Mass transport in a thin layer of power-law mud under surface waves

Abstract

The mass transport velocity in a two-layer system is studied theoretically. The wave motion is driven by a periodic pressure load on the free water surface, and mud in the lower layer is described by a power-law rheological model. Perturbation analysis is performed to the second order to find the mean Eulerian velocity. A numerical iteration method is employed to solve the non-linear governing equation at the leading order. The influence of rheological properties on fluid motion characteristics including the flow field, the surface displacement, the mass transport velocity, and the net discharge rates are investigated based on theoretical results. Theoretical analysis shows that under the action of interfacial shearing, a recirculation structure may appear near the interface in the upper water layer. A higher mass transport velocity at the interface does not necessarily mean a higher discharge rate for a pseudo-plastic fluid mud.

Appendices

Appendix 1

1.1 Numerical solutions

1.1.1 The first-order problem

Equation (56) is a non-linear differential equation, which has to be solved numerically. It is known that ${\widehat{u}}_f^{(1)}$ is 2π-periodic inξ, namely ${\widehat{u}}_f^{(1)}\left(\xi \right)={\widehat{u}}_f^{(1)}\left(\xi +2\pi \right)$, which means it is sufficient to find a solution in the computational domain 0 ≤ ξ ≤ 2π and $-1\le \widehat{z}\le 0$. A pseudo-transient iterative method is adopted, and the pseudo-time step Δt is introduced to adjust the iteration progression and to improve the quality of the convergence and calculation. Thus, the following diffusion equations are obtained for the upper and the lower layer, respectively

$$ \frac{-{\widehat{u}}_w^{(1)\left(c+1\right)}+{\widehat{u}}_w^{(1)(c)}}{\varDelta t}+{\lambda}_w^2{\left|\frac{\partial {\widehat{u}}_w^{(1)(c)}}{\partial \widehat{z}}\right|}^{n_w-1}\frac{\partial^2{\widehat{u}}_w^{(1)\left(c+1\right)}}{\partial {\widehat{z}}^2}+\frac{\partial {\widehat{u}}_w^{(1)(c)}}{\partial \xi }=-\frac{1}{Fr}{\widehat{p}}_s \sin \xi $$

(74)

$$ \frac{-{\widehat{u}}_m^{(1)\left(c+1\right)}+{\widehat{u}}_m^{(1)(c)}}{\varDelta t}+{\lambda}_m^2{\left|\frac{\partial {\widehat{u}}_m^{(1)(c)}}{\partial \widehat{z}}\right|}^{n_m-1}\frac{\partial^2{\widehat{u}}_m^{(1)\left(c+1\right)}}{\partial {\widehat{z}}^2}+\frac{\partial {\widehat{u}}_m^{(1)(c)}}{\partial \xi }=-\frac{\kern0.1em {\rho}_w\kern0.1em }{\rho_m}\frac{{\widehat{p}}_s}{Fr} \sin \xi $$

(75)

where ${\widehat{u}}_c^{(1)}$ is the cth iteration of the solution.

The standard second-order implicit central-difference scheme is used to discretize these equations. The computational domain is discretized into a uniform mesh of N × N = 201 × 201 grid points, as shown in Fig. 10.

Fig. 10

Sketch of the mesh grids

The discretization of the governing formulations yields

$$ \begin{array}{l}\frac{1}{\Delta {\mathrm{z}}^2}{\widehat{u}}_{a,b+1}^{(1)\left(c+1\right)}-\left(\frac{2}{\Delta {\mathrm{z}}^2}+\frac{1}{\varDelta t{\lambda}_w^2}\right){\widehat{u}}_{a,b}^{(1)\left(c+1\right)}+\frac{1}{\Delta {\mathrm{z}}^2}{\widehat{u}}_{a,b-1}^{(1)\left(c+1\right)}\\ {}=\frac{1}{\lambda_w^2}\left(-\left(\frac{1}{2\varDelta \xi}\left({\widehat{u}}_{a+1,b}^{(1)(c)}-{\widehat{u}}_{a-1,b}^{(1)(c)}\right)\right)-\frac{{\widehat{p}}_s}{Fr} \sin \left[\left(a-1\right)\varDelta \xi \right]-\frac{1}{\varDelta t}{\widehat{u}}_{a,b}^{(1)(c)}\right)\end{array} $$

