esa-sceva/satcom-chunk-collection huggingface dataset:
# AWEncoder: Adversarial Watermarking Pre-Trained Encoders in Contrastive Learning
## 3 Proposed Method
### Watermark Verification  
When the defender verifies whether the suspicious encoder infringes the intellectual property of the watermarked encoder, white-box scenario and black-box scenario can be considered. In the white-box scenario, the defender has the access to the target encoder \(E_{\theta}^{-}\approx E_{\theta}^{+}\). Namely, the defender can directly obtain the output of \(E_{\theta}^{-}\) for watermark verification. We use the average Jensen-Shannon (JS) divergence between a set of clean images and the corresponding adversarial images for similarity analysis:  
\[T_{\text{sim}}=1-\frac{1}{|\mathcal{D}^{\prime\prime}|}\sum_{i=1}^{|\mathcal{D }^{\prime\prime}|}\text{JS}(\sigma(E_{\theta}^{-}(\mathbf{x}_{i}^{\prime \prime})),\sigma(E_{\theta}^{-}(\mathbf{x}_{i}^{\prime\prime}+\mathbf{w}_{ \text{adv}}))), \tag{6}\]  
where \(\mathcal{D}^{\prime\prime}\) is the clean dataset used for watermark verification. The watermark will be successfully verified if \(T_{\text{sim}}\) is less than a threshold \(t_{s}\), namely, we hope to keep \(T_{\text{sim}}\) as high as possible. In our experiments, \(|\mathcal{D}^{\prime\prime}|\) is set to 1000 by default.  
In a black-box scenario, given a suspicious downstream model \(\mathcal{M}\), the defender wants to verify whether \(\mathcal{M}\) is developed from \(E_{\theta}^{+}\). The defender should build a clean dataset \(\mathcal{D}^{*}\) related to the downstream task. Then, the classification performance for the downstream task is analyzed by:  
\[T_{\text{cls}}=1-\frac{1}{|\mathcal{D}^{*}|}\sum_{i=1}^{|\mathcal{D}^{*}|} \mathbb{I}[\mathcal{M}(\mathbf{x}_{i}^{*})\
eq\mathcal{M}(\mathbf{x}_{i}^{*}+ \mathbf{w}_{\text{adv}})], \tag{7}\]  
where function \(\mathbb{I}\) returns 1 when the inside condition is true, or 0 otherwise. The watermark will be successfully verified if \(T_{\text{cls}}\) is less than a threshold \(t_{c}\), namely, we hope to keep \(T_{\text{cls}}\) as high as possible. In our experiments, \(|\mathcal{D}^{*}|\) is set to 1000 by default.  
**Remark 1**.: _As with many existing adversarial attacking methods, \(\mathbf{w}_{\text{adv}}\) is generated by using back-propagation. The strength of \(\mathbf{w}_{\text{adv}}\) is constrained by a threshold \(\epsilon\). To ensure that \(\|\mathbf{w}_{\text{adv}}\|_{\infty}\leq\epsilon\), \(\mathbf{w}_{\text{adv}}\) is projected on the \(\ell_{\infty}\) norm-ball around \(\mathbf{x}_{i}\) with radius \(\epsilon\), we set \(\ell_{\infty}\)-norm and \(\epsilon=15\) by default. With this perturbation strength, it can effectively cluster the perturbed images around the key image \(\mathbf{x}_{\text{tar}}\) in the embedding space. The implementation can be found in the released source code._
## 4 Results and Discussion
### Baseline Methods  
In order to assess the performance of the proposed EDRL model, the existing methods, such as random forest (RF), CNN, GRU, and DQN, are employed and simulated to make a fair comparison. The selected models indicate a diverse set of machine and deep learning paradigms, each with distinct strengths in handling network data. The parameters utilized in the comparison experiment are associated with the settings stated in the original publications of the respective methods, guaranteeing fairness in the comparison. The summary of these existing methods is discussed below as follows:  
RF: The RF classification is an ensemble method that trains several decision trees constantly using bootstrapping, averaging, and bagging. Several independent decision trees can be built simultaneously on various training sample segments by using different subsets of accessible features. The RF variance is reduced since the decision tree within the random forest is unique. The RF classifier has a significant generalization since it integrates many decision trees to reach its final decision. Also, it offers interpretability and robustness and is particularly fit for structured network data.  
