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CFD calculation of convective heat transfer coefficients and validation – Part 2: Turbulent flow

Annex 41 – Kyoto, April 3rd to 5th, 2006

Adam Neale1, Dominique Derome1, Bert Blocken2 and Jan Carmeliet2,3

1) Dep. of Building, Civil and Environmental Engineering, Concordia University, 1455 de Maisonneuve blvd West, Montreal, Qc, H3G 1M8, corresponding author e-mail: aneale@sympatico.ca

2) Laboratory of Building Physics, Department of Civil Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 40, 3001 Heverlee Belgium

3) Building Physics and Systems, Faculty of Building and Architecture, Technical University

Eindhoven, P.O. box 513, 5600 MB Eindhoven, The Netherlands Abstract

When a fluid flows over a wall and heat is exchanged, the boundary layer (BL)

velocity profile will to a large extent determine the value of the convective heat

transfer coefficient (hc). Using the Computational Fluid Dynamics (CFD) software

Fluent, the turbulent flow field and heat transfer is simulated for forced convection

over a smooth flat plate and the performance of the most commonly used turbulence

models is evaluated. The BL velocity and temperature profiles are compared with

semi-empirical near-wall data and the heat transfer coefficients calculated in Fluent

are validated by comparison with correlations from literature. In addition, a

simulation of natural convection is provided as an example of when wall function

theory fails.

1. Introduction The surface coefficients for heat and mass transfer (hc and hm, respectively)

are parameters that are generally not easily calculated analytically and difficult to

derive from experimental measurements. The values of surface coefficients depend

on many variables – flow field, boundary conditions, material properties, etc. In

addition, despite the fact that the two transfer processes are mutually dependent,

they are often solved as uncoupled phenomena. Finally, although existing

correlations relating hc and hm are only valid for specific cases, such correlations are

applied widely throughout literature.

This paper is the second part of a two part study of the option to solve for hc

using Computational Fluid Dynamics (CFD). In Part I, the CFD code Fluent was

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validated for convective heat transfer in the laminar regime for fully developed flow

between parallel plates. Part II is a validation and comparative study of heat transfer

coefficients calculated using different turbulence models implemented in Fluent. The

validity of using wall functions for natural convection cases is also examined.

The paper starts with a brief outline of the methodology that will be used to

validate the results of the simulations. A brief overview of wall function theory is

provided as a necessary basis for the following sections. In Section 4 the heat

transfer theory is presented for forced convection over a smooth flat plate. The

simulation results for both forced and natural convection are in Sections 5 and 6,

respectively, followed by some conclusions and remarks in Section 7.

2. Methodology

In the interest of validating the turbulent models within Fluent, it was

necessary to obtain experimental data to use as a basis for comparison. While

experiments have been performed in this area, it is often difficult to establish whether

the simulation truly matches all of the experimental parameters. However, one area

that has been focused on extensively in the past is the universal “law-of-the-wall”

that describes turbulent boundary layer flow. Through analytical derivations of

equations and experimental data fitting, the boundary layer velocity profile (and

temperature profile, if applicable) has been subdivided into three regions: the laminar

sublayer, the buffer region, and log-law region (Chen & Jaw 1998, Blocken 2004).

Semi-empirical relationships have been developed for the laminar sublayer and log-

law regions, and empirical equations exist for the buffer region as well (e.g. Spalding

1961). The (semi-)empirical equations will be used to validate the simulation results

from Fluent.

3 Near-wall modelling Boundary layer velocity and temperature profiles are generally described

using dimensionless parameters. Before the BL regions can be discussed in proper

detail, some dimensionless terms must be introduced:

ν

*yuy ≡+ (1)

where y+ is the dimensionless distance from the wall, y is the distance from the wall,

ν is the kinematic viscosity, and u* is the friction velocity defined as:

3

ρτ wu ≡* (2)

where wτ is the wall shear stress and ρ is the fluid density. The wall shear stress is

based on the velocity gradient in the direction normal to the surface of the wall, or in

equation form:

0=∂

∂≡

yw y

Uµτ (3)

where U is the fluid velocity along the wall, and µ is the dynamic viscosity. The

velocity can be described in a dimensionless form as a function of the fluid velocity

and the friction velocity:

*uUu ≡+ (4)

For cases with heat transfer, the dimensionless temperature may be calculated using

the following equation:

*TTTT w−≡+ (5)

where Tw is the wall temperature at a certain point, T is the fluid temperature, and T*

is defined as

**

ukqT wα

≡ (6)

where α is the thermal diffusivity, qw is the wall heat flux and k is the thermal

conductivity.

