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.

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.

Large Eddy Simulation and Study of the Urban Boundary Layer

Based on a pseudo-spectral large eddy simulation (LES) model, an LES model with an anisotropy turbulent kinetic energy (TKE) closure model and an explicit multi-stage third-order Runge-Kutta scheme is established. The modeling and analysis show that the LES model can simulate the planetary boundary layer (PBL) with a uniform underlying surface under various stratifications very well. Then, similar to the description of a forest canopy, the drag term on momentum and the production term of TKE by subgrid city buildings are introduced into the LES equations to account for the area-averaged effect of the subgrid urban canopy elements and to simulate the meteorological fields of the urban boundary layer (UBL). Numerical experiments and comparison analysis show that: (1) the result from the LES of the UBL with a proposed formula for the drag coefficient is consistent and comparable with that from wind tunnel experiments and an urban subdomain scale model; (2) due to the effect of urban buildings, the wind velocity near the canopy is decreased, turbulence is intensified, TKE, variance, and momentum flux are increased, the momentum and heat flux at the top of the PBL are increased, and the development of the PBL is quickened; (3) the height of the roughness sublayer (RS) of the actual city buildings is the maximum building height (1.5-3 times the mean building height), and a constant flux layer (CFL) exists in the lower part of the UBL.

Selective enhancement of oblique waves caused by finite amplitude second mode in supersonic boundary layer

Nonlinear interactions of the two-dimensional （2D） second mode with oblique modes are studied numerically in a Mach 6.0 fiat-plate boundary layer, focusing on its selective enhancement effect on amplification of different oblique waves. Evolution of oblique modes with various frequencies and spanwise wavenumbers in the presence of 2D second mode is simulated successively, using a modified parabolized stability equation （PSE） method, which is able to simulate interaction of two modes with different frequen- cies efficiently. Numerical results show that oblique modes in a broad band of frequencies and spanwise wavenumbers can be enhanced by the finite amplitude 2D second mode instability wave. The enhancement effect is accomplished by interaction of the 2D second mode, the oblique mode, and a forced mode with difference frequency. Two types of oblique modes are found to be more amplified, i.e., oblique modes with frequency close to that of the 2D second mode and low-frequency first mode oblique waves. Each of them may correspond to one type of transition routes found in transition experiments. The spanwise wavenumber of the oblique wave preferred by the nonlinear interaction is also determined by numerical simulations.

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 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.

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.

Similarity solutions of Prandtl mixing length modelled two dimensional turbulent boundary layer equations

The exact similarity solutions of two dimensional laminar boundary layer were obtained by Blasius in 1908,however,for two dimensional turbulent boundary layers,no Blasius type similarity solutions(special exact solutions)have ever been found.In the light of Blasius’pioneer works,we extend Blasius similarity transformation to the two dimensional turbulent boundary layers,and for a special case of flow modelled by Prandtl mixing-length,we successfully transform the two dimensional turbulent boundary layers partial differential equations into a single ordinary differential equation.The ordinary differential equation is numerically solved and some useful quantities are produced.For numerical calculations,a complete Maple code is provided.

Development of a Single-Column Model in RegCM4 and Its Preliminary Application for Evaluating PBL Schemes in Simulating the Dry Convection Boundary Layer

A single-column model(SCM)is developed in the regional climate model RegCM4.The evolution of a dry convection boundary layer(DCBL)is used to evaluate this SCM.Moreover,four planetary boundary layer(PBL)schemes,namely the Holtslag-Boville scheme(HB),Yonsei University scheme(YSU),and two University of Washington schemes(UW01,Grenier-Bretherton-Mc Caa scheme and UW09,Bretherton-Park scheme),are compared by using the SCM approach.A large-eddy simulation(LES)of the DCBL is performed as a benchmark to examine how well a PBL parameterization scheme reproduces the LES results,and several diagnostic outputs are compared to evaluate the schemes.The results show that the SCM is properly constructed.In general,with the DCBL case,the YSU scheme performs best for reproducing the LES results,which include well-mixed features and vertical sensible heat fluxes;the simulated wind speed,turbulent kinetic energy,entrainment flux,and height of the entrainment zone are all underestimated in the UW09;the UW01 has all those biases of the UW09 but larger,and the simulated potential temperature is not well mixed;the HB is the least skillful scheme,by which the PBL height,entrainment flux,height of the entrainment zone,and the vertical gradients within the mixed layer are all overestimated,and a inversion layer near the top of the surface layer is wrongly simulated.Although more cases and further testing are required,these simulations show encouraging results towards the use of this SCM framework for evaluating the simulated physical processes by the RegCM4.

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.

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.

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.

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).

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.

Tomographic PIV investigation of coherent structures in a turbulent boundary layer flow

Tomographic particle image velocimetry was used to quantitatively visualize the three-dimensional co- herent structures in the logarithmic region of the turbulent boundary layer in a water tunnel. The Reynolds number based on momentum thickness is Reo = 2 460. The in- stantaneous velocity fields give evidence of hairpin vortices aligned in the streamwise direction forming very long zones of low speed fluid, which is flanked on either side by high- speed ones. Statistical support for the existence of hairpins is given by conditional averaged eddy within an increasing spanwise width as the distance from the wall increases, and the main vortex characteristic in different wall-normal re- gions can be reflected by comparing the proportion of ejec- tion and its contribution to Reynolds stress with that of sweep event. The pre-multiplied power spectra and two-point cor- relations indicate the presence of large-scale motions in the boundary layer, which are consistent with what have been termed very large scale motions （VLSMs）. The three dimen-sional spatial correlations of three components of veloc- ity further indicate that the elongated low-speed and high- speed regions will be accompanied by a counter-rotating roll modes, as the statistical imprint of hairpin packet structures, all of which together make up the characteristic of coherent structures in the logarithmic region of the turbulent boundary layer （TBL）.