sustainability

Article The Numerical Diffusion Effect on the CFD Simulation Accuracy of Velocity and Temperature Field for the Application of Sustainable Architecture Methodology

Vladimíra Michalcová 1 and Kamila Kotrasová 2,*

1 Department of Structural Mechanics, Faculty of Civil Engineering, VSB—Technical University of Ostrava, Ludvíka Podéštˇe1875/17, 708 33 Ostrava-Poruba, Czech Republic; [email protected] 2 Institute of Structural Engineering, Faculty of Civil Engineering, The Technical University of Košice, Vysokoškolská 4, 042 00 Košice, Slovakia * Correspondence: [email protected]



Received: 9 November 2020; Accepted: 3 December 2020; Published: 5 December 2020


Abstract: Numerical simulation of fluid flow and heat or mass transfer phenomenon requires numerical solution of Navier–Stokes and energy-conservation equations, together with the continuity equation. The basic problem of solving general transport equations by the Finite Volume Method (FVM) is the exact calculation of the transport quantity. Numerical or false diffusion is a phenomenon of inserting errors in calculations that threaten the accuracy of the computational solution. The paper compares the physical accuracy of the calculation in the Computational Fluid Dynamics (CFD) code in Ansys Fluent using the offered discretization calculation schemes, methods of solving the gradients of the transport quantity on the cell walls, and the influence of the mesh type. The paper offers possibilities on how to reduce numerical errors. In the calculation area, the sharp boundary of two areas with different temperatures is created in the flow direction. The three-dimensional (3D) stationary flow of the fictitious gas is simulated using FVM so that only advective transfer, in terms of momentum and heat, arises. The subject of the study is to determine the level of numerical diffusion (temperature field scattering) and to evaluate the values of the transport quantity (temperature), which are outside the range of specified boundary conditions at variously set calculation parameters.

Keywords: CFD; discretization scheme; numerical diffusion; transport equation

1. Introduction Aerodynamics deals with the movement of the air and the interaction between airflow and solid objects. Aerodynamics of buildings study the physical problems of airflow effects on buildings and their surroundings. The motion of the air—the wind—affects not only the design of the load-bearing parts of the building structures, but the dimensioning and construction of their non-load-bearing parts. One of the important areas of sustainable architecture is the knowledge of airflow effects on the surrounding objects [1]. The wind significantly affects the energy efficiency of buildings [2], associated with the general phenomenon of air filtration [3] (see Figure1), the details, elements, and systems of the packaging structures [4].

Sustainability 2020, 12, 10173; doi:10.3390/su122310173 www.mdpi.com/journal/sustainability Sustainability 2020, 12, 10173 2 of 19 Sustainability 2020, 12, x FOR PEER REVIEW 2 of 18 Sustainability 2020, 12, x FOR PEER REVIEW 2 of 18

Figure 1. Illustration of air filtration filtration and numerical simulation [[3].3]. Figure 1. Illustration of air filtration and numerical simulation [3]. TheThe aerodynamics aerodynamics of of a a building building examines the effect effect of the windwind onon thethe structurestructure itselfitself [[5],5], thethe airair velocityvelocityThe near nearaerodynamics the the structure structure of a [6–10] [building6–10] (see examines Figure 22), ),the theth effecte pressurepressure of the on onwind the the structureon structure the structure [ [5,11],5,11], itself thethe turbulenceturbulence [5], the air aroundvelocityaround the thenear building building the structure [12], [12], and and [6–10] the the influence (see influence Figure of of meteorological2), meteorological the pressure conditions conditionson the structure [13]. [13 ].The The [5,11], mutual mutual the grouping turbulence grouping of thearoundof thebuildings buildingsthe building modifies modifies [12], the and theairflow, the airflow, influence which which createsof meteorological creates the thewindy windy conditionsclimate climate in [13].their in their Thesurroundings mutual surroundings grouping [14], [ 14and of], affectstheand buildings aff theects human–wind the modifies human–wind the interactio airflow, interaction,n, which human humancreates safety, safety,the and windy thermal and thermalclimate comfort in comfort their [15]. surroundings [By15 ].studying By studying the[14], wind theand movementaffectswind movementthe human–windin relation in relation to the interactio wider to the topographicaln, wider human topographical safety, units, and it units,isthermal possible it is comfort possibleto positively [15]. to positively Byregulate studying the regulate efficiencythe wind the ofmovementeffi theciency ventilation of in therelation ventilation of tothe the urban wider of the units, topographical urban the units, scatte the units,ring scattering itof is thepossible of exhaust the to exhaust positively fumes, fumes, and regulate and the the theformation formation efficiency of snowdriftsofof snowdriftsthe ventilation around around theof thebuildings buildingsurban andunits, and th ethe line linescatte transport transportring structuresof structuresthe exhaust [16–18]. [16 fumes,–18 ]. and the formation of snowdrifts around the buildings and the line transport structures [16–18].

Figure 2. The Computational Fluid Dynamics (CFD) simulationsimulation of the wind around buildings [[10].10]. Figure 2. The Computational Fluid Dynamics (CFD) simulation of the wind around buildings [10]. TheThe results results of of the interaction of air movement and constructionconstruction cancan bebe obtainedobtained fromfrom realreal The results of the interaction of air movement and construction can be obtained from real measurementsmeasurements [19], [19], from from building building scale scale models models in the wind tunnel [[20–22],20–22], asas wellwell asas byby usingusing measurements [19], from building scale models in the wind tunnel [20–22], as well as by using computercomputer simulations. simulations. Computational Computational Fluid Fluid Dynamics Dynamics (CFD), (CFD), also also called called CFD CFD flow flow analysis, analysis, is isone one of computer simulations. Computational Fluid Dynamics (CFD), also called CFD flow analysis, is one of theof thebasic basic methods methods in innumerical numerical modeling modeling concerning concerning fluid fluid flow, flow, and and it it is is used as aa solutionsolution toto the basic methods in numerical modeling concerning fluid flow, and it is used as a solution to engineeringengineering problems problems in in almost almost all all industries industries [2 [233,24],,24], especially especially in in building building construction [4,10]. [4,10]. engineering problems in almost all industries [23,24], especially in building construction [4,10]. TheThe numerical numerical solution solution of of the general transport equations in thethe AnsysAnsys FluentFluent softwaresoftware byby thethe The numerical solution of the general transport equations in the Ansys Fluent software by the FiniteFinite Volume MethodMethod (FVM) (FVM) uses uses the the discretization discretization process, process, in which in which the basic the problem basic problem is the accurate is the Finite Volume Method (FVM) uses the discretization process, in which the basic problem is the accuratecalculation calculation of the transport of the transport quantity throughquantity thethrough walls the of the walls particular of the volumeparticular and volume its advective and its accurate calculation of the transport quantity through the walls of the particular volume and its advectiveflow across flow these across boundaries these boundaries [25]. When [25]. calculating, When calculating, it is necessary it tois countnecessary with to the count occurrence with the of advective flow across these boundaries [25]. When calculating, it is necessary to count with the occurrencethe so-called of “numericalthe so-called di “numericalffusion”, often diffusion”, referred often to in referred literature to asin “diliteratureffusion as error” “diffusion or “numerical error” or occurrence of the so-called “numerical diffusion”, often referred to in literature as “diffusion error” or “numericalviscosity” [ 26viscosity”], and with [26], the and occurrence with the of occurrence values that of are values outside that the are range outside of the the correct range of solution the correct [27]. “numerical viscosity” [26], and with the occurrence of values that are outside the range of the correct solutionThis non-physical [27]. This non-physical CFD artifact impairsCFD artifact the accuracy impairs ofthe the accuracy discrete of solutions the discrete of the solutions equations of the in solution [27]. This non-physical CFD artifact impairs the accuracy of the discrete solutions of the equationsdescribing in the describing advection the transport advection of the transport scalar [ 28of]. the It is scalar known [28]. that It the is numericalknown that di fftheusion numerical occurs equations in describing the advection transport of the scalar [28]. It is known that the numerical diffusionmainly in occurs the case mainly where in the the flow case directionwhere the is flow not parallel direction to is the not grid parallel walls to [29 the]. grid walls [29]. diffusion occurs mainly in the case where the flow direction is not parallel to the grid walls [29]. However,However, the the optimal states (parallel flow)flow) can only bebe achievedachieved byby calculatingcalculating straightstraight sectionssections However, the optimal states (parallel flow) can only be achieved by calculating straight sections ofof pipeline pipeline without without the the obstacles obstacles using using the the hexahedral hexahedral cells cells [30]. [30]. The The direction direction of of the the flow flow is isalways always in of pipeline without the obstacles using the hexahedral cells [30]. The direction of the flow is always in thein thegeneral general direction direction with with respect respect to the to cell the walls cell (hexahedral, walls (hexahedral, tetrahedral, tetrahedral, and polyhedral) and polyhedral) in the most in the general direction with respect to the cell walls (hexahedral, tetrahedral, and polyhedral) in the most flowsthe most [31]. flows When [31 evaluating]. When evaluating the advective the advective state, it state,is necessary it is necessary to consider to consider the numerical the numerical error flows [31]. When evaluating the advective state, it is necessary to consider the numerical error (numericalerror (numerical diffusion) diffusion) [32]. [This32]. Thiscauses causes considerab considerablele problems problems in inthe the numerical numerical solving solving of severalseveral (numerical diffusion) [32]. This causes considerable problems in the numerical solving of several technicaltechnical problems, problems, including including sustainable sustainable architecture. architecture. technical problems, including sustainable architecture. ThereThere are aa number number of of studies studies improving improving the numericalthe numerical solution. solution. Total Total Variation Variation Diminishing Diminishing (TVD) There are a number of studies improving the numerical solution. Total Variation Diminishing (TVD)schemes schemes have been have [33 been] a widely [33] applieda widely group applied of monotonicity-preserving group of monotonicity-preserving advection di ffadvectionerencing differencing(TVD) schemes schemes have for been partial [33] di fferentiala widely equations applied ingroup numerical of monotonicity-preserving computational fluid dynamics advection and heatdifferencing transfer schemes since the for last partial century. differential Many scientequationsific teams in numerical continue computational to develop this fluid TVD dynamics method. and It heat transfer since the last century. Many scientific teams continue to develop this TVD method. It Sustainability 2020, 12, 10173 3 of 19

