CFD modelling pond dynamic processes
Eric L. Peterson *, Jonathan A. Harris, Lal C. Wadhwa
School of Engineering and the Cooperati6e Research Centre for Aquaculture,James Cook Uni6ersity,Towns6ille 4811, Australia
Received 11 August 1999; accepted 27 September 1999
Abstract
The parametric pond simulation methodology AUTOPOND was developed as a tool to evaluate the effect of aerators on the sediment quality in an aquaculture pond. The methodology is capable of simulating any combination of paddlewheels and propeller-aspira-tors in a single pond. Pond bathymetry is modelled with a smooth bottom and a piece-wise series of inclined banks, to generally represent any convoluted shoreline. Simulations predict that a paddlewheel imparts more circulation into a pond than a propeller-aspirator of the same motor horsepower. The propeller-aspirator concentrates an intensive effect in a localized scourhole, surrounded by relatively weak bottom stress. In contrast, paddlewheel simulations predicted a wide swath of high stress, followed by an equally extensive region of moderate stress. A simulation of the combined effect of two paddlewheels and four propeller-aspirators was produced to represent a ‘real world’ earthen mariculture pond stocked withP.monodon. Water currents and sediment conditions reported in the real pond were comparable with the results of the simulation. Given sufficient computational resources, it is now possible to investigate the interactions of pond geometry and aerator operation with a computer. Portions of pond bottoms found to experience high shear stress are now identified for reinforcement to prevent the sedimentation of sludge in otherwise productive zones. This would result in improved feed conversion and reduced ammonia levels. Ponds and aerators can now be objectively coordinated to achieve more productive culturing conditions. © 2000 Elsevier Science B.V. All rights reserved.
Keywords:Shrimp; Ponds; Aerators; Sediment; Simulation; CFD
www.elsevier.nl/locate/aqua-online
* Corresponding author. Tel.: +61-7-47814420; fax:+61-7-47751184. E-mail address:[email protected] (E.L. Peterson)
1. Introduction
This paper presents the results of computational fluid dynamics (CFD) simula-tions of the aerator driven circulation and sediment shear stress in an intensively aerated shrimp growout pond. Such physical processes strongly affect the biological productivity of a pond. Understanding and controlling these processes may help to mitigate some of the problems which plague the aquaculture industry: electrical power load; bank erosion; anoxic pond sediment conditions; wasteful feed-conver-sion; and high water exchange rates.
CFD is the generic term for numerical schemes which account for the flow of mass and momentum throughout a fluid continuum. CFD has the advantage that it does not generally require calibration, because it is derived from universal equations which govern fluid flow. To the authors’ knowledge, CFD has not been applied in research on aquaculture ponds. The method has only recently been applied to wastewater treatment ponds.
Wood et al. (1995) simulated water exchange through wastewater oxidation ponds using a commercially available CFD code. They first attempted to model the problem in two-dimensions for laminar flow, neglecting the effects of bottom friction. Wood et al. (1997) published results of simulations of three-dimensional water exchange through an effluent retention pond, using the standardk–o turbu-lence model. This simulation was validated with observations of hydraulic retention time in a real pond, using a tracer dye.
The present paper introduces a methodology to build CFD simulations of aerator driven aquaculture ponds and interprets the effect of these machines on the banks and sediments. Methods and computer source code are detailed in the thesis by Peterson (1999a), which are referred to collectively as the automatic pond simulation methodology, AUTOPOND. The methodology works with the fluid dynamics analysis package, FIDAP, to build computer simulations of aerated ponds. The methodology is capable of simulating a wide variety of pond geometries and aerator specifications, given access to a high performance computer running FIDAP. Details of FIDAP were published by FDI (1993, 1995).
Simulation results presented herein are interpreted in the context of the theory of Peterson (1999b), whereby water currents are assessed with regard to the underlying shear stress exerted in the particular pond of interest. The magnitude of the bottom shear stress is related to the sediment condition, and the pond bottom is classified into six zones on that basis. Results of simulations have been compared with the experimental data obtained by Peterson (2000) from an intensive shrimp aquacul-ture pond.
