Validation of Turbulence Models for Numerical Simulation of Fluid Flow and Convective Heat Transfer

CHAPTER ONE

Validation of Turbulence Models for Numerical Simulation of Fluid Flow and Convective Heat Transfer Ephraim M. Sparrow*, John M. Gorman*, John P. Abraham†, Wally Minkowycz‡ *University of Minnesota, Minneapolis, MN, United States † School of Engineering, University of St. Thomas, St. Paul, MN, United States ‡ University of Illinois at Chicago, Chicago, IL, United States

Contents 1. Introduction 2. Plan of the Presentation 3. Case (1): Turbulent Swirl Flow in a Diffuser 3.1 Two-Equation Turbulence Models for Case (1) 3.2 LES Turbulence (WALE) Model for Case (1) 3.3 Dimensionless Constants for the Selected Turbulence Models 3.4 Numerical Approach for Case (1) 3.5 Evaluation of the κ
ε Model for Case (1) 3.6 Evaluation of the RNG κ
ε Model for Case (1) 3.7 Evaluation of the κ
ω Model for Case (1) 3.8 Evaluation of the SST κ
ω Model for Case (1) 3.9 Evaluation of the LES (WALE) Model for Case (1) 3.10 Summary of the Turbulence Model Evaluations for Case (1) 4. Case (2): Turbulent Flow in a Pipe Bend 4.1 Physical Situation for Case (2) 4.2 Numerical Approach for Case (2) 4.3 Evaluation of the SST κ
ω Model for Case (2) 4.4 Summary of the SST κ
ω Model Evaluation for Case (2) 5. Case (3): Turbulent Flow in Round Pipes 5.1 Physical Situation for Case (3) Round Pipes 5.2 Numerical Approach for Case (3) Round Pipes 5.3 Evaluation of the SST κ
ω Model for Velocity Profiles in Case (3) 5.4 Evaluation of the SST κ
ω Model for Heat Transfer in Case (3) 5.5 Summary of the SST κ
ω Model Evaluation for Case (3) 6. Case (4): Turbulent Flow in a Rectangular Duct 6.1 Physical Situation for Case (4) Rectangular Duct Advances in Heat Transfer, Volume 49 ISSN 0065-2717 https://doi.org/10.1016/bs.aiht.2017.09.002

#

2017 Elsevier Inc. All rights reserved.

3 5 6 7 10 11 11 12 13 13 14 14 16 16 17 19 19 20 20 20 21 23 25 27 27 28 1

2

Ephraim M. Sparrow et al.

6.2 Numerical Approach for Case (4) Rectangular Duct 6.3 Evaluation of the SST κ
ω Model in Case (4) 6.4 Summary of the SST κ
ω Model Evaluation for Case (4) 7. Case (5): Turbulent Flow in Perforated Plates 7.1 Physical Situation for Case (5) 7.2 Numerical Approach for Case (5) 7.3 Evaluation of the SST κ
ω Model for Case (5) 7.4 Summary of the SST κ
ω Model Evaluation for Case (5) 8. Concluding Remarks References

28 29 30 30 30 32 32 33 33 34

Abstract In this chapter, several physical situations are examined: fluid flows (1) with secondary swirl, (2) in a pipe bend, (3) in round pipes, (4) in a rectangular duct, and finally (5) through perforated plates. In each of these cases, validation was performed using accurate and complete experimental results available in the published literature to be compared with appropriate numerical simulations. Different quantities are used for validation metric depending on the availability of experimental data, i.e., comparing velocity profiles, local Nusselt numbers, and pressure drop. Several popular turbulence models, κ
ε, RNG κ
ε, κ
ω, SST κ
ω, and WALE LES were investigated. The first two cases, (1) and (2), utilized available velocity and turbulence profiles at the inlet of the solution domain. From Case (1), the SST κ
ω turbulence model clearly was the preferred choice because of the good agreement with experimental data and the shorter CPU time required as compared to the LES model. The remaining cases (2)–(5) were primarily investigated with the SST κ
ω turbulence model. Overall, the SST κ
ω turbulence model performed very well compared to the available experimental data for all of the cases investigated here.

NOMENCLATURE Cp specific heat Cw LES model constant Cε1, Cε2, Cε2RNG turbulence model constants Cμ, CμRNG turbulence model constants D diameter F1, F2 blending functions in the SST model k thermal conductivity kturb turbulent thermal conductivity Nu Nusselt number p pressure Pk production term for the turbulent kinetic energy Prtur turbulent Prandtl number r radial coordinate R radius S shear strain rate

Validation of Turbulence Models for Numerical Simulation

3

U mean axial velocity u0 , v0 , w0 fluctuating velocity component u velocity component or local velocity x coordinate y+ nondimensional near-wall mesh quality metric y coordinate z axial coordinate

GREEK SYMBOLS α1, α3 turbulence model constants β1, β, βRNG turbulence model constants Δ element volume ε turbulence dissipation κ turbulent kinetic energy μ molecular viscosity μsgs small-scale eddy viscosity μt turbulent eddy viscosity ρ fluid density σ Prandtl-number-like diffusion coefficient σ ε, σ εRNG turbulence model constants σ κ, σ κ3, σ κRNG turbulence model constants σ ω, σ ω2 turbulence model constants τ shear stress ω specific rate of turbulence dissipation ϕ angle Ω vorticity tensor

SUBSCRIPTS c centerline i index notation for Cartesian coordinates j index notation for Cartesian coordinates k index notation for Cartesian coordinates m index notation n index notation

1. INTRODUCTION The rapid development of readily applicable software and the simultaneous increase in raw computer power has propelled numerical simulation forward as a viable tool for design of heat transfer and fluid flow devices. In order to obtain the full benefit of these advances, it still remains to foster the

4

Ephraim M. Sparrow et al.

attitude changes that will persuade potential users to abandon prior, more traditional and less efficient approaches. One of the key needs is to demonstrate that the use of numerical simulation will, in all likelihood, lead to the attainment of more accurate outcomes. This need is succinctly expressed by the term validation. Validation, as a component of engineering activity, is presently coming into its own, as witnessed by its becoming a focus of its own at national and international meetings of major professional engineering societies. Validation may be pursued along numerous specific pathways to achieve various specific goals, all of which may be collected under the heading correctness. Here, the specific goal to be addressed is the identification of turbulence models that are accurate in the engineering sense and are also readily applicable without the demand of super high sophistication on the part of the user. These characteristics fit very neatly into what is needed for the utilization of numerical simulation as an advantageous design tool. The fulfillment of the aforementioned goal of this chapter would provide advantage not only with regard to its use in numerical simulation, but also for any approach to the solution of real-world problems involving convective heat transfer and/or fluid flow. The validation methodology to be used here has already found considerable favor in the recent past. It is critically dependent on the availability of highly accurate and complete experimental data for a well-defined physical situation. The need for accuracy is, of course, evident, but the completeness of the data is equally important and worthy of elaboration. For the elaboration, it is advantageous to consider the simple case of turbulent flow in a pipe. Except for the fully developed flow far downstream of the pipe inlet, the state of the flow at any cross section is not unique unless considerable and detailed information is known about the state of the flow at the inlet cross section. In particular, it is necessary to know, at minimum, the quantitative nature of the turbulence at the inlet. Since it is rare that the aforementioned details are measured in the majority of experiments, the pool of suitable experiments is moderate. Different physical problems may involve flows of a distinct character. For example, a flow in which there are separated regions may not be well described by the use of a turbulence model which gives rise to a highly accurate description of a turbulent flow with swirl. This realization suggests that the proof of suitability of a turbulence model for a given physical situation cannot be regarded as a universal proof. On this basis, it seems appropriate to investigate turbulence models in several physical situations in order to determine if any model has broad applicability.

Validation of Turbulence Models for Numerical Simulation

5

Several researchers have worked on the validation issues that have been identified in the foregoing paragraphs. The topic has been of particular interest to the present authors because of their major concern with real-world engineering problems where turbulent flow is the primary regime of fluid flow. This nature of involvement has fostered contact with a wide range of turbulent fluid flow and convective heat transfer. With this broad range as background, the authors have decided that this presentation would be well served if they shared their own experiences.

