Order Boundary Layer Research Articles

A Reduced Order Series Expansion for the BEM Incorporating the Boundary Layer Impedance Condition

When modeling sound waves in fluids it can be important to include the viscous and thermal losses originating from the fluids’ interaction with boundaries. In the audible frequency range, the thickness of the boundary layers is between a micrometer and a millimeter. As such the viscous and thermal losses are important when simulating the properties of small acoustical devices such as e.g. hearing aids or transducers. However, the inclusion of viscous and thermal losses is a computationally demanding task as it requires a fine discretization of the boundary layer in order to fully capture the complicated physical phenomena happening on the microscale. Recently, there has been developments to ease the computational demands using both the Finite Element Method and the Boundary Element Method, by approximating the losses using the Boundary Layer Impedance (BLI) boundary condition. In this paper, we extend previous developments for multi-frequency analysis using the Reduced Order Series Expansion Boundary Element Method to handle the BLI condition. This model follows a two-step procedure: Using a series expansion to decrease the assembly time of the BEM matrices and a projection to reduce the overall memory consumption of the model. Results from two acoustic interior problems show that the model decreases the total computational time by around 96% while using less than 15% of the memory. For both test setups the limiting factor of the accuracy was the reduction and not the series expansion.

On nonlinear instability of Prandtl's boundary layers: The case of Rayleigh's stable shear flows

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to O(ν1/4) order terms in L∞ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces.In addition, we also prove that monotonic boundary layer profiles, which are stable when ν=0, are nonlinearly unstable when ν>0, provided ν is small enough, up to O(ν1/4) terms in L∞ norm.

A simple method to account for thermal boundary layers during the estimation of CTmax in small ectotherms

As temperatures rise, understanding how ectotherms will become impacted by thermal stress is of critical importance. In this context, many researchers quantify critical temperatures – these are the upper (CTmax) and lower (CTmin) thermal limits at which organisms can no longer function. Most studies estimate CTs using bath-based methods where organisms are submerged within a set thermal environment. Plate-based methods (i.e. hot plates), however, offer huge opportunity for automation and are readily available in many lab settings. Plates, however, generate a unidirectional thermal boundary layer above their surface which means that the temperatures experienced by organisms of different sizes is different. This boundary layer effect can bias estimates of critical temperatures. Here, we test the hypothesis that biases in critical temperature estimation on hot plates are driven by organism height. We also quantify the composition of the boundary layer in order to correct for these biases. We assayed four differently sized species of UK ants for their CTmax in dry baths (with no boundary layer) and on hot plates (with a boundary layer). We found that hot plates overestimated the CTmax values of the different ants, and that this overestimate was larger for taller species. By statistically modelling the thickness of the thermal boundary layer, and combining with estimates of species height, we were able to correct this overestimation and eliminate methodological differences. Our study provides two main findings. First, we provide evidence that organism height is positively related to the bias present in plate-based estimates of CTmax. Second, we show that a relatively simple statistical model can correct for this bias. By using simple corrections for boundary layer effects, as we have done here, researchers could open up a new possibility space in the design and implementation of thermal tolerance assays using plates rather than restrictive dry or water baths.

A wall model for external laminar boundary layer flows applied to the Wall-Modeled LES framework

Wall-Modeled Large Eddy Simulation (WMLES) is a prominent technique for obtaining high-fidelity solutions of turbulent, high Reynolds number flows, with reasonably acceptable computational costs. However, for external flows, the very thin laminar boundary layer developing near the body leading edge imposes quite restrictive mesh resolution requirements, leading to prohibitively high computational costs for practical Reynolds numbers. We propose a wall-modeling approach for the laminar portion of the boundary layer in order to alleviate these costs by reducing the aforementioned mesh resolution requirements. The wall model is based on local self-similar solutions of the boundary layer, and is implemented in the same context of wall-stress models in the WMLES approach. A thorough assessment of the proposed wall model is performed in terms of the distribution of both pressure and skin friction coefficients along the surface, for a fully laminar flow around a NACA 0012 airfoil geometry, with a chord Reynolds number of Rec=5×103 and Mach number of M∞=0.5. Simulations are carried out to evaluate the effects of the wall model height on the solution. The influence of mesh refinement and mesh element distributions along the surface are also analyzed and discussed. The results obtained in the simulations using the proposed wall model are in good agreement with the reference solution, demonstrating the suitability of the model for external laminar flows.

