Sidewall Boundary Layer Research Articles

The structure of sidewall boundary layers in confined rotating Rayleigh–Bénard convection

AbstractTurbulent rotating convection is usually studied in a cylindrical geometry, as this is its most convenient experimental realization. In our previous work (Kunnen et al., J. Fluid Mech., vol. 688, 2011, pp. 422–442) we studied turbulent rotating convection in a cylinder with the emphasis on the boundary layers. A secondary circulation with a convoluted spatial structure has been observed in mean velocity plots. Here we present a linear boundary-layer analysis of this flow, which leads to a model of the circulation. The model consists of two independent parts: an internal recirculation within the sidewall boundary layer, and a bulk-driven domain-filling circulation. Both contributions exhibit the typical structure of the Stewartson boundary layer near the sidewall: a sandwich structure of two boundary layers of typical thicknesses ${E}^{1/ 4} $ and ${E}^{1/ 3} $, where $E$ is the Ekman number. Although the structure of the bulk-driven circulation may change considerably depending on the Ekman number, the boundary-layer recirculation is present at all Ekman numbers in the range $0. 72\times 1{0}^{- 5} \leq E\leq 5. 76\times 1{0}^{- 5} $ considered here.

Sidewall Boundary Layer of Transverse Waves in Simulated Liquid Rocket Engines

This work seeks to provide a closed-form analytical solution for the transverse vortical wave generated at the sidewall of a circular cylinder with headwall injection. This particular configuration mimics the conditions leading to the onset of traveling radial and tangential waves in an idealized liquid rocket engine chamber. Assuming a short cylindrical enclosure with axisymmetric injection, regular perturbations are used to linearize the problem’s conservation equations. Flow decomposition is subsequently applied to the first-order disturbance equations, thus giving rise to a compressible, inviscid, acoustic set responsible for driving the unsteady motion and to an incompressible, viscous, vortical set driven by virtue of coupling with the acoustic mode along both the sidewall and headwall. While the acoustic mode is readily recovered from the wave equation, the induced vortical mode is resolved using boundary-layer theory and an expansion of the rotational equations with respect to a small viscous parameter, . At the outset, an explicit formulation for the leading-order vortical field is derived and verified numerically. A radial penetration number akin to the Stokes or Womersley numbers is identified and found to control the penetration depth of the viscous boundary layer forming above the inert sidewall. This parameter is based on the transverse oscillation mode frequency and scales with the squared ratio of the Stokes layer and the chamber’s characteristic radius.

Study of instabilities and quasi-two-dimensional turbulence in volumetrically heated magnetohydrodynamic flows in a vertical rectangular duct

We consider magnetohydrodynamic (MHD) rectangular duct flows with volumetric heating. The flows are upward, subject to a strong transverse magnetic field perpendicular to the temperature gradient, such that the flow dynamics is quasi-two-dimensional. The internal volumetric heating imitates conditions of a blanket of a fusion power reactor, where a buoyancy-driven flow is imposed on the forced flow. Studies of this mixed-convection flow include analysis for the basic flow, linear stability analysis and Direct Numerical Simulation (DNS)-type computations. The parameter range covers the Hartmann number (Ha) up to 500, the Reynolds number (Re) from 1000 to 10 000, and the Grashof number (Gr) from 105 to 5 × 108. The linear stability analysis predicts two primary instability modes: (i) bulk instability associated with the inflection point in the velocity profile near the “hot” wall and (ii) side-wall boundary layer instability. A mixed instability mode is also possible. An equation for the critical Hartmann number has been obtained as a function of Re and Gr. Effects of Ha, Re, and Gr on turbulent flows are addressed via nonlinear computations that demonstrate two characteristic turbulence regimes. In the “weak” turbulence regime, the induced vortices are localized near the inflection point of the basic velocity profile, while the boundary layer at the wall parallel to the magnetic field is slightly disturbed. In the “strong” turbulence regime, the bulk vortices interact with the boundary layer causing its destabilization and formation of secondary vortices that may travel across the flow, even reaching the opposite wall. In this regime, the key phenomena are vortex-wall and various vortex-vortex interactions. Flow and magnetic field effects on heat transfer are also analyzed.

