Boundary Layer Terms Research Articles

Temperature statistics in turbulent Rayleigh–Bénard convection with a Prandtl number of Pr = 12.3

The transport of plumes in turbulent convective systems must be understood to study the mantle and various industrial applications. We measured the probability density function P(T) of the temperature at various radial and vertical positions in the bulk of a convection cell. The asymmetric-shaped distribution was decomposed into a turbulent background and plumes. The temperature of the turbulent background was fitted by a Gaussian function according to the peak of P(T). We proposed a simple quantity A ≡ (⟨T⟩ − Tbg) to describe the effective strength of the plume, where ⟨T⟩ is the time-averaged value of the local temperature. The hot plume diminishes as it rises in the cell. The plume strength varies logarithmically with the vertical position. For larger Ra, the plume along the centerline has a longer travel distance in terms of the thermal boundary layer. For a given Ra, the strength and travel distance of the plume increase as the measurements move closer to the sidewall. At the cell center, the temperature fluctuations can be decomposed into fluctuations due to the turbulent background σbg and fluctuations due to the plume. The value of σbg is so small that the relation between σbg and the vertical position can be fitted by a logarithmic function or a power law. The Ra dependence on these two fluctuations was also investigated. The measurements were collected in a cylindrical cell with a unity aspect ratio of 1, and FC72 was used as the working fluid.

Investigation of the stress-strain state of a hollow cylinder with a coating based on the gradient model of thermoelasticity

The study of the stress-strain state of a thermoelastic hollow cylinder with a homogeneous coating is carried out taking into account the scale effects. Aifantis' one-parameter gradient model is used to account for scale effects. Equilibrium equations and boundary conditions for a composite hollow thermoelastic cylinder are obtained on the basis of the Lagrange variational principle. In comparison with the classical formulation of the problem, additional boundary conditions and conjugation conditions are set for moment stresses and displacement gradients. The dimensionlessness of the task of thermoelasticity has been carried out. Solving the problem of uncoupled thermoelasticity begins with finding the radial temperature distribution of a layered cylinder on the basis of solving the problem of heat conduction in the classical formulation. The solution of the problem in displacements is presented as a sum of solutions in the classical formulation of the problem and additional boundary layer terms found on the basis of the asymptotic properties of the modified Bessel functions. Simplified analytical expressions are obtained for finding radial displacements, radial and circumferential Cauchy stresses, nonzero components of the tensor of moment and total stresses. On specific examples, calculations of the radial distribution of displacements and stresses of a composite cylinder in the case of both mechanical and thermal loading are carried out. The limits of applicability of the asymptotic solution of the problem are investigated. The difference between the radial distribution of displacements and stresses found on the basis of solutions to the problem in the classical formulation and in the gradient formulation is shown. It was found that the Cauchy radial stresses experience a jump at the boundary of the cylinder and the coating, which is explained by the continuity of radial displacements and their first derivatives. The components of the moment stress tensor either take on peak values or experience a jump at the interface. The moment stresses are proportional to the square of the gradient parameter, at small values of which they have values that are much less than the values of the total stresses. With an increase in the dimensionless scale parameter, the values of radial displacements and total circumferential stresses decrease, but moment stresses increase.

Large-eddy simulation and streamline coordinate analysis of flow over an axisymmetric hull

A new perspective on the analysis of turbulent boundary layers on streamlined bodies is provided by deriving the axisymmetric Reynolds-averaged Navier–Stokes equations in an orthogonal coordinate system aligned with streamlines, streamline-normal lines and the plane of symmetry. Wall-resolved large-eddy simulation using an unstructured overset method is performed to study flow about the axisymmetric DARPA SUBOFF hull at a Reynolds number of $Re_L = 1.1 \times 10^{6}$ based on the hull length and free-stream velocity. The streamline-normal coordinate is naturally normal to the wall at the hull surface and perpendicular to the free-stream velocity far from the body, which is critical for studying bodies with concave streamwise curvature. The momentum equations naturally reduce to the differential form of Bernoulli's equation and the $s$ – $n$ Euler equation for curved streamlines outside of the boundary layer. In the curved laminar boundary layer at the front of the hull, the streamline momentum equation represents a balance of the streamwise advection, streamwise pressure gradient and viscous stress, while the streamline-normal equation is a balance between the streamline-normal pressure gradient and centripetal acceleration. In the turbulent boundary layer on the mid-hull, the curvature terms and streamwise pressure gradient are negligible and the results conform to traditional analysis of flat-plate boundary layers. In the thick stern boundary layer, the curvature and streamwise pressure gradient terms reappear to balance the turbulent and viscous stresses. This balance explains the characteristic variation of static pressure observed for thick boundary layers at the tails of axisymmetric bodies.

