Procedia Engineering 126 ( 2015 ) 275 – 279

Available online at www.sciencedirect.com

1877-7058 Crown Copyright © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM)doi: 10.1016/j.proeng.2015.11.241

ScienceDirect

7th International Conference on Fluid Mechanics, ICFM7

Fluid flow in a cavity driven by an oscillating lid by an improved incompressible SPH

Zhenhong Hua, Xing Zhenga,*, Qing-Wei Maa,b, Wen-Yang Duana

a. College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China b. Schools of Engineering and Mathematical Science, City University, London EC1V 0HB, UK

Abstract

In this paper, it adopts the improved incompressible smoothed particle hydrodynamics (SPH) method based the Rankine source decrease order technique, and some important numerical handling techniques will also be included, like solid boundary discrete scheme and higher order derivative approximation. The newly improve method will be applied to modeling lid drive flow. Although SPH has been established for simulation of lid drive flow inside a cavity, it has not yet been used to solve periodic flow situations in such case. We first validated the improved SPH simulation against benchmark solutions with experimental results and other numerical results with different Reynolds numbers. Then we analyzed the flow phenomena in a square cavity with an oscillating lid. According various comparisons, some features of this new SPH method will be given at last. © 2015 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM).

Keywords: SPH; ISPH; Meshless particle method; Lid-driven flow

1. Introduction

Flow inside a cavity is interesting as it involves a number of unique fluid dynamic phenomena and has applications in numerous engineering systems. Such flows can be driven by buoyancy effects or by the motion of one or more walls that define the cavity. A common problem for the latter is lid-driven flow (LDF) in which the motion of the top wall of the cavity induces a circulatory movement of the fluid. LDF in a cavity is relevant for short dwells, flexible blade coaters, and mixing and drying technologies. LDF in a cavity exhibits many fluid dynamic

* Corresponding author. Tel.: +86-451-82569123; fax: +86-451-82518443.

E-mail address: zhengxing@hrbeu.edu.cn

Crown Copyright © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM)

276 Zhenhong Hu et al. / Procedia Engineering 126 ( 2015 ) 275 – 279

phenomena such as longitudinal vortices, corner eddies, turbulence, and transitions. Such intricacies have attracted the attention of researchers over the years for investigation of different aspects of lid-driven cavity flow.

Early efforts were made several decades ago to establish numerical techniques for solving fluid flow problems in such geometries. Burggarf [1] presented analytical and numerical solutions for 2D LDF in a square cavity. As computational power has increased, a number of numerical methodologies to solve fluid LDF problems in cavities have been suggested. Benjamin et al. used the alternative direction implicit (ADI) method to solve 2D steady incompressible LDF in a cavity for Re up to 10,000 [2]. Ghia et al. [3] provided a stream function and a vorticity formulation to solve 2D incompressible Navier–Stokes equations for flow in a square cavity at Re ≤ 10,000. Schreiber et al. [4] conducted numerical experiments using a finite difference method with higher-order accuracy at different Re. Calhoon and Roach [5] used the finite volume method to predict LDF phenomena in a cavity at different Re.

Although most investigations of LDF in a cavity have been conducted for steady unidirectional motion of the cavity walls, interest has also been shown in fluid motion caused by oscillatory movement of the lid in transient situations. Soh and Goodrich [6] studied the flow inside a square cavity with an impulsively starting lid and an oscillating lid using a time-accurate finite difference method to solve unsteady incompressible Navier–Stokes equations in the form of primitive variables. Siva Subrahmanyam Mendu [7] used the lattice Boltzmann method (LBM) to simulate two dimensional fluid flow in a square cavity driven by a periodically oscillating lid.

