2.1. The DNS data

The data employed by this work came from an open-access DNS database (https://torroja.dmt.upm.es/) for high-Reynolds-number turbulent boundary layer (TBL). A full description of the algorithm and the computational set-up could be found in the papers of Simens et al. (Reference Simens, Jiménez, Hoyas and Mizuno2009) and Borrell, Sillero & Jiménez (Reference Borrell, Sillero and Jiménez2013). Basically, their simulation bypassed the transition stage for boundary layer and directly generated a developed TBL with accurate inflow conditions. The inflow for this simulation was fed by an auxiliary simulation, whose calculation domain is located at the upstream of the main simulation domain. The inflow for the auxiliary simulation was obtained from one plane in its own domain using a rescaling–recycling scheme (Lund, Wu & Squires Reference Lund, Wu and Squires1998).

The DNS data (Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2013) have been widely validated by the Jiménez group (Borrell et al. Reference Borrell, Sillero and Jiménez2013; Sillero et al. Reference Sillero, Jiménez and Moser2013; Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2014). Particularly, Sillero et al. (Reference Sillero, Jiménez and Moser2013) showed that the skin friction and shape factor from this simulation agree well with the empirical fit of Monkewitz, Chauhan & Nagib (Reference Monkewitz, Chauhan and Nagib2007), with the mean percentage deviations less than 1 %. This indicates that the canonical asymptotic state is attained after the relaxation of the auxiliary simulation. A number of investigators (Marusic, Baars & Hutchins Reference Marusic, Baars and Hutchins2017; Wang, Wang & He Reference Wang, Wang and He2017; Wang et al. Reference Wang, Pan, Gao and Wang2018; Chen & Sreenivasan Reference Chen and Sreenivasan2021; Wang, Pan & Wang Reference Wang, Pan and Wang2021) cited the DNS dataset in their investigation works. In our previous work (Wang et al. Reference Wang, Gao, Wang, Wang and Pan2019), we compared the DNS data with 3-D PIV data regarding several aspects, including the p.d.f. of the vortex orientation angles and the conditionally averaged vortex structures. The DNS results and the experimental results were remarkably consistent, which further emboldens us to use the DNS data.

The production DNS field corresponds to a spatially evolving zero pressure gradient TBL from $R{e_\tau } \approx \textrm{1000}$ to $R{e_\tau } \approx \textrm{2000}$. The calculation domain contains 15361, 4096 and 535 grid nodes for the streamwise, spanwise and wall-normal directions, respectively. The separation between adjacent collocation points along the wall-normal direction is non-uniform, which is determined by local Kolmogorov scale (Borrell et al. Reference Borrell, Sillero and Jiménez2013). The DNS dataset is fully resolved and covers a vast scale range, which provides an ideal database for the implementation and validation of the V2V reconstruction.

In this work, only the data segments for $R{e_\tau } = 1148 \sim 1250$ were truncated from the whole DNS field. The average boundary thickness, determined by the height with 99 % of the free stream velocity, is approximately 1200 wall units, i.e. $\delta^{+} \approx 1200$. The selection for this boundary thickness was based on two considerations. From the perspective of the V2V reconstruction, the logarithmic layer of this data segment covers a wall-normal range of approximately 160 wall units (from 80 wall units to $0.2\delta^{+}$), corresponding to an acceptable number of computational nodes for the V2V reconstruction. On the other hand, investigations (Tanahashi et al. Reference Tanahashi, Kang, Miyamoto, Shiokawa and Miyauchi2004; del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006; Herpin et al. Reference Herpin, Stanislas, Jean and Coudert2013; Wang et al. Reference Wang, Gao, Wang, Wang and Pan2019) have validated that the fine-scale vortices populated in TBL present universal behaviours, independent of the Reynolds number. The truncated DNS data segment has a streamwise dimension of $8\delta^{+}$, a spanwise dimension of $13.4\delta^{+}$, and a wall-normal range from the viscous sublayer to $\textrm{0}\textrm{.3}\delta^{+}$, corresponding to a Cartesian grid of $1412 \times 4096 \times 91$. The variation range of the viscous wall unit in the DNS data segments is ±0.8 %, which is small enough to be viewed as a constant. Totally, twelve frames of DNS fields were downloaded, providing sufficient data for deriving the statistical quantities. To minimize the processing errors, all the calculations in the present work were based on the original DNS grid. The derivative computations were performed via a 5-node central differential scheme. More parameters about the DNS data segments employed in this work are collected in table 1.

Table 1. A collection of parameters for the DNS data employed in this work.

Throughout this article, the coordinate origin is located on the wall, and $x,y,z$ align with the streamwise, spanwise and wall-normal directions, respectively. The variable t indicates the time dimension. $\boldsymbol{u}$ denotes the fluctuating velocity vector, obtained by subtracting the mean streamwise velocity component from the original DNS vector field; $u,v,w$ designate the streamwise, spanwise and wall-normal fluctuating velocity components, respectively. A superscript of ‘+’ indicates that the quantity is normalized by the wall unit or the friction velocity. Only the fluctuating velocity field is considered in the following analysis.

2.2. The vortex identification

Various vortex identification criteria were developed to extract filamentary (tube-like) vortices from turbulence. Most of them are based on local velocity gradient tensor ($\boldsymbol{\nabla }\boldsymbol{u}$), including the second invariant Q (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988), the discriminant $\varDelta$ (Chong, Perry & Cantwell Reference Chong, Perry and Cantwell1990), the swirl strength ${\lambda _{ci}}$ (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999), the ${\lambda _2}$ criterion (Jeong & Hussain Reference Jeong and Hussain1995) and so on. Commonly, the iso-surfaces of the criterion for one prescribed value are displayed as recognized vortices. Chakraborty et al. (Reference Chakraborty, Balachandar and Adrian2005) revealed the relationships between these identification criteria and demonstrated that results from different criteria are approximately equivalent for turbulent flows. Among these criteria, the swirl strength $({\lambda _{ci}})$ corresponds to the imaginary part of the complex eigenvalue of the velocity gradient tensor, which has the same dimension as vorticity. A lot of investigators (Ganapathisubramani et al. Reference Ganapathisubramani, Longmire and Marusic2006; Hambleton, Hutchins & Marusic Reference Hambleton, Hutchins and Marusic2006; Lee & Sung Reference Lee and Sung2009) used ‘signed swirl’ as the representative variable for the in-plane component of a vortex, which is obtained by multiplying ${\lambda _{ci}}$ by the sign of the out-of-plane component of vorticity. For eduction purposes, Pirozzoli et al. (Reference Pirozzoli, Bernardini and Grasso2010) defined the strength of vortex tubes by introducing a ‘vorticity-like’ variable, which is equal to $2{\lambda _{ci}}$.

As for the vortex orientation, Gao et al. (Reference Gao, Ortiz-Dueñas and Longmire2011) and Tian et al. (Reference Tian, Gao, Dong and Liu2018) suggested using the real eigenvector of $\boldsymbol{\nabla }\boldsymbol{u}$ (denoted as ${\boldsymbol{\varLambda }_r}$). Tian et al. (Reference Tian, Gao, Dong and Liu2018) gave a mathematical discussion on the definition of the vortex vector based on the Schur decomposition theory (Gentle Reference Gentle1998). According to the viewpoint of Tian et al. (Reference Tian, Gao, Dong and Liu2018), the vortex orientation $({\boldsymbol{\varLambda }_r})$ corresponds to a direction with zero rotation speed. As indicated by the definition of velocity gradient tensor $(\boldsymbol{\nabla }\boldsymbol{u})$, the relative velocity $\delta \boldsymbol{u}$ for two points separated by a small position vector $\delta \boldsymbol{r}$ could be expressed as

(2.1)\begin{equation}\delta \boldsymbol{u} = \boldsymbol{\nabla }\boldsymbol{u}\boldsymbol{\cdot }\delta \boldsymbol{r}.\end{equation}

The non-rotation requirement means that the relative velocity should be along $\delta \boldsymbol{r}$, which means

(2.2)\begin{equation}\boldsymbol{\nabla }\boldsymbol{u}\boldsymbol{\cdot }\delta \boldsymbol{r} = \mu \delta \boldsymbol{r},\end{equation}

where $\mu $ should be a real constant. The above equation implies that $\delta \boldsymbol{r}$ is the real eigenvector of $\boldsymbol{\nabla }\boldsymbol{u}$, viz. ${\boldsymbol{\varLambda }_r}$.

Considering the eigenvector $({\boldsymbol{\varLambda }_r})$ contains no sign information about rotation, the local vorticity $\boldsymbol{\omega }$ is used as a reference. Specifically, the eigendirection (${\boldsymbol{\varLambda }_r}$ or $- {\boldsymbol{\varLambda }_r}$) forming an acute included angle with $\boldsymbol{\omega }$ is viewed as the vortex orientation. In this work, for the convenience of numerical implementation, the swirl strength ${\lambda _{ci}}$ and the orientation vector ${\boldsymbol{\varLambda }_r}$ are combined to define a vector field as

(2.3)\begin{equation}\boldsymbol{\varLambda } = {\lambda _{ci}}\textrm{sign}({\boldsymbol{\varLambda }_r}\boldsymbol{\cdot }\boldsymbol{\omega }){\boldsymbol{\varLambda }_r},\end{equation}

where ‘sign’ is the sign function, returning the sign of the bracketed variable. Throughout this article, the vortex field refers to $\boldsymbol{\varLambda }$ as default, which is consistent with the definition by Wang et al. (Reference Wang, Gao, Wang, Wang and Pan2019). It should be remembered that (2.3) is not a widely accepted expression since no mathematical definition for a vortex is available so far. In this work, the vortex field serves as a carrier for the geometrical and strength information of tubular vortices, which is necessary for the V2V reconstruction. In the following § 3, we will inspect the definition of (2.3) from the respect of the geometrical property of the recognized vortices.

2.3. The correlation coefficient and the complex coherence spectrum

A successful V2V reconstruction should return a velocity field sharing a high degree of similarity with the original DNS field, which can be measured by the correlation coefficient (CC). The CC is an indicator of the linear correlation degree between two input signals. In the current investigation, the CC for the reconstructed/original streamwise velocity components at one wall-normal position $({z_0})$ is calculated by

(2.4)\begin{equation}CC({z_0}) = \frac{{{{\langle {u_{rec}}(x,y,{z_0},t){u_{DNS}}(x,y,{z_0},t)\rangle }_{x,y,t}}}}{{\sqrt {{{\langle {u_{rec}}{{(x,y,{z_0},t)}^2}\rangle }_{x,y,t}}{{\langle {u_{DNS}}{{(x,y,{z_0},t)}^2}\rangle }_{x,y,t}}} }}.\end{equation}

Herein, ${u_{rec}}(x,y,z,t)$ and ${u_{DNS}}(x,y,z,t)$ denote the reconstructed streamwise velocity component field and the corresponding DNS field; ${\langle \cdot \rangle _{x,y,t}}$ represents the ensemble average, which is implemented by averaging the quantities in the brackets along the homogeneous dimensions, including the temporal dimension (t) and the wall-parallel dimensions (x and y), as indicated by the subscript $x,y,t$. The CCs regarding other velocity components (v and w) are calculated based on similar expressions with (2.4).

