prediction using a

simplified correlation-based model

* Corresponding author. Tel.: +86 571 87951391.E-mail addresses: xcc578394953@163.com (C. Xia), chenwf-

nudt@163.com (W. Chen).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2015.12.0031000-9361 � 2015 The Authors. Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Xia Chenchao, Chen Weifang *

School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China

Received 27 January 2015; revised 19 May 2015; accepted 21 June 2015

Available online 23 December 2015

KEYWORDS

Boundary-layer transition;

Computational fluid

dynamics;

Correlation;

Transition model;

Turbulence model

Abstract This paper describes a simplified transition model based on the recently developed

correlation-based c� Reht transition model. The transport equation of transition momentum thick-

ness Reynolds number is eliminated for simplicity, and new transition length function and critical

Reynolds number correlation are proposed. The new model is implemented into an in-house com-

putational fluid dynamics (CFD) code and validated for low and high-speed flow cases, including

the zero pressure flat plate, airfoils, hypersonic flat plate and double wedge. Comparisons between

the simulation results and experimental data show that the boundary-layer transition phenomena

can be reasonably illustrated by the new model, which gives rise to significant improvements over

the fully laminar and fully turbulent results. Moreover, the new model has comparable features of

accuracy and applicability when compared with the original c� Reht model. In the meantime, the

newly proposed model takes only one transport equation of intermittency factor and requires fewer

correlations, which simplifies the original model greatly. Further studies, especially on separation-

induced transition flows, are required for the improvement of the new model.� 2015 The Authors. Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. This is an

open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Boundary-layer transitions arise in the majority of applica-tions of aeronautics and astronautics, such as the airfoil of asubsonic passenger plane, the turbofan engine and the hyper-sonic reentry vehicle. Usually, skin friction and heat transfer

rate increase significantly when transition occurs, which couldlead to the increase of drag or the aggravation of aerodynamic

heating. These troublesome uncertainties on aerodynamic oraerothermodynamic characteristics of aircraft necessitate theaccurate prediction of boundary-layer transition, not only

from the view point of economy, but also safety. Complexthough the physics of transition is, study of boundary-layertransition has been the topic of great importance and urgency

over the past several decades, and substantial numbers of sat-isfactory findings have been achieved.

As an effective means of exploring the mechanism of fluid,experiment plays an important role in revealing instability phe-

nomena of boundary layer and finding new flow scenarios.1

Typical works can be traced from Lee and Wu, who reviewedplenty of experimental results for wall-bounded flows.2 Besides

Boundary-layer transition prediction using a simplified correlation-based model 67

experiment, several methods have been developed to predictthe phenomena of transition. Simply, these methods could beclassified as four main categories: empirical correlation

approaches, methods based on stability theory, transportequation models based on Reynolds averaged Navier–Stokes(RANS) solver and high-accuracy methods such as large eddy

simulation (LES) or direct numerical simulation (DNS). Theempirical correlation approaches3 which originate from exper-imental data are the simplest ways of transition onset predic-

tion. However, they usually encounter the problem ofnumerical implementation, as well as the inadequacy of gener-ality in three-dimensional flows. The DNS method4 is mostaccurate and universal for the overwhelming majority of flows,

while it is extremely time-consuming and may be unrealistic atpresent for applications of complicated flows. By contrast, theLES method offers lower cost and has been applied to many

transitional flows.5,6 With the development of high-performance computing, the LES has been used for simulationof complex flows and has been one of the most promising

methods in engineering problems.7 Relatively, the methodsbased on stability theories and RANS solvers are affordablechoices for engineering problems, considering their merits of

accuracy and efficiency. In practice, it is widely accepted bythe boundary-layer transition community that the eN

method,8,9 which is based on the linear stability theory, isone of the most effective methods for aeronautical flows for

the moment. Although the eN method has been introducedfor approximately half a century and plenty of successfulapplications have been achieved, challenges still exist when try-

ing to apply it to three-dimensional complex flows. Precisely,the method demands the integral of the growth of disturbancealong the streamline, which is not an easy task for complex

grid systems of modern CFD. Additionally, the method issemi-empirical in nature, since the ‘N’ value is not universaland should be calibrated for different experimental data from

wind tunnels, which limits its generality. Transition modelsbased on RANS solvers, which has been proposed in the pastfew years, is the combination of transport equations withRANS framework. Pioneer works can be traced to Steelant

