Multi-scale physics-informed neural networks for solving high Reynolds number boundary layer flows based on matched asymptotic expansions

Multi-scale system remains a classical scientific problem in fluid dynamics,biology,etc.In the present study,a scheme of multi-scale Physics-informed neural networks is proposed to solve the boundary layer flow at high Reynolds numbers without any data.The flow is divided into several regions with different scales based on Prandtl's boundary theory.Different regions are solved with governing equations in different scales.The method of matched asymptotic expansions is used to make the flow field continuously.A flow on a semi infinite flat plate at a high Reynolds number is considered a multi-scale problem because the boundary layer scale is much smaller than the outer flow scale.The results are compared with the reference numerical solutions,which show that the msPINNs can solve the multi-scale problem of the boundary layer in high Reynolds number flows.This scheme can be developed for more multi-scale problems in the future.

Sidewall influence of varying free stream Mach number in ramp induced shock wave boundary layer interactions

This study investigates the three-dimensional(3D)effects introduced by the end walls for an aspect ratio of1 in ramp-induced shock wave boundary layer interactions.The simulations are performed using a symmetry boundary condition in the spanwise direction at free-stream Mach numbers in 3D.The simulations are performed using an in-house compressible supersonic solver“Open SBLIFVM”.Two free stream Mach numbers 2.5,and3 are used in the current work,and the simulated results are compared with the aspect ratio 1 simulations by Mangalagiri and Jammy.The inflow is initialized with a similarity solution;its Reynolds number based on the boundary layer thickness is adjusted such that the Reynolds number at the start of the ramp is kept at 3×10^(5)for all simulations.From the results,it is evident that the introduction of sidewalls resulted in a shorter centerline separation length when compared with the two-dimensional(2D)simulations.This contradicts the results at Mach 2 by Mangalgiri and Jammy where the vortex observed at Mach 2 in the central separation region disappeared with increasing free-stream Mach number.Additionally,the topology of interaction shifted from owl-like separation of the second kind to the first kind when the freestream Mach number increased from2 to 2.5.It can be concluded that the interaction topology is crucial to the increase or decrease of the central separation length when compared to 2D simulations.

Analytical solutions of turbulent boundary layer beneath forward-leaning waves

As a typical nonlinear wave,forward-leaning waves can be frequently encountered in the near-shore areas,which can impact coastal sediment transport significantly.Hence,it is of significance to describe the characteristics of the boundary layer beneath forward-leaning waves accurately,especially for the turbulent boundary layer.In this work,the linearized turbulent boundary layer model with a linear turbulent viscosity coefficient is applied,and the novel expression of the near-bed orbital velocity that has been worked out by the authors for forward-leaning waves of arbitrary forward-leaning degrees is further used to specify the free stream boundary condition of the bottom boundary layer.Then,a variable transformation is found so as to make the equation of the turbulent boundary layer model be solved analytically through a modified Bessel function.Consequently,an explicit analytical solution of the turbulent boundary layer beneath forward-leaning waves is derived by means of variable separation and variable transformation.The analytical solutions of the velocity profile and bottom shear stress of the turbulent boundary layer beneath forward-leaning waves are verified by comparing the present analytical results with typical experimental data available in the previous literature.

NUMERICAL SIMULATION AND ANALYSIS OF HORSESHOE-SHAPED VORTEX IN NEAR-WALL REGION OF TURBULENT BOUNDARY LAYER

The low-Reynolds-number full developed turbulent flow in channels is simulated using large eddy simulation（LES）method with the preconditioned algorithm and the dynamic subgrid-scale model,with a given disturbance in inlet boundary,after a short development section.The inlet Reynolds number based on momentum thickness is 670.The computed results show good agreement with direct numerical simulation（DNS）,which include root mean square fluctuated velocity distribution and average velocity distribution.It is also found that the staggered phenomenon of the coherent structures is caused by sub-harmonic.The results clearly show the formation and evolution of horseshoe vortex in the turbulent boundary layer,including horseshoe vortex structure with a pair of streamwise vortexes and one-side leg of horseshoe vortex.Based on the results,the development of the horseshoe-shaped coherent structures is analyzed in turbulent boundary layer.

