Analytical method of predicting turbulence transition in pipe flow

T he validity of the ‘‘theorem’’ of minimum entropy production (per unit time) was assessed in Refs 1 and 2 for the well documented case of single – phase fluid tube flow bifurcation from laminar to turbulent regimes. The analysis led to the following two main conclusions: 1) This theorem is not generally valid. 2) The stable type of flow regime is associated with a maximum specific (per unit mass) entropy change. More general broader statements of optimality in thermodynamics are now available in. In order to avoid the range of Re in the so called ‘‘transition region’’ the analysis in 2 was limited to Reynolds numbers one order of magnitude above the estimated critical Reynolds number Rcr for the transition to turbulent flow and the kinetic energy term of the first law flow equation was neglected in line with normal engineering practice. In the present report the analysis is carried out down to Re‘,Recr and based on the complete flow equation i.e. it includes the kinetic term. As a result a new independent variable: the relative tube length or its inverse L/D or D/L as well as two parameters: the laminar and turbulent kinetic energy factorsMl andMt respectively, appear in the analysis and its solution. As the kinetic energy factors, or multipliersM play a key role in our analysis, a few words may be useful to clarify the definition of these parameters and provide a brief explanation on the methods adopted to assess their values under fully developed flow conditions for Newtonian fluids flowing in straight horizontal circular pipes. The kinetic energy term V/2g is only valid in the idealized world of inviscid fluid mechanics; due to the absence of friction the average velocity V also represents the local velocity that remains constant within the whole tube flow area. With real Newtonian fluids the local velocity is no longer constant and diminishes to zero at the pipe innerwall for both flow regimes. In the specific case of laminar regimes the actual local velocities could be determined accurately on the basis of Newton’s laws of mechanics and from this a constant factor Ml 5 2 determined to compute the correct kinetic energy term according to 2 (V/2g). For turbulent regimes no exact theories are available to determine Mt (as well ft) and we depend entirely on experiments. Fortunately Mt < 1 and its impact on the prediction of Recr is negligible compared to the laminar kinetic energy factor Ml. The same end results are obtained in and the present report; the basic difference between these two communications lies in the method adopted to predict the critical Reynolds number Recr. In a quasi ‘‘empirical’’ approach was selected based on numerous plots produced with EXCEL. In the present report a purely analytical method was adopted based on a single premise: the extension of the validity of the Entropy Maximizing Principle proposed in 2 to any Reynolds number, including the singularity at Re‘ where D/L 5 O and L/D R‘. Although this topic may be of interest to some physicists, the theoretical prediction of the critical Reynolds number should prove particularly useful to scientists and engineers engaged in fluid mechanics and heat transfer mainly for several reasons that can be highlighted as follows. Soon after he demonstrated the existence of a ‘‘sinuous flow’’ regime and proposed an excellent estimate of Recr5 2200 for smooth circular tubes, Reynolds derived what became known as the’’Reynolds analogy’’, This shows that the internal hear transfer coefficient h for a circular pipe is directly proportional to the friction factor ft.. Since the ht is usually much larger than hl it is advantageous to operate in turbulent regimes. Therefore an advanced knowledge of Recr for instance in such radically new applications as micro-channel heat exchangers is quite important. SUBJECT AREAS: