2.1. Problem setting

In the present study we predict $v_{10}$ from a spatial distribution of wall variables ($\chi _w$) in a turbulent channel flow, where a CNN is used to extract hidden features of $\chi _w$ which may closely represent $v_{10}$. We consider three different wall variables ($\chi _w=p_w$, $\partial u/\partial y \vert _w$ and $\partial w/\partial y \vert _w$) that are measurable quantities in real systems (Kasagi, Suzuki & Fukagata Reference Kasagi, Suzuki and Fukagata2008). Each of these wall variables is used to predict $v_{10}$ (figure 1) and is used for the control. Since Bewley & Protas (Reference Bewley and Protas2004) and Chevalier et al. (Reference Chevalier, Hœpffner, Bewley and Henningson2006) showed that using more wall variables improved the prediction performance, all three wall variables are also used to predict $v_{10}$ and the results are given in § 4.3. A region on the wall (coloured in yellow) in figure 1 is an example of the sensing region of the wall variable $\chi _w$ whose streamwise and spanwise lengths are approximately 90 wall units. The size of each sensing region is selected considering those of previous studies in which at least 90 wall units in the spanwise direction was required for $\partial w/\partial y \vert _w$ (Lee et al. Reference Lee, Kim, Babcock and Goodman1997), and 90 wall units in the streamwise direction was sufficient for $p_w$ (Yun & Lee Reference Yun and Lee2017). One of the wall variables is the input of the present CNN (see below), and the output is $v_{10}^{pred}$ at the centre location of each sensing region. As we show below, this size is not big enough to include the influence of $v_{10}$ on the wall variables, but is still sufficient to have a high correlation between $v_{10}$ and $v_{10}^{pred}$.

Figure 1.

Schematic diagram on the relation between the predicted $v$ at $y^+=10$ ($v_{10}^{pred}$) and wall-variable distribution with a CNN in a turbulent channel flow. The input ($\chi _w$) of the CNN is one of $p_w$, $\partial u/\partial y \vert _w$ and $\partial w/\partial y \vert _w$, and the output is $v_{10}^{pred}$.

The two-point correlation coefficient $\rho$ between $v_{10}$ and $\chi _w$ in a turbulent channel flow is defined as

(2.1)\begin{equation} \rho \left(\Delta x, \Delta z \right)=\frac{\langle v_{10} (x,z,t)\chi_w (x+\Delta x, z+ \Delta z,t) \rangle} {v_{10,rms}\chi_{w,rms}}, \end{equation}

where $\langle v_{10} \rangle =0$, and $\Delta x$ and $\Delta z$ are the separation distances in the streamwise and spanwise directions, respectively. Figure 2 shows the contours of the two-point correlations for three different flows: ($a$–$c$) uncontrolled flow at $Re_{\tau } = 178$, ($d$–$f$) controlled flow with opposition control (Choi et al. Reference Choi, Moin and Kim1994) at $Re_{\tau } = 178$, and ($g$–$i$) uncontrolled flow at $Re_{\tau } = 578$, where $Re_{\tau } = u_{\tau _o} \delta / \nu$ and $u_{\tau _o}$ is the wall shear velocity of the uncontrolled flow. These correlation contours indicate that there are distinct regions of close relations between $v_{10}$ and $\chi _w$. For the uncontrolled flow at $Re_{\tau } = 178$ (figure 2$a$–$c$), the wall pressure has the highest correlation on the downstream of $v_{10}$, but has the lowest maximum correlation among three wall variables investigated in this study. The streamwise wall shear rate $\partial u/\partial y \vert _w$ has the highest correlation at the upstream of $v_{10}$, whereas the correlation with the spanwise wall shear rate $\partial w/\partial y \vert _w$ is highest at slightly downstream but sideways locations. The two-point correlation is highest for the spanwise wall shear rate, but this correlation magnitude ($\rho = 0.56$) is not high enough to accurately predict $v_{10}$. Also, these correlation contours themselves do not provide how one can construct $v_{10}$ from this information. Hence, in the present study, we construct $v_{10}$ from the wall-variable information in $-45 < \Delta x^+ < 45$ and $-45 < \Delta z^+ < 45$ using a CNN, and discuss how high correlations can be obtained from this approach.

Figure 2.

Contours of the correlation coefficients between $v_{10}$ and $\chi _w$: ($a$–$c$) uncontrolled flow at $Re_{\tau } = 178$; ($d$–$f$) controlled flow at $Re_{\tau } = 178$ by opposition control; ($g$–$i$) uncontrolled flow at $Re_{\tau } = 578$. ($a{,}d{,}g$) $\chi _w = p_w$, ($b{,}e{,}h$) $\chi _w = \partial u /\partial y \vert _w$ and ($c{,}f{,}i$) $\chi _w = \partial w / \partial y \vert _w$. Solid circles at the centre denote the location of $v_{10}$ ($\Delta x = \Delta z = 0$), and cross symbols are the locations of the maximum correlation magnitude. The values of $\rho$ at these locations are given at the bottom of each figure. Here, $\Delta x^+ = \Delta x u_{\tau _o}/\nu$ and $\Delta z^+ = \Delta z u_{\tau _o}/\nu$.

