Sidewall Boundary Conditions Research Articles

Wall modes and the transition to bulk convection in rotating Rayleigh-Bénard convection

We investigate states of rapidly rotating Rayleigh-Bénard convection in a cylindrical cell over a range of Rayleigh numbers 3×105≤Ra≤5×109 and Ekman numbers 10−6≤Ek≤10−4 for Prandtl number Pr=0.8 and aspect ratios 1/5≤Γ≤5 using direct numerical simulations. We characterize, for perfectly insulating sidewall boundary conditions, the first transition to convection via wall mode instability and the nonlinear growth and instability of the resulting wall mode states, including a secondary transition to time dependence. We show how the radial structure of the vertical velocity uz and the temperature T is captured well by the linear eigenfunctions of the wall mode instability where the radial width of uz is δuz∼Ek1/3r/H whereas δT∼e−kr (k is the wave number of a laterally infinite wall mode state). The disparity in spatial scales for Ek=10−6 means that the heat transport is dominated by the radial structure of uz since T varies slowly over the radial scale δuz. We further describe how the transition to a state of bulk convection is influenced by the presence of the wall mode states. We use temporal and spatial scales as measures of the local state of convection and the Nusselt number Nu as representative of global transport. Our results elucidate the evolution of the wall state of rotating convection and confirm that wall modes are strongly linked with the boundary zonal flow being the robust remnant of nonlinear wall mode states. We also show how the heat transport (Nu) contributions of wall modes and bulk modes are related and discuss approaches to disentangling their relative contributions. Published by the American Physical Society 2024

Temporal characteristics of hypersonic flows over a double wedge with Reynolds number

Laminar hypersonic flows at Mach 7.10 with unit Reynolds numbers of 5.2×104, 1.04×105, and 4.14×105 m−1 over a 30°/55° double-wedge configuration were studied to investigate the spatial–temporal characteristics of the flow in a time-accurate manner. Close comparisons were made between previous kinetic and current continuum approaches to test the validity of the continuum assumption, especially considering the presence of large gradients associated with shock–shock and shock–boundary layer interactions, as well as spanwise instabilities. Previous results from direct simulation Monte Carlo, which inherently predicts rarefied effects such as velocity slip and temperature jumps, were found to be in very close agreement with the current work, even for the lowest Reynolds number. The impact of velocity slip and temperature jumps on flow and surface parameters was investigated, and comparisons were made with a no-slip and constant temperature wall model. The temporal and spatial variation of two- and three-dimensional flows were thoroughly investigated using two-dimensional (2D), three-dimensional (3D) periodic sidewall boundary conditions, and a full 3D configuration consistent with existing experimental data. Close comparisons among the 2D and 3D cases were made. The characteristics of 2D periodic oscillations were reported for the moderate Reynolds number case for the first time. The presence of spanwise instabilities, even at a relatively low free stream pressure of about 100 Pa, establishes that the flow field depends on spanwise effects and is fully 3D. High-fidelity numerical schlieren videos captured strong spanwise oscillations for 3D configurations.

Effects of spanwise length and side-wall boundary condition on plunging breaking waves

A systematic study of the effect of the spanwise length and the sidewall boundary condition of a numerical wave flume (NWF) on direct numerical simulation of a plunging breaking wave is performed. To deal with the topological changes of free surfaces, a high-fidelity numerical model is employed to solve the Navier–Stokes equations together with the volume of fluid function. After verification by two-dimensional (2D) simulations of a plunging breaker on a sloping beach, ten NWFs with different spanwise extents and sidewall boundary conditions are studied. Special attention is devoted to the three-dimensionality of the plunging breaker. Compared with three-dimensional (3D) models, the 2D model accurately reproduces the dynamics of a breaking solitary wave in the early stage, but it is inadequate for the study of the post-breaking process. For a 3D NWF with nonslip sidewall boundary condition, the wave domain can be divided into two regions with different physical properties. In the near-wall region, the nonslip boundary condition on the sidewall plays a crucial role in the wave hydrodynamics, while in the central region, the properties of the breaking wave are similar to those for the periodic boundary condition, which provide a closer representation of the real sea environment. The spanwise length of the NWF plays only a minor role in simulations under the periodic boundary condition. Furthermore, lateral boundaries and spanwise length show more influences on a plunging breaker with larger incident wave steepness. This study provides valuable support for the design of numerical simulations of wave breaking.

