1 Introduction

As the foundation of turbulence research, the Richardson-Kolmogorov theory^[1-4] leads to two conclusions in the inertial range of high-Reynolds number turbulence, i.e., the scaling law of k^-5/3 where k is the wavenumber for the energy distribution at an instant and the equilibrium energy transfer process which leads to the equilibrium dissipation scaling

(1)

in which the dissipation coefficient C_ϵ is constant, ϵ is the dissipation rate, is the turbulent kinetic energy, and is the integral length scale. The same resultant dissipation scaling has also been suggested by simplifying the generic Kármán-Howarth equation (without assumptions of homogeneity and isotropy)^[5-8]. In fact, in some situations, the assumptions involved in the above approaches are not satisfied, and this might result in non-constant C_ϵ, which has been considered as a consequence of the non-equilibrium turbulent kinetic energy transfer process. This anomalous dissipation scaling has been found in many experiments and numerical simulations for fractal grid turbulence^[9-13], turbulent wakes^[14-16], homogeneous isotropic turbulence (HIT) with unsteady forcing^[17-18], transitional channel flows^[19], etc. In general, the non-equilibrium dissipation scaling can be described as C_ϵ~ Re_I^m/Re_Lⁿ or C_ϵ~Re_I^m/2/Re_λⁿ with m≈1≈ n, where Re_λ is the local Reynolds number defined as in which λ is the Taylor scale defined as , ν is the viscosity, Re_L is the integral scale Reynolds number defined as , and Re_I the inlet Reynolds number.

In recent years, the skewness of longitudinal velocity derivative^[20-21] defined by

(2)

where u is the fluctuating velocity component in the x-direction and ⟨·⟩ is the ensemble average, is suggested to indicate the similar non-equilibrium phenomena as C_ϵ. In the experiments of Hearst and Lavoie^[22], the consistent revolution tendency between C_ϵ and S_k along the downstream of the grid was observed. This definition has also been extended to the resolved-scale skewness to enable the investigation of multi-scale non-equilibrium phenomena^[23].

Though the two quantities described above, i.e., C_ϵ and S_k, can describe non-equilibrium turbulence phenomena, they are limited in simple geometries and are not applicable for complex turbulent flows. Indeed, the calculation of C_ϵ needs to calculate the integral scale involving spacial integration, which by definition is only valid in homogeneous turbulence. Meanwhile, S_k is defined by only the local variables, and can be used in inhomogeneous turbulence, which cannot be simply extended to anisotropic turbulence. In order to extend these definitions to complex flows, Fang et al.^[24] used the equation of Lagrangian velocity gradient correlation to estimate the non-equilibrium phenomena. Specifically, the equation is

(3)

where the superscript ' denotes fluctuation, is the fluctuating velocity gradient tensor, d/dt represents the Lagrangian derivative^[24-25], and

(4)

are the production, pressure, and viscosity terms, respectively. This equation can also be extended to the large-eddy simulation (LES) databases containing only the information of the resolved scale, by adding a subgrid-scale term according to the expression of the subgrid stress^[24]. In equilibrium HIT, the constant value of S_k implies that Φ_ijij'=0, illustrating that Φ_ijij' can be considered as a new quantity for estimating non-equilibrium turbulence^[24]. In Subsection 3.4 of Ref. [24], the consistency was discussed among the criteria of C_ϵ, S_k, and Φ_ijij' in an interior flow in a compressor, but the results were still quite preliminary.

In order to study the non-equilibrium turbulence under complex geometries and to find more different mechanisms of non-equilibrium turbulence, in the present contribution, we focus on a direct-numerical simulation (DNS) database of a backward-facing ramp to resolve the whole range of energetic eddies of turbulence. Such "numerical experiments" have been widely adopted in the studies in turbulence control^[26-30] and turbulence physics^[31-34]. In this study, the non-equilibrium phenomena raised by two typical flow conditions, i.e., the boundary layer flow and the wake with a separation vortex downstream the ramp, are investigated, respectively, showing different underlying mechanisms. In Section 2, we will introduce the basic settings of the DNS database and its validation. In Section 3, we will show the consistency among the three indicators of non-equilibrium turbulence mentioned above, explain the mechanisms of energy and transfer spectra evolutions, and quantitatively show the non-equilibrium turbulent phenomena at different filtering scales. Concluding remarks are given in Section 4 finally.

