Constrained large-eddy simulation of laminar-turbulent transition in compressible channel flows

Abstract

An improved approach for constrained large-eddy simulations (CLES) of wall-bounded compressible transitional flows is proposed by introducing an intermittency factor. The improved model is tested and validated with compressible channel flows at various Mach numbers and Reynolds numbers that are transitioning from laminar to turbulent states. The improved model is compared against traditional dynamic Smagorinsky model (DSM) and Direct Numerical Simulations (DNS), where the improved model is in better agreement with DNS results than traditional DSM model, in terms of mean velocity profiles, total Reynolds stress and total heat flux. Therefore, the proposed method can be used to accurately predict the temporal laminar-turbulent transition process of compressible wall-bounded flows.

1 Introduction

Accurate evaluation of the laminar-turbulent transition in compressible wall bounded flows is critical to the advancement of supersonic and hypersonic aircraft design. There is a significant difference between laminar and turbulent flows [1], for instance, the friction and heat flux under turbulent state can be 3 to 5 times higher than in laminar flows [2]. Considerable efforts have been placed in understanding the physical mechanisms of the laminar turbulent transition using Direct Numerical Simulations (DNS) over the past years to improve the prediction of the onset of transition in large-eddy simulations (LES) and Reynolds-averaged Navier-Stokes (RANS) methods [1, 3].

DNS method resolves all relevant length scales without invoking any turbulence models, making it the most suitable approach to further our understanding on the transition process. Orszag and Kells [4] utilized DNS to investigate the effects of Reynolds number on the transition process in two difference incompressible configurations with finite amplitude disturbance: (i) plane Poiseuille flow and (ii) plane Couette flow. Subsequently, Sayadi, Hamman, and Moin [5] studied the H- and K- type of transition process and found that there is a correlation between the location of maximum skin-friction and the onset of self-sustaining mechanism in near wall turbulence DNS. Zhao and Yang [6] investigated the relationship between vortex reconnection and the sharp increase of skin-friction in the K-type temporal transition in a channel flow, where they found that the surge of skin-friction coefficient occurs at the identified reconnection time. This is due to the rapid reconnection of vortex lines that could induce a velocity that is opposite to the mean flow, accelerating the near-wall fluid motion in the flow with constant mass flux. While the DNS is a powerful tool to further our understanding on the transition mechanism, its usefulness in engineering applications is hindered by the extensive requirements in numerical grid points and computational power.

In contrary to DNS, the LES resolves only the large-scale eddies, while relying on the sub-grid stress (SGS) model to accurately predict the small scale. Hence, the grid resolution requirement for LES is significantly less than DNS, allowing LES, combined with proper wall models, to be used in large industrial applications. Nevertheless, since LES relies heavily on the SGS model to capture accurately the small scale, the accuracy of the LES method lies within the sub-grid scale model. Ducros, Comte and Lesieur [7] investigated the suitability of LES on predicting the laminar-turbulent transition process in a spatially developing turbulent boundary layer over a flat plate, where they found that LES could capture the linear growth of Tollmien-Schlichting (TS) waves and the secondary instability. Sayadi and Moin [8] assessed the performance of three different SGS models, namely, the dynamic Smagorinsky eddy viscosity model [9], dynamic scale similarity model [10], and dynamic one equation model based on the SGS kinetic energy [11], where they found these models underpredicted the overshoot in skin friction profile at the end of the laminar-turbulent transition. Most importantly, the deviation from DNS results increases as the grid resolution gets coarser. Recently, an SGS eddy viscosity model that is based on the relations of eddy-viscosity and the product of the large-scale velocity strain-rate tensor with the symmetric part of the large-scale vorticity gradient [12, 13] is proposed to tackle the laminar turbulent transition process. The theoretical analysis showed that the model is capable of distinguishing between laminar and turbulent states. In addition, their simulation results further demonstrated the capability of the model to predict accurately the onset of transition, friction peak, mean velocity profile, mean temperature profile and turbulence intensities. However, the large grid requirements in the near-wall region for these models to accurately predict the laminar-turbulent transition process have hindered its usefulness for practical industrial problems [14].

