Nature of turbulence inside a cubical lid-driven cavity: Effect of Reynolds number

doi:10.1016/j.ijheatfluidflow.2019.108498

International Journal of Heat and Fluid Flow

Volume 80, December 2019, 108498

https://doi.org/10.1016/j.ijheatfluidflow.2019.108498 Get rights and content

Highlights

•
Large eddy simulation of cubic lid-driven cavity flow has been performed at Reynolds number varying from 1000 to 15000.
•
Flow is laminar but unsteady at Re = 3200; with increase in Re, flow becomes turbulent and crosscorrelation value decreases.
•
Fourier spectra of auto-correlation shows a slope of $- 5 / 3$ in the sub-inertial range irrespective of the Reynolds number.
•
The turbulence is non-homogeneous even at low Re (= 3200) and inhomogeneity increases with Reynolds number.
•
AIM shows that turbulence is anisotropic, specially at near wall and corner regions.

Abstract

Large eddy simulation of incompressible flow inside a cubic lid driven cavity for a range of Reynolds number $(Re = 1000, 3200, 5000, 10000, 15000)$ is carried out using dynamic Smagorinsky model (DSM). Centerline average velocity profiles are compared with the existing experimental and numerical results. Except for Re = 1000, the boundary layer slope at wall remains almost same up to the point of inflexion on the mid-bottom wall and on the mid-upstream wall. With increase in Re, the point of inflexion adjusts in such a way so as to make larger and larger volume of fluid a sluggishly-rotating central core. The turbulent quantities such as turbulent kinetic energy and turbulent dissipation are found to be increasing with increase in Re. The auto and cross-correlation coefficient of velocity components are studied and it is found that low Re (= 3200) flow shows periodic nature and as Re increases it becomes aperiodic and turbulent. The cross-correlation decreases with increase in Re. The Fourier spectra of auto-correlation coefficient follows Kolmogorov’s $\frac{- 5}{3}$ slope in the inertial subrange irrespective of the Re, but the length of inertial subrange increases as Re increases. Kolmogorov length scale (η) and Taylor micro scale (λ) are calculated and it is found that both decrease towards the wall as dissipation is high at the walls. Anisotropic invariant map (AIM) has been plotted; nature of turbulence is found to be non-homogeneous and anisotropic even at low Re (= 3200) and non-homogeneity increases as Re increases. With increasing Re, the turbulence at the core region of the cavity shifts closer to isotropic.

Introduction

In a rectangular three-dimensional cavity, tangential movement of a plane wall with constant velocity representing Dirichlet boundary condition evokes interesting fluid flow physics. Despite simple in geometry, the flow inside the cavity is very much involved, specially at high Reynolds number (Re). Evolution and sustenance of turbulence inside the cavity makes it very difficult to analyze with the presence of complex phenomena like flow separation and reattachment, vortex formation (demonstration of Prandtl-Batchelor theorem), vortex stretching, recirculation, flow bifurcation, transition from laminar to turbulent flow etc. Even though a large number of researcher studied the lid-driven-cavity (LDC) before, it is difficult to analyze all the aspects of the flow. Any change in the flow conditions like Reynolds number or geometrical conditions like aspect ratio, orientation of the cavity, shape of the cavity makes the flow to behave differently. A schematic of an LDC is shown in Fig. 1.

The central main vortex as presented on the plane of symmetry $(y = \frac{W}{2})$ is called the primary vortex (PV). Three other secondary vortices viz. downstream secondary vortex (DSV), upstream secondary vortex (USV) and upper upstream secondary vortex (UUSV) are formed at the corners of the symmetry plane. In addition to these symmetry-plane vortices, Taylor-Görtler like (TGL) vortices are also present in the cavity on the transverse plane.