Boundary-layer evolution over long wind farms

The structure of the internal boundary layer above long wind farms is investigated experimentally. The transfer of kinetic energy from the region above the farm is dominated by the turbulent flux of momentum together with the displacement of kinetic energy operated by the mean vertical velocity: these two have comparable magnitude along the farm opposite to the infinite-farm case. The integration of the energy equation in the vertical highlighted the key role of the energy flux, and how that is balanced by the growth of the internal boundary layer in terms of energy thickness with a small role of the dissipation. The mean velocity profiles seem to follow a universal structure in terms of velocity deficit, while the Reynolds stress does not follow the same scaling structure. Finally, a spectral analysis along the farm identified the leading dynamics determining the turbulent activity: while behind the first row the signature of the tip vortices is dominant, already after the second row their coherency is lost and a single broadband peak, associated with wake meandering, is present until the end of the farm. The streamwise velocity peak is associated with a nearly constant Strouhal number weakly dependent on the farm layout and free stream turbulence condition. A reasonable agreement of the velocity spectra is observed when the latter are normalised by the velocity variance and integral time scale: nevertheless the spectra show clear anisotropy at the large scales and even the small scales remain anisotropic in the inertial subrange.

ASYMPTOTIC EXPANSION OF SOLUTION OF BVP WITH INITIAL JUMPS FOR SINGULARLY PERTURBED INTEGRO-DIFFERENTIAL EQUATION

We consider the undivided boundary value problem for singularly perturbed linear integro-differential equations of th order with Fredholm’s integral terms have an initial jump of the phenomenon th order on the left end of the segment. Defined regular and boundary layer members of the asymptotic expansions of the solutions. Regular members of the asymptotics constructed in the form of integro-differential equations, which different from the usual unperturbed equations by the presence of additional term called initial jump of integral terms. The values of the initial jumps are defined. Boundary conditions for the regular members of the asymptotics also contain additional term, called an initial jump of the derivatives th order. Thus, for the determination of the regular members of the asymptotics obtained boundary value problems for linear integro-differential with additional parameter. To determine the boundary layer of the asymptotic expansion of solution we obtained initial problems for homogeneous an inhomogeneous ordinary differential equations with constant coefficients. Exponential estimates for boundary layer terms of the asymptotics are obtained. A theorem of existence and uniqueness and the asymptotic representations of solution with an estimate of the remainder term of the asymptotics. It is found that the constructed asymptotics approximation to the solution of the original singularly perturbed integro-differential boundary value problem is uniformly throughout the considered interval.

Time-Resolved Ship Airwake Measurements in a Simulated Atmospheric Boundary Layer

The unsteady flows produced over the stern of a Simple Frigate Shape 2 ship model are studied in a low-speed wind tunnel. Time-resolved particle image velocimetry (TR-PIV) measurements were performed in several streamwise and cross planes. Two grids of Cowdrey rods were used to simulate the basic characteristics of an atmospheric boundary layer in terms of upstream velocity profile and turbulence intensity. The TR-PIV measurements showed large regions of unsteady flow separation, dominant vortical flows, and significant wall-normal flows over the flight deck regions caused, in part, by the irregular shedding of vorticity and turbulence from the upstream funnel and superstructure of the ship. Turbulence intensities were found to be particularly high over the flight deck. The measured flows also suggested the existence of asymmetric, intermittent flow in the near-wall regions of the deck, and time-varying fluctuations in the recirculation region behind the hangar. The crossplane measurements showed the development of shear layers at the corners of the flight deck on both the port and starboard sides, and sets of counterrotating vortices at the edges of the flight deck. An energy spectrum analysis showed dominant frequencies in the regions where the shear layer was developed behind the funnel and above the flight deck. Spectral proper orthogonal decomposition revealed that the physically relevant coherent structures in the airwake were low-frequency modes near the flight deck and at large-scale of the order of the length of the deck.

Near-critical reflection of internal waves

Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes' laws, and it is expected to be singular if the slope has the same inclination as the group velocity. In this paper, we prove that in this critical geometry the weakly viscous and weakly nonlinear wave equations have actually a solution which is well approximated by the sum of the incident wave packet, a reflected second harmonic and some boundary layer terms. This result confirms the prediction by Dauxois and Young, and provides precise estimates on the time of validity of this approximation.

