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Fluid Mechanics

Thermal boundary layer

The constructive model of turbulence was developed to estimate the turbulent transport of heat in the boundary layer. This model is based on two assumptions:

1)      the turbulent mixing parameter c depends on the transition layer scales;

2)      the mean temperature profile depends on the thermal sublayer scale  as well as on the transition layer scales (here Pr is the Prandtl number, kh  is the constant that approximately equals to the von Karman's constant).

 

With this assumptions the mean temperature gradient in a steady turbulent flow with the heat flux applied to the wall can be written in the form:

 

 

where , Tg is the wall temperature, T is the turbulent flow temperature,  is the scale of temperature, qH is the heat flux from the wall to the flow, cp is the specific heat at constant pressure of the gas,  is the parameter dependent on the Prandtl number. As it was established the parameter Pt(Pr) can be approximated in the range of the Prandtl number 2<Pr<1000 as follows: .

The boundary conditions on smooth wall and for the long distance from the wall are given by

The estimated mean temperature profiles are shown in Figure 1 by solid lines together with the profiles computed on the model of Sebeci (1972) - they are shown by square symbols. The agreement between two models in general is good. The constant kh  and the temperature profiles also were calculated in the range 0.4<Pr<1.5 - see Figure 2.

Figure 1. The estimated mean temperature profiles (solid lines) in the turbulent boundary layer at small Prandtl number (left) and at large Prandtl number (right). The square symbols -the model of Sebeci (1972).

 

Figure 2. The ratio  versus the molecular Prandtl number and the estimated mean temperature profiles (solid lines).

 

Turbulent boundary layers in roughened tubes 

 

 

Figure 3. Roughness density effect on the turbulent flow in a case of 2D roughness elements: the experimental data of various authors listed in Table 1 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 is given by   where

 . Here b1 =0.12.

 

Table 1

Authors

Year

Geometry

Symbol

Möbius

1940

Tube

10.0-29.22

0.3-2.20

3

Chu & Streeter

1949

Tube

1.95-7.57

0.93

4

Sams

1952

Tube

2.0-2.3

0.88-1.37

9

Nunner

1956

Tube

16.36

0.8

16

Koch

1958

Tube

9.8-980

1.0-5.0

5

Fedynskii

1959

Annulus

6.67-16.7

1.0

10

Draycott & Lawther

1961

Annulus

2.0

1.0

2

Skupinski

1961

Annulus

Tube

2.0-41.0

22.2-133.4

1.0

2.0

6

Savage & Myers

1963

Tube

3.66-43.72

1.33-2.67

13

Perry & Joubert

1963

Wind tunnel

4.0

1.0

19

Sheriff, Gumley & France

1963

Annulus

2.0-10.0

1.0

14

Gargaud & Paumard

1964

Tube

Annulus

1.8-16.0

10.0-16.0

1.0-1.67

1.0

1

 

Bettermann

1966

Wind tunnel

2.65-4.18

1.0

20

Massey

1966

Annulus

7.53-30.15

1.06

15

Kjellström & Larson

1967

Annulus

2.02-38.52

0.086-4.08

12

Fuerstein & Rampf

1969

Annulus

2.91-25.04

0.42-2.50

8

Lawn & Hamlin

1969

Annulus

7.61

1.0

17

Watson

1970

Annulus

6.49-7.22

1.0

11

Stephens

1970

Annulus

7.20

1.0

18

Webb, Eckert & Goldstein

1971

Tube

9.70-77.63

0.97-3.88

7

Antonia & Luxton

1971

Wind tunnel

4.0

1.0

21

Antonia & Wood

1975

Wind tunnel

2.0

1.0

22

Dalle Donne & Meyer

1977

Annulus

4.08-61.5

0.25-2.0

24

Pineau, Nguyen, Dickinson & Belanger

1987

Wind tunnel

4.0

1.0

23

 

 

 

 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 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.

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