As a refined variant for CC, the complex coherence spectrum (CCS) measures the scale-specific linear coherence of two input signals (Stoica & Moses Reference Stoica and Moses2005; Ramirez, Via & Santamaria Reference Ramirez, Via and Santamaria2008). In the current work, the CCS formula takes a form resembling the correlation coefficient, but in the spectral representation. Specifically, the streamwise CCS between ${u_{rec}}(x,y,z,t)$ and ${u_{DNS}}(x,y,z,t)$ at a given wall-normal position $({z_0})$ could be calculated by

(2.5)\begin{align}CCS(k\textrm{)} = \frac{{{{\langle {{\tilde{u}}_{rec}}(k,y,{z_0},t){{\tilde{u}}_{DNS}}{{(k,y,{z_0},t)}^\ast }\rangle }_{y,t}}}}{{\sqrt {{{\langle {{\tilde{u}}_{rec}}(k,y,{z_0},t){{\tilde{u}}_{rec}}{{(k,y,{z_0},t)}^\ast }\rangle }_{y,t}}{{\langle {{\tilde{u}}_{DNS}}(k,y,{z_0},t){{\tilde{u}}_{DNS}}{{(k,y,{z_0},t)}^\ast }\rangle }_{y,t}}} }}.\end{align}

In (2.5), ${\tilde{u}_{rec}}(k,y,{z_0},t)$ and ${\tilde{u}_{DNS}}(k,y,{z_0},t)$ denotes the streamwise (1-D) Fourier-transform coefficient of ${u_{rec}}(x,y,z,t)$ and ${u_{DNS}}(x,y,z,t)$, respectively; k denotes the corresponding wavenumber. For convenience, ${\tilde{u}_{rec}}(k,y,{z_0},t)$ and ${\tilde{u}_{DNS}}(k,y,{z_0},t)$ are simply represented by ${\tilde{u}_{rec}}(k)$ and ${\tilde{u}_{DNS}}(k)$ in the following discussions. Notably, both ${\tilde{u}_{rec}}(k)$ and ${\tilde{u}_{DNS}}(k)$ are complex numbers, carrying the scale-specific amplitude and phase information of the input velocity fields. The asterisk designates the complex conjugate operation; ${\langle \cdot \rangle _{y,t}}$ indicates averaging the quantities in the brackets along the temporal dimension and the spanwise direction, which is simply denoted as $\langle \cdot \rangle$ in the following discussions. Equation (2.5) results in a complex number, which is different from the linear coherence spectrum (LCS) used by Baars & Marusic (Reference Baars and Marusic2020a). In their work, the LCS was defined as the squared magnitude of the current CCS, which measures the scale-by-scale coupling degree for two-point velocity signals. The phase angle of the CCS indicates the scale-by-scale phase deviation, which is also considered as the indicator for accessing the V2V reconstruction.

To illuminate the physical implication of (2.5), we express ${\tilde{u}_{rec}}(k)$ and ${\tilde{u}_{DNS}}(k)$ by the exponential forms, viz. ${\tilde{u}_{rec}}(k)\textrm{ = }{A_{rec}}(k)\,{\textrm{e}^{\textrm{i}{\gamma _{rec}}(k)}}$ and ${\tilde{u}_{DNS}}(k)\textrm{ = }{A_{DNS}}(k)\,{\textrm{e}^{\textrm{i}{\gamma _{DNS}}(k)}}$. Herein, ${A_{rec}}(k)$ and ${A_{DNS}}(k)$ stand for the amplitudes of ${\tilde{u}_{rec}}(k)$ and ${\tilde{u}_{DNS}}(k)$, respectively; ${\gamma _{rec}}(k)$ and ${\gamma _{DNS}}(k)$ indicate the corresponding phase angles. By substituting ${\tilde{u}_{rec}}(k)$ and ${\tilde{u}_{DNS}}(k)$ in (2.5) with their exponential forms, it could be easily derived that

(2.6)\begin{equation}CCS(k) = \langle {\hat{A}_{rec}}(k){\hat{A}_{DNS}}(k)\,{\textrm{e}^{\textrm{i}\Delta \gamma (k)}}\rangle .\end{equation}

In (2.6), ${\hat{A}_{rec}}(k) = {{{A_{rec}}(k)} / {\sqrt {\langle {A_{rec}}{{(k)}^2}\rangle } }}$ and ${\hat{A}_{DNS}}(k) = {{{A_{DNS}}(k)} / {\sqrt {\langle {A_{DNS}}{{(k)}^2}\rangle } }}$, which could be viewed as the normalized amplitudes; $\Delta \gamma (k) = {\gamma _{rec}}(k) - {\gamma _{DNS}}(k)$ represents the phase deviation between ${\tilde{u}_{rec}}(k)$ and ${\tilde{u}_{DNS}}(k)$; ${\textrm{e}^{\textrm{i}\Delta \gamma (k)}}$ is the polar representation for the phase deviation $\Delta \gamma (k)$. Equation (2.6) means that the CCS equals the weighted average of the sample data for ${\textrm{e}^{\textrm{i}\Delta \gamma (k)}}$, and the corresponding weighting coefficients are ${\hat{A}_{rec}}(k){\hat{A}_{DNS}}(k)$.

For the sake of clarity, we first disregard the weighting coefficient and consider the special case of $LCS(k) = \langle {\textrm{e}^{\textrm{i}\Delta \gamma (k)}}\rangle$. In this case, the CCS is determined by the phase deviation between ${\tilde{u}_{rec}}(k)$ and ${\tilde{u}_{DNS}}(k)$. If ${\tilde{u}_{rec}}(k)$ and ${\tilde{u}_{DNS}}(k)$ are extremely incoherent in phase, their phase deviation, deemed as a random variable, would present a uniform probability distribution between 0 and 2${\rm \pi}$. In the complex coordinate system (as shown by figure 1a), the samples of ${\textrm{e}^{\textrm{i}\Delta \gamma (k)}}$ (blue circles) would be spread evenly on the unit circle around the origin. Thus, the centre of these scattered numbers (i.e. the statistical average, CCS) would occur very close to the origin. On the contrary, if the ${\tilde{u}_{rec}}(k)$ and ${\tilde{u}_{DNS}}(k)$ are coupled in phase, these hollow circles would cluster towards one specific orientation (as shown in figure 1b). The central point (the red dot) would be in close proximity to the unit circle, giving a CCS with a magnitude close to one. The above explanation can be extended to the general cases of (2.6), as long as these scattered sampling points are given different weighting coefficients when calculating the central point. In this sense, the CCS incorporates both the phase and magnitude information from ${\tilde{u}_{rec}}(k)$ and ${\tilde{u}_{DNS}}(k)$, and provides a comprehensive diagnosis on their scale-specific coherence relation.

Figure 1. Schematic plots for illustrating the relation between CCS and the distribution of the phase deviations $(\Delta \gamma )$ for a set of input signal samples. The sample data of ${\textrm{e}^{\textrm{i}\Delta \gamma }}$ are schematically displayed in the complex coordinate system by hollow blue circles. The red dot, which is the centre of the blue circles, marks the position of CCS ($CCS\textrm{(}k) = \langle {\textrm{e}^{\textrm{i}\Delta \gamma }}\rangle$). Panel (a) corresponds to the situation with a small CCS magnitude; (b) corresponds to the situation with a large CCS magnitude.

4.1. The general theory for the V2V reconstruction

Denote ${\hat{u}_i}$ as the estimation variable for ${u_i}$, and suppose that ${\hat{u}_i}$ could be expressed by linear operators $({\mathcal{L}_{ij}})$ acting on ${\varLambda _j}$, which yields

(4.1)\begin{equation}{\hat{u}_i} = {\mathcal{L}_{ij}}{\varLambda _j},\end{equation}

where $[{\mathcal{L}_{ij}}]$ is a 3 × 3 matrix of linear operators. The linear operators can be viewed as continuous transformations on ${\varLambda _j}$, and no limitations on their realization forms are needed in this general analysis. (4.1) is a general form for the V2V reconstruction, and ${\mathcal{L}_{ij}}$ needs to be determined by minimizing the reconstruction deviations.

Considering both ${\hat{u}_i}$ and ${u_i}$ are 3-D vector fields, their deviation (D) can be defined by an integral form as

(4.2)\begin{equation}D = {{\iint\!\!\!\int_\varOmega {({{\hat{u}}_i} - {u_i})({{\hat{u}}_i} - {u_i})\,\textrm{d}\varOmega } } / {\iint\!\!\!\int_\varOmega {\textrm{d}\varOmega } }}.\end{equation}

Facilitated by the above definition, the optimal ${\mathcal{L}_{ij}}$ can be determined by minimizing the deviation in a statistical sense, i.e.

(4.3)\begin{equation}\min \langle D\rangle ,\end{equation}

where $\langle \cdot \rangle$ represents an ensemble average.

According to the calculus of variation (Appendix B), the above optimization problem is equivalent to the following equations,

(4.4)\begin{equation}{\mathcal{L}_{ij}}\langle {\varLambda _j}{\varLambda _m}(\boldsymbol{r^{\prime}})\rangle = \langle {u_i}{\varLambda _m}(\boldsymbol{r^{\prime}})\rangle \quad for\;\forall \,{\varLambda _m}(\boldsymbol{r^{\prime}})\textrm{ (}\boldsymbol{r^{\prime}} \in \varOmega \textrm{;}\;m = 1,2,3).\end{equation}

Equations (4.4) constitute the governing equations for the linear operators ${\mathcal{L}_{ij}}$. Note that both ${\varLambda _j}$ and ${u_i}$ are 3-D functions of the implicit position vector $\boldsymbol{r}$, and $\langle {\varLambda _j}{\varLambda _m}(\boldsymbol{r^{\prime}})\rangle$ should also be regarded as a 3-D function of $\boldsymbol{r}$ rather than $\boldsymbol{r}^{\prime}$ when the linear operators $({\mathcal{L}_{ij}})$ act on it. For any given ${\varLambda _m}(\boldsymbol{r^{\prime}})$, the form of $\langle {\varLambda _j}{\varLambda _m}(\boldsymbol{r^{\prime}})\rangle$ or $\langle {u_i}{\varLambda _m}(\boldsymbol{r^{\prime}})\rangle$ indicates a conditional average for the event ${\varLambda _m}(\boldsymbol{r^{\prime}})$. Here, $\langle {\varLambda _j}{\varLambda _m}(\boldsymbol{r^{\prime}})\rangle$ can be understood as the averaged vortex distribution given an event of ${\varLambda _m}$ occurring at $\boldsymbol{r}^{\prime}$, and similarly, $\langle {u_i}{\varLambda _m}(\boldsymbol{r^{\prime}})\rangle$ could be viewed as the conditionally averaged velocity distribution. The operators ${\mathcal{L}_{ij}}$ always act on the local statistical vortex distribution and return the corresponding velocity distribution.

Equation (4.4) are comparable to the Yule–Walker equation from classical LSE (Adrian Reference Adrian1994). The latter was established to determine the unknown parameters in the estimating expression, the number of which is usually small or finite at least. In this work, (4.4) is established to determine the linear operators ${\mathcal{L}_{ij}}$, which have infinite dimensions. The starting point of the current method is to estimate one 3-D field based on another given 3-D field, which is named field-based linear stochastic estimation (FLSE) in this work.

For a general case, the discretizing of (4.4) leads to a linear system with huge numbers of variables. For examples, if the spatial domain is discretized into ${N_x} \times {N_y} \times {N_z}$ computation nodes, numbers of the undetermined variables in ${\mathcal{L}_{ij}}$ would be ${(3 \times {N_x} \times {N_y} \times {N_z})^2}$, which might be huge for a typical case. Fortunately, for wall-bounded turbulence flows such as the turbulent channel flow and TBL, the flow can be regarded as homogeneous in the wall planes, at least for a limited streamwise range. The homogeneous property would significantly reduce the numbers of variables and will be discussed in the following subsection.

For the convenience of the following discussion, denote that

(4.5)\begin{gather}{{\mathsf{R}}_{jm}}(\boldsymbol{r},\boldsymbol{r^{\prime}}) = \langle {\varLambda _j}{\varLambda _m}(\boldsymbol{r^{\prime}})\rangle ,\end{gather}

(4.6)\begin{gather}{{\mathsf{q}}_{im}}(\boldsymbol{r},\boldsymbol{r^{\prime}}) = \langle {u_i}{\varLambda _m}(\boldsymbol{r^{\prime}})\rangle .\end{gather}

Thus (4.4) can be rewritten as

(4.7)\begin{equation}{\mathcal{L}_{ij}}{{\mathsf{R}}_{jm}}(\boldsymbol{r},\boldsymbol{r^{\prime}}) = {{\mathsf{q}}_{im}}(\boldsymbol{r},\boldsymbol{r^{\prime}})\quad \textrm{for}\;\forall \,\boldsymbol{r^{\prime}} \in \varOmega \textrm{;}\;m = 1,2,3.\end{equation}

4.2. FLSE for wall-bounded turbulence

The estimated velocity ${\hat{u}_i}$ could be viewed as a result of the joint inducing effects caused by the neighbouring vortices. Suppose that the inducing effect of a vortex field could be expressed as an integral of the inducing effects of vortex vectors at different locations, which yields a form of integral transformation as

(4.8)\begin{equation}{\hat{u}_i}(\boldsymbol{r}) = \iint\!\!\!\int_\varOmega {{\varphi _{ij}}(\boldsymbol{r};\boldsymbol{r^{\prime}}){\varLambda _j}(\boldsymbol{r^{\prime}})\,\textrm{d}x^{\prime}\,\textrm{d}y^{\prime}\,\textrm{d}z^{\prime}} .\end{equation}

In the equation, ${\varphi _{ij}}(\boldsymbol{r};\boldsymbol{r^{\prime}})$ is a kernel function, which indicates the contribution of ${\varLambda _j}(\boldsymbol{r^{\prime}})$ with unit magnitude to the ith velocity component at the position $\boldsymbol{r}$. Equation (4.8) provides a specific realization form for the linear operator ${\mathcal{L}_{ij}}$.

