REVIEW pubs.acs.org/IECR
On the Gradient Diffusion Hypothesis and Passive Scalar Transport in Turbulent Flows Daniel P. Combest, Palghat A. Ramachandran, and Milorad P. Dudukovic* The Chemical Reaction Engineering Laboratory, Department of Energy, Environmental, and Chemical Engineering, Washington University, St. Louis, Missouri 63130, United States ABSTRACT: A discussion of modeling passive scalar transport in turbulent flows is given. Several methods employed to close the scalarflux term Æuuφuæ that arises during Reynolds averaging are provided. Alternatives and improvements to the gradient diffusion hypotheses are addressed, most notably, the need for an alternative to the global constant turbulent Schmidt and Prandtl numbers. The reader is given a brief history covering methods used to predict turbulent Schmidt and Prandtl numbers, along with recommendations for future research, based partially on studies by Professor Stuart Churchill. More detailed formulations of turbulent Schmidt or Prandtl numbers will enable better approximations of the influence of turbulence in models of passive scalar flows using the gradient diffusion hypothesis.
1. INTRODUCTION The development of accurate mathematical descriptions and computationally efficient numerical methods representing reacting turbulent flow is paramount to the future of reactor modeling in chemical reaction engineering (CRE). As computational capabilities increase with the improvement of numerical algorithms and computing hardware, CRE moves toward more-detailed multiscale models of chemical reactors. The methods and complexity of such technology aim to maximize description with the constraint of computational cost and practicality. This begs the question: How can an existing method be employed more efficiently to produce more accurate and useful results? This paper strives to address this question, with regard to the modeling of passive scalar transport in a turbulent flow using a Reynoldsaveraging approach, the use of the gradient-diffusion hypothesis to close the scalar-flux term, and the turbulent Schmidt (Sct = νt /Dt ) and Prandtl (Prt = νt /rt) number-based approximations of the turbulent mass and thermal diffusivity. The following discussion in this paper is mainly aimed at the graduate student or researcher, providing a brief background, building up to the suggested improvement of Sct and Prt estimation techniques. Section 2 defines the scalar and gives a discussion on scalar mixing in turbulent flows. Section 3 reviews turbulence modeling using the Reynolds-averaged NavierStokes (RANS) equations, enabling the reader to become familiar with the turbulent closure problem. Section 4 introduces the transport of passive scalars, while several methods of closing the scalar-flux term are addressed in section 5. Section 6 discusses the meaning of the Sct and Prt, with a review of the numerical determination of Sct and Prt. Finally, section 7 discusses a reinterpretation of the Prt by Professor Stuart Churchill and a look toward the near future of Sct and Prt prediction. Overall, the reader should be left with an understanding of the typical usage of Sct and Prt in computational fluid dynamics (CFD) simulations. Conclusions and recommendations for future research that fully leverages Sct and Prt to more accurately predict passive scalar transport in turbulent flows using Reynolds-averaged models are given in the final section. r 2011 American Chemical Society
2. CONSERVED SCALARS IN TURBULENT FLOWS A scalar is defined as a rank-zero tensor representing a simple physical quantity (e.g., volume fraction, mass, temperature, or species concentration) in a system. Subsequently, transport equations are used to describe the conservation of a scalar within a system to represent inflow, outflow, generation, and accumulation of a scalar quantity. If scalar transport does not interact with concurrent transport processes in the system, it is appropriately named a passive scalar. Conversely, if the scalar affects other simultaneous transport processes, it is referred to as an active scalar (e.g., thermal interactions with momentum through a buoyancy term in the momentum conservation equation). Scalar transport (passive or active) is present throughout many scientific fields, from basic science in the study of transport phenomena to industrial engineering processes. In the case of a scalar in a flowing fluid, turbulence is often encountered and requires additional consideration. Although our understanding of turbulent flow has grown