Application of hybrid Computational Fluid Dynamics turbulence model, STRUCT-[epsilon], on heated flow cases

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Application of Hybrid CFD Turbulence Model, STRUCT-ε, on

Heated Flow Cases

by

Ka-Yen K. Yau

S.B., Massachusetts Institute of Technology (2019)

SUBMITTED TO THE DEPARTMENT OF NUCLEAR SCIENCE AND ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF

MASTER OF SCIENCE IN NUCLEAR SCIENCE AND ENGINEERING AT THE

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

JUNE 2019

Signature of Author:

Ka-Yen K. Yau S.B., Massachusetts Institute of Technology (2019) Department of Nuclear Science and Engineering May 18, 2019

Certified by:

Emilio Baglietto Associate Professor of Nuclear Science and Engineering and Thermal Hydraulics Focus-Area Lead for the Consortium for Advanced Simulation of Lightwater Reactors (CASL) Thesis Supervisor

Certified by:

Michael Short Class of ’42 Career Development Associate Professor of Nuclear Science and Engineering Thesis Reader

Accepted by:

Ju Li Battelle Energy Alliance Professor of Nuclear Science and Engineering and Professor of Materials Science and Engineering Chairman, NSE Committee for Graduate Students

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Application of Hybrid CFD Turbulence Model, STRUCT-

ε

, on Heated

Flow Cases

by

Ka-Yen K. Yau

S.B., Massachusetts Institute of Technology (2019)

Submitted to the Department of Nuclear Science and Engineering on May 18, 2019 in Partial Fulfillment of the Requirements for the Degree of

Master of Science in Nuclear Science and Engineering

ABSTRACT

Computational Fluid Dynamics (CFD) modeling is a powerful numerical method that can be used to characterize fluid flow, pressure drop, and thermal transient behavior in complex flow geometries. However, with current CFD simulation techniques, the accu-rate modeling of turbulence structures is often prohibitively expensive or time-intensive. Therefore, a new hybrid turbulence model, STRUCT-

ε

, was developed to more accu-rately and quickly resolve the formation and propagation of unsteady turbulence struc-tures. STRUCT-

ε

introduces a source term that implicitly reduces fluid eddy viscosity, which in turn reduces Reynolds stresses which are traditionally over-predicted with two-equation models. Most notably, STRUCT-

ε

’s method of implicit hybrid activation is uniquely simple to implement while remaining grid-size and inlet turbulence condition independent.

STRUCT-

ε

has previously demonstrated improved accuracy in the prediction of flow topology and velocity compared to results produced by URANS models for several inter-nal and exterinter-nal flow cases. This study seeks to extend understanding of STRUCT-

ε

’s capability by benchmarking the model’s performance with the experimental or Direct Numerical Simulation (DNS) results from a variety of flow cases, including an impinging jet [1], film cooling [2], and an infinite wire-wrapped nuclear fuel assembly [3]. For each case, computed parameters from each CFD simulation were evaluated and compared both numerically and qualitatively, through the computation of root mean square error and identification of characteristic flow features. The improved performance of the RANS turbulence model could have large implications on the practicality and applicability of CFD modeling in the design and qualification of numerous technologies.

Thesis Supervisor: Emilio Baglietto

Title: Associate Professor of Nuclear Science and Engineering and Thermal Hydraulics Focus-Area Lead for the Consortium for Advanced Simulation of Lightwater Reactors (CASL)

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Acknowledgements

I would first like to thank my research advisor, Professor Baglietto. His teachings, sup-port, and guidance were invaluable for the completion of this project. I admire and am inspired by his seemingly unending industry experience, work ethic, and fresh perspective on engineering.

I also must thank the other members of his research group, the self-proclaimed "Bubble Bros," who always seem to be there to offer help, advice, and encouragement. Special thanks to Liangyu Xu, the developer of the STRUCT-

ε

model.

Thanks to TerraPower, LLC. for their support of this project. From Terrapower, I owe special thanks to the members of their thermal-hydraulics team, who adopted me as an intern and first introduced me to CFD.

Finally, I would like to thank my family, friends, and boyfriend, who have all sup-ported, cared for, and inspired me throughout my time at MIT. I promise, now that this is all done, I will be better about answering your text messages :)

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List of Figures

2.1 Kolmogorov turbulent spectrum depicting relation between turbulence en-ergy and lengthscales [4] . . . 20

4.1 Half of the impinging jet CAD geometry used to mimic the fluid volume of the experimental test facility of Cooper et. al. Notable geometric di-mensions are labeled. . . 35

4.2 Experimentally determined Nusselt number distribution on base plate of impinging jet when Re = 23,750 for geometries with jet outlets located at various heights of Z/D [5] . . . . 37 4.3 Expected flow structures in the axisymmetric impinging jet flow case.

No-table flow regions are labeled. 1) potential core region, 2) developing flow 3) stagnation region, 4) wall jet region [6] . . . 38 4.4 Mesh refinement applied to region representing jet pipe. . . 41

4.5 Mesh refinement applied to free volume region. . . 42

4.6 View of the mesh created to simulate the impinging jet geometry on two-dimensional cross section of the geometry. . . 42

4.7 Boundary conditions used in axisymmetric impinging jet simulations. Im-age from [7] . . . 43

4.8 Temporal and spatial average of wall-temperature at location r/D = 3 plotted against the cube root of mesh elements for each mesh used in the grid sensitivity study. . . 46

4.9 Average temperature plotted as a function of r/D for all meshes used in mesh sensitivity analysis. . . 47

4.10 Max temperature difference at each radial location between the results computed using the coarse, medium, and fine meshes. . . 47

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4.11 Circumferentially and time averaged values of scaled Nusselt number at the impinging jet wall computed with the STRUCT-

ε

, Standard k-

ε

, and k-ω SST models. . . . 48 4.12 Percent difference between CFD and experimentally measured average

Nusselt number plotted as a function of r/D for all turbulence models used. . . 49 4.13 Mean velocity magnitude profile at impinging jet cross section for the

tested turbulence models. . . 50 4.14 Radial velocity near wall of impinging jet at various radial locations . . . 51 4.15 Time averaged turbulent kinetic energy values at wall computed by the