Investigation of wind characteristics of typhoon boundary layer through field experiments and CFD simulations

High-resolution observations of typhoon boundary layer above 100 m are rare as traditional wind towers are generally below 100 m, which limits the study of typhoon boundary layer and engineering applications such as wind-resistant design of tall buildings and wind turbines in typhoon-prone regions. In this study, boundary layer winds of super typhoon Lekima (2019) are observed, simulated and analyzed. Together with traditional wind tower, Doppler wind lidar is utilized for observations of typhoon boundary layer in order to obtain measured data above 100 m. Besides, Computational Fluid Dynamics (CFD) simulation based on Large Eddy Simulation (LES) method is conducted to further investigate the impact of complex terrain on the near-surface wind characteristics. The results show that the power law fits the mean wind speed profile well below 100 m. However, before and after the typhoon lands, a local reverse or low-level jet occurs in the mean wind speed profile at the height of 100–300 m, which cannot be depicted by the power law. Meanwhile, the turbulence intensity increases with height and experiences larger fluctuations. In addition, there is a significant negative correlation between the ground elevation and power exponents of the fitted mean wind speed profiles. This study provides useful information to better understand wind characteristics of the typhoon boundary layer.

A Single-Column Model Evaluation of Mixing Length Formulations and Constraints for the sa-TKE-EDMF Planetary Boundary Layer Parameterization

Abstract Parameterizing boundary layer turbulence is a critical component of numerical weather prediction and the representation of turbulent mixing of momentum, heat, and other tracers. The components that make up a boundary layer scheme can vary considerably, with each scheme having a combination of processes that are physically represented along with tuning parameters that optimize performance. Isolating a component of a PBL scheme to examine its impact is essential for understanding the evolution of boundary layer profiles and their impact on the mean structure. In this study we conduct three experiments with the scale-aware TKE eddy-diffusivity mass-flux (sa-TKE-EDMF) scheme: 1) releasing the upper limit constraints placed on mixing lengths, 2) incrementally adjusting the tuning coefficient related to wind shear in the modified Bougeault and Lacarrere (BouLac) mixing length formulation, and 3) replacing the current mixing length formulations with those used in the MYNN scheme. A diagnostic approach is adopted to characterize the bulk representation of turbulence within the residual layer and boundary layer in order to understand the importance of different terms in the TKE budget as well as to assess how the balance of terms changes between mixing length formulations. Although our study does not seek to determine the best formulation, it was found that strong imbalances led to considerably different profile structures both in terms of the resolved and subgrid fields. Experiments where this balance was preserved showed a minor impact on the mean structure regardless of the turbulence generated. Overall, it was found that changes to mixing length formulations and/or constraints had stronger impacts during the day while remaining partially insensitive during the evening.

Closure scheme for stably stratified turbulence without critical Richardson number

This paper presents a new modification of the second order turbulence closure that removes the critical gradient Richardson number limitation typically found in Mellor-Yamada style models. The mean wind speed and potential temperature profiles are derived for the newly modified model in terms of similarity and structure functions depending on the gradient Richardson number. The derivation is based on a second order boundary layer approximation in neutral to very stable stratification conditions. Some recent closure assumptions for pressure-temperature and heat flux are considered. Variances and covariances of the turbulent fluctuations are also investigated with respect to the gradient Richardson number. The new model predictions are confronted with some well known models.

Closed-form meromorphic solutions of some third order boundary layer ordinary differential equations

We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic methods recently developed, for the generic cases, it is shown that all their meromorphic solutions must be rational, or rational in one exponential, and then we find all of them explicitly. For a few non-generic cases, some solutions, which are meromorphic or singlevalued, are also obtained. Our results also explain why it is so difficult to obtain new closed-form solutions of the Falkner-Skan equation.

Local laminarization at negligible light gas injection into the compressible weakly accelerated boundary layer

