PARTICLE-LADEN FLOW - ERCOFTAC SERIES Phần 3 ppsx

Heat and water vapor transport 77

3 Numerical model implementation

The model proposed here makes use of the transient heat conduction equa￾tion (9) and the general gamma distribution (16). The one-dimensional flow

equation can be solved by the implicit finite difference method as discussed

below in section 3.1, whereas the generation of random deviates for arrival

times and approach distances is described in section 3.2.

3.1 Finite difference model setup

The one-dimensional transient system of equation (9) with boundary con￾ditions (10) can easily be written in finite differences [15]. In this case the

implicit system of equations can be written as a tridiagonal matrix equation

that is easily and quickly solved with the Thomas algorithm.

In order to make the model as realistic as possible, a grid was used consist￾ing of 1000 elements with properties as summarized in Table 1. A temperature

solution array of 1000 elements is produced at each time step and an average

H was determined for each set of parameters after each simulation run.

3.2 Gamma distribution

A basic procedure to generate random deviates with a gamma distribution

is given by [16]. However, since this procedure assumes that β = 1, a more

general procedure gamdev(α, β) was developed which returns random deviates

as a function of both α and β. The average approach distance hp,avg was used

instead of β as a parameter during the simulation runs. Parameter β is then

calculated as hp,avg/α.

The Brutsaert model [1] can be implemented by drawing the arrival rates

with gamdev(1, 1) and resetting the entire temperature T array to the lower

temperature upon arrival of the eddies. The more general procedure consists

of drawing arrival rates as in the Brutsaert model with gamdev(1, 1) and

then, after selecting hp,avg and α, drawing an approach distance hp with

gamdev(α, β).

4 Results

Table 1 below shows the basic parameter set with their chosen values. The

parameters such as κ, ρ, Cp and ν depend to a minor extent on temperature.

However, this has been ignored in the simulations. The following parameters

were varied during the simulations: the friction velocity u∗, the average ap￾proach distance hp,avg and the gamma distribution parameter α. In section

4.1 the model validation against the analytical Brutsaert solution is briefly

described, after which the simulations with variable approach distance are

summarized in section 4.2.

78 A.S.M. Gieske

Table 1: Model parameters with their selected values.

4.1 Model validation with the Brutsaert analytical model

As mentioned before, the Brutsaert model [1] can be implemented by drawing

the arrival rates with gamdev(1, 1) and resetting all temperature values to

the constant air temperature at z = L immediately after arrival of the eddies.

This offers the opportunity to validate the stochastic numerical model against

the analytical solution of the simple case with approach distance zero. The

analytical solution is given by (13) with the renewal rate s given by (15). This

renewal rate s depends mainly on the friction velocity u∗ because z0 is taken

as a constant equal to 0.001 m. Figure 3 below shows the roughness Stanton

number Stk as a function of the surface roughness number Re∗ with a range

from 6 to 200, corresponding to a range in friction velocity from 0.1 to 2 ms−1.

It is clear that the stochastic numerical model results compare well with the

analytical approach by Brutsaert [1, 3, 4]. It should be noted that both the

analytical and numerical model make use of relation (15) with constant C2

having a value of 4.84 based on reported experimental values [1].

4.2 Model simulations with variable approach distances

In addition to varying Re∗ as in Figure 3, α and hp,avg were also changed

systematically. Parameter α was given the values 1, 2, 4, 9, 16. Increase in

α means a decrease in the variance of the gamma distribution. The average

approach distance was assigned the values 0.0001 m, 0.0002 m, 0.0005 m,

0.0010 m, 0.0015 m, 0.0020 m and 0.0040 m and finally, the gamma distribu￾tion parameter β was calculated as hp,avg/α.

Some results are illustrated in Figures 4 and 5 below. Figure 4 shows the

simulation results for the inverse roughness Stanton number St−1

k as a function

of the approach distance at Re∗ = 13.34 (u∗ = 0.2 ms−1). The curves show

a marked increase in the St−1

k value when the approach distances become

larger. The curves also indicate that the heat transfer coefficient does not

depend strongly on α, especially at low values of the approach distance hp.

Heat and water vapor transport 79

Fig. 3. Inverse Stanton number (St−1) as a function of surface roughness (Re∗) for

both the numerical model simulation and the analytical solution by Brutsaert [1].

This seems to be the case for all values of u∗. Because it appears that changes

in α only have a minor influence on the heat transfer coefficients, a value of

α = 1 is chosen to show the general response of Stk to Re∗ and hp.

Fig. 4. Simulation results for the inverse roughness Stanton number St−1

k as a

function of the approach distance (thickness interfacial boundary layer) at u∗ =

0.2 ms−1 (Re∗ = 13.3).

Figure 5 shows the simulation results for St−1

k versus Re∗ for α = 1.

The curves show that a strong decrease of the heat transfer coefficient St

(increase in St−1) occurs with larger approach distance. All variations show

80 A.S.M. Gieske

Fig. 5. The figure shows the inverse roughness Stanton number as a function of Re∗

for several model approach distances. The shaded bar indicates the range of reported

experimental results, for simplicity only shown at Re∗ = 10 [1, 3, 10, 20, 21, 22]. The

solid line shows the results obtained with the Brutsaert analytical model (Equation

20). The black square indicates the offset from the Brutsaert line resulting from the

analysis by Trombetti et al. [10].

a decrease from the simple Brutsaert model with hp = 0 (section 4.1). The

reported experimental/theoretical results are shown in figure 5 where the solid

line shows the results obtained with the Brutsaert analytical model (as in

Fig. 3) while the shaded rectangle indicates the range of reported results.

These have been indicated for simplicity at Re∗ = 10 only. The wide range of

results appears to be caused partly by the nature of the different experiments,

partly by the different definitions and conventions with regard to the Stanton

numbers B, Stk and the drag coefficient Cd (relations 2, 3, 4 and 5). The most

important reviews were made by [1, 3, 10, 20, 21, 22].

It appears that the stagnant interfacial layer thickness (as modeled here

with the approach distance) may perhaps explain the variability in reported

experimental results. The stagnant layer thickness would then be related to

the type of surface roughness used in these experiments. Inspection of Fig. 5

suggests that the approach distance lies on average between 0.0002 and 0.0005

m based on the experimental evidence. The Brutsaert model [1] is

St−1 = 7.3 Re1/4 ∗ P r1/2 (Brutsaert) (17)

where the constant 7.3 is mainly based on the experiments reported by [1].

However, the value of the constant is probably as high as 9.3 based on the

review by [10] and therefore it is suggested to adapt relation (20) to the

following relation which is also more in accordance with [22]