2 Descriptions of backward-facing ramp and DNS settings

In this section, the settings of DNS simulation are described in Subsection 2.1 in three aspects: the governing equations, the numerical method, and the computation setup, respectively. The simulation validations will be found in Subsection 2.2. The averaged flow condition will be described in Subsection 2.3.

2.1 Methodology 2.1.1 Governing equations

The three-dimensional (3-D) unsteady compressible Navier-Stokes (N-S) equations in a general time-invariant system are solved numerically. This set of N-S equations can be written in a strong conservation non-dimensional form as follows:

(5)

where Q=(ρ, ρu, ρv, ρw, E)^T is the solution vector, in which ρ is the density, u, v, and w are components of the velocity, and E is the total energy. The static temperature T and the pressure P are related to the density via the equation of state of the ideal gas as P=ρRT. The convection terms E_i and the diffusion terms F_i in Eq. (5) are, respectively, expressed as

(6)

with the Einstein summation convention. The indices i=1, 2, 3 correspond to the streamwise, wall-normal, and spanwise directions, respectively. The notation x_i=(x₁, x₂, x₃) is adopted to represent (x, y, z), and u_i=(u₁, u₂, u₃)= (u, v, w) corresponds to the streamwise, wall-normal, and spanwise velocity components, respectively. δ_ij is the standard Kronecker delta. The total energy E is expressed as follows:

(7)

The stress tensor and the heat flux vector are expressed as follows:

(8)

(9)

The dynamic viscosity coefficient μ is calculated via the Sutherland law as follows:

(10)

where T_ref=298.15 K, and T_S=110.4 K.

The N-S equations are non-dimensionalized with the reference density ρ_ref, the reference velocity u_ref, the temperature at wall T_ref, and the dynamic viscosity μ_ref as well as the height of the ramp H. The resulting dimensionless parameters are the Reynolds number Re=ρ_refu_refH/μ_ref and the Mach number A constant Prandtl number Pr=μc_p/k=0.72 is used, where c_p=γR/(γ-1) is the specific heat capacity of gas at constant pressure, and k is the thermal conductivity. R and γ are the gas constant and the specific heat capacity ratio, which are set to be R=287.1 J·kg^-1·K^-1 and γ=1.4, respectively.

2.1.2 Numerical method

The in-house code ASTR, which has been applied in DNS^[35-36] and LES^[37] of wall bounded turbulent flows, is used to solve Eq. (5). The N-S equations are projected to the computational coordinate system and solved with the classic sixth-order compact central scheme^[38]. To remove the small-scale wiggles due to the aliasing errors resulting from the discrete evaluation of the nonlinear convection terms, the 10th-order compact filter is incorporated, which limits the filter being implemented only at high wave numbers^[39]. After all the spatial terms are solved, the 3rd-order three-step total variation diminishing Runge-Kutta method^[40] is used for the time integration.

2.1.3 Computation setup

The considered geometry corresponds to the configuration of Lardeau and Leschziner^[41] and Bentaleb et al.^[42] (see Fig. 1). The Reynolds number, based on the height of the ramp H and the incoming free-stream velocity, is Re=7 106. The corresponding friction Reynolds number is Re_τ=755, which is based on the wall friction velocity u_τ at the inlet plane x=-30.0. The freestream Mach number is M=0.2, defining a weakly compressible flow. The size of the computational domain L_x× L_y×L_z ranges from 55×8×5 to 55×9×5, which are discretized with a mesh of 1 290×200×300 points.

