Experimental Study of Local Scour around Tripod Foundation in Combined Collinear Waves-Current Conditions

A series of laboratory experiments were conducted in a wave-current flume to investigate the scour evolution and scour morphology around tripod in combined waves and current. The tripod model was made using the 3D printing technology, and it was installed in seabed with three installation angles α = 0°, 90°and 180° respectively. In the present study, the scour evolution and scour characteristic were first analyzed. Then, the equilibrium scour depth Seq was investigated. Furthermore, a parametric study was carried out to study the effects of Froude number Fr and Euler number Eu on equilibrium scour depth Seq respectively. Finally, the effects of tripod’s structural elements on Seq were discussed. The results indicate that the maximum scour hole appeared underneath the main column for installation angle α = 0°, 90° and 180°. The Seq for α = 90° was greater than the case of α = 0° and α = 180°, implying the tripod suffered from more severe scour for α = 90°. When KC was fixed, the dimensionless time scale T* for α = 90° was slightly larger than the case of α = 0° and α = 180° and the T* was linearly correlated with Ucw in the range of 0.347 < Ucw < 0.739. The higher Fr and Eu both resulted in the greater scour depth for tripod in combined waves and current. The logarithmic formula can depict the general trend of Seq and Fr (Eu) for tripod in combined waves and current.

Topological acoustic triple point

Acoustic phonon is a classic example of triple degeneracy point in band structure. This triple point always appears in phonon spectrum because of the Nambu–Goldstone theorem. Here, we show that this triple point can carry a topological charge $${\mathfrak{q}}$$ q that is a property of three-band systems with space-time-inversion symmetry. The charge $${\mathfrak{q}}$$ q can equivalently be characterized by the skyrmion number of the longitudinal mode, or by the Euler number of the transverse modes. We call triple points with nontrivial $${\mathfrak{q}}$$ q the topological acoustic triple point (TATP). TATP can also appear at high-symmetry momenta in phonon and spinless electron spectrums when Oh or Th groups protect it. The charge $${\mathfrak{q}}$$ q constrains the nodal structure and wavefunction texture around TATP, and can induce anomalous thermal transport of phonons and orbital Hall effect of electrons. Gapless points protected by the Nambu–Goldstone theorem form a new platform to study the topology of band degeneracies.

Deep multi-task mining Calabi-Yau four-folds

We continue earlier efforts in computing the dimensions of tangent space cohomologies of Calabi-Yau manifolds using deep learning. In this paper, we consider the dataset of all Calabi-Yau four-folds constructed as complete intersections in products of projective spaces. Employing neural networks inspired by state-of-the-art computer vision architectures, we improve earlier benchmarks and demonstrate that all four non-trivial Hodge numbers can be learned at the same time using a multi-task architecture. With 30 % (80 %) training ratio, we reach an accuracy of 100 % for h(1,1) and 97 % for h(2,1) (100 % for both), 81 % (96 %) for h(3,1), and 49 % (83 %) for h(2,2). Assuming that the Euler number is known, as it is easy to compute, and taking into account the linear constraint arising from index computations, we get 100 % total accuracy.

Scour Characteristics and Equilibrium Scour Depth Prediction around Umbrella Suction Anchor Foundation under Random Waves

A series of numerical simulation were conducted to study the local scour around umbrella suction anchor foundation (USAF) under random waves. In this study, the validation was carried out firstly to verify the accuracy of the present model. Furthermore, the scour evolution and scour mechanism were analyzed respectively. In addition, two revised models were proposed to predict the equilibrium scour depth Seq around USAF. At last, a parametric study was carried out to study the effects of the Froude number Fr and Euler number Eu for the Seq. The results indicate that the present numerical model is accurate and reasonable for depicting the scour morphology under random waves. The revised Raaijmakers’s model shows good agreement with the simulating results of the present study when KCs,p < 8. The predicting results of the revised stochastic model are the most favorable for n = 10 when KCrms,a < 4. The higher Fr and Eu both lead to the more intensive horseshoe vortex and larger Seq.

Cyclone pressure drop reduction and its effect on gas–particle separation capability: principle, performance, and assessment

Cyclone separators have been widely used for gas–particle separation in chemical engineering. However, their enhancement in separation performances usually increases the pressure drop, which inevitably leads to an increase in operating energy consumption. One of the challenging issues is how to reduce the cyclone pressure drop while improving separation performances. To gain insight into the pathways and impacts of cyclone pressure drop reduction, this work reviews the state-of-the-art technical principles, performances and effects, focusing on the processes, mechanisms, and characteristics of pressure drop reduction by inlet/outlet variations, additional auxiliary devices, local cyclone dimension improvement, and global optimization based on intelligent algorithms. The cyclone performances are assessed using a proposed index that combines the Euler number and the square-root particle cut-off Stokes number. It is suggested that the pressure drop and separation capability usually have a dynamic compromise. Considering the comprehensive performances, the technology using helical roof inlet, cross cone, increasing cylindrical height with h/D = 4.3 (H/D = 6.35), and globally optimized design by Sun et al. (2017) are respectively considered to the others. Particularly, the last one is recommended to be more representative in practice. Finally, the key issues to be considered in further research were also prospected.

An arithmetic count of the lines on a smooth cubic surface

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$ , generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$ . In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$ . There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$ , \[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \] where $\operatorname {Tr}_{L/k}$ denotes the trace $\operatorname {GW}(L) \to \operatorname {GW}(k)$ . Taking the rank and signature recovers the results over ${\mathbf {C}}$ and ${\mathbf {R}}$ . To do this, we develop an elementary theory of the Euler number in $\mathbf {A}^1$ -homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

On a class of Hilbert-type inequalities in the whole plane related to exponent function

By introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality.

Numerical Analysis of Flow across Brush Elements Based on a 2-D Staggered Tube Banks Model

In order to improve efficiency in turbomachinery, brush seal replaces labyrinth seals widely in the secondary air system. A 2-d staggered tube bank model is adopted to simulate the gas states and the pressure character in brush seal, and computational fluid dynamics (CFD) is used to solve the model in this paper. According to the simulation results, the corrected formula of the Euler number and dimensionless pressure are given. The results show that gas expands when flow through the bristle pack, and the gas expansion closes to an isotherm process. The dynamic pressure increases with decreasing static pressure. The Euler number can reflect the seal performance of brush seals in leakage characteristics. Compared with increasing the number of rows, the reduction of the gap is a higher-efficiency method to increase the Euler number. The Euler number continually increases as the gap decreases. However, with the differential pressure increasing, Euler number first increases and then decreases as the number of rows increases. Finally, the pressure distribution on the surface of end rows is asymmetric, and it may increase the friction between the bristles and the back plate.