For the controlled flow at $Re_{\tau } = 178$ (figure 2$d$–$f$), the correlations with $p_w$ and $\partial w/\partial y \vert _w$ are very similar to those for the uncontrolled flow. This suggests that a CNN trained with the uncontrolled flow can be applied to predict $v_{10}$ for the controlled flow and also to control the flow in a feedback manner even without requiring training data of the controlled flow. On the other hand, the correlations with $\partial u/\partial y \vert _w$ have opposite signs in many places to those for the uncontrolled flow. This is because the blowing and suction at the wall from opposition control changes $\partial u/\partial y \vert _w$ to be approximately $180^{\circ }$ out-of-phase different from $v_{10}$. For the uncontrolled flow at $Re_{\tau } = 578$ (figure 2$g$–$i$), the correlations are very similar to those at $Re_{\tau } = 178$, as the near-wall flow is well scaled in wall units, which suggests that the CNN trained at a lower Reynolds number should be applicable to the flow at a higher Reynolds number.

Note that near-wall flow structures are significantly changed by opposition control (Choi et al. Reference Choi, Moin and Kim1994; Hammond et al. Reference Hammond, Bewley and Moin1998), and a higher Reynolds number flow contains smaller scales than those at $Re_{\tau } = 178$. Therefore, the success of the present control based on a CNN trained with uncontrolled flow at $Re_{\tau } = 178$ relies on the proper selection of wall sensing variable that maintains a similar correlation coefficient with $v_{10}$ for controlled and higher Reynolds number flows. For the present turbulent channel flow, the wall sensing variables satisfying this requirement are $p_w$ and $\partial w/\partial y \vert _w$, but $\partial u/\partial y \vert _w$ fails to satisfy this requirement. The details of the CNN used are provided in § 2.3. Other machine learning techniques such as the Lasso, random forest and FCNN are also tested, and comparisons of the prediction performance by different machine learning techniques are given in appendix A.

2.2. The dataset

The dataset ($v_{10}^{true}$, $\chi _w$) for training a CNN is obtained from direct numerical simulation of a turbulent channel flow at $Re_{\tau } = 178$, where $v_{10}^{true}= v_{10}$. The governing equations for the continuity and incompressible Navier–Stokes equations are

(2.2)\begin{gather} \frac{\partial {{u}_{i}}}{\partial {{x}_{i}}}=0, \end{gather}

(2.3)\begin{gather} \frac{\partial {u_i}}{\partial t}+\frac{\partial {u_i}{u_j}}{\partial {x_j}}=-\frac{\textrm{d}P}{\textrm{d}{x_1}}{{\delta }_{1i}}-\frac{\partial p}{\partial {x_i}}+\frac{1}{Re}\frac{{{\partial }^{2}}{u_i}}{\partial {x_j}\partial {x_j}}, \end{gather}

where $x_i\, (= (x,y,z))$ are the Cartesian coordinates, $u_i\, (= (u,v,w))$ are the corresponding velocity components, $p$ is the pressure fluctuation, $-\textrm {d}P/\textrm {d}x_1$ is the mean pressure gradient to maintain a constant mass flow rate in a channel. The Reynolds number is $Re= 5600$ based on the bulk velocity ($u_b$) and channel height ($2 \delta$), and is 178 based on the wall shear velocity of the uncontrolled flow ($u_{\tau _o}$) and channel half-height ($\delta$). A semi-implicit fractional step method is used to solve (2.2) and (2.3), where a third-order Runge–Kutta and the Crank–Nicolson schemes are used for the convection and diffusion terms, respectively. For spatial derivatives, the second-order central difference scheme is used. The no-slip condition is applied to the upper and lower walls, and periodic boundary conditions are used in the wall-parallel directions. The computational domain size is $3{\rm \pi} \delta (x) \times 2\delta (y) \times {\rm \pi}\delta (z)$ and the number of grid points is $192(x) \times 129(y) \times 128(z)$. In the wall-normal direction a non-uniform grid is used with ${\Delta }{y^+} \approx 0.2 - 7.0$ (dense grids near the wall). Uniform grids are used in the wall-parallel directions with ${\Delta }{x^+} \approx 8.7$ and ${\Delta }{z^+} \approx 4.4$.

The simulation starts with a laminar velocity profile with random perturbations and continues until the flow reaches a fully developed state. Then, 740 instantaneous fields of $v_{10}^{true}$ and $\chi _w$'s ($=p_w$, $\partial u/\partial y \vert _w$, and $\partial w/\partial y \vert _w$) are stored during $T^+=Tu_{\tau _o}^2/\nu =29\,560$ with an interval of ${\Delta }T^+=40$, where $v_{10}^{true}$ is the label for output of a CNN ($v_{10}^{pred}$), and $\chi _w$'s are the input whose domain size is approximately $90 (l_x^+) \times 90 (l_z^+)$ in wall units (corresponding to $11 \times 21$ grid points, respectively), as shown in figure 1. Here, one instantaneous field contains the information of $\chi _w$'s and $v_{10}^{true}$ at both sides of the channel. The $\chi _w$ and $v_{10}^{true}$ are normalized with their root-mean-square (subscript $rms$) values as