Sustainabilityschemes for 2020 partial, 12, x FOR diff erentialPEER REVIEW equations in numerical computational fluid dynamics and heat transfer3 of 18 since the last century. Many scientific teams continue to develop this TVD method. It allows the allowsimplementation the implementation of the whole of the spectrum whole ofspectrum TVD schemes of TVD into schemes unstructured into unstructured networks, whilenetworks, their while exact theirformulation exact formulation was restored was on restored structured on networksstructured [34 networks]. The authors [34]. The [35] authors analyzed [35] the analyzed TVD diff erencingthe TVD differencingon unstructured on three-dimensionalunstructured three-dimensional meshes, focusing meshes, on the non-linearityfocusing on ofth TVDe non-linearity differencing of and TVD the differencingextrapolation and of thethe virtual extrapolation upwind node.of the Furthermore,virtual upwind they node. proposed Furthermore, a novel monotonicity-preserving they proposed a novel monotonicity-preservingcorrection method for the correction TVD schemes method that for significantly the TVD reducesschemes thethat numerical significantly diffusion reduces caused the numericalby mesh skewness.diffusion caused The authors by mesh [36 skewness.] analyzed The the authors causes [36] of the analyzed numerical the causes errors, of in the terms numerical of the errors,numerical in terms diffusion of the and numerical the compression diffusion arising and fromthe compression the use of the arising explicit from second-order the use totalof the variation explicit second-orderdiminishing schemestotal variation in the diminishing one-dimensional schemes advection in the simulation.one-dimensional advection simulation. TheThe presented presented work work offers offers possibilities possibilities on on how how to reduce the numerical errors inin AnsysAnsys FluentFluent softwaresoftware using using the the correct correct calculation settings. It It uses uses the available discretization schemes inin thethe software,software, together together with with the the available available solutions solutions methods of transport quantityquantity gradients.gradients. TheThe physicalphysical accuracyaccuracy of of the the calculations, calculations, for for the the various various combination-listed combination-listed parameters, is monitored on three typestypes ofof mesh. mesh. TheThe conclusions conclusions will will recommend recommend suitable suitable variants variants of calculations and, thus, contribute toto betterbetter numericalnumerical simulations simulations in in th thee field field of of construction. construction.

2.2. Method Method TheThe three-dimensional three-dimensional stationary stationary virtual virtual gas gas flow flow of the computational domain withwith dimensiondimension 11 × 11 × 0.250.25 m m (x( x× y ×y z) zis) issimulated simulated by by the the Finite Finite Volume Volume method method (FVM). (FVM). The The fictitious fictitious gas gas density density is × × 3 × × 1 1 is ρ = 1 kg−3 m . The values of the thermal conductivity λ −(W1 −m1 K ) and the dynamic viscosity ρ = 1 kg·m ·. The− values of the thermal conductivity λ (W·m ·K· ) −and· −the dynamic viscosity μ (Pa·s) µ (Pa s) of the gas are close to zero. The pressure-velocity coupling algorithm is determined by of the gas· are close to zero. The pressure-velocity coupling algorithm is determined by the segregated SIMPLEthe segregated method, SIMPLE which is method, suitable which for steady-state is suitable ca forlculations steady-state [37]. calculationsThe boundary [37 conditions]. The boundary are set soconditions that the identical are set sovectors that the enter identical at the two vectors mutually enter perpendicular at the two mutually walls with perpendicular the velocity walls vx and with vy. Onethe velocityof these wallsvx and hasvy .the One temperature of these walls T1 = has300 theK and temperature the secondT T12 == 300400 K. K andThe thepressure second outletT2 = is400 on theK. Thetwo pressureopposite outletwalls (the is on static the two pressure opposite p = 0 walls Pa). (theAs at static the entrance, pressure thep = outlet0 Pa). temperature As at the entrance, is also onethe wall outlet T temperature1 = 300 K and is the also second one wall wallT1 T=2 300= 400 K andK, see the Figure second 3. wall Thereby,T2 = 400 the K, sharp see Figure boundary3. Thereby, of the twothe sharpdomains boundary with the of temperature the two domains difference with the∆T temperature= 100 K is created difference in the∆T calculation= 100 K is createddomain. in This the boundarycalculation is domain.in the direction This boundary of the isflow in the (in direction the domain of the at flowthe (inangle the of domain 45°). Only at the the angle advective of 45◦). transmission,Only the advective in terms transmission, of the momentum in terms of and the momentum the heat, should and the occur heat, shouldfor the occur accurate for the numerical accurate calculation.numericalcalculation. The diffuse Thetransfer diffuse should transfer not shouldoccur. The not occur.output The on outputthe two on side the opposite two side walls opposite (in the walls xy plane)(in the isxy theplane) zero is flow the zeroof all flow quantities of all quantities across the across border the (symmetry border (symmetry boundary boundary condition, condition, normal velocitynormal is velocity zero). These is zero). two These sides two of the sides opposite of the oppositewalls are walls not shown are not in shown Figure in 3. Figure 3.

Figure 3. The calculation domain scheme 1 × 11 × 0.250.25 m m and and the the boundary boundary conditions. conditions. × × The solution of the problem is independent of the domain dimensions, gas density, and velocity. It was verified on pilot calculations, whereby gradually changing all of these values in the thousands. These test tasks were also performed for porous domain, with a wide range of permeability values corresponding to building materials. The transport quantity does not have to be only the temperature, Sustainability 2020, 12, 10173 4 of 19

The solution of the problem is independent of the domain dimensions, gas density, and velocity. It was verified on pilot calculations, whereby gradually changing all of these values in the thousands. These test tasks were also performed for porous domain, with a wide range of permeability values corresponding to building materials. The transport quantity does not have to be only the temperature, but also, for example, the concentration of substances. In the presented paper, the size of the calculation area is set so that the calculations did not have the problem with above-standard number of cells and the scattering of the temperature field (numerical error—diffusion) was obvious at the same time. The air density is close to the air density. The velocity is chosen so that the calculations converged in an acceptable time. The Ansys Fluent software uses the FVM to convert the general transport equations to the system of linear equations that are solved numerically by the Gauss–Seidel iteration method. This solution consists in integrating the equations in each control volume (cell), where the result is the discrete equations presented the flow equilibrium (the conservation laws of the transport quantity Φ in given volume). Its mathematical description for the stationary flow in the integral form is: Z Z Z ρ →v Φd→A = Γ Φd→A + S dV, (1) · · Φ·∇ Φ A A V where →A (m s 1) is the surface vector, V (m3) is the control volume (cell volume), ρ (kg m 3) is the · − · − density of the flowing medium, →v (m s 1) is the velocity vector, Φ (K) is the value of the transport · − quantity (temperature), Φ (K m 1) the gradient of the transport quantity Φ, Γ is the diffusion ∇ · − Φ coefficient of the transport quantity Φ (K kg m 1 s 1), S (K kg m 3 s 1) is the source term of the · · − · − Φ − − quantity Φ per unit of the volume. Equation (1) is applied to all control volumes of the calculation area. Equation (2) is obtained by the discretization of the Equation (1) in the given cell:

N N Xf aces Xf aces ρ →v Φ →A = Γ Φ →A + S V, (2) f · f · f · f Φ·∇ f · f Φ· f n

→ 1 where N s is the number of the faces surrounding of the cell, ρ →v A (kg s ) is the mass flow over face f · f · f · − → 2 the surface f, A f (m ) is the surface vector f, Φf (K) is the value of the transport quantity flowing over the surface f (the face value), Φ (K m 1) is the gradient of the transport quantity Φ on the surface f. ∇ f · − The left side in both equations represents the advective transfer of the quantity Φ, the right side expresses the diffuse transfer and the source term of the transport quantity Φ (its decrease or increase). The basic problem in the discretization of the advective term is the exact calculation of the transport quantity on the face of the specific volume Φ and its gradient Φ . The diffusion process f ∇ f affects the transfer of the transport quantity along its gradient in all directions, while the advective transfer pervades only in the direction of the flow. It is very difficult to find the exact discretization computational scheme for the solving of the advective term in the Equation (2). The software Ansys Fluent stores the discrete values of the scalar quantity Φ in the center of the cell. The values of the scalar quantity Φf on the cell face are required for the calculation of the advective term in the equations and they are determined by the interpolation from the values in the centers of the adjacent cells. The number of the surrounding cells depends on the type of the grid, but, in most cases, the amount is the same as the number of the faces forming of the interest cell. The discretization “upwind” schemes are used for this process; it means that the value Φf is derived from the value of the next cell in the flow direction. The most upwind schemes require the determination of the transport quantity gradient Φ ∇ for their solution. The gradients are necessary for the calculation of the scalar values on the cells faces not only for discretization the advective but also for the diffusion term in the Equation (2). Sustainability 2020, 12, 10173 5 of 19

3. The Parameters Influencing the Accuracy of the Calculation The level of the physical accuracy of the numerical calculation is influenced by the mesh type, the choice of discretization schemes for the conversion of the general transport equations to the linear equations, and also the choice of the calculating method of the transport quantity gradient Φ ∇ (here temperatures). Sustainability 2020, 12, x FOR PEER REVIEW 5 of 18 3.1. The Mesh Type and the Mesh Density Three types of the grids: hexahedral, tetrahedral, and polyhedral are used for solving the problems,Three see types Figure of the 4. grids: All grid hexahedral, types have tetrahedral, double anddensity. polyhedral The coarse are used hexahedral for solving and the tetrahedral problems, meshsee Figure were4 formed. All grid from types 25 havecells doublewith the density. length Theof 1 coarsem at all hexahedral longitudinal and and tetrahedral vertical edges mesh wereof the domainformed ( fromx, y-axis 25 direction), cells with thesee Figure length 5a. of 1The m 6 at cells all longitudinalper 0.25 m are and formed vertical on the edges domain of the edges domain in z- axis(x, y direction.-axis direction), The fine see hexahedral Figure5a. and The tetrahedral 6 cells per 0.25mesh m were are formed formed on from the 100 domain cells per edges 1 m in inz the-axis x, ydirection.-axis directions, The fine see hexahedral Figure 5b. and The tetrahedral 25 cells pe meshr 0.25 were m formedare formed from on 100 the cells domain per 1 m edges in the inx, yz-axis-axis direction.directions, The see polyhedral Figure5b. Themesh 25 was cells formed per 0.25 directly m are formedin the Ansys on the Fluent domain from edges the in tetraz-axis cells. direction. The six calculationThe polyhedral areas mesh with wasthe identical formed directly dimensions in the (Figure Ansys Fluent3) and from the parameters the tetra cells. listed The in six Table calculation 1 were created.areas with the identical dimensions (Figure3) and the parameters listed in Table1 were created.

(a) (b) (c)

FigureFigure 4. 4. ThreeThree mesh mesh types types and and axis scheme onon whichwhich thethe transporttransport quantities quantities are are evaluated; evaluated; the coarse themesh: coarse (a) mesh: hexahedral; (a) hexahedral; (b) tetrahedral; (b) tetrahedral; (c) polyhedral. (c) polyhedral.

(a) (b)

Figure 5. TwoTwo different different densities of the tetrahedral mesh: ( a) coarse; (b) fine.fine.