tremendously over the last century, fundamental knowledge of the mechanism of conserved scalar transport in a turbulent flow is still a growing field of research. Recent reviews by Dimotakis,1 Warhaft,2 and Tominaga and Stathopoulos3 provide valuable background covering the experimental, theoretical, and modeling work concerning scalar transport during the 20th century. Dimotakis reviewed turbulent mixing, with a thorough discussion into what has been categorized as level-1, level-2, and level-3 mixing,1 where • level-1 mixing involves passive scalars, such that the combination of a fluid and tracer or fluids of similar properties result in no effect on overall flow dynamics. Special Issue: Churchill Issue Received: January 10, 2011 Accepted: May 4, 2011 Revised: April 12, 2011 Published: May 04, 2011 8817
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• level-2 mixing is indicated by the interaction of two fluids causing a change in flow dynamics (e.g., RayleighTaylor instability flows). • level-3 mixing is categorized by mixing that produces changes in the overall fluid-intensive properties (density or composition), resulting in a change in flow dynamics (e.g., buoyancy-driven flow). Dimotakis noted that studies on level-2 and level-3 mixing are very much open research topics, while level-1 mixing research has been limited to canonical flows (e.g., pipe flow, free shear layers, and jets) and relations to empirical data.1 The conclusions drawn by Dimotakis1 end on the hope for experimental studies to produce more-detailed data needed to develop large eddy simulation (LES) subgrid scale (SGS) models based on observations of scalar mixing, rather than on low-order statistics. Warhaft focused on the turbulent passive scalar, touching deeply on passive scalar anisotropy and the direct connection of turbulent length scales, rather than the cascading mechanisms of turbulent interactions.2 Mainly giving an experimental perspective, cornerstones of turbulence theory are addressed in an effort to understand intermittency. However, Warhaft’s discussion,2 which is fundamentally important in understanding turbulence, is beyond the general remarks of this review and left to the motivated reader. Lastly, the work by Tominaga and Stathopoulos offers a discussion of the Reynolds-averaged approach to modeling passive scalars, with an emphasis on gas-phase urban and building diffusion problems.3 The overall conclusion from Tominaga and Stathopoulos3 is that there is a need for optimal global Sct numbers based on dominant effects, instead of the generally accepted values of the turbulent Schmidt (Sct) number in the range of 0.70.9. The implications of a turbulent Schmidt number are provided later in this discussion. The cited reviews provide an appropriate introduction into the topic of passive scalar transport in turbulent flows, but leave room for further improvement on the subject in terms of Reynoldsaveraged modeling approaches.
3. THE REYNOLDS-AVERAGED NAVIERSTOKES EQUATION In the Reynolds-averaged NavierStokes (RANS) equations, turbulent momentum conservation is described using the Navier Stokes equations coupled with a perturbation in the velocity and pressure variables.4 The approach uses a time- and space-dependent velocity variable U(x,t) decomposed into an ensemble-averaged velocity ÆUæ and a fluctuating velocity uu component, where 0
Uðx, tÞ ¼ ÆUðx, tÞæ þ u ðx, tÞ
ð1Þ
Similarly, the pressure can be represented in terms of the Reynolds decomposition of the original variable p(x,t), so that pðx, tÞ ¼ Æpðx, tÞæ þ p0 ðx, tÞ
ð2Þ
By substituting eqs 1 and 2 into the NavierStokes equations, the continuity equation becomes r 3 ðÆUæ þ u0 Þ ¼ 0
ð3Þ
and momentum conservation is described by DÆUi æ DÆUi æ 1 D 0 0 1 DÆpæ þ ÆUj æ ¼ νr2 ÆUi æ Æu u æ Dt Dxj F Dxj i j F Dxj
ð4Þ
In eqs 3 and 4, ÆUæ and uu are solenoidal vector fields,5,6 ν is the molecular kinematic viscosity, and F is the fluid density. Equation
