CFD Prediction of Airfoil Drag in Viscous Flow Using the Entropy Generation Method · 2019. 7. 30. · CFD Prediction of Airfoil Drag in Viscous Flow Using the Entropy Generation

Research ArticleCFD Prediction of Airfoil Drag in Viscous Flow Usingthe Entropy Generation Method

WeiWang JunWang Hui Liu and Bo-yan Jiang

School of Energy and Power Engineering Huazhong University of Science and Technology Wuhan 430074 China

Correspondence should be addressed to Jun Wang wangjhusthusteducn

Received 15 January 2018 Revised 6 March 2018 Accepted 13 March 2018 Published 16 May 2018

Academic Editor Eusebio Valero

Copyright copy 2018 Wei Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new aerodynamic force of drag prediction approach was developed to compute the airfoil drag via entropy generation rate inthe flow field According to the momentum balance entropy generation and its relationship to drag were derived for viscous flowModel equations for the calculation of the local entropy generation in turbulent flows were presented by extending the RANSprocedure to the entropy balance equation The accuracy of algorithm and programs was assessed by simulating the pressurecoefficient distribution and dragging coefficient of different airfoils under different Reynolds number at different attack angleNumerical data shows that the total entropy generation rate in the flow field and the drag coefficient of the airfoil can be related bylinear equation which indicates that the total drag could be resolved into entropy generation based on its physical mechanism ofenergy loss

1 Introduction

Accurate prediction of the aerodynamic force is a criticalrequirement in aircraft design CFD methods have beenwidely applied in aerodynamic design and optimization ofaircraft However even for airfoils with attached flow atrelatively low angles of attack the predicted drag based onintegration of the surface pressure and skin-friction distribu-tions can be off bymore than 100 even though the computedsurface pressure and skin friction are in good agreement withthe experimental data [1] Therefore designers might need acalculation method which could resolve the drag accordingto its mechanism of production

There are two common approaches to predict total dragforce of airfoils or wings a standard surface integrationmethod and a wake integration method The surface inte-gration method relies on calculations of pressures and skinfriction over a series of flat surfaces However the surfaceintegration was met with difficulties especially for the com-plex configuration due to the need to approximate the curvedsurfaces of the body with flat faces which can be affectedby significant errors introduced by the ldquonumerical viscosityrdquoand ldquodiscretizationrdquo error of the numerical solution [2] This

has led various researchers to look at the experimental wakeintegral method which is derived from the momentumequation of the governing equations of fluid mechanicsand to attempt to apply them to CFD computations [3 4]The main advantage of this technique is that no detailedinformation on the surface geometry of the configuration isrequired and also has the drag decomposition capability intowave profile and induced drag component Zhu et al [5]applied the wake integrationmethod to predict drags of someexamples including airfoil a variety of wings and wing-bodycombinationThey introduced the cutoff parameters methodto reduce the computational time required for integratingthe wake cross plane In the method cells in the wakecross plane which contain small proportions of vorticity andentropy are not included in the integration Numerical resultswere compared with those of traditional surface integrationmethod showing that the predicting drag values with thewake integration method are closer to the experimental data

Oswatitsch [6] derived a far-field formula of the entropydrag considering first-order effects in which the drag isexpressed as the flux of a function only dependent onentropy variations In [7 8] Oswatitschrsquos formula is usedfor computing the entropy drag in RANS solutions by

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4347650 15 pageshttpsdoiorg10115520184347650

2 Mathematical Problems in Engineering

limiting the far-field flux computation to a box enclosingthe aircraft By applying the divergence theorem of Gaussthe surface integral in Oswatitschrsquos formula can be replacedwith a volume integral Therefore the integrand can be setto zero a priori in regions where it is known that physicalentropy variations should be zero thus removing spuriouscontributions to drag In a viscous domain all drag on anairfoil is eventually realized as entropy generation so wecan establish a relationship between total entropy generationof the flow field and drag on the airfoil Li and Stewartreported on 2D validation studies for predicting drag bymeans of calculating the generation both for airfoils andfor viscous pipe flows [9] Stewart [10] and Monsch et al[11] calculated the drag force of a 3D wing by performinga volume integration within the numerical domain and asurface integral over the outlet of the domain

It is pointed out that a representation of losses in terms ofentropy generation offers significant insight into the flow andthermal transport phenomena over the airfoil and providesan effective tool for drag prediction Entropy analysis is amethod to evaluate a process based on the second law of ther-modynamics It is basically calculating entropy generation ina system and its surroundings and using it as a proxy forthe evaluation of the energy loss [12] Mahmud and Fraser[13] analyzed the mechanism of entropy generation and itsdistribution through fluid flows in basic channel configura-tions including two fixed plates and one fixed and onemovingplates by considering simplified or approximate analyticalexpressions for temperature and velocity distributions andderived analytically general expressions for the number ofentropy generation and Bejan number A detailed review ofentropy and its significance in CFD is presented by Kock andHerwig [14]

In this paper entropy generation and its relationshipto drag were derived for viscous flow we present modelequations for the calculation of the local entropy generationrates in turbulent flows by extending the RANS procedureto the entropy balance equation This equation serves toidentify the entropy generation sources without need to solvethe equation itself The main objective of this paper is todemonstrate the viability of entropy-based drag calculationmethod and to compare the consistency of predicting thedrag of single-element airfoils using surface integration wakeintegration and entropy generation integration

2 Alternative Methods forAirfoil Drag Prediction

The conservation law of momentum to the control volumeenclosing the configuration makes it possible to predict thetotal aerodynamic drag force by an integration of stresseson the aircraft configuration (surface integration) or byan integration of momentum flux on a closed surface farfrom the configuration (wake integration) We consider asteady flow with freestream velocity 119880infin pressure 119901infin andtemperature 119879infin around the configuration the only externalforce acting on the body is due to the fluid Neglecting bodyforces the fundamental formula for the total aerodynamic

force acting on the configuration in the Cartesian referencesystem can be derived as follows [1]

int119878[minus119901119899 + 120591 ∙ 119899 minus 120588 ( ∙ 119899)] 119889119878 = 0 (1)

where 119878 represents the entire control surface 119899 = 119899119909+119899119910+119899119911denotes the outward unit vector normal to 119878 = 119906 + V+ 119908denotes the velocity vector 120588 represents the fluid density and120591 represents the shear stress tensor Note that 119878 consists of 119878farand 119878body with 119878body specifies the aircraft surface and 119878far theclosed surface far from the configuration Consequently (1)can be written as

int119878body

[minus119901119899 + 120591 ∙ 119899 minus 120588 ( ∙ 119899)] 119889119878+ int119878far

[minus119901119899 + 120591 ∙ 119899 minus 120588 ( ∙ 119899)] 119889119878 = 0 (2)

Themomentumflux term 120588(∙119899) in the first integral is zeroin the general case of solid body surface Thus (2) reduces to

minus int119878body

[minus119901119899 + 120591 ∙ 119899] 119889119878= int119878far

[minus119901119899 + 120591 ∙ 119899 minus 120588 ( ∙ 119899)] 119889119878 (3)

where the integral on the left-hand side represents the totalaerodynamic drag force by an integration of stresses on theaircraft configuration which can be written as

119863 = minusint119878body

[minus119901119899 + 120591 ∙ 119899] 119889119878 (4)

The far-field expression of drag prediction is given by theright-hand side of (3)

