# Method

When using any CFD method, we first need to make a grid over the do­main. The number of grid nodes in each direction depends on the type of flow, and what accuracy or reliability the user wants for the predictions. The gov ­erning equations have been discretized on this grid, and the finer the grid, the more accurate the results. If the user only wants an overall picture of the flow field, a 30 x 30 x 30 grid could be sufficient. The grid should be refined in re­gions where large gradients are expected. This normally means that the grid should be made finer near walls and in the region of the inlet.

It should also be remembered that the discretization scheme influences the accuracy of the results. In most CFD codes, different discretization schemes can be chosen for the convective terms. Usually, one can choose between first­order schemes (e. g., the first-order upwind scheme or the hybrid scheme) or second-order schemes (e. g., a second-order upwind scheme or some modified QUICK scheme). Second-order schemes are, as the name implies, more accu­rate than first-order schemes. However, it should also be remembered that the second-order schemes are numerically more unstable than the first-order schemes. Usually, it is a good idea to start the computations using a first-order scheme. Then, when a converged solution has been obtained, the user can con­tinue the calculations with a second-order scheme.

When a coarse grid is used, w’all functions are used for imposing bound­ary conditions near the walls (Section 11.2.3.3). The nondimensional wall dis­tance should be 30 < Y+ < 100, where Y+ = u*y/v. We cannot compute the friction velocity w. before doing the CFD simulation, because the friction ve­locity is dependent on the flow. However, we would like to have an estimation of Y* to be able to locate the first grid node near the wall at 30 < Y+ < 100. It we can estimate the maximum velocity in the boundary layer, the friction ve­locity can be estimated as 0.04f/max. After the computation has been car­ried out, we can verify that 30 < Y+ < 100 is satisfied for the grid nodes adjacent to the walls.

If the user wants higher accuracy in the predictions, and if he/she is willing to pay for the increase in accuracy in terms of increased computing cost, more grid nodes should be used. In general, more grid nodes should be located where the flow is complex. For an empty room without furniture or persons, this normally means that more grid nodes should be placed near walls and in

 Ax,

 Ax, ! FIGURE 11.4 Aspect ratio and biasing for control volumes. Aspect ratio is defined as Ax/Ay for cell /. Biasing in the X direction is defined as A^+j/Ax^,

The inlet region. If heat transfer at walls is to be predicted, it is usually very im­portant to refine the grid near the wails. When using a fine grid near the walls, low-Reynolds-number (LRN) models should be used (see Section 11.2.3.3). In this case the grid nodes should be located so that Y+ — 1 near the wall. As pre­viously explained, we cannot compute the friction velocity U-, in advance, but it can be estimated. After the computation has been carried out, the friction velocity can be computed by using the fact that the velocity in the viscous sub­layer is linear (valid for y+ < 5), given by

A high aspect ratio (e. g., above 100) usually does not pose any problems (Fig. 11.4). Biasing causes larger problems, both in terms of convergence and accuracy. Preferably, biasing should be smaller than approximately 1.1, that is, the cell size between two adjacent cells in one direction should not increase or decrease by more than 10%.

I 1.2.2.2 Equations

The incompressible, time-averaged continuity and the Navier-Stokes equations can be written as

 (11.t) 3^ = 0 OXj

 ,0U, DU V{ dxf + dxt (11.2) —! + — r-(U U,) = +

Dt oXj 1 ‘ pdXj dXj

+ r-7ret)

 ‘jlsLL —, Prdx, Ur (11.3) — + —(U-T) = — Bt Bar, I ‘ dx:

Where

Pressed in °C then /3 = 1/(273 + Tref). The X2 coordinate increases vertically upwards. Turbulent stresses and turbulent heat fluxes Ujt appear on the right-hand side of Eqs. (11.2) and (11.3). They act as additional diffusion terms due to correlations between fluctuating velocities and temperature, which are unknown. Transport equations can be derived for Utuj and N !. The higher-order unknown correlations in the resulting equations are then modeled; this type of model is called a Reynolds stress model.

 + Tref) In ventilation problems, it is often sufficient to use simpler turbu­lence models, such as eddy-viscosity models. and Ujt are then re­placed by introducing a turbulent viscosity; Eqs. (11.2) and (11.3) can be rewritten as (11.5)

(11.6)

Commonly used eddy-viscosity turbulence models are the K-e model and the K-o) model. The eddy viscosities for these models have the form

 (11.7) K2 , , , = *~e model

Il, = C..Q— K-a> model.

^^ <o

A derivation can be found in the literature.2^

If the flow is isothermal, there is no need to solve for the temperature equation (Eq. (11.6)). In this case the last term in Eq. (11.5) is also dropped. If, however, the thermal comfort is simulated, then the tempera­ture equation must be solved. In ventilation the temperature variations are normally small, which means that it is sufficient to account for density vari­ation only in the gravitation term (the last term in Eq. (11.5)). The gravita­tion term acts in the vertical direction, and in Eq. (11.5) it is assumed that the X2 coordinate is directed vertically upward. Tref denotes a reference tem­perature, which should be constant. It does not influence the predicted re­sults, except that the pressure level is changed. It could, however, affect convergence rate (i. e., increase the number of required iterations required to reach a converged solution), and it should be chosen to a reasonable value, such as the inlet temperature.

The temperature equation is derived from the energy’ equation, in which the units of each term is joules per unit volume per second, J/(m3 s) = W/m5. The temperature equation above has been divided by the specific heat Cf) and density P (assumed to be constant), and thus the units of each term in Eq.

(11.6) is °C/s. If a heat source Q is to be added in a cell, it should be divided by