Boundary Layer Profiles Research Articles

Gain in efficiency of jet receptivity to acoustic disturbances with increasing nozzle-lip thickness

The gain in efficiency of the receptivity of jets to acoustic disturbances as the nozzle lip is thicker is investigated using numerical simulations. For that, axisymmetric acoustic pulses are introduced in jets with Blasius laminar boundary-layer profiles at Mach numbers $M=0.4$ , 0.6, 0.9 and 1.3 for nozzle-lip thicknesses between 1 % and 93 % of the nozzle radius. They are located on the jet axis or outside the jet with incidence angles $\varphi$ between $5^{\circ }$ and $90^{\circ }$ with respect to the downstream direction. Instability waves develop in the jet shear layer after the acoustic disturbances hit the nozzle. In all cases except for $\varphi \geq 75^{\circ }$ , their amplitudes and hence the efficiency of the jet receptivity to the disturbances increase with the nozzle-lip thickness. The gains in efficiency are greater for a pulse inside the jet, generating upstream-travelling pressure waves resembling guided jet waves, than for a pulse outside the jet, producing free-stream sound waves. In the second case, the gains are significant for $\varphi =5^{\circ }$ and decrease with the incidence angle, especially for $\varphi >30^{\circ }$ . Moreover, the gains are stronger for a higher Mach number, and roughly double between $M=0.4$ and $1.3$ , thus reaching, for a pulse inside the jet, values close to 6 between the thinnest and thickest lips. Finally, according to additional simulations for $M=0.9$ , the gains in receptivity efficiency do not change appreciably when different azimuthal mode numbers of the acoustic disturbances, widths of the pulse and shapes of the boundary-layer profile are considered.

Stability analysis of dual solutions for nonlinear radiative magnetohydrodynamic flow of [formula omitted] hybrid nanofluid over a nonlinearly shrinking surface

The present work explores the existence and temporal stability of dual solutions with the consequence of nonlinear thermal radiation on hybrid nanofluid flow (including Silver and Titanium dioxide nanoparticles in water) past a permeable shrinking sheet. The analysis also considers the united outcomes of suction/injection, Joule heating, viscous dissipation, and magnetohydrodynamics. Employing realistic assumptions and suitable similarity transformations, the governing nonlinear partial differential equations are formulated and altered into a nonlinear ordinary differential equations system. These equations are then solved numerically using the Lobatto IIIA technique. It is observed that dual solutions can occur within a precise range of the shrinking surface parameter. Analyzing the temporal stability of these dual solutions under small disturbances shows that the upper solution branch is stable and represents a physically realistic solution to the problem. In contrast, the lower solution branch is unstable and not physically realizable. No solution exists beyond the critical λc=−1.3915 for the shrinking parameter. The critical values of shrinking parameter λc=−1.2565,−1.3188, and −1.4161 correspond to volume fractions of hybrid nanofluid with φAg−TiO2 of 2%,4%, and 8%, respectively. No solution exists beyond this critical point. The graphical representation and quantitative explanation demonstrate how various emergent characteristics affect momentum and thermal boundary layer profiles, Nusselt number, and skin friction. The velocity and temperature profile of hybrid nanofluid can be improved by adding a small volume of nanoparticle volume fraction. Thermal boundary layer thickness is amplified by an accumulation in the shrinking parameter, power index of velocity component, thermal radiation parameter, and temperature difference ratio. Sheering stress and heat transfer coefficient improve with adding more nanoparticles in the base fluid for the first solution braches, but opposite effects are found in the second solutions. The contributions of this research are significant both theoretically, in terms of advancing the mathematical modeling of hybrid nanofluid flow with heat transfer in engineering systems, and practically, in terms of applications to engineering cooling systems.