and Dick,10 Suzen and Huang,11 Langtry and Menter,12 whoselected the intermittency factor as a transport equation, aswell as Walters and Leylek,13 who proposed a transport equa-

tion for the laminar kinetic energy. Among these works, thelocal correlation-based c� Reht transition model proposed byLangtry and Menter has gained much attention and has been

validated for a wide range of applications, such as airfoils, tur-bomachineries, and even high speed flows.14–17 By now, themodel has been widely considered to possess the advantagesof high accuracy, simple implementation and good generality,

and has been incorporated into many pieces of popular com-mercial software, such as Fluent, CFX, STAR-CCM+ andCFD++. Recent studies related to c� Reht model include

the extension of the model to one-equation Spalart–Allmarasturbulence model,18 the development of a new transition modelbased on stability theory,19 and the extension of the model for

simulation of transition caused by crossflow instability.20

Themajor goal of this work is to simplify the implementationof the c� Reht transition model by neglecting the transport

equation of transition momentum thickness Reynolds number.The idea was inspired by the work of Coder and Maughmer,21

in which the transport equation was replaced by an algebraiccorrelation with no loss of accuracy and generality of the origi-

nal model. In this study, new correlations for Flength andRehc are

introduced to control the length of transition region and transi-tion onset respectively, and the single intermittency factor trans-port equation is coupled with the two-equation shear stress

transportation (SST) turbulence model, resulting in a three-equation transition model. The new model is implemented intoan in-house CFD solver, followed by several simulations andanalyses for the evaluation of its performance. Finally, conclu-

sions and recommendations are made at the end of this paper.

2. Turbulence and transition modeling

2.1. Two-equation SST turbulence model

The two-equation SST turbulence model, originally developedby Menter,22 is the combination of k� x and k� e turbulencemodels. It can be switched from k� x model near the wall tok� e model away from the wall through well-designed blend-ing functions, which will be defined in the following text. The

SST model has been applied to large quantities of flows andshows excellent performances of accuracy and robustness.Overall, it has been regarded as one of the most successful tur-bulence models, not only in the area of aeronautics, but also in

the industrial community. Hence, all the fully turbulent simu-lations in this work are performed by SST model. The trans-port equations for turbulent kinetic energy and specific

dissipation rate are as follows:

@ðqkÞ@t

þ @

@xj

qujk� ðlþ rkltÞ@k

@xj

� �¼ Pk � b�qxk ð1Þ

@ðqxÞ@t

þ @

@xj

qujx� ðlþ rxltÞ@x@xj

� �¼ v

mtPk � bqx2

þ 2ð1� F1Þqrx2

x� @k@x@xj@xj

ð2Þ

where q is density, t is time, k is turbulent kinetic energy, uj is

velocity component, xj is coordinate component, x is specific

dissipation rate. l and lt are laminar and turbulent eddy vis-

cosity respectively, mt is kinematic eddy viscosity, b�, b, v, rk,

rx, and rx2 are model constants. F1 is the blending functionand defined as:

F1 ¼ tanhðarg41Þ

arg1 ¼ min max

ffiffiffik

p

0:09xy;500lqy2x

!;4qrx2

CDkxy2

" #

CDkx ¼ max 2qrx21

x� @k@xj

� @x@xj

; 10�10

� �

The variable y is the distance from the cell to the nearestwall. Instead of computing the production term exactly, thevorticity magnitude is adopted for approximation:

Pk ¼ ltX2 � 2

3qkdij

@ui@xj

; Pk ¼ minðPk; 10b�qxkÞ

where X is the magnitude of vorticity and defined as:

X ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2WijWij

p; Wij ¼ 1

2

@ui@xj

� @uj@xi

� �The kinematic eddy viscosity is calculated by

mt ¼ a1k

maxða1x;SF2Þ

68 C. Xia, W. Chen

where the constant a1 ¼ 0:31, S is the invariant of strain rate

and defined as:

S ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi2SijSij

p; Sij ¼ 1

2

@ui@xj

þ @uj@xi

� �

The blending function F2 is defined as

F2 ¼ tanhðarg22Þarg2 ¼ max 2

ffiffik

p0:09xy ;

500my2x

� �Coefficients of the model are calculated from the formula

/ ¼ F1/1 þ ð1� F1Þ/2

The subscript ‘‘1” is for the k� x model and subscript ‘‘2”

is for the k� e model. Two sets of coefficients used at presentare: rk1 ¼ 0:85, rx1 ¼ 0:50, b1 ¼ 0:075, v1 ¼ 5=9, rk2 ¼ 1:0,rx2 ¼ 0:856, b2 ¼ 0:0828, v2 ¼ 0:44.

