Fig. 7. Paving around SW paddlewheel in Pond X.
4. Solution quality
The multiple aerator ‘real-world’ simulation required a compromise of mesh resolution, owing to the finite capacity of computer memory. The ‘real-world’
model produced less confident results than single-aerator simulations because of the reduced mesh and increased number of forcers. Before interpreting a simulation, it
must be certain that it is numerically stable and practically meaningful. Simulation quality may be quantified with four measures:
1. convergence relative difference between iterations; 2. boundary element y-plus number y
+
; 3. integrated shear stress acting on the banks and bottom net force-reaction; and
4. mesh independence. CFD modellers should ensure that convergence and y-plus criteria are satisfied
before simulation results are reported. The net force-reaction test provides a global audit of momentum in the pond. Mesh independence is established with a series of
simulations to determine if results are biased as a consequence of the discretization methodology.
4
.
1
. Con6ergence The convergence of the segregated solver is measured by the relative difference
from iteration to iteration. A relative error f
i
− f
i − 1
f
i
is assessed at each outer iteration for each field variable f
i
u,6,w,p,k,o. Notation f
i
indicates the magnitude of a particular field variable vector at iteration i. Ideally all field
variables should converge below a specified criterion, which is nominally set at 10
− 3
also expressed as 0.1 for the segregated-iterative scheme. In practice this criterion may only be achievable in the case of a single
paddlewheel where computer resources were available to intensify the density of mesh. Results for ‘real world’ deployments of several aerators produced less
convincing results. Table 3 lists alternative simulations and the relative errors of velocity and other variables actually achieved are presented in Table 4 for all of the
simulations described in the present research. Table 5 lists the computational costs associated with each simulation, by reference to the memory and time used in each
case. Note that restarts caused by system crashes derange the operation of dynamic solvers.
The various simulations of the northwest NW paddlewheel have been compared to evaluate the effects of mesh resolution. Most successful were simulations where
llayers was increased to 32, meshspacing along banks set at 1 m, and the one simulation involving higher-order quadratic elements. Strangely the low resolution
specification of 22 alayers and 16 llayers also produced a convergence approaching the desired 0.1 criteria. The nominal mesh parameters did not achieve the 0.1
criteria for pressure and turbulence, although velocities were well within the convergence criteria.
Table 3 Description of alternative simulations
Description Mesh
Deployment Example
SW paddle Nominal meshrule, with paddle in SW corner
Nominal meshrule, with paddle in shallow corner Nominal
NW paddle a22l16
22 layers around and 16 layers along jets alays32
32 layers around jets nominally 24 32 layers along jets nominally 18
llays32 Eight layers across paddlejets nominally 6
slays8 1 m spacing along banks nominally 2
meshsp1 plays16
16 layers throughout pond nominally 8 Eight layers of quadratic elements throughout pond
plays8 –Q
a
N aero Nominal
Nominal meshrule, with single propeller-aspirator all6
Nominal meshrule, with six aerators. The real pond was observed to ‘Real-world’
have this particular arrangement of two paddlewheels and four propeller-aspirators
a
Quadratic simulation, comprised of the same number of nodes as a linear simulation, having 16 pond layers, but the quadratic arrangement forms eight layers.
Table 4 Convergence achievements of simulations
a
6 w
p Deployment
k Mesh
o u
0.34 0.08
0.46 Example
b
0.68 0.46
0.62 SW paddle
0.08 0.02
0.32 0.27
0.20 NW paddle
Nominal 0.04
0.04 0.01
0.10 0.04
0.08 0.10
a22l16 0.12
alays32 0.16
0.03 0.29
0.31 0.33
0.03 llays32
0.04 0.01
0.05 0.06
0.10 0.35
0.04 0.75
0.16 0.93
0.43 slays8
0.08 meshsp1
0.08 0.01
0.06 0.07
0.10 0.72
0.42 0.53
0.58 0.69
0.53 plays16
0.06 0.01
0.09 0.06
0.07 plays8 –Q
0.06 0.30
0.19 0.29
Nominal 0.30
0.33 0.13
N aero 0.99
0.36 0.91
0.82 ‘Real-world’
0.43 all6
0.73
a
Values are percentages. Degrees of freedom are velocity u, 6, w, pressure and turbulence parameters k and o. Refer to Table 3 for description of alternative simulations.
b
The example may have converged further if given more computer time.