Table 1. Number of cells in computational areas. Table 1. Number of cells in computational areas. Mesh Type Hexahedral Tetrahedral Polyhedral Mesh type Hexahedral Tetrahedral Polyhedral Values in Values in Thousands25 Cells 25 Cells 100 100 Cells Cells 25 25 Cells Cells 100 CellsCells 25 25 Cells Cellstetra 100100 Cells Cellstetra Thousands tetra tetra Cells number 3.8 250 28.4 1660 5.9 294 CellsFaces number number 3.8 12.2 250 765 28.4 58.9 16603355 38.8 5.9 2023 294 Faces number 12.2 765 58.9 3355 38.8 2023 Nodes number 4.7 265 5.8 294 32.9 1729 NodesNodes number number 4.7 4.7 265 265 5.8 5.8 294 32.9 32.9 1729 1729

3.2. The Discretization Scheme 3.2.3.2. The The Discretization Discretization Scheme Scheme It is possible that the choice from the five upwind discretization schemes for the given problem. ItIt is is possible possible that that the the choice choice from from the the five five upwind discretization schemes for the given problem.problem. Their brief description and the scheme are in Table 2. Their brief description and the scheme are in Table2.

Sustainability 2020, 12, x FOR PEER REVIEW 6 of 18

Sustainability 2020, 12, x TableFOR PEER 2. The REVIEW discretization schemes for the adventive flow calculation. 6 of 18

SustainabilityFlow Direction 2020, 12, x Table→ FOR PEER 2. The REVIEW discretization Description schemes for the of adventiveDiscretization flow calculation. Schemes 6 of 18 Sustainability 2020, 12, 10173 6 of 19 Sustainability 2020, 12, x FOR PEERFirst-order REVIEW upwind scheme is based on the assumption that the value6 ofof 18 Flow Φ(x)Direction interpolated →Table 2. The discretizationDescription schemes for the of adventiveDiscretization flow calculation. Schemes ΦP value the quantity in the cell center corresponds to the average value in the Φ ef First-order upwind scheme is based on the assumption that the value of whole control volume and the face value Φf is set equal to the cell-center Flow Φ(x)Direction interpolatedTableΦE 2. Table→The discretization2. The discretization schemes Descriptionschemes for the adventivefor the of adventiveDiscretization flow calculation. flow calculation. Schemes W P E the quantity in the cell center corresponds to the average value in the ΦP e value value of Φ in the upstream cell. The first-order upwind scheme is not Φ ef First-order upwind scheme is based on the assumption that the value of Flow Direction → whole control volumeDescription and the offace Discretization value Φf is set Schemes equal to the cell-center Flow Φ(x) Direction interpolatedΦE depended on the gradientDescription ∇Φ. of Discretization Schemes W Φ P P e value E→ the quantity in the cell center corresponds to the average value in the Φ First-ordervalue of Φ inupwind the upstream scheme cell. is based The first-orderon the assumption upwind schemethat the isvalue not of Φ(x) interpolated ef First-order upwind scheme is based on the assumption that the Φ whole control volume and the face value Φf is set equal to the cell-center Φ Evalue thedependedPower quantity law on scheme in the the gradient cellis based center ∇ onΦ corre. the analyticalsponds to solutionthe average the valueone-dimensional in the W P P e E value of the quantity in the cell center corresponds to the average Φef value of Φ in the upstream cell. The first-order upwind scheme is not advection-diffusion equation. The face valuef Φf is determined from the Φ(x) interpolatedΦE whole controlvalue in volume the whole and control the volumeface value and Φ the is face set valueequalΦ tof is the set cell-center equal W P E Powerdepended law onscheme the gradient is based ∇ onΦ. the analytical solution the one-dimensional ΦP e value valueexponential ofto Φ the in profile cell-center the upstream by valueusing cell. of theΦ The invalues the first-order upstream of the cell cell.upwind in The their first-orderscheme center. is The not advection-diffusion equation. The face value Φf is determined from the Φ(x) interpolated Φef ΦE upwind scheme is not depended on the gradient Φ. dependedexponentialPower law on scheme profile the gradient is baseddepended ∇ onΦ. the on analytical the Peclet solution number∇ thePe (ratioone-dimensional of the W Φ P e value E P exponentialconvectionPower heat profile law flow scheme by (advection) usingis based the onvalues and the the analytical of heat the cellflow solution in by their the the center. mechanism The of advection-diffusion equation. The face value Φf is determined from the Φ(x) interpolated Φef ΦE one-dimensional advection-diffusion equation. The face value Φf is Powerexponentialthe convection law scheme profile (diffusion)). is baseddepended Theon the solutionon analytical the Peclet is the solution number same with thePe (ratioone-dimensionalthe first-order of the W Φ P P e value E exponentialdetermined profile from by theusing exponential the values profile of the by cell using in thetheir values center. of the The advection-diffusionconvection heat flow equation. (advection) The and face the value heat Φ flowf is determined by the mechanism from the of Φ(x) interpolated upwind scheme for |Pe| ≥ 10. Φef ΦE exponentialcell in profile their center. is depended The exponential on the profile Peclet is number depended Pe on (ratio the Peclet of the ΦP value W P e E theexponential convectionnumber profile Pe (diffusion)). (ratio by ofusing the convectionThe the solutionvalues heat of is flowthe the cell (advection)same in theirwith and thecenter. the first-order heat The convection heat flow (advection) and the heat flow by the mechanism of Φef ΦE upwindexponentialSecond-orderflow scheme by profile theupwind for mechanism is|Pe| depended scheme: ≥ 10. of the the on convection face the Pecletvalue (di Φnumberffusion)).f is determined Pe The (ratio solution from of the is the Φ W interpolatede W P E cellthe convectionvaluesthe same in the (diffusion)). with two the cells first-order upstream The solution upwind of the schemeis face.the same forThis|Pe withscheme| 10. the requiresfirst-order the Φ(x) value convection heat flow (advection) and the heat flow by ≥the mechanism of Second-orderupwindSecond-order scheme upwind for upwind|Pe| scheme: ≥ 10. scheme the :face the facevalue value Φf isΦ determinedis determined from the Φ interpolated thedetermination convection of(diffusion)). the gradient The ∇ Φsolution in each is cell. the Thissamef is with more the accurate first-order than W ΦP Φef ΦE cell valuesfrom in the the cell two values cells in upstream the two cells of upstreamthe face. ofThis the scheme face. This requires the W Φ(x)P e value E upwindthe first-order scheme upwind for |Pe| scheme, ≥ 10. but in the regions with the strong determinationSecond-orderscheme requiresupwindof the gradient the scheme: determination ∇ theΦ in face each of value the cell. gradient Φ Thisf is determined is moreΦ in each accurate cell.from thethan Φ W interpolated ∇ ΦP Φef ΦE gradients,cell valuesThis it in is can morethe result two accurate cells in the thanupstream face the va first-orderlues of the that face. upwind are This outside scheme, scheme of butthe requires in range the of the Φ(x)e value Second-orderthe first-order upwindupwind scscheme,heme: thebut face in the value regions Φf is withdetermined the strong from the W P E the correctregions cell with values. the strong gradients,∇ it can result in the face values that Φ W interpolated cellgradients,determination values itin can the of result twothe gradientcells in the upstream face Φ va inlues ofeach the that cell. face. are This This outside is schememore of accuratethe requires range than ofthe Φ(x)ΦP Φ evaluef ΦE are outside of the range of the correct cell values. e the first-order upwind scheme, but in the regions with the strong W P E determinationtheQuadratic correctQuadratic Upwindcell ofvalues. Upwindthe Interpolation gradient Interpolation ∇Φ for in eachConvective for Convective cell. This Kinetics is Kinetics more (QUICK) accurate than ΦP Φef ΦE scheme.gradients, The it canquadratic result incurve the faceis fitted values with that two are upstream outside ofnodes the rangeand one of W P e E the first-order(QUICK) upwind scheme .scheme, The quadratic but in curve the regions is fitted withwith twothe upstreamstrong ΦW interpolated Quadraticdownstreamthe correctnodes Upwind cell andnode. values. one ThisInterpolation downstream scheme node.requires for Convective This the scheme determination Kinetics requires (QUICK) the of the gradient Φ(x) value gradients, it can result in the face values that are outside of the range of scheme.determination The quadratic of the curve gradient is fittedΦ inwith each two cell. upstream This is a very nodes accurate and one ΦP the∇Φ incorrect each cellcell. values. This is a very accurate∇ scheme, but it can lead to stability ΦW interpolatedΦef ΦE downstreamQuadraticscheme, Upwind node. but it ThisInterpolation can lead scheme to stability requires for Convective problems the determination in theKinetics calculation (QUICK) of the in the gradient Φ(x) value problems in the calculation in the regions with strong gradients. W P e E scheme.regions The quadratic with strong curve gradients. is fitted QUICK with scheme, two upstream documented nodes in [and37], one ΦP Quadratic∇Φ in each Upwind cell. This Interpolation is a very accurate for Convective scheme, but Kinetics it can lead(QUICK) to stability ΦW interpolatedΦef ΦE QUICKdownstreamis scheme, mainly node. suitable documented This for scheme the proin requires[37], structured is mainly the hexahedral determination suitable meshes, for the of but thepro it gradient Φ(x) value problemsscheme. The in thequadratic calculation curve in is the fitted regions with with two upstreamstrong gradients. nodes and one W P e E structuredcan behexahedral used for the meshes, unstructured but it orcan the be hybrid used meshes.for the unstructured or Φ W interpolated ΦP ∇Φ in each cell. This is a very accurate scheme, but it can lead to stability Φef ΦE QUICKdownstream scheme, node. documented This scheme in requires[37], is mainly the determination suitable for the of thepro gradient Φ(x) value the hybridThird-order meshes. Monotone Upstream-centered Scheme for W P e E problems in the calculation in the regions with strong gradients. ΦP structured∇Φ in eachConservation hexahedralcell. This Laws is meshes,a very (MUSCL). accurate but This it can scheme, third-order be used but convectionfor it thecan unstructuredlead scheme to stability or Φef ΦE Third-orderQUICKwas scheme, created Monotone documented from theUpstream-centered original in [37], MUSCL is mainly [28 Scheme], by suitable mixing for Conservation thefor centralthe pro Laws W P- e E theproblems hybrid in meshes. the calculation in the regions with strong gradients. differentiation scheme and the second-order winding scheme. This QUICK(MUSCL).structured scheme, Thishexahedral third-order documented meshes, convection in but [37], it canis scheme mainly be used wassuitable for created the for unstructured thefrom pro the or Third-orderscheme Monotone requires the Upstream-centered determination of the Scheme gradient forΦ Conservationin each cell. It Laws - originalthe hybrid MUSCL meshes. [28], by mixing the central differentiation∇ scheme and the structuredis usable hexahedral for all grid meshes, types. but it can be used for the unstructured or (MUSCL).second-order This winding third-order scheme. convection This sc hemescheme requires was created the determination from the of - theThird-order hybrid meshes. Monotone Upstream-centered Scheme for Conservation Laws originalthe gradient MUSCL ∇Φ in[28], each by cell. mixing It is the usable central for alldifferentiation grid types. scheme and the 3.3. The Solution of the Transport Quantitysecond-orderThird-order(MUSCL). Gradients This Monotone winding third-order scheme.Upstream-centered convection This scheme scheme Scheme requires was forcreated the Conservation determination from the Laws of - the(MUSCL).original gradient MUSCL This ∇Φ third-order in[28], each by cell. mixing convection It is theusable central scheme for alldifferentiation gridwas createdtypes. schemefrom the and the The3.3. gradients The Solution are of needed, the Transport not only Quantity for the Gradients constructing values of the scalar at the cell faces, but - originalsecond-order MUSCL winding [28], by scheme. mixing This the centralscheme differentiation requires the determination scheme and the of also for computing the secondary diffusion terms∇ and the velocity derivatives. The gradient Φ of 3.3. TheThe Solution gradients of the are Transport needed,second-orderthe gradient Quantitynot only windingΦ Gradientsfor in theeach constructischeme. cell. It isThisng usable valuesscheme for of allrequires the grid scalar types. the at determination the cell∇ faces, of but a given the variable Φ is used to discretization of the advection and the diffusion terms in the flow also for computing the secondarthe gradienty diffusion ∇Φ in termseach cell. and It the is usablevelocity for derivatives. all grid types. The gradient ∇Φ of a conservation equations. The gradients are computed in Ansys Fluent, according to the three following given3.3. TheThe the Solution gradients variable of theareΦ isTransport needed, used to Quantitynot discretization only Gradientsfor the of constructi the advengction values and of the the diffusion scalar at termsthe cell in faces, the flow but methods:also for computing the secondary diffusion terms and the velocity derivatives. The gradient ∇Φ of a 3.3.conservation The Solution equations. of the Transport The gradients Quantity are Gradients computed in Ansys Fluent, according to the three following givenThe the gradients variable Φare is needed, used to not discretization only for the of constructi the advengction values and of the the diffusion scalar at termsthe cell in faces, the flow but 1. Green–Gaussmethods: cell-based; ∇ conservationalso forThe computing gradients equations. arethe needed, secondarThe gradients noty diffusion only are for computed theterms constructi and in Ansystheng velocity values Fluent, derivatives.of according the scalar toThe at the the gradient three cell followingfaces, Φ ofbut a 2. Green–Gaussalsomethods:1.given forGreen–Gauss the computing variable node-based; cell-based; Φthe is secondarused to ydiscretization diffusion terms of theand adve the ctionvelocity and derivatives. the diffusion The terms gradient in the ∇Φ flowof a 3. Leastgiven2.conservation Green–Gauss Squares the variable cell-based.equations. node-based;Φ is Theused gradients to discretization are computed of the in adve Ansysction Fluent, and theaccording diffusion to the terms three in following the flow 1. Green–Gauss cell-based; conservation3.methods: Least Squares equations. cell-based. The gradients are computed in Ansys Fluent, according to the three following The2. firstGreen–Gauss two methods node-based; use the Green–Gauss theorem to compute the Φ at the cell center c0: methods: ∇ 1. TheGreen–Gauss first two methods cell-based; use the Green–Gauss theorem to compute the ∇Φ at the cell center c0: 3. Least Squares cell-based. X 2. Green–Gauss node-based; 1 → 1. Green–Gauss cell-based; ( Φ)c0 = Φ f A f , (3) 3. TheLeast first Squares two methods cell-based. use the∇ Green–GaussV theorem1 · to compute the ∇Φ at the cell center c0: 2. Green–Gauss node-based; ∇Φ f = Φ ∙ 𝐴⃗ , 𝑉 (3) 3. LeastThe first Squares two methods cell-based. use the Green–Gauss theorem1 to compute the ∇Φ at the cell center c0: The face value Φ by default of the Green–Gauss cell-based is⃗ taken from the arithmetic average f ∇Φ = Φ ∙ 𝐴, (3) 𝑉 ∇ of the valuesThe at first the neighboringtwo methods cell use centers: the Green–Gauss theorem1 to compute the Φ at the cell center c0: ∇Φ = Φ ∙ 𝐴⃗ , 𝑉 (3) Φ + Φ1 Φ =∇Φc0 = ci Φ.∙ 𝐴⃗ , (4) f 2 𝑉 (3) Sustainability 2020, 12, x FOR PEER REVIEW 7 of 18