As shown in Fig. 2, the mesh is refined above the ramp along the streamwise direction, stretched towards the walls in the wall-normal direction, and evenly distributed in the spanwise direction. A non-slip isothermal boundary condition is applied at the wall. The digital filter method proposed by Touber and Sandham^[43] is used to generate the synthetic inflow turbulence, and a transitional region of 12δ₀ is incorporated to let synthetic fluctuations evolve into fully developed turbulence. The generated artificial turbulent fluctuations are super-imposed onto the turbulent mean velocity and temperature profiles. The inflow condition is then adopted to prescribe the flow variables, and the pressure is extrapolated from the inner grid points. At the upper and outlet planes, the generalized non-reflecting boundary conditions^[44-45] are used. The periodic boundary condition is prescribed in the spanwise direction. Due to the use of periodic boundary condition, it is important to examine the influence of the computational domain size on the simulation results^[31]. There are different criteria for choosing the computational domain size. The most strict criterion is to contain all energetic eddies at different scales^{[32, 46]}, which usually requires a very large computational domain size and requires a large amount of computer sources. This criterion is usually used when there exist very large scale coherent structures in the flow^[47-51]. A more practical criterion is to ensure that the statistical results are insensitive to the change in the computational domain size. This criterion is used in the present study. It is found that reducing the spanwise computational domain size by 1/3 results in the change of less than 5% in the mean velocity. Based on this test, it is believed that the results presented in this paper are not affected by the periodic boundary condition. The mesh settings at x=-10.0, which is located in the fully developed turbulent boundary layer upstream of the ramp, for the present simulation are as follows:

where Δy_δ refers to the mesh spacing in the wall-normal direction at y=δ, and δ is the local boundary layer thickness at x=-10.0.

The mesh resolution scaled with the local Kolmogorov length scale^[52] is checked (see Fig. 2). is the effective mesh spacing, where Δx, Δy, and Δz refer to the local grid spacing in the streamwise, wall-normal, and spanwise directions, respectively. η is the Kolmogorov scale calculated by

where ε is the local dissipation rate of the turbulence kinetic energy^[53]. The superscript "-" stands for the time-averaged operator, and ⟨·⟩_z stands for the spanwise-averaged operator. The ratio of the effective mesh spacing to the local Kolmogorov length scale is less than 4.2 throughout the computational zone, indicating that the mesh matches the resolution requirement of DNS.

After the transient period, the flow becomes fully turbulent and reaches a statistically steady state after about 133 time units. The data samples are collected for about 182 time units for the post-process and analysis.

2.2 Validation

The mean velocity profile in the equilibrium zone (x=-10.0) is compared with the classic law of wall and the incompressible DNS data at Re_τ≈1 000 in Ref. [54] in Fig. 3. An agreement in both the linear sub-layer and log-law layer is seen to be close and the difference in the wake layer is attributed to the Reynolds number effects^[36]. The root mean square (RMS) velocity fluctuations and Reynolds shear stress at the same streamwise position are compared with the DNS data in Refs. [54] and [55] in Fig. 4. Good agreements for RMS velocity fluctuations and Reynolds shear stress are obtained in the near-wall region.

2.3 Averaged flow condition

As shown in Fig. 5, in front of the ramp, a steady flat plate boundary flow condition is established with a thickness of δ/H≈1. In the wake of the ramp, the flow separates around x/H≈1 and reattaches around x/H≈5 within which a recirculating vortex appears. Meanwhile, streamlines are bent toward the lower wall to refill the area after the expansion after the ramp. This expansion causes a larger low speed region and a lower main flow velocity.

3 Analysis of non-equilibrium turbulence

In this section, we first demonstrate the consistency among C_ϵ, S_k, and Φ_ijij' by the analysis of their distributions in space, along a streamline and their statistical correlations, respectively. Then, we show the distributions of energy and transfer at different scales by presenting energy spectra and transfer spectra at different points. At the end of this section, we quantitatively show the non-equilibrium phenomena at different filtering scales by applying filter operations to the DNS database.

