Implementación del sistema de actividades en la práctica educativa y sus resultados.

A more generalised approach to enforcing volumetric flux is very briefly outlined in Chu

et al. (1992), which makes use of an efficient Green’s function method in order to impose the flux condition. Whilst this method will be expanded upon here in the context of pipe flow in a domain of lengthL, it is trivially extended to other fluid problems. For example, in the

next section we demonstrate how it can be implemented in the context of a spectral element code.

We begin by taking the incompressible Navier-Stokes equations written in the form of equation (2.24), so thatuis a solution of

@u

@t D rpCL.u/CN.u/; (3.6)

u.x; 0/_Dun.x/: (3.7)

One timestep of this system therefore gives the solution obtained at timetnC1 D.nC1/t.

In addition, let us assume periodic boundary conditions

u.x_D0; r; ; t /_Du.x_DL; r; ; t /; ₈r₂Œ0; R; ₂Œ0; 2/; t >0

and impose a no-slip boundary condition on the pipe wall. Under this formulation, in order to uniquely determinepwe necessarily assume that the initial conditionu0monotonically

decays to the trivial zero solution since no external forcing term is added to the system. An alternative approach to imposing the volumetric flux is through the use of a correction

Q

u.r; ; x; t /, so thatu WDuC Quwith associated pressure fieldp is a solution of (3.6) and

has fluxQ.u/fixed at some desired valueQ. In many cases this approach is hindered by

the nonlinear operatorN.u/. However the use of a splitting scheme, such as the three-step

high-order splitting scheme in section 2.3.6, offers a solution to this problem.

Let us consider a general two-step splitting scheme in a continuous setting, so that the equations to solve are:

8 ˆ < ˆ : @u @t DN.u/ u.x; 0/_Dun.x/ O u.x; t / ! 8 ˆ < ˆ : @u @t D rpCL.u/ u.x; 0/_{D O}u.x; t /

where the solution of the first equation is the intermediate solutionuO and of the second is the

desired fieldunC1. In this setting, the first set of equations performs the non-linear advection,

and the second provides the pressure correction and linear dissipation. This second set of equations is in fact the definition of the linear Stokes problem, and so we may append any other solution of these equations without penalty (up to roundoff error), therefore bypassing the nonlinearity of the original equations. Consider then a correction fielduQ which satisfies

the forced Stokes equations

@u_Q

@t D r QpCL.uQ/C˛

n_xO

and is subject to the usual incompressibility condition and identical boundary conditions to the original equation. ˛nrepresents a (yet undetermined) body forcing term which is

calculated at each time leveln. In order to satisfy (3.7),u_Q necessarily has initial condition Q

u.x; 0/D0.

The dependence ofuQ on˛nmust be removed for the scheme to be computationally useful,

since otherwise the Stokes solution must be computed at each timestep. Assuming that the flow is always driven (and hence˛n_¤0), definevD Qu=˛nso thatvsolves the equations

@v

@t D r OpCL.v/C Ox; (3.8)

v.x; 0/_D0I (3.9)

E2

E1

singular point

p _bfs

Figure 3.4:Geometry of the backwards facing step, for which the Stokes field possesses a singular point at the step edge.

condition, we then calculateun D unC Qun D unC˛nv1, and so we may compute v1

before the start of the simulation. Finally, in order to determine˛n, taking the flux of both

sides of the previous equation yields

Q.un/_DQ.un/_C˛nQ.v1/₎˛n_D Q Q.u

n_/

Q.v1/ : (3.10)

The procedure then is to initially solve (3.8) to obtainv1.x/Dv.x; t /and its fluxQ.v1/

before the main simulation occurs. Given a solutionunC1which solves (3.6) and (3.7) and

is obtained through a splitting scheme, we calculate˛nC1by

˛nC1D Q Q.u

n_C1_/

Q.v1/ :

The corrected velocity fieldunC1 DunC1C˛nC1v1then has has flowrate exactly equal to

Q. We then useunC2instead ofunC1as the initial condition in (3.7) at timestepnC1to

obtain a solution with desired flowrate at timestepn_C2, and so on. This method is extremely

efficient; bothv1andQ.v1/can be pre-computed, and so all that is needed at each timestep

is to calculateQ.unC1/and perform a simple vector addition operation.

In certain domains which possess sharp corners, a solution to the Stokes equation cannot be guaranteed to exist even if the Navier-Stokes equations have a solution, and thus this flowrate correction technique cannot be used. The simplest example of this is the backwards facing step, depicted in figure 3.4, for which the Stokes solution has a singularity at the step edge. One possible solution, proposed by Lundet al. (1998) and demonstrated in Kopera (2011) is the use of a so-called copy boundary condition. The domain is partitioned into two

pieces; the regular backwards facing step geometrybfsand an extended inletp.p is a

self-contained periodic channel, with fluid exiting atE2re-entering atE1. The flow inp

is driven using the constant flowrate condition. The velocity field atE2is then ‘copied’ to

form the inlet condition ofbfs. This is particularly useful for simulations which require a

turbulent inflow condition, since a turbulent initial field in the inlet channel is easily obtained. Care must be taken, however, to ensure that the channel is sufficiently long so that any periodic features of the flow dissipate before entering the main domain. The implementation of this boundary condition inSemtexis discussed in Cantwell (2009).