International Conference Nuclear Energy for New Europe 2004 Portorož • Slovenia • September 6-9

[email protected] www.drustvo-js.si/port2004 +386 1 588 5247, fax +386 1 561 2276 PORT2004, Nuclear Society of Slovenia, Jamova 39, SI-1000 Ljubljana, Slovenia

The Smallest Thermal Scales in a Turbulent Channel Flow at Prandtl Number 1

Robert Bergant, Iztok Tiselj “Jožef Stefan” Institute Jamova 39, SI-1000 Ljubljana, Slovenia [email protected], [email protected]

ABSTRACT

For describing the turbulent heat transfer from a wall to a fluid at low Reynolds (Re < 10000) and low Prandtl numbers (Pr < 20) a direct numerical simulation (DNS) can be used, which describes all the length and time scales of the phenomenon. The object of this paper is to find out the influence of the smallest temperature scales on the largest ones, which are responsible for the “macro” behavior of the near-wall heat transfer. Simulation, performed at Re = 2650 and Pr = 1, was calculated for velocity field with the DNS accuracy and for three different temperature fields. First temperature field, calculated with the DNS accuracy, was used as a reference to the second and third temperature fields where the highest Fourier coefficients in streamwise and spanwise directions were filtered and damped. It means, that the smallest temperature scales were not described with DNS accuracy anymore. New approach shows that results, for at least first and second order statistics, are comparable to the DNS ones without filtering and damping.

1 INTRODUCTION

Over the past 15 years Direct Numerical Simulation (DNS) has become an important research tool in understanding of the near-wall turbulent heat transfer. A DNS means precise solving of Navier-Stoke’s equations without any extra turbulent models. In 1987, Kim et al [1] performed first DNS of velocity field at low Reynolds number (Re = 5700). The scope of their DNS study was investigating velocity field and observing turbulent structures in the channel. Later, they [2] also added energy equation to the equations of velocity field for the heat transfer calculations. When considering heat transfer, the temperature field between two walls and the coherent structures near the walls were studied. But this was an unusual approach, because uniform volumetric heat generation was assumed, where the walls represented a heat sink. Kasagi et al [3] also researched the temperature field, but in this case the top and bottom walls of the channel heated the fluid. All these simulations were performed at low Reynolds and Prandtl numbers. Later, Kawamura et al [4], Na et al [5] raised the limit of Prandtl number to ten, while Kawamura et al [6] studied the influence of Reynolds numbers (up to Re = 14000) and Prandtl numbers (up to Pr = 5). They found a weak influence of the Reynolds number and stronger influence of the Prandtl number near the wall for turbulent heat transfer (velocity profiles, velocity fluctuations, turbulent heat fluxes). Theoretically, the grid spacing for DNS of heat transfer based on Eulerian method at Prandtl numbers higher than one should be inversely proportional to the square root of the

128.1 128.2 Prandtl number [7]. This requirement was taken into account in the DNS studies of Kawamura et al [4] and Tiselj et al [8, 9]. However, as shown by Na et al [5] and Bergant et al [10] this requirement is too stringent. Papavassiliou et al [11] performed the DNS of turbulent channel flow for Prandtl or Schmidt numbers that span five orders of magnitude (up to 2400). They described temperature or concentration fields without increasing the resolution of the numerical grid. This is possible due to Lagrangian method whose idea is based on the system of reference that moves with heat or mass markers. In other words, heat markers, which are released from the infinitesimal source on the wall, are monitored in space and time as they move in the hydrodynamic field created by DNS. The focus of this paper is to describe the velocity field at a low Reynolds number, Re = 2760, with DNS accuracy, and contemporarily to describe the temperature field at a low Prandtl number, Pr = 1, but with the coarser computational grid, which does not allow DNS accuracy. The velocity field was calculated with the grid of 128x64x97 computational points, whereas the temperature field was calculated at three different grids. The first temperature field used the same grid as the velocity field. Therefore, it was calculated with the DNS accuracy and it was needed as the reference to the second and third temperature fields. Streamwise and spanwise grid density was reduced for factor 2 and 4 in the second and third temperature fields, respectively. In such a manner, the number of computational points of the temperature field was reduced by 4 and 16 times. Such numerical simulations can be called under-resolved DNS. The remaining Fourier coefficients were damped to prevent rising of the temperature spectra in the dissipation and partly in the inertial range [12]. Nevertheless, the results show that the temperature fields obtained with a reduced number of computational points are highly correlated to the “reference” DNS.

