Predicting crossflow induced transition with laminar kinetic energy transition model

International Journal of Heat and Fluid Flow 81 (2020) 108522

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Predicting crossflow induced transition with laminar kinetic energy transition model Yupei Qin, Xin Yu

T

⁎

Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

ARTICLE INFO

ABSTRACT

Keywords: Crossflow Boundary layer transition Laminar kinetic energy Hypersonic

The hypersonic laminar kinetic energy transition model is developed to be appropriate for crossflow induced boundary layer transition prediction. A crossflow timescale is constructed and incorporated in the kT-kL-ω transition model to reflect crossflow effect during three-dimensional boundary layer transition. The stream-wise vorticity is selected as the indicator of crossflow strength. Regarding the inviscid unstable characteristic of crossflow instability, the crossflow timescale is constructed by reference to the second mode timescale. To eliminate inappropriate development of the crossflow timescale where the effective length scale is large enough while the crossflow strength remains at a quite low level, a crossflow velocity limit function is proposed. The revised kT-kL-ω transition model has been applied to HIFiRE-5 and blunt cone with 1°angle of attack test cases. Results show good correspondence with the experimental data and DNS data, which demonstrates that the constructed crossflow timescale makes the revised transition model capable of reproducing crossflow induced transition behavior with a reasonable degree of accuracy.

1. Introduction For aircraft operating at sustained supersonic or hypersonic speeds, transition is responsible for the prominent increase in the friction drag and aerodynamic heat compared to laminar flow. As a result, it is of critical importance to predict boundary layer transition accurately for the design of aerodynamic configurations, thermal protection systems and so on. Transition to turbulence can be induced by the presence of any one of several flow instabilities which allow small disturbances to grow to sufficiently high amplitude to cause breakdown. For many hypersonic boundary layers, while Mack modes (Mack, 1975) play dominant role in the transition procedure, crossflow instability (Hassan et al., 2018) could be the significant mechanism in three-dimensional flow fields. On account of a pressure gradient in the boundary layer, which is misaligned with the velocity vector on the boundary layer edge, a component of flow in the span-wise direction, normal to the edge vector is induced. Fig. 1 presents a velocity profile which is decomposed into a component parallel with the boundary layer edge velocity and a crossflow component. In Fig. 1, xt, yt and zt are the Cartesian coordinates. ut and wt denote the tangential component direction and the crossflow component direction of the velocity profile, respectively. For the crossflow profile, the non-slip boundary condition on the wall and

⁎

the asymptotic return to zero on the boundary layer edge necessitates the existence of an inflection point in the crossflow velocity. Thus, the inviscid unstable inflection point results in the crossflow instability. For the transition mechanisms induced by travelling crossflow vortices, Wassermann and Kloker (2002, 2003, 2005) present a detailed description in their benchmark Direct Numerical Simulation (DNS) works. For computational fluid dynamics (CFD), methods based on flow mechanisms possess more superiorities among varieties of boundary layer transition prediction approaches, and these methods are usually more reliable within a broad range of applications. As one of the refined flow simulation methods, DNS (Kuwata and Kawaguchi, 2019) has made significant contributions to revealing transition mechanisms. For example, DNS of hypersonic crossflow instability on an elliptic cone has been conducted by Dinzl and Candler (2017), and a steady physical mechanism is introduced for the sharp increase in the wall heat flux. However, regarding the tremendous computational resource consumption, there is still a considerable distance for DNS to be applied in the engineering level. For example, the mesh requirement for an accurate three dimensional channel flow is proportional to Re9/4 (Wilcox, 1993). Comparatively, based on Kolmogorov's Universal Equilibrium Theory, the universal assumption reduces the computational requirements for Large Eddy Simulation (LES) (Xu et al., 2015) by modeling the turbulence below the inertial sub-

Corresponding author. E-mail addresses: [email protected] (Y. Qin), [email protected] (X. Yu).

https://doi.org/10.1016/j.ijheatfluidflow.2019.108522 Received 17 October 2019; Received in revised form 28 November 2019; Accepted 30 November 2019 0142-727X/ © 2019 Elsevier Inc. All rights reserved.

International Journal of Heat and Fluid Flow 81 (2020) 108522

Y. Qin and X. Yu

Nomenclature xj Uj ρ p E T dij S Ω kT kL w kT,s kT,l PkT DkT PkL DkL μ μT,l μT,s νT,l νT,s νT

effective length scale λeff τT,l characteristic timescale first mode characteristic timescale τnt1 τnt2 second mode characteristic timescale crossflow characteristic timescale τc ΩStreamwise stream-wise vorticity μnt1 first mode viscosity μnt2 second mode viscosity μntc crossflow mode viscosity fν viscous damping function fINT intermittency factor shear-sheltering damping function fss wmax maximum crossflow velocity f(w) crossflow velocity limit function turbulent viscosity coefficient Cμ αT effective diffusivity fω kinematic damping function model term for bypass transition RBP RNAT model term for natural transition blend function F2 Tu free stream turbulence intensity Marel local relative Mach number Ma mach number free stream Reynolds number Re∞ Tw wall temperature φ azimuthal angle

