ANALYTICAL VELOCITY PROFILE IN TUBE FOR LAMINAR AND ...

1 Engineering MECHANICS, Vol. 21, 2014, No. 6, p. 371 379371 ANALYTICAL VELOCITY PROFILEIN TUBE FOR LAMINAR AND TURBULENT FLOWJ aroslav Stigler*A new ANALYTICAL formula of the VELOCITY PROFILE for both the LAMINAR and turbulentflow in a tube with a circular cross-section will be introduced in this article. Thisformula is rather simple and easy to use. The advantage of this VELOCITY PROFILE isthat one formula can be used for LAMINAR and turbulent flow. This new formula willbe compared with power law VELOCITY PROFILE and with the law of the wall also calledas the :power-law, ANALYTICAL VELOCITY PROFILE , vorticity, law of the wall, log-law,turbulent shear stress1.

2 IntroductionThe author is dealing with a fluid flow in a straight tube with a circular cross-sectiongoverned by the pressure gradient in this paper. Some fundamental ideas of the new ana-lytical VELOCITY PROFILE derivation and its comparison with the power law VELOCITY PROFILE andwith the log-law will be presented and discussed formula for the LAMINAR VELOCITY PROFILE has to be mentioned first. The derivation ofit is possible to find in every book dedicated to the fluid mechanics. The LAMINAR velocityprofile of the fluid flow governed by the pressure gradient is (max) 1 rR 2 (1)whereRisthetuberadius,v(max)is the maximal VELOCITY or the centerline VELOCITY of thevelocity PROFILE .

3 It is assumed that there is only one VELOCITY component in the tube axisdirection. This VELOCITY PROFILE expressioncan be also rewritten as a function of averagevelocityv(av).v=2v(av) 1 rR 2 .(2)This expression is more suitable for the practical use because the average VELOCITY can beeasy expressed from the flow rate and the radius of tube. Both these expressions can berewritten to the dimensionless (max)=1 rR 2,(3)* Stigler, , brno university of technology , Faculty of Mechanical Engineering, EnergyInstitute, Victor Kaplan Department of Fluid Engineering, Technick a 2896/2, Brno372 Stigler J.

4 : ANALYTICAL VELOCITY PROFILE in Tube for LAMINAR and Turbulent Flowvv(av)=2 1 rR 2 .(4)The expressions with the average VELOCITY will be preferred in this case of the turbulent flow it is more complicated to find some ANALYTICAL researches have been trying to find it. One of the well-known expressions of theturbulent VELOCITY PROFILE is the power law VELOCITY PROFILE . The power law VELOCITY PROFILE isforexamplementionedin[3] (max) 1 rR 1n.(5)It is also possible to rewrite this formula as a function of the average velocityv(av)insteadof the maximal velocityv(max).

5 V=v(av)2 1n+1 1n+2 1 rR 1n(6)wherenis a coefficient which is a function of the Reynolds number. It has to be determinedon the basis of the experimental data. It is possible to find this dependence on a Reynolds number in [3]. Some other expressions are mentioned in [2]. For examplen=1+6 Re50(7)orncan be also expressed by this ln(Re) (8)The valuen= 7 is reasonable for many practical flow approximations as it is mentioned byMunson in [3].This power law VELOCITY PROFILE has two fundamental discrepancies. First of them appearsnear the tube wall.

6 It consists in the infinite derivative value on the wall which means thatthere is the infinite shear stress on the wall what represents infinite friction losses in discrepancy is related to the VELOCITY PROFILE smoothness at the tube center. Thefirst derivative of this formula is not smooth in the tube center. The utilizing of theseformulas is then restricted because of these fundamental problem near the wall has to be solveda different way by using other empiricalformulas which are valid only near the wall. The near wall region of the flow is called theboundary layer or the shear layer.

7 The boundary layer is an area of the flow near the wallwhere magnitude of the friction (viscous) forces and dynamic forces are comparable. It canbe also defined as the area near the wall with the not zero vorticity. It means that curlvisnot zero. This boundary layer can be divided into three layers. The first of them is the onewhich is nearest to the wall. It is called viscous sub-layer. The second one is the transitionarea and the third one is the turbulent boundary is necessary to define some quantities in order to be able to describe the VELOCITY profilein the boundary = w (9)Engineering MECHANICS373wherev is the shear VELOCITY , wis shear stress on the wall, is fluid density.

8 Then it ispossible to define the dimensionless VELOCITY (v+) and the dimensionless distance from thewall (y+).v+=vv ,(10)y+=v y (11)whereyis the distance from the dimensionless VELOCITY PROFILE in viscous sublayer can be then expressed this wayv+=y+.(12)The dimensionless VELOCITY PROFILE in the turbulent boundary layer can be expressed bythe +=1 log(y+)+B.(13)Different authors are using different values of the constants andB. For example theused coefficients are : = andB= in the book [1], = [2].2. Introduction of a new VELOCITY profileThe new VELOCITY PROFILE will be introduced after the previous brief overview of velocityprofiles in a tube with circular cross-section.

9 It is based on the vorticity density distributionover the tube cross-section. This vorticity induces the VELOCITY . The vorticity density isclosely related with vorticity vector i. The relationship between the induced VELOCITY andthe vorticity density is through the Biot-Sawart law. Biot-Sawart law is derived for a vortexfilament element dsj. Vortex filament is represented by curve s with the circulation around it. The situation is depicted in fig. 1. The Einstein summation convection will beused in all mathematical 4 r3 ijkdsjrk= 4 r3 ijkdsj(x k xk).

10 (14)For instance the Biot-Sawart law can be applied on the infinite straight vortex filamentparallel to thex3axis with the constant circulation . This situation is depicted in fig. : Biot-Sawart : Biot-Sawart law the infinite straight vortexfilament parallel with the axisx3374 Stigler J. : ANALYTICAL VELOCITY PROFILE in Tube for LAMINAR and Turbulent FlowThe VELOCITY induced by such vortex filament at the pointx kcan be expressed by 4 r2(0) i3kr(0)k= 4 r2(0) i3k(x k x(0)k).(15)Now the aim is to solve the VELOCITY induced by the single circular vortex filament with theconstant circulation around it.