An Adaptive Formulation of the Sliding Innovation Filter

In this paper, an adaptive formulation of the sliding innovation filter (SIF) is presented. The SIF is a recently proposed estimation strategy that has demonstrated robustness to modeling errors and uncertainties. It utilizes a switching gain that is a function of the innovation (measurement error) and sliding boundary layer term. In this paper, a time-varying sliding boundary layer is derived based on minimizing the state error covariance. The resulting solution creates an adaptive formulation of the SIF. The adaptive SIF is applied on a linear aerospace system, and is compared with the well-known Kalman filter (KF) and the standard SIF. The results demonstrate the robustness of the new estimation strategy in the presence of modeling uncertainties and system faults.

Solution of the gradient thermoelasticity problem for a cylinder with a heat-protected coating

The investigation of the stress-strain state of an infinitely long thermoelastic cylinder is carried out taking into account the scale effects. A thermal protective coating is applied to the outer side surface of the cylinder, the thermomechanical characteristics of which are functions of the radial coordinate. Thermal boundary conditions of the first kind are set on the stress-free side surfaces of the cylinder. To take into account the scale effects, the one-parameter gradient theory of thermoelasticity proposed by Aifantis is used. Additional boundary conditions and conjugation conditions for couple stresses are specified. The displacements and stresses are represented as the sum of the solutions of the thermoelasticity problem in the classical formulation and the gradient parts. After finding the radial temperature distribution, the thermoelasticity problem in the classical formulation with respect to radial displacements and stresses is solved numerically by the shooting method. Additional boundary layer terms for radial displacements at small values of the gradient parameter are found using the asymptotic method for solving linear differential equations with spatially varying coefficients (the Wentzel-Kramers-Brillouin method - WKB method). Calculations of radial displacements, Cauchy stresses, couple stresses and total stresses in the case of both homogeneous and inhomogeneous coatings are carried out using specific examples. It has been found that the Cauchy stresses and total stresses experience a jump at the boundary between the cylinder and the coating. The couple stresses at small gradient parameters are much less than the total stresses. An increase of the scale parameter reduces the values of radial displacements and total stresses. Deformations are continuous in both cases of coating (homogeneous and inhomogeneous). A comparative study of the influence of the value of the inhomogeneity parameter on the distribution of displacements and total stresses is carried out.

Uniform Approximation to the Solution of a Singularly Perturbed Boundary Value Problem with an Initial Jump

In this study, a third-order linear integro-differential equation with a small parameter at two higher derivatives was considered. An asymptotic expansion of the solution to the boundary value problem for the considered equation is constructed by considering the phenomenon of an initial jump of the second degree zeroth order on the left end of a given segment. The asymptotics of the solution has been sought in the form of a sum of the regular part and the part of the boundary layer. The terms of the regular part are defined as solutions of integro-differential boundary value problems, in which the equations and boundary conditions contain additional terms, called the initial jumps of the integral terms and solutions. Boundary layer terms are defined as solutions of third-order differential equations with initial conditions. A theorem on the existence, uniqueness, and asymptotic representation of a solution is presented along with an asymptotic estimate of the remainder term of the asymptotics. The purpose of this study is to construct a uniform asymptotic approximation to the solution to the original boundary value problem over the entire considered segment.

Intercomparison of Large-Eddy Simulations of the Antarctic Boundary Layer for Very Stable Stratification

In polar regions, where the boundary layer is often stably stratified, atmospheric models produce large biases depending on the boundary-layer parametrizations and the parametrization of the exchange of energy at the surface. This model intercomparison focuses on the very stable stratification encountered over the Antarctic Plateau in 2009. Here, we analyze results from 10 large-eddy-simulation (LES) codes for different spatial resolutions over 24 consecutive hours, and compare them with observations acquired at the Concordia Research Station during summer. This is a challenging exercise for such simulations since they need to reproduce both the 300-m-deep convective boundary layer and the very thin stable boundary layer characterized by a strong vertical temperature gradient (10 K difference over the lowest 20 m) when the sun is low over the horizon. A large variability in surface fluxes among the different models is highlighted. The LES models correctly reproduce the convective boundary layer in terms of mean profiles and turbulent characteristics but display more spread during stable conditions, which is largely reduced by increasing the horizontal and vertical resolutions in additional simulations focusing only on the stable period. This highlights the fact that very fine resolution is needed to represent such conditions. Complementary sensitivity studies are conducted regarding the roughness length, the subgrid-scale turbulence closure as well as the resolution and domain size. While we find little dependence on the surface-flux parametrization, the results indicate a pronounced sensitivity to both the roughness length and the turbulence closure.

