Ind. Eng. Chem. Res. 1995,34, 3359-3367
3359
The Use of Lagrangian Methods To Describe Turbulent Transport of Heat from a Wall Dimitrios V. Papavassiliou and Thomas J. Hanratty* Department of Chemical Engineering, University of Illinois, Urbana, Illinois 61801
Tracking of heat markers released instantaneously from a point source on a wall boundary in a numerically simulated turbulent channel flow is used to describe the transfer of heat. The trajectories of 16 129 heat markers are monitored in time and space. These results are used to calculate temperature fields generated by continuous line or planar sources and the temperature profile in a channel with a heated bottom wall and a cooled top wall. The good agreement between the calculated fully-developed profile and the profile generated by a direct numerical simulation illustrates the equivalence of the Lagrangian and Eulerian approaches. The main point to be made is that dispersion from an instantaneous point source is a basic element. Its understanding could provide a better foundation for establishing the physics of wall heat transfer than is supplied by Eulerian analyses. 1. Introduction
The central problem in describing turbulent transfer of heat and mass is to relate the turbulent scalar field to the turbulent velocity field. The classical approach has been to describe the spatial variation of the average value of a scalar quantity in an Eulerian framework and to use the Reynolds analogy, which assumes that the eddy diffisivity of heat or mass is directly proportional to the eddy viscosity. A fundamental diEculty with this approach is that it does not directly recognize that turbulent transport is a large scale event that depends on the previous history of the velocity field with which the diffising material is associated. A particular difficulty is that the turbulent Prandtl number is not a constant but varies with molecular Prandtl number and with distance from a wall in a manner that is not currently explained. The present paper explores the possibility of using Lagrangian methods to describe turbulent transport from a wall. An approach is taken which recognizes that turbulent transport of a scalar is a linear process. Statistical properties of the temperature field at a given location are described as resulting from the sum of contributions of a number of different sources and sinks at the wall. Within this framework the basic physical problem is to describe the behavior of one of these sources or sinks. This description of turbulent transport has been hampered because of the difficulty of carrying out studies of wall sources in the laboratory. However, the recent availability of high-speed supercomputers allows a Lagrangian analysis to be pursued in a meaningful way. This paper carries out experiments with a velocity field that is generated by a direct numerical simulation (DNS) of turbulent flow in a channel. Heat (or mass) markers are released, at a given instant, from the bottom wall, and their trajectories are followed in space and time. Results from these experiments are used to describe the temperature field that results from a heat transfer situation in which the bottom wall is heated and the top wall is cooled. The calculated fully-developed temperature profile is compared with a direct numerical simulation. The main motivation for this work is the expectation
* Author to whom correspondence should be addressed. E-mail:
[email protected].
that the physics of turbulent transport evolve in a more natural way in a Lagrangian framework. 2. Background
The dispersion of a molecule of a thermal species from a point source can be described in terms of the x component - of the mean-squared displacement from the source, p. Einstein (1905) developed the following relation for molecular dispersion in a nonturbulent field:
dx2= m dt where D is the molecular diffisivity. The most influential contribution to the theory of diffision in a turbulent field is Taylor’s (1921) description of the dispersion of fluid particles from a point source in a homogeneous, isotropic turbulence:
where u2 is the mean square of the x component of the velocity of fluid particles and RL is the Lagrangian correlation coefficient. The influence of history on the rate of dispersion is defined by specifying RL(t). At small t the Lagrangian correlation is unity and the rate of dispersion increases linearly with time. For large times the Lagrangian correlation goes to zero. The rate of dispersion is a constant given by
dx2-
-= 2u2? dt
(3)
where the Lagrangian time scale is defined as
8 = I R L ( z )dz
(4)
Reviews of advances since the publication of Taylor’s paper have been given by Batchelor and Townsend (1956), Monin and Yaglom (1971),and by Hinze (1987). An additional complication arises in describing the dispersion of heat (or mass) since the marker can move off a fluid particle as a result of molecular diffusion. As
0 1995 American Chemical Society
3360 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
a consequence, markers with different Prandtl, Pr, numbers can exhibit different behavior; the influence of molecular diffision increases with increasing Pr. Saffnan (1960) defined a material autocorrelation function, Ri, and showed that
