Full-scale subjective and manikin studies are both complicated, expensive and time consuming. The use of CFD makes it possible to solve the fluid flow as well as heat transfer and get satisfactory simulation results for most indoor airflows. In order to design a comfortable indoor environment with MANIKIN3, it is important to get relevant input data of the airflow pattern, velocity, and temperature around the virtual manikin. The majority of flows in the indoor environment are turbulent. Unfortunately no turbulence model exists for general use. Every model must be employed with care and its results treated with caution (Sørensen et al., 2003). A short description of some models used, beginning in the most detailed end of the turbulence simulation methods:
DNS, LES, VLES and RANS
Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES), are compu- tational tools with which almost all scales of turbulence can be simulated. (DNS)
simulates all the scales of turbulence at the highest level of detail, at the expense of huge computer costs. LES computes only the large or very large (VLES) scales of turbulent fluid flow. At a slightly less complex level there is also Reynolds Averaged Navier- Stokes (RANS) methods that solve time-averaged equations, trying to model all the scales of turbulence in order to solve flows over more complex geometries at higher Reynolds numbers.
Drawbacks of DNS and LES is that they are limited to fairly simple geometries such as channel flows, simple boundary layer flows and the use of large number of cells and many time steps that are well beyond current personal computational capabilities used by engineers today. For engineering use is it important to use a more simple turbulence model for the simulation of the indoor environment, that could calculate the airflow, velocity and temperature fields in an adequately correct way.
Standard k-e model
The ‘standard’ model in which the high (turbulent) Reynolds number forms of the k and e equations are used in together with algebraic ‘law of the wall’ representations of flow, heat and mass transfer for the near-wall region (Launder et al., 1974). The standard k-e model is consequently often called a two-equation model. The standard k-e is appro- priate for fully turbulent flows where the Reynolds number (Re) is high.
Low-Re number k-e model
The low Reynolds numbers model in which general transport equations for k and e are solved everywhere, including the near-wall regions. ‘Law of the wall’ functions are therefore not required. For low Reynolds number flows the standard k-e model overestimates the turbulent diffusivity. The standard k-e model has been modified to give better results with these flows. In low-Reynolds number turbulence models, correction functions are introduced in the calculation.
RNG-k-e model
The ‘ReNormalisation Group’ (RNG) version of the k-e model is denoted as RNG k-e. This is used in high Reynolds number form together with ‘law of the wall’ functions. From a comparison by Chen, 1995, the RNG k-e model was recommended for the simulation of indoor environments. The RNG-k-e model was better than the standard k-e model for mixed convection flow and impinging jets.
Zero-equation turbulence model
When the details of the turbulence are not so important, rather the general mixing behaviour, then it is often possible to use a constant turbulent (eddy) viscosity µt in
stead of the molecular viscosity (Nielsen, 1998). This turbulence model is called the zero-equation model and uses a constant or an algebraic function to express the turbulent viscosity. It does not require the solution of any additional differential equations beyond the Navier-Stokes equations. This turbulence model calculates the turbulent viscosity empirically by:
H u t =0.04×r× 0 ×
m Equation 20
Where
mt turbulent (eddy) viscosity (Pa s)
Fluid density (kg/m3)
u0 Characteristic velocity, inlet velocity (m/s)
H Characteristic length, inlet min length (m)
The length scale is a characteristic length, in this thesis the case specific min length of the inlet is used. In the same way the inlet velocity is used as the characteristic velocity for each case. The empirical constant suitable for different indoor airflows is a number between 0.038 and 0.040. This model is often sufficient for predicting the total
characteristic of a turbulent flow; it may not always be suitable for predicting local details. One benefit of this method is that the time used for calculations with the zero- equation model is much less compared to the more complicated models. Further more the use of this turbulence model does not need extensive grid refinement or the use of special wall functions, two factors that significantly speeds up the working process. Consequently the computer power needed to calculate indoor airflow is less and can be realised with an ordinary personal computer.
The next level of complexity with the zero-equation models is to compute an effective viscosity that is a function of local conditions. This is made on the basis of Prandtl’s mixing-length hypothesis, which gives that the viscosity is proportional to the local shear rate. Results calculated by Chen et al., 1998 shows that the accuracy of the zero-equation model is acceptable for indoor airflow design purpose. More experiments have been done to validate the zero-equation model ability to predict indoor airflow (Srebric et al., 1999). The results from these studies show that zero-equation model predicts the main flow reasonably well even with thicker boundary layers, due to the larger cell size used. Although there are some differences, the important air velocity and temperature profile results are good.
In this thesis such a simple model for indoor airflow simulation is used; the zero- equation turbulence model. Below table 16 with the turbulence viscosities used.
