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On turbulent secondary flows in pipes of noncircular cross-section
OmO-7225/82/07086NBl3.0010 PergmonPressLtd.
ht. 1. Engng Sci Vol. 20. No. 7. pp. 863-872, 1982 Printed inGreatBritnio.
ON TURBULENT SECONDARY FLOWS IN PIPES OF NONCIRCULAR CROSS-SECTIONt CHARLES G. SPEZIALE Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ 07030,U.S.A. (Communicated by G. A. MAUGIN)
Abstract-Theorigin of turbulent secondary flow in pipes of noncircular cross section is examined from a theoretical standpoint. It is proven mathematically that secondary flows result from a nonzero difference in the normal Reynoldsstresseson planesperpendicular to the axialflowdirection.Furthermore, it is shown that the K-c model of turbulence has no natural mechanism for the development of secondary flow while the currently popular second-order closure models do. The implications that this has on turbulence modeling are discussed briefly.
I. INTRODUCTION THE INABILITY
to maintain a unidirectional mean turbulent flow in straight pipes of noncircular cross section was first recognized by Nikuradse[l]. He discovered that for the fully-developed turbulent flow of a Newtonian fluid in rectangular and triangular pipes a nonzero mean flow existed in the transverse planes of the pipe which he referred to as secondary flow (Fig. 1). This secondary flow, which like the axial mean velocity depends only on the cross sectional coordinates of the pipe and, thus, exists independent of end effects, is small in magnitude (approximately 1% of the magnitude of the axial mean velocity) but has a profound effect on the overall flow. More specifically, the secondary flow substantially distorts the mean axial isostachs (i.e. lines of constant velocity) and can lead to considerable friction losses in pipes (Nikuradse [l]). Of course, for the corresponding laminar case no secondary flow exists. Approximately 40yr after the initial discovery of secondary flow by Nikuradse, some extensive measurements of the turbulence structure in rectangular pipes were made. Hoagland[2] measured the complete 3-dimensional mean velocity distribution and local wall shear stress distribution for turbulent flow in a square pipe. Leutheusser[3] made similar measurements for rectangular pipes. Subsequently, Brundrett and Baines [4] examined the structure of turbulent secondary flow by an analysis of the mean vorticity transport equation and made careful measurements of the Reynolds stresses in a rectangular pipe. Gessner and Jones[S] made more extensive measurements of turbulent flow in a rectangular pipe which included the effect of the Reynolds number on the structure of the secondary flow and the
secondary 5, ,8,
flow
Fii. I. Fully-developed turbulent flow in a rectangular pipe. Wedicated to Professor A. C. Eringen on the occasion of his 60th birthday. 863
864
C. G. SPEZIALE
directional characteristics of the local wall shear stresses. In addition, from a simplified analysis of the Reynolds equation along a secondary flow streamline, Gessner and Jones[S] concluded that “secondary flow is the result of small differences in magnitude of opposing forces exerted by the Reynolds stresses and static pressure gradients in planes normal to the axial flow direction”. In this paper, it will be proven that secondary flow occurs when the axial mean velocity gives rise to a nonzero normal Reynolds stress difference on planes perpendicular to the axial flow direction. While this result is compatible with the results of Gessner and Jones (as well as with those of Brundrett and Baines) it is more concise and provides a direct means for determining the origin of secondary flow in turbulent closure models. By utilizing this basic result it will be shown that the K-E model of turbulence has no driving mechanism for the creation of secondary flow in pipes of noncircular cross section while second-order closure models do. Furthermore, it will be shown which of the higher-order turbulence correlations in certain second-order closure models lead to the onset of secondary flow. The implications that this has on turbulence modeling will be discussed in a later section. 2. THE DRIVING MECHANISM
FOR SECONDARY
FLOW
The problem to be considered is that of the fully-developed turbulent flow of an incompressible Newtonian fluid in a straight pipe of noncircular cross section (Fig. 2). The mean axial pressure gradient afiaz = - G is constant and is maintained by external means. The pipe is sufficiently long so that there exists a section of the pipe where the flow properties are independent of the axial coordinate I. For the case of laminar flow the steady state velocity field is of the unidirectional form
v = {O,O, 4(x, Y,}.
(2.1)
In the absence of body forces, U, is determined by the Poisson equation (Batchelor[6])
(2.2) (where p is the shear viscosity) with the boundary condition that u, = 0 on the wall of the pipe. However, for the case of fully-developed turbulent flow the mean velocity field is 3-dimensional, i.e. f is of the form if =
{fi*,(x,Y), qix,YL v,k Y,l
(2.3)
where V, and CYconstitute the secondary flow. Here, f is a solution of the Reynolds equation
Fig. 2. Fullydeveloped turbulent flow in an arbitrary pipe of noncircular cross-section.
On turbulent secondary flows in pipes
865
and mean continuity equation which, neglecting body forces, are given by (Hinze[7])
V-T=0
(2.4)
where p is the fluid density, P is the mean pressure, I is the Reynolds stress tensor, and the usual summation convention applies on repeated indices. Of course, (2.4) is not a closed system of equations for the determination of t unless a turbulent closure scheme is used that ties 7 to the mean velocity field. More specifically, r must be taken to be a functional in the history of the mean velocity field of all points of the fluid volume V, i.e. 7(x, t) = 7[V(x’, t’); x, t]
x’ E “v,
t’ E (-m,
t).
