REYNOLDS NUMBER EFFECTS ON STATISTICS AND STRUCTURE OF AN ISOTHERMAL REACTING TURBULENT WALL-JET

REYNOLDS NUMBER EFFECTS ON STATISTICS AND STRUCTURE OF AN ISOTHERMAL REACTING TURBULENT WALL-JET Zeinab Pouransari and Arne V. Johansson Linné FLOW Centre, Dept. of Mechanics, Roal Institute of Technolog SE- 44 Stockholm, Sweden zeinab@mech.kth.se ABSTRACT In this stud, three-dimensional direct numerical simulation (DNS) is used to investigate a reacting turbulent walljet. The flow is compressible and a single-step global reaction is considered. At the inlet, fuel and oxidizer enter the domain separatel in a non-premixed manner. Two different simulations with different Renolds numbers, Re = and Re = 6 in terms of the inlet quantities are considered. The DNS-database of Pouransari et al. () with the lower Renolds number is used here for comparison and a new DNS is performed, in which the bulk Renolds number is increased b a factor of three. This results in a comparable increase in the friction Renolds number, which allows the stud of Renolds number effects. The main objective of this stud is to compare the influences of changing the Renolds number of the flow with the heat-release effects caused b the chemical reaction, that we studied earlier in Pouransari et al. (). Results primaril show that, the Renolds number effects can clearl be observed both in turbulent structures and in the flame characteristics. While, both turbulent and flame structures become finer at the higher Renolds number, the effect of decreasing the Renolds number and adding the combustion heat release are not alwas the same. INTRODUCTION Man different numerical studies have been performed on various aspects of turbulent combustion in several flow configurations. However, in most of these studies, in particular when direct numerical simulation is emploed as the computational tool, the Renolds number of the flow is ver limited and much lower than in real phsical situations. Thus, some of the conclusions drawn in these cases, ma be subject to both qualitative and quantitative changes at higher Renolds numbers. Effects of Renolds number in most canonical flows have been addressed both numericall and experimentall. Such as in turbulent channel flow (Hoas & Jiménez, 8; Wei & Willmarth, 989), turbulent boundar laers, (DeGraaff et al., 998), and pipe flows (Den Toonder & Nieuwstadt, 997). Some studies on the scaling properties have also been performed to capture these dependencies. Toda, the Renolds number of experimental facilities has increased substantiall and both smooth and rough wall flows are still of great interest, see Hong et al. (); Baile et al. (). It is onl in the last few ears that DNS studies have reached Renolds numbers where significant scale separation (between inner and outer scales) can be claimed. Man other efforts in the stud of the Renolds number effects are directed towards the degree of local isotrop and the fine structures of turbulence. For these purposes, experiments and numerical investigations with different Renolds numbers are considered and in general at higher Renolds numbers a high degree of universalit of small scales and local isotrop have been found, see Ferchichi & Tavoularis (); Yiu et al. (4); Schumacher (8). Renolds number effects on scalar dissipation rate transport and its modeling in turbulent premixed combustion is addressed b Chakrabort & Swaminathan (). The have used scaling arguments to explain the effects of Renolds number on the turbulent transport, scalarturbulence interaction and the combined reaction and molecular dissipation terms. The found that the modeling of the term arising from densit variation has no significant Renolds number dependence (within the range examined). DNS is a reliable tool for the stud of various turbulent flows. However, some cases such as compressible and reacting flows are computationall more demanding and hence numerical studies of turbulent combustion flows are usuall restricted to lower Renolds number ranges, (Peters, ; Vervisch & Poinsot, 998). In our previous studies of wall-jet flows, we have addressed the turbulent flow effects on the structure of the flame both in the absence and presence of combustion heatrelease, (Pouransari et al.,, ). However, in those studies, we kept the bulk Renolds number constant and focused on the heat-release and turbulence interactions. In the present stud, the bulk Renolds number is increased b a factor of three, which results in a comparable increase in the friction Renolds number. In the present investigation, the Renolds number effects in a reacting turbulent wall-jet are considered. In particular, we are interested in comparing these effects which we observed in the stud of the heatrelease effects at the higher Renolds number where the scale separation starts to become significant. In general, including the combustion heat-release in a reacting turbulent flow decreases the local Renolds number. Thus, in our previous studies on the effects of heat-release in a reacting turbulent wall-jet; we observed different changes in the mixing scales of the flow. There, we examined how the presence of heat-release could affect the scales of turbulence and in return how it can affect the global reaction. Here, we dis-