4 is referred to as the unsteady Reynolds-averaged Navier Stokes equation (or URANS). For steady-state turbulent flows, the time derivative in eq 4 is zero and the remaining equation is referred to as the Reynolds-averaged NavierStokes (or RANS) equation. Equation 4 presents a Reynolds stress term, F Æuuu i uæ j as a cross correlation of uiu and uju, leaving more unknowns than equations. This is referred to as the turbulence closure problem. To deal with the issue of closure, the problem can be approached in several different ways, three of which are presented in this discussion. The first method is to approximate the unclosed Æuiuujuæ term in the RANS and URANS equations with the Boussinesq eddy viscosity hypothesis,7 given as8 ! DÆUi æ DÆUj æ 2 DÆUk æ 0 0 FÆui uj æ ¼ μt þ þ Fk δij μt Dxj 3 Dxk Dxi ð5Þ where k is the turbulent kinetic energy (k = uuu δij is the i u/2), j Kronecker delta,4 and μt is the turbulent viscosity scalar determined by a turbulent viscosity model. Over the last century, numerous turbulent viscosity models have been devised, including kε, kω, SpalartAllmaras, and RNG kε. These models, along with numerous other formulations, are covered in detail in many books on fluid dynamics and computational continuum mechanics.8,9 The second method for determining the unclosed stress F Æuiuujuæ involves modeling the Reynolds stresses with transport equations. The model derivation for the Reynolds stresses transport equations is thoroughly covered in many books on computational fluid dynamics.8,10 The final equation11 of the transport of Reynolds stresses is expressed as 0
0
DÆui uj æ Dt
0
þ ÆUk æ
0
DÆui uj æ Dxk
0
þ
0
0
DÆui uj uk æ Dxk
0
0
¼ Pij þ Πij þνrÆui uj æ εij ð6Þ
with the production term being defined as 0 0 DÆUj æ 0 0 DÆUi æ þ Æuj uk æ Pij ¼ Æui uk æ Dxk Dxk
the velocitypressure gradient term being defined as 0 1 0 Dp0 0 Dp Πij ¼ ui þ uj F Dxj Dxi and the turbulent dissipation term 0 0 Dui Duj εij ¼ 2ν Dxk Dxk
ð7Þ
ð8Þ
ð9Þ
The triple-correlation term Æuiuujuuku æ is closed with an assumption similar to the Boussinesq eddy viscosity hypothesis. An additional approach used to close the Reynolds stress term involves algebraic Reynolds stress models (ASMs). These types of models assume a nonlinear constitutive model to relate the Reynolds stresses and the rate of mean strain rather than additional transport equations for each of the Reynolds stresses.8 These relationships are held to physical and mathematical constraints including Galilean invariance and “realizability” to ensure physically meaningful results. Lastly, the basis of ASMs require that adjustments be made to adequately predict turbulent flow 8818
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and generally do not perform as well as a full Reynoldsstress model. The RANS equation and subsequent closure relations provide valuable insight that has been applied to Reynolds-averaged passive scalar transport discussed in later sections. What is important to note is that gradient diffusion hypothesis approximations are used extensively throughout RANS model derivation in the Boussinesq eddy viscosity hypothesis (eq 5) and throughout ASMs. What is also integral to the discussion has been addressed by Tannehill, Anderson, and Pletcher,8 stating that transport equations can be written to close almost any term, but none of them can be solved exactly. This idea, along with closure techniques for the Reynolds-stress term Æuiuujuæ, are applied and discussed in the next section concerning the Reynolds-averaged passive scalar transport.
4. THE REYNOLDS-AVERAGED PASSIVE SCALAR EQUATION Inasmuch the RANS equation requires closure to account for the effect of chaotic turbulent fluctuations, so does the transport of a passive scalar in a turbulent flow field.5,11 The conservation equation describing the scalar transport of species R is DφR þ r 3 ðUφR Þ ¼ r 3 DR rφR þ SR ðφÞ ð10Þ Dt where SR(φ) is the chemical source term (i.e., rate of production) for species R. Using Reynolds decomposition,4 a scalar quantity φR is decomposed to an ensemble-averaged value ÆφRæ and a fluctuating component φRu as 0
φR ðx, tÞ ¼ ÆφR ðx, tÞæ þ φR ðx, tÞ
ð11Þ
Substituting eq 11 into the scalar transport equation for species R in eq 10, we are left with DÆφR æ þ r 3 ÆUφR æ ¼ r 3 DR rÆφR æ þ ÆSR ðφÞæ Dt
ð12Þ
where the second term on the left-hand side of eq 12 requires further decomposition to r 3 ÆUφR æ ¼ ¼
0
r 3 ÆðÆUæ þ u0 ÞðÆφR æ þ φR Þæ 0 r 3 ðÆUæÆφR æ þ Æu0 φR æÞ
ð13Þ
Back substitution of eq 13 into eq 12 results in the unsteady Reynolds-averaged passive scalar equation DÆφR æ 0 þ r 3 ðÆUæÆφR æÞ ¼ r 3 DR rÆφR æ r 3 Æu0 φR æ þ ÆSR ðφÞæ Dt ð14Þ As it was seen in the RANS equation, the unclosed ÆuuφRu æ scalarflux term in eq 14 involves fluctuating components that must be reconciled. In the case of production (i.e., nonzero chemical source SR(φ)), additional numerical treatment is required to close the ensemble-averaged chemical source term ÆSR(φ)æ. Further discussion of the chemical source closure can be found in works by Pope5 and Fox.11 The closure of the ÆuuφuRæ term in eq 14 is discussed in the next section.