CNN: Learning appropriate feature representations of the input data is the aim of CNN, a neural network variant. The two primary differences between CNNs and MLPs are weight sharing and pooling. Numerous convolution kernels, which are used to create various feature maps, can make up each layer of a CNN. A neuron in the feature map of the next layer is related to each region of nearby neurons. Moreover, the kernel is shared by all input spatial locations in order to create the feature map. One or more fully connected layers are utilized for classification after a few convolution and pooling layers. CNNs are useful for network traffic representations owing to their exceptional capability to capture spatial and hierarchical patterns.  
GRU: One innovative memory cell that has been shown to work well in a range of applications is a GRU. A GRU can perform similarly to LSTM and can be seen as an upgrade and simplicity of LSTM. The GRU has fewer gates than the LSTM. This is due to the factthat the GRU lacks a cell state and merges the input and forget gates into an update gate. Therefore, the GRU has a significant advantage over the LSTM in terms of performance and convergence because it is structurally simpler and uses fewer parameters.  
DQN: DQN is a multi-layered neural network that generates a vector of action values for a given state. The deployment of a target network and experience replay are two crucial components of the DQN algorithm. With the exception of copying its parameters at each step from the online network, the target network, with parameters \(\theta-\), is identical to the online network. In order to update the network for the experience replay, observed transitions are evenly sampled from this memory bank after being kept for some time. The algorithm's performance is significantly enhanced by both the target network and the experience replay.
## 3 Results
### Relationship between Topological Indicators and Carbon Sequestration Capacity of Ecological Nodes  
The results of the correlation matrix between topological indicators and NBP for ecological nodes of all vegetation types showed that only the clustering coefficients were significantly correlated with NBP (\(p<0.05\), Figure 13a). However, even at a larger regional scale, we believe that the plant functional types are still the decisive factor in productivity. Therefore, the results of the correlation matrix after the ecological nodes were divided into forest nodes and grassland nodes according to vegetation type showed that the betweenness centrality of forest nodes was significantly correlated with NBP (\(p<0.05\)), and the clustering coefficient of grassland nodes was significantly correlated with NBP (\(p<0.01\), Figure 13a). The results of the subsequent linear regression analysis showed that the betweenness centrality of forest nodes showed a negative linear correlation with NBP (\(R^{2}=0.59\), \(p<0.05\); Figure 13b), and the clustering coefficient of grassland nodes showed a positive linear correlation with NBP (\(R^{2}=0.49\), \(p<0.01\); Figure 13c).  
Betweenness centrality describes the proportion of ecological nodes present on the shortest path of any two nodes in the network, and it measures the intermediary properties of ecological nodes. Some studies refer to nodes (regions) with high betweenness properties as \"stepping stones\" [38]. For the forest nodes in the YRB, as the betweenness centrality increased, the NBP decreased linearly. In the process of decreasing NBP, a value of 0 represented the cutoff point for the forest node being in the state of carbon source or carbon sink. In other words, as the net carbon sequestration capacity of the forest ecosystem decreases, the carbon sink node will be transformed into a carbon source node. The clustering coefficient reflects the proportion of connectivity between neighboring nodes around a node. For the grassland nodes in the YRB, the NBP gradually increased as the clustering coefficient increased. This means that the net carbon sequestration capacity of the grassland ecosystem will gradually increase, and the carbon source will be transformed into a carbon sink.  
Figure 12: Comparison of NBP and NPP trends for all vegetation nodes; \(r\) indicates the Pearson’s correlation coefficient.
## 3 Description of Numerical Algorithms
### Simultaneous Vector Iteration Methods  
Simultaneous vector iteration or subspace method is a vector iteration method that calculates multiple eigenvectors simultaneously in one iteration. Suppose that we want to compute the \(p\) LM eigenvalues of the LEP (8) with pencil \(s\mathbf{I}_{r}-\mathbf{A}\). Extending the idea of the power method, consider the subsequence (11), where now \(\mathbf{b}\) is not a vector, but \(\mathbf{b}\in\mathbb{C}^{r\times p}\). As it is, the subsequence does not converge, as we would desire, to a matrix with columns the \(p\) eigenvectors of \(s\mathbf{I}_{r}-\mathbf{A}\). However, convergence to these eigenvectors can be achieved by ensuring that the columns of the \(k\)-th element \(\mathbf{A}^{k}\mathbf{b}\) are orthonormal vectors. Subsequently, we say that the columns of \(\mathbf{A}^{k}\mathbf{b}\) span the subspace:  
\[\mathcal{V}_{p}=\mathrm{span}\{\mathbf{A}^{k}\mathbf{b}\}\;. \tag{17}\]  
The steps of the subspace iteration algorithm are the following.  