Cartesian coordinates velocity vector fluid density pressure total energy temperature Kronecker delta function strain rate, |Sij| vorticity, |Ωij| turbulence kinetic energy laminar kinetic energy specific turbulence dissipation rate effective small-scale turbulence effective large-scale turbulence production term of kT equation dissipation term of kT equation production term of kL equation dissipation term of kL equation molecular viscosity large-scale dynamic viscosity coefficient small-scale dynamic viscosity coefficient large-scale kinematic viscosity coefficient small-scale kinematic viscosity coefficient turbulent kinematic viscosity coefficient

range, while the computation is still quite beyond affordability for engineering applications. Linear Stability Theory (LST) has achieved great success in the crossflow investigations historically (Mankbadi, 1994), while the assumptions of linear prediction are not strictly appropriate for crossflow because disturbances grow large enough to have nonlinear effects (Reed et al., 1996). For transition scenarios resulting from nonlinear disturbances growth with strong amplification of high-frequency secondary instabilities, the more sophisticated Nonlinear Parabolized Stability Equations (NPSE) methods would be preferable (Herbert, 1994). Transition models, based on Reynolds Averaged Navier–Stokes (RANS) equations, provide a reasonable compromise between accuracy and calculation cost, and have become the most practical methods for engineering practice.

In 2004, Menter and Langtry proposed the correlation based and completely local γ-Reθt transition model (Menter et al., 2004a, b), which can be implemented into a finite volume based CFD solver directly and compatible with massively parallel execution. The γ-Reθt transition model is able to capture transition induced by T-S instability and separation. Various test cases (Robitaille et al., 2015; Grabe and Krumbein, 2013) manifest the successful prediction of boundary layer transition with such a model, while the most severe limitation is that any three-dimensional transition mechanism is not included. Inspired by Owen and Randall's work (Owen and Randall, 1952), in which a criterion for the development of the crossflow instability on a sweptback wing is proposed, Medida and Baeder (2013) present a crossflow transition criterion incorporating a modified crossflow Reynolds number. Discarding the crossflow Reynolds number criterion, Langtry et al. (2015) describe a new measure of crossflow strength based on the stream-wise vorticity to develop a correlation for stationary crossflow. The improved transition model has shown very encouraging results for transitional flows dominated by crossflow. Taking unstable modes in hypersonic flows into account, Warren and Hassan (2001) put forward the non-turbulent viscosity μnt, a function of non-turbulent timescale related to the unstable modes. Making use of the non-turbulent viscosity μnt, Wang and Fu (2009, 2001) and Wang et al. (2016) proposed the k-ω-γ transition model appropriate for both subsonic and supersonic/hypersonic flows. Based on crossflow velocity and crossflow Reynolds number, a crossflow timescale is developed and incorporated in the k-ω-γ transition model by Zhou et al. (2017). Test cases results demonstrate the ability of the improved k-ω-γ transition model to predict boundary layer transition induced by crossflow, however for the sharp cone at different angles of attack, the transition onsets on the windward centerline predicted by the improved model are not consistent with the experimental data (Zhou et al., 2017). Without transition estimation correlations, phenomenological or physics-based models (Medina et al., 2018; Wang and Perot, 2002) are remarkably advantageous in transition prediction because the transition mechanisms are taken into consideration directly. However, regarding the fact that mechanisms of transition are not fully understood,

Fig. 1. Velocity profile in three-dimensional boundary layer decomposed into tangential component and crossflow component (White and Saric, 2005). 2

International Journal of Heat and Fluid Flow 81 (2020) 108522

Y. Qin and X. Yu

the construction of phenomenological or physics-based models is faced with enormous challenges. Nonetheless, it is of great value that Walters and Cokljat (2008) propose the kT-kL-ω transition model to describe the magnitude development of low-frequency velocity fluctuations which are identified as the precursors to transition in the pretransitional boundary layers. Introducing the influence of hypersonic unstable modes, Qin et al. (2017) improved the kT-kL-ω model to be applicable for hypersonic flows. Test cases indicate the capacity of the revised model to reproduce transition behavior with a favorable degree of accuracy and reflect the effect of Reynolds number successfully. As a further step, and inspired by the work of Kubacki and Dick (2016) and Qin et al. (2018) proposed an algebraic intermittency factor to make the kT-kL-ω model possess the ability to capture overshoots in the late transition region. However, crossflow receives little attention in the previous work for the kT-kL-ω model, which restricts the application of the transition model to three-dimensional boundary layer transition predictions. To achieve predictions for boundary layer transition induced by crossflow within the phenomenological or physics based kT-kL-ω framework, a crossflow timescale is proposed in this paper.