119863 = int119878far

[minus119901119899 + 120591 ∙ 119899 minus 120588 ( ∙ 119899)] 119889119878 (5)

The wake expression for drag is derived from (5) bymoving the inlet and side faces of the control volume toinfinity and the following wake integration for the drag isobtained

119863 = minusint119878exit

[(119901 minus 119901infin) + (1205881199062 minus 1205881198802infin) minus 120591119909119909] 119889119878 (6)

where 119878exit is perpendicular to the freestream direction Thethird integral on the right-hand side of (6) is often neglectedif an integration plane sufficiently far down stream [4]Considering the conservation of the mass flux we have


[(119901 minus 119901infin) + 120588119906 (119906 minus 119880infin)] 119889119878 (7)

In a two-dimensional flow as we move the boundary 119878fartowards infinity (119878far rarr 119878infin) the pressure term vanishes asproved by Paparone and Tognaccini [4]

lim119878farrarr119878infin

int119878far

(119901119899) 119889119878 = 0 (8)

Mathematical Problems in Engineering 3

and in the far-field drag expressions the viscous stresses areusually neglected for a high Reynolds number flow (5) canbe simplified to

119863 = minusint119878far

[120588 ( ∙ 119899) ] 119889119878 (9)

For ideal gas the module of the velocity can be expressedin terms of variations of total enthalpy (Δℎ = ℎminusℎinfin) entropy(Δ119904 = 119904 minus 119904infin) and static pressure (Δ119901 = 119901 minus 119901infin) [4]

119906119880infin = radic1 + 2( Δℎ1198802infin) minus 2[(120574 minus 1)1198722infin] ((Δ119901119901infin + 1)(120574minus1)120574 exp (Δ119904119877) [(120574 minus 1) 120574] minus 1) = 119891(Δ119901119901infin Δ119904119877 Δℎ1198802infin) (10)

where 120574 is the ratio between the specific heats of the fluid and119877 is the gas constant for air Hence the drag can be expressedas the flux of 119891(Δ119901119901infin Δ119904119877 Δℎ1198802infin) across 119878far

119863 = minus119880infin int119878infin

119891(Δ119901119901infin Δ119904119877 Δℎ1198802infin)( ∙ 119899) 119889119878 (11)

Equation (10) can be expanded in Taylorrsquos series ignoringsecond-order term119906119880infin = 1 + 1198911199011 (Δ119901119901infin) + 1198911199041 (Δ119904119877 ) + 1198911198671 ( Δℎ1198802infin)

+ I[(Δ119901119901infin)2 (Δ119904119877 )2 ( Δℎ1198802infin)

2 ] (12)

the coefficients of the series expansion depend on 120574 and thefreestreamMach number (119872infin) they are

1198911199011 = minus 11205741198722infin 1198911199041 = minus 11205741198722infin 1198911198671 = 1

(13)

After substitution of expression (12) into (10) the contri-butions of pressure entropy and total enthalpy are isolatedMoreover all terms with Δ119901119901infin vanish on 119878infin Hence thefar-field drag expression around a body in two-dimensionalflows becomes119863

= minus119880infin int119878infin

[1198911199041 (Δ119904119877 ) + 1198911198671 ( Δℎ1198802infin)]120588 ( ∙ 119899) 119889119878+ I[(Δ119904119877 )2 ( Δℎ1198802infin)

2 ] (14)

The term depending on Δℎ is negligible in the caseof power-off condition and the far-field drag expression isrepresented by the entropy


[1198911199041 (Δ119904119877 )] 120588 ( ∙ 119899) 119889119878 (15)

Finally (15) can be expressed in volume integral form byapplying Gaussrsquos theorem in the domainΩ

119863 = minus119880infin intΩnabla ∙ 120588 [1198911199041 (Δ119904119877 )] 119889Ω

= minus119880infin1198911199041119877 intΩnabla ∙ [120588 (119904 minus 119904infin) ] 119889Ω (16)

The differential balance equation of the entropy 120588 ∙ nabla119904 =120588 119904gen leads to119863 = minus119880infin1198911199041119877 int

Ω119904gen119889Ω = 119879infin119880infin intΩ 119904gen119889Ω (17)

According to Gouy-Stodola theorem the relationshipbetween the exergy destruction and the entropy generationis defined by

119864119883des = 1198790 (119878gen) (18)

where 1198790 is the absolute temperature of the environment inKelvin [15] As a result of this theorem the amount of irre-versibility is directly proportional to the entropy generationand is responsible for the inequality sign in the second lawof thermodynamics Based on (17) the drag and entropygeneration rate can be related by a linear balance equationThis is essential because drag can be directly estimated fromentropy generation without other effects and the balance iscorrect only when the numerical volume completely enclosesthe entropy changes caused by the airfoil Equation (17)provides amethod to predict total drag force of airfoil directlyby performing a volume integration of entropy generationrate within the numerical domain

3 The CFD Procedure for Entropy Generation

31TheDirectMethod of Calculating EntropyGeneration RateThe entropy is a state variable and the transport equationfor entropy per unit volume in Cartesian coordinates can beexpressed as

120597119905 (120588119904) + 120597119895 (120588119906119895119904) + 120597119895 119902119895119879 = minus1199021198951205971198951198791198792 + 120591119894119895120597119895119906119894119879 (19)

where 119894 stands for the three directions (119894 = 1 2 3) and 119906119895denotes the velocity component in the direction [14]


There are basically two methods how entropy generationcan be determined [16] In the direct method the entropygeneration rate is connected with the entropy transportequation

119904gen = 1119879120591119894119895 120597119906119894120597119909119895 minus 1199021198961198792 120597119879120597119909119896 (20)

Equation (19) consists of two terms The first term denotesthe entropy generation due to viscous dissipation (119904gen119889)while the second term denotes the entropy generation dueto heat conduction (119904gen119888) The entropy generation terms arecalculated in the postprocessing phase of a CFD calculationbased on (20) That means they are determined by using theknown field quantities velocity and temperature Integrationof these field quantities over the whole flow domain results inthe overall entropy generation rate

In the indirect method the entropy generation is calcu-lated by equating it to the rest of (18)

119904gen = 1119879120591119894119895 120597119906119894120597119909119895 minus 1199021198961198792 120597119879120597119909119896⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟GENERATION

= 120597119905 (120588119904) + 120597119895 (120588119906119895119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟CONVECTION

+ 120597119895 119902119895119879⏟⏟⏟⏟⏟⏟⏟MOLECULAR FLUX

(21)

Since one is interested in the total entropy generation rate ofthe flow field (20) must be integrated over the entire flowdomain This corresponds to the fact that the global balancecan be cast into the following form

intΩ119904gen119889Ω

= intΩ(120597119905 (120588119904) + 120597119895 (120588119906119895119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

CONVECTION

+ 120597119895 119902119895119879⏟⏟⏟⏟⏟⏟⏟MOLECULAR FLUX

)119889Ω (22)

Obviously the direct method is superior and should beapplied in complex flow situations And there is one moreimportant advantage of this method [16] from the directmethod we get the information of how the overall entropygeneration is distributed an information which the indirectmethod cannot provide It may however help to understandthe physics of the complex process and be important infinding ways to reduce the overall entropy generation in atechnical device