Wall-Modeled Large-Eddy Simulation Method for Unstructured-Grid Navier–Stokes Solvers

This paper reports on the implementation and assessment of a wall-modeled large-eddy simulation (WMLES) methodology in an unstructured-grid, node-centered flow solver, FUN3D, that is developed and supported at the NASA Langley Research Center. Finite-volume (FV) and finite-element (FE) discretization schemes considered in the study provide formal second-order spatial accuracy. Large-eddy simulations (LES) resolve large-scale turbulent-flow features, and small-scale effects are modeled using the Vreman subgrid-scale model. At solid-wall boundaries, a shear-stress model is employed to provide a proper boundary-flux closure. The nonlinear equations are integrated in time using either an optimized backward difference formula or an implicit multistage Runge–Kutta temporal scheme. The implicit equations at each time step are solved by strong nonlinear iteration schemes. WMLES demonstrations are shown for two high-lift configurations, namely, the McDonnell Douglas 30P30N multi-element airfoil and a NASA High-Lift Common Research Model. Results show that the WMLES approaches implemented in the FV and FE discretization methods produce consistent solutions and are capable of capturing key aerodynamic characteristics and flow structures for high-lift configurations at a wide range of angles of attack, including maximum-lift conditions. In the 30P30N example, correct trends in the variations of integrated aerodynamic forces and moments, surface pressure distributions, and boundary-layer profiles are captured as the Reynolds number is increased.

Design and characterisation of an open-jet pressure gradient test rig for an aeroacoustic wind tunnel

This paper summarises an approach to generating controllable pressure gradients within an open-jet configuration for aeroacoustics research. A novel open-jet pressure gradient test rig has been designed for the UNSW Anechoic Wind Tunnel with the help of RANS simulations. A range of pressure gradients is created by adjusting the inclination angle of the top plate to change the cross-sectional area along the streamwise direction gradually. The test model mounting point is located on the bottom plate adjacent to partially opened side walls to allow far-field noise measurements. A comprehensive characterisation of flow quality, pressure gradient parameters and acoustic data quality has been carried out using Particle Imaging Velocimetry (PIV), surface pressure taps, and a microphone array. The test rig produces near-uniform pressure gradient flows at the model mounting point, with momentum Reynolds numbers (Reθ) ranging from 3014 to 11853 and Clauser's pressure gradient parameters (β) from -0.24 to 1.66. The pressure gradients generated by this facility are approximately linear, approaching the model mounting point, and the boundary layer profiles compare favourably with those from conventional hard-walled enclosed pressure gradient wind tunnel facilities. Measurements of airfoil trailing-edge noise from this test rig compare well with classical semi-empirical model predictions. Simultaneous acoustic and flow measurements on a square finite wall-mounted cylinder showcase the capability of this facility for coupled acoustic-flow diagnosis.

Roughness constant selection for atmospheric boundary layer simulations using a k-ω SST turbulence model within a commercial CFD solver

Transit of an atmospheric boundary layer (ABL) profile within computational fluid dynamics (CFD) simulations is often hindered through unintended streamwise gradients in the flow resulting in ABL inhomogeneity. Within Reynolds-Averaged Navier–Stokes (RANS) turbulence modeling in Ansys Fluent, user-defined wall functions are not available for the k–ω class of RANS models as a remedy to this problem. Instead, the practitioner is required to use the sand–grain roughness property for ground surface roughness calibration, and specify a roughness height and roughness constant, Cs, accordingly. To-date, no clear guidance on their selection is available that can accommodate both high and low roughness terrains. To overcome this, a straightforward and practical calculation is presented for the roughness constant based on the standard wall function implementation. When this calculation is synthesized with other best-practice guidelines, it is possible to reliably model transiting ABL profiles based on wind-tunnel data whilst minimizing effects due to ABL inhomogeneity.

Streamwise-Elongated Sinusoidal Roughness Elements with Enhanced Laminarizing Effect on Three-Dimensional Boundary Layer

As a laminar flow control device for delaying the crossflow-induced transition of a three-dimensional boundary layer, sinusoidal roughness elements (SREs) are placed in a Falkner–Skan–Cooke boundary layer, and the resultant laminarizing effect is numerically investigated in comparison with discrete roughness elements (DREs). Because SREs are elongated in the streamwise direction and designed to avoid flow tripping, the critical height of SREs is much higher than that of DREs. Moreover, the wake flow behind SREs efficiently generates and sustains crossflow vortices that are not dangerously unstable against secondary instabilities but able to strongly distort the mean crossflow profile into a less unstable one. By measuring this mean flow distortion by SREs and DREs, the laminarizing effect is compared among them. It is shown that the effect of SREs is higher than that of DREs and can be enhanced by choosing the appropriate height, angle, and wavelength depending on the local boundary-layer profile.