Tidal Modulation of a Lateral Shear Margin: Priestley Glacier, Antarctica

We use high resolution, ground-based observations of ice displacement to investigate ice deformation across the floating left-lateral shear margin of Priestley Glacier, Terra Nova Bay, Antarctica. Bare ice conditions allow us to fix survey marks directly to the glacier surface. A combination of continuous positioning of a local reference mark, and repeat positioning of a network of 33 stakes installed across a 2 km width of the shear margin are used to quantify shear strain rates and the ice response to tidal forcing over an 18-day period. Along-flow velocity observed at a continuous Global Navigation Satellite Systems (GNSS) station within the network varies by up to ∼30% of the mean speed (±28 m a−1) over diurnal tidal cycles, with faster flow during the falling tide and slower flow during the rising tide. Long-term deformation in the margin approximates simple shear with a small component of flow-parallel shortening. At shorter timescales, precise optical techniques allow high-resolution observations of across-flow bending in response to the ocean tide, including across-flow strains on the order of 10–5. An elastodynamic model informed by the field observations is used to simulate the across-flow motion and deformation. Flexure is concentrated in the shear margin, such that a non-homogeneous elastic modulus is implied to best account for the combined observations. The combined pattern of ice displacement and ice strain also depends on the extent of coupling between the ice and valley sidewall. These conclusions suggest that investigations of elastic properties made using vertical ice motion, but neglecting horizontal displacement and surface strain, will lead to incorrect conclusions about the elastic properties of ice and potentially over-simplified assumptions about the sidewall boundary condition.

Flow states and heat transport in Rayleigh–Bénard convection with different sidewall boundary conditions

This work addresses the effects of different thermal sidewall boundary conditions on the formation of flow states and heat transport in two- and three-dimensional Rayleigh–Bénard convection (RBC) by means of direct numerical simulations and steady-state analysis for Rayleigh numbers ${\textit {Ra}}$ up to $4\times 10^{10}$ and Prandtl numbers ${\textit {Pr}}=0.1,1$ and $10$ . We show that a linear temperature profile imposed at the conductive sidewall leads to a premature collapse of the single-roll state, whereas a sidewall maintained at a constant temperature enhances its stability. The collapse is caused by accelerated growth of the corner rolls with two distinct growth rate regimes determined by diffusion or convection for small or large ${\textit {Ra}}$ , respectively. Above the collapse of the single-roll state, we find the emergence of a double-roll state in two-dimensional RBC and a double-toroidal state in three-dimensional cylindrical RBC. These states are most prominent in RBC with conductive sidewalls. The different states are reflected in the global heat transport, so that the different thermal conditions at the sidewall lead to significant differences in the Nusselt number for small to moderate ${\textit {Ra}}$ . However, for larger ${\textit {Ra}}$ , the heat transport and flow dynamics become increasingly alike for different sidewalls and are almost indistinguishable for ${\textit {Ra}}>10^{9}$ . This suggests that the influence of imperfectly insulated sidewalls in RBC experiments is insignificant at very high ${\textit {Ra}}$ – provided that the mean sidewall temperature is controlled.

