M. Valero et al. (Eds.): ISHPC 2000, LNCS 1940, pp. 502-513, 2000. Springer-Verlag Berlin Heidelberg 2000

Direct Numerical Simulation of Coherent Structure inTurbulent Open-Channel Flows with Heat Transfer

Yoshinobu Yamamoto, Tomoaki Kunugi, and Akimi Serizawa

Department of nuclear engineering, Kyoto University,Yoshida, Sakyo, Kyoto, Japan

{yyama,kunugi,serizawa}@nucleng.kyoto-u.ac.jp

Abstract. In the present study, turbulent heat transfer in open-channelflows has been numerically investigated by means of a DirectNumerical Simulations (DNSs) with a constant temperature at both freesurface and bottom wall. The DNSs were conducted for two Prandtlnumber, 1.0 and 5.0 with a neutral (i.e., zero gravity) or stablestratification (Richardson number; 27.6), while a Reynolds number of200, based on the friction velocity and flow depth. As the results, thecoherent turbulent structures of fluid motion and thermal mixing, andthe influence of Pr change for heat transfer, buoyancy effect forturbulent structures and hear transfer, and relationship among them, arerevealed and discussed.

1 Introduction

Free surface turbulent flows are very often found in the industrial devices such as anuclear fusion reactor and a chemical plant, not to speak of those in river and ocean.Therefore, to investigate the turbulent structures near free surface is very important tounderstand the heat and mass transport phenomena across the free surface. From likethis viewpoint, some DNSs with scalar transport including a buoyancy effect werecarried out [1], [2]. The interesting information about the relationship betweenturbulent motion, so-called "surface renewal vortex" [3], and the scalar transport, andalso the interaction between buoyancy and turbulence were obtained. However, inthese studies, molecular diffusivity of scalar was comparable to that of momentum, itis questionable whether these results can be used for higher Prandtl or Schmidtnumber fluid flows. The accuracy of experimental or DNS database near free-surfacehas not been enough to make any turbulence model at the free-surface boundary untilnow, especially considered the effects of the Prandtl number of fluids, the buoyancyand the surface deformation on the flow, heat and mass transfer.The aims of this study are to clarify the buoyancy effect on the turbulent structureunder various Prandtl number conditions and to reveal the turbulent heart transfermechanism in open-channel flows, via DNS.

Direct Numerical Simulation 503

2 Numerical Procedure

2.1 Governing Equations

Governing equations are incompressible Navier-Stokes equations with the Boussinesqapproximation, the continuity equation and the energy equation:

jj

i

ii

j

ij

ixx

uPx

gTxu

ut

u∂∂

∂+

∂∂

−∆=∂∂

+∂∂ *2*

2*

**

*ν

ρδθβ , (1)

0*

=i

ixu

∂∂ , (2)

jjj

j xxxu

t ∂∂∂

=∂∂

+∂

∂ *2**

* θαθθ . (3)

where ui* is i th-component of velocity (i=1, 2, 3), x1(x) is a streamwise direction,

x2(y) is a vertical direction, x3(z) is a spanwise direction, t is time, β is the thermalcoefficient of volumetric expansion, a super script * denotes the instantaneous value,g is the gravitational force, p* is the pressure, ρ is the fluid density, ν is the kineticviscosity, and the dimensionless temperature is defined as TTT wall ∆−= /)( **θ ,temperature difference is defined as wallsurface TTT −=∆ , Tsurface denotes free surfacetemperature, Twall denotes wall temperature, α is the thermal diffusivity, respectively.

2.2 Numerical Method and Boundary Condition

Numerical integration of the governing equations is based on a fractional stepmethod [4] and time integration is a second order Adams-Bashforth scheme. A secondorder central differencing scheme [5], [6] is adapted for the spatial discretization. Thecomputational domain and coordinate system are shown in Fig. 1.

As the boundary conditions for fluid motion, free-slip condition at the free surface,no-slip condition at the bottom wall and the cyclic conditions in the stream- and thespanwise- directions are imposed, respectively. As for the equation of energy,temperatures at the free surface and the bottom wall are kept constant (Tsurface > Twall).