RANS method is still a popular choice across all types of industrial applications due to its high efficiency and satisfactory accuracy, especially after the introduction of the local-correction-based transition $\gamma -Re_\theta$ model by Menter and Langtry et al. [15,16,17]. The main advantage of the model is that it uses experimental data to associate the transition momentum thickness Reynolds number with physical parameters such as local turbulence intensity and pressure gradient, where the transition criterion is determined by the ratio of local vorticity Reynolds number and critical momentum thickness Reynolds number. Hence, the transition model is compatible with modern CFD approaches, i.e., unstructured grids and massive parallel execution. However, the model has no physical mechanism of the transition process and it is rather an ad-hoc model determined through brute force. The $\gamma -Re_\theta$ model has successfully been mainly used in the simulation of low-speed boundary layer transition flow with further improvement done to the model by Krause, Behr and Ballmann [18] for hypersonic boundary layer transition simulations. Krause, Behr and Ballmann [18] constructed a relationship between $F_{length}$ and $Re_{\theta c}$ by considering freestream turbulence intensities, allowing the simulation results to agree well with the experimental data in the cases of hypersonic double ramp. Bensassi, Lani and Rambaud [19] tested the model by simulating a Mach 8 flow over a 7$^\circ$ half angle sharp cone at three different freestream turbulence intensity levels, where they reported that the heat flux in the transition regime shows excellent agreement with experimental data. Zhang et al. [20, 21] calibrated the basic $\gamma -Re_{\theta }$ empirical correlation with different samples and results in different cases are satisfactory.

Therefore, the couple of RANS and LES methods could combine the best of both worlds to achieve a good balance between computational efficiency and accuracy to capture important features of unsteady flows with large separations in large industrial applications [22,23,24]. The RANS-LES hybrid methods usually decompose the computational domain into inner and outer regions with the RANS method handling the inner region and the LES method resolving the outer region. Detached-eddy simulation (DES) is a representative of the RANS-LES hybrid method, which was first proposed in 1997 and has been widely used in wall turbulence at high Reynolds numbers [25]. Although the DES method has achieved great success in engineering flows, some non-physical problems such as log-layer mismatch (LLM) still exist in the wall turbulence. Delayed DES (DDES) [23] and improved delayed DES (IDDES) [26] are subsequently developed to address these issues by altering the formulation of the length-scale function, but these schemes introduced other non-physical factors.

The main reason for the existence of LLM is the obvious difference in the fluctuation information near the interface. Chen et al. [27] proposed a new method, namely the constrained large-eddy simulation (CLES), that simulates the whole flow domain by LES while enforcing a Reynolds stress constraint on the mean part of SGS model in the inner layer. It has been demonstrated to achieve remarkable results in the incompressible channel flow and the flow with separation. Jiang et al. [28] further extended the CLES method for compressible wall-bounded turbulence by decomposing the SGS stress and heat flux into their mean part and fluctuation part in the near-wall region, with the mean SGS stress and heat flux being constrained by the prescribed Reynolds stress model and turbulent heat flux model, respectively. Zhang et al. [29, 30] simulated a turbulent flow over rough walls in the framework of CLES by invoking the rough-wall-like mean shear model without actually resolving the roughness elements. The statistical results from the work of Zhang et al. [29, 30] show that the effect of rough-wall-like mean shear on the turbulent flow is similar to that of roughness elements and the approach based on CLES can be used as a new wall model for the simulation of the rough-wall flows. Zhao et al. [31] incorporated an intermittency factor into the CLES method, where their improved model is capable of predicting accurately the transition onset of laminar-turbulent transition in incompressible channel flow. However, their model is unable to capture the maximum skin friction at later stage of the transition. Wang et al. [32] introduced a transition-predictive RANS model as the controlling condition to the CLES (TrCLES). The TrCLES method performed excellent well in predicting the laminar separation bubble and separation-induced transition process in both the Eppler 387 (E387) airfoil and the ultra-high-lift low-pressure turbine T106C flows.

In this paper, we extend Zhao et al.’s [31] method by constructing an intermittency factor with CLES that is capable of predicting the transition process in wall bounded compressible flows. In the next section, required mathematical background and the simulation parameters are provided. Numerical results obtained from DNS, DSM-based LES and CLES are presented in Section 3, followed by concluding remarks.

2 Problem formulation

2.1 Governing equations

The filtered conservation of mass, momentum, and energy equations in the context of LES employed in this study to describe the compressible flows are formulated as [33]:

$$\begin{aligned} \frac{\partial {\bar{\rho }}}{\partial {t}}+\frac{\partial {\left( \bar{\rho }\tilde{u}_i \right) } }{\partial {x_i}}=0, \end{aligned}$$

(1)

$$\begin{aligned} \frac{\partial {\left( \bar{\rho }\tilde{u}_i\right) } }{\partial t}+\frac{\partial {\left( \bar{\rho }\tilde{u}_i \tilde{u}_j +\bar{p}\delta _{ij} \right) }}{\partial {x_j}}=\frac{1}{Re}\frac{\partial \tilde{\sigma }_{ij} }{\partial x_j}+\frac{\partial {\tau }_{ij}^{LES}}{\partial x_j}, \end{aligned}$$

