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Direct Numerical Simulation of Open Channel Flow with Buoyancy.

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Dev. Chem. Eng. Mineral Process., I1 (5/6), pp. 465-479, 2003.
Direct Numerical Simulation of Open
Channel Flow with Buoyancy
Itsuro Hondal, Naotaka Kanagawa and Yosuke
Dept. of Chemical Engineering, Himeji Institute of Technology,
21 67 Syosha, Himeji, Hyogo, Japan
Free surface flow is important in chemical reactions, heat exchange,
atmospheric science and industrial technology. In terms of fundamental
research, turbulence near a free surface is important in transport
phenomenon. However, free surface effects exhibit some nonlinear behavior,
so that numerical methods are usually employed instead of analytical
approaches. In this investigation, direct numerical simulation is applied to
turbulent channel flow with a free surface. The Navier-Stokes equations are
solved using a finite difference method, and an interpolation scheme is applied
to the convective term. Grid resolution is 128x65~128and Reynolds number
based on the friction velocity and channel elevation is 150. Slip wall
approximations have been used extensively. This means that the free surface
has no stress, velocity or deformation. In order to solve such problems, low
Froude number approximations have recently been presented and these are
used to calculate the free surface flow. In turbulent flow, dissipation and
production are essential qualities, Reynolds-stresses balances and pressurestrain correlations are examined numerically. It is shown that the main
difference between free surface and slip wall approximations is in pressurestrain correlations near the surface.
*Authorfor correspondence (pitot@mech.
I. Honda,N.Kanagawa and Y Kawashima
Free surface flows are important in thermal science, mechanical engineering
and industrial technology. Most of these flows are turbulent, and have surface
waves that play an important role in the interaction of materials across the
surface by means of turbulent air-water interfaces and wind-wave interactions.
For vertical flows near a free surface, an important interaction mechanism is
when vortex lines disconnect and subsequently terminate at the free surface. In
terms of fundamental research, turbulence near a free surface is important in
transport phenomena. However, current knowledge of transport mechanisms
near free surfaces is neither complete nor satisfactory. This is because
measurement of fluctuations of turbulent quantities in a layer near a free
surface is very difficult. In a various equipment, control of thermo-fluid
phenomena near free surfaces is crucial in maintaining high quality operation.
Since free surface effects are dominated by nonlinear behavior, numerical
methods are usually employed instead of analytical approaches.
In the early 1980s, advances in computer architecture made direct
numerical simulation [ 13 of turbulent flows possible. Many numerical studies
have investigated wall turbulence. Moin and Kim [ 11 calculated turbulent
channel flow by large eddy simulations. In their calculations, 64 and 128
uniform spacing grid points were used in the streamwise and spanwise
directions respectively, while 63 non-uniform spacing grid points were used in
the direction normal to the wall. Most experimentally observed wall-layer
structures were reproduced using this setup.
Kim et al. [2] calculated the channel flow by direct numerical simulation
using 192x129~160grid points. Their results have been used to clearly explain
wall turbulent phenomena and reproduce statistical turbulent properties with a
high degree of accuracy. Lam and Banerjee [3] treated free surfaces as a flat
slip wall, giving a Froude number of zero. This method has been widely used.
However, the lack of variations in the surface shape precludes accurate
computation of turbulence statistics. To address this problem, Komori et al.
[4] considered variations in the free surface height, giving a non-zero Froude
number. They applied an upwind scheme to the convective term in the NavierStokes equations. However, adding numerical viscosity may have a significant
impact on turbulence statistics. Nagaosa [5] applied a consistent scheme
developed by Kajishima to the convective term to avoid this effect, but also
employed the slip wall approximation.
Direct Numerical Simulation of Open Channel Flow with Buoyancy
In this paper we study open channel flow with buoyancy using numerical
techniques and the nonzero Froude number approximation proposed by Borue
et al. [ 6 ] .The purpose of this investigation is to find the effects of buoyancy
and Froude number on statistical characteristics of free surface flow.