There are two common near-wall modeling techniques employed in CFD:

Low-Reynolds-number modelling and Wall function theory.

3.1 Low-Reynolds-number modelling (Low-Re) If the boundary layer is meshed sufficiently fine so that the first cells are

placed entirely in the laminar sublayer of the BL, the approach used is generally

referred to as Low-Re Modelling. The laminar sublayer is valid up to y+ < 5 and, in

dimensionless coordinates, the height of the first cell is generally taken to be

approximately y+ = 1. In the range of 5 < y+ < 30 there exists a buffer region between

the laminar sublayer and the log-law region of the boundary layer. It is generally not

advisable to have meshes where the first cell lies within the buffer region, though

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often it is unavoidable in CFD. For meshes with a y+ > 30, wall function theory may

be applied.

3.2 Wall function theory Fluid flow over a smooth flat plate is referred to as the simplest case for

analytical fluid dynamics (Schetz 1993). There has been a significant amount of

work done in experiments for boundary layer flow evaluation (summarized in Bejan

1984, Schlichting 1987, Schetz 1993, Chen & Jaw 1998, etc). That work was

subsequently transformed into the wall function concept (e.g. Spalding 1961). Wall

functions allow CFD models to interpret behaviour near a wall without the need for a

very fine mesh that also discretises the generally quite thin laminar sublayer at the

surface of the wall. The wall function equations are based on an analytical solution

of the transport equations in combination with experimental data fitting. The result is

a reduction in computation time and a relatively accurate representation of what

happens within the BL, at least under the conditions for which the wall functions were

derived. Wall functions are recommended for cases where the domain is that large

and the Reynolds numbers are that high that the low-Re number modelling approach

would lead to a mesh that is too large and can no longer be solved economically. On

the other hand, wall functions may cease to be valid in complex situations.

Nevertheless, they are often used – even when not valid – for complex calculations,

which can be responsible for considerable errors in near-wall flow and the related

convective heat transfer coefficients (Blocken 2004).

Wall functions are generally described as having two regions: the laminar

sublayer and the log-law layer. It is commonly accepted in CFD that the laminar

sublayer is said to be valid in the region where y+ < (5 to 10) (Chen & Jaw 1998).

The equations for the dimensionless velocity and temperature within this region are

(Fluent Inc. 2003):

++ = yu (7)

++ = yT Pr (8)

where Pr is the Prandtl number (Pr = ν/α). The region above the laminar sublayer

(y+ > 30) is the log-law layer, which is generally described in the form of:

ByAn += ++ ln (9)

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where n is either the dimensionless velocity or dimensionless temperature. The

constants A and B in Equation (9) are usually fitted to experimental data, and are

consistent throughout the literature. For the purpose of this report, the following

equations will be used (Fluent Inc. 2003):

45.5ln5.2 += ++ yu (10)

+= ++ PEyT t )ln(1Prκ

(11)

where Prt is the turbulent Prandtl number (= 0.85 for air), E is an experimentally

determined constant (= 9.793), and P is described by the following equation:

+

−

=

−

tePt

PrPr007.04

3

28.011PrPr24.9 (12)

Spalding (1961) suggests an equation that will cover the entire y+ range of values for

the dimensionless velocity u+ (including the buffer region):

( ) ( ) ( )

−−−−−+= +++++++ 432

241

61

211exp BuBuBuBuBuAuy (13)

where A=0.1108 and B=0.4. The equations for the dimensionless velocity and

temperature in the laminar sublayer, the buffer layer and the logarithmic layer are

illustrated in Figure 1.

0

5

10

15

20

25

1 10 100 1000

y+

u+

0

5

10

15

20

25

T+

u+ Equation (7)

u+ Equation (10)

u+ Spalding Equation (13)

T+ Equation (8)

T+ Equation (11)

Figure 1. Wall function dimensionless velocity and temperature distributions

6

It is important to note that Fluent uses a slightly different dimensionless height

y*, but for the cases that were examined the values of y+ and y* are equivalent

(Fluent Inc. 2003).

4. Heat transfer in turbulent flow Convective heat transfer coefficients are generally expressed in the form of

dimensionless correlations that are based on experimental data. There are a

number of works in literature that summarize the numerous correlations that exist for

different types of flows (e.g. Bejan 1984, Saelens 2002, Lienhard & Lienhard 2006).