2. Mathematical model
feasible to resolve all the scales of motion operating in such an aerated pond. The practical approach is to consider only the time-averaged flow field. The product of fluid density and the six time-averaged correlations of mutually orthogonal velocity-fluctuations
r ·u%iu%j
is viewed as a stress tensor, termed theReynolds Stress. Eq. (1) is the incompressible continuity equation for the mean flow, written in vector notation, and simply states that fluid mass is neither created nor destroyed.
9 · u=0 (1)
Eq. (2) is a statement of momentum conservation for each component of the time-averaged flow, and is commonly called the Reynolds averaged Navier – Stokes (RANS) equation. The term on the far right represents the driving force field, fb.
(u · 9)u= −9p/r+9 · [(n+n
t)9u]+fb (2)
Eqs. (1) and (2) give a total of four equations for four flow variables (velocity components and pressure,p). Solution of the RANS equation is not possible unless the magnitude of eddy viscosity,nt, can be quantified throughout the domain of the
problem. The eddy viscosity is a variable relating the Reynolds stress to the gradient of the time-averaged flow. Eddy viscosity is not a constant fluid property, but varies throughout the flow field. Our pond model makes use of the standard
k-epsilon turbulence model, whereby nt=0.09(k2/o).
Ferziger and Peric (1996) as well as the ASCE’s Task Committee on Turbulence Models in Hydraulic Computations (ASCE, 1989) reviewed the standardk–omodel of eddy viscosity. Isotropic turbulent kinetic energy, k, represents the energy of turbulence (J/kg or m2
/s2), and
o represents the dissipation rate of turbulent kinetic energy (m2
/s3). The
k–o model accounts for the generation, advection, diffusion, and destruction of k ando.
Solution of the conservation equations depends on the specification of boundary conditions. No-slip conditions were applied at solid boundaries of the pond, and wall elements parameterized the effect of boundary layers (Ferziger and Peric review thelaw of the wall). Our pond model invokes the special case of hydrauli-cally rough walls, as implemented by FIDAP. Nikuradse roughness of 1 cm was applied at the banks of the pond, and the bottom roughness was taken to be 1 mm and 100mm in the peripheral and central portions of the pond, respectively. The
surface was free to flow horizontally, but with a fixed elevation.
As an important digression, we explain how we may neglect the effect of waves. According to Wetzel (1983) the maximum wave height in a lake is estimated from the fetch, Y, with wave height=0.0105 m1/2
The flow in an aquaculture pond may be forced by aerators, wind, water exchange, and natural convection (solar). Of these, only aerators were considered, as the whole purpose of the present research has been to evaluate the effect of these machines. Aerators are intended to supply dissolved oxygen and strip carbon dioxide, but they also impart momentum as required to circulate and turning over the water column. The present research has focussed on understanding the thrust effect of paddlewheel and propeller-aspirator aerators. Our approach was to impose the propulsive thrust of each machine as a body force, uniform within the small parcel of water swept by blades. The body force is applied in Eq. (2) with the fb
term. Stern et al. (1991) used a similar approach to model the propeller of a ship. Airborne droplets, waves and submerged bubbles produced by aerators are not detailed in our model, since the total momentum of these effects is lumped in the specification of propulsive thrust ascribed to each individual aeration machine.
The discretised forms of the governing equations were solved using the finite element method as implemented in FIDAP (FDI, 1993, 1995). RANS and k– ep-silon equations must be linearized each iteration, based upon successive estimates of field variables (u,6,w,p,k,o). Gaussian elimination would simply invert the global equation system at each step, but this was not possible due to the enormous size of our problem. We adopted the segregated-iterative solver reviewed by Haroutunian et al. (1993) because it was the only feasible approach to handling large CFD problems. In this approach, the equations for each field variable are solved sequentially in an outer iteration, and an iterative method is used to solve the set of linearized equations as they appear at each sub-step. The process could be repeated endlessly, but should be programmed to conclude when certain conver-gence criteria are satisfied, or stopped if the exercise is found to be futile.
3. Methods
The simulation of pond hydrodynamics involves a number of steps. First the geometry of the pond and layout of aerators must be specified and represented by a mesh of finite elements. Then governing equations and boundary conditions must be specified. Finally the system of equations must be numerically solved and then results processed to yield useful information.
The AUTOPOND methodology provides a concentration of computational mesh around jets emanating from different types of aerators, and three-dimensionally transitions these into a sparse mesh filling the vast majority of the pond bathyme-try. This is a crucial innovation. The high accelerations and steep gradients in the near-field around an aerator necessitate a fine-scaled mesh as small as 1 cm3.