Koseff and Street (1984c) have conducted experiments in a cavity of L × H × W of 150 mm × 150 mm × 450 mm i.e. 1:1:3 cavity. Their focus was directed towards the presence of turbulence, three-dimensionality of flow, effect on DSV and the start-up flow for TGL vortices. The Reynolds number was varied from 1000 to 10000. They reported that the turbulent features appear at Re in the range of 6000–8000. The symmetry-plane velocity distribution and the DSV size were found to be smaller when compared with the two-dimensional (2D) numerical solutions of Ghia et al. (1982). The decrease was manifested due to the presence of the end-walls. The initial 30 s showed the formation of TGL for all the Reynolds number considered. Koseff and Street (1984b) also investigated the effect of the end-walls and found the DSV to decrease in size for span-wise aspect ratio (SAR) 1:1:1 and 1:1:2 whereas the size increased for 1:1:3 when Re is increased up to 10000. They attributed this phenomenon to the interaction of the DSV with the corner vortices. The time-mean averaged velocity measurements were provided by Koseff and Street (1984a) along with the turbulence measurements. They contend that the flow is concave in the DSV which produces maximum turbulence; end walls in a 3D flow play a significant role, which completely change the flow physics within the cavity. Freitas et al. (1985) made one of the pioneering 3D numerical study to predict the presence of TGL vortices. However, their work was limited to Re = 3200. A most comprehensive experimental study was conducted by Prasad and Koseff (1989). They carried out experiment on lid-driven cavity flow at a wide range of Reynolds number (3200 ≤ Re ≤ 10000) with varying span-wise aspect ratios (SAR = 0.25:1, 0.5:1, 0.75:1 and 1:1). They found that SAR has a dominant effect on flow field and this effect is different at higher and at lower Re. Reduced SAR causes drop in average and root mean square (rms) velocities and the Taylor-Görtler like (TGL) vortices are mostly responsible for velocity fluctuation at lower Re. Iwatsu et al. (1989) have examined the three-dimensional flow structure in detail over a wide range of Reynolds number Re up to 4000 and reported that the flow becomes unsteady at higher values when Re exceeds approximately 2000. Zang et al. (1993) have numerically simulated the LDC for Re = 3200 at 1:1:1 aspect ratio (i.e. cubic) and for Re = 7500 and 10000 at 1:1:0.5 span-wise aspect ratio. They reported that flow becomes unstable near the DSV at Re higher than 6000. Depth-wise aspect ratio (DAR) study along with cubic cavity study was numerically carried out by Cortes and Miller (1994) for Re up to 5000. However, a laminar flow formulation was used. In another numerical study, Chiang, Hwang, Sheu, 1996, Chiang, Sheu, Hwang, 1997, Chiang, Sheu, Hwang, 1998 have considered Re up to 3200 and focused on the transition from steady to unsteady flow. For example, transition occurs at much lower Re in 3D flows. Deshpande and Milton (1998) performed DNS on lid-driven cavity flow where they covered all the flow regime, starting from Stokes flow (Re = 0.1) to turbulent flow at Re = 10000. They noticed that instantaneous dissipation or total dissipation is very high near the moving lid and has high amplitude. The dissipation becomes lower towards the primary vortex core due to high Kolmogorov length scale (η). They also found that the dissipation due to fluctuation or turbulent kinetic energy (TKE) dissipation contributes more to the total dissipation as compared to the dissipation due to mean velocity except for the near wall region. Shankar and Deshpande (2000) in a review article have presented that there exists a considerable difference between a 2D flow and a 3D flow. They have discussed in details about turbulence and their length scales in a cubic cavity.

Leriche and Gavrilakis (2000) carried out DNS in a cubic lid-driven cavity at Re = 12000. Even at this Re, they found that at most part of the cavity, flow is laminar but unsteady and more than 70% of the total energy dissipation in the flow occurs in a thin region close to the moving lid. The flow is not span-wise homogeneous on the symmetrical mid plane which divides the flow into two statistically symmetrical halves. Bouffanais et al. (2007) repeated the work of Leriche and Gavrilakis (2000) but using large eddy simulation (LES) with dynamic Smagorinsky model (DSM) and dynamic mixed model (DMM). They found that the maximum turbulent production occurs at the DSV region just above the bottom plate and the turbulent energy dissipation is significant in the impingement region of the wall jets. Liberzon et al. (2011) conducted experiment on cubic LDC using particle image velocimetry to validate the numerical prediction of steady-oscillatory transition at lower Re. Their study showed that a stable steady state occurs for Re  ≤  1700. The steady-unsteady transition occurs in the range 1700  <  Re  <  1970 and for Re  >  1970, flow becomes oscillatory. The coherent structure passing through the maximum turbulent production zone have been discussed by Patel, Das, Roy, 2013, Patel, Das, Roy, 2014. In this study, λ₂ criterion is used to identify the shearing and the swirling structures on the statistically symmetric plane. Using LES, Samantaray and Das (2018) have computed the LDC for aspect ratio (AR) 1:1:0.5 and 1:1:1.5, 1:1:1 and DAR 1:0.5:1 and 1:2:1 where equivalent Re (Re_m) =10030. Their computed results are very close to the experimental results of Prasad and Koseff (1989). They concluded that DSV size increases with increase in SAR. Turbulent kinetic energy and turbulent production increases with lower DAR. The anisotropy in turbulence and type of anisotropy are also discussed at length.