5. THE CLOSURE OF THE SCALAR-FLUX TERM ÆuuφRu æ The closure of the ÆuuφRu æ term in eq 14 is generally accomplished with either a gradient-diffusion hypothesis (GDH)
model that is similar to the eddy viscosity model (eq 5), an algebraic moment (AM) model, or a scalar-flux transport model (SFM). Each of these approaches differ in formulation, as well as complexity of transport equations, and has strengths and weaknesses that will be covered in this discussion. The full derivations and applications of each of these methods are available in the cited literature. 5.1. Gradient Diffusion Models. By far, the simplest method to account for ÆuiuφRu æ is to use a gradient diffusion hypothesis (GDH), in which νt 0 Æu0 φR æ ¼ Dt rÆφR æ ¼ rÆφR æ ð15Þ Sct with Sct being the turbulent Schmidt (or turbulent Prandtl) number and Dt being the turbulent mass diffusivity. The gradient diffusion hypothesis assumes isotropic turbulence for simplicity. As a result, the GDH is known to inaccurately predict turbulent effects in cases where the scalar flux is not aligned with the mean scalar gradient (i.e., highly anisotropic flows). The benefit of the GDH closure is that there are no additional transport equations, making it relatively simple to implement numerically. 5.2. Algebraic Models. A more rigorous approach than the GDH involves determining an anisotropic turbulent diffusivity tensor Dtij, which was first introduced by Batchelor12 as 0
0
Æui φR æ ¼ Dtij
DÆφR æ Dxj
ð16Þ
with the difficulty being observed in directly determining the components of Dtij. Younis13 noted that the simplest rational algebraic model (AM) was developed by Daly and Harlow,14 by setting Dtij directly proportional to the Reynolds stresses, with k 0 0 0 0 DÆφ æ Æui φR æ ¼ Cθ ð17Þ Æui uj æ R ε Dxj where Cθ is set as a positive constant. This formulation overcomes one of the shortcomings of GDH due to misalignment of scalar flux and mean scalar gradient but, as Younis13 comments, eq 17 predicts the wrong magnitude of the scalar flux in the direction normal to the mean scalar gradient. The conclusion by Younis was drawn through comparison of the ratio of streamwise to cross-stream heat fluxes in fully developed channel flow with heated walls, noting the slight differences of the DalyHarlow model (eq 17), compared to the DNS results by Kim.15 A similar model that was discussed by Fox11 reveals a slight variation, where 0
0
Æui φR æ ¼
k 0 0 DÆφR æ Æu u æ Sct ε i j Dxj
ð18Þ
which is used in conjunction with the kε or Reynolds-stress models. Equation 18 correspondingly overcomes the flaw of scalar flux and mean gradient misalignment of the GDH, because of its use of the anisotropic Æuuu i uæ j term. Both eqs 17 and 18 draw criticism11,13 for their lack of accuracy in predicting the scalar flux; yet, they both imply a strong influence of Reynolds stress on simple scalar-flux models. This is a more agreeable description of physical phenomena in turbulent flows than those relying on the simplifying assumptions of turbulent isotropy. The theme of Reynolds-stress influence on scalar flux is recurrent in commentaries by Churchill,16 related directly to Prt, and the discussions in classic literature by Levich17 on diffusion rates in turbulent 8819
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boundary layers. Not surprisingly, more extensive models used to close the scalar-flux yield more detailed descriptions of turbulent effects on scalar transport. Hence, one may more accurately resolve scalar flux by utilizing additional transport equations, which are discussed in the next section. 5.3. Scalar-Flux Models. Similar to the Reynolds-stress equations, in a scalar-flux model (SFM) approach, the ÆuuφRu æ is treated directly11,13 with the transport equation: 0 0 0 0 DÆui φR æ DÆu φ æ D 1 0 0 0 0 þ ÆUj æ i R ¼ Jij Æuj ui φR æ Æp0 φR æδij Dt Dxj Dxj F þ Pi þ Ri εi
ð19Þ
where Jij is the molecular diffusion component (which is often neglected, since molecular diffusion will be small compared to turbulent diffusion effects in higher Reynolds number flows); Pi is the closed production term; Ri is the unclosed pressure-scalar-gradient term; and εi is the scalar-flux dissipation term (which is often neglected if small-scale isotropy is assumed). The remaining ÆujuuiuφRu æ and ÆpuφRu æ flux terms are generally closed by relations similar to the gradient-diffusion hypothesis.11 The triple correlation term Æuuu j uφ i uRæ in eq 19 should be of much lower magnitude, compared to the double correlation term ÆuiuφRu æ in eq 14, making the use of GDH to describe ÆujuuiuφRu æ, which is an acceptable approximation of less-dominant phenomena in the SFM. The SFM is one of the most-detailed Reynolds-averaged models for scalar-flux transport, and it is the most computationally intensive method, compared to the GDH and AM presented in this discussion.