For the purpose of this paper, two correlations were selected from Lienhard &

Lienhard (2006) that correspond to the geometry and flow conditions for the forced

convection cases that were simulated. They are given by Equations (14) and (15)

below. The dimensionless parameters required for the equations are as follows

(further described in Appendix C):

Reynold’s Number: µ

ρ xUx

∞≡Re

Prandtl Number: αν

≡Pr

Stanton Number: ∞

≡UchStp

c

ρ

Nusselt Number: kxhNu c

x ≡

Skin Friction Coefficient: ( )[ ]2Re06.0ln

455.0

xfxC = (White 1969)

( ) 21Pr8.1212

68.0fx

fxx C

CSt

−+= ; Pr > 0.5 (14)

43.08.0 PrRe032.0 xxNu = (15)

The heat transfer coefficients for the natural convection case are not compared in

the framework of this paper.

5. Turbulent Forced Convection Simulations 5.1 Computational domain

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The domain used to represent fluid flow over a flat plate is shown in Figure 2.

The boundary condition (BC) for the top of the domain was chosen to be a symmetry

condition in order to reduce the computation time of the simulation. If a pressure

outlet BC is chosen instead of symmetry it can lead to convergence problems when

modelling turbulence. The height of the domain was selected to be high enough to

reduce the influence of the symmetry condition on the boundary layer.

Figure 2. Computational domain and boundary conditions (BC)

As explained in Section 3.1, the Low-Re modelling approach recommends

that the first grid cell has a dimensionless height of y+ ≈ 1, which means that it is

submerged in the laminar sublayer. (Fluent Inc. 2003). For simulations with Wall

Functions (WF), a y+ between 30 and 60 is recommended. The y+ value is based on

the flow conditions at the surface (see Equation (1)) and therefore requires an

iterative procedure to properly size the first cell. After a number of grid adjustments

the mesh fulfilled the requirements for Low-Re modelling, and is shown below in

Figure 3. The mesh used for wall function cases is shown in Figure 4. An

exponential relationship was used to mesh the vertical direction and a uniform

spacing was used for the horizontal direction. The grid dimensions are shown in

Table 1.

Table 1. Grid parameters and dimensions – forced convection case

Grid #Cells in X-direction

#Cells in Y-direction

Smallest Cell Width

Smallest Cell Height

Total number of cells

Low-Re 500 100 0.01 m 1.285x10-3 m 50000

WF 100 13 0.03 m 4.653x10-2 m 1300

H = 1m

L = 5m

X

Y U∞ = 0.5m/s

T∞ = 283 K

Symmetry BC

Wall BC - Constant Heat Flux

qw = 10 W/m2

Velocity Inlet BC

Pressure Outlet BC

Note : Not to scale.

8

Figure 3. Grid used for simulations with low-Reynolds-number modelling

Figure 4. Grid used for simulations with wall functions

Note that the grids for the Low-Re modelling and wall function cases can have

the same spacing near the symmetry boundary, since the boundary layer solution

will not be not affected by the grid resolution near the top region of the domain.

The simulations were all initialized with a uniform velocity profile of 0.5 m/s.

The simulations were iterated until the scaled residuals for all parameters were

below 10-7. The outlet velocity profile and turbulence conditions were then used as

the new inlet conditions and the simulation was repeated. The thermal conditions at

the outlet were not used as new inlet conditions. Instead, the same inlet profiles were

used for each simulation. This means that the flow was always thermally developing

from the start of the domain. This procedure was continued until the inlet and outlet

velocity profiles were approximately the same, resulting in a fully developed flow

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profile. The original uniform velocity profile ensured that the bulk velocity was 0.5

m/s for all cases. A sample of the Fluent solution parameters is provided in

Appendix A. The material properties used for the fluid region in the simulations are

provided below in Table 2. Table 2. Material properties for air

Density ρ 1.225 kg/m3

Dynamic Viscosity µ 1.7894 x 10-5 kg/m•s

Thermal Conductivity k 0.0242 W/m•K

Heat Capacity cp 1006.43 J/kg•K

5.2 Simulation results for forced convection The simulation results are compared at x= 4.5 m for all cases. Simulations

were performed with the following turbulence models with Low-Re Modelling:

1) Spalart-Allmaras Model

2) Standard k-ε Model

3) RNG k-ε Model

4) Realizable k-ε Model

5) Standard k-ω Model

6) SST k-ω Model

7) Reynold’s Stress Model (RSM)

Simulations were performed with the following models with Wall Functions (WF):

1) Standard k-ε Model

2) Standard k-ω Model

Note that the Standard k-ω Model will automatically interpret whether Low-Re or WF

will be used based on the y+ of the first cell. The default settings for each model

were used for all cases unless otherwise specified.

