A higher-order finite-difference technique is developed to calculate the developing-flow field of steady incompressible laminar flows in the entrance regions of circular pipes. Navier-Stokes equations governing the motion of such a flow field are solved by using this new finite-difference scheme. This new technique can increase the accuracy of the finite-difference approximation, while also providing the option of using unevenly spaced clustered nodes for computation such that relatively fine grids can be adopted for regions with large velocity gradients. The velocity profile at the entrance of the pipe is assumed to be uniform for the computation. The velocity distribution and the surface pressure drop of the developing flow then are calculated and compared to existing experimental measurements reported in the literature. Computational results obtained are found to be in good agreement with existing experimental correlations and therefore, the reliability of the new technique has been successfully tested

Boddy, Douglas E.

Gladden, Herbert J.

Ko, Ching L.

English

NASA Technical Reports Server

N95- 27352
ANALYSISOFDEVELOPINGLAMINARFLOWSINCIRCULARPIPESUSING
HIGHER-ORDERFINITE-DIFFERENCETECHNIQUE
Ching L. Ko and Douglas E. Boddy
Oakland University
Rochester, Michigan
and
Herbert J. Gladden
NASA Lewis Research Center
Cleveland, Ohio
"->7?-
SUMMARY
A higher-order finite-difference technique is developed to calculate the developing-flow
field of steady incompressible laminar flows in the entrance regions of circular pipes. Navier-
Stokes equations governing the motion of such a flow field are solved by using this new finite-
difference scheme. This new technique can increase the accuracy of the finite-difference
approximation, while also providing the option of using unevenly spaced clustered nodes for
computation such that relatively fine grids can be adopted for regions with large velocity .gradients.
The velocity profile at the entrance of the pipe is assumed to be uniform for the computauon. The
velocity distribution and the surface pressure drop of the developing flow then are calculated and
compared to existing experimental measurements reported in the literature. Computational results
obtained are found to be in good agreement with existing experimental correlations and therefore,
the reliability of the new technique has been successfully tested.
INTRODUCTION
Due to the effect of viscous dissipation, velocity and pressure distributions in fluid flows
normally vary non-uniformly. The flow velocity typically has a large spatial variation near a wall
and a relatively small variation in a region far away from wall surfaces. To calculate flow
characteristics, the classical finite-difference method discretizes the mathematical domain into
uniform-size meshes. In order to obtain accurate results without resolving to using extremely fine
meshes, the physical domain is preferable to be discretized into unenvenly spaced clustered nodes
such that fine meshes can be adopted in regions with large velocity gradients and coarse meshes
can be used in regions with small velocity gradients. As was discussed by Anderson et al (ref. 1),
a physical domain discretized by using unevenly spaced clustered nodes can be transformed into a
mathematical domain consisting of evenly spaced nodes by using the method of coordinate
transformation such that the classical finite-difference technique can be applied. However, to find a
proper mathematical function to transform coordinates of a physical domain discretized by using
arbitrarily clustered nodes into new coordinates for a mathematical domain with uniformly spaced
meshes is practically infeasible because of the complex nature of this type of transformation. In
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addition,transformed governing equations for the mathematical domain can become highly
transcendental because of the nonlinear behavior of fluid flows. Hence, it is highly desirable to
develop a finite-difference technique which can be applied directly to a mathematical domain
discretized by using arbitrarily spaced clustered nodes such that the transformation of governing
equations can be avoided.
Several existing methods which utilize the classical finite-difference formulation to solve
partial differential equations have been major tools in computational fluid mechanics. The