Simulation of a single propeller-aspirator was not fully converged, but better than the ‘real-world’ case of multiple aerators of both types, which only achieved
1 relative error between iterations. The real-world case was by far the most expensive simulation in terms of processing time CPU as well as demand for
random access memory RAM.
4
.
2
. Wall elements An ideal model would involve adaptive meshing, where wall element thickness
would be adjusted during the solution of the problem to keep the dimensionless element thickness within acceptable bounds. Unfortunately that is presently beyond
the capabilities of FIDAP and other general purpose finite element CFD codes. The simulation results have been checked to ensure that most of the bottom and bank
surfaces of the pond simulation have y
+
values within the recommended range, as shown in Table 6.
4
.
3
. Force reaction Prediction of the distribution of wall shear stress is the main purpose of the
present research. The classification scheme for reporting benthic shear stress was delineated in Peterson 1999b in terms of where cells, feed, clay, silt and sand
particles are expected to scour or accumulate. Simulations have been assessed on the basis of how closely the total integrated stress on the banks and bottom of the
pond approaches the total applied thrust of the aerators, being the only forcing agent within the pond microcosm. This is offered as a global measure of the
accuracy of the computed bed shear stress. Note that the shear stresses are
74
E .L
. Peterson
et al
. Aquacultural
Engineering
23 2000
61 –
93
Table 5 Computational cost of simulations
a
Total iterations Deployment
CPU h Mesh
RAM Mb; 64 bit Initial siter. 99 iter.
Number of restarts Later siter.
994 276
1033 2
SW paddle 688
b
156 Example
1 930
552 135
NW paddle Nominal
170 667
549 145
1013 a22l16
1 675
147 1076
alays32 614
176 180
814
b
4 674
222 llays32
212 975
3 1220
791 219
1030 765
170 5
slays8 1640
meshsp1 516
224 288
1220 4
4 2511
876 585
plays16 291
1565 358
94 plays8 –Q 179
693 1
1042 686
184 3
795 172
994 N aero
Nominal ‘Real-world’
518 all6
907 743
4687
b
1 6697
a
Refer to Table 3 for description of alternative simulations.
b
Initial 99 iterations log file lost such that CPU time is partially estimated.
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Table 6 Percentage of pond bottom and banks classified by y-plus status
a
Deployment Rather thick
Mesh Too thick
Too thin Rather thin
Thin to ideal Ideal to thick
1000–3000 Over 3000
300–1000 100–300
30–100 Under 30
SW paddle 27.7
Example 0.7
1.7 2.1
23.7 44.1
15.3 0.9
19.3 NW paddle
Nominal 37
19.4 8.1
a22l16
b
27.8 21.7
1.1 11.6
25.6 12.2
16.3 26.6
25.4 1.2
alays32
b
5.6 24.8
llays32
b
29.3 20.4
1.4 11.4
25.9 11.5
26.7 1.4
25.4 25.3
6.0 15.1
slays8
b
25.2 meshsp1
b
22.4 1.1
9.2 28.4
13.7 19.1
29.0 18.6
1.2 plays16
b
6.4 25.6
24.2 7.7
plays8 –Q
b
2.1 28.8
14.0 23.2
2.1 0.1
47.7 30.2
2.3 17.6
N aero Nominal
‘Real-world’ 3.9
all6 0.1
8.3 19.7
46.7 21.2
a
Values are percentages. Refer to Table 3 for description of alternative simulations.
b
Simulations share the same default set of meshing parameters meshrule, except for the seven simulation alternatives following the nominal NW paddle case.
computed as a post-processing operation from the derivative of the velocity field, and so these results are sensitive to numerical error.
Table 7 reports the relati6e error of magnitude, F
applied
− F
reaction
F
applied
of our simulations. Based upon the superior relative error of reaction magnitude, it is
likely that the single simulation based upon quadratic elements provides a ‘truer’ prediction than any of the linear simulations. Integrated reaction error was reduced
by 72 going from the nominal linear to the quadratic simulation, while the relative error of velocity averaged over u,6,w,p,k,o was reduced by 62.