(2)

$$\begin{aligned} \frac{\partial {\left( \bar{\rho }\tilde{e}\right) }}{\partial t}+\frac{\partial {\left[ \tilde{u}_i \left( \bar{\rho }\tilde{e}+\bar{p} \right) \right] }}{\partial x_i}=\frac{\partial \tilde{q}_i}{\partial x_i}+\frac{\partial {\left( \tilde{\sigma }_{ij}\tilde{u}_j \right) }}{\partial x_i}+\frac{\partial {q^{LES}_{i}} }{\partial x_i}+\frac{\partial {\left( \tau ^{LES}_{ij} \tilde{u}_j \right) }}{\partial x_i}, \end{aligned}$$

(3)

where $\bar{\cdot }$ and $\tilde{\cdot }$ represent the spatial and Favre filter, respectively, with a filter width of $\bar{\Delta }$. The Favre averaging operation of a flow field variable $\mathcal {F}$ is defined as:

$$\begin{aligned} \widetilde{\mathcal {F}}(x_i,t) = \frac{\overline{\rho \mathcal {F}}}{\bar{\rho }} . \end{aligned}$$

(4)

Here, $\bar{p}=\bar{\rho }R\widetilde{T}$ is the static pressure, $\bar{\rho }$ is the density, $\tilde{u}_i$ is the velocity vector, $\widetilde{T}$ is the temperature, R is the specific gas constant, $\tilde{e}=C_v\widetilde{T}+\frac{1}{2} \tilde{u}_i \tilde{u}_i$ is the total energy per unit mass, $C_v=1/\left[ \alpha \left( \alpha -1\right) M^2_\infty \right]$ is the specific heat at constant volume, $M_\infty =U_\infty /\sqrt{\alpha R T_\infty }$ is the Mach number, and $\alpha$ is the ratio of specific heat with a value of 1.4 adopted in this study. The viscous stress tensor $\tilde{\sigma }_{ij}$ is described as

$$\begin{aligned} \tilde{\sigma }_{ij}=\mu \left( \frac{\partial \tilde{u}_i}{\partial x_j}+\frac{\partial \tilde{u}_j}{\partial x_i} \right) -\frac{2}{3}\mu \frac{\partial \tilde{u}_k}{\partial x_k}\delta _{ij}, \end{aligned}$$

(5)

and the heat flux vector $\tilde{q}_i$ is determined as

$$\begin{aligned} \tilde{q}_i=-\frac{C_p \mu }{Pr} \frac{\partial \widetilde{T}}{\partial x_i}, \end{aligned}$$

(6)

where $C_p=\alpha C_v$ is the specific heat at constant pressure and Pr is the molecular Prandtl number. $Pr = 0.7$ is adopted in this study. The molecular viscosity is computed using Sutherland’s law:

$$\begin{aligned} \mu =\widetilde{T}^{3/2} \frac{1+S}{\widetilde{T}+S}, \end{aligned}$$

(7)

with $S=110.3\,\mathrm{K}/T_\infty$ . The unclosed terms in Eqs. (2) and (3) are the SGS stress tensor and the SGS heat flux vector, respectively, and they are given as:

$$\begin{aligned} \tau ^{LES}_{ij} = -\bar{\rho }\left( \widetilde{u_i u_j}-\tilde{u}_i\tilde{u}_j \right), \end{aligned}$$

(8)

$$\begin{aligned} q^{LES}_i=\bar{\rho }C_p\left( \widetilde{Tu_i}-\widetilde{T}\tilde{u}_i \right) , \end{aligned}$$

(9)

where the closures for the two terms are provided in the following section.

2.2 CLES

In the CLES method, the flow field is divided into near-wall inner region and outer region, and the governing equations introduced in Section 2.1 are numerically solved in both the inner and outer regions. In the outer region, the Smagorinsky model introduced by Yoshizawa [34] for compressible turbulence is employed in this study as follows:

$$\begin{aligned} \tau ^{SGS}_{ij}=2C^2_S\bar{\rho }\bar{\Delta }^2\Vert \tilde{S}\Vert \left( \tilde{S}_{ij}-\frac{1}{3}\tilde{S}_{kk}\delta _{ij} \right) -\frac{2}{3}C_I\bar{\rho }\bar{\Delta }^2\Vert \tilde{S} \Vert ^2\delta _{ij}, \end{aligned}$$

(10)

$$\begin{aligned} q^{SGS}_i=-C^2_S\frac{\bar{\Delta }^2\bar{\rho }C_p\Vert \tilde{S}\Vert }{Pr_T}\frac{\partial \widetilde{T}}{\partial x_j}, \end{aligned}$$

(11)

where $\bar{\Delta }$ is the grid scale and $\Vert \tilde{S}\Vert =\left( 2\tilde{S}_{ij}\tilde{S}_{ij} \right) ^{1/2}$ is the magnitude of $\tilde{S}_{ij}$. The constants, $C_S$, $C_I$ and $Pr_T$, in the above equations are computed dynamically following ref. [35]. The boundary separating both constrained and non-constrained regions is determined automatically using the DES criterion [25, 28].

