ELSEVIER Energy and Buildings 23 (1995) 33-39

AND 6UILDII IG

CFD modelling of the air and contaminant distribution in rooms

Youchen Fan VTT Building Technology, Indoor Environment and Systems, Box 1804, 02044 VT"T, Espoo, Finland

Received 12 August 1994; in revised form 18 December 1994; accepted 26 December 1994

Abstract

The k-E model is a wid!ely used model in engineering practice in handling indoor air quality problem. However, difficulties may arise when using the high Reynolds number k-e model to simulate air flow patterns close to the boundaries of air and the stagnant component as well as the low air flow fluctuation elsewhere in a room. When using the k-E model for low Reynolds number cases, the correlations between turbulent coefficients and turbulent Reynolds number must also be defined. By using the so-called Kolmogorov micro scale method, a new set of turbulent coefficient functions was deduced in this paper for the k-e model in a case of low Reynolds number flow. Using the standard wall function leads to large differences between the measured and calculated heat transfer coefficient. A special wall function valid for a viscous sublayer, a buffer zone and a fully turbulent log-law zone is recommended in this paper. In addition, the modelling of air terminal devices in CFD simulations is summarized by using a literature collection.

Keywords: Computational fluid dynamics modelling; Air contaminants; Indoor environment

1. Introduction

Computational fluid dynamics (CFD) modelling and scale (or physical) modelling are, in principle, the two alternative approaches for evaluating the global indoor environments with the .aim to obtaining a proper design of building service systems. Both approaches have their advantages and limitations. CFD has, nevertheless, be- come a widely accepted alternative to the scale modelling in recent twenty years for predicting the air motion in buildings and indoor air quality. The main obstacles in using CFD modelling', for HVAC research are assumed to be on how to handle the mix-forced (momentum and buoyancy) air flow and simulate the occupant- behaviour-related (OER) factors. Numerical problems relating to the boundary layer flow are of secondary importance in CFD application of HVAC engineering. This is due to the fact that the CFD models were originally developed by assuming the flow to be of a high Reynolds number (HRN) type. This is not the case particularly in modern buildings. Although a few low Reynolds number (LRN) models already exist, there is still a need to develop these models. For simulating the OBR factors, wider use of statistical mathematics will be required. At the moment, the simulation research of OBR factors based on statistical mathematics is totally missing.

0378-7788/95/$09.50 1995 Elsevier Science S.A. All rights reserved SSDI 0378-7788(95)00916-L

The k-e model has been used worldwide since it was created. However, for the HRN model, the so-called wall function must be properly designed to correlate the physical parameters between the LRN and HRN flow zones. It has so far been believed that the wall function has a large influence on the heat transfer near to the solid boundaries. Thus, the problem is now to find the correct wall function if it is needed. In addition, for the LRN model, the relationship between turbulent coefficients and turbulent Reynolds number was not totally known until now. Another consideration with respect to the LRN model for room air flow is that these kinds of models, in fact, are trying to handle the low turbulent Reynolds number effect both on the boundary flow and on low fluctuation flow elsewhere within the room. New findings for the above problems will be addressed in the following sections. The research method involved theoretical analysis and literature col- lections.

2. Theoretical analysis of the LRN model

The specific property of the k-e model refers to its closure equations of Reynolds stress, i.e., the turbulence kinetic energy and its dissipation rate transportation equations. The general form of transportation equations of k and e which are derived from Navier-Stokes (N-S)

34 Y. Fan I Energy and Buildings 23 (1995) 33-39

equations are as follows [1]: k-equation

-- r ( Dk 1_0 /x, +/~ + Dt p axj I_\Cr~

/3 ~, OH + - - - -g i - - e ar,

e-equation

. , 0u, o il

(1)

D--7 = p~ ~+/x ax, j

where

d /3 e iz, OH -c2 +c3 p--C k g' (3)

Each modified k-E model is concentrated on the se- lection of a set of turbulence parameters, i.e., C,, Ca, Cz and C3. These modified models are called the LRN model, which are developed mainly for phasing out the wall functions. By analysing the calculation of these parameters, some observations can be drawn.

