EFFECTS OF STRONG TEMPERATURE GRADIENT ON A COMPRESSIBLE TURBULENT CHANNEL FLOW

th International Symposium on Turulence and Shear Flo Phenomena (TSFP, Chicago, USA, July, 7 EFFECTS OF STRONG TEMPERATURE GRADIENT ON A COMPRESSIBLE TURBULENT CHANNEL FLOW Mitsuhiro Nagata Mechanical Engineering Dept. I TOYOTA Central R&D Las., Inc. -, Yoomichi, Nagaute, Aichi 8-9, Japan nagata@mos.tytlas.co.jp Maoto Nagaoa Mechanical Engineering Dept. I TOYOTA Central R&D Las., Inc. -, Yoomichi, Nagaute, Aichi 8-9, Japan nagaoa@mos.tytlas.co.jp ABSTRACT A direct numerical simulation of a compressile turulent channel flo at lo-mach numer (Ma.3 suject to a strong temperature gradient is conducted. The all temperature ratios, hich are defined y the temperature on the upper all divided y that on the loer all, are set to and 3. It is shon that the flo on the high-temperature side is laminarized as the all temperature ratio increases. Moreover, compressile and dilatational motions are induced on the lo- and high-temperature all sides, respectively. The identity equation of the friction coefficient originally derived y Fuagata et al. ( is extended to those of the friction coefficient and Nusselt numer in a compressile channel flo. The viscous variation component is shon to reach % of the friction coefficient in the case of a high temperature ratio. Furthermore, the pressure or component occupies approximately % of the overall Nusselt numer for all cases. The pressure or has a redistriutive effect eteen mean inetic and internal energies. The energy transfer on the lo-temperature side is from the mean inetic energy to the internal energy, hereas the energy transfer on the hightemperature side is from the internal energy to the mean inetic energy, even in the present lo-mach numer flo. INTRODUCTION High-temperature all turulence commonly appears in comustion chamers ith cooling system and so forth. In order to achieve high-efficiency industrial instruments, it is important to clarify this phenomenon. Numerical simulation of a compressile turulent channel flo ith a high-temperature gradient induced y a temperature difference eteen to alls has een performed y relatively fe researchers. Wang and Pletcher (99 examined this prolem ith a preconditioning method in order to avoid encountering the stiffness prolem in a lo-mach numer flo. In their calculation, the spatial Favre-filtered variales are otained ecause a dynamic sugrid-scale model is adopted. Moreover, pressure and viscous ors, hich are intrinsic features of the compressile flo, are neglected. Nicoud ( also simulated this prolem under the lo-mach numer assumption. His research as devoted primarily to the proposal of an energy conservative scheme under the lo-mach numer assumption, therefore fe computational results ere presented. Tamano and Morinishi ( simulated a fully compressile turulent channel flo ith a high temperature gradient at Ma.5 using a high-accuracy spectral method. Their result revealed that the region in hich the mean temperature exceeded the high all temperature appeared as a result of the viscous or of the high-speed flo. To the est of our noledge, in the previous studies, the DNS of the lo-mach numer turulent channel flo ith a high temperature gradient, in hich the pressure and viscous ors ere taen into account, as not seen. In the present study, the DNS of a fully compressile turulent channel flo ith a high temperature gradient at lo- Mach numer is investigated in order to reveal the momentum and heat transfer mechanisms. Furthermore, the identity equation of the friction coefficient originally derived y Fuagata et al. ( is extended to the friction coefficient and Nusselt numer in a compressile channel flo in order to clarify hich component is dominant in the all friction or heat transfer at the all. COMPUTATIONAL CONFIGURATION AND NUMERICAL METHODOLOGY Figure shos the computational domain, in hich x, y, and z (or x, x, and x 3 represent the streamise, all-normal, and spanise coordinates, respectively, and the velocity components are u, v, and (or u, u, and u 3 in a similar order. The temperature at the upper all is represented y T H, and that at the loer all is represented y T L. The computational conditions are listed in Tale. Hereinafter, terms ith superscripts H and L represent values at the high- and lo-temperature alls, respectively. Tale shos the non-dimensional parameters. The Reynolds numer Re is defined y the ul velocity U, the channel half idth h, the ul density ρ, and the viscosity at the P-