On the other hand, the $\tau ^{LES}_{ij}$ and heat flux $q^{LES}_i$ in the near-wall region are split into the mean and fluctuating parts as [28]:

$$\begin{aligned} \tau ^{LES}_{ij}=\left< \tau ^{LES}_{ij} \right> + \left( \tau ^{LES}_{ij} \right) ^\prime \end{aligned}$$

(12)

and

$$\begin{aligned} q^{LES}_i=\left<q^{LES}_i \right>+\left( q^{LES}_i \right) ^\prime , \end{aligned}$$

(13)

respectively, where $\left<\cdot \right>$ represents ensemble averaging over the statistically homogeneous direction. The density weighted Reynolds averaging (also referred to as Favre averaging) is defined as:

$$\begin{aligned} \left\{ \mathcal {F} \right\} =\frac{\left<\rho \mathcal {F} \right> }{\left<\rho \right> }. \end{aligned}$$

(14)

The mean SGS stress and heat flux are constrained by Reynolds stress and heat flux approximately by RANS models through the following relations:

$$\begin{aligned} \left<\tau ^{LES}_{ij} \right>=\tau ^{RANS}_{ij}+\left<\bar{\rho } \right>\left( \left\{ \tilde{u}_i\tilde{u}_j \right\} -\left\{ \tilde{u}_i \right\} \left\{ \tilde{u}_j \right\} \right) \end{aligned}$$

(15)

and

$$\begin{aligned} \left<q^{LES}_i \right>=q^{RANS}_i-C_p\left<\bar{\rho } \right>\left( \left\{ \tilde{u}_i\widetilde{T} \right\} -\left\{ \tilde{u}_i \right\} \left\{ \widetilde{T} \right\} \right) . \end{aligned}$$

(16)

Here,$-\left<\bar{\rho } \right>\left( \left\{ \tilde{u}_i\tilde{u}_j \right\} -\left\{ \tilde{u}_i \right\} \left\{ \tilde{u}_j \right\} \right)$ and $\left<\bar{\rho } \right>C_p\left( \left\{ \tilde{u}_i\widetilde{T} \right\} -\left\{ \tilde{u}_i \right\} \left\{ \widetilde{T} \right\} \right)$ are the resolved Reynolds stress and heat flux respectively. The expressions of $\tau _{ij}^{RANS}$ and $q_i^{RANS}$ can be obtained from the governing equations of RANS [28]:

$$\begin{aligned} \tau ^{RANS}_{ij} = -\left\langle \rho \right\rangle \left( \left\{ u_iu_j\right\} -\left\{ u_i\right\} \left\{ u_j\right\} \right) \end{aligned}$$

(17)

and

$$\begin{aligned} q^{RANS}_i = \left\langle \rho \right\rangle C_p\left( \left\{ u_iT\right\} -\left\{ u_i\right\} \left\{ T\right\} \right) . \end{aligned}$$

(18)

$\tau ^{RANS}_{ij}$ and $q^{RANS}_{i}$ can be modeled from different models for the specific flow geometry, such as $k-\epsilon$ model, $k-\omega$ model [36], Baldwin-Lomax (B-L) model [37], SST model [38], etc. In the present work, the turbulent viscosity $\mu _t$ is calculated from the one equation Spalart-Allmaras (S-A) model [39].

The fluctuation parts of the SGS stress tensor and heat flux vector are given by

$$\begin{aligned} \left( \tau ^{LES}_{ij}\right) ^\prime = \tau ^{SGS}_{ij}-\left<\tau ^{SGS}_{ij} \right>, \end{aligned}$$

(19)

$$\begin{aligned} \left( q^{LES}_i \right) ^\prime =q^{SGS}_i-\left<q^{SGS}_i \right>, \end{aligned}$$