Therefore, we perform direct numerical simulations employing a consistent
scheme and allowing free surface fluctuations.
Numerical Method
Channel flow can be treated as an incompressible Newtonian fluid. Governing
equations are the equation of continuity and Navier-Stokes equations with
Boussinesq approximation:
O = 'au.
The equations are shown in non-dimensional form. Variables in these
equations are normalized to H , u , and A T . Cartesian axis, being either
streamwise, normal to wall or spanwise while; S is Kronecker's delta. Nondimensional parameters in the above equations are defined as follows:
The flow geometry and coordinate system of the channel is shown in
Figure1 . The two walls are kept at different but constant temperatures, without
fluctuations. The Reynolds number ( R e ) was set to 150. The Prandtl number
(Pr ) was assumed to be 1, and the Richardson number ( Ri ) was set to 10.
This value for the Prandtl number is much smaller than that for water. If the
Prandtl number is larger than 1, then a large number of grid points are
necessary for DNS computation. It is thought that Prandtl number effects are
not significant in large-scale motion.
I. Honda, N. Kanagawa and I! Kawashima
Figure 1. Low Froude number flow.
Spatial terms excluding convective terms are approximated by a secondorder central finite difference scheme. Convective terms in the Navier-Stokes
and scalar transport equations are discretized using the second-order
interpolation scheme proposed by Kajishima [7] on a staggered grid:
where Sxi denotes central difference and
denotes mean value.
Since this scheme accurately reproduces the continuity equation, results are
in good agreement with those of the spectral method [B]. For integration over
time, second-order Adams-Bashforth and Crank-Nicolson methods are applied
to the nonlinear and viscous terms respectively [9].
Direct Numerical Simulation of Open Channel Flow with Buoyancy
pn+l= P“
+ Q - -AttS,,
2 Re
s,, Q
In the above equations, Q is a scalar potential and Ai is a convective term.
The dynamic boundary conditions on the free surface are:
2 du,
Re ax2
Coupling Algorithm
Time Advancement
Spatial Scheme
Boundary Conditions
Boundary Conditions
Staggered Grid
Fractional Step Method
2nd-OrderAdams Bashforth Method
(Convective Term)
2nd-OrderCrank Nicolson Method
(Viscous Term)
2nd-OrderConsistent Scheme
(Convective Term)
2nd-OrderCentral Scheme
(Other Terms)
Periodic ( x , ,x , direction)
Slip Wall ( x2direction, x2 = H )
Zero Froude ( x2direction, x 2 = H )
Low Froude ( x2 direction, x2 = H )
x2 = O / T = O , x , = l / T = l
x, =OIT=l,x, = 1 / T = O
Fr is the Froude number, defined as Fr = u,/(gh)”’, and h is the height of
the free surface above the mean height. Equation ( 1 1 ) represents a type of
hyperbolic. Since the equations need to be solved to a high degree of accuracy,
a third-order Adams-Bashforth method with sixth-order central discretization
I. Honda, N. Kanagawa and X Kawashima
is applied to spatial derivatives in this study. The case of a slip-wall
approximation is the Fr = 0 limit of the present problem.
Boundary conditions are shown in Tablel, a fully developed flow field was
assumed so that periodic boundary conditions could be applied in both
streamwise and spanwise directions.
Time increments in this investigation were O . O 7 5 ~ l ( u , ) ~Results
were from
time steps)
ensemble averages over space and time taken 7 5 0 0 v l ( ~ , )(100,000
after the flow fields were judged to have reached a fully developed state.
Computations were carried out on a personal computer with Pentium 4 CPU,
which took approximately 4 seconds of CPU time per time-step.
Results and Discussion
First, channel flow was computed by applying the non-zero Froude number
approximation without buoyancy. Figure 2 shows mean flow velocity as a
function of vertical position compared with Nagaosa’s data, which was
computed using slip-wall approximations. Figure 3 shows root mean square
(r.m.s.) fluctuations in the three velocity components (u, v and w represent
velocities in the streamwise xl, vertical x2 and crosswise x3 directions,
respectively) as a function of vertical position. The results are in good
agreement with Nagaosa’s data near the wall, but show differences near the
free surface arising from the different boundary conditions. The vertical
component of turbulent intensity at the free surface is zero because the
boundary condition forces the vertical component of velocity (v) to be zero at
the surface in both results.
