transfer in fluid flow
2.10 Application of the NIMO Higher-Order Scheme to the Turbulent Flow in Pipes
For constant-density steady turbulent flow in pipes, the modeled time-mean momentum equation can be written as follows:
∂(ρuiuj)
where ui, uj, and p are time-mean values. The turbulent viscosity and turbulent kinetic energy are denoted by µt and k; in addition to time-mean momentum equations, the time-mean mass continuity must be included.
∂(ρui)
∂xi = 0 (76)
The turbulent viscosity can be computed with the help of the two-equation model of turbulence for the kinetic energy of turbulence (k) and its dissipation rate (ε). The governing equation for the kinetic energy of turbulence and its dissipa-tion rate are written here such that they can be applied to the standard k-ε model (Launder & Spalding [14]) and its low Reynolds number version of Launder and Sharma [15].
is the turbulent kinetic energy generation term and is given by Launder and Spalding [14]. The turbulent viscosity µtis thus given as:
µt = Cµfµρk2/ε (79)
The high Reynolds number standard k-ε model is specified with the effec-tive viscosity µe= µt, D= E = 0.0, and f1= f2= fµ= 1.0. Moreover, the high Reynolds number k-ε model logarithmic wall functions that prescribe at the wall the shear stress, the generation term (G), and the value of ε are given by Launder and Spalding [14].
However, the low Reynolds number k-ε model of Launder and Sharma [15]
takes the solution right to the pipe wall. The common constants between the two versions of the k-ε model are given by Launder and Sharma [15] as cµ= 0.09, c1= 1.44, c2= 1.92, σk= 1.0, and σe= 1.3. Moreover, the low Reynolds number additional functions are defined as follows [15]:
fµ= exp [−3.4/(1 + Rt/50)2] (80) where the turbulent Reynolds number Rtis defined as [15]:
Rt= ρk2/(µε) (85)
Equations (83) and (84) are given specifically for the turbulent flow in pipes where r is the radial distance and u is the axial velocity.
In the low Reynolds number model, µeis defined as µ+ µtand the wall bound-ary condition is defined as ∂p/∂r= u = v = k = ε = 0.0. The boundary condition at the centerline is defined as ∂φ/∂r= v = 0, where φ stands for all dependent vari-ables except v. At the exit section, the axial gradients of all dependent varivari-ables are equal to zero. At the inlet section, u= u0and v= ∂p/∂x= 0, where u0is the inlet uniform axial velocity.
The assessment of the accuracy of the results of the NIMO scheme is achieved with the help of the experimental data of Laufer [16] at Re= 500,000 and the semi-empirical relation obtained by Schlichting [17] for a large number of experimental data for the fully developed turbulent pipe flow. The Schlichting 1/7th power law is given as [17]:
u/uCL=
R− r R
1/7
0 < r < 0.96R (86) where uCLis the fully developed centerline axial velocity.
The numerical predictions for turbulent flow in pipes, using the NIMO higher-order scheme, are shown in Figures 2.30–34. Air at 1.0 bar and 300 K enters the pipe at a uniform axial velocity of 29 m/s. The pipe diameter is 24.7 cm and its dimensionless length (L/D) is equal to 30. For this flow, the Reynolds number is 500,000. For the low and high Reynolds number turbulent models, a (100× 100) grid is used. The grid of the high Reynolds number model is uniform, while that for the low Reynolds number model converses toward the pipe wall. For both models the number of iterations approximates 1,000, giving errors less than 0.1 percent in the difference equations.
The developing turbulent flow in pipes is depicted in Figure 2.30 for axial distances (x/D) equal to 2 and 5. The dimensionless axial velocity (u/um) numerical results of the main grid are essentially indistinguishable from those of the x-grid, as can be seen from Figure 2.30. The near-wall region is shown in Figure 2.30b, to facilitate the comparison of the wall effect as predicted by the low and high Reynolds number models. The wall effect of the low Reynolds number model propagates into the core of the turbulent flow in pipes, while the high Reynolds number model has its wall effect restricted to (r/R) > 0.95 for (x/D) < 5. More developing flow numerical results are shown in Figure 2.31 for (x/D) equal to 10 and 20. The high Reynolds number k-ε model wall effect is still confined to (r/R) > 0.75, while the low Reynolds number counterpart has its wall effect all over the turbulent flow for (x/D)= 20. The dimensionless radial velocity (v/um) profiles are depicted in Figure 2.32, at (x/D)= 5 and 20 and at the exit section.