The smoothed particle hydrodynamics (SPH) method is a meshless, purely Lagrangian technique, which was originally developed by Lucy and Gingold and Monaghan. It has been successfully applied in a wide range of problems subsequently, e.g. fluid mechanics, explosion mechanics, fluid structure interaction and so on. In SPH, the particles are scattered by moving nodes and carry field variables such as pressure, density and velocity. The smoothing kernels are used to approximate a continuous field. The incompressibility of fluid has two ways. In the most common approach, weakly compressible SPH (WCSPH) uses an equation of state with a large sound speed, and the results of the WCSPH can induce a noisy pressure field and spurious oscillation of pressure in time history for wave impact problem simulation. An alternative approach, the truly incompressible SPH (ISPH) technique uses a pressure Poisson equation to calculate the pressure. Although the pressure distribution in the whole field obtained by ISPH is smooth, the stability of the techniques is still an open discussion.

An exhaustive survey of LDF in a cavity revealed that there has not been much work on the periodic nature of fluid flow in a cavity. Furthermore, most researchers have used conventional CFD methods. In this paper, a new solid boundary handling method is introduced to improve the accuracy of ISPH. This modified ISPH is used to study two-dimensional fluid flow in a square cavity driven by a periodically oscillating lid.

2. SPH methodology

In the method of incompressible SPH, the incompressibility is enforced by way of setting 0D Dt at each particle on each computing time step. So in ISPH, it can be calculated by 0u . The computation of the ISPH method is composed of two basic steps. The first step is the prediction one in which the velocity filed is computed without enforcing incompressibility. In the second step, which is the correction step, incompressibility is enforced in the calculations through Poisson equation of pressure. So using the combined method of divergence-free and density- invariance can get more smooth results, it can be shown as

*2 *

1 2( 1)tP

t t

u . (1)

The parameter can be adjusted according to different problems. In this paper, 0.01 . Particles on solid boundary should satisfy the solid boundary condition, it can include two factors, velocity and pressure. These particles which is near the solid boundary should satisfy the no penetration on normal direction and free slip on tangent direction, which can be shown as

u n U n (2) Furthermore, the pressure of particles on solid boundary should still satisfy the solid bound condition, which can

be shown as ( )pn n g n U (3)

277 Zhenhong Hu et al. / Procedia Engineering 126 ( 2015 ) 275 – 279

The key problem for solving pressure on solid boundary particles is how to calculate the first order derivative of pressure. In this paper, it adopts the SFDI method to compute the pressure spatial derivative, more details can be found in Ma [8]. In 2D case, the formulas of pressure derivative can be shown as

0

0,1 0,12 0,2,

0,12 0,21

( )1x r

C a Cp

a a ,

0

0,2 0,21 0,2,

0,12 0,21

( )1y r

C a Cp

a a , (4)

2, ,

0, 2

( )( )k k

k

Nj x i x

x j ij i

j i

r rn W r r

r r , (5)

, 0,0, 0 0

0, 0

( )1[ ( ) ( )] ( )m m

m

Nj x x

m j jjx j

r rC p r p r W r r

n r r , (6)

, 0, , 0,0, 0

0, 0

( )( )1( )m m k k

m

Nj x x j x x

mk jjx j

r r r ra W r r

n r r , (7)

where 1, 2m k or 2, 1m k . The solid boundary condition of pressure is applied directly to the wall particle on solid boundary, and more details of this boundary handling can be found in Zhou et al [9].

3. Numerical tests

3.1. Problem description

Fig. 1 presents a schematic diagram of the 2D square cavity of height H and width L considered in the present study. The two side walls and the bottom wall of the cavity are stationary. The top wall of the cavity experiences an oscillating motion. The velocity of the top wall in the x-direction is given by

)cos( tUu (8) where U, the maximum velocity of the top wall, is the oscillation amplitude, t is time, ω is the oscillation frequency and T is a time period. We assume that the fluid is Newtonian and the flow is laminar. Furthermore, we assume that the flow is isothermal, incompressible and 2D with constant fluid properties.

Fig. 1 Schematic representation of cavity flow Fig. 2 comparison of normalized u-velocity profiles along y-axis through the geometric center of the cavity for various particle numbers at Re=100

For simulations, we consider Re values of 100, 400 and 1000, and oscillating frequencies of 2π/6.