conventional explicit method, Crank-Nicholson method, and the Box method of Keller (refs. 2 to
4) are major finite-difference techniques that have been widely used in computational fluid
mechanics. However, theses methods are only second-order accurate and are not appropriate to be
applied to cases with unevenly spaced clustered nodes without using the coordinate transformation
technique. In view of these shortcomings, the objective of this work is to develop a fourth-order
explicit finite-difference scheme such that clustered nodes can be directly used in a mathematical
domain. In addition to having the capability of allowing untransformed governing equations be
applied directly to unevenly spaced clustered nodes in a physical domain, this new technique
determines the first four derivatives of dependent variables with respect to any independent variable
consistently to the fourth-order accuracy. Therefore, it should be more accurate than the classical
second-order f'mite difference method.
In order to test the reliability of the new explicit finite-difference technique, it is used to
solve the flow-development problems of fluid flows in the entrance region of a circular tube as well
as in the leading-edge region between parallel plates. The well known solutions of Couette flow
and of plane Poiseuille flow are applicable to the fully developed regions of these problems. As
were described by Sparrow et al (ref. 5), several analytical techniques such as linearized methods
and boundary-layer approaches have also been developed to approximately model flows in the
entrance regions of these two problems. Sparrow et al (ref. 5) used a linearized method to solve
the developing flow problems for both cases. In addition, Bodoia and Osterle (ref. 6) also utilized
the classical finite-difference method to solve these flow-development problems. However, they
applied Prandtl's momentum equation for the boundary layer instead of Navier-Stokes equations to
these problems. In order to properly model the actual developing process, the present analysis
applies Navier-Stokes equations to the entire domain and utilizes an iterative sweeping technique to
calculate nonlinear terms. Theffore, the present mathematical approach is different from any
existing analyses.
HIGHER-ORDER FINITE-DIFFERENCE FORMULATION
Five different types of higher-order finite-difference formulations which allow the usage of
clustered nodes can be developed by using Taylor's series expansion of functions up to the fourth-
order accuracy. Nodal intervals for these five types of formulation, namely, the central difference,
the partially forward difference, the fully forward difference, the partially backward difference, and
the fully backward difference, are shown in Figure 1. All five types can be used for different nodes
in the same domain and their selection for each node depends upon the distribution of unknown
dependent variables surrounding the node under consideration. The first, the second, the third, and
the fourth derivatives of a dependent variable with respect to an independent variable evaluated at
this particular node (node i) can be expressed algebraically in terms of the nodal values of the same
dependent variable associated with the five neighboring clustered nodes. The coefficients of these
linear algebraic relationships can be calculated for any values of nodal intervals by solving four
simultaneous linear algebraic equations relating to Taylor's series expansion. As an example,
central-difference relationships for the case with uniform intervals (hi = h2 - h3 = h4 = h) can be
expressed as
108
_xx] = ._L_ (ui. 2 . 8ui_ 1 + 8Ui+l - ui+2)
x=xi 12h (I)
d2u] 1 (-Ui-2 + 16ui.1 30ui + 16ui+l - ui+2)dx2 x--xi'- --12h 2
x-xS- --2h3 (-ui-2 + 2Ui_l - 2Ui+l + ui+2)
[d4u]dx4 x=x,= "L (ui'2 - 4ui-1 + 6ui " 4Ui+l +ui+2)h4
(2)
(3)
(4)
where Ui-2, Ui-1, Ui, Ui+l and Ui+2 are values of u evaluated at xi_ 2, Xi.l, xi, Xi+l and Xi+2,
respectively.
FORMULATION OF THE PROBLEM
Two physical problems are considered to test the higher-order finite-difference technique.
one is the axisymmetric developing incompressible laminar flow in a circular pipe and the other is
the two-dimensional developing incompressible laminar flow between parallel plates. Flow
velocity and the pressure at the inlet region are assumed to be uniform and the no-slip boundary
conditions are imposed on all wall surfaces. By utilizing a switching constant m for both the two-