For wall-bounded turbulence, as discussed before, the flow could be treated as a homogeneous field in the x–y plane. Thus, the inducing effect of a vortex should keep invariable when its position is shifted in the x–y plane, which yields

(4.9)\begin{equation}{\varphi _{ij}}(\boldsymbol{r};\boldsymbol{r^{\prime}}) = {\varphi _{ij}}(x,y,z;x^{\prime},y^{\prime},z^{\prime}) = {\varphi _{ij}}(x - x^{\prime},y - y^{\prime},z;0,0,z^{\prime}).\end{equation}

For simplicity, it is denoted that

(4.10)\begin{equation}{\varphi _{ij}}(x - x^{\prime},y - y^{\prime},z;0,0,z^{\prime}) = {\varphi _{ij}}(x - x^{\prime},y - y^{\prime},z;z^{\prime}).\end{equation}

By the way, such a shifting-invariable property also holds for ${{\mathsf{R}}_{jm}}(\boldsymbol{r},\boldsymbol{r^{\prime}})$ and ${{\mathsf{q}}_{im}}(\boldsymbol{r},\boldsymbol{r^{\prime}})$, thus we also have ${{\mathsf{R}}_{jm}}(x,y,z;z^{\prime})$ and ${{\mathsf{q}}_{im}}(x,y,z;z^{\prime})$ as the simplifications for ${{\mathsf{R}}_{jm}}(x,y,z;0,0,z^{\prime})$ and ${{\mathsf{q}}_{im}}(x,y,z;0,0,z^{\prime})$.

Combining (4.8) and (4.9), we have

(4.11) \begin{align}\begin{aligned} {{\hat{u}}_i}(\boldsymbol{r}) & = \iint\!\!\!\int_\varOmega {{\varphi _{ij}}(x - x^{\prime},y - y^{\prime},z;z^{\prime}){\varLambda _j}(\boldsymbol{r^{\prime}})\,\textrm{d}x^{\prime}\,\textrm{d}y^{\prime}\,\textrm{d}z^{\prime}},\\ & = \int_{z^{\prime}} {\left( {\iint_{x^{\prime} - y^{\prime}} {{\varphi_{ij}}(x - x^{\prime},y - y^{\prime},z;z^{\prime}){\Lambda_j}(\boldsymbol{r^{\prime}})\,\textrm{d}x^{\prime}\,\textrm{d}y^{\prime}} } \right)\,\textrm{d}z^{\prime}} . \end{aligned}\end{align}

The bracketed term is a 2-D convolution operation in the x–y plane, which could be denoted as an asterisk. Thus, (4.11) can be simplified as

(4.12)\begin{equation}{\hat{u}_i}(\boldsymbol{r}) = {\mathcal{L}_{ij}}{\varLambda _j} = \int_{z^{\prime}} {{\varphi _{ij}}({\cdot} , \cdot ,z;z^{\prime}) \ast {\varLambda _j}({\cdot} , \cdot ,z^{\prime})\,\textrm{d}z^{\prime}} ,\end{equation}

where the position variables x and y are substituted as dots, indicating the convolution works in the x–y plane. Equation (4.12) is the formula for the so-called V2V reconstruction, where ${\varphi _{ij}}(x,y,z;z^{\prime})$ needs to be determined first. By definition, ${\varphi _{ij}}(x,y,z;z^{\prime})$ reflects the inducing effect of an individual vortex vector at different wall-normal positions, which is named the inducing model function in this work.

Substituting the linear operators in (4.7) by the explicit expression of (4.12), the governing equations for ${\varphi _{ij}}(x,y,z;z^{\prime})$ are obtained as

(4.13)\begin{equation}\int_{z^{\prime\prime}} {{\varphi _{ij}}({\cdot} , \cdot ,z;z^{\prime\prime}) \ast {{\mathsf{R}}_{jm}}({\cdot} , \cdot ,z^{\prime\prime};z^{\prime})\,\textrm{d}z^{\prime\prime}} = {{\mathsf{q}}_{im}}(x,y,z;z^{\prime})\quad \textrm{for}\;\forall \,z^{\prime}\textrm{;}\;m = 1,2,3.\end{equation}

Note that only the equations for $x^{\prime} = 0$ and $y^{\prime} = 0$ in (4.7) are collected into (4.13) since the number for unknown variables have been significantly reduced by using the homogeneous condition.

4.3. Discretization and implementation for FLSE

Equations (4.13) need to be solved in the discrete form. Suppose that the domain is discretized as ${N_x} \times {N_y} \times {N_z}$ computation nodes, and we use the symbols $a,b,c,c^{\prime},c^{\prime\prime}$ as the indices for discretized variables of $x,y,z,z^{\prime},z^{\prime\prime}$, respectively. The integral operation could be discretized as a summation implied by the repeated subscript of $c^{\prime\prime}$. Thus, we have

(4.14)\begin{equation}{\varphi _{ij}}({\cdot} , \cdot ,{z_c};{z^{\prime\prime}_{c^{\prime\prime}}}) \ast {{\mathsf{R}}_{jm}}({\cdot} , \cdot ,{z^{\prime\prime}_{c^{\prime\prime}}};{z^{\prime}_{c^{\prime}}}) = {{\mathsf{q}}_{im}}({x_a},{y_b},{z_c};{z^{\prime}_{c^{\prime}}})\quad \textrm{for}\;\forall \,c^{\prime},m.\end{equation}

Equation (4.14) is a linear system with underdetermined variables of ${\varphi _{ij}}({x_a},{y_b},{z_c};{z^{\prime\prime}_{c^{\prime\prime}}})$ for all the realizations of $i,j,a,b,c,c^{\prime\prime}$. The asterisk herein indicates the discrete convolution operation. Consistent with the former regulation, ${x_a},{y_b}$ are omitted in (4.14) in order to indicate the working dimensions for the convolution operation. The regulation also avoids the occurrence of repeated $a,b$, which would cause confusion by erroneously triggering the automatic summation.

The number of underdetermined variables corresponding to all the possible realizations of $i,j,a,b,c,c^{\prime\prime}$ is $3 \times 3 \times {N_x} \times {N_y} \times {N_z} \times {N_z}$, which might be of the order of $\textrm{1}{\textrm{0}^\textrm{9}}$. Fortunately, these variables do not need to be solved simultaneously. Closer observation on the indices occurring in the left term of (4.14) reveals that the indices i and c occur only once, which implies that ${\varphi _{ij}}({x_a},{y_b},{z_c};{z^{\prime\prime}_{c^{\prime\prime}}})$ could be independently solved for the two indices. Specifically, for any prescribed indices i and c, the equations from (4.14) constitute a closed linear system with respect to all the realizations of ${\varphi _{ij}}({x_a},{y_b},{z_c};{z^{\prime\prime}_{c^{\prime\prime}}})$ for $\forall \,j,a,b,c^{\prime\prime}$, which could be written as a matrix form.

(4.15)\begin{equation}\boldsymbol{\mathsf{R}}{\boldsymbol{\varphi }^{i,c}} = {\boldsymbol{q}^{i,c}},\end{equation}

where $\boldsymbol{\mathsf{R}}$ is a matrix formed by arranging the realizations of ${{\mathsf{R}}_{jm}}({x_a},{y_b},{z^{\prime\prime}_{c^{\prime\prime}}};{z^{\prime}_{c^{\prime}}})$; ${\boldsymbol{\varphi }^{i,c}},{\boldsymbol{q}^{i,c}}$ are vectors formed by collecting all the realizations of ${\varphi _{ij}}({x_a},{y_b},{z_c};{z^{\prime\prime}_{c^{\prime\prime}}})$ and ${{\mathsf{q}}_{im}}({x_a},{y_b},{z_c};{z^{\prime}_{c^{\prime}}})$ for prescribed indices i and c. In this way, equations (4.14) could be divided into ${N_z} \times 3$ sets of reduced equations based on the different combinations of indices i and c. Each set of equations contains $3 \times {N_x} \times {N_y} \times {N_z}$ variables and can be solved independently.

Besides directly solving (4.15), the problem could also be efficiently solved by using fast Fourier-transform (FFT) technique. We perform 2-D discrete FFT in the x–y plane on each term of (4.14), and apply the well-known convolution theorem of FFT. Let ${\tilde{{\mathsf{R}}}_{jm}},{\tilde{\varphi }_{ij}}$ denote the FFT results of ${{\mathsf{R}}_{jm}},{\varphi _{ij}}$, and ${k_\alpha },{k_\beta }$ indicate the discrete wavenumbers along the $x,y$ direction. Thus we have

(4.16)\begin{equation}\sum\limits_{c^{\prime\prime},j} {{{\tilde{{\mathsf{R}}}}_{jm}}({k_\alpha },{k_\beta },{{z^{\prime\prime}}_{c^{\prime\prime}}};{{z^{\prime}}_{c^{\prime}}}){{\tilde{\varphi }}_{ij}}({k_\alpha },{k_\beta },{z_c};{{z^{\prime\prime}}_{c^{\prime\prime}}})} = {\tilde{{\mathsf{q}}}_{im}}({k_\alpha },{k_\beta },{z_c};{z^{\prime}_{c^{\prime}}})\quad \textrm{for}\;\forall \,c^{\prime},m.\end{equation}

Einstein's summation convention is abandoned in this equation since the repeated indices $\alpha ,\beta$ do not imply a summation here. For numerical implementation, (4.16) should also be written as the matrix form. In this case, we can see ${\tilde{\varphi }_{ij}}({k_\alpha },{k_\beta },{z_c};{z^{\prime\prime}_{c^{\prime\prime}}})$ is independent with the indices of $i,\alpha ,\beta ,c$. Therefore, for fixed indices of $i,\alpha ,\beta ,c$, we can collect all realizations for ${\tilde{{\mathsf{R}}}_{jm}}({k_\alpha },{k_\beta },{z^{\prime\prime}_{c^{\prime\prime}}};{z^{\prime}_{c^{\prime}}})$ and arrange them into a matrix $({\boldsymbol{\mathsf{A}}^{\alpha ,\beta }})$. Similarly, ${\tilde{\varphi }_{ij}}$ and ${\tilde{{\mathsf{q}}}_{im}}$ are arranged as vector forms as ${\boldsymbol{\varphi }^{i,\alpha ,\beta ,c}}$ and ${\boldsymbol{b}^{i,\alpha ,\beta ,c}}$ for prescribed indices $i,\alpha ,\beta ,c$.

Facilitated by these conventions, (4.16) could be rewritten as

(4.17)\begin{equation}{\boldsymbol{\mathsf{A}}^{\alpha ,\beta }}{\boldsymbol{\varphi }^{i,\alpha ,\beta ,c}} = {\boldsymbol{b}^{i,\alpha ,\beta ,c}}.\end{equation}

Einstein's summation convention is abandoned in this expression. For each set of indices $i,\alpha ,\beta ,c$, the equations of (4.17) constitute a closed linear system for ${\boldsymbol{\varphi }^{i,\alpha ,\beta ,c}}$, which contains ${N_z} \times 3$ variables. After all the equations are solved, ${\tilde{\varphi }_{ij}}({k_\alpha },{k_\beta },{z_c};{z^{\prime\prime}_{c^{\prime\prime}}})$ is recovered based on all the results. Inverse FFT operation is subsequently performed on ${\tilde{\varphi }_{ij}}({k_\alpha },{k_\beta },{z_c};{z^{\prime\prime}_{c^{\prime\prime}}})$ to return ${\varphi _{ij}}({x_a},{y_b},{z_c};{z^{\prime\prime}_{c^{\prime\prime}}})$. For clarify, a procedure card illustrating the FFT implementation has been provided in Appendix C. This card also incorporates the strategy of dealing with the numerical instability issue, which will be introduced in the next subsection.