Figures 5 and 6 and 7 compare the empirical and semi-empirical dimensionless

velocity and temperature profiles on one hand with the calculated dimensionless

profiles on the other hand. The empirical and the calculated convective heat transfer

coefficients are shown in Figure 7.

10

0

5

10

15

20

25

1 10 100 1000

y+

u+

Semi-Empirical Equation - Laminar SublayerSemi-empirical Equation - Log-lawEmpirical Equation - Spalding (1961)k-e standardk-e RNGk-e realizablek-w standardk-w SSTSpalart-AllmarasRSMWF - keWF - kw

Figure 5. Comparison of empirical, semi-empirical and calculated dimensionless velocity

profiles for the turbulent simulations

0

4

8

12

16

1 10 100 1000

y+

T+

Semi-empirical Equation - Laminar SublayerSemi-empirical Equation - Log-lawk-e standardk-e RNGk-e realizablek-w standardk-w SSTSpalart-AllmarasRSMWF - keWF - kw

Figure 6. Comparison of empirical, semi-empirical and calculated dimensionless

temperature profiles for the turbulent simulations

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0

1

2

3

4

5

6

0 1 2 3 4 5

X Position (m)

h c (W

/m2 K

)k-e standardk-e RNGk-e realizablek-w standardk-w SSTSpalart-AllmarasRSMk-e WFk-w WFLienhard (2006) Eq. 6.111Lienhard (2006) Eq. 6.115

Figure 7. Comparison of empirical and calculated convective heat transfer coefficients

for the turbulent simulations 5.3 Forced convection results summary

The velocity profiles shown in Figure 5 indicate a good agreement between

the simulations and the “universal” law-of-the-wall relationships and the “universal”

Spalding curve, which were both developed based on experimental data. The

laminar sublayer and the log-law region are well defined for all of the turbulence

models, though some models (RSM) tend to under predict the velocity near the

upper boundary (for large values of y+). This can be explained by the fact that the

law-of-the-wall relationship ceases to be valid beyond a certain point (roughly y+ >

500, but the actual boundary depends on the situation) (Blocken 2004).

The temperature profiles in Figure 6 are also consistent with the expected

boundary layer profile, though at upper regions of y+ (>200) the curves begin to

diverge from the log-law equation. The same remark can be made here concerning

the failure of the law-of-the-wall theory at high y+-values.

The empirical correlations for heat transfer are shown in red on Figure 7. The

heat transfer coefficients are consistent between the turbulence models and the

correlations, including the solutions using wall functions. However, in the thermally

developing region (approximately 0 m < x < 1 m), the wall function solutions differ

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from the other curves. The result is an important underprediction of heat transfer for

cases where there is thermally developing flow. This is due to the fact that the wall

function approach is not valid under these conditions.

6. Natural convection simulations 6.1 Computational domain The domain used to represent a case of natural convection is shown in Figure

8. The boundary conditions (BC) for the four sides of the domain are impermeable

walls. The vertical wall at x= 0 m is maintained at a constant temperature of 298K,

while the opposite wall at x= 3.0 m is maintained at 288K. The remaining two walls

are considered isothermal. The fluid temperature was initialized to 293K.

The domain is divided into two regions denoted as the “Hot Side” and the

“Cold Side”. The velocity profile and temperature profile results will be described in

terms of which wall they are incident upon, either the “Hot” wall of 298K or the “Cold”

wall of 288K. The simulation results will be compared at three reference heights: y =

0.75 m, y = 1.50 m, and y = 2.25 m.

Figure 8. Natural convection computational domain

Two meshes were created to represent the domain in Figure 8: one for Low-Re

modelling and one for wall functions. The grid guidelines for Low-Re modelling and

wall functions described in Section 3 were followed where applicable. The grid

Tw1 = 298K

0m

Y Note : Not to scale.

Tw2 = 288K

Isothermal

Isothermal X = 3m 0m

y = 3m Air, Tinitial = 288K

“Hot Side” “Cold Side”

y = 1.50m

y = 0.75m

y = 2.25m

X

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parameters are outlined below in Table 3, and the two meshes are illustrated in

Figure 9.