dimensional problem (m - 0) and the axisymmetric problem (m = 1), governing equations for both
problems in different regions can be expressed as follows:
(A) General Region of Fluid Flow (0 < y* < 1)
(1) Continuity Equation
bbU* +rnv* +bY* =0
bx* y* by* (5)
(2) Momentum Equations
__ bu'=.abbP'+b 262u" ¢. rn bu', 1ebzu_.__*_*bu* bu* + v*
bx* _y* bx* Re _ y'Re by* by .2 (6)
__ bv* -abP*+b 262v * m bv* my* +_] b2v*bu* by* + v* -- = I- -- -
bx* by* by* Re bx,2 y'Re by* y'Re Re by,2 (7)
(B) Centerline Region (y*= 0)
(1)Continuity Equation
b _ + (m+1) bv___*.*= 0by* (8)
(2) Momentum Equations
109
b2 O2u* (l+m) O2u*bu* Ou._.._*= -ab ¢)P" + +
0x* i0x* Re OX*2 Re 0y,2 (9)
V*--0 (10)
(C) Wall-Surface Region (y* = 1)
(1) No-Slip Conditions
U*'-V* --0 (11)
(2) Momentum Equation
ab _P* + a_P* = b2 _2u* /}u"
._+ m
_x* _y* Rc _x*2 y*Rc_y*
. mv____*+ 1 _2v*
y'Re Rc by,2
+ m _v*
y'Rc Oy*
(12)
where
b-L_- ' x*m x YL, Y* ---H,
8
u
PN
a=-- pHU.
pU 2" Re ffi tt
Here, x and y arc either the Cartesian coordinates in the longitudinal and the n'ansvcrse directions
for the parallcl-platc problem or the cylindrical coordinates in the axial and the radial directions for
the circular-pipe problem. The origins of these coordinate systems arc located at the flow inlets.
The x axis is located eithcr at the ccnterlinc of the parallel planes for the two-dimensional problem
or at the center of the pipe for the axisymmctric problem. The flow domain in the x and y directions
are denoted as L and H, respectively. Hence, H represents either the half distance between two
parallel planes or the radius of the circular pipe. The axial velocity and the pressure at the inlet arc
specified as U- and p.., respectively. Velocity components in the x and the y directions as well as
the pressure, me density, and the viscosity of the flow axe denoted as u, v, p, p, and P.,
respectively. The x-momentum equation applied at x = 0 (ccntcrline region) is derived by applying
LTIospital rule to the Navier-Stokes equation. In addition, equation (12) is obtained by combining
both the x and the y components of the momentum equations such that the number of unknowns
can bc identical to the number of equations at each surface node.
Governing equations for the entire domain, equations (5-12), are discretized in the y*
direction by using the present higher-order finite-difference technique to adapt to unevenly spaced
clustered nodes. However, they arc discretized in the x* direction by applying the classical finite-
difference formulation to evenly spaced nodes such that an iterative sweeping technique can bc
used to calculate nonlinear terms. The initial value of a nonlinear term associated with a given node
can bc estimted by using values of terms evaluated at upstream nodes at the same y* location such
that.governing equations can bc solved line by line from the upstream to the downstream by using
the classical backward difference formulation in the x* direction. To reduce the inaccuracy due to
this approximation, results calculated for each line arc iterated to a convergent value before the
110
nextmarchingsequenceof thesweepingprocessis undertaken.To starttheprocess,thefirst two
lines haveto be solvedsimultaneouslyby usingthe centraldifferencefor the first line andthe
backwarddifference for the secondline in the x° direction. For all subsequentsweeping
calculations,only or_elineof nodesareinvolvedin the process.
NUMERICAL RESULTS
The normalizeddomainsfor both the parallel-plateand the circular pipe problems are
discretized by dividing the domain in the x* direction into 30 evenly spaced intervals and the
domain in the y* direction into 20 uneven clustered intervals. These unevenly speced intervals in
the y* direction vary from 0.0025 at the wall to 0.1 at the centerline. Table 1 compares non-
dimensional axial velocities of the circular pipe between those obtained by the present analysis and
those measured by Nikuradse (ref. 7) as well as those calculated by Sparrow et al (ref. 5). Table 2
shows the comparison of non-dimensional longitudinal velocities for the parallel-plate problem
between those calculated by using the present technique and those determined by Sparrow et al as
well as by Bodia and Osterle (refs. 5 and 6). Table 3 summarizes the comparison of the non-
dimensional centerline pressure drop, [2(p** - p)]/(pU_, of the pipe flow between those calculated
by using the present analysis and the Schiller's experimental correlation reported by Prandtl and