3.1 Consistency among C_ϵ, S_k, and Φ'_ijij

It is well-known that under homogeneity and isotropy conditions, C_ϵ, S_k, and Φ_ijij' are consistent with each other in the description of non-equilibrium phenomena. Here, we provide evidence by comparing the distributions in space, along a streamline and the statistical correlation, respectively. In the present sample, as the flow is homogeneous in the spanwise direction and in time, the ensemble average involved in the definitions of all the three quantities represents the average over the spanwise direction and over time. The integral scale in Eq. (1) is defined as

where R_ww(r)=⟨w(z)w(z+r)⟩, along the spanwise direction, and in Eq. (1) is adopted as the RMS of the spanwise fluctuating velocity w. The streamwise components are used to calculate the values of S_k. The values of Φ_ijij' are normalized by 1/⟨τ_K³⟩_s in which τ_K is the Kolmogorov time scale defined by

(11)

where ⟨·⟩_s denotes a mass-flow-rate-weighted average operator^[24]. We select a streamwise cross-section x/H=-10 to represent the incoming flow with a fully developed flat plate boundary and without being disturbed by the ramp to perform the normalization process.

3.1.1 Consistency in space

As shown in Fig. 6, in the regions far from the wall and the ramp, the flow is under equilibrium condition with a typical value of C_ϵ≈0.6. This is in agreement with the traditional results in HIT^[56]. The non-equilibrium turbulent phenomena concentrate in the boundary, the recirculation zone, and a long distance downstream of the ramp, where C_ϵ departs from the equilibrium value. In the boundary layer upstream of the ramp, non-equilibrium turbulent phenomena stem from the wall and spread to about 0.2 times the boundary layer thickness, which corresponds to the thickness of the inner layer of a flat plate boundary layer, with a relatively higher value of C_ϵ than the equilibrium region. In the wake behind the ramp, a lower value of C_ϵ is presented in the center of the recirculation vortex, where C_ϵ shows a similar value within the equilibrium region. This can be explained as that the flow is slow, and the non-equilibrium turbulence has enough time to reestablish an equilibrium state. This reestablishment is a slow process as argued in our previous study^[24]. Therefore, the results in a long distance of non-equilibrium turbulence phenomena are downstream the ramp till more than twenty times the ramp height.

According to traditional values in HIT, the typical value of S_k in the equilibrium region should be around -0.45^[23]. In Fig. 7, it is shown that, in the main flow above the beginning point of the ramp, S_k shows a higher value than that in equilibrium regions. We remark that, this is not the result of the non-equilibrium energy transfer process, since, in this area, the distributions of both C_ϵ and Φ_ijij' indicate an equilibrium turbulence state (see Figs. 6 and 8(a)). Indeed, this phenomenon should be related to the curvature of streamlines. Since the isotropy condition is not satisfied in this area, the traditional values of S_k in HIT are not satisfied. Similar to C_ϵ, in the boundary layer upstream of the ramp, the flow shows non-equilibrium transfer properties with a smaller value of S_k compared with that in the equilibrium region. Also, the effects of the non-equilibrium region continue till about 0.2 times the boundary layer thickness. Downstream the ramp, S_k shows a higher value, and the non-equilibrium region is larger than that shown by C_ϵ, which should be the combined effects of the non-equilibrium phenomenon and the streamline curvature. The non-equilibrium region lasts for a long distance similar to C_ϵ.

As shown in Fig. 8(c), Π_ijij' is much smaller than Q_ijij' and V_ijij' for about one decade. It only affects a small region at the beginning of the ramp with a diffusion effect, which is not obvious downstream the ramp. Φ_ijij' is mainly the balance between the generating Q_ijij' and the dissipating V_ijij'. In the boundary layer upstream the ramp, both Q_ijij' and V_ijij' affect till about 0.2 times the boundary layer thickness. By contrast, their resultant balance Φ_ijij' is only observed within 0.05 times the boundary layer, which corresponds to y⁺≈15 and is located in the buffer layer. The results of Φ_ijij' are negative, indicating the decrease in the dissipation rate when following the fluid particles. Downstream the ramp, the influence regions of Q_ijij' and V_ijij' are smaller than those of C_ϵ and S_k. The influence region of their balance Φ_ijij' is even much smaller.