(2.5)
Here, the term functional is used in its broadest mathematical sense (namely, any quantity determined by a function) and it is tacitly assumed that there is an implicit dependence of r on its initial and boundary conditions
4x7f-v,[4x, Olav
(2.6)
Specific forms of (2.5) that are currently popular in the turbulence literature will be discussed in the next section. In order to determine the driving mechanism of secondary flow we will examine the mean vorticity transport equation which is obtained by taking the curl of (2.4),. This equation takes the form p
=ti,s+Ew at ax, (~%+~,ao, >
a2hm pv20, m--Iax,ax,
axI
(2.7)
where 6 is the mean vorticity vector, i.e. ii =
vxv,
and eklmis the permutation tensor. As a result of (2.3) the mean continuity eqn (2.4)2reduces to
$+ EY=O
ay
(2.8)
where (XI9x2,
x3)
=
6%
Y, z).
Consequently, there exists a secondary flow stream function I+% such that
(2.9) where
v2* = (3,.
(2.10)
From (2.9) and (2.10), it is quite clear that the secondary flow i&, fiYemanates directly from the axial mean vorticity 0, The axial mean vorticity is determined from the z-component of (2.7)
C. G. SPEZIALE
866
which takes the following form
(2.11) where r is independent of the axial coordinate .z and possesses two continuous partial derivatives with respect to x and y. The axial mean velocity can be obtained from the z-component of the Reynolds eqn (2.4) which reduces to (2.12) where G is the constant pressure gradient that forces the fluid through the pipe. Once a closure scheme is chosen, (2.9)-(2.12) represent a closed set of equations for the determination of the mean velocity V and Reynolds stress tensor 7. Of course, here we are interested in the steady state solution of the system since we are considering fully-developed turbulence. However, for complete generality we have not omitted the time derivatives in (2.11) and (2.12) since 7 can have a history dependence in time (e.g. hereditary integrals over time) for which the steady state solution might not be directly obtainable by the simple setting of time derivatives to zero. Since, as a result of (2.10), the development of secondary flow is directly tied to the presence of a nonzero axial mean vorticity & it is quite clear that the axial vorticity source term in (2.11) given by (2.13) is the cause of secondary flow. This term was analyzed in detail by Brundrett and Baines [4] for the case of fully-developed turbulent flow in rectangular pipes. More specifically, Brundrett and Baines utilized (2.13) to determine the lines of symmetry of the secondary flow and, thus, establish that the secondary flow consists of eight vortices (Fig. 1). However, here we will utilize (2.13) to establish a simple criterion concerning the structure of the normal Reynolds stresses which in this case will serve as a sufficient condition for the development of secondary flow. In order to accomplish this task we first note that by simple symmetry arguments there cannot be an undirectional mean turbulent flow (i.e. an absence of secondary flow) in a pipe unless (Nikuradse [ 11) 7x2 =
(2.14)
0.
Hence, as a result of (2.13), secondary flow will develop if the condition
a2(TyY - 7,) =
0
axay
(2.15)
is violated. This is quite clear since if (2.15) is violated, a nonzero rXYcould not be generated to balance it (i.e. that is not permissible in an unidirectional mean turbulent flow). A simple integration of (2.15), subject to the continuity assumptions stated previously, yields ?YY
-
7xX =
constant.
(2.16)
However, since the Reynolds stresses vanish at the pipe wall, the constant in (2.16) must be zero, i.e. Tyy -
Tu
=
0.
(2.17)
867
On turbulent secondary flows in pipes
Consequently, if the axial mean velocity gives rise to a nonzero normal Reynolds stress difference, i.e. if 17~yields -
TYY
(2.18)
7,# 0,
secondary flow will occur. We thus have the following sufficient condition for the development of secondary flow: Secondary flow will develop in pipes of noncircular cross section if the axial mean velocity gives rise to any nonzero difference in the normal Reynolds stresses on planes perpendicular to the axial flow direction. This condition represents the simplest criterion for the
development of secondary flow that has yet to be proven in the turbulence literature. In the next section, we will show how this criterion can be used in a direct manner to deduce what the origins of secondary flow are in certain specific turbulent closure models. 3. TURBULENT
CLOSURE
MODELS AND SECONDARY
FLOW
One of the simplest turbulent closure models that is currently popular is the so called K--E (or K-l) model of turbulence which is a direct generalization of the eddy viscosity concept. Here, the Reynolds stress tensor for incompressible flow is taken to be of the form
7k,
=
$tr
&
+
CplK”2
2 +3 (ax, ax,>
(3.1)
where K = - 6 tr r,
C is a dimensionless constant, tr (e) denotes the trace, and 1 represents the length scale of turbulence. The turbulent kinetic energy K is determined from a modeled version of the turbulent kinetic energy transport equation (Mellor and Herring[8]) which takes the form
P
--E+ vV2K
(3.2)
where p is the fluctuating pressure, u is the fluctuating velocity, v = p/p is the kinematic viscosity, and E is the turbulent dissipation rate given by
auk
‘=%j
9 ax,
(3.3)
In (3.2) and (3.3) an overbar is used to indicate an averaged quantity (for steady turbulence time averages would be used). The scalar length scale 1 is determined by a separate transport equation. For the K-e model, 1 is taken to be of the form
l=F
(3.4)
where l is determined from a modeled version of the scalar dissipation rate equation (Launder, Reece and Rodi[9]). For the K-l model of turbulence, 1 is determined by a modeled version of the contracted form of the transport equation for the two-point velocity correlations c&x’, d’) = uk(x’)u,(f) x1=x-r/2,
f=x+r/2
obtained by integrating over a sphere of radius r (Mellor and Herringlg]).
(3.5)
C.G.SPEZIALE
868
From (3.1), it is quite clear that for the K-E or K-l model of turbulence the axial mean velocity V, gives rise to the following Reynolds stress distribution rXv= 0,
(3.6)
and, thus, we have (3.7) Consequently, as a result of the criterion derived in the previous section, it is quite clear that the K--E and K-1 model of turbulence have no built in mechanism for the development of secondary flow (i.e. they possess unidirectional mean flow solutions contrary to what is observed experimentally). It is, therefore, unlikely that either model can serve as an adequate model for 3-dimensional wall turbulence. In so far as secondary flows are concerned, this defect in the K-r and K-l model can be remedied by taking the anisotropic part of the Reynolds stress tensor to be a nonlinear function of the mean velocity gradients, i.e. we can set
where (3.9) Then, invariance considerations yield (Speziale [ lo]) (3.10) where (3.11) is the mean rate of deformation tensor. Here, aa. al and a2 are functions of K, I, p and the nonvanishing invariants of h, i.e. tr 5*, tr ;S3. Equation (3.10) represents a generalized “Reiner-Rivlin fluid” which is known to yield secondary flows in pipes of noncircular cross section (Eringen[l I]). We will use the simple sufficient criterion derived in the previous section to show that this is the case. The axial mean velocity & in (3.10) gives rise to the following normal Reynolds stress difference
?YY -
Txx
=‘a2[($2_(g] 4
(3.12)
which is not zero for this case and, hence, secondary flows will occur in pipes of noncircular cross section. For rectangular and elliptical pipes, (3.10) has been shown to yield secondary flows with the correct qualitative features. However, for turbulent channel flow in the y, z-plane where tiZ= V,(y), (3.10) predicts that rzz= Tyy
(3.13)
869
On turbulent secondary flows in pipes
which is severely contradicted by experiments (Hinze[7] and Speziale[lO]). Since secondary flow is a normal Reynolds stress phenomenon, it is, thus, doubtful that any simple generalization of the K-e or K-l model of the form (3.8) can quantitatively describe secondary flow in an accurate fashion. At this point, it should be mentioned that a nonlocal generalization of (3.8), i.e. the nonlocal Stokesian fluids of Eringen, alleviates this difficulty (3.13) and has been shown to yield secondary flows that are in qualitative agreement with experiments (Speziale[lO] and Eringen[l2]). This theory is relatively new and is more complicated than (3.8) and, therefore, research is still being conducted on its fundamental properties. Nevertheless, this theory appears to be quite promising and warrants much future research. Now, we will examine the occurrence of secondary flow in certain second-order closure models. In second-order closure models, closure is based on the Reynolds stress transport equations which are a rigorous consequence of the Navier-Stokes equations and are given by (Hinze [7])
DTk/- Dt
T,,,k
2 -
T,,,/
2 +p-&(u,uku,) +
m
m
auk) 7
m
1
(3.14) where
g=$+f.v.
(3.15)
Closure is usually obtained by algebraically tying the higher-order turbulence correlations that appear on the r.h.s. of (3.14) to the mean velocity gradients, the Reynolds stresses (as well as their spatial gradients), and the length scale of turbulence. More specifically, it is assumed that A’“’= A@(r, VT,VV,1,p),
(Y= 1,2,3,4,
(3.16)
where A!$‘,= Al:‘=
u,,,uk,,I,
p(%+z),
Af'
=
Ag’x
puk
~22
(3.17)
are referred to, respectively, as the third-order diffusion correlation, the pressure diffusion correlation, the pressure-strain correlation, and the dissipation rate correlation. It should be noted that only the diffusion correlations are assumed to depend on Vr. In the Rotta-Kolmogorov model (Mellor and Herring[8]), the following closure relations are assumed to be valid
(3.18)
+ C4pK(2+z)
where C,, C2, C, and C, are dimensionless constants which are determined from experiments. A Reynolds stress closure model is obtained when (3.18) is substituted into (3.14) and
870
C. G. SPEZIALE
supplemented by a transport equation for the length scale. This equation, which is obtained by modeling the contracted from of the transport equation for the two-point velocity correlations as discussed earlier, assumes the form (Mellor and Herring[8])
vt
cp21)-&(Kl) t c(jIP21$] k
+
C,
1 P
7k,
%
_
ax,
C
k
K3/2
(3.19)
a
where C,, C,, C, and C’sare dimensionless constants. It is a straightforward procedure to show that in the Rotta-Kolmogorov model the axial mean velocity gives rise to a steady state normal Reynolds stress difference determined by the equation
vV2(Tyr
C,K1'"l-&-7,)
-
+;
1
C,K1~21-&y-~xJ
C,K1’21$f
-2-$!,K”21~)-2~(C2pK1~21$) t 2 5 ( C2pKu21g)
- C, 7
(3.20) (7yr - r,,) = 0.
