Problem Statement   

      Following is the schematic of the 2-D step-wall diffuser with FEMLAB boundary numbering

 

The problem statement requires that DM and DP be addressed. The non-dimensional forms for these conservation principles for incompressible flow are 

            DM:  

            DP:   

where P = p/r is kinematic pressure. The lecture detailed the transformation to streamfunction-vorticity variables, which produces a very stable PDE system with tractable BCs. For the definitions  and , one obtains 

            DM                  :    identically

                   :  

            kinematics        :  

 The resultant streamfunction-vorticity Navier-Stokes elliptic boundary value problem statement is

           

           

            BCs:  ¶Win          :   u(y, xin) Þ win, yin  via definitions

¶Wout       :  

¶Wwall     :   y = constant by definition

                                        

      The Galerkin weak statement process 

            GWSN Þ GWSh = Se{WSe} º 0

 leads to the template pseudo-code statements 

           

            

 Since GWSh is explicitly non-linear, the Newton iterative algorithm form is 

            [JAC]{dQ}p+1 = - {GWSh}

            {Q}p+1 = {Q}p + {dQ}p+1 

for {Q} = {OMG, PSI} and iterative convergence occurs when max|dQ| £ e

Setting-up the problem in FEMLAB 

      Open the 2D, general non-linear stationary PDE mode in the FEMLAB model navigator. Set the number of dependent variables to 2 and their names to psi (y) and omega (w). This mode is selected when a built-in mode is not available, hence one must define the PDE system, which is done under subdomain settings. The subdomain settings, which define the PDE terms are 

Subdomain

1

G(1)

-omegax   -omegay

G(2)

-psix   -psiy

F(1)

Re*(psix*omegay - psiy*omegax)

F(2)

omega

       Applying the w BC  is a default in FEMLAB. BCs for y, referring to the schematic, must be applied on the inflow boundary 1 and no-slip boundaries 2, 3, 4 and 5. Applying the BCs in FEMLAB amounts to 

Boundary

1

2,4,5

3

6

R(1)

-psi+(3*s^2/2-s^3)

-psi

-psi+1/2

0

R(2)

0

0

0

0

 In the solver option, the parametric mode is chosen to enable executing the study for
100 £ Re £ 600. FEMLAB self-generates the initial triangle element mesh for Re = 100. 

      You are required to compare the reattachment length with the experimental data. Table 1 lists the experimentally determined primary recirculation reattachment intercept non-dimensionalized by the step height s for various Reynolds number. 

Table 1. Dimensionless reattachment intercept vs. Re.

Re

L/s (Exp. data)

100

3.10

200

4.95

300

6.80

400

8.60

500

10.40

600

11.60