Contextualizing Polarimetric Retrievals of Boundary Layer Height Using State-of-the-Art Boundary Layer Profiling

Abstract Knowledge about the depth of the planetary boundary layer (PBL) is crucial for a variety of applications, but direct observations of PBL depth are spatiotemporally sparse. Recent studies have proposed using operational dual-polarization weather radars to observe the evolution of PBL depth by capitalizing on unique differential reflectivity (ZDR) signatures of Bragg scatter at the top of the PBL. While this approach appears promising and cost-effective, uncertainties remain about the representativeness of these estimates and how its efficacy may vary by geography and climatology. To address these outstanding uncertainties, this study compares collocated observations collected from two WSR-88D radars and two state-of-the-art mobile boundary layer profiling systems and evaluates the proposed methodology over the full diurnal cycle. Results indicate good overall correspondence between the profiling- and radar-based PBL depth estimates, with an abrupt divergence during the early evening transition and large discrepancies overnight. Relatively large root-mean-square-deviations (RMSDs) coupled with small biases match expectations when comparing spatially averaged data with point observations during PBL growth, which capture frequent fluctuations. A qualitative examination of the radar data reveals signatures of elevated residual layers, clouds, and ground clutter, all of which can obfuscate the desired surface-based PBL signal but which may have their own utility. The prominence of the Bragg scatter signal is found to be correlated with the observed moisture gradient at the top of the PBL, reflecting climatological variability that should be considered. These findings motivate further work to improve the automated detection of Bragg scatter layers from polarimetric radar data. Significance Statement Knowledge of the height of the planetary boundary layer matters for weather forecasting, air quality, and renewable energy production. Currently, boundary layer height measurements are taken at select locations twice a day. However, a method to use the existing national network of polarimetric weather radars for this purpose has been proposed. This work evaluates this method against specialized boundary layer measurements. The results show that the method is generally reliable during the daytime and could be used for a variety of applications including climatologies and model evaluation. There remain a number of situational caveats, including residual turbulence, clouds/precipitation, ground clutter, and certain meteorological environments, that may require modification of the approach and need to be considered in future work.

Reduced-Order Modeling of Steady and Unsteady Flows with Deep Neural Networks

Large-eddy and direct numerical simulations generate vast data sets that are challenging to interpret, even for simple geometries at low Reynolds numbers. This has increased the importance of automatic methods for extracting significant features to understand physical phenomena. Traditional techniques like the proper orthogonal decomposition (POD) have been widely used for this purpose. However, recent advancements in computational power have allowed for the development of data-driven modal reduction approaches. This paper discusses four applications of deep neural networks for aerodynamic applications, including a convolutional neural network autoencoder, to analyze unsteady flow fields around a circular cylinder at Re = 100 and a supersonic boundary layer with Tollmien–Schlichting waves. The autoencoder results are comparable to those obtained with POD and spectral POD. Additionally, it is demonstrated that the autoencoder can compress steady hypersonic boundary-layer profiles into a low-dimensional vector space that is spanned by the pressure gradient and wall-temperature ratio. This paper also proposes a convolutional neural network model to estimate velocity and temperature profiles across different hypersonic flow conditions.