2. PLAN OF THE PRESENTATION The presentation is organized according to the physical situation being investigated. All told, five physical cases will be dealt with: flows (1) with secondary swirl, (2) inside of a pipe bend, (3) inside round pipes, (4) inside rectangular ducts, and (5) through perforated plates. In each of these situations, validation of turbulence models is to be performed utilizing highly accurate and complete experimental results in conjunction with high-fidelity numerical simulations. As to be discussed in the following, the details of the validation approach differ from case-to-case among those investigated. In one approach, performed in the first of the cases enumerated in the preceding paragraph, four different two-equation turbulence models were employed to solve for an evolving turbulent swirl flow in a diffuser. A fifth model, large eddy simulation (LES), was also employed. This approach was used to establish the clear differences between the relatively simple and less resource intensive, two-equation models and a sophisticated and high-resource-requiring turbulence model. The outcome of that detailed investigation demonstrated that the use of the SST κ
ω turbulence model [1] gave rise to substantially better agreement with the experimental data than did any of the other two-equation models. In fact, the quality of the SST κ
ω model, based on numerical predictions, was as good as those provided by the LES model. However, the CPU time required to perform the LES-based solution was approximately 11 times the CPU time required for the SST κ
ω solution. This outcome was persuasive in convincing the investigators that the sought-for characteristics of modest sophistication and moderate computational resources was better satisfied by an accurate two-equation turbulence model than a more-complex and encompassing turbulence model. The authors had already evaluated the SST κ
ω model in other lessencompassing studies of turbulent flows and in all cases had found the

6

Ephraim M. Sparrow et al.

corresponding numerical predictions to be virtually congruent with appropriate experimental data. Also, based on extensive literature searches, it was found that the SST κ
ω model consistently bested all other two-equation turbulence models whenever more than one model was used competitively. In this regard, reference may be made to [2–5] for successful applications of the SST κ
ω model by other investigators in very diverse situations. With this outcome, the presentation of information for cases (2)–(5) of the antepenultimate paragraph will be, in each case, a comparison of results between a numerical simulation based on the SST κ
ω model and a corresponding experiment.

3. CASE (1): TURBULENT SWIRL FLOW IN A DIFFUSER The selection of this case to begin the validation of candidate turbulence models was based on the completeness and accuracy of the available experimental data [6]. The focus of that research was the evolution of an entering-swirled axial turbulent flow in a stationary-walled conical diffuser. In essence, the experimental setup consisted of an upstream cylindrical swirl chamber made up of a honeycomb whose outer members were rigidly attached to the inner wall of a round pipe rotating around its own axis. The swirled axial flow discharging from the rotating section was passed through a short length of equal-diameter stationary round pipe and thence into the diffuser. In general, the rotating honeycomb created a steady and regular secondary swirling flow in the pipe. The complete set of details and results for Case (1) can be found in Ref. [7]. Experimental measurements in Ref. [6] of the nature of the flow in the straight nonrotating pipe, just upstream of the diffuser, made it possible to specify all the necessary inlet values (velocity profiles and turbulence quantities) in the numerical simulation to match the experiments. The measured quantities in Ref. [6] included the cross-sectional profiles of the axial and tangential (swirl) velocities. In addition, profiles of the turbulent velocity fluctuations u0 , v0 , and w0 were also measured. For the proper implementation of the simulation model, it was necessary to have quantitative information for turbulence quantities at the diffuser inlet to serve as boundary conditions. In fact, it is the common absence of such information from experimental data that, in general, limits the ability to either reproduce or validate the numerical simulations. In the work presented in Ref. [7], the profiles of the individual turbulent velocity fluctuations u0 , v0 , and w0 were converted into profiles of the turbulence kinetic energy κ using

Validation of Turbulence Models for Numerical Simulation

7

 1
02 u + v0 2 + w 0 2 (1) 2 The other turbulence quantity needed to properly characterize the inlet flow is the turbulence dissipation ε. This quantity is not measured directly, but again was calculated by the authors in Ref. [7] from the equation κ¼

κ 3=2 (2) 0:3D where D is the diameter of the pipe. By use of the velocity and turbulent quantity profiles created from the data presented in Ref. [6], it is possible to numerically reproduce the fluid flow conditions present in the experiment at the inlet of the diffuser which can be used with any turbulence model. ε¼

3.1 Two-Equation Turbulence Models for Case (1) Five turbulence models were selected for evaluation by comparison with the swirl-flow experimental data of [6]. This selection was based on a number of factors. The κ
ε and κ
ω models were chosen both because of their wide popularity and because they have provided platforms for more refined twoequation models. The best of these refinements are, respectively, conveyed by the RNG κ
ε and the SST κ
ω models. The aforementioned two-equation turbulence models are used in conjunction with the RANS turbulent-flow momentum equations and the equation of continuity for mass conservation. The fifth of the selected models, an LES model, is not related to the RANS equations and was chosen because it provides a different approach to turbulence modeling. The details of the two-equation models used in the evaluation of Case (1) will now be presented for completeness, followed directly by the description of the chosen LES model. The steady-state, constant-property form of the RANS equations, written in Cartesian tensor form, is     ∂uj ∂uj ∂p ∂ ρ ui + ðμ + μt Þ ¼
∂xj ∂xi ∂xi ∂xi (3) i ¼ 1,2,3 and j ¼ 1,2,3 The quantity μt that appears in Eq. (3) is called the turbulent viscosity. It is a standard feature of all turbulence models that are connected with the RANS equations. The various two-equation models utilize different approaches to determining this quantity. For mass conservation, the continuity equation is used for all constant density models.

8

Ephraim M. Sparrow et al.

∂ui ¼0 ∂xi

(4)

In what follows, turbulence-model equations for steady, constantproperty flows are presented along with their sources and with explanatory notes when appropriate. A listing of the constants and coefficients that appear in the various models is provided after the presentation of the models. 3.1.1 κ 2 ε Model for Case (1) The κ
ε model [8] was the first two-equation turbulence model based on the RANS equations. The relationship between κ, ε, and the turbulent viscosity μt is μ t ¼ Cμ ρ

κ2 ε

(5)

where Cμ is a model constant. The governing equations for κ and ε are, respectively,    ∂ ∂ μt ∂κ ðρui κÞ ¼ μ+ (6) + Pκ
ρε ∂xi ∂xi σ κ ∂xi    ∂ ∂ μ ∂ε ε + ðCε1 Pκ
Cε2 ρεÞ ðρui εÞ ¼ μ+ t (7) ∂xi ∂xi κ σ ε ∂xi in which Cε1, Cε2, σ κ, and σ ε are model constants. The quantity Pκ is a turbulence production term due to viscous forces and is determined from     ∂ui ∂uj ∂ui ∂uκ ∂uκ Pκ ¼ μt +
3μt + ρκ (8) ∂xj ∂xi ∂xj ∂xκ ∂xκ 3.1.2 RNG κ 2 ε Model for Case (1) The RNG approach was developed in Ref. [9], where RNG stands for renormalized group. The equations are the same as the regular κ
ε model, but the model constants are different. The relationship between κ, ε, and the turbulent viscosity μt is μt ¼ CμRNG ρ

κ2 ε

(9)

The turbulence dissipation transport equation with the new constants is    ∂ ∂ μt ∂ε ε + ðCε1RNG Pκ
Cε2RNG ρεÞ (10) ðρui εÞ ¼ μ+ ∂xi ∂xi κ σ εRNG ∂xi

Validation of Turbulence Models for Numerical Simulation

where


η  η 1
4:38 Cε1RNG ¼ 1:42
ð1 + βRNG η3 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pκ η¼ ρCμRNG ε

9

(11) (12)

3.1.3 κ 2 ω Model for Case (1) The κ
ω model [10] equations are of a similar form as the previous models, but the model constants are different. The relationship between κ, ω, and the turbulent viscosity μt is μt ¼ ρ