Table 3. Grid parameters and dimensions – natural convection case

Grid #Cells in X-direction

#Cells in Y-direction

Smallest Cell Dimension

Largest Cell Dimension

Total number of cells

Low-Re 150 150 1.784x10-3 m 7.580x10-2 m 22500

WF 30 30 0.1 m 0.1 m 900

Figure 9. Low-Re modelling mesh (left) and wall function mesh (right)

6.2 Simulation results for natural convection The velocity profiles that are the result of natural convection flows are very

different from those predicted by wall functions. A typical example of a velocity

profile near the surface of a wall (from the Low-Re simulation results) is compared to

the Spalding profile (converted to dimensional units) in Figure 10. The shape of the

simulation velocity profile is consistent with velocity profiles from other natural

convection simulations and experiments (Zitzmann et al 2005). The Spalding profile

represents the velocity profile that would be used when wall functions are employed.

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.05 0.1 0.15

Position (m)

Velo

city

(m/s

)

Low-Re Velocity profile at y = 2.25 m , Hot s ide

Spalding wall function equation

Figure 10. Velocity profiles for natural convection (simulation) and universal law-of-the-wall

The two simulation cases resulted in different velocity patterns within the

domain. The velocity fields for both simulation domains are shown in Figure 11.

Figure 11. Velocity contours for the Low-Re simulation (left) and the wall function simulation

(right). (Note that the scale is in m/s)

Using wall functions will impose a velocity gradient along every wall surface more or less

uniformly. In the Low-Re contour plot the boundary layer velocity profile in the upper right

hand corner is clearly not the same as the upper left hand corner. As described in Figure 8,

the velocity profile data was output for three heights at the “hot” and “cold” walls. The

profiles were non-dimensionalized with the method described in Section 3, and the results

are shown below in Figure 12. The Spalding curve is plotted on Figure 12 to compare the

simulation results with the law-of-the-wall theory.

15

0

5

10

15

20

25

30

1 10 100 1000 10000y+

u+

Low-Re: y=0.75m , Hot s ide Low-Re: y=0.75m , Cold s ideLow-Re: y=1.50m , Hot s ide Low-Re: y=1.50m , Cold s ideLow-Re: y=2.25m , Hot s ide Low-Re: y=2.25m , Cold s ideWF: y=0.75m , Hot Side WF: y=0.75m , Cold SideWF: y=1.50m , Hot Side WF: y=1.50m , Cold SideWF: y=2.25m , Hot Side WF: y=2.25m , Cold SideSpalding Wall Function Equation

Figure 12. Non-dimensionalized velocity profiles from the Low-Re and wall function (WF)

simulations

6.3 Natural convection results summary The results for the natural convection simulations indicate that the standard

wall functions are simply not valid for these cases. The velocity profiles along the

“hot” and “cold” walls found in the Low-Re simulation are consistent with profiles for

similar cases in literature, both simulated and experimental (Zitzmann et al 2005).

When compared with the velocity profile described by wall functions, there is a

significant difference in both shape and magnitude. In addition to boundary layer

differences, the velocity field in the domain is very different when wall functions are

used (refer to Figure 11). It is counteradvised to use standard wall functions for the

natural convection cases described in this paper.

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7. Conclusions

Semi-empirical relationships developed using experimental data and

analytical theory were used to validate CFD simulation results for fully developed

forced convection over a smooth flat plate. The results indicate a good agreement

between (semi-)empirical equations and simulation boundary layer velocity and

temperature profiles for all of the turbulence models studied.

The heat transfer coefficients calculated from the forced convection

simulations are consistent for all of the turbulence models studied, and also coincide

closely with selected correlations from literature. The results for simulations with wall

functions indicate that the heat transfer coefficients calculated in the thermally

developing region of the domain are not consistent with the Low-Re simulation

results or with the correlations. It is concluded that the wall functions are not valid for

thermally developing regions.

Two simulations of natural convection were performed to investigate the

validity of wall functions for natural convection simulations. The Low-Reynolds-

number (Low-Re) simulation resulted in flow field and local velocity profiles

consistent with results seen in literature. The local velocity profiles were very

different from profiles obtained by the wall function equations, and the flow field

resulting from the wall function simulation differed significantly from the Low-Re case.

The dimensionless velocity profiles from the wall function simulation were very

different from natural convection results from literature. It is concluded that the

standard wall functions are not valid for cases involving natural convection and that

instead low-Re number modelling should be used.