2 EQUATIONS AND NUMERICAL PROCEDURE

When studying turbulent heat transfer near the wall, channel (Fig. 1) is the most frequently used geometry form due to its simplicity and possibility of performing very accurate numerical simulations. The channel is bounded by top and bottom walls, which are heated by a constant heat source, while the pressure gradient drives the fluid flows between them. The flow in the channel is assumed to be fully developed. The dimensionless Navier- = τ ρ Stokes equations normalized by the channel half height h, the friction velocity uτ w / , = ( ρ ) τ and the friction temperature Tτ qw uτ f c pf were used, where w stands for the wall shear τ = −µ stress defined as w (du / dy) w . Such scaling and dimensionless equations can be found in the papers of Kasagi et al [3] or Kawamura et al [4]:

∇ ⋅ uv + = 0 (1) ∂v + u = −∇ ⋅ ()v + v + + 1 ∇ 2 v + − ∇ + v u u u p l x (2) ∂t Reτ + + ∂θ + + 1 + u = −∇ ⋅ ()uv θ + ∇ 2θ + x (3) ∂ + t Reτ Pr u B

Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, Sept. 6-9, 2004

128.3

Figure 1. Flow geometry of the channel.

v + + Terms lx (unit vector in streamwise direction) and u x u B appear in Eqs. (2) and (3) due to the nature of the spectral numerical scheme, which requires periodic boundary conditions in streamwise and spanwise directions. Reτ is the friction Reynolds number and is defined as

uτ h Reτ = (4) ν where h is channel half height. The friction Reynolds number Reτ should not be confused with = ⋅ ν the “standard” Reynolds number, which is defined as Re uB h / . The Reynolds number in the channel can be obtained from the friction Reynolds number multiplied by the bulk velocity uB. Friction Reynolds number Reτ = 170.8 corresponds to the Reynolds number Re = 2650. The Prandtl number is defined as usually as

ν Pr = (5) α where α is the thermal diffusivity. Dimensionless wall units denoted with superscript + are based on the friction Reynolds number. By definition, the height of the channel in wall units is equal to two times the friction Reynolds number. The meaning of the wall units is in comparison of the turbulent flows near the wall at different Reynolds numbers. The dimensionless wall temperature difference is defined as

 T −T()x, y, z, t  θ + ()x, y, z, t =  w  . (6)    Tτ 

The dimensionless velocity components at the wall-fluid interface are set to zero (no- slip boundary condition) as well as the dimensionless temperature:

======u x (y y w ) u y (y y w ) u z (y y w ) 0 (7) θ + ()= = y y w 0 (8)

As can be seen from Eqs. (1-3) the temperature is assumed to be a passive scalar. This assumption introduces two approximations: 1) neglected buoyancy, 2) neglected temperature Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, Sept. 6-9, 2004

128.4 dependence of the material properties - especially viscosity and heat conductivity. It should be emphasized that the same approximations are used in a large number of DNS studies performed by different researchers: Kasagi et al [3], Kawamura et al [4], Na et al [5], and Tiselj et al [8]. The numerical procedure and the code of Gavrilakis et al [13] modified by Lam et al [14, 15] are used to solve the continuity and momentum equations. The code was later upgraded with an energy equation [4] and improved to solve several energy equations with different boundary conditions and different Prandtl numbers paralleled with a single velocity field solution [9]. This approach reduced the CPU time but increased the required physical memory of the computer.