The paper presents a numerical modeling of a compressible xenon boundary layer over a permeable wall with various concentrations of helium in a helium-xenon mixture. We set zero intensity of the gas injection at the wall and a mild streamwise pressure gradient with acceleration parameter K = 4 × 10−7. A local laminarization may occur near the wall under such conditions. This phenomenon is not well-studied, especially in the helium-xenon mixture flow and has a noticeable effect on heat and mass transfer coefficients at the wall. The numerical model consists of the Prandtl equations, which describe the dynamic, thermal and diffusion processes within the boundary layer, and the k-ω-γ turbulence model, which is able to reasonably simulate a laminar-turbulent transition as well as a suppression of turbulence. In some simulated cases we blocked the turbulence model to obtain parameters of exclusively laminar flow. Also, the numerical model allows blocking the term of viscous dissipation in the energy equation and setting the constant temperature at the outer border of the boundary layer in order to evaluate a compressibility effect on the flow. Throughout, we applied finite-difference methods for solving the partial differential equations. The simulation demonstrated that a local laminarized flow may occur near the wall under the considered conditions, while an outer part of the boundary layer keeps a turbulent regime. Such character of the stream leads to levels of friction and mass transfer intermediate to laminar and turbulent flows at the same conditions. We obtained that the laminarized area is placed at y+<30, and this value does not depend on the helium concentration at the wall.

Embedded LES of thermal stratification effects on the airflow and concentration fields around an isolated high-rise building: Spectral and POD analyses

Non-isothermal stratification conditions can alter the airflow pattern and pollutant dispersion process within urban areas. The present study is focused on the impact of various stratification conditions, namely, stable, isothermal (neutral), and unstable, on the airflow and concentration fields around an isolated high-rise building. Zonal Reynolds-averaged Navier-Stokes (RANS)-large eddy simulation (LES), also known as embedded large eddy simulation (ELES), is employed for simulating the airflow and concentration fields under non-isothermal boundary layers in order to make a balance between computational costs and accuracy. Comparing the results predicted by the present ELES with an available LES study, with almost similar computational settings (i.e., the inflow turbulence generation method, grid resolution, etc.) shows better performance of ELES in predicting the concentration field. The findings also illustrate that the impact of the unstable stratification condition on turbulence statistics is more pronounced than that of the stable stratification condition. The present article also investigates the effect of thermal stratification conditions on the mechanisms of pollutant dispersion, namely, convective and turbulent diffusion fluxes. The findings reveal that an increase in turbulence kinetic energy (TKE), caused by the unstable thermal stratification condition, increases the concentration fluctuations, which causes the pollutant concentration to be decreased. Furthermore, spectral and proper orthogonal decomposition (POD) analyses are performed for all stratification scenarios. The results show that by altering the thermal condition from isothermal to non-isothermal, either stable or unstable conditions, the contribution of the primary dominant modes to total TKE increases.

Shedding vortex simulation method based on viscous compensation technology research

Owing to the influence of the viscosity of the flow field, the strength of the shedding vortex decreases gradually in the process of backward propagation. Large-scale vortexes constantly break up, forming smaller vortexes. In engineering, when numerical simulation of vortex evolution process is carried out, a large grid is needed to be arranged in the area of outflow field far from the boundary layer in order to ensure the calculation efficiency. As a result, small scale vortexes at the far end of the flow field cannot be captured by the sparse grid in this region, resulting in the dissipation or even disappearance of vortexes. In this paper, the effect of grid scale is quantified and compared with the viscous effect through theoretical derivation. The theoretical relationship between the mesh viscosity and the original viscosity of the flow field is established, and the viscosity term in the turbulence model is modified. This method proves to be able to effectively improve the intensity of small-scale shedding vortexes at the far end of the flow field under the condition of sparse grid. The error between the simulation results and the results obtained by using fine mesh is greatly reduced, the calculation time is shortened, and the high-precision and efficient simulation of the flow field is realized.

Testing the Accuracy of the Cell-Set Model Applied on Vane-Type Sub-Boundary Layer Vortex Generators

Vortex Generators (VGs) are applied before the expected region of separation of the boundary layer in order to delay or remove the flow separation. Although their height is usually similar to that of the boundary layer, in some applications, lower VGs are used, Sub-Boundary Layer Vortex Generators (SBVGs), since this reduces the drag coefficient. Numerical simulations of sub-boundary layer vane-type vortex generators on a flat plate in a negligible pressure gradient flow were conducted using the fully resolved mesh model and the cell-set model, with the aim on assessing the accuracy of the cell-set model with Reynolds-Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES) turbulence modelling techniques. The implementation of the cell-set model has supposed savings of the 40% in terms of computational time. The vortexes generated on the wake behind the VG; vortical structure of the primary vortex; and its path, size, strength, and produced wall shear stress have been studied. The results show good agreements between meshing models in the higher VGs, but slight discrepancies on the lower ones. These disparities are more pronounced with LES. Further study of the cell-set model is proposed, since its implementation entails great computational time and resources savings.