4.4. Issues and considerations for numerical implementation

Although the FFT implementation significantly improves the computation efficiency, some considerations need to be addressed. In theory, the convolution theorem for FFT exactly holds for the circular convolution operation, which imposes a periodic extension on the input functions. If the computation domain $\varOmega$ is not large enough, the magnitudes of ${{\mathsf{R}}_{jm}}$ or ${{\mathsf{q}}_{im}}$ would not decay sufficiently at borders. The periodic extension leads to strong disconnections on borders, which would cause numerical instability in the FFT results. Therefore, the windowed FFT technique is suggested to deal with this issue. In this situation, the FFT method would be viewed as an approximate method, whose accuracy depends on the computation domain and needs to be further validated by the results.

The numerical stability for solving (4.15) or (4.17) requires that $\boldsymbol{\mathsf{R}}$ and ${\boldsymbol{\mathsf{A}}^{\alpha ,\beta }}$ should be sufficiently positive definite, which means all the eigenvalues should be larger than a positive critical number. In practical implementation, both $\boldsymbol{\mathsf{R}}$ and ${\boldsymbol{\mathsf{A}}^{\alpha ,\beta }}$ have zeros or very small eigenvalues, which cause ill-posed systems. In order to avoid this problem, Tikhonov regularization (Kress Reference Kress2014) is suggested to improve the numerical stability. Specifically, (4.15) and (4.17) are reshaped into the following forms

(4.18)\begin{gather}(\boldsymbol{\mathsf{R}} + s\boldsymbol{\mathsf{I}}){\boldsymbol{\varphi }^{i,c}} = {\boldsymbol{q}^{i,c}},\end{gather}

(4.19)\begin{gather}({\boldsymbol{\mathsf{A}}^{\alpha ,\beta }} + s\tilde{\delta }({k_\alpha },{k_\beta })\boldsymbol{\mathsf{I}}){\boldsymbol{\varphi }^{i,\alpha ,\beta ,c}} = {\boldsymbol{b}^{i,\alpha ,\beta ,c}},\end{gather}

where $\boldsymbol{\mathsf{I}}$ is the identity matrix and s is the regularization parameter. Einstein's summation convention is cancelled in the equation. Here, $\tilde{\delta }({k_\alpha },{k_\beta })$ is the 2-D FFT result for a discrete 2-D Dirac function $\delta ({x_a},{y_b})$, which is defined as

(4.20) \begin{equation}\delta ({x_a},{y_b}) = \left\{ {\begin{array}{@{}ll} 1&{\textrm{for}\;{x_a} = 0,\;{y_b} = 0\textrm{ }}\\ 0&{\textrm{otherwise}} \end{array}} \right..\end{equation}

In (4.18) and (4.19), s plays a role in improving the statistical convergence and controlling the smoothness of the solution, which will be tested in § 5.

The FFT implementation of FLSE is a reminiscence of spectral linear stochastic estimation (SLSE) proposed by Tinney et al. (Reference Tinney, Coiffet, Delville, Hall, Jordan and Glauser2006). Baars, Hutchins & Marusic (Reference Baars, Hutchins and Marusic2016) applied SLSE to predict the velocity signals in wall-bounded turbulence and refined the inner–outer interaction model proposed by Marusic, Mathis & Hutchins (Reference Marusic, Mathis and Hutchins2010). More recently, Baars & Marusic (Reference Baars and Marusic2020a) developed a data-driven decomposition scheme to decouple inner–outer coherence based on SLSE. Basically, SLSE reconstructs the prediction for 1-D signal by minimizing the estimation error in spectral space, which avoids the phase mismatch issue. While expanding SLSE from one dimension to two dimensions in x-y plane is straightforward, further expanding to 3-D application needs special treatment for the wall-normal dimension. In this work, FLSE offers a general routine to estimate one field based on another, regardless of the dimensions. As the implementation of FLSE, the FFT method expands spectra-based LSE to 3-D application in wall-bounded turbulence.

5.1. The correlation functions: ${{\mathsf{R}}_{jm}}$ and ${{\mathsf{q}}_{im}}$

Before performing FLSE, the self-correlation function for vortex fields $({{\mathsf{R}}_{jm}})$ and vortex–velocity correlation function $({{\mathsf{q}}_{im}})$ need to be derived from the DNS data. In this work, these correlation functions are calculated based on twelve large-size instantaneous DNS fields (as shown in table 1). The ensemble average is implemented by averaging along the statistically uniform dimensions, including the x–y plane and the temporal dimension. Concretely, the two correlation functions are calculated by

(5.1)\begin{gather}{{\mathsf{R}}_{jm}}(x,y,z;z^{\prime}) = {\langle {\varLambda _j}({x_\textrm{0}} + x,{y_\textrm{0}} + y,z){\varLambda _m}({x_\textrm{0}},{y_\textrm{0}},z^{\prime})\rangle _{{x_\textrm{0}},{y_\textrm{0}},t}},\end{gather}

(5.2)\begin{gather}{{\mathsf{q}}_{im}}(x,y,z;z^{\prime}) = {\langle {u_i}({x_0} + x,{y_0} + y,z){\varLambda _m}({x_0},{y_0},z^{\prime})\rangle _{{x_0},{y_0},t}}.\end{gather}

Noting that ${\langle \cdot \rangle _{{x_0},{y_0},t}}$ denotes averaging the bracketed quantity for all the realizations of ${x_0},{y_0},t$.

In order to avoid the resolution issue, the calculation is carried out on the original computational grid of DNS. The statistical range for correlation functions is set as $x^{+} \in [ - 0.5\delta^{+},0.5\delta^{+}]$, $y^{+} \in [ - 0.25\delta^{+},0.25\delta^{+}]$, $z^{+} \in [\textrm{5},0.3\delta^{+}]$, resulting in a computation grid of 179 × 155 × 91. The investigation of del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2006) on the size of attaching clusters supported the aspect ratio prescribed in this work. The wall-normal range covers the whole buffer layer and logarithmic layer, which contains most of the vortex populations in TBL. It should be pointed out that the statistical range for correlation functions also determines the calculation domain of ${\varphi _{ij}}$, which is correlated with the reconstruction accuracy. More arguments for setting this statistical range will be emphasized in the following discussions.

Figure 5 displays contours of ${{\mathsf{R}}_{11}}$ and ${{\mathsf{q}}_{11}}$ as functions of x and y for three wall-normal positions $(z^{\prime + }\textrm{ = }{z^ + }\textrm{ = 15}\textrm{.4, 119}\textrm{.4, 359}\textrm{.3)}$. At first glance, these contours are reasonably smooth and symmetric, indicating that the statistical results are well converged. At $z^{+} = 15.4$, streak patterns occur nearby the centre points for both ${{\mathsf{R}}_{11}}$ and ${{\mathsf{q}}_{11}}$, which are imprints of the quasi-streamwise vortices and low-speed streaks populated in this region. When $z^{+}$ increase from 119.4 to 359.3, both ${{\mathsf{R}}_{11}}$ and ${{\mathsf{q}}_{11}}$ display a self-similar increase in size, supporting the famous attached eddy hypothesis. Distinctive characteristics for ${{\mathsf{R}}_{11}}$ and ${{\mathsf{q}}_{11}}$ are also observed: while ${{\mathsf{R}}_{11}}$ remains a sharp peak at the central point, ${{\mathsf{q}}_{11}}$ shows two flat peaks, filling a large portion of the area in the plots. At the edge of the wall-normal range considered $(z^{+} = 359.3)$, bulks of the two ${{\mathsf{q}}_{11}}$ peaks still remain in the calculation range although they appear to be truncated at the boundaries to some extent. A much larger statistical range is needed if one attempts to cover the whole pattern for ${{\mathsf{q}}_{11}}$ in the x–y plane at $z^{+} = 359.3$, which poses a severe challenge for both the number of sampling DNS fields and the computing resources. The statistical range employed in this work is based on the comprehensive considerations on the computation efficiency we can bear and the reconstruction accuracy we have expected. In fact, most of the results in this work will focus on the regions below the logarithmic layer, particularly for $z^{+} < 120$, where the correlation peaks for ${{\mathsf{R}}_{11}}$ is satisfyingly covered in the calculation range. More importantly, the resulting reconstruction accuracy in this region is good enough to promote the FLSE method and to draw some revealing conclusions as we can see in the following discussions.

Figure 5. Contours of ${\mathsf{R}}_{11}^ + (x,y,z;z^{\prime})$ (the first row) and ${\mathsf{q}}_{11}^ + (x,y,z;z^{\prime})$ (the second row). From left to right, the contours correspond to ${z^ + } = {z^{\prime+}} = 15.4$, ${z^ + } = {z^{\prime+}} = 119.4$ and ${z^ + } = {z^{\prime+}} = \textrm{359}\textrm{.3}$, respectively.

Once ${{\mathsf{R}}_{jm}}$ and ${{\mathsf{q}}_{im}}$ for $i,j,m = 1,2,3$ are obtained, the vortex model function ${\varphi _{ij}}(x,y,z;z)$ can be solved based on (4.18) or (4.19). Considering the number of computation nodes involved is quite large (approximately 2.5 million) in this case, an FFT implementation scheme is employed based on the processing procedures shown in Appendix C. To avoid the interruptions at boundaries, both ${{\mathsf{R}}_{11}}$ and ${{\mathsf{q}}_{11}}$ are weighted by a Hanning window before the FFT operation. Weighting a signal by a Hanning window procedure will force the input signal to decay to zero at boundaries, which benefits the numerical stability. Testing on the Hanning window and the regularization parameter will be introduced in the following subsection.

5.2. The regularization parameter and the Hanning window

The performance of FLSE is determined by the inducing model functions. Figure 6 shows ${\varphi _{13}}(x,y,z;z)$ in the x–y plane, which reflects the in-plane inducing effect of the individual wall-normal vortex on the streamwise velocity component. Figure 6 displays the results from three calculation cases: two cases passing the typical processing routines with $s = 0.82 \times {10^{ - 6}}{\mu _{max}}$ and $s = 0.82 \times {10^{ - 4}}{\mu _{max}}$ respectively, and one case skipping Hanning window weighting before FFT with $s = 0.82 \times {10^{ - 4}}{\mu _{max}}$. Herein, ${\mu _{max}}$ is the maximum eigenvalue for all the realizations of ${\boldsymbol{\mathsf{A}}^{\alpha ,\beta }}$. To make it clear, only a small region with a streamwise range of ±150 and a spanwise range of ±75 is displayed. It shows that while figures 6(a,d) and 6(b,e) show similar contours, the contour maps of figure 6(c,f) suffer from numerical issues and present spurious patterns. It indicates that the Hanning window is necessary for the FFT implementation of FLSE in order to obtain credible results. In figure 6(a,d) or figure 6(b,e), ${\varphi _{13}}$ shows two streamwise-extending streaks at ${z^ + } = 15.4$, which reflects the basic flow feature in this region. At ${z^ + } = 119.4$, the streak patterns shrink along the streamwise direction and form a velocity distribution resembling an intense swirl in the x–y plane. The wall-normal variation of the inducing model function gives signs of adaption to local flow features, which is vital for the success of V2V reconstruction. By comparing figures 6(a,d) and 6(b,e), it can be found that adjusting the value of s works as a dilation or erosion effect on the ${\varphi _{13}}$ pattern. Larger s helps to obtain smoother and more symmetrical contour patterns, which implies a good convergence. For a very large s, the equation system of (4.19) will result in a solution very close to ${{\mathsf{q}}_{im}}$, which shows a quite flat pattern in the x–y plane as shown in figure 5. At last, although ${\varphi _{13}}$ from case 1 and case 2 shows a clear difference in magnitude, the corresponding reconstructed velocity fields are quite similar. Quantitative examination shows that the averaged CC between the reconstructed velocity fields based on the two parameters is approximately 0.995 for two wall-normal positions considered, which validates the robust performance of FLSE.

Figure 6. Value of $\varphi _{13}^ + (x,y,z;z^{\prime})$ as a function of ${x^ + }$ and ${y^ + }$ for ${z^ + } = {z^{\prime+}} = 15.4$ (the first row) and ${z^ + } = {z^{\prime+}} = 119.4$ (the second row). The left, middle and right columns correspond to three calculation cases: (a,d) calculated by a typical routine with $s = 0.82 \times {10^{ - 6}}{\mu _{max}}$; (b,e) calculated by a typical routine with $s = 0.82 \times {10^{ - 4}}{\mu _{max}}$; (c,f) calculated by a routine neglecting the window-weighting procedure with $s = 0.82 \times {10^{ - 4}}{\mu _{max}}$.