regard the heat-release effects, and instead change the bulk Renolds number of the flow. Therefore, we can stud the Renolds number effects on the isothermal reaction and on the structure and statistics of the flow, in the absence of thermal effects. Governing equations The conservation equations of total mass, momentum and energ read compact finite difference scheme for spatial discretization and a third-order Runge-Kutta method for the temporal integration. Table. Parameters of direct numerical simulations of reacting turbulent wall jet flows; Pr =.7, Sc =.7 and Da = in both of the simulations Case L x L L z N x N N z Re ρe t ρeu j ρu i t ρu iu j ρ t ρu j = () = p x i τ i j () = q i x i (u i(τ i j pδ i j )). () Here ρ is the total mass densit, u i are the velocit components, p is the pressure and E = e u iu i is the total energ. The summation convention over repeated indices is used. The heat fluxes are approximated b the Fourier s law according to q i = λ T x, where λ is the coefficient of i thermal conductivit and T is the temperature. The viscous ( stress tensor is defined as τ i j = µ ui x u j µ j u k x δ k i j where µ is the dnamic viscosit. The fluid is assumed to be caloricall perfect and to obe the ideal gas law according to e = c v T, p = ρrt, and a ratio of specific heats of γ = c p /c v =.4 is used. The viscosit is determined ( ) through the Sutherland s law µ / µ j = TTj Tj S TS, where T is the local temperature, T j is the jet center temperature at the inlet and S is a reference value, taken as S =.4K. Conservation of the species masses is governed b ρθ k t ( ρθk u j ) = x i ) ( ρd θ ) k ω x k, (4) j where θ k and ω k are the mass fractions and the reaction rate of the oxidizer, fuel and passive scalar species. An equal diffusion coefficient D for all scalars, is used to approximate the diffusive fluxes. The simulated reaction is modeled b a single-step irreversible reaction between oxidizer species O and fuel species F that react to form a product P, which is described as ν O O ν F F ν P P. Stoichiometric coefficients of one are used for all species and the molecular weights are also assumed to be equal for the reactants. The reaction mass rate is formulated as ω = k r ρ θ o θ f. The reaction rate is prescribed b a Damköhler number of Da = τ conv τ react = U h j k r ρ j =, where k r is the reaction rate and U j and ρ j are the centerline velocit and the densit of the jet at the inlet. No heat release is used in the present simulation and onl the oxidizer and fuel mass fractions are computed since the product does not enter in the reaction rate expression. Since the flow is uncoupled from the reactions, the influence of increasing the Renolds number can be studied in the absence of temperature effects. Apart from reacting species, a passive scalar equation is also solved for comparison. Numerical method and parameters A full compressible Navier-Stokes solver is emploed for simulation of the wall-jet. The code uses a sixth-order Re τ C f I 5 7 7. 9 8 II 5 4 7. 96 48 84 6 The computational domain is a rectangular box of size [L x L L z = 5h 7h.6h] where L x, L and L z denote dimensions in the streamwise, wallnormal and spanwise directions, respectivel, and h is the inlet jet height. The number of grid points are [N x N N z = 96 48 84] and the inlet based Renolds number of the wall-jet is Re = 6. In the remaining of this paper, this case is referred to as case II, while the DNS data in paper Pouransari et al. () is tagged as case I, in which the grid points were [N x N N z = 9 8] and Re =. The densit and temperature are varied over the jet profile and the wall temperature is kept constant and equal to the ambient flow temperature. The specifications are summarized given in Table. 7 6 5 4 5 5 5 Figure. Downstream development of the friction Renolds number; solid line: case I (Re = ) and dashed line: case II (Re = 6). 5 x - 4 5 5 5 Figure. Downstream development of the skin friction coefficient.

