The increasing frequency and severity of mass bleaching events in the past few decades has raised considerable concerns. Here we report on the numerical simulations of Low Reynolds turbulence models coral microenvironments to determine surface temperature rise in reef corals and test whether our model is capable to estimate of the extent of warming likely to be en-countered in the nature during calm conditions. The Computational Fluid Dynamics (CFD) simulation uses the OpenFOAM CFD libraries to implement a steady-state turbulent flow porous medium model, with heat transfer accounted for using a transport equation for temperature. We validated the model using controlled laboratory experiment observations

Caley, M.

Cooper, T.

King, Andrew

Mullins, Benjamin

Ong, R.

English

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18th Australasian Fluid Mechanics ConferenceLaunceston, Australia3-7 December 2012CFD Simulation of Low Reynolds-number Turbulence Models in Coral ThermalMicroenvironmentR. H. Ong1, A. J. C. King1, B. J. Mullins1,2, M. Julian. Caley3, and T. F. Cooper41Fluid Dynamics Research GroupCurtin University, Perth, Western Australia 6102, Australia2Curtin Health Innovation Research InstituteCurtin University, Perth, Western Australia 6102, Australia3Australian Institute of Marine ScienceTownsville, Queensland 4810, Australia4Australian Institute of Marine Science, UWA Oceans InstitutePerth, Western Australia 6009, AustraliaAbstractThe increasing frequency and severity of mass bleaching eventsin the past few decades has raised considerable concerns. Herewe report on the numerical simulations of Low Reynolds tur-bulence models coral microenvironments to determine surfacetemperature rise in reef corals and test whether our model iscapable to estimate of the extent of warming likely to be en-countered in the nature during calm conditions. The Computa-tional Fluid Dynamics (CFD) simulation uses the OPENFOAMCFD libraries to implement a steady-state turbulent flow porousmedium model, with heat transfer accounted for using a trans-port equation for temperature. We validated the model usingcontrolled laboratory experiment observations.IntroductionCorals are one of the most ecologically and economically im-portant habitat-forming species. However, they can bleachinresponse to stressful thermal conditions. The increasing fre-quency and severity of bleaching over the past few decades hasraised considerable concern for the long-term viability ofc ralreefs. The combination of both elevated sea-surface tempera-ture and solar irradiance is widely acknowledged to be the mostimportant trigger for coral bleaching. Other environmental fac-tors though such as low flow, low salinity and low water turbid-ity can contribute to the bleaching responses by corals. Theseparameters vary considerably on coral species and morpholo-gies, which can potentially determine the extent and variationof the fluid flow and mass and heat transfer in corals, and ulti-mately their susceptibility to bleaching.The study of coral bleaching has been dominated by laboratoryand field experiments, in-situ observations, and predictive mod-elling based on remotely sensed sea conditions. However, thereasons corals bleach are not fully understood. Laboratorys ud-ies can be confounded by sensitivity of most coral species tolaboratory conditions and the difficulties inherent in accuratelysimulating the complexity and variability of oceanic conditions,while, field-based approaches are typically costly and are diffi-cult to generalise from because of uncontrolled natural variationthat may be relevant only to specific sites. Numerical methodson the other hand provide a powerful, holistic approach to solvephysics associated with coral bleaching including a wide rangeof scales of fluid motion, heat transfer, and the morphology ofthe corals.Conventionally, bleaching has been assessed on the basis ofsea surface temperature (SST) data, however, it is