(2.4a,b)\begin{equation} \chi_w^{\ast} =\frac{\chi_w - \langle \chi_w \rangle}{\chi_{w, rms}}, \quad v_{10}^{\ast} =\frac{v_{10}^{true}}{v_{10, rms}^{true}}, \end{equation}

where $\langle \chi _w \rangle$ denotes the mean value of $\chi _w$. The dataset of $\chi _w^{\ast }$ and $v_{10}^{\ast }$ is divided into three sets of different sizes, i.e. training, validation and test sets. Only the training set is used for optimizing a CNN. The validation set is used for checking the optimization process at each training iteration, and the prediction performance is evaluated with the test set after the whole training procedure is finished. We use 700 instantaneous fields (containing 34 406 400 pairs of $\chi _w$'s and $v_{10}^{true}$) for the training, and extract data at every third grid point in the streamwise and spanwise directions (resulting in approximately 3.8 million pairs of $\chi _w$'s and $v_{10}^{true}$), respectively, to exclude highly correlated data. Twenty instantaneous fields (containing 983,040 pairs of $\chi _w$'s and $v_{10}^{true}$) are used for each validation and test set. Here, we use the number of training data of $N_{train} \approx 3.8 \times 10^6$ which is approximately three times that used in the ImageNet large-scale visual recognition challenge (ILSVRC) for developing convolutional neural networks (Krizhevsky, Sutskever & Hinton Reference Krizhevsky, Sutskever, Hinton, Pereira, Burges and Bottou2012; Simonyan & Zisserman Reference Simonyan and Zisserman2014; Szegedy et al. Reference Szegedy, Liu, Jia, Sermanet, Reed, Anguelov, Erhan, Vanhoucke and Rabinovich2014; He et al. Reference He, Zhang, Ren and Sun2015; Russakovsky et al. Reference Russakovsky, Deng, Su, Krause, Satheesh, Ma, Huang, Karpathy, Khosla and Bernstein2015). This is because the present training searches for the spatial correlations of $\chi _w$'s and $v_{10}^{true}$ and, thus, it possesses some similarity with that of image recognition in the ILSVRC, but it may require more training data due to the unsteady characteristics of the present problem than that used in the ILSVRC. In appendix B we show that $N_{train} \approx 3.8 \times 10^6$ is sufficient for the present problem.

2.3. Convolutional neural network

The CNN is a class of neural network, composed of input, hidden and output layers with artificial neurones. The CNN uses a discrete convolution operation with filters to construct the next layer keeping spatially two-dimensional feature maps. Therefore, unlike a FCNN whose inputs to a neurone are outputs from all neurones in the previous layer, local outputs from the previous layer in the CNN are inputs to a neurone, and neurones share the same weights (LeCun et al. Reference LeCun, Boser, Denker, Henderson, Howard, Hubbard, Jackel and Touretzky1989, Reference LeCun, Bengio and Hinton2015). Figure 3 shows the architecture of the CNN used in the present study. We use 17 hidden layers, one average pooling layer and one linear layer adopting a residual block proposed by He et al. (Reference He, Zhang, Ren and Sun2015). For the hidden layers without downsampling, we use a filter size of $3\times 3$ or $5\times 5$, with a stride of 1 for the convolution, where the stride is the magnitude of movement between applications of the filter to the input feature map (Singh & Manure Reference Singh and Manure2019). After the first and second downsampling layers, the height ($h_m$) and width ($w_m$) of the feature maps are reduced by half, and the depth ($d_m$) is doubled, as in He et al. (Reference He, Zhang, Ren and Sun2015). We use a convolution operation with a stride of 2 and a filter size of $2\times 2$ for the first and second downsampling layers. After $h_m$ or $w_m$ of the feature map becomes equal to $h_f$ or $w_f$ of the filter, respectively, we use global average pooling for the last downsampling (average pooling layer in figure 3), where the feature map is averaged while keeping the depth unchanged. After the average pooling layer, the feature map is connected to the linear layer to print out $v_{10}^{pred}$ without an activation function. In the present CNN Relu (Nair & Hinton Reference Nair, Hinton, Fürnkranz and Joachims2010) is used as the activation function, and a batch normalization (Ioffe & Szegedy Reference Ioffe and Szegedy2015) is applied after each convolution operation. All weights ($w_j$) in the filters are initialized by the Xavier method (Glorot & Bengio Reference Glorot, Bengio, Teh and Titterington2010), and they are optimized to minimize a given loss function defined as

(2.5)\begin{equation} L = \frac{1}{2N}\sum_{i=1}^{N}{{{\left( \frac{v_{10\, i}^{pred}-v^{true}_{10 \,i}}{v^{true}_{10, rms}} \right)}^{2}}} + 0.025\sum_{j}{w_j^2}, \end{equation}

where $N$ is the number of mini-batch data (256 in this study following He et al. Reference He, Zhang, Ren and Sun2015). An adaptive moment estimation (Kingma & Ba Reference Kingma and Ba2014), which is a variant of gradient descent, is used for updating the weights, and the gradients of the loss function with respect to the weights are calculated through the back-propagation algorithm (Rumelhart, Hinton & Williams Reference Rumelhart, Hinton and Williams1986). We conduct early stopping to prevent overfitting (Bengio Reference Bengio2012). There are many user-defined parameters in constructing a CNN. A study on these parameters is conducted and its results are given in appendix B.