where R equals 287 J/(kg•K). The governing equations presented above are discretized by the second-order finite volume method with multi-block structured meshes. The inviscid flux is selected to be the Roe scheme, the left and right state variables of which are reconstructed with a minmod limiter. The viscous fluxes are discretized by the second order central difference scheme. The time items are discretized by the Euler backward first order scheme, and for time marching, the implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) scheme is employed. 3. Transition model 3.1. kT-kL-ω transition model The kT-kL-ω transition model was firstly proposed by Walters and Cokljat (2008) in 2008. Plenty of corrections and developments have been carried out to the original transition model. For example, based on the Klebanoff mode dynamics, Jecker et al. (2019) proposed a new formation of the kT-kL-ω transition model. Since bypass transition is beyond the research scope of this paper and based on the previous work (Qin et al., 2018), the intermittency factor weighted laminar kinetic energy transition model is taken into consideration. Details about the transition model can be found in reference (Qin et al., 2018). The kT-kL-ω transition model is composed of three transport equations: the equation for turbulent kinetic energy kT, the equation for laminar kinetic energy kL and the equation for specific turbulence dissipation rate ω. The equations can be interpreted as follows:

2. Numerical methods The governing equations in this paper are compressible NavierStokes equations including conservations of mass, momentum and energy. By Reynolds averaging and using the Boussinesq approximation and the assumption of constant Prandtl numbers, the RANS equations can be written as:

+

t

xj

( Uj ) = 0

Ui + ( Ui Uj ) = t xj

D ( kT ) = (P kT + RBP + RNAT Dt

(1)

p + xi

E + ( Uj H ) = (Ui ij ) t xj xj

µt

xj

µ + Pr Prt

T xj

(2)

D ( kL) = (P kL Dt

(3)

D( ) Dt

where ρ is density, Uj is the jth component of velocity and xj stands for Cartesian coordinates. p denotes pressure and E is total energy. Meanwhile,

H=E+

DT ) +

xj

µ+

T k

kT xj (9)

ij

xj

kT

=

C

1k

+C 3 f

RBP

T

P kT +

2 T fW

RNAT

(

kT d3

)

C R fW

+

DL) +

1

kT

µ

xj

kL xj

(RBP + RNAT )

(µ + )

(10)

C

2

2

T

xj

xj

(11)

where PkT and PkL are the production terms for kT and kL. DT and DL are the anisotropic (near-wall) dissipation terms for kT and kL. They are modeled as

p (4)

In addition, σij is the stress tensor:

2

(12)

2

(13)

P kT =

T ,s S

(5)

P kL =

T ,l S

where μ and μt are molecular viscosity and turbulent viscosity, respectively. For Newtonian fluids, μ is calculated with the Sutherland's law:

DT =

kT xj

kT xj

(14)

DL =

kL xj

kL xj

(15)

ij

= 2(µ + µt ) Sij

µ µ0

T T0

1.5

1 Skk 3

ij

T0 + Ts T + Ts

where T is temperature, and T0 = 273.16 μ0 = 1.7161 × 10−5Pa•s, Ts = 124 K. Sij is the strain rate tensor in the form as follows:

Sij =

1 2

Uj Ui + xj xi

(6) K.

For

air,

νT,s and νT,l denote the small-scale and large-scale eddy viscosities in the production terms, and defined as follows:

(7)

= fint f ·min[kT / , a1 kT /( F2)]

(16)

T ,l

= Cµ kL

(17)

T ,l

For νT,s, γ represents the intermittency.

δij is the Kronecker delta function defined as δij = 1 for i = j and δij = 0 for i≠j. The laminar Prandtl number Pr and the turbulent Prandtl number Prt are set to 0.72 and 0.9 respectively. Besides, the air is assumed to obey the perfect gas equation of state

p = RT

T ,s

= min max

kT C

1, 0 , 1

(18)

where δ is the boundary layer thickness. Note that the grid-reorder method for massively parallel execution, proposed by Hao et al. (2017)

(8) 3

International Journal of Heat and Fluid Flow 81 (2020) 108522

Y. Qin and X. Yu

is adopted to obtain boundary layer parameters. ν is the kinematic viscosity. Amendment on γ is needed to capture the overshoot accurately. Specific practice is available in reference (Qin et al., 2018). fINT and fν are damping functions. F2 is a blending function. The concrete forms are as follows:

fINT = min

f =1

kT ,1 CINT (kT + kL)

fss is a damping function to incorporate the shear-sheltering effect (Jacobs and Durbin, 1998).

2

(21)

RBP = CR

where d is the distance to the wall. The effective turbulence Reynolds number ReT in the damping function fν takes the form

ReT =

fW2 kT

(22)

Marel 1 Marel > 1 nt1 + nt 2,

BP

1.5 eff /

= Cl1 [

nt 2

= Cl2 [2

eff / U

BP

T

nt

= min(

=

nt ,

(24)

(ys )]

(25)

kT

= d2

U

= Cµ, std f

kT , s

eff

exp

NAT

(36)

ANAT

kT

= max

Re =

CBP, crit , 0

(37)

Re

CNAT , crit ,0 fNAT , crit

(38)

d2

(39)

fNAT,crit is constructed to preserve an appropriate manner in which the amplitude of the pre-transitional fluctuations influence the initiation of natural transition.

fNAT , crit = 1

exp

CNC

kL d (40)

Finally, the turbulent viscosity used in the RANS equations is the intermittency factor weighted sum of the large-scale and small-scale contributions.