6. THE TURBULENT SCHMIDT AND PRANDTL NUMBERS Throughout the last century, the turbulent Schmidt (Sct) and Prandtl (Prt) numbers have played a key role in turbulent passive scalar transport modeling that utilizes the gradient diffusion hypothesis (eq 15). Specifically, Sct or Prt are used to relate the turbulent mass or thermal diffusivity in eq 15 to the turbulent viscosity (νt) determined by a turbulent-viscosity model (eq 5) in the RANS or URANS equations. This leads to the key question of what determines Sct and Prt and why are they important? This is briefly addressed in the following section. In general, Sct or Prt is treated as a global parameter in complex fluid simulations; the parameter is usually set to a default value of 0.7 or unity. The treatment of Sct and Prt numbers in this sense is ubiquitous: they are used in both open source18 and commercial19 CFD packages; are accepted in scientific literature;3,2022 and detailed in the scientific literature by Pope,5 Ranade,23 and Fox11 devoted to the subject of reacting turbulent flows and chemical reaction engineering. Estimates of a global turbulent Schmidt number in homogeneous turbulent flows were discussed thoroughly by Corrsin24 by comparing Taylor microscales for the scalar φu and velocity uu components. As noted by Corrsin, values of Sct were approximately unity or less, in agreement with the work of Batchelor.25,26 Although the results of Corrsin were valuable for homogeneous turbulence, inhomogeneous turbulence is often encountered in practice and requires transport equations to model its effects. Nevertheless, the use of a constant global turbulent Schmidt number in packed-bed simulations,22 urban diffusion problems,3 and combustion modeling27,28 is seen throughout computational fluid dynamics, ranging in value from 0.1 to 2.2. This large range alludes to the fact that prescribing a global value is
problem-dependent. In all cases, a sensitivity study of a particular variable correlated to Sct should be performed.28 In contrast to the use of a global Sct and Prt to lump the complex relationship of turbulent fluctuations and turbulent mixing through values of νt and Dt or Rt, very little work has been done over the last few decades to elucidate the factors influencing Sct and Prt. As addressed in a 1974 article by A. J. Reynolds,29 the previous 25 years of predictions of the turbulent Schmidt number (Sct = νt/Dt) and Prandtl number (Prt = νt/Rt) were reviewed. The discussion was restricted to temperature and concentration ranges that do not significantly interact with the flow (i.e., passive scalar flows and level-1 mixing), and assumed an almost-perfect analogy between Prt and Sct numbers, with minor exception in gases. This exception is based on the differences between actual and assumed enthalpy fluxes seen in the flows, where they are negligible and within measured experimental variability. Reynolds noted that Prt and Sct depended on the corresponding molecular value of Pr and Sc, the position within the flow (i.e., distance from the wall), and the local turbulent intensity of the flow in question. In general, Reynolds summarized these observations into a single general formula: " n # νt ð20Þ Sct ¼ C1 exp C2 Scm ν where C1, C2, m, and n are all positive constants. In eq 20, the position within the flow is taken into account implicitly through the νt term, which is spatially variable and highly influenced by the wall and turbulent boundary layer. Equation 20 is consistent with the observation that Sct decreases as νt/ν increases, indicating greater turbulent mixing in more-turbulent regions. For νt/ν ratios approaching zero, Sct approaches C1. This is inconsistent with the definition of Sct = νt/ Dt, because Sct should be undefined, since νt = 0 and Dt = 0 in laminar regions. The boundedness of eq 20 in laminar regions makes its use practical, from a modeling perspective, albeit incorrectly overstating turbulent mass flux in the laminar sublayer of boundary layers. Finally, all of the relationships presented by Reynolds classified the Sct and Prt numbers into Prandtl mixing-length-based