b. Overview Of Turbulence Models

Whenever turbulence is present in a certain flow it appears to be the dominant over all other flow phenomena. That is why successful modeling of turbulence greatly increases the quality of numerical simulations. All analytical and semi-analytical solutions to simple flow cases were already known by the end of 1940s.

On the other hand there are still many open questions on modeling turbulence and properties of turbulence it-self. (Sodja, 2007).

FLUENT^® Package problem can solve turbulent flow problems with different turbulence models. Here some of them will be explained.

Computational fluid dynamics has become a popular tool in performance analysis and design, but one of the major challenges of CFD is turbulent flow. There are many turbulence models available. (Neel, 1997).

Neel (1997), summarized the turbulence model’s history as follows;

In the past century turbulence models have been invented and researched in great detail. The need for turbulence models came out of solving some forms of the Navier- Stokes (N-S) equations. The physical phenomenon behind turbulent flow is described completely by the Navier-Stokes equations, but the problem is not in the equations themselves, but in solving them. Only a few exact solutions exist for the N-S equations, and these consist of relatively simple flows with limited applications.

In order to solve the complete N-S in a more general sense, numerical methods have to be employed. Solving the full N-S equations or some reduced form of it by numerical means is the nature of CFD. As computer processing speeds become faster and memory capabilities grow, analysis and design using CFD continues to become more important. But there still remains a large gap between the resolution needed to resolve all the small scale turbulent motion and the available computer resources to accomplish this. A few solutions have been done that have resolved all the scales of motion by solving the full N-S equations. This method of solving the Navier-Stokes equations without making any approximations is called the Direct Navier-Stokes (DNS) method. The few calculations that have been done using DNS have been valuable to the research community, providing insight into the physics of turbulence and turbulence modeling. The ability to use DNS for general problems of interest is not seen for the near future. An example of the required resources needed for a DNS calculation for a simple channel flow can be found in (Kim et. al., 1987). Because of this and the growing importance of CFD, turbulence modeling needs to be seriously studied to further improve numerical calculations.

The use of turbulence models became important after the assumption was first made that the N-S equations could be solved for the mean flow parameters.

Turbulent motion is time dependent, so the mean flow represents the flow field after the turbulent fluctuations are time-averaged. This approach has become widely accepted and is now the most common approach to solving the equations. The concept was first thought up by Reynolds in the late nineteenth century, and is named after him (Reynolds averaged Navier-Stokes equations). When the N-S equations are time-averaged, the unknown variables become time-averaged quantities. This eliminates the need to resolve the time dependence of turbulent fluctuations, but also

introduces new unknowns into the governing equations. The new unknowns are referred to as the turbulent Reynolds stresses. This trade-off allows solutions to turbulent flows to be attained using current computer resources. The unknowns create a closure problem, which is taken care of by using a turbulence model.

The first category for turbulence models comes from the Boussinesq assumption. Boussinesq suggested in 1877 that the apparent turbulent shearing stresses are related to the rate of mean strain through an apparent scalar turbulent viscosity. This apparent scalar viscosity is known as the eddy viscosity, and forms the basis for the majority of turbulence models today. Turbulence models that fall into this category can be further classified according to the number of supplementary differential equations that must be solved. These supplementary equations are solved for modeling parameters, the number of which varies according to the model. Due to these extra equations, the different turbulent models are described as n-equation models where n refers to the number of differential equations that must be solved for a given model. For the case of n = 0, the model is referred to as a zero-equation or algebraic model. The appearance of one-equation models came next as a result of trying to develop a more realistic description of the flow. A model for the eddy viscosity was postulated which depended upon the kinetic energy of the turbulence.

An equation for the kinetic energy was developed, which when solved, allowed the eddy viscosity to be more sensitive to the surrounding flow by taking into account the flow history. Before this, the more simple algebraic models were influenced only by the mean flow properties at a given location. By adding additional equations, for example an equation for the turbulent mixing length scale, two-equation models came into existence. Along with the added number of differential equations to be solved came increased numerical complexity. One and two-equation models are not as simple to program as algebraic models and require more computational time to solve.

