The calculation of mechanical energy loss for incompressible steady pipe flow of homogeneous fluid

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2013,25(6):912-918 DOI: 10.1016/S1001-6058(13)60440-0

The calculation of mechanical energy loss for incompressible steady pipe flow of homogeneous fluid* LIU Shi-he (刘士和), XUE Jiao (薛娇), FAN Min (范敏) State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China (Received June 4, 2013, Revised October 25, 2013) Abstract: The calculation of the mechanical energy loss is one of the fundamental problems in the field of Hydraulics and Engineering Fluid Mechanics. However, for a non-uniform flow the relation between the mechanical energy loss in a volume of fluid and the kinematical and dynamical characteristics of the flow field is not clearly established. In this paper a new mechanical energy equation for the incompressible steady non-uniform pipe flow of homogeneous fluid is derived, which includes the variation of the mean turbulent kinetic energy, and the formula for the calculation of the mechanical energy transformation loss for the non-uniform flow between two cross sections is obtained based on this equation. This formula can be simplified to the Darcy-Weisbach formula for the uniform flow as widely used in Hydraulics. Furthermore, the contributions of the mechanical energy loss relative to the time averaged velocity gradient and the dissipation of the turbulent kinetic energy in the turbulent uniform pipe flow are discussed, and the contributions of the mechanical energy loss in the viscous sublayer, the buffer layer and the region above the buffer layer for the turbulent uniform flow are also analyzed. Key words: mechanical energy loss, energy equation, pipe flow

Introduction The mechanical energy is the sum of the potential energy and the kinetic energy. The calculation of the mechanical energy loss between two sections along the main flow direction is an important problem in practical engineering, as well as an important research topic in Engineering Fluid Mechanics and Hydraulics. In the field of Hydraulics, the mechanical energy loss is usually called the resistance loss, which is divided into the friction loss and the local loss, and the resistance coefficient is mainly determined by experiments. For example, Reynolds carried out experiments to study the differences of the resistance coefficients in laminar and turbulent pipe flows, Nikuradse conducted experiments for flows in artificially roughened pipes and obtained the variation of the resistance loss against the Reynolds number and the relative roughness of the pipe wall. Many studies of the resistance loss were based on these two fundamental studies[1-5]. * Biography: LIU Shi-he (1962-), Male, Ph. D., Professor

In the field of Fluid Mechanics, many studies focused on the mechanisms of turbulent flow[6-9], flow stability[10] and the related spatial or temporal distributions of the flow quantities. With regards to pipe flows, Samanta et al.[11] carried out the experimental investigation of the laminar turbulent intermittency in pipe flows, Liu et al.[12] studied the velocity distributions in the transition region of pipes with Particle Image Velocimetry (PIV), Van Doorne and Westerweel[13] investigated the flow characteristics of laminar, transitional and turbulent pipe flows by using Stereoscopic Particle Image Velocimetry (SPIV), Genç et al[14] studied the Reynolds stresses in a swirling turbulent pipe flow with Laser-Doppler Anemometer (LDA), Wagner et al.[15], Wu and Moin[16] studied the turbulent pipe flow by Direct Numerical Simulation (DNS). However, the relation between the mechanical energy loss in the volume between two cross sections and the kinematical and dynamical characteristics of the flow field was not addressed for a non-uniform flow in previous studies. This paper derives a new mechanical energy equation for incompressible steady non-uniform pipe flows of homogeneous fluid, including the variation of the mean turbulent kinetic energy,

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and the theoretical results for uniform flows are used for comparison. Furthermore, the contributions of the mechanical energy loss are analyzed relative to the time averaged velocity gradient and the dissipation of the turbulent kinetic energy in a uniform flow, in different regions, especially in the radial direction. 1. Formulation of the mechanical energy equation Considering the flow of homogeneous incompressible fluid in the gravitational field, the second order tensor of the surface force[6] Tij can be expressed as

 (Ti j ui ) x j

 Du  = Ti j si j +  ui  i  f i   Dt 

two cross-sections A1 , A2 at the upstream and the downstream with distance L and the pipe wall, and the flows in this volume are gradually varied. Integrating Eq.(2) over V , and using the Gaussian Theorem to transform the volume integral into the surface integral for the left hand side and the first term on the right hand side, then 

    g x

3

 1 + ui ui  (u j n j )d A =  2 

p

+

A

  

ij

A

(1)

(ui n j )d A    i j si j dV

(3)

