A Large-Eddy Simulation of the Shear-Driven Cavity Flow Using Dynamic Modeling

This paper presents the development of the three- dimensional prolonged adaptive finite element equation solver for the Navier–Stokes equations. The finite element used is the tetrahedron with quadratic approximation of the velocities and linear approximation of the pressure. The equation system is formulated in the basic variables. The grid is adapted to the solution by the element Reynolds number. An element in the grid is refined when the Reynolds number of the element exceeds a preset limit. The global Reynolds number in the investigation is increased by scaling the solution for a lower Reynolds number. The grid is refined according to the scaled solution and the prolonged solution for the lower Reynolds number constitutes the start vector for the higher Reynolds number. Since the Reynolds number is the ratio of convection to diffusion, the grid refinements act as linearization and symmetrization of the equation system. The linear equation system of the Newton formulation is solved by CGSTAB with coupled node fill-in preconditioner. The test problem considered is the three-dimensional driven cavity flow. © 1997 John Wiley & Sons, Ltd.

A time-accurate, additive-correction multigrid method for the prediction of two-dimensional unsteady flows is presented in this article. The method makes use of the additive-correction multigrid strategy, which, originally proposed for steady-flow problems, is extended to the calculation of time-dependent and chaotic flows at high Reynolds or Rayleigh numbers. The numerical algorithm guarantees absolute, to machine accuracy, mass conservation at every time step, and it is characterized by second-order accuracy in space and time. In a companion article [1], the method is applied to the calculation of unsteady flow in cavities, i.e., the lid-driven cavity problem at high Reynolds number, and the buoyant flow in differentially heated cavities at high values of the Rayleigh number. Although the method has been implemented and tested for two-dimensional flows, it can also be extended to three-dimensional problems.

Fluid flow, heat transfer and entropy generation characteristics of micro-pipes are investigated computationally by considering the simultaneous effects of pipe diameter, wall heat flux and Reynolds number in detail. Variable fluid property continuity, Navier-Stokes and energy equations are numerically handled for wide ranges of pipe diameter (d = 0.50–1.00 mm), wall heat flux (q''= 1000–2000 W/m2) and Reynolds number (Re = 1 – 2000), where the relative roughness is kept constant at e/d = 0.001 in the complete set of the scenarios considered. Computations indicated slight shifts in velocity profiles from the laminar character at Re = 500 with the corresponding shape factor (H) and intermittency values (γ) of H = 3.293→3.275 and γ = 0.041→0.051 (d = 1.00→0.50 mm). Moreover, the onset of transition was determined to move down to Retra = 1,656, 1,607, 1,491, 1,341 and 1,272 at d = 1.00, 0.90, 0.75, 0.60 and 0.50 mm, respectively. The impacts of pipe diameter on friction mechanism and heat transfer rates are evaluated to become more significant at high Reynolds numbers, resulting in the rise of energy loss data at the identical conditions as well. In cases with low pipe diameter and high Reynolds number, wall heat flux is determined to promote the magnitude of local thermal entropy generation rates. Local Bejan numbers are inspected to rise with wall heat flux at high Reynolds numbers, indicating that the elevating role of wall heat flux on local thermal entropy generation is dominant to the suppressing function of Reynolds number on local thermal entropy generation. Cross-sectional total entropy generation is computed to be most influenced by pipe diameter at high wall heat flux and low Reynolds numbers.

The main objectives of this work are the formulation, implementation and applications of a numerical algorithm to simulate turbulent, incompressible, isothermal flows. The transient three-dimensional flow is analyzed using an explicit Taylor-Galerkin scheme and the finite element method with hexahedrical eight-node element. Turbulence is simulated using Large Eddy Simulation. For sub-grid scales two models were implemented, the classical Smagorinsky's model and the dynamic eddy viscosity model. For the second filtration, which is necessary in the dynamic model, a new method was developed based on independent finite elements that involve each node in the original mesh. The implemented scheme is efficient and good results with low additional computational cost were obtained. Results for two classical problems, the driven cavity and the backward facing step are presented. Comments about the model applicability for flows with high Reynolds numbers are also presented.