κ ω 

  ∂ ∂ μt ∂κ ðρui κÞ ¼ μ+ + Pκ
β1 ρκω ∂xi ∂xi σ κ ∂xi    ∂ ∂ μ ∂ω ω + α1 Pκ
βρω2 ðρui ωÞ ¼ μ+ t ∂xi ∂xi κ σ ω ∂xi

(13) (14) (15)

Here, α1, β1, β, σ k, and σ ω are model constants, and the quantity Pκ is a turbulence production term which is calculated in the same way as for the regular κ
ε model. 3.1.4 SST κ 2 ω Model for Case (1) The SST κ
ω model [1] is a blending of the original κ
ε and κ
ω models and has additional functions not present in the previous models. The original κ
ε model is known to provide acceptable results away from bounding walls, and the κ
ω model has been shown to be better in the neighborhood of bounding walls. In the SST model, the turbulent viscosity μt is given by μt ¼

α1 ρκ max ðα1 ω, SF2 Þ

(16)

in which   pffiffiffi 2 2 κ 500μ , F2 ¼ tan h max β1 ωy ρy2 ω    ∂ðρui κ Þ ∂ μt ∂κ ¼ μ+ + Pκ
β1 ρκω σ κ3 ∂xi ∂xi ∂xi

(17) (18)

10

Ephraim M. Sparrow et al.

   ∂ðρui ωÞ ∂ μt ∂ω 1 ∂κ ∂ω ρα3 ¼ μ+ + Pκ
βρω2 (19) +2ρð1
F1 Þ ∂xi ∂xi σ ω2 ω ∂xi ∂xi μt σ ω ∂xi where F1 is a blending functions given by   4  pffiffiffi  4ρκ κ 500μ , F1 ¼ tan h min max , β1 ωy ρy2 ω CDκω σ ω2 y2 where y is the distance from a wall boundary, and   1 ∂κ ∂ω
10 , 10 CDκω ¼ max 2ρ σ ω2 ω ∂xi ∂xi

(20)

(21)

3.2 LES Turbulence (WALE) Model for Case (1) The governing equations for the LES (WALE) model [11] are obtained from filtering the time-dependent Navier–Stokes equations. The filtering eliminates eddies smaller than the size of the physical mesh elements and uses an eddy viscosity approach for scales not directly solved for. This leads to Navier–Stokes equations taking the form      ∂ρ ui uj ∂τij ∂ ∂p ∂ ∂ ui ∂ uj ¼
+ ðρ ui Þ + + μ + ∂t ∂xj ∂xi ∂xi ∂xj ∂xi ∂xi i ¼ 1,2, 3 andj ¼ 1,2,3

(22)

where τij is the small-scale stress defined as 1 τij ¼
ρui uj + ρ ui uj ¼ 2μsgs Sij + δij τkk 3

(23)

and Sij is the large-scale strain rate tensor. The small-scale eddy viscosity μsgs is found from
3=2 Sijd Sijd μsgs ¼ ρðCw ΔÞ2 (24)

5=2
d d 5=4 Sij Sij + Sij Sij in which Cw is a constant and Δ ¼ (element volume)1/3. The tensor Sdij can be written in terms of the strain-rate and vorticity tensors:  kj
1 δij ðSmn Smn
Ω  mn Þ  ik Ω  mn Ω Sijd ¼ Sik Skj + Ω 3

(25)

11

Validation of Turbulence Models for Numerical Simulation

Table 1 Dimensionless Constants Used for the Models Shown in Sections 3.1 and 3.2 Model Constants Values Model Constants Values

κ
ε

κ
ω

LES (WALE)

Cε1

1.44

Cε2

RNG κ
ε

Cε2RNG

1.68

1.92

CμRNG

0.085

Cμ

0.09

βRNG

0.012

σκ

1.0

σ κRNG

0.7179

σε

1.3

σ εRNG

0.7179

α1

0.5555

α1

0.31

β1

0.09

α3

0.8

β

0.075

β1

0.09

σk

2.0

β

0.075

σω

2.0

σω

2

Cw

0.5

SST κ
ω

 ij is defined as where the vorticity tensor Ω   1 ∂ui ∂uj  Ωij ¼ + 2 ∂xj ∂xi

(26)

3.3 Dimensionless Constants for the Selected Turbulence Models For each of the foregoing turbulence models presented, there are corresponding dimensionless constants. Those constants used for Case (1) are presented in Table 1.

3.4 Numerical Approach for Case (1) Numerical solutions for each of the five turbulence models were obtained by means of ANSYS-CFX 16.1 software. This package discretizes the governing partial differential equations by the finite-volume method to create algebraic equations. As described in Ref. [7], a mesh/grid independency study was done to ensure the final mesh was capable of providing highly accurate results. The mesh used for the final solutions of the governing equations consisted of approximately 7 million nodes. This mesh led to an average y + value between 0.53 and 0.63 in the solution domain, depending on the turbulence model. The y + value serves as an important nondimensional

12

Ephraim M. Sparrow et al.

mesh-quality metric. In general, smaller values of y +, less than one, are desired to accurately resolve fluid motion near wall boundaries for the SST model. The numerical solutions were regarded as converged when the normalized residuals became 10
6. Constant property air at 25°C was used as the working fluid. The general validation approach for Case (1) was based on comparing experimentally determined profiles of the axial velocities at several streamwise locations from Ref. [6] with numerical predictions for those same locations. These predictions were made for each of the five individual turbulence models that have been described in Sections 3.1–3.2. These comparisons are presented in the following sections, respectively, for the κ
ε, RNG κ
ε, κ
ω, SST κ
ω, and LES models. The most appropriate comparisons between the simulation results and experimental data are made for axial velocity profiles at selected streamwise locations. The results of the numerical predictions from Ref. [7] are displayed by solid lines in the figures. In contrast, the experimental data from Ref. [6] are conveyed data symbols.

3.5 Evaluation of the κ 2 ε Model for Case (1) Attention is first directed to the numerical predictions based on the original κ
ε model and the comparisons with the corresponding experimental data. These results displayed in Fig. 1 are for several axial velocity profiles. In the figure, the velocity profiles are made dimensionless using the mean axial velocity U and are plotted as a function of the dimensionless cross-sectional

Local axial velocity/U

1.0

x/D = 0.385

0.8

x/D = 0.673

0.6

0.4 x/D = 0.962 0.2 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

y/D

Fig. 1 Comparison of axial velocity profiles at selected cross sections in a diffuser presented in Ref. [7]. The numerical results, represented by solid lines, were produced by the κ
ε turbulence model [8], and the experimental data were extracted from Ref. [6].

13

Validation of Turbulence Models for Numerical Simulation

coordinate y/D. A value of y/D ¼ 0 represents the wall of the diffuser. The inlet diffuser diameter D is used to make the locations dimensionless at selected axial cross sections x/D. The x-locations were chosen based on where experimental information was available [6]. The location of x/D ¼ 0 represents the diffuser inlet and D was equal to 0.26 mm. Inspection of Fig. 1 reveals considerable deviations between the experimentally measured values and the predicted axial velocity profiles. In the near-wall region, the data fall below the predicted curves, while an opposite relationship is in evidence away from wall. In general, it appears that the κ
ε model does not handle this situation very well.

3.6 Evaluation of the RNG κ 2 ε Model for Case (1) Next, attention is turned to the predictions based on the RNG κ
ε turbulence model, which are displayed in Fig. 2. When the results conveyed in this figure are compared with Fig. 1, it is astonishing that the figures are almost identical. It would appear that the RNG κ
ε model has the same difficulty in obtaining accurate predictions for swirling flows in the diffuser as did the original κ
ε model.

3.7 Evaluation of the κ 2 ω Model for Case (1) The comparisons between the predictions based on the original κ
ω model and the experimental data are presented in Fig. 3. In the figure, it is seen that there are again noticeable differences between the numerical predictions

Local axial velocity/U

1.0 x/D = 0.385

0.8

x/D = 0.673

0.6

0.4 x/D = 0.962 0.2 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

y/D

Fig. 2 Comparison of axial velocity profiles at selected cross sections in a diffuser presented in Ref. [7]. The numerical results, represented by solid lines, were produced by the RNG κ
ε turbulence model [9], and the experimental data were extracted from Ref. [6].