One-hectare of pond averaging 1 m deep has a volume of 1010 cm3. Since
computation cost increases faster than the square of the number of elements, non-transitional meshing would create unmanageable matrices. Transitional mesh-ing brmesh-ings the number of computational elements in the pond problem down to a fraction of 106. By exploiting the segregated solver provided with FIDAP the pond
problem may be simulated in 64 bit precision within one gigabyte of random access memory (RAM).
3.1.Pond modelling data structure
Given bathymetric survey data and the aerator deployment specification for a particular pond, AUTOPOND will generate a governing command script and a series of FIDAP command files which contain all of the instructions necessary to sequentially build a CFD model of an aerated pond. Table 1 outlines the default values of AUTOPOND parameters. Results also depend upon the specification of pond geometry and aerator deployment.
3.1.1. Pond geometry
Pond geometry is specified with the Cartesian coordinates (x,y,z) of the bottom and banks. At least six survey data files are required to describe pond boundaries (for example surface, bottom, north, east, south, and west). Additional bank-seg-ments may be used to describe a convoluted shoreline in piece-wise fashion. We then apply a linear regression analysis to the bottom and each bank. The result is an array of best fit equations in the form of expression 3.
z=C1+xC2+yC3 (3)
The standard estimate of error is used to report the goodness of fit (m height) with Eq. (4).
SEE=
'
%(z−f(x,y))/(n−2) (4)AUTOPOND reports on the bathymetric analysis, and then determines lines of intersection between adjoining bank-segments and the bottom. These lines defini-tively bound the model.
3.1.2. Aerator deployment
The coordinate variablesxandy (m) specify the point where the aero2 shaft enters the water, while they represent the central gearbox of the four-rotor paddlewheel. The variableorientation is the horizontal angle of aerator jet. The variableplunge
specifies the downward angle (degrees) of an aero2 and the depth (m) of scooping of a paddlewheel.
3.2.Discretisation of the problem
The problem must be properly meshed to obtain a meaningful solution. This was achieved by investing most computational resources within the jet(s) produced by an aerator. A scheme was developed to gradually transition the mesh into the
Table 1
Default values of parameters defined inmeshruleandpropertyfiles Value Description
Parameter meshrule
MESHSPACING 2.0 m per element
Elements per corner 8
CORNEREDGE
MINMESHSPACING 0.5 m per element 18
THRESHOLD m (aerator’s radius of influence)
Fraction of bank (end length to remain unsplit) 0.15
MARGINS
8
PLAYERS Pond layers (water column)
24
ALAYERS Aerator layers (orbiting macro) 6
SLAYERS Streamwise layers (at outlet of propeller) Lengthwise layers (alongside of jets)
LLAYERS 18
Bank layers (paver guidance at boundary) 2
BANKLAYERS
BANKFIRSTLENGTH 0.125 m (first layer out from bank) BANKSECONDLENGTH 0.250 m (thickness of next layer)
1.5 m (further layers) BANKGROWTH
0.1
HYPOLIMLENGTH m (lower water column thickness) property
SEDDENSITY 1075.0 kg/m3
Table 2
Default values foraero2andpaddle4parameter file Parameter Nominal value Description
paddle4 aero2
Number of propellers or wheels
ROTORS 1 4
m, shaft length from water to propeller or spacing between 1.667
1.0 SHAFTL
paddle wheels
m, outside diameter of propeller or paddlewheel blades 0.650
DIAMETER 0.140
m, inside diameter of propeller or paddlewheel blades ANNULAR 0.040 0.350
m, axial length of propeller or width of paddlewheel blades 0.21
AXIAL 0.025
2.042
0.440 m, downstream displacement per revolution PITCH
4 Number of motor poles MPOLES
50.0 50.0 Revolutions per second (Hz) HZ –NOMINAL
1.9 kW, electrical power requirementa
1.9 KWE –
NOMINAL 2.0
BHP – 2.0 hp, mechanical power requirementa
NOMINAL
200.0
THRUST – 200.0 Newtons of force NOMINAL
rms (velocity)/mean (velocity) 0.2
aVariable not used by the present version of AUTOPOND.
surrounding water column. It was also necessary to control the element thickness along the banks and bottom to meet the requirements of the wall functions. Furthermore, the strategy was developed to accommodate an arbitrary number and arrangement of aerators within a single pond of any geometric shape.
Hexahedron brick elements were used, each having front, back, left, right, top, and bottom faces. Fig. 1 illustrates a sectional profile through the depth of a pond, schematically discretised with four layers of finite elements. The elements adjoining banks and bottom must be structured so that the first node within the fluid continuum is situated within the logarithmic portion of the turbulent boundary layer. The height of these special wall elements must be adapted so that their dimensionless thickness,y+
=u* · Ln/nis between 30 and 1000. Ferziger and Peric, as well as FDI, warn that serious errors will result if the first computational node above the wall has a dimensionless thickness less than 30. The height of the finite element measured normal to the wall surface isLn=y
+
Fig. 2 shows the schematic approach to three-dimensional transitioning with brick elements in the vicinity of an aerator. The aerator’s rotor is represented by the small zone of dense mesh in the centre of the illustration. Some number (players) of mesh layers surrounds the aerator, and then transitions in the local region of the aerator. The small packet of fluid swept by the blades of the aerator is assumed to have a force of acceleration equal to the propulsive thrust divided by volume and divided by density, represented by body force vectors. The aerator is orbited by a succession of finite elements enveloping the aerator (alayers), like the layers of an onion. The aerator is also dissected by slayersand llayerslayers of finite elements in the streamwise and lateral directions.
Meshing variables are controlled in each run of AUTOPOND with themeshrule
file detailed in Table 1. Parametric mesh generation ensures that the network of
Fig. 1. Diagrammatic section through pond mesh.
Fig. 3. Long section of meshing scheme for a paddlewheel aerator.
Fig. 4. Cross section of meshing scheme for a paddlewheel aerator.
mesh is coordinated so that the entire pond volume is filled with finite elements, without gaps or discontinuities.
Fig. 3 presents a long-section through the paddlewheel mesh. This is an applica-tion of the conceptual scheme of Fig. 2. Both illustraapplica-tions indicate a concentraapplica-tion of force in the region of water swept by rotors, but the latter provides for the gradual expansion of the jet at a ratio of 1:7 (diagram not to scale), while the transition at the inlet is abrupt. The arrows on Fig. 3 were drawn along mesh elements, but they are also thought to be similar to streamlines in the vicinity of the paddlewheels. The paddles are repeated four times as illustrated in Fig. 4. The
paddle4 macro is detailed in Peterson (1999a).
Pa6ing provides a transitional layer of brick elements adjoining a surface of any shape, with any number of voids. Paving is evoked on the bottom of our pond model, by going around each aerator macro block as illustrated in Fig. 7. Brick elements were then mapped upwards through the water column players times, transitioning in conformance with banks.
Fig. 5. Long section of meshing scheme for a propeller-aspirator aerator.
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.
4.1.Con6ergence
The convergence of the segregated solver is measured by the relative difference from iteration to iteration. A relative error fi−fi−1/fi is assessed at each outer iteration for each field variable fi (u,6,w,p,k,o). Notation fi 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 22alayersand 16llayersalso 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.
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 –Qa
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
aQuadratic simulation, comprised of the same number of nodes as a linear simulation, having 16
Table 4
Convergence achievements of simulationsa
6 w p
Deployment Mesh u k o
0.34 0.08 0.46
Exampleb 0.46 0.68 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
aValues are percentages. Degrees of freedom are velocity (u,6,w), pressure and turbulence parameters
(kando). Refer to Table 3 for description of alternative simulations.
bTheexamplemay 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 havey+ values within the recommended range, as shown in Table 6.
4.3.Force reaction
E
Computational cost of simulationsa
Total iterations
Deployment Mesh RAM (Mb; 64 bit) Initial s/iter. (99 iter.) Number of restarts Later s/iter. CPU h
994 276 1033
2 SW paddle Example 156 688b
1 930 552 135
NW paddle Nominal 170 667
549 145 1013
a22l16 147 675 1
1076
alays32 180 814b 4 614 176
674 222 llays32 212 975 3 1220
791 219
meshsp1 288 1220 4 516 224
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’ all6 743 4687b 1 6697 518 907
aRefer to Table 3 for description of alternative simulations.
75
Percentage of pond bottom and banks classified byy-plus statusa
Deployment Mesh Too thin Rather thin Thin to ideal Ideal to thick Rather thick Too thick 1000–3000 Over 3000 300–1000
100–300 30–100
Under 30
SW paddle Example 1.7 2.1 23.7 44.1 27.7 0.7
15.3 0.9 19.3
NW paddle Nominal 8.1 19.4 37
a22l16b 11.6 25.6 12.2 27.8 21.7 1.1
16.3 26.6 25.4 1.2 alays32b 5.6 24.8
llays32b 11.4 25.9 11.5 29.3 20.4 1.4
26.7 1.4
meshsp1b 9.2 28.4 13.7 22.4 1.1
19.1 29.0 18.6 1.2
‘Real-world’ all6 8.3 19.7 46.7 21.2 3.9 0.1
aValues are percentages. Refer to Table 3 for description of alternative simulations.
bSimulations share the same default set of meshing parameters (meshrule), except for the seven simulation alternatives following the nominal NW paddle
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, Fapplied−Freaction/Fapplied 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 overu,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 thexandydirections 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
77
Comparison of forcing and resulting reactions
Magnitude (N) Relative error of magnitude (%) Direction (deg) Error of angle (deg) Deployment Mesh Fx(N) Fy(N)
200 −12
SW paddle
169.6 1.6 Example −233.6 42.7 237.5 18.8
200 −89.0
NW paddle
90.6 0.4 5.1
Nominal −2.1 189.7 189.7
6.0
a22l16 −8.2 212 212.1 92.2 1.2 213.6 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 −20 210.7 211.7
5.9
meshsp1 −3.2 211.7 211.7 90.9 0.1 3.4
plays16 −11.6 206.4 206.7 93.2 2.2 90.8 0.2
aThe magnitude and direction in bold face indicate the net applied force of aerators. Refer to Table 3 for description of alternative simulations. b
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
Simulation results of individual aerators are presented in Figs. 8 – 11, where the plotted zones of benthic shear stress provide a measure of pond condition as hypothesized by Peterson (1999b). The plot axes have units of metres in the north and west directions, respectively, while contours represent the magnitude of benthic shear stress in N/m2, tangential to bottom and bank surfaces. Wall
functions were disrupted within those finite elements which adjoin both the bottom and a bank, and so these conflicts resulted in the appearance of certain zigzag contours.
Table 8
Bottom stress predictions for NW paddlewheel in Pond Xa
Clay
meshsp1 13.1 20.1 8.1 3.6
2.7
Fig. 8. Single paddlewheel deployed in SW corner of Pond X.
5.1.Paddlewheel
Alternative simulations of a paddlewheel deployed in the southwest (SW) and northwest (NW) corner of the pond are illustrated in Figs. 8 and 9, respectively. A smaller zone of sand scour (greater than 0.1 N/m2) was found in the SW corner
deployment than resulted from the NW corner deployment. The SW deployment caused the aerator jet to travel somewhat inclined offshore towards the middle of the next bank, with only 1.7% of the pond bottom experiencing enough stress to mobilize sand, whereas the NW deployment would cause sand to be scoured from 2.9% of the bottom area.
Both cases demonstrate that a paddlewheel effectively transmits momentum into the water column, causing a swath of excessive stress across the pond, followed by reasonably good circulation around the pond periphery. Between 55 and 60% of the pond bottom would experience shear stress in thecells-feed-clay range of 0.001 – 0.03 N/m2, which is thought to be preferable to higher and lower stress conditions.
5.2.Propeller-aspirator
Fig. 10 illustrates the shear stress distribution resulting from a propeller-aspira-tor. Stress is very intense within the impingement crater, and much weaker in the surrounding region. A close-up of the region around the impingement crater is detailed in Fig. 11, while the pond wide picture illustrates how minimal this disturbance is. The patch of sand (t\0.1 N/m2) corresponds to the conditions observed in the real pond, where there is a localized scour-hole. A single propeller-aspirator appears to be unable to cause sufficient circulation in a 1-ha pond, as the simulation predicts that most of the bottom would be dead (less than 0.001 N/m2),
without the benefit of forced convection.