Figure 3.

Architecture of the CNN used in the present study. Each box and arrow after the input and before the average pooling layer represent a hidden layer and flow of the feature maps, respectively. Dimensions of the feature maps, denoted as [height ($h_m$), width ($w_m$), depth ($d_m$)], are given next to the arrows, and the size and number of filters ($h_f\times w_f\times d_{input}$, $d_{output}$, respectively) are given inside each box. The $h_m$ and $w_m$ are the numbers of grid points of the feature maps in the $z$ and $x$ directions, respectively. The $h_f$ and $w_f$ are the numbers of filter weights in the $z$ and $x$ directions, respectively. Zero paddings are used to adjust the sizes of $h_m$ and $w_m$ of the feature maps after convolution operations. Grey-coloured boxes are the downsampling layers. A residual block without a downsampling layer (lower left figure) consists of two hidden layers, and its output is the sum of the output from the last hidden layer $f(x)$ and the input of the residual block $x$. For a residual block with a downsampling layer (lower right figure), its output is the sum of the output from the last hidden layer $f(x)$ and the downsampled input $x^{\ast }$, where downsampling ($D^{\ast }$) is carried out with the same filter size and stride as those of the downsampling layer ($D$). For downsampling ($D$ and $D^{\ast }$), zero padding is applied on the bottom row or right column of a feature map when $h_m$ or $w_m$ of the input $x$ is an odd number.

3.1. Multiple input (spatial distribution of $\chi _w$) and single output ($v_{10}^{pred}$ at a point)

The correlation coefficients between $v_{10}^{true}$ and $v_{10}^{pred}$'s by the CNN with $\chi _w=p_w$, $\partial u/\partial y \vert _w$ and $\partial w/\partial y \vert _w$ are $\rho _{v_{10}} = 0.95$, 0.90 and 0.95, respectively, where $\rho _{v_{10}} = \langle v_{10}^{true}(x,z,t)v_{10}^{pred}(x,z,t) \rangle / ({v_{10,rms}^{true}}{v_{10,rms}^{pred}})$. These magnitudes are much bigger than the maximum two-point correlations described before ($\rho = 0.36$, 0.50 and 0.56, respectively) and also those from other machine learning techniques considered (appendix A). Figure 4 shows the instantaneous fields of $v_{10}^{true}$ and $v_{10}^{pred}$'s reconstructed by the CNN, together with $\chi _w$'s. Although the distributions of $\chi _w$'s are very different from that of $v_{10}^{true}$, the CNN captures most of the $v_{10}$ field from all the wall variables investigated, indicating that the CNN is an adequate tool to predict $v_{10}$. To understand how $v_{10}^{pred}$ is correlated with $\chi _w$, we compute the saliency map proposed by Simonyan, Vedaldi & Zisserman (Reference Simonyan, Vedaldi and Zisserman2013), and provide the results in appendix C.

Figure 4.

Contours of the instantaneous $v_{10}^{true}$, $v_{10}^{pred}$'s and instantaneous $\chi _w$'s: ($a$) $v_{10}^{true}$ (DNS); ($b$) $v_{10}^{pred}$ from $\chi _w=p_w$; ($c$) $v_{10}^{pred}$ from $\chi _w=\partial u/\partial y \vert _w$; ($d$) $v_{10}^{pred}$ from $\chi _w=\partial w/\partial y \vert _w$; ($e$) $p_w$; ($f$) $\partial u/\partial y \vert _w$; ($g$) $\partial w/\partial y \vert _w$.

3.2. Multiple input and multiple output (spatial distributions of $\chi _w$ and $v_{10}^{pred}$)

Although the CNN in § 3.1 performed well, the reconstructed flow field $v_{10}^{pred}$ (figure 4) contained spatial oscillations that might provide numerical instability during feedback control. To understand the source of these oscillations, we (i) try even numbers of grid points $(24 \times 12)$ for $\chi _w$ to see if they came from zero paddings at the downsampling layers owing to the use of odd numbers $(21 \times 11)$; (ii) use a continuous activation function, $y = \tanh (x)$, since we used a discontinuous activation function (Relu), $y = \max (0, x)$; (iii) apply a linear regression model (Lasso) with $21 \times 11$ grid points for $\chi _w$. The spatial oscillations in $v_{10}^{pred}$ still exist for (i) and (ii), but disappear for (iii) (not shown in this paper). This may indicate that the spatial oscillations in $v_{10}^{pred}$ occur because it is nonlinearly determined with $\chi _w$ by the CNN. Therefore, to obtain a smoother distribution of $v_{10}^{pred}$ in space, we consider another CNN in this section in which multiple output (a spatial distribution of $v_{10}^{pred}$) is produced from multiple input (a spatial distribution of $\chi _w$). We call this CNN an MP-CNN, whereas the CNN in § 3.1 is called 1P-CNN.