Experimental Investigation of the Compressor Cascade under an Active Flow Control

The paper is concerned with flow past compressor blade cascade (NACA 65 with thickened trailing edge) at off-design regimes, which are characteristic by partial or complete flow separation on the suction surface of the blades. An attempt has been made to moderate the flow separation using continuous or periodic blowing from the sidewalls. The flow field was visualized using schlieren technique and surface paint visualization. The visualizations were complemented by measurement of the static pressure distribution on the suction surface of the blades. In agreement with the literature, the visualizations confirmed a complexity of the 3-dimensional flow separation, which was intensified by influence of the sidewall boundary layers developing from upstream parts of the test section. Furthermore, it was found out that the effect of both continuous and periodic blowing was rather minor. Finally, the results agree with the available literature showing that it is highly difficult to considerably control the complex 3-dimensional flow separation in the compressor cascade by control jets issuing (only) from the sidewalls

Effects of aspect ratio in a transonic shock tube airfoil flow

In the present study, the flow visualizations were performed around the NACA 0012 models which differ in aspect ratios. We discussed the effects of the aspect ratio in the test models. Additionally the unsteady, two-dimensional, compressible Euler equations were solved for the NACA 0012 airfoil. Experiments were performed utilizing the conventional gas driven shock tube as the intermittent transonic wind tunnel. The aspect ratios of the models are about 0.86 and 1.5, respectively. The Mach numbers M 2 are about 0.84. The Reynolds numbers of the present experimental conditions were constant that Re based on chord length is about 4.0×105. The results are as follows: in different aspect ratios, the difference of the shock wave location is confirmed though the Mach number and Reynolds number are same. It indicates the different correction Mach number by the effects of the side wall boundary layer though the nominal Mach number measured the same value. Also, on the difference of shock wave location for the effects of the aspect ratio, the tend of CFD shows the qualitative agreement with the result of an experiment.

Three-dimensional features of a Mach 2.1 shock/boundary layer interaction

2D particle image velocimetry was used to study the three-dimensionality of the shock-boundary layer interaction generated by a small 20° compression ramp in a low aspect ratio continuously operated wind tunnel. High-resolution data were taken in four streamwise-wallnormal planes: three planes located in the sidewall boundary layer and one near the tunnel centerline. The incoming boundary layer was found to show three-dimensionality, with significant overshoot in the velocity profiles observed near the sidewall. The size of the wedge influenced the interaction, which was weaker than that observed in the case of a large compression wedge. The flow turning angle was ≈8° near the tunnel centerline and changed significantly across the span. Measurements behind the compression wedge in the centerline plane showed that both velocity and turbulence properties were nearly fully recovered ≈14δ behind the compression corner. The shock angle varied with spanwise position, and a multi-shock structure was observed in the sidewall planes. The size of the interaction decreased in the sidewall boundary layer. Non-monotonic variations in both velocity and turbulence profiles across the sidewall planes suggest the presence of significant spanwise flows, possibly corner vortices.

Fluid flows in a librating cylinder

The flow in a cylinder driven by time-harmonic oscillations of the rotation rate, called longitudinal librations, is investigated. Using a theoretical approach and axisymmetric numerical simulations, we study two distinct phenomena appearing in this librating flow. First, we investigate the occurrence of a centrifugal instability near the oscillating boundary, leading to the so-called Taylor-Görtler vortices. A viscous stability criterion is derived and compared to numerical results obtained for various libration frequencies and Ekman numbers. The strongly nonlinear regime well above the instability threshold is also documented. We show that a new mechanism of spontaneous generation of inertial waves in the bulk could exist when the sidewall boundary layer becomes turbulent. Then, we analyse the librating flow below the instability threshold and characterize the mean zonal flow correction induced by the nonlinear interaction of the boundary layer flow with itself. In the frequency regime where inertial modes are not excited, we show that the mean flow correction in the bulk is a uniform rotation, independent of the Ekman number and cylinder aspect ratio, in perfect agreement with the analytical results of Wang [“Cylindrical tank of fluid oscillating about a state of steady rotation,” J. Fluid. Mech. 41, 581 (1970)]. When inertial modes are resonantly excited, the mean flow correction is found to have a more complex structure. Its amplitude still scales as the square of the libration amplitude but now depends on the Ekman number.