The present investigation of cavity flow in the subsonic and transonic speed regimes gives attention to the effect of Reynolds number on the flow inside the cavity. A rectangular, three-dimensional cavity was tested over Mach 0.3-1.05, and at Reynolds number/foot from 1 million to 4.2 million; two different cavity sizes were tested. For the deeper of the two cavities, at low Reynolds number, and for all conditions tested with the shallow cavity, the data showed much less steadiness. The results also show that the shallow cavity static pressure measurements were sensitive to the thickness of the boundary layer entering the cavity.

Local and global effects of cylindrical vortex generators on the mass-transfer distributions over the four active walls of a square, rib-roughened rotating duct with a sharp 180-deg bend are investigated. Cylindrical vortex generators (rods) are placed above, and parallel to, every other rib on the leading and trailing walls of the duct so that their wake can interact with the shear layer and recirculation region formed behind the ribs, as well as the rotation-generated secondary e ows. Local increases in near-wall turbulence intensity resulting from these interactions give rise to local enhancement of mass (heat)transfer. Measurements are presented for duct Reynolds numbersReintherange5 £ 10 3‐3 £ 10 4 andforrotationnumbersin therange0 ‐0.3. Therib height-to-hydraulic diameter ratio e/Dh is e xed at 0.1, and the rib pitch-to-rib height ratio P/e is 10.5. The vortex generator rods have a diameter-to-rib height ratio d/e of 0.78, and the distance separating them from the ribs relative to the rib height s/e is 0.55. Mass-transfer measurements of naphthalene sublimation have been carried out using an automated acquisition system and are correlated with heat transfer using the heat/mass transfer analogy. The results indicate that the vortex generators tend to enhance overall mass transfer in the duct, compared to the case where only ribs are present, both before and after the bend at high Reynolds and rotation numbers. Local enhancements of up to 30%are observed on all fourwalls oftheduct. At low Reynolds numbers, forexample, 5 ££ 10 3 , the insertion of the rods often leads to mass-transfer degradation. At high Reynolds numbers, for example, 3 ££ 10 4 , the enhancement due to the rods occurs on the surfaces stabilized by rotation (trailing edge on the inlet pass and leading edge on the outlet pass )and the side walls. The enhancement is more pronounced as the rotation number is increased. The detailed measurements in a ribbed duct with vortex-generator rods clearly show localized regions of enhanced mass (heat) transfer at Reynolds and rotation numbers within the envelope of practical interest for gas-turbine blade cooling applications.

Solving steady-state lid-driven square cavity flows at high Reynolds numbers via a coupled improved element-free Galerkin–reduced integration penalty method

Roughness Based Crossflow Transition Control for a Swept Wing Airfoil Design

The numerical simulation of incompressible turbulent flow at Re = 11 800 (equivalent Reynolds number Rem = 10 030) inside lid-driven cavities with varying spanwise aspect ratio and depthwise aspect ratio (DAR) has been carried out using the dynamic Smagorinsky model. A better comparison of central line averaged velocities and second order turbulent properties is obtained with the existing experimental result of Prasad and Koseff [“Reynolds number and end-wall effects on a lid-driven cavity flow,” Phys. Fluids A 1, 208–218 (1989)] than with the one provided by Leriche and Gavrilakis [“Direct numerical simulation of the flow in a lid-driven cubical cavity,” Phys. Fluids 12, 1363–1376 (2000)]. It is noticed that the size of the downstream secondary vortex increases with an increase in the aspect ratio. Lower DAR cavities are found to be having the maximum turbulent kinetic energy (TKE) and turbulent production, which is further confirmed by the TKE spectra shown at the point of maximum turbulent production on the symmetrical mid-plane. The power spectra maintain a slope of −53 in the inertial subrange as proposed by Kolmogorov and show a steeper slope in the dissipating range. The time integral scale shows that in low DAR cavities, eddies are formed and undergo complete distortion more frequently. The joint probability density function confirms the anisotropy in turbulence, and the anisotropy invariant map (Lumley triangle) shows the type of turbulent anisotropy which shows whether the turbulence is either one-component or two-component near the walls.