th International Symposium on Turulence and Shear Flo Phenomena (TSFP, Chicago, USA, July, 7 T H x x3 x h flo T L Figure. Computational domain. Tale. Computational conditions. Case Case T L (K 93 93 T H (K 58 879 T H /T L 3 Tale. Non-dimensional parameters. Re, Pr.7 Ma.3 γ. loer all µ L. The Mach numer is defined as Ma U / γrt L, here γ is the specific heat ratio, and R is the gas constant. The oring fluid is assumed as air, and the temperature dependency of viscosity is expressed y the Sutherland la. The variales are non-dimensionalized y the values used for the definition of the non-dimensional parameters, unless otherise stated. On the other hand, values ith superscript + represent values that are non-dimensionalized y all-variales. Flo is driven y a constant mass flo rate condition. The periodic oundary conditions in the to homogeneous directions (x and z are applied, hereas non-slip oundary conditions are applied at the to alls. The computational domain is.5πh h πh (in the x, y, and z directions ith 9 8 9 grid points. Fully compressile Navier-Stoes equations are discretized y the sixth-order compact finite difference scheme (Lele, 99 in the x and z directions. For the y direction, a second-order central finite difference scheme is applied. No numerical diffusion scheme, such as upinding or spatial filtering, is used ecause numerical staility is ensured y adopting a staggered grid system (Nagarajan et al., 3. Time integration is performed y the Wray-type third-order Runge-Kutta scheme. The computation is conducted at least for h/u. After confirming that the flo reaches a statistically steady state, the computation continues for 3h/U in order to collect the statistical quantities. INSTANTANEOUS FLOW STRUCTURES Figure shos the vortical structures and temperature distriution. The second invariant of the velocity gradient tensor is adopted as an indicator of the vortical structures: u ( i j II x j xi The value of the second invariant is set to IIh /U throughout the visualizations. In the case of T H /T L (Fig. (a, vortical structures are tilted toard the all and are coherently distriuted on the lo-temperature side. This distriution is significantly similar to that in the incompressile case. On the other hand, vortex motion is significantly suppressed on the hightemperature side. As the temperature ratio T H /T L increases, the vortical structure disappears on the high-temperature side (Fig. (. Figs. 3 and sho top-don vies of vortical structures and all heat flux. On the lo-temperature side (Figs. 3(a and (a, vortical structures coincide ith high all heat flux regions, as shon in the incompressile flo ith the passive scalar. On the other hand, vortical structures are scarcely seen and high/lo heat flux regions are elongated almost linearly on the hightemperature side. These features are attriuted to the laminarizing effect over the high-temperature all as a result of a decrease in density (shon later and an increase in viscosity. TURBULENCE STATISTICS The friction coefficient and Nusselt numer are defined as follos: C f τ ρu ( T y α y y (3 Nu λ T T here τ, λ, and α are the all shear stress, the thermal conductivity, and the heat transfer rate, respectively, T is the ul temperature, and y corresponds to the all-normal position, here the mean temperature is equal to T. Figure 5 shos the temperature ratio dependency of C f and Nu. Each value is normalized y the incompressile/passive scalar value (C fi or Nu i in Fig. 5 evaluated y Kasagi and Iida (999. Moreover, C f gradually ecomes larger as the temperature ratio increases. On the other hand, Nu decreases significantly, hereas C f is approximately constant on the high-temperature all. This is due primarily to the decrease in the temperature gradient at the all as a result of the laminarizing effect, as shon in Fig.. The first invariant of the mean velocity gradient tensor is expressed as I ( The value of I is less than for compressile motion, is greater than for dilatational motion, and is equal to for an incompressile flo. The first invariant distriution is shon in Fig.. The compressile and dilatational motions are induced over the lo- and high-temperature