2.2. Description of c� Reht and simplified transition model

The c� Reht transition model is a local correlation-basedmodel, in which the localization is achieved by the definition

of a vorticity Reynolds number and the formulation of a trans-port equation for transition momentum thickness Reynoldsnumber. Moreover, the intermittency factor is used for the

control of production of turbulent kinetic energy. The trans-port equation for intermittency factor is listed as

@ðqcÞ@t

þ @

@xj

qujc� ðlþ lt

rcÞ @c@xj

� �¼ Pc þ Ec ð3Þ

where c is the intermittency factor.

The production and destruction terms of intermittency fac-tor are

Pc ¼ Flengthca1qSðcFonsetÞ0:5ð1� ce1cÞ ð4Þ

Ec ¼ ca2cqXFturbð1� ce2cÞ ð5Þwhere Flength is the transition length function, Fonset controls the

onset of transition and defined as:

Fonset ¼ maxðFonset2 � Fonset3; 0Þ

Fonset1 ¼ Rev2:193Rehc

Fonset2 ¼ minðmaxðFonset1;F4onset1Þ; 2:0Þ

Fonset3 ¼ max 1� RT

2:5

� �3

; 0

!

with

RT ¼ qklx

; Rev ¼ qy2Sl

The transport equations for transition momentum thick-ness Reynolds number is formulated as

@ðq ~RehtÞ@t

þ @

@xj

quj ~Reht � rhtðlþ ltÞ@ ~Reht@xj

� �¼ Pht ð6Þ

where ~Reht is the transition momentum thickness Reynoldsnumber, the source term is

Pht ¼ chtqtðReht � ~RehtÞð1:0� FhtÞ ð7Þ

with

Fht ¼min max Fwake exp � y

d

� �4� �;1:0� c�1=ce2

1:0�1=ce2

� �2" #

;1:0

( )

t ¼ 500l

qU2; Fwake ¼ exp � Rex

1� 105

� �2 !

; Rex ¼ qxy2

l

d ¼ 50XyU

dBL; dBL ¼ 15

2hBL; hBL ¼

~RehtlqU

where U is the magnitude of velocity.The transition Reynolds number in the source term is com-

puted from freestream turbulence intensity and local pressure

gradient parameter:

Reht¼ð1173:51�589:428TI1þ0:2196TI�2

1 ÞFðkhÞ TI161:3

331:5ðTI1�0:5658Þ�0:671FðkhÞ TI1>1:3

(

ð8Þwith

FðkhÞ¼1�ð�12:986kh�123:66k2h�405:689k3hÞexp � Tu

1:5

� �1:5 !

kh 6 0

1þ0:275ð1� e�35kh Þ expð�2TuÞ kh > 0

8>><>>:

ð9Þwhere TI1 is the freestream turbulence intensity and kh is thepressure gradient parameter, they are calculated by

TI1 ¼ 100

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k1=3

pU1

; kh ¼ qh2

l� dUds

Correlations for critical Reynolds number and transition

length function in the original c� Reht transition model are

Rehc ¼~Reht � ð3:96� 1:21� 10�2 ~Reht þ 8:68� 10�4 ~Re2ht

�6:97� 10�7 ~Re3ht þ 1:74� 10�10 ~Re4htÞ ~Reht 6 1870

~Reht � ½593:11þ 0:482ð ~Reht � 1870:0Þ� ~Reht > 1870

8><>:

ð10Þ

Flength¼39:8�1:19�10�2 ~Reht�1:33�10�4 ~Re2ht

~Reht<400

263:4�1:24 ~Rehtþ1:95�10�3 ~Re2ht�1:02�10�6 ~Re3ht 4006 ~Reht<596

0:5�ð ~Reht�596:0Þ�3:0�10�4 5966 ~Reht<1200

0:3188 ~RehtP1200

8>>><>>>:

ð11Þwhere Rehc is the critical Reynolds number and Flength is the

length function.The combination of SST turbulence model and c� Reht

transition model is achieved by multiplying the effective inter-

mittency factor to the production and destruction terms of theturbulent kinetic energy equation, shown as follows:

