Ind. Eng. Chem. Res. 1993,32, 1504-1508
1504
Asymptotic Rate of Decay of Turbulence in a Tube following a Combustion-Induced Step in Temperature L a n c e R.Collins'*+a n d Stuart W.Churchill Department
of
Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Combustion in a ceramic tube produces a nearly discontinuous change in temperature of the premixed fuel and air at the flame front, from room temperature up to the adiabatic flame temperature (-2100 K). The upstream Reynolds number for a stable flame in a 9.5-mm tube is in the range of 3000-6000, corresponding to turbulent flow. Owing to property changes that accompany the severe increase in temperature at the flame front, the downstream Reynolds number is reduced below the transitional value (-2100); consequently the turbulence decays while the velocity profile approaches the parabolic one characteristic of laminar flow. A previous study of ours revealed that, far downstream from the flame front, the turbulent energy decayed exponentially with downstream distance. This paper examines the asymptotic behavior of the k-r model and compares the results to that for two-dimensional (axisymmetric) disturbances in a laminar flow. Both analyses predict exponential decay; however the exponent predicted by the k-t model is substantially larger than the equivalent one for a two-dimensional disturbance. Differences in the two exponents highlight differences in the respective mechanisms for decay. The k-c model is effective when the turbulence is almost fully developed, but is unable to predict the rate of decay far downstream where the continuous spectrum of turbulent energy has given way to a discrete one. REGION 2
Introduction Previous work on combustion in a ceramic tube (Collins and Churchill, 1992) demonstrated that the essentially abrupt change in the gas temperature at the flame front causes a correspondingly abrupt change in the Reynolds number,from the 3000-6000rangedownto650-1550.This is easily demonstrated by consideringthe Reynoldsnumber for flow in a pipe, which is defined as Re = pCJD/p = 4W/ r D p . The mass rate of flow at steady state and the diameter are uniform throughout the tube. Themolecular viscosity, however, increases with temperature resulting in a reduction in the downstream Reynolds number by a factor p(T4/p(300 K), where Tr is the adiabatic flame temperature. Asaresultofthe reduced Reynoldsnumber, the turbulent fluctuations inherited from the upstream flow decay with increasing distance from the front, while simultaneously the velocity field develops into the parabolicprofilecharacteristicoflaminarflow.Earlier studies investigated the turbulent-laminar transition using a k-t model (CollinsandChurchill,1992),itseffectonpostflame reactions (Collinsand Churchill, 1990),and even a second transition due to downstream cooling (Strenger and Churchill, 1992). The k-f model used in these studies was a low-Reynolds-numberversion adapted from the model of Jones and Launder (1972) (see also Jones and Launder (1973)). It should be noted that HanjaliE and Launder (1976)subsequently developed acomplete Reynolds stress turbulence model for low Reynolds number flows;however it was deemed appropriate that the earlier study of thermally stabilized combustion begin with the simpler k-c representation. Furthermore, the strategy for effectively accounting for low Reynolds number effects is consistent between the two models; hence the results we present would agree qualitatively with those from the Reynolds stress model proposed by HanjaliEand Launder. For convenience the thermally stabilized burner was divided into three regions, as shown in Figure 1. Region 'Author to whom correspondence should be addressed. Current address: Department of ChemicalEngineering,The Pennsylvania State University, University Park, PA 16802.
E-mail address: COLLINS@I WILBUR.PSU.EDU. 0888-5885/93/2632-1504$04.00/0
-
REGION 3 T - 2100K
Re=3WO-MOO
R e = M x ) - 1500
".*.,."",
Figure 1. Schematic of three regions in ceramic burner. IO 0
ClD
Figure 2. Longitudinal profile of the root mean square velocity and length scale evaluated at the centerline for a downstream Reynolds number of 653
1 consisted of the nearly isothermal flow of the cold reactant gases upstream from the flame front, Region 2 was the very thin flame zone itself, and Region 3 consisted of the nearly isothermal flow of hot product gases downstream from the flame front. The turbulent kinetic energy and length scale were observed to decrease with distance in Region 3 as expected. The rate of decay was initially complex near the flame, but far downstream from the flame both k and r were predicted to decay exponeutially with distance (see Figure 2). The section of region 3 in which the turbulent energy decreases exponentially,
0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1505 referred to herein as the region of final decay, is the focus of this paper. The region of final decay is somewhat reminiscent of the final period of decay in homogeneous turbulence, first described theoretically by Batchelor (1949),and observed experimentally by Batchelor and Townsend (1948) (see also Batchelor (1953)). The primary mechanism for decay in homogeneous turbulence occurs via a cascade in the inertial range and viscous dissipation at the very smallest scales. Batchelor observed that as the turbulent energy was depleted by dissipation, the system should eventually reach a state in which the fluctuations were too weak to continue the cascade process. The remaining turbulence would then decay slowly due to viscous dissipation only, without the assistance of the nonlinear cascade mechanism. Batchelor referred to this as the final period of decay. On the basis of scaling arguments, he predicted a power-law relationship for the turbulent energy as shown below:
12ko=
-
u = 2U[1-
U--
ax
(r/R)'I
(6)
rar
Introducing the following dimensionless variables, X F k e x* = r* = P k* = 12,' e* = - Re = 2UR2/u' k,ulR''
2uR(9) Y
('>"
(1)
to
where p is approximately 2.5 (as compared to 1.35 for fully developed turbulence). The analysis described herein is similar in spirit to that for the final period of decay, but the problem is somewhat more complex because of the geometry. This paper compares the rate of decay predicted by the k-c model in the region of final decay to that for axisymmetric disturbances. The discrepancy between the two rates highlights a limitation of the k-e model (and other single-scalemodels) for low-level turbulence.
k-e Representation of the Region of Final Decay Collins and Churchill (1992) (see also Collins (1987)) studied the decay of turbulence and consequent rearrangement of the velocity profile downstream from the flame front using the k-c model of Jones and Launder (1972, 1973). The pertinent results for the flow within region 3 are summarized in Figure 2, which shows a semilog plot of the centerline value of 0 and 1' as a function of axial distance. It is readily apparent that although the behavior is complex near the flame front, the decay approaches an exponential form for large axial distances. It is this exponential part of Region 3 that is referred to as the region of final decay. The equations that govern the flow in Region 3, including the k-e turbulence model, have been summarized in a previous study (see eqs 32-38 in Collins and Churchill (1992)). The assumptions for the region were (1) axisymmetry, (2) steady state, and (3) isothermal flow. The results summarized in Figure 2 indicate that beyond an initial distance near the flame front, the velocity and turbulence fields approach the following limits:
-
u(x,r)= 2U[1- (r1R)'I
(2)
u(x,r) = 0
(3)
k(x,r) = f ( r )exp(-ax)
(4)
e(x,r) = g(r) exp(-ax)
(5)
-
k-e model approach their limiting constant values cf2 = 0.7 and f, = 0.08). The behavior in the region of final decay is therefore governed by the following simplified equations,
This model implies that both k and e decay exponentially with x , although the ratio kle remains finite throughout the region of final decay. A second consequence is that the turbulent Reynolds number (Rt)is exponentially small, implying that the mean-flow equations are completely uncoupled from the equations of turbulence (i.e., p J p