alls, respectively. The enhancement of the asolute value of I is seen as increasing the temperature ratio. Therefore, more strong compressile /dilatational motions appear. As a consequence of these motions, the mean vertical velocity from the high-temperature all to the lo-temperature all is induced, as shon in Fig. 7. Moreover, an increased vertical velocity is seen as increasing the temperature ratio. Figure 8 represents the mean temperature distriution. The symols indicate the results reported y Na et al. (999 for the turulent flo ith a passive scalar, and each result is rescaled to correspond ith the present case. Although the difference eteen the present result and the result reported y Na et al. (999 is limited in the region near the high-temperature all in the case of T H /T L, a large region in hich the temperature P-

th International Symposium on Turulence and Shear Flo Phenomena (TSFP, Chicago, USA, July, 7 ( x x Figure. Instantaneous flo structures: side vie, (a TH/TL ; ( TH/TL 3. White, vortical structure; color contour, temperature distriution hich ranges from lue (T TL to red (T TH. (a x x3 Figure 3. Instantaneous flo structures at TH/TL : top vie, (a lo temperature side; ( high temperature side. White, vortical structure; color contour, all heat flux ith high value (q 5 represented y red and lo one (q y lue. Figure. Instantaneous flo structures at TH/TL 3: top vie, (a lo temperature side; ( high temperature side. White, vortical structure; color contour, all heat flux ith high value (q represented y red and lo one (q y lue. Figure 5. Friction coefficient and Nusselt numer. Figure. First invariant distriution. Figure 7. Vertical velocity profiles. ( referred to as ISK in the figure. The pea of u on the high-temperature all side moves toard the center of the channel as the temperature ratio increases and oth v and are extensively eaened. These characteristics are common features hen the flo ecomes laminarized. On the other hand, the turulence intensities on the lo-temperature all side are similar for all cases and correspond ell to those for the incompressile flo. distriution is loer than that of the passive scalar case is oserved at TH/TL 3. In addition, the temperature gradient at the high-temperature all is consideraly decreased. Since pressure is approximately constant across the channel in the present cases, the density is inversely proportional to the temperature. As shon in Fig. 9, the mean density taes the loest value on the hightemperature all and gradually varies on that side. This density distriution affects the value of Cf as descried in the later section. Turulence intensities are shon in Fig. ith the incompressile channel dataase reported y Iamoto et al. 3 P-

th International Symposium on Turulence and Shear Flo Phenomena (TSFP, Chicago, USA, July, 7 Figure 8. Mean temperature distriution. Figure 9. Mean density profiles. Figure. Turulence intensities. COMPONENTIAL DECOMPOSITION OF THE FRICTION COEFFICIENT Fuagata et al. ( derived the identity equation of C f on the incompressile all turulence, hich is descried y the linear comination of laminar and turulent components ( FIK identity. In the present study, this identity is extended to those of C f and Nu on the compressile turulent channel flo, and, according to these identities, each component of C f and Nu is quantified. Identity Equation of the Friction Coefficient in a Compressile Channel Flo By triple integration normal to the all of the mean streamise momentum equation, e otain the folloing identity of C f for the compressile channel flo: C fa 3µ L Re 3 + Re udy + 3 ( y( V + V ( y ( ρ{ u v } here { } and ( ˮ represent the Favre average and the fluctuation from the Favre averaged value, respectively, and V and V are the mean and fluctuation components, respectively, of the viscous variance: dy dy (5 V ( µ µ L (a y v V µ + µ ( y Note that each C fl or C fh previously shon in Fig. 5 is not ale to e decomposed into components as C fa decomposition (eq. (5. The friction coefficients are presented in Tale 3. It is confirmed that C fa evaluated y eq. (5 preferaly coincides ith the directly computed value otained y eq. (7. The laminar, turulence, and viscous variance components correspond to the first, second, and last terms in eq. (5, respectively. Evaluation of Friction Coefficient Components The results