with
where vi(x0,t) is the ith component of the fluid velocity a t the location X(x0,t) of a marker which was released at xo at t = 0. The material autocorrelation differs from the Lagrangian correlation, RL,in that it correlates fluid velocity components along the trajectories of markers instead of fluid particles. Taylor’s and Saffman’s formulations refer to homogeneous and isotropic turbulence. Corrsin (1953,1959) examined the diffusion from a line source in a homogeneous &ear flow with a constant mean velocity gradient, dU/dy. Lagrangian displacement in the direction of the mean flow, X,is different than displacement in the direction of the velocity gradient, Y,
where u is the velocity of the particle - in they direction. The mean-squared displacement, p,is the same as in Taylor’s analysis. For large times the dispersion is (Corrsin, 1959; Monin and Yaglom, 1971)
where is the Lagrangian time scale in the y direction. The theory of Taylor was used by Hanratty (1956) to describe heat transfer in a channel as resulting from an infinite number of line sources of heat along one wall and an infinite number of line sinks of heat along another wall. A spatial variation of the eddy conductivity was found to result from the time dependency of turbulent diffusion. Temperature gradients close to the wall were found to be associated with thermal markers that had been in the field for small periods of time. In the center of the channel the thermal markers have - long lifetimes so the eddy diffusivity is equal to u2r, A difficulty with this analysis is that a homogeneous isotropic velocity field with no mean velocity gradients was assumed. The attempt by Eckelman and Hanratty (1972)to take account of these effects points to the need to obtain experimental information on the behavior of wall sources. Laboratory studies have been carried out by Poreh and Cermak (19641, Raupach and Legg (1983), Parantheon et al. (1983), and Incropera et al. (1986). None of these give the space-time behavior of an instanta-
neous source, needed t o demonstrate the use of Lagrangian methods to describe Eulerian temperature fields. 3. Computational Method
In order to calculate trajectories of individual heat markers, detailed instantaneous information about the turbulent field is required. A DNS gives the evolution of the fluctuating Eulerian velocity field at a large number of spatial locations by solving the threedimensional time-dependent Navier-Stokes equations without any modeling assumptions. A code developed by McLaughlin and discussed by Lyons et al. (1991a) is used in this work; it simulates fully-developed turbulent flow in a channel at Re = 2660, on the basis of the bulk velocity and half channel height. The fluid is Newtonian, has constant physical properties, and is incompressible. The simulation is done on a 128 x 65 x 128 grid in x , y, z. The streamwise direction is x , the spanwise is z , and the direction normal to the channel walls is y. The dimensions of the computational box are 4nh x 2h x 2nh, where h = 150 is the half height of the channel in wall units. The flow is regarded as periodic in the streamwise and spanwise directions,with periodicity lengths equal to the size of the computational box in these directions. No-slip boundary conditions are imposed at the rigid channel walls. The Navier-Stokes equations are integrated in time using the pseudospectral fractional step method developed by Orszag and Kells (1980) with a pressure field correction suggested by Marcus (1984). All variables are made dimensionless with the use of wall parameters: the kinematic viscosity, v , and the friction velocity, u*. The numerical accuracy of these simulations was verified by a comparison with laboratory measurements at the same conditions (Niederschulte, 1989). Statistics for particles released at the wall of the channel can be calculated using the tracking algorithm developed by Kontomaris et al. (1992). The basic assumption is that a heat marker at each time has the velocity of the fluid particle on which it “rides”. The equa_tion of particle motion is integrated numerically. Let VEo,t)be the Lagrangian velocity of a particle which y a s at location 20at time t = to = 0. Then the position X(20,t)of the marker at time t is given by the equation of motion
(10) The Lagangian velocity is related to the Eulerian velocity U by
V(Z0,t)= az(20,t),t1
(11)
By using a mixed Lagrangian-Chebyshev interpolation scheme to calculate the velocity vector between grid points, it is possible to track heat or mass markers. Molecular diffision is simulated by imposing a random walk on the particle motion; it is added to the convective part of the motion after each time step At and takes values from a Gaussian distribution with zero mean and standard deviation (T = ((2At)/Pr)1’2, in wall units (Kontomaris and Hanratty, 1994). The description of the diffusion process by random walk of a large number of particles in a three-dimensional space has been reviewed in detail by Chandrasekhar (1943).