Simulation of the SPARC plasma boundary with the UEDGE code

In this work, the UEDGE edge transport code is used to examine conditions in the SPARC divertor and edge plasma for various levels of carbon impurity and power from the core (P SOL). A double-null magnetic configuration is simulated assuming up-down symmetry in geometry and physics. The anomalous heat and particle transport coefficients are tuned to match empirical predictions for SPARC’s midplane density profiles, target plate heat flux profiles, and inner/outer divertor power sharing. Convective transport is included on the low-field side, while on the high-field side the transport is modeled as purely diffusive. Hydrogen neutrals are modeled as a fluid with inertial effects, and a carbon impurity is included using the fixed-fraction model. We find that detachment induced by impurity seeding could significantly reduce the heat flux to the divertor surfaces in the SPARC tokamak. At P SOL = 28 MW (the value predicted for SPARC’s full-power H-mode scenario) cases with both divertor legs detached were obtained with a carbon impurity fraction between 0.3%–1.4%, far below Z eff limits for SPARC. When the plasma in the outer leg is detached, the peak heat flux density perpendicular to the target plate is below 1 MW m−2, electron and ion temperatures are less than 1.5 eV, and momentum detachment is observed. However, the detachment state is found to be sensitive to the side-wall boundary conditions, the level of neutral pumping, and the target plate tilt. Finally, a broadly similar SOLPS simulation of SPARC is used to assess the appropriateness of the simpler impurity and neutral models used in UEDGE.

Side wall boundary effect on the Rayleigh–Taylor instability

Rayleigh–Taylor instability (RTI) is a common phenomenon in daily life and in industrial application. Despite the long history of investigating the RTI with periodic boundary condition, there are only few studies for the RTI with no-slip side wall boundary condition. In fact, the RTI with the side wall boundary condition is more realistic physics. In this work, we investigate the side wall boundary effect on the RTI using a phase-field model. No-slip conditions are employed on all boundaries. The governing equations are the coupled Navier–Stokes (NS) and Cahn–Hilliard (CH) system. We temporally solve the NS equation using Chorin’s projection method and the CH equation using Eyre’s nonlinear splitting scheme. A multigrid method is employed to solve the discrete system of CH equation. By various numerical experiments, we find that the side wall will delays the evolution of bubble and spike, causes a longer interface length and complex interface shape. Besides that, some other parameters, such as Reynolds number and aspect ratio of domain etc., can also cause significantly different phase evolutions.

Controlling flow reversal in two-dimensional Rayleigh–Bénard convection

In this paper, we report that reversals of large-scale circulation in two-dimensional Rayleigh–Benard convection could be suppressed or enhanced by imposing local constant-temperature control on sidewalls. When the control area is away from the centre of the sidewalls, the control can successfully eliminate the flow reversal if the size of the control region is large enough. With a proper location, the width can be as small as 1 % of the system size. When the control region is located around the centre, the control may enhance the flow reversal. It may also stimulate the occurrence of a double-roll mode when the control is located in the centre. Explanations are also discussed based on the twofold effects of the control region on the nearby plumes and the concept of symmetry. The present work provides a new way to control the flow reversals in Rayleigh–Benard convection through modifying sidewall boundary conditions.

Effect of side wall conditions and aspect ratio on convective pattern formation in Rayleigh-Benard convection

Effect of side wall boundary conditions and aspect ratio in very small boxes on patterns formed during Raleigh-Benard (RB) convection is studied using an indigenously developed flow solver based on the finite difference method. While keeping the bottom (hot) and top (cold) walls isothermal, the side wall thermal conditions are kept either constant heat flux or isothermal type. For constant heat flux side wall condition, the considered value of heat flux lies between 0 and maximum corresponding to conductive heat flux between top and bottom walls. For isothermal side wall condition, the chosen value of temperature is between the top and bottom wall temperatures. The computational domain is a cube or a square prism with aspect ratio in longitudinal direction (ARx) varied between 1.2 and 2 in steps of 0.2. For fixed value of ARx = 1.6, aspect ratio in transverse direction (ARz) has been varied for adiabatic side wall conditions. Effect of side wall condition on orientation of roll axis and the number of rolls has been investigated with observations well supported by stream traces and temperature contours. Moreover, for selected parameters, based on nature of pattern formation for steady flow, onset of oscillatory convection is explained.

Chaotic Rayleigh-Bénard convection with finite sidewalls.