(20)

where $\tau ^{SGS}_{ij}$ and $q^{SGS}_i$ are calculated from traditional SGS stress and heat flux vector in LES. Hence, the full expression of the SGS stress tensor and heat flux vector in near-wall regions reads:

$$\begin{aligned} \tau ^{LES}_{ij}=\tau ^{RANS}_{ij}+\left<\bar{\rho } \right>\left( \left\{ \tilde{u}_i\tilde{u}_j \right\} -\left\{ \tilde{u}_i \right\} \left\{ \tilde{u}_j \right\} \right) +\left( \tau ^{SGS}_{ij}-\left<\tau ^{SGS}_{ij} \right> \right), \end{aligned}$$

(21)

$$\begin{aligned} q^{LES}_i=q^{RANS}_i-C_p\left<\bar{\rho } \right>\left( \left\{ \tilde{u}_i\widetilde{T} \right\} -\left\{ \tilde{u}_i \right\} \left\{ \widetilde{T} \right\} \right) +\left( q^{SGS}_{i}-\left<q^{SGS}_{i} \right> \right) . \end{aligned}$$

(22)

2.3 Intermittency factor

The original unmodified CLES method has been demonstrated by Zhao et al. [31] to be unable to capture the transition process accurately. Hence, they introduced an intermittency factor to the CLES method to improve its predictability in an incompressible channel transition problem. Intermittency factor is widely used in RANS models and we have included it in the theoretical framework of CLES, allowing us to select any available transitional RANS model as a constraint within the CLES method for transitional flows. In the present work, the intermittency factor based on the one equation S-A eddy viscosity model is selected due to its simplicity and without introducing any additional extra equations.

The turbulent eddy-viscosity $\mu _t$ used in the Reynolds stress constraint in the CLES of transitional flows can be reformulated by introducing an intermittency factor $\gamma$ such that $\mu ^{\gamma }_t=\gamma \mu _t$. Following the works of Piomelli et al. [40] and Zhao et al. [31], we extend the intermittency factor $\gamma$ to be used in compressible transitional flows.

$$\begin{aligned} \gamma =\frac{H_l-H_{12}}{H_l-H_t}. \end{aligned}$$

(23)

Here, $H_l$ and $H_t$ are the shape factors of laminar and turbulent channel flows, respectively. In this present study, $H_t$ is fixed at approximately 1.7 and $H_l$ varies from 2.2 to 2.7, respectively. $H_{12}$ is the shape factor and its value varies within the transition layer as:

$$\begin{aligned} H_{12}=\frac{\delta _1\left( x \right) }{\delta _2\left( x\right) }, \end{aligned}$$

(24)

where $\delta _1\left( x\right)$ and $\delta _2\left( x\right)$ are the displacement thickness and momentum thickness, respectively [41]:

$$\begin{aligned} \delta _1\left( x\right) =\int _{0}^{H/2} \left( 1-\frac{\rho u}{\rho _\infty U_\infty } \right) dy, \end{aligned}$$

(25)

$$\begin{aligned} \delta _2\left( x \right) =\int _{0}^{H/2}\frac{\rho u}{\rho _\infty U_\infty }\left( 1-\frac{u}{U_\infty } \right) dy, \end{aligned}$$

(26)

with $\rho _\infty$ and $U_\infty$ being the density and speed of the free-stream, respectively.

2.4 Test cases

The suitability and performance of the CLES method for transitional flows are evaluated on three-dimensional compressible channel flows with temporal mode at different Mach numbers and Reynolds numbers. The simulation results obtained from the CLES method are compared to conventional DNS and DSM-based LES methods in three cases with $M_\infty =1.5$ and $Re_\infty =3000$, $M_\infty =2.0$ and $Re_\infty =3500$, $M_\infty =3.0$ and $Re_\infty =4880$, respectively. The LES based on the dynamic Smagorinsky model (DSM) will be referred hereafter as LES. All three methods have identical simulation setup and settings with only the exception of the number of grid points. The number of grid points in CLES and traditional LES methods are identical while the number of grid points in DNS is significantly higher to resolve all necessary length scales. The simulation parameters are reported in Table 1, where $M_\infty =U_\infty /c_\infty$ is the free stream Mach number with $c_\infty$ being the sound speed at the far field, $Re_\infty =\rho _\infty U_\infty /\mu _\infty$ is the Reynolds number, $T_\infty =288.15\,\mathrm{K}$ is the free stream temperature, and $T_w$ is the temperature at the wall boundary. The transport equations are discretized using a Cartesian grid of $N_x \times N_y \times N_z$ with uniform distribution in the streamwise and spanwise directions and hyperbolic tangent distribution in the wall-normal direction in a rectangular computational domain of $L_x \times L_y \times L_z =4\pi \times 2.0\times 4\pi /3$. $\Delta x^+$, $\Delta y^+_w$ and $\Delta z^+$ are the first grid cell length in streamwise, wall-normal and spanwise directions, respectively, and they are normalized by the viscous length scale $\delta _v$.