STRUCT-

ε

model. . . 52 4.16 Time averaged wall temperature computed by the STRUCT-

ε

model. . . 53 4.17 Q-Criterion isosurface of Q = 5000 s-2 _{visualizing turbulence structures}

generated by the STRUCT-

ε

model. Highlighted in orange is r/D = 2.75. 54 4.18 Wall shear stress magnitude on impingement plate from three turbulence

models used. . . 55 4.19 Time averaged turbulent kinetic energy values computed by the Standard

k-

ε

model. . . 56 4.20 Q-Criterion value at Q = 5,000 s-2 _{visualizing computed flow field of}

im-pinging jet using the SKE model. Highlighted in orange is r/D = 2.75. . 57 4.21 Time averaged turbulent kinetic energy values at wall computed by the

k-ω SST model. . . . 58 4.22 Q-Criterion value at Q = 5,000 s-2 _{visualizing computed turbulence}

struc-tures of impinging jet using the k-ω SST model. Highlighted in orange is r/D = 2.75. . . . 59 5.1 Schematic of film cooling test configuration used by Sinha et al (1991) [2]. 61 5.2 The wind tunnel width was truncated such that only the region highlighted

in blue was simulated. . . 62 5.3 CAD geometry created to simulate the fluid volume of the film cooling test

facility. . . 62 5.4 Illustration of turbulent structures that form at the jet exit of film cooling

flow[8]. . . 64 5.5 Jet wall mesh and resulting y+ _values. _{. . . .} ₆₉

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5.7 Zones used to mesh the wind tunnel continuum. . . 70 5.8 Example of meshed wind tunnel portion. Pictured is mesh created with

base mesh size of 1.5625 mm. . . 71 5.9 Merged jet and wind tunnel regions. . . 72 5.10 Mean velocity profile in jet pipe computed using LES from experimentation

of Ziefle et al. Location of the jet cross sections pictured are detailed below. 74 5.11 Centerline cooling efficiency plotted as function of normalized streamwise

distance from the jet outlet for each mesh used in the grid sensitivty study. 76 5.12 Film cooling efficiency plots from each mesh used in the grid-sensitivity

study with the aforementioned features labeled for each figure . . . 77 5.13 Time and centerline averaged cooling efficiency plotted against the cube

root of cell count for each mesh used in the grid sensitivity study. . . 78 5.14 Q = 10,000 s-2 _{isosurface at t = 10s for each of the meshes compared in}

the mesh sensitivity study. . . 79 5.15 Time average turbulent kinetic energy at the base of of the geometry from

each of the meshes compared in the mesh sensitivity study. . . 80 5.16 Normalized time average velocity magnitude of the film cooling wall

cen-terline for each of the meshes compared in the mesh sensitivity study. Velocity normalized by T∞ = 300 K. . . 81

5.17 Time averaged centerline cooling efficiency plotted as function of normal-ized streamwise distance from the jet outlet for the three models tested. . 82 5.18 Time and centerline averaged cooling efficiency plotted as function of

nor-malized lateral distance for the three models tested. To reduce the effects of flow asymmetry, the cooling efficiency values were averaged across the centerline. . . 82 5.19 Time and centerline averaged percentage difference in cooling efficiency

plotted as function of normalized lateral distance for the three models tested. . . 83 5.20 Time averaged cooling efficiency at the base of the wall predicted using

the STRUCT-

ε

model. . . 84 5.21 Instantaneous temperature monitored at points along the base of the

geom-etry. The locations of the points at which the temperature was monitored is pictured in Figure 5.22 . . . 84 5.22 Points at which temperature was monitored during the transient simulation. 85

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5.23 Cross sectional comparison of velocity profile of jet pipe resulting from use of LES and the STRUCT-

ε

model. . . 86 5.24 Mean velocity vectors at lateral cross section of x/D = 1.0 produced by

the STRUCT-

ε

model. . . 87 5.25 Q-Criterion = 500 s-2 isosurface picturing coherent turbulence structures

produced by the STRUCT-

ε

model. . . 87 5.26 Velocity profile and contours of lateral cross section of x/D = 1.0 produced

by the STRUCT-

ε

model. . . 88 5.27 Velocity profile and contours of lateral cross section of z/D = 15.0 produced

by the STRUCT-

ε

model. . . 89 5.28 Comparison of mean velocity profile of cross section at streamwise

center-line for all turbulence models used. . . 90 5.29 Average turbulent kinetic energy of at wall of STRUCT-

ε

and k-ω SST

modeling methods. . . 90 5.30 Time averaged cooling efficiency at the base of the wall predicted using

the Standard k-

ε

model. . . 91 5.31 Average turbulent kinetic energy of at wall of the Standard k-

ε

model. . 92 5.32 Q-Criterion = 500 s-2 _{isosurface picturing coherent turbulence structures}

produced by the Standard k-

ε

model. . . 92 5.33 Cross sectional comparison of velocity profile and contours of jet inlet pipe

resulting from use of LES and Standard k-

ε

model. . . 93 5.34 Mean velocity vector image at lateral cross section of x/D = 1.0 produced

by the Standard k-

ε

by the Standard k-

ε

by the Standard k-

ε

model. . . 95 5.37 Time averaged cooling efficiency at the base of the wall predicted using

the k-ω SST model. . . . 95 5.38 Cross sectional comparison of velocity profile and contours of jet inlet pipe

resulting from use of LES and the k-ω model. . . . 97 5.39 Mean velocity vector image at lateral cross section of z/D = 1.0 produced

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5.40 Q-Criterion = 500 s-2 _{isosurface picturing coherent turbulence structures}

produced by the k-ω model. . . . 98 5.41 Velocity profile and contours of lateral cross section of z/D = 1.0. . . . . 99 5.42 Velocity profile and contours of lateral cross section of z/D = 15.0. . . . . 99 6.1 Truncated fluid geometry used in CFD simulations with fluid boundaries

labeled. . . 101 6.2 Predicted structures from cross-flow across a wire. Image from Ranjan et

al (2010) [9]. . . 103 6.3 Normalized axial velocity and temperature results of the infinite

wire-wrapped fuel assembly predicted using various RANS methods completed by Dovizio et al (2019) [10]. . . 104 6.4 Sample of mesh constructed for simulation of infinite wire-wrapped fuel

assembly. . . 109 6.5 Connected fluid interfaces labeled above with numbered arrows such that

similar numbers indicate interfaces that are connected. . . 110 6.6 Temperature profile at line probe with endpoints (0.0, 0.0075, 0.131) m

and (0.0, 0.001, 0.131) m plotted for the each meshes as a function of s+ = x/D. . . 112 6.7 CFD computed uτ plotted against the cube root of total mesh elements

contained in the coarse, medium, and fine meshes. . . 112 6.8 Velocity profile plotted for line probe with endpoints (0.0, 0.0075, 0.131)

m and (0.0, 0.001, 0.131) m for the coarse, medium, and fine meshes. . . 113 6.9 Line probes at which q-DNS data was extracted for benchmarking purposes 114 6.10 Mean T+ profiles computed using each model at each of the line probes. 115 6.11 Mean umag+ profiles computed using each model at each of the line probes. 116

6.12 Mean T +∗ profiles computed using each model at each of the line probes. 117 6.13 Mean Umag+∗ profiles computed using each model at each of the line probes. 118

6.14 Percentage difference between turbulence modeled and q-DNS results of

T +∗ profiles at each of the line probes. . . 119 6.15 Percentage difference between turbulence modeled and q-DNS results of

Umag+∗ profiles at each of the line probes. . . 120

6.16 Comparison of turbulent kinetetic between STRUCT-

ε

and q-DNS models 121 6.17 Comparison of normalized temperature between STRUCT-

ε

and q-DNS

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6.18 STRUCT-

ε

computed wall shear stress . . . 123 6.19 STRUCT-esource distribution computed using STRUCT-

ε

. Regions of

el-evated activation circled. . . 124 6.20 Comparison of turbulent viscosity, µτ, between the three models used. . . 124