According to the distributions of the three indicators shown above, the three indicators, i.e., C_ϵ, S_k, and Φ_ijij', show qualitative agreement, and thus can be used to estimate the non-equilibrium phenomena.

3.1.2 Consistency along a streamline

The definition of Φ_ijij' is the Lagrangian derivatives of the pseudodissipation, which gives another way to think about the question. The changes of Φ_ijij' by following a fluid particle may reveal the non-equilibrium process it experienced. The advection term is originally expanded as follows:

(12)

The two terms on the right-hand side of Eq. (12) are the advection effects induced by the average velocity and fluctuating velocity, respectively. We denote the latter one as -T_ijij' to represent the turbulent advection of the pseudodissipation when following a fluid particle along a statistical streamline. The distribution is shown in Fig. 9. It is only evident in a small region at the beginning of the ramp, and acts as a diffusion effect similar to the pressure term. By contrast, in other regions, it is relatively small, which indicates the fact that the turbulent advection is not essential in the analysis of non-equilibrium phenomena and the investigation along the streamlines can be considered to use the statistical streamlines in the regions away from the beginning of the ramp. Though the turbulent advection cannot be neglected in some regions, we can still approximately analyze along a statistical streamline with the understanding that the values of the indicators for non-equilibrium phenomena at one single point in these regions, where the fluid particles pass through, represent only the statistical results at this point but not the particles experienced along a streamline, since the turbulent advection effect is too obvious to be ignored and the statistical streamline cannot be considered as the streamline along which the particles go through.

We select a streamline passing through a point inside the inner layer upstream the ramp whose coordinate is x/H=-10, and y/H=1.1 (see Fig. 10). The variations of C_ϵ, S_k, and Φ_ijij' and the variations of Φ_ijij' and its components versus x/H are shown in Fig. 11. As shown in Fig. 11(a), the three quantities show good consistency with each other. In the inner layer upstream the ramp (x/H ≤ 0), the flow is under the non-equilibrium state, but is steady with a high level of C_ϵ and a low level of S_k; similar results can also be found in Ref. [19]. This non-equilibrium phenomenon should be caused by the influence of the wall. When the flow passes the ramp (0 ≤ x/H ≤ 3), all the three quantities fluctuate greatly, indicating a severe non-equilibrium transfer phenomenon in this region. Note that in this region, the results along the streamline are not experienced by the fluid particles on this statistical streamline, but are the average results on each single point as explained before. Around about 20 times the ramp height downstream the ramp, the flow returns to the equilibrium state gradually. Though the transfer from the non-equilibrium state to the equilibrium state takes a long distance downstream the disturbance, the flow will always reestablish an equilibrium transfer state, unless it is disturbed all the time. We stress here that the values of the three quantities deviate to different sides of the equilibrium value in the boundary layer and the wake of the ramp, which reveals the different non-equilibrium phenomena in the boundary layer and the wake.

As shown in Fig. 11(b), the diffusing Π_ijij' has only a little effect at the separation point, and is one order smaller than Q_ijij' and V_ijij'. Φ_ijij' is mainly the balance between the generating Q_ijij' and the dissipating V_ijij'.