Equation (3.20) is obtained by differencing the yy and xx components of (3.14) after utilizing (3.18) with the axial mean velocity
where all of the turbulence quantities are assumed to depend only on the x and y coordinates as discussed earlier. From (3.20) it is quite clear that the normal Reynolds stress difference 7Yu- T~ will not, in general, be zero unless
c, = c, = 0
(3.21)
which is not the case. Hence the Rotta-Kolmogorov model will yield secondary flows in pipes of noncircular cross section as a result of the criterion derived in the previous section. A simple comparison of (3.21) with (3.18) demonstrates that in the Rotta-Kolmogorov model secondary flows originate from the diffusion terms (i.e. the third-order diffusion correlation and the pressure diffusion correlation). Another second-order closure model that is currently popular is that due to Launder, Reece and Rodi[9]. Here, the following closure relations for the higher-order turbulence correlations are postulated
(3.22)
Onturbulentsecondaryflowsin pipes
871
(3.22) auk au, i v--=-b&, axmax, 3
where BI, B2 and B3 are dimensionless constants. The scalar dissipation rate E is determined from a modeled version of its transport equation which takes the form (Launder, Reece and Rodi [9]) (3.23)
where B4, B5 and B6 are dimensionless constants. A straightforward, although somewhat tedious, calculation shows that the axial mean velocity gives rise to a steady state normal Reynolds stress difference determined by the equation
(3.24)
- B2;
(ryy- T,,) = 0.
Equation (3.24) was obtained by the same method used to obtain (3.20). It is now quite clear that the axial mean velocity will, in general, give rise to a nonzero normal Reynolds stress difference rYy- T~ unless B
-0 I--
8B3-2_0 9
11
(3.25)
which is not the case. Consequently, the second-order closure model of Launder, Reece and Rodi will lead to the development of secondary flows in pipes of noncircular cross section. A comparison of (3.25) and (3.22) shows that in this model the origin of the secondary flow is in the third-order diffusion correlation and the pressure-strain correlation. To date, there have been no complete turbulence calculations of secondary flow in pipes of noncircular cross section using second-order closure models. While such calculations can, in principle, be carried out numerically on a high speed digital computer they represent a formidable task. This results from the fact that the problem is 3-dimensional and, thus, requires the solution of eleven coupled nonlinear partial differential equations for the eleven unknowns T, 7, I,+and 1. 4.CONCLUSION
It has been proven that turbulent secondary flow in pipes of noncircular cross section results when there is a nonzero difference in the normal Reynolds stresses on planes that are perpendicular to the axial flow direction. In addition, a simple sufficient condition was derived for the development of secondary flow which is useful in the analysis of turbulent closure models. More specifically, it was proven that if the axial mean velocity gives rise to any nonzero normal Reynolds stress difference on planes perpendicular to the axial flow direction, UES Vol. 20. No. 7-F
812
C. G. SPEZIALE
secondary flow will occur. While these results are consistent with previous results on the subject they are more concise and represent a more basic mechanism for the generation of secondary flow. Utilizing these results, it was shown that the K-E (or K-l) model of turbulence has no built in mechanism for the generation of secondary flow. Consequently, it is doubtful that this model can adequately be applied to most types of 3-dimensional wall turbulence. In addition, the onset of secondary flow was examined for certain second-order closure models that are widely used in the turbulence literature. It was proven that both the Rotta-Kolmogorov model and the Launder, Reece and Rodi model predict the occurrence of secondary flows in pipes of noncircular cross section. However, there is a disparity between the two models in so far as where the secondary flow is generated. In the Rotta-Kolmogorov model the secondary flow is generated by the third-order diffusion correlation and the pressure diffusion correlation. On the other hand, in the Launder, Reece and Rodi model the secondary flow is generated by the third-order diffusion correlation and the pressure-strain correlation. Furthermore, there are two serious questions regarding the applicability of these second-order closure models to the study of secondary flow. The dissipation terms, in both models, play no role in the generation of secondary flow while it is well known that the dissipation terms play an important role in the near wall region of the pipe where secondary flows originate. In addition, there is a serious question as to whether these second-order closure models can accurately predict wall turbulence that is highly anisotropic which is the case for secondary flows (Speziale[l3]). As mentioned earlier, to date, there have been no detailed quantitative studies of secondary flow in pipes of noncircular cross section without ad hoc empiricisms. In fact, most of the previous calculations have been qualitative in nature in that the Reynolds stresses have not been accurately calculated and shown to be in agreement with experimental data. Future research is needed in this direction to better understand the role that various turbulence correlations play in the development and maintenance of secondary flow. Acknowledgement-This work was supported by the National Science Foundation under Grant No. ENG 79-08180.
REFERENCES [l] J. NIKURADSE, PD.1 Forschungsheft 70, 1229(1926). [2] L. C. HOAGLAND, Ph.D. Thesis, Massachusetts Institute of Technology (l%O). 131H. J. LEUTHEUSSER, A.SC.E I. Hyd. Diu.89, 1 (1%3). 141E. BRUNDRETT and W. D. BAINES. .I Fluid Mech. 19.375 (1964). i5j F. B. GESSNER and J. B. JONES, J. &id Mech. 23,684 (I%$. [6] G. K. BATCHELOR, Introduction to J&id Dynamics. Cambridge University Press, London (1%7). [7] J. 0. HINZE, Turbulence.McGraw-Hill Book Company, New York (1975). [8] G. L. MELLOR and H. J. HERRING, AIAA J. 11,590 (1973). 191B. LAUNDER, C. REECE and W. RODI, 1. Fluid Me&. 68,537 (1975). [lo] C. SPEZIALE, Ph.D. Thesis, Princeton University (1978). [ll] A. C. ERINGEN, Mechanics of Continua. Krieger, New York (1980). [12] A. C. ERINGEN, Crystal Lattice Dejects 7, 109(1977). [13] C. SPEZIALE, Phys. Fluids 23,459 (1980). (Received 20 July 1981)
ht. 1. Engng Sci Vol. 20. No. 7. pp. 863-872, 1982 Printed inGreatBritnio.