@ðqkÞ@t

þ @

@xj

qujk� ðlþ rkltÞ@k

@xj

� �¼ ~Pk � ~Dk ð12Þ

~Pk ¼ ceffPk; ~Dk ¼ min½maxðceff; 0:1Þ; 1:0�Dk

where Pk and Dk are the production and destruction terms of

the original SST turbulence model. Specially, a correction isproposed for separation-induced transition:

csep ¼ min 2maxRev

3:235Rehc

� �� 1; 0

� �Freattach; 2

� �Fht ð13Þ

Boundary-layer transition prediction using a simplified correlation-based model 69

Freattach ¼ exp � RT

20

� �4 !

the effective intermittency factor is defined as ceff ¼ maxðc; csepÞ. Model coefficients are ca1 ¼ 2:0, ca2 ¼ 0:06, ce1 ¼ 1:0,

ce2 ¼ 50:0, cht ¼ 0:03, rc ¼ 1:0 and rht ¼ 2:0.

The transport equation of transition momentum thicknessReynolds number plays the role of diffusing the freestreamvalue into the boundary-layer. The work of Coder and Maugh-mer21 indicated that the transport equation can be eliminated

if a locally-defined pressure gradient parameter is available.They proposed the parameter Hc ¼ yX=U to form the correla-

tions for ~Reht and showed reasonable results. Actually, thetransition momentum thickness Reynolds number is used to

calculate the critical Reynolds number Rehc and transitionlength function Flength, as shown in Eqs. (10) and (11). If proper

correlations for Rehc and Flength can be supplied, the transport

equation can also be removed. According to Ge,23 we proposea correlation for the critical Reynolds number using theparameter Tw and freestream turbulence intensity:

Rehc ¼ 900� 810min ½T1ð0:285TI0:5261 þ 0:35Þ=4:0; 1:0� ð14Þwhere

T1 ¼ RT

Xx

ð15Þ

The T1 parameter behaves, to some extent, similarly to theHc parameter, due to the existence of the magnitude of vortic-

ity. The critical Reynolds number decreases with the increaseof T1 and has a value between 90 and 900. The transitionlength function controls the length of transition region and

has some effect on the transition onset location. Previousresearch24 indicated that it is appropriate to formulate thefunction only by the freestream turbulence intensity and a

value between 0.1 and 100 is suitable for the function. Thisleads to the new transition length function as

Flength ¼ max ð0:1; 30:0 lnðTI1Þ þ 89:97Þ ð16ÞAll of the constants in the correlations in Eqs. (14) and (16)

are empirically determined by numerical calculations based onthe transitional flat plate. It is noteworthy that the abandon of~Reht may result in the inability of prediction for separation-

induced transition, as shown in Eq. (13). Although cases withseparation are conducted in this paper, no special treatment atpresent has been made for separation correction, which will be

studied in the succeeding work. In addition, the transport equa-tion for intermittency factor and the connectionwith SST turbu-lence model are identical to the original c� Reht model.

For clear and complete acquaintance of the model, the gov-

erning equations and empirical correlations for the simplifiedtransition model are summarized as

@ðqkÞ@t

þ @@xj

qujk� ðlþ rkltÞ @k@xj

h i¼ Pk � b�qxk

@ðqxÞ@t

þ @@xj

qujx� ðlþ rxltÞ @x@xj

h i¼ c

mtPk � bqx2 þ 2ð1� F1Þ qrx2

x � @k@x@xj@xj

@ðqcÞ@t

þ @@xj

qujc� ðlþ ltrcÞ @c@xj

h i¼ Pc þ Ec

8>>>>>>><>>>>>>>:with

Rehc ¼ 900:0� 810:0min ½T1ð0:285TI0:5261 þ 0:35Þ=4:0; 1:0�

Flength ¼ max ð0:1; 30:0 lnðTI1Þ þ 89:97Þ

3. Computational method

The new transitionmodel in this work is implemented into an in-house CFD solver, which is based on finite-volume method and

uses multi-block structured grid for simulations. The solver,which is parallelized by massage passing interface (MPI), pro-vides Euler, Navier–Stokes, RANS and even Burnett equations

for bothperfect gas and chemical reacting gas. In this paper, onlythe RANS equations and perfect gas assumption are used forcalculations. Concretely, the advection upstream splittingmethod by pressure-based weight functions (AUSMPW+)25 is