3.1.3 Consistency by statistical analysis

As first found by Hearst and Lavoie^[22], C_ϵ and S_k are consistent with each other in the wake of fractal grids under the non-equilibrium state. We calculate the joint probability distribution function (joint-PDF) between C_ϵ and S_k in three typical regions, i.e., the equilibrium region as reference (set as -20 < x/H < -10, 1.3 < y/H < 1.7), the wake region downstream the ramp (set as 0 < x/H < 20, y/H < 1.2), and the inner layer region (set as -20 < x/H < -10, y/H < 1.3). As shown in Fig. 12, C_ϵ and S_k obviously correlate. In the wake, C_ϵ and the absolute value of S_k are lower than those in the equilibrium region. The results in the wake and in the equilibrium region show good agreement with the results of Hearst and Lavoie^[22]. By contrast, in the inner layer, C_ϵ and the absolute value of S_k are higher than those in the equilibrium region, which seems to be a new phenomenon which, to our knowledge, has not been reported in the literature. This high value appears in the region near the wall, showing great similarity to the result in the named equilibrium region of a transitional channel flow obtained in Ref. [19]. The distribution on the phase space may indicate a different non-equilibrium energy transfer mechanism. Indeed, in the boundary layer, the turbulence structures grow from the wall and form large structures like the streaks under non-linear interaction, which is a procedure of generating large scales from small scales. As a result, the portion of the energy transfer from large scales to small scales should be lower than that in the equilibrium region. However, in the wake, the process is on the contrary, and the large-scale flow structures break down to small vortices, indicating a process of generation of small scales from large scales.

From Fig. 13, similar consistency between C_ϵ and Φ_ijij' can be found. In the inner layer region, high values of C_ϵ correspond to negative values of Φ_ijij'. In the wake, the phenomenon is on the contrary.

We calculate the joint-PDF between C_ϵ and Re_λ^-1 in three typical regions, respectively. The results are shown in Fig. 14. As shown in Fig. 14(b), inside the equilibrium region, the values of C_ϵ and Re_λ^-1 remain nearly unchanged. For the third region, i.e., the inner layer shown in Fig. 14(c), the value of C_ϵ varies in a large range from a typical value under the equilibrium state to higher values. We observe an obvious proportional correlation between C_ϵ and Re_λ^-1. However, in the wake, the present calculation domain is not long enough for the flow to recover a steady equilibrium state with constant C_ϵ and S_k. Therefore, the results in Fig. 14(a) only concentrate in the lower left quarter without evolution toward equilibrium typical values. However, a similar proportional correlation should be expected according to the results found by Seoud and Vassilicos^[9]. In short, C_ϵ is inversely proportional to Re_λ in the wake and inner layer. According to the Richardson-Kolmogorov theory, in turbulence with equilibrium energy transfer, a higher Reynolds number implies a wider range of scales that the turbulent energy has to travel through from large to small scales, which results in a larger ratio of to λ. Also, from the definitions of C_ϵ and λ, one can write immediately

(13)

We can get the same conclusion from Eq. (13) that if the flow is under the equilibrium state with a constant C_ϵ, a higher Reynolds number results in a larger ratio of to λ^[56]. As can be seen from Fig. 13, both in the boundary layer and the wake, the local Reynolds number Re_λ is not constant in the whole region. From Eq. (13), we can see that inconstant Re_λ should lead to an inconstant ratio of to λ under the equilibrium energy transfer condition. On the contrary, according to Eq. (13), the inversely proportional correlation between C_ϵ and Re_λ^-1 indicates a constant ratio of to λ. This conflict reveals a turbulent energy transfer phenomenon different from the equilibrium state described by the Richardson-Kolmogorov theory, and should imply a non-equilibrium energy transfer phenomenon.