1. The iteration starts with an initial matrix \(\mathbf{b}^{(0)}\), \(\mathbf{b}^{(0)}\in\mathbb{C}^{r\times p}\), with orthonormal columns, i.e., \((\mathbf{b}^{(0)})^{\mathrm{H}}\mathbf{b}=\mathbf{I}_{p}\).
2. At the \(k\)-th step of the iteration, \(k=1,2,3,\ldots\): 1. Matrix \(\mathbf{C}\), \(\mathbf{C}\in\mathbb{C}^{r\times p}\), is formed as: \[\mathbf{C}^{(k)}=\mathbf{A}\mathbf{b}^{(k-1)}\;.\] 2. Matrix \(\mathbf{b}^{(k)}\) is updated by maintaining column-orthonormality, through the QR decomposition of matrix \(\mathbf{C}\): \[\begin{split}\mathbf{C}^{(k)}&=\mathbf{Q}\mathbf{R} \;,\\ \mathbf{b}^{(k)}&=\mathbf{Q}\;,\end{split}\] (19) where \(\mathbf{Q}\) is a unitary matrix and \(\mathbf{R}\) is an upper triangular matrix.  
The columns of \(\mathbf{b}^{(k)}\) eventually converge to \(p\) right eigenvectors that correspond to the \(p\) LM eigenvalues.  
Note that the LM eigenvalues may not be the important ones for the needs of SSSA. Similarly to the discussion of the inverse power method, finding the eigenvalues of smallest magnitude is possible by forming an inverse subspace iteration.  
The convergence of the subspace iteration is, in general, slow and, thus, its use is avoided for complex problems. Still, acceleration of the speed of convergence is possible by combining the method with the Rayleigh-Ritz procedure. Because the utility of the Rayleigh-Ritz procedure goes far beyond the subspace iteration, we describe it separately in Appendix A.
## 3 Suppression Methods for Low-Frequency Pressure Fluctuation
### Vortex Generators
#### 3.1.3 Corner Vortex Generator  
[PERSON] et al. [23; 49] recently discovered that the placement of vortex generators at the corners of the rectangular nozzle in the LAWT wind tunnel effectively suppressed low-frequency pressure fluctuations. In contrast, placing vortex generators at the center of the inner wall of the rectangular nozzle increased the low-frequency pressure fluctuations. In response to these findings, [PERSON] et al. [23; 49] designed a corner vortex generator (CVG) with a circular upper surface and a flat rectangular lower surface (shown in Figures 10 and 11), which was able to control the root-mean-square (RMS) level of low-frequency pressure  
Figure 8: Schematic of passive rectangular flaps for the S2A wind tunnel nozzle.  
Figure 9: Schematic of the three-dimensional vortex generator FKFA besst.  
fluctuations in the test section of the LAWT wind tunnel to less than 0.4% and maintain favorable acoustic characteristics [23].  
By using numerical simulations, [PERSON] [49] found that the vortices generated by the CVG at the corners of the square jet have a pumping effect on the turbulence at the lateral centers of the jet. As a consequence, the jet expansion angle decreased, enabling the collector to wrap more effectively, as shown in Figure 12. As will be discussed in Section 3.2.2 of this paper, increasing the inlet size of the collector can effectively mitigate the impact of the transverse standing wave. The oscillation caused by the width-direction standing wave of the plenum was suppressed by reducing the jet expansion angle, which is equivalent to widening the collector inlet.  
The function of the CVG is not to increase the stability of the shear layer by introducing small-scale vortices into the jet shear layer, as traditional vortex generators do, but rather to regulate the jet expansion angle through the suction effect of corner vortices on turbulence in the lateral center of the jet. Consequently, the effect of the CVG and traditional nozzle vortex generators on jets is the opposite: while traditional vortex generators promote the development of turbulence in the jet shear layer, thereby increasing its expansion angle, the CVG reduces this angle by drawing turbulence toward the corners. Keeping this in mind, it is clear why [PERSON] et al. [23; 49] witnessed amplified low-frequency pressure fluctuations by adding vortex generators at the center of the inner wall of the rectangular nozzle.  
Figure 11: CVG mounted on the LAWT nozzle.  
Figure 10: Schematic of CVG installation location.