used to calculate the inviscid fluxes and the viscid fluxes are cen-trally discretized. The monotone upstream centered scheme forconservation laws (MUSCL), along with the van Albada limiter

function is adopted for achievement of second-order accuracy.Notably, the turbulence model and transition model equationsare also solved by the second-order scheme for the consideration

of high-resolution. Steady-state solutions are finally obtained bya fully coupled implicit lower–upper symmetric Gauss–Seidel(LU-SGS)26 time-marching method, which solves all governing

equations simultaneously at each iteration step. Additionally,the source terms of turbulence model and transition model areimplicitly treated for the diminution of stiffness problem. Pre-cisely, the negative part of the source terms can be written as:

Snþ1� ¼ Sn

� þ f@S�@Q

� �DQ ð17Þ

where S� is the source term and Q the conservative variables.

The Jacobian matrix of source term is defined as

T ¼ @S�@Q

¼T11

T22

T33

264

375 ð18Þ

with

T11 ¼ �2b�x ð19Þ

T22 ¼ �2bx� 2F1

rx2

x2� @k@xi

� @x@xi

ð20Þ

T33 ¼ Flengthca1SðFonsetÞ0:5ð0:5c�0:5 � 1:5ce1c0:5Þ

þ Fturbð1:0� 2:0ce2cÞca2X ð21Þ

fðxÞ ¼ x x < 0

0 x P 0

ð22Þ

The boundary conditions for turbulent kinetic energy and

specific dissipation rate at inlet are k1 ¼ 1:5ðTI1=100U1Þ2and x1 ¼ q1k1=lt1, while their values on the wall are

k ¼ 0 and x ¼ 60ll=qb1y2 respectively. The freestream values

for c and ~Reht are given as 1 and 0.0001 respectively, and both

of them have a zero flux (@c=@n ¼ 0) on the wall. Besides, allfreestream conditions are set as the initial conditions for thewhole computational domain.

4. Results and discussion

4.1. Zero pressure gradient flat plate

The commonly used Schubauer and Klebanoff case27 is a zero

pressure gradient flat plate with a Mach number of 0.15. It is a

70 C. Xia, W. Chen

standard test case for the evaluation of performance of newtransition models in natural transition. The freestream velocityis 50.1 m/s, freestream turbulence intensity is 0.18%, and unit

Reynolds number is 3:4� 106. Final results of skin frictioncoefficient along the flat plate from three sets of grid andexperimental data are shown in Fig. 1. Dimensions for Grid

1 to Grid 3 are 154� 50, 194� 90 and 234� 130 respectively.All grids are clustered near the solid wall, and the spacing of

the first grid to the wall is 1� 10�6 m, which corresponds toa y-plus value of about 0.15. Similarly, the y-plus values for

all the following cases are guaranteed to be less than one.We can see from Fig. 1 that grid convergence is obtained, sincethe transition location changes little with the refinement ofgrid. The present new transition model predicts the transition

location, skin friction values in laminar part and in turbulentpart accurately except the transition length. The deficiency oftransition length may largely relate to the empirical correlation

shown in Eq. (16). In general, the transition model performsmuch better than the fully laminar and fully turbulent results,especially in the latter part of the flat plate, where the turbu-

lence model under-estimates the skin friction with a value ofabout 20%.

4.2. Aerospatial-A airfoil

The Aerospatiale-A airfoil, shown in Fig. 2, is another typicaltest case for transition models. The experimental resultsobtained from ONERA F1 wind tunnel at 13.1� angle of

attack are used in this work.28 Simulations are carried out at

Mach number 0.15, Reynolds number 2:1� 106 and free-stream turbulence intensity 0.2%. Experiment turns out that

a laminar separation bubble forms at 12% of the suction sideof the chord, resulting in a turbulent boundary-layer down-stream. Details of the flow field can be witnessed from theMach number contour in Fig. 3(a), and the separation-

induced transition can be clearly seen from Fig. 3(b), in whichthe intermittency factor c develops rapidly when turbulenceoccurs. Specifically, separation occurs at about 80% of the

chord and intermittency factor increases significantly whentransition occurs. The computed skin friction coefficient Cf

along the suction side of the airfoil is shown in Fig. 4(a). We

Fig. 1 Skin friction coefficient distribution along Schubauer and

Klebanoff flat plate for different grids.