3.2 Distributions of energy and transfer at different scales

In order to investigate more about the underlying mechanism of the non-equilibrium transfer process, in this subsection, we examine the scale-dependent energy and transfer behaviors. The energy spectrum is defined as

(14)

where k=|k| in which k is the wave number vector, denotes the spectral quantities, the superscript ^* denotes the conjugate of a complex quantity, and dA(k) is the area element of the sphere whose radius equals k in the spectrum space. Also, the transfer spectrum is defined as follows:

(15)

where i is the imaginary unit. Here, we denote the convection term as follows:

(16)

where x is the position vector in the physic space. Then, its spectral expression is

(17)

Combining Eq. (17) with Eq. (15), we can obtain

(18)

Because of the existence of convolution calculation in the original definition of transfer spectra, which is caused by the convection term, we use Eq. (18) instead of Eq. (15) to calculate the transfer spectra, in which the convection term is calculated in the physical space with a pseudo-spectral method. In the present case, homogeneity is only valid in the spanwise direction. Therefore, the results are presented as one-dimension spectra in the z-direction. E(k) is normalized by (ϵν⁵)^1/4, T(k) is normalized by ϵ/k, and the wave number k is normalized by 1/η where η is the dissipation scale.

We calculate the energy spectra and transfer spectra at six points, respectively. The wavenumber is normalized by the local dissipation scale, the energy spectra are normalized by the local dissipation rate and viscosity, while the transfer spectra are normalized by the wavenumber and local dissipation rate. Four of the points are selected downstream the ramp along the streamline discussed in Subsection 3.1.2 and marked in Fig. 10. The other two points are selected inside the boundary layer and in the equilibrium region upstream the ramp. Point 1 is located near the separation point. Point 2 is the middle point between the non-equilibrium state of the boundary layer and the non-equilibrium state of the wake of the ramp reflected by C_ϵ. Point 3 is around the middle region of the wake, while Point 4 is located near the reattachment point. We select (x/H=-10, y/H=1.5) as the reference equilibrium point and (x/H=-10, y/H=1.1) as the inner layer point.

The energy spectra at these points are shown in Fig. 15(a). All spectra overlap with each other in the inertial range, indicating that even for non-equilibrium turbulence, the energy distribution is similar to those of equilibrium cases. Within the region 0.1 < kη < 0.2, the energy distribution approximately coincides with Kolmogorov's -3/5 law and can be considered as the inertial range. Meanwhile, this region is the same as the interval between the wave numbers of the integral scale and the Taylor scale (see Table 1), indicating that the energy distribution between the integral scale and the Taylor scale is coherent with the distribution of the inertial region described by Kolmogorov's law. As shown in Fig. 15(b), the transfer spectra at each point show a similar tendency that negative values are at small wave numbers while positive values are at large wave numbers, which indicates the fact that turbulent kinetic energy is transferred from large scales to small scales. However, the transfer spectra do not show exactly the similar distributions like energy spectra, which implies the different presences of the distribution of energy and its transfer under different non-equilibrium states. This can be explained since the non-equilibrium turbulence can stem from the phase de-coherence, which is sensitive to the different perturbations raised from complex geometries, leading to the non-injection facts from energy to transfer^[19].

It is interesting to see the detailed evolutions of energy and transfer in the boundary layer and the wake. At the inner layer point, the energy is relatively lower in large scales as well as small scales than that at the equilibrium point. For the energy transfer, the starting point and the peak of energy output move toward smaller scales with a larger output peak value. Meanwhile, energy is also gained at smaller scales compared with that at the equilibrium point but with a similar input peak value. Because the physical process inside the boundary is the generation of turbulent flow from small structures to large structures with limitations on the size of the largest structures, it is natural that energy is lower in large scales while the energy output moves to smaller scales. Also, the portion of dissipated energy is relatively smaller than that under the equilibrium state for the generating process. This leads to a lower energy distribution in small scales. At Point 1, i.e., the separation point, the energy distribution remains nearly the same as that in the inner layer point. However, the transfer spectrum changes a lot. At large scales, it loses energy at a smaller scale than that in the boundary, while the output peak value is still low, which results in the remaining of energy in large scales. However, at small scales, the transfer spectrum presents a similar distribution at the boundary layer point, at which the energy transfer process is changed at large scales while remains nearly the same in small scales, indicating the scale effect of non-equilibrium disturbance. This means that the non-equilibrium disturbance firstly affects certain scales, which are usually the large scales, since the disturbance originates mostly from the complex geometry, e.g., the ramp, the backward step, the grid, and the cascade, whose scale is relatively larger than that of the turbulence. Then, the non-equilibrium effect will spread to smaller scales with the evolution of the turbulence. From Point 2 to Point 3, the energy at large scales increases gradually and the energy at small scales keeps nearly the same inside the wake. While from Point 3 to Point 4, the large scale energy rests unchanged, while increases at small scales. For the energy transfer from Point 2 to Point 4, the energy output and input move toward large scales. This is because of the gradual increase in the energy at large scales. In fact, the physical process inside the wake is a gradual breaking down of large scale structures formed inside the boundary. The breaking down process gives large scales more energy from the main flow. Then, it will be transferred toward small scales with the turbulence evolution.