ON TURBULENT SECONDARY FLOWS IN PIPES OF NONCIRCULAR CROSS-SECTIONt CHARLES G. SPEZIALE Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ 07030,U.S.A. (Communicated by G. A. MAUGIN)
Abstract-Theorigin of turbulent secondary flow in pipes of noncircular cross section is examined from a theoretical standpoint. It is proven mathematically that secondary flows result from a nonzero difference in the normal Reynoldsstresseson planesperpendicular to the axialflowdirection.Furthermore, it is shown that the K-c model of turbulence has no natural mechanism for the development of secondary flow while the currently popular second-order closure models do. The implications that this has on turbulence modeling are discussed briefly.
I. INTRODUCTION THE INABILITY
to maintain a unidirectional mean turbulent flow in straight pipes of noncircular cross section was first recognized by Nikuradse[l]. He discovered that for the fully-developed turbulent flow of a Newtonian fluid in rectangular and triangular pipes a nonzero mean flow existed in the transverse planes of the pipe which he referred to as secondary flow (Fig. 1). This secondary flow, which like the axial mean velocity depends only on the cross sectional coordinates of the pipe and, thus, exists independent of end effects, is small in magnitude (approximately 1% of the magnitude of the axial mean velocity) but has a profound effect on the overall flow. More specifically, the secondary flow substantially distorts the mean axial isostachs (i.e. lines of constant velocity) and can lead to considerable friction losses in pipes (Nikuradse [l]). Of course, for the corresponding laminar case no secondary flow exists. Approximately 40yr after the initial discovery of secondary flow by Nikuradse, some extensive measurements of the turbulence structure in rectangular pipes were made. Hoagland[2] measured the complete 3-dimensional mean velocity distribution and local wall shear stress distribution for turbulent flow in a square pipe. Leutheusser[3] made similar measurements for rectangular pipes. Subsequently, Brundrett and Baines [4] examined the structure of turbulent secondary flow by an analysis of the mean vorticity transport equation and made careful measurements of the Reynolds stresses in a rectangular pipe. Gessner and Jones[S] made more extensive measurements of turbulent flow in a rectangular pipe which included the effect of the Reynolds number on the structure of the secondary flow and the
secondary 5, ,8,
flow
Fii. I. Fully-developed turbulent flow in a rectangular pipe. Wedicated to Professor A. C. Eringen on the occasion of his 60th birthday. 863
864
C. G. SPEZIALE
directional characteristics of the local wall shear stresses. In addition, from a simplified analysis of the Reynolds equation along a secondary flow streamline, Gessner and Jones[S] concluded that “secondary flow is the result of small differences in magnitude of opposing forces exerted by the Reynolds stresses and static pressure gradients in planes normal to the axial flow direction”. In this paper, it will be proven that secondary flow occurs when the axial mean velocity gives rise to a nonzero normal Reynolds stress difference on planes perpendicular to the axial flow direction. While this result is compatible with the results of Gessner and Jones (as well as with those of Brundrett and Baines) it is more concise and provides a direct means for determining the origin of secondary flow in turbulent closure models. By utilizing this basic result it will be shown that the K-E model of turbulence has no driving mechanism for the creation of secondary flow in pipes of noncircular cross section while second-order closure models do. Furthermore, it will be shown which of the higher-order turbulence correlations in certain second-order closure models lead to the onset of secondary flow. The implications that this has on turbulence modeling will be discussed in a later section. 2. THE DRIVING MECHANISM
FOR SECONDARY
FLOW
The problem to be considered is that of the fully-developed turbulent flow of an incompressible Newtonian fluid in a straight pipe of noncircular cross section (Fig. 2). The mean axial pressure gradient afiaz = - G is constant and is maintained by external means. The pipe is sufficiently long so that there exists a section of the pipe where the flow properties are independent of the axial coordinate I. For the case of laminar flow the steady state velocity field is of the unidirectional form
v = {O,O, 4(x, Y,}.
(2.1)
In the absence of body forces, U, is determined by the Poisson equation (Batchelor[6])
(2.2) (where p is the shear viscosity) with the boundary condition that u, = 0 on the wall of the pipe. However, for the case of fully-developed turbulent flow the mean velocity field is 3-dimensional, i.e. f is of the form if =
{fi*,(x,Y), qix,YL v,k Y,l
(2.3)
where V, and CYconstitute the secondary flow. Here, f is a solution of the Reynolds equation
Fig. 2. Fullydeveloped turbulent flow in an arbitrary pipe of noncircular cross-section.
On turbulent secondary flows in pipes
865
and mean continuity equation which, neglecting body forces, are given by (Hinze[7])
V-T=0
(2.4)
where p is the fluid density, P is the mean pressure, I is the Reynolds stress tensor, and the usual summation convention applies on repeated indices. Of course, (2.4) is not a closed system of equations for the determination of t unless a turbulent closure scheme is used that ties 7 to the mean velocity field. More specifically, r must be taken to be a functional in the history of the mean velocity field of all points of the fluid volume V, i.e. 7(x, t) = 7[V(x’, t’); x, t]
x’ E “v,
t’ E (-m,
t).