Table 1: Computational conditions at different dimensions of the grids for temperature fields calculations: + + + + Case Grid (N1xN3xN2 ) ∆t ∆x ∆z ∆y Averaging time 1 (DNS) 128x64x97 9.2 4.2 ½ 64x32x97 0.085 18.4 8.4 0.02-2.45 8540 ¼ 32x16x97 36.8 16.8 * Grid for velocity field is 128x64x97 for all three cases.

Eqs. (1-3) are solved with a pseudo-spectral scheme using Fourier series in streamwise x and spanwise z directions and Chebyshev polynomials in the wall-normal y direction:

N1 2 N 2 N3 2 i()k x+k z f ()x, y, z = a (k , n , k )e 1 3 T ()y , (9) ∑∑∑ 1 2 3 n2 = n1 n2 0 n3 − + ≤ ≤ N1 2 1 n1 N1 2 − + ≤ ≤ N 3 2 1 n3 N 3 2 where f represents the velocity components u or the temperature θ, and T ()y represents the i n2 Chebyshev polynomial of the order n2. Numbers N1, N3 and N2 are the numbers of the Fourier terms and the number of the Chebyshev polynomials used in the simulation (see Table 1), and () () a k1 ,k3 stands for Fourier coefficients and a n2 for Chebyshev coefficients. The numbers of Fourier coefficients for temperature θ in the streamwise and spanwise directions were different as shown in Table 1. Case 1 represents DNS, meanwhile case ½ and case ¼ used less Fourier coefficients as needed for DNS accuracy, and are therefore named as under- resolved DNS. A cut-off filter was used to filter out the Fourier coefficients at high wave numbers. After filtering, the last halves of the remaining Fourier coefficients were damped by following weight factors:

= − wi i N r / 2 (10) where Nr represents the number of the remaining Fourier coefficients in streamwise or spanwise directions and i represents index running from Nr/2 to Nr. Increasing damping effect, when approaching the dissipation range or the last remaining Fourier coefficient Nr, is important due to the transfer of the turbulent diffusion into the internal heat, which has the strongest effect at the highest wave numbers where the smallest temperature scales appear. It forces the transformation of the turbulent diffusion into the internal heat. Without damping a

Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, Sept. 6-9, 2004

128.5 “pile-up” phenomenon occurs, which means that the smallest scales are not properly modeled. The weighting function (Eq. 10) was chosen among several testing functions as the most suitable one. The computational domain of all simulations was 4π x π x 2 (2146 x 537 x 342 wall units in x, z and y directions). Results were averaged after the fully developed turbulent flow was achieved, which means that the flow did not change from statistical point of view. Statistical quantities were sampled at every fourth step and averaged for about 8540 dimensionless time wall units.

3 RESULTS

Fig. 2 shows average dimensionless temperatures for three different filtered and damped temperature fields. The dimensionless temperature θ is averaged in planes parallel to the heated wall and in time. It should be emphasized that the temperature θ is a negative dimensionless difference, which means that a maximum temperature difference appears in the middle of the channel. Despite different numbers of Fourier coefficients in streamwise and spanwise directions, and consecutively coarser computational grid, no differences larger than statistical uncertainties in mean temperature profiles can be seen.

20 18 16 14 12 1 0.5 θ θ θ θ 10 0.25 8 6 4 2 0 0.1 1 10y+ 100 1000

Figure 2. Dimensionless profiles of mean temperature.

Fig. 3 shows the temperature fluctuations. Temperature fluctuations RMS_θ are obtained as the root mean square difference of instantaneous and averaged temperatures averaged in planes parallel with heated wall and in time. Temperature fluctuations are zero near the wall, afterwards they begin to grow up through the conductive sublayer. In the buffer sublayer, between turbulent and conductive sublayer, the maximum is reached, afterwards fluctuations are decreasing towards the middle of the channel. Once again, no important differences are seen among temperature fields with different numbers of Fourier coefficients. In a study of the velocity and temperature scales in the turbulent flow the spectra are usually used to see how different length scales are presented. It is well known that a turbulent flow consists of vortices of the different dimensions. The largest vortices are defined by the flow geometry, while the smallest ones are defined by the viscous forces. High dissipation of the turbulent kinetic energy into heat is typical for the smallest vortices. Viscous shear stress makes deformation work which transforms turbulent kinetic energy into internal energy of the fluid. In other words, larger vortices represented by lower wave number modes diffuse into Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, Sept. 6-9, 2004

128.6 smaller vortices represented by higher wave number modes in the spectrum diagrams. The energy of the smaller vortices is smaller than the energy of the larger vortices, therefore spectra decrease with increasing wave number modes.