Effects of the Plasmapause on the Radial Propagation of Fast Magnetosonic Waves: An Analytical Approach

AbstractIn this study, analytical approximation is used to solve the wave equations near the plasmapause boundary layer in order to examine the validity of ray tracing approach for fast magnetosonic (MS) wave propagations, and then analytical solutions for electromagnetic fields of MS waves through the plasmapause boundary layer are provided for the first time. Both theoretical calculations from the analytical expressions and observations of Van Allen Probes have indicated that electric fields of MS waves decrease rapidly but magnetic fields increase rapidly when propagating across the plasmapause boundary layer from the outside. Considering effects of the plasmapause width and wave frequency, parameter analysis has shown that when the width of the plasmapause boundary layer is narrow in comparison with the wavelength of MS waves, a significant part of waves will be reflected. In these circumstances, the WKB approximation and then ray tracing method might become invalid, and analytical approach provided in this paper could be utilized to solve the wave equation.

Discrete-time sliding mode control and observation via an adjustable accuracy disturbance estimation

Across the industrial sector, precision engineering represents a set of technologies that are the key to successful international competitive development. Ultra precision machine tools demand to control the point of contact of tool and workpiece within the range 100 nm to 0.1 nm. This work presents the novel Adjustable Accuracy (AA) approximation of a DT function based on user-defined previous steps values. Three control and observation problems are addressed with this novel AA Disturbance Estimation that allows to achieve an adjustable accuracy boundary layer of order O(Tβ) of the corresponding sliding manifold of the closed loop system in the presence of disturbances while avoiding the undesired chattering phenomenon with an ultimate bound that decay exponentially w.r.t. the used number of previous steps for the disturbance approximation. The corresponding simulations show the effectiveness of each scheme.

Radiated sound control from a smart cylinder subjected to piezoelectric uncertainties based on sliding mode technique using self-adjusting boundary layer

This approach employs a robust controller on the basis of sliding mode control to propose a novel strategy nominated as self-adjusting boundary layer in order to prevent occurrence of chattering phenomenon. Since the boundary layers and the controller parameters are adjusted just for the special conditions, it is possible that the system losses its desirable performance and leads to this event. In order to better understand the applied procedure, in addition to classification of the paper in two sections, a closed loop block diagram for a system equipped with sliding mode controller is configured, in detail. Therefore, in the first step, on the basis of considering sliding mode control technique, a robust controller is designed in order to control the radiated noise from an intelligent cylinder. In fact, this procedure is employed to extend the offered new strategy. To cover this issue, a series of formulations are developed. Accordingly, the vibration equations of the construction subjected to piezoelectric patches are derived and discretized according to Rayleigh-Ritz procedure. Additionally, by the aid of using the effective control signal for each mode and extracting uncertainties of the system, the robust control signals for uncertainties in sensor and actuator are determined. The results prove that the considered methodology either suppresses the noise transmitted or keeps the system consistency. Moreover, it establishes a compromise between error and chattering that preserves the stability and admissible performance of the system in a wide range of disturbances and uncertainties. As another consequence, this study also develops a new approach to show the effect of using controller on the acoustic pressures at various coordinates taking account self-adjusting boundary layer.

Tracing Extreme Updrafts in Hurricane Wilma (2005) to Their Boundary Layer Roots Using Trajectories: A Thermodynamic and Structural Analysis

Abstract This study uses a recently developed trajectory model to trace eyewall updrafts in a high-resolution Hurricane Wilma (2005) prediction to their roots in the maritime boundary layer (MBL) in order to better understand their thermodynamics and how they interact with the swirling winds. Out of 97 020 four-hour backward trajectories seeded from the upper troposphere, the 45% of them originating from the MBL are stratified into five subsamples binned by peak vertical velocity wMAX. Of particular interest are the thermodynamic characteristics of parcels belonging to the wMAX-Extreme subsample (i.e., those with wMAX exceeding 20 m s−1) that ascend through Wilma’s strongest convective burst (CB) cores. A vertical momentum budget computed along a selected wMAX-Extreme trajectory confirms that the parcel possesses large positive buoyancy that more than compensates for negative hydrometeor loading to yield an upper-tropospheric wMAX ~ 30 m s−1. Comparing all 1170 wMAX-Extreme trajectories with all 19 296 secondary circulation trajectories shows that the former tends to originate from the MBL where equivalent potential temperature θe and ocean surface heat and moisture fluxes are locally enhanced. The wMAX-Extreme parcels become further differentiated from the background ascent in terms of their (i) greater updraft width and smaller θe reduction while ascending into the midtroposphere, implying lower environmental entrainment rates, and (ii) less hydrometeor loading in the z = 3–5-km layer. The Lagrangian analysis herein bridges two previous studies that focused separately on the importance of high SSTs and fusion latent heat release to the development of CBs, the latter of which may facilitate upper-level warm core development through their compensating subsidence.