Scaling between Wind Tunnels–Results Accuracy in Two-Dimensional Testing

The establishment of exact two-dimensional flow conditions in wind tunnels is a very difficult problem. This has been evident for wind tunnels of all types and scales. In this paper, the principal factors that influence the accuracy of two-dimensional wind tunnel test results are analyzed. The influences of the Reynolds number, Mach number and wall interference with reference to solid and flow blockage (blockage of wake) as well as the influence of side-wall boundary layer control are analyzed. Interesting results are brought to light regarding the Reynolds number effects of the test model versus the Reynolds number effects of the facility in subsonic and transonic flow.

Instabilities and inertial waves generated in a librating cylinder

AbstractA librating cylinder consists of a rotating cylinder whose rate of rotation is modulated. When the mean rotation rate is large compared with the viscous damping rate, the flow may support inertial waves, depending on the frequency of the modulation. The modulation also produces time-dependent boundary layers on the cylinder endwalls and sidewall, and the sidewall boundary layer flow in particular is susceptible to instabilities which can introduce additional forcing on the interior flow with time scales different from the modulation period. These instabilities may also drive and/or modify the inertial waves. In this paper, we explore such flows numerically using a spectral-collocation code solving the Navier–Stokes equations in order to capture the dynamics involved in the interactions between the inertial waves and the viscous boundary layer flows.

Régimes of sloping thermal convection in a rotating liquid “annulus”

Wide-ranging laboratory investigations of thermally-driven flows in a baroclinic liquid of low viscosity and thermal conductivity in apparatus bounded by concentric cylindrical side-walls have (a) identified the dynamically-significant dimensionless parameters in terms of which the impressed conditions can be expressed, (b) stimulated advances in theoretical research on nonlinear dynamical systems, (c) established that under most (mechanical and thermal) boundary conditions, the action of gyroscopic (Coriolis) forces ensures that the dominant mode of heat transfer is “sloping thermal convection” (STC), and (d) provided insights into flows occurring in natural rotating fluid systems on length-scales up to thousands of kilometres. STC is characterised by relative flow that is nearly horizontal nearly everywhere, in wide-ranging patterns of waves and eddies which exhibit jet streams and fronts and which in some cases are spatially and temporally highly regular and in others are highly irregular (“chaotic”). Some of these manifold laboratory flows have their counterparts in the gaseous atmospheres of the Earth, Jupiter, Saturn and other spinning planets. When Coriolis forces are not quite strong enough to promote baroclinic instability leading to STC, heat transfer and associated generation of kinetic energy by buoyancy forces is effected by axisymmetric meridional overturning involving inter alia radial flow in end-wall Ekman boundary layers and vertical flow in side-wall boundary layers. Otherwise Coriolis forces promote quasi-geostrophic non-axisymmetric STC in the main body of the fluid. “Barotropic stability” due to enstrophy constraints renders this STC regular (i.e. spatially and temporally periodic) when the azimuthal scale of the flow exceeds a critical value found to satisfy a remarkably simple criterion. At shorter scales, enstrophy constraints weaken and no longer suffice to prevent inter-mode kinetic energy exchanges characteristic of “geostrophic turbulence”. The present article on rôle of boundary layers and on other aspects of annulus flows is largely intended to guide future investigations of crucial details of the dynamical processes that underlie fully-developed STC, some made possible by the ever-improving techniques of computational fluid dynamics.