The face value Φf by default of the Green–Gauss cell-based is taken from the arithmetic average of the values at the neighboring cell centers:

Φ +Φ Φ = . (4) 2 Sustainability 2020, 12, 10173 7 of 19 The face value Φf by default of the Green–Gauss node-based is taken by the arithmetic average of the nodal values on the face: The face value Φf by default of the Green–Gauss node-based is taken by the arithmetic average of the nodal values on the face: , N 1 Xf ,nodes Φ = 1 Φ, (5) Φ f = 𝑁, Φn, (5) N f ,nodes n where Nf,nodes are the number of the nodes on the face. The nodal values Φn are determined from the where Nf,nodes are the number of the nodes on the face. The nodal values Φn are determined from the weightedweighted average average of of the the cell values surrounding the nodes. The node-basednode-based gradientgradient methodmethod isis notnot availableavailable with with the the polyhedral polyhedral meshes. meshes. TheThe third third least least squares squares cell-based cell-based method method is is based on assumption the change inin thethe cellcell valuesvalues between the cell c0 and ci along the vector 𝑟⃗ from the center of the cell c0 to ci is expressed as: between the cell c0 and ci along the vector →r i from the center of the cell c0 to ci is expressed as:

∇Φ ∙∆𝑟⃗ = Φ Φ. (6) ( Φ) ∆→r = (Φ Φc ). (6) ∇ c0· i ci − 0 4.4. Results Results TheThe influence influence of of the the setting setting combination combination the above parameters on the accuracy ofof thethe calculationcalculation isis tested tested in in this this paper. paper. The The level level of of the the temperature temperature field field scattering (numerical didiffusion)ffusion) isis comparedcompared toto the the original original sharp sharp boundary boundary of of two two areas with the temperature didifferencefference ∆ ∆TT= = 100 K.K. TheThe studystudy alsoalso monitorsmonitors in in which which cases cases the the temperature temperature values a appearppear outside the range ofof thethe specifiedspecified boundaryboundary conditions, i.e., outside the range T∈ (300 K, 400 K). conditions, i.e., outside the range T ∈(300 K, 400 K). 4.1. Evaluation the Mesh Density and the Mesh Type on the Numerical Diffusion 4.1. Evaluation the Mesh Density and the Mesh Type on the Numerical Diffusion It is generally known that the mesh density has a significant effect on the magnitude of the It is generally known that the mesh density has a significant effect on the magnitude of the numerical diffusion. Figure6 shows the temperature fields in the median plane of the calculation numerical diffusion. Figure 6 shows the temperature fields in the median plane of the calculation area area (plane z = 0.125 m, see Figure3) calculated using the hexahedral meshes, the coarse also the (plane z = 0.125 m, see Figure 3) calculated using the hexahedral meshes, the coarse also the fine grids. fine grids. The difference in the temperature scattering with the mesh density of 25 cells/m versus The difference in the temperature scattering with the mesh density of 25 cells/m versus the mesh the mesh density of 100 cells/m is shown. Figure6a,b presents the results of the solutions using the density of 100 cells/m is shown. Figure 6a,b presents the results of the solutions using the first-order first-order upwind scheme and Figure6c,d using the second-order upwind scheme. The similar e ffect upwind scheme and Figure 6c,d using the second-order upwind scheme. The similar effect has the has the grid density on the numerical diffusion using the first-order upwind scheme for the tetrahedral grid density on the numerical diffusion using the first-order upwind scheme for the tetrahedral and and polyhedral mesh types. Using the second-order upwind, QUICK, and the third-order MUSCL polyhedral mesh types. Using the second-order upwind, QUICK, and the third-order MUSCL discretization schemes, the scatterings of the temperature fields and their changes (by changing of the discretization schemes, the scatterings of the temperature fields and their changes (by changing of the grid density) are smaller in all meshes, but there are problems with the temperature values outside the grid density) are smaller in all meshes, but there are problems with the temperature values outside the range of the correct solution (more Section 4.4). range of the correct solution (more Section 4.4).

(a) (b) (c) (d)

FigureFigure 6. 6. HexahedralHexahedral elements, elements, influence influence of ofthe the mesh mesh de densitynsity on onthe the temperature temperature field field scattering: scattering: (a) 25(a )cells/meter, 25 cells/meter, first-order; first-order; (b) 100 (b )cells/meter, 100 cells/meter, first-order; first-order; (c) 25 cells/meter, (c) 25 cells /second-order;meter, second-order; (d) 100 cells/meter,(d) 100 cells second-order./meter, second-order.

The grid type also has influence on the numerical diffusion of the transport quantity, although not as significant as its density. The comparison of the temperature fields scatterings for all three types of the coarse grids (25 cells/m) for the identically selected calculation scheme (second-order upwind) and the method of the solution of the transport quantity gradients (Green–Gauss node-based) is shown in Figure7. Figure7a–c document the temperature field from the Ansys Fluent software and Figure7d the temperature diagram in the horizontal axis of the calculation area (y = 0.5 m, z = 0.125 m). As it SustainabilitySustainability 20202020,, 1212,, xx FORFOR PEERPEER REVIEWREVIEW 88 ofof 1818

The grid type also has influence on the numerical diffusion of the transport quantity, although not as significant as its density. The comparison of thethe temperaturetemperature fieldsfields scatteringsscatterings forfor allall threethree typestypes of the coarse grids (25 cells/m) for the identically selected calculation scheme (second-order upwind) andSustainability the method2020, 12 of, 10173the solution of the transport quantity gradients (Green–Gauss node-based) is shown8 of 19 inin FigureFigure 7.7. FigureFigure 7a–c7a–c documentdocument thethe temperaturetemperature fieldfield fromfrom thethe AnsysAnsys FluentFluent softwaresoftware andand FigureFigure 7d the temperature diagram in the horizontal axis of the calculation area (y == 0.50.5 m,m, zz == 0.1250.125 m).m). AsAs itit cancan bebe seenseen from from Figure Figure7 7,7,, the thethe smallest smallestsmallest scattering scatteringscattering is isis achieved achievedachieved using usingusing the thethe tetrahedral tetrtetrahedral elements elements using using the the coarsecoarse meshmesh and and the the described described calculationcalculation setting.setting.setting. Contrariwise,Contrariwise, the thethe biggest biggestbiggest numerical numericalnumerical di diffusiondiffusionffusion arises arisesarises usingusing the the hexahedral hexahedral elements. elements. The The ideal ideal variant variant (“ideal” (“ideal”))) isis thethe is the illustrativeillustrative illustrative fictitiousfictitious fictitious casecase case withwith with thethe zero thezero numericalzero numerical diffusion diffusion in Figure in Figure 7d. 7d.