2.1. Turbulent coefficients

For the case of the boundary layer, the e transportation equation of the k-e model has the form [2] as follows

-

+Ca-~ -C2 ~ (4)

In the region close to a wall, the shear stress is uniform and the length scale increases linearly with distance from the wall (fully turbulent log-law region). The convective terms can be omitted [3], which follows that Eq. (4) reduces to

By considering equilibrium turbulence, the term m(au/ay)2/p can be expressed as

/L' (~)z= e P (6)

(7)

and

= C~, k2 p e

by using the Boussinesq assumption.

Substituting Eqs. (6) and (7) into Eq. (5), gives K 2

C1 = C2 r,C,f 2 (8)

in which K = Karman constant ( = 0.4), p, = 1.3 [3] and 2.0 > C2 > 1.71 [2].

By considering the influence of the turbulent Reynolds number R,, the coefficient C2 has been interpreted as a function of Rt [4] as follows

C2 = C20[1 - 0.3 exp( -Rt2)] (9)

Substituting Eq. (9) into Eq. (8), Ca can then be expressed by

K 2 Ca=C2o o.C~a -0.3C2o exp(-R, 2) (10)

or

C~ = C~o - 0.3(72o exp( -R, 2) (11)

If the turbulent constants in Eqs. (9) and (10) or (11) are selected to be K=0.41, or,= 1.3, C~, = 0.09, Cm= 1.85, then, we can have the following expressions of Ca and C2 modified with respect to R,

Ca = 1.42 - 0.56 exp( -n t 2) (12)

Ca = 1.85 - 0.56 exp( -Rt 2) (13)

Some researchers [5] have introduced stronger non- linear modification functions as follows

[ (Acl'~ 3 ] (14) CI=Clo 1+\ f~, ] 3

C2 = C2o[1 + exp(-Rt) z] (15)

where

At f ~, = [1- exp( -A~,Rk) ]2( l + -l~, ) (16)

It can be easily found that the derivatives of C~ with respect to e and k are

aCl aCl a-~" (~(f/'L)--4; "~ ~(f~)--4 (17)

Thus, even a small variation in R t will produce dramatic changes in the damping function value, which increases the numerical stiffness of the model. Moreover, the damping function for C2 will be effective only in the region of y+

Y. Fan / Energy and Buildings 23 (1995) 33-39 35

not suitable for reproducing the laminar behaviour or laminarization.

In addition, the production term of the e-equation (see Ref. [1]) should include three parts, which are shear, gradient and turbulent production. By using the Kolmogorov micro scale theory, the shear and gradient production parts were dropped under the consideration of high turbulent Reynolds number R,. Their relations are as follows

Ps = C~R,- ~z2p, (18)

P~=C~2R,-S/4P, (19)

Eqs. (18) and (19) can be interpreted in a way that if R, is not very high, then, P~ and Pg should not be neglected. In this case, the turbulent coefficient C~ can be expressed as

C1 ~-- C1 + f l iR t -1/2 + Cl2Rt -5/4 (20)

Constants Cm~ and C~2 are difficult to determine. Never- theless, the coefficient C~ should be a function of Rt, when the turbulent Reynolds number is not sufficiently high.

Based on the foregoiing analysis, it can be concluded that the turbulent 'constants' have to be expressed as a function of R, in the, air flow simulation of room(s) where R, will not be very high. The difficulties caused by near wall regions may be solved by the wall function method even in the LRN model if the computation time and computer memory are primitive issues.

3. Calculation of the partial differential equations

For solving the governing equations of air and con- taminant motion in room(s), Baker et al. [9] recently presented a new method, called the Taylor weak state- ment (TWS). The basis of the method is similar to the method of weighted residuals. The typical partial dif- ferential equation (PDE) has been expressed by:

L(q) =0 (21) Using Taylor's expansion technique, a PDE companion was defined in the form of

LC(q)-L(q)- [3 At -~j { A/I~ (22)

Further, an approximate solution qN which contains a number of undetermined parameters Qj(t) was assumed, the solution is then expressed by Baker et al. [9] as:

N

qfv(y, t) = ~ ~.(xOQj(t ) (23) j - -1

Substituting the approximate solution into Eq. (21) leaves a residual. Then let the sum of the residual (TWS) over a sub-domain (control volume or element) of interest be minimized as follows

TWS = |~bi(x~)LC(q N) dr= 0 (24) i f l

From the foregoing introduction of the TWS method, it can be found that the weighted residual method was executed to a resultant PDE by introducing an artificial diffusive term to the governing PDEs (see Eq. (22)). It was found by Baker et al. [9] that the stability of the solution can be improved by this diffusive term. This term causes a filtering effect to the high frequency fluctuation of air flow, because the low frequency fluc- tuation is dropped first (Eq. (22)). After that, the resultant governing equation which contains high order fluctuation is minimized with respect to the approximate solution (Eq. (24)).

In the TWS method, two problems arise. One is to determine the/3 values and the other one is to choose the 'known' functions of ~i and ~ in Eqs. (22)-(24). It is obvious from the definition of Eq. (22) that the 13 value should be tightly connected with the grid-point distribution. Each problem must have its own/3 value, which can handle the physical phenomenon properly. In the case of selected or known functions, q~i and ~, considerations must be made to avoid the physically unrealistic. Linear interpolation would result in a similar problem as in the central-difference scheme. When the Peclet number is large for convection-diffusion prob- lems, unrealistic results may be obtained [10]. Patankar [10] has proposed the following exponential function for a two-dimensional case:

pUX =A+B exp - -~ +CY (25)

The proper selection of 'known-functions' for three- dimensional cases still needs further model tests and demonstrations.

The bench mark analysis made by Baker and co- workers [9,11,12] using the TWS method as limited to the LRN (in the order of 9 X 108) and large air change rate (30 ach) flow which is far from that found in practice. The reason for the good agreement of their results was because the air flow can be identical in the whole domain except for the region near the solid boundaries. But this is not the case in reality. Most difficulties in modelling room air flow reside in the handling of multi-state (laminar, turbulent and tran- sition flow) flow and transformation from one flow state to another.

4. Modelling of air terminal devices (ATD) in rooms

In fact, both CFD simulation and scale modelling are procedures of similarity. It is not difficult for a CFD model to guarantee thermal and momentum sim- ilarity within the fluid domain. This is not the case when considering the momentum similarity on bound-

36 Y. Fan / Energy and Buildings 23 (1995) 33-39

aries, particularly the ATD. The momentum similarity on boundaries can mainly embody at the air supply and exhaust devices in a ventilated space. This is because the influence of air leakage (e.g. through windows) on air flow similarity is limited typically. And on the other hand, the momentum similarity is automatically satisfied for impermeable walls due to the no slip law. Modelling of air terminal devices has been becoming increasingly important. In the following, the approaches in modelling ATD will be summarized and discussed based on a literature review.

4.1. Momentum similarity at ATD

In this method, a complicated diffuser is modelled by a simple opening which lets the supply air have the same momentum and same direction as reality.

Two versions to the basic approach exist. In the first, a simple opening has the same effective area as the real diffuser, then the same velocity as each nozzle (if the diffuser has many nozzles). This version has been used by Heikkinen [13], and Skovgaard and Nielsen [14]. From Nielsen's results [15], it can be found that a simple opening can fairly simulate a complicated diffuser, especially the jet decay far from the ATD. The measurements made by Heikkinen [13] also con- firmed Nielsen's conclusion. However, the jet spread in horizontal and thickness in vertical obtained by this simplified ATD model can be very different in com- parison to the measured air jet generated by a com- plicated diffuser (see Ref. [13]).

The second version lets the momentum force defined by Eq. (26) and the supply air direction be the same between the simple opening and the complicated dif- fuser. The opening area can, somewhat freely, be chosen.

F = JpU z dA (26)

This version has been used earlier by Chen et al. [16].