Figure 5 shows the schematic diagrams of 1P-CNN and MP-CNN. For MP-CNN, we keep the architectures of all hidden layers of 1P-CNN (17 hidden layers), and then add three additional hidden layers. The sizes of the input wall variable $\chi _w$ and output $v_{10}^{pred}$ are $l^+_x\times l^+_z\approx 270\times 135$ and $130\times 65$, respectively, and the corresponding numbers of grid points for the input and output are $32\times 32$ and $16\times 16$, respectively. The centre positions of the input and output are the same. The input size in space should be taken to be larger than the output size, because $v_{10}$ at a point is correlated with the wall variables nearby. As shown in figure 2, the maximum correlations between $v_{10}$ and $\chi _w$'s occur at $|\Delta x^+| \le 45$ and $|\Delta z^+| \le 15$, and, thus, the input size, which is twice the output size, should be enough to produce high performance of MP-CNN. The choice of the output size, $l^+_x\times l^+_z \approx 130\times 65$, is rather arbitrary, but this size is at least comparable to the size of a region of rapidly varying $v_{10}$ (see, for example, figure 4). A dataset of $\chi _w$ and $v_{10}^{true}$ are obtained from direct numerical simulation of a turbulent channel flow as before. We apply the generative adversarial networks (GAN; Goodfellow et al. Reference Goodfellow, Pouget-Abadie, Mirza, Xu, Warde-Farley, Ozair, Courville and Bengio2014) to optimize MP-CNN, because previous studies (Ledig et al. Reference Ledig, Theis, Huszár, Caballero, Cunningham, Acosta, Aitken, Tejani, Totz and Wang2016; Lee & You Reference Lee and You2019) showed that a CNN trained with GAN produces more realistic images than using only the quadratic error as a loss function. The details about GAN and loss function are described in appendix D.

Figure 5.

Schematics of the present convolutional neural networks: ($a$) 1P-CNN; ($b$) MP-CNN. The detail of 1P-CNN is given in figure 3. For MP-CNN, the size and number of filters are given in each box, and the dimensions of the feature maps are given next to each arrow. The grey-coloured box is the deconvolution layer where the height and width of the feature map increase twice.

Figure 6 shows $v_{10}$, $\partial v_{10} / \partial x$ and $\partial v_{10} / \partial z$ from 1P-CNN and MP-CNN with $\chi _w=\partial u/\partial y \vert _w$, respectively, together with those from DNS. The correlation coefficients between the true (DNS) and predicted values with MP-CNN are $\rho = 0.92$, 0.87 and 0.91 for $v_{10}$, $\partial v_{10} / \partial x$ and $\partial v_{10} / \partial z$, respectively, whereas those with 1P-CNN are $\rho = 0.90$, 0.81 and 0.89, respectively. The results of the correlation coefficients and reconstructed fields (figure 6) indicate that the prediction performance is improved both quantitatively and qualitatively with MP-CNN. Note that oscillations observed with 1P-CNN nearly disappear with MP-CNN. For $\chi _w=p_w$, the correlation coefficients for $v_{10}$, $\partial v_{10} / \partial x$ and $\partial v_{10} / \partial z$ are $\rho = 0.96$, 0.92 and 0.96 with MP-CNN, respectively, whereas those with 1P-CNN are $\rho = 0.95$, 0.89 and 0.95, respectively. For $\chi _w=\partial w/\partial y \vert _w$, $\rho = 0.96$, 0.90 and 0.96 with MP-CNN, whereas $\rho = 0.95$, 0.89 and 0.95 with 1P-CNN. The reconstructions with $\chi _w=p_w$ and $\partial w/\partial y \vert _w$ show results similar to those with $\chi _w=\partial u/\partial y \vert _w$.

Figure 6.

Contours of the instantaneous $v_{10}, \partial v_{10} / \partial x$ and $\partial v_{10} / \partial z$: ($a$–$c$) $v_{10}$; ($d$–$f$) $\partial v_{10} / \partial x$; ($g$–$i$) $\partial v_{10} / \partial z$. ($a{,}d{,}g$) are from DNS, and ($b{,}e{,}h$) and ($c{,}f{,}i$) are from 1P-CNN and MP-CNN with $\chi _w =\partial u/\partial y \vert _w$, respectively. Here, $\partial v_{10} / \partial x$ and $\partial v_{10} / \partial z$ are calculated using the second-order central difference. Flow variables are normalized with $u_{\tau _o}$ and $\delta$.

Figure 7 shows the streamwise and spanwise energy spectra of $v_{10}^{pred}$ from $\chi _w = p_w,$$\partial u/\partial y \vert _w$ and $\partial w/\partial y \vert _w$, together with those of $v_{10}^{true}$. Overall, both 1P-CNN and MP-CNN predict the energy spectra very well. At high wavenumbers, 1P-CNN exhibits severe energy pile up both in the streamwise and spanwise wavenumbers, whereas MP-CNN reduces the energy pile up in the streamwise wavenumber and matches the spanwise energy spectrum nearly perfectly at all wavenumbers. This indicates that small-scale motions of $v_{10}$ is better predicted by MP-CNN than by 1P-CNN.