Supersonic Fin-Body Interactions in an Internal Scramjet Flow

This paper reports on the study of a fin-body junction flow in the context of an internal scramjet flowpath. Threedimensional Reynolds-averaged Navier-Stokes numerical simulations of the flowfield inside a particular scramjet configuration have been performed using the commercial computational fluid dynamics code CFD++. The junction is formed by a central strut fuel injector spanning the entrance to a direct-connect combustor. The local flow Mach number is 2.5. The bow shock generated by the blunt strut injector causes flow separation on the side wall, which leads to the formation of horseshoe vortices. The computed results in the symmetry plane and on the injector surface near the leading edge of the strut are in good agreement with the solutions obtained by other investigators for isolated fin-body junctions. Further downstream, reflections of the strut bow shock at the internal flow top wall and then the strut surface have significant effect on the propagation of the fin-body interaction. The shock-vortex interaction was found to despin the primary horseshoe vortex; i.e., oppositely orientated vorticity is induced, and hence dissipate vorticity. Further reduction of the vorticity, associated with the observed vortices, occurs due to viscous dissipation of the horseshoe vortices in the internal flow side wall boundary layer.

Sidewall boundary layer instabilities in a rapidly rotating cylinder driven by a differentially corotating lid

The flow in an enclosed completely filled rapidly rotating cylinder that is driven by the differential corotation of the top lid is studied numerically. Although the flow is in a very simple geometry, the fast background rotation and large differential rotation of the lid lead to very thin boundary layers with a variety of instability modes with very fine spatial scales as well as inertial waves that are sustained in the fast rotating interior flow and that interact with the viscous modes in the sidewall boundary layer, leading to complex spatiotemporal dynamics. The numerical simulations are compared and contrasted to experimental visualizations of the sidewall boundary layer instabilities reported by Hart and Kittelman [“Instabilities of the sidewall boundary layer in a differentially driven rotating cylinder,” Phys. Fluids 8, 692 (1996)]. The experiments report observing axisymmetric rolls propagating down the sidewall layer for small differential corotation of the top lid. As the differential rotation was increased, backward tilted diagonal rolls that precess slightly retrograde with respect to the rotating sidewall and forward tilted rolls with prograde precession significantly faster than the sidewall rotation were observed. For still larger differential rotation, a wavy turbulent state that has backward tilted structures erupting from deep within the sidewall layer into the interior and is riding on the forward tilted diagonal rolls in the deep layer was observed. Our simulations capture all of these states and strongly suggest that the observed axisymmetric rolls are unstable and were only transiently observed due to the slow and continuous increase in the differential rotation employed in the experiments. The influence of inertial waves driven by the sidewall instabilities on the three-dimensional wavy turbulent state is discussed.

Velocimetry Measurements of Unstart of an Inlet-Isolator Model in Mach 5 Flow

The dynamics of unstart in a floor-mounted inlet-isolator model in a Mach 5 flow are investigated experimentally using particle image velocimetry and fast-response wall pressure measurements. The inlet compression is obtained with a 6-deg ramp and the isolator is a rectangular straight duct that is 25.4 mm high by 50.8 mm wide by 242.3 mm long. Unstart is initiated from the scramjet mode (fully supersonic in the isolator) by deflecting a motorized flap at the downstream end of the isolator. With the flap fully down, the particle image velocimetry data of the started flow capture the characteristics of the isolator boundary layers and the initial inlet reflected shock system. During unstart, the unstart shock system propagates upstream through the inlet-isolator. The particle image velocimetry data reveal a complex, three-dimensional flow structure that is strongly dependent on viscous mechanisms. Particularly, the unstart shock system propagates upstream and induces significant boundary-layer separation. Side-view particle image velocimetry data show that the locations of strongest separation during unstart correlate with the impingement locations of the initial inlet shock as it reflects down the isolator. For example, in the middle of unstart, the unstart shock system is associated with massive separation of the ceiling boundary layer that begins where the first inlet shock reflection impinges on the ceiling. The observation that separation increases at the inlet shock reflection impingement locations is likely due to the fact that the boundary layers in these locations are subject to larger adverse pressure gradients, thus making them more susceptible to separation. During the unstart process, large regions of separated flow form near the floor and ceiling with reverse flow velocities up to about 0.4U ∞ . These regions of separated, subsonic flow appear to extend to the isolator exit, creating a path by which the isolator exit boundary condition can be communicated upstream. Plan-view particle image velocimetry data show the unstart process begins with separation of the isolator sidewall boundary layers. Overall, the unstart flow structure is highly three-dimensional.