3.2. Results and discussion

In any numerical technique, the simulated results depend on the grid size. It is necessary to verify the astringency of code. To test the astringency of the numerical results, we conducted simulations for three different grid sizes (100×100, 200×200, 300×300) for cavity flow driven by uniform motion of the lid at Re=100, as shown in Fig. 2. It is clear that the predicted results converge quickly towards the experimental results as the grid size increases. The accuracy of result of grid size of 300×300 is sufficient for engineering computations. Thus, a grid size of 300×300 is considered for simulations.

y/L

u/U

278 Zhenhong Hu et al. / Procedia Engineering 126 ( 2015 ) 275 – 279

The validity of our SPH code was examined against some benchmark solutions of LDF in a square cavity and experimental results from the literature for various Reynolds numbers. Fig. 3 a,b shows centerline u- and v-velocity profiles for different Reynolds numbers. Our results obtained are in good agreement with those of Ghia et al. [3].

(a) (b)

Fig. 3 Comparison of velocity profiles for our SPH method and results reported by Ghia et al. [3] for various Reynolds numbers. (a) Normalized u-velocity profile along the y-axis through the geometric center of the cavity. (b) Normalized v-velocity profile along the x-axis through the

geometric center of the cavity.

Vector plots in Fig. 4 shows the location of the vortex center. Because of space constraints, vector plots are only presented for Re=100 and 1000. The vortex center is near the top of square cavity at Re=100, and it moves to the center of square cavity at Re=1000.

(a) (b)

Fig. 4 Vector plots of cavity flow driven by uniform motion of the top lid for (a) Re=100 and (b) Re=1000.

After the SPH code was validated for cavity flow driven by uniform motion of the top lid, we used it to simulate fluid dynamics for the oscillating lid case. Under the action of the oscillating lid, the flow field becomes periodic with a frequency identical to that of the lid. Results presented in this section represent the steady periodic state of the fluid motion. Here lid motion of Re=100 and ω=2π/6 was adopted. Fig. 5a,b shows centerline u-velocity profiles during the first and second halves of the cycle for Re = 100 and ω = 2π/6. As the top lid undergoes a cosinusoidal oscillation, the adjacent fluid particles also experience the same motion due to the no-slip condition, whereas the velocity at the stationary bottom wall is zero at any instant of the cycle. Therefore, the centerline velocity changes from a typical non-zero value at the topmost point to zero at the bottom-most point depending on the time (Fig. 5a,b).

y/L

u/U

v/U

x/L

x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

279 Zhenhong Hu et al. / Procedia Engineering 126 ( 2015 ) 275 – 279

(a) the first half cycle (b) the second half cycle

Fig. 5 The u-velocity profile along the vertical centerline of the cavity at Re =100 and ω = 2π/6. Fig. 6 Vector plot at Re =100 and ω = 2π/6.

A better appraisal of the fluid dynamics caused by oscillation of the lid can be made from the vector patterns observed inside the cavity. Fig. 6 shows vortices in the vector plots for Re = 100 and ω = 2π/6. The vortexes in the cavity are more complicated than the uniform motion case. There are many vortexes in the square cavity. The fluid flow in the cavity appears periodical variation.

4. Conclusions

This paper presents a modified ISPH for simulating flows generated by uniform motion and oscillation of the top lid in a square cavity. With the help of a high order solid boundary discretized method, modified ISPH can get stable and reliable results for fluid flow in cavity. The results obtained by modified ISPH can get a good agreement with available experimental data. Velocity profiles and vector plots revealed details of fluid flow in the cavity at various Reynolds numbers. For the flow generated by uniform motion of the top lid in a square cavity, the location of vortex center is more close to the center of cavity as the Reynolds number increases. The fluid flow generated by oscillation of the top lid in the cavity appears periodical variation.

Acknowledgements

This work is sponsored by The National Natural Science Funds of China (51009034, 51279041), Foundational Research Funds for the central Universities (HEUCDZ1202, HEUCF120113), Foundational Research Funds of HEU (HEUFP05001, HEUFT05023) and Defense Pre Research Funds program (9140A14020712CB01158), to which the authors are most grateful.

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y/L

u/U

y/L

u/U x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1