The second category of turbulence modeling is referred to as Reynolds stress or stress-equation models. These models start with transport equations derived for the Reynolds stresses. The work for this class of turbulence modeling was first done by Rotta (Rotta, 1951). As a result of the new transport equations, more unknowns

were introduced which must then be modeled based upon experimental observations.

With the additional unknowns to predict, these types of turbulence models became complex, and have not demonstrated great advantages over the eddy viscosity models. A review of this class of turbulence modeling is given by Launder, (1979).

The problem with turbulence models may be understood better by realizing that no one turbulence model has been found to be superior. Depending upon the problem at hand, different models may be more suitable to use than others. The increased amount of computational time and memory for the more complex models do not always produce better accuracy over the simpler models. Until more is learned about the process of turbulent flow aiding in the development of a universal turbulence model, scientists and engineers will continue to pursue the different avenues of turbulence modeling described above. (Neel, 1997).

k-ε Turbulence Model is the most common model used in Computational Fluid Dynamics to simulate mean flow characteristics for turbulent flow conditions.

It is a two-equation model, which gives a general description of turbulence by means of two transport equations (PDEs). The original impetus for the k-ε model was to improve the mixing-length model; as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows.

The first transported variable determines the energy in the turbulence and is called turbulent kinetic energy (k). The second transported variable is the turbulent dissipation (ε), which determines the rate of dissipation of the turbulent kinetic energy.

Unlike earlier turbulence models, k-ε model focuses on the mechanisms that affect the turbulent kinetic energy. The mixing length model lacks this kind of generality. The underlying assumption of this model is that the turbulent viscosity is isotropic, in other words, the ratio between Reynolds stress and mean rate of deformations is same in all directions.

The exact k-ε equations contain many unknown and unmeasurable terms. For a much more practical approach, the standard k-ε turbulence model (Launder and Spalding, 1974) is used which is based on our best understanding of the relevant

processes, thus minimizing unknowns and presenting a set of equations which can be applied to a large number of turbulent applications.

For turbulent kinetic energy k

ij

ui represents velocity component in corresponding direction Eij represents component of rate of deformation

μt represents eddy viscosity

k2

t C (35)

After determining the suitable grid structure, generated mesh was analyzed with k- Standard, k- RNG, k- Realizable, k- Standard and Spallart-Almaras turbulence models under the same conditions.

Firstly analysis was conducted with k- Standard Turbulence model under the conditions indicated in Table 3.39 until the residuals converged to 10^-5 level.

Resultant graphic of residuals and results table can be seen in Figure 3.69 and Table 3.42 respectively.

Figure 3.69. Graphics of residuals converged to 10^-5 Table 3.42. Results of residuals

Iteration Continuity X-velocity Y-velocity Z-velocity k Epsilon 800 1.867*10^-05 1.887*10^-05 1.243*10^-05 1.137*10^-05 2.744*10^-05 4.279*10^-05

In order to check whether the convergence criteria 10^-5 is proper or not, changes of area weighted average velocities on the outlet surfaces of Exit 1 and 14, which are the exits at the ends of the channel, with iterations were observed.

Graphics of area weighted average exit velocities of Exit 1 and 14 can be seen in Figure 3.70 and 3.71 respectively.

Figure 3.70. Graphics of area weighted average exit velocity of Exit 1

Figure 3.71. Graphics of area weighted average exit velocity of Exit 14

As can be seen from the graphics, level of changes in area weighted average exit velocities are so minor and nearly constant after 750^th iteration for Exit 1 and 650^th iteration for Exit 14, which corresponds 10^-5 level. So, it was decided that, iterating the analyses until the residuals were converged to 10^-5 level with default Under-Relaxation factors, tabulated in Table 3.43, gives precise enough results for this application.