14

Ephraim M. Sparrow et al.

Local axial velocity/U

1.0

x/D = 0.385

0.8

x/D = 0.673

0.6

0.4 x/D = 0.962 0.2 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

y/D

Fig. 3 Comparison of axial velocity profiles at selected cross sections in a diffuser presented in Ref. [7]. The numerical results, represented by solid lines, were produced by the κ
ω turbulence model [10], and the experimental data were extracted from Ref. [6].

using this model and the experimental data. Although difficult to discern, the disparities in this model’s predictions are somewhat greater than those provided by the RNG κ
ε model. Near the wall, the predictions are above the measured data, whereas moving away from the wall, the values overpredict the velocities. Overall, these results indicate that the original κ
ω model is less capable than the RNG κ
ε model as a means of predicting these velocity profiles.

3.8 Evaluation of the SST κ 2 ω Model for Case (1) The next comparisons bring together the predictions of the SST κ
ω model from Ref. [7] and the experimental data from Ref. [6]. The comparisons are made in Fig. 4 for the axial velocity profiles. It can be seen that the numerical predictions are in very good agreement with the experimental data in the near wall region, which did not occur in the other investigated turbulence models. In addition, away from the wall, the SST κ
ω predictions are also in better agreement with the data from the other models.

3.9 Evaluation of the LES (WALE) Model for Case (1) The last comparison for Case (1) between the experimental data and the results of the numerical simulations is made in Fig. 5 for the simulations performed by means of the LES turbulence model. This is an especially useful comparison because it is not a two-equation turbulence model like the previous models. The LES model is not based on isotropic turbulence nor on

15

Validation of Turbulence Models for Numerical Simulation

Local axial velocity/U

1.0

x/D = 0.385

0.8

x/D = 0.673

0.6

0.4 x/D = 0.962 0.2 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

y/D

Fig. 4 Comparison of axial velocity profiles at selected cross sections in a diffuser presented in Ref. [7]. The numerical results, represented by solid lines, were produced by the SST κ
ω turbulence model [1], and the experimental data were extracted from Ref. [6].

Local axial velocity/U

1.0 x/D = 0.385

0.8

x/D = 0.673

0.6

0.4 x/D = 0.962 0.2 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

y/D

Fig. 5 Comparison of axial velocity profiles at selected cross sections in a diffuser presented in Ref. [7]. The numerical results, represented by solid lines, were produced by the LES (WALE) turbulence model [11], and the experimental data were extracted from Ref. [6].

the RANS equations. It is widely regarded as a higher-order turbulence model compared with the RANS-based models. As seen in Fig. 5, the agreement between the simulation and experiment is very good. A comparison of Fig. 5 with the results based on the SST κ
ω model in Fig. 4 shows that the LES predictions match the experimental results slightly better. This better agreement is small. It is important to note that, in Ref. [7], the CPU time needed to obtain the SST κ
ω results was much shorter than the time for the LES solutions. The SST κ
ω model CPU time was 14.2 days, whereas the LES results took

16

Ephraim M. Sparrow et al.

155.3 days of CPU time. Other investigators in the literature have similarly noted that LES simulations are far more demanding than simulations with two-equation turbulence models. Just based on the difference in computational time, it seems that the small differences between the two sets of predictions is not worth the extra time and recourses needed to execute the LES solutions.

3.10 Summary of the Turbulence Model Evaluations for Case (1) It was found that among the investigated two-equation turbulence models in Ref. [7], the SST κ
ω model predicted axial velocity profiles that best agreed with the experimental results of [6]. The velocity profile predictions from the LES turbulence model were in slightly better agreement with the experimental data. However, there was an enormous difference in the CPU times required to obtain a solution by use of these two respective methods. Because of the good agreement and shorter simulation time, it is believed that is the SST κ
ω model is the most efficient of those investigated for this physical situation. Since convective heat transfer is very sensitive to the near-wall velocity field, this finding augurs well for the determination of heat transfer predictions in swirl-flow situations when the SST κ
ω is used. The only limitation of Case (1) is that the swirling flow was generated by the rotating honeycomb structure which created a very regular and steady fluid flow. Other applications may encounter very unsteady secondary flows.

4. CASE (2): TURBULENT FLOW IN A PIPE BEND Among the fittings that are encountered in piping systems, 90-degree bends are the most common. The fluid flow passing through a pipe bend is made complicated by the presence of a secondary flow superposed on the main axial flow. The secondary flow is a circumferential flow created by pressure variations which occur in each cross section of the bend. Due to the curvature of the bend, the pressure in the fluid adjacent to the outer radius of the bend is greater than that in the fluid adjacent to the inner radius of the bend. The aforementioned secondary flow is sometimes referred to as swirl. This situation is dissimilar to Case (1) because the secondary flow is superimposed on the axial flow that is introduced in a controlled manner. The practical importance of pipe-bend flow, the availability of complete and accurate experimental data for it, and the complexity of the pattern of

Validation of Turbulence Models for Numerical Simulation

17

fluid flow make the pipe-bend problem an attractive subject for the evaluation of the efficacy of turbulence models. In this regard, note may be taken of the outcome of the study described earlier in Case (1). There, the clear superiority of the SST κ
ω turbulence model among other two-equation models was clearly established. If this outcome is given credence, a different approach may be adopted for the pipe-bend problem compared to the case of turbulent flow in a diffuser (Case 1). In particular, for the pipe-bend problem, the evaluation will be restricted to the investigation of the suitability of the SST κ
ω turbulence model for predicting the available experimental results.

4.1 Physical Situation for Case (2) A schematic diagram of the fluid flow situation that was studied experimentally in Refs. [12,13] and that was used for validation purposes here is conveyed in Fig. 6. As seen in the figure, a pipe bend is connected to a short length of pipe at the upstream end and at its downstream end to a somewhat greater length of pipe. The pipe diameter D is 48 mm, and the downstream straight section of pipe extended 480 mm (10 pipe diameters) beyond the exit of the bend, i.e., ϕ ¼ 90 degrees. The inlet boundary for the numerical simulation was located 0.58 diameters upstream of the start of the bend (ϕ ¼ 0 degree) because the available data (velocity profiles and turbulence intensities) were reported at that location. The bend radius of curvature is 2.8 times the pipe diameter. Since the present focus is on turbulent flow, only the relevant axial velocity profile and turbulence intensity profile were extracted from Refs. [12,13] and are displayed, respectively, in Figs. 7 and 8. In Fig. 7, a dimensionless

f

Inlet profiles

Fig. 6 Schematic representation of the experimental situation of [12,13]. The figure displays the end-to-end physical piping system.

18

Ephraim M. Sparrow et al.

1.4 1.2

u/U

1 0.8 0.6 0.4 0.2 0 −1

−0.75

−0.5

−0.25

0 r/R

0.25

0.5

0.75

1

Fig. 7 Axial inlet velocity profiles used for the numerical simulation for Re ¼ 43,000 based on the data extracted from Refs. [12,13] at a location x/D ¼ 0.58 before the pipe bend.

0.08

Turbulence intensity

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 −1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

r/R

Fig. 8 Inlet turbulence intensity profile used for the numerical simulation for Re ¼ 43,000 based on the data extracted from Refs. [12,13] at a location x/D ¼ 0.58 before the pipe bend.

velocity profile is conveyed for a pipe Reynolds number of 43,000. Not unexpectedly, the velocity profile corresponding to a turbulent Reynolds number is flatter than a profile for a laminar case. In Fig. 8, it can be seen that higher values of turbulence intensity are confined to a region slightly displaced inward from the pipe wall.