V

where n j is the component of the unit normal vector

where  is the fluid density, fi is the mass force per unit volume and ui is the velocity components. De-

of the surface A . For a simple laminar shear flow, the potential energy g x3 + p /  (which is called the pie-

stress and si j is the rate of deformation. For an incom-

zometric head in Hydraulics) on either A1 or A2 are constant since the flows across these sections are the gradually varied flows[6]. Noting that the velocity on the pipe wall is zero we have

pressible steady flow of homogeneous fluid in the gravitational field (let x3 be the vertical coordinate), we have

    g x

compose the surface force tensor as Ti j =  p i j +  i j , where p is the pressure,  i j = 2  si j is the viscous

  ui f i =

3

Dui  = Dt xi



 U1 A1  g z1 + 

p1   

(4)

where A is the cross-sectional area and U is the mean velocity at a section, and z is substituted for x3 for the vertical coordinate to conform with the symbols used in Hydraulics and Engineering Fluid Mechanics. The indexes 1 and 2 are used to represent the corresponding quantities at cross sections A1 and

1    u j u j ui  2 

therefore Eq.(1) can be simplified as     ( i j ui )   p 1   i j si j   ui  g x3 + + u j u j   =  2 xi  xi  

p2   p  u j n j d A =  U 2 A2  g 2 z2 +      

+

A

 (  g x3ui ) , Ti j si j =  i j si j xi

and

 ui



A2 . Define the kinetic energy correction coefficients (2)

In Eq.(2), g x3 + p /  + u j u j / 2 is the mechanical energy, therefore this equation is the differential form of the mechanical energy equation for an incompressible steady flow of homogeneous fluid in the gravitational field, the integral form of the corresponding mechanical energy equation for a steady pipe flow in laminar and turbulent states (corresponding to the statistical quantities in ensemble average) will be discussed afterwards based on this equation. 1.1 Laminar flow Consider a control volume V as shown in Fig.1, where the entire surface is expressed as A , including

1 and  2 such that 1U13 A1 =   ui ui u j n j d A A1

and

 2U 23 A2 =  ui ui u j n j d A A2

and use hw to represent the mechanical energy transformation (the second and third terms on the right hand side of Eq.(5)) and the loss (the first terms on the right hand side of Eq.(5)) per unit weight in unit time, i.e.,

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flow in the sense of ensemble averaged quantities we have

hw  gAU =   i j si j dV    i j (ui n j )d A  V



ij

A1

(ui n j )d A

(5)

A2

the total mechanical energy equation Eq.(3) for a laminar flow then becomes z1 +

p1 U2 p U2 + 1 1 = z2 + 2 +  2 2 + hw g 2g g 2g

(6)

It can be seen from Eq.(6) that a part of the mechanical energy is consumed owing to the viscous effect when the fluid flows from the cross section A1 to A2 .

p = p + p , ui = ui + ui and

 i j =  i j +  ij therefore, Eq.(8) can be obtained by taking ensemble average  xi

   p 1 1   ui  g x3 + + u j u j + u j u j   =  2 2    





  i j si j +  ij sij +  

  i j ui +  ij ui + pui + xi   

1 2

  u j uiu j + uiu j u j 

Fig.1 Sketch of pipe flow

If the shape and the size of the cross sections A1 and A2 are identical, and

 i j , ui have the same dis-

tributions on them, we have

Integrating Eq.(8) over the control volume V , and using the Gauss Theorem to transform the volume integral into the surface integral for the left hand side and the second term on the right hand side, we have 

    g x

3

+

A



ij

A1

(ui n j )d A +   i j (ui n j )d A = 0

 1 1 + u j u j + u j u j  ui ni d A =  2 2 



i.e., there would be no mechanical energy transformation while the fluid flows from cross section A1 to A2 . In this case the mechanical energy loss becomes   ui  u j 1   i j si j dV = +     gAU 2 gAU xi 1 1 V 1 1 V  x j   dV 

    (7a)

Furthermore, if the flow included between the cross sections A1 and A2 is uniform, the mechanical energy loss can be simplified as hw =

L 2 gAU 1 1

  ui

  x A1



j

+

 u j   ui  u j +  xi   x j xi



   i j si j +  ij sij dV +    i j ui +  ij ui +

A2

  ui  u j +   x j xi

 

p

V

hw =

(8)

  d A 

(7b)

1.2 Turbulent flow Apply the Reynolds decomposition to Eq.(2), and then take the ensemble average. For a steady turbulent

A

1   pui +   u j uiu j + uiu j u j  ni d A 2  

(9)

For a non-circular turbulent pipe flow, the hydrostatic assumption (the hydrodynamic pressure is equal to the hydrostatic pressure at the cross section of gradually varied flow) would not be hold, strictly speaking, owing to the non-homogeneity of the Reynolds stresses and the secondary flow at the cross section. Using ps to represent the hydrostatic pressure such that g x3 + ps /  = C , and using  F1 and  F2 to represent the deviations between the hydrodynamic pressures p1 and p2 and the hydrostatic pressures ps1 and ps 2 at the cross-sections A1 and A2 , by using the method of Green function[6], we have

p1 = ps1 +  F1

(10a)

p2 = ps 2 +  F2

(10b)