models, simple empiricism, and statistical calculations. Reynolds concluded that the mixing length models were not fundamentally advantageous, since they do not elucidate transport phenomena but are practical in the sense that they are easy to use and contain realistic information. The remaining models were neither practical nor fundamentally advantageous, since they merely served to “remind us how little confidence can be placed in any limited group of measurements”.29 Although the mixing-length models were considered most desirable at the time, Reynolds concluded that more work should be done to account for position from the wall separately from the turbulent intensity. Since the review of A. J. Reynolds,29 further research related to the prediction of Sct and Prt numbers has been more empirical. Work by Jischa30 described the parameters Prt and Sct, using simple empirical correlations, Pr t ¼ A þ
B Pr
Sct ¼ A þ
B Sc
ð21Þ
with no influence of wall distance or turbulent intensity taken into account. A review by Kays31 examined experimental data on the Prt number for 2D turbulent boundary layer flows in circular pipes or flat ducts. Kays discussed empirical models based on DNS and 8820
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experimental data to predict Prt in different regions of the boundary layer for various fluids (i.e., air, water, oil, liquid metals). The conclusions by Kays further support the conclusions by A. J. Reynolds29 in that the Prt and Sct numbers are dependent on the molecular Pr and Sc numbers and the distance from the wall. More recent work by Koeltzsch32 investigated the height dependence of Sct in boundary layers through wind-tunnel experiments. As a result, a power series approximation of Sct was used to fit experimental data, only to further show the wall dependence of Sct in turbulent boundary layer flows. In a different approach, Guo et al.33 applied the genetic algorithm to optimize a variable Sct proportional to three constants to control the magnitude of the Sct, the relative importance of turbulent frequency scale, and the affect of asymmetry in the stresses to agree with experimental results for a jet in crossflow. Validated against three different cases, Guo et al. showed qualitative and quantitative agreement with previously published data. For completeness, a comparison against cases of constant global Sct was performed. The work by Guo et al. addresses the need for new spatially and temporally variable Sct (rather than a global constant Sct) and shows that a variable Sct number fits numerical data to experimental data better than using a constant Sct number, but fails to elucidate phenomena influencing the Sct in turbulent flows. The overall importance of the Sct and Prt parameters in CFD is that the value of these two numbers strongly influence the level of turbulent mixing in the modeled system. If small global Sct and Prt numbers are assumed, then a large turbulent mass or thermal diffusivity is defined. For instance, in combustion, a lower Sct number intensifies combustion due to enhanced species diffusion and turbulent mixing, while a higher Sct number may create mixtures that are unable to sustain combustion.27 Without a more fundamental understanding of the Sct and Prt numbers, more uncertainty is introduced into CFD simulations. The implications of an improved ability to predict Prt and Sct reach beyond CFD to more classical fields of mass- and heattransfer correlations using the Sherwood and Nusselt numbers (Sh and Nu, respectively). For the turbulent Prt number, a review by Churchill in 200034 noted that “The development of a comprehensive predictive or correlative expression for the turbulent Prandtl number is the principle remaining challenge with respect to the prediction of turbulent forced convection.” This can be equally stated for the Sct number in highly complex, boundary layer, and multiscale flows. Dudukovic and Pjanovic35 found that “it is precisely this assumption of Sct = (constant) that leads to misrepresentation of the Reynolds number dependence”, referring to a Sherwood number correlation, rffiffiffiffiffiffi f Sc Sh ¼ a1 Re Sct
ð22Þ
for boundary layers in pipe flows and falling films. The conclusions by Dudukovic and Pjanovic further stated that Sct is not constant and is dependent on turbulent spectra, which agrees with the conclusions by Reynolds29 concerning the importance of the turbulent intensity in predicting the Sct value.