19

Validation of Turbulence Models for Numerical Simulation

4.2 Numerical Approach for Case (2) As before, the solutions of the discretized forms of the governing equations and the SST κ
ω model from Section 3.1.4 were carried out by means of ANSYS CFX 16.1 software. A mesh independence study was performed with 5.5 and 14 million nodes. The overall pressure drop in the piping system was used as a metric for this mesh independence study, defined as the pressure drop between the inlet and the downstream outlet of the system. The percent difference in pressure drop between the two node counts was 0.3%. The maximum y + value was 4.3 with an area average y + value below one. The numerical solutions were regarded as converged when the normalized residuals became 10
6. The working fluid for this case was constant property water.

4.3 Evaluation of the SST κ 2 ω Model for Case (2) As has already been noted, a necessary requisite for the confident use of a numerical simulation model is to verify the predictions provided therefrom with data from well-executed experiments. That search for suitable information yielded Refs. [12,13], where suitable careful experiments are described. In Fig. 9, comparisons are made between the results from the present numerical simulations and the experimental results from Ref. [12,13]. 1.4 1.2

u/U

1 0.8 0.6 0.4 0.2 0 −1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

r/R Simulation at 30 degrees Experiment at 30 degrees

Simulation at 60 degrees Experiment at 60 degrees

Fig. 9 Comparison of simulation predictions with experimental turbulent flow velocity measurements [12,13] at ϕ ¼ 30 and 60 degrees in a pipe bend, where Re ¼ 43,000, U ¼ 0.91 m/s, and D is 48 mm.

20

Ephraim M. Sparrow et al.

Two angular stations within the bend were selected for the measurements, specifically ϕ ¼ 30 and 60 degrees. The information is presented as a normalized velocity profile along a diametral line extending from the inner edge to the outer edge of the bend cross section and passing through the center of the pipe. Fig. 9 displays a very high degree of agreement between the predictions of the SST κ
ω turbulence model and the experimental data at both angular locations.

4.4 Summary of the SST κ 2 ω Model Evaluation for Case (2) It was found that the SST κ
ω model accurately predicted the velocity profiles when compared with the experimental results of [12,13]. This case was uniquely different from Case (1) since the secondary flow was not imparted on the axial flow in a controlled manner. Instead, the secondary flow was a result of the bend geometry.

5. CASE (3): TURBULENT FLOW IN ROUND PIPES The previous sections were comprised of situations where experimentally measured velocity and turbulence profiles were used as the inlet boundary conditions for a flow that experiences a secondary flow. This section focuses on cases where the inlet values are not prescribed but are a result obtained from the numerical simulation. Since the published literature has sufficient experimental data regarding round pipes and rectangular ducts, comparisons will be focused on these situations.

5.1 Physical Situation for Case (3) Round Pipes The literature on turbulent pipe flows was examined to obtain experimental data for comparison with the numerical results obtained here. Two types of comparisons are possible: (a) direct comparison of fully developed velocity profiles and (b) fully developed heat transfer results. Fig. 10 schematically A B

Flow direction D

x

L

Fig. 10 Schematic diagram of the round-pipe scenario used for validation.

21

Validation of Turbulence Models for Numerical Simulation

illustrates the round pipe used for the present cases where D is the inner pipe diameter and L is the length of the pipe. The straight pipe (x ¼ 0) is attached to an upstream bellmouth that draws constant property air (at 25°C) into the pipe from an environment with stagnant air at ambient pressure and a uniform temperature. The bellmouth design is characterized by the ratio of A to B, in this case the ratio was chosen to be 8 to 1. To ensure fully developed flow conditions, an L/D > 200 was chosen with D ¼ 2 cm. At the downstream end of the pipe, a uniform pressure boundary condition was specified to get the desired Reynolds number. Since the geometry is axisymmetric, only a 3-degree wedge around the central axis, as illustrated by the dashed line in Fig. 10, was numerically modeled.

5.2 Numerical Approach for Case (3) Round Pipes The solutions of the discretized forms of the governing differential equations from Section 3.1.4 were carried out by means of ANSYS CFX 16.1 software. Solutions were deemed to be converged to sufficient accuracy when the RMS residuals for all of equations were less than 10
6. Attention was given to the cells aligned in the wall-normal direction in order to ensure that at least 40 elements were situated within 1 mm from any bounding wall. The corresponding area-averaged y+ values for the highest Reynolds numbers under investigation ranged from 0.675 to 0.715, and the corresponding maximum y + values were between 0.703 and 1.09. Table 2 Mesh/Grid Independent Study (SST κ
ω Turbulence Model) Reynolds Number Nodes (Million) f

8000

400,000

2,000,000

1.6

0.0354

40

0.0355

80

0.0355

4.8

0.0124

9.6

0.0124

19

0.0124

24

0.0107

40

0.0109

80

0.0109

22

Ephraim M. Sparrow et al.

Mesh/grid independence was established by use of various discretizations consisting of nodal number variations, and the results are shown in Table 2 for a range of investigated Reynolds numbers. The node number was increased until the difference between successive simulations was 0.2% or less for the fully developed friction factor f. The definition of the friction factor relevant for current needs is   dp
D dx f¼ (27) 1 2 ρU 2 where (
dp/dx) denotes the pressure gradient for fully developed flow in a round pipe of diameter D, U is the mean velocity, and ρ is the air density. The requirement of fully developed flow is often unmet in experiments and has caused erroneous data to be collected and mislabeled as corresponding to fully developed flow. Therefore, the friction factor f was only evaluated for locations (L/D > 200) where the pressure gradient was constant and the velocity profiles were unchanging. Table 2 shows that the variations of the friction factor f with the number of nodes are limited to the third significant figure. The table also shows that accuracies to the third figure required node numbers as large as 40 million for some of the Reynolds numbers. Although such high node numbers are very uncommonly found in the literature dealing with numerical computation of fluid flows, it is clear that such numbers are relevant to the present goal of validation. Another noteworthy comparison is the sensitivity of the fully developed friction factor compared with two of the other popular two-equation models; RNG κ
ε and κ
ω model. Table 3 shows the resulting fully Table 3 Comparison of Friction Factors From Different Turbulence Models Reynolds Number Turbulence Model f

8000

2,000,000

RNG κ
ε

0.0268

κ
ω

0.0363

SST κ
ω

0.0355

RNG κ
ε

0.00993

κ
ω

0.0109

SST κ
ω

0.0109

23

Validation of Turbulence Models for Numerical Simulation

developed friction factor for two very different Reynolds numbers, Re ¼ 8000 and 2 million. The RNG κ
ε model and the κ
ω model were simulated in the same geometry with the same mesh as the SST κ
ω model. At the lower Reynolds number, the RNG κ
ε model prediction differs significantly from the other two models, whereas the original κ
ω is about 2% difference from the SST κ
ω result. At the higher Reynolds number of 2 million, the RNG κ
ε model prediction differs by 9% when compared to the other two models.

5.3 Evaluation of the SST κ 2 ω Model for Velocity Profiles in Case (3) For the velocity profile comparisons, several Reynolds numbers were chosen (5000, 25,000, 45,000, and 3.2 million) based on available experimental data [14–17]. The outcome of the comparisons is presented in Figs. 11–14, 1.2 1 u/Uc

0.8 0.6

Simulation Re = 5000

0.4

Experiment [14] Re = 5450

0.2

Experiment [15] Re = 4900

0 0.00

0.10

0.20

0.30

0.40

0.50

r/D

Fig. 11 Comparison of numerically predicted fully developed velocity profiles to published experimental data from Refs. [14,15] for a Reynolds number of approximately 5000.

1.2 1 u/Uc

0.8 0.6 Simulation Re = 25,000

0.4

Experiment [15] Re = 24,600

0.2 0 0.00

0.10

0.20

0.30

0.40

0.50

r/D

Fig. 12 Comparison of numerically predicted fully developed velocity profiles to published experimental data from Ref. [15] for a Reynolds number of 25,000.

24

Ephraim M. Sparrow et al.