It is presumed that high accelerations and gradients in this region were responsi-ble for numerical noise which prevented force-equilibrium. Benthic shear stress in
Fig. 10. Single propeller-aspirator deployed at north end of Pond X.
the pond would need to average 0.008 N/m2higher than the simulation predicts to
balance the applied aerator thrust. It is not known if the unaccounted force actually presents itself near the aerator or over a wide region of the pond benthos. Three-dimensional curvature of the scour crater has been neglected, exaggerating the deceleration at impact. Numerical instabilities might not be so severe if the jet were modelled within the curvature imposed by the real crater.
5.3.Real-world simulation
5.3.1. Circulation
Figs. 12 and 13 compare the simulated speed contours with observation vectors at a depth of 150 mm below the surface and 200 mm above the bottom, respectively. The simulation predicted velocities at over 500 000 computational nodes on an irregular grid, while there were only 68 observation points. The nodes and measurements points were not coincident, and so the two data sets have been interpolated onto the same rectangular grid. Note that the ragged contours along the shorelines are artifacts of this interpolation. Vectors from the simulation were sampled every 10 m (marked as thin gray lines) to enable comparison with observation vectors (bolded black).
The layout of survey stakes in Pond X is shown in Table 9 and the pond surface circulation observations and simulation are compared in Table 10. Speed dis-crepancy and directional differences indicate a conglomeration of many factors, including simulation and experimental errors, and possibly some transient events. The discrepancy between surface speed observations and simulation results cer-tainly include turbulence and wind effects. The simulation is not necessarily invalid, as the theoretical basis is the long time Reynolds-averaged flow field. A linear regression analysis of all surface data gives a correlation coefficient of 0.1049, as
Fig. 12. Near surface speed simulation contours and observation vectors.
plotted on Fig. 14. The poor correlation indicates a very weak signal to noise ratio. Partitioning of the zero-wind data gives a correlation coefficient of 0.5228 with a slope of 0.7949, which suggests that the simulation slightly underpredicts actual surface speed.
Comparisons of simulation direction and all observation data are plotted in Fig. 15. Flotsam was visually observed and subjectively averaged over a period longer than wind events. The simulation results provide a reasonable comparison with the observations.
The lower water column circulation speed and direction are given in Table 11, comparing the results of the simulation with speed measurements. Data have not been partitioned according to the nil-wind instances noted in Table 10. Near bottom speed correlation of simulations and observations is shown in Fig. 16. The speed observation data presented here are generally accepted with greater confi-dence than was the case with the near surface data. This is due to the fact that the bottom of the pond is protected by inertia from the transient breezes which first impact upon the surface.
5.3.2. Sediment condition
Fig. 18 shows the benthic shear stress predicted by the simulation of a ‘real-world’ deployment of two paddlewheel and four propeller-aspirator aerators in Pond X. The simulation results were interpolated onto the 20×20 m grid described in Peterson (2000), so that sediment and shear stress magnitude are sampled at the same resolution. The interpolation was the result of kriging from the thousands of simulation nodes surrounding each stake. The interpolation of simulated shear stress was then paired with observations of sediment conditions at the correspond-ing samplcorrespond-ing locations. Foremost of interest is the mean size of sediment deter-mined by laser diffraction. Assuming all samples were principally silica allows plotting on the Shields curve given in Fig. 19. This clearly illustrates that most locations were below the Shields curve, and therefore subjected to the process of sedimentation.
Calculated shear stress provides an important measure of the sediment condition. Particle size and stress together form a complete picture which illustrates the suspended sediment transport in the pond microcosm. In each case, the conjugate of particle size and shear stress plotted on or below the Shields curve of incipient motion. Note that conditions plotting above the curve would be expected for water column suspended solids, not core samples.
Table 9
Layout of survey stakes in Pond Xa
aDimensions are schematic distances from the southwest corner (upper left above), northward along
the west bank (to the right across the top above) and easterly on the south bank (down on the left). Refer to Peterson (2000) for a detailed survey of Pond X.
It is believed that the principal source of suspended solids were the banks and patches of the bottom directly impinged upon by aerator jets, of which stake c3 was the only representative sample. Stake c2 did not exactly plot as a scour site because it had completely lost all material other than cohesive clay, and it was slightly missed by the simulated jet streamlines. Stake c18 is the best representa-tive of the central sedimentation dead-spot, where anaerobic flocs had accumulated in a deep drift. Other sampling locations were distributed at varying distances below the Shields curve, while they experienced similarly varying rates of sedimen-tation, as documented in Peterson (1999a).