Boundary Layer Characteristics and Numerical Simulation Analysis of Winter Dense Fog in Nanjing

Based on the boundary layer data of winter dense fog in 2007 from Nanjing University of Information Science & Technology,the profile characteristics of temperature,wind direction,wind speed and humidity in a dense fog weather on December 13-14 in 2007 were analyzed,as well as their evolution laws in the formation and dispersion of fog,and the boundary layer characteristics of winter dense fog in Nanjing were revealed,while the development of fog was simulated by means of mesoscale numerical model.The results showed that the formation and dispersion of fog was greatly affected by inversion and humidity in the surface layer,and the wind direction in the surface layer also had effect on the formation and dispersion of advection fog.Mesoscale numerical model could preferably simulate the evolution of temperature,humidity,vertical speed in the development of fog,and the simulation of water vapor content in the fog could forecast the formation and dispersion of fog.

Test Research on the Characteristics of Pollution Boundary Layer in the River Outlet Area of Grand Liao River

By using the meteorological data in the pollution boundary layer which was observed in two ground observation sites:coast and land in the river outlet area of Grand Liao River during January-February in 2007,the daily change characteristics of pollute boundary layer in winter in the area were discussed. The results showed that the pollute boundary layer in the river outlet area of Grand Liao River was affected by the sea and land. In the certain weather condition,maybe the sea-land breeze appeared in the low altitude which was below 200 m in the coastal zone. The stability change in the different height in the coastal zone was more stable than in the land zone,and the wind field change in the area was mainly in 300 m low altitude. At night,the temperature inversion often appears in the area,and the thickness of temperature inversion layer is stably during 200-300 m. The thermal internal boundary layer penetrated deeply into the land about 10 km,and the height could reach 800 m. The atmospheric diffusion ability in the coastal area was weaker and stronger in the land area.

Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations

The perfectly matched layer （PML） is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the second- order wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finite- element matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media.

Numerical Simulation of Two-Dimensional Shock/Boundary-Layer Interaction between a Rocket and Booster

A two-dimensional Reynolds-averaged Navier-Stokes solver is applied to analyze the aerodynamic behavior of the Shock/Boundary-Layer interaction of rocket with a boosted The K-ε turbulence model and a finite volume method in a unstructured body-fitted curvilinear coordinates have been used. The results indicate that the separation and the reattachment occur in the Boundary-Layer of the main rocket because of the shock interaction. The shape of the booster nose effects the flow field obviously. In the case of the hemisphere booster nose the pressure has complicate distributions and the separation is very clear. The distance between the booster and main rocket has the evident effect on the flow field. If the distance is smaller the pressure coefficient is bigger the separation zone even the separation bubble occurs.

NONLINEAR EFFECT OF LOCAL ROUGH WALL ON LARGE EDDY STRUCTURE IN BOUNDARY LAYER

Based on the building of a theoretical model for the large eddy structure, the nonlinear effect of the local rough wall on the large eddy structure in the boundary layer is studied by direct numerical simulation. Numerical results show that factors of the local rough feature, the distributing structure and the intensity, etc. play an important role in the evolution of the large eddy structure in the boundary layer.