(26)

µT = (1

(41)

) µT , l + µT , s

Model constants are listed in Table 1. 3.2. Construction of the crossflow timescale

(28)

The crossflow induced transition has not been taken into consideration in the original kT-kL-ω transition model, which might cause limitations on the transition prediction in three-dimensional boundary layers. To extend the application scope of the kT-kL-ω transition model, a crossflow timescale is proposed. The timescale is based on the concept of stream-wise vorticity

(29)

Table 1 Model constants.

An effective diffusivity αT is included in the transport equations for kT and ω, defined as T

(35)

ABP

ReΩ is defined as

(27)

T)

=1

= max

NAT

where cr is the disturbance phase velocity, equal to U(ys). In addition, a is the local sound speed. It is worth attention that the model constants Cl1 and Cl2 may need to be calibrated before the transition model is utilized. But for a specific configuration, recalibration is not always required. The meaning of these constants as well as the general range are discussed in reference (Qin et al., 2018). The kinematic wall effect is taken into account by the effective (wall-limited) turbulence length scale λeff . eff

BP

exp

(23)

2Eu u/ ]

cr )/ a

(34)

NAT kL

where

where Eu is the average kinetic energy relative to the wall, interpreted as 0.5|U|2. U(ys) denotes the local mean velocity at the generalized inflection point which is equal to 0.94 times of the boundary layer edge velocity Ue approximately. Marel is the local relative Mach number to demarcate the affected region of different unstable modes and interpreted as

Marel = (U

=1

NAT

where τnt1 and τnt2 denote the characteristic timescales of the first and second unstable modes disturbances, respectively. They can be expressed as nt1

(33)

/ fW

The model forms of the threshold functions βBP and βNAT are as follows

nt1,

=

BP kL

RNAT = CR, NAT

On account of the adjustment on the effective length scale in reference (Qin et al., 2017), fw has been set to 1. Similarly, the kinematic damping function fω in the transport equation for ω is replaced by a constant of 0.34. For the large-scale viscosity νT,l, the characteristic timescale τT,l incarnates the unstable modes in hypersonic flows. T ,l

(32)

The model terms RNAT and RBP represent natural and bypass transition, respectively. With opposite signs in the transport equations for kT and kL, they describe a transfer of energy from laminar kinetic energy to turbulent kinetic energy, implying the mechanism of transition from laminar flow to turbulent flow.

(20)

F2 = tanh({max[2 kT /(0.09 d ), 500µ/( d 2 )]} )

2

Css kT

fss = exp

(19)

ReT A

exp

(31)

kT , s = fss fW kT

A = 6.75 CBP, crit = 1.2 CR,NAT = 0.02 C 2 = 0.92

(30)

Cµ = 0.02

kT,s stands for the effective small-scale turbulence, modeled as 4

ABP = 0.6 CNAT ,crit = 1250 CR = 0.12 C 3 = 0.3 k

=1

ANAT = 200 CINT = 0.75 CSS = 1.5 C R = 1.5 = 1.17

ATS = 200 CNC = 0.1 C 1 = 0.44 Cµ, std = 0.09

C = 12

International Journal of Heat and Fluid Flow 81 (2020) 108522

Y. Qin and X. Yu

ΩStreamwise, also known as helicity. As an indicator of crossflow strength, the form defined by Langtry et al. (2015) is adopted, i.e.

U =

=

u u2 + v 2 + w 2 w y

Streamwise

v

,

v u , z z

u2 + v 2 + w 2 w v , x x

,

(42)

selected. It is noted that all the test cases are calculated by an in-house solver developed by the authors, and the solver has been verified in previous works (Qin et al., 2017, 2018; Jecker et al., 2019). Meanwhile, the initial values and boundary values for kt, kL and ω are the same as reference (Qin et al., 2017).

(43)

4.1. HIFiRE-5

(44)