The low Reynolds number model predicts higher absolute radial velocities, while both models predict a vanishing radial velocity at the fully developed flow near the exit section of (x/D)= 30. The fully developed turbulent time-mean axial velocity radial profiles are depicted in Figure 2.33, for the high and low Reynolds number models. On the same figure, the measured fully-developed dimensionless axial
0 0.2 0.4 0.6 0.8 1 1.2
0 0.2 0.4 0.6 0.8 1
Main grid, x /D = 2, lowRe x-shifted, x /D = 2, low Re Main grid, x /D = 5, lowRe x-shifted, x /D = 5, low Re Main grid, x /D = 2, high Re x-shifted, x /D = 2, high Re Main grid, x /D = 5, high Re x-shifted, x /D = 5, high Re
r /R u/um
0 0.2 0.4 0.6 0.8 1 1.2
0.75 0.8 0.85 0.9 0.95 1
r /R
u/um Main grid, x /D = 2, low Re x-shifted, x /D = 2, low Re Main grid, x /D = 5, low Re x-shifted, x /D = 5, low Re Main grid, x /D = 2, high Re x-shifted, x /D = 2, high Re Main grid, x /D = 5, high Re x-shifted, x /D = 5, high Re
(a)
(b)
Figure 2.30. Dimensionless axial velocity (u/um) at different axial locations for turbulent flow in pipes with Re= 500,000. (a) r/R from 0 to 1.
(b) r/R from 0.75 to 1.
velocity [16] and the similar profile that can be calculated by the semi-empirical relation of Schlichting [17] are also plotted. The NIMO scheme predicts excellent agreement with the fully developed axial velocity profile given by the 1/7th power law of Schlichting [17] when the low Reynolds number k-ε model is used. However, the present numerical results for u/uCLfall nicely between the experimental data of Laufer [16] and the Schlichting’s 1/7th power law [17]. On the other hand, the high Reynolds number model poorly predicts the fully developed turbulent pipe flow and hence should not be adopted for this type of flow. Finally, the pressure axial distributions, predicted using the low and high Reynolds number k-ε models, are shown in Figure 2.34. This figure shows clearly that the absolute axial pressure
0 0.2 0.4 0.6 0.8 1 1.2
0 0.2 0.4 0.6 0.8 1
r /R
u/um Main grid, x/D = 10, low Re x-shifted, x/D = 10, low Re Main grid, x/D = 20, low Re x-shifted, x/D = 20, low Re Main grid, x/D = 10, high Re x-shifted, x/D = 10, high Re Main grid, x/D = 20, high Re x-shifted, x/D = 20, high Re
Figure 2.31. Dimensionless axial velocity at different axial locations for turbulent flow in pipes, Re= 500,000.
−0.004
−0.003
−0.002
−0.001 0.000
0 0.2 0.4 0.6 0.8 1
r/R v/um Main grid, x/D = 5, low Re
x-shifted, x/D = 5, low Re Main grid, x/D = 20, low Re x-shifted, x/D = 20, low Re Main grid, exit, low Re x-shifted, exit, low Re Main grid, x/D = 5, high Re x-shifted, x/D = 5, high Re Main grid, x/D = 20, high Re x-shifted, x/D = 20, high Re Main grid, exit, high Re x-shifted, exit, high Re
Figure 2.32. Dimensionless radial velocity profiles at different axial locations for turbulent pipe flows, Re= 500,000.
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
r/R u/uCL
Numerical data, low Re Numerical data, high Re Power law
Experimental data
Figure 2.33. Dimensionless time-mean axial velocity for turbulent pipe flows using low and high Reynolds number models, Re= 500,000.