1.2 1

u/Uc

0.8 0.6

Simulation Re = 44,000

0.4

Experiment [16] Re = 44,000

0.2 0 0.00

0.10

0.20

0.30

0.40

0.50

r/D

Fig. 13 Comparison of numerically predicted fully developed velocity profiles to published experimental data from Ref. [16] for a Reynolds number of 44,000. 1.2 1 u/Uc

0.8 0.6

Simulation Re = 3.2 million

0.4

Experiment [17] Re = 3.2 million

0.2 0 0.00

0.10

0.20

0.30

0.40

0.50

r/D

Fig. 14 Comparison of numerically predicted fully developed velocity profiles to published experimental data from Ref. [17] for a Reynolds number of 3.2 million.

respectively, for the aforementioned Reynolds number ranges. In each figure, fully developed dimensionless velocity profiles are plotted as a function of the dimensionless radial coordinate. The velocity profiles are made dimensionless by the centerline velocity Uc in each case. In the first figure, Fig. 11, the predicted velocity profile based on the SST κ
ω turbulence model for Re ¼ 5000 is compared with two separate experimental results with similar Reynolds numbers, respectively, for Re ¼ 5450 [14] and 4900 [15]. It is seen from the figure that the predictions match very closely with the former data set and are satisfactory agreement with the latter. In Figs. 12 and 13 (Re ¼ 25,000 and 44,000), there is, in general, satisfactory agreement but not perfect agreement between the predicted and experimental results of [15,16]. For these ranges, the best agreement occurs

Validation of Turbulence Models for Numerical Simulation

25

near the centerline and near the pipe wall. For both of the figures, the numerical results at any radial location where within 5% of the experimentally measured data points. In the last figure of the set, Fig. 14, corresponding to a Reynolds number of 3.2 million, extremely good agreement is seen at all radial locations between the numerically predicted values and experimentally determined values [17]. In recognition of normal experimental uncertainties, the outcomes seen in Figs. 11–14 can be regarded as a validation of numerical predictions made with the SST κ
ω turbulence model.

5.4 Evaluation of the SST κ 2 ω Model for Heat Transfer in Case (3) In addition to the governing differential equations and the SST κ
ω model from Section 3.1.4 for three-dimensional, steady, incompressible, constantproperty turbulent flow, the energy equation is needed if heat transfer is being considered.   ∂ðui T Þ ∂ ∂T ρcp (28) ¼ ðk + kt Þ ∂xi ∂xi ∂xi For the heat transfer problem governed by Eq. (28), the turbulent thermal conductivity kt is required. Its value is closely linked to that of the turbulent viscosity μt by means of the turbulent Prandtl number Pr t ¼

cp μt kt

(29)

It has been shown that a constant value of Prt ¼ 0.85 gives rise to highly accurate heat transfer results [18,19]. In addition to the uniform upstream temperature of the incoming fluid, the straight pipe wall, (immediately after the bellmouth section) had a specified constant temperature that had a value greater than the fluid. The presentation of the heat transfer results will be given in dimensionless form using the local Nusselt number Nu, where the characteristic length is the inner pipe diameter D and the thermal conductivity k of air was 0.0261 W/m-°C. The first comparison, shown in Fig. 15, is for fully developed Nusselt numbers over a range of Reynolds numbers. The figure shows excellent agreement between the present, numerically determined values with the widely accepted Gnielinski Nusselt number correlation presented in Ref. [20]. Of particular note is the virtually exact agreement of the numerical predictions with the correlation. For comparison at Re ¼ 25,000, 75,000, and

Fully developed Nusselt number

26

Ephraim M. Sparrow et al.

250 200 150 100 Simulation result Gnielinski correlation [20]

50 0 0

20,000

40,000

60,000

80,000

100,000

120,000

Reynolds number

Fig. 15 Comparison of fully developed Nusselt numbers for Case (3) from the present numerical simulation results to data conveyed in the experimentally based correlation of [20].

Local Nusselt number

300 250 200 150 Simulation Re = 105,000

100

Experiment [21] Re = 102,600

50 0 0

5

10

15

20

x/D

Fig. 16 Comparison of developing Nusselt numbers for Case (3) from the present numerical simulation results to data reported in the experimental data reported in Ref. [21].

100,000, the percent difference compared to the correlation was 0.4%, 0.6%, and 0.9%, respectively. Another useful comparison is in the thermal entrance region of the pipe in question with available experimental data, as presented in Ref. [21]. Fig. 16 has been prepared to compare the present entrance region Nusselt numbers with experimental values for a pipe with a “long calming region” as described in Ref. [21]. The presently predicted Nusselt numbers differ only slightly from the data, with deviations of such magnitude as to be readily attributable to the geometric differences between the experiment and the simulation model or to experimental uncertainty. In all likelihood,

Validation of Turbulence Models for Numerical Simulation

27

experimental scatter is possibly responsible for the deviant data point at x/D ¼ 10. Overall, there is good agreement between these numerical results and the data reported in Ref. [21] in Fig. 16.

5.5 Summary of the SST κ 2 ω Model Evaluation for Case (3) Case (3) examined how well the SST κ
ω turbulence model accurately predicted the fully developed velocity profiles for a variety of turbulent Reynolds numbers in a straight pipe connected downstream of a bellmouth inlet. These numerically determined velocity profiles where then compared with several different investigators from the published literature [14–17] and in general, very good agreement was found to prevail. In addition, heat transfer results in the form of Nusselt numbers from the numerical simulation of Case (3) could be compared with available literature [20,21]. The fully developed Nusselt numbers displayed very good agreement with the widely accepted Gnielinski Nusselt number correlation [20]. This outcome supports what the present authors have shown previously, that is, where excellent agreement is possible between the numerically obtained turbulent Nusselt numbers with the SST κ
ω turbulence model and experimental data [22]. Similarly, the experimental data reported in Ref. [21] for a heated pipe with a bellmouth-style inlet also showed very good agreement with numerical predictions despite some of the differences between the geometry of the simulation and the physical experiment. This positive outcome with heat transfer is important because of the link between the turbulent viscosity μt from the SST κ
ω turbulence model and the turbulent thermal conductivity kt in the energy equation.

6. CASE (4): TURBULENT FLOW IN A RECTANGULAR DUCT The previous section was focused on both velocity profiles and Nusselt number values for flows inside of round pipes where initial inlet profiles were not specified. This section continues with another case study, where inlet values are not specified but are a result of the numerical simulation. This is possible when the upstream boundary in the solution domain is placed very distant from the inlet to the apparatus where the fluid is believed to be nearly stationary. Here, the apparatus in question is a rectangular duct with a baffle plate at the inlet cross section and a vacuum being applied at the downstream end. The focus here is to compare pressure drop results from the numerical simulation using the SST κ
ω turbulence model to experimental data presented in Ref. [23].

28

Ephraim M. Sparrow et al.

R

Flow direction H

x

L

Fig. 17 Schematic diagram of the rectangular duct with baffle plate used for Case (4).

6.1 Physical Situation for Case (4) Rectangular Duct The physical situation to be addressed here is the development of turbulent flow in a smooth rectangular duct. An overall schematic diagram of the apparatus is displayed in pictorial view in Fig. 17. The main component of the apparatus, which extends from its upstream end to its downstream end, is a flat, horizontal rectangular duct whose aspect ratio (width-toheight) is 4.33. The upstream end of the duct is framed by a large circular baffle plate whose radius R is 91.44 cm. The baffle serves to ensure that the air entering the duct originates from the space upstream of the inlet and not from lateral spaces. Once entered, the air passes through a velocity development length that consists of a horizontal rectangular duct. The duct height H was 1.905 cm, the width was 8.255 cm, and the length L was 172.72 cm. The corresponding hydraulic diameter Dh for the rectangular duct was 3.1 cm. In the work performed in Ref. [23], at a location downstream of the development region, the airflow enters an 18-rows deep, in-line tube bank situated in the downstream continuation of the duct. The tube bank was made up of vertically deployed, in-line cylindrical pin fins, the height of each pin fin being approximately half the vertical height of the duct, thereby providing a clearance space for flow above the tube bank. The airflow passes over the pin fins in crossflow. After traversing the tube bank, the air exited the rectangular duct to the suction side of a centrifugal blower that exhausted the air at the roof of the building. In this work, only results pertaining to the development of turbulent flow in the rectangular duct will be presented. To match the experimental results of [23], the Reynolds number was 15,000 and the incoming fluid was constant property air at 25°C.