The Hypersonic International Flight Research Experimentation (HIFiRE) program is a hypersonic flight test program executed by the U.S. Air Force Research Laboratory and Australian Defense Science and Technology Organization. HIFiRE-5 is devoted to aerothermodynamics experiments, in particular transition on a three-dimensional geometry. The HIFiRE-5 vehicle is an elliptic cone with a 2:1 aspect ratio. The full length of HIFiRE-5 is 0.86 m with a 7° half angle on the minor axis. The nose bluntness is 2.5 mm on the minor axis. Detailed descriptions on the configuration are available in reference (Kimmel et al., 2010, 2013). A 38.1% scale model of HIFiRE-5 vehicle was tested in the Boeing/ AFOSR Mach 6 Quiet Tunnel (Juliano and Schneider, 2010). The identical configuration is calculated and the experiment results are selected as comparisons. For flow conditions, the free stream Mach number Ma∞ is 5.8, the angle of attack α is 0°, the wall temperature Tw is 300 K, and the free stream turbulence intensity Tu∞ is set to 0.4%. The free stream Reynolds number Re∞ includes 6.1 × 106/m, 8.1 × 106/m and 10.2 × 106/m. The other flow parameters are the same as the experiments in reference (Juliano and Schneider, 2010). Grid convergence analysis is conducted to preserve high grid resolution inside the boundary layer. The same grid topology and the same boundary conditions as in reference (Zhou et al., 2018) are adopted. Four grids with y+ ranging from 0.7 to 5 are included. The detailed information of the grids is listed in Table 2. To learn the basic flow structures for the HIFiRE-5 configuration, Fig. 2 presents the laminar result for the Re∞=10.2 × 106/m condition with G2 grid. Streamline patterns and contours of Ma and heat flux are illustrated. Regarding the aspect ratio of HIFiRE-5, different oblique angles of the shock surface along the minor axis (the centerline) and the major axis (the attachment line) are generated. The discrepancy in oblique angles induces a pressure gradient which transports fluid from the attachment line towards the centerline. As a result, the boundary layer thickness increases from the attachment line to the centerline, and the stream-wise vortex forms. Compared with the centerline, heat flux is much higher near the attachment line on account of a thinner boundary layer. The profile of the heat flux agrees qualitatively with the base flow solution of the DNS result (Dinzl and Candler, 2017). Due to the thicker boundary layer and the stream-wise vortex, the computed flow field on the centerline is more susceptible to the grid resolution. Therefore, the velocity profiles on the centerline computed by the revised transition model are compared. Fig. 3 presents the velocity profiles computed with the four different grids for Re∞ = 10.2 × 106/m condition. By comparison, it is obvious that at x = 60 mm, the velocity profiles computed with the four grids are in good correspondence. However, as the flow develops downstream, the discrepancy among the velocity profiles becomes more and more evident. When x = 180 mm, x = 240 mm and x = 300 mm, because of a remarkable decrease in the cell number, the velocity profiles computed with grid G3 and G4

w u2 + v 2 + w 2

u y

= U·

Similar to a pressure gradient parameter typically used in an empirical correlation (Langtry et al., 2015), the non-dimensional crossflow strength is defined as

HCrossflow =

d

Streamwise

(45)

U

To satisfy the dimensional consistency and also consider the inviscid unstable characteristic of crossflow, analogous to the form of the second mode timescale, the crossflow timescale is constructed through dimensional analysis and takes the form as follows: c

= Clc

eff

Ue

·Hcrossflow ·f (w )

(46)

where Clc is a constant and equal to 5.0 after calibration. It is noted that although the crossflow timescale τc has a similar form as the second mode timescale τnt2, it does not mean that crossflow instability possesses the same frequency characteristics as the second mode. f(w) is a crossflow velocity limit function and interpreted as

f (w ) =

1 w · sgn 1.0, max 2 Ue

Crosscrit + 1

(47)

where sgn is the sign function. wmax is the maximum crossflow velocity in the wall normal direction. Crosscrit is a constant, and equal 2% according to the experiment data by Reed (Reed and Haynes, 1994). The crossflow velocity limit function f(w) is proposed to eliminate inappropriate development of the crossflow timescale τc where the effective length scale λeff is large enough while the crossflow strength remains at a quite low level. Taking the crossflow timescale into consideration, the total characteristic timescale (Eq. (23)) can be interpreted as T ,l

=

nt1 nt1

+ +

c, nt 2

+

c,

Marel 1 Marel > 1

(48)

Relying on the total characteristic timescale τT,l, the large-scale viscosity νT,l in formula (17) includes the crossflow effect and promotes the development of the laminar kinetic energy. As the energy transfers from laminar kinetic energy to turbulent kinetic energy, transition takes place. The purpose of this paper is to render the kT-kL-ω transition model the capability to achieve the prediction for crossflow induced boundary layer transition. 4. Test cases To investigate the performance of the revised kT-kL-ω transition model to predict boundary layer transition induced by crossflow, the elliptic cone HIFiRE-5 and blunt cone with 1° angle of attack are Table 2 Grid resolutions for convergence analysis. Grid

Cell number (stream-wise × circumferential × normal)

Wall normal distance of the first grid cell (mm)