# Physical-Layer Security Analysis over M-Distributed Fading Channels
## 2 System Model  
In this paper, we consider a wireless communication system with three entities, i.e., a transmitter ([PERSON]), a legitimate receiver ([PERSON]), and an eavesdropper ([PERSON]), as depicted in Figure 1. In the system, [PERSON] transmits data to [PERSON]. Due to the broadcast feature of the wireless channel, both [PERSON] and [PERSON] can receive the signal. For [PERSON] and [PERSON], the received signals \(Y_{n}\) (\(n=\text{B}\) for Alice-Bob link and \(n=\text{E}\) for Alice-Eve link) are given by  
\[Y_{n}=H_{n}X+Z_{n} \tag{1}\]  
where \(X\) is the transmit signal, and it satisfies \(E(X^{2})=P\), where \(P\) is the average power of the transmit signal. \(Z_{n}\sim\mathcal{N}(0,\sigma_{n}^{2})\) stands for the AWGN, where \(\sigma_{n}^{2}\) is the noise variance. \(H_{n}\) denotes the M-distributed channel fading, which is a seven-parameter distribution and can be denoted as \(H_{n}\sim\mathcal{M}(\alpha_{n},\beta_{n},\rho_{n},\Omega_{n},\zeta_{n},\phi_{ 1,n},\phi_{2,n})\). The PDF of \(H_{n}\) is given by [19]  
\[f_{H_{n}}(h)=A_{n}\sum_{k=1}^{\beta_{n}}a_{k,n}h_{n}^{\frac{\alpha_{n}+k}{2}- 1}K_{\alpha_{n}-k}\Bigg{(}2\sqrt{\frac{\alpha_{n}\beta_{n}h}{\zeta_{g,n}\beta _{n}+\Omega_{n}^{\prime}}}\Bigg{)}\,,\,\,h\geq 0 \tag{2}\]  
where  
\[\left\{ \begin{array}{l}A_{n}=\frac{2a_{n}^{\frac{\alpha_{n}}{2}}}{\frac{1+a _{n}^{\frac{\alpha_{n}}{2}}}{\zeta_{g,n}^{\frac{1}{2}+\frac{\alpha_{n}}{2}}} \Gamma(a_{n})}\Big{(}\frac{\zeta_{g,n}\beta_{n}}{\zeta_{g,n}\beta_{n}+\Omega_ {n}^{\prime}}\Big{)}^{\beta_{n}+\frac{\alpha_{n}}{2}}\\ a_{n,n}=\left(\begin{array}{c}\beta_{n}-1\\ k-1\end{array}\right)\frac{\left(\zeta_{g,n}+\Omega_{n}^{\prime}\right)^{1- \frac{1}{2}}}{\Gamma(k)}\Big{(}\frac{\zeta_{g,n}^{\prime}}{\zeta_{g,n}}\Big{)} ^{k-1}\Big{(}\frac{\alpha_{n}}{\beta_{n}}\Big{)}^{\frac{k}{2}}\\ \Omega_{n}^{\prime}=\Omega_{n}+\zeta_{c,n}+2\sqrt{\zeta_{c,n}\Omega_{n}}\cos( \phi_{1,n}-\phi_{2,n})\\ \tilde{\zeta}_{c,n}=\rho_{n}\tilde{\xi}_{n}\\ \tilde{\zeta}_{g,n}=(1-\rho_{n})\xi_{n}\end{array} \right. \tag{3}\]  
Moreover, \(\alpha_{n}>0\) is a parameter related to the effective number of large-scale cells of the scattering process. \(\beta_{n}\in\mathbb{N}\) denotes the amount of fading parameter. \(\Omega_{n}^{\prime}\) denotes the average power of the LOS  
Figure 1: A wireless communication system with a transmitter ([PERSON]), a legitimate receiver ([PERSON]), and an eavesdropper ([PERSON]). component and the coupled-to-LOS scattering term. \(\phi_{1,n}\) and \(\phi_{2,n}\) are the deterministic phases of the LOS and the coupled-to-LOS scatter components, respectively. \(\xi_{n}\) is the average power of the total scatter components, and \(\rho_{n}\) is a parameter in the range of \([0,1]\).
