(7)

(8)

(9)

(6)

Exercise 1 CFD in Process Engineering

Zero gradient boundary condition

The aim of this paper is to evaluate the effects of the no-slip condition on the pressure gradient. The no-

slip condition in general holds for many applications in engineering. However in some cases the no-slip

condition may fail such as extremely smooth surfaces, which can be solved with a partial-slip condition.

For the 2D flow the Navier-Stokes- and the continuity equation are given in equation (1), (2) and (3).

Rearranging (1) and (2) according to the pressure gradient leads to (4) and (5):

+ = 0

=

+

+ +

=

+

+ +

In general the flow through the wall is zero for all times t and thus:

| = 0 | 0 + | 1 = 0

| = 0 | = 0

| = 0

(4)

(5)

(3)

(15)

(10)

(11)

(12)

And the no-slip condition (10) also holds for all times t and along the whole surface:

| = 0 | = 0

| = 0

With the following assumptions (i=x,y):

steady state: !" = 0

fully developed ! = 0

The equations (3), (4) and (5) become:

=

=

= 0

Evaluating (13) and (14) at the wall y=0 with (7) and (15) the pressure gradient follows:

| =

|

| = 0

And thus it follows for the zero gradient boundary condition (19):

$%&'|( |( = 0

| 0 +

| 1 = 0

(13)

(14)

(16)

(17)

(18)

The evaluation of equation (3) - (5) at the boundary with the equations (7) - (12) leads to the zero

gradient boundary condition of equation (18) without the aforechosen assumptions. Thus the steady

state condition and the assumption of a fully developed flow is not necessary. The zero gradient

boundary condition is given by the velocities at the wall (no-slip). The zero gradient boundary condition

can be interpreted as a Neumann boundary condition.