G1 G2 G3 G4

255 225 195 175

0.7 1.0 2.0 5.0

× × × ×

310 300 290 280

× × × ×

171 151 131 111

5

× × × ×

10−3 10−3 10−3 10−3

y+ 0.7 1 2 5

International Journal of Heat and Fluid Flow 81 (2020) 108522

Y. Qin and X. Yu

original kT-kL-ω transition model and the revised kT-kL-ω transition model as well as the wind tunnel experiment results (Juliano and Schneider, 2010) are compared in Fig. 4. In consideration that our inhouse code does not take the material properties of the wall into account, the computational results cannot show the wall temperature distributions. As a result, the intermittency γ distributions are presented instead. Thereinto, Fig. 4(a1)–(c1) show the intermittency γ distributions predicted by the original kT-kL-ω transition model for the three conditions Re∞ = 6.1 × 106/m, Re∞ = 8.1 × 106/m and Re∞ = 10.2 × 106/m, separately. Fig. 4(a2)–(c2) present the intermittency γ distributions predicted by the revised kT-kL-ω transition model including the constructed crossflow timescale for different conditions. Fig. 4(a3)–(c3) illustrate the temperature sensitive paint (TSP) images of the wind tunnel experiment results at corresponding Reynolds numbers (Juliano and Schneider, 2010). It is noted that for the three test conditions, the constants Cl1 and Cl2 are equal to 3.5. If the position where the intermittency γ and temperature begins to rise evidently is defined as transition onsets, transition zones predicted by the original kT-kL-ω transition model are mainly concentrated near the centerline when Re∞ = 6.1 × 106/m and Re∞ = 8.1 × 106/m. For the test case Re∞ = 10.2 × 106/m, when x < 250 mm, transition zone predicted by the original transition model is centralized near the centerline as well. When x > 250 mm, transition zone begins to extend to the mid-span region between the centerline and the attachment line. In contrast, transition zone distributions predicted by the revised kT-kLω transition model involve both the centerline and the mid-span region. Overall, both the transition onsets and the transition zone shapes calculated with the revised transition model are in better agreement with the experimental results for the three different Reynolds number conditions. Therefore, it can be concluded that with the constructed crossflow timescale, the revised kT-kL-ω transition model is practicable with satisfactory accuracy in terms of the prediction of hypersonic three-dimensional boundary layer transition. Besides, the revised transition model is capable of reflecting the effect of Reynolds number on transition, which is that the transition onset moves forward as the Reynolds number increases. To explain how the constructed crossflow timescale makes the revised kT-kL-ω transition model possess the ability to predict boundary layer transition induced by crossflw, specific analysis is carried out in according to the Re∞ = 10.2 × 106/m condition. First of all, the laminar result of the maximum crossflow velocity magnitude normalized by the velocity on the boundary layer edge wmax/Ue is depicted in Fig. 5. Because of the flow symmetry, the crossflow velocity on the centerline and attachment line remains zero. Whereas, the maximum crossflow velocity in the large mid-span region of the configuration reaches up to approximately 8% of the boundary layer edge velocity. The value of wmax/Ue is much larger than the critical value 2% in formula (47). Thus, boundary layer transition induced by crossflow could happen. Consistent with the linear BiGlobal modal instability analysis result by Paredes and Theofilis (2015) and Paredes et al. (2016), the elliptic cone generates significant crossflow instability under hypersonic conditions. Fig. 6 presents different non-dimensional large-scale viscosity distributions computed with the revised transition model for the Re∞ = 10.2 × 106/m condition at x = 150 mm. Expanding formula (17) with the corresponding timescales, the first mode, second mode and crossflow large-scale viscosities are respectively calculated by the formulas as follows,

Fig. 2. Streamline patterns and contours of Ma and heat flux. (Laminar result, Re∞ = 10.2 × 106/m).

Fig. 3. Velocity profiles in different locations on the centerline (transition result, Re∞ = 10.2 × 106/m).

deviate from the other profiles significantly. Meanwhile, the less the cell number is, the more significant the discrepancy is. Regarding the velocity profiles computed with G1 and G2 are quite in agreement, it can be concluded that the grid convergence is well established. To reduce the calculation burden, the following analysis is based on grid G2. In addition, it is noted that as the stream-wise vortex develops downstream, inflection points emerge in the velocity profile. According to LST, the existence of inflection point would lead to inviscid instability, which manifests that for the HIFiRE-5 configuration, the most unstable inviscid second mode is expected to dominate the transition process on the centerline as described in the DNS study (Dinzl and Candler, 2017) and experiment investigations (Juliano and Schneider, 2010; Paredes and Theofilis, 2015). The transition zone distributions on HIFiRE-5 calculated with the

µnt1 = Cµ kL

nt1

(49)

µnt 2 = Cµ kL

nt 2

(50)

µntc = Cµ kL

c

(51)

As shown in Fig. 6, both the first mode large-scale viscosity μnt1 and the second mode large-scale viscosity μnt2 are mainly distributed in the 6

International Journal of Heat and Fluid Flow 81 (2020) 108522

Y. Qin and X. Yu

Fig. 4. Comparison of transition zone distributions on HIFiRE-5 between computational and experimental results (Juliano and Schneider, 2010) at different Reynolds numbers.