We explore the role of finite sidewalls on chaotic Rayleigh-Bénard convection. We use large-scale parallel spectral-element numerical simulations for the precise conditions of experiment for cylindrical convection domains. We solve the Boussinesq equations for thermal convection and the conjugate heat transfer problem for the energy transfer at the solid sidewalls of the cylindrical domain. The solid sidewall of the convection domain has finite values of thickness, thermal conductivity, and thermal diffusivity. We compute the Lyapunov vectors and exponents for the entire fluid-solid coupled problem. We quantify the chaotic dynamics of convection over a range of thermal sidewall boundary conditions. We find that the dynamics become less chaotic as the thermal conductivity of the sidewalls increases as measured by the value of the fractal dimension of the dynamics. The thermal conductivity of the sidewall is a stabilizing influence; the heat transfer between the fluid and solid regions is always in the direction to reduce the fluid motion near the sidewalls. Although the heat interaction for strongly conducting sidewalls is only about 1% of the heat transfer through the fluid layer, it is sufficient to reduce the fractal dimension of the dynamics by approximately 25% in our computations.

Effects of the wall boundary conditions of a showerhead plasma reactor on the uniformity control of RF plasma deposition

Technical difficulties hinder the formation of uniform deposition profiles near the electrode edge during a deposition in a showerhead capacitively coupled plasma (CCP) reactor. The discharge structure, gas flow, and radial distribution of the source gas outward from the electrode edge are subject to change significantly, which in turn affects the spatial distributions of the radical fluxes toward the electrode. To control the local non-uniformity of deposition profiles in the SiH4/NH3/N2/He CCP discharges for a hydrogenated silicon nitride (SiNxHy) film, the effects of the reactor components―including the sidewall boundary condition, electrode spacing, and showerhead design―were investigated using an axisymmetric fluid model. When the sidewall is electrically grounded, the deposition rate profiles of the SiNxHy film remain consistently convex (in which the deposition rate at the reactor center is locally much higher than that near the electrode edge), regardless of electrode spacing. However, when the sidewall surface is dielectric, the deposition rate profile can be transformed between a convex and a concave shape (in which the deposition rate at the reactor center is locally much lower than that near the electrode edge) by varying electrode spacing. The showerhead design also enables the modification of edge deposition profiles by redistribution of the local depletion rate of radicals. The simulation results agree very well with the experimental measurement.

The upper Bay of Bengal salinity structure in a high-resolution model

Salinity in the Bay of Bengal is highly heterogeneous, with extremely fresh waters found at the surface in the Northern part of the basin, and saltier waters at subsurface as well as to the south. This paper investigates the seasonal structure of sea surface salinity of the Bay in a regional high-resolution model forced by ERA-Interim reanalysis and various precipitation products. Surface circulation is believed to drive the spreading of northern Bay of Bengal fresh waters to the rest of the Indian Ocean. We first present a series of experiments to infer the sensitivity of modeled circulation to various numerical choices. Surface circulation is found to be sensitive to the horizontal resolution of the model, with the 1/12° version appearing much more realistic than the 1/4° version. The sidewall boundary condition is also drastically influencing the characteristics of the western boundary current simulated. We then investigate the sensitivity of the salinity response to the various precipitation products. We observe that ERA-Interim excess precipitation induces a fresh bias in the surface salinity response. Spaceborne precipitation products are more satisfactory. We then identify the pathways of the northern Bay freshwater mass, based on passive tracers experiments. Our model suggests that over timescales of a few months, vertical exchanges between the upper fresh layer and the underlying saltier layer appear to be the main export pathway for the freshwater. The horizontal circulation within the mixed layer also acts to convey fresh waters out of the Bay at these timescales, but in a lesser quantity compared to the vertical export. Beyond its intrinsic interest for the understanding of Bay of Bengal physics, this study highlights the need for a careful design of any realistic numerical model, in three key aspects: the choice of the resolution of the model, the choice of the sub-grid scale parameterizations, and the choice of the forcing fluxes.