When the turbulent flow is considered the derivation ofthe entropy generation rate is carried out in terms of theRANS equations which splits the velocities 119906119894 and tempera-ture 119879 into time-mean and fluctuating components that is119906119894 = 119906119894 + 1199061015840119894 119879 = 119879 + 1198791015840 Note that average entropygeneration rate per unit volume in turbulent flow can beexpressed as follows

119904gen = 1119879120591119894119895 120597119906119894120597119909119895 minus 1199021198961198792 120597119879120597119909119896 (23)

the eddy viscosity-type assumption is adopted which isconsistent with the RANS turbulence model there [12]

119904gen = 120583eff119879 120597119906119894120597119909119895 + 120597119906119895120597119909119894 minus 23 120597119906119896120597119909119896 120575119894119895 120597119906119894120597119909119895+ 119896eff1198792 120597119879120597119909119896 120597119879120597119909119896

(24)

where 120583eff = 120583 + 120583119905 and 120583 and 120583119905 are laminar and turbulentviscosity respectively And the effective thermal conductivity119896eff is also replaced by 119896eff = 119896 + (119888119901120583119905Pr119905) where 119888119901 isthe specific heat at constant pressure and Pr119905 is the turbulentPrandtl number

32 Flow Field Simulation In the flow field calculation anin-house code was usedThe RANS based turbulence modelsare used in conjunction with the Navier-Stokes equationsfor viscous flow simulations The numerical simulationsdiscussed herein use the general steady viscous transportequations in conservative form which can be casted into thefollowing compact notation form120597120597119909119894 (120588119906119894120601) = 120597120597119909119894 (Γ 120597120601120597119909119894) + 119878120601 (25)

where 120601 is the general transport variables Γ is the nominaldiffusion coefficient and 119878120601 is the source term which canbe expressed as a linear function of 120601119875 [17] Thus 119878120601 can bewritten as follows 119878120601 = 119878119862 + 119878119875120601119875 (26)

where 119878119862 stands for the constant part of 119878120601 while 119878119875 is thecoefficient of 120601119875

We discrete the transport equations by FVM (FiniteVolume Method) on a nonorthogonal collocated grid thatall transport variables are stored at cell centers and theintegration and discretization about the control volume Ωyields

sum119895=119899119904119908119890

int119878119895

(120588119906119895120601 minus Γ120601 120597120601120597119909119895) ∙ 119889119878 = intΩ 119878120601119889119881 (27)

where the summation is over the faces of the control volumeThe deferred correction method [18] is used to discrete

the convection term while it can be expressed as

119862119895 = int119878119895

120588120601U ∙ n 119889119878 asymp 119865119895120601119895= max (119865119895 0) 120601119875 +min (119865119895 0) 120601119875119895+ 120574CDS (119865119895120601119895 minusmax (119865119895 0) 120601119875 minusmin (119865119895 0) 120601119875119895)

(28)

where 119865119895 is the mass flow rate (defined to be positive if flow isleavingΩ) through the face

The diffusion term at the face is

119863119895 = int119878119895

Γ120601nabla120601 ∙ n 119889119878 asymp (Γ120601)119895 (nabla120601)119895 S119895 (29)


minus10 0 5 10 15 20minus5xc

minus10

minus5

0

5

10

15yc

(a) (b)

Figure 1 Computational domain around the airfoil (a) complete grid and (b) fine grid around airfoil

where S119895 is the area vector The form of this term in nonor-thogonal grid schemes can be decomposed into orthogonaland nonorthogonal terms [19] Thus

119863119895 = (Γ120601)119895 (120601119875119895 minus 120601119875) S119895 ∙ S119895S119895 ∙ d119895

+ (Γ120601)119895 ((nabla120601)119895 ∙ S119895 minus (nabla120601)119895 ∙ d119895 S119895 ∙ S119895S119895 ∙ d119895) (30)

where (nabla120601)119895 at the face is taken to be the average of thederivatives at the two adjacent cells The first term on theright-hand side of (29) represents the primary gradient andthe discretization is equivalent to a second-order centraldifferent representation which is treated implicitly and leadsto a stencil that includes all neighboring cells while the secondterm is the secondary or cross-diffusion termwhich is treatedexplicitly

For evaluating (nabla120601)119875 the cell derivative of 120601 Gaussrsquos the-orem is adopted and the reconstruction gradient is estimatedas (nabla120601)119875 = 1119881sum

119895

(120601119895S119895) (31)

where the face value 120601119895 can be linear reconstructed from thecell neighbors of the face120601119895 = (119891120601119875 + (1 minus 119891) 120601119875119895) (32)

For steady flows strongly implicit procedure (SIP) [20]iteration technique is adopted to solve the algebraic equationsand to accelerate the rate of convergence Since pressure andvelocity components are stored at cell centers computingface mass flow rate by averaging the cell velocity is proneto checker boarding In this paper momentum interpolationmethod (MIM) [21] is adopted to overcome this

4 Numerical Examples and Discussion

41 Entropy Generation Calculation Validation So far wehave shown how the total entropy generation in turbulentflows can be calculated in a postprocess In this sectionentropy generation calculation has been carried out usinga turbulent flow over NACA standard series airfoils tovalidate the eddy viscositymodel used for entropy generationcalculations in the present paper details of which are pro-vided below We discussed the relationship between the totalentropy generation in the flow field and the drag coefficientat various angle-of-attack under different Reynolds numberwith different turbulence models

The nonorthogonal structured mesh is generated byalgebraic method As shown in Figure 1 a 240 times 60 gridis used and 120 grid points are distributed over the airfoilThe top and bottom far-field boundaries are located at 125chord lengths from the airfoil The upstream velocity inletboundary is 125 chord length away from the airfoil trailingedge while the downstream outflow boundary is located 21chord lengths away from the airfoil The upstream boundaryis set to be velocity inlet and the downstream boundary isset to be outflow while the velocity profile normal to the exitplane is adjusted to satisfy the principle of global conservationof mass flow rate

411 Turbulence Model Comparison Shuja et al [22]reported a dependency between the various turbulencemodels used and the entropy generation estimate on animpinging jet flow This is a consequence of the differingestimates of the effective viscosity used in the differentmodels In this paper five turbulence models 119896-120576 RNG119896-120576 119896-120596 SST 119896-120596 and S-A which are based on eddyviscosity-type assumptions are used to model the flowaround NACA0012 airfoil at 119877119890 = 288 times 106 under different


Experiment

k- modelRNG k- modelk- model

SST k- modelS-A model

1

08

06

04

02

0

minus02

minus04

minus06C

p

02 04 06 08 10xc

(a) 120572 = 0∘Experiment



02 04 06 08 10xc

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Cp

(b) 120572 = 10∘

Experiment



0

minus2

minus4

minus6

minus8

minus10

Cp

02 04 06 08 10xc

(c) 120572 = 15∘

Figure 2 Surface pressure coefficient distribution profiles for NACA0012 airfoil at different turbulence models

turbulence models in order to discuss the effect of theturbulence models on entropy generation calculation

The surface pressure distribution at different angle-of-attack comparedwith the experimental data [23 24] is plottedin Figure 2 In general all models performed quite well inthe simulation yielding predictions which are in excellentagreement with measurements as shown in Figure 2 Thuswe thought the numericalmethods and discretization schemefor flow simulation are justified

