INTRODUCTION
Lid-driven cavity flow is widely used as a benchmark case in computational fluid dynamics (CFD) research due to its simple geometry and easily imposed boundary conditions. This makes it an ideal model for validating numerical simulation techniques. The phenomenon has various industrial applications, including film melt spinning processes for manufacturing microcrystalline materials and continuous drying systems. Additionally, similar flow patterns are observed in nature, such as sediment transport modeling in riverbeds.
This study presents both an experimental and numerical investigation of lid-driven cavity flow for laminar and turbulent flow regimes. The experimental setup consists of two non-intrusive velocity measurement techniques: Particle Image Velocimetry (PIV), which captures global velocity data, and Laser Doppler Anemometry (LDA), which measures local velocities. PIV data is calibrated using LDA results to enhance measurement accuracy.
Numerically, the study employs a commercial CFD solver, using the Reynolds Averaged Navier-Stokes (RANS) equations with the k-ฮต (K-epsilon) turbulence model. Validation of the numerical solver is conducted by comparing its results with experimental data, ensuring accuracy. Additionally, the study examines the effects of lid acceleration on the development of flow patterns and circulation within the cavity.
LITERATURE REVIEW
Several experimental and numerical studies have been conducted on lid-driven cavity flow, classified based on measurement techniques, aspect ratio, Reynolds number range, mathematical formulations, and numerical methods.
One of the most widely referenced studies is Ghia et al. (1982), which employed a strongly implicit multigrid method to solve the vorticity and stream function formulation of the Navier-Stokes equations for Reynolds numbers ranging from 1,000 to 10,000. This work is considered a benchmark for numerical studies in this field.
Other notable studies include:
- Erturk & Gokcol, who performed steady 2D simulations for incompressible cavity flow up to Re = 21,000 using a finite volume solver.
- Pradhan & Kumaran, who conducted Direct Simulation Monte Carlo (DSMC) simulations to analyze 2D and 3D lid-driven cavity flow with opposite-moving walls at different velocities for compressible and transitional flow.
- Deshpande & Milton, who directly simulated 3D lid-driven cavity flow and compared the results with experimental data at Re = 3200 and 10,000.
- Jordan & Ragab, who used Direct Numerical Simulations (DNS) to predict 3D cavity flows at moderate Reynolds numbers and Large Eddy Simulations (LES) for higher Reynolds numbers.
Additionally, the review discusses various turbulence models applied to Reynolds Averaged Navier-Stokes (RANS) simulations, including Spalart-Allmaras, Shear Stress Transport (SST), and k-ฯ models, evaluating their effectiveness in predicting internal flows.
EXPERIMENTAL STUDY
This chapter details the experimental setup, measurement techniques, and calibration processes for the lid-driven cavity flow investigation.
3.1 Experimental Setup
The experimental setup consists of a square cavity made of plexiglass, with a 1:1 aspect ratio and a depth of 5 inches, ensuring a two-dimensional flow pattern. The cavity is filled with deionized water, seeded with silver-coated hollow glass spheres for enhanced visibility in velocity measurements. The setup includes an impermeable rigid lid mounted on a rail system, allowing controlled motion.
3.2 PIV Settings
A 2D PIV system manufactured by TSI is used for global velocity measurements. The test section is illuminated by a 532 nm Nd-YAG laser with a 500 mm lens, positioned perpendicular to the cavity. The laser beam is directed using a mirror placed at a 45-degree angle, ensuring uniform illumination of the cavity. A 2048 ร 2048 pixel CCD camera captures the velocity field.
Figure 4 โ Laser Sheet Projected Inside the Cavity with Water and Seeding
3.3 LDA Settings
The 2D LDA system, also by TSI, measures local velocity using an Argon Ion Laser (600 mW). The laser beam is split into four beams using fiber optics, which are then directed into the cavity through a 120 mm lens. The system detects Doppler frequency shifts, converting them into velocity values. The LDA is set to backscatter mode, with a PMT voltage of 450V and a burst threshold of 30mV.
Figure 5 โ Schematic of the LDA System
3.4 Experimental Results
The post-processing stages of PIV data collection are illustrated in Figures 6โ8:
Figure 6 โ Raw Image Captured by the CCD Camera, showing illuminated seeding particles.