400400 TT [K][K] SecondSecond orderorder 350350 2525 cells/mcells/m xx [m][m] 300300 0.250.25 0.375 0.375 0.5 0.5 0.625 0.625 0.75 0.75 IdealIdeal hexahedhexahed tetrahedtetrahed polyhedpolyhed ((a)) ((b)) ((cc)) ((d))

FigureFigure 7. 7. CoarseCoarse mesh, mesh, 25 25 cells/meter, cells/meter, second-order, influenceinfluence of of the the grid grid type type on on the the temperature temperature field field scattering:scattering: (( aa))) hexahedral;hexahedral; ( (b)) tetrahedral; tetrahedral; ((cc)) polyhedral;polyhedral; (((dd))) thethethe temperaturetemperaturetemperature diagram diagramdiagram in inin the thethe direction directiondirection ofof horizontal horizontal axis axis ( (yy == 0.50.50.5 m,m, m, zzz === 0.1250.1250.125 m).m). m). TheThe The legendlegend legend isis is identiidenti identicalcalcal withwith with legendlegend legend inin in FigureFigure Figure 6.6.6.

TheThe temperature fields fields using thethe finefine meshmesh withwith thethe unchangedunchanged calculationcalculation parametersparameters areare in in FigureFigure 88.. ItIt isis obviousobvious thatthat thethe didifferencesfferences inin thethe individualindividual resultsresults are are smaller. smaller. The The tetrahedral tetrahedral mesh mesh stillstill achievesachieves thethe mostmost accurateaccurate result. result.

400400 TT [K][K] SecondSecond orderorder 350350 100100 cells/mcells/m xx [m][m] 300300 0.250.25 0.375 0.375 0.5 0.5 0.625 0.625 0.75 0.75 IdealIdeal hexahedhexahed tetrahedtetrahed polyhedpolyhed

((a)) ((b)) ((cc)) ((d))

FigureFigure 8. 8. FineFine mesh, mesh, 100 100 cells/meter, cells/meter, second-order,second-order, influenceinfluence ofof thethe gridgrid type type on on the the temperature temperature field field scattering:scattering: (( aa))) hexahedral;hexahedral; ( (b)) tetrahedral; tetrahedral; ((cc)) polyhedral;polyhedral; (((dd))) thethethe temperaturetemperaturetemperature diagram diagramdiagram in inin the thethe direction directiondirection ofof horizontal horizontal axis axis ( (yy == 0.50.50.5 m,m, m, zzz === 0.1250.1250.125 m).m). m). TheThe The legendlegend legend isis is identiidenti identicalcalcal withwith with legendlegend legend inin in FigureFigure Figure 6.6.6.

However,However, it it is is important important to to remember remember that that the the over overallall evaluation evaluation of of the the mesh mesh type type is is necessary necessary to performto perform in inparallel parallel with with the the choice choice of of the the discre discretizationtizationtization schemescheme scheme andand the the methodmethodmethod ofofof calculating calculatingcalculating the thethe gradientgradient of of the the transport transport quantity quantity on the control volume face (see next chapter).

4.2.4.2. Evaluation Evaluation of of the the Discretizatio Discretizationn Schemes Schemes in in the the Numerical Numerical Diffusion Diffusion TheThe discretization discretization schemes schemes listed listed inin in SectionSection Section 3.2,3.2, 3.2 exceptexcept, except thethe the powerpower power lawlaw law scheme,scheme, scheme, areare aretested.tested. tested. TheThe solutionsolutionThe solution ofof thethe of powerpower the power lawlaw lawschemescheme scheme isis identicalidentical is identical withwith with thethe thefirst-orderfirst-order first-order upwindupwind upwind schemescheme scheme inin thisthis in this typetype type ofof the ofthe problemthe problem (|Pe| (|Pe > |10). > 10). FigureFigure 99 showsshows thethe temperaturetemperature diagramsdiagrams forfor all alll upwind upwindupwind schemes schemesschemes using usingusing the thethe hexahedral hexahedralhexahedral mesh. mesh.mesh. TheThe Green–Gauss node-based method was chosenchosen forfor thethethe transporttransporttransport quantityquantityquantity gradients gradientsgradients in inin all allallcases casescases presentedpresented here. here. The The results results using using the the coarse coarse and and fine fine gridsgrids grids areare are comparedcompared compared here.here. here. TheTheThe resultsresults results ofof thethe of first-first- the orderfirst-order upwind upwind scheme scheme are significantly are significantly diffusive, diffusive, yett thethe yet differencesdifferences the differences areare slightslight are slight inin otherother in other cases.cases. cases. TheThe smallestsmallestThe smallest temperaturetemperature temperature fieldfield scatteringsscatterings field scatterings inin thisthis incasecase this isis shown caseshown is byby shown thethe QUICKQUICK by the schemescheme QUICK usingusing scheme thethe using finefine mesh.mesh. the fine mesh. Sustainability 2020, 12, 10173 9 of 19

Sustainability 2020, 12, x FOR PEER REVIEW 9 of 18

400 400 Hexahedral Hexahedral T [K] Hexahedral T [K] Hexahedral 25 cells/m 100 cells/m 350 Node-based 350 Node-based

x [m] x [m] 300 300 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1

First-order QUICK First-order QUICK Second-order Third-order Second-order Third-order 400 400 T [K] T [K]

350 350

x [m] x [m] 300 300 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.35 0.4 0.45 0.5 0.55 0.6 0.65

(a) (b)

FigureFigure 9. 9. InfluenceInfluence of of the the discretization discretization scheme scheme choice on the temperaturetemperature fieldfield scattering,scattering, hexahedralhexahedral elements,elements, node-based node-based method method for for the the transport transport quantity quantity gradients, gradients, y = y0.5= m,0.5 z m,= 0.125z = 0.125m: (a) m:25 points/meter;(a) 25 points/ meter;(b) 100 ( bpoints/meter.) 100 points/meter.

FigureFigure 1010 showsshows the the temperature temperature diagrams diagrams only only for the for fine the tetrahedral fine tetrahedral mesh (all mesh upwind (all schemes) upwind schemes)but the record but the on record the horizontalon the horizontal axis in axis the in higher the higher layer layer is also is also added added (y = (y0.9 = 0.9 m, m,z z= =0.125 0.125 m).m). TheThe Green–Gauss Green–Gauss node-based node-based method method was was chosen for the transporttransport quantityquantity gradientsgradients inin allall casescases presentedpresented here. here. It It is is in in this this case, case, when the differen differentt temperatures limitlimit isis atat thethe greatergreater distancedistance fromfrom thethe entrance entrance to to the the calculation calculation area (x = 0.90.9 m). m). It It is is evident evident from from the graphs that the mutual didifferencesfferences inin the the results results between between the the selected selected schemes remain very similarsimilar alongalong thethe entireentire lengthlength ofof thethe sharpsharp boundaryboundary of of the the different different temperatures. The The numeri numericalcal diffusion diffusion increases slightly atat thethe greatergreater distancedistance from from the the entrance entrance to to the area (y == 0.9 m). The results give onlyonly slightslight didifferencesfferences usingusing thethe tetrahedraltetrahedral mesh mesh except except of of the the first-order first-order upwind scheme.scheme. The smallest temperaturetemperature fieldfield scatteringscattering is is nownow shown shown the the third-order third-order MUSCL, MUSCL, and and this positivity is slightly highlighted at the greatgreat distance.distance. ThisThis also also applies applies to to the the polyhedral polyhedral mesh. It It will will be be discussed discussed in more detail in the following chapters.

400 400 400 y = 0.5 m; Tetra 400 y = 0.9 m; Tetra T [K] y = 0.5 m; Tetra T [K] y = 0.9 m; Tetra 100 cells/m 100 cells/m 350 350 Node-based 350 Node-based x [m] 300 x [m] x [m] 300 300 0.45 0.475 0.5 0.525 0.55 0.45 0.475 0.5 0.525 0.55 0.85 0.875 0.9 0.925 0.95 First-order 0.5 QUICK 0.5 First-order 0.9 QUICK 0.9 Second-order 0.5 Third-orderL 0.5 Second-order 0.9 Third-order 0.9 Second-order 0.5 Third-orderL 0.5 Second-order 0.9 Third-order 0.9 (a) (b)

FigureFigure 10. 10. InfluenceInfluence of of the the discretization discretization scheme scheme choice choice on the temperature fieldfield scattering,scattering, tetrahedraltetrahedral elements,elements, node-based node-based method method for for the the transp transportort quantity quantity gradients, gradients, 100 100 points/meter, points/meter, z = 0.125z = 0.125 m: (a m:) y =( a0.5) y m;= 0.5 (b)m; y = ( b0.9) y m.= 0.9 m.

ItIt is is apparent apparent that that the the calculation in all casescases usingusing thethe first-orderfirst-order upwindupwind schemescheme createscreates aa significantlysignificantly larger larger scattering scattering of the transport quantityquantity inin comparingcomparing withwith thethe otherotherschemes. schemes. AlthoughAlthough thethe higher-order higher-order discretization discretization schemes schemes give give the the sm smallall mutual didifferences,fferences, butbut thethe resultsresults areare moremore pronouncedpronounced for for the the different different choice of thethe methodsmethods ofof thethe transporttransport quantityquantity gradients.gradients. ThisThis willwill bebe evaluatedevaluated in in the the next next section. section.

Sustainability 2020, 12, x FOR PEER REVIEW 10 of 18 SustainabilitySustainability 2020 2020, ,12 12, ,x x 10173 FOR FOR PEER PEER REVIEW REVIEW 10 1010 of of of 18 1819

4.3.4.3. EvaluationEvaluation ofof thethe TransportTransport QuantiQuantityty GradientsGradients onon thethe NumericalNumerical DiffusionDiffusion 4.3. Evaluation of the Transport Quantity Gradients on the Numerical Diffusion TheThe influenceinfluence ofof thethe methodmethod choicechoice forfor thethe transptransportort quantityquantity gradientsgradients solvingsolving ofof thethe numericalnumerical diffusiondiffusionThe is influenceis presented presented of theherehere method for for all all fine fine choice mesh mesh for types types the transport (100 (100 cells/meter). cells/meter). quantity gradientsThe The temperature temperature solving change change of the numerical record record for for yydi ==ff 0.9;0.9;usion zz == is 0.120.12 presented mm isis shownshown here in forin FiguresFigures all fine 11–15. mesh11–15. types TheThe first-first- (100orderorder cells / upwindmeter).upwind The schemescheme temperature doesdoes notnot change dependdepend record onon thethe ∇ gradientgradientfor y = 0.9; ofof z thethe= 0.12 transporttransport m is shown quantityquantity in Figures ∇∇ΦΦ,, asas 11 TableTable–15. The 22 waswas first-order shown.shown. upwind TheThe temperaturetemperature scheme does fieldfield not scatteringscattering depend on isis significantthe gradient (it of was the also transport shown quantity in SectionΦ 4.2.), as Tableand it2 wasvaries shown. only due The to temperature the mesh type field change scattering (Figure is significant (it was also shown in Section∇ 4.2.) and it varies only due to the mesh type change (Figure 11).11).significant (it was also shown in Section 4.2.) and it varies only due to the mesh type change (Figure 11).