By carrying out a detailed literature survey, it is observed that all the previous studies are classified under following broad categories: (a) mean flow distribution for laminar and turbulent regimes, (b) laminar to turbulent transition, (c) turbulence statistics at high Re, viz. at 10000. However, the effect of gradual increase of Re on the passage from laminar flow (i.e. absence of turbulence) to appearance of turbulence to change in the turbulent length scales, anisotropic nature of turbulence was not studied in details. The present work is a large eddy simulation of incompressible fluid flow in a cubic lid-driven cavity at a wide range of Reynolds number (1000 ≤ Re ≤ 15000) using dynamic Smagorinsky model. Though Leriche (2006) conducted a DNS study at Re as high as 18000 and 22000, limited information such as mean and rms velocities were given in that paper. The present study includes a detailed review of all the mean, rms and Reynolds shear stresses for Re 1000, 3200, 5000, 10000 and 15000. Other turbulent characteristics such as turbulent kinetic energy dissipation, auto- and cross-correlation are covered. The anisotropic nature of turbulence has been analyzed with the help of anisotropic invariant map (AIM). The study covers all the way from laminar to turbulent flow at high Re with in-depth description of the physics of the flow.

Section snippets

Governing Equation and Mathematical Formulation

The filtering process is carried out by averaging over a volume △_x × △_y × △_z and is given by the expression below. $\bar{f} (x, y, z, t) = \frac{1}{▵_{x} ▵_{y} ▵_{z}} \int_{D} f (x, y, z, t) d x d y d z$ Here $\bar{f}$ is the filtered variable, Davidson (2011).

The non-dimensionalized continuity and Navier-Stokes equation for incompressible flow filtered by a filtering process (Davidson, 2011) as shown in Eq. (1) are given in Eqs. (2) and (3) respectively. Here, position, time, velocity and pressure are normalized by the scales L, $\frac{L}{U_{0}},$ U₀ and $ρ U_{0}^{2}$

Discretization and Numerical procedure

By substituting Eqs. (4) and (5) in Eq. (3) and rearranging, we get $\frac{\partial {\bar{u}}_{i}}{\partial t} = H_{i} - \frac{\partial \bar{p}}{\partial x_{i}}$ where $H_{i} = \frac{\partial}{\partial x_{j}} [(\frac{1}{Re} + ν_{s g s}) \frac{\partial {\bar{u}}_{i}}{\partial x_{j}}] + \frac{\partial}{\partial x_{j}} (ν_{s g s} \frac{\partial {\bar{u}}_{j}}{\partial x_{i}}) - \frac{\partial}{\partial x_{j}} ({\bar{u}}_{i} {\bar{u}}_{j})$ Second order central difference scheme is used for the spatial derivatives of Eq. (11). They are discretized on a staggered grid arrangement. The velocity field for the first time step is obtained from one-step Euler method (Eq. (13)) and the subsequent steps are obtained from a second order time accurate Adams-Bashforth method (Eq. (14)) $\frac{{\bar{u}}_{i}^{*} - {\bar{u}}_{i}^{n}}{δ t} = H_{i}$

Computational domain and boundary condition

The computational domain is cubic space and the flow inside the cavity is assumed to be Newtonian and incompressible with uniform temperature. The numerical simulation is carried out using a structured grid with a grid distribution of hyperbolic tangent transformation (Eq. (17)), which gives almost uniform grid structure in the middle and denser grid size near the walls. $x_{i + 1} = \frac{l}{2} (1 + \frac{tanh [ζ (\frac{2 i}{N - 1} - 1)]}{tanh [ζ]}) for 0 \leq i \leq N - 1$