To quantitatively investigate the influence of s in the reconstruction accuracy, a series of s ranging from $s = 0.82 \times {10^{ - 7}}{\mu _{max}}$ to $0.82 \times {10^{ - 1}}{\mu _{max}}$ (logarithmically spaced) are trialled. The minimum s employed in this test is very close to the lower limit for keeping numerical stability. The solved inducing model functions from (4.19) are used to reconstruct a large DNS field (as detailed by table 1) based on (4.12). The CCs between reconstructed velocity fields and original velocity fields at three wall-normal positions are displayed in figure 7 as functions of s. The results show that smaller s corresponds to higher reconstruction accuracy, although the results for $s/{\mu _{max}} < {10^{ - 3}}$ are quite close. The best performance of FLSE achieves a CC of larger than 0.9 for all the three velocity components in figures 7(a) and 7(b), which indicates that the reconstructed velocity field is very similar to the real one. For figure 7(c), the reconstruction accuracy is lower since it corresponds to the edge of the reconstructed domain, and only the vortices below this position are considered in the reconstruction. Based on the discussions for figures 6 and 7, this work will adopt $s = 0.82 \times {10^{ - 4}}{\mu _{max}}$ in the following analysis, which corresponds to a good performance in both convergence and accuracy. It is worthy to point out that the reconstruction results for $s/{\mu _{max}} < {10^{ - 3}}$ are quite close to each other, and thus prescribing any other s in this range brings no substantial change in the following results of this work.

Figure 7. Correlation coefficients for reconstructed/original velocity fields at ${z^ + } = \textrm{15}\textrm{.4, 119}\textrm{.4, 359}\textrm{.3}$ (a-c, respectively) as functions of the regularization parameter.

5.3. Comparing with the Biot–Savart law

The Biot–Savart (BS) law provides an explicit formula for reconstructing velocity fields based on vorticity fields. According to Batchelor (Reference Batchelor1967), the BS law returns the exact velocity fields only if the vorticity has zero normal component at each point of the integration boundary. For wall turbulence, we numerically examined the performance of the BS law in the vorticity-to-velocity reconstruction, and found its performance is excellent even at the buffer layer $({z^ + } = 15.4)$: the correlation coefficient between reconstructed and DNS velocity fields is larger than 0.99. This means that the classical BS law works well for the vorticity-to-velocity reconstruction of wall turbulence.

Based on the inherent correlation between vorticity and vortex, the BS law could be employed to reconstruct the velocity field directly based on the vortex field. Several investigators (Perry & Marusic Reference Perry and Marusic1995; Marusic Reference Marusic2001; de Silva et al. Reference de Silva, Hutchins and Marusic2016a) have employed the BS law to reconstruct wall-bounded turbulence based on the random arrangement of typical vortex tubes. The BS law with respect to the vortex-based reconstruction can be expressed as

(5.3)\begin{equation}\boldsymbol{u}(\boldsymbol{r}) = \frac{\mu }{{4{\rm \pi}}}\iint\!\!\!\int_\varOmega {\frac{{\boldsymbol{\varLambda }(\boldsymbol{r^{\prime}}) \times (\boldsymbol{r} - \boldsymbol{r^{\prime}})}}{{|\boldsymbol{r} - \boldsymbol{r^{\prime}}{|^3}}}\,\textrm{d}\varOmega } .\end{equation}

In (5.3), $\boldsymbol{r^{\prime}}$ is the running position vector for the volume integration and $\boldsymbol{r}$ is the reference position vector; $|\cdot |$ represents the magnitude of the vector quantity in the middle; $\mu $ is an artificial multiplier. For vorticity-to-velocity reconstruction, the multiplier equals one by theory. However, for current V2V reconstruction, the multiplier should be adjusted to make the magnitude of reconstructed velocity comparable to the real velocity magnitude. Pirozzoli et al. (Reference Pirozzoli, Bernardini and Grasso2010) employed a multiplier of two when dealing with this issue, which was derived based on the situation of a rigidity rotation. In this work, the multiplier is determined by minimizing the mean squares of the deviations between reconstructed results and original DNS results at given wall-normal positions.

For convenience in numerical implementation, (5.3) is reformed as

(5.4) \begin{equation}{u_i}(\boldsymbol{r}) = \iint\!\!\!\int_\varOmega{} {{{\mathsf{P}}_{ij}}(\boldsymbol{r} - \boldsymbol{r^{\prime}}){\varLambda _j}(\boldsymbol{r^{\prime}})\,\textrm{d}\varOmega } = {{\mathsf{P}}_{ij}} \ast {\varLambda _j}.\end{equation}

Herein, ${\ast} $ indicates a 3-D convolution operation, different from the 2-D convolution in (4.12). And the convolution kernel function ${{\mathsf{P}}_{ij}}$

(5.5) \begin{equation}[{{\mathsf{P}}_{ij}}] = \frac{\mu }{{4{\rm \pi}{{({x^2} + {y^2} + {z^2})}^{3/2}}+ \varepsilon }} \begin{bmatrix} 0 & z & - y\\ - z & 0 & x\\ y& - x & 0 \end{bmatrix}\end{equation}

where $\varepsilon$ is a small parameter, added to avoid the singularity at the origin. In this work, we prescribe ${\varepsilon ^ + } = {10^{ - 8}}$, which makes ${{\mathsf{P}}_{ij}}(0,0,0) = 0$ and brings only a tiny change on the values of ${{\mathsf{P}}_{ij}}$ at other positions. The convolutional kernel function of the BS law $({{\mathsf{P}}_{ij}})$ resembles the inducing model function ${\varphi _{ij}}$ in FLSE. Differently, the BS law employs a fixed and isotropic model function while FLSE adopts a data-driven inducing model, which incorporates the statistical imprints of flows. For implementation, ${{\mathsf{P}}_{ij}}$ is truncated and discretized on the same grid as ${\varphi _{ij}}$.

The reconstructed u fields based on both the BS law and FLSE are displayed in figure 8. The u field displayed in the figure covers an area of $2\delta \times \delta$ in the streamwise–spanwise plane, which is four times as large as the calculation domain of ${\varphi _{ij}}$. Results show that FLSE performs surprisingly well in both the near-wall region and the logarithmic region. The turbulent motions with scales ranging from the spacing of low-speed streaks to the boundary thickness are well recovered. As emphasized in the foregoing discussion, vortices are very sparse in space, and the vortex field contains almost as less information as a scalar field, which poses great challenges for the reconstruction. The success in recovering velocity fields based on vortex structure should be owing to the inherent coherence of vortex structures which could be well modelled as empirical functions as ${\varphi _{ij}}$. The BS law performs as well as FLSE for the logarithmic region but fails to reconstruct the low-speed streaks in the near-wall region. The comparatively poor performance of the BS law in the near-wall region is expected since the distributions of vortices and vorticities are quite different below the buffer layer, as shown in figure 2. The discussion reminds us that the BS law cannot be employed below the logarithmic region for the V2V reconstruction.

Figure 8. Instantaneous ${u^ + }$ fields from ${z^ + } = 15.4$ (the first row) and ${z^ + } = 119.4$ (the second row). (a,d) Original DNS data, (b,e) reconstructed ${u^ + }$ based on the BS law, (c,f) reconstructed ${u^ + }$ based on FLSE.

Quantitative comparisons between the performances of FLSE and the BS law are provided in figure 9. Figure 9(a) shows the CCs for reconstructed/original $u,v,w$ as functions of z. The most attractive point indicated from this plot is the good performance of FLSE for ${z^ + } < 100$. In this region, the CCs from FLSE remain a steady variation above 0.9 while the CCs from the BS law encounter a dramatic drop for $u,v,w$. In the logarithmic layer, both FLSE and the BS law perform well, achieving CCs of larger than 0.9 for $u,v,w$. The BS law performs slightly better than LSE for ${z^ + } > 150$, which supports the application of the BS law in the logarithmic layer (Perry & Marusic Reference Perry and Marusic1995; Marusic Reference Marusic2001; de Silva et al. Reference de Silva, Hutchins and Marusic2016a) and makes the BS law very competitive in this region considering its simplicity in implementation. The imperfect performance of FLSE for $z^+>150$ is attributed to the limited calculation domain for ${\varphi _{ij}}$, which will be further analysed in the following scale-specific assessment. Finally, the performances of both methods deteriorate at the upper boundary of the reconstruction domain, consistent with the results of figure 7.

Figure 9. Quantitative comparisons between the reconstruction results of FLSE and the BS law. (a) Correlation coefficients for reconstructed/original fields as functions of ${z^ + }$; (b) premultiplied streamwise energy spectra of ${u^ + }$ from the original DNS data and the reconstructed results as functions of $k_x^ + $; (c) magnitudes of streamwise CCS for reconstructed/original ${u^ + }$; (d) phase angles of streamwise CCS for reconstructed/original ${u^ + }$.

The premultiplied spectra shown in figure 9(b) provide scale-by-scale comparisons between the reconstructed results and original DNS data. Only the streamwise spectra for the u component (denoted as ${\phi _{uu}}$) are taken as examples in this discussion. At ${z^ + } = 15.4$, the spectra from the BS law show a plateau at $k_x^ +{=} 1.5 \times {10^{ - 3}} \sim 2 \times {10^{ - 2}}$ ($k_x^ + $ is the streamwise wavenumber), which is obviously different from the prominent peak pattern of the real spectra. The spectra from FLSE show slightly lower energy for all the wavenumbers, but the shape of the spectral curve resembles the real spectra. At ${z^ + } = 119.4$, the spectra from FLSE are very close to the real spectra except for the very low wavenumbers, where the spectra from both FLSE and the BS law are lower than the real spectra. For high wavenumbers, the spectra from FLSE tend to be lower than the real spectra while the spectra from the BS law tend to be higher to more extent. The results show that FLSE performs better in recovering the energy spectra.

Generally, two signals sharing the same spectra might be quite different since the spectra are integral quantities, which are robust to local events. To further evaluate the reconstruction, the CCS between reconstructed and original u along the streamwise direction is calculated. As introduced in § 2.3, the magnitude and phase angle of the CCS indicate the scale-by-scale correlation degree and scale-by-scale phase deviation between two input signals, respectively. The CCS offers a strict evaluating indicator for the performance of FLSE in reconstructing turbulent motions with different scales. One particular scale deserving more attention is the streamwise length of the calculation domain for ${\varphi _{ij}}$ (denoted as ${L_x}$), which corresponds to $k_x^ +\approx 5 \times {10^{ - 3}}$.

The magnitudes and phase angles of CCS are displayed in figure 9(c,d). A lot of information can be drawn from these two plots. Firstly, the overall performance and the effective scale range for FLSE are focused. Figure 9(c) shows that for both ${z^ + } = 15.4$ and ${z^ + } = 119.4$ the magnitudes of the CCS for FLSE-reconstructed/original u remain larger than 0.9 for the scale range of $k_x^ + < 2 \times {10^{ - 2}}$, which corresponds to a bulk of the total energy. The good performance for FLSE is also observed from the phase angles of the CCS, which fluctuate in a small range of ±5° for the whole wavenumber range considered. The most striking point is that FLSE performs well even for the largest scale considered, which is eight times as large as ${L_x}$. Another interesting point is that, for the most part, the CCS magnitude decreases with the increase of the wavenumber, which indicates that the large-scale motions are easier to reconstruct than the small-scale fluctuations. This trend is expected since the large structures are composed of large numbers of independent vortex tubes. While the individual vortex tubes suffer from strong random fluctuations, their joint inducing effects are more robust due to the offset effect of independent random fluctuations.

On the other hand, figure 9(c,d) also allows a further comparison between FLSE and the BS law. For the results of ${z^ + } = 15.4$, the advantages of FLSE are observed from both figures 9(c) and 9(d). FLSE achieves a larger CCS magnitude and smaller CCS angles for almost all wavenumbers. For the results of ${z^ + } = 119.4$, the BS law has slightly better performance for $k_x^ + < {10^{ - 2}}$, which corresponds to a length scale of approximately $0.5{L_x}$. A similar demarcation with $k_x^ +\approx {10^{ - 2}}$ is also observed in the performance of FLSE in figure 9(d). For $k_x^ + > {10^{ - 2}}$, the phase angle for the CCS fluctuates around zero; yet for $k_x^ + < {10^{ - 2}}$ the phase angle deviates from zero to a negative or positive number. The explanation is clear since the governing equations for ${\varphi _{ij}}$ are limited in the calculation domain. The intention of minimizing the reconstruction deviations is not enforced for the scales larger than ${L_x}$.