(2.5)
Here, the term functional is used in its broadest mathematical sense (namely, any quantity determined by a function) and it is tacitly assumed that there is an implicit dependence of r on its initial and boundary conditions
4x7f-v,[4x, Olav
(2.6)
Specific forms of (2.5) that are currently popular in the turbulence literature will be discussed in the next section. In order to determine the driving mechanism of secondary flow we will examine the mean vorticity transport equation which is obtained by taking the curl of (2.4),. This equation takes the form p
=ti,s+Ew at ax, (~%+~,ao, >
a2hm pv20, m--Iax,ax,
axI
(2.7)
where 6 is the mean vorticity vector, i.e. ii =
vxv,
and eklmis the permutation tensor. As a result of (2.3) the mean continuity eqn (2.4)2reduces to
$+ EY=O
ay
(2.8)
where (XI9x2,
x3)
=
6%
Y, z).
Consequently, there exists a secondary flow stream function I+% such that
(2.9) where
v2* = (3,.
(2.10)
From (2.9) and (2.10), it is quite clear that the secondary flow i&, fiYemanates directly from the axial mean vorticity 0, The axial mean vorticity is determined from the z-component of (2.7)
C. G. SPEZIALE
866
which takes the following form
(2.11) where r is independent of the axial coordinate .z and possesses two continuous partial derivatives with respect to x and y. The axial mean velocity can be obtained from the z-component of the Reynolds eqn (2.4) which reduces to (2.12) where G is the constant pressure gradient that forces the fluid through the pipe. Once a closure scheme is chosen, (2.9)-(2.12) represent a closed set of equations for the determination of the mean velocity V and Reynolds stress tensor 7. Of course, here we are interested in the steady state solution of the system since we are considering fully-developed turbulence. However, for complete generality we have not omitted the time derivatives in (2.11) and (2.12) since 7 can have a history dependence in time (e.g. hereditary integrals over time) for which the steady state solution might not be directly obtainable by the simple setting of time derivatives to zero. Since, as a result of (2.10), the development of secondary flow is directly tied to the presence of a nonzero axial mean vorticity & it is quite clear that the axial vorticity source term in (2.11) given by (2.13) is the cause of secondary flow. This term was analyzed in detail by Brundrett and Baines [4] for the case of fully-developed turbulent flow in rectangular pipes. More specifically, Brundrett and Baines utilized (2.13) to determine the lines of symmetry of the secondary flow and, thus, establish that the secondary flow consists of eight vortices (Fig. 1). However, here we will utilize (2.13) to establish a simple criterion concerning the structure of the normal Reynolds stresses which in this case will serve as a sufficient condition for the development of secondary flow. In order to accomplish this task we first note that by simple symmetry arguments there cannot be an undirectional mean turbulent flow (i.e. an absence of secondary flow) in a pipe unless (Nikuradse [ 11) 7x2 =
(2.14)
0.
Hence, as a result of (2.13), secondary flow will develop if the condition
a2(TyY - 7,) =
0
axay
(2.15)
is violated. This is quite clear since if (2.15) is violated, a nonzero rXYcould not be generated to balance it (i.e. that is not permissible in an unidirectional mean turbulent flow). A simple integration of (2.15), subject to the continuity assumptions stated previously, yields ?YY
-
7xX =
constant.
(2.16)
However, since the Reynolds stresses vanish at the pipe wall, the constant in (2.16) must be zero, i.e. Tyy -
Tu
=
0.
(2.17)
867
On turbulent secondary flows in pipes
Consequently, if the axial mean velocity gives rise to a nonzero normal Reynolds stress difference, i.e. if 17~yields -
TYY
(2.18)
7,# 0,
secondary flow will occur. We thus have the following sufficient condition for the development of secondary flow: Secondary flow will develop in pipes of noncircular cross section if the axial mean velocity gives rise to any nonzero difference in the normal Reynolds stresses on planes perpendicular to the axial flow direction. This condition represents the simplest criterion for the
development of secondary flow that has yet to be proven in the turbulence literature. In the next section, we will show how this criterion can be used in a direct manner to deduce what the origins of secondary flow are in certain specific turbulent closure models. 3. TURBULENT
CLOSURE
MODELS AND SECONDARY
FLOW
One of the simplest turbulent closure models that is currently popular is the so called K--E (or K-l) model of turbulence which is a direct generalization of the eddy viscosity concept. Here, the Reynolds stress tensor for incompressible flow is taken to be of the form
7k,
=
$tr
&
+
CplK”2
2 +3 (ax, ax,>
(3.1)
where K = - 6 tr r,
C is a dimensionless constant, tr (e) denotes the trace, and 1 represents the length scale of turbulence. The turbulent kinetic energy K is determined from a modeled version of the turbulent kinetic energy transport equation (Mellor and Herring[8]) which takes the form
P
--E+ vV2K
(3.2)
where p is the fluctuating pressure, u is the fluctuating velocity, v = p/p is the kinematic viscosity, and E is the turbulent dissipation rate given by
auk
‘=%j
9 ax,
(3.3)
In (3.2) and (3.3) an overbar is used to indicate an averaged quantity (for steady turbulence time averages would be used). The scalar length scale 1 is determined by a separate transport equation. For the K-e model, 1 is taken to be of the form
l=F
(3.4)
where l is determined from a modeled version of the scalar dissipation rate equation (Launder, Reece and Rodi[9]). For the K-l model of turbulence, 1 is determined by a modeled version of the contracted form of the transport equation for the two-point velocity correlations c&x’, d’) = uk(x’)u,(f) x1=x-r/2,
f=x+r/2
obtained by integrating over a sphere of radius r (Mellor and Herringlg]).