Contours of entropy generation rate around theNACA0012 airfoil for angle-of-attack 0∘ 10∘ and 20∘ at119877119890 = 288 times 106 under different turbulence models can beseen in Figure 3 Because the order of magnitude of theentropy generation rate is 10minus15 sim 104 this paper performsa logarithmic process to clearly show the source of theentropy generation For all turbulence models it can beconcluded from the present results that the bad regions thatmost of the entropy generated are the front the near wall


k- model

RNG k- model

k- model

SST k- model

S-A model

minus14 minus125 minus11 minus95 minus8 minus65 minus5 minus35 minus2 minus05 1 25





(a) 120572 = 0∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(b) 120572 = 10∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(c) 120572 = 20∘

Figure 3 Streamlines and contours of lg( 119904gen) at attack angle 0∘ (a) 10∘ (b) and 20∘ (c) for different turbulence models

the turbulent wake and the recirculation zones The highentropy generation in the near wall region is due to the largevelocity gradients that develop to establishing a smoothvelocity transition from the wall values while the turbulentwake region is due to high eddy viscosity

Figure 4 shows the total (ie integral) entropy generationrate (on the left) of the flow field and drag coefficient (onthe right) curves of the airfoil for the five turbulence modelsunder the attack angle from 0∘ to 20∘ The value of theentropy generation rate and drag coefficient is different dueto the different turbulence models for the calculation of theturbulence viscosity coefficient The increase of angle-of-attack had a direct effect to the total entropy generation inthe flow over both airfoils for all turbulence models It can beseen from the figure that the variation of the drag coefficientand entropy generation rate calculated by the correspondingturbulence model has the same trend

Figure 5 shows the relationship between the entropygeneration rates in the flow field and the drag coefficientof the airfoil under different turbulence models Because ofturbulent modeling the evaluation of shear stress in CFD isdifferent which result in differences in drag calculation It canbe seen that the entropy generation rate is linear with the dragcoefficient for all turbulence models with almost uniformslope

412 Reynolds Number Effect To study the relationshipbetween the Reynolds number effect and the entropy gener-ation we simulated the flow over NACA0012 airfoil underdifferent Reynolds number with 119896 minus 120596 turbulence model

Contours of entropy generation rate around the NACA0012airfoil for angle-of-attack 0∘ 10∘ and 20∘ at different Reynoldsnumber can be seen in Figure 6

Figure 7 shows the axial variation of entropy generationrate along the NACA0012 airfoil for angle-of-attack 0∘ 10∘and 20∘ under different Reynolds number When Reynoldsnumber is increased from 205times106 to 377times106 the entropygeneration rate increased correspondingly Note that whenthe angle-of-attack is 20∘ the recirculation bubble occurredwhich results in value of entropy generation dropping to verylow values The results show good agreement with [25]

Figure 8 shows the effect of Reynolds number on thetotal (ie integral) entropy generation of the flow field(on the left) and drag coefficient (on the right) by surfaceintegration The value of entropy generation rate increaseswith Reynolds number for thewhole range of angle-of-attackIt also can be seen from the figure that the variation of thedrag coefficient and entropy generation rate calculated underthe corresponding Reynolds number have the same trend

We consider a dimensionless version of (17) for theexpression of drag in order to highlight the relations betweenthe drag total entropy generation rate andReynolds number

119862119889 = 2119877119890 intΩ 119904gen119889Ω (33)

where the superscript ldquosimrdquo means nondimensional form andaverage entropy generation rate can be expressed in nondi-mensional form as follows119904gen = 120583eff120583 120597119906119894120597119909119895 + 120597119906119895120597119909119894 minus 23 120597119906119896120597119909119896 120575119894119895 120597119906119894120597119909119895




5 10 15 200 (∘)

5

10

15

20

25

S ＡＨ

(WK

)

(a)



5 10 15 200 (∘)

005

01

015

02

Cd

(b)

Figure 4 Entropy generation rate (a) and drag coefficient (b) versus angle-of-attack for NACA0012 airfoil at different turbulence models



5

10

15

20

25

S ＡＨ

(WK

)

01 015 02005Cd

Figure 5 Drag coefficient versus entropy generation rate at differentturbulence models

+ 119896eff1205832 120597120597119909119896 120597120597119909119896 (34)

Equation (33) and Figure 9 clearly reflect the relationshipbetween the total entropy generation in the flow field and

the drag coefficient of the airfoil under different Reynoldsnumber for NACA0012 airfoil The entropy generation rateis linear with the drag coefficient under all Reynolds numberwhile the slope decreases with Reynolds number

413 Airfoil Model Testing Moreover in order to study therelationship between the airfoil configuration and the entropygeneration we simulated the flow over different standardseries NACA airfoils under 119877119890 = 288 times 106 with 119896 minus 120596turbulence model Figure 10 shows a comparison betweenthe total entropy generation and drag coefficient for fiveNACA airfoils The increase of angle-of-attack has a directeffect to the total entropy generation in the flow over bothairfoils However as shown in Figure 10 each airfoil hasa different entropy generation signature for different angle-of-attack NACA0012 airfoil yields less entropy generationthan NACA0024 at low angle-of-attack while such trendis reversed when higher angles were considered It alsocan be seen from the figure that the variation of the dragcoefficient and entropy generation rate calculated under thecorresponding Reynolds number have the same trend

Figure 11 shows the relationship between the entropygeneration in the flow field and the drag coefficient of the fiveNACA airfoils The entropy generation rate is linear with thedrag coefficient for all airfoils with almost uniform slope

In this section the entropy generation and its relationshipto drag is validated by numerical simulating the total entropygeneration rate and drag coefficient of different airfoilsunder different Reynolds number at different attack angleTherefore we are confident to say that entropy generationin turbulent flow using the effective viscosity leads to a goodapproximation of aerodynamic drag of airfoil


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(c) 120572 = 20∘

Figure 6 Streamlines and contours of lg( 119904gen) at attack angle 0∘ (a) 10∘ (b) and 20∘ (c) at different Reynolds number

42 Drag Prediction Comparison Next we chose NACA0012airfoil as the benchmark airfoil as it is well documentedin the open literature By comparing drag from alternativenumerical methods we are able to verify and validate thecomputational procedure of airfoil drag based on entropygeneration method The free stream conditions are set to be119877119890 = 288 times 106 at different angle-of-attack The mesh isa single-block O-type grid which is shown in Figures 12(a)and 2(b) In order to cut an exit plane perpendicular to thefreestream direction in this paper we adopted a multivariateinterpolation using radial basis functions (RBFs) [26] Itprovides a direct mapping between the control points thesurface geometry and the locations of grid points in the CFDvolumemeshThe deformedmesh of NACA0012 is plotted inFigure 12(c) when the angle-of-attack is 15∘

In order to ensure that the numerical model is free fromnumerical diffusion and artificial viscosity errors severalgrids are tested to estimate the number of grid elementsrequired to establish a grid independent solution Table 1shows the specifications of different grids used in such testFigure 13 plots the nondimensional normal distance fromthe first grid point to the wall along the airfoil Noting that119910+ is reduced with the mesh refinement increase the gridconvergence study is summarized in Table 1 where dragcoefficient values (direct integration of pressure and frictionforces at the airfoil surface) as well as the entropy generationrate are presented All the drag coefficients are given in dragcounts (one drag count is 110000 of drag coefficient) Therewas a logical decrease of the pressure drag and skin frictiondrag value from the coarse to the medium fine grid This is

due to a better discretization of the computational domainwhich leads to a more accurate solution For the entropygeneration an obviously increase is observed from the coarseto the fine grid Figure 14 shows the pressure coefficientdistribution on the upper and lower surfaces of the airfoil ascomputed by the four grids In general the results are veryconsistent and themedium-fine gird is chosen to conduct theanalysis presented hereafter