These results constitute an all-round assessment of the reconstruction performance of FLSE and the BS law, which benefits the application of both methods. Mostly, FLSE achieves better reconstruction accuracy, particularly for the buffer layer.

5.4. Full energy spectra of FLSE-reconstructed TBL

Figure 10 provides the full pre-multiplied spectra for original velocity fields and the FLSE-reconstructed ones, varying with the wall-normal position. The pre-multiplied streamwise spectra ${k_x}{\phi _{uu}},{k_x}{\phi _{vv}},{k_x}{\phi _{ww}}$ are shown as functions of z and ${\lambda _x}$ in the first row, and the pre-multiplied spanwise spectra ${k_y}{\Phi _{uu}},{k_y}{\Phi _{vv}},{k_y}{\Phi _{ww}}$ are displayed as functions of z and ${\lambda _y}$ in the second row. Herein, ${\lambda _x}$ and ${\lambda _y}$ denote the streamwise and spanwise wavelength, respectively; and they are linked to the corresponding wavenumber ${k_x}$ and ${k_y}$ by ${\lambda _x} = 2{\rm \pi}/{k_x}$ and ${\lambda _y} = 2{\rm \pi}/{k_y}$. The energy spectra shown here are very consistent with the results of several investigators such as Hutchins & Marusic (Reference Hutchins and Marusic2007) for ${k_x}{\phi _{uu}}$, Wang et al. (Reference Wang, Wang and He2017, Reference Wang, Pan, Gao and Wang2018) for ${k_y}{\Phi _{uu}}$. The spectra from FLSE agree well with the original spectra for the bulk of the total energy. The performance for buffer layer $({z^ + } < 80)$ is excellent, recovering a satisfying energy distribution for almost the whole scale range considered. The success in the buffer layer is attributed to the relatively smaller inducing range of vortices in this region, which is completely covered in the calculation domain for ${\varphi _{ij}}$ as shown in figure 5. For ${z^ + } > 80$, deviations between reconstructed spectra and the original spectra are observed for the large scales. Generally, the reconstructed spectra for ${\phi _{vv}},{\phi _{ww}}$ are more accurate than those of ${\phi _{uu}}$, which is consistent with the fact that u corresponds to comparatively larger coherence scales. A larger calculation domain of ${\varphi _{ij}}$ will account for the structures with larger scales embedded in the logarithmic layer but also significantly burden the calculation resources, which is outside the scope of the work. All in all, the result of FLSE is satisfying for the scale range covered in the current calculation domain.

Figure 10. Contour maps showing the variation of premultiplied spectra for ${u^ + },{v^ + },{w^ + }$ with wall-normal position. The first row: the premultiplied streamwise spectra $k_x^ + \phi _{uu}^ + ,k_x^ + \phi _{vv}^ + ,k_x^ + \phi _{ww}^ +$ as functions of ${z^ + }$ and $k_x^ + $; the second row: the premultiplied spanwise spectra $k_y^ + \Phi _{uu}^ + ,k_y^ + \Phi _{vv}^ + ,k_y^ + \Phi _{ww}^ + $ as functions of ${z^ + }$ and $k_y^ + $. Contour lines show 1/6 to 5/6 of the maximum premultiplied energy, with an increment of 1/6. The shaded and solid contour lines show the results of original DNS data while the dashed contour lines show the results of reconstructed velocity fields based on FLSE.

Energy spectra are among the primary considerations in modelling works of wall-bounded turbulence. To cover the whole scale range of energy spectra, Perry & Marusic (Reference Perry and Marusic1995) suggested that three types of vortices need to be involved in the vortex model: two types of detached eddies accounting for the largest energy-containing motions and small-scale inertial motions, respectively, and one type of the attached eddy accounting for the self-similar motions in middle scales. As for the representation of the attached eddies, controversy appears. Earlier investigators treated the attached eddies or detached eddies as hierarchies of vortex tubes with $\varLambda $ or $\varOmega $ shapes (Perry & Chong Reference Perry and Chong1982) or vortex clusters (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006) while recent works tend to recognize them by a pattern of velocities, such as the connected sets of points by Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014) and the iso-surface of u by Lee et al. (Reference Lee, Lee, Choi and Sung2014) and Yoon et al. (Reference Yoon, Hwang, Yang and Sung2020). Recently, there are also some efforts to extract the contribution of attached eddies from the streamwise spectra of u (Baars & Marusic Reference Baars and Marusic2020a,Reference Baars and Marusicb; Hu, Yang & Zheng Reference Hu, Yang and Zheng2020).

Results of figure 10 show that the bulk range of energy spectra can be covered in the induced motions of the vortex fields. Certainly, the contribution of attached eddies could also be attributed to the joint inducing effects of vortex groups. While the vortices with random sizes and strength characterize the local turbulence, the large-scale component of the vortex field explains the large-scale coherent structures (Gad-el-Hak & Bandyopadhyay Reference Gad-el-Hak and Bandyopadhyay1994). The large-scale component of the vortex field might be attributable to the clustered organization of CFSEs (del Álamo et al. Reference del Álamo, Jiménez, Zandonade and Moser2006; Kang et al. Reference Kang, Tanahashi and Miyauchi2007). Up to present, the modelling works based on random CFSEs are mostly limited to the inertial range for isotropic turbulence (Townsend Reference Townsend1951; Corrsin Reference Corrsin1962; Lundgren Reference Lundgren1982; Pullin & Saffman Reference Pullin and Saffman1993). Of particular note is the strained-spiral vortex model proposed by Lundgren (Reference Lundgren1982), which derives the Kolmogorov spectra based on the randomly oriented collections of non-axisymmetric and strained vortex models. For wall turbulence, the non-isotropy nature and the large-scale coherence hinder the extension of these models. The CFSE-based modelling works should incorporate both the random geometries for individual vortex tubes and the large-scale clustering trend of vortex groups. Perhaps a combination of the attached eddy model with randomly placed vortex tubes could be attempted in future works.

Another topic related to the energy spectra is the inner–outer interaction in wall-bounded turbulence. A lot of investigations (Hutchins & Marusic Reference Hutchins and Marusic2007; Marusic et al. Reference Marusic, Mathis and Hutchins2010; Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2011) have shown that large-scale motions in the logarithmic layer have modulation and superposition effects on the velocity fluctuation in the near-wall region. And only the superposition effect contributes to the energy of the near-wall region, accounting for the increasing trend of inner peaks with the increase of the Reynolds number. The superposition effect could be understood as the inducing effect of the vortices at the logarithmic layer, and FLSE allows us to have an insight into this aspect.

In order to separate the inducing effects of vortices from different wall-normal positions on a reference plane, a V2V reconstruction based on height-filtered vortices is performed. Specifically, only the vortices located at a given height range $z \in [{z_c} - \Delta ({z_c}),{z_c} + \Delta ({z_c})]$ are considered in the reconstruction of u at the reference plane (e.g. $z_{ref}^ +{=} 15.4$), and the result is denoted as $u(x,y,{z_{ref}};{z_c})$. Herein, ${z_c}$ is the central position for the wall-normal range of the vortices participating in the reconstruction; $\Delta ({z_c})$ denotes the half-span, which is set as twice the local interval of the original wall-normal grid. The pre-multiplied streamwise spectra for $u(x,y,{z_{ref}};{z_c})$ are calculated, which are represented by ${k_x}{\phi _{uu}}({k_x},{z_{ref}};{z_c})$. Contours of ${k_x}{z_c}{\phi _{uu}}({k_x},{z_{ref}};{z_c})/(2\Delta ({z_c}))$ are shown as functions of ${k_x}$ and ${z_c}$ in logarithmic coordinates in figure 11(a), where the multiplier ${z_c}$ is added to balance the squeezing effect of logarithmic coordinates. Basically, this contour map shows the contribution of vortices at different wall-normal positions to the streamwise spectra of u at the reference plane of $z_{ref}^ +{=} 15.4$, which are called the contribution spectra for ${\phi _{uu}}$ regarding the reference plane of $z_{ref}^ +{=} 15.4$. Similarly, the contribution spectra for the streamwise spectra of $v,w$ regarding the reference plane of $z_{ref}^ +{=} 15.4$, and the contribution spectra for the streamwise spectra of $u\textrm{,}v\textrm{,}w$ regarding the reference plane of $z_{ref}^ +{=} 119.4$ are also calculated and displayed in the first and second rows of figure 11, respectively.

Figure 11. Contour maps showing the contribution of vortices at $z_c^ +$ to $\phi _{uu}^ + $, $\phi _{vv}^ + $, $\phi _{ww}^ + $ (a–f, respectively) at the reference plane of $z_{ref}^ +{=} 15.4$ (the first row) and $z_{ref}^ +{=} 119.4$ (the second row), as marked by the red dashed lines. The contours show the local energy contribution premultiplied by $k_x^ + z_c^ +$ and normalized by the maximum values.

Figure 11 is revealing in several aspects. Firstly, it shows that, compared with figures 11(b) and 11(c), the contribution spectra for ${\phi _{uu}}$ in figure 11(a) are flatter along the ${z_c}$ direction, which indicates the u component is more sensitive to the vortices below or above the reference positions. The sensitivity of u to vortices may be correlated to the lift-up mechanics caused by streamwise vortices, which tends to produce low-speed structures in the near-wall region (Schoppa & Hussain Reference Schoppa and Hussain2002). In figure 11(a), the contribution from $z_c^ + > 100$ can be observed at ${\lambda _x} > 900$ for ${\phi _{uu}}$, supporting the well-documented superposition effect of large-scale structures on near-wall fluctuations. In figures 11(b) and 11(c), the energy-contributing regions for ${\phi _{vv}}$ and ${\phi _{ww}}$ regarding the reference plane of $z_{ref}^ +{=} 15.4$ are limited below $z_c^ + < 100$ and $z_c^ + < 50$, respectively. For the contribution spectra regarding the reference plane of $z_{ref}^ +{=} 119.4$ (the second row), while the energy-contributing regions for ${\phi _{uu}}$ and ${\phi _{vv}}$ cover the bulk of the whole wall-normal range considered, the energy-contributing region for ${\phi _{ww}}$ remains local. In previous investigations (Woodcock & Marusic Reference Woodcock and Marusic2015), the special behaviour of the w component was also noticed in the scaling law of $\langle {w^2}\rangle$, which remains constant in the logarithmic region while both $\langle {u^2}\rangle$ and $\langle {v^2}\rangle$ show a logarithmic variation with the wall-normal position. The special behaviour of the w component is owing to the bounding effect of the wall, which weakens the inducing effects of vortices on the w component. In fact, the wall-bounding effect was incorporated as the boundary condition in the theoretical works of Perry & Marusic (Reference Perry and Marusic1995) and Woodcock & Marusic (Reference Woodcock and Marusic2015), which yielded the correct scaling law for $\langle {u^2}\rangle ,\langle {v^2}\rangle ,\langle {w^2}\rangle$. Figure 11(f) provides some evidence for this condition.

5.5. High-order moments of FLSE-reconstructed TBL

The energy spectra discussed above only involve the second-order moments of velocities. The higher-order moments offer further indicators to distinguish the reconstructed velocity fields and original DNS velocity fields. This subsection focuses on higher-order moments, including the skewness and flatness for the three velocity components. The skewness and flatness factors for u (denoted as ${S_u}$ and ${F_u}$) are defined as $\langle {u^3}\rangle /{\langle {u^2}\rangle ^{3/2}}$ and $\langle {u^4}\rangle /{\langle {u^2}\rangle ^2}$, respectively. While the skewness factor reflects the asymmetry degree of fluctuating signals, the flatness factor is more correlated to the peaky signals associated with the intermittent turbulence events. Gad-el-Hak & Bandyopadhyay (Reference Gad-el-Hak and Bandyopadhyay1994) suggested that the third moment retains the sign information and is more revealing in inferring the characteristics of coherent structures; $\langle {u^3}\rangle $ could also be explained as the streamwise flux for the streamwise turbulent energy, and such an explanation is also suitable for $\langle {v^3}\rangle $ or $\langle {w^3}\rangle $. Besides the moments of individual velocity components, the correlation term $\langle uw\rangle $ (the kinematic Reynolds shear stress with an opposite sign) and its streamwise or wall-normal flux (i.e. $\langle {u^2}w\rangle $, $\langle u{w^2}\rangle $) (Bandyopadhyay & Watson Reference Bandyopadhyay and Watson1988) are also analysed, considering that they are important terms in the energy balance equation (Bradshaw Reference Bradshaw1967) or the transportation equations of the shear stress in wall turbulence.