3.3 Non-equilibrium turbulence at different filtering scales

In order to analyze the non-equilibrium turbulence phenomena at different scales, we perform filter operations on the DNS database with different cutoff wave numbers (see Figs. 16-20). We remark that, in the spectral space, the filter operators are easy to be implemented, but in physical space, it is more difficult. Indeed, the sharp cutoff filters correspond to the convolution calculations, and they are implemented by applying a template operation to take the average of several layers of points around the selected one and traverse all the mesh points in the computational domain.

With the filter operator, the evolution equation of Lagrangian velocity gradient correlation should be modified as follows:

(19)

where the superscript < is the resolved scale, F_(n) is the results obtained in the filtering process of n layers of points around the center point, and the subgrid scale term is

with the subgrid stress

After the filter operations, the value of Φ'_ijij increases greatly and the non-equilibrium turbulence regions in large scales are larger than those in small scales. With the increases in the filtering layers, in the wake region, the non-equilibrium turbulence region is separated into two parts gradually, one is located above the wake, and the other is located inside the wake and extends toward downstream. Inside the inner layer, Φ'_ijij changes from negative to positive, and affects the outer region of the boundary.

The distributions of the production term and viscosity term remain similar to those under the non-filtered situation upstream the ramp and in the wake, but the absolute values decrease greatly with the increase in the filtering level. However, downstream the wake, the non-equilibrium turbulence region is further extended, which indicates a slower evolution speed of the non-equilibrium turbulence at large scales. Similarly, the absolute value of the pressure term also decreases. The affecting regions of pressure term extend toward the main flow above the wake. By contrast, Φ'_ijij is the balance between the production and the viscosity term before the filter operation. After the filter operation, it is dominated by the subgrid term, reflecting the fact that non-equilibrium turbulence phenomena in large scales originate from small scales.

Downstream the ramp, near the boundary around 2 < x/H < 3, there is a region under a relatively equilibrium state before the filter operation. However, with the increase in the filter level, this region diminishes gradually and becomes invisible in the end. As we have mentioned before, the non-equilibrium disturbance should be transferred from scale to scale. As a result, the reason why it shows an equilibrium state before the filter operation is that the non-equilibrium disturbance at large scales have not reached the mesh scale.

4 Conclusions

In this paper, we present the non-equilibrium turbulence phenomena in the boundary layer and the wake of a backward ramp flow. The contributions are as follows:

(ⅰ) Based on the distributions of C_ϵ, S_k, and Φ_ijij' in the physical space and their joint distributions, the consistency among them is verified in the wake and in the boundary layer. C_ϵ is inversely proportional to the local Reynolds number, indicating the non-equilibrium nature in these regions. It is further shown that the mechanisms of non-equilibrium transfer differ between the wake and the boundary layer.

(ⅱ) From the energy spectra and transfer spectra, it is shown that the energy transfer at each scale varies greatly, indicating that the underlying mechanism of non-equilibrium turbulence might be a phase de-coherence phenomenon.

(ⅲ) By analyzing the filtered results, it is shown that, at different scales, the non-equilibrium turbulence phenomena are quantitatively different, indicating the inter-scale spread of non-equilibrium disturbances.