**The friction parameter of the turbulent boundary layer over
the rough surface with 2D and 3D roughness elements. **

**Turbulent
boundary layer in pressure gradient. **

** **

** **

Figure
1. The scheme of the turbulent flow over a rough surface (left), and the
roughness elements geometry: spheres, spherical segments, conical elements (3D)
and transverse rectangular roods (2D).

* *

Figure 2. Roughness density effect on the turbulent flow in a case of 3D
roughness elements: a) spheres; b) spherical segments; c) cones; d) the
generalised correlation.

The mean velocity profile is given by . The roughness density parameter proposed by Dvorak (1969)
is given by , where S is the total roughness area, is the whole
area. The universal roughness density parameter for 3D roughness elements
depends on the Dvorak's roughness density parameter , where *a * is the geometrical parameter
defined as coefficient in the equation for the averaged roughness element
height , *b _{0}*

Figure 3. Roughness density effect on the turbulent flow in a case of 2D roughness
elements: the experimental data of various authors are shown with symbols *1-24*.
The universal roughness density parameter depends on the Dvorak's parameter as
follows , here *f* is
the re-circulation zones parameter: .

The mean velocity profile
parameter is given by where *b _{1 }*=0.12.

To read more
about the turbulent flows over rough surfaces please copy Chapter3.zip file
556K with MS Word97 doc Model
of turbulent flows over rough surfaces by A P Trunev .

The
constructive model of turbulence has been developed for the case of
turbulent boundary layer flow in pressure gradient as follows:

where is the dynamic
Reynolds number.

This model is based on three
assumptions:

1)
the
turbulent mixing parameter *c* depends on the transition layer scale *l** _{t}*;

2)
the
mean velocity profile in the inner layer depends on the transition layer scales
and on the dimensionless pressure gradient ;

3)
the mean
velocity profile in the outer region depends on the outer layer variable , where *y _{0}* is the middle position of the
mixing layer,

*Figure 4. The estimated mean velocity profiles in the turbulent boundary
layer in adverse pressure gradients for p+=0.00898; 0.0251 - the solid lines
1-2 accordingly, and the experimental data (symbols) by Nagano et al (1992).*

The function of pressure gradient
parameter estimated from the experimental data by Nagano *et al* (1992) for the turbulent boundary layer in adverse pressure
gradients at low Reynolds number as follows

.

The mixing layer parameters also
have been established from the experimental data and they shown to be* z** _{*}* =0.27,

The computed mean velocity profiles
together with the experimental data by Nagano *et al* (1992) are shown in Figure 4.

To read more about the constructive theory of turbulence copy ZIP file (261K) with MS Word2000 doc: "Theory and constants of wall turbulence" by A P Trunev.

Trunev A. P. *Theory of Turbulence and Turbulent Transport in the Atmosphere. *WIT
Press, 2001.

Trunev A. P. *Theory
of Turbulence, *Russian Academy of Sciences, Sochi, 1999.

Dvorak , F.
A. Calculation
of Turbulent Boundary Layer on Rough Surface in Pressure Gradient. *AIAA Journal* **7**, 9, 1969.

Nagano, Y., Kasagi, N., Ota, T., Fujita, H., Yoshida, H. & Kumada,
M. 1992 *Data-Base on Turbulent Heat
Transfer*, Department
of Mechanical Engineering, Nagoya Institute of Technology, Nagoya, DATA No. FW BL004.