Shallow Cavity Flows Research Articles

Numerical Investigation of Flow and Heat Transfer over a Shallow Cavity: Effect of Cavity Height Ratio

Flow over shallow cavities is used to model the flow field and heat transfer in a solar collector and a variety of engineering applications. Many studies have been conducted to demonstrate the effect of cavity aspect ratio (AR), but very few studies have been carried out to investigate the effect of cavity height ratio (HR) on shallow cavity flow behavior. In this paper, flow field structure and heat transfer within the 3-D shallow cavity are obtained numerically for two height ratio categories: HR = 0.0, 0.25, 0.5, 0.75, and 1.0 and HR = 1.25, 1.5, 1.75, 2.0, 2.25, and 2.5. The governing equations, continuity, momentum, and energy are solved numerically and using the standard (K-ε) turbulence model. ANSYS FLUENT 14 CFD code is used to perform the numerical simulation based on the finite volume method. In this study, the cavity aspect ratio, AR = 5.0, and Reynolds number, Re = 3 × 105, parameters are fixed. The cavity’s bottom wall is heated with a constant and uniform heat flux (q = 740 W/m2), while the other walls are assumed to be adiabatic. For the current Reynolds number and cavity geometry, a single vortex structure (recirculation region) is formed and occupies most of the cavity volume. The shape and location of the vortex differ according to the height ratio. A reverse velocity profile across the recirculation region near the cavity’s bottom wall is shown at all cavity height ratios. Streamlines and temperature contours on the plane of symmetry and cavity bottom wall are displayed. Local static pressure coefficient and Nusselt number profiles are obtained along the cavity’s bottom wall, and the average Nusselt number for various height ratios is established. The cavity height ratio (HR) is an important geometry parameter in shallow cavities, and it plays a significant role in the cavity flow behavior and heat transfer characteristics. The results indicate interesting flow dynamics based on height ratio (HR), which includes a minimal value in average Nusselt number for HR ≈ 1.75 and spatial transitions in local Nusselt number distribution along the bottom wall for different HRs.

Finite Difference and Cubic Interpolated Profile Lattice Boltzmann Method for Prediction of Two-Dimensional Lid-Driven Shallow Cavity Flow

In this paper, two-dimensional lid-driven cavity flow phenomena at steady state were simulated using two different scales of numerical method: the finite difference solution to the Navier–Stokes equation and the cubic interpolated pseudo-particle lattice Boltzmann method. The aspect ratio of cavity was set at 1, 2/3, 1/2 and 1/3 and the Reynolds number of 100, 400 and 1,000 for every simulation condition. The results were presented in terms of the location of the center of main vortex, the streamline plots and the velocity profiles at vertical and horizontal midsections. In this study, it is found that at the simulation of Reynolds numbers 100 and 400, both methods demonstrate a good agreement with each other; however, small discrepancies appeared for the simulation at the Reynolds number of 1,000. We also found that the number, size and formation of vortices strongly depend on the Reynolds number. The effect of the aspect ratio on the fluid flow behavior is also presented.

Jet-cavity interaction: effect of the cavity depth

A two-dimensional turbulent wall jet over a shallow cavity is studied numerically. The present paper investigates the wall jet outer layer effect on a shallow cavity flow structure. This layer is an important turbulence source which is added to that of the inner shear layer. It was found that the reattachment process seems to be accelerated in the turbulent wall jet incoming flow compared to that of the cavity in duct flow. The present study shows that the flow structure is very sensitive to the cavity depth to nozzle height ratio; the increasing of this ratio induces a reattachment length decrease.

Self-Sustained Oscillations of Turbulent Flow in an Open Cavity

Direct numerical and large eddy simulations of incompressible turbulent flows over deep and shallow cavities were performed in the range of 600 ≤ Re D ≤ 12, 000 to investigate the influence of the incoming turbulent boundary layer on self-sustained oscillations of the shear layer. When the turbulent boundary layer approached the open cavity with Re D = 3000, the energy spectra of the pressure fluctuations showed spectral peaks in the range of 0.15 ≤ω θ ≤0.3. Conditionally averaged flowfields disclosed that the spectral peaks arise from the separation of high-speed streaky structures rather than from a geometric singularity of the cavity. The same spectral peaks were observed in a backward-facing step flow as well as in deep and shallow cavity flows, despite the different geometries of these systems. In the turbulent cavity flow of R eD = 12, 000, however, the peak frequencies of the energy spectra at cavity lengths of L/D = 1 and 2 were found to correspond to the Nth modes with N = 2 and 3, respectively. These Nth modes were very similar to the frequency characteristics of self-sustained oscillations reported for laminar cavity flows. Inspection of instantaneous pressure fluctuations as well as spanwise-averaged pressure fluctuations revealed that regular shedding of quasi-two-dimensional vortical structures was responsible for the peak frequency in the energy spectra.