(76)

$$ \begin{array}{l}\frac{1}{\Delta {\mathrm{z}}^2}{\widehat{u}}_{a,b+1}^{(1)\left(c+1\right)}-\left(\frac{2}{\Delta {\mathrm{z}}^2}+\frac{1}{\varDelta t{\lambda}_m^2}{\left|\frac{1}{2\Delta \mathrm{z}}\right|}^{1-{n}_m}{\left|{\widehat{u}}_{a,b+1}^{(1)(c)}-{\widehat{u}}_{a,b-1}^{(1)(c)}\right|}^{1-{n}_m}\right){\widehat{u}}_{a,b}^{(1)\left(c+1\right)}\\ {}+\frac{1}{\Delta {\mathrm{z}}^2}{\widehat{u}}_{a,b-1}^{(1)\left(c+1\right)}=\frac{1}{\lambda_m^2}{\left|\frac{1}{2\Delta \mathrm{z}}\right|}^{1-{n}_m}{\left|{\widehat{u}}_{a,b+1}^{(1)(c)}-{\widehat{u}}_{a,b-1}^{(1)(c)}\right|}^{1-{n}_m}\\ {}\cdot \left(-\left(\frac{1}{2\varDelta \xi}\left({\widehat{u}}_{a+1,b}^{(1)(c)}-{\widehat{u}}_{a-1,b}^{(1)(c)}\right)\right)-\frac{\rho_w\kern0.1em }{\rho_m}\frac{{\widehat{p}}_s}{Fr} \sin \left(\left(a-1\right)\varDelta \xi \right)-{\widehat{u}}_{a,b}^{(1)(c)}\right)\end{array} $$

(77)

where a, b = 1, 2, …, N. The computational domain is divided into 200 grid cells, with a spacing of Δξ = 2π/(N − 1) in the x-direction and Δz = 1/(N − 1) in the z-direction, respectively.

On the free water surface (a = 1, 2, …, N; b = N ), the dynamic condition yields

$$ {\widehat{u}}_{\mathrm{a},N}^{(1)\left(c+1\right)}-{\widehat{u}}_{\mathrm{a},N-1}^{(1)\left(c+1\right)}=0 $$

(78)

Along the upper/lower fluid interface (a = 1, 2, …, N; b = (N-1)/2 + 1), the normal stress is continuous

$$ {\lambda}_w^2{\rho}_w\frac{1}{\Delta \mathrm{z}}{\widehat{u}}_{\mathrm{a},\mathrm{b}+1}^{(1)\left(c+1\right)}-\left({\lambda}_w^2{\rho}_w\frac{1}{\Delta \mathrm{z}}+{\lambda}_m^2{\rho}_m{\left|\frac{1}{2}\right|}^{1-{n}_m}{\left|\frac{1}{\Delta \mathrm{z}}\right|}^{2-{n}_m}{\left|{\widehat{u}}_{\mathrm{a},\mathrm{b}+1}^{(1)(c)}-{\widehat{u}}_{\mathrm{a},\mathrm{b}-1}^{(1)(c)}\right|}^{1-{n}_m}\right){\widehat{u}}_{\mathrm{a},\mathrm{b}}^{(1)\left(c+1\right)}+{\lambda}_m^2{\rho}_m{\left|\frac{1}{2}\right|}^{1-{n}_m}{\left|\frac{1}{\Delta \mathrm{z}}\right|}^{2-{n}_m}{\left|{\widehat{u}}_{\mathrm{a},\mathrm{b}+1}^{(1)(c)}-{\widehat{u}}_{\mathrm{a},\mathrm{b}-1}^{(1)(c)}\right|}^{1-{n}_m}{\widehat{u}}_{\mathrm{a},\mathrm{b}-1}^{(1)\left(c+1\right)}=0 $$

(79)

At the grid bottom (i = 1, 2, …, N; j = 1), the horizontal velocity equal zero

$$ {\widehat{u}}_{\mathrm{i},1}^{(1)\left(c+1\right)}=0 $$

(80)

The numerical calculation has six input parameters: n _f , ρ _f , ${\widehat{h}}_f$, ${\lambda}_f^2$, Fr, and ${\widehat{p}}_s$. We are here primarily interested in examining the effects of power-law flow index n _m and the pressure load ${\widehat{p}}_s$ on the flow field and mass transport velocity. For the upper water layer, the flow index n _w  = 1.0; thus, we only consider the value of n _m . To make the comparison clear, the following values of n _m have been selected in turn: 1.0, 0.9,…, 0.4, 0.3. The analytical solutions for the case n _f  = 1.0 (Ng 2004a) are used to provide an initial guess values for the iteration of solutions. When the maximum relative change between two consecutive iterates falls below 10⁻⁴, the iteration will be ended.