Table 1 Summary of computational parameters. The computational domains $L_{x}$, $L_{y}$ and $L_{z}$ are non-dimensionalized by channel half-height $\delta$

Periodic boundary conditions are applied to streamwise and spanwise directions. The non-slip boundary and isothermal conditions are set at the walls in the transverse direction. The convection terms are discretized by a seventh-order upwind scheme with the Steger-Warming flux splitting [42], and the viscous terms are approximated by an eighth-order central difference scheme. A third-order total variation diminishing type of Runge-Kutta method is applied for time advancing [43]. The flow is driven by a uniform body force, which is time-dependent and maintains a constant mass flux [44]. Laminar flow with random disturbance is used as the initial condition for the CLES and LES methods. In contrary to the CLES and LES methods, the statistically steady state solution from CLES is subsequently interpolated onto the DNS grid points to be used as an initial condition for the DNS method.

3 Results

3.1 Intermittency factor and transition process prediction

To evaluate the grid effect of the CLES method, we perform two more simulations with different grid resolutions for the C2-CLES case at $M_\infty =2.0$ and $Re_\infty =3500$. The grid settings of the coarser and the finer are $N_x\times N_y\times N_z=48\times 49\times 48$ and $96\times 97\times 96$ respectively. As shown in Fig. 1, the evolution of $Re_\tau$ with time is identical for the C2-CLES case and the case with finer mesh configuration in the laminar and transition stage. While the $Re_\tau$ from the case with coarser mesh is a little bit larger than results from the other two CLES cases throughout the flow stage, which may be caused by higher dissipation of the coarser mesh. Furthermore, the $Re_\tau$ for C2-CLES and the case with finer mesh are almost the same in the fully developed turbulent stage. Therefore, the grid resolution of C2-CLES is enough for the CLES method and it is adopted in the following analysis.

Fig. 1

Temporal evolution of the $Re_\tau$ from the C2-CLES case with different grid resolutions. t is normalized as: $t^* = tU_b/\delta$

Figure 2 shows the evolution of the intermittency factor $\gamma$ from the three CLES cases, where it is equal to 0 in the laminar state and progresses towards unity in the turbulent state, indicating the intermittency factor to agree well with the transition process for all cases. It is worth noting that $\gamma$ may be slightly larger or slightly smaller than 1 for different cases, and subsequent analysis shows that this has little effect on the flow, which is also reported in Zhao et al.[31]. The Friction Reynolds numbers ($Re_\tau = \delta /\delta _v$) computed from the three different methods (LES, CLES, and DNS) of three cases are shown in Fig. 3. In the laminar stage, the predicted $Re_\tau$ from CLES and LES methods are slightly higher than the DNS results (eg. $t^*<50$ for the $M_\infty =1.5$). The transition from laminar to turbulent is found to occur at $t^*\approx 100$ for the C1 case with DNS and CLES is able to capture the transition period accurately (see Fig. 3(b)). However, the LES predicted a much later onset of the transition for the case of $M_\infty =1.5$. While both LES and CLES methods can capture the rapid rise of $Re_\tau$ right after the transition to its peak, the CLES method has better agreement with the DNS results for all three cases. This is further reflected in the time-averaged values of $Re_\tau$ obtained from DNS, LES, and CLES (see Table 1), where the value predicted by CLES is in much closer agreement with the DNS result for all cases. It is worth mentioning here that the reported time-averaged value of $Re_\tau$ is computed after the flow becomes fully developed turbulence. It should also be noted that both CLES and LES underestimated the peak value of $Re_\tau$ at the late transition stage obtained from DNS data for all cases.

Fig. 2

Temporal evolution of the intermittency factor $\gamma$ for C1-CLES, C2-CLES, and C3-CLES. t is normalized as: $t^* = tU_b/\delta$

Fig. 3

Temporal evolution of $Re_\tau$ obtained from DNS, CLES, and LES methods for all three cases a, b C1, c, d C2, e, f C3

As shown in Section 2.2, the total Reynolds shear stress $\tau _{ij}^{RANS}$ for CLES and LES is the sum of the mean SGS stress $\left\langle \tau _{ij}^{LES} \right\rangle$ and the resolved Reynolds shear stress $-\left\langle \rho \tilde{u}^{\prime \prime }_i \tilde{u}^{\prime \prime }_j \right\rangle$, while the total Reynolds shear stress is expressed as $-\left\langle \rho u^{\prime \prime }_i u^{\prime \prime }_j \right\rangle$ in DNS. Reynolds stresses from different methods of the C2 case in the laminar and transition stages are shown in Fig. 4. In the laminar period of $t^*=120$, both CLES and LES methods show great agreement in the near wall and channel center regions, and the CLES result also matches well to the DNS result in the area of $0.3< y_w<0.5$ while LES is worse than CLES. In the transition stage of $t^*=180$, 200, and 220, both CLES and LES methods can only match the Reynolds stress in the region of $y_w<0.15$ and show a significant discrepancy with the DNS result in the far from wall region. To sum up, CLES shows better performance than the traditional LES method near the wall, which is induced by the effect of the constrained model prescribed in the inner region of the computational domain [27].

