multilayer.h File Reference

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Functions
void	vertical_viscosity (Point point, double h, vector *ul, double dt)
	For stability, we discretise the viscous friction term implicitly as.
void	vertical_fluxes (vector *evolving, vector *updates, scalar *divl, scalar dh)

Variables
vector *	ul = NULL
scalar *	wl = NULL
double *	layer
	attribute
	The index of the layer is set as a field attribute.
double	nu = 0.
const scalar	lambda0 [] = 0
const scalar	dut0 [] = 0
const scalar	u_b0 [] = 0
const scalar	lambda_b = lambda0
const scalar	dut = dut0
const scalar	u_b = u_b0

Function Documentation

vertical_fluxes()

void vertical_fluxes	(	vector *	evolving,
		vector *	updates,
		scalar *	divl,
		scalar	dh
	)

Fluxes between layers

The relative vertical velocity between layers \(l\) and \(l+1\) is defined as (eq. (2.22) of Audusse et al, 2011)

\[ G_{l+1/2} = \sum_{j=0}^{l}(\mathrm{div}_j + \mathrm{layer}_j\mathrm{dh}) \]

with

\[ \mathrm{div}_l = \nabla\cdot(h_l\mathbf{u}_l) \]

\[ \mathrm{dh} = - \sum_{l=0}^{nl-1} \mathrm{div}_l \]

To compute the vertical advection term, we need an estimate of the velocity at \(l+1/2\). This is obtained using simple upwinding according to the sign of the interface velocity \(\mathrm{Gi} = G_{l+1/2}\) and the values of the velocity in the \(l\) and \(l+1\) layers. Note that the inequality of upwinding is consistent with equs. (5.110) of Audusse et al, 2011 and (77) of Audusse et al, 2011b but not with eq. (2.23) of Audusse et al, 2011.

The flux at \(l+1/2\) is then added to the updates of the bottom layer and substracted from the updates of the top layer.

To compute the vertical velocity we use the definition of the mass flux term (eq. 2.13 of Audusse et al, 2011):

\[ \mathrm{w}(\mathbf{x},z_{l+1/2}) = \partial_t z_{l+1/2} - G_{l+1/2} + \mathbf{u}_{l+1/2} \cdot \nabla z_{l+1/2} \]

We can write the vertical position of the interface as:

\[ z_{l+1/2} = z_{b} + \sum_{j=0}^{l} h_{j} \]

so that the vertical velocity is:

\[ \mathrm{w}(\mathbf{x},z_{l+1/2}) = \mathrm{dh}\sum_{j=0}^{l}\mathrm{layer}_{j} - G_{l+1/2} + \mathbf{u}_{l+1/2} \cdot \left[\nabla z_{b} + \nabla h \sum_{j=0}^{l}\mathrm{layer}_{j}\right] \]

Definition at line 202 of file multilayer.h.

References _i, dh, dimension, evolving, flux, h, l, layer, nl, w, x, and zb.

Referenced by update_saint_venant().

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vertical_viscosity()

void vertical_viscosity	(	Point	point,
		double	h,
		vector *	ul,
		double	dt
	)

For stability, we discretise the viscous friction term implicitly as.

\[ \frac{(hu_l)_{n + 1} - (hu_l)_{\star}}{\Delta t} = \frac{\nu}{\mathrm{layer}_l} \left( \frac{u_{l + 1} - u_l}{h_{l + 1 / 2}} - \frac{u_l - u_{l - 1}}{h_{l - 1 / 2}} \right)_{n + 1} \]

which can be expressed as the linear system

\[ \mathbf{Mu}_{n + 1} = \mathrm{rhs} \]

where \(\mathbf{M}\) is a tridiagonal matrix. The lower, principal and upper diagonals are a, b and c respectively.

The rhs of the tridiagonal system is \(h_lu_l = h\mathrm{layer}_lu_l\).

The lower, principal and upper diagonals \(a\), \(b\) and \(c\) are given by

\[ a_{l > 0} = - \left( \frac{\nu \Delta t}{h_{l - 1 / 2}} \right)_{n + 1} \]

\[ c_{l < \mathrm{nl} - 1} = - \left( \frac{\nu \Delta t}{h_{l + 1 / 2}} \right)_{n + 1} \]

\[ b_{0 < l < \mathrm{nl} - 1} = \mathrm{layer}_l h_{n + 1} - a_l - c_l \]

For the top layer the boundary conditions give the (ghost) boundary value

\[ u_{\mathrm{nl}} = u_{\mathrm{nl} - 1} + \dot{u}_t h_{\mathrm{nl} - 1}, \]

which gives the diagonal coefficient and right-hand-side

\[ b_{\mathrm{nl} - 1} = \mathrm{layer}_{\mathrm{nl} - 1} h_{n + 1} - a_{\mathrm{nl} - 1} \]

\[ \mathrm{rhs}_{\mathrm{nl} - 1} = \mathrm{layer}_{\mathrm{nl} - 1} (hu_{\mathrm{nl} - 1})_{\star} + \nu \Delta t \dot{u}_t \]