In Eq. (17), ζ is the stretching function and has the value of 1.0; l and N are

Results and discussion

The non-dimensional integration times are 1250, 1350, 1250, 1045 and 875 for Re =1000, 3200, 5000, 10000 and 15000 respectively. The time increment, δt is taken as 0.0025 in case of Re = 1000, 3200, 5000 and 0.002 in case of Re = 10000 and 15000. These time increment values are well below maximum δt that is calculated using the CFL condition (Eq. (15)) to ensure the numerical stability. Samples are collected at every 0.5 non-dimensional time and the samples that are collected after the

Conclusion

The large eddy simulation of a cubic lid-driven cavity flow for a range of Re (Re = 1000, 3200, 5000, 10000, 15000) has been performed. It is seen that the dynamic Smagorinsky model (DSM) is good at resolving the subgrid-scale stress tensor in the LES. The mean and the fluctuating flow fields have been validated by comparing center line velocities with the experimental results of Prasad and Koseff (1989) (for Re = 3200 and 5000) and Kato and Tanahashi (1992) (for Re = 5000). Conducting discrete

Declaration of Competing Interest

None.

Acknowledgment

The helpful comments and suggestions of the reviewers are gratefully acknowledged by the authors.

References (38)

L. Davidson
Fluid Mechanics Turbulent Flow and Turbulence Modeling
Technical Report
(2011)
M. Emory et al.
Visualizing turbulence anisotropy in the spatial domain with componentality contours
Cent. Turbul. Res. Annu. Res. Briefs
(2014)
C.J. Freitas et al.
Numerical simulation of three-dimensional flow in a cavity
Int. J. Numer. MethodsFluids
(1985)
H. Görtler
On the Three-Dimensional Instability of Laminar Boundary Layers on Concave Walls
(1954)
Y. Kato et al.
Finite-element method for three-dimensional incompressible viscous flow using simultaneous relaxation of velocity and Bernoulli function: flow in a lid-driven cubic cavity at Re = 5000
JSME Int. J. Ser. 2
(1992)
D.K. Patel et al.
LES of turbulent flow in a cubical cavity with two parallel lids moving in opposite direction
Int. J. Heat Mass Transf.
(2014)
U. Piomelli et al.
Large-eddy simulation of rotating channel flows using a localized dynamic model
Phys. Fluids
(1995)
R.S. Rogallo et al.
Numerical simulation of turbulent flows.
Annu. Rev. Fluid Mech.
(1984)
S. Banerjee et al.
Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches
J. Turbul.
(2007)
M. Bisoi et al.
Turbulent statistics in flow field due to interaction of two plane parallel jets
Phys. Fluids
(2017)

R. Bouffanais et al.

Large-eddy simulation of the flow in a lid-driven cubical cavity

Phys. Fluids

(2007)

T.P. Chiang et al.

Finite volume analysis of spiral motion in a rectangular lid-driven cavity

Int. J. Numer. MethodsFluids

(1996)

T.P. Chiang et al.

Three-dimensional vortex dynamics in a shear-driven rectangular cavity

Int. J. Comput. Fluid Dyn.

(1997)

T.P. Chiang et al.

Effect of Reynolds number on the eddy structure in a lid-driven cavity

Int. J. Numer. MethodsFluids

(1998)

K.-S. Choi et al.

The return to isotropy of homogeneous turbulence

J. Fluid Mech.

(2001)

M. Clauser et al.

The Effect of Curvature on the Transition from Laminar to Turbulent Boundary Layer

(1937)

A.B. Cortes et al.

Numerical experiments with the lid driven cavity flow problem

Comput. Fluids

(1994)

M.D. Deshpande et al.

Kolmogorov scales in a driven cavity flow

Fluid Dyn. Res.