Analysis of the spectral relationships of cavity tones in subsonic resonant cavity flows

The understanding of the self-sustained flow-acoustic coupling mechanism in flows over shallow rectangular cavities is of great interest owing to its various practical applications. The ability to understand and predict the resonant frequencies in such flows has recently been advanced through contributions from signal processing theory and by viewing the Rossiter tones as the product of an amplitude modulation process between a fundamental aeroacoustic loop frequency (fa) and a modulating lower frequency. The results obtained using this approach applied to detailed and high-quality spectral data of shallow cavity flow over the Mach number range of 0.20–0.65 are presented and discussed. The new approach, while not a predictive technique, is used to clearly identify all the tones (Rossiter modes, their harmonics, and harmonics of fa) observed in the pressure spectra and to show relationships between the tones. The asymptotic growth with Mach number of fa and the small-step changes of the modulating lower frequency over the Mach number range studied provide insight into the variation of the Rossiter mode parameters. The results also indicate that the empirical parameters in the Rossiter equation vary with Mach number for fixed cavity geometry.

An investigation into supersonic shallow swept cavity flows

AbstractAn experimental investigation was conducted to determine the flow characteristics of shallow swept cavities at a free-stream Mach number of 2. The investigation described herein focused on obtaining information on both time-averaged and time-dependent flow features. The data consisted of steady and unsteady pressure measurements as well as some surface oil and schlieren flow visualisation. The effective length-to-depth ratiosl/dof cavities investigated ranged between 5·65 and 8 for shallow ‘open’ cavities and between 11·31 and 16 for ‘closed’ cavities. The cavity sweep angles were 0°, 15° and 45°. The results of the swept cavities, when compared to the datum cases of the straight (0°) cavities, showed some distinct differences. With regard to time-averaged flow characteristics, in an open swept cavity (5·65 <l/d< 8), the flow displayed quasi-open flow behaviour, distorted by the spanwise cross flow within the cavity. In the case of the closed swept cavity (11·31 <l/d< 16), flow features resembling the ‘closed’ to ‘transitional closed’ cavity flow types were seen to exist simultaneously across the span. Unsteady pressure data indicated that for an open cavity at 15° sweep angle, the discrete frequencies observed were similar to those of a straight open cavity. In contrast to this, at 45° sweep angle, the frequencies were broadband with no discrete frequency across the cavity length. For the closed cavity, the frequencies were all broadband irrespective of the sweep angle. The investigation also showed that the influence of the sweep angle on the pressure drag of the cavity strongly depends on the type of cavity flow: a sweep angle increase from 0° to 45° approximately doubled the pressure drag for an open cavity, while it led to a drag reduction of about 37% for the closed cavity.

Neural network-based modelling of subsonic cavity flows

A fundamental problem in the applications involved with aerodynamic flows is the difficulty in finding a suitable dynamical model containing the most significant information pertaining to the physical system. Especially in the design of feedback control systems, a representative model is a necessary tool constraining the applicable forms of control laws. This article addresses the modelling problem by the use of feedforward neural networks (NNs). Shallow cavity flows at different Mach numbers are considered, and a single NN admitting the Mach number as one of the external inputs is demonstrated to be capable of predicting the floor pressures. Simulations and real time experiments have been presented to support the learning and generalization claims introduced by NN-based models.