4.2. Momentum similarity in front of the ATD

This approach is based on well-developed wall-jet techniques. The air velocity field within a small space in front of a diffuser is self-similar. Thus, the momentum similarity of the ATD can be defined away from the ATD by using the well-developed wall-jet technique. The approaches developed in the above way were referred by Nielsen [17] as the Box method and the prescribed velocity method. These are sketched in Figs. 1 and 2. At surface b there is no flux. In addition, universal or dimensionless physical parameters' (~b) profiles for a given diffuser at surface a must be pre- defined. This method is more difficult than the method described above. However, it was believed that better results could be achieved by using it [17].

(a)

1.00 0.80

0.40

0.20

0. I( 0.0(

0.0,

0.02

0.00' 1

(c)

(b)

1

* Ful l sca le . Diffuser type D ~ Model. Nozzle

~ ,; ; ;/o 20 do~o6'o8o

Fig. 1. Modelling of ATD [15]. (a) End wall mounted diffuser; (b) nozzle directed against the ceiling in 45; (c) velocity decay in wall jet along the ceiling in a room and in a model.

Y Fig. 2. Modelling of ATD [17].

X0

b

-~y'0

r

C .e-

All the above methods have been used in CFD simulation. The first one seems to be more user-friendly, particularly in assessing a system design. Whereas, for the second one, measurements are normally required.

5. Wall function

The wall function was originally a method to deal with the boundary conditions for the HRN model. The HRN model was based on the assumption that the g t value is so high that the turbulent parameters C1, C2, C~, and C3 could be treated as constant. This assumption does not hold in the region near the solid boundaries at any time. Hence, a 'known' function (so-called wall function), is needed to connect the boundary conditions with the physical parameters in the domain where the

Y. Fan / Energy and BuiMings 23 (1995) 33-39 37

influence of Rt on turbulent parameters is negligible. The velocity and temperature distributions (VTD) of the fiat plate boundary layer flow or the Couette flow have been selected as the wall function. In a model test, Chen [1] found that if the VTD within the boundary layer were expressed by linear and logarithmic equations, the computed convective heat transfer coefficient near a wall of a ventilated room could be about 40-50% lower than the measured one. Later he improved the wall function by dividing the boundary layer into three regions, the viscous sublayer, buffer zone and fully turbulent log-law zone then evaluated the VTD by using different formulas for each region. It was con- cluded [1] that modified wall function could reduce the differences between computed and measured con- vective heat transfer coefficient to about 20%, but it is still high. The velocity distribution in the whole boundary layer over a flat plate had been deduced by Spalding [18] more than 30 years ago, which is expressed by Eq. (27). Unfortunately, Spalding's equation seems to have not been noticed by researchers of CFD mo- delling. A comparison of different wall functions is shown in Fig. 3.

y+ =u + +0.1108

X [ exp '4" + -

u + - u(p/rw)lr2;

(0.4u+) 2 (0.4~ +) 3 ] 1-0 .4u + - 2!

y+ -y(,r,,, p//z) 'a (27)

The air flow over the walls is seldom in parallel to the wall surface. Therefore, each wall function, in fact, is based on the assumption that the flow is similar. Hence, any model plus the wall function has more or less the same difficulties as the mixing-length model.

The LRN model has phased out the usage of the wall function. Though it still has some problems that have not yet been solved, it offers an opportunity to handle the room air flow.

6. Conc lud ing remarks

The k-e model in simulating the room air flow was reviewed. It was concluded that the turbulent parameters should be functions of the turbulent Reynolds number even in the case where the wall function was to be used. It is evident that the air flow in modem buildings is very slow. Thus, the basic assumption of fully turbulent flow in the region far from the walls for the HRN model will no longer hold. The recommended corre- lation, Eq. (20), is based on the exact time-averaged N-S equations and Kolmogorov micro scale theory. If the constants in this equation could be defined by CFD analysis or by experiments, better results could be expected. It was found that the damping function of turbulent coefficients which contain the wall parameter y+ fails to handle the room air flow. The damping function of turbulent coefficients should affect not only

U

2,0

2:8

26

24

22

20

' , 8

16

14

~2

10

8

6

4

2

0

u + = 0.25 In(y ) + 5.5 j ~

/ /

0 2 4 6

Fig. 3. Comparison of different wall functions.

f . /

.~

y+= 1;y~

U + - U

I !