(1998)

M. Emory et al.

Visualizing Turbulence Anisotropy in the Spatial Domain with Componentality Contours

Annual Research Briefs. Technical Report

(2014)

Cited by (12)

Numerical analysis of turbulent flow characteristics with the influence of speed ratio in a double-sided cavity
2024, MethodsX
The present study numerically investigates the two-dimensional steady incompressible turbulent flow characteristics in an enclosed cavity. The finite volume method (FVM) is used to discretize the governing equations, and k-ε turbulence models are adopted to predict the flow characteristics. The turbulent flow behavior is studied by varying the speed ratio (0.05 ≤ S ≤ 1.0), aspect ratio (0.5 ≤ K ≤ 2.0), and Reynolds number (1 × 10⁴ ≤ Re ≤ 2 × 10⁵). The flow characteristics are analyzed using stream function (ψ), Reynolds stresses (u'v'), and turbulent quantities. Results show the Reynolds number and speed ratio significantly influence the formation of vortices over the selected range of operating parameters. With the speed ratio, the turbulent kinetic energy reduces considerably by increasing the Reynolds number and aspect ratio. Similarly, for S = 0.05 and K = 0.5, the turbulent kinetic energy and dissipation rate are decreased by 89.16% and 42.28%, respectively. When Re is increased from 1 × 10⁴ to 2 × 10⁵, the turbulent viscosity increases by 92.10%. By comparing the results, average turbulent quantities are decreased by increasing the flow parameters.
- •
  Turbulent flow behavior is investigated by using the FVM near-wall treatment approach.One of the unique parameters called speed ratio is emphasized.
- •
  Contours of turbulence kinetic energy, dissipation, and viscosity are examined.
- •
  The average intensity of turbulent quantities is decreased by increasing the speed ratio.
The steady and unsteady regimes in a cubic lid-driven cavity with viscoplastic fluid solved with the lattice Boltzmann method
2024, Journal of Non-Newtonian Fluid Mechanics
This work builds upon the previously published analysis of a lid-driven cavity filled with viscoplastic fluid. We extend the study from a two-dimensional case to a three-dimensional one, employing the moment representation of the lattice Boltzmann method to obtain numerical results. The findings expand the existing dataset, which can potentially serve as benchmark results for inertial regimes of viscoplastic flows. In this study, we investigate the Reynolds and Bingham numbers until the flow transition from stationary to a transient regime. The results reveal that, similarly to the Newtonian case, there is an effective Reynolds number for the bifurcation, approximately Re^⁎ = Re₀ (1 + Bn), where Re₀ represents the bifurcation point for a Newtonian fluid. Like the Newtonian cases, there were instances where the Taylor-Görtler-like vortices moved toward the cavity's side periodically. In other cases, more than two vortices simultaneously formed, with their number changing over time. Finally, similar to the two-dimensional case, the bifurcation initiated after the Moffat eddies in the downstream corner broke down into plugs.
A three-dimensional high-order flux reconstruction lattice boltzmann flux solver for incompressible laminar and turbulent flows
2023, Computers and Fluids
A three-dimensional (3D) high-order flux reconstruction lattice Boltzmann flux solver (FR-LBFS) is developed in this paper for accurate and efficient simulations of incompressible laminar and turbulent flows. The original lattice Boltzmann flux solver (LBFS) is a second-order finite volume method (FVM) which combines the advantages of conventional Navier-Stokes (N-S) solvers and lattice Boltzmann equation (LBE) solvers. But it is not convenient to construct high-order FVM, and the compactness is always a bottleneck. The present work separates LBFS from FVM and combines LBFS with high-order FR scheme. Thus, the FR-LBFS inherits the superiority of LBFS and FR and shares the following attractive features: (i) a fully-explicit arbitrary order compact method, (ii) weakly compressible (WC) model for unsteady incompressible flows and (iii) evaluating inviscid and viscous fluxes simultaneously and no particular technique requirement, such as local discontinuous Galerkin method (LDG), for viscous term discretization. To validate the current method, various 3D incompressible laminar isothermal and thermal numerical experiments are presented first. Good agreements are achieved between the current results and the previous experimental and numerical data whilst the present high-order solver adopt coarse meshes. Furthermore, the ability of the proposed method to perform accurate and stable computations of incompressible decaying homogeneous isotropic turbulent flows and turbulent channel flows with different mesh resolutions and accuracy orders are investigated. The results show that the present 3D FR-LBFS is a promising tool for under-resolved or implicit large eddy simulation (ILES) of incompressible turbulent flows.
Driven flow motion by a dually moving lid of a square cavity
2022, European Journal of Mechanics, B/Fluids
Citation Excerpt :
In the very recent years, attempts have been made towards analyzing the engineering interest within the lid-driven cavity problems. The researches on cross-shaped lid-driven cavity [37], entropy generation of an alumina–water nanofluid [38], Marangoni convection [39], presence of a triangular block [40], dynamic Smagorinsky model [41], thermal chamfer effects [42], the oscillatory wall induced vortex dynamics [43], elastoviscoplastic flows [44] and partially heated hexagon cavity [45] are only a few of recently published intriguing aspects. As cited above, the traditional lid-driven cavity problem has been abundantly studied in the literature to explore the mixing process within the cavity as a consequence of single, double or quadruple moving lids of the enclosure [1].
One of the important prototype problems of fluid mechanics, the lid-driven cavity flow is revisited such that the single lid is separated into two joint active/passive walls representing potential stirrers during a chemical mixing process. The right part of the wall is released freely to move right with a uniform velocity, while the left part attached to the adjacent wall at a point of dislocation is considered stationary or movable freely with a constant speed. The novel cavity wall mechanism introduced in this manner has the capability of driving the pressure of the fluid and resulting in formation of the primary/secondary vortex structures. This sort of cavity setup might be useful if mixing process of a fluid is demanded to be in different sections of the cavity or if a mixture is desired to be purified in different sections of the cavity. Special attention should be drawn here that this kind of wall action is dissimilar to the familiar lid-driven cavity flow problem due to moving one, two or more parallel/crossing sides of the cavity studied earlier in the literature. Having conveniently made the usual equations of motion non dimensional, three dimensionless parameters are arisen governing the physical phenomenon, namely the point linking the lid parts, the velocity of left part of the lid and the Reynolds number associated with the right part. A finite element approach is adopted to simulate the resulting pressure and flow fields affected by variation of these parameters. From the simulations, the mixing properties of flow as a result of the triggered pressure are elucidated at three distinct parts of the lid; (a) when the lid is separated close to the left portion of the wall, (b) when the lid separation is set at the middle of the wall and (c) when the parts of the lid are connected next to the right portion of the lid. Depending on the above scenarios, a quite rich phenomenon of pressure distribution and the resultant recirculating primary/secondary vortex flow in the form of streamline patterns is observed. Assisting and opposing influences of the dually moving lid and its control over the small and high Reynolds number driven flow regimes are explored through graphical illustrations as well as tabular forms.
Multigrid accelerated projection method on GPU cluster for the simulation of turbulent flows
2023, Journal of Mechanics
Analytical solution and flow topology in a lid-driven S-shaped cavity
2022, Physics of Fluids

