Studies of vortex breakdown and its stability in a confined cylindrical container 3

CHAPTER 3 NUMERICAL SIMULATION METHOD 3.1 Introduction To better understand the flow phenomenon in the confined cylindrical container with one rotating end, a numerical simulation cod

CHAPTER 3 NUMERICAL SIMULATION METHOD

3.1 Introduction

To better understand the flow phenomenon in the confined cylindrical container with one rotating end, a numerical simulation code was developed The equations governing the flow are the axisymmetric Navier-Stokes equations, together with the continuity equation and appropriate boundary and initial conditions Gelfgat et al (2001) showed numerically that the onset of unsteadiness of the flow is via a supercritical axisymmetric Hopf bifurcation for H/R in the range of 1.6 to 2.8 Nonlinear computations (Blackburn and Lopez 2000, 2002, Blackburn 2002) have shown that this oscillatory state remains stable to three-dimensional perturbations for

Re up to about 3400 That numerical finding is consistent with the experimental observations of Stevens et al (1999) Thus, the problem under the conditions studied here (for most of the cases, the Reynolds number is less than 3000) can be solved by axisymmetric numerical simulations The approach adopted follows the method which has been extensively used by Lopez (1990), Stevens et al (1999), i.e solving the axis-symmetric Navier-Stokes and continuity equations in streamfunction / vorticity / circulation forms using a predictor-corrector finite difference method A brief description of the numerical scheme is presented in this Chapter It should be noted that with the exception of the numerical results in Chapters 5 and 6, which were

performed and provided by Prof J.M Lopez using a different numerical scheme as part

of the collaborative project, all the numerical results reported here are performed by the author using the axisymmetric scheme The corresponding numerical method used

by Prof J.M Lopez will be introduced in the corresponding chapters

3.2 Governing Equations and Boundary Conditions

For the problem studied here, the system is axisymmetric, and the equations governing the flow are the axisymmetric Navier-Stokes equations, together with the continuity equation and appropriate boundary and initial conditions A cylindrical container with radius R and height H is completely filled with an incompressible fluid

of constant density ρ and kinematic viscosity ν (see Fig, 3.1) A cylindrical polar coordinate system (r, θ, z) is adopted, with the origin at the centre of the rotating end wall and the positive-z axis pointing towards the stationary endwall

Fig 3.1 Flow configuration in a confined cylinder with one rotating end

r

z

H R

Two kinds of rotating motions were considered: a constant rotation only and a constant rotation with a superimposed sinusoidal modulation In all cases, the bottom endwall is impulsively started from rest For the sinusoidal modulation, the bottom disk rotates at a modulated rate of Ω(1+Asin(Ωft*)), where Ω (rad/s) is the mean constant rotation speed and Ωf (rad/s) is the angular modulation frequency, A is the relative amplitude of the modulation, t* is dimensional time in seconds The angular modulation frequency can also be written as Ωf = 2πff = 2π/Tf, where ff and Tf are the frequency and period of modulation, respectively

In the present study, the system is non-dimensionalized using R as the length scale, and the dynamic time 1/Ω as the time scale The flow thus can be specified by the following non-dimensional parameters: Reynolds number Re = ΩR2/ ν, aspect ratio Λ

= H/R, forcing amplitude A, and forcing frequency ωf = Ωf /Ω

The velocity vector (u, v, w) in this coordinate system is:

( ) ⎟

⎠

⎞

⎜

⎝

r r r w

v

u , , 1 ψ , 1 , 1 ψ , (1)

where ψ is the Stokes streamfunction Γ is defined asΓ =vr Subscripts denote partial differentiation with respect to the subscripted variables The corresponding vorticity field is:

( ) ⎟

⎠

⎞

⎜

⎝

1 , r

1 , r

1 ,

ξ , (2)

where

2 ( ) ( )zz rr ( )r

r

1

− +

=

∇∗

Γ 2Γ

Re

1

D = ∇∗ (3)

( )4 z

2

r

2

r r

r

2 r Re

1 r

⎭

⎬

⎫

⎩

⎨

⎧

⎟

⎠

⎞

⎜

⎝

⎛ +

⎟

⎠

⎞

⎜

⎝

⎛

∇

=

⎟

⎠

⎞

⎜

⎝

∇2∗ψ =−rη (5)

where ( )t z( )r r( )z

r

1 r

1

r

1

+ +

=

∇

The boundary and axis conditions corresponding to the flow are:

ψ = Г = η = 0, (r = 0, 0≤ z≤ H / R),

2

r r

1

∂

∂

−

η (r = 1, 0≤ z≤H / R), (6)