Upper-ocean Ekman current dynamics: a new perspective

The work examines upper-ocean response to time-varying winds within the Ekman paradigm. Here, in contrast to the earlier works we assume the eddy viscosity to be both time and depth dependent. For self-similar depth and time dependence of eddy viscosity and arbitrary time dependence of wind we find an exact general solution to the Navier–Stokes equations which describes the dynamics of the Ekman boundary layer in terms of the Green’s function. Two basic scenarios (a periodic wind and an increase of wind ending up with a plateau) are examined in detail. We show that accounting for the time dependence of eddy viscosity is straightforward and that it substantially changes the ocean response, compared to the predictions of the models with constant-in-time viscosity. We also examine the Stokes–Ekman equations taking into account the Stokes drift created by surface waves with an arbitrary spectrum and derive the general solution for the case of a linearly varying with depth eddy viscosity. Stability of transient Ekman currents to small-scale perturbations has never been examined. We find that the Ekman currents evolving from rest quickly become unstable, which breaks down the assumed horizontal uniformity. These instabilities proved to be sensitive to the model of eddy viscosity, they have small ( ) spatial scales and can be very fast compared to the inertial period, which suggests spikes of dramatically enhanced mixing localized in the vicinity of the water surface. This picture is incompatible with the Ekman paradigm and thus prompts radical revision of the Ekman-type models.

Bulk models of sheared boundary layer convection

The paper discusses approaches to the construction of integral models of the convective boundary layer (CBL), based on the concept of complete mixing. To test analytic bulk models and the basic hypotheses of similarity, we use the results of eddy modeling (LES – Large Eddy Simulation). The empirical constants of the CBL integral models obtained according to the LES data for the case of free convection, are in good agreement with the previously published data of laboratory experiments. It is also shown that the flow of kinetic energy from the upper boundary of the CPS, carried out by gravitational waves, is small compared with other components of the balance of turbulent kinetic energy (TKE) in the convective layer. Parametrization of TKE generation for the case of sheared convective boundary layer in terms of the friction velocity and the average wind velocity in the CBL derived; resulting dimensionless constants are obtained from LES data. The results of the work allow us to formulate an integral model of the shear KPS suitable for practical use.

Bulk Models of Sheared Boundary Layer Convection

The paper discusses approaches to the construction of integral models of the convective boundary layer (CBL), based on the concept of complete mixing. To test analytic bulk models and the basic hypotheses of similarity, we use the results of eddy modeling (LES – Large Eddy Simulation). The empirical constants of the CBL integral models obtained according to the LES data for the case of free convection, are in good agreement with the previously published data of laboratory experiments. It is also shown that the flow of kinetic energy from the upper boundary of the CPS, carried out by gravitational waves, is small compared with other components of the balance of turbulent kinetic energy (TKE) in the convective layer. Parametrization of TKE generation for the case of sheared convective boundary layer in terms of the friction velocity and the average wind velocity in the CBL derived; resulting dimensionless constants are obtained from LES data. The results of the work allow us to formulate an integral model of the shear KPS suitable for practical use.

High-resolution modeling study of an isolated convective storm over Seoul Metropolitan area

The ability of a high-resolution (500 m) Weather Research and Forecasting (WRF) model to simulate an isolated convective precipitation event over Seoul metropolitan area on August 16, 2015 was investigated. To understand the effects of micro- and mesoscale forcing on the initiation of convective rainfall under large-scale conditions, we performed sensitivity tests using different initial times. Despite simulating the same case, the quantitative precipitation forecast and the timing of moist convection varied widely among the experiments. Mesoscale features such as outflow and low-level convergence are different in location and intensity among the experiments. When assimilation of surface observations and radar data was performed, the simulation reproduced the low-level convergence and, hence, the location and amount of rainfall reasonably well within the first 6 h of simulation period. The timing differences of convective rainfall among the experiments were examined in terms of the atmospheric boundary layer (ABL) growth. Rapid growth of the ABL enabled moist convection to occur early in the presence of an outflow. An overestimated maximum ABL height by the model also led to earlier collapse of the ABL as compared to observations, which contributed to a reduction of convective available potential energy over the urban area in late afternoon. The results of this study demonstrate that accurate simulation of ABL growth is important for predicting the timing and intensity of isolated convective storms.