6.21 Comparison of Umag+∗ between STRUCT-

ε

and q-DNS models . . . 125

6.22 Comparison of U +∗ between STRUCT-

ε

and q-DNS models . . . 125 6.23 Normalized U +∗ velocity component of all turbulence models tested

plot-ted at each of the line probes. . . 126 6.24 Normalized U +∗ velocity component of all turbulence models tested

plot-ted at each of the line probes. . . 127 6.25 Comparison of velocity vector scenes between STRUCT-

ε

and q-DNS models128 6.26 Comparison of Q-criterion = 10,000.0/s-2 _{of the three models tested . . .} ₁₂₈

6.27 Wall shear stress computed with the Standard k-

ε

model. . . 129 6.28 Comparison of T +∗ between Standard k-

ε

and q-DNS. . . 130 6.29 Comparison of umag+ between Standard k-

ε

and q-DNS. . . 130

6.30 Comparison of Lateral RMS+ values between STRUCT-

ε

and q-DNS models 131 6.31 Comparison of turbulent kinetetic between Standard k-

ε

and q-DNS. . . 132 6.32 Comparison of velocity vector image between the Standard k-

ε

and q-DNS

models . . . 132 6.33 Comparison of turbulent viscosity between k-ω SST and Standard k-

ε

. . 133 6.34 Comparison of T +∗ profiles between k-ω SST and Standard k-

ε

results . 134 6.35 Comparison of Umag+∗ profiles between k-ω and Standard k-

ε

results. . . 134

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List of Tables

4.1 Thermal Physical Properties of Impinging Jet Simulant Fluid . . . 36

4.2 Meshing Details for Coarse, Medium, and Fine Mesh of Impinging Jet . . 43

4.3 Features of Impinging Jet Inlet Flow Profiles Computed with CFD . . . . 44

4.4 Total Average Nusselt Number RMSE per Turbulence Model . . . 48

4.5 Total Average Radial Velocity RMSE per Turbulence Model . . . 49

5.1 Thermal Physical Properties of Film Cooling Simulant Fluid . . . 63

5.2 Mesh Details for Wind Tunnel Meshes Used for Mesh Sensitivity Study . 72 5.3 Total Time Averaged Film Cooling Efficiency RMSE per Turbulence Model 83 6.1 Flow Geometry Properties of the MYRRHA Reactor . . . 101

6.2 Thermo-Physical Properties of Simulant Fluid . . . 102

6.3 Line Probe Locations at z = 0.131 m of Infinite Wire-Wrapped Fuel Assembly 113 6.4 uτ Resulting from Each Simulation Methodology (m/s) . . . 115

6.5 RMSE of T +∗ Results for Each Line Probe . . . 117

6.6 RMSE of Umag+∗ Results for Each Line Probe . . . 118

6.7 ∆P D of T +∗ for Each Line Probe . . . 119

6.8 ∆P D of Umag+∗ for Each Line Probe . . . 120

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Chapter 1 Introduction

Accurate and high resolution characterization of the flow distribution and heat transfer in nuclear reactor systems is crucial to their safe and efficient operation. However, reactor flow systems are complex and remain difficult to predict. Yet, the full understanding of these systems is necessary to prevent local phenomena that could lead to core damage, such as critical heat flux, dryout, thermal striping, or two-phase flow instabilities. Unlike most current thermal-hydraulic analysis methods used in the nuclear industry, computa-tional fluid dynamics (CFD) modeling can provide 3-D, highly detailed solutions and is therefore becoming an increasingly important and powerful method used in the design and licensing of nuclear reactors.

Using CFD, flow field parameters are computed by iteratively solving the discretized Navier-Stokes equations. However, the accuracy and scope of CFD modeling is limited by the computationally intensive task of modeling fluid turbulence. Turbulent structures are highly chaotic, random, and a multi-length-scale phenomenon, thus, complete modeling of turbulent flow can only be accurately computed using direct numerical solution (DNS) methods. However, DNS, while rigorous and extremely accurate, is too costly to be a practical tool for engineering design purposes. Therefore, to expedite and lower the demands of CFD turbulence computations, turbulence models are used to approximate fluid flow characteristics.

Within industry, two popular, simple, and easily implementable turbulence models are the k-

ε

and k-ω models, both of which which are classified as two-equation Reynolds Averaged Navier-Stokes (RANS) models. These models also have unsteady counterparts (classified as Unsteady RANS or URANS models) for the computation of unsteady flow cases. However, because the (U)RANS methodology is based on a simplified treatment

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of eddy viscosity, they are often incapable of accurately resolving a number of complex or strongly fluctuating turbulent behavior, such as flow separation and reattachment, swirling flows, or jet spreading. These modeling restrictions limit the usefulness and applicability of CFD simulations in determining the safety of nuclear energy applications.

Therefore, a hybrid turbulence model, STRUCT-

ε

was developed to improve upon the accuracy of existing models while simultaneously maintaining the computing efficiency of (U)RANS. STRUCT-

ε

achieves hybridization through the introduction of a source term in the Cubic k-

ε

turbulence dissipation rate equation. This reduction factor implicitly limits typically over-predicted fluid eddy viscosity values.

The use of the STRUCT-

ε

model has demonstrated improved prediction of flow topology and velocity for a variety of internal and external flow cases. However, to fur-ther characterize the model’s performance, three heated flow cases that are traditionally challenging for classic RANS closure models were simulated using this hybrid model.

• The impinging jet case by Cooper et al (1992) was simulated to determine the model’s capability of resolving an axisymmetric jet and the resulting heat transfer [1].

• The film cooling geometry from Sinha et al (1991) was simulated to understand the performance of STRUCT-

ε

on a more complex geometry [2].

• A q-DNS of an infinite, wire-wrapped fuel assembly from Shams et al (2018) was re-simulated using STRUCT-

ε

in order to directly benchmark its performance on a nuclear-specific flow case [3].

For each case, the calculated results from each simulation were compared to the corresponding experimentally collected/DNS results both numerically and qualitatively. Additionally, the performance and accuracy of STRUCT-

ε

were compared to that of the Standard k-

ε

and k-ω SST models. The computation of root mean square error and local percentage error for available measurements provide quantitative information about model agreement. Simultaneously, the identification of characteristic flow features and evaluation of likenesses between the results of each modeling methods tested serve as a qualitative demonstrator of the validity and fidelity of STRUCT-

ε

and can be used to understand the model’s accuracy in representing physical systems.

Increased understanding of the behavior and modeling uncertainties associated with the STRUCT-

ε

model is necessary for its widespread adoption and use. The development of higher accuracy, computationally simple, and lower cost CFD simulation methods could widen the design space of both nuclear and other engineering industries by enabling the

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use of robust numeric modeling for complex geometries and providing an alternative to costly, time-intensive, and regulation burdened experimental testing.

1.1 Objectives

The objective of this work is to evaluate the performance of the STRUCT-

ε

turbulence model in the simulation of heated flow cases. This goal will be accomplished using the following steps.

1. Identify and construct flow cases that contain heat transfer and turbulence struc-tures that are traditionally difficult to model.

2. Perform simulations using the STRUCT-

ε

, Standard k-

ε

, and k-ω SST turbulence models.

3. Compare results with experimental/DNS data to evaluate modeling performance and quantify modeling uncertainty.

1.2 Structure

This thesis is divided into six chapters. Each flow case tested is given a chapter, in which model geometry, previous simulation work, the modeling process, and an analysis of the simulation results are detailed.