of C fa decomposition are shon in Fig. along ith the results for an incompressile flo reported y Fuagata et al. (. The laminar component has a similar value for all cases. Moreover, the turulence component also varies only slightly for to temperature ratios. This feature is due primarily to the increase in mean density on the lo-temperature side (shon in Fig. 9, y hich the turulent shear stress is multiplied in the turulence component (as seen in the second term of eq. (5, hereas the turulent motion is significantly suppressed on the high-temperature all side as the temperature ratio increases (see Fig.. On the other hand, the viscous variance component is enhanced as the temperature ratio increases. Fig. shos the fraction of C fa components. Each component is normalized y the overall value of C fa. Although the turulence component decreases as the temperature ratio increases, this is a virtual appearance oing to the overall increase in C fa. The viscous variance component reaches -3 viscous variance.. viscous variance In eq. (5, C fa is the overall friction coefficient, hich is defined y C fa ( τ L + τ / C fl + C (7 H fh ρ U Tale 3. Friction coefficients. Cf a 8 turulence laminar Cf a fraction.8.... turulence laminar C fa 3 T H /T L T H /T L 3 Evaluated y eq. (5 9. 9.7 Directly computed 9.9 9.7 Figure. Componential decomposition of C fa. Figure. Fraction of C fa components. P-

th International Symposium on Turulence and Shear Flo Phenomena (TSFP, Chicago, USA, July, 7 (a diffusion(variance ( 7 pressure or (a.. (.. Nu L 8 viscous or pressure or turulence conduction Nu H 5 3 - turulence conduction viscous or diffusion(variance Nu fraction Nu L.8.... Nu fraction Nu H.8.... -. Figure 3. Componential decomposition of Nu: (a Nu L ; ( Nu H. approximately % and % of C fa for T H /T L and 3, respectively. DECOMPOSITION OF THE NUSSELT NUMBER A similar procedure for C fa decomposition as descried in the previous section is also applicale to the Nusselt numer. In this section, the identity of the Nusselt numer is derived, and each component is evaluated precisely. Figure. Fraction of Nu components: (a Nu L ; ( Nu H. Colors as in Figure 3. in hich λ is the thermal conductivity at the all, and σ ij is a viscous stress tensor. Good agreement eteen the values of Nu evaluated through the identity (eq. (8 and those computed directly is confirmed (not shon here. The first, second, third, fourth, and last terms on the right-hand side of eq. (8 correspond to the conduction (value of, turulence, pressure or, viscous or, and diffusion (variance, respectively. The third through last terms in eq. (8 vanish for the incompressile/passive scalar flo. Identity of the Nusselt Numer on the Compressile Channel Flo The identity of the Nusselt numer is otained through doule integration normal to the all of the mean internal energy equation. Nu + y ( r{ v T } dy γλ T T ( (8 RePr γλ ( T T RePr + γλ ( T T RePr y y y y y ( y y P dy λ ( T T ( y y Vdy + y y Ddy here P, V, and D are the constitutions of the pressure or, viscous or, and diffusion coefficient variance, respectively, given y i P ( γ ρt i Ma i V γ ( γ σ ij Re D ( λ j (9a (9 T λ (9c y Evaluation of the Nusselt Numer Components Fig. 3 shos the componential decomposition of Nu. The turulence component taes a similar value to Nu L eteen the cases of T H /T L and 3. On the other hand, the turulence component decreases noticealy as the temperature ratio increases on the high-temperature all, as shon in Fig. 3(. This tendency is consistent ith the attenuation of turulence intensities shon in Fig.. The pressure or component has a large positive contriution to oth of Nu L and Nu H. The viscous or and diffusion components are not negligile on Nu and have a negative contriution on the high temperature all (Fig. 3( ecause the direction of heat flux, hich is aay from the all, is defined as positive on the high-temperature all. The fraction of Nu components is depicted in Fig.. It is confirmed that the pressure or component occupies approximately % of the overall Nu for all cases. Role of Pressure Wor In the previous section, the pressure or component is oserved to occupy approximately % of the overall Nusselt numer. The role of pressure or is examined in this section. The mean internal energy udget (for instance, Huang et al. (995 is given as follos: 5 P-