Feedback control of subsonic cavity flows using reduced-order models

Development, experimental implementation, and the results of reduced-order model based feedback control of subsonic shallow cavity flows are presented and discussed. Particle image velocimetry (PIV) data and the proper orthogonal decomposition (POD) technique are used to extract the most energetic flow features or POD eigenmodes. The Galerkin projection of the Navier–Stokes equations onto these modes is used to derive a set of nonlinear ordinary differential equations, which govern the time evolution of the eigenmodes, for the controller design. Stochastic estimation is used to correlate surface pressure data with flow-field data and dynamic surface pressure measurements are used to estimate the state of the flow. Five sets of PIV snapshots of a Mach 0.3 cavity flow with a Reynolds number of 105 based on the cavity depth are used to derive five different reduced-order models for the controller design. One model uses only the snapshots from the baseline (unforced) flow while the other four models each use snapshots from the baseline flow combined with snapshots from an open-loop sinusoidal forcing case. Linear-quadratic optimal controllers based on these models are designed to reduce cavity flow resonance and are evaluated experimentally. The results obtained with feedback control show a significant attenuation of the resonant tone and a redistribution of the energy into other modes with smaller energy levels in both the flow and surface pressure spectra. This constitutes a significant improvement in comparison with the results obtained using open-loop forcing. These results affirm that reduced-order model based feedback control represents a formidable alternative to open-loop strategies in cavity flow control problems even in its current state of infancy.

An application of Gappy POD

An iterative procedure, based on the proper orthogonal decomposition (POD), first proposed by Everson and Sirovich (J Opt Soc Am A 12(8):1657–1664, 1995) is applied to marred particle image velocimetry (PIV) data of shallow rectangular cavity flow at Mach 0.19, 0.28, 0.38, and 0.55. The procedure estimates the POD modes while simultaneously estimating the missing vectors in the PIV data. The results demonstrate that the absolute difference between the repaired vectors and the original PIV data approaches the experimental uncertainty as the number of included POD modes is increased. The estimation of the dominant POD modes is also shown to converge by examining the subspace spanned by the POD eigenfunctions.

Shallow cavity flow in a porous medium driven by differential heating

This paper describes steady flow through a porous medium in a shallow two-dimensional cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated. The main emphasis is on the situation where the Darcy–Rayleigh number are also discussed.

Experimental Study of Linear Closed-Loop Control of Subsonic Cavity Flow

A study is presented of the modeling and implementation of different concepts for linear feedback control of a single-mode resonance shallow cavity flow. When a physics-based linear model is used for cavity pressure oscillations, an H ∞ controller was designed and tested experimentally. It significantly reduced the main Rossiter mode for which it was designed, while leading to strong oscillations at other Rossiter modes. Other linear control methods such as Smith predictor controller and proportional integral derivative (PID) controller exhibited similar results. The ineffectiveness of using fixed linear models in the design of controllers for the cavity flows is discussed. A modification of the PID design produced a parallel-proportional with time-delay controller that remedied this problem by placing zeros at the frequencies corresponding to other resonance states. Interestingly, it was observed that introducing the same zero to the H ∞ controller can also successfully avoid the strong oscillations at other Rossiter modes otherwise observed in the single-mode-based design. The parallel-proportional with time-delay controller was compared to a very effective open-loop method for reducing cavity resonance and exhibited superior robustness with respect to departure of the Mach number from the design conditions. An interpretation is presented for the physical mechanisms by which the open-loop forcing and the parallel-proportional with time-delay controllers reduce the cavity flow noise. The results support the idea that both controls induce in the system a rapid switching between modes competing for the available energy that can be extracted from the mean flow.

Numerical Simulation of Incompressible Laminar Flow over Three-Dimensional Rectangular Cavities

This paper presents results of investigations of unsteady incompressible flow past three-dimensional cavities, where there is a complex interaction between the external flow and the recirculating flow inside the cavity. A computational fluid dynamics approach is used in the study. The simulation is based on the solution of the unsteady Navier-Stokes equations for three-dimensional incompressible flow by using finite difference schemes. The cavity is assumed to be rectangular in geometry, and the flow is assumed to be laminar. Typical results of computation are presented, showing the effects of the Reynolds number, cavity geometry, and inflow condition on the cavity flow fields. The results show that high Reynolds numbers, with deep cavity and shallow cavity flows can become unsteady with Kelvin-Helmholtz instability oscillations and exhibiting a three-dimensional nature, with Taylor-Go¨rtler longitudinal vortices on the floor and longitudinal vortex structures on the shear layer. At moderate Reynolds numbers the shallow cavity flow is more stable than deep cavity flows. For a given Reynolds number the flow structure is affected by the thickness of the inflow boundary layer with a significant interaction between the external flow and the recirculating flow inside the cavity.