(3.5)
C.G.SPEZIALE
868
From (3.1), it is quite clear that for the K-E or K-l model of turbulence the axial mean velocity V, gives rise to the following Reynolds stress distribution rXv= 0,
(3.6)
and, thus, we have (3.7) Consequently, as a result of the criterion derived in the previous section, it is quite clear that the K--E and K-1 model of turbulence have no built in mechanism for the development of secondary flow (i.e. they possess unidirectional mean flow solutions contrary to what is observed experimentally). It is, therefore, unlikely that either model can serve as an adequate model for 3-dimensional wall turbulence. In so far as secondary flows are concerned, this defect in the K-r and K-l model can be remedied by taking the anisotropic part of the Reynolds stress tensor to be a nonlinear function of the mean velocity gradients, i.e. we can set
where (3.9) Then, invariance considerations yield (Speziale [ lo]) (3.10) where (3.11) is the mean rate of deformation tensor. Here, aa. al and a2 are functions of K, I, p and the nonvanishing invariants of h, i.e. tr 5*, tr ;S3. Equation (3.10) represents a generalized “Reiner-Rivlin fluid” which is known to yield secondary flows in pipes of noncircular cross section (Eringen[l I]). We will use the simple sufficient criterion derived in the previous section to show that this is the case. The axial mean velocity & in (3.10) gives rise to the following normal Reynolds stress difference
?YY -
Txx
=‘a2[($2_(g] 4
(3.12)
which is not zero for this case and, hence, secondary flows will occur in pipes of noncircular cross section. For rectangular and elliptical pipes, (3.10) has been shown to yield secondary flows with the correct qualitative features. However, for turbulent channel flow in the y, z-plane where tiZ= V,(y), (3.10) predicts that rzz= Tyy
(3.13)
869
On turbulent secondary flows in pipes
which is severely contradicted by experiments (Hinze[7] and Speziale[lO]). Since secondary flow is a normal Reynolds stress phenomenon, it is, thus, doubtful that any simple generalization of the K-e or K-l model of the form (3.8) can quantitatively describe secondary flow in an accurate fashion. At this point, it should be mentioned that a nonlocal generalization of (3.8), i.e. the nonlocal Stokesian fluids of Eringen, alleviates this difficulty (3.13) and has been shown to yield secondary flows that are in qualitative agreement with experiments (Speziale[lO] and Eringen[l2]). This theory is relatively new and is more complicated than (3.8) and, therefore, research is still being conducted on its fundamental properties. Nevertheless, this theory appears to be quite promising and warrants much future research. Now, we will examine the occurrence of secondary flow in certain second-order closure models. In second-order closure models, closure is based on the Reynolds stress transport equations which are a rigorous consequence of the Navier-Stokes equations and are given by (Hinze [7])
DTk/- Dt
T,,,k
2 -
T,,,/
2 +p-&(u,uku,) +
m
m
auk) 7
m
1
(3.14) where
g=$+f.v.
(3.15)
Closure is usually obtained by algebraically tying the higher-order turbulence correlations that appear on the r.h.s. of (3.14) to the mean velocity gradients, the Reynolds stresses (as well as their spatial gradients), and the length scale of turbulence. More specifically, it is assumed that A’“’= A@(r, VT,VV,1,p),
(Y= 1,2,3,4,
(3.16)
where A!$‘,= Al:‘=
u,,,uk,,I,
p(%+z),
Af'
=
Ag’x
puk
~22
(3.17)
are referred to, respectively, as the third-order diffusion correlation, the pressure diffusion correlation, the pressure-strain correlation, and the dissipation rate correlation. It should be noted that only the diffusion correlations are assumed to depend on Vr. In the Rotta-Kolmogorov model (Mellor and Herring[8]), the following closure relations are assumed to be valid
(3.18)
+ C4pK(2+z)
where C,, C2, C, and C, are dimensionless constants which are determined from experiments. A Reynolds stress closure model is obtained when (3.18) is substituted into (3.14) and
870
C. G. SPEZIALE
supplemented by a transport equation for the length scale. This equation, which is obtained by modeling the contracted from of the transport equation for the two-point velocity correlations as discussed earlier, assumes the form (Mellor and Herring[8])
vt
cp21)-&(Kl) t c(jIP21$] k
+
C,
1 P
7k,
%
_
ax,
C
k
K3/2
(3.19)
a
where C,, C,, C, and C’sare dimensionless constants. It is a straightforward procedure to show that in the Rotta-Kolmogorov model the axial mean velocity gives rise to a steady state normal Reynolds stress difference determined by the equation
vV2(Tyr
C,K1'"l-&-7,)
-
+;
1
C,K1~21-&y-~xJ
C,K1’21$f
-2-$!,K”21~)-2~(C2pK1~21$) t 2 5 ( C2pKu21g)
- C, 7
(3.20) (7yr - r,,) = 0.