It is known from Eq. (27) that the value of ${\lambda}_w^2/{\lambda}_m^2$ depends on the ratio of the square of the Stokes boundary layer thickness in the upper-layer water (${\delta}_{ws}^2$) to that in the lower-layer viscous fluid (${\delta}_{ms}^2$), and the higher value of ${\delta}_{ms}^2$ corresponds to a highly viscous fluid. According to the rheological experiments (Hsu and Hwung 2013; Sakakiyama and Bijker 1989; Tian 2010; Pang 2011), muds of viscosity vary over a wide range. Referring to the related literatures (Piedra-Cueva 1995; Ng 2004c), ${\lambda}_w^2/{\lambda}_m^2=1/50$ and ρ _w /ρ _m  = 0.77 have been adopted in the calculations. The values of other inputs have been fixed as follows: ${\widehat{h}}_w$ = 0.8, ${\widehat{h}}_m$ = 0.2, Fr = 1.0, and ${\widehat{p}}_s$ = 1.0 and 2.0. The flowchart of the numerical solution procedures is shown in Fig. 11.

Fig. 11

Flowchart of the numerical solution procedures

The accuracy of the computer codes has been examined by comparing numerical with analytical solutions for the Newtonian case, as shown in Fig. 4.

1.1.2 The second-order problem

Similar to the first-order problem, the discrete governing equations for the second-order problem can be obtained from Eqs. (69) to (70)

$$ \begin{array}{l}\frac{1}{2\varDelta \xi}\left({\widehat{u}}_{\mathrm{a},\mathrm{b}+1}^{(2)}-{\widehat{u}}_{i-1,j}^{(2)}\right)+{\lambda}_w^2\frac{1}{\Delta {\mathrm{z}}^2}\left({\widehat{u}}_{\mathrm{a},\mathrm{b}+1}^{(2)}-2{\widehat{u}}_{\mathrm{a},\mathrm{b}}^{(2)}+{\widehat{u}}_{\mathrm{a},\mathrm{b}-1}^{(2)}\right)\\ {}={\widehat{u}}_{\mathrm{a},\mathrm{b}}^{(1)}\left(\frac{1}{2\varDelta \xi}\left({\widehat{u}}_{\mathrm{a}+1,\mathrm{b}}^{(1)}-{\widehat{u}}_{i-1,j}^{(1)}\right)\right)+{\widehat{w}}_{\mathrm{a},\mathrm{b}}^{(1)}\left(\frac{1}{2\varDelta z}\left({\widehat{u}}_{\mathrm{a},\mathrm{b}+1}^{(1)}-{\widehat{u}}_{\mathrm{a},\mathrm{b}-1}^{(1)}\right)\right)\end{array} $$

(81)

$$ \frac{1}{2\varDelta \xi}\left({\widehat{u}}_{\mathrm{a}+1,\mathrm{b}}^{(2)}-{\widehat{u}}_{\mathrm{a}-1,\mathrm{b}}^{(2)}\right)+{\lambda}_m^2{n}_m{2}^{1-{n}_m}{\left|\frac{1}{\varDelta z}\right|}^{n_m+1}{\left|{\widehat{u}}_{\mathrm{a},\mathrm{b}+1}^{(1)}-{\widehat{u}}_{\mathrm{a},\mathrm{b}-1}^{(1)}\right|}^{n_m-1}\left({\widehat{u}}_{\mathrm{a},\mathrm{b}+1}^{(2)}-2{\widehat{u}}_{\mathrm{a},\mathrm{b}}^{(2)}+{\widehat{u}}_{\mathrm{a},\mathrm{b}-1}^{(2)}\right)=\frac{1}{2\varDelta \xi }{\widehat{u}}_{\mathrm{a},\mathrm{b}}^{(1)}\left({\widehat{u}}_{\mathrm{a}+1,\mathrm{b}}^{(1)}-{\widehat{u}}_{\mathrm{a}-1,\mathrm{b}}^{(1)}\right)+\frac{1}{2\varDelta z}{\widehat{w}}_{\mathrm{a},\mathrm{b}}^{(1)}\left({\widehat{u}}_{\mathrm{a},\mathrm{b}+1}^{(1)}-{\widehat{u}}_{\mathrm{a},\mathrm{b}-1}^{(1)}\right) $$