All these statistical quantities from original DNS data and the reconstructed results are shown in figure 12. It shows that the second-order moments from FLSE are lower than the original DNS results, which is consistent with the observations on the energy spectra in figures 9 and 10. As for the skewness and flatness of u (figure 12b,c), the results from FLSE present distinctive behaviours at the near-wall region compared with the DNS results. For the skewness of u, the FLSE results from $15 < {z^ + } < 100$ are mostly positive, which are on the opposite of the corresponding DNS results. For the flatness of u, the FLSE results from $15 < {z^ + } < 50$ stay above or very close to three, while the corresponding DNS results are always smaller than three. Generally, a positive skewness indicates that the extreme events corresponding to a large fluctuation magnitude are more likely positive, and vice versa. A flatness of three is the behaviour of a standard Gaussian distribution, and larger or smaller flatness corresponds to a super-Gaussian or sub-Gaussian behaviour, respectively. It can be inferred from the deviations that the reconstructed u tends to be flatter and contains weak negative fluctuations compared with the original field. In the logarithmic region, the deviations for both skewness and flatness are reduced: both results show a negative skewness and sub-Gaussian behaviour for flatness. Different from the results of u, the skewness and flatness of $v,w$ from FLSE agree well with the original DNS data for the whole wall-normal range considered, which further validates that u poses the most significant challenge for the accurate reconstruction.

Figure 12. Various moments for ${u^ + },{v^ + },{w^ + }$ as functions of wall-normal positions. (a) Fluctuation energy, (b) skewness, (c) flatness, (d) $\langle {u^ + }{w^ + }\rangle $ and its wall-normal/streamwise flux quantity ($\langle {u^ + }{w^ + }^\textrm{2}\rangle $ and $\langle {u^ + }^\textrm{2}{w^ + }\rangle $).

The skewness and flatness factors have also been considered in the attached eddy model. The model predicts that the flatness of u, is invariably larger than three (Woodcock & Marusic Reference Woodcock and Marusic2015), which is different from the well-accepted experimental result of 2.8 (Fernholz & Finleyt Reference Fernholz and Finleyt1996; Stanislas et al. Reference Stanislas, Perret and Foucaut2008). To build on this aspect, de Silva et al. (Reference de Silva, Woodcock, Hutchins and Marusic2016b) introduced the spatial exclusion effect in attached eddy model and found that the improved attached eddy model gives a correct prediction on the flatness of u. It validates the importance of the spatial exclusion effect in deriving accurate predictions. The current work reconstructs the velocity fields based on the original vortex fields, which embodies the spatial exclusion effects. The reconstruction results retain the correct behaviour for the flatness of u in the logarithmic layer but not in the near-wall region. This indicates that, besides incorporating the spatial exclusion effect, further reform of the ideal vortex model is needed in order to explain the flatness in the near-wall region.

In figure 12(d), the reconstructed results retain the same signs with the corresponding DNS results for all the three quantities, which is promising considering that the sign of $\langle u{w^2}\rangle $ could be related to the wall-ward or outward motion induced by dominant vortex structures (Bandyopadhyay & Watson Reference Bandyopadhyay and Watson1988). On the other hand, significant deviations are noticed from the results of $\langle {u^2}w\rangle$ at the buffer region. This means that the reconstructed turbulence contains weaker streamwise flux of $\langle uw\rangle $, which might be correlated with the relatively weaker low-speed structures observed in the reconstructed results. The reconstructed $\langle uw\rangle $ has a steeper slope above ${z^ + } = 50$, resulting in a slightly lower magnitude at the logarithmic region. The discount amount for $\langle uw\rangle $ might be attributable to the very large structures, which are not covered in the domain of the current calculation. Finally, of all the three quantities, the reconstructed results for $\langle u{w^2}\rangle $ is most satisfying, it is remarkably consistent with the DNS results.

In order to reveal the reason that FLSE fails to recover the higher-order moments in the near-wall region, the inducing model employed by FLSE needs to be reconsidered. FLSE expressed the inducing effects of vortices as linear operators, which implies that the inducing effects of two vortices with opposite orientations can be modelled by one function with adjustable signs or multipliers. However, the real situation might be more complex, sometimes beyond the capacity of the linear operators. To support this argument, figure 13 shows the conditionally averaged u field around a given positive streamwise vortex at ${z^ + } = 37.8$, which is calculated by ${\langle u(\boldsymbol{r} + \Delta \boldsymbol{r})\varLambda _x^p(\boldsymbol{r})\rangle _{\boldsymbol{r},t}}$. Herein, the superscript p indicates a positive-pass filter, which retains positive values of the input field and sets the negative values as zeros. The wall-normal position for the conditional average corresponds to the largest deviation of ${S_u}$ in figure 12(b). As we can see, the u field around a positive vortex shows an asymmetric pattern. In this contour map, the upper side of the central vortex corresponds to negative u with a larger magnitude compared with that from the lower side, which is consistent with the fact that a positive streamwise vortex is distributed below a low-speed streak in this coordinate system. In the implementation of FLSE, the inducing effects of a positive streamwise vortex and a negative streamwise vortex are averaged in one correlation function (as shown in figure 5), which results in a symmetric pattern for inducing models. Such an averaging operation will reduce the intensity of low-speed streaks in reconstructed fields and result in a smaller skewness magnitude and a larger flatness, and also a smaller streamwise flux for $\langle uw\rangle $, which reasonably explains the deviations of the FLSE results displayed by figure 12. To deal with this issue, the reconstruction scheme might be improved by distinguishing the inducing effects of vortices with different signs, which is expected to be the topic of future work.

Figure 13. Conditionally averaged ${u^ + }$ field around a given positive streamwise vortex at ${z^ + } = 37.8$. (a) Contour map of the conditionally averaged ${u^ + }$ field, (b) the conditionally averaged ${u^ + }$ as a function of $y^{+}$ along a line extracted from the contour map, as marked by the red dash-dot line.

5.6. The influence of the $\lambda _{ci}^ + $ threshold on the reconstructed results

Recalling the analysis in § 3 of this article, a threshold of $\lambda _{ci}^ + $ was prescribed to isolate the typical tube-like vortex structures from the vortex fields. Moreover, the correlated discussions about the geometrical properties were mostly limited to these isolated tube-like vortices. In contrast, in the numerical tests of §§ 5.3–5.5, the velocity fields were reconstructed based on the whole vortex fields $(\boldsymbol{\varLambda })$, without any threshold. Thus, we need to perform an additional numerical test involving the influence of the threshold issue, with the purpose of bridging the divide between the discussions about the isolated vortex tubes and the V2V reconstruction results in §§ 5.3–5.5.

To show the isolated vortices for prescribed thresholds, figure 14 provides the contour maps for the vortex strength $(\lambda _{ci}^ + )$ at two wall-normal positions. Two sets of contour lines for $\lambda _{ci}^ +{=} 0.05$ and $\lambda _{ci}^ +{=} 0.1$ are highlighted by the solid lines with different colours. As we can see, the $\lambda _{ci}^ + $ field presents a visually fragmented pattern. Most of the isolated vortices have a thickness of less than 50 wall units. Comparatively, $\lambda _{ci}^ +{=} 0.1$ recognizes stronger and finer vortex cores, which covers only a small portion of the total area. The iso-surfaces of $\lambda _{ci}^ +{=} 0.1$ were used to display the vortex tubes in figure 2. It is close to the threshold of $\lambda _{ci}^ +{=} 0.08$ employed by Jodai & Elsinga (Reference Jodai and Elsinga2016), also for the vortex visualization purposes. On the other hand, for the sake of extracting the vortex properties, such as the radius or circulation, investigators tended to use smaller thresholds in order to count in the low-swirl regions surrounding the vortex cores. For example, Ganapathisubramani et al. (Reference Ganapathisubramani, Longmire and Marusic2006) and Gao et al. (Reference Gao, Ortiz-Dueñas and Longmire2011) adopted a threshold of approximately 0.035 in their investigations, close to $\lambda _{ci}^ +{=} 0.05$ shown in the contour maps. Outside the solid lines, weak vortices with $\lambda _{ci}^ + < 0.05$ can also be observed in the contour maps. These weak vortices are usually disregarded in experimental investigations because they are vulnerable to measurement uncertainties. However, their roles in the V2V reconstruction might be non-negligible since they cover a large portion of the total area and have larger scales.

Figure 14. The contour maps of the vortex strength $(\lambda _{ci}^ + )$ in the streamwise–spanwise plane at ${z^ + } = 15.4$ (a) and ${z^ + } = 119.4$ (b). Two sets of contour lines for $\lambda _{ci}^ +{=} 0.05$ and $\lambda _{ci}^ +{=} 0.1$ are highlighted by white and red lines, respectively.

Based on these discussions, we will trial three specially filtered vortex fields in the V2V reconstructions: (a) the weak vortex fields ${\boldsymbol{\varLambda }_W}$, obtained by retaining the vortex vectors satisfying $\lambda _{ci}^ + < 0.05$ and setting all the other vortex vectors as zeros; (b) the strong vortex fields ${\boldsymbol{\varLambda }_{S\textrm{1}}}$, acquired by filtering the original vortex fields based on the criterion $\lambda _{ci}^ + \ge 0.05$; and (c) the strong vortex fields ${\boldsymbol{\varLambda }_{S\textrm{2}}}$, acquired by filtering original vortex fields based on $\lambda _{ci}^ + \ge 0.1$. The original vortex fields without any filtering process are denoted as ${\boldsymbol{\varLambda }_O}$. Obviously, the above definitions yield that ${\boldsymbol{\varLambda }_O} = {\boldsymbol{\varLambda }_W} + {\boldsymbol{\varLambda }_{S1}}$.

In the following numerical tests, the velocity fields are separately reconstructed based on ${\boldsymbol{\varLambda }_W}$, ${\boldsymbol{\varLambda }_{S\textrm{1}}}$ or ${\boldsymbol{\varLambda }_{S\textrm{2}}}$ using the same method with the previous reconstruction for ${\boldsymbol{\varLambda }_O}$. Specifically, a convolution operation is performed on these vortex fields, and the kernel functions are the inducing model functions $({\varphi _{ij}})$ calculated in § 5.2. The reconstruction formula could be found in (4.12), where the input vortex fields should be replaced accordingly. To make it convenient for the following discussions, the reconstructed streamwise velocity components based on ${\boldsymbol{\varLambda }_O}$, ${\boldsymbol{\varLambda }_W}$, ${\boldsymbol{\varLambda }_{S\textrm{1}}}$ and ${\boldsymbol{\varLambda }_{S\textrm{2}}}$ are represented by ${u_{O,rec}}$, ${u_{W,rec}}$, ${u_{S\textrm{1},rec}}$ and ${u_{S\textrm{2},rec}}$, respectively. The original streamwise velocity component from the DNS data is represented by ${u_{DNS}}$. Based on the linear property of the reconstruction, we can derive ${u_{O,rec}} = {u_{W,rec}} + {u_{S1,rec}}$ from ${\boldsymbol{\varLambda }_O} = {\boldsymbol{\varLambda }_W} + {\boldsymbol{\varLambda }_{S1}}$. The linear reconstruction also implies that ${u_{W,rec}}$, ${u_{S\textrm{1},rec}}$ and ${u_{S\textrm{2},rec}}$ reflect the separate contributions of ${\boldsymbol{\varLambda }_W}$, ${\boldsymbol{\varLambda }_{S\textrm{1}}}$ and ${\boldsymbol{\varLambda }_{S\textrm{2}}}$ in the ${\boldsymbol{\varLambda }_O}$-based reconstruction results $({u_{O,rec}})$.