Fig. 4

Reynolds stresses in the laminar and transition stages of case C2. a $t^*=120$; b $t^*=180$; c $t^*=200$; d $t^*=220$

3.2 Statistics of fully developed turbulence

The normalized mean streamwise velocity profiles for fully developed turbulent state of case C1 obtained from DNS, CLES and LES are compared to Morinishi [45] in Fig. 5(a). The LES method overestimated the mean velocity profile at large $y^+$ values, while the CLES method agrees extremely well with the results from the present DNS and also previously obtained DNS by Morinishi et al. [45]. In addition, the normalized temperature $\langle u\rangle ^+ - T^+_w$, where $T_\tau =-\left\langle q_w\right\rangle /\left( \left\langle \rho _w\right\rangle C_p u_\tau \right)$, obtained from present DNS, CLES, and LES are shown in Fig. 5(b). Similar to the results of mean streamwise velocity profile, the temperature distribution predicted by CLES is almost identical to the DNS results, but LES results deviate from DNS results at large $y^+$ ($y^+>10$).

Fig. 5

Comparison of the a normalized mean streamwise velocity profiles $\langle u\rangle ^+=\langle u \rangle /u_\tau$ and b normalized temperature vs. normalized $y^+=y/\delta _v$ obtained from present and previous studies

Figure 6 shows the comparison of the Reynolds stress computed from DNS, LES, and CLES at $Re_\infty =3000$ and $M_\infty =1.5$. It is found that the peak value of the total Reynolds stress is within the viscous wall region ($y^+<50$), indicating the region with the most vigorous turbulent activity [46] and momentum exchange. In addition, the total Reynolds stress obtained from CLES agrees extremely well with the DNS data, except for the region near the interface. Nevertheless, the results obtained in CLES are far more superior than the traditional LES method in the near-wall constrained region.

Fig. 6

The normalized total Reynolds stress vs. the normalized distance to the wall with the half height of the channel $\delta$

The components of the normalized total Reynolds stress from CLES and LES ($-\left\langle \rho \tilde{u}^{\prime \prime }\tilde{v}^{\prime \prime } \right\rangle /\left\langle \tau _w \right\rangle$) and DNS ($-\left\langle \rho u^{\prime \prime }v^{\prime \prime } \right\rangle /\left\langle \tau _w \right\rangle$) are shown in Fig. 7(a). The Reynolds stresses derived from CLES and LES are almost identical, but they deviate significantly from the DNS results in the viscous wall region. This is due to the lack of small-scale fluctuations in LES and CLES, where the introduction of the SGS stress models could improve the Reynolds stress predictions. As shown in Fig. 7(b), the mean modeled SGS stress is significantly different in the near-wall region due to the prescribed RANS model within the CLES framework. The value of the normalized modeled Reynolds stress grows to its peak rapidly in the buffer layer along the wall normal direction in the CLES method, yielding an order of magnitude larger than the DSM results. The value of $\left\langle \tau ^{LES}_{12} \right\rangle$ decreases rapidly to 0 near the edge of the buffer layer and outer layer as expected.

Fig. 7

The components of the normalized total Reynolds stress $\left\langle \tau _{12}^{RANS} \right\rangle /\left\langle \tau _w \right\rangle$ vs. the distance to the wall $y_w$: a the resolved Reynolds stress $-\left\langle \rho u^{\prime \prime }v^{\prime \prime } \right\rangle /\left\langle \tau _w \right\rangle$ and b the average of modeled SGS stress $\left\langle \tau _{12}^{LES} \right\rangle /\left\langle \tau _w\right\rangle$

Similarly to the total Reynolds stress $\tau _{ij}^{RANS}$, the total turbulent heat flux is defined as the sum of the mean modeled SGS heat flux $\left\langle q_i^{LES} \right\rangle$ and the resolved turbulent heat flux $\left\langle \rho C_p \tilde{u}^{\prime \prime }_i\tilde{T}^{\prime \prime } \right\rangle$, and the modeled and the resolved heat flux are compared in Fig. 8. The turbulent heat flux reaches its peak in the buffer layer, indicating that the turbulent energy exchange in this region is energetic. The CLES method has an overall better agreement to the DNS result than the LES method, but some noticeable deviations from the DNS result can be observed near the interface of constrained and non-constrained regions.