• Chapter 2 introduces computational fluid dynamics and the turbulence modeling methods used for this line of work. The mathematical approaches of both the STRUCT-

ε

and classic RANS turbulence models are detailed.

• Chapter 3 details the methodology that was used to perform and analyze the CFD modeling of each test case.

• Chapter 4 details the impinging jet case. • Chapter 5 details the film cooling case.

• Chapter 6 details the wire-wrapped fuel assembly.

• Chapter 7 concludes the thesis, and the performance of the STRUCT-

ε

model is evaluated.

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Chapter 2 Background

To support this thesis, Chapter 2 provides a brief introduction to computational fluid dynamics (CFD) modeling and fluid turbulence, an incomprehensive review of existing CFD turbulence models, and detail on the novel STRUCT-

ε

method. Understanding of the functionality, advantages, and limitations of existing turbulence modeling methods provides context and motivation for the development of the STRUCT-

ε

.

2.1 Computational Fluid Dynamics

Single-phase CFD is a numerical modeling method in which the discretized Navier-Stokes equations are solved to calculate the properties and behavior of a continuous fluid medium. CFD can be a uniquely useful tool for understanding fluid flow behavior, because it is able to provide information about the fluid at high spatial and temporal resolution.

To model a fluid volume, the geometry is subdivided into a spatial mesh consisting of smaller fluid volumes called cells. For each of these cells, the Navier-Stokes equations are solved, ensuring that the conservation of momentum, mass, and energy are maintained across each individual fluid volume. The conservative differential forms of the formerly mentioned equations are as follows:

Mass: ∂ρ ∂t + ∇ · (ρ ~U ) = 0 (2.1) Momentum: ∂(ρu) ∂t + ∇ · (ρu ~U ) = − ∂P ∂x + ∇ · (µ∇u) + SM x (2.2)

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∂(ρv) ∂t + ∇ · (ρv ~U ) = − ∂P ∂y + ∇ · (µ∇v) + SM y (2.3) ∂(ρw) ∂t + ∇ · (ρw ~U ) = − ∂P ∂z + ∇ · (µ∇w) + SM z (2.4) Energy: ∂(ρcpT ) dt + ∇ · (ρcpT ~U ) = −P ∇ · ~V + ∇ · (k∇T ) + φ + Si (2.5)

whereu, v, and w are the x, y, and z components of the velocity, ~U [12]. While there are a variety of discretization methods, for this work and most commercial CFD codes, the finite volume methodology (FVM) was used. All computations for this set of work were completed using STAR-CCM+ v11.06.011.

2.2 Turbulence

Turbulence is the unsteady, non-periodic motion of a fluid in which the velocity, pressure, and temperature fluctuate. Unlike in laminar flow, where the fluid flows along uniform, steady streamlines, turbulent flow is irregular and dissipative, dominated by distinctive rotating vortex structures called eddies. The Reynolds Number,

Re = ρuL

µ (2.6)

where ρ is the fluid density, u is bulk fluid velocity, L is the characteristic length scale, and µ is the kinematic viscosity, is an important non-dimensional number that can be used to predict the level of turbulence in a flow.

Turbulent flow has proven to be challenging to numerically model. The complexity of modeling turbulence stems from the wide range of vortex structure length scales that are simultaneously present in turbulent flow. The ratio of the largest and smallest length scales that exist in flow simultaneously is proportional to Re3/4_{. In general, three main}

length scales are used to characterize turbulent flow: the integral scale, Taylor microscale, and Kolmogorov scale [12].

Larger scale eddies carry most of the energy and have length scales based on the geometry of the fluid domain. These eddies break down into the following smaller struc-tures that are diffused by viscosity. The integral scale is a measure of the average size of the turbulent eddies within a flow field. Turbulence at this length scale has large velocity fluctuations, low frequency, and high anisotropy. The Taylor microscale is an

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interme-diate length scale representing the inertial subrange in which turbulence is isotropic. Finally, the Kolmogorov length scale describes the smallest eddy structures which are in a regime in which the kinetic energy of turbulent structures is converted to thermal energy through viscous dissipation. Structures at the Kolmogorov scale are the smallest present turbulent structures and are structurally isotropic.

Figure 2.1: Kolmogorov turbulent spectrum depicting relation between turbulence energy and lengthscales [4]

.

Therefore, to fully reproduce turbulent flow behavior, a CFD simulation would have to be fully spatially resolved down to the Kolmogorov length scale, which is defined as

η = v

3

ε

!1₄

(2.7)

where v is the kinematic viscosity and

ε

is the dissipation of turbulent kinetic energy. The time scale used to describe the motion and dissipation of these small scale turbulent structures is [12] tη = v ε !1 2 (2.8)

The methodology of solving the Navier-Stokes equations with the aforementioned spatial and temporal resolution is called Direct Numerical Simulation (DNS). DNS is the most direct method of solving turbulence and, when computed correctly, yields incredibly high accuracy results.

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2.3 Turbulence Modeling

However, DNS is both computationally intractable and time intensive. The number of mesh grid points that would be required to resolve the smallest turbulence scales in a three dimensional DNS simulation scales by Re9/4_{, and the overall cost scales by Re}3

[13]. If DNS is computed on too coarse of a mesh, significant errors would arise from insufficient modeling of small scale structures. Therefore, even with modern computa-tional capabilities, only small scale flow cases of relatively low Reynolds numbers can be reasonably simulated.

To widen modeling capability, save time, and reduce computational demands, a vari-ety of turbulence modeling methods have been developed. These models utilize various assumptions and equation closures to account for the effect of small scale turbulence, therefore eliminating the need to deterministically simulate the flow features which are generally the most computationally demanding.

2.3.1 Reynolds Averaged Navier Stokes (RANS)

Within industry, the most commonly utilized method of modeling turbulent flow is the Reynolds Averaged Navier Stokes (RANS) equations. The RANS equations are time-averaged equations of fluid motion. The use of these time-time-averaged equations greatly decreases necessary computational time, while still providing adequate information for most industrial and engineering purposes.

In the RANS equations, velocity and pressure are decomposed into mean and fluctu-ating parts. This process is called Reynolds Decomposition.

U = U + U0 (2.9)

P = P + P0 (2.10)

These decomposed values are then input into the Navier Stokes equations and the equations are averaged such that,

ρ∂U

∂t + ρU · ∇U = −∇P + ∇(2µS − ρu

0

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where the Reynolds stress tensor is ρu0

iu0j, and the mean strain tensor, S, is defined as,

S = 1 2 _∂U ∂x + ∂U ∂y (2.12)

In this set of equations, the Reynolds stress tensor is a symmetric matrix that accounts for the effects of turbulent motions on the mean stresses, with the diagonal components representing normal stresses and the off-diagonal components representing shear stresses. However, the Reynolds stress tensor is not a closed system, containing six additional independent unknowns [12].