(82)

On the free water surface (a = 1, 2,…, N; b = N), the dynamic condition yields

$$ {\widehat{u}}_{(2)}^{\mathrm{i},N}-{\widehat{u}}_{(2)}^{\mathrm{i},N-1}=0 $$

(83)

Along the interface (a = 1, 2, …, N; b = (N − 1)/2 + 1), the normal stress is continuous

$$ \begin{array}{l}{\lambda}_w^2{\rho}_w\frac{1}{\Delta \mathrm{z}}{\widehat{u}}_{\mathrm{a},b+1}^{(2)}-\left({\lambda}_w^2{\rho}_w\frac{1}{\Delta \mathrm{z}}+{\lambda}_m^2{\rho}_m{n}_m{\left(\frac{1}{2}\right)}^{1-{n}_m}{\left|\frac{1}{\Delta \mathrm{z}}\right|}^{1-{n}_m}{\left|{\widehat{u}}_{\mathrm{a},b+1}^{(1)}-{\widehat{u}}_{\mathrm{a},b-1}^{(1)}\right|}^{1-{n}_m}\right){\widehat{u}}_{\mathrm{a},b}^{(2)}\\ {}+{\lambda}_m^2{\rho}_m{n}_m{\left(\frac{1}{2}\right)}^{1-{n}_m}{\left|\frac{1}{\Delta \mathrm{z}}\right|}^{2-{n}_m}{\left|{\widehat{u}}_{\mathrm{a},b+1}^{(1)}-{\widehat{u}}_{\mathrm{a},b-1}^{(1)}\right|}^{1-{n}_m}{\widehat{u}}_{\mathrm{a},b-1}^{(2)}=0\end{array} $$

(84)

At the fixed grid bottom (a = 1, 2, …, N; b = 1), the horizontal velocities are zero

$$ {\widehat{u}}_{\mathrm{a},1}^{(2)}=0 $$

(85)

The numerical calculation results of the first-order problem are used to solve the second-order problem. Because Eqs. (79)–(80) are linear, an improved chasing method has been developed to calculate the five-diagonal sparse matrix.

Appendix 2

f :

$\left\{{}_{m\kern0.7em for\kern0.4em \mathrm{muddy}\kern0.24em \mathrm{fluid}\kern0.35em \mathrm{in}\kern0.5em -{h}_m<\kern0.3em z<0}^{w\kern0.7em for\kern0.2em \mathrm{water}\kern0.3em \mathrm{in}\kern0.5em 0\kern0.4em <\kern0.4em z\kern0.4em <\kern0.4em {h}_w}\right.$

η _f :

Displacement on the free surface or on the interface

h _f :

Depth of water or mud layer

h :

Total thickness of water and mud

ρ _f :

Density of water or mud

u _f :

Horizontal component of velocity of water or mud

w _f :

Vertical components of velocity of water or mud

p _f :

The pressure of water or mud

p :

External periodic pressure

p _s :

Amplitude of the external periodic pressure

x :

Horizontal coordinate

z :

Vertical coordinate

k :

Wave number

g :

Gravitational acceleration

n _f :

Flow index of power-law model

L :

Wavelength

Fr :

Froude number

u _L :

Mass transport velocity

N :

Grid points

1.1 Greek letters

σ:

Angular frequency

τ _fij :

Stress component of water or mud

μ_f :

Dynamic viscosity of water or mud

γ _fij :

Shear rate

γ _f :

Shearing stress amplitude

ε :

Wave steepness

${\lambda}_f^2$ :

Parameter derived from dimensionless equations

δ _fs :

Stokes boundary layer thickness

ξ :

Phase position

Δξ :

Grid spacing in horizontal coordinate

Δz :

Grid spacing in vertical coordinate

Δt :

Pseudo-time step

Q _f :

The net discharge rate in the water or in the mud layers

1.2 Subscripts

(1):

First order

(2):

Second order

i, j :

Equals x or z

E :

Eulerian coordinate

L :

Lagrangian coordinate

S :

Stokes drift

a:

Grid position in horizontal coordinate

b:

Grid position in vertical coordinate

c:

Iteration step