Chien-physics-informed neural networks for solving singularly perturbed boundary-layer problems

A physics-informed neural network(PINN)is a powerful tool for solving differential equations in solid and fluid mechanics.However,it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives.In this paper,we introduce Chien's composite expansion method into PINNs,and propose a novel architecture for the PINNs,namely,the Chien-PINN(C-PINN)method.This novel PINN method is validated by singularly perturbed differential equations,and successfully solves the wellknown thin plate bending problems.In particular,no cumbersome matching conditions are needed for the C-PINN method,compared with the previous studies based on matched asymptotic expansions.

Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.

THE PREDICTIONS OF CONVECTIVE HEAT TRANSFER ON TURBINE BLADE AIRFOIL BY USING LOW－REYNOLDS NUMBER TURBULENCE MODEL

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The influence of thermal radiation on MHD station-point flow past a stretching sheet with heat generation

This letter is concerned with the plane and axisymmetric stagnation-point flows and heat transfer of an electrically-conducting fluid past a stretching sheet in the presence of the thermal radiation and heat generation or absorption. The analytical solutions for the velocity distribution and dimensionless temperature profiles are obtained for the various values of the ratio of free stream velocity and stretching velocity, heat source parameter, Prandtl number, thermal radiation parameter, the suction and injection velocity parameter and magnetic parameter and dimensionality index in the series form with the help of homotopy analysis method （HAM）. Convergence of the series is explicitly dis- cussed. In addition, shear stress and heat flux at the surface are calculated.

Transport mechanisms of contaminants released from fine sediment in rivers

Contaminants released from sediment into rivers are one of the main problems to study in environmental hydrodynamics. For contaminants released into the overlying water under different hydrodynamic conditions, the mechanical mechanisms involved can be roughly divided into convective diffusion, molecular diffusion, and adsorption/desorption. Because of the obvious environmental influence of fine sediment （D90 = 0.06 mm）, non-cohesive fine sediment, and cohesive fine sediment are researched in this paper, and phosphorus is chosen for a typical adsorption of a contaminant. Through theoretical analysis of the contaminant release process, according to different hydraulic conditions, the contaminant release coupling mathematical model can be established by the N-S equation, the Darcy equation, the solute transport equation, and the adsorption/desorption equation. Then, the experiments are completed in an open water flume. The simulation results and experimental results show that convective diffusion dominates the contaminant release both in non-cohesive and cohesive fine sediment after their suspension, and that they contribute more than 90 % of the total release. Molecular diffusion and desorption have more of a contribution for contaminant release from unsuspended sediment. In unsuspension sediment, convective diffusion is about 10-50 times larger than molecular diffusion during the initial stages under high velocity; it is close to molecular diffusion in the later stages. Convective diffusion is about 6 times larger than molecular diffusion during the initial stages under low velocity, it is about a quarter of mole- cular diffusion in later stages, and has a similar level with desorption/adsorption. In unsuspended sediment, a seepage boundary layer exists below the water-sediment interface, and various release mechanisms in that layer mostly dominate the contaminant release process. In non-cohesive fine sediment, the depth of that layer increases linearly with shear stress. In cohesive fine sediment, the range seepage boundary is different from that in non-cohesive sediment, and that phenomenon is more obvious under a lower shear stress.

Mathematical modeling of tornadoes and squall storms

Recent advances in modeling of tornadoes and twisters consist of significant achievements in mathematical calculation of occurrence and evolution of a violent F5-class tornado on the Fujita scale, and four-dimensional mathematical modeling of a tornado with the fourth coordinate time multiplied by its characteristic velocity. Such a tornado can arise in a thunderstorm supercell filled with turbulent whirlwinds. A theory of the squall storms is proposed. The squall storm is modeled by running pertur- bation of the temperature inversion on the lower boundary of cloudiness. This perturbation is induced by the action of strong, hurricane winds in the upper and middle troposphere, and looks like a running solitary wave （soliton）; which is developed also in a field of pressure and velocity of a wind. If a soliton of a squall storm gets into the thunderstorm supercell then this soliton is captured by supercell. It leads to additional pressure fall of air inside a storm supercell and stimulate amplification of wind velocity here. As a result, a cyclostrophic balance inside a storm supercell generates a tornado. Comparison of the radial distribution of wind velocity inside a tornado calculated by using the new formulas and equations with radar observations of the wind velocity inside Texas Tornado Dummit in 1995 and inside the 3 May 1999 Oklahoma City Tornado shows good correspondence.

EXPERIMENTAL STUDY ON TURBULENT BOUNDARY LAYER CHARACTERISTICS OVER STREAMWISE RIBLETS

Measurements of characteristics by means of a two-component Laser DopplerVelocimeter (LDV) were carried out in turbulent boundary layers over both a symmetric V-shapedribbed plate and a smooth one in a low speed wind tunnel. The present results clearly indicate thatthe logarithmic velocity profile over the riblets surface is shifted upward with a 30. 9% increasein the thickness of the viscous sublayer. Also a change in the log-law region is found. And themaximum value of streamwise velocity fluctuations is reduced by approximately 17%. The skewness andflatness factors do not show any change besides those in the region of y^+ 【 0. 6 . It is evidentthat the Reynolds shear stress over the riblets is reduced. Further more, in log-law region, theReynolds shear stress has a larger reduction of up to 18%.

Symmetry Analysis and Exact Solutions of the 2D Unsteady Incompressible Boundary-Layer Equations

To find intrinsically different symmetry reductions and inequivalent group invariant solutions of the 2D unsteady incompressible boundary-layer equations, a two-dimensional optimal system is constructed which attributed to the classification of the corresponding Lie subalgebras. The comprehensiveness and inequivalence of the optimal system are shown clearly under different values of invariants. Then by virtue of the optimal system obtained, the boundary-layer equations are directly reduced to a system of ordinary differential equations(ODEs) by only one step. It has been shown that not only do we recover many of the known results but also find some new reductions and explicit solutions, which may be previously unknown.