DIRECT NUMERICAL SIMULATION OF VIBRATIONINDUCED T-S WAVE ON BOUNDARY LAYER WALL

The spatial growth of the disturbance in the boundary layer is directly numerically simulated, and the receptivity of the Blasius basic flow to the local two-dimensional （2-D） sustainable micro-vibration is investigated. Results show that the disturbance velocity presents the sine vibration features with the change of time, and the vibration period is identical to the vibration of the local wall. The disturbance velocity presents the fluctuation feature downstream, and the streamwise wave length approximates to the results from the Orr-Sommerfeld equation （OSE）. The growth rate from direct numerical simulation（DNS） is a little greater than that from OSE, and their trends are almost consistent. Under the condition of Re= 2 800, the disturbance amplitude gradually grows in the given computational region with the period T=30. However, it firstly increases and then decreases with the period T= 20. The disturbance harmonic of the former is obviously larger than that of the latter. The maximum streamwise and vertical disturbance velocities from DNS do not fully coincide with those from OSE at the vicinity of the local vibration wall, but coincide well with the former when they travel downstream. The 2-D disturbance induced by the local micro-vibration represents the form of Tollmien-Schlichting （T-S） wave on the boundary layer.

STABILITY OF 3-D DISTURBANCE WAVES ON BOUNDARY LAYER WITH DISTRIBUTED LOCAL ROUGHNESS

The numerical solution of the stable basic flow on a 3-D boundary layer is obtained by using local ejection, local suction, and combination of local ejection and suction to simulate the local rough wall. The evolution of 3-D disturbance T-S wave is studied in the spatial processes, and the effects of form and distribution structure of local roughness on the growth rate of the 3-D disturbance wave and the flow stability are discussed. Numerical results show that the growth of the disturbance wave and the form of vortices are accelerated by the 3-D local roughness. The modification of basic flow owing to the evolvement of the finite amplitude disturbance wave and the existence of spanwise velocity induced by the 3-D local roughness affects the stability of boundary layer. Propagation direction and phase of the disturbance wave shift obviously for the 3-D local roughness of the wall. The flow stability characteristics change if the form of the 2-D local roughness varies.

NONLINEAR EVOLUTION OF TWO-DIMENSIONAL T-S WAVE IN LOCALIZED ROUGH BOUNDARY LAYER

Numerical simulations are carried out to investigate the mechanism of the nonlinear evolution of two- dimensional （2-D） Tollmien-Schlichting （T-S） wave on a localized rough boundary layer. The three-dimensional （3-D） numerical solution of a base flow on a boundary layer is obtained for the localized rough wall with the local- ized ejection, the localized suction and the combination of ejection and suction. Based on numerical simulations, the processes of stable and the most instable nonlinear evolution of the 2-D disturbance T-S wave are studied. The effects of the form on the localized roughness, the intensity, and the distribution structure on the nonlinear evolution of 2-D T-S wave and the growth rate are discussed. Results show that the basic flow induced by the lo- calized rough wall is a key factor causing the fast growth of the disturbance wave. Due to the change of the aver- age flow profile and the existence of the spanwise velocity, the localized rough wall enhances the instability of the flow. Consequently, the instable region of the neutral curve is enlargened, and the maximnum growth rate of the 2-D T-S wave is increased. In the process of the nonlinear evolution of 2-D disturbance T-S wave, with the in- crease of the nonlinear interaction, the most instable 2-D disturbance wave triggers the appearance of the 3-D dis- turbance wave and the high-frequency harmonic wave. Its streamwise wave number and the frequency are the same as those of 2-D disturbance wave. The spanwise velocity can excite the growth of the 2-D disturbance wave, the instability of 2-D wave, the formation of the streamwise vortex, and the generation of 3-D disturbance wave. Simulation results agree well with experimental results.

NONPARALLEL BOUNDARY LAYER STABILITY IN HIGH SPEED FLOWS

The parabolized stability equations （PSEs） for high speed flows, especially supersonic and hypersonic flows, are derived and used to analyze the nonparallel boundary layer stability. The proposed numerical techniques for solving PSE include the following contents： introducing the efficiently normal transformation of the boundary layer, improving the computational accuracy by using a high-order differential scheme near the wall, employing the predictor-corrector and iterative approach to satisfy the important normalization condition, and implementing the stable spatial marching. Since the second mode dominates the growth of the disturbance in high Mach number flows, it is used in the computation. The evolution and characteristics of the boundary layer stability in the high speed flow are demonstrated in the examples. The effects of the nonparallelizm, the compressibility and the cooling wall on the stability are analyzed. And computational results are in good agreement with the relevant data.