## 4 Discussion  
### Comparison of the Total Gas Content Tests Using Different Methods  
#### 4.1.1 Traditional Method  
For highly over-mature shale reservoirs, shale gas mainly consists of adsorbed gas and free gas. The adsorbed gas is usually characterized via methane isothermal adsorption experiments at formation temperature and pressure conditions. However, free gas in shale reservoirs is calculated through the parameters of porosity, bulk density, water saturation, formation temperature, and pressure of shale reservoirs; we call this method the traditional method. The specific calculation formula can be expressed as below [52]:  
\[n_{free}=\frac{\text{PT}_{sc}S_{\rm g}\varphi}{\text{ZTP}_{sc}\rho_{r}} \tag{7}\]  
where S\({}_{\rm g}\) is the gas saturation, %; T\({}_{\rm sc}\) is the ground standard temperature, 298.15 K; Z is the gas compressibility factor; \(\varphi\) is the porosity, %; P\({}_{\rm sc}\) is the ground standard pressure, 0.101 MPa; \(\rho_{\rm r}\) is the density of gas-bearing rocks, in g/cm\({}^{3}\); T is the formation temperature, K; P is the formation pressure, MPa.  
The free gas content of the L214 shale sample under formation conditions was calculated to be 9.82 m\({}^{3}\)/t using traditional methods, and its adsorption capacity was 2.50 m\({}^{3}\)/t. Therefore, the total gas content of the L214 shale sample was 11.82 m\({}^{3}\)/t.  
Figure 7: Absolute adsorption curves of the shale samples from the Zhaotong, Weiyuan, and Luzhou areas.
## 4 Discussion
### Ablation Studies
#### 4.1.3 Global Spectral Restoration and Feature Enhancement  
The GSRFE module significantly enhances spatiotemporal fusion by adaptively adjusting spectral features. To validate the effectiveness of the GSRFE module, we designed two ablation experiments: one without spectral adjustment and one with only spectral adjustment but no feature enhancement. Specifically, we named the model without spectral adjustment as BCSR-STF-NA (No Spectral Adjustment) and the model with only spectral adjustment but no feature enhancement as BCSR-STF-OA (Only Spectral Adjustment). In BCSR-STF-NA, we removed the GSRFE module and directly used the spatiotemporal fusion result \(F_{st}^{(i)}\) as the output. For BCSR-STF-OA, we only used the result \(F_{st}^{(i)^{\prime\prime}}\) that had undergone spectral adaptive adjustment without any feature enhancement. Other settings remained unchanged.  
The results of the ablation experiments, shown in Figure 13, indicate that BCSR-STF outperforms both BCSR-STF-NA and BCSR-STF-OA. This improvement can be attributed to the combination of spectral adaptive adjustment and feature enhancement in the GSRFE  
Figure 11: Quantitative evaluation results of BCSR-STF and BCSR-STF-S.  
module. The spectral adaptive adjustment helps correct spectral inconsistencies caused by sensor errors, while feature enhancement further improves the expressive capability of the spatiotemporal fusion features.
## 2 Numerical Method
### Turbulence Model  
The Large eddy simulation (LES) has inherent advantages over the steady-state and unsteady RANS turbulence models in physical simulation, and is well suited for the simulation of turbulence and nonlinear features of complex terrain wind. The use of LES for complex mountainous terrain requires large computing resources. However, for wind field modeling over complex terrain, RANS remains the main turbulence model adopted in CFD simulation up to present, where it is often being applied with a satisfactory degree of success. Many researchers ([PERSON] et al. [40]; [PERSON] et al. [38]; [PERSON] et al. [21]; [PERSON], et al. [41] and [PERSON] et al. [42]) found that results obtained by RNG \(k-\varepsilon\) agree well with these of experiments. Therefore, the RNG \(k-\varepsilon\) model is adopted to numerical simulation in this paper.  
Figure 1: Schematic diagram of Artificial Cliffs.  