Similar to the discussions in Section 1.1, define the

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mean kinetic energy correction coefficients 1 ,  2 and the mean turbulent energy correction coefficients 1 and  2 at the cross-sections A1 and A2 such that

1U13 A1 =   ui ui u j n j d A , A1

 2U 23 A2 =  ui ui u j n j d A , A2

1U1k1 =   2U 2 k2 =

1 u j u j ui ni d A , 2  A1

1 u j u j ui ni d A 2  A2

2 1

ps1 p U k U + 1 + 1 1 = z2 + s 2 +  2 + g g 2g g 2g

(11)

Equation (11) is the integral form of the mechanical energy equation for the incompressible steady nonuniform pipe flow of homogeneous fluid, which includes the variation of the mean turbulent kinetic energy from A1 to A2 . In Eq.(11)





1   i j si j +  ij sij dV + F  gAU 1 1 V

(12)

and F is relative to the local loss in Hydraulics, which can be calculated by

F =

 1    i j ui +  ij ui + pui +  u j uiu j +  gAU 1 1   A1



  1 2

 

 ij ui + pui +   u j uiu j + uiu j u j  ni d A +

 



  F2 ui ni dA A2



averaged shear stress and the time averaged rate of deformation,   u u 1   i + j   x 2 gAU xi 1 1 V    j

 u u j  i +  x j xi 

   

   + 2  dV   

(14a)

Furthermore, if the flow between the cross sections A1 and A2 is uniform (for example, the flow in a long and straight pipe), the mechanical energy loss can be further simplified as

hw =

  u u j L   i +   2 gAU xi  1 1 A1    x j

 u u j  i +  x j xi 

   + 2  d A   

   

(14b)

2. Mechanical energy loss for steady uniform laminar flow in a circular pipe For the steady uniform laminar flow in a circular pipe the longitudinal velocity u z only changes in the radial direction r such that[6]

1  uiu j u j  ni d A +   F1 ui ni d A +   i j ui + 2  A1 A2

 

2 Sij Sij ) and using the relationship between the time

hw =

2 2

k  2 2 + hw g

hw =

 g )  ( z2 + ps 2 /  g ) ) is in balance with the sum of the variation of the mean kinetic energy, the variation of the mean turbulent energy and the mechanical energy transformation and loss. If the shape and the size of the cross sections A1 and A2 are identical and the integrands of the first and the third terms on the right hand side of Eq.(13), the second and the forth terms on the right hand side of Eq.(13) have the same distributions, then F = 0, i.e., there would be no mechanical energy transformation while the fluid flows from the cross section A1 to A2 . In this case the mechanical energy loss hw per unit weight in unit time can be expressed as follows after introducing the definition of the dissipation rate of the turbulent energy  ( =

and use hw to represent the transformation(the second term on the right hand side of Eq.(12)) and loss (the first terms on the right hand side of Eq.(12)) of the mechanical energy. Equation (9) can be simplified by using the no-slip condition on the pipe wall such that z1 +

It can be seen from Eq.(11) that the variation of the potential energy between two sections ( ( z1 + ps1 /

u z = C[(0.5d ) 2  r 2 ]

(13)

(15)

where d is the diameter of the circular pipe and C is a constant related to the mean pressure gradient in the

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longitudinal direction. Substituting Eq.(15) into Eq.(7b) the following equation can be obtained finally after some simplifications hw = 

L U12 d 2g

(16)

Equation (16) is the same as the Darcy-Weisbach formula for the resistance loss of a steady uniform laminar flow in a circular pipe in Hydraulics, and the resistance coefficient  = 64 / Red , where Red = U1d / is the Reynolds number. 3. Mechanical energy loss for uniform turbulent flow in a circular pipe For the uniform turbulent flow in a circular pipe the longitudinal velocity u z also only changes in the radial direction, using the condition of the axial symmetry Eq.(14b) can be rewritten as 8L   hw = gU d 2  



0.5 d 0

 duz   dr

2   0.5 d  r d r +  0  r d r    

(17)

It can be seen from Eq.(17) that the mechanical energy loss has two parts, both from the fluid viscosity for the uniform turbulent flow in a circular pipe, the former is related to the time averaged velocity gradient and the latter is related to the dissipation of the turbulent kinetic energy. For the uniform turbulent flow in a circular pipe, the total dissipation of the turbulent energy is equal to the total generation of the turbulent energy at section A1 , i.e., 

 ui   d A =   d A j  A1

  uu  x A1



i

j

(18)

Therefore Eq.(17) can be simplified considering that 0.5 d d 2 U1 = 2   u z r d r 0 4

nt for the mechanical energy loss as  = 8(u* / U1 )2 , Eq.(19) can be simplified further as hw = 