The propeller-aspirator simulation was found to under-predict the net reaction by 71, as only 29 of the applied force appeared in the net reaction integrated across
the bottom and banks. Possible reasons for the observed discrepancy are discussed in Section 5.2.
The propeller aerator simulation used the same meshing parameters as the nominal paddlewheel simulation. Inequity of force was not improved with further
iterations, and no better convergence was obtained. This probably indicates that the solution had converged as far as it was capable, but that finer mesh resolution
would be likely to produce better results. The angle of reaction force tended to oscillate a few degrees with each iteration on alternative sides of the expected
bearing. This suggests an instability or ‘wiggle’ in the solution.
It should be noted that the multi-aerator deployment resulted in a low net force because they were oriented almost symmetrically around the pond periphery. Thus
the simple summation of components of force in the x and y directions gives a near neutral result, with a rather unpredictable resulting angle.
4
.
4
. Mesh independence The effects of mesh density variation were evaluated by conducting a series of
simulations of the NW paddlewheel where one parameter was varied from the nominal set in each simulation case. Table 4 indicated that the simulations ‘a22l16’,
‘llays32’, ‘meshsp1’, and ‘plays8 –Q’ quadratic achieved 10
− 3
convergence on all degrees of freedom, and so it is reasonable to assume that these were the ‘best’
solutions. According to Table 7 only the quadratic simulation plays8 –Q achieved a net force balance of less than 1.5. Table 8 compares these and other simulations
in terms of the predicted zones of benthic shear stress resulting from the NW paddle deployed in pond X. The quadratic elements of case ‘plays8 –Q’ were
assembled from 27-node elements with higher-order shape functions, as opposed to the eight-node linear elements. The quadratic model proved to converge faster and
yielded a more accurate force balance because it modelled fluid gradients truer to reality.
The nominal linear element case comprised 24 aerator layers alayers, 18 layers along jets llayers, six cross stream layers in a jet slayers, and eight layers
throughout the pond players. The mesh spacing parameter was nominally set so that nodes were no more than 2 m apart along shorelines. One parameter was
adjusted in each alternative linear case, except that case ‘a22l16’ slightly decreased alayers to 22 and llayers to 16.
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61 –
93
Table 7 Comparison of forcing and resulting reactions
a
Magnitude N Relative error of magnitude
Direction deg Error of angle deg
Deployment Fx N
Mesh Fy N
200 −12
SW paddle 169.6
1.6 Example
18.8 237.5
42.7 −
233.6
200 −89.0
NW paddle 90.6
0.4 5.1
Nominal −
2.1 189.7
189.7 6.0
a22l16 92.2
1.2 −
8.2 212
212.1 213.6
6.8 93.7
2.7 alays32
− 14
213.1 93.6
2.6 llays32
− 13.6
215 215.5
7.8 95.4
4.4 5.9
slays8 211.7
210.7 −
20 5.9
meshsp1 90.9
0.1 −
3.2 211.7
211.7 3.4
plays16 93.2
2.2 −
11.6 206.4
206.7 90.8
0.2 plays8 –Q
b
1.4 197.2
197.2 −
2.6
216 N aero
200
50.7 14.7
37 70.8
58.4 45.2
Nominal
141 ‘Real-world’
98.9
122 23
105 144
− 31.1
all6 118
a
The magnitude and direction in bold face indicate the net applied force of aerators. Refer to Table 3 for description of alternative simulations.
b
plays8 –Q had been the only simulation using quadratic elements.
Strangely the various higher resolution linear mesh schemes vary more from the quadratic case than the nominal case. That may be because each test case
only provided mesh refinement in one direction at a time, while the quadratic simulation provided second-order element shape functions in all directions.
The most consistently predictable zones were the extremes: where sand would be eroded and where sludge flocs settle. Almost as consistent was the prediction
of the area where silt would be scoured. The extent of the regions of silt scouring was slightly over-predicted with respect to the quadratic simulation. Fortunately
such error would provide conservative pond engineering if erosion control were to be provided in response to these simulations.
It was found that nominal mesh parameters produced reasonably indicative results when applied to form linear non-quadratic finite elements. Given a finite
capacity of random access memory on our computer, coarser mesh rules were accepted for the ‘real-world’ simulation of multiple-aerators.
5. Results
Kategori