8 10 In (y*)

38 Y. Fan / Energy and Buildings 23 (1995) 33-39

the near wall region air flow but also the air flow in the central zones of the room.

To predict the momentum and heat energy distri- bution over a finite cell will form the basis of a calculation method of the partial differential equations. These kinds of distributions are typically considered to be linear. Patankar has proposed an exponential distribution in his two-dimensional convection--diffusion simulation [10]. If this function works for three-dimensional cases, further model tests and demonstration will be needed.

Each method mentioned in Section 4 shows the benefit from evaluating the jet decay, but the prediction on jet spread and thickness may be far from reality. This could be indicated by unbelievably uneven computed temperature and contaminant distributions. Further research is required.

The k-e model (LRN and HRN models) together with wall function has more or less the same problem as the mixing length model. This point can be proved by using the von Karman's similarity hypothesis [19]. In order to improve the ability of the k-6 model in handling the circulated flow, the wall function must be phased out. It is true that the use of the LRN model without wall function will increase computational cost. Each turbulent model, which includes the k-e model, Reynolds stress model and vortex model may be an- alogized by the Taylor's expansion. The differences between them are in how many terms have been kept in the expansion. The k-e model is the simplest one. If the turbulent coefficients can be correlated correctly to the turbulent Reynolds number, then use of the model in engineering will be more promising.

7. Nomenclature

A Aj Acl. A. , A t B C C1, C2, C3 C~o

C. F gi

H k P es Pt P, q(xj, t) R,

constant (-) kinematic flux vector jacobian (J/m 2) constants (-) constant (-) constant (-) turbulence parameters (-) turbulence parameters (-) constant (-) turbulence parameter (-) momentum force (N) gravitative acceleration vector in direction i (m/s 2) time-averaged enthalpy (J/kg) kinetic energy (J/kg) pressure (Pa) shear production (J/kg s 2) turbulent production (J/kg s 2) gradient production (J/kg s 2) a vector (={u 1, 19, ~b, p}T, j= 1, 2, 3) (--) turbulent Reynolds number (=k2p/lze) (-)

gk U u

X Y Y

Greek

p6 IZt P Ok, O" H

w

O

,(x)

turbulent Reynolds number ( = kl/2yp/lz) (-) time-averaged velocity (m/s) velocity near wall (m/s) coordinate (-) coordinate normal to X (-) distance from the wall (m)

coefficient of thermal expansion (l/K) dynamic viscosity (N s/m 2) turbulent viscosity (=C~pkZ/E) (N s/m 2) density (kg/m 3) equivalent turbulent Prandtl number (-) shear stress at the wall surface (Pa) resultant variable which connects velocity field and pressure through continuity equa- tion (-) temperature (K) known function (-) known function (-)

Acknowledgement

The author thanks Professor Dr Markku Virtanen of our laboratory for his intelligent comments and editorial help with this paper.

References

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[3] B.E. Launder and D.B. Spalding, The numerical computation of turbulent flow, Comput. Methods Mech. Eng., 3 (1974) 269-289.

[4] W.P. Jones and B.E. Launder, The calculation of low-Reynolds- number phenomena with a two-equation model of turbulence, Int. Z Heat Mass Transfer, 16 (1973) 1119-1130.

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[9] A.J. Baker, P.T. Williams and R.M. Kelso, Development and validation of a robust CFD procedure for predicting indoor

Y. Fan / Energy and BuiMings 23 (1995) 33-39 39

room air motion, Proc. Indoor Air '93, HelMnki, Finland, 1993, Vol. 5, pp. 183-188.

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[15] P.V. Nielsen, Model experiments for the determination of airflow in large spaces, Proc. Indoor Air '93, Helsinki, Finland, 1993, Vol. 5, pp. 253-258.

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[17] P.V. Nielsen, Description of supply openings in numerical models for room air distribution, ASHRAE Trans., 98 (1) (1992) 963-971.

[18] D.B. Spalding, A single formula for the 'law of the wall', J. AppL Mech., (1961) 455-458.

[19] L.C. Burmeister, Convective Heat Transfer, Wiley, New York, 1983, Ch. 9.