On the asymptotic behaviour of the kernel of an adjoint convection-diffusion operator in a long cylinder

This paper studies the asymptotic behaviour of the principal eigenfunction of the adjoint Neumann problem for a convection diffusion operator defined in a long cylinder. The operator coefficients are 1-periodic in the longitudinal variable. Depending on the sign of the so-called longitudinal drift (a weighted average of the coefficients), we prove that this principal eigenfunction is equal to the product of a specified periodic function and of an exponential, up to the addition of fast decaying boundary layer terms.

Free vibrations of axisymmetric shells: Parabolic and elliptic cases

Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with laterally clamped boundary conditions (Lam{\'e} system) are determined by an asymptotic analysis as the thickness ($2\varepsilon$) tends to zero. The departing point is the Koiter shell model that we reduce by asymptotic analysis to a scalar modelthat depends on two parameters: the angular frequency $k$ and the half-thickness $\varepsilon$. Optimizing $k$ for each chosen $\varepsilon$, we find power laws for $k$ in function of $\varepsilon$ that provide the smallest eigenvalues of the scalar reductions.Corresponding eigenpairs generate quasimodes for the 3D Lam{\'e} system by means of several reconstruction operators, including boundary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpair corresponds to the first eigenpair of the Lam{\'e} system.Geometrical conditions are necessary to this approach: The Gaussian curvature has to be nonnegative and the azimuthal curvature has to dominate the meridian curvature in any point of the midsurface. In this case, the first eigenvector admits progressively larger oscillation in the angular variable as $\varepsilon$ tends to $0$. Its angular frequency exhibits a power law relationof the form $k=\gamma \varepsilon^{-\beta}$ with $\beta=\frac14$ in the parabolic case (cylinders and trimmed cones), and the various $\beta$s $\frac25$, $\frac37$, and $\frac13$ in the elliptic case.For these cases where the mathematical analysis is applicable, numerical examples that illustrate the theoretical results are presented.

A comparison of opposition control in turbulent boundary layer and turbulent channel flow

A comparison between classical opposition control applied in the configuration of a fully developed turbulent channel flow and applied locally in a spatially developing turbulent boundary layer is presented. It is found that the control scheme yields similar drag reduction rates if compared at the same friction Reynolds numbers. However, a detailed analysis of the dynamical contributions to the skin friction coefficient reveals significant differences in the mechanism behind the drag reduction. While drag reduction in turbulent channel flow is entirely based on the attenuation of the Reynolds shear stress, the modification of the spatial flow development is essential for the turbulent boundary layer in terms of achievable drag reduction. It is shown that drag reduction due to this spatial development contribution becomes more pronounced with increasing Reynolds number (up to Reτ = 660, based on friction velocity and boundary layer thickness) and even exceeds drag reduction due to attenuation of the Reynolds shear stress. In terms of an overall energy balance, it is found that opposition control is less efficient in the turbulent boundary layer due to the inherently larger fluctuation intensities in the near wall region.

Selection of boundary conditions for modeling thermowave processes in multilayer objects

The process of nonstationary heat transfer in layered objects is considered. The propagation of thermal waves in these systems is analyzed for various parameters of adjacent regions and the interface. It is shown that proper allowance for variation of the thermal flux via a boundary layer in terms of the capacitive thermal impedance significantly increases the accuracy of calculations and simplifies the simulation of nonstationary thermal fields in multilayer objects.

What is the smallest physically acceptable scale for 1D turbulence schemes?

In numerical weather prediction (NWP) models, at mesoscale, the subgrid convective boundary-layer turbulence is dominated by the uni-directional (1D) vertical thermal production. In Large-Eddy Simulations (LES), the thermal plumes are resolved and the residual subgrid turbulent motions are homogeneous and isotropic, thus three-dimensional (3D), resulting from the dynamical production. This article sets the critical horizontal resolution for which the usually 1D turbulence schemes of NWP models must be replaced by 3D turbulence schemes. LES from five dry and cumulus-topped free convective boundary layers and one forced convective boundary layer are performed. From these LES data, the thermal production and vertical and horizontal dynamical productions are calculated at several resolutions from LES to mesoscale. It appears that the production terms of both dry and cumulus-topped free convective boundary layers have the same behavior. A pattern emerges whenever data are ranked by the resolution scaled by the size of thermal plumes, (h + hc , where h is the boundary-layer height and hc is the depth of the cloud layer). In free onvective boundary layers, the critical horizontal resolution for which the horizontal motions must be represented is 0.5(h + hc ). However, the critical horizontal resolution in the forced convective boundary layer case is 3(h + hc ).