Table 3.43. Default Under-Relaxation Factors

Pressure : 0,3

Density : 1

Body Forces : 1

Momentum : 0,7

Turbulent kinetic Energy : 0,8 Turbulent Dissipation Rate : 0,8 Turbulent Viscosity : 1 3.3.3. Channel Design Studies

Before starting designing by means of numerical analyses, a study to determine the suitable inlet velocity boundary condition, which is evaporator fan exit velocity, was conducted. According to this study, target surface exit velocity of the air nozzles was assumed as 5 m/s. (Temsa, 2007). The net surface area of the nozzle

(Figure 3.72), where air flows through, was calculated as 1,96*10^-3 m². So, the volumetric flow rate of the air flowing through 1 nozzle at this situation is 9,8*10^-3 m³/s.

Figure 3.72. Air nozzle employed in the test vehicle

Since each of the exit channels supplies air to 2 nozzles, volumetric flow rate passing through 1 exit channel was calculated as 19,6*10^-3 m³/s. And since there are 14 exit channels on designed air channel, total volumetric flow rate to be supplied to the channel is 274,4*10^-3 m³/s dir. This amount of air will be supplied from an inlet area having dimensions 58X118 mm, so the inlet velocity calculated as 6,68 m/s.

So suitable fan speed level must be determined by measuring the evaporator outlet fan velocities in different fan levels, and real inlet velocity boundary condition must be determined.

Selected A/C unit has 6 fan speed levels. In the velocity measurements, outlet velocities at different fan speeds were measured in 3 points (left, center, and right) of each of the evaporator exits than the mean of 3 measurements were noted to the Table 3.44. Measurement points are shown in Figure 3.73.

(a) (b) (c)

Figure 3.73. Evaporator outlet velocities measurement points a: Left, b: Center, c: (d) Right, d: Exits of evaporator fan

Table 3.44. Measurement results of evaporator outlet velocities

Evaporator

So, as Table 3.44 was investigated, it was seen that 3^rd level of fan speed with a mean velocity of 6,61 m/s is very near to 6,68 and this value can be taken as the velocity inlet boundary condition for the analyses.

Since channel geometry is restricted by vehicle length, roof structure, luggage rack section details, and seat layout etc. Control parameters of air channel were determined as main channel diameter, exit channel diameter, channel geometry, and zoning condition. Flow rate to be transferred through and air velocity values inside the channels were considered to determine the starting values of main and exit channel diameters.

An important issue that must be considered in design of air channels is the noise created by airflow through the channel. So, it is desired to keep the velocities inside the channels at a certain level (Carrier, 2004). In this study, starting diameter of the exit channels were calculated as 50 mm, to keep the velocity around 10 m/s while 19,6*10^-3 m³/s flow rate flows through the exit channels. With a similar approach, main channel diameter was calculated as 150 mm to keep the velocity around 10 m/s.

By considering flow rates, and velocity limits, it was decided that starting points for main and exit channel’s diameters were 150mm and 50 mm respectively.

Front and side views of the first channel having 150 mm main and 50 mm exit channel diameters can be seen in Figure 3.74.

(a): Front view

(b): Side view

Figure 3.74. Front (a) and side (b) views of the channel having 150 mm main, 50 mm exit diameters and single part geometry

FLUENT^® 6.3.26 has been used for the computations. A steady state formulation has been applied. Standard k- ε turbulence model has been used to model viscous flow. A first order upwind scheme has been chosen for initial set of iterations

on momentum equations and is then switched to a second order upwind to achieve a faster convergence (Somani, 2008).

Analyses were run with previously specified grid and under conditions specified in Table 3.45.