Shallow cavity flow tones: transformation from large- to small-scale modes

The generation of flow tones in an axisymmetric cavity is addressed for the case where the inflow is fully turbulent and the acoustic wavelength is much longer than the cavity length. Emphasis is on the nature of flow tones in relation to variations of cavity length. For a sufficiently long cavity, the fully evolved axisymmetric instability yields a large-scale mode. When the cavity length is decreased to progressively smaller values, the mode of oscillation transforms from the large-scale mode to a small-scale mode; it occurs at small values of cavity length, and scales on the momentum thickness of the separating shear layer. At a minimum value of cavity length, distinct oscillations do not occur. The existence of pronounced oscillations, i.e., flow tones, is also related to the dimensionless cavity depth. Oscillations are suppressed when the cavity depth is sufficiently small relative to the momentum thickness of the inflow boundary layer.

Logic-Based Active Control of Subsonic Cavity Flow Resonance

We present the results of an experimental investigation for controlling a shallow cavity flow in the Mach number range 0.25‐0.5. The flow exhibits the characteristic staging behavior predicted by the semi-empirical Rossiter formula with multiple modes in the Mach number range 0.32‐0.38 and a single strong mode in other Mach numbers. A survey of the velocity at the exit of the zero net mass flow compression-driver actuator used for control reveals that its amplitude is frequency dependent and its behavior is little influenced by the main flow. Forcing the Mach 0.3 flow indicated that the actuator has good authority over a large range of frequencies, with reduction of spectral peaks observed at some forcing frequencies. We took advantage of this phenomenon to develop a logicbased controller that searches the frequency space and maintains the optimal forcing frequency for peak reduction at each Mach number. Optimal frequencies and the corresponding reduced resonance have been obtained for all of the flow conditions explored. The physics of optimal frequency forcing as well as some characteristics of the logic-based controller are discussed. I. Introduction A PPLICATION of closed-loop control in fluid dynamics is by its nature a challenging and fascinating problem. Although many significant results have been obtained with open-loop flow control, this technique lacks the responsiveness or the flexibility needed for application in dynamic flight environments. In contrast, closed-loop flow control, although in its infancy, appears to be the ideal technique for the successful management of flow in many applications due to its adaptability to variable conditions and to its potential for significantly reducing the power required for controlling the flow. Fo re xample, Cattafesta et al. 1 found that closed-loop control of cavity tones requires an order-of-magnitude less power than open-loop control. The results presented here are part of a larger multidisciplinary effort to develop tools and methodologies that can apply closed-loop aerodynamic flow control for manipulating the flow over maneuvering air vehicles. The first step was to select a particular flowfield relevant to Air Force applications and to utilize it in the development of various components of closed-loop flow control techniques. The case study chosen is the control of shallow cavity flow pressure fluctuations that are characterized by a strong resonance produced by a natural feedback mechanism similar to that occurring in other flows with self-sustained oscillations (e.g., impinging jet, screeching jet). In all of these cases, shear layer structures impacting a discontinuity or obstacle in the flow (e.g., the cavity trailing edge) scatter acoustic waves that propagate upstream and reach the shear layer receptivity region where they tune and enhance the development and growth of shear layer structures. In the case of flow over a weapon bay cavity, these fluctuations can lead to structural damage to the air vehicle or to the stores carried within the bay. Rossiter 2 first developed an empirical formula for predicting the cavity flow resonance frequencies, today referred to as Rossiter frequencies or modes. He also investigated the concept of a dominant mode of oscillation. Later Rockwell and Naudascher 3 ob

Shallow cavity flow tone experiments: onset of locked-on states

Fully turbulent inflow past a shallow cavity is investigated for the configuration of an axisymmetric cavity mounted in a pipe. Emphasis is on conditions giving rise to coherent oscillations, which can lead to locked-on states of flow tones in the pipe–cavity system. Unsteady surface pressure measurements are interpreted using three-dimensional representations of amplitude–frequency, and velocity; these representations are constructed for a range of cavity depth. Assessment of these data involves a variety of approaches. Evaluation of pressure gradients on plan views of the three-dimensional representations allows extraction of the frequencies of the instability (Strouhal) modes of the cavity oscillation. These frequency components are correlated with traditional models originally formulated for cavities in a free-stream. In addition, they are normalized using two length scales: inflow boundary-layer thickness and pipe diameter. These scales are consistent with those employed for the hydrodynamic instability of the separated shear layer, and are linked to the large-scale mode of the shear layer oscillation, which occurs at relatively long cavity length. In fact, a simple scaling based on pipe diameter can correlate the frequencies of the dominant peaks over a range of cavity depth. The foregoing considerations provide evidence that pronounced flow tones can be generated from a fully turbulent inflow at very low Mach number, including the limiting case of fully developed turbulent flow in a pipe. These tones can arise even for the extreme case of a cavity having a length over an order of magnitude longer than its depth. Suppression of tones is generally achieved if the cavity is sufficiently shallow.