area above the centerline. The concentrated distribution characteristic of these two large-scale viscosities brings about that the transition zones predicted by the original kT-kL-ω transition model in Fig. 4 are mainly restricted in the near centerline region. Besides, compared with μnt1, a larger value of μnt2 demonstrates that the transition process on the centerline is dominated by the second mode instability, which is consistent with the existing research achievements (Dinzl and Candler, 2017; Juliano and Schneider, 2010; Paredes and Theofilis, 2015). As for the crossflow large-scale viscosity μntc, the distribution covers the most mid-span region between the centerline and the attachment line. By comparison, it is obvious that the region with larger wmax/Ue in Fig. 5 corresponds to the area with larger μntc in Fig. 6(c). The actuality that the crossflow transition near the centerline of the cone falls off can be explained by the reduction of the streamwise vorticity resulting from curvature-sign alteration, and a detailed description can be obtained in references (Wassermann and Kloker, 2005; Balakumar and Owen, 2010). Affected by the crossflow large-scale viscosity, the laminar kinetic energy develops quickly in the mid-span region. As the energy transfers from laminar kinetic energy to turbulent kinetic energy, transition induced by crossflow can be predicted by the revised kTkL-ω transition model. However, two points need to be stated here. Firstly, the first mode and second mode large-scale viscosities also diminish from the center line to the attachment line, and remains at a quite small level in the

Fig. 5. Contours of maximum crossflow velocity magnitude normalized by boundary layer edge velocity (wmax/Ue). (Laminar result, Re∞ = 10.2 × 106/ m).

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Fig. 6. Comparison of different large (Re∞ = 10.2 × 106/m, x = 150 mm).

scale

viscosity

windward is larger than the leeward and crossflow is generated. Comparison of the skin friction coefficient Cf calculated with the original and revised kT-kL-ω transition model as well as the DNS result (Li et al., 2010) is presented in Fig. 8. The blunt cone surface is unfolded into a rectangle via the azimuthal angle φ. The definition of φ can be obtained in reference (Li et al., 2010). The windward centerline and the leeward centerline correspond to φ = 0° and = 180°, separately. It is noted that for this test case, the constants Cl1 and Cl2 are equal to 2.0. The dashed line in Fig. 8(c) denotes the transition onsets where Cf increases remarkably. On account of the 1° angle of attack, transition onset profile on the blunt cone is no more asymmetric. On the windward centerline (φ = 0°), both the original and revised kT-kL-ω transition model predict accordant transition onsets with the DNS result at x ≈ 750 mm. For the transition zone predicted by the original transition model, in the range of 0° ≤ φ ≤ 160°, transition onset moves downstream monotonously. On the contrary, close to the leeward centerline, in the range of 160° ≤ φ ≤ 180°, transition onset moves forward. With regard to the revised transition model result, as φ increases from 0° to 40°, transition onset moves downstream but with a relatively smaller magnitude compared with the original transition model result. A significant discrepancy comes into occurrence between the original result and the revised result where 40° ≤ φ ≤ 160°. The revised kT-kL-ω transition model predict the prominent forward transition zone similar to the DNS result. Fig. 9 presents the laminar result of the maximum crossflow velocity magnitude normalized by the velocity on the boundary layer edge wmax/Ue. By comparison, the region with larger wmax/Ue in Fig. 9 corresponds to the area with earlier transition frontier in Fig. 8(b). Although there are still differences in detail, transition zone predicted by the revised kTkL-ω transition model shows much more agreement with the DNS data than the original transition model. It can be concluded that the revised kT-kL-ω transition model is capable of reproducing boundary layer transition induced by crossflow. Therefore, for three-dimensional boundary layer transition prediction, the revised kT-kL-ω transition model would be more preferable. With respect to the flow structure, the 1° angle of attack results in a thicker boundary layer near the leeward centerline (φ = 180°). The angle of attack effect brings about the earlier boundary layer transition near the leeward centerline (φ = 180°) than the windward centerline (φ = 0°). Many experimental and computational investigations have verified this phenomena, and relative explanations can be obtained in reference Horvath et al., 2002; Papp and Dash, 2008). In Fig. 8, both the original and the revised transition model predict such phenomena. However, compared with the original transition model result, the revised transition model brings forward the transition onset near the leeward centerline (φ = 180°). In this region, the maximum crossflow velocity magnitude normalized by boundary layer edge velocity wmax/ Ue is no larger than 2%. According to formula ((47), the constructed crossflow time scale τcross should be zero theoretically, which means crossflow cannot take effect.