can see that boundary layer transition is accurately predictedfrom the sharp increase of skin friction obtained by transitionmodels, and good agreement with experimental data is

obtained by both the original c� Reht and new transition mod-els. The fully turbulent calculation, however, fails to capturethe transition onset feature. Fig. 4(b) shows the pressure coef-

ficient Cp around the airfoil. We can see that the transitionmodels outperform the fully turbulence model significantlyexcept in the trailing edge. The under-prediction of pressure

near the trailing edge region can be attributed to the inabilityof the transition model for simulation of strong flow separa-tion, which occurs at about 83% of the chord as observed inexperiment. A closer look at the figure shows that the pre-

sented new transition model performs a bit better for the skinfriction and the pressure coefficient in the trailing edge thanthe original model. Nevertheless, the difference between the

two transition models is slight in this case.

4.3. S809 airfoil

The S809 airfoil, designed by the National Renewable EnergyLaboratory (NREL), is a primary airfoil used for wind turbineapplications. Detailed experimental data, including lift L, drag

D and transition locations for different angles of attack a canbe found from Somers.29 The computational grid is shown inFig. 5, and simulations are performed at Mach number 0.1,

Reynolds number 2:0� 106 and freestream turbulence inten-

sity 0.2%. Fig. 6 shows the transition locations x/c on the pres-sure side and suction side of the airfoil. Specifically, thetransition location on the pressure side moves downstream

steadily from about 50% to 60% of the chord as the angleof attack increases, while the transition location on the suctionside moves forward sharply at approximate 5� angle of attackdue to the adverse pressure gradient. Computed lift coefficient

CL and drag coefficient CD are depicted in Fig. 7, in which wecan see that the transition models show apparent improvementfor both of the coefficients, especially for the drag. As

expected, transition models possess no distinct advantage overturbulence model at high angles of attack, due to the strongseparation. On the whole, good agreement with experimental

data is achieved by transition models, and the simplified modelseems to perform better than the original model.

Fig. 2 Computational grid for Aerospatial-A airfoil.

Fig. 3 Contours of Mach number and intermittency factor for

the Aerospatial-A airfoil.Fig. 4 Skin friction and pressure coefficients distribution along

the Aerospatial-A airfoil.

Fig. 5 Computational grid for S809 airfoil.

Boundary-layer transition prediction using a simplified correlation-based model 71

4.4. Hypersonic flat plate

The test case considered for validation of the new transition

model in high speed flow is from Mee.30 Experiments were car-ried out in the T4 free-piston shock tunnel at University ofQueensland for a 1.5 m long and 0.12 m wide flat plate. Frau-

holz et al.31 studied the case extensively based on the c� Rehttransition model with modified correlations, and satisfactoryresults were obtained. The freestream conditions for computa-

tion are listed in Table 1. The Stanton number is defined as

St ¼ qwq1U1ðT0;1 � TwÞ

where q1,U1 and T0;1 are the freestream density, velocity and

total temperature respectively, qw and Tw are the heat flux andtemperature on the wall. It should be noted that the Stantonnumber measured in experiment has an uncertainty of

±18%, as estimated by Mee.30 Since the real freestream turbu-lence intensity in the wind tunnel is unknown, preliminary sim-ulations are performed to tune the transition location, as

shown in Fig. 8, where TI1 represents the freestream turbu-lence intensity, and the increase of TI1 promotes the transitionof boundary layer. We can also find that the transition locationand transition length are strongly affected by the freestream

turbulence intensity. The four computed conditions are sum-

marized in Fig. 9, and significant improvement of predictionaccuracy has been achieved by the transition models, togetherwith the results obtained by Frauholz et al.,31 in which theauthors predicted the transitions by using a modified

Fig. 6 Transition location for S809 airfoil.

Fig. 7 Lift and drag coefficients at different angles of attack for

S809 airfoil.

Table 1 Freestream parameters under different conditions.

Parameter Condition

1 2 3 4

Ma1 5.3 6.2 6.8 12.4

T1 (K) 570 690 800 1560

Re1 1:7� 106 2:6� 106 4:9� 106 1:6� 106

TI1 (%) 2.6 2.0 1.7 2.2

Fig. 8 Stanton number distribution of hypersonic flat plate

under Condition 1.