6.2 Numerical Approach for Case (4) Rectangular Duct The governing equations and turbulence model used for the numerical simulations to be performed for the physical situation defined in the foregoing section are the same as those used previously for the SST κ
ω turbulence model, in Section 3.1.4. With the baffle plate in place, the total number of

29

Validation of Turbulence Models for Numerical Simulation

nodes was 17 million where roughly 11 million of the nodes were located in the rectangular duct. A mesh/grid independence study was performed to establish the necessary accuracy of the numerical results. Different node numbers and deployments were used for this study to establish threesignificant-figure accuracy in the results. The numerical simulations were executed by means of ANSYS CFX 16.1 software. The present situation is again unique in that the fluid flow is created by means of the suction mode. This different approach to flow creation has a marked effect on the manner in which the boundary conditions are applied. For a blown flow, it is necessary to specify both the fluid velocity and the state of turbulence at the inlet of the flow space. In contrast, if the air is drawn from a totally quiescent upstream space, there is neither a specifiable inlet velocity nor a specifiable inlet state of turbulence. In actuality, the treatment of a suction flow requires that the solution domain be extended far upstream of the inlet, well into the quiescent region. At the far upstream boundary (opening) of the extended solution domain, the velocity is virtually zero and the pressure is the uniform atmosphere value. Downstream of the pin-fin array at the end of the duct, a constant exiting mass flow rate was specified.

6.3 Evaluation of the SST κ 2 ω Model in Case (4) The numerical predictions (solid line) for the pressure distribution corresponding to Case (4) is shown in Fig. 18 along with the corresponding

(p - po)/(0.5rU 2 )

−0.005 −0.007

Simulation Re = 15,000

−0.009

Experiment [23] Re = 15,000

−0.011 −0.013 −0.015 −0.017 −0.019 −0.021 −0.023 −0.025

0

5

10

15

20

25

30

35

x/Dh

Fig. 18 Comparison of dimensionless pressure drop from the simulation and experimental data from Ref. [23] for Case (4), turbulent flow entering a rectangular duct, Re ¼ 15,000.

30

Ephraim M. Sparrow et al.

experimental data [23]. The local pressure p is subtracted by the atmospheric pressure p0 and divided by 0.5ρU2 to make it dimensionless. A value of zero for the dimensionless quantity would represent atmospheric pressure. The data are plotted as a function of x/Dh from the inlet x ¼ 0 to the duct length of x ¼ 34Dh, which is the region before the pin-fin array. As can be seen from the figure, the numerical results corresponding to this case are in excellent agreement with the experimental data in this region upstream of the pin-fin array. This level of agreement can be attributed to the realistic treatment of the inlet condition, adequacy of the mesh, and ability of the SST κ
ω turbulence model. For this velocity development portion of the duct, the predicted pressures are, at maximum, 1.5% lower than the experimental data points in Fig. 18.

6.4 Summary of the SST κ 2 ω Model Evaluation for Case (4) Case (4) was focused on numerically reproducing the exact situation of the experiments in Ref. [23] that are being used for validation with the SST κ
ω turbulence model. The rectangular duct with upstream baffle plate was modeled starting far upstream of the inlet (x ¼ 0), well into the quiescent region. The airflow was allowed to develop in the rectangular duct and pressure results were extracted from the simulation. These predictions for Re ¼ 15,000 were carefully compared with the experimental data, and it was found that the most extreme deviation of the predictions was 1.5%. Overall, the SST κ
ω turbulence model had very good agreement with the given experimental data.

7. CASE (5): TURBULENT FLOW IN PERFORATED PLATES The previous case focused on the development region of a rectangular duct, and attention will now be turned to modeling fluid flow through a perforated plate located in the fully developed region of a square duct [24]. Perforated plates commonly serve as flow control devices where the plates are placed perpendicular to an oncoming freestream flow in a duct, or they are used as a means of homogenizing a nonuniform flow. For many practitioners, being able to predict the pressure drop caused by a perforated plate is crucial to the design of flow systems.

7.1 Physical Situation for Case (5) A schematic diagram for Case (5) is shown in Fig. 19 where a perforated plate is located in a square duct, downstream of the flow development region.

31

Validation of Turbulence Models for Numerical Simulation

Flow direction

Perforated plate H L

x

Fig. 19 Schematic diagram of a square duct with a perforated plate for Case (5).

Single aperture stream tube Single aperture

Fig. 20 Illustration of a single aperture stream tube used in Case (5).

The nature of the flow that takes place within the open apertures (holes) of perforated plates is complex because of the separated flow that occurs. In particular, the separated flow has a dominant impact on the plate-based pressure drop. Downstream of the perforated plate, after the disturbance, the flow eventually redevelops. The experimental setup used in Ref. [24] was a square duct of height H ¼ 30.48 cm and length L ¼ 243.8 cm. Four pressure taps were installed 45.7 cm upstream of the perforated plate, and 10 were located downstream of the plate at 2.54 cm increments. The exit end of square duct was attached to a plenum chamber and operated in suction mode. In order to reduce the computational resources required to solve the problem, a numerical model was devised that focused on a representative orifice of the perforated plate, as illustrated in Fig. 20. That solution domain is sometimes referred to as a stream tube, and the velocity at the upstream end of the stream tube was measured by redundant Pitot tubes. The use of the available geometric symmetry enabled the numerical model to be further simplified to 1/8th of the orifice. This allowed for a greater density of nodes to be placed in a smaller solution domain. The porosity (open area) of the model in Fig. 20 was controlled by the orifice diameter D and the height and width of the stream tube. The plate

32

Ephraim M. Sparrow et al.

thickness t was fixed at t ¼ 0.5D, and the correspond porosities of 19.6% and 34.9% were set. The duct Reynolds number Re∞ is linked to the orifice Reynolds number Re0 using the porosity ξ, i.e., Re∞ ¼ ξRe0. To match the experiment, a uniform velocity was set 45.7 cm upstream of the orifice. The turbulence quantities were not measured in the experiment; therefore, a 5% turbulence intensity with μt/μ ¼ 10 was chosen. Downstream of the aperture, an outlet boundary condition was specified (atmosphere pressure) at a location corresponding to the last pressure tap in the experiment (25.4 cm). Aside from the no-slip boundary condition on the plate walls, the other surfaces were symmetry planes.

7.2 Numerical Approach for Case (5) There were several different solution domains in Ref. [24] depending on the perforated plate geometry (porosity, plate thickness, and whether it was a square or staggered array of holes). For each of the individual cases, a mesh independence study was performed, and the focus was directed to the overall dimensionless pressure drop caused by the presence of the perforated plate. The mesh was determined to be sufficient when this metric varied by less than 0.2%. Because of the small size of the solution domain, the nodal count ranged from 100,000 to 500,000 depending on the case. The numerical solutions were regarded as converged when the normalized residuals became 10
6. The working fluid for this case was constant property air at 25°C. The solutions of the discretized forms of the governing equations were based on the SST κ
ω model from Section 3.1.4 and were carried out in a manner similar to the previous cases.

7.3 Evaluation of the SST κ 2 ω Model for Case (5) The relevant outcomes of the work extracted from Ref. [24] are displayed in Fig. 21. The figure shows a variation of Reynolds number based on the orifice diameter and orifice velocity (as opposed to the duct Reynolds number). The pressure drop is presented in a dimensionless form normalized by ρU2, where U is the mean orifice velocity. The numerical data points represent both square and staggered arrays. Each set of results is grouped with respect to the porosity values of the perforated plate. Not unexpectedly, the pressure drop is higher for the lower porosity. A simple linear trend line was used to connect the data points for each set of numerical results. Inspection of Fig. 21 indicates that there is very good agreement between the predictions of the numerical simulation and where the experimental data points appear in the

33

Validation of Turbulence Models for Numerical Simulation

0.9 0.85 19.6%

Dp/(rU2)

0.8 0.75

Simulation [24] Square array Simulation [24] Staggered array Experiment [24] Staggered array

0.7 0.65 34.9% 0.6 0.55 0.5 2700

2900

3100

3300

3500 Reo

3700

3900

4100

Fig. 21 Dimensionless pressure drops from Ref. [24] for a perforated plate thickness of t/D ¼ 0.5, for both square and staggered hole arrangements, and for porosities of 19.6% and 34.9%.

data extracted from Ref. [24]. This level of agreement fully validates the choice of the turbulence model for this case.