7. CHURCHILL’S REINTERPRETATION AND A LOOK TO THE FUTURE Re-examination from a different perspective often leads to improvements in fundamental knowledge and understanding.
This is true with respect to the Prt number in a published work by Churchill in 2002.16 Churchill provided a reinterpretation of the Prt number, for a fully developed turbulent flow in a round tube, in terms of a local fraction of shear stress and fraction of heat flux density due to fluctuations in velocity. The essential steps in the derivation require the stresses in the radial direction, the transport of energy in the negative radial direction, and a formulation of dimensionless shear and heat flux density. As a result, Churchill was able to formulate ðu0 v0 Þþþ 1 ðT 0 v0 Þþþ Pr t ¼ ð23Þ Pr ðT 0 v0 Þþþ 1 ðu0 v0 Þþþ Equation 23 expands the Prandtl numbers in terms of dimensionless shear stresses, 0
ðu0 v0 Þþþ
¼
0
FÆui uj æ τij
ð24Þ
and dimensionless heat flux density, ðT 0 v0 Þþþ ¼
FcÆT 0 u0 æ q
ð25Þ
where τij is the total shear in the radial direction, q the heat flux density in the y-direction, and F the fluid density, and c the heat capacity. Equation 23 is an important starting point for relating Prandtl numbers in fully developed turbulent flow in channels and pipes. One can see the possibilities of extending this methodology to spatially variable Prt (or Sct) in more-complex geometries. The crux of using such an approach is the determination of (uuvu)þþ and (Tuvu)þþ. Based on the work by Papavassiliou and Hanratty,36 Churchill noted that a Lagrangian form of DNS provided an accurate calculation of these quantities without empiricism. Also, a recent work by Srinivasan and Papavassiliou37 explored the use of Lagrangian DNS to determine Prt for classical Poiseulle channel and plane Couette flows. Prt was found to be a function of Pr, which agrees with the observations presented earlier in this paper. The extension of determination of Prt for curved geometries (where transport is not necessarily unidirectional) remains an open challenge and is the subject of ongoing research in our laboratory that will be addressed in upcoming publications.
8. CONCLUSIONS As a precursor to passive scalar transport modeling, the Reynolds-averaged NavierStokes (RANS) equations, along with the turbulent closure issue, were provided as a starting point. Inasmuch the RANS equation requires closure to account for the effect of chaotic turbulent fluctuations, so does the transport of a passive scalar in a turbulent flow field. Using Reynolds averaging, a scalar is broken into ensemble-averaged and fluctuating components (eq 11), while creating an additional unclosed scalar-flux term ÆuuφuRæ. We presented several methods used to close the scalar-flux term, including the gradient diffusion hypothesis (GDH) (eq 15), algebraic models (eqs 17 and 18), and a scalarflux transport model (eq 19). Of the models presented, the GDH is the simplest and least computationally intensive. The only requirement is the determination of a turbulent mass or thermal diffusivity, via a turbulent Schmidt Sct or Prandtl number Prt and a known turbulent 8821
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Industrial & Engineering Chemistry Research viscosity (νt) calculated by an eddy viscosity turbulence model. It is common practice to define a constant global Sct or Prt as a flowspecific (e.g., combustion, urban diffusion, jet in crossflow) value usually in the range of 0.1 to >1 for both dimensionless numbers. However, by setting a constant global Sct or Prt, additional uncertainty is introduced into the numerical CFD simulation through assumptions about levels of turbulent mass or heat diffusion and turbulent mixing in the system. For example, combustion simulations have shown that an assignment of lower Sct intensifies combustion, because of enhanced species diffusion and turbulent mixing, while a higher Sct number may create mixtures that are unable to sustain combustion. What has been shown in previous literature is that both the Sct and Prt numbers are strongly influenced by three factors: the corresponding molecular value of the Pr and Sc parameters, the position within the flow (i.e., distance from the wall), and the local turbulent intensity of the flow. This indicates that spatially