L U12 d 2g

(20)

Equation (20) is the same as the Darcy-Weisbach formula for the resistance loss for the steady uniform turbulent flow in a circular pipe in Hydraulics. Now we discuss the compositions of the mechanical energy loss based on Eq.(17). From Eq.(20) and Eq.(17)  can be expressed as 8  0.5 d  d u  = 3    z U1 R0  0  d r 

2   0.5 d  r d r +  0  r d r    

(21)

Using 1 and 2 to represent the contributions to the coefficients of the mechanical energy loss relative to the time averaged velocity gradient and the dissipation of the turbulent kinetic energy, respectively, we have 8 1 = 3 U1 R0

2 =   1



0.5 d 0

 duz   dr

2

  r d r 

(22a) (22b)

To calculate the time averaged velocity u z , the Van Driest model[17] and the Spalding formula[17] are used to determine the mean velocity distribution in this paper. In Fig.2, the calculated velocity by the numerical simulation is compared with the experimental data of Ref.[2] and Ref.[17], where y+ = [u* (0.5d  r ) / ] and u + = u z / u . It can be seen that the two results are in good agreement with each other. Therefore using these two theoretical results for the time averaged velocity, the variation of 1 against the Reynolds number Red can be obtained by the numerical integration of Eq.(22a), and 2 can be obtained further by Eq.(22b), the variations of 1 /  and 2 / 

and

against Red can be obtained as well and the related

 du  2r  rz =   z  uru z  =  u*2 d  dr 

results are shown in Fig.3. It can be seen that: (1) 1 /

 decreases with the increase of Red , but 2 /  in-

for the uniform turbulent flow in a circular pipe such that hw =

where u is the shear velocity. Defining the coefficie-

4 Lu*2 gd

(19)

creases with the increase of Red . (2) when Red is equal to 3×104, the contributions of those two parts are nearly the same, i.e., when Red is less than 3×104, the contribution from the term related to the time averaged velocity gradient is larger, when Red is larger

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than 3×104, the contribution from the term related to the dissipation of the turbulent kinetic energy is larger.

Fig.2 Comparison of theoretical velocity distribution with experimental data

Fig.3 Variation of 1 /  and 2 /  with Red

Fig.4 Contributions of viscous sublayer, buffer layer and the region above buffer layer to the mechanical energy loss

In the radial direction, the uniform turbulent flow in a circular pipe can be divided into a viscous sublayer (r  (0.5d  5 / u*  0.5d )) , a buffer layer (r  (0.5d  30 / u*  0.5d  5 / u* )) , and the region above the buffer layer which contains the logarithmic region[6,17]. The contributions of the above three regions to the mechanical energy loss are calculated based on the time averaged velocity distributions relative to the Van Driest model and the Spalding formula, as

shown in Fig.4. It can be seen that: (1) when Red is less than 2.5×104, the contributions of the above three regions to the mechanical energy loss are in the following order: buffer layer>viscous sublayer>the region above the buffer layer. (2) when Red is larger than 2.5×104 and less than 1.3×105, the contributions of the above three regions to the mechanical energy loss are in the following order: buffer layer>the region above the buffer layer>viscous sublayer. (3) when Red is larger than 1.3×105, the contributions of the above three regions to the mechanical energy loss are in the following order: the region above the buffer layer>buffer layer>viscous sublayer. 4. Conclusions (1) A new energy equation for incompressible steady non-uniform pipe flow of homogeneous fluid is derived, which includes the variation of the mean turbulent kinetic energy, and the formula for the calculation of the transformation and loss of the mechanical energy for the non-uniform flow between two cross sections is obtained based on this equation. This formula can be simplified to the Darcy-Weisbach formula widely used in Hydraulics for uniform flows. (2) The mechanical energy loss of a turbulent flow is resulted from the fluid viscosity in a circular pipe and can be divided into two parts for a uniform flow, the first is related to the time averaged velocity gradient, and the other is related to the dissipation of the turbulent kinetic energy. The former decreases with the increase of the Reynolds number, and the latter increases with the increase of the Reynolds number. When the Reynolds number is larger than 3×104, the contribution from the term related to the dissipation of the turbulent kinetic energy is dominant. (3) In the radial direction, the uniform turbulent flow in a circular pipe can be divided into the viscous sublayer, the buffer layer and the region above the buffer layer. With the increase of the Reynolds number, the contributions of the mechanical energy loss in the buffer layer and the viscous sublayer decrease, while the contribution from the region above the buffer layer increases. When the Reynolds number is larger than 1.3×105, the contribution of the mechanical energy loss in the region above the buffer layer becomes dominant. References [1]

[2]

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