Equation (3.20) is obtained by differencing the yy and xx components of (3.14) after utilizing (3.18) with the axial mean velocity
where all of the turbulence quantities are assumed to depend only on the x and y coordinates as discussed earlier. From (3.20) it is quite clear that the normal Reynolds stress difference 7Yu- T~ will not, in general, be zero unless
c, = c, = 0
(3.21)
which is not the case. Hence the Rotta-Kolmogorov model will yield secondary flows in pipes of noncircular cross section as a result of the criterion derived in the previous section. A simple comparison of (3.21) with (3.18) demonstrates that in the Rotta-Kolmogorov model secondary flows originate from the diffusion terms (i.e. the third-order diffusion correlation and the pressure diffusion correlation). Another second-order closure model that is currently popular is that due to Launder, Reece and Rodi[9]. Here, the following closure relations for the higher-order turbulence correlations are postulated
(3.22)
Onturbulentsecondaryflowsin pipes
871
(3.22) auk au, i v--=-b&, axmax, 3
where BI, B2 and B3 are dimensionless constants. The scalar dissipation rate E is determined from a modeled version of its transport equation which takes the form (Launder, Reece and Rodi [9]) (3.23)
where B4, B5 and B6 are dimensionless constants. A straightforward, although somewhat tedious, calculation shows that the axial mean velocity gives rise to a steady state normal Reynolds stress difference determined by the equation
(3.24)
- B2;
(ryy- T,,) = 0.
Equation (3.24) was obtained by the same method used to obtain (3.20). It is now quite clear that the axial mean velocity will, in general, give rise to a nonzero normal Reynolds stress difference rYy- T~ unless B
-0 I--
8B3-2_0 9
11
(3.25)
which is not the case. Consequently, the second-order closure model of Launder, Reece and Rodi will lead to the development of secondary flows in pipes of noncircular cross section. A comparison of (3.25) and (3.22) shows that in this model the origin of the secondary flow is in the third-order diffusion correlation and the pressure-strain correlation. To date, there have been no complete turbulence calculations of secondary flow in pipes of noncircular cross section using second-order closure models. While such calculations can, in principle, be carried out numerically on a high speed digital computer they represent a formidable task. This results from the fact that the problem is 3-dimensional and, thus, requires the solution of eleven coupled nonlinear partial differential equations for the eleven unknowns T, 7, I,+and 1. 4.CONCLUSION
It has been proven that turbulent secondary flow in pipes of noncircular cross section results when there is a nonzero difference in the normal Reynolds stresses on planes that are perpendicular to the axial flow direction. In addition, a simple sufficient condition was derived for the development of secondary flow which is useful in the analysis of turbulent closure models. More specifically, it was proven that if the axial mean velocity gives rise to any nonzero normal Reynolds stress difference on planes perpendicular to the axial flow direction, UES Vol. 20. No. 7-F
812
C. G. SPEZIALE
secondary flow will occur. While these results are consistent with previous results on the subject they are more concise and represent a more basic mechanism for the generation of secondary flow. Utilizing these results, it was shown that the K-E (or K-l) model of turbulence has no built in mechanism for the generation of secondary flow. Consequently, it is doubtful that this model can adequately be applied to most types of 3-dimensional wall turbulence. In addition, the onset of secondary flow was examined for certain second-order closure models that are widely used in the turbulence literature. It was proven that both the Rotta-Kolmogorov model and the Launder, Reece and Rodi model predict the occurrence of secondary flows in pipes of noncircular cross section. However, there is a disparity between the two models in so far as where the secondary flow is generated. In the Rotta-Kolmogorov model the secondary flow is generated by the third-order diffusion correlation and the pressure diffusion correlation. On the other hand, in the Launder, Reece and Rodi model the secondary flow is generated by the third-order diffusion correlation and the pressure-strain correlation. Furthermore, there are two serious questions regarding the applicability of these second-order closure models to the study of secondary flow. The dissipation terms, in both models, play no role in the generation of secondary flow while it is well known that the dissipation terms play an important role in the near wall region of the pipe where secondary flows originate. In addition, there is a serious question as to whether these second-order closure models can accurately predict wall turbulence that is highly anisotropic which is the case for secondary flows (Speziale[l3]). As mentioned earlier, to date, there have been no detailed quantitative studies of secondary flow in pipes of noncircular cross section without ad hoc empiricisms. In fact, most of the previous calculations have been qualitative in nature in that the Reynolds stresses have not been accurately calculated and shown to be in agreement with experimental data. Future research is needed in this direction to better understand the role that various turbulence correlations play in the development and maintenance of secondary flow. Acknowledgement-This work was supported by the National Science Foundation under Grant No. ENG 79-08180.
REFERENCES [l] J. NIKURADSE, PD.1 Forschungsheft 70, 1229(1926). [2] L. C. HOAGLAND, Ph.D. Thesis, Massachusetts Institute of Technology (l%O). 131H. J. LEUTHEUSSER, A.SC.E I. Hyd. Diu.89, 1 (1%3). 141E. BRUNDRETT and W. D. BAINES. .I Fluid Mech. 19.375 (1964). i5j F. B. GESSNER and J. B. JONES, J. &id Mech. 23,684 (I%$. [6] G. K. BATCHELOR, Introduction to J&id Dynamics. Cambridge University Press, London (1%7). [7] J. 0. HINZE, Turbulence.McGraw-Hill Book Company, New York (1975). [8] G. L. MELLOR and H. J. HERRING, AIAA J. 11,590 (1973). 191B. LAUNDER, C. REECE and W. RODI, 1. Fluid Me&. 68,537 (1975). [lo] C. SPEZIALE, Ph.D. Thesis, Princeton University (1978). [ll] A. C. ERINGEN, Mechanics of Continua. Krieger, New York (1980). [12] A. C. ERINGEN, Crystal Lattice Dejects 7, 109(1977). [13] C. SPEZIALE, Phys. Fluids 23,459 (1980). (Received 20 July 1981)