Shallow cavity flow tone experiments: Onset of locked-on states

Fully turbulent inflow past a shallow cavity is investigated for the configuration of an axisymmetric cavity mounted in a pipe. Emphasis is on conditions giving rise to coherent oscillations, which can lead to locked-on states of flow tones in the pipe–cavity system. Evaluation of pressure gradients on plan views of three-dimensional representations allows extraction of the frequencies of the instability (Strouhal) modes of the cavity oscillation. These frequency components are correlated with traditional models originally formulated for cavities in a free stream. In addition, they are normalized using two length scales; inflow boundary-layer thickness and pipe diameter. These scales are consistent with those employed for the hydrodynamic instability of the separated shear layer, and are linked to the large-scale mode of the shear layer oscillation, which occurs at relatively long cavity length. The foregoing considerations provide evidence that pronounced flow tones can be generated from a fully turbulent inflow at very low Mach number. These tones can arise even for the extreme case of a cavity having a length over an order of magnitude longer than its depth. Suppression of tones is generally achieved if the cavity is sufficiently shallow.

On the evolution of thermally driven shallow cavity flows

The temporal evolution of thermally driven flow in a shallow laterally heated cavity is studied for the nonlinear regime where the Rayleigh numberRbased on cavity height is of the same order of magnitude as the aspect ratioL(length/height). The horizontal surfaces of the cavity are assumed to be thermally insulating. For a certain class of initial conditions the evolution is found to occur over two non-dimensional timescales, of order one and of orderL2. Analytical solutions for the motion throughout most of the cavity are found for each of these timescales and numerical solutions are obtained for the nonlinear time-dependent motion in end regions near each lateral wall. This provides a complete picture of the evolution of the steady-state flow in the cavity for cases where instability in the form of multicellular convection does not occur. The final steady state evolves on a dimensional timescale proportional tol2/κ, wherelis the length of the cavity,κis the thermal diffusivity of the fluid and the constant of proportionality depends on the ratioR/L.

Convection in a shallow laterally heated cavity with conducting boundaries

Shallow cavity flows driven by horizontal temperature gradients are analysed over a range of Rayleigh numbersR and Prandtl numbers σ, whereR is comparable in size to the aspect ratioL(≫1). Eigenvalue calculations show the existence of a critical Prandtl number ∂R > ∂R∂c(σ), below which the parallel core-flow structure is destroyed for Rayleigh numbersR>Rc(σ). For other Rayleigh numbers and Prandtl numbers the horizontal scale of influence of the end walls of the cavity is determined.

Thermally driven shallow cavity flows in porous media: the merged-layer régime

An analysis is given of convective motions generated in a shallow rectangular cavity by an applied horizontal temperature gradient; the cavity is filled with a fluid-saturated porous medium. For large Rayleigh numbers R , and for large aspect ratios L , a distinguished limit arises when R = O ( L 2 ). In this limit the boundary layers on the horizontal walls merge and completely fill the cavity. The régime is characterized by a non-parallel core flow that provides the dominant structure over a surprisingly wide range of the parameter R / L 2 . New results for the merged layer core are compared with the parallel flow convection-dominated boundary layer limit R / L 2 → ∞, and with the parallel flow diffusion-dominated limit R / L 2 → 0. Heat transfer calculations, which define the dependence of the Nusselt number on R / L 2 , have also been carried out.

Onset of multicellular convection in a shallow laterally heated cavity

Shallow cavity flows, driven by horizontal thermal gradients, are analysed over a range of Rayleigh numbers R and Prandtl numbers σ. Emphasis is placed on the limit in which R is comparable in size to the cavity aspect ratio L . Departures from two-dimensional steady Hadley cells, defined by the limit L → ∞ at fixed R and σ, are shown to be associated with nonlinear end effects. Eigenvalue calculations indicate the existence of a critical Prandtl number σ c , below which the core structure is not necessarily parallel with the horizontal boundaries. For σ < σ c the parallel flow core is destroyed at Rayleigh numbers R > R c (σ). Results for the stability boundary R c (σ) are presented.