.5 u rms uτ.5 v rms uτ.5.5.5.5 6 (c).5 (d) 4 w rms uτ.5 K 8 ρ uτ 6 4.5 Figure. Streamwise, wall-normal, intensit and (d) Mean p (c) spanwise fluctuation p turbulent kinetic energ using (µ /ρ ) (µ /µ ) semi-local normalization defined as uτ = ρ w /ρ uτ and l = u = ρ /ρ w l, where uτ = τw /ρ w and l = ν w /uτ are the w τ friction velocit and length scales, and wall conditions are denoted b a subscript w. Profiles at = 5. /h case I compared to case II. In the full turbulent region, at = 5, it is about 5% lower. These observations can, in some sense, be seen in analog with the reductions in C f observed when heat release effects were added (to the low Re case) and hereb further reduced the Renolds number, see Pouransari & Johansson (). /h The averaged turbulence intensit for the three components of the velocit are plotted in figure (a-c) using semi-local inner scaling. The turbulent kinetic energ is plotted in figure (d) using the same normalization. With this scaling we observe a good collapse in the near-wall region, up to = 7 8. In order to make the comparison easier, the same scaling method is used later for plotting the two-dimensional spectra. For details of this normalization see Pouransari et al. (). Figure shows that increasing the Renolds number, will increase the fluctuation intensities in the streamwise direction both in the inner and outer regions. It is a clear increase in the amplitude of the two peaks of the fluctuation intensit and also as a assign of increase in the Renolds number the get further separated from each other. For the spanwise component, figure (c), the peak of the fluctuation profile slightl increases with increasing of the Renolds number and also gets further awa from the wall. Figure shows that the wall-normal velocit fluctuations in some regions decreases when the Renolds number increases, this behavior is the same if the absolute values are considered without an scaling. The peak value seems to be unaffected for the two Renolds numbers considered. What is consistent, using different scalings, is that due to increasing the Renolds number, the outer maximum is shifted further outward from the wall region and the locations of the two peaks get further separated from each other for all components of the fluctuations. As a result, the same trend is observed in the turbulent kinetic energ profile (figure (d)). The advantage of using the semi-local scaling is the good collapse of Figure 4. Instantaneous snapshots of the streamwise velocit fluctuations at 8 for case I and case II; Light and dark colors represent positive and negative fluctuations. RESULTS AND DISCUSSIONS Turbulent statistics and flow structure To get a general insight into the two cases with different bulk Renolds number we examine the friction Renolds number Reτ = uτ δ /νw, see figure. In this figure a dashed line is used for case II and a solid line for case I. Note that these line stles are used throughout the paper unless otherwise stated. When an appropriate outer lengthscale δ is used, the friction Renolds number can be seen as an estimate of the outer-laer to inner-laer length-scale ratio. Around = 5, for case II (Re = 6) the friction Renolds number reaches the value of Reτ = 54 and for case I (Re = ) it reaches Reτ =. The accurate experimental determination of the skin friction still remains to be a major challenge in turbulent wall flows. The values of the skin friction coefficients, C f = (uτ /Um ) are shown in figure for cases I and II. At the downstream position = 5, for instance, in the transition region, the C f coefficient is more than 55% lower for

the profiles in the near-wall region for the two cases with different Renolds numbers. To identif and explain the influences of Renolds number effects on the structure of the turbulent wall-jet and to have a general idea about the flow field, the streamwise velocit fluctuations are plotted. Visualizations of the instantaneous fields indicate that the jet is full turbulent beond 5. Figure 4 shows the instantaneous fluctuations of the streamwise velocit in the xz plane at 8. Elongated streamwise streaks, tpicall present in the viscous sublaer of wall flows are also seen in the turbulent wall-jets presented here. It is clearl observed that due to increasing the Renolds number the distances between the streaks decreases and therefore the two point correlation profiles of velocities are modified. The turbulent flow structure near the walls is significantl different at high Renolds number compared to the lower Renolds number. For instance, the hairpin and vorticit structures and their interactions will be evolved and modified going from low Renolds number to high Renolds number cases. However, for the range of Renolds numbers considered in this stud, the general form of the structures remain fairl similar between the cases. The size of the streaks is changing with the Renolds number. To determine the size of the turbulence structures in the near-wall region, two-point correlations, R ui u i (x,,r) =u i (x,,z,t)u i (x,,z r,t), of the fluctuating velocities in the spanwise direction, at = 8 are calculated and the corresponding one dimensional spectra in each direction are obtained. -5 number. The spectrum of the streamwise velocit is clearl influenced at all wave-numbers. The general increase in the turbulence energ of figure (d), is also clearl reflected in the spectra. However, increasing the Renolds number has affected different components in an anisotropic manner. While the energ content at low wave numbers both for the streamwise and spanwise components is substantiall increased, it is almost unaffected for the wall-normal component. A more comprehensive insight can also be gained b examining the two-dimensional spectra, see del Álamo & Jiménez (). The two-dimensional premultiplied spectra, Euu z = λ z Φ uu /u rms of the streamwise velocit fluctuations as a function of spanwise wavenumber and wall-normal