The Total Flow Integral Model Of Steady Pipe Flow And The Primary Application

Pipe flow is a common flow pattern in the field of hydraulics and engineering fluid mechanics.The flow characteristics of steady pipe flow have been investigated deeply from the view of fluid mechanics and the total flow characteristics have been studied systematically from the view of hydraulics.However,two descriptions differ in the selection of parameters and variation of flow characteristics.The existing energy equation is deduced from the Bernoulli equation of the ideal incompressible fluid,so that there is neither explicit expression of energy loss nor general relation between flow resistance and energy loss and effect of turbulence cannot be considered directly,which cause the differences.The flow characteristic description and total flow description cannot be connected.Therefore,the existing total flow integral model of steady pipe flow and following results are acquired in this paper.(1)The total flow integral model of steady pipe flow is established from the mathematical model of incompressible viscous fluid through theoretical analysis.This model realizes the unified description of the characteristics of flow field and total flow.There are continuity equation and energy equation for laminar flow while continuity equation,energy equation and momentum equation for turbulent flow.The continuity equation is consistent with the one in hydraulics.The energy equation is similar to the one in hydraulics and the differences are:1)the surface force potential energy difference between static liquid and dynamic liquid is put forward to reflect the potential energy variation so that the flow in the calculation cross section can be rapidly varied flow;2)the energy loss can be calculated directly.The momentum equation is more exquisite than the one in hydraulics and the general relation between flow resistance and energy loss is provided.(2)The velocity distributions of laminar flow in pipes or ducts of various cross section forms are analyzed through analytical solutions.Both the total flow integral model and the hydraulic method are used to calculate the total flow characteristics and the results of these two methods are consistent,which illustrates that the model is correct and it can realize the accurate unified description of the characteristics of flow field and total flow.The energy loss coefficients decrease with the increase of Reynolds number,and the energy loss coefficients of elliptical pipe,rectangular pipe and annulus pipe is also related to the shape factors.With the same areas of cross sections and the same Reynolds number,the energy loss coefficient of circular pipe is the minimum.Meanwhile,the energy loss density decreases with the increase of the distance from the wall and become zero in the pipe center in the circular pipe.(3)The characteristics of flow field and total flow in a long straight circular pipe are discussed with the theoretical solution of turbulent flow and the integral model.The flow field is calculated by Spalding formula and Van-Driest formula.Based on this,the energy loss coefficient is calculated by the integral model.The results match well with the existing theoretical results and experimental results.It is shown that the contribution of mean velocity gradient on the energy loss decreases with the increase of Reynolds number and the contribution of turbulent dissipation increases with the increase of Reynolds number.In the radial direction,the contribution of region above buffer layer on the energy loss increases with the increase of Reynolds number,the contributions of viscous sublayer and buffer layer on the energy loss decrease with the increase of Reynolds number.The energy loss mainly comes from the near wall region.The larger the Reynolds number is,the larger the energy loss near the wall is.(4)The flow in long straight pipes,curved pipes and pipes with sudden expansion is simulated with Reynolds averaged equations which are closed by Reynolds stress model.The flow characteristics are obtained and the total flow characteristics are discussed by the total flow integral model.In long straight pipes,the Reynolds number has little effect on the distribution of flow characteristics but affects the size.The energy loss coefficient decreases with the increase of Reynolds number.The energy loss coefficient of rectangular pipe is also related to the width-height ratio.With the same Reynolds number,the energy loss coefficient increases when the Reynolds number increases.The variation of energy loss coefficient against the width-height ratio can be neglected when the Reynolds number is large enough.In long straight pipe,the contribution of mean velocity gradient on the energy loss of turbulent flow decreases with the increase of Reynolds number and the contribution of turbulent dissipation increases with the increase of Reynolds number.In the circular pipe with a 90-degree bend,the flow characteristics in the upstream of the bend are the same with the flow characteristics in long straight pipes.When the flow enter the bend,the distributions of flow characteristics and wall stress change,the main stream deviate from the pipe center,the secondary flow appears and the intensity continues to grow in the bend.The flow adjusts gradually in the downstream of the bend exit and become uniform in the region which is 50 to 100 times of diameter far away from the bend exit.In curved pipes,the Reynolds number affects the size of flow characteristics while the bending radius and pipe diameter ratio affects the flow characteristics distribution.The energy loss coefficient is obtained by the total flow integral model.It decreases with the increase of Reynolds number.It is also related to the bending radius and pipe diameter ratio.With the same Reynolds number,it decreases with the bending radius and pipe diameter ratio.In a two dimensional pipe with sudden expansion,the lines of flow velocity contour and the streamlines are parallel to each other before expansion.In the downstream of sudden expansion cross section,the distributions of flow characteristics change.There is a recirculation zone in which the flow adjusts gradually and become uniform finally and it enlarges when the expansion ratio increases.The wall shear stress undertakes a sudden change on the cross section of expansion.In the pipe with sudden expansion,the Reynolds number affects the size of flow characteristics while the expansion ratio affects the distributions.In the pipe with sudden expansion,the energy loss of the rapidly varied flow region intensifies sharply compared with the straight pipe flow.It increases with the increase of the expansion ratio.