Center for Turbulence Research Proceedings of the Summer Program 2000 ...

Center for Turbulence Research

109

Proceedings of the Summer Program 2000

Particle distributions in the flow over a wavy wall

By Bendiks Jan Boersma

In this paper we will present the results of direct numerical simulation (DNS) of the flow over a small amplitude wavy wall. The evolution in space and time of particles are released in this flow will be examined. It will be shown that small waves on the channel bottom can generate large longitudinal vortices similar to Langmuir vortices that are observed in flows with waves at the free-surface. The simulation results show that the concentration of the particles is maximal on the downstream side of the wave crest.

1. Introduction

Water flow over a rippled bottom is an important flow geometry in civil engineering applications. Knowledge of flow statistics and sediment transport in such a geometry is useful for the maintenance of coastal structures, harbors, and rivers. Moreover, pollutants such as heavy metals and pesticides tend to chemically bind to the sediment particles. Ripples on river beds have in general a complicated three-dimensional shape. In this paper we will study a slightly simplified problem, namely the sediment transport over a smooth two-dimensional nearly sinusoidal bottom with a small amplitude.

In the recent literature, various experimental and theoretical/numerical results are reported for such a geometry (see for instance De Angelis et al. (1997), Cherukat et al. (1998), and Gong et al. (1996)). De Angelis et al. (1997) and Cherukat et al. (1998) report results obtained from DNS. The amplitude of the wave in these simulations is relatively large. which results in a separated flow downstream of the wave crest. Gong et al. (1996) report wind tunnel experiments for the flow over a wavy surface with a relative small amplitude. They shown that in a flow over a wavy wall without flow separation, large longitudinal vortices similar to Langmuir vortices (Leibovich (1983)) are generated. This observation is supported by a theoretical analysis performed by Philips et al. (1996).

In this study we will use DNS to simulate the flow over a low amplitude wavy wall. The amplitude of the waves on the channel bottom in the DNS is comparable to those used in the wind tunnel experiments of Gong et al. (1996). The Reynolds number based on channel height and bulk velocity is 3, 500. The wave length of the waves on the channel bottom is equal to the channel height, and the amplitude of the waves is 5% of the wavelength (or channel height). The flow solver used for the DNS is very similar to the one used by Van Haarlem et al. (1998). To study sediment transport, small spherical particles are placed in the flow and their motion is tracked in space and time.

In ?2, we will give the governing equations and we will briefly discuss the solution techniques. In ?3, we shortly discuss the equation for the sediment particles. Finally, in ?4 we will present results both for the flow and sediment particles. In ?5 we will give some conclusions.

Delft University of Technology, Laboratory for Aero- and Hydrodynamics, The Netherlands

110

0 -0 .1 -0 .2 -0 .3 -0 .4 Y -0.5 -0 .6 -0 .7 -0 .8 -0 .9

-1

B. J. Boersma

0.5

1

1.5

2

2.5

3

3.5

X

Figure 1. The computational domain.

2. Governing equations

In this section we will give the governing equations for the flow over a wavy wall and briefly discuss the solution technique which has been used to solve these equations.

The physical domain is shown in Fig. 1. The wave on the bottom is nearly sinusoidal. This is for computational reasons only. A fully sinusoidal bottom in combination with a flat surface can not be calculated using an orthogonal grid. From a computational point of view, an orthogonal grid is preferable over an non-orthogonal grid. With help of the following two-dimensional orthogonal coordinate transformation, the physical domain is mapped onto a rectangular computational domain.

x = -x + A sinh(-kz ) sin(kx ), y=y, z = -z - A cosh(-kz ) cos(kx ),

(2.1) (2.2) (2.3)

where x, y, and z denote the coordinates in the physical domain (Fig. 1) and x , y , and z denote the coordinates in the rectangular (computational) domain, with A the amplitude and k the wave number. With help of vector algebra (see for instance Morse and Feshback (1953)), the Navier-Stokes equations in the transformed computational domain can be written as (we have dropped the primes for convenience):

u 1 t + h2

huu h2uv huw

+

+

x

y

z

w + h2

uh - wh z x

=

1 P 1 - h x + h2

hxx + h2xy + hxz

x

y

z

+

xz h2

h z

-

zz h2

h x

+

Fx,

(2.4)

v 1 huv h2vv hvw

t + h2

+

+

x

y

z

=

1 P 1 - y + h2

hxy + h2yy + hyz

x

y

z

+ Fy,

(2.5)

w 1 t + h2

huw h2vw hw2

+

+

x

y

z

u + h2

wh - uh x z

=

1 P 1 - h z + h2

hxz + h2yz + hzz

x

y

z

+

xz h2

h x

-

xx h2

h z

+

Fz ,

(2.6)

Particle distributions in the flow over a wavy wall

111

with the geometric scale factor

h = 1 - 2Ak sinh(kz) cos(kx) - A2k2 cos2(kx) + A2k2 cosh2(kz),

and u, v, w the velocity components in the x, y, and z direction respectively, with p the pressure and ij the Newtonian stress term. The components of ij are given as follows:

u u h w h xx = 2 x h + h2 x + h2 z ,

u 1 v

xy =

+ y h x

,

u w

xz = z

h

+ x

h

,

v yy = 2 y ,

1 v w

yz =

+ h z y

,

w u h w h zz = 2 z h h2 x + h2 z .