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Appendix A: Sample forced convection Fluent solution parameters k-ε Turbulent model parameters: Model Settings Space 2D Time Steady Viscous Standard k-epsilon turbulence model Wall Treatment Standard Wall Functions Heat Transfer Enabled Solidification and Melting Disabled Radiation None Species Transport Disabled Coupled Dispersed Phase Disabled Pollutants Disabled Soot Disabled Equation Solved Flow yes Turbulence yes Energy yes Numerics Enabled Absolute Velocity Formulation yes Relaxation:

Variable Relaxation Factor Pressure 0.3 Density 1 Body Forces 1 Momentum 0.7 Turbulent kinetic energy 0.8 Turbulent dissipation rate 0.8 Turbulent viscosity 1 Energy 1

Solver Termination Residual Reduction Variable Type Criterion Tolerance Pressure V-Cycle 0.1 X-Momentum Flexible 0.1 0.7 Y-Momentum Flexible 0.1 0.7 Turbulence Kinetic Energy Flexible 0.1 0.7 Turbulence Dissipation Rate Flexible 0.1 0.7 Energy Flexible 0.1 0.7 Discretization Scheme Variable Scheme Pressure Standard Pressure-Velocity Coupling SIMPLE Momentum First Order Upwind Turbulence Kinetic Energy First Order Upwind Turbulence Dissipation Rate First Order Upwind Energy First Order Upwind Solution Limits Quantity Limit Minimum Absolute Pressure 1 Maximum Absolute Pressure 5000000 Minimum Temperature 1 Maximum Temperature 5000 Minimum Turb. Kinetic Energy 1e-14 Minimum Turb. Dissipation Rate 1e-20 Maximum Turb. Viscosity Ratio 100000

18

Appendix B: Sample natural convection Fluent solution parameters Model Settings Space 2D Time Steady Viscous Standard k-epsilon turbulence model Wall Treatment Enhanced wall treatment Heat Transfer Enabled Solidification and Melting Disabled Radiation None Species Transport Disabled Coupled Dispersed Phase Disabled Pollutants Disabled Soot Disabled Equation Solved Flow yes Turbulence yes Energy yes Numerics Enabled Absolute Velocity Formulation yes Relaxation:

Variable Relaxation Factor Pressure 0.3 Density 1 Body Forces 1 Momentum 0.7 Turbulent kinetic energy 0.8 Turbulent dissipation rate 0.8 Turbulent viscosity 1 Energy 1

Solver Termination Residual Reduction Variable Type Criterion Tolerance Pressure V-Cycle 0.1 X-Momentum Flexible 0.1 0.7 Y-Momentum Flexible 0.1 0.7 Turbulence Kinetic Energy Flexible 0.1 0.7 Turbulence Dissipation Rate Flexible 0.1 0.7 Energy Flexible 0.1 0.7 Discretization Scheme Variable Scheme Pressure Body Force Weighted Pressure-Velocity Coupling SIMPLE Density First Order Upwind Momentum First Order Upwind Turbulence Kinetic Energy First Order Upwind Turbulence Dissipation Rate First Order Upwind Energy First Order Upwind Solution Limits Quantity Limit Minimum Absolute Pressure 1 Maximum Absolute Pressure 5000000 Minimum Temperature 1 Maximum Temperature 5000 Minimum Turb. Kinetic Energy 1e-14 Minimum Turb. Dissipation Rate 1e-20 Maximum Turb. Viscosity Ratio 100000

19

Appendix C: Nomenclature a Order of the discretization error scheme (-)

b Distance between parallel plates (m)

cp Specific heat (J/kgK)

Dh Hydraulic diameter (m)

hc Convective heat transfer coefficient (W/m2K)

k Thermal conductivity (W/m-K)

L Length of domain (m)

NuDh Nusselt number calculated with the hydraulic diameter (-)

P Heated perimeter of the domain (m)

q Heat flux (W/m2)

u Velocity component in the x-direction (m/s)

v Velocity component in the y-direction (m/s)

T Temperature (K)

U Velocity magnitude (m/s)

Greek symbols µ Dynamic viscosity (kg/m-s)

ρ Density (kg/m3)

Subscripts AV Average property

b Bulk property

c Property taken at the centerline of the domain

f Fluid property

i Property of an element i

ref Reference property

x Property taken at a location x

∞ Free stream property

20

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