View all citing articles on Scopus

View full text

Nature of turbulence inside a cubical lid-driven cavity: Effect of Reynolds number

Highlights

Abstract

Introduction

Section snippets

Governing Equation and Mathematical Formulation

Discretization and Numerical procedure

Computational domain and boundary condition

Results and discussion

Conclusion

Declaration of Competing Interest

Acknowledgment

Cent. Turbul. Res. Annu. Res. Briefs

Int. J. Numer. MethodsFluids

JSME Int. J. Ser. 2

Int. J. Heat Mass Transf.

Phys. Fluids

Annu. Rev. Fluid Mech.

Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches

J. Turbul.

Turbulent statistics in flow field due to interaction of two plane parallel jets

Phys. Fluids

Large-eddy simulation of the flow in a lid-driven cubical cavity

Phys. Fluids

Finite volume analysis of spiral motion in a rectangular lid-driven cavity

Int. J. Numer. MethodsFluids

Three-dimensional vortex dynamics in a shear-driven rectangular cavity

Int. J. Comput. Fluid Dyn.

Effect of Reynolds number on the eddy structure in a lid-driven cavity

Int. J. Numer. MethodsFluids

The return to isotropy of homogeneous turbulence

J. Fluid Mech.

The Effect of Curvature on the Transition from Laminar to Turbulent Boundary Layer

Numerical experiments with the lid driven cavity flow problem

Comput. Fluids

Kolmogorov scales in a driven cavity flow

Fluid Dyn. Res.

Visualizing Turbulence Anisotropy in the Spatial Domain with Componentality Contours

Annual Research Briefs. Technical Report