Figure 7.

Energy spectra of $v_{10}$ from $\chi _w = p_w,$$\partial u/\partial y \vert _w$, and $\partial w/\partial y \vert _w$ (uncontrolled flow): ($a$–$c$) streamwise wavenumber; ($d$–$f$) spanwise wavenumber. $(a,d)\,\chi _w = p_w$; $(b,e)\,\chi_w=\partial u/\partial y \vert _w; (c, f)\,\chi_w=\partial w/\partial y \vert _w$. Black circle, $v_{10}^{true}$ (DNS); black line, $v_{10}^{pred}$ with 1P-CNN; red line, $v_{10}^{pred}$ with MP-CNN.

An additional advantage of MP-CNN is a significant reduction of the computational cost. Total of $192 (x) \times 128 (z)$ prediction processes should be required to reconstruct an entire $v_{10}$ field with 1P-CNN, where one prediction process means one operation of the CNN to print out $v_{10}^{pred}$ with given $\chi _w$. Because the number of the grid points of $v_{10}^{pred}$ is $16(x) \times 16(z)$ for MP-CNN, it requires only $192/16 (x) \times 128/16 (z)$ prediction processes to reconstruct an entire field. Although the computational cost for one prediction process is greater for MP-CNN due to larger input and output sizes than for 1P-CNN, the computational time required to reconstruct the entire field is approximately 40 times smaller with MP-CNN than that with 1P-CNN.

4.1. Control with $v_{10}^{pred}$

Figure 8 shows the scatter plots of $v_{10}^{true}$ (from opposition control) and $v_{10}^{pred}$ by MP-CNN trained with the uncontrolled flow. The MP-CNN trained with the uncontrolled flow completely loses its prediction performance for $\chi _w=\partial u / \partial y \vert _w$ (figure 8$b$), whereas it still maintains approximately linear relations (but with slopes different from 1) for $\chi _w=p_w$ and $\partial w / \partial y \vert _w$. Therefore, we modify the magnitude of $v_{10}^{pred}$ at each control time step such that the control becomes

(4.1)\begin{equation} v_w=-\sigma v_{10}^{pred}\quad \text{with}\ \sigma = 0.5v_{10, rms}^{true} (\text{uncontrolled}) / v_{10, rms}^{pred}, \end{equation}

because $v_{10, rms}^{true}\,(\textrm {controlled}) \approx 0.5 v_{10, rms}^{true}\,(\textrm {uncontrolled})$ under opposition control (Kim & Choi Reference Kim and Choi2017). Instead of using $\sigma$ in (4.1), the fitting lines given in figure 8 may be used to determine $\sigma$. However, as these relations are a priori unknown in practical situations, we use a simple relation (4.1) for the control. The correlation coefficients between $v_{10}^{true}$ (controlled) and $\sigma v_{10}^{pred}$'s from $\chi _w =p_w$, $\partial u/\partial y \vert _w$ and $\partial w/\partial y \vert _w$ are $\rho _{v_{10}} = 0.85$, $-0.08$ and 0.84, respectively, and they are lower than those from the uncontrolled flow (0.96, 0.92 and 0.96, respectively). This is expected because the current MP-CNNs are trained with the uncontrolled flow. Nevertheless, those correlation coefficients for $\chi _w=p_w$ and $\partial w/\partial y \vert _w$ are high enough to reconstruct $v_{10}$ even for the controlled flow (figure 9). Therefore, we perform a feedback control based on (4.1) and $v_{10}^{pred}$ by MP-CNN trained with the uncontrolled flow. Note that when MP-CNN is trained with the controlled flow, $\rho _{v_{10}} = 0.97$, 0.98 and 0.98 for $\chi _w =p_w$, $\partial u/\partial y \vert _w$ and $\partial w/\partial y \vert _w$, respectively.

Figure 8.

Scatter plots of $v_{10}^{true}$ and $v_{10}^{pred}$: ($a$) $\chi _w = p_w$; ($b$) $\chi _w = \partial u/\partial y \vert _w$; ($c$) $\chi _w =\partial w/\partial y \vert _w$. Here, $\chi _w$'s and $v_{10}^{true}$ are from the controlled flow with original opposition control (Choi et al. Reference Choi, Moin and Kim1994), while MP-CNN is trained with the uncontrolled flow. A black line in each figure denotes the slope of 1, and a red line is a fitting line for each scatter plot and is given at the bottom of each figure.

Figure 9.

Contours of the instantaneous $v_{10}$ from the controlled flow by opposition control: ($a$) $v_{10}^{true}$; ($b$) $\sigma v_{10}^{pred}$ from $\chi _w = p_w$; ($c$) $\sigma v_{10}^{pred}$ from $\chi _w =\partial w/\partial y \vert _w$.