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3361 Four tracking experiments were performed with the same velocity field simulation. Prandtl or Schmidt numbers of 0.1, 1,10, and 100 were studied with 16 129 markers released instantaneously for each case. The initial position of the markers was on a uniform 127 x 127 grid that covered the bottom wall of the channel. The trajectories of these markers were traced, and their velocities and positions were stored every wall time unit for statistical post-processing. The simulation was carried out on one processor of a CONVEX C3880 supercomputer up to 2000 wall time units and required about 27 CPU h for 500 time iterations. It is noted here that Eulerian computations at Pr = 100 are not handled by the present generation of supercomputers due to memory and speed limitations, since extremely fine grids and very small time steps are necessary to resolve the steep temperature gradients close to the wall.
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4. Instantaneous Single Point Source at the
Wall Homogeneity in the streamwise and spanwise directions of the Eulerian velocity field removes any statistical dependence on the initial location of the markers. The paths of the 16 129 markers for each experiment can be viewed as an ensemble of 16 129 independent trajectories. Figure la-d presents initial stages of individual trajectories for typical markers with Pr = 0.1, 1, 10, and 100, which were released from the same position at the wall. At small times the markers move away from the wall by molecular diffusion. Eventually they become entrained in the turbulence, and turbulent motions dominate their dispersion from the wall. As Pr decreases, the distance from the wall a t which the particles become entrained increases, but the time period for which the markers are dispersed only by molecular diffusion decreases. Three zones are observed: a region close t o the wall where molecular diffusion dominates, an intermediate region where there is an interaction between molecular and turbulent diffusion, and a region where turbulent dispersion dominates. For very small Pr the first time zone is very small, The magnitude of the molecular jump of the particles decreases with Pr, so the marker trajectories become smoother with increasing Pr. The ensemble of markers can be viewed as a cloud released instantaneously from the bottom wall of the channel a t t = 0 and x = 0. Let Pl(x - xoy,t - t o l t o ~ o ) be the joint and conditional probability density function for a marker to be at a location ( x y ) a t time t, given that the particle was released at xo a t time to. In the present case t o = 0 and xo = 0. Figure 2 shows contours of P1 for the location of a Pr = 1marker a t six different times. The contour lines can be interpreted as concentration contours and, thus, as snapshots of a cloud of contaminants released instantaneously from xo = 0. Note that in Figure 2 parts a-c and d-f are plotted on a different scale. At small times the cloud is more dense and stays close to the source; the markers move due to molecular diffusion while they are in the viscous wall region (t = 25). When the edge of the cloud enters the outer region of the flow (t = 50), markers are swept by eddies and the cloud expands in the normal direction (t = 100). Turbulent diffusion is dominating the dispersion of the cloud for t > 100. As a result of the normal extension of the cloud, the mean velocity that the markers experience varies with the distance from the wall. The markers closer to the center of the channel "see" higher mean velocity and thus move faster down-
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Y Figure 1. Typical trajectories of heat or mass markers released at the same position at the wall. Particle no. 16 was released a t x = 0 and z = 118.75, and particle no. 16 000 was released at 3c = 1865 and z = 927. Point A denotes the end of the region dominated by motion due to molecular diffusion. From top to bottom: (a)Pr = 0.1, (b)Pr = 1, (c) Pr = 10, and (d) Pr = 100.
stream than the markers closer to the wall. Figure 3 shows the behavior of the cloud for different Pr at the same time instant (t = 100). For Pr = 0.1 the contami-
3362 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 -50. -60
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Figure 3. Pr effects on the joint probability function for a heat or mass marker to be at a location (yp)at t = 100, when it was released at zo= 0 and t o = 0, (a) Pr = 0.1, (b)Pr = 1, (c) Pr = 10, and (d) Pr = 100, (y = -150 corresponds to the bottom wall of the channel, and y = 0 corresponds to the center).
nants are seen to have moved far downstream, where the cloud extends almost to the center of the channel.