Evidence of quasiequilibrium in pressure-gradient turbulent boundary layers

Two sets of measurements utilizing hot-wire anemometry and oil-film interferometry for flat-plate turbulent boundary layers, exposed to various controlled adverse and favourable pressure gradients, are used to evaluate history effects of the imposed and varying free-stream gradients. The results are from the NDF wind tunnel at Illinois Tech (IIT) and the MTL wind tunnel at KTH, over the range $800 < Re_\tau < 22\,000$ (where $Re_{\tau }$ is the friction Reynolds number). The streamwise pressure-gradient parameter $\beta \equiv (-\ell /\tau _{w}) \cdot (\partial P_{e}/\partial x)$ varied between $-2 < \beta < 7$ , where $\ell$ is an outer length scale for boundary layers equivalent to the half-height of channel flow and the radius of pipe flow, and is estimated for each boundary-layer profile; note that $\tau_w$ is the wall-shear stress and $P_e$ is the free-stream static pressure. Extracting from each profile the three parameters of the overlap region, following the recent work of Monkewitz & Nagib (J. Fluid Mech., vol. 967, 2023, p. A15) that led to an overlap region of combined logarithmic and linear parts, we find minimum history effects in the overlap region. Thus, the overlap region in this range of pressure-gradient boundary layers appears to be in ‘quasiequilibrium’.

Improved Prediction of the Flow in Cylindrical Critical Flow Venturi Nozzles Using a Transitional Model

Critical flow Venturi nozzles (CFVNs) are a state-of-the-art secondary standard widely used for gas flow measurements with high precision. The flow rate correlates with the type and thickness of the boundary layer (BL) inside the nozzle throat. In the cylindrical type—one of the two standard designs of CFVNs—the nozzle throat encompasses a defined axial length in which the BL develops. This numerical study is concerned with the BL effects in a cylindrical CFVN by means of two turbulence models. Compared to experimental data, the k-ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega$$\\end{document} SST model predicts the discharge coefficient well for high and low Reynolds numbers, but not in the intermediate regime. The γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma$$\\end{document}-Reθ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re_{\ heta }$$\\end{document} model, on the contrary, agrees well with experimental data in the entire flow range. Relevant quantities and profiles of the BL are separately investigated in the laminar, turbulent, and transitional region. The calculated laminar and turbulent BL thicknesses correspond to predictions based on integral methods for solving the BL equations. Simple representations are proposed for the Zagarola-Smits scaled laminar and turbulent deficit BL profiles removing the effects of axial position, Reynolds number, and pressure gradient. Furthermore, the shape factor is investigated as a characteristic parameter for determining the transitional region.

Unsteady cylinder wakes from arbitrary bodies with differentiable physics-assisted neural network.

This work describes a hybrid predictive framework configured as a coarse-grained surrogate for reconstructing unsteady fluid flows around multiple cylinders of diverse configurations. The presence of cylinders of arbitrary nature causes abrupt changes in the local flow profile while globally exhibiting a wide spectrum of dynamical wakes fluctuating in either a periodic or chaotic manner. Consequently, the focal point of the present study is to establish predictive frameworks that accurately reconstruct the overall fluid velocity flowfield such that the local boundary layer profile, as well as the wake dynamics, are both preserved for long time horizons. The hybrid framework is realized using a base differentiable flow solver combined with a neural network, yielding a differentiable physics-assisted neural network (DPNN). The framework is trained using bodies with arbitrary shapes, and then it is tested and further assessed on out-of-distribution samples. Our results indicate that the neural network acts as a forcing function to correct the local boundary layer profile while also remarkably improving the dissipative nature of the flowfields. It is found that the DPNN framework clearly outperforms the supervised learning approach while respecting the reduced feature space dynamics. The model predictions for arbitrary bodies indicate that the Strouhal number distribution with respect to spacing ratio exhibits similar patterns with existing literature. In addition, our model predictions also enable us to discover similar wake categories for flow past arbitrary bodies. For the chaotic wakes, the present approach predicts the chaotic switch in gap flows up to the mid-time range.

The Wind Profile Characteristics of Super Typhoon Lekima Based on Field Measurement

Many cities in coastal areas are prone to typhoon disasters due to their location on the Pacific storm path, and the direct effect of catastrophic winds can lead to the destruction of low-rise buildings and severe damage to high-rise structures. The purpose of this study was to enhance the understanding of boundary layer wind profiles of strong typhoons in coastal areas and reduce property losses and casualties caused by wind disasters. Based on the field measurements of wind profile acoustic radar in coastal areas, the variation characteristics of the boundary layer wind profile during the passage of super typhoon Lekima were first studied in depth, and the evolution law of the typhoon boundary layer profile was summarized. Then, the effects of typhoon horizontal structure, topography, wind speed, and time distance on the characteristics of the typhoon profile were discussed, respectively. Finally, the evolution characteristics of wind profile parameters were obtained by fitting three wind profile theoretical models. Due to the strong variability of typhoon profile morphology, the theoretical model of wind profile is only applicable to the wind profile from the bottom to the low-level jet height of typhoons, while wind parameters are closely related to the spatial location of the typhoon wind field.