Figure 7 โ Raw Vector Image Generated from PIV Software, showing velocity vectors before processing.
Figure 8 โ Final Processed Vector Plot, displaying refined velocity field data.
3.5 Calibration of PIV Measurements with LDA
Calibration is performed by comparing PIV global velocity measurements with LDA local velocity data along the vertical centerline of the cavity. The study finds that:
- The horizontal velocity profiles predicted by PIV and CFD simulations are in close agreement.
- However, LDA measurements tend to be significantly lower than PIV and CFD results.
- At higher Reynolds numbers, the discrepancy between PIV and CFD velocities decreases, but CFD continues to underpredict PIV measurements near the cavity center.
- LDA measurements align better with PIV data near the moving lid at higher Reynolds numbers.
Velocity profile comparisons at different Reynolds numbers are illustrated in:
Figure 9 โ Comparison of CFD, PIV, and LDA Velocities at Re = 1250
Figure 10 โ Comparison of CFD, PIV, and LDA Velocities at Re = 2030
Figure 11 โ Comparison of CFD, PIV, and LDA Velocities at Re = 3050
Chapter 4: Mathematical Formulation and Numerical Approach
This chapter presents the mathematical framework and numerical methodology used to simulate lid-driven cavity flow. The study employs a Reynolds-Averaged Navier-Stokes (RANS) approach with a k-ฮต turbulence model to capture the dynamics of both laminar and turbulent regimes. The equations governing fluid motion are solved using the Finite Volume Method (FVM) with an implicit discretization scheme.
4.1 Governing Equations
The fundamental equations governing incompressible, viscous flow are the Navier-Stokes equations, which describe the conservation of mass and momentum. The formulation is expressed as follows:
Continuity Equation (Mass Conservation):
Momentum Conservation (Navier-Stokes Equations):
For turbulent flow cases, the velocity field is decomposed into mean and fluctuating components using Reynolds decomposition. The Reynolds-Averaged Navier-Stokes (RANS) equations are then derived by time-averaging the Navier-Stokes equations:
The Reynolds stress term is approximated using the k-ฮต turbulence model, which introduces two additional transport equations:
Turbulent kinetic energy equation (k):
Turbulent dissipation equation (ฮต):
where P_k represents the production of turbulent kinetic energy, and C_1, C_2, ฯ_k, ฯ_ฮต are empirical constants.
4.2 Numerical Discretization and Solution Procedure
The governing equations are solved using the Finite Volume Method (FVM), where the computational domain is divided into discrete control volumes. The numerical procedure follows these steps:
- Spatial Discretization: The convective terms are discretized using a second-order upwind scheme to minimize numerical diffusion.
- Temporal Discretization: A second-order implicit time-stepping method ensures stability for transient simulations.
- Pressure-Velocity Coupling: The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm is employed to iteratively solve for pressure and velocity fields.
A structured mesh is used for all simulations, with refined grid resolution near the cavity walls to accurately capture boundary layer effects. The computational grid is tested for grid independence to ensure numerical accuracy.
Chapter 5: Numerical Validation and Verification
This chapter validates the numerical solver by comparing its predictions with experimental results and established benchmark studies. The accuracy of the numerical model is assessed using velocity profile comparisons, vortex location analysis, and grid independence tests.
5.1 Comparison with Benchmark Data
To verify the accuracy of the numerical model, simulation results are compared with:
- Experimental PIV and LDA measurements conducted in this study.
- Benchmark computational studies, particularly Ghia et al. (1982), which provide reference velocity profiles for various Reynolds numbers.
Key validation parameters include:
- U and V velocity profiles along the vertical and horizontal centerlines.
- Location and strength of primary and secondary vortices.
- Turbulent intensity predictions in the cavity.
The results indicate that velocity distribution and turbulence intensity vary significantly with changing Reynolds numbers, impacting pressure drop and overall efficiency. Higher Reynolds numbers lead to more pronounced turbulence, affecting flow separation and vortex formation. Contour plots visualize these effects, showing the relationship between flow dynamics and system performance. A strong correlation between numerical and experimental data supports the modelโs accuracy, though minor discrepancies arise due to turbulence modeling assumptions. These findings contribute to optimizing system design and enhancing fluid dynamics understanding for engineering applications.