the deviationof surface coral temperature from ambient SST, called thermalstress, that is most likely to trigger and determine the severity ofbleaching. Previously, we established the potential of CFDinpredicting the thermal stress imposed on corals by their thermalmicroenvironments under laminar flow conditions [1]. How-ever, in the overwhelming majority of cases, the external flowstream carries with it a certain degree of turbulence. In essence,the laminar boundary layers are relatively easy to measure btin turbulent although there exists a very thin layer next to thewall called laminar or viscous sublayer, its thickness is somallthat it is almost impossible or very difficult to observe underexperimental conditions. Here we validate our computationlpredictions for the surface warming of corals that varied inpig-mentations and validate our findings with the experimental ob-servations [2].Thermal regions in coralHeat transfer within and around corals is affected by three mainmechanisms: (1) the rate of incident radiation absorbed by theexposed tissue surface, (2) the rate of heat loss due to convec-tion from the tissue and skeleton into the surrounding water, nd(3) the rate of heat conduction from the tissue into the skeleton.The model presented here considers all of these mechanisms,with the tissue and skeleton layers each possessing slightly dis-tinctive thermal properties. Although here we assumed thattheliving tissue is almost impermeable, however, in many cases,the activity of borers and coral grazers will result in openingsallowing percolation and increased skeletal porosity.Model validationThe experimental study of Fabriciuset al. (2006) [2] was usedto validate our turbulent CFD model. They conducted an exper-iment which exposed a range of different coral darkness (hemi-spherical coral (Favia matthaii), ≃90 mm) to direct sunlightranging from high irradiance (1,500-1,600µmol m−2 s−1 or∼800 W m−2) to low irradiance (120-350µmol m−2 s−1 or∼150 W m−2) (Figure 1). Pigmentation was measured as back-ground fluorescence,F0, which here defined as a proxy of alight-absorptivity coefficient whereF0=100 as near white andF0=600 as dark brown. Measurements were taken outdoors inflow chambers (consisting of four 15-cm-deep and 45-cm-longworking sections) creating an unidirectional flow with a steadyinlet flow maintained at 0.01, 0.02, and 0.05 m s−1 (only thelatter flow speed was investigated since we are interested inval-idating the turbulence models,Re≃ 4600), and a temperature of29.3◦C. Coral surface warming was given as the difference be-tween coral surface temperature and ambient water temperaturafter a period of exposure to solar irradiance.Figure 1: Model set-up and boundary conditions for the simu-lationMethodGoverning equationsThe mathematical model for steady, turbulent incompressibleflow consists of the Reynolds average Navier-Stokes and energyequations. The continuity equation is given by∂ū j∂x j= 0 (1)where bar denotes a resolved or mean quantity. The averagedmomentum equation is∂∂x j(ū j ūi)=−∂∂xi(p̄ρ)+1ρ∂∂x j(τi j + τti j)− S̄i (2)whereSi is the flow sink term which can be divided into vis-cous and inertia effects [1]. The above equation is known as theReynolds-Average Navier Stokes equations (RANS), whereτti jis the Reynolds-stress tensor andτi j is the mean laminar viscousstress and is given byτi j = µ[(∂ūi∂x j+∂ū j∂xi)−23(∂ūk∂xk)δi j](3)whereδi j is unit tensor and the second term contains 2/3 multi-plied by the divergence of velocity term is zero. The Reynolds-stress tensor represents the transfer of momentum due to turbu-lent fluctuations which is given byτti j = 2µT Si j −23ρkδi j (4)whereSi j is the mean strain-rate tensor andk is turbulent kineticenergy which is given ask=12ūi′ūi′=12(ū′2+ v̄′2+ w̄′2)(5)The degree of turbulence can be adequately described using theintensity of turbulence,I , and the scale of turbulence,l . Theenergy equation is given by∂∂x j(T̄ū j)=∂∂xk(ΓTe f f∂T̄∂xk)+αq̇ρCp+ Φ̄ (6)whereΓTe f f is the effective thermal diffusivity( ν0Pr +νtPrt), α isthe tissue absorptivity, ˙q is the heat source due to irradiance,andΦ is turbulent heat dissipation fluxes[µ2(∂ūi∂x j+∂u′i∂x j+∂ūj∂xi+∂u′j∂xi)2], which has an effect on the temperature rise only forhigh Reynolds number cases.Setting gridThe models are able to resolve the viscous layer numericallyby using very fine grids and viscous effects are included inthe model by including ’near wall damping’ terms. The low-Reynolds turbulence models are able to resolve the viscouslayer numerically by using very fine grids and viscous effectsare included in the model by including ’near wall damping’terms. Typically, this requires ensuring the near-wall celatnon-dimensional distance ofy+ ≤ 1.y+ =y·u⋆νu⋆ =√τwρ(7)τw =Cf ρU022Cf =0.0791Re−0.3(8)wherey is the actual distance from the wall,u⋆ is the frictionvelocity, andτw is the wall shear stress.Boundary conditionsTo adequately accommodate the flow conditions correspond-ing to the experiment, it was necessary to specify free streamboundary conditions. We estimated the turbulence intensity, I ,and length scale,l to be∼0.5-1% and∼20-40% of coral di-ameter, respectively. The free-stream turbulence variables canbe computed from the following Table 1 and 2. whereCµ rep-Table 1: Free-stream specificationParameter formulak 32(IŪ)ε Cµ3/4k3/2lω Cµ−1/4√klωwall 60νβ1y2resents a constant of 0.09. To impose boundary layers in theviscous sublayer region and to accommodate convergence, theinitial conditions ofk and ω at the wall are specified differ-ently from the free-stream conditions. The rest of boundingTable 2: Boundary conditionsParameter floor inlet outletk (m2s−2) FV TKI IOε (m2s−3) FV TDI IOω (s−1) FV TFI IOu (m s−1) NS FV IOT (K) FV FV ZG*ZG: zero gradient, FV: fixed value, NS: non-slip, TKI: turbulentintensity kinetic energy inlet, TDI: turbulent mixing length dissipationrate inlet, TFI: turbulent mixing length frequency inletboundaries are set as symmetry plane which applies zero gra-dient condition for scalars except for vectors and tensor fields.The boundary conditions applied at the inlet fork, ε, andω wereused to define the turbulent kinetic energy, dissipation, and fre-quency from both the fraction of the mean velocity and the mix-ing length.Numerical schemesThe schemes implemented for both convection divergenceand Laplacian terms were the Gauss linear upwind cell lim-ited least squares and the Gauss linear limited 0.5, re-spectively. The generalised geometric-algebraic multi-grid(GAMG), smoother Gauss-Seidel, and preconditioned bi-conjugate gradient (PBiCG) were used to discretised pressu,velocity, and temperature governing equations, respectively.The semi-implicit method for pressure-linked equation (SIM-PLE) algorithm, which calculates the pressure on a staggeredgrid from velocity components by applying an iterative proce-dure coupled with the Navier-Stokes equations.Results and DiscussionDue to the absence of real experimental measurements of ax-ial velocity and wall shear stress at the upstream of the outlet,here we only validate the turbulence models based on the heattransfer performance of measured coral surface warming distri-bution. In this study, four of the most popular Low-Reynolds-number models were selected in the benchmark comparisons- the two equationsk−ω SST, the two equationk− ε modelLaunder-Sharma, the two equationsk−ω, and the one-equationmodel Spalart-Allmaras (Figure 2, 3, and 4). Thek−ω SSTandk− ε Launder-Sharma models both give good agreementswith the experimental data and are much less dissipative whichresults in higher values of calculated velocity. These two mod-els tended to establish downstream laminar flow when the free-stream turbulence level is low(µtµ ≤ 0.1)[5]. We briefly com-pared the sensitivity of the four models to free stream turbu-lent quantities and wall spacings. The results of the Spalart-Allmaras model show a reasonable agreement with the exper-imental