For the bottom layer, the boundary conditions give the (ghost) boundary value \(u_{- 1}\)

\[ u_{- 1} = \frac{2 h_0}{2 \lambda_b + h_0} u_b + \frac{2 \lambda_b - h_0}{2 \lambda_b + h_0} u_0, \]

which gives the diagonal coefficient and right-hand-side

\[ b_0 = \mathrm{layer}_0 h_{n + 1} - c_0 + \frac{2 \nu \Delta t}{2 \lambda_b + h_0} \]

\[ \mathrm{rhs}_0 = \mathrm{layer}_0 (hu_0)_{\star} + \frac{2 \nu \Delta t}{2 \lambda_b + h_0} u_b \]

We can now solve the tridiagonal system using the Thomas algorithm.

Definition at line 91 of file multilayer.h.

References a, b, c, dimension, dt, dut, h, l, lambda_b, layer, max, min, nl, nu, u, u_b, ul, vector::x, and x.

Referenced by advance_saint_venant(), and event_viscous_term().

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Variable Documentation

attribute

attribute

Initial value:

{
  int l

The index of the layer is set as a field attribute.

Definition at line 49 of file multilayer.h.

dut

const scalar dut = dut0

Definition at line 72 of file multilayer.h.

Referenced by vertical_viscosity().

dut0

const scalar dut0[] = 0

Definition at line 71 of file multilayer.h.

lambda0

const scalar lambda0[] = 0

Definition at line 71 of file multilayer.h.

lambda_b

const scalar lambda_b = lambda0

Definition at line 72 of file multilayer.h.

Referenced by vertical_viscosity().

layer

double* layer

Definition at line 44 of file multilayer.h.

Referenced by event_cleanup(), event_defaults(), update_saint_venant(), vertical_fluxes(), and vertical_viscosity().

nu

double nu = 0.

Viscous friction between layers

Boundary conditions on the top and bottom layers need to be added to close the system for the viscous stresses. We chose to impose a Neumann condition on the top boundary i.e.

\[ \partial_z u |_t = \dot{u}_t \]

and a Navier slip condition on the bottom i.e.

\[ u|_b = u_b + \lambda_b \partial_z u|_b \]

By default the viscosity is zero and we impose free-slip on the top boundary and no-slip on the bottom boundary i.e. \(\dot{u}_t = 0\), \(\lambda_b = 0\), \(u_b = 0\).

Definition at line 70 of file multilayer.h.

Referenced by vertical_viscosity().

u_b

const scalar u_b = u_b0

Definition at line 72 of file multilayer.h.

Referenced by vertical_viscosity().

u_b0

const scalar u_b0[] = 0

Definition at line 71 of file multilayer.h.

ul

vector* ul = NULL

Multilayer Saint-Venant system with mass exchanges

Note that the multilayer solver provides the same functionality and should be prefered for most applications.

The Saint-Venant system is extended to multiple layers following Audusse et al, 2011 as

\[ \partial_th + \partial_x\sum_{l=0}^{nl-1}h_lu_l = 0 \]

with

\[ h_l = \mathrm{layer}_lh \]

with \(\mathrm{layer}_l\) the relative thickness of the layers satisfying

\[ \mathrm{layer}_l >= 0,\;\sum_{l=0}^{nl - 1}\mathrm{layer}_l = 1. \]

The momentum equation in each layer is thus

\[ \partial_t(h\mathbf{u}_l) + \nabla\cdot\left(h\mathbf{u}_l\otimes\mathbf{u}_l + \frac{gh^2}{2}\mathbf{I}\right) = - gh\nabla z_b + \frac{1}{\mathrm{layer}_l}\left[\mathbf{u}_{l+1/2}G_{l+1/2} - \mathbf{u}_{l-1/2}G_{l-1/2} + \nu\left(\frac{u_{l+1} - u_l}{h_{l+1/2}} - \frac{u_{l} - u_{l-1}}{h_{l-1/2}}\right)\right] \]

where \(G_{l+1/2}\) is the relative vertical transport velocity between layers and the second term corresponds to viscous friction between layers.

These last two terms are the only difference with the one layer system.

The horizontal velocity in each layer is stored in ul and the vertical velocity between layers in wl.

Definition at line 42 of file multilayer.h.

Referenced by event_cleanup(), event_defaults(), runge_kutta(), update(), and vertical_viscosity().

wl

scalar* wl = NULL

Definition at line 43 of file multilayer.h.

Referenced by event_cleanup(), event_defaults(), and update_saint_venant().