Numerical Study of the Thermal Performance of the combined effect of solar energy and variable density around a Laminar Vertical jet

A computational analysis due to the thermal performance of combined effects of solar radiation and variable density around a laminar vertical jet is carried out. For this purpose, a mathematical model in terms of nonlinear coupled partial differential equations, keeping in view the coordinate system of the proposed problem, is established. Further, to get insight to the physical significance of the proposed problem, numerical data is obtained. For this purpose, the finite difference method is used to integrate the primitive form of the partial differential equations. The coefficient matrices of the unknown variables involved in the model are solved by using the Gaussian elimination technique with the help of Fortran-Laher 90. The obtained numerical results for different parameters involved in the flow model, that is density parameter m, Prandtl number Pr, radiation parameter Rd, and fluid's absorption parameter D, are highlighted graphically by using Techplot-360. The primary findings show that enhancing values of the density variation parameter m leads to an increase in the velocity profile and a decrease in the thermal profile. On the other hand, for increasing values of Rd, the boundary layer profiles decline. Also, the heat transfer rate prominently declines with the rise in numeric value of m, Rd, and D. The main novelty of the current work is to improve the thermal performance of natural convection heat transfer with the inclusion of solar energy and variable density.

Experimental and numerical investigation of porous bleed control for supersonic/subsonic flows, and shock-wave/boundary-layer interactions

This paper presents a comprehensive experimental and numerical investigation into the control of boundary layers using porous bleed systems. The study focuses on both supersonic and subsonic flow regimes, as well as on the control of shock-wave/boundary-layer interactions. By simplifying this complex problem into two separate scenarios, the research establishes the necessity of individually characterizing bleed systems for both supersonic and subsonic flow regimes due to variations in the flow topology within the bleed holes. In supersonic conditions, a strong agreement between experimental and numerical results validates the effectiveness of porous bleed in controlling boundary layers. Notably, three-dimensional effects are experimentally demonstrated, and variations of the boundary-layer profiles along the span are the first time experimentally proven. The study extends its scope to subsonic flows, revealing that while boundary-layer bleeding enhances flow momentum near the wall, mass removal induces a decrease in momentum in the outer boundary layer and external flow. The research also explores shock-wave/boundary-layer interaction control, achieving a remarkable alignment between simulations and experiments. The findings endorse the use of Reynolds-Averaged Navier-Stokes simulations for studying porous bleed systems in various flow conditions, providing valuable insights to enhance bleed models, especially in the context of shock-wave/boundary-layer interaction control.

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.

Tollmien–Schlichting waves in the subsonic regime

AbstractThe Tollmien–Schlichting (T‐S) waves play a key role in the early stages of boundary layer transition. In a breakthrough work, Grenier, Guo, and Nguyen gave the first rigorous construction of the T‐S waves of temporal mode for the incompressible fluid. Yang and Zhang recently made an important contribution by constructing the compressible T‐S waves of temporal mode for certain boundary layer profiles with Mach number . In this paper, we construct the T‐S waves of both temporal mode and spatial mode to the linearized compressible Navier–Stokes system around the boundary layer flow in the whole subsonic regime , including the Blasius profile. Our approach is based on a novel iteration scheme between the quasi‐incompressible and quasi‐compressible systems, with a key ingredient being the solution of an Orr–Sommerfeld‐type equation using a new Airy–Airy–Rayleigh iteration instead of Rayleigh–Airy iteration introduced by Grenier, Guo, and Nguyen. We believe that the method developed in this work can be applied in solving other related problems for subsonic flows.