Figure 4 shows the flatness factor as a function of vertical position in the
flow. A flatness factor of 3 means that velocity fluctuations are distributed
normally. Factors greater than 3 are indicative of highly intermittent
phenomena. Flatness factors of the vertical component of velocity near the
free surface were larger than those found by Nagaosa. These large flatness
factors seem to be the result of highly intermittent phenomena near the free
surface which are allowed by the non-zero Froude number approximation.
Flatness of the vertical component was found to decrease rapidly near the free
surface, but in slip-wall calculations remained large at the free surface.
4 70
Direct Numerical Simulation of Open Channel Flow with Buoyancy
Figure 2. Mean velocity.
Figure 3. Turbulent intensity.
I, Honda, N.Kanagawa and I! Kawashima
Figure 4. Flatness factors.
Figure 5 shows skewness of the flow. Nagaosa found large positive
skewness at the free surface in slip-wall calculations, indicating large vertical
velocities. However, skewness of vertical velocity in non-zero Froude number
calculations decreased sharply near the free surface, indicating that the
distribution of vertical velocity fluctuations becomes the normal distribution at
the free surface. Velocity fluctuations found at the surface in non-zero Froude
number calculations are lower than in slip-wall calculations, because the shape
of the free surface is allowed to change. Vertical skewness factors confirm
highly intermittent phenomena in the slip-wall case. These results seem to
indicate that streamwise vortices organize coherent structures near the surface.
These vortices are transported from the lower wall.
Figure 6 shows pressure-strain correlations. The streamwise component
was negative throughout the channel, but the other two components were
positive, indicating that streamwise turbulent energy is redistributed into
vertical and spanwise turbulence through pressure-strain effects, with more
energy transferred to spanwise turbulence than to vertical turbulence. While
the results of non-zero Froude number calculations agree with slip-wall
calculations near the free surface, vertical components differ at the surface.
4 72
Direct Numerical Simulation of Open Channel Flow with Buoyancy
Figure 5. Skewness.
t t
Nagaosa Present
4 33
Figure 6. Pressure-Strain correlation.
4 73
I. Honda. N. Kanagawa and L Kawashima
Figure 7. Mean velocity on heated plane.
Figure 8. Turbulent intensity.
4 74
50 100
Direct Numerical Simulation of Open Channel Flow with Buoyancy
l o A, I, ,
Figure 9. Flatness on the heated plane.
We concluded that free surface deformation causes a decrease in energy of
the vertical component of pressure-strain. Figure 7 shown mean velocity
profiles at two different boundary temperature conditions, lower surface
heating and upper surface heating, as compared to Nagaosa's results. It should
be noted that the velocity profile for a heated lower surface is much larger
than for a heated upper surface. The velocity profile heated under the bottom
wall is smaller in the region from x; = 55 to 150 than Nagaosa's result. In the
case of a heated lower surface, the friction velocity at that surface does not
change. It is clear that the turbulent boundary layer becomes relaminarized.
Figure 8 shows r.m.s. velocity fluctuation components for the two different
surface heating schemes compared to Nagaosa's results. Components
calculated across the flow in the region from x2+ = 90 to the free surface are
greater than that in slip-wall calculations. It is thought that longitudinal
vortices become stronger with respect to surface splatting. This tendency is
also seen in flatness and skewness, as shown in Figure 9 and 10 respectively.
Vertical -components of flatness decrease sharply near the surface, but are
larger between x2+ = 100 and the surface.
4 75
I. Honda. N Kanagawa and X Kawashima
Figure 10. Skewness on the heatedplane.