Velocity overshoot within the accelerated subsonic boundary layer over the heated wall

The paper presents results on the numerical investigation of the accelerated compressible flow near the hot surface with constant heat flux. Such conditions may generate a phenomenon of the overshoot, when the streamwise velocity within the boundary layer is higher than the velocity of the main flow. We simulated properties of helium-xenon mixture and superheated water vapor. Numerical model consist of the system of Prandtl equations, which describe dynamic and thermal processes in the boundary layer, and k-ω-γ turbulence model that allows simulate reasonably as turbulence as laminar-turbulent transition. Finite difference methods were applied for solving of differential equations. The numerical model was able to block artificial the term of viscous dissipation in the energy equation and set a constant temperature at the outer border of the boundary layer in order to evaluate the effect of compressibility on the overshoot phenomenon and other flow characteristics. It is shown that the unambiguous relationship between skin friction coefficient and the overshoot magnitude is violated for Mach number from 0.4 to 1.0. Maximal values are disappeared at representation of velocity through Mach number.

Large‐eddy simulations of the evolution of imposed turbulence in forced boundary layers in a very long domain

AbstractThe technique of using imposed turbulence in combination with a forced boundary layer in order to model the atmospheric boundary layer is analyzed for a very long domain using large‐eddy simulations with different combinations of prescribed velocity profiles and pregenerated turbulence fields based on the Mann model. The ambient flow is first studied in the absence of wind turbines. The velocity profiles undergo a transition throughout the domain with a velocity increase of 10% to 15% close to the ground far downstream in the domain. The turbulence characteristics close to the turbulence plane are, as expected, similar to those of the added Mann turbulence. The turbulence will then undergo a transition throughout the domain to finally reach a balance with the shear profile at a certain downstream distance. This distance is found to depend on the turbulence level of the added Mann turbulence planes. A lower Mann turbulence level generally results in a shorter “balancing” distance. Secondly, a row of 10 turbines is imposed in the simulations at different distances from the plane of turbulence in order to determine how the distance affects wake conditions and power production levels. Our results show that a “balancing” distance is needed between the turbulence plane and the first turbine in the row in order to ensure nonchanging ambient conditions throughout the turbine row. This introduces an increase in the computational costs. The computational cost for the forced boundary technique is normally lower compared with using precursor simulations, for longer domains; however, this needs to be verified further.

Sliding Mode Control with Gaussian Process Regression for Underwater Robots

Sliding mode control is a very effective strategy in dealing not only with parametric uncertainties, but also with unmodeled dynamics, and therefore has been widely applied to robotic agents. However, the adoption of a thin boundary layer neighboring the switching surface to smooth out the control law and to eliminate the undesired chattering effect usually impairs the controller’s performance and leads to a residual tracking error. As a matter of fact, underwater robots are very sensitive to this issue due to their highly uncertain plants and unstructured operating environments. In this work, Gaussian process regression is combined with sliding mode control for the dynamic positioning of underwater robotic vehicles. The Gaussian process regressor is embedded within the boundary layer in order to enhance the tracking performance, by predicting unknown hydrodynamic effects and compensating for them. The boundedness and convergence properties of the tracking error are analytically proven. Numerical results confirm the improved performance of the proposed control scheme when compared with the conventional sliding mode approach.

$$L^\infty $$ Instability of Prandtl Layers

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz was that the solution of Navier Stokes equations can be described as a solution of Euler equations, plus a boundary layer corrector, plus a vanishing error term in $$L^\infty $$ in the inviscid limit. In this paper we prove that, for a class of smooth solutions of Navier Stokes equations, namely for shear layer profiles which are unstable for Rayleigh equations, this Ansatz is false if we consider solutions with Sobolev regularity, in strong contrast with the analytic case, pioneered by Sammartino and Caflisch (Commun Math Phys 192(2)433–461, 1998; Commun Math Phys 192(2)463–491, 1998). Meanwhile we address the classical problem of the nonlinear stability of shear layers near a boundary and prove that if a shear flow is spectrally unstable for Euler equations, then it is non linearly unstable for the Navier Stokes equations provided the viscosity is small enough.