Among many existing formulations, there are two leading appraches used to determine the unknowns in the Reynolds stress tensor: Reynolds stress models (RSM) and eddy viscosity models (EVM). Using a RSM, each component of the tensor is individually solved, enabling the complex interactions in turbulent flow fields to be accounted for, such as directional effects, heat flux, etc. However, since the basis of the STRUCT-

ε

model is the eddy viscosity model, the most commonly used EVMs are detailed in the following sections and RSM will not be discussed further (though the interested reader can refer to Chapter 11 of Turbulent Flows by Pope (2015).

2.3.2 Linear Eddy Viscosity Models

Linear eddy viscosity models are the most commonly used closure models in industry and for engineering applications. The fundamental concept of linear EVMs is that the Reynolds stresses are modeled using the Boussinesq hypothesis, which states that,

τij = 2µtSij −

2

3ρkδij (2.13)

where µt is the eddy viscosity,k is the mean turbulent kinetic energy, and Sij is the mean

strain rate [12]. The Boussinesq hypothesis presents a simplified model of turbulence, in which the effects of turbulence on mean flow are treated in a way similar to the effects of molecular viscosity on laminar flow.

There are a variety of zero-, one-, and two-equation models used to solve for the eddy viscosity, µt. The most commonly used within industry are the two-equation

mod-els, which add two additional transport equations. The turbulent kinetic energy, k, a measure of the energy contained within the turbulence, is commonly paired with either the turbulence dissipation rate,

ε

, or the specific turbulence dissipation rate, ω, which

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determines the scale of the turbulence, to form the two transported variables for the two-equation models. These equations comprise the k-

ε

and k-ω models respectively [14].

k-

ε

Model

The Standard k-

ε

model, popularized by Jones and Launder (1972), is the basis of and will serve as a point of comparison for the STRUCT-

ε

model. The k-

ε

model uses the turbulent kinetic energy (TKE), k, and the turbulent dissipation rate,

ε

, to model the eddy viscosity. Equations A.1 through A.4 in Appendix 2.3.2 describe the Standard k-

ε

model. It is important to note that the terms in the

ε

equation were not derived, but rather were selected to be analogous to the TKE equation in efforts to reduce the complexity of the two-equation model [15]. The Standard k-

ε

also makes use of the Boussinesq hypothesis to compute eddy viscosity.

As one of the most commonly used turbulence models for engineering purposes due to its simplicity, numerical robustness, and predictability, the Standard k-

ε

’s performance has been extensively documented and understood. However, through the utilization of the linear eddy-viscosity model, the k-

ε

class of models are limited when describing secondary flow structures, flows with high streamline curvature, swirling or rotational flows, transitional flow, unsteady flows, or flow in stagnation regions.

However, numerous various formulations have been developed, including the Realiz-able k-

ε

which modifies the turbulent viscosity and

ε

formulas, resulting in generally higher fidelity performance of jet spreading, boundary layers under strong adverse pres-sure gradients, rotation, and strong streamline curvature, and the RNG based k-

ε

model, which modifies the computation of TKE with through the addition of a term in the

ε

equation [15].

k-ω Model

The k-ω family of two-equation models instead uses the specific turbulence dissipation rate, ω, which is the rate of turbulence dissipation per unit volume where

ω = ε

βk (2.14)

There are a number of accepted k-ω models, however, the Wilcox k-ω is one of the most widely used and understood variants (see Appendix for details about the model

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formulation). Unlike the k-

ε

model, the Wilcox k-ω is physically valid in the resolved boundary layer. It also performs more favorably in wall-bounded flows with mild and lightly adverse pressure gradients than the k-

ε

. However, the k-ω has demonstrated overestimated levels of shear stress, underestimated levels of turbulent kinetic energy in simple strains, and over-sensitivity to the freestream inlet boundary conditions for turbulence levels [12].

k-ω SST Model

An additional popular variant of the linear EVMs relies on a hybridized form of the two aforementioned models. This model is the k-ω SST (also known as Menter’s Shear Stress Transport) which models fluid using the k-ω model in the boundary layer near the wall and switches to k-

ε

models in regions of free shear flow.

This combination of modeling methods is claimed to offer various improvements over traditional linear EVMs. The use of the k-ω formulation in the innermost regions of the boundary layer enables direct low Reynolds turbulence modeling without the use of extra damping functions. The transition to the k-

ε

model in the free-stream then minimizes the effects of k-ω’s over-sensitivity to inlet free-stream turbulence properties and takes advantage of k-

ε

’s better performance in adverse pressure gradients and separating flow. The method by which this switching is implemented can be found in Appendix 2.3.2.

The k-ω SST model has demonstrated improved capability of resolving flows in in adverse pressure gradients, separating flow, and near wall flow. However, the improved performance in these cases are attributable to the clever use of model limiters, rather than improved physical representation. Tests have also shown that the model still over-predicts turbulence levels in regions with large normal strain, such as regions of stagnation or strong acceleration. Finally, there is concern that the use of a blending function has the potential to obscure critical turbulence features, particularly in internal flow applications, in which the transition region between near- and far-wall is less well defined [16].

2.3.3 Non-Linear Eddy Viscosity Models

The use of the Boussinesq eddy viscosity, which simplifies the complex fluid stress-strain relationship to a non-physical linear relation, limits the accuracy of the two-equation models. In flows with complex features, such as strong gradients, adverse pressure fields, and streamline curvature, the Boussinesq assumption is invalid, due to poor prediction of turbulent viscosity from the use of this linear relaation. In response, non-linear eddy

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viscosity models (NLEVM) using higher order relations for the Reynolds stress tensors were developed to achieve higher accuracy solutions [12].

For example, the Cubic k-

ε

model developed by Baglietto and Ninokata (2006) is a cubic eddy viscosity model that demonstrates higher solution robustness than its linear counterparts, and it serves as the basis of the STRUCT-

ε

model. To more accurately represent flow anisotropy, Baglietto and Ninokata optimized their formulation using DNS test data from a variety of flow cases (see Appendix 2.3.2. The transport equations for

k and

ε

are the same as those from the Standard k-

ε

model of Launder and Spalding [11], however, a cubic stress-strain relationship is used to include sensitivity to turbulence anisostropy, curvarture, and swirling[11].

2.3.4 Large Eddy Simulation

Large Eddy Simulation (LES) is a CFD turbulence modeling technique in which spatial filtering is used to separate the treatment of large and small scale eddy structures. In LES, large, energetic, and geometry dependent eddies are explicitly solved using the Navier Stokes equations, leaving smaller, sufficiently isotropic eddies to be implicitly modeled. In LES, the velocity field is separated into a resolved component, ui, and a sub-grid

component, u0_i, such that

ui = ui+ u0i (2.15)

Sub-grid-scale (SGS) models are used to account for the momentum exchange between resolved and modeled structures. While there are a variety of SGS closures, most SGS techniques employ the Boussinesq hypothesis to calculate the SGS stresses as

τij −

1

3τkk∂ij = −2µtSij (2.16)

where µt is the subgrid scale turbulent viscosity and Sij is the rate-of-strain tensor for

the resolved scale, such that.

Sij = 1 2 ∂u i ∂xj + ∂uj ∂xi (2.17)

Because the large scale eddies are directly modeled in LES, residual stresses and flow parameters are not smeared as they would be by the URANS ensemble averaging processes. These flow structures are able to retain their physically consistent random-like behavior, enabling overall higher accuracy flow descriptions. Simultaneously, through the filtering of the Navier-Stokes equations, LES can demand lower computational costs than

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DNS by using models to represent the flow’s smallest length-scale structures [12].

