I am confused regarding some basic relations and their approaches in basic fluid mechanics. I tried to look at these in text but couldn't find appropriate answer. If someone may kindly guide me in this regard.
To my understanding the following holds. (Kindly correct whereever I am wrong)
1. The darcy equation is valid for all flows, laminar or turbulent.
2. Further tau is (0.5*rho*V^2)*f where f is friction factor. Here, tau is sheer stress at the wall. Also tau is mu * del(u)/del(y) at y =0. Both these hold for laminar and turbulent flows.
3. the relation we 4f=64/Re is obtained for laminar flows in a circular tube by solving momentum equation in NVS under appropriate boundary condition, then obtaining expression for gradient of pressure and comparing it with Darcy equation and hence we get the appropriate relation.
4. The query is how it goes for turbulent flow. Does it follow the similar procedure to solve RANS under appropriate boundary conditions and then comparing the same?
5. We use shear velocity u*=sqrt(tau/rho), which is obtained by putting for delta P in darcy equation from the relation Pdrop*Area=tau*P*L
P is perimeter, L is length for flow.
Does this relation holds any importance in case of laminar flow? In case of turbulent flow, beyond sub-laminar range, we use velocity deficit law to obtain relation between u and Vavg etc. in terms of ‘f’. This f is friction factor which is obtained as described in point 3.
1. It is not clear what you mean by Darcy equation, if you are referring to hf = f L/D V^2/(2g), then you are right. Notice that your f is different from this f by a factor of 4.
2. You can show that shear at wall is tau = f (rho V^2/8).
3. f = 64/Re can be derived from Poiseuille laminar flow.
4. For turbulent flow, f is not only a function of Re, but also of the surface roughness. Check out Moody's chart for this dependence or Haaland's relation f=f(Re, eps/d). The correlation for turbulent flow as obtained by combining domensional analysis and experimental measurements.
Issam , you are in the correct thing. The problem is that they use erroneously the hydraulic diameter as transverse area of the section divided between the humid perimeter giving as four (4) times the radius. Some authors use this criterion. In this case there is necessary to correct the equations in order that they meet compatible on the usable ones
I also agree with the submissions of Issam Lakkis.
I will like to add the following points:
In turbulent flow, friction coefficient depends on both the Reynolds number, Re and relative roughness of the pipe. The relative roughness of the pipe is a measure of the ratio of the average height of the asperities (irregularities) on the surface of the pipe to the diameter of the pipe. Its dependence on Reynolds number has limitation. Please see Moody's chart.
When you look at the relations (i) 4f=64/Re and (ii) f=64/Re and go through the formula suggested by Henri Darchy for evaluation of “head lost to friction” and the derivations that led to 4f=64/Re, one could see that the difference is just the bases of the derivations. One is based on hydraulic radius while the other is based on hydraulic diameter.
I will also suggest that you read the paper of V.V. Mikhailov on “universal velocity defect law for turbulent boundary layer”. You may learn one thing or the other about the effect of Reynolds number on the asymptotic correctness of the law and the use of some heuristic scaling that can be of help
You are rigth, that use 64/Re for laminar flow. It is absolute.
The second absolut for developed turbulent flow is a Blasius - 0.316/(Re)^(1/4).
The gap in Re Number between this two laws is depends on the yours installation.
Odinary it is from Re=1800 and to Re=3000-5000.
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