Sidewall Boundary Layers of the Bidirectional Vortex

This study seeks to resolve the sidewall boundary layers forming in the axial and radial directions of a bidirectional vortex chamber. Our analysis is initiated by the formulation of the laminar boundary-layer equations via an order of magnitude reduction of the incompressible Navier-Stokes equations at the wall. Asymptotic theory is then applied to linearize and systematically truncate the governing equations, thus converting them from partial differential equations to more manageable ordinary differential equations. Scaling transformations are additionally applied to resolve the rapid changes arising near the sidewall. Because of the spatial character of the outer solutions, further transformations of the dependent variables are undertaken to secure the axially changing outer conditions. Through the use of matched-asymptotic expansions, we recover similar boundary-layer structures in all three orthogonal directions: the axial and radial components presented here, and the wall-tangential boundary layer obtained previously. This behavior is consistent with the resultant velocity being dominated by its tangential component and with the tangential boundary layer being axially invariant. These factors cause the axial layer to remain uniform in the streamwise direction. Based on the ensuing asymptotic results, viscous corrections at the wall are seen to be mainly dependent on the vortex Reynolds number, V. The latter combines the swirl number, the Reynolds number, and the chamber aspect ratio. Having obtained the three components of the velocity, essential flow characteristics, such as pressure, vorticity, swirling intensity, and wall shear stresses, are evaluated and discussed.

Numerical Simulations of Effects of Micro Vortex Generators Using Immersed-Boundary Methods

This work presents an immersed-boundary technique for compressible, turbulent flows and applies the technique to simulate the effects of micro vortex generators in controlling oblique-shock/turbulent boundary-layer interactions. The Reynolds-averaged Navier-Stokes equations, closed using the Menter k-ω turbulence model, are solved in conjunction with the immersed-boundary technique. The approach is validated by comparing solutions obtained using the immersed-boundary technique with solutions obtained on a body-fitted mesh and with experimental laser Doppler anemometry data collected at Cambridge University for Mach 2.5 flow over single micro vortex generators. Simulations of an impinging oblique-shock boundary-layer interaction at Mach 2.5 with and without micro vortex-generator flow control are also performed, considering the development of the flow in the entire wind tunnel. Comparisons are made with experimental laser Doppler anemometry data and surface-pressure measurements from Cambridge University and an analysis of the flow structure is performed. The results show that three dimensional effects initiated by the interaction of the oblique shock with the sidewall boundary layers significantly influence the flow patterns in the actual experiment. The general features of the interactions with and without the micro vortex-generator array are predicted to good accord by the Reynolds-averaged Navier-Stokes/ immersed-boundary model.

On steady rotational cyclonic flows: The viscous bidirectional vortex

This study is focused on the tangential boundary layers of a bidirectional vortex, specifically those forming at the core and the sidewall of a swirl-driven cyclonic chamber. Our analysis is based on the regularized, tangential momentum equation which is rescaled in a manner to capture the forced vortex near the chamber axis and the no slip requirement at the sidewall. After identifying the coordinate transformations needed to resolve the rapid changes in the regions of nonuniformity, two inner expansions are constructed. These expansions are then matched with the outer, free vortex solution that is sandwiched between the core and the hard wall. By combining inner and outer expansions, uniformly valid approximations are subsequently obtained for the swirl velocity, vorticity, and pressure. These are shown to be strongly influenced by a dimensionless grouping that we refer to as the vortex Reynolds number, V. This keystone parameter appears as a ratio of the mean flow Reynolds number and the product of the swirl number and the chamber aspect ratio. Based on V, several fundamental features of the bidirectional vortex are quantified. Among them are the thicknesses of the viscous core and sidewall boundary layers; these decrease with V1/2 and V, respectively. The converse may be said of the peak velocity which increases with V1/2. In the same vein, the angular speed of the rigid-body rotation of the forced vortex is found to be linearly proportional to V. Our laminar swirl velocity is reminiscent of Sullivan’s two-cell vortex except for its additional dependence on the aspect ratio of the chamber. For the purpose of verification, theoretical predictions are compared to particle image velocimetry measurements and Navier–Stokes simulations at high vortex Reynolds numbers. By properly accounting for the turbulent eddy viscosity in the analytical model, local agreement is obtained with both laboratory measurements and computer simulations.