Chapter 6: Numerical Prediction of Lid-Driven Cavity Flow for Laminar and Turbulent Regimes
This chapter focuses on numerical predictions using the validated method from Chapter 5. Simulations are conducted for both laminar and turbulent flows using calibrated parameters.
6.1 Numerical Prediction for Laminar Flow
Simulations are performed at a low Reynolds number (Re = 1000) with a lid velocity of u = 0.03956 m/s. The results are presented in the form of:
- Velocity vector plots, illustrating the primary circulation pattern in the cavity center.
- Pressure and velocity contours, depicting the flow distribution.
- U and V velocity distributions along the vertical and horizontal centerlines of the cavity.
Figures to be included:
- Figure 32 โ Velocity Vector Plot for Re = 1000
- Figure 33 โ Streamline Plot for Re = 1000
6.2 Turbulence Model Calibration
To improve simulation accuracy for transitional and turbulent flows, the k-ฮต turbulence model is calibrated by refining its coefficients. This adjustment is crucial, as standard turbulence models often fail to capture the intricate dynamics of transitional flows.
6.3 Numerical Prediction for Turbulent Flow
Simulations are conducted for higher Reynolds numbers (Re = 5000 – 10,000), with the following key findings:
- Vortex patterns and velocity contours indicate increased flow instability at higher Reynolds numbers.
- Comparison with previous studies demonstrates high accuracy after turbulence model calibration.
Figures to be included:
- Figure 42 โ Contour Plot of Turbulent Viscosity at Re = 10,000
- Figures 43-44 โ U velocity profile on the vertical centerline for Reynolds numbers ranging from 6000 to 10,000
Chapter 7: Study of Unsteady Lid-Driven Cavity Flow
This chapter explores how the flow evolves from stagnation to a steady-state condition across different Reynolds numbers.
7.1 Evolution of Circulation Patterns
The study examines how primary circulation develops within the cavity due to lid movement.
Key findings:
- For Re = 1000 – 3000, circulation gradually stabilizes into a steady-state pattern.
- For Re โฅ 4000, significant variations in vortex patterns occur before reaching stability.
- For Re = 10,000, the primary circulation center shifts further toward the upper right of the cavity.
Figures to be included:
- Figures 45-50 โ Vector Plots Showing Development of Circulation Patterns from Stagnation to Steady-State
- Figure 51 โ Final Position of Primary Circulation at t = 6.0 sec
7.2 Effect of Lid Acceleration on Flow Development
This section analyzes how different lid acceleration profiles affect flow formation. Three acceleration scenarios are simulated:
- Step acceleration
- Linear acceleration
- Sinusoidal acceleration
Key findings:
- Sinusoidal acceleration produces a smoother transition compared to step acceleration.
- Step acceleration generates stronger but more unstable vortices.
- U and V velocity distributions differ significantly based on acceleration type.
Figures to be included:
- Figures 60-68 โ Comparison of Velocity Distributions for Different Lid Acceleration Methods
Chapter 8: Conclusion
This chapter summarizes the main findings from the experimental and numerical simulations conducted in this study.
Key conclusions:
- Numerical validation confirms that the CFD method used is highly accurate, particularly after turbulence model calibration.
- Flow patterns in the cavity strongly depend on the Reynolds number, where laminar flow exhibits a more stable evolution compared to turbulent flow.
- PIV (Particle Image Velocimetry) and LDA (Laser Doppler Anemometry) measurements align well with numerical results, although LDA tends to measure slightly lower velocities compared to PIV and CFD.
- Lid acceleration significantly impacts flow formation, with sinusoidal acceleration producing a more uniform velocity distribution.
Recommendations for Future Research:
Use more advanced turbulence models, such as Large Eddy Simulation (LES) or Direct Numerical Simulation (DNS), for better turbulence prediction.
Conduct experiments at higher Reynolds numbers to further understand the transition from turbulent to chaotic flow.
Improve PIV measurement accuracy through more precise LDA calibration.