The wake of a large wind turbine in stable atmospheric boundary layer flow, simulated in the EnFlo stratified-flow wind tunnel

The EnFlo stratified-flow wind tunnel is described, and the parameterization of stable atmospheric boundary layers in the context of wind turbines and stability classification is given, as are the scaling constraints for stratified-flow wind tunnel experiments. Wake measurements of mean velocity, Reynolds stresses, and turbulent heat flux were made for three mild stable states, including overlying inversion, and the neutral state. The depth of the atmospheric boundary layer was kept constant, and so the results show the effect of change in stability alone, without the change in scale that would also arise in full scale measurements, an advantage provided by wind tunnel experiments. The simulated boundary-layer wind-speed profile is the same in each case, its height slightly exceeding the blade-tip top height, and the velocity deficit at the turbine is also the same in each, implying a constant thrust coefficient. In the near wake the momentum deficit rises more rapidly in the stable cases, and stays higher further downstream where it is subsequently reduced by turbulent mixing. The wake grows less rapidly in the vertical direction in the stable cases, both above and below the hub height, the height above growing still less rapidly with an imposed inversion, while the height below is unaffected by the inversion. The wake width is largely unaffected by stability. Stability reduces the Reynolds stresses in the wake over and above the reduction in the undisturbed flow; there is not a simple superposition. In the lower part of the wake the stresses are not affected by the inversion, while they are in the upper part. Turbulent heat flux is increased in the bulk of the wake, more so with an inversion, but is reduced to the surface and unaffected by an inversion.

Shear flow effects in a 2D duct: Influence on wave propagation and direct impedance eduction

Impedance eduction of acoustic liners with flow is commonly performed under the assumption of a uniform velocity profile. In practice, the flow profile is not uniform and a significant mean flow shear is present close to the duct wall. The present paper aims to assess the validity of the uniform flow assumption, and more generally to study the effect of the mean flow shear on the impedance eduction process. This is particularly relevant when considering large ducts, high frequencies, high-order acoustic modes and flow velocities representative of aircraft engine nacelles. A numerical mode-matching model is used to compute the sound field in a 2D lined duct with different flow profiles. A detailed parametric study evaluates the influence of the mean flow shear on the axial wavenumbers of the acoustic modes and on the educed impedance. Various parameters are considered: the flow velocity profile (boundary layer thickness and mean Mach number), the direction of acoustic propagation relative to the flow, two different liner configurations and the presence of noise in the pressure data. A number of recommendations are given to achieve robust impedance eduction in large-duct facilities and at high-frequencies.

Improving Three-Dimensional Synthetic Jet Modeling in a Crossflow

Abstract Three different circular synthetic jet modeling inlet conditions are studied for a turbulent crossflow. The study examines the differences when modeling the whole synthetic jet actuators (SJA), neck-only or jet-slot-only under constant actuation frequency (f = 300 Hz), and crossflow blowing ratio (CB = 0.67). Phase-averaged and time-averaged results reveal that both whole SJA and neck-only methods generated nearly identical flow fields. For the neck-only case, a notable reduction in computational cost is achieved through the implementation of an analytical jet profile. The jet-slot-only method, on the other hand, introduces reversed flow during the ingestion cycle, leading to the injection of false-momentum into the crossflow. However, the false-momentum primarily affects the flow immediately downstream of the jet exit, with the boundary layer profile recovering rapidly. A parametric study highlights the importance of maintaining a volume ratio less than 1 of ingested to modeled neck volume to prevent the creation of false-momentum.

Characteristic boundary layers in the vanishing viscosity limit for the Hunter-Saxton equation

The Hunter-Saxton equation models the propagation of weakly nonlinear orientation waves in a massive director field of the nematic liquid crystal. In this paper, we study the vanishing viscosity limit for an initial boundary value problem of the Hunter-Saxton equation with the characteristic boundary condition. By the formal multiscale analysis, we first derive the characteristic boundary layer profile, which satisfies a nonlinear parabolic equation. On the base of the Galerkin method along with a compactness argument, we then establish the global well-posedness of the boundary layer equation. Finally, we prove the global stability of the boundary layer profiles together with the optimal convergence rate of the vanishing viscosity limit by the energy method.