Table 3.45. Analyses conditions

Solver : Pressure Based, Implicit, Steady, 3D Turbulence Model : k- , Standard

Near-Wall

Treatment : Standard Wall Functions (First 500 Iterations) Enhanced Wall Treatment (up to converge 10^-5) Discretization : First Order Upwind (First 500 Iterations)

Second Order Upwind (up to converge 10^-5) Boundary

Conditions : Velocity Inlet : Velocity Magnitude : 6,61 m/s

: Turbulent Intensity : 10 %

: Hydraulic Diameter : 78,7 mm

: Pressure Outlet : Turbulent Intensity : 10 %

Hydraulic Diameter : 50 mm

: Wall : No slip condition

Operating

Conditions : Fluid Material : Air

: Operating Pressure : 101.325 Pa

: Gravity : No

4. RESULTS AND DISCUSSION 4.1. Cooling Load Calculations

4.1.1. Temperature Distribution Under The Passenger Compartment

Results of temperature determination test can be seen as follows. After investigating the data taken from the thermo-couples, average temperature values for each of the region and test condition were calculated as shown in the following graphs (Figures 4.1 to 4.11).

TEMPERATURES ABOVE THE RADIATOR

50 60 70 80 90

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.1. Graphic of the temperatures above the radiator (FL1)

TEM PERATURES ABOVE THE ENGINE

50 60 70 80 90

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.2. Graphic of the temperatures above the engine (FL2)

TEMPERATURES ABOVE THE EXHAUST

60 70 80 90 100

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.3. Graphic of the temperatures above the exhaust (FL3)

TEMPERATURES ABOVE THE REAR LEFT LUGGAGE

40 50 60 70

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.4. Graphic of the temperatures above the rear left luggage (FL4)

TEMPERATURES ABOVE THE TRANSMISSION

40 45 50 55 60 65 70

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.5. Graphic of the temperatures above the transmission (FL5)

TEMPERATURES ABOVE THE BATTERY

40 50 60 70

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.6. Graphic of the temperatures above the battery (FL6)

TEMPERATURES ABOVE THE REAR AXLE

30 40 50 60

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.7. Graphic of the temperatures above the rear axle (FL7)

TEMPERATURES ABOVE THE LUGGAGE ROOM

25 35 45

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.8. Graphic of the temperatures above the luggage room (FL8)

TEMPERATURES ABOVE THE FRONT AXLE

25 35 45

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.9. Graphic of the temperatures above the front axle (FL9)

TEM PERATURES ABOVE THE FUEL TANK

25 30 35 40 45

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.10. Graphic of the temperatures above the fuel tank (FL10)

TEMPERATURES BELOW THE DRIVER PLATFORM

25 30 35 40 45

1

TIME

TEMP (C)

100 km/h Ramp Idle

Figure 4.11. Graphic of the temperatures below the driver platform (FL11)

As result of test data examinations and assumptions, the average temperatures of the regions under the vehicle floor were determined as shown in Table 4.1.

Table 4.1. Average temperatures of the regions under the vehicle floor

4.1.2. Cooling Load by Conduction and Convection

Amount of heat transfer by conduction and convection from the vehicle floor, right side wall, left side wall, front surface, rear surface and roof regions are listed in the following Tables respectively. (Table 4.2 to 4.7)

Table 4.2. Amount of heat transferred from vehicle floor region (FL)

Detail Region Name Detail Region Code Heat Gain (Watt)

Above the Radiator FL1 21,4

Above the Engine FL2 26,0

Above the Exhaust FL3 24,1

Above the Rear Left Luggage FL4 80,9

Above the Transmission FL5 105,5

Above the Battery FL6 11,6

Above the Rear Axle FL7 32,2

Above the Luggage Compartment FL8 325,4

Above the Front Axle FL9 140,1

Above the Fuel Tank FL10 53,9

Below the Driver Platform FL11 73,3

Vehicle Floor FL 894,4

Table 4.3. Amount of heat transferred from right sidewall region (RS)

Detail Region Name Detail Region Code Heat Gain (Watt)

Front Door RS1 62,3

Rear Door RS2 116,0

Side Panel RS3 436,9

Side Glass RS4 299,4

Panorama Glass Above Front

Door RS5 14,8

Front Door Glass RS6 23,5

Right Side Wall RS 952,9

Table 4.4. Amount of heat transferred from left sidewall region (LS)

Detail Region Name Detail Region Code Heat Gain (Watt)