Figure 10 shows the time histories of the mean pressure gradient for the controls based on MP-CNNs with $\chi _w = p_w$ and $\partial w/ \partial y \vert _w$, together with those based on a neural network (NN). Here, we show the results from two different MP-CNNs, one trained with the uncontrolled flow and the other trained with the controlled flow by opposition control, respectively. For the first MP-CNN, we apply $v_w = - \sigma v_{10}^{pred}$, and for the latter, $v_w = - v_{10}^{pred}$. The NN considered has one hidden layer and one neurone (the output of this NN is one point $v_{10}$), which is the same model as that of Lee et al. (Reference Lee, Kim, Babcock and Goodman1997). With opposition control, the drag is reduced by approximately 20 % from that of the uncontrolled flow. Note that this amount of drag reduction is smaller than 25 % reported by Choi et al. (Reference Choi, Moin and Kim1994). This difference may come from a numerical set-up such as the spatial discretization method and grid resolution. In particular, Chang et al. (Reference Chang, Collis and Ramakrishnan2002) and Chung & Talha (Reference Chung and Talha2011) also reported approximately 20 % drag reduction with $v_w = - v_{10}$ at the same Reynolds number.

Figure 10.

Time histories of the mean pressure gradient in a turbulent channel flow at $Re_{\tau } = 178$: ($a$) MP-CNN and NN trained with the uncontrolled flow; ($b$) MP-CNN trained with the controlled flow.

When MP-CNN trained with the controlled flow is applied, the drag is reduced by 11 % and 16 % with $\chi _w = p_w$ and $\partial w/\partial y \vert _w$, respectively (figure 10b). By the NN trained with the controlled flow, $v_{10}^{pred}$ continues to grow with $v_w = - v_{10}^{pred}$, and the simulation eventually diverges unless the magnitude of $v_w$ is forced to be smaller than a predetermined constant. In the case of MP-CNN trained with the uncontrolled flow, the amounts of drag reduction are 10 % and 15 % with $\chi _w = p_w$ and $\partial w/\partial y \vert _w$, respectively (figure 10a), which are slightly lower than those by MP-CNN trained with the controlled flow. This result indicates an excellent capability of reducing drag by the present MP-CNN trained with the uncontrolled flow. By the NN trained with the uncontrolled flow, the amounts of drag reduction are approximately 0 % and 12 %, respectively, for $\chi _w = p_w$ and $\partial w/\partial y \vert _w$, suggesting that the control performance of the NN depends more on the choice of the input variable than that of MP-CNN, and is not better than that of MP-CNN. We also conduct controls using MP-CNN trained with the uncontrolled flow for $\chi _w =\partial u/\partial y \vert _w$, and obtain 5 % drag reduction. Although $v_{10}^{pred}$ with $\chi _w = \partial u/\partial y \vert _w$ is not quite similar to $v_{10}^{true}$, drag reduction still occurs albeit its small amount. On the other hand, the amounts of drag reduction with $\chi _w = p_w$ and $\partial w/\partial y \vert _w$ are different from each other, even though their MP-CNN's have similar prediction performance for the controlled flow. This is due to the different sensitivity of $p_w$ and $\partial w/\partial y \vert _w$ to the wall actuation. That is, as shown in figure 8, the slope of a fitting line for $\chi _w=\partial w/\partial y \vert _w$ is closer to 1 than that for $\chi _w = p_w$. Also, $v_{10, rms}^{pred}/v_{10, rms}^{true}$ from $\chi _w = \partial w/\partial y \vert _w$ is 1.6, but it is 2.2 from $\chi _w = p_w$.

4.2. Improving the control performance by filtering small to intermediate scales

Although the control based on the present MP-CNN performs quite well, the drag-reduction performance is still lower than that of opposition control. In this section we explain the reason for this lower drag reduction by MP-CNN and suggest a way to improve the drag-reduction performance.

Figure 11 shows the streamwise and spanwise energy spectra of $v_{10}^{true}$ and $v_{10}^{pred}$'s for the controlled flow by MP-CNN trained with the uncontrolled flow. As shown, the energy spectra of $v_{10}^{pred}$'s for $p_w$ and $\partial w/\partial y \vert _w$ at low wavenumbers are quite similar to those of $v_{10}^{true}$, but the energy spectra of $v_{10}^{pred}$ for $\partial u/\partial y \vert _w$ at all wavenumbers and for $p_w$ and $\partial w/\partial y \vert _w$ at intermediate to high wavenumbers are quite different from those of $v_{10}^{true}$. Therefore, the intermediate to high wavenumber components of $v_{10}^{pred}$ may degrade the drag-reduction performance of the MP-CNN control. To test this conjecture, we remove some length scales of $v_{10}^{pred}$ by applying three different low-pass filters, where the cut-off wavenumbers are $(k_{x,c}^+, k_{z,c}^+) \approx (0.150, 0.540)$, (0.075, 0.270) and (0.038, 0.135), respectively (hereafter, $v_{10}$ with a low-pass filter is called $\tilde v_{10}$). The opposition control, $v_w=-\tilde {v}_{10}^{true}$, with the smallest cut-off wavenumbers $(k_{x,c}^+, k_{z,c}^+) = (0.038, 0.135)$ provides 18 % drag reduction, as opposed to 20 % drag reduction by the control with all the wavenumber components. Lorang et al. (Reference Lorang, Podvin and Le Quéré2008) also showed that opposition controls with the POD- or Fourier-truncated $v_{10}$ provided 15 % and 8 % drag reductions, respectively, where the first streamwise and first three spanwise modes of the POD and Fourier coefficients were used. The control by MP-CNN with a low-pass filter is $v_w = -\tilde {\sigma }\tilde {v}_{10}^{pred}$, where $\tilde \sigma = 0.5 v_{10, rms}^{true} (\text {uncontrolled})/\tilde {v}_{10, rms}^{pred}$. Table 1 shows the variation of the drag-reduction rate with the low-pass filter, together with that of opposition control (Choi et al. Reference Choi, Moin and Kim1994). The low-pass filter at $(k_{x,c}^+, k_{z,c}^+) = (0.038, 0.135)$ enhances the control performance for all three input wall variables. Especially, the amount of drag reduction by $\partial w/\partial y \vert _w$ is quite comparable (18 %) to that by opposition control. Although the controls with low-pass filters at higher cut-off wavenumbers are less effective, the amounts of drag reduction are still meaningfully large. These results indicate that the intermediate to high wavenumber components of $v_{10}^{pred}$ degrade the drag-reduction performance, and an elimination of those components enhances the drag-reduction performance.

