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4. Roughness wall functions

4.1 Sand-grain roughness

In PHOENICS the effect of roughness is taken into account by allowing the roughness parameter E, in the logarithmic law for the near-wall velocity, to vary according to the empirical laws proposed by Jayatilleke [1969] for "sand-grain" roughness.

This enhancement is available for use with both equilibrium and non- equilibrium logarithmic wall functions, whether applied at wall-type patches, or at solid-fluid interfaces if EGWF is set to T in Q1 (see WALL_functions and EARTH_Generated_Wall_Functions entries in PHENC).

For a smooth wall, E is a constant, denoted Em, taken as 8.6; and for a rough wall E is expressed as a function of the roughness Reynolds number, Rer, defined as:

Rer = UTAU*Hr/ENUL (18)

in which Hr is the absolute value of the equivalent sand-grain roughness height.

The formula for E is as follows:

• when Rer <3.7: E="Em"
• when 3.7 <Rer < 100: E="1./SQRT(a*(Rer/b)**2+(1-a)/Em**2)"
• and when Rer > 100: E = b/Rer
  where: b = 29.7; a = (1.+2*X**3-3*X**2); and X = 0.02248*(100.-Rer)/Rer**0.564 (Equation 19)

Allowance is made for rough walls in the calculation of the Stanton number by replacing the sublayer resistance function for smooth walls, Pm, with the empirical formula of Jayatilleke [1969]:

P = 3.15*Prl**0.695*(1/E-1/Em)**0.359+Pm*(E/Em)**0.6 (Equation 20)

where Pm is the value of P under fully smooth conditions.

The "sand-grain roughness" height is the roughness height which gives the same actual resistance coefficient as that caused by the real non-uniform wall roughness. The height of this equivalent "sand-grain roughness" is often specified in terms of a relative roughness (e.g. equivalent protrusion height-to-diameter ratio in pipe flows).

Further information giving recommendations and formulae for the effective "sand-grain roughness" height can be found in the literature (see for example 'Boundary-Layer Theory', H.Schlichting, McGraw Hill [1968]).

4.2 Fully-rough logarithmic law

PHOENICS GRND5 wall functions are suitable for a fully-rough near-wall layer in local equilibrium defined in terms of the effective roughness height, as for example in the atmospheric boundary layer.

These wall functions are the same as those described above for GRND2 equilibrium turbulent wall functions, except for the form of the logarithmic wall law, which is now given by:

Ur/UTAU = ln(Y/Y0)/k .... (Equation 21)

where Y0 is the effective roughness height. This height is related to the size of the roughness elements on the surface, e.g. a typical value for grass is 0.01m, whereas for forest it is about 1m.

Equation (21) implies that the absolute value of the wall shear stress can be computed from:
TAUW = s*RHO*Ur**2 .... (Equation 22)
where
s = (0.41*ln(Y0/Y))**2 .... (Equation 23)

If x is a coordinate direction aligned with the surface, the stress in that direction is:
TAUWx = s*RHO*Ur*Ux .... (Equation 24)
where
Ux is the x-direction velocity at the near-wall grid node at distance Y from the wall.

For heat and mass transfer, the Stanton number is computed from a modified Reynolds analogy, i.e. St=s/Prt.

4.3 Specification of the Roughness Height

In PHOENICS there are two methods by which the user can set the roughness height for walls:

1. The roughness height can be set through the PIL variable WALLA stored in the common block /RSG/ of the GRDLOC include file.

   This roughness height is then applied to all walls in the solution domain (whether defined by wall-type patches, or found by EARTH if e.g.WF=T), unless the roughness for a particular wall is defined by the alternative method described below.

   The default value of WALLA is 0.0, which defines a hydrodynamically smooth wall.

2. The second method is applicable only to walls defined by wall-type patches and employs the PIL command SPEDAT, as follows:
   
   PATCH(NAME,WWALL,.,.,.,.,.,.,.,.)
   SPEDAT(SET,ROUGHNESS,NAME,R,0.001)
   COVAL(NAME,U1,GRND2,0.0); etc.
   
   Here NAME is the name of wall-type patch for which the roughness height is to be specified via the SPEDAT command. The argument ROUGHNESS must always be used, and the last argument of SPEDAT defines the roughness height, for use in wall-functions. For example, the above PIL-commands set a roughness height of 0.001 for use in the equilibrium wall-functions for the wall patch NAME.
   
   If SPEDAT is not defined, or its last argument is 0.0, the roughness height is defined by WALLA.
   
   By default, the roughness height set via SPEDAT is uniform across the patch. It is stored for all cells covered by the patch in the patch-wise variable PVROGH allocated to this patch.

   This allows the user to modify the roughness height in GROUND coding; and, for example, introduce an arbitrary distribution of roughness height across a patch. The appropriate subroutine can be called from any convenient place in GROUND, except Group 1 Section 1.

   For details of the work with EARTH patch-wise variables see entry PATCH-WISE VARIABLES entry to PHENC.

Note that the coding for wall functions can be found in the file GXWALL which comprises the subroutines GXWFUN, FNSKIN, FNCOEF and GXYPLS.