Driver Glass LS1 57,8

Side Panel LS2 475,3

Side Glass LS3 258,5

Panorama Glass Above Driver

Glass LS4 12,0

Left Side Wall LS 803,6

Table 4.5. Amount of heat transferred from front surface region (FR)

Detail Region Name Detail Region Code Heat Gain (Watt)

Front Glass FR1 404,7

Front Mask FR2 75,8

Front Surface FR 480,5

Table 4.6. Amount of heat transferred from rear surface region (RR)

Detail Region Name Detail Region Code Heat Gain (Watt)

Rear Glass RR1 149,2

Rear Mask RR2 89,9

Rear Surface RR 239,1

Table 4.7. Amount of heat transferred from roof region (RO)

Detail Region Name Detail Region Code Heat Gain (Watt)

Roof RO1 1059,8

Emergency Exit RO2 98,6

Roof RO 1158,4

As a result of summation of the cooling loads of each main region, the total cooling load originated from convection and conduction is calculated as 4.530 Watt and listed in Table 4.8.

Table 4.8. Total convection and conduction heat transfer to the passenger compartment

Main Region Name Main Region Code Heat Gain (Watt)

Vehicle Floor FL 894,4

Right Side Wall RS 952,9

Left Side Wall LS 803,6

Roof RO 1158,4

Front Surface FR 480,5

Rear Surface RR 239,1

Total Surfaces 4.530

4.1.3. Heat Transfer by Radiation

By employing the software program prepared by Büyükalaca and Yılmaz, (2004) and considering the 8 different calculation options for sun position, cooling loads were calculated for all times of a day. Finally it is found that the maximum cooling load can be reached with calculation option F (Right side of the vehicle subjected to sun light) in 15^th hour of the day as 11.257,4 Watt.

4.1.4. Cooling Load by Internal Sources

According to the considerations in Section 3.1.5, cooling load by people is determined as 7.280Watt. Cooling load originated by lightening and equipment was determined as 1.350 W from the specification of the vehicle.

4.1.5. Cooling Load by Ventilation

According to the considerations in section 3.1.6 and ASHRAE standards, cooling load by ventilation was calculated as 5.822,1 W. Details of cooling load by ventilation are shown in Table 4.9.

Table 4.9. Cooling load by ventilation calculation table

Fresh Air Requirement For Passengers

Number of Adults in the vehicle 52 Number of Children in the vehicle 0

Fresh Air Requirement For Each Person Adult (m³/h) 25 Child (m³/h) 15 Total Fresh Air Requirement m³/h 1300 Total Fresh Air Requirement m³/s 0,361

Air Properties At a Mean Temperature

Property Unit Value

Desired Temperature inside the vehicle. °C 24

Outdoor Temperature °C 38

Mean Temperature °C 31

Delta T °C 14

Specific Heat (cp) 1.005,8

Density kg/m³ 1,15

Total Ventilation Mass Flow Rate kg/s 0,413 Total Heat Gain by Ventilation W 5.822,1

4.1.6. Cooling Load by Infiltration

As a result of assumptions described in Section 3.1.7 ventilation-cooling loads were calculated as 1.163,5 Watt. Details of the calculation are shown in Table 4.10:

Table 4.10. Details of infiltration cooling load

COOLING LOAD BY INFILTRATION

Front Door Height (m) m 2,09

Front Door Infiltration Volume Flow Rate m³/h 143,93 Middle Door Infiltration Volume Flow Rate m³/h 115,86

Air Properties At a Mean Temperature

Property Unit Value

Desired Temerature inside the vehicle. °C 24

Outdoor Temperature °C 38

Total Infiltration Volume Flow Rate m³/h 259,8 Total Infiltration Volume Flow Rate m³/s 0,072 Total Infiltration Mass Flow Rate kg/s 0,083 Total Heat Gain by Infiltration W 1.163,5

As a result, by the summation of all types of cooling loads, total cooling load

As a result, by the summation of all types of cooling loads, total cooling load

Belgede ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND APPLIED SCIENCES (sayfa 119-0)