Table 2 collected some statistical information about ${\boldsymbol{\varLambda }_W}$, ${\boldsymbol{\varLambda }_{S\textrm{1}}}$, ${\boldsymbol{\varLambda }_{S\textrm{2}}}$ and their reconstruction results. The third column of this table provides the volume fractions corresponding to different vortex fields, defined by the ratio of the volumes occupied by the selected vortices (with non-zero magnitudes) to the total volume considered. The fourth and fifth columns give the mean squares of the reconstructed u, i.e. $\langle {u_{M,rec}}^2\rangle$, $\langle {u_{S1,rec}}^2\rangle$ and $\langle {u_{S2,rec}}^2\rangle$ (collectively denoted as $\langle {u_{{\cdot} ,rec}}^2\rangle$) at ${z^ + } = 15.4$ and ${z^ + } = 119.4$, respectively. The last two columns provide the correlation coefficients between the reconstructed u and ${u_{DNS}}$ at the same wall-normal positions.

Table 2. Statistical information about different vortex fields and their reconstruction results.

Among the three kinds of vortex field, ${\boldsymbol{\varLambda }_W}$ owns the largest volume factions, occupying nearly half of the total volume. The correlation coefficient between ${u_{W,rec}}$ and ${u_{DNS}}$ at ${z^ + } = 15.4$ is as large as 0.83, which is the largest among these results. But at ${z^ + } = 119.4$, the corresponding correlation coefficient decreases to 0.77, much lower than that of ${\boldsymbol{\varLambda }_{S\textrm{1}}}$. In contrast to ${\boldsymbol{\varLambda }_W}$, the volume fraction for ${\boldsymbol{\varLambda }_{S\textrm{2}}}$ is only approximately 5 %, the lowest among the three vortex fields. The shortcoming in volume faction causes the lowest reconstruction correlation coefficients for ${\boldsymbol{\varLambda }_{S\textrm{2}}}$. Compared with ${\boldsymbol{\varLambda }_W}$, the advantage of ${\boldsymbol{\varLambda }_{S\textrm{2}}}$ for the V2V reconstruction is in the vortex strength. Stronger vortices induce larger fluctuation magnitudes, which explains why $\langle {u_{S2,rec}}^2\rangle$ is much larger than $\langle {u_{M,rec}}^2\rangle$; ${\boldsymbol{\varLambda }_{S\textrm{1}}}$ has a volume fraction three times larger than that of ${\boldsymbol{\varLambda }_{S\textrm{2}}}$ and contains more strong vortices, which make it achieve larger $\langle {u{{_{S1,rec}^ + }^2}} \rangle$ and correlation coefficients at the same time. The data in the table also reflect the trend that the reconstruction accuracy drops with the increase of the filtering threshold.

Figure 15(a) shows the premultiplied energy spectra for ${u_{W,rec}}$, ${u_{S\textrm{1},rec}}$ and ${u_{S\textrm{2},rec}}$. At first glance, we can see the integral energy of ${u_{S1,rec}}$ is much larger than that of ${u_{W,rec}}$ or ${u_{S2,rec}}$, consistent with the $\langle {u_{{\cdot} ,rec}}^2\rangle$ data shown in table 2. Both the spectra of ${u_{S\textrm{1},rec}}$ and ${u_{S\textrm{2},rec}}$ present obvious peaks at $k_x^ +\approx {10^{ - 2}}$ and $k_x^ +\approx 6 \times {10^{ - 3}}$, for ${z^ + } = 15.4$ and ${z^ + } = 119.4$, respectively. But for the spectra of ${u_{W,rec}}$, the curve shows a plateau at $k_x^ + < 7 \times {10^{ - 3}}$. The distribution of the spectra for ${u_{W,rec}}$ indicates that it spares a larger proportion of the total energy for small wavenumbers. Particularly for the results at ${z^ + } = 15.4$, the curve for ${u_{W,rec}}$ is above that of ${u_{S\textrm{2},rec}}$ at $k_x^ + \mathrm{\ < 2} \times {10^{ - 3}}$. However, for the spectra of larger wavenumbers, the contribution from ${\boldsymbol{\varLambda }_W}$ is much lower compared with ${\boldsymbol{\varLambda }_{S1}}$ and ${\boldsymbol{\varLambda }_{S\textrm{2}}}$.

Figure 15. Spectral analysis on ${u_{W,rec}}$, ${u_{S\textrm{1},rec}}$ and ${u_{S\textrm{2},rec}}$. (a) The premultiplied energy spectra for ${u_{W,rec}}$, ${u_{S\textrm{1},rec}}$ and ${u_{S\textrm{2},rec}}$; (b) the magnitude of streamwise CCS between ${u_{DNS}}$ and ${u_{{\cdot} ,rec}}$ (including ${u_{W,rec}}$, ${u_{S\textrm{1},rec}}$ and ${u_{S\textrm{2},rec}}$).

The magnitude of the streamwise CCS between ${u_{DNS}}$ and the corresponding reconstructed result (${u_{W,rec}}$, ${u_{S\textrm{1},rec}}$ or ${u_{S\textrm{2},rec}}$) is displayed in figure 15(b). As mentioned in § 2, the CCS magnitude provides the scale-by-scale correlation degree between two input velocity fields. As shown by figure 15(b), the CCS magnitude for ${u_{S\textrm{1},rec}}$ stays above 0.8 for $k_x^ + \mathrm{\ < 3} \times {10^{ - \textrm{2}}}$ and the corresponding curve shows a decreasing trend with the increase of wavenumbers. The CCS magnitude for ${u_{S\textrm{2},rec}}$ is obviously lower than that of ${u_{S\textrm{1},rec}}$ for all the wavenumbers considered, presenting a decreasing trend similar to the curve for ${u_{S\textrm{1},rec}}$. Comparatively, the curves for ${u_{W,rec}}$ decrease faster than those of ${u_{S\textrm{1},rec}}$ or ${u_{S\textrm{2},rec}}$. At ${z^ + } = 15.4$, the curve for ${u_{W,rec}}$ stays above 0.9 for $k_x^ + < {10^{ - 3}}$, achieving the largest CCS magnitude. However, for $k_x^ + > 3 \times {10^{ - 2}}$, the curve for ${u_{W,rec}}$ drops below that of ${u_{S\textrm{2},rec}}$, which is the lowest.

To summarize, ${\boldsymbol{\varLambda }_{S\textrm{1}}}$ contributes significantly to the bulk energy spectra and the corresponding CCS magnitudes maintain high levels for a wide wavenumber range; ${\boldsymbol{\varLambda }_W}$ mainly builds on the low-wavenumber spectra, and its contribution is more prominent in the near-wall region. This indicates that the large-scale velocity structures could be formed by the joint inducing effects of the clusters containing both weak vortices and stronger vortices. The advantage of ${\boldsymbol{\varLambda }_W}$ for smaller wavenumbers might be attributable to the larger volume fraction, which means more vortex vectors participating in the reconstructions.

Besides ${\boldsymbol{\varLambda }_W}$ and ${\boldsymbol{\varLambda }_{S\textrm{1}}}$, the performance of ${\boldsymbol{\varLambda }_{S2}}$ deserves more attention because the vortex tubes corresponding to ${\boldsymbol{\varLambda }_{S2}}$ have been highlighted for the geometrical property in § 3 of this work. The above tests show that the spectrum of ${u_{S\textrm{2},rec}}$ is much lower than that of ${u_{S\textrm{1},rec}}$ and the corresponding CCS magnitudes for most wavenumbers are approximately 0.7, which is not satisfying. It is important to keep in mind that the above tests emphasized the separate contributions of ${\boldsymbol{\varLambda }_W}$, ${\boldsymbol{\varLambda }_{S\textrm{1}}}$ and ${\boldsymbol{\varLambda }_{S2}}$ to the total (${\boldsymbol{\varLambda }_O}$-based) reconstruction. Specifically, the inducing model functions used in the above reconstructions were obtained in § 5.2 for the purpose of ${\boldsymbol{\varLambda }_O}$-based reconstruction. Therefore, the above results cannot reflect the utmost ability of ${\boldsymbol{\varLambda }_{S2}}$ in independently reconstructing the original velocity fields. To determine this, we need to recalculate the inducing model functions based on the correlation functions of ${\boldsymbol{\varLambda }_{S2}}$ and ${u_{DNS}}$.

Based on these considerations, we will refine the reconstruction based on ${\boldsymbol{\varLambda }_{S2}}$ via the standard FLSE. Firstly, the correlation function of ${{\mathsf{R}}_{jm}} = \langle {\varLambda _{S\textrm{2},j}}{\varLambda _{S\textrm{2},m}}\rangle$ and ${{\mathsf{q}}_{im}} = \langle {\varLambda _{S\textrm{2},i}}{u_{DNS,m}}\rangle$ are statistically calculated. Then, equations (4.13) are solved by using the FFT scheme, which returns new inducing model functions ${\varphi _{S\textrm{2},ij}}$. The regularization parameter s is prescribed as the same value as that used in § 5.2. At last, the reconstructed velocity fields are obtained by (4.12). To make a distinction with the former reconstructed result $({u_{S\textrm{2},rec}})$, we denote the newly reconstructed u (via the standard FLSE) as ${u_{S\textrm{2},FLSE}}$.

Figures 16(a) and 16(b) shows the contour maps for ${u_{S\textrm{2},FLSE}}$ at ${z^ + } = 15.4$ and ${z^ + } = 119.4$. These maps can be compared with the contour maps of ${u_{DNS}}$ shown in figure 8. To facilitate the comparison, we overlay the contour maps in figure 16(a,b) with the contour lines of ${u_{DNS}} = 0$. As can be seen from the maps, while the low-speed streak in ${u_{S\textrm{2},FLSE}}$ is weaker than that of ${u_{DNS}}$ (in figure 8), the larger flow patterns are reasonably consistent. Comparatively, ${u_{S\textrm{2},FLSE}}$ for ${z^ + } = 119.4$ is more consistent with ${u_{DNS}}$. The premultiplied spectra for ${u_{S\textrm{2},FLSE}}$ and ${u_{DNS}}$ are shown in figure 16(c). At ${z^ + } = 119.4$, the curve for ${u_{S\textrm{2},FLSE}}$ is close to that of ${u_{DNS}}$ nearby the peak position $(k_x^ + \textrm{ = 3} \times {10^{ - 3}} \sim \textrm{4} \times {10^{ - 3}})$, which corresponds to a length scale of $\textrm{1}\textrm{.3}\delta \sim \textrm{1}\textrm{.7}\delta $. This implies that the large-scale motions in the logarithmic region could be approximately explained as the inducing effects of the strong vortices. For smaller wavenumbers of $k_x^ + \mathrm{\ < 2} \times {10^{ - 3}}$, the deviations between the spectra of ${u_{S\textrm{2},FLSE}}$ and ${u_{DNS}}$ become larger. This might be attributable to the limited calculation domain for solving the inducing functions, as we discussed in § 5.1. At ${z^ + } = 15.4$, the maximum magnitude of the spectrum of ${u_{S\textrm{2},FLSE}}$ is approximately 30 % lower than that of ${u_{DNS}}$. The worse performance in the near-wall region could be caused by the viscous diffusion effect, which makes the vortex distribution not as concentrated as that in the logarithmic region. Thus, a smaller threshold should be adopted to count in more regions around the vortex centres. Comparing the spectra shown in figures 16(c) and 15(a), we can see that the refined reconstruction scheme significantly improves the spectrum magnitude. The magnitude of streamwise CCS between ${u_{S\textrm{2},FLSE}}$ and ${u_{DNS}}$ is provided in figure 16(d). The improvement brought by the refined reconstruction strategy can also be found by comparing the curves in figure 16(d) with those in figure 15(b). For the wavenumber range of $k_x^ + \mathrm{\ < }{10^{ - 2}}$ $(\lambda _x^ + > 0.52{\delta ^ + })$, the curves in figure 16(d) stay at the level of 0.75–0.8. The results are mostly encouraging since these vortices participating in the reconstruction only occupy 5 % of the volume considered. It demonstrates that the strong tube-like vortices are theoretically important for explaining the large-scale motions in TBL.

Figure 16. The reconstructed results based on ${\boldsymbol{\varLambda }_{S\textrm{2}}}$ via the refined reconstruction strategy. (a–b) The contour maps of ${u_{S\textrm{2},FLSE}}$ at ${z^ + } = 15.4$ and ${z^ + } = 119.4$, respectively. To provide a reference, the contour lines of ${u_{DNS}} = 0$ are also shown by the white dashed lines. (c) The premultiplied energy spectra for ${u_{S\textrm{2},FLSE}}$ and ${u_{DNS}}$. (d) The magnitude of streamwise CCS between ${u_{S\textrm{2},FLSE}}$ and ${u_{DNS}}$.