(2.7)

In which is the kinematic viscosity of the fluid. The equations given above are nondimensionalized with the mean friction velocity U at the channel bottom and the channel height H. In the streamwise and spanwise direction, periodic boundary conditions are used. At the channel bottom no-slip boundary conditions are used, and at the surface freeslip conditions are used. The normal component of the velocity is also set to zero at the free surface. The flow in the channel is driven by a constant pressure gradient (2U2/H) in the x-direction. The spatial derivatives in Eqs. (2.4)-(2.7) are integrated with help of a fully central second-order finite-volume method on a staggered grid. The time integration of Eq. (2.4)-(2.6) has been carried out with a second-order Adams-Bashforth method. The pressure-correction method is used to satisfy the incompressibility constraint.

3. Particle equation

The motion of a small spherical particle, under the assumption that the lift force, Basset history force, added mass force, and pressure gradient force are small (see Maxey and Riley (1983)), can be described by the following equation (Maxey and Riley (1983))

4 3

p

a3

dup dt

=

6?(uf

-

up)

+

4 3 (p

- f )a3g,

dxp dt

= up,

(3.1)

where up is the particle velocity, xp the particle position, a the particle radius, uf the fluid velocity, p the particle density, f the fluid density, and g the gravity. The first term on the right-hand side of Eq. (3.1) denotes the Stokes drag and the second term

the gravity on the particle. Equation (3.1) can be rewritten as

dup dt

=

1 (uf

- up) +

p - f g, p

dxp dt

=

up,

(3.2)

112

B. J. Boersma

1312 10 12

12

10 8

11

12

9

9 10

11

11

12

8

12

8

14

9

10

10 11

14

11

1211

12 13

9 10

10

11 12

11

Level Wm

15 0.81

14 0.57

12 11 10 10

11 1143

10 11

12 11

10 7 7

13 1112

8 9

1098 7 9

10 11

12 12

11

7 54

86 10 11

14

13 0.34

11 9 87

10

98

13 10

12 0.10 11 -0.14 10 -0.37

9

8

12 11 9

6 2 43 6 9 11 13 97 5

5 13

10

8

9

14

13 14

7

12

13

8 9

12

9 -0.61

6

8 -0.85

8

7 -1.08

6 -1.32

5 -1.56

8

5 111079 13

8

15

12

12

10

9 10

15

10

12

4 -1.79 3 -2.03 2 -2.26 1 -2.50

Figure 2. The wall normal velocity in the wavy channel. (The flow is going from left to right.) W ranges from -2.50 (Level 1) to 0.81 (Level 15).

Um

15 11.75

14 10.98

13 10.22

12 9.45

11 8.69

10 7.92

9

7.16

8

6.39

7

5.63

6

4.86

5

4.10

4

3.33

3

2.57

2

1.80

1

1.04

Figure 3. The streamwise velocity u in the cross flow plane.

where is the particle relaxation (or response) time given by

= 2f a2 . 9p

(3.3)

Typical radii of sand grains are 10-5 to 10-3 meters, and typical densities are 3000kg/m3, which results in 10-2 and (p - f )/p 0.7. The time integration of Eq. (3.2) is performed with the same method and time step as the Navier-Stokes equations. The particle velocities at the particle positions xp are obtained with help of a quadratic interpolation using 27 neighboring velocity points. Periodic boundary conditions for the particles are used the stream- and spanwise directions. When a particle reaches the bottom of the channel, the sign of the vertical velocity is changed, i.e. the particle bounces back. When a particle reaches the surface of the channel, its vertical velocity component is set to zero and the gravity force will pull it back into the flow.

4. Results

In this section we will present some results obtained from the DNS. All simulations have been performed on a computational grid with 256 points in the streamwise direction, 128 points in the spanwise direction, and 96 points in the wall normal direction (all uniformly spaced). The Reynolds number based on U was equal to 250. The grid size in the wall normal direction is this 250/96 2.5Y +. The domain size in the streamwise and spanwise direction was 5H and 3H respectively, which corresponds to a grid spacing of 5Y + in the streamwise direction and 6Y + in the spanwise direction. This grid should be fine enough

Particle distributions in the flow over a wavy wall

113

14

12

10

U(z)

8

wave trough

wave crest 6

4

2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

z

Figure 4. The mean velocity profile as a function of the wall normal direction, at various

positions along the wave. Wave trough:

and

; Wave crest:

and

.

2.5

2

1.5

rms

1

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

z

Figure 5. The streamwise (

), spanwise (

), and wall normal (

averaged over the streamwise and spanwise direction.

1

) rms profiles

to capture all important scales of motion. The statistics we will present in this section are calculated over 20 independent samples, each separated by 0.1H/U in time.

Figure 2 shows an instantaneous plot of the vertical velocity in the wavy channel. Just upstream of the wave crest we observe a rather large positive (upward) velocity. Behind the wave crest the velocity is negative.

In Figure 3, we show an isosurface plot of the averaged axial velocity in the cross flow plane. The two large structures visible in this figure are due to a Langmuir type circulation, Leibovich (1983), which is induced by the waves on the channel bottom. Numerical experiments with smaller and larger spanwise domains also show this phenomena. A de-

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