position are shown in figure 5, in inner units, for two cases. Here, we have used the semi-local inner scaling as, Euu z = λ z Φ uu /u τ ; A corresponding wavelength, as λz = (π/κ z)u τ /ν, could not be defined, due to the densit variation along the wall-normal axis, see Pouransari et al. (). Here, λ z = π/κ z is the wavelength and is scaled with the corresponding local half-height of the jet for each case. The spectra have two peaks. The inner peak, which corresponds to the near-wall streaks, is located almost at the same wall-distance, in terms of semi-local inner scaling ( ) for the two cases. The outer-peak, which corresponds to the large energetic structures of the outer-laer, is located at different distances from the wall for the two cases, when inner scaling is used. This behavior is consistent with the observations made in the profiles of the turbulence kinetic energ. The peaks would essentiall coincide for outer scaling (/ / ). The figure confirms that the inner peak location is shifted to smaller scales. Though, λz cannot be used for the whole figure. One can estimate the locations of the peak in terms of λz would be roughl the same for two cases. In outer scaling λ z / /, the outer peak for the higher Renolds number is onl slightl shifted to higher values. Φii Scalar statistics and flame structure - z/h k z λ z / /.5 4 5 Figure 5. One-dimensional spanwise spectra, Φ ii, at = 8. Color code as follows, streamwise: Φ (black), wall-normal: Φ (can) and spanwise: Φ (magenta); Premultiplied spanwise spectra, Euu z = λ z Φ uu /u τ, of the streamwise velocit fluctuations, case I (black, solid) and case II (red, dashed), scaled in the semi-local inner scaling; Contours at = 5. z/h Figure 6. Instantaneous snapshots of the reaction rate fields for case I and case II; The lighter color indicates higher reaction rates. Renolds number influences on turbulence scales in the near-wall region of the wall-jet flow can be illustrated b examining the one-dimensional spanwise spectra close to the wall. Figure 5 shows the one-dimensional spectra at = 8 of the three velocit components as function of the spanwise wavenumber in outer units. As we should expect, one can observe an increase in the range of scales that is essentiall proportional to the increase in Renolds The snapshots of the reaction term for both cases are presented in figure 6. These plots show that the reaction mainl occurs in the upper shear laer in thin sheet-like structures, but also takes place near the wall. Figure 6 confirms that the flame gets much more wrinkled as the Renolds number increases. The structure of the flame is affected both in the outer laer, close to the half-height of 4

M o.9 M o, j.8 M f.7 M f, j.6.5 5 5 5.5 /.5 /.5.5..4.6.8 Θ f Θ f,m, Θ o Θ o,m, Θ Θ m. θ f,rms.5 Θ f,m. θ o,rms.5 Θ o,m. θ rms Θ m.5 (c).5.5.5 / /. v θ f U m Θ f.5,m v θ U m Θ m v θ -.5 o U m Θ o,m -. (d) -.5.5.5.5 / / Figure 8. Reactants mass flux and cross-stream profiles of the mean values, (c) fluctuation intensit and (d) wall-normal fluxes of the scalars; oxidizer: blue, fuel: red, passive: black; Subscript m refers to the maximum value. /h /h log (χ) Figure 7. Iso-contours of the logarithm of the instantaneous scalar dissipation rate, case I and caseii. the jet, and in the near-wall region. However, even though that the flame gets much more wrinkled and folded in different regions, in general also gets somewhat thinner and becomes more intensified and concentrated. The scalar dissipation rate is a quantit of prominent importance in non-premixed combustion. The Renolds number effects on the scalar dissipation rate are illustrated in figure 7 where the iso-contours of the scalar dissipation rate for two cases are shown. The fields are plotted on a logarithmic scale to accentuate their wide dnamic range. The flow is full turbulent in both cases and long thin sheetlike structures are observed, similar to what is reported b Hawkes et al. (7). These figures show much finer scale structures in the dissipation rate field of case II, indicating generation of more small scales due to the increase in the Renolds number. In the stud b Pouransari et al. (), we observed how the combustion heat release can damp the scales and perhaps merge them to larger structures, the higher Renolds number effects is clearl the opposite and the structures are getting finer at the higher Renolds number. In a non-reacting flow, one expect that the scalar dissipation rate fluctuations increase with the Renolds number, however, here in the reacting turbulent wall-jet flow we need to examine the fluctuations of the scalars. Different statistics of the reacting and passive scalars are shown in figure 8. The downstream development of the fuel and oxidizer mass fluxes are plotted in figure 8, which shows that the fuel species are consumed faster at the lower Renolds number case, at = for example about 4 % less fuel is consumed in the higher Renolds number case compared to case I. Thus, the rate of fuel consumption is decreased b increasing the Renolds number. Moving further downstream, the oxidizer mass flux in both cases increase after = 7, as a result of the oxidizer influx at the top of the domain. The cross-stream profiles of the scalar concentrations are displaed in figure 8. This plot confirms that for both non-reacting and reacting scalar, the profiles of different cases collapse when outer scaling is used. As seen in figure 8(c), the fluctuation level of the scalars has onl weakl been affected b increasing the Renolds number. The peaks of the profiles are almost unchanged. This is clearl contrar to the heat-release effects, where the influences were considerable, see Pouransari et al. (). The wall-normal fluxes, shown in figure 8(d), exhibit a similar behavior. REFERENCES Baile, S., Hultmark, M., Mont, J., Alfredsson, P.H., Chong, M., Duncan, R., Fransson, J. H. M., Hutchins, 5

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