3.5

3 1 0.5 2.5 0.25 θ θ θ θ 2

RMS_ 1.5

1

0.5

0 0.1 1 10y+ 100 1000

Figure 3. Dimensionless temperature fluctuations.

Fig. 4 shows streamwise spectra of temperature fluctuations for three different cases at two different distances from the wall (y+=3.3 and y+=171). It can be seen that the spectra near the wall decrease faster than spectra in the middle of the channel. Different numbers of Fourier coefficients can be seen in different lengths of the curves. The longest curves, marked as 1 in the legend, represent temperature fields calculated with DNS accuracy, whereas the others, marked as ½ and ¼ in the legend, represent temperature fields calculated with coarser grid due to the filtering of the Fourier coefficients. Damping effects can be seen in the last halves of the Fourier coefficients for case ½ and case ¼ where spectra begin to decrease faster. Damping is important in order to prevent “pile-up” phenomenon, which means that the temperature dissipation in conductive scale cannot change the whole turbulent diffusion into internal energy. “Pile-up” phenomenon can be seen as rising of the spectrum at high wave number modes. It can be concluded, that the smallest temperature structures, which were not captured with the numerical simulation, do not have considerable influence on the temperature field.

1.00E+00

1-3.7 1.00E-01 0.5-3.7 0.25-3.7 1.00E-02 1-171 0.5-171 0.25-171 θθ θθ θθ θθ 1.00E-03 E

1.00E-04

1.00E-05

1.00E-06 0.001 0.01kx+ 0.1 1

Figure 4. Spectra of temperature fluctuations in streamwise direction at y+ = 3.3 and y+ = 171 wall units from the heated wall.

Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, Sept. 6-9, 2004

128.7

1.00E+00

1-3.7 1.00E-01 0.5-3.7 0.25-3.7 1-171 1.00E-02 0.5-171 0.25-171 θθ θθ θθ θθ E 1.00E-03

1.00E-04

1.00E-05 0.01 0.1kz+ 1

Figure 5. Spectra of temperature fluctuations in spanwise direction at y+ = 3.3 and y+ = 171 wall units from the heated wall.

Fig. 5 shows spanwise spectra of temperature fluctuations at the same distance from the wall as in the streamwise spectra (y+ = 3.3 and y+ = 171). The same conclusion as in streamwise direction can be found here.

4 CONCLUSIONS

The numerical simulations of the turbulent heat transfer at a friction Reynolds number, Reτ = 170.8 and at a Prandtl numbers Pr = 1 for three different temperature fields were simultaneously performed. The first temperature field was the “real” DNS, meanwhile the second and the third temperature fields used, comparing to the “real” DNS, only one half and one quarter of the Fourier coefficients in the streamwise and spanwise directions, respectively. Furthermore, the last half of the remaining Fourier coefficients was additionally damped in order to prevent the pile-up phenomenon. The number of computational nodes was reduced by factors of 4 and 16, respectively. Therefore, latter simulations did not describe all temperature scales. The obtained results revealed negligible discrepancies between the “real” DNS and under-resolved DNS. The only visible difference can be seen in the spectra of the temperature fluctuations. Our last results at Pr = 100 confirm predictions that the smallest temperature scales do not play an important role in the turbulent heat transfer. We estimate, that applied method of filtering and damping can be useful for additional researches at high Prandtl numbers, i.e. Pr = 100, Pr = 200, Pr = 500, where DNS studies cannot be performed due to the very dense grids required for such computations.