Despite these advantages, accurate use of LES methods remains difficult. Because large flow structures are deterministically calculated, large eddy simulations are highly sensitive to boundary and inlet conditions. Ill-defined boundary conditions have the potential to introduce significant error into computed solutions. As a result, large eddy simulations require very fine grids near the wall to accurately resolve the formed boundary layer, which limits the computational cost savings of LES.

Additionally, the spatial threshold selected for the turbulent structure filtering has large effect on the simulations’ accuracy. The cutoff between "small" and "large" eddies is defined by the user, however, the improper selection of this threshold can introduce significant error into the solution. If the spatial mesh size is too large, flow defining turbulent structures cannot be accurately represented by the Navier-Stokes equations. Fortunately, as the spatial mesh decreases, the solution is able to converge to DNS results. However, in many cases, due to the high resolution demanded by LES, LES still remains impracticable for most industrial CFD simulations [12].

2.3.5 Hybrid Models

Hybrid turbulence models combine URANS and LES modeling methods in attempt to achieve the accuracy of LES simulations while maintaining the speed and cost-efficiency of URANS simulations. There have been wide variety of hybrid models developed over time, however, a summary of three important classes of hybrid modeling methods, their implementation, performance, and challenges, will be briefly discussed to provide context for STRUCT-

ε

.

Detached Eddy Simulation (DES)

DES, proposed by Spalart et al (1997) is one of the first successful hybrid CFD turbulence modeling methods to be introduced and it remains as one the most widely utilized. To reduce computational costs, DES switches between a URANS and LES computation of eddy viscosity depending on grid resolution with respect to a locally defined characteristic length scale. In general, URANS computations are applied at the wall while LES is used in the far flow field, which mitigates the costly task of computing fine-grid resolution at the wall with LES. DES has demonstrated higher accuracy performance in prediction of time-averaged values in many test cases, particularly in cases with large separations.

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However, the model is strongly wall grid-size dependent and is ill defined for internal flows or flows with thick boundary layers [17].

Partially Averaged Navier-Stokes (PANS)

The PANS model, developed by Girimaji, Srinivasan, and Jeong (2003) is a general hybridization approach that uses, fk,

fk =

km

k (2.18)

the ratio between modeled and resolved turbulent kinetic energy and fε,

fε =

εm

ε (2.19)

the ratio between modeled and resolved turbulent dissipation rate to control the level of resolution. When km = k and εm = ε, the model behaves as URANS because the

smallest scales in PANS are the smallest scales in URANS. For values of 0 < fk < 1,

model hybridization occurs, and the turbulence dissipation rate is modified using this ratio. Using PANS, the computational cost of completing a CFD simulation is minimized, because it enables the use of a grid coarser than required for LES. However, the main difficulty in the implementation of the model is the fact that the determination of the PANS control parameter depends on prior, user-defined physical resolution [18].

Scale Adaptive Simulation (SAS)

SAS was proposed by Menter et al (2003) and uses the von Karman length-scale,

Lvk = κ

u0

u00 (2.20)

where the von Karmann constant (κ) is 0.41, to identify optimal regions of hybrid activa-tion. This length-scale is proportional to the size of the resolved eddies and is compared with turbulence length scales to distinguish between regions with stationary flow and regions with strong instabilities. This comparison enables SAS to dynamically adjust turbulence flow qualities (such as ω in k-ω SST) based on the resolution of the turbu-lent flow through a URANS simulation. The main advantage of using the von Karman length, a value based on intrinsic flow properties rather than the grid, to determine acti-vation of turbulence hybridization is its wall distance and cell size independence. SAS has

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demonstrated the ability to simultaneously improve flow predictions while reducing grid sensitivity. However, the model has difficulty accurately modeling the physics in regions that contain mild separation or complex strain and is poorly applicable for internal flows.

2.3.5.1 STRUCT-T

STRUCT-T is a second-generation URANS model (Fröhlich and Von Terzi, 2008) devel-oped by Lenci and Baglietto (2014) that aims to improve the ease of use and robustness of previously developed hybrid turbulence models. STRUCT-T uses the NLEVM developed by Baglietto and Ninokata (2007) as the baseline URANS closure. The use of this Cubic k-

ε

model enables STRUCT-T to better resolve turbulence anisotropy and unsteadiness. In addition, STRUCT-T hybridizes the computational regime by reducing the overall eddy viscosity in specific regions of the flow using a reduction parameter, r, such that

νt = Cµ

k2

ε r (2.21)

In STRUCT-T, the averaging and selection of resolution parameters is developed in real time based on a comparison of resolved and modeled time scals. As a result, the TKE ratio and the turbulence resolution are reduced in regions where the modeled and the resolved scales are the most similar and the large scale separation assumption of URANS is invalid. Further detail about the STRUCT-T formulation can be found in Appendix 2.3.2.

Unlike most existing hybrid approaches, STRUCT-T uses a hybrid activation method-ology that is independent of the computational grid size, a requirement of second gen-eration URANS models. Instead, STRUCT-T identifies regions to hybridize based on locally computed flow quantities.

The STRUCT-T model was previously tested on a variety of classic but challenging flow cases, including the flow past a square cylinder [19], turbulent mixing in a T-junction [20], mild separation in an asymmetric diffuser [21], etc. In all of the aforementioned simulations, STRUCT-T yielded improved numerical agreement with experimental values of mean and RMS velocity components. However, like many hybrid turbulence models, the formulation suffers when improper boundary conditions are applied. STRUCT-T has demonstrated undesirable hybrid activation in external flow applications when improper inlet conditions are specified, because values of k and

ε

are able to transport through the whole domain in open flows. The explicit dependence of the hybrid activation on the modeled time scale, tm, which is a funtion ofk and

ε

, therefore propagates error through

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the geometry. An interested reader can refer toA Methodology Based on Local Resolution

of Turbulent Structures for Effective Modeling of Unsteady Flows by Lenci and Baglietto

(2013) for more information and details about these effects [22].

2.3.5.2 STRUCT-

ε

While many hybrid models have demonstrated increased accuracy when applied to specific test cases, the successful use of most hybrid models depends heavily on the appropriate activation of hybridization. However, many turbulence models lack a robust, accurate, or physically consistent method of identifying optimal hybridization methods. In addition, the behavior of most hybrid models are not well tested or easily predictable, thus limiting their practical application in industrial or high risk uses.

Therefore, STRUCT-

ε

is a turbulence model in development by Xu and Baglietto (2018) that further enhances the robustness and performance of STRUCT-T. STRUCT-

ε

reduces the effects of error introduced at boundaries through the use of a source term, esource, in the

ε

transport equation of the Cubic k-

ε

equations developed by Baglietto

and Ninokata. This source term,

esource= C3ερkQabs (2.22)

implicitly reduces the simulation domain’s eddy viscosity, reducing artificially over-predicted Reynolds stresses from two-equation models. In this source term formulation, C3ε was

chosen to be 1.5 through a series of sensitivity studies conducted on several test cases.