2.3 Numerical Method and Boundary Condition

Numerical conditions are tabled in Table 1, where νττ /huR = is a turbulentReynolds number based on a friction velocity of the neutral stratification and the flowdepth h, and Ri= 2/ τβ uhTg ∆ is a Richardson number. The computations were carried

out for about 2000 non-dimensional time units ( ντ /2tu ) and all statistical values werecalculated by time and spatial averages over horizontal planes (homogeneousdirections), after flows reached to a fully developed one. However, in case of thestable stratification (Ri=27.6) for Pr=1.0, a laminarization of the flow was appeared,

504 Yoshinobu Yamamoto et al.

so the computation was stopped at the 1200 non-dimensional time units from theinitial turbulent condition of a fully-developed neutral stratification case for thepassive scalar. All quantities normalized by the friction velocity of the neutralstratification, kinetic viscosity and the mean heat flux at the free surface, are denotedby the super script +.

3 Results and Discussion

3.1 Coherent Turbulent Structures in Open-Channel Flow

Figures 2-4 show the visualization of coherent turbulent structures in case of theneutral stratification. Figure 2 shows the iso-surface representation of a secondinvariant velocity gradient tensor Q+= )//(2/1 **

ijji xuxu ∂∂⋅∂∂ . The iso-surfaceregions are corresponding to the strong vorticity containing regions. Near the bottomwall, the streamwise vortex stretched out the streamwise direction can be seen. Thisindicates that turbulence is generated near the wall. However, the free surface has nocontribution to the turbulence generation in open-channel flows at low Reynoldsnumber.

Figure 3-(b) shows a top view of fluid markers being generated along a line to thez-axis at y+=12.35. Alternating high and low speed regions can be seen near thebottom wall. In these like, turbulence structure near the wall is as well as the wallturbulence of ordinary turbulent channel flows. Figure 3-(a) shows a side view offluid markers being generated along a line to the y-axis at z+=270. As well as the wallturbulence, the lift-up of low-speed streaks, so-called the "burst", are depicted.However, in open-channel flows, if this burst reaches to the free surface, a typicalturbulence structure affected by the free surface could be appeared underneath thefree surface. Near the free surface, the effect of the velocity gradient on the turbulentstructure is reduced by a very large horizontal vortex as shown in Fig. 3-(c) as well asthe effect of the flow depth scale. This horizontal vortex impinges onto the freesurface and turns toward the wall. This motion is in good agreement with the flowvisualization experiment [7], i.e., it may correspond to the "surface renewal vortex." Itis also consistent with turbulent statistic results of neutral stratification [2], [3], [9].The mean velocity, the turbulent statistics and the budget of Reynolds stresses arepublished in elsewhere [9].

Figure 4 shows an instantaneous temperature field. Since the lifted-up cold fluidsnear the bottom wall and the down-drafted warm fluids near the free surface causedby this vortex motion are observed, a thermal mixing between these motions has beenconducted. These typical fluid motions could be the main reason of heat transferenhancement in the turbulent open-channel flows despite the neutral or stablestratification.

Direct Numerical Simulation 505

3.2 Statistics of Turbulence

Mean velocity profiles are shown in Fig. 5. In the stable stratification and high Pr(=5.0) case, the flow laminarization was clearly observed near the free surface while itis no difference from the neutral case near wall region. On the other hand, in thestable stratification and low Pr (=1.0) case, the turbulence throughout the flow cannotbe maintained.

The turbulent intensity profiles are shown in Fig. 6. Near the free surface, allcomponents of turbulent intensity are constrained by the stable stratification.Reynolds-stress profiles are shown in Fig. 7. There is a slightly negative value nearthe free surface in case of the stable stratification.

Mean temperature profiles are shown in Fig. 8. The mean temperature gradient forthe stable case (Pr=5.0, dotted line) is compared with the neutral case (Pr=5.0, solidline). This might indicate that a local heat transfer for the stable case may bepromoted by the buoyancy effect. However, a bulk mean temperature of the stablecase (Pr=5.0) is the lowest of all cases, total heart transfer itself seems to beconstrained by the stable stratification.

Figure 9 shows the scalar flux and fluctuation profiles in case of neutralstratification. It can be seen that the scalar fluctuation is produced by the meanvelocity gradient near the wall, and the mean scalar gradient near the free surface.However, these profiles are distinct from each other. Especially, in the neutral case(Pr=5.0), the turbulent scalar statistics amount to maximums at near free surfacewhere typical turbulence structures are existent. These may suggest that if the Prandtlnumber is higher, the heat transfer is enhanced by turbulent structures near the freesurface.