Streamwise and vertical components of skewness in the heated lower wall
case are very different from those of the slip-wall case. Calculated streamwise
skewness was smaller than in slip-wall calculations, suggesting that velocity
fluctuations are lower. It is thought that a low velocity core will break up the
wall by buoyancy. The v component is larger than the Nagaosa result in the
region from x2+ = 80 to the free surface. This is because the energy of
buoyancy is transferred from the energy of u component to the v component.
Figure 11 shows pressure-strain correlations. Calculated pressure-strain
correlation components $ 1 1and $22 were smaller than those of channel flow.
For channel flow, $ 1 1 9 2 2 are zero at the free surface, meaning that no energy
redistributions exist at the surface. Streamwise pressure-strain correlation is
negative overall, indicating that streamwise energy is redistributed to vertical
and spanwise components. Vertical component $22 decreases rapidly near the
free surface, with its sign changing from positive to negative, while spanwise
component $33 increases near the surface. These distributions suggest that
vertical turbulent energy is transferred into spanwise turbulence. Surface
deformation derived from the pressure gradient hastens the redistribution of
4 76
Direct Numerical Simulation of Open Channel Flow with Buoyancy
a -
Nagaosa Present
, ,
- - -I - -
" - I
, , , ,
Figure 11. Pressure-Strain correlation.
We have performed a series of numerical simulations of channel flow using
slip-wall and low Froude number approximations. The relation between the
turbulent structures and energy redistributions in open channel flows with
buoyancy were investigated.
Vertical velocity fluctuations of slip-wall calculations are lower than those
found using low Froude number approximations at the free surface, while
vertical flatness factors computed by low Froude number calculations are
smaller than those for slip-wall.
Simulations of two boundary temperature conditions were performed.
These indicate that the flow is relaminarized by free surface heating. In the
case of a heated lower wall, longitudinal vortices seem to become strong with
respect to surface splatting, while energy redistributions become larger near
the free surface. Furthermore, it appears from numerical results that a very
thin layer near the free surface prevents fluctuations from increasing in the
non-zero Froude number calculations. These results appear to be necessary to
explain surface energy redistribution mechanisms.
I. Honda, N. Kanagowa and L Kawashima
Courant number
Froude number
gravitational acceleration
free surface height
reference length
space index
Prandtl number
Reynolds number
Richardson number
time increment
xI(x)component of velocity
wall friction velocity
x2 (y)component of velocity
xj (w) component of velocity
velocity vector
Cartesian reference
length divisions
thermal diffusion
bulk modulus of elasticity
Kronecker’s delta
scalar potential
kinematic viscosity
shear stress
[ l ] Moin, P. and Kim, J. 1982. Numerical investigation of turbulent channel flow, J. Fluid
Mech.. 118, 341-377.
[2] Kim, J. et a]. 1978. Turbulent statistics in fully developed channel flow at low Reynolds
number, J. Fluid Mech., 117, 133-166.
[3] Lam. K. and Banerjee, S. 1992. On the condition of streak formulation in bounded
turbulent flow, Phys. Fluids, 4(2), 306-320.
Komori, S. et al. 1993. Direct numerical simulation of three-dimensional open-channel
flow with zero-shear gas-liquid interface, Phys. Fluids, 5(1). 115-125.
4 78
Direct Numerical Simulation of Open Channel Flow with Buoyancy
[ 5 ] Nagaosa,
R. and Saito, T. 1997. Turbulence Structure and Scalar Transfer in Stratified
Free-Surface Flows, AICnE J., 43(10), 2393-2404.
(61 Borue, V. et al. 1995. Interaction of surface waves with turbulence: direct numerical
simulations of turbulent open-channel flow, J. Fluid Mech., 286, 1-23.
[7] Kajishima. T., 1994. Conservation Properties of Finite Difference Method for Convection
(in Japanese), Trans. JSME, 60(574), 2058-2063.
[S] Honda, I. et al. 2002, A Study on Turbulent Channel Flow With Buoyancy, Proceeding of
JSME, 24-1.31 (In Japanese)
[9] Kim, J. and Moin, P. 1985. Application of a fractional-step method to incompressible
Navier-Stokes equations, J. Comput. Phys., 59,308-323.
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