Using the obtained velocity field drag force on the airfoilfrom entropy generation is calculated and compared withsurface and wake integration methods The drag can beexpressed in the following forms


[minus119901119899 + 120591 ∙ 119899] 119889119878= minusint119878exit

[(119901 minus 119901infin) + 120588119906 (119906 minus 119880infin)] 119889119878= 119879infin119880infin intΩ 119904gen119889Ω

(35)

where 119878body is aircraft surface 119878exit is the wake line and Ωdenotes the numerical domain

Table 2 gives the comparison of drag values of NACA0012airfoil between calculated and experimental data From thetable it can be seen that the calculated drag values by wakeintegration method increase when the integration is donenear the trailing edge (119909119888means the downstream position ofwake plane after the wing) Nevertheless with the wake line


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

0

2

4

6

8

10

In(S

ＡＨ

(WK

))

(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

minus4

minus2

0

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(c) 120572 = 20∘

Figure 7 Entropy generation rate distribution along airfoil for attack angle 0∘ (a) 10∘ (b) and 20∘ (c) at different Reynolds number

Table 1 Specification of grid and convergence study (119877119890 = 288 times 106 120572 = 0∘)Grid Number of elements First cell wall distance Growth rate 119884 Plus 119862119889 119862119889119901119903 119862119889119904119891 119878gen (WK)Coarse grid 400 times 45 16 times 10minus3 115 50sim200 124 39 85 0804Medium grid 400 times 60 20 times 10minus4 115 5sim40 103 23 80 0919Medium-fine gird 400 times 80 12 times 10minus5 115 lt5 94 21 73 1352Fine gird 400 times 120 50 times 10minus6 105 lt1 101 24 77 1385


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)

Figure 8 (a) Entropy generation rate and (b) drag coefficient versus angle-of-attack for NACA0012 airfoil at different Reynolds number

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)

Figure 9 Drag coefficient versus entropy generation rate for NACA0012 at different Reynolds number

Table 2 Comparison of drag coefficients from various methods (NACA0012 airfoil 119877119890 = 288 times 106)Angle-of-attack 119878gen (WK) Drag from

experimentDrag from surface

integrationDrag from wake integration Drag from entropy

generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739


NACA0012NACA0015NACA0018

NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)

Figure 10 (a) Entropy generation rate and (b) drag coefficient versus angle-of-attack for different airfoils


NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)

Figure 11 Drag coefficient versus entropy generation rate fordifferent airfoils

placed beyond 119909119888 the predicted drag values remains nearlyconstant

Figure 15 gives the comparison of drag values obtainedfrom different methods with the measured drag It depictsthat the calculation drag values from entropy generationare much lower than both the surface integration and wake

integration methods The wake integration is performedwith the wake-line placed at four chord lengths down-stream of the airfoil trailing edge At 120572 = 0∘ the dragcoefficient from surface integration and wake integration is942 and 921 drag counts which is 188 and 151 higherthan the experimental value respectively This is due to awell known phenomenon of the over prediction of dragpresent in all CFD solvers However if we calculate thedrag coefficient from entropy generation it brings down thenumber to 891 drag counts Although it still overpredictsthe drag it is closer to the experimental data and muchbetter than the traditional surface integration and wakeintegration methods The test case verifies that drag can beestimated from entropy generation with very high accura-cies

5 Conclusions and Extensions

Traditionally surface integration of the pressure and stresstensor on the body surface of aircraft which is calledsurface integration is used for the drag prediction in CFDcomputations But it is pointed out that the drag computedby the near-field method has inaccuracy relating to thenumerical diffusion and error An advanced drag predic-tion method (wake integration) based on the momentumconservation theorem around aircraft is watched with keeninterests which can compute the drag components from thesurface integration on the wake plane of the downstream ofaircraft The method has the drag decomposition capabilityinto wave profile and induced drag component Howeverthe method is closely related to the wake cross surfaceIn this paper a drag prediction method based on entropy


50 10 15minus5minus10xc

minus10

minus5

0

5

10

yc

(a)

(b) (c)

Figure 12 Computational domain around the NACA0012 airfoil (a) complete grid (b) fine grid and (c) deformed grid around airfoil

generationwas integrated into a RANS-basedCFD solver andan approach was developed to compute the airfoil drag viaentropy generation rate in the flow field which is a volumeintegral method derived from far-field method by applyingthe divergence theorem of Gauss Present paper has beendevoted to the analysis of some RANS calculations aroundtwo-dimensional airfoils at subsonic freestream conditionsThe main objective of this paper is to compare the consis-tency of predicting the drag of single-element airfoils usingsurface integration wake integration and entropy generationintegration Overall entropy generation integration showspotential as a simpler method than surface integration andwake integration for calculating drag The main advantageof this technique is that no detailed information on thesurface geometry of the configuration is needed The resultsshow that drag prediction using CFD and entropy generationintegration is possible within engineering accuracy and

that the proposed method also have the drag visualizationcapability in the flow field for designers to take measuresto minimize drag The main conclusions are summarized asfollows

(1) The drag and entropy generation in 2D domain canbe related by a linear balance equation so the drag ofthe airfoil can be directly estimated from total entropygeneration without other effects

(2) Entropy generation consists of two parts heat transferand viscous dissipation In turbulent flow the fluctu-ating velocity and temperature contribution can beaccounted for by using the effective viscosity andeffective thermal conductivity

(3) In the 2D turbulent flow of airfoil most of the entropyis generated in the front region the near wall regionand the turbulent wake region The total entropy


Fine girdCoarse gridMedium grid

Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus

Figure 13 The nondimensional normal distance from the first grid point to the wall along the airfoil

Coarse gridMedium gridMedium-fine gird

Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp

Figure 14 Pressure coefficient plotted on the normalized airfoil cordfor different grid resolutions

generation of flow field increases with the angle-of-attack and Reynolds number

(4) Drag prediction using CFD and entropy generationintegration is possible within engineering accuracyFuture work should concentrate on 3D airfoil dragand compressible turbulent flow to demonstrate theuniversality of entropy-basedmethod in CFD predic-tion of airfoil drag

Drag from surface integrationDrag from wake integrationDrag from entropy generationExperiment

5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4

Figure 15 NACA 0012 airfoil Computed drag coefficient versusangle-of-attack

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] H Vinh C P Dam D Yen et al ldquoDrag prediction algorithmsfor Navier-Stokes solutions about airfoilsrdquo in Proceedings of the13th Applied Aerodynamics Conference San Diego CA USA


[2] C D Nicola B Mele and R Tognaccini ldquoAirfoil drag calcula-tions in stall and post-stall conditionsrdquo inProceedings of the 43rdAIAA Aerospace Sciences Meeting and Exhibit pp 683ndash693 usaJanuary 2005

[3] C P V Dam ldquoRecent experience with differentmethods of dragpredictionrdquoProgress inAerospace Sciences vol 35 no 8 pp 751ndash798 1999

[4] L Paparone and R Tognaccini ldquoComputational fluid dynam-ics-based drag prediction and decompositionrdquo AIAA Journalvol 41 no 9 pp 1647ndash1657 2003

