A Turbulence Scaling Factor that converts laminar solutions to turbulent solutions


 The quest to understand turbulent flows continues to be as important as it was during the previous century.
Present work shows that if a 'laminar' solution to Navier -Stokes equations can be found then skin friction and heat transfer coefficients for the turbulent case can readily be obtained. There is no need for Reynolds averaging and turbulence modelling.
This can be done by defining a turbulence scaling factor which converts 'laminar' diffusivities to turbulent diffusivities. Using turbulent diffusivities in the laminar skin friction coefficient formula and laminar heat transfer coefficient formula gives the corresponding turbulent formula.
Five different test cases with credible experimental measurements have been used to show the success of the present approach. This work also gives the lengths of internally generated turbulent eddies and roughness created
turbulent eddies. If main flow mixes the turbulent eddies , smaller eddies are merged by the larger ones and this is the suggested model for roughness effects which dominates at large Reynolds numbers. A single effective roughness which determines the friction factor has also been obtained and the fractal dimension of turbulence is given as power to Reynolds number. This fractal dimension is in accord with literature for turbulent/non-turbulent interfaces.

I take a cube of volume L 3 in the fluid and suppose that there are N eddies in it then N=L 3   3 where L is the characteristic length of flow geometry and  is the largest characteristic eddy length. It is an experimentally observed fact that increasing Re number makes  smaller so N increases with increasing Re number. A simple plausible way to compare laminar and turbulent friction surface areas is to look at the ratio of total external areas of small cubes to the parent cube's area ; I will use pipe flow data in obtaining  as it is one of the most intensively studied fluid dynamics problem. Traditionally friction factor  ,  =p / (0.5  u 2 D/ l ) is plotted against Re number where p is the pressure drop along the distance l through pipe , u is the mean velocity ,  is the density of fluid and D is the diameter of pipe. Re = u D /  ,  being the kinematic viscosity. Some researchers used fanning friction factor C f = /(0.5  u 2 ) , being the shear stress on the pipe wall. It is easy to see that =  / 4 .
There are two famous graphs : Nikuradse (1) and Moody (2)  Where Rec is the critical Re number , erf and erfc are the standart error function and complementary error function respectively. This fit is not designed to be the best fit of data and performance of it will be seen by the application of the turbulent viscosity based It is worth noting that equation (6) also gives the dimention of turbulence since N 1/3 = L / is defined as a stress ratio with eqn.
(2) and stresses act on planes which intersects with turbulent fractal surface hence N= K , D=D 2 is the appropriate form for N. In order to obtain turbulent skin friction coefficients eqn.(7) will be used in place of (laminar) viscosity for cases for which there are solutions for laminar flows and calculated turbulent friction coefficients will be compared with available experimental data.

A -Fully developed flow in rectangular channels
There is an analytical solution for fully developed laminar velocity profile in rectangular ducts (3) . Friction factor is given in terms of aspect ratio and Reynolds number which is based on hydraulic diameter and mean velocity. Ref. (7) and Ref. (8)  In order to obtain turbulent friction coefficient  , I replace  with  t given with eqn. (7 12) where u is the free stream velocity , x streamwise distance from a reference point .
I replace  with  t given with eqn. (7) and get a formula for turbulent local friction coefficient as follows ; where Nu x = h x / k , local Nusselt number , Pr =  C p / k , Prandtl number , h heat transfer coefficient , k thermal conductivity , C p heat capacity at constant pressure , x streamwise distance from a reference point .

E-Rotating cylinder in a quiescent fluid
There is an exact analytical solution (13) and skin friction formula is as follows ; I believe that strict one dimentionality of this flow prevents merging of turbulent eddies , this situation corresponds very high Reynolds numbers for multi-dimentional flows.Following paragraphs on rough flows will shed some ligh on this matter I believe.      Conclusions : A turbulence scaling factor has been defined and has been used in obtaining a turbulent viscosity and a turbulent heat conductivity which convert a laminar solution to a turbulent solution in obtaining skin friction and heat transfer from a solid surface . Turbulent solutions are in good agreement with experiments .
Present work predicts the lengths of internaly generated turbulent eddies and roughness created turbulent eddies . Using this new knowledge and available experimental works it is shown that largest eddies determine skin friction and heat transfer . If there is mixing of eddies smaller eddies are merged by larger ones and this is the suggested model for roughness effects which dominates at large enough Reynolds numbers. A single effective roughness which determines the friction factor has also been defined . Friction factor depends only on this effective roughness irrespective of flow regimes . Fractal dimention of turbulence is also given which is in accord with measurements for turbulent/non-turbulent interfaces.