7.4 Summary of the SST κ 2 ω Model Evaluation for Case (5) This case examined how well the pressure drop results from a numerical simulation using the SST κ
ω turbulence model compared with experimental measurements for a perforated plate situated in a square duct [24]. Two different plate porosities (open area) were modeled over a range of orifice Reynolds numbers for both square and staggered perforation arrays. Two experiments were performed with staggered perforation arrays at a single Reynolds number within the same range of the simulations for comparison. It is seen from the results that the experimentally measured values of pressure drop were in total agreement with the numerically predicted values.

8. CONCLUDING REMARKS In summary, five physical cases were examined: flow with secondary swirl, flow in a pipe bend, flow in round pipes, flow in a rectangular duct, and finally flow through a perforated plate. In each of these cases, turbulence model validation was performed using highly accurate and complete experimental results available in the published literature compared with appropriate numerical simulations. In each section of this chapter, a different validation approach was used depending on the availability of experimental

34

Ephraim M. Sparrow et al.

results. Initially, this took the form of velocity profile comparisons and was later extended to include heat transfer and pressure drop comparisons. The first two cases, (1) and (2), utilized available velocity and turbulence profile data at the inlet of the solution domain. From Case (1), the SST κ
ω turbulence model clearly was the preferred choice because of the good agreement with experimental data and the shorter CPU time required as compared to the LES model. Case (2) was a useful comparison because the flow had a secondary flow caused by the physical presence of a pipe bend and was not artificially created as in Case (1). Cases (3) and (4) were more focused on allowing the fluid flow to develop and not specifying inlet velocity and turbulence profiles. Again, the SST κ
ω turbulence model performed well at predicting the velocity profiles in Case (3) for various Reynolds numbers. In addition, the heat transfer results from Case (3) matched the experimentally based results from the published literature. Case (4) focused on a different approach, where pressure drop information was compared, and again the simulation based on the SST κ
ω turbulence model compared extremely well to the experimental data. Case (5) was based on a different form of comparison because a perforated plate, which is a common flow control device, abruptly disrupts the fluid flow in a duct. As the flow passes through the orifices in the plate, separated regions are formed and the flow has to redevelop downstream. None of the other physical cases presented here experienced a similar abrupt change. For this case, there was an overall agreement between the simulation results and the experimental data. Overall, the SST κ
ω turbulence model performed very well compared to the available experimental data for the cases investigated here. This finding is in agreement with other investigations where the SST κ
ω turbulence model had excellent agreement or the best results compared to experimental data [2–6,22,24]. As final note, it is the belief of the authors that if a turbulence model has been shown to be effective for numerical predictions of fluid flow, it is highly probable that model will be effective for heat transfer predictions.

REFERENCES [1] F.R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA J. 32 (8) (1994) 1598–1605. [2] A. Li, X. Chen, L. Chen, R. Gao, Study on local drag reduction effects of wedge-shaped components in elbow and T-junction close-coupled pipes, in: Building Simulation, vol. 7(2), Tsinghua University Press, Beijing, China, 2014, pp. 175–184.

Validation of Turbulence Models for Numerical Simulation

35

[3] W. Li, J. Ren, J. Hongde, Y. Luan, P. Ligrani, Assessment of six turbulence models for modeling and predicting narrow passage flows, part 2: pin fin arrays, Numer. Heat Transf. Part A Appl. 69 (5) (2016) 445–463. [4] U. Engdar, J. Klingmann, in: Investigation of two-equation turbulence models applied to a confined axis-symmetric swirling flow, ASME 2002 Pressure Vessels and Piping Conference, American Society of Mechanical Engineers, Vancouver, BC, Canada, 2002, January, pp. 199–206. [5] J. Park, S. Park, P.M. Ligrani, Numerical predictions of detailed flow structural characteristics in a channel with angled rib turbulators, J. Mech. Sci. Technol. 29 (11) (2015) 4981–4991. [6] P.D. Clausen, S.G. Koh, D.H. Wood, Measurements of a swirling turbulent boundary layer developing in a conical diffuser, Exp. Thermal Fluid Sci. 6 (1) (1993) 39–48. [7] J.M. Gorman, E.M. Sparrow, J.P. Abraham, W.J. Minkowycz, Evaluation of the efficacy of turbulence models for swirling flows and the effect of turbulence intensity on heat transfer, Numer. Heat Transf. Part B Fundam. 70 (6) (2016) 485–502. [8] B.E. Launder, D.B. Spalding, The numerical computation of turbulent flows, Comput. Methods Appl. Mech. Eng. 3 (2) (1974) 269–289. [9] V.S. Yakhot, S.A. Orszag, S. Thangam, T.B. Gatski, C.G. Speziale, Development of turbulence models for shear flows by a double expansion technique, Phys. Fluids A 4 (7) (1992) 1510–1520. [10] D. Wilcox, Multiscale model for turbulent flows, AIAA J. 26 (11) (1988) 1311–1320. [11] F. Nicoud, F. Ducros, Subgrid-scale stress modelling based on the square of the velocity gradient tensor, Flow Turbul. Combust. 62 (3) (1999) 183–200. [12] M.M. Enayet, M.M. Gibson, A.M.K.P. Taylor, M. Yianneskis, Laser-Doppler measurements of laminar and turbulent flow in a pipe bend, Int. J. Heat Fluid Flow 3 (4) (1982) 213–219. [13] M.M. Enayet, M.M. Gibson, A.M.K.P. Taylor, M. Yianneskis, Laser Doppler measurements of laminar and turbulent flow in a pipe bend, NASA Contractor Report 3551, (1982) 1–58. [14] J.G.M. Eggels, F. Unger, M.H. Weiss, J. Westerweel, R.J. Adrian, R. Friedrich, F.T.M. Nieuwstadt, Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment, J. Fluid Mech. 268 (1994) 175–210. [15] J.M.J. Den Toonder, F.T.M. Nieuwstadt, Reynolds number effects in a turbulent pipe flow for low to moderate Re, Phys. Fluids 9 (11) (1997) 3398–3409. [16] X. Wu, P. Moin, A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow, J. Fluid Mech. 608 (2008) 81–112. [17] J. Nikuradse, Laws of flow in rough pipes, in: VDI Forschungsheft, 1933. [18] S.W. Churchill, A reinterpretation of the turbulent Prandtl number, Ind. Eng. Chem. Res. 41 (25) (2002) 6393–6401. [19] W.M. Kays, Turbulent Prandtl number—where are we? J. Heat Transfer 116 (2) (1994) 284–295. [20] V. Gnielinski, New equations for heat and mass transfer in turbulent flow through pipes and ducts, Forsch. Ingenieurwes. 41 (1) (1975) 359–368. [21] A.F. Mills, Experimental investigation of turbulent heat transfer in the entrance region of a circular conduit, J. Mech. Eng. Sci. 4 (1) (1962) 63–77. [22] E.M. Sparrow, J.M. Gorman, J.P. Abraham, Quantitative assessment of the overall heat transfer coefficient U, J. Heat Transfer 135 (6) (2013) 061102. [23] E.M. Sparrow, J.W. Ramsey, C.A.C. Altemani, Experiments on in-line pin fin arrays and performance comparisons with staggered arrays, ASME J. Heat Transfer 102 (1) (1980) 44–50. [24] Y. Bayazit, E.M. Sparrow, D.D. Joseph, Perforated plates for fluid management: plate geometry effects and flow regimes, Int. J. Therm. Sci. 85 (2014) 104–111.