Characteristics of the Boundary Layer Structure of Sea Fog on the Coast of Southern China

Using boundary layer data with regard to sea fog observed at the Science Experiment Base for Marine Meteorology at Bohe,Guangdong Province,the structure of the atmospheric boundary layer and the characteristics of the tops of the fog and the clouds were analyzed.In addition,the effects of advection,radiation,and turbulence during sea fog were also investigated.According to the stability definition of saturated,wet air,the gradient of the potential pseudo-equivalent temperature equal to zero was defined as the thermal turbulence interface.There is evidence to suggest that two layers of turbulence exist in sea fog.Thermal turbulence produced by long-wave radiation is prevalent above the thermal turbulence interface,whereas mechanical turbulence aroused by wind shear is predominant below the interface.The height of the thermal turbulence interface was observed between 180 m and 380 m.Three important factors are closely related to the development of the top of the sea fog：（1） the horizontal advection of the water vapor,（2） the long-wave radiation of the fog top,and（3） the movement of the vertical turbulence.Formation,development,and dissipation are the three possible phases of the evolution of the boundary-layer structure during the sea fog season.In addition,the thermal turbulence interface is the most significant turbulence interface during the formation and development periods;it is maintained after sea fog rises into the stratus layer.

Urban Heat Island and Boundary Layer Structures under Hot Weather Synoptic Conditions:A Case Study of Suzhou City,China

A strong urban heat island （UHI） appeared in a hot weather episode in Suzhou City during the period from 25 July to 1 August 2007. This paper analyzes the urban heat island characteristics of Suzhou City under this hot weather episode. Both meteorological station observations and MODIS satellite observations show a strong urban heat island in this area. The maximum UHI intensity in this hot weather episode is 2.2℃, which is much greater than the summer average of 1.0℃ in this year and the 37–year （from 1970 to 2006） average of 0.35℃. The Weather Research and Forecasting （WRF） model simulation results demonstrate that the rapid urbanization processes in this area will enhance the UHI in intensity, horizontal distribution, and vertical extension. The UHI spatial distribution expands as the urban size increases. The vertical extension of UHI in the afternoon increases about 50 m higher under the year 2006 urban land cover than that under the 1986 urban land cover. The conversion from rural land use to urban land type also strengthens the local lake-land breeze circulations in this area and modifies the vertical wind speed field.

Observations of boundary layer parameters and suspended sediment transport over the intertidal flats of northern Jiangsu, China

A current-turbidity monitoring system (CTMS) was deployed on the intertidal flat at Wanggang, northern Jiangsu during October 16-17, 2000, to measure the tidal current speeds and seawater turbidities at 5 levels above the seabed. Based upon the logarithmic-profile equation, the boundary layer parameters, i.e., u, z0 and C60, were obtained for 247 tidal flow velocity profiles. Around 90% of the profiles were logarithmic according to the critical correlation coefficient. Internal consistency analysis shows that these parameters derived by different methods are consistent with each other. In addition, the height of the bedforms observed is close to the seabed roughness lengths calculated from the velocity profiles, indicating that the boundary layer parameters obtained can reveal the conditions at the sediment-water interface on the intertidal flats. Suspended sediment concentrations were obtained from the 5 CTMS turbidity meters using laboratory and in-situ calibrations. The results show that the in-situ calibrated SSCs have a much higher accuracy than the laboratory calibrated ones. Calculation of suspended sediment fluxes on the intertidal flats, with a magnitude of 104 kg/m per spring tidal cycle, indicates that suspended sediment moves towards the northwest, which is reversal to the transport pattern controlled by the southward Northern Jiangsu Coastal Current in the sub-tidal zone and adjacent shallow waters.