ψ = 0, Г = 0, 2

2

z r

1

∂

∂

−

η (z = H/R, 0 ≤r≤1),

2

z r

1

∂

∂

−

η , and Г =

The boundary condition at r = 0 is due to the axial symmetry of the flow, the boundaries at z = H/R and r = 1 are rigid and stationary, while at z = 0, the rigid endwall is in rotation for t > 0

rv, (z = 0, 0≤r≤1), constant rotation speed

r2[1+Asin(ωft)] (z = 0, 0≤ r≤1), sinusoidal modulation

3.3 Method of Solution

Equations 3-5 can be rewritten in explicit expressions as:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂ +

∂

∂

−

∂

∂

=

∂

∂

∂

∂ +

∂

∂

∂

∂

−

∂

∂

2

2 2

2

z r r

1 r Re

1 z r r

1 r

z

r

1

t

Γ Γ Γ

Γ ψ Γ

ψ

⎠

⎞

⎜⎜

⎝

⎛

∂

∂ +

−

∂

∂ +

∂

∂

=

∂

∂

−

∂

∂ +

∂

∂

∂

∂ +

∂

∂

∂

∂

−

∂

∂

2

2 2 2

2 2

3

2

z r r r

1 r Re

1 z

r

1 z r z r r

1 r

z

r

1

t

η η η η

Γ ψ

η η ψ η

ψ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂ +

∂

∂

−

∂

∂

−

z r r

1 r

r

η (9)

Combined with the boundary conditions, a second order central difference scheme can

be used to solve the above well-posed equations

Mesh generation

Uniform mesh was used for the present study Assuming N+1 and M+1 are the number of grid points in the r and z direction, respectively, then Δr and Δz can be calculated as:

1 M

R H

and

+

= +

1

N

1

Δ

so,

1 M 0,1, ,

1 N 1, , 0,

i

+

=

∗

=

+

=

∗

=

j , z j z

, r i r

i

i

Δ

Δ

Solution of vorticity and angular momentum equations

Equations 7-8 are discretized at all interior points 1 ≤ i ≤ N, 1 ≤ j ≤ M with the second order finite difference method:

⎟

⎠

⎞

⎜

⎜

⎝

⎟

⎟

⎠

⎞

⎜

⎜

⎝

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎟

⎟

⎠

⎞

⎜

⎜

⎝

z 2 r

2 r

1 r

2 z

2 r

1

dt

i

j , 1 i j , 1 i 1 j , i 1 j ,

i

i

j

,

i

Δ

Γ Γ

Δ

ψ ψ

Δ

Γ Γ

Δ

ψ ψ

Γ

⎟

⎟

⎠

⎞

⎜

⎜

⎝

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

2

1 j , i j , i 1 j , i j

, 1 i j , 1 i

i 2

j , 1 i j , i j

,

1

i

z

2 r

2 r

1 r

2

Re

1

Δ

Γ Γ Γ

Δ

Γ Γ

Δ

Γ Γ

Γ

(10.1)

Setting RHS equals to G1 gives i , j G 1

dt

d

=

Γ

(10.2)

Similarly, for vorticity η equation, we have:

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎟

⎟

⎠

⎞

⎜

⎜

⎝

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎟

⎟

⎠

⎞

⎜

⎜

⎝

z 2 r

2 r

1 r

2 z

2 r

1

dt

i

j , 1 i j , 1 i 1 j , i 1 j ,

i

i

j

,

i

Δ

η η Δ

ψ ψ

Δ

η η Δ

ψ ψ

η

⎟

⎟

⎠

⎞

⎜

⎜

⎝

+

⎟⎟

⎠

⎞

⎜⎜

⎝

z 2 r

1 z

2

r

2 1 j , i

2 1 j , i 3 i

1 j , i 1

j

,

i

2

i

j

,

i

Δ

Γ Γ

Δ

ψ ψ

η

⎞

⎜

⎜

⎝

+

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

2

1 j , i j , i 1 j , i 2 i

j , i j , 1 i j , 1 i

i 2

j , 1 i j , i j

,

1

i

z

2 r

r 2 r

1 r

2

Re

1

Δ

η η η

η Δ

η η Δ

η η η

(11.1)

Setting RHS equals to G2 gives i , j G 2

dt

d

=

η

(11.2)

For equations 10 and 11, a second order predictor-corrector scheme was employed:

¾ Predictor step:

k 1 k

j , i j

,

i∗ =η +0 5Δt⋅G

η (12.1)

2

k j , i j

,

i∗ =Γ +0 5Δt⋅G

Γ (12.2)

¾ Corrector step:

+ = k + ⋅ 1∗

j , i 1

k

j

,

i η t G

η (12.3)

+ = k + ⋅ 2∗

j , i 1

k

j

,

i Γ t G

Γ (12.4)

In between predictor step and corrector step, the boundary conditions for vorticity

η need to be updated by solving the streamfunction equation, which will be introduced

in the following part After the corrector step, the boundary conditions for vorticity

also need to be updated for the next time step calculation

Solution of stream function equation

Applying the central difference scheme to discretize the stream function equation

gives:

j , i i

r i

r

Δ

ψ ψ

−

=

− +

− + +

−

− +

−

− +

−

+

2 Δz

1 j i, ψ j i, 2ψ 1 j i, ψ r

2

j 1, i ψ j 1, i 2

Δr

j 1, i ψ j

i,

2ψ

j

1,

This difference equation can be solved by normal Gauss-Seidel (SOR or SLOR)

method, but we prefer a direct method here, since the matrix is not too large and the

direct method will save much computational time

Since r i =i×Δr, the left two terms can be expressed as:

⎦

⎤

⎢

⎣

⎡

+

⎟

⎠

⎞

⎜

⎝

⎛ − +

−

−

⎟

⎠

⎞

⎜

⎝

⎛ +

j 1, i j

i, 2ψ j 1, i

ψ

2

i 2

1 1 i

2

1

1

triangular matrix:

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

− +

−

− +

−

−

j , N

j , 2

j , 1

2

2 N

2

1 1 0 0

0 0

N 2

1 1 0

0 0

0 0

0

0 0

0

0 0

0 2

4

1 1

0 0

0 0

2

1 1 2

r

1

ψ

ψ ψ

Similarly, for the third term in Equation 13.1, we have another triangular matrix:

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

−

−

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

2 1 0 0 0 0

1 0

0 0

0 0

0

0 0 0

0 0 0 2

1

0 0 0 0 1 2

r

1

M , i

2 , i

1 , i

2

ψ

ψ ψ

Hence, the vorticity equation can be written as:

A NNΨNM +ΨNM B MM =F NM (13.2) where F NM =−r iηi , j

Setting A NN Z NN =Z NN E NN,

where E NN is diagonalized, and its entities are the eigenvalues of A NN The columns

of Z NN are the corresponding eigenvectors

Now, let ΨNM =Z NN V NM (13.3) where V is to be determined Equation 13.2 can be written as: NN

A NN Z NN V NM +Z NN V NM B MM = F NM

Multiply the above equation by 1

NN

Z− , we have

1 NM

NN MM NM NM

NN V V B Z F

Taking transpose of the above equation, we get

T MN

NM MM NN

T

NM E B V H

NN

T NM

MN F Z

Let vK be the vectors of T

NM

V , e the eigenvalues of i A , h NN G the columns of H MN, then

(B MM +e i I MM)vG =i hKi for i =1,…N (13.4)

Once initial vorticity values are determined, hG can be calculated Then, solving Equation 13.4 gives vK and Equation 13.3 gives the streamfunction The LAPACK Routine was used in our code for solving Equations 13

Implementation of boundary conditions

Since the boundary conditions for the stream function and the angular momentum are Dirichlet type, the boundary value can be used directly However, the boundary conditions for the vorticity are Neumann type, and they need to be updated; this will be used for the computation of vorticity and stream function at interior points in the next time level The top, bottom and wall boundary conditions for vorticity are second order

of derivative of stream function, hence an one-sided second order finite difference scheme was used to approximate this derivative condition:

2 j , 2 N j , 1 N

i

j

,

i

r 2

8

r

1

Δ

ψ ψ

−

= (r = 1, 0≤ z≤H / R),

2 2 , i 1

,

i

i

j

,

i

z 2

8

r

1

Δ

ψ ψ

η =− − (z = 0, 0≤r≤1), (14)

2 2 M , i 1 M

,

i

i

j

,

i

z 2

8

r

1

Δ

ψ ψ

−

= (z = H/R, 0≤r≤1),

3.4 Method Verification

The quality of the numerical simulation code was verified by the comparison with numerical simulation results obtained by independent calculations in other studies and with experimental results Table 1 presents the values and locations of three local maxima and minima of ψ and η at H/R = 2.5 and Re = 2494 with constant rotating

speed The finite differential results by Lopez and Shen(1999) are also included in the table for comparison, and it can be seen that our results show good agreement with theirs

Figure 3.2 shows the effects of grid density on the solution, in terms of a global value–the kinetic energy in the flow domain–Ek, which is defined as:

( u v w ) rdrdzd θ

k

2 2 2 1

0

1

0 0 2

= ∫ ∫ ∫

It can be seen from the figure that the solutions asymptotically approach a fixed value

as the grid density is increased Various tests have shown that the 160 x 400 uniform grid solutions is sufficiently accurate with acceptable computing time Hence, this

highly dense grid was used in this study, and the time-step δt = 0.01 was chosen to satisfy both the Courant–Friedrichs–Lewy condition and the diffusion requirement (Lopez 1990),

Δt < 1/8 Re Δr2

Table 1 Local maximum and minimum of ψ, η and their locations for Re = 2494, Λ = 2.5

at t = 3000

N, M (r, z) ψ max (r, z) ψ min (r, z) η max (r, z) η min

60, 150 (t = 0.025) 7.0776 × 10(0.183, 1.95) -5 -7.0723 × 10(0.767, 0.800) -3 (0.233, 2.033) 0.52227 (0.333, 2.283) -0.50656

90, 225 (t = 0.025) 7.2530 × 10

-5

(0.178, 1.956)

-7.0842 × 10-3 (0.767, 0.800)

0.53006 (0.233, 2.033)

-0.51090 (0.333, 2.278)

120, 300 (t = 0.01) (0.175, 1.958)7.3326 × 10-5 -7.1017 × 10(0.758, 0.808) -3 (0.233, 2.033) 0.53471 (0.333, 2.275) -0.51378

160, 400 (t = 0.01) (0.181, 1.956)7.4323 × 10-5 -7.1161 × 10(0.763, 0.800) -3 (0.231, 2.038) 0.53774 (0.331, 2.281) -0.51656

60, 150 (t = 0.05)

(Lopez and Shen, 1999)

7.1706 × 10-5

(0.183, 1.95)

-7.0783 × 10-3

(0.767, 0.800)

0.52433 (0.233, 2.033)

-0.50879 (0.333, 2.28) 120,300 (t = 0.01)

(Lopez and Shen, 1999)

7.3988 × 10-5 (0.183, 1.95) -7.1075 × 10

-3

(0.758, 0.825) (0.233, 2.03) 0.53590 (0.333, 2.280) -0.51547

Steady state

Typical simulation results for steady state with the contours of ψ, Γ, and η are

shown in Fig 3.3 for Re = 1918, 1994, 2126, and 2494 These results agree well with the numerical results of Lopez (1990) and the well established experimental results of Escudier (1984)

(a) Re = 1918

(b) Re = 1994

(c) Re = 2126

(d) Re = 2494

Fig 3.3 Contours of ψ, Γ and η for the axisymmetric steady-state solution at H/R = 2.5

and Reynolds number as indicated; there are 20 positive and negative contour levels determined by c-level (i) = [min/max]x(i/20)3 respectively

Unsteady state

As Reynolds number is increased beyond a critical value, the flow becomes a time-periodic axisymmetric state, which is characterized by a large double vortex breakdown bubble undergoing large amplitude pulsations along the axis (Lopez, 1990) Figure 3.4 shows part of time history of kinetic energy Ek at Λ = 2.5, Re =

2765, from which the period can be determined to be about 36.2 The corresponding

Fig 3.5, with the time indicated in Fig 3.4 with filled squares These results agree well with those of Lopez (1990)

From the above comparisons, it can be seen that the solutions calculated from this axisymmetric code are in good agreement with other independent numerical results and experimental results, allowing us to have confidence to explore the flow behavior,

at least at the conditions of the Reynolds number below 3000, where the flow is still in

an axisymmetric state Note this numerical code is applied in Chapter 6 only, while in other studies, the numerical calculations were performed by Lopez with a more advanced 3-Dimensional calculation

0.01450

0.01454

0.01458

0.01462

0.01466

0.01470

t

Fig 3.4 Time history of kinetic energy Ek at Λ = 2.5, Re = 2765, showing the time-periodic flow state The filled squares and the alphabets correspond to the images in Fig 3.5

a b

c d

e f g h

i j k

l

(a) t = 6229.78 (b) t = 6232.92 (c) t = 6236.06

(d) t = 6239.20 (e) t = 6242.34 (f) t = 6245.49

(g) t = 6248.63 (h) t = 6251.77 (i) t = 6254.91

(j) t = 6258.05 (k) t = 6261.19 (l) t = 6264.34

Fig 3.5 Instantaneous streamline contours of ψ, for the axisymmetric time-periodical solution at Λ = 2.5, Re = 2765; there are 20 positive and negative contours determined

by c-level (i) = [min/max] x (i/20)3, with ψ ∈[-0.007, 0.0002]