DISTANCE from the WALL (WDIS), and between walls (WGAP)

1. The LTLS feature
2. Examples
3. The differential equation for L
4. The calculation of WDIS and WGAP
5. The parallel-wall situation
6. The deduction of the turbulent length scale
7. Further examples
8. Activation and use in PHOENICS

1. The LTLS feature

What it does

It must be understood that WDIS and WGAP have precise meanings only for especially simple circumstances, such as when the space between two wide parallel surfaces is considered. How they should be defined for points within a room cluttered with furniture is far from being obvious.

All that is claimed for the LTLS method is that it produces precisely correct values in the wide-parallel-plate circumstances; and results which are plausible in all others

Use for turbulence models

Many turbulence models require the calculation, for every cell in the domain, of the distance from the nearest wall. They may also require the distance between the two nearest walls.

The Nikuradze mixing-length formula is an example.

Two-equation turbulence models such as k-epsilon have been invented for the purpose of making prescription of length scales unnecessary; but even they, in their low-Reynolds-Number forms, may require independent knowledge of distance from the wall for the evaluation of some terms.

Use for the IMMERSOL model of radiation

One of the most convenient and (in PHOENICS) widely used if the IMMERSOL model. This also requires knowledge of the distribution in space of the distance between walls.

IMMERSOL therefore involves the solution of the feature.

2. Examples

The next three pictures represent:

Their inspection may lend credibility to the claims of plausibility which have just been made.

A relevant core-library case

It is a convenient starting point for newcomers to the subject.

3. The differential equation for L

The LTLS feature involves the solution for a scalar variable, L, which:

• obeys the differential equation:
  div grad L = -1

  within the fluid; and

• equals zero within solid materials and at thin surfaces to which no-slip boundary conditions are applied to the velocity equations.

This equation is similar to that for temperature within a uniformly-conducting medium, having a uniform heat source, and in contact with solids and other surfaces at which the temperature is held at zero.

It is of course easily solved by the linear-equation solver of PHOENICS, whether the geometry is one-, two- or three-dimensional. This solution needs to be determined once only, if the solids do not change position.

4. Deduction of the distance from and between walls

The variable L is not itself the distance from the wall, even though it is proportional to that distance at locations which are near a wall. Its dimensions are indeed those of length squared.

However, that distance can be deduced from the solution for L, as can also a plausible estimate of the effective distance between walls. The method is to derive a relationship between:

• the distance from the wall W_dis and
• the distance between walls W_gap, on the one hand, and
• the local value of L and
• the local value of its gradient, on the other,

from consideration of a simple geometry, namely that between two parallel walls, and then to presumethat it has general validity.

5. The parallel-wall situation

Let the distance measured from one wall be y, and the distance to the opposite wall y₁.

The differential equation takes the one-dimensional form:

d²L/dy² = -1 , which can be integrated to give:
dL/dy = - y + A, where A is a constant, and then further to:
L = - y²/2 + Ay + B , where B is another constant.

Insertion of the boundary condition L=0 at y=0 and y=y₁ yields:

B = 0 , and A = y₁/2

with the result:

Elimination of y₁ from the first equation by use of the second yields:

It is these equations which are employed generally.

They have been found to give plausible results in all situations; but of course, although W_dis has the unequivocal meaning of the distance to the nearest wall, the significance of W_gap has no clear meaning, in a corner for example.

It is interesting in this connection to consider a duct of circular cross-section; for it is again easy to obtain an analytical solution for L. The above equations then show that the above expression for W_dis is exactly correct in the immediate vicinity of the wall; and it rises to a maximum equal to radius divided by the square root of 2 at the centre of the duct.

W_gap, correspondingly, has the same value as the radius near the wall; and it rises at the centre of the duct to the radius times the square root of 2, i.e. exactly twice the reported distance from the wall.

6. The deduction of the turbulent length scale from W_dis and W_gap

How (and whether) the length scale of turbulence, LTURB, is to be deduced from WDIS and/or WGAP is of course a quite separate matter.

The model of Nikuradze, in which LTURB/WDIS is expressed as a poly-nomial in (WDIS/W_gap), is only one of many possibilities, of which another is to use the smaller of the Nikuradze formula and a length proportional to:

(maximum velocity - minimum velocity) / velocity gradient.

7. Further examples

The following pictures illustrate the use of the model for the flow in a small element of a "labyrinth", i.e. a space in which fluid is forced to follow a winding path by the presence of solid obstacles.

8. Activation and use in PHOENICS

See PHENC entries: DISWAL, LVEL, LAM-Bremhorst, and LOW-REYNOLDS NO.