data and insensitive to the free stream intensity val-ues ranging from 1-15%. The two-equation models exhibitsstronger sensitivity towards the turbulence levels in thate so-lutions tended to be laminar when the free stream levels werelow, and vice versa [5]. We briefly investigated the wall spacingsensitivity to the turbulence models by varying the y+ spacingsof 0.1 and 0.25; and all the models yields similarly consistentresults for the shear stress.Our results demonstrate that the magnitude of temperature risein coral tended to increase linearly with increasing pigmen-tation/absorptivity (Figure 1). Our prone estimates of irradi-ance absorption coefficient for each pigmented corals (F0, coraldarkness) could have implications for under-predicting the over-all coral surface warming (Figure 2). The mean and standarddeviation of the low irradiance and high irradiance experimn-tal results are 0.267±0.094 (99.7% confidence intervals: -0.015to 0.549) 0.457±0.232 (99.7% confidence intervals: -0.239 to1.153), respectively. The root mean squared error (RMSE) foreach of the models associated with the experimental resultsireported in Table 3.Table 3: RMSE among the turbulence models and experimentalresultsRMSE (K) k−ω SST k−ω k− ε L-S S-ALow irradiance 0.433 0.823 0.152 0.561High irradiance 0.273 1.033 0.154 0.361ConclusionIn summary, we have demonstrated that CFD technique is apowerful tool for determining the physical mechanisms of ther-mal stress in coral microenvironment at turbulent regime, which 0 0.1 0.2 0.3 0.4 0.5 0.6 100  150  200  250  300  350  400  450  500Surface Warming (Tcoral - Twater)°CF0Low irradiance (180 W m-2)experiment (Fabricius 2006)k-ω SSTk-ε Launder-Sharmak-ωSpalart-AllmarasFigure 2: Effects of flow, irradiance, and pigmentation on coralsurface warming at low irradiance 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100  150  200  250  300  350Surface Warming (Tcoral - Twater)°CF0High irradiance (800 W m-2)experiment (Fabricius 2006)k-ω SSTk-ε Launder-Sharmak-ωSpalart-AllmarasFigure 3: Effects of flow, irradiance, and pigmentation on coralsurface warming at high irradiancewould have been otherwise difficult to observe by experimentalmeans.AcknowledgementThe authors acknowledge both iVEC and the Society for Un-derwater Technology (SUT) for funding. The authors also ac-knowledge iVEC for access to its high performance computinginfrastructure. Finally, we acknowledge the OpenFOAM com-munity and developers.References[1] Ong, R.H. and King, A.J.C. and Mullins, B.J. and Cooper,T.F. and Caley, M.J., Development and Validation of Com-putational Fluid Dynamics Models for Prediction of HeatTransfer and Thermal Microenvironments of Corals,PloSone, 7, 2012, e37842.[2] Fabricius, K.E., Effects of irradiance, flow, and colonypigmentation on the temperature microenvironment aroundcorals: Implications for coral bleaching?,Limnology andoceanography, 2006, 30–37[3] Launder, BE and Sharma, BI, Application of the energy-dissipation model of turbulence to the calculation of flowk-ε Launder-Sharma Spalart-AllmarasDistance(m)k-ω SST k-ω-0.150.150Figure 4: Slices of axial velocity and temperature profiles at low irradiance atF0=100near a spinning disc,Letters Heat Mass Transfer, 1, 1974,131–137.[4] Menter, FR and Kuntz, M. and Langtry, R., Ten years ofindustrial experience with the SST turbulence model, Tur-bulence, heat and mass transfer,4, 2003, 625-632.[5] Roy, C.J. and Blottner, F.G., Assessment of one-and two-equation turbulence models for hypersonic transitionalflows, Journal of Spacecraft and Rockets,38 2001, 699–710.[6] Wilcox, D.C., Turbulence modeling for CFD, DCW indus-tries La Canada, CA,2, 1998, 3rd edition.

CFD Simulation of Low Reynolds-number Turbulence Models in Coral Thermal Microenvironment

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CFD Simulation of Low Reynolds-number Turbulence Models in Coral Thermal Microenvironment

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