−300
−250
−200
−150
−100
−50 0
0 10 20 30
x/D P − P0 (N/m2)
Low Re High Re
Figure 2.34. Centerline axial pressure profiles for turbulent flow in pipes, Re= 500,000.
gradient of the low Reynolds number model is approximately five times higher than that of the high Reynolds number model. Alternatively, the wall shear stress in the low Reynolds number model is five times that of the high Reynolds number k-ε model.
2.11 Conclusions
This chapter describes a novel numerical procedure–the NIMO finite-difference procedure. The NIMO numerical procedure is free from any interpolations of con-vective fluxes that normally produce high levels of false diffusion in most known finite-difference schemes. The NIMO system involves four grids in 2D flows. The four grids are located in space such that each grid is displaced midway with respect to one of the remaining grids. The main grid and the xy-grid interconnect indi-rectly through the x- and y-grids and vice versa. Similar interconnection between the four NIMO grids is valid in the cylindrical-polar solution domain. The percent-age sum of the absolute residual errors of the main grid and xy-grid behaves in the same fashion, which is different from the similar trend of the other two grids.
The NIMO scheme is applied to two test problems. The first test problem is the oblique flow in the Cartesian coordinates, while the second test problem is repre-sented by a rotating radial flow in an annular sector. The NIMO finite-difference scheme can produce numerical profiles of φ that capture the analytical solution of the test problem for 45-degree and 30-degree oblique flows for high Peclet num-bers. Moreover, NIMO can essentially capture the analytical solution of the two test problems for square, rectangular, and nonuniform grids. The second test prob-lem is also computed with a very high accuracy, which confirms the validity of the NIMO scheme to the cylindrical-polar coordinates as well as to the Cartesian coordinates. Most of the single-grid numerical procedures utilizing rectangular or nonuniform grids could not reach that level of accuracy due to false diffusion, overshooting, or undershooting. This could be concluded from the comparison of the data of these schemes with the corresponding present numerical results. The NIMO finite-difference equations contain convective terms that slightly resemble the upwind schemes. However, unlike the upwind scheme, the incoming convective fluxes to a particular grid are supplied by two of the remaining three grids of NIMO.
A new feature emerges that allows the outgoing fluxes to leave behind positive or negative traces in a particular grid control volume, which is completely ignored in the upwind scheme. These outgoing convective fluxes constitute the incoming convective fluxes to two of the remaining three grids of NIMO. Results obtained for different values of the Peclet number showed that the diagonal core of the test prob-lem is surrounded by narrow strips bounded by φ= 0.99 and 10−2. These narrow strips become essentially nonexistent as the Peclet number is increased to 1,000. The converged numerical solutions of the four NIMO grids for a passive scalar variable at a fixed value of the Peclet number are consistent and identical. The NIMO higher-order scheme has further been tested against laminar and turbulent flow in pipes as well as recirculating flow behind a fence. The laminar flow in pipes is predicted with a very high accuracy for a (100× 100) grid. However, the recirculating laminar flow behind a fence can be predicted accurately with a finer (150× 150) grid. The turbulent flow in pipes is well predicted when the low Reynolds number k-e model is used.
Nomenclatures
A surface area of control volume a finite-difference coefficients c convective transport rates
ˆc, ˜c modified convective transport rates
cµ, c1, c2, σk, σe common coefficients in high and low Reynolds number models
D, E, f1, f2, fµ coefficients in low Reynolds number model
d diffusive transport rate
k turbulent kinetic energy
n number of iterations
Pe Peclet number
p pressure correction
Rt turbulent Reynolds number
Su, Sp source term coefficients in linear form u velocity component along x direction uCL fully developed centerline axial velocity uk fluid velocity along coordinate direction xi
v velocity component along y direction
xk coordinate direction
δx distance between central node and neighboring node ε dissipation rate of turbulent kinetic energy
φ scalar variable
φ molecular diffusivity of φ
µe effective viscosity
µt turbulent viscosity
ρ density
Superscript
E, W east and west directions
p value at center of control volume S, N south and north directions t turbulent flow
i, j grid node indices Superscript
x x-grid y y-grid xy xy-grid
* uncorrected values
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