th International Symposium on Turulence and Shear Flo Phenomena (TSFP, Chicago, USA, July, 7 GAIN LOSS (a.3. turulent diffusion. pressure or. -. viscous or -. -.3 molecular diffusion -...5. y/h (.3. molecular diffusion. viscous or. -. pressure or -. turulent diffusion -.3 -...5. y/h Figure 5. Budgets of internal energy at T H /T L : (a lo-temperature side; ( hightemperature side. GAIN LOSS Figure. Schematic of energy transfer y the pressure or. ρ t { T} + ( ρ{ u T } ( γ ρt ( γ ( ρt ( ( γ γma i µσ + ( µσ Re γ + Re Pρ ij T λ j ij T + λ i j The first term on the right-hand side of eq. ( is the turulent diffusion, the second is the pressure or, the third is the pressure-dilatation correlation, the fourth is the viscous or, and the last term is the molecular diffusion. Pressure or exhiits a redistriutive effect eteen the internal and mean inetic energies ecause pressure or having the opposite sign to that in eq. ( also appears in the mean inetic energy udget. Fig. 5 represents the udgets of internal energy at T H /T L. On the lo-temperature side (Fig. 5(a, the main source of internal energy is turulent diffusion, hereas the sin is molecular diffusion. Since the sign of pressure or is positive, the energy flo from mean inetic to internal energy arises over the lotemperature all. On the other hand, pressure or has a negative value on the high-temperature all side (Fig. 5(. Therefore, energy flo from internal to mean inetic energy is induced over the high-temperature all. These energy flos are depicted schematically in Fig.. The pressure or has a major effect on Nu, as shon in Fig. 3, through the aove-mentioned mechanism. CONCLUSION Direct numerical simulation of a compressile turulent channel flo suject to a strong temperature gradient at lo Mach numer as performed. An increase in temperature ratio caused the flo to laminarize over the high-temperature all. Compressile and dilatational motions ere induced on the loand high-temperature all sides, respectively. Consequently, the mean vertical flo occurs from the high-temperature all to the lo-temperature all. The identity of the friction coefficient for the compressile channel flo as derived through the procedure proposed y Fuagata et al. (. The viscous variance component as enhanced as the temperature ratio increased. The identity of the Nusselt numer as also derived. The viscous or and diffusion components ere not negligile, and the pressure or as shon to reach approximately % of the overall Nusselt numer. The energy transfer eteen the mean inetic and internal energies through the pressure or as clarified, even in the present lo-mach numer flo. REFERENCES Fuagata, K., Iamoto, K., and Kasagi, N.,, "Contriution of Reynolds Stress Distriution to the Sin Friction in Wall-Bounded Flos", Physics of Fluids, Vol., pp. L73- L7. Huang, P. G., Coleman, G. N., and Bradsha, P., 995, "Compressile Turulent Channel Flos: DNS Results and Modelling", Journal of Fluid Mechanics, Vol. 35, pp. 85-8. Iamoto, K., Suzui, Y., and Kasagi, N.,, "Reynolds Numer Effect on Wall Turulence: toard Effective Feedac Control", International Journal of Heat and Fluid Flo, Vol. 3, pp. 78-89. Kasagi, N., and Iida, O., 999, "Progress in Direct Numerical Simulation of Turulent Heat Transfer", Proceedings of the 5th ASME/KSME Joint Thermal Engineering Conference, ASME, San Diego, Paper No. AJTE99-3, pp. -7. Lele, S. K., 99, "Compact Finite-Difference Schemes ith Spectral-lie Resolution", Journal of Computational Physics, Vol. 3, pp. -. Na, Y., Papavassiliou, D. V., and Hanratty, T. J., 999, "Use of Direct Numerical Simulation to Study the Effect of Prandtl Numer on Temperature Fields", International Journal of Heat and Fluid Flo, Vol., pp. 87-95. Nagarajan, S., Lele, S. K., and Ferziger, J. H., 3, "A Roust High-Order Compact Method for Large Eddy Simulation", Journal of Computational Physics, Vol. 9, pp. 39-9. Nicoud, F.,, "Conservative High-Oder Finite-Difference Schemes for Lo-Mach Numer Flos", Journal of Computational Physics, Vol. 9, pp. 39-9. Tamano, S., and Morinishi, Y.,, "Effect of Different Thermal Wall Boundary Conditions on Compressile Turulent Channel Flo at M.5", Journal of Fluid Mechanics, Vol. 58, pp. 3-373. Wang, W. K., and Pletcher, R. H., 99, "On the Large Eddy Simulation of a Turulent Channel Flo ith Significant Heat Transfer", Physics of Fluids, Vol. 8, pp. 335-33. P-