Figure 9 shows the components of the total turbulent heat flux obtained using the CLES and DSM methods with the DNS results. The turbulent heat flux of DNS is expressed as $\left\langle \rho C_p {u}^{\prime \prime }_i{T}^{\prime \prime } \right\rangle$, which is significantly larger than the $\left\langle \rho C_p \tilde{u}^{\prime \prime }_i\tilde{T}^{\prime \prime } \right\rangle$ of CLES and DSM in the buffer layer. This further indicates that the coarse mesh filtered out many small scale fluctuations of the temperature field. In addition, the results obtained from the CLES and LES methods are consistent through the inner and outer layers, except for regions near the interface. The CLES modeled SGS heat flux is significantly larger than the DSM method in the inner region because of the effect of the constrained heat flux model as shown in Fig. 9(b).

Fig. 8

The normalized total turbulent heat flux $\left\langle q_{2}^{RANS} \right\rangle /\left\langle q_w \right\rangle$ vs. the distance to the wall $y_w$

Fig. 9

The components of the normalized total Reynolds stress $\left\langle q_{2}^{RANS} \right\rangle /\left\langle q_w \right\rangle$ vary with the distance to the wall $y_w$: a the resolved Reynolds stress $-\left\langle \rho C_p v^{\prime \prime }T^{\prime \prime } \right\rangle /\left\langle q_w \right\rangle$ and b the average of modeled SGS stress $\left\langle q_{12}^{LES} \right\rangle /\left\langle q_w\right\rangle$

The time averaged flow fields of streamwise velocity u at fully developed state in $x-z$ (x and z are normalized by the half height of the channel $\delta$) plane at $y^+=10$ and $y^+=30$ from DNS, CLES, and LES are shown in Figs. 10 and 11, respectively. As shown in Fig. 10, the low speed streaks distribute uniformly in the DNS results, with the CLES results to be in better agreement to the DNS results than the LES results. In addition, the low speed streaks of DNS feature a thinner width and more meandering paths along the streamwise direction than the CLES and LES results, and this is mainly due to the lower grid resolution used in the CLES and LES methods. Nevertheless, the CLES performs much better in predicting the width and path of low speed streaks when compared to LES even at an identical grid resolution. Figure 11 exhibits similar characteristics to Fig. 10, but the velocity is higher in $y^+=30$ than $y^+=10$, suggesting the CLES to be more accurate in predicting the time averaged flow field.

Fig. 10

The time averaged velocity field of u in $x-z$ plane at $y^+=10$ from DNS, CLES and DSM

Fig. 11

The time averaged velocity field of u in $x-z$ plane at $y^+=30$ from DNS, CLES and DSM

4 Conclusion

In this paper, an intermittency factor based on the shape factor is introduced to improve the existing CLES method for compressible channel flows involving the temporal laminar-turbulent transition period. The proposed method is critically evaluated by comparing its performance on simulating channel flows at $M_\infty =1.5$ and $Re_\infty =3000$, $M_\infty =2.0$ and $Re_\infty =3500$, and $M_\infty =3.0$ and $Re_\infty =4880$ with the widely used LES-DSM model and the DNS method. The time history of the Reynolds friction demonstrated that the CLES with intermittency factor could better capture the transition onset and the transition process than the traditional LES with DSM model. The Reynolds stresses during the transition show that both the LES with DSM and the improved CLES methods match well with the DNS result in the near wall region.

The Reynolds stresses and turbulent heat flux from CLES are further investigated and compared with DNS and LES methods, where the CLES method performs better than the traditional LES method in the inner region. Unfortunately, both CLES and LES methods failed to capture accurately the maximum Reynolds stress and turbulent heat flux near the buffer layer. However, the mean streamwise velocity and the mean temperature profiles obtained from CLES are in excellent agreement with the DNS result, while the LES method overestimated the streamwise velocity profiles in the log-law region and the temperature profiles at large $y^+$ ($y^+>10$).

All in all, the current CLES model coupled with the intermittency factor is capable of predicting accurately the transition process and onset in compressible channel flows. Most importantly, the proposed method maintained the original characteristics of CLES in terms of accurate predictions of the statistical properties of fully developed turbulence. Therefore, CLES shows great potential in simulating transitional flows, but more studies are warranted to develop better intermittency factor to accurately capture the physics near the interface, as well as in the spatial laminar-turbulent transition.