Qabs, also referred to as the Q-Criterion, is the the second invariant of the resolved velocity

gradient tensor, or Q = −1 2 ∂ui ∂xj ∂uj ∂xi (2.23)

The Q-criterion is a value useful for defining a turbulent structure as it represents the local balance between shear strain rate and vorticity magnitude [23].

This additional term’s effect is similar to that of the reduction parameter in the original STRUCT-T model (Equation A.29). Both models identify hybridization regions as regions where the Q-criterion is larger than _kε. However, instead of modifying the eddy viscosity at selected regions, it modifies the

ε

equation everywhere. Therefore, with this new model, the hybridization region no longer depends on inlet turbulence conditions. This implicit hybrid activation is also one of the simplest hybrid models to implement currently available.

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These features of the STRUCT-

ε

model have high potential for addressing the lim-itations of existing EVMs and hybrid turbulence models. The STRUCT-

ε

model has already been applied to a variety of other internal and external flow cases, including the Ahmed body [24], the periodic hill [25], and asymmetric diffuser [21], and flow past a square cylinder [19]. For these simulations, the STRUCT-

ε

model shows more accurate prediction of flow topology and velocity values than those produced by URANS models.

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Chapter 3 Methodology

To understand the performance of the STRUCT-

ε

model when simulating heated internal flow cases, the model was applied to a variety of relevant turbulent flow geometries, each characterized by unique turbulence structures: the impinging jet [1], film cooling [2], and flow through an infinite wire-wrapped assembly [3]. These geometries will each be described in further detail in their corresponding sections.

3.1 Implementation of STRUCT-ε

One of the primary benefits of STRUCT-

ε

is its simple implementation that makes the model accessible for all users and feasible for industry purposes. The basis of the STRUCT-

ε

model is the Cubic k-

ε

model. The model utilizes the coefficients proposed by Baglietto and Ninokata (tabulated in Table A.1). Implementation of the model simply requires the activation of an additional turbulence dissipation rate term (Equation 2.22) in the

ε

equation.

3.2 Modeling Parameters

To fully justify the benefits of using the STRUCT-

ε

model, a rigorous comparison of the STRUCT-

ε

turbulence model with previously developed models was performed. All simulations in this study were completed as transient simulations so that STRUCT-

ε

’s ability to resolve unsteady flow structures can be explored. In addition, the effects of resolving these unsteady structures on the overall fluid property predictions is important in understanding modeling accuracy and performance. The unsteady Standard k-

ε

and

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the k-ω SST were selected for this comparison study, because they are two of the most commonly used and well understood turbulence models. To provide a consistent basis for comparison, the three turbulence models were applied to the chosen test geometry using otherwise identical parameters (mesh, time-step, convection schemes, etc.)

For each simulation, the time step was selected such that the Courant Number,

C = k1/2∆t

∆x, (3.1)

throughout the entire geometry was maintained at less than or equal to 1.0 to ensure numerical accuracy during the advancement of the solution through time. The details of all other parameters selected for each flow geometry are detailed in their respective sections.

3.3 Mesh Convergence

To ensure that simulations solutions are mesh-independent, a mesh convergence study was performed for each flow geometry case tested in this work. For each case, multiple meshes of varying refinement were generated, using a constant mesh refinement coefficient to step between the mesh sizes. The refinement factor both influences the base mesh size, as well as any refinement regions and the number of wall prism layers. For each case, the mesh refinement coefficient chosen such that the variation between each meshes’ elements were large enough to clearly exhibit convergence behavior. Simultaneously, if the mesh refinement factor was too large, the resulting simulations would quickly become computationally intractable. A refinement factor no less than 1.25 is recommended by the Best Practice Guideline for the Use of CFD in Nuclear Reactor Safety Applications by the Nuclear Energy Agency [26].

Simulations for each grid sensitivity mesh were run using otherwise identical modeling parameters. A value that is both sensitive and critical to the flow geometry was selected and compared between all three simulations to determine meshing convergence. The most refined mesh at which the solution appropriately stabilizes will be chosen as the converged mesh with which all other turbulence models will be tested.

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3.4 Performing Validation

Experimental or DNS data for each of the selected flow cases was used to benchmark the performance of the STRUCT-

ε

turbulence model. However, since each test case’s comparison data was collected from a different experimental group, the set of information varies from case-to-case. The details of each geometries’ available dataset is detailed in their respective sections.

However, the diversity of available data and format complicates the determination of a consistent figure of merit. Therefore, in this work, the root mean square error (RMSE) between simulation and experimental results for all available parameters will serve as the primary basis of comparison.

RM SE = v u u u t N X n=1 (yCF D− yexp)2 N (3.2)

Finally, the results of the simulations must be compared both quantitatively and qualitatively with expected fluid behavior to fully understand the performance of the model. Therefore regardless of the numerical fidelity, the appropriate representation of turbulence topology will be heavily considered in determining model validity.

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Chapter 4 Impinging Jet

Due to its simple implementation, cost-effectiveness, and ability to produce some of the highest Nusselt numbers of all single-phase heat transfer mechanisms, the turbulent im-pinging jet flow configuration is extensively used for a wide variety of industrial purposes, including the cooling of fusion reactor systems, de-icing airplanes, and heat exchange for many automotive and aeronautical applications. However, the accurate prediction of local heat transfer from the jet has historically been difficult to model. Therefore, the impinging jet is one of the underlying flow regimes curated and studied by the European Research Community on Flow, Turbulence, and Combustion [1].

For this thesis, ERCOFTAC’s documented results of the Re = 23,000 impinging jet experiments conducted by Cooper et al (1992) was replicated and simulated using the STRUCT-

ε

turbulence model. The data included in the ERCOFTAC database that characterize the impinging jet flow geometry are the mean velocity profiles and Reynolds stress components in the near vicinity of the plate at positions of r/D = 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 from Cooper et al as well as the normalized Nusselt number as a function of r/D from Baughn and Shimizu (1989). The absolute accuracy of the experimental data set was evaluated to be 2% for the maximum mean velocities, 4% for u’, 6% for v’, and 9% for u’v’. [1].

It should be noted that the while the experiments conducted by Cooper et al (1992) chose flow conditions such that Re = 23,000, Baughn and Shimizu (1989) completed ex-perimentation at Reynolds number of 23,300 to measure wall temperature distributions. The following CFD simulated results were completed using a Reynolds number of 23,000 to match the flow conditions used by Cooper et al. However, this constraint becomes a source of error when benchmarking heat transfer behavior, as the lower Reynolds

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num-ber used will yield Nusselt numnum-ber predictions that are lower than the experimentally reported values, since lower turbulence values lead to less effective heat transfer.

4.1 Flow Geometry

This fundamental flow case is comprised of an axisymmetric jet impinging orthogonally onto a smooth, flat, heated plate. The jet is issuing air at ambient temperature from a circular pipe of inner diameter D. The jet is positioned at an axial discharge distance of

H = 2D. In this flow case, the inner diameter of the pipe is 26 mm, and the pipe from

which the jet is issuing has an outer diameter of 29.1 mm. In the experimental facility, the pipe had a length of 80D through which the flow develops. However, to reduce the size of the computational domain, this length was shortened to 2D in the CFD geometry (see Section 4.6 for the specification of flow inlet conditions such that inlet conditions remain fully developed despite this geometric reduction). The pipe discharges fluid into an effectively infinitely large wind tunnel of still, ambient temperature fluid [7].