Wall-normal turbulent scalar flux profiles are shown in Fig. 10. In the neutral andlower Pr (=1.0) case, the profile of turbulent heat flux is almost symmetry, and in theneutral and higher Pr (=5.0) case, the profile leans toward the bottom wall. In thestable case (Pr=5.0), it leans toward the free surface caused by the buoyancy effect.

A scale difference between the neutral and the stable cases may be concerned witha normalization method based on the friction velocity of the neutral stratification, etc.Figures 11 and 12 show the budgets of the Reynolds shear stress and turbulent kineticenergy in the stable case (Pr=5.0). As for the Reynolds stress as shown in Fig. 11, abuoyancy production (solid line) is actively conducted near the wall. In the turbulentkinetic energy as shown in Fig. 12, a stable stratification does not affect the turbulentenergy budget near the wall. It is shown the reason why the momentum boundarylayer thickness is thinner than that of the thermal boundary layer caused by the abovelocal heat transfer mechanism. These are consisting with the results of mean velocityand scalar profiles.

In the neutral case, instantaneous turbulent temperature fields near the free surfaceare shown in Fig. 13. A scalar field is transferred with the fluid motions, so-called a"surface renewal vortex." However, in case of Pr=5.0, the filamentous hightemperature fragments are kept because the time scale of the fluid motion is so fastcompared with the thermal diffusion time scale. This filamentous structure might beclosely concerned with the local heat transfer and Counter-Gradient Flux (CGF) [10]as shown in Fig. 14. This indicates that we have to pay attention whether theBoussinesq approximation for high Prandtl or Schmidt number fluids can be assumed.

506 Yoshinobu Yamamoto et al.

4 Conclusions

In this study, Direct Numerical Simulations of two-dimensional fully developedturbulent open-channel flows were performed. The main results can be summarized asfollows:

(1) According to the flow visualization, Near the free surface, a large horizontalvortex as well as the flow depth scale affected by the presence of free surface isenhanced the heat transfer.

(2) If the Prandtl number is higher, turbulent structures near the free surface greatlyimpact on the scalar transport and the Reynolds analogy between a momentum and ascalar transports could not be applied in near free surface region. The reason is thatthe filamentous high temperature fragments are kept because the time scale of thefluid motion is so fast compared with the heat diffusion time scale.

(3) By the stable stratification effect, if the Prandtl number is lower, the flow couldnot maintain the turbulence and be impacted on the turbulence structures near the freesurface.

(4) By the buoyancy effect, the wall-normal turbulent scalar flux in the stable caseis locally enhanced near the wall and its statistical scalar profile is the opposite one ofthe neutral stratification case. Eventually, the total heat transfer itself was constrained.

Acknowledgments

This work was supported by Japan Science and Technology Corporation (JST) andJapan Atomic Energy Research Institute, Kansai-Establishment (JAERI).

References

1. Nagaosa and Saito, AIChE J., vol. 43, 1997.2. Handler et al, Physics of Fluids, vol. 11, No.9, pp.2607-2625, 1999.3. Komori et al., Int. J. Heat Mass Transfer, 25, pp.513-522, 1982.4. A. J. Chorin, Math. Comp., vol. 22, pp.745-762, 1968.5. Kawamura, The Recent Developments in Turbulence Research, pp.54-60, 1995.6. Kajishima, Trans. JSME, Ser. B, vol. 60, 574, pp.2058-2063, 1994 (In Japanese).7. Banerjee, Appl. Mech, Rev., vol.47, No. 6, Part 2, 1994.8. Handler et al., AIAA J., vol.31, pp.1998-2007, 1993.9. Yamamoto et al., Proc. of 8th European Turbulence conference, pp.231-234,

2000.10. Komori et al., J. Fluid Mech., vol. 130, pp.13-26, 1983.

Direct Numerical Simulation 507

Table 1 Numerical condition

τR Grid Number(x, y, z)

Resolution( , , )∆ ∆ ∆x y z+ + + Pr Ri

200 128,108,128 10, 0.5-4, 5 1.0 -200 128,108,128 10,0.5-4, 5 1.0 27.6200 256,131,256 5, 0.5-2, 2.5 5.0 -200 256,131,256 5, 0.5-2, 2.5 5.0 27.6