NOMENCLATURE

EE spectrum u, w, v velocity components in x, y and z directions h channel half height uB bulk velocity r 1x unit vector in x direction (1,0,0) uτ dissipative velocity K heat transfer coefficient α thermal diffusivity k wave number θ dimensionless temperature difference L1, L3 streamwise and spanwise length of box λ thermal conductivity p pressure γ kinematic viscosity

Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, Sept. 6-9, 2004

128.8

Pr Prandtl number ( Pr=ν α ) ρ density

Re Reynolds number ()f fluid

uτ h ()w wall Reτ friction Reynolds number ( Re= ) τ ν

T temperature ()τ dissipation T time ()+ normalized by uτ , Tτ , ν x, y, z streamwise, spanwise, wall normal ()' (fluctuations) distance

REFERENCES

[1] Kim, J., Moin, P., Moser, R. D., 1987, Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., Vol. 130, pp. 133-166. [2] Kim, J., Moin, P., 1989, “Transport of Passive Scalars in a Turbulent Channel Flow”, Turbulent Shear Flows VI, Springer-Verlag, Berlin. [3] Kasagi, N., Tomita, Y., Kuroda, A., Direct Numerical Simulation of Passive Scalar Field in a Turbulent Channel Flow, J. Heat Transfer - Transactions of ASME, 114, 1992, pp. 598-606. [4] Kawamura, H., Ohsaka, K, Abe, H., Yamamoto, K., DNS of Turbulent Heat Transfer in Channel Flow with low to medium-high Prandtl number fluid, Int. J. Heat and Fluid Flow, 19, 1998, pp. 482-491. [5] Na, Y., Hanratty, T.J., 2000, ”Limiting Behavior of Turbulent Scalar Transport Close to a Wall”, Int. J. Heat and Mass Transfer, Vol. 43, pp. 1749-1758. [6] Kawamura, H., Abe, H., Matsuo, Y., DNS of turbulent heat transfer in channel flow with respect to Reynolds and Prandtl number effects, Int. J. Heat and Fluid Flow, 20, 1999, pp. 196-207. [7] Tennekes, H., Lumley, J.L., 1972, “A First Course in Turbulence”, MIT Press, Cambridge, MA. [8] Tiselj, I., Pogrebnyak, E., C. Li, Mosyak, A., Hetsroni, G., 2001, “Effect of wall boundary condition on scalar transfer in a fully developed turbulent flume”, Physics of Fluids, 13 (4), pp.1028-1039. [9] Tiselj, I., Bergant, R., Mavko, B., Bajsic, I., Hetsroni, G., 2001, DNS of turbulent heat transfer in channel flow with heat conduction in the solid wall, J. Heat Transf., 123, 849-857. [10] Bergant R., Tiselj I., Vpliv Prandtlovega števila na turbulentni prenos toplote ob ravni steni = The influence of Prandtl number on near-wall turbulent heat transfer. Stroj. vestn., 2002, letn. 48, št. 12. [11] Papavassiliou, D., Hanratty, T.,J., Transport of a passive scalar in a turbulent channel flow, Int. J. Heat and Mass Transfer, 40, 1997, pp. 1303-1311. [12] Pope S.B., Turbulent Flows, University Press, Cambridge, 2000. [13] Gavrilakis S., Tsai, H.M., Voke, P.R., Leslie, D.C., Direct and Large Eddy Simulation of Turbulence, Notes on Numerical Fluid Mechanics, 15, Vieweg, Braunschweig, D.B.R. , 1986, pp. 105. [14] Lam, K.L., Banerjee, S., Investigation of Turbulent Flow Bounded by a Wall and a Free Surface, Fundamentals of Gas-Liquid Flows, Vol. 72, 29-38, ASME, 1988, Washington DC. [15] Lam K.L., Numerical Investigation of Turbulent Flow Bounded by Wall and a Free - Slip Surface, Ph.D. thesis, University of California, Santa Barbara, 1989. [16] Smith, C.R., Walker, J.D.A., Turbulent Wall-Layer Vortices, poglavje iz knjige Fluid Vortices, Kluwer Academic Publishers, Dordrecht, 1995.

Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, Sept. 6-9, 2004