[5] Z Zhu X Wang J Liu and Z Liu ldquoComparison of predictingdrag methods using computational fluid dynamics in 2d3dviscous flowrdquo Science in China Series E Technological Sciencesvol 50 no 5 pp 534ndash549 2007

[6] K Oswatitsch Gas Dynamics Academic Press New York NYUSA 1956

[7] D Hue and S Esquieu ldquoComputational drag prediction of theDPW4 configuration using the far-field approachrdquo Journal ofAircraft vol 48 no 5 pp 1658ndash1670 2011

[8] R C Lock ldquoPrediction of the drag of aerofoils andwings at highsubsonic speedsrdquoThe Aeronautical Journal vol 90 no 896 pp207ndash226 2016

[9] H Li and J Stewart ldquoExergy based design methodology forairfoil shape optimization and wing analysisrdquo in Proceedings ofthe Aiaa Thermophysics Conference 2006

[10] J T Stewart An exergy-based study of three dimensional viscousflow over wings[D] MS thesis Clemson University Clemson2005

[11] S Monsch R Figliola E D Thompson and J A CamberosldquoComputation of induced drag for 3D wing with volumeintegral (Trefftz plane) techniquerdquo in Proceedings of the 45thAIAA Aerospace Sciences Meeting 2007 pp 13006ndash13014 usaJanuary 2007

[12] K Alabi F Ladeinde M vonSpakovsky D Moorhouse andJ Camberos ldquoAssessing CFD Modeling of Entropy Genera-tion for the Air Frame Subsystem in an Integrated AircraftDesignSynthesis Procedurerdquo in Proceedings of the 44th AIAAAerospace Sciences Meeting and Exhibit Reno Nevada

[13] S Mahmud and R A Fraser ldquoThe second law analysis infundamental convective heat transfer problemsrdquo InternationalJournal of Thermal Sciences vol 42 no 2 pp 177ndash186 2003

[14] F Kock and H Herwig ldquoEntropy production calculation forturbulent shear flows and their implementation in cfd codesrdquoInternational Journal of Heat and Fluid Flow vol 26 no 4 pp672ndash680 2005

[15] A Bejan ldquoFundamentals of exergy analysis entropy generationminimization and the generation of flow architecturerdquo Inter-national Journal of Energy Research vol 26 no 7 pp 545ndash5652002

[16] H Herwig and F Kock ldquoDirect and indirect methods ofcalculating entropy generation rates in turbulent convectiveheat transfer problemsrdquo Heat and Mass Transfer vol 43 no 3pp 207ndash215 2007

[17] S V Patankar ldquoA calculation procedure for two-dimensionalelliptic situationsrdquoNumerical Heat Transfer Part B Fundamen-tals vol 4 no 4 pp 409ndash425 1981

[18] P K Khosla and S G Rubin ldquoA diagonally dominant second-order accurate implicit schemerdquo Computers amp Fluids vol 2 no2 pp 207ndash209 1974

[19] S Murthy R and Y Murthy J ldquoA pressure-based methodfor unstructured meshesrdquo Numerical Heat Transfer Part B

Fundamentals An International Journal of Computation ampMethodology vol 31 no 2 pp 195ndash215 1997

[20] H L Stone ldquoIterative solution of implicit approximations ofmultidimensional partial differential equationsrdquo SIAM Journalon Numerical Analysis vol 5 pp 530ndash558 1968

[21] C M Rhie and W L Chow ldquoNumerical study of the turbulentflow past an airfoil with trailing edge separationrdquoAIAA Journalvol 21 no 11 pp 1525ndash1532 1983

[22] S Z Shuja B S Yilbas and M O Budair ldquoLocal entropygeneration in an impinging jetMinimumentropy concept eval-uating various turbulence modelsrdquo Computer Methods AppliedMechanics and Engineering vol 190 no 28 pp 3623ndash36442001

[23] N Gregory and OrsquoReilly ldquoLow-speed aerodynamic character-istics of NACA0012 aerofoil section including the effects ofupper-surface roughness simulating hoar frostrdquo ChemInformvol 23 no 48 pp 6697ndash6700 1970

[24] C L Ladson ldquoEffects of independent variation of Mach andReynolds numbers on the low-speed aerodynamic charac-teristics of the NACA 0012 airfoil sectionrdquo NASA TechnicalMemorandum no 4074 1988

[25] E Abu-Nada ldquoNumerical prediction of entropy generation inseparated flowsrdquo Entropy vol 7 no 4 pp 234ndash252 2005

[26] M Buhmann Radial basis functions Cambridge UniversityPress Cambridge UK 1st edition

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


limiting the far-field flux computation to a box enclosingthe aircraft By applying the divergence theorem of Gaussthe surface integral in Oswatitschrsquos formula can be replacedwith a volume integral Therefore the integrand can be setto zero a priori in regions where it is known that physicalentropy variations should be zero thus removing spuriouscontributions to drag In a viscous domain all drag on anairfoil is eventually realized as entropy generation so wecan establish a relationship between total entropy generationof the flow field and drag on the airfoil Li and Stewartreported on 2D validation studies for predicting drag bymeans of calculating the generation both for airfoils andfor viscous pipe flows [9] Stewart [10] and Monsch et al[11] calculated the drag force of a 3D wing by performinga volume integration within the numerical domain and asurface integral over the outlet of the domain

It is pointed out that a representation of losses in terms ofentropy generation offers significant insight into the flow andthermal transport phenomena over the airfoil and providesan effective tool for drag prediction Entropy analysis is amethod to evaluate a process based on the second law of ther-modynamics It is basically calculating entropy generation ina system and its surroundings and using it as a proxy forthe evaluation of the energy loss [12] Mahmud and Fraser[13] analyzed the mechanism of entropy generation and itsdistribution through fluid flows in basic channel configura-tions including two fixed plates and one fixed and onemovingplates by considering simplified or approximate analyticalexpressions for temperature and velocity distributions andderived analytically general expressions for the number ofentropy generation and Bejan number A detailed review ofentropy and its significance in CFD is presented by Kock andHerwig [14]

In this paper entropy generation and its relationshipto drag were derived for viscous flow we present modelequations for the calculation of the local entropy generationrates in turbulent flows by extending the RANS procedureto the entropy balance equation This equation serves toidentify the entropy generation sources without need to solvethe equation itself The main objective of this paper is todemonstrate the viability of entropy-based drag calculationmethod and to compare the consistency of predicting thedrag of single-element airfoils using surface integration wakeintegration and entropy generation integration

2 Alternative Methods forAirfoil Drag Prediction

The conservation law of momentum to the control volumeenclosing the configuration makes it possible to predict thetotal aerodynamic drag force by an integration of stresseson the aircraft configuration (surface integration) or byan integration of momentum flux on a closed surface farfrom the configuration (wake integration) We consider asteady flow with freestream velocity 119880infin pressure 119901infin andtemperature 119879infin around the configuration the only externalforce acting on the body is due to the fluid Neglecting bodyforces the fundamental formula for the total aerodynamic

force acting on the configuration in the Cartesian referencesystem can be derived as follows [1]

int119878[minus119901119899 + 120591 ∙ 119899 minus 120588 ( ∙ 119899)] 119889119878 = 0 (1)

where 119878 represents the entire control surface 119899 = 119899119909+119899119910+119899119911denotes the outward unit vector normal to 119878 = 119906 + V+ 119908denotes the velocity vector 120588 represents the fluid density and120591 represents the shear stress tensor Note that 119878 consists of 119878farand 119878body with 119878body specifies the aircraft surface and 119878far theclosed surface far from the configuration Consequently (1)can be written as

int119878body

[minus119901119899 + 120591 ∙ 119899 minus 120588 ( ∙ 119899)] 119889119878+ int119878far

[minus119901119899 + 120591 ∙ 119899 minus 120588 ( ∙ 119899)] 119889119878 = 0 (2)

Themomentumflux term 120588(∙119899) in the first integral is zeroin the general case of solid body surface Thus (2) reduces to

minus int119878body

[minus119901119899 + 120591 ∙ 119899] 119889119878= int119878far

[minus119901119899 + 120591 ∙ 119899 minus 120588 ( ∙ 119899)] 119889119878 (3)

where the integral on the left-hand side represents the totalaerodynamic drag force by an integration of stresses on theaircraft configuration which can be written as


[minus119901119899 + 120591 ∙ 119899] 119889119878 (4)

The far-field expression of drag prediction is given by theright-hand side of (3)

119863 = int119878far

[minus119901119899 + 120591 ∙ 119899 minus 120588 ( ∙ 119899)] 119889119878 (5)

The wake expression for drag is derived from (5) bymoving the inlet and side faces of the control volume toinfinity and the following wake integration for the drag isobtained


[(119901 minus 119901infin) + (1205881199062 minus 1205881198802infin) minus 120591119909119909] 119889119878 (6)

where 119878exit is perpendicular to the freestream direction Thethird integral on the right-hand side of (6) is often neglectedif an integration plane sufficiently far down stream [4]Considering the conservation of the mass flux we have


[(119901 minus 119901infin) + 120588119906 (119906 minus 119880infin)] 119889119878 (7)

In a two-dimensional flow as we move the boundary 119878fartowards infinity (119878far rarr 119878infin) the pressure term vanishes asproved by Paparone and Tognaccini [4]

lim119878farrarr119878infin

int119878far

(119901119899) 119889119878 = 0 (8)




[120588 ( ∙ 119899) ] 119889119878 (9)





119891(Δ119901119901infin Δ119904119877 Δℎ1198802infin)( ∙ 119899) 119889119878 (11)



2 ] (12)



(13)




2 ] (14)



[1198911199041 (Δ119904119877 )] 120588 ( ∙ 119899) 119889119878 (15)







119864119883des = 1198790 (119878gen) (18)




120597119905 (120588119904) + 120597119895 (120588119906119895119904) + 120597119895 119902119895119879 = minus1199021198951205971198951198791198792 + 120591119894119895120597119895119906119894119879 (19)




119904gen = 1119879120591119894119895 120597119906119894120597119909119895 minus 1199021198961198792 120597119879120597119909119896 (20)




= 120597119905 (120588119904) + 120597119895 (120588119906119895119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟CONVECTION

+ 120597119895 119902119895119879⏟⏟⏟⏟⏟⏟⏟MOLECULAR FLUX

(21)



= intΩ(120597119905 (120588119904) + 120597119895 (120588119906119895119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

CONVECTION

+ 120597119895 119902119895119879⏟⏟⏟⏟⏟⏟⏟MOLECULAR FLUX

)119889Ω (22)



119904gen = 1119879120591119894119895 120597119906119894120597119909119895 minus 1199021198961198792 120597119879120597119909119896 (23)


119904gen = 120583eff119879 120597119906119894120597119909119895 + 120597119906119895120597119909119894 minus 23 120597119906119896120597119909119896 120575119894119895 120597119906119894120597119909119895+ 119896eff1198792 120597119879120597119909119896 120597119879120597119909119896

(24)






sum119895=119899119904119908119890

int119878119895

(120588119906119895120601 minus Γ120601 120597120601120597119909119895) ∙ 119889119878 = intΩ 119878120601119889119881 (27)



119862119895 = int119878119895


(28)



119863119895 = int119878119895




minus10

minus5

0

5

10

15yc

(a) (b)



119863119895 = (Γ120601)119895 (120601119875119895 minus 120601119875) S119895 ∙ S119895S119895 ∙ d119895




119895

(120601119895S119895) (31)








Experiment



1

08

06

04

02

0

minus02

minus04

minus06C

p

02 04 06 08 10xc




02 04 06 08 10xc

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Cp

(b) 120572 = 10∘

Experiment



0

minus2

minus4

minus6

minus8

minus10

Cp

02 04 06 08 10xc

(c) 120572 = 15∘






k- model

RNG k- model

k- model

SST k- model

S-A model






(a) 120572 = 0∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(b) 120572 = 10∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(c) 120572 = 20∘










119862119889 = 2119877119890 intΩ 119904gen119889Ω (33)





5 10 15 200 (∘)

5

10

15

20

25

S ＡＨ

(WK

)

(a)



5 10 15 200 (∘)

005

01

015

02

Cd

(b)




5

10

15

20

25

S ＡＨ

(WK

)

01 015 02005Cd


+ 119896eff1205832 120597120597119909119896 120597120597119909119896 (34)







２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(c) 120572 = 20∘









(35)




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

0

2

4

6

8

10

In(S

ＡＨ

(WK

))

(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

minus4

minus2

0

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(c) 120572 = 20∘




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)





generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739



NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018











[120588 ( ∙ 119899) ] 119889119878 (9)





119891(Δ119901119901infin Δ119904119877 Δℎ1198802infin)( ∙ 119899) 119889119878 (11)



2 ] (12)



(13)




2 ] (14)



[1198911199041 (Δ119904119877 )] 120588 ( ∙ 119899) 119889119878 (15)







119864119883des = 1198790 (119878gen) (18)




120597119905 (120588119904) + 120597119895 (120588119906119895119904) + 120597119895 119902119895119879 = minus1199021198951205971198951198791198792 + 120591119894119895120597119895119906119894119879 (19)




119904gen = 1119879120591119894119895 120597119906119894120597119909119895 minus 1199021198961198792 120597119879120597119909119896 (20)




= 120597119905 (120588119904) + 120597119895 (120588119906119895119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟CONVECTION

+ 120597119895 119902119895119879⏟⏟⏟⏟⏟⏟⏟MOLECULAR FLUX

(21)



= intΩ(120597119905 (120588119904) + 120597119895 (120588119906119895119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

CONVECTION

+ 120597119895 119902119895119879⏟⏟⏟⏟⏟⏟⏟MOLECULAR FLUX

)119889Ω (22)



119904gen = 1119879120591119894119895 120597119906119894120597119909119895 minus 1199021198961198792 120597119879120597119909119896 (23)


119904gen = 120583eff119879 120597119906119894120597119909119895 + 120597119906119895120597119909119894 minus 23 120597119906119896120597119909119896 120575119894119895 120597119906119894120597119909119895+ 119896eff1198792 120597119879120597119909119896 120597119879120597119909119896

(24)






sum119895=119899119904119908119890

int119878119895

(120588119906119895120601 minus Γ120601 120597120601120597119909119895) ∙ 119889119878 = intΩ 119878120601119889119881 (27)



119862119895 = int119878119895


(28)



119863119895 = int119878119895




minus10

minus5

0

5

10

15yc

(a) (b)