Figure 4.1: Half of the impinging jet CAD geometry used to mimic the fluid volume of the experimental test facility of Cooper et. al. Notable geometric dimensions are labeled.

Cooper et al’s experimental facility was recreated in CAD, and a cross sectional view of the resulting geometry is pictured in Figure 4.1. Despite knowing that the impinging jet flow case should yield an axisymmetric flow profile, to fully capture the unsteady, three dimensional turbulence structures resulting from the impingement requires the entire domain (all 360◦) to be modeled. The bottom plate and outer edge of the fluid domain were chosen to be a disk of radiusR = 10D as suggested by ERCOFTAC’s best modeling

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practices, because it positions the outlet adequately far from the impingement location to reduce the effect of outlet conditions on the flow domain of interest [27].

4.2 Modeling Parameters

The fluid released from the jet is characterized by a Reynold’s number of 23,000, with

Re = ubD

ν (4.1)

where ub is the average bulk velocity of the jet, D is the internal diameter of the pipe,

and ν is the kinematic viscosity. The simulant fluid in this set of experiments is air, and the default STAR-CCM+ thermal physical properties were selected for this set of CFD simulations (values tabulated in Table 4.1).

Table 4.1: Thermal Physical Properties of Impinging Jet Simulant Fluid

Density (ρ) 1.184 kg/m3

Dynamic Viscosity (µ) 1.855 ∗ 10−5 Pa-s

Kinematic Viscosity (ν = µ_ρ) 1.567−5 Pa-s

Thermal Conductivity (K) 0.0260 W/mK

Specific Heat Capacity (cp) 1003.62 J/kg-K

Using these values, the Prandtl number (Pr), which is the ratio of momentum diffu-sivity to thermal diffudiffu-sivity, is equivalent to 0.71, where

P r = cpµ

K (4.2)

4.3 Expected Flow Topology

The turbulent structures that develop from a jet impingement are fundamentally different from those that develop from flow that is primarily parallel to the wall. Local turbulent length scales are mostly defined by the length scales of the incoming jet rather than by the physical proximity of the fluid to the wall [6]. This difference makes the impinging jet an uniquely useful test case for the validation of turbulence models, because it contributes to the diversity of verified simulated flow characteristics.

Observing the experiments conducted by Cooper et al., the impingement of the jet on the base wall is essentially axisymmetric. The resulting temperature and velocity

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distribution of flow on the wall is a function of pipe diameter and outlet height. However, when the jet is close to the surface (Z/D= 2), maximum heat transfer is observed at the stagnation point, followed by a local minimum at r/D of approximately 1.3 and another local maximum around 1.8 [5]. The cause of this local maximum that occurs for the Z/D = 2 flow configuration corresponds to a peak in wall turbulent kinetic energy as the jet travels away from the stagnation region. The fluid in this region experiences high levels of shear (Behnia). In their experiments, Baughn and Shimizu (1989) determined that maximum heat transfer at the stagnation point occurs when the pipe outlet is positioned at a height of Z/D = 6 [5].

Figure 4.2: Experimentally determined Nusselt number distribution on base plate of impinging jet when Re = 23,750 for geometries with jet outlets located at various heights of Z/D [5]

.

There are four main turbulence regions of note in the axisymmetric impinging jet flow: the potential core region, the developing zone, the stagnation region, and the wall jet. These regions are pictured in Figure 4.3 and detailed below:

1. Potential Core Region As the jet fluid leaves the pipe, the surrounding fluid’s mass, momentum, and energy are entrained into the jet. Turbulence is generated by normal straining from the surrounding fluid, and the fluid’s axial velocity decreases

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Figure 4.3: Expected flow structures in the axisymmetric impinging jet flow case. Notable flow regions are labeled. 1) potential core region, 2) developing flow 3) stagnation region, 4) wall jet region [6]

.

with increasing distance from the exit nozzle. However, the jet flow maintains a high velocity core with velocities similar to the initial jet velocity, because entrainment interactions are minimal in the center of the jet.

2. Developing Zone As the jet flow continues through the medium, the fluid contin-ues to decrease in velocity. The profile of the jet fully develops, and the high velocity core begins to dissipate. However, this developing zone is negligible when the pipe exit is only two hydraulic diameters above the wall. Therefore, this turbulence region is not expected to be present in this study’s computations.

3. Stagnation Region The point at which the jet strikes the wall is known as the stagnation region. At this point, the axial velocity quickly diminishes, there is large total strain along the streamline, and there is a a consequent rise of static pressure. The most efficient cooling is expected to occur at this stagnation point, after which, the bulk flow is redirected radially and axisymmetrically outward. It is important to note that the velocity of the fluid in the stagnation region is dominated by fluid pressure interactions, not Reynolds stresses.

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4. Wall Jet Zone After the jet hits the wall, the jet is deflected radially such that the mean flow changes to be parallel to the wall. In this region, the flow is decelerated by strong shear with the wall, and ring-shaped vortices form [28].

The maximum Nusselt number, the ratio of convective to conductive heat transfer, occurs at the stagnation point. The Nusselt number is defined as,

N u = hD

K (4.3)

where h is the heat transfer coefficient, D is the diameter of the jet pipe, and K is the fluid thermal conductivity. However, in the impinging jet case,h can be computed as,

h = qw Tw − Tf

(4.4)

where qw is the heat flux at the wall, Tw is the temperature at the wall, and Tf is the

temperature of the incoming jet. Therefore, the Nusselt number is also equivalent to

N u = qwD K(Tw− Tf)

(4.5)

4.4 Literature Review

Heat transfer at the wall of the impinging jet has shown to be difficult to accurately predict using CFD turbulence models. In this flow geometry, the accurate prediction of the Nusselt number depends heavily on the accurate representation of Reynold’s stresses at the walls, because these values dictate the diffusion of momentum and heat between the wall and the fluid. However, errors arise when traditional eddy viscosity models, such as the Standard k-

ε

and k-ω model, are used. These linear EVMs make use of the Boussinesq equation, which was designed to model flows parallel rather than orthogonal to the wall. As a result, overestimation of both wall-normal fluctuations at the impinge-ment region and wall-normal diffusion processes have been witnessed. In addition, the computations traditionally overproduce Reynold’s stress values. This overestimation of turbulent fluctuations leads to inaccurate prediction of both the heat transfer and veloc-ity distribution at the impingement wall. Below is an incomprehensive literature review of the CFD modeling methods used to compute this impinging jet case.

Behnia et al (1997) used both the Standard k-

ε

and an elliptic normal-velocity re-laxation model (V2F). Behnia et al found that the k-

ε

model does not properly resolve

Application of hybrid Computational Fluid Dynamics turbulence model, STRUCT-[epsilon], on heated flow cases

Application of Hybrid CFD Turbulence Model, STRUCT-ε, on

Heated Flow Cases

Application of Hybrid CFD Turbulence Model, STRUCT-

ε

, on Heated

Flow Cases

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Acknowledgements

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Contents

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Chapter 1

Introduction

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