400 400400 T [K] First-order 390 TT [K] [K] First-orderFirst-order 390390 upwind scheme 380 upwindupwind scheme scheme 380380 370 370370 360 360360 350 350350 340 340340 330 330 Cell Hexa Cell Terta Cell Poly 320 CellCell Hexa Hexa CellCell Terta Terta CellCell Poly Poly 320320 Node Hexa Node Terta Node Poly NodeNode Hexa Hexa NodeNode Terta Terta NodeNode Poly Poly 310 Least Hexa Least Terta Least Poly 310 LeastLeast Hexa Hexa LeastLeast Terta Terta LeastLeast Poly Poly 300 x[m] 300300 x[m]x[m] 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 0.80.8 0.82 0.82 0.84 0.84 0.86 0.86 0.88 0.88 0.9 0.9 0.92 0.92 0.94 0.94 0.96 0.96 0.98 0.98 1 1 Figure 11. First-order upwind scheme: influence of the method choice for the solving of the transport FigureFigure 11.11. First-orderFirst-order upwind upwind scheme: scheme: infl influenceinfluenceuence of ofof the the methodmethod choicechoice forfor thethe solving solvingsolving of ofof the thethe transport transporttransport quantity gradients and the mesh type on the numerical diffusion; 100 cells/m; y = 0.9; z = 0.125 m. quantityquantity gradients gradients and and the the mesh mesh type type on on the the numerical numerical diffusion; didiffusion;ffusion; 100 100 cells/m; cellscells/m;/m; y y = = 0.9; 0.9; z z = = 0.125 0.1250.125 m. m. m.

400 400400 T [K] 390 TT [K] [K] Hexahedral mesh 390390 HexahedralHexahedral mesh mesh 380 380380 370 370370 360 360360 350 350350 340 340340 330 330 Sec. Cell QUICK Cell Third Cell 320 Sec.Sec. Cell Cell QUICKQUICK Cell Cell ThirdThird Cell Cell 320320 Sec. Node QUICK Node Third Node Sec.Sec. Node Node QUICKQUICK Node Node ThirdThird Node Node 310 Sec. Least QUICK Least Third Least 310 Sec.Sec. Least Least QUICKQUICK Least Least ThirdThird Least Least 300 x[m] 300300 x[m]x[m] 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 0.80.8 0.82 0.82 0.84 0.84 0.86 0.86 0.88 0.88 0.9 0.9 0.92 0.92 0.94 0.94 0.96 0.96 0.98 0.98 1 1 Figure 12. Hexahedral mesh: influence of the method choice for the solving of the transport quantity FigureFigure 12. 12. HexahedralHexahedral mesh: mesh: influenceinfluence influence of of the the method methodmethod choichoicechoicece forfor thethe solvingsolving of ofof the thethe transport transporttransport quantity quantityquantity gradients and the mesh type on the numerical diffusion; 100 cells/m; y = 0.9; z = 0.125 m. gradientsgradients and and the the mesh mesh type type on on th thethee numerical numerical diffusion; didiffusion;ffusion; 100 100 cells/m; cellscells/m;/m; y yy = = 0.9; 0.9; z z = = 0.125 0.1250.125 m. m. m.

400 400400 390 T [K] Tetrahedral mesh 390390 T [K] TetrahedralTetrahedral mesh mesh 380 380380 370 370370 360 360360 350 350350 340 340340 330 330330 Sec. Cell QUICK Cell Third Cell 320 Sec. Cell QUICK Cell Third Cell 320320 Sec. Node QUICK Node Third Node 310 Sec. Node QUICK Node Third Node 310310 Sec. Least QUICK Least Third Least 300 Sec. Least QUICK Least Third Leastx[m] 300300 x[m]x[m] 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 0.80.8 0.82 0.82 0.84 0.84 0.86 0.86 0.88 0.88 0.9 0.9 0.92 0.92 0.94 0.94 0.96 0.96 0.98 0.98 1 1 Figure 13. Tetrahedral mesh:mesh: influenceinfluence ofof thethe methodmethod choicechoice forfor thethesolving solving of of the the transport transport quantity quantity FigureFigure 13.13. TetrahedralTetrahedral mesh:mesh: influenceinfluence ofof thethe methodmethod choichoicece forfor thethe solvingsolving ofof thethe transporttransport quantityquantity gradients and the mesh type on thethe numerical didiffusion;ffusion; 100100 cellscells/m;/m; yy == 0.9; z = 0.1250.125 m. m. gradientsgradients and and the the mesh mesh type type on on th thee numerical numerical diffusion; diffusion; 100 100 cells/m; cells/m; y y = = 0.9; 0.9; z z = = 0.125 0.125 m. m. Sustainability 2020, 12, 10173 11 of 19

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10 10 [%] Hexahedral mesh [%] Second-order scheme 8 8

6 6

4 4

2 2 x [m] 0 x [m] 0 0.8 0.825 0.85 0.875 0.9 0.925 0.95 0.975 1 0.8 0.825 0.85 0.875 0.9 0.925 0.95 0.975 1 -2 -2 Sec. Cell QUICK Cell Third Cell Hexa Cell Tetra Cell Poly Cell Sec. Node QUICK Node Third Node Hexa Node Tetra Node Poly Node Sec. Least QUICK Least Third Least Hexa Least Tetra Least Poly Least

10 10 [%] Tetrahedral mesh [%] QUICK scheme 8 8

6 6

4 4

2 2 Sustainability 2020, 12, x FOR PEER REVIEW x [m] 12x[m] of 18 0 0 0.8 0.825 0.85 0.875 0.9 0.925 0.95 0.975 1 0.8 0.825 0.85 0.875 0.9 0.925 0.95 0.975 1 achieved-2 using the third-order MUSCL with the Green–Gauss-2 node-based method option. The second- Sec. Cell QUICK Cell Third Cell Hexa Cell Tetra Cell Poly Cell order upwindSec. Node and the QUICKQUICK Node schemes withThird Node options, the Green–GaussHexa Node node-based,Tetra Node and thePoly Nodeleast squares Sec.cell-based Least methods,QUICK usingLeast this gridThird Leastshow good, almostHexa Least identical Tetraresults, Least in termsPoly of Least the 10 10 diffusivity[%] on their accuracy, and is Polyhedralcomparable mesh to the QUICK[%] hexahedral mesh scheme.Third-order scheme 8 The percentage expression of the of the transport quantity8 error value from the ideal sharp limit in6 the line y = 0.9 m is shown in Figure 14. The area in the6 immediate vicinity of this limit x∈ (0.88 m, 0.924 m) is not evaluated. The reason is the discontinuous4 function of ideal values. Figure 14a (left)

presents2 the effect of different combinations of the calculation2 parameters on the specific meshes. The percentage expression of the error from the idealx [m] value is in accordance with the real temperaturex [m] 0 0 diagram0.8 0.825 in Figures 0.85 0.87512, 13, 0.9or 15. 0.925 Figure 0.95 14b 0.975 (right) 1 shows 0.8the effect 0.825 0.85of the 0.875 mesh 0.9 type 0.925 and 0.95 the 0.975 method 1 for-2 the solving of the transport quantity gradient on the -2results of the chosen upwind scheme. Sec. Cell QUICK Cell Third Cell Hexa Cell Tetra Cell Poly Cell PolyhedralSec. Node mesh (FiguresQUICK Node 14 and 15): Thirdthe Nodenumerical diffusionHexa Node using theTetra polyhedral Node meshPoly is Node not affected Sec.by Least the choice of QUICKthe methodsLeast for Thirdthe Leastsolving of the Hexatransport Least quantityTetra Leastgradient. ThePoly smallLeast (a) (b) differences arise only by changing of the discretization scheme. The second-order upwind and the QUICKFigure schemes 14. Percentage show the values same of results, the error where from the the te idealmperature sharp limitfield ofscattering the temperature is larger field than in the solvingthe immediate used the vicinitythird-order of this MUSCL. limit: (a )Compared combination to ofother calculation meshes, parameters the results on show specific the meshes; average diffusivity.(b) combination of calculation parameters for a specific upwind scheme; 100 cells/m; y = 0.9; z = 0.125 m.

400 T [K] 390 Polyhedral mesh 380 370 360 350 340 330 320 Sec. Cell QUICK Cell Third Cell Sec. Node QUICK Node Third Node 310 Sec. Least QUICK Least Third Least 300 x[m] 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 FigureFigure 15. 15.Polyhedral Polyhedral mesh:mesh: influenceinfluence of the method choi choicece for for the the solving solving of of the the transport transport quantity quantity gradientsgradients and and the the mesh mesh typetype onon thethe numerical diffusion; diffusion; 100 100 cells/m; cells/m; y y= =0.9;0.9; z =z 0.125= 0.125 m. m.