The transport equations for RNG \(k-\varepsilon\) model are given as follows,  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{i})= \frac{\partial}{\partial x_{j}}(\alpha_{k}\mu_{eff}\frac{\partial k}{\partial x _{j}})+G_{k}-\rho\varepsilon+S_{k} \tag{3}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}} (\rho\varepsilon u_{i})=\frac{\partial}{\partial x_{j}}(\alpha_{\varepsilon} \mu_{eff}\frac{\partial\varepsilon}{\partial x_{j}})+C_{1\varepsilon}\frac{ \varepsilon}{k}G_{k}-C_{2\varepsilon}\rho\frac{\varepsilon^{2}}{k}-R_{ \varepsilon}+S_{\varepsilon} \tag{4}\]  
where \(G_{k}\) is the turbulent kinetic energy caused by mean speed gradient whose definition is \(G_{k}=-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\partial u_{j}/\partial x_ {i}\) and the equation is \(G_{k}=\mu_{i}S^{2}\) using Boussinesq hypothesis where \(S\) is the strain rate tensor of mean speed, i.e., \(S=\sqrt{2S_{ij}S_{ij}}\). \(\alpha_{k}\) and \(\alpha_{\varepsilon}\) are reciprocals of turbulent kinetic energy, \(k\) and dissipation rate, \(\varepsilon\) respectively. \(S_{k}\) and \(S_{\varepsilon}\) are source terms defined by users. \(R_{\varepsilon}\) is the addition item of RNG \(k-\varepsilon\) model compared with \(k-\varepsilon\) model and its expression is  
\[R_{\varepsilon}=\frac{C_{\mu}\rho\eta^{3}(1-\eta/\eta_{0})}{1+\beta\eta^{3}} \frac{\varepsilon^{2}}{k} \tag{5}\]  
where \(\eta=Sk/\varepsilon\) is the ratio of turbulence and mean flow time scale. \(\eta_{0}\) is the typical value of \(\eta\) in uniform shear flow. \(C_{\mu}\) and \(\beta\) are constants of the turbulence model. \(\rho\) is the air density.
# Application of a Local Dynamic Model of Large Eddy Simulation to a Marine Propeller Wake
## 3 Verification and Comparison
### Mean Force Comparison  
Mean non-dimensional thrust coefficient \(\langle K_{T}\rangle\) and torque coefficient \(\langle K_{Q}\rangle\) of LES-LDKSGS model, RANS-kEpsilon model and LES-GASGS model are compared with the three physical experiment results, as shown in Table 4. For physical experiments, both mean thrust and torque coefficients in the water tunnel (WT) are slightly smaller than in the open water towing tank (OW), possibly due to tunnel blocking effects. LES-LDKSGS results show good agreements with the WT ([PERSON] et al. 2006) results, but less than RANS-kEpsilon results under the same refinement, which implies that if your concern is only for mean force of the propeller the LES method may be not your best choice considering cost. The results of LES-LDKSGS with a lower computational cost is more accurate than the results of LES-DASGS, which shows the superiority of the LES-LDKSGS model to some extent.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Experiment** & \(Re\) & \(Re_{e}\) & \(\langle K_{T}\rangle\) & \
\begin{tabular}{c} **Relative Error** \\ **(\(e_{1},e_{2},e_{3}\))\%** \\ \end{tabular} & \(\langle K_{Q}\rangle\) &
\begin{tabular}{c} **Relative Error** \\ **(\(e_{1},e_{2},e_{3}\))\%** \\ \end{tabular} * \\ \hline LES-LDKSGS (case1) & \(8.9\times 10^{5}\) & \(8.3\times 10^{5}\) & \(-0.201\) & (0.0, 11.6, 4.7) & 0.041 & (2.6, 7.8, 2.3) \\ RANS-kEpsilon (case3) & \(8.9\times 10^{5}\) & \(8.3\times 10^{5}\) & \(-0.194\) & (3.5, 7.8, 8.1) & 0.042 & (0.2, 10.5, 0.0) \\ LES-DASGS (3) & \(8.9\times 10^{5}\) & \(8.3\times 10^{5}\) & \(-0.21\) & (4.5, 16.7, 0.5) & 0.041 & (2.6, 7.8, 2.3) \\ OW ([PERSON] et al., 2004) & \(1.1\times 10^{6}\) & \(1.02\times 10^{6}\) & \(-0.201\) & - & 0.0421 & - \\ WT ([PERSON] et al., 2006) & \(8.9\times 10^{5}\) & \(8.3\times 10^{5}\) & \(-0.18\) & - & 0.038 & - \\ OW ([PERSON] and Remmers, 1971) & \(6.47\times 10^{5}\) & \(6\times 10^{5}\) & \(-0.211\) & - & 0.042 & - \\ \hline \hline \end{tabular}  
* \(e_{1},e_{2},e_{3}\) respectively correspond to the errors relative to WT, OW2006, and OW1971.  
\end{table}
Table 4: Mean thrust and torque coefficients.  
Figure 6: Axial velocity at \(J=0.889\) for LES-LDKSGS: (**a**) At \(r/R=6\) in streamwise direction. (**b**) At \(x/D=-1.8\) in radial direction.