For Pr = 1 the cloud is compact; it is closer to the source because it has been exposed, on average, to smaller
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3363 7 h
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streamwise velocities. The compactness of the cloud a t
t = 100 increases with further increases in Pr. The spreading of the cloud for each Pr is related to the amount of time that the dispersion is dominated by turbulent diffusion. Markers with lower Prandtl numbers undergo larger molecular jumps and become free from the viscous wall region a shorter time after their release. Consequently, at t = 100, the Pr = 0.1 markers have been under the influence of turbulent diffusion longer than the markers for the other Pr. However, at the same time the Pr = 100 markers are still in the viscous wall region and move predominantly due to molecular effects. Probability function PI is used as the building block for the synthesis of mean concentration profiles in the next section of this paper. Figure 4a-c presents the mean displacement of the markers for all four Pr until t = 2000. The average is calculated over the trajectories of all markers:
where n = 16 129 is the total number of markers and (xy)i(t)is the location a t time t of the ith marker which was released at (x0yo)i a t t o = 0. Since the markers are released at the wall, yo = 0. For Pr = 0.1 the markers leave the laminar sublayer close to the wall shortly after their release and rapidly disperse toward the center of the channel. For Pr = 100, markers stay in the viscous sublayer for the first 250 time units. There they see a low mean velocity. After about 800 viscous time units they are entrained in large scale eddies and swept to the center of the channel. At large enough times the average position of the markers in they direction will be the center of the channel, since the markers eventually are distributed uniformly across the channel. In this sense, Figure 4b is an indicator as to whether a stationary state has been reached for each experiment. For low Pr (Pr = 0.1 and 1)average displacements in the normal direction seem t o be close to the asymptotic limit a t t = 2000. The high Pr (Pr = 10 and 100) experiments appear t o be in a transient state, even a t t = 2000. At every time instant, t , a marginal probability density function can be defined, P(y), for a marker to be at distance y from the wall when it was released a t to and xo. Probability density P(y) is interpreted as a concentration profile (Cermak, 1963; Hunt, 1985). Figure 5 presents P(y) a t different times for Pr = 1. As time progresses the probability density function tends to become more uniform across the channel. Second moments of the marker displacement,
(13)
are shown in Figure 6a. Values ofz2, calculated from P(z), are presented in Figure 6b. The walls of the channel confine the motion of the markers and force P(y) to become a uniform distribution at large times. The variance of Pot) then should equal that of a uniform distribution between y = 0 and y = 2h, u2 = (300- 0)2/ 12 = 7500, -and the mean square of the displacement should be y2 = u2 T2 = 7500 1502= 30 000. Thus, as time increases and P(y) tends toward uniformity,
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3364 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 m
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y2 is expected to reach the value 30 000 for all Pr, as shown in Figure 6a. The expected value of the squared displacement in they direction can be another criterion for the achievement of a stationary state. The meansquared displacement in the spanwise direction, z2, is closer to Taylor's model due to the homogeneity in this direction. A quantitative measure of the time for which the trajectories are dominated by molecular motion can be obtained by comparing the simulation results to Einstein's relation for molecular diffusion (eq 1). This is done in Figure 7a,b, where values of 7, 12, 15,and 45 viscous time units are estimated for Pr = 0.1, 1, 10, and 100. These times are indicated in the individual marker trajectories shown in Figure 1. The rate of dispersion may be considered to be the sum of contributions from molecular diffusion and turbulent mixing. Thus
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Figure 7. Comparison of mean squared particle displacement in the normal direction a t the initial stages of the simulation with Einstein's predictions for molecular diffision. (a, top) Low Pr (Pr = 0.1, and Pr = 1). (b, bottom) High Pr (Pr = 10, and Pr = 100). -Pr=O.l
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