72 C. Xia, W. Chen

c� Reht model. Good agreement with experimental data canbe witnessed clearly, and it is not an easy task to judge betweenthe two transition models at present. A notable improvement

should be mentioned that the new transition model not only

predicts the transition onset accurately, but also gets closerto experimental data than the turbulence model on the fullyturbulent part of the flat plate.

4.5. Hypersonic double wedge

A more complicated case for the transition model is the hyper-sonic double wedge, tested by Neuenhahn and Olivier in the

TH2 shock tunnel.32 The shape considered at present has aslightly blunted leading edge of 0.5 mm. The first ramp is178 mm long and has an angle of 9�, while the second ramp

is 200 mm long and has an angle of 20.5�. Two-dimensionalflow is guaranteed with a width of 270 mm for the geometry.The freestream flow conditions are listed in Table 2 and com-

putational grid is shown in Fig. 10. Unlike the previous cases,the hypersonic double wedge encounters severe shock wave/boundary-layer interaction, which can be demonstrated bythe Mach number contour obtained by the new transition

model in Fig. 11. Details of the flow can be noted that adetached bow shock from the blunted leading edge interactswith the separation-induced shock in the corner, followed by

a combination with the reattachment shock.Results of pressure coefficient and Stanton number are

shown in Fig. 12. We can see that generally good agreement,

especially of the pressure coefficient, is obtained by the transi-tion model. The fully laminar simulation under-estimates theStanton number significantly, and has a slightly larger separa-tion zone. On the contrary, the fully turbulent calculation

over-estimates the Stanton number and has no separation at

Fig. 9 Stanton number distribution of hypersonic flat plate under different conditions.

Boundary-layer transition prediction using a simplified correlation-based model 73

all. For comparison, the Stanton number predicted by theoriginal c� Reht transition model is extracted from You14

and shown in Fig. 12(b). We can see that results of the new

transition model outperform that of the fully laminar and tur-bulent assumptions. Specifically, the new simplified model hasa relatively lower value of Stanton number and bigger separa-

tion zone as compared with the original model. However, dis-crepancies with experiment still exist for both of the transitionmodels. The absence or inappropriate correction in the transi-

tion model for separation flow may be one possible explana-tion, since the research of You14 demonstrated that a morephysical meaningful effective intermittency involved pressuregradient performed better. Moreover, some of our previous

Table 2 Freestream parameters for hypersonic double wedge.

Parameter Value

Ma1 8.1

Re1 3:8� 106

u1 (m/s) 1635

p1 (Pa) 520

T1 (K) 106

TI1 (%) 0.9

numerical attempts indicated that the transition length func-tion and critical Reynolds number correlations have non-negligible effects on the flow features of transition. Anyway,

these aspects are of great importance and will be the key pointsof the further work.

Fig. 10 Computational grid for double wedge.

Fig. 11 Contour of Mach number for double wedge.

Fig. 12 Pressure coefficient and Stanton number distribution

along double wedge.

74 C. Xia, W. Chen

5. Conclusions

(1) A simplified local-correlation-based transition modelhas been developed by the removal of the momentumthickness Reynolds number equation in the origin

c� Reht transition model, along with the new transitionlength function and critical Reynolds numbercorrelation.

(2) The new transition model is implemented into an in-house CFD solver and tested for some typical cases,ranging from low speed flows to hypersonic flows.

Results from simulations in this work show that theaccuracies of the flows are significantly improved bythe proposed transition model, compared with the fully

laminar and fully turbulent calculations.(3) It is shown that the simplified transition model is com-

parable to the original c� Reht model for the simula-tions presented in this work. The new model appears

to be promising and deserves further research due toits satisfactory results and less computational cost thanthe original model.

(4) It is, however, necessary to validate the new model withmore complex test cases for the verification of its practi-cability. Future attempts will be made to the separation-

induced transition correction. Besides, the correlationsin the model need to be calibrated in detail by wind tun-nel data.

Acknowledgement

This study was supported by the State Key DevelopmentProgram for Basic Research of China (No. 2014CB340201).

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Xia Chenchao is a Ph.D. candidate at School of Aeronautics and

Astronautics, Zhejiang University. His main research intersects are

computational fluid dynamics and aerodynamic shape optimization.

Chen Weifang is a professor and Ph.D. supervisor at School of

Aeronautics and Astronautics, Zhejiang University. His current

research interests are gas dynamics and aircraft design.