Response of mantle heat flux to plate evolution

By analyzing convection characteristics and surface and basal heat flux we investigate the influence on mantle convection of finite thickness plates with dynamically determined velocities and mobile boundaries. Our 6 × 6 × 1 Cartesian geometry numerical model features periodic sidewall boundary conditions, a lower mantle that increases in viscosity and a Rayleigh number of 5 × 106 (based on the uniform viscosity upper mantle). A calculation with 9 plates and fixed plate boundaries is compared with a second calculation incorporating evolving plate boundaries. The number of plates in the latter case fluctuates between 6 and 9. We find that mean heat flux output decreases during periods characterized by the presence of larger plates and that evolving plate boundaries cause surface (and basal) heat flux output to vary much more than the heat flux observed with either fixed plate boundaries or a free‐slip surface.

Effect of sidewall boundary conditions on unsteady high speed cavity flow and acoustics

The effect of sidewall boundary conditions on the computed unsteady flow and sound pressure level is investigated in a transonic open cavity. The hybrid approach used for modeling turbulence combines a Reynolds averaged mode in the boundary layer, and a large eddy simulation mode in the massively separated flow region within the cavity to resolve the wide dynamic range involved. Computational results are presented for the instantaneous vorticity and for the sound pressure level spectra. Comparison of the results obtained using inviscid and periodic sidewall boundary conditions show the sensitivity of the computed SPL spectra and autocorrelation to the conditions enforced at the sidewalls. The computed SPL spectra are also compared with available experimental results, with LES computational results, and with prior investigations based on the same hybrid turbulence model without the wall function used in the current investigation. The comparisons show that the current results obtained using inviscid sidewall boundary conditions are closest to the experimental sound pressure level spectra and that agreement is achieved at considerable saving in required computational resources.

Free-surface granular flows down heaps

The present paper extends the granular-flow constitutive model of Savage (1998 J Fluid Mech 377:1–26) to treat spherical particles. Savage accounted for both quasi-static and collisional stresses by considering: (i) strain-rate fluctuations embodied in a critical state plasticity model, as well as, (ii) individual particle velocity fluctuations modelled by granular-flow kinetic theory. In the present work, the governing equations of the kinetic theory of Jenkins (1998 In: Hermann HJ, Luding S (eds) Physics of Dry Granular Media. Kluwer Academic pp. 353–370) for identical spherical, smooth, inelastic particles are supplemented with additional quasi-static terms that have forms patterned after the corresponding terms in the equations of Savage for two-dimensional disk-like particles. The resulting equations along with side-wall and free-surface boundary conditions are applied to examine free-surface granular flow down a heap contained between two frictional vertical side walls. Width-averaged equations of motion are integrated to obtain depth profiles of mean velocity, granular temperature, solids fraction and the Savage–Jeffrey parameter. Detailed comparisons are made with particle-tracking experiments. When the gap between the vertical side walls is fairly narrow, good agreement is found between the predicted and the measured profiles of mean velocity and granular temperature.

Asymptotic theory of inertial convection in a rotating cylinder

Inertial convection in a fluid contained in a rotating cylinder heated uniformly from below is investigated on the basis of the assumption that convection at leading order can be represented by a single or several inertial wave modes which propagate either in the prograde or retrograde direction. Buoyancy forces appear at the next order to drive inertial convection against the effect of viscous damping. Asymptotic expressions for inertial convection for four different combinations of the sidewall boundary condition are derived for a cylinder of arbitrary aspect ratio. New convection patterns in rotating cylinders are revealed by the asymptotic analysis. A fully numerical solution of the same problem is also carried out, demonstrating a quantitative agreement between the asymptotic and numerical analysis.