and temporally variable Prt and Sct are more appropriate and accurate in approximating the turbulent diffusivity in eq 15. Furthermore, research has shown that a spatially variable Sct model produced simulation results in much closer agreement with experimental data than simulations using a global Sct number. Until recently, models predicting the Sct or Prt number have been limited to mixing-length models, empirical relations, statistical calculations, or optimization algorithms. Work by Professor Stuart Churchill has provided further insight that supports the knowledge that fluctuations in velocity—more importantly, the amount of dimensionless shear—play an important role in heat transport and may be used to relate molecular and turbulent Prandtl numbers within fully developed turbulent flows in a pipe. What can be reiterated is that an improved knowledge of Sct and Prt is necessary to fully exploit the GDH, especially in situations where moreintricate modeling techniques (i.e., large eddy simulation (LES), direct numerical simulation (DNS), scalar-flux models) are intractable, because of computational domain complexities and/or lack of computational resources. In summary, the development of a comprehensive approach for evaluation of the turbulent Prandtl (or Schmidt) number remains as much of a principal challenge, with respect to the prediction of scalar transport in turbulent forced convection, today as it was a decade ago, as indicated by Churchill. The increasing demands for improved efficiency in the energy and chemical and materials processing areas provide the incentives for renewed efforts.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We are grateful for the work by Professor Churchill and his unique insight into turbulence modeling. His work has been useful to us in our research in multiphase modeling as well as the teaching of transport phenomena to the future generation of engineers. We would also like to thank the industrial partners of the Chemical Reaction Engineering Laboratory (CREL) at Washington University for their generous support of our research projects. ’ NOMENCLATURE A = model constant used in eq 21 (from ref 30)
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a1 = model constant used in eq 22 (from ref 35) B = model constant used in eq 21 (from ref 30) C1 = model constant used in eq 20 (from ref 29) C2 = model constant used in eq 20 (from ref 29) Cθ = DalyHarlow model constant used in eq 17 (from ref 13) c = heat capacity at constant pressure D = mass diffusivity Dtij = anisotropic turbulent mass diffusivity f = model constant used in eq 21 (from ref 35) J = molecular flux (eq 19) k = turbulent kinetic energy Pr = Prandtl number; Pr = ν/R P = production term p = pressure q = heat flux density R = pressure-scalar-gradient term S = scalar source term (eq 14) Sc = Schmidt number; Sc = ν/D Sh = Sherwood number (eq 22) t = time U = velocity vector u = fluctuating velocity component (uuvu)þþ = dimensionless shear stresses; (uuvu)þþ = FÆuuvuæ/τ (eq 24) þþ (Tuvu) = dimensionless heat flux density; (Tuvu)þþ = FcÆTuuuæ/q (eq 25) x = Cartesian vector space x = independent space variable Æ 3 æ = ensemble average Greek Symbols
r = thermal conductivity R = species δij = Kronecker delta ε = turbulent or scalar-flux dissipation μ = viscosity ν = kinematic viscosity Π = velocitypressure gradient F = density τ = total shear stress tensor φ = a conserved scalar ω = specific turbulent dissipation rate Subscripts
i, j, k = roman indices notation t = turbulent T = total Accents/Superscripts
u = fluctuations m, n = model exponents used in eq 20 (from ref 29) t = anisotropic turbulence Acronyms
CFD = computational fluid dynamics DNS = direct numerical simulation GDH = gradient diffusion hypothesis (eq 15) LES = large eddy simulation RANS = Reynolds-averaged NavierStokes equations RNG kε = renormalization group kε turbulence model SGS = LES subgrid scale models URANS = unsteady Reynolds-averaged NavierStokes equation (see eq 4) 8822
dx.doi.org/10.1021/ie200055s |Ind. Eng. Chem. Res. 2011, 50, 8817–8823
Industrial & Engineering Chemistry Research URAPS = Reynolds-averaged passive scalar equation (see eq 14) kω = shear stress transport (SST) k-omega model
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