Figure 11.

Energy spectra of $v_{10}$ from the controlled flow by MP-CNN: ($a$) streamwise wavenumber; ($b$) spanwise wavenumber. Here, $v_{10}^{pred}$ and $\chi _w$'s are from the controlled flow by MP-CNN trained with the uncontrolled flow, whereas $v_{10}^{true}$ is from the controlled flow by opposition control (Choi et al. Reference Choi, Moin and Kim1994). Dashed lines in ($a$) denote $k_{x}^+ = 0.038, 0.075$ and 0.150, and those in ($b$) correspond to $k_{z}^+ = 0.135$, 0.270 and 0.540, respectively.

Table 1.

Variation of the drag-reduction rate (DR) by MP-CNN with the low-pass filter applied to $v_{10}^{pred}$, together with that by opposition control. $k_{x,c}^+$ and $k_{z,c}^+$ are the streamwise and spanwise cut-off wavenumbers, respectively.

4.3. Control based on all three wall-variable sensing

In this section we train MP-CNN using all three wall variables ($p_w$, $\partial u/\partial y \vert _w$ and $\partial w/\partial y \vert _w$) with the uncontrolled flow at $Re_{\tau } = 178$ (called MP-CNN3 hereafter) instead of using one of them as the input. The only difference in this MP-CNN3 from MP-CNN is the input size, i.e. $32 \times 32 \times 3$ instead of $32 \times 32$. For the uncontrolled flow, the correlation from MP-CNN3 is $\rho _{v_{10}}= 0.99$, which is higher than those from MP-CNN using single $\chi _w$ ($\rho _{v_{10}} = 0.96$, 0.92 and 0.96 for $\chi _w = p_w$, $\partial u/\partial y \vert _w$ and $\partial w/\partial y \vert _w$, respectively), indicating that more input wall variables provide a higher correlation with $v_{10}^{true}$ for the uncontrolled flow. On the other hand, for the controlled flow, the scatter plot of $v_{10}^{true}$ and $v_{10}^{pred}$ from MP-CNN3 (figure 12a) demonstrates that its correlation ($\rho _{v_{10}}= 0.83$) is very similar to those ($\rho _{v_{10}}= 0.85$ and 0.84) from MP-CNNs with $\chi _w = p_w$ and $\partial w/\partial y \vert _w$, respectively, but its slope of the fitting line is larger than that from MP-CNN with $\chi _w = \partial w/\partial y \vert _w$ and slightly smaller than that with $\chi _w = p_w$ (see also figure 8).

Figure 12.

Scatter plot of $v_{10}^{true}$ and $v_{10}^{pred}$ from MP-CNN3 (controlled flow) and mean pressure gradient at $Re_{\tau } = 178$: ($a$) scatter plot; ($b$) mean pressure gradient. In ($a$) a black line denotes the slope of 1, and a red line is a fitting line for the scatter plot and is given at the bottom of the figure. In ($b$), black line, uncontrolled flow; black dashes, opposition control (Choi et al. Reference Choi, Moin and Kim1994); red line, control by MP-CNN3; blue dashes, control by MP-CNN with $p_w$; blue line, control by MP-CNN with $\partial w/\partial y \vert _w$.

We apply MP-CNN3 to the control with $v_w=-\sigma v_{10}^{pred}$ (4.1). The amount of drag reduction by this MP-CNN3 is 12 %, which is higher and lower than those obtained by MP-CNNs trained with $\chi _w =p_w$ and $\partial w/\partial y \vert _w$, respectively (figure 12b). This result indicates that a good prediction of the correlation by a CNN for uncontrolled flow does not guarantee a good performance in the present feedback control. A similar inconsistency has been also observed with a priori and a posteriori tests of a subgrid-scale model in large eddy simulation (Park, Yoo & Choi Reference Park, Yoo and Choi2005).