distributions

mid-span region as shown in Fig. 6(a) and (b). If the free stream Reynolds number is large enough, transition zone predicted by the original transition model could involve the mid-span region just like Fig. 4(c1). But results like Fig. 4(c1) does not indicate that the original transition model can reflect the effect of crossflow physically. Secondly, the normalized maximum crossflow velocity magnitude wmax/Ue near the centerline illustrated in Fig. 5 exceeds the critical value 2% in formula (47). Consequently, the stream-wise vortex also results in a moderate value of the crossflow large-scale viscosity μntc above the centerline as shown in Fig. 6(c), which brings about a larger laminar kinetic energy production term in formula (13). As a result, a streaky structure of the intermittency γ on the centerline in Fig. 4(b2) and (c2) emerges. In consequence, the applicability of the critical value in the crossflow velocity limit function f(w) still needs further consideration. 4.2. Blunt cone with 1° angle of attack In addition to the elliptic cone HIFiRE-5, circular cone with angle of attack is another typical test case for crossflow induced transition. Li et al. (2010) have conducted a direct numerical simulation on the boundary layer transition over a 5° half-cone-angle blunt cone with 1° angle of attack. The cone is 1000 mm long with a 1 mm nose radius. For the flow conditions, Ma∞ = 6, T∞ = 79 K, Re∞ = 1 × 107/m, the angle of attack α = 1°, Tu∞ = 0.1% and Tw = 294 K. Identical configuration and flow conditions are adopted for the transition prediction in this section. Detailed description on the blunt cone model and the flow conditions can be obtained in reference (Li et al., 2010). As for the calculation grid, the distance from the first grid point to the wall is set to preserve y+< 1. Fig. 7 presents the laminar result of the blunt cone. Streamline patterns and contours of Mach number as well as surface pressure are illustrated. The surface pressure is non-dimensioned by the free stream value. Because of the 1° angle of attack, surface pressure on the

Fig. 7. Streamline patterns and contours of Ma and non-dimensional pressure (laminar result). 8

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Fig. 10. Comparison of effective length scale calculated by the original and revised transition model (x = 700 mm).

from the respect of crossflow timescale construction. Although the revised kT-kL-ω transition model reproduces the crossflow induced transition in this paper, the test case is restricted to rather simple configurations. The applicability for complex geometries still needs further investigations. In addition, the crossflow induced transition prediction of the revised kT-kL-ω transition model draws lessons from the stream-wise vorticity ΩStreamwise proposed by Langtry et al. (2015). And the crossflow timescale τcross is constructed by reference to the inviscid unstable second mode timescale τnt2. In terms of transition prediction accuracy, there is still potential room for improvements. Based on a deeper understanding of the crossflow mechanism, a more sophisticated physics-based description of transition would be more preferable.

Fig. 8. Comparison of Cf distributions on the blunt cone with 1° angle of attack.

5. Conclusion In this paper, based on the hypersonic kT-kL-ω transition model, a crossflow timescale is constructed to render the transition model the capacity to predict boundary layer transition induced by crossflow. Two test cases are selected to analyze the performance of the revised transition model, and several conclusions are drawn as follows. The crossflow timescale borrows from the experience of the streamwise vorticity proposed by Langtry et al. (2015) and is constructed by reference to the inviscid unstable second mode timescale through dimensional analysis. The HIFiRE-5 test case and the blunt cone with 1° angle of attack test case demonstrate that, with the constructed crossflow timescale, the revised kT-kL-ω transition model is capable of crossflow induced transition prediction. For the HIFiRE-5 test case, the stream-wise vortex on the centerline results in a moderate value of crossflow timescale, which brings about an improper laminar kinetic energy production. The applicability of the critical value in the crossflow velocity limit function f(w) still needs further consideration.

Fig. 9. Maximum crossflow velocity magnitude normalized by boundary layer edge velocity (wmax/Ue) (laminar result).

Fig. 10 presents the profiles of the effective length scale λeff calculated by the original and the revised kT-kL-ω transition model at x = 700 mm. It can be found that the introduction of crossflow timescale increases the effective length scale in the range of 40°≤φ≤160° remarkably. This is because boundary layer transition induced by crossflow makes a larger turbulent kinetic energy kT. As a result of the turbulent kinetic energy transport effect, the profile of kT in the adjacent area 160°≤φ≤180° would become larger, which results in a larger effective length scale λeff in return. The larger effective length scale λeff leads to a larger laminar kinetic energy kL caused by the first and the second unstable modes in the region 160°≤φ≤180° Finally, transition occurs forward near the leeward centerline as the energy transfer from laminar kinetic energy kL to turbulent energy kT. In a word, the crossflow timescale may lead to a quicker increase of the first and second modes in the adjacent region, and this needs to be corrected

(1) For the blunt cone with 1° angle of attack test case, the introduction of crossflow timescale makes the revised kT-kL-ω transition model predicts an earlier transition on the leeward centerline. Analysis manifests that, for the revised transition model, the crossflow may lead to a quicker increase of the first and second modes in the adjacent region, which needs to be corrected from the respect of crossflow timescale construction. 9

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(2) For the rather simple test cases, the revised transition model predicts the crossflow induced transition in good correspondence with the experimental data and DNS data. However, the applicability for complex configurations still needs further investigations. Besides, the practice of the crossflow induced transition prediction originates from the perspective of transition model construction, a more sophisticated physics-based description of crossflow induced transition would be more preferable. Additionally, the computational results are affected to some extent by the model constants Cl1 and Cl2, maybe introducing more transition mechanisms into the kT-kLω transition model will improve this situation.

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