In spite of suspended sediment fallout interactions, there was a positive trend between applied shear stress and sandiness. The converse is true for the content of silt and organic carbon, which appear to be inversely related to shear stress. Only silt demonstrated a correlation coefficient better than 0.25 with respect to benthic shear stress, as plotted in Fig. 20.
Table 10
Near surface observations compared with the simulationa
Difference Stake Observed Simulation Discrepancy Direction Simulation
(°) West of 7 0.095 0.088
0.013 −0.261
aMany observations were taken about 5 m away from the fixed survey stakes. These are denoted with
a prefix indicating the direction of offset, i.e. ‘South of 1’.
Fig. 14. Surface speeds correlation of simulation with observations.
6. Discussion
It has been found that the multi-aerator simulation cannot be expected to converge much better than 1% maximum relative error of velocity. The inability to converge may be associated with chaotic interactions between so many jets of high velocity water. Certainly convergence would have improved if the simulation mesh had not been restricted by finite computer resources. In a single paddlewheel simulation it was found that 27-node quadratic elements provide faster and better
Table 11
Near bottom observations compared with the simulationa
Simulation North of 8 0.091 0.022
0.010 West of 14 0.093 0.088
0.165
0.123 0.041 172 174 2
14
19 0.101 0.154 0.053 164 170 7
177 170 −6
aMany observations were taken about 5 m away from the fixed survey stakes. These are denoted with
Fig. 16. Near bottom speed correlation of simulations and observations.
convergence, and apparently more accurate results with nearly the same memory usage as the nominal simulation scheme.
In spite of imperfect convergence the nominal linear brick meshing scheme resulted in simulated velocities comparable to those observed 200 mm above the bottom of Pond X. This largely verifies the vertical gradient at the bottom, which is essential to the estimation of benthic shear stress.
The ‘real-world’ simulation of shear stress and the observed particle size form a sensible pattern when conjugate-pairs are plotted on the non-dimensional Shields diagram. Material sampled from locations known to experience scouring plot directly on the incipient motion curve, while samples from sites subject to sediment deposition plot far below the Shields curve. The shear stress and particle size from the sampling location closest to the centre of the pond quantify a state of sediment condition which is further below the curve than any other place in the pond. The deepest drifts of anaerobic sludge flocs were observed to form at the same location with each production cycle of the pond. Sedimentation was generally greater at sites where the benthic shear stress and particle size plot farther below the Shields curve. It is expected that an advection – diffusion analysis would confirm the spatial variability detected with the sediment traps described in Peterson (1999a), although that is beyond the scope of the present research.
It has now been demonstrated that the extent of silt and sand scouring may be determined with more confidence than lower stress zones. Portions of pond bottom found to experience such high shear stress should be physically armored to prevent scouring of silt and sand. Scouring of these materials would otherwise be trans-ported via a ‘conveyor-belt’ process, to fall out on surfaces experiencing less shear stress. Sedimentation forms an anaerobic sludge as organic matter is trapped in soil interstices.
7. Conclusion
Taken together, the benthic shear stress and size of sediment found at a particular location describe the local state of the sediment – water interface. Shear stress alone acts as an indicator of likely conditions, but there is some uncertainty about sediment quality without information about mineral deposition from the water column above. Since the stated objective is to avoid scouring mineral soil, then fallout would be less significant in a well managed pond, and so benthic shear stress would act as the singular governor of sediment condition. This could be termed the ‘well managed pond hypothesis’.
Fig. 18. Sediment shear stress resulting from all of the aerators in Pond X.
Given sufficient computational resources, it is now possible to parametrically investigate the interactions of pond geometry, aerator design, and management. Within the limitations of the present research there are some interesting results. Propeller-aspirators spend most of their thrust effect at the point where they impact into the bottom of a pond. It is suggested that a diffuser should be fitted onto these machines. The wide area of paddlewheel induced scour on banks demands armor-ing with stone or geotextiles. Patterns of sediment conditions suggest that aerator usage should be reduced during daylight hours when photosynthesis delivers sufficient oxygen. The immediate outcome would be reduced benthic shear stress, so that feed pellets would not be swept into the pond centre and covered by siltation.
Acknowledgements
Fig. 19. Shields curve with sample-simulation pairs plotted as points.
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