Incident angle-dependent effective surface impedance (Z/sub S/) of Electromagnetic Crystal (EMXT)

Electromagnetic crystal (EMXT) surfaces are modeled with effective surface impedance method. Effects of oblique incidence are studied using a technique similar to the waveguide simulator method for analyzing infinite phased array. Numerical results are compared with an analytical model. EMXT samples are characterized by open-ended waveguide reflection measurements using different aperture sizes (WR28 and WR22). The measurements confirm the angular dependence of the EMXT surface impedance. A rectangular waveguide with EMXT sidewalls is modeled by using the angle-dependent effective surface impedance as the sidewall boundary condition. Excellent agreement between this model and full-wave finite-element simulation of the entire waveguide structure is obtained.

Wake Flow of Single and Multiple Yawed Cylinders

An experimental and computational study is performed of the wake flow behind a single yawed cylinder and a pair of parallel yawed cylinders placed in tandem. The experiments are performed for a yawed cylinder and a pair of yawed cylinders towed in a tank. Laser-induced fluorescence is used for flow visualization and particle-image velocimetry is used for quantitative velocity and vorticity measurement. Computations are performed using a second-order accurate block-structured finite-volume method with periodic boundary conditions along the cylinder axis. Results are applied to assess the applicability of a quasi-two-dimensional approximation, which assumes that the flow field is the same for any slice of the flow over the cylinder cross section. For a single cylinder, it is found that the cylinder wake vortices approach a quasi-two-dimensional state away from the cylinder upstream end for all cases examined (in which the cylinder yaw angle covers the range 0⩽ϕ⩽60°). Within the upstream region, the vortex orientation is found to be influenced by the tank side-wall boundary condition relative to the cylinder. For the case of two parallel yawed cylinders, vortices shed from the upstream cylinder are found to remain nearly quasi-two-dimensional as they are advected back and reach within about a cylinder diameter from the face of the downstream cylinder. As the vortices advect closer to the cylinder, the vortex cores become highly deformed and wrap around the downstream cylinder face. Three-dimensional perturbations of the upstream vortices are amplified as the vortices impact upon the downstream cylinder, such that during the final stages of vortex impact the quasi-two-dimensional nature of the flow breaks down and the vorticity field for the impacting vortices acquire significant three-dimensional perturbations. Quasi-two-dimensional and fully three-dimensional computational results are compared to assess the accuracy of the quasi-two-dimensional approximation in prediction of drag and lift coefficients of the cylinders.

Confined turbulent convection

New measurements of the Nusselt number have been made in turbulent thermal convection confined in a cylindrical container of aspect ratio unity. The apparatus is essentially the same as that used by Niemela et al. (2000), except that the height was halved. The measurement techniques were also identical but the mean temperature of the flow was held fixed for all Rayleigh numbers. The highest Rayleigh number was . Together with existing data, the new measurements are analysed with the purpose of understanding the relation between the Nusselt number and the Rayleigh number, when the latter is large. In particular, the roles played by Prandtl number, aspect ratio, mean wind, boundary layers, sidewalls, and non-Boussinesq effects are discussed. Nusselt numbers, measured at the highest Rayleigh numbers for which Boussinesq conditions hold and sidewall forcing is negligible, are shown to vary approximately as a 1/3-power of the Rayleigh number. Much of the complexity in interpreting experimental data appears to arise from aspects of the mean flow, including complex coupling of its dynamics to sidewall boundary conditions of the container. Despite the obvious practical difficulties, we conclude that the next generation of experiments will be considerably more useful if they focus on large aspect ratios.

Numerical simulation of the cooling of a hot disk rapidly subjected to combined convective and radiant heat losses

The development of a transient method for measuring the spectral directional emissivity of solid materials at high temperature is currently being completed by the analysis of the influence of various sources of error. A hot disk is subjected to mixed convective and radiant heat losses on one face after the rapid withdrawal of a shutter. The cooling is investigated during a short period of observation through numerical simulations of the transient temperature. The thermal problem is simulated using 3D and axisymmetric 2D codes. The progressive withdrawal of the shutter results in a delay for the transient temperature in comparison with the instantaneous withdrawal. For an insulating material, the influence of the side wall boundary conditions remains negligible at the center of the front face. The strong influence of the convective heat transfer is demonstrated.