Diurnal Variations of Air Pollution and Atmospheric Boundary Layer Structure in Beijing During Winter 2000/2001

The diurnal variations of gaseous pollutants and the dynamical and thermodynamic structures of the atmospheric boundary layer (ABL) in the Beijing area from January to March 2001 are analyzed in this study using data from the Beijing City Air Pollution Observation Field Experiment (BECAPEX). A heavy pollution day (22 February) and a good air quality day (24 February) are selected and individually analyzed and compared to reveal the relationships between gaseous pollutants and the diurnal variations of the ABL. The results show that gaseous pollutant concentrations exhibit a double-peak-double-valley-type diurnal variation and have similar trends but with different magnitudes at different sites in Beijing. The diurnal variation of the gaseous pollutant concentrations is closely related to (with a 1-2 hour delay of) changes in the atmospheric stability and the mean kinetic energy in the ABL.

The Influence of Convergence Movement on Turbulent Transportation in the Atmospheric Boundary Layer

Classical turbulent K closure theory of the atmospheric boundary layer assumes that the vertical turbulent transport flux of any macroscopic quantity is equivalent to that quantity's vertical gradient transport flux. But a cross coupling between the thermodynamic processes and the dynamic processes in the atmospheric system is demonstrated based on the Curier-Prigogine principle of cross coupling of linear thermodynamics. The vertical turbulent transportation of energy and substance in the atmospheric boundary layer is related not only to their macroscopic gradient but also to the convergence and the divergence movement. The transportation of the convergence or divergence movement is important for the atmospheric boundary layer of the heterogeneous underlying surface and the convection boundary layer. Based on this, the turbulent transportation in the atmospheric boundary layer, the energy budget of the heterogeneous underlying surface and the convection boundary layer, and the boundary layer parameterization of land surface processes over the heterogeneous underlying surface are studied. This research offers clues not only for establishing the atmospheric boundary layer theory about the heterogeneous underlying surface, but also for overcoming the difficulties encountered recently in the application of the atmospheric boundary layer theory.

The Influences of Boundary Layer Parameterization Schemes on Mesoscale Heavy Rain System

The mesoscale numerical weather prediction model (MM4) in which the computations of the turbulent exchange coefficient in the boundary layer and surface fluxes are improved, is used to study the influences of boundary layer parameterization schemes on the predictive results of the mesoscale model. Seven different experiment schemes (including the original MM4 model) designed in this paper are tested by the observational data of several heavy rain cases so as to find an improved boundary layer parameterization scheme in the mesoscale meteorological model. The results show that all the seven different boundary layer parameterization schemes have some influences on the forecasts of precipitation intensity, distribution of rain area, vertical velocity, vorticity and divergence fields, and the improved schemes in this paper can improve the precipitation forecast. Key words Boundary layer parameterization - Mesoscale numerical weather prediction (MNWP) - Turbulent exchange coefficient - Surface fluxes - Heavy rain This paper was supported by the National Natural Science Foundation of China (Grant No. 49875005 and No. 49735180).

Development of the Convective Boundary Layer Capping with a Thick Neutral Layer in Badanjilin:Observations and Simulations

In this study, the development of a convective boundary layer （CBL） in the Badaujilin region was investigated by comparing the observation data of two cases. A deep neutral layer capped a CBL that occurred on 30 August 2009. This case was divided into five sublayers from the surface to higher atmospheric elevations： surface layer, mixed layer, inversion layer, neutral layer, and sub-inversion layer. The development process of the CBL was divided into three stages： S1, S2, and S3. This case was quite different from the development of the three-layer CBL observed on 31 August 2009 because the mixed layer of the five-layer CBL （CBL5） eroded the neutral layer during S2. The specific initial structure of the CBL5 was correlated to the synoptic background of atmosphere during nighttime. The three-stage development process of the CBL5 was confirmed by six simulations using National Center for Atmospheric Research （USA） large-eddy simulation （NCAR-LES）, and some of its characteristics are presented in detail.