Interacting oscillatory boundary layers and wall modes in modulated rotating convection

Thermal convection in a rotating cylinder near onset is investigated using direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation in a regime dominated by the Coriolis force. For thermal driving too small to support convection throughout the entire cell, convection sets in as alternating pairs of hot and cold plumes in the sidewall boundary layer, the so-called wall modes of rotating convection. We subject the wall modes to small amplitude harmonic modulations of the rotation rate over a wide range of frequencies. The modulations produce harmonic Ekman boundary layers at the top and bottom lids as well as a Stokes boundary layer at the sidewall. These boundary layers drive a time-periodic large-scale circulation that interacts with the wall-localized thermal plumes in a non-trivial manner. The resultant phenomena include a substantial shift in the onset of wall-mode convection to higher temperature differences for a broad band of frequencies, as well as a significant alteration of the precession rate of the wall mode at very high modulation frequencies due to the mean azimuthal streaming flow resulting from the modulations.

Stability of a stratified fluid with a vertically moving sidewall

We present the results of a combined theoretical and experimental study of the stability of a uniformly stratified fluid bounded by a sidewall moving vertically with constant velocity. This arrangement is perhaps the simplest in which boundary effects can drive instability and, potentially, layering in a stratified fluid. Our investigations reveal that for a given stratification and diffusivity of the stratifying agent, the sidewall boundary-layer flow becomes linearly unstable when the wall velocity exceeds a critical value. The onset of instability is clearly observed in the experiments, and there is good quantitative agreement with some predictions of the linear stability analysis.

Influence of wall modes on the onset of bulk convection in a rotating cylinder

The onset of thermal convection in an enclosed rotating cylinder is greatly influenced by the interaction between the Coriolis force and the cylinder sidewall. For temperature differences between the hot bottom and the cool top that are too small to sustain convection throughout the entire cylinder, convection sets in as pairs of wall-bounded hot thermal plumes ascend and cold thermal plumes descend in the sidewall boundary layer, the so-called wall modes of rotating convection. Over an extensive range of parameter space, several wall modes with different numbers of thermal pairs coexist stably, and this multiplicity of states leads to very rich nonlinear dynamics as the temperature difference is increased to a level supporting thermal convection throughout the bulk of the cylinder. The bulk convection takes on characteristics of Küppers–Lortz spatiotemporal chaos, but its interaction with the persistent wall modes also leads to further flow complications which are explored here via numerical simulations.

Direct numerical simulations of bifurcations in an air-filled rotating baroclinic annulus

Three-dimensional Direct Numerical Simulation (DNS) on the nonlinear dynamics and a route to chaos in a rotating fluid subjected to lateral heating is presented here and discussed in the context of laboratory experiments in the baroclinic annulus. Following two previous preliminary studies by Maubert and Randriamampianina, the fluid used is air rather than a liquid as used in all other previous work. This study investigated a bifurcation sequence from the axisymmetric flow to a number of complex flows. The transition sequence, on increase of the rotation rate, from the axisymmetric solution via a steady, fully-developed baroclinic wave to chaotic flow followed a variant of the classical quasi-periodic bifurcation route, starting with a subcritical Hopf and associated saddle-node bifurcation. This was followed by a sequence of two supercritical Hopf-type bifurcations, first to an amplitude vacillation, then to a three-frequency quasi-periodic modulated amplitude vacillation (MAV), and finally to a chaotic MAV\@. In the context of the baroclinic annulus this sequence is unusual as the vacillation is usually found on decrease of the rotation rate from the steady wave flow. Further transitions of a steady wave with a higher wave number pointed to the possibility that a barotropic instability of the side wall boundary layers and the subsequent breakdown of these barotropic vortices may play a role in the transition to structural vacillation and, ultimately, geostrophic turbulence.