Lx+=12

80

Ly +=200

Flow

FLOW x, U+u

y, v

z, w

Fig.1 Computational domain and coordinate system

� Æ*=1 at y=h

� Æ*=0 at y=0

U : Mean velocity

h 6.4h

3.2h

g

Fig. 2 Surfaces of second invariant velocity gradient tensor Q+=0.03

(a) Side view

508 Yoshinobu Yamamoto et al.

Lz +=640

Flow

Lz+=64

0

Ly+=20

0

Flow

Lx+=12

80

Lx+=12

80

(b) Top view

(c) Bird view

Fig. 2 (continue) Surfaces of second invariant velocity gradient tensorQ+=0.03

Direct Numerical Simulation 509

0 400 800 12000

100

200

x+

y+

0

0

200

400

600

z+

0

0

200

400

600

z+

(a) Fluid markers are generated along a line to the y-axis

400 800 1200x+

400 800 1200x+

(c) Fluid markers are generated along a line to the z-axis Near free-surface, y+=194.1 (Top view)

Fig. 3 Visualization of coherent structures

(b) Fluid markers are generated along a line to the z-axis Near bottom wall, y+=12.35 (Top view)

510 Yoshinobu Yamamoto et al.

Lx+=1

Lx+=

Lx+=1

Lx+=1

(a) Side view 0.0(Black)<θ*<1.0(White)

(b) Top view 0.1(Black)<θ*<0.9(White) y

(c) Top view 0.7(Black)<θ*<1.0(White) y

Fig.4 Instantaneous sc

Surfa

Surface renewal vortex

280

Lz +=640

1280

Ly +=200

280

Lz +=640

280

z+=270

+=12.35 Near bottom wall

+=196�Near free-surface

alar fields Pr=1.0

ce renewal vortex region

Direct Numerical Simulation 511

0.1 1 10 1000

5

10

15

20

25

y+

U+

Fig.5 Mean velocity profiles

Neutral Stable(Pr=5.0)

Stable(Pr=1.0)

U+=1/0.41*ln(y+)+5.5

U+=y+

0 50 100 150 2000

1

2

3

u+ rms

, v+ rm

s ,w

+ rms

y+

Neutral (Pr=5.0)

Fig.6 Turbulent intensity profiles

Stable (Pr=5.0)u+

rms

v+rms

w+rms

0 100 2000

0.20.40.60.8

11.2

y+

- u+v+ Stable (Pr=5.0) - u+v+ Neutral (Pr=5.0)

Total shear stress (Neutral)

Shea

r stre

ss

Fig.7 Shear stress profiles

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

y+

ɦ

Fig. 8 Mean scalar profiles

Neutral (Pr=5.0)

Neutral (Pr=1.0)

Stable (Pr=5.0)

0 50 100 150 2000

5

10 u+ÉÆ+ (Pr=5.0)

u+ÉÆ+ (Pr=1.0)

ÉÆrms+ (Pr=5.0)

ÉÆrms+ (Pr=1.0)

u+ ÉÆ+ ,É

Æ rms+

y+

Fig.9 Scalar flux and fluctuation profiles(Neutral stratification)

0 50 100 150 2000

0.5

1

1.5

- v+ É

Æ+

y+

Neutral(Pr=5.0)

Neutral(Pr=1.0)

Stable (Pr=5.0)

Fig.10 Wall-normal scalar profiles

512 Yoshinobu Yamamoto et al.

0 20 40 60 80 100-0.1

-0.05

0

0.05

0.1 Velocity-pressure gradient

Production

Turbulent diffusion Viscous diffusion

DissipationGai

n

Loss

y+

Fig.11 Budget of u+v+ (Pr=5.0)

Buoyancy production

0 20 40 60 80 100-0.3-0.2-0.1

00.10.20.3

Loss

G

ain

y+

Fig. 12 Budget of turbulent energy (Pr=5.0)

Production Turbulent diffusion Viscous diffusion

Pressure diffusion Dissipation Buoyancy production

Lx+=12

Lz

(a) Pr=1.0

Lx+=12

Lz

(b) Pr=5.0

Fig.13 Turbulent scalar fields (Top view), -0.27 (Black)<θ�0.27 (White)

Direct Numerical Simulation 513

Lx+=1280

Lz+=640

Lx+=1280

Ly +=20

0

Ly +=20

0L

z +=640

(a) Side view 0.1 (Black ) < θ* < 0.9 (White )

(b) End view 0.1 (Black ) < θ* < 0.9 (White )

(C) Top view 0.4 (Black ) < θ* < 0.9 (White )

Fig.14 Instantaneous scalar fields (Pr=5.0, Neutral stratification case)