Tietjens (ref. 7). Results obtained by using the present technique are shown to be in good
agreement with those determined by existing techniques. The non-dimensional entrance length of
the circular pipe estimated by locating the cross section with its centerline velocity being 99% of the
fully developed centerline velocity is found to be approximately equal to x/(DReD) -0.0593 or
x/(HRe) = 0.237 which is roughly consistent with the value of x/(DReD) = 0.05 suggested by
Kays (ref. 8) or the value of x/(HRe) = 0.26 reported by Prandtl and Tietjens (ref. 7) for pipe
flows.
CONCLUSIONS
A higher-order finite-difference technique which allows the usage of clustered nodes has
been successfully developed. Numerical results obtained in the present analysis have also verified
the reliability of this technique. Therefore, it can be a very useful tool in computational fluid
mechanics because of its accuracy and the need to use unevenly spaced clustered nodes for
modeling fluid flows.
REFERENCES
1. Anderson, D.A.; Tannehill, J.C.; and Pletcher, R.H.: Computational Fluid Mechanics and Heat
Tansfer. Hemisphere, 1984.
2. Bradshaw, P.; Cebeeei, T.; and Whitelaw, J.H.: Engineering Calculation Methods for
Turbulent Flow, Academic Press, 1984.
3. Crank, J.; and Nicholson, P.: A Practical Method for Numerical Evaluation of Solutions of
Partial-Differential Equations of the Heat-Conduction Type. Proc. Cambridge Phil. Soc., vol.
43, 1947, pp. 50-67.
4. Keller, H.B.: A New Difference Scheme for Parabolic Problems. Numerical Solutions of
Partial Differential Equations. (edited by Bramble, J.) Academic Press, 1970.
5. Sparrow, E.M.; Lin, S.H.; and Laudgren, T.S.: Flow Development in the Hydrodynamic
Entrance Region of Tubes and Ducts. Phys. Fluids, vol. 7, no. 3, Mar. 1964, pp. 338-347.
6. Bodoia, J.R.; and Osterle, J.F.: Finite Difference Analysis of Plane Poiseuille and Couette
Flow Developments. Appl. Sci. Res. see. A, vol. 10, 1961, pp. 265-276.
7. Prandtl, L; and Tietjens, O.G.: Applied Hydro- and Aeromechanics. Dover, 1957, p. 25.
8. Kays, W.M.: Convetive Heat and Mass Transfer. McGraw-Hills, 1966, p. 63.
111
(a) Fully Forward Difference
I h_ I h2 I
i i+l i+2
h3 I
i+3
h4
Co) Partially Forward Difference
I h, I h_ I
i- 1 i i+l
I
i+4
(c) Central Difference
[ h4 [h3 [ h, [
i-2 i- I i i+l
h_ I h3 I
i+2 i+3
(d) Partially Backward Difference
I h3 I h_
h2
i-3 i-2 i-1
(e) FullyBackward Difference
I h, I h_ I
i-4 i-3 i-2
h2
hl I h4
I
i+2
I
i+l
[ hi [
i-I i
Figure 1. Higher-order finite-difference nodes
I12
TABLE 1 - Comparison of the non-dimensional axial velocity (u/U**) of pipe flows
0.01
0.02
0.03
0.04
0.05
0.10
0.20
Present Analysis
y/H
0.0 0.4 0.8 0.9
1.34 1.32 0.99 0.63
1.44 1.41 0.92 0.53
1.53 1.48 0.87 0.48
1.60 1.52 0.84 0.46
1.66 1.56 0.81 0.44
1.84 1.63 0.76 0.40
1.97 1.67 0.73 0.38
Nikuradse's Expermental
Measurement (ref.7)
y/H
0.0 0.4 0.8 0.9
1.30 1.32 1.12 0.71
1.43 1.44 0.95 0.58
1.53 1.53 0.89 0.52
1.60 1.58 0.86 0.50
1.66 1.60 0.83 0.48
1.83 1.65 0.77 0.41
1.97 1.67 0.72 0.38
Analysis by Sparrow et al
(ref. 5)
y/H
0.0 0.4 0.8 0.9
1.32 1.31 1.00 0.59
1.44 1.42 0.92 0.51
1.53 1.49 0.87 0.47
1.60 1.53 0.84 0.45
1.66 1.57 0.81 0.44
1.85 1.63 0.76 0.40
1.97 1.67 0.72 0.38
TABLE 2 - Comparison of the non-dimensional axial velocity (u/U..) of flows between parallel plates
X
HRe
0.01
0.02
0.03
0.04
0.05
0.10
0,20
Present Analysis
y/H
0.0 0.5 0.7 0.9
1.20 1.15 1.00 0.50
1.25 1.18 0.96 0.42
1.30 1.18 0.91 0.37
1.33 1.18 0.88 0.35
1.36 1.17 0.86 0.34
1.44 1.15 0.80 0.30
1.49 1.13 0.77 0.29
Finite-Difference Calculation
by Bodia and Osterle (ref. 6)
y/H
0.0 0.5 0.7 0.9
1.16 1.16 1.05 0.50
1.22 1.18 0.99 0.41
1.27 1.18 0.93 0.37
1.31 1.18 0.89 0.35
1.34 1.17 0.86 0.33
1.44 1.14 0.80 0.30
1.49 1.13 0.77 0.29
Calculation by Sparrow et al
(ref. 5)
y/H
0.0 0.5 0.7 0.9
1.18 1.16 1.03 0.48
1.24 1.18 0.96 0.40
1.29 1.18 0.91 0.37
1.33 1.17 0.88 0.35
1.36 1.16 0.86 0.33
1.44 1.14 0.80 0.30
1.49 1.13 0.77 0.29
113
TABLE 3 - Comparisonof thenon-dimensionalcenterlinepressuredrop,
2(p**-p)/pU2**,of pipeflows.
X
HRe
0.01
0.02
0.03
0.04
0.05
0.10
0.12
0.16
Present Analysis
0.89
1.21
1.49
1.74
1.96
2.94
3.30
3.98
Schiller's Correlation
(ref. 7)
0.66
1.03
1.27
1.48
1.80
2.68
3.18
3.70
114


Analysis of developing laminar flows in circular pipes using a higher-order finite-difference technique

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