The application of the Kutta condition to unsteady flows has been controversial over the years, with increased research activities over the 1970s and 1980s. This dissatisfaction with the Kutta condition has been recently rejuvenated with the increased interest in low-Reynolds-number, high-frequency bio-inspired flight. However, there is no convincing alternative to the Kutta condition, even though it is not mathematically derived. Realizing that the lift generation and vorticity production are essentially viscous processes, we provide a viscous extension of the classical theory of unsteady aerodynamics by relaxing the Kutta condition. We introduce a trailing-edge singularity term in the pressure distribution and determine its strength by using the triple-deck viscous boundary layer theory. Based on the extended theory, we develop (for the first time) a theoretical viscous (Reynolds-number-dependent) extension of the Theodorsen lift frequency response function. It is found that viscosity induces more phase lag to the Theodorsen function particularly at high frequencies and low Reynolds numbers. The obtained theoretical results are validated against numerical laminar simulations of Navier–Stokes equations over a sinusoidally pitching NACA 0012 at low Reynolds numbers and using Reynolds-averaged Navier–Stokes equations at relatively high Reynolds numbers. The physics behind the observed viscosity-induced lag is discussed in relation to wake viscous damping, circulation development and the Kutta condition. Also, the viscous contribution to the lift is shown to significantly decrease the virtual mass, particularly at high frequencies and Reynolds numbers.

A theory is described for the nonlinear waves on the surface of a thin film flowing down an inclined plane. Attention is focused on stationary waves of finite amplitude and long wavelength at high Reynolds numbers and moderate Weber numbers. Based on asymptotic equations accurate to the second order in the depth-to-wavelength ratio, a third-order dynamical system is obtained after changing to the frame of reference moving at the wave propagation speed. By examining the fixed-point stability of the dynamical system, parametric regimes of heteroclinc orbits and Hopf bifurcations are delineated. Extensive numerical experiments guided by the linear analyses reveal a variety of bifurcation scenarios as the phase speed deviates from the Hopf-bifurcation thresholds. These include homoclinic bifurcations which lead to homoclinic orbits corresponding to well separated solitary waves with one or several humps, some of which occur after passing through chaotic zones generated by period-doublings. There are also cases where chaos is the ultimate state following cascades of period-doublings, as well as cases where only limit cycles prevail. The dependence of bifurcation scenarios on the inclination angle, and Weber and Reynolds numbers is summarized.

The flow in a spherical cavity on a curved round pipe is a canonical flow that describes well the flow inside a sidewall aneurysm on an intracranial artery. Intracranial aneurysms are often treated with a flow-diverting stent (FDS), a low-porosity metal mesh that covers the entrance to the cavity, to reduce blood flow into the aneurysm sac and exclude it from mechanical stresses imposed by the blood flow. Successful treatment is highly dependent on the degree of reduction of flow inside the cavity, and the resulting altered fluid mechanics inside the aneurysm following treatment. Using stereoscopic particle image velocimetry, we characterize the fluid mechanics in a canonical configuration representative of an intracranial aneurysm treated with a FDS: a spherical cavity on the side of a curved round pipe covered with a metal mesh formed by an actual medical FDS. This porous mesh coverage is the focus of Part 2 of the paper, characterizing the effects of parent vessel Re, De and pulsatility, Wo, on the fluid dynamics, compared with the canonical configuration with no impediments to flow into the cavity that is described in Part 1 (Chassagne et al., J. Fluid Mech., vol. 915, 2021, A123). Coverage with a FDS markedly reduces the flow Re in the aneurysmal cavity, creating a viscous-dominated flow environment despite the parent vessel Re > 100. Under steady flow conditions, the topology that forms inside the cavity is shown to be a function of the parent vessel De. At low values of De, flow enters the cavity at the leading edge and remains attached to the wall before exiting at the trailing edge, a novel behaviour that was not found under any conditions of the high-Re, unimpeded cavity flow described in Part 1. Under these conditions, flow in the cavity co-rotates with the direction of the free-stream flow, similar to Stokes flow in a cavity. As De increases, the flow along the leading edge begins to separate, and the recirculation zone grows with increasing De, until, above De ≈ 180, the flow inside the cavity is fully recirculating, counter-rotating with respect to the free-stream flow. Under pulsatile flow conditions, the vortex inside the cavity progresses through the same cycle - switching from attached and co-rotating with the free-stream flow at the beginning of the cycle (low velocity and positive acceleration) to separated and counter-rotating as De reaches a critical value. The location of separation within the harmonic cycle is shown to be a function of both De and Wo. The values of aneurysmal cavity Re based on both the average velocity and the circulation inside the cavity are shown to increase with increasing values of De, while Wo is shown to have little influence on the time-averaged metrics. As De increases, the strength of the secondary flow in the parent vessel grows, due to the inertial instability in the curved pipe, and the flow rate entering the cavity increases. Thus, the effectiveness of FDS treatment to exclude the aneurysmal cavity from the haemodynamic stresses is compromised for aneurysms located on high-curvature arteries, i.e. vessels with high De, and this can be a fluid mechanics criterion to guide treatment selection.