4.4.The Values results of the of Transport the other Quantities discretization outside schemesthe Range proveof the Input the certain Parameters dependence from the choice of the methodThe choices for the of solving the discretization transport quantityscheme and gradients the method and theyfor the di ffsolvinger according of the transport to the mesh quantity type. gradient affect to the formation of the transport quantity values outside the range of the input parameters; it is also affected in the accuracy of the numerical calculations. The lowest temperature deflections occur using of the hexahedral mesh. Their values outside the boundary conditions T∈ (300 K, 400 K) are documented in Figure 16. The results differ significantly in comparing with the results using the other mesh; therefore, the scale in this case is ten times smaller than the following images. The highest deflections occurred using the dense mesh (100 cells/m) using the third-order MUSCL with the Green–Gauss cell-based method. The frequency of these deflections in the median plane of the calculation area (plane z = 0.125 m, see Figure 3) is shown in Figure 17a. Although the frequency is relatively high, these deflections can be considered negligible due to their minimum values. Figure 17b shows the frequency of the deflections using the Second-order upwind scheme with the Green–Gauss node-based; this shows the second highest values using the hexahedral mesh. Sustainability 2020, 12, 10173 12 of 19

Hexahedral mesh (Figures 12 and 14): the choice of the method for the solving transport quantity gradients affects the results in the combination using the Second-order upwind and the third-order MUSCL. The Green–Gauss node-based method shows significantly larger scattering of the temperature field in comparing with the other two methods. The greatest numerical diffusion arises using the second-order upwind scheme and all methods for solving the transport quantity gradients. The highest diffusivity using the hexahedral mesh (second-order upwind scheme and Green–Gauss node-based method) is comparable to the same calculation using the tetrahedral mesh (Figure 13). The calculation using the QUICK scheme, using the hexahedral mesh, is one of the most accurate, and the choice method for solving the transport quantity gradients in this case will not affect the results. Tetrahedral mesh (Figures 13 and 14): the method for solving the transport quantity gradient using the tetrahedral mesh affects the results for all discretization schemes, and the differences are more pronounced than used the hexahedral mesh. In the case of the Green–Gauss cell-based method choice, the numerical calculations have almost identical results in all discretization schemes and their diffusivity is the most pronounced. Conversely, the smallest scattering of the temperature field is achieved using the third-order MUSCL with the Green–Gauss node-based method option. The second-order upwind and the QUICK schemes with options, the Green–Gauss node-based, and the least squares cell-based methods, using this grid show good, almost identical results, in terms of the diffusivity on their accuracy, and is comparable to the QUICK hexahedral mesh scheme. The percentage expression of the of the transport quantity error value from the ideal sharp limit in the line y = 0.9 m is shown in Figure 14. The area in the immediate vicinity of this limit x (0.88 m, 0.92 m) is not evaluated. The reason is the discontinuous function of ideal values. Figure 14a ∈ (left) presents the effect of different combinations of the calculation parameters on the specific meshes. The percentage expression of the error from the ideal value is in accordance with the real temperature diagram in Figure 12, Figure 13, or Figure15. Figure 14b (right) shows the e ffect of the mesh type and the method for the solving of the transport quantity gradient on the results of the chosen upwind scheme. Polyhedral mesh (Figures 14 and 15): the numerical diffusion using the polyhedral mesh is not affected by the choice of the methods for the solving of the transport quantity gradient. The small differences arise only by changing of the discretization scheme. The second-order upwind and the QUICK schemes show the same results, where the temperature field scattering is larger than the solving used the third-order MUSCL. Compared to other meshes, the results show the average diffusivity.

4.4. Values of the Transport Quantities outside the Range of the Input Parameters The choices of the discretization scheme and the method for the solving of the transport quantity gradient affect to the formation of the transport quantity values outside the range of the input parameters; it is also affected in the accuracy of the numerical calculations. The lowest temperature deflections occur using of the hexahedral mesh. Their values outside the boundary conditions T (300 K, 400 K) are documented in Figure 16. The results differ significantly ∈ in comparing with the results using the other mesh; therefore, the scale in this case is ten times smaller than the following images. The highest deflections occurred using the dense mesh (100 cells/m) using the third-order MUSCL with the Green–Gauss cell-based method. The frequency of these deflections in the median plane of the calculation area (plane z = 0.125 m, see Figure3) is shown in Figure 17a. Although the frequency is relatively high, these deflections can be considered negligible due to their minimum values. Figure 17b shows the frequency of the deflections using the Second-order upwind scheme with the Green–Gauss node-based; this shows the second highest values using the hexahedral mesh. Sustainability 2020, 12, 10173 13 of 19 Sustainability 2020, 12, x FOR PEER REVIEW 13 of 18 Sustainability 2020,, 12,, xx FORFOR PEERPEER REVIEWREVIEW 13 13 of of 18 18

T [K] T [K] T [K] Hexahedral mesh ; 25 cells/meter T [K] Hexahedral mesh ; 100 cells/meter 0.7T [K] Hexahedral mesh ; 25 cells/meter 0.7T [K] Hexahedral mesh ; 100 cells/meter 0.7 0.7 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 First; Cell First; Cell Third; Cell Third; Cell First; Cell First; Cell First; Least First; Least First; Cell First; Node First; Cell First; Node Third; Cell Third; Cell QUICK; Cell QUICK; QUICK; Cell QUICK; Third; Least Third; Least Third; Cell Third; Node Third; Cell Third; Node First; Least First; Least First; Node Second; Cell First; Node Second; Cell First; Least QUICK; Cell QUICK; First; Least QUICK; Cell QUICK; First; Node First; Node Third; Least Third; Least QUICK; Least QUICK; QUICK; Least QUICK; Third; Node QUICK; Cell QUICK; Third; Node QUICK; Node QUICK; QUICK; Cell QUICK; QUICK; Node QUICK; Second; Cell Third; Least Second; Cell Third; Least Third; Node Third; Node Second; Least Second; Least Second; Cell Second; Cell Second;Node Second;Node QUICK; Least QUICK; QUICK; Least QUICK; QUICK; Node QUICK; QUICK; Node QUICK; QUICK; Least QUICK; QUICK; Least QUICK; QUICK; Node QUICK; QUICK; Node QUICK; Second; Least Second; Least Second;Node Second;Node Second; Least Second; Least Second;Node Extreme below 300 K Extreme above 400 K Extreme below 300 Second;Node K Extreme above 400 K Extreme below 300 K Extreme above 400 K Extreme below 300 K Extreme above 400 K (a) (b) (a) (b) Figure 16. Hexahedral mesh: the extreme values of the transport quantities outside the range of the FigureFigure 16. 16. HexahedralHexahedralHexahedral mesh:mesh: mesh: thethe the extremeextreme extreme valuesvalues ofof the the transport quantities outsideoutside thethe rangerange ofof thethe correct solution: (a) coarse mesh ; (b) fine mesh. correctcorrect solution: solution: (a (a) )coarse coarse mesh mesh; ; ( (bb)) fine fine mesh. mesh.

(a) (b) (c) (d) (a) (b) (c) (d) Figure 17. Frequency of the transport quantity deflections outside the range of input parameters (white FigureFigure 17. 17. FrequencyFrequencyFrequency ofof of thethe the transporttransport transport quantityquantity quantity deflectiondeflection deflectionss outside the range ofof inputinput parametersparameters (white(white field), fine mesh: (a) hexahedral, third-order MUSCL; (b) hexahedral, second-order; (c) tetrahedral; (d) field),field), fine fine mesh: mesh: (a ()a hexahedral,) hexahedral, third-order third-order MUSCL; MUSCL; (b) ( bhexahedral,) hexahedral, second-order; second-order; (c) (tetrahedral;c) tetrahedral; (d) polyhedral. polyhedral.(d) polyhedral.

TheThe significant significant extreme extreme deflections deflections of the specifiedspecified temperature fieldfield valuesvalues forfor solvingsolving thethe The significant extreme deflections of the specified temperature field values for solving the transporttransport quantity quantity gradient gradient using using the the tetrahedral tetrahedral me meshsh occur, occur, the third-order MUSCL for all methods. transport quantity gradient using the tetrahedral mesh occur, the third-order MUSCL for all methods. ItIt is is markedly markedly for for the the coarse mesh, see Figure 1818a.a. The deflectiondeflection values using thethe finefine meshmesh areare inin It is markedly for the coarse mesh, see Figure 18a. The deflection values using the fine mesh are in FigureFigure 18b.18b. The The first first and and second-order second-order upwind and the QUICK schemesschemes havehave thethetargeted targeted 10 10 × smallersmaller Figure 18b. The first and second-order upwind and the QUICK schemes have the targeted 10× × smaller scalescale (identical (identical with with the the hexahedral), hexahedral), becaus becausee the the temperature temperature values values outside outside the the range range T∈ T(300(300 K, 400 K, scale (identical with the hexahedral), because the temperature values outside the range T∈ (300∈ K, 400 K)400 are K) in are order in order of the of magnitude the magnitude lower lower (maximum (maximum ±0.4 K)0.4 than K) thanusing using the third-order the third-order MUSCL. MUSCL. This K) are in order of the magnitude lower (maximum ±0.4 ±K) than using the third-order MUSCL. This schemeThis scheme has the has scale the scaleidentical identical to the to coarse the coarse mesh. mesh. scheme has the scale identical to the coarse mesh.

T [K] Tetrahedral mesh ; 25 cells/meter T [K] Tetrahedral mesh ; 100 cells/m 7.0T [K] Tetrahedral mesh ; 25 cells/meter 0.7T [K] Tetrahedral mesh ; 100 cells/m 7.0 7.0 0.7 7.0 7.0 0.6 6.0 6.0 0.6 6.0 6.0 0.5 5.0 5.0 0.5 5.0 5.0 0.4 4.0 4.0 0.4 4.0 4.0 0.3 3.0 3.0 0.3 3.0 3.0 0.2 2.0 2.0 0.2 2.0 2.0 0.1 1.0 1.0 0.1 1.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 First; Cell Third; Cell First; Cell First; Least First; Cell First; Node Third; Cell QUICK; Cell QUICK; First; Cell Third; Least Third; Cell Third; Node First; Least First; Node Second; Cell Third; Cell QUICK; Cell QUICK; First; Least First; Cell First; Node Third; Least QUICK; Least QUICK; Third; Node QUICK; Cell QUICK; First; Least QUICK; Node QUICK; First; Cell First; Node Second; Cell Third; Least Third; Cell Third; Node QUICK; Cell QUICK; Second; Least Second; Cell Second;Node Third; Least QUICK; Least QUICK; Third; Cell Third; Node First; Least QUICK; Node QUICK; First; Node Second; Cell QUICK; Least QUICK; QUICK; Cell QUICK; First; Least QUICK; Node QUICK; Second; Least First; Node Second;Node Third; Least QUICK; Least QUICK; Third; Node QUICK; Cell QUICK; QUICK; Node QUICK; Second; Least Second; Cell Second;Node Third; Least Third; Node Second; Least Second; Cell Second;Node QUICK; Least QUICK; QUICK; Node QUICK; QUICK; Least QUICK; QUICK; Node QUICK; Second; Least

Second;Node Extreme above 400 K Second; Least Second;Node Extreme above 400 K Extreme below 300 K Extreme above 400 K Extreme below 300 K Extreme below 300 K Extreme above 400 K Extreme below 300 K (a) (b) (a) (b) FigureFigure 18. 18. TetrahedralTetrahedral mesh: mesh: the the extreme extreme va valueslues of the transport quantitiesquantities outsideoutside thethe rangerange ofof thethe Figure 18. Tetrahedral mesh: the extreme values of the transport quantities outside the range of the correctcorrect solution: solution: (a (a) )coarse coarse mesh; mesh; (b (b) )fine fine mesh. mesh. correct solution: (a) coarse mesh; (b) fine mesh. Sustainability 2020, 12, 10173 14 of 19 Sustainability 2020, 12, x FOR PEER REVIEW 14 of 18

TheThe absolute absolute highest highest deflections deflections of the temperature values occuroccur forfor thethe calculationcalculation usingusing thethe Green–GaussGreen–Gauss node-based node-based method, method, both both the the fine fine and the coarse mesh, andand thisthis isis thethe highesthighest valuevalue exceedanceexceedance of all variants ofof TT= = 400400 K,K,see see Figure Figure 18 18.. The The frequency frequency of of the the deflections deflections for for this this case case is in is inFigure Figure 17 17c.c. Using the polyhedral mesh gives almost zero exceeding values in the range of T ∈ (300 K, 400 K) Using the polyhedral mesh gives almost zero exceeding values in the range of T∈ (300 K, 400 K) exceptexcept for for the the third-order MUSCL. This applies to allall methodsmethods forfor solvingsolving thethe transporttransport quantityquantity gradient;gradient; see see Figure Figure 19. 19 The. The highest highest deflections deflections occur occur for the for calculation the calculation using using the Green–Gauss the Green–Gauss node- basednode-based method. method. Figure Figure17d shows 17d showsthe frequency the frequency of the deflections. of the deflections.