Table 16. Turbulence viscosities used in the 3 validation cases. Zero-equation model
case s w rdis10 rdis20 rdis30 omix odis23
Density (kg/m3) 1.205 1.205 1.205 1.205 1.205 1.205 1.205
Inlet velocity (m/s) 0.03 0.03 0.08 0.17 0.25 3.38 0.15
Inlet characteristic min length (m) 0.50 0.50 0.20 0.20 0.20 0.10 0.30
Constant (ND) 0.04 0.04 0.04 0.04 0.04 0.04 0.04
Turbulence viscosity (Pa s) 0.00072 0.00072 0.00077 0.00164 0.00241 0.016 0.0022
Turbulence viscosity k-e model typical (Pa s) 0.00047 0.00047 0.00034 0.00034 0.00034 0.015 0.0019
case odis40 odis55 cclearg cclearp ccolop creflg
Density (kg/m3) 1.205 1.205 1.205 1.205 1.205 1.205
Inlet velocity (m/s) 0.27 0.37 4.40 4.40 4.40 4.40
Inlet characteristic min length (m) 0.30 0.30 0.05 0.05 0.05 0.05
Constant (ND) 0.04 0.04 0.04 0.04 0.04 0.04
Turbulence viscosity (Pa s) 0.0039 0.0054 0.011 0.011 0.011 0.011
General conservation equations
The conservation of mass, momentum and energy can describe the airflow field. If the boundary conditions are known, the flow pattern can be determined by solving the combined Navier-Stokes and energy equations. STAR-CD solves the general mass, momentum and scalar conservation equations, with different meanings of f:
( )
f f f r f rf u S x x t j j j ÷÷ø+ ö ç ç è æ - ¶ ¶ G ¶ ¶ = ¶ ¶ Equation 21 Where t Time (s) r Fluid density (kg/m3)f Represents any mean scalar variable
uj Fluid velocity component in direction xj (m/s)
Sf Source term
Gf Diffusion coefficient
The term to the left of the equal sign is the convection term, the first term to the right is the diffusion term and (Sf) is the source term.
Numerical solution algorithms and discretisation
STAR-CD permits three different solution algorithms: · SIMPLE method
· PISO method.
· SIMPISO algorithm (combines elements of both the other methods).
The solution is complicated by the pressure term in the momentum equation. One way to overcome this is a procedure in which the pressure field is obtained via the continuity equation, the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) (Patankar 1980). Given an initial pressure field, the momentum equations are solved. A pressure correction is obtained from the revised continuity equation and the velocity component values are corrected.
After calculation of the flow field variables, temperature and turbulent values, the calculated pressure is used as the new pressure field and the calculation is repeated until the solution is converged. SIMPLE differs from the PISO algorithm in two principal respects; it is mainly suitable for steady-state calculations and it employs only one corrector stage. Under-relaxation is required in iterative steady-flow calculations with SIMPLE; it is therefore essential for stability reasons, to under-relax velocity and pressure as well as other variables, such as the turbulence parameters.
Buoyancy-driven flows and natural convection
STAR-CD contains built-in functions for different kinds of body forces, including buoyancy. To analyse problems with buoyancy is it necessary to specify the pressure and gravity forces in the momentum equation. A correction term for the buoyancy force is
then incorporated in the revised continuity equation (Computational Dynamics 1999). This feature is used in all calculations since effects of buoyancy can be expected.
Differencing schemes
Discretisation in space requires the flow field to be divided in small control volumes. First-order schemes select the nearest upwind neighbour value. This form of inter- polation preserves the correct physical bounds on under all conditions, but can lead to numerical diffusion. This is way only second order schemes are used in the following calculations.
MARS (Monotone Advection and Reconstruction Scheme) is a multidimensional second-order scheme that works in two steps. MARS does not rely on any problem dependent parameters to work properly. The user can control the ability of the advection scheme to accurately capture sharp discontinuities in the flow by setting the scheme’s compression level to a value between 0 and 1. Low values for this parameter result in a computationally efficient scheme at the expense of sharpness of resolution. High values improve the resolution but result in an increased number of iterations when steady flows are computed. The default value for this parameter is 0.5 that is the best compromise between accuracy and convergence rate.
Convergence scheme
In order to avoid numerical instability, under-relaxation is introduced. A relaxation factor controls the change of a variable as calculated at each iteration. The solution for the selected variable(s) is taken as a weighted mean of the previous and current
iteration. In principle, under-relaxation may be used on any of the dependent variables as well as quantities like density and viscosity. The convergence is checked by several criteria: the mass and heat conservation should be balanced; the residuals of the discretisised conservation equations must steadily decrease; and the change in field values between two iterations should be very small, below 0.01.
Grid technology
CFD calculations are performed on a grid that fills the experimental volume, in the three dimensional case. The shape of the grid, over which the equations are solved, is very important. The simplest grid is called the Cartesian grid, in the 3-D case the cells are
cubes. When a coarse grid is used, the cubical cells sometimes are severely deformed. A grid with smaller cells will solve this problem to some extent, but will on the other hand make the processing time longer.
One commonly used solution is to refine the grid only in areas where it is absolutely necessary. That is close to the manikin, walls and in/outlets. Another alternative is to use a grid based on tetrahedron cells. This solves some of the problem with distorted cubical cells. To take the best from these two grid solutions sometimes a combination of these two types is used. To avoid any prolongation of the calculation time the standard Cartesian 3-D grid used in the following analyses.
Figure 41. A tetrahedron surface grid for the first case, made with CFX-Build 5.4. The inlet and outlet is positioned on the left wall opposite the manikin. The manikin is sitting on a net chair with the power supply behind. In the final calculations it was decided to use a standard cartesian grid topology.
When a virtual thermal manikin is placed in a room it usually requires grid refinement close to the manikin surface. Fine grids are sometimes required to determine the flow field up to the smallest length scale close to the body (Murakami et al., 1989). These requirements sometimes create difficulties to reach convergence of the calculation domain as well as requiring high computational power.
Methods of creating a fine mesh by internal subdivision of a coarser mesh are usually called “embedded mesh refinement methods”. Adaptation can result in a grid that is refined or coarsened when applying user-defined adaptation parameters is also solution dependent. In this thesis it was decided that no grid adaptation should be used, as this mostly increases the number of cells and hence the calculation time.