A suite of large-eddy simulations (LESs) of decaying homogeneous isotropic turbulence at high Reynolds numbers is performed and compared to wind-tunnel experiments in the tradition of Comte-Bellot and Corrsin. The error-landscape approach is used for the evaluation of the Smagorinsky model, and the results are used to identify an optimal combination of model parameter and resolution in a statistically robust fashion. The use of experimental reference data in the error-landscape approach allows to evaluate the optimal Smagorinsky coefficient at high Reynolds numbers and to perform detailed comparisons with analytical predictions. We demonstrate, using a pseudospectral discretization, that the optimal so-called Smagorinsky trajectory obtained from the error-landscape analysis converges at high simulation resolutions to the high-Re theoretical Lilly prediction for the Smagorinsky coefficient. Using modified wavenumbers in the same spectral code, the current study also presents error-landscape results based on LES with “second-order” discretization errors. By slightly revising Lilly’s analysis, we show that including the effect of numerical discretization when evaluating the strain-rate tensor needed in the subgrid-scale model leads to a good prediction of the optimal Smagorinsky parameter obtained from the corresponding error-landscape. Using similar analytical tools, we further demonstrate that the dynamic procedure can also be adapted to better account for the effects of discretization and test-filter shape.

The mechanisms of formation of three-dimensional jet flows inside large-scale vortex structures in a closed cubic cavity have been considered. The vortex structure of an incompressible viscous-fluid flow in this cavity has been investigated using the qualitative theory of ordinary differential equations. Singular points and their types and positions were determined at different Reynolds numbers on the basis of numerical calculations.

In turbulent flows over streamwise-aligned superhydrophobic surface (SHS) textures, the percent drag reduction is dependent on Reynolds number. This dependence is examined using direct numerical simulations of channel flow over SHS texture at three bulk Reynolds numbers, R e b = 2800, 6785 and 10975. Simulations of regular no-slip channel flows at the same bulk velocities are also performed for comparison. Changes in the flow due to the SHS texture are examined with particular focus on phase averaged statistics and coherent turbulent motions. As the Reynolds number is increased, the texture performance, or the percent drag reduction, is enhanced. In terms of turbulent motions, the weakened shear stresses are manifest as a decrease in the population of the small-scale vortical structures and a weakening of large-scale motions. The influence of stochastic fluctuations near the large-scale motions diminishes at higher Reynolds number. The enhanced performance is therefore due to a significant drop in the strength of small- and large-scale vortical structures. The weaker vortical motions yield a considerable reduction in the Reynolds shear stress and the transport of mean momentum. The SHS texture introduces a performance penalty in the form of a local increase in drag at the no-slip free-slip edge. This adverse effect is caused by the inhomogeneity of the near-wall flow field. However, the affected region narrows in width at higher Reynolds numbers. As a result, the percent drag reduction approaches the gas fraction of the texture.