T [K] Polyhedral mesh ; 25 cells/meter T [K] Polyhedral mesh ; 100 cells/meter 7.0 7.0 6.0 6.0 5.0 5.0 4.0 4.0 3.0 3.0 2.0 2.0 1.0 1.0 0.0 0.0 First; Cell First; Cell Third; Cell Third; Cell First; Least First; Least First; Node First; Node QUICK; Cell QUICK; QUICK; Cell QUICK; Third; Least Third; Least Third; Node Third; Node Second; Cell Second; Cell QUICK; Least QUICK; QUICK; Least QUICK; QUICK; Node QUICK; QUICK; Node QUICK; Second; Least Second; Least Second;Node Second;Node Extreme below 300 K Extreme above 400 K Extreme below 300 K Extreme above 400 K (a) (b)

FigureFigure 19. 19. PolyhedralPolyhedral mesh: mesh: the the extreme extreme values values of the transport quantitiesquantities outsideoutside thethe rangerange ofof thethe correctcorrect solution: solution: ( (aa)) coarse coarse mesh; mesh; ( (bb)) fine fine mesh. mesh.

5.5. Discussion Discussion 5.1. Mesh Type 5.1. Mesh Type Hexahedral mesh: Hexahedral mesh: Using the correct combination of the calculation parameters achieve the most accurate results of •• Using the correct combination of the calculation parameters achieve the most accurate results of all tested tasks, but the results are diffusive with incorrect settings (for more see Section 5.2). all tested tasks, but the results are diffusive with incorrect settings (for more see Section 5.2). The lowest deflection of the entered values of the transport quantity Φ occur on this mesh. •• The lowest deflection of the entered values of the transport quantity Φ occur on this mesh. • The calculations are less sensitive to the choice of methods for solving the transport quantity • The calculations are less sensitive to the choice of methods for solving the transport quantity gradientgradient than than using using the the tetrahedral tetrahedral mesh, unfortunatel unfortunately,y, inin manymany cases,cases, itit isis notnot possiblepossible toto createcreate thethe hexahedral hexahedral cells cells for for comp complexlex geometries geometries separately. separately. TetrahedralTetrahedral mesh: mesh: • The disadvantage is the large number of cells and, thus, the greater complexity of calculations. • The disadvantage is the large number of cells and, thus, the greater complexity of calculations. • The sensitivity to the choice of calculation parameters is high, but with the correct combination, • The sensitivity to the choice of calculation parameters is high, but with the correct combination, cancan achieve achieve good good results results (fo (forr more more see see Section Section 5.2).5.2). PolyhedralPolyhedral mesh: mesh: • InIn many many cases, cases, it it is is recommended recommended for great savings in the number ofof cellscells and,and, thus,thus, lessless didifficultyfficulty • inin the the calculations. calculations. • TheThe accuracy accuracy of of the the solutions solutions is average to belowbelow average inin thisthis submittedsubmitted studystudy (see(see SectionSection 5.2 5.2 • forfor more more information). information).

5.2.5.2. Discretization Discretization Scheme Scheme and and the the Method Method for for Solving Solving the the Transport Transport Quantity Quantity Gradients First-orderFirst-order upwind upwind scheme: scheme: • ProducesProduces very very diffusive diffusive results, but it is easy to implement and its calculations are very stable. • • This scheme is convenient for starting calculations. Sustainability 2020, 12, 10173 15 of 19

This scheme is convenient for starting calculations. • Second-order upwind scheme: It is more accurate than the first-order upwind scheme, but it is generally stated that it can lead to • the values—that they are outside the range of the input in the regions with the strong gradients [38]. It is reflected in both the hexahedral and the tetrahedral mesh, but the deflections of the values are very low (see Figures 16 and 18), and the frequency of the occurrence does not exceed the other cases (see Figure 17b). This scheme achieves the weakest results among the higher order schemes using the • hexahedral mesh. Conversely, the solution using the tetrahedral mesh with the choice of the node-based and the • least methods, together with the QUICK scheme, is relatively accurate, and comparable to the most accurate, using the hexahedral mesh (see Figures 12–14). QUICK scheme: The scheme, stated in [37], is suitable mainly for the structured hexahedral meshes, but it can be • used for the unstructured or hybrid meshes. Using the hexahedral mesh, this scheme presents the most accurate results of all tested variants, • regardless of the choice of the method for solving the transport quantity gradient. The QUICK scheme, using the tetrahedral mesh, gives the results in terms of the size of the • numerical diffusion comparable to the hexahedral mesh with the combination the Green–Gauss node-based or the least squares cell-based methods (Figures 12–14), but there are slightly higher values of the temperature field value deflections. (Figures 16 and 18). The combination of the QUICK scheme with the Green–Gauss cell-based method for solving the • transport quantity gradient is not suitable for the tetrahedral mesh; the results are very diffusive. Third-order MUSCL: The scheme has the potential to improve the spatial accuracy for all types of grids by reducing • the numerical diffusion and it is available for all transport equations. This is confirmed only by the numerical calculations using the tetrahedral mesh in combination with the Green–Gauss node-based method. However, the occurrence of the values outside the correct solution is high (see Figure 13, Figure 14 or Figure 18), and the high deflections are also for the other methods. This scheme shows the high deflections of the values of the transport quantity for the tetrahedral • and the polyhedral meshes in all methods for solving the transport quantity gradient (see Figures 18 and 19). The recommended combinations of the calculation parameters are in Table3. The unrecommended settings are listed in Table4.

Table 3. Recommended combinations of the calculation parameters settings for three mesh types.

Method for Discretization Numerical Mesh Type Solving Transport Out Range Values Scheme Diffusion Quantity Gradient Method does not Lowest of all Hexahedral QUICK scheme Significantly low affect variants Green–Gauss Second-order and node-based, or Tetrahedral Small Low QUICK schemes least squares cell-based Second-order and Method does not United Low Polyhedral QUICK schemes affect Method does not Third-order Average High affect Sustainability 2020, 12, 10173 16 of 19

Table4. Inappropriate combinations of the calculation parameter for hexahedral and tetrahedral meshes.

Discretization Mesh Type Explanation Scheme High diffusion, the largest for Green–Gauss node-based choice, Second-order scheme Hexahedral values outside the input range are low, Figurs 12, 14, and 16. High diffusion for Green–Gauss node-based choice, values Third-order scheme outside the input range significantly high, Figurs 12, 14, and 16. Height diffusion for Green–Gauss cell-based choice, Figurs 13 Tetrahedral All schemes and 14.

6. Conclusions The presented work compares the physical accuracy of the calculation in the CFD code in the Ansys Fluent software using the offered discretization calculation schemes, the methods of the transport quantity gradients solving on the cell faces, and the influence of the mesh type. The sharp boundary of two areas with the different temperatures is created in the direction of the flow direction in the calculation area. The FVM simulates the 3D stationary flow of the fictitious gas so that only the advective transfer in the terms of the momentum and the heat arises. Ideally, the diffuse transmission should not occur. The level of the scattering of the temperature field (numerical diffusion) against the original sharp boundary of the two areas is monitored and also in which cases the values of the transport quantity (temperature) appear outside the range of the specified boundary conditions, i.e., outside the T range (300 K, 400 K). The frequencies of the deflections and the value deflections from ∈ the correct solution are evaluated. The article offers the options for reducing of the numerical errors by setting the correct combination of the calculation parameters. The recommended combinations of the calculation parameters settings for the numerical modeling of the airflow effect on the buildings, including the partial results, are described in detail in the Results section. Their global summary is commented in the Discussion section. The results of the presented study contribute to the development of the methodology for the numerical studies focused on the sustainable architecture. The conclusions will be used in solving specific problems of construction engineering practices, for example, in solving the velocity of substances in the porous domain of building materials and soils. Final practical summary: The coarse mesh is to be clearly more accurate. • The first-order upwind scheme does not depend on the choice of the methods for the solving • of the transport quantity gradient, it produces the very diffusive results, but the calculation is stable, and it is suitable for starting calculations that are more complex. When calculating with the first-order upwind scheme, the values of the transport quantity Φ do not occur outside the input range. The QUICK scheme shows the best results with the low diffusivity, even with the low values Φ • outside the input range. The effect of the choice of the methods for solving the transport quantity gradient is manifested only on the tetrahedral mesh. Although the third-order MUSCL is able to calculate the acceptable diffusivity, the values Φ • outside the input range are significantly high compared to other discretization schemes. The Green–Gauss cell-based method for the solving the transport quantity gradient is not suitable • for the tetrahedral mesh; it shows the diffusive results for all discretization schemes except the first-order upwind. The Green–Gauss node-based method for the solving of the transport quantity gradient is not • suitable for the hexahedral mesh using the second-order upwind and the third-order MUSCL. It shows the diffusive results with these schemes. The advantage of the polyhedral mesh with the low number of cells and, thus, less computational • complexity, is negatively balanced by the higher diffusivity in all discretization schemes. The effect Sustainability 2020, 12, 10173 17 of 19

of the method for solving the transport quantity gradient option has no effect on the results on this mesh.

Author Contributions: V.M. performed the numerical simulations and analysis of results. V.M. and K.K. participated in the design of this study and drafted the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by VŠB-TUO by the Ministry of Education, Youth, and Sports of the Czech Republic. Acknowledgments: Financial support from VSB-Technical University of Ostrava by means of the Czech Ministry of Education, Youth and Sports through the Institutional support for conceptual development of science, research, and innovations for the year 2020 and from the Scientific Grant Agency of the Ministry of Education of Slovak Republic and support from the Slovak Academy of Sciences the project VEGA 1/0374/19 are gratefully acknowledged. Conflicts of Interest: The authors declare no conflict of interest.

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