119863119895 = (Γ120601)119895 (120601119875119895 minus 120601119875) S119895 ∙ S119895S119895 ∙ d119895




119895

(120601119895S119895) (31)








Experiment



1

08

06

04

02

0

minus02

minus04

minus06C

p

02 04 06 08 10xc




02 04 06 08 10xc

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Cp

(b) 120572 = 10∘

Experiment



0

minus2

minus4

minus6

minus8

minus10

Cp

02 04 06 08 10xc

(c) 120572 = 15∘






k- model

RNG k- model

k- model

SST k- model

S-A model






(a) 120572 = 0∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(b) 120572 = 10∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(c) 120572 = 20∘










119862119889 = 2119877119890 intΩ 119904gen119889Ω (33)





5 10 15 200 (∘)

5

10

15

20

25

S ＡＨ

(WK

)

(a)



5 10 15 200 (∘)

005

01

015

02

Cd

(b)




5

10

15

20

25

S ＡＨ

(WK

)

01 015 02005Cd


+ 119896eff1205832 120597120597119909119896 120597120597119909119896 (34)







２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(c) 120572 = 20∘









(35)




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

0

2

4

6

8

10

In(S

ＡＨ

(WK

))

(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

minus4

minus2

0

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(c) 120572 = 20∘




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)





generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739



NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018










119904gen = 1119879120591119894119895 120597119906119894120597119909119895 minus 1199021198961198792 120597119879120597119909119896 (20)




= 120597119905 (120588119904) + 120597119895 (120588119906119895119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟CONVECTION

+ 120597119895 119902119895119879⏟⏟⏟⏟⏟⏟⏟MOLECULAR FLUX

(21)



= intΩ(120597119905 (120588119904) + 120597119895 (120588119906119895119904)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

CONVECTION

+ 120597119895 119902119895119879⏟⏟⏟⏟⏟⏟⏟MOLECULAR FLUX

)119889Ω (22)



119904gen = 1119879120591119894119895 120597119906119894120597119909119895 minus 1199021198961198792 120597119879120597119909119896 (23)


119904gen = 120583eff119879 120597119906119894120597119909119895 + 120597119906119895120597119909119894 minus 23 120597119906119896120597119909119896 120575119894119895 120597119906119894120597119909119895+ 119896eff1198792 120597119879120597119909119896 120597119879120597119909119896

(24)






sum119895=119899119904119908119890

int119878119895

(120588119906119895120601 minus Γ120601 120597120601120597119909119895) ∙ 119889119878 = intΩ 119878120601119889119881 (27)



119862119895 = int119878119895


(28)



119863119895 = int119878119895




minus10

minus5

0

5

10

15yc

(a) (b)



119863119895 = (Γ120601)119895 (120601119875119895 minus 120601119875) S119895 ∙ S119895S119895 ∙ d119895




119895

(120601119895S119895) (31)








Experiment



1

08

06

04

02

0

minus02

minus04

minus06C

p

02 04 06 08 10xc




02 04 06 08 10xc

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Cp

(b) 120572 = 10∘

Experiment



0

minus2

minus4

minus6

minus8

minus10

Cp

02 04 06 08 10xc

(c) 120572 = 15∘






k- model

RNG k- model

k- model

SST k- model

S-A model






(a) 120572 = 0∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(b) 120572 = 10∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(c) 120572 = 20∘










119862119889 = 2119877119890 intΩ 119904gen119889Ω (33)





5 10 15 200 (∘)

5

10

15

20

25

S ＡＨ

(WK

)

(a)



5 10 15 200 (∘)

005

01

015

02

Cd

(b)




5

10

15

20

25

S ＡＨ

(WK

)

01 015 02005Cd


+ 119896eff1205832 120597120597119909119896 120597120597119909119896 (34)







２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(c) 120572 = 20∘









(35)




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

0

2

4

6

8

10

In(S

ＡＨ

(WK

))

(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

minus4

minus2

0

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(c) 120572 = 20∘




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)





generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739



NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018










minus10

minus5

0

5

10

15yc

(a) (b)



119863119895 = (Γ120601)119895 (120601119875119895 minus 120601119875) S119895 ∙ S119895S119895 ∙ d119895




119895

(120601119895S119895) (31)








Experiment



1

08

06

04

02

0

minus02

minus04

minus06C

p

02 04 06 08 10xc




02 04 06 08 10xc

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Cp

(b) 120572 = 10∘

Experiment



0

minus2

minus4

minus6

minus8

minus10

Cp

02 04 06 08 10xc

(c) 120572 = 15∘






k- model

RNG k- model

k- model

SST k- model

S-A model






(a) 120572 = 0∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(b) 120572 = 10∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(c) 120572 = 20∘










119862119889 = 2119877119890 intΩ 119904gen119889Ω (33)





5 10 15 200 (∘)

5

10

15

20

25

S ＡＨ

(WK

)

(a)



5 10 15 200 (∘)

005

01

015

02

Cd

(b)




5

10

15

20

25

S ＡＨ

(WK

)

01 015 02005Cd


+ 119896eff1205832 120597120597119909119896 120597120597119909119896 (34)







２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(c) 120572 = 20∘









(35)




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

0

2

4

6

8

10

In(S

ＡＨ

(WK

))

(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

minus4

minus2

0

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(c) 120572 = 20∘




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)





generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739



NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018









Experiment



1

08

06

04

02

0

minus02

minus04

minus06C

p

02 04 06 08 10xc




02 04 06 08 10xc

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Cp

(b) 120572 = 10∘

Experiment



0

minus2

minus4

minus6

minus8

minus10

Cp

02 04 06 08 10xc

(c) 120572 = 15∘






k- model

RNG k- model

k- model

SST k- model

S-A model






(a) 120572 = 0∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(b) 120572 = 10∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(c) 120572 = 20∘










119862119889 = 2119877119890 intΩ 119904gen119889Ω (33)





5 10 15 200 (∘)

5

10

15

20

25

S ＡＨ

(WK

)

(a)



5 10 15 200 (∘)

005

01

015

02

Cd

(b)




5

10

15

20

25

S ＡＨ

(WK

)

01 015 02005Cd


+ 119896eff1205832 120597120597119909119896 120597120597119909119896 (34)







２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(c) 120572 = 20∘









(35)




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

0

2

4

6

8

10

In(S

ＡＨ

(WK

))

(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

minus4

minus2

0

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(c) 120572 = 20∘




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)





generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739



NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018









k- model

RNG k- model

k- model

SST k- model

S-A model






(a) 120572 = 0∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(b) 120572 = 10∘






k- model

RNG k- model

k- model

SST k- model

S-A model

(c) 120572 = 20∘










119862119889 = 2119877119890 intΩ 119904gen119889Ω (33)





5 10 15 200 (∘)

5

10

15

20

25

S ＡＨ

(WK

)

(a)



5 10 15 200 (∘)

005

01

015

02

Cd

(b)




5

10

15

20

25

S ＡＨ

(WK

)

01 015 02005Cd


+ 119896eff1205832 120597120597119909119896 120597120597119909119896 (34)







２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(c) 120572 = 20∘









(35)




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

0

2

4

6

8

10

In(S

ＡＨ

(WK

))

(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

minus4

minus2

0

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(c) 120572 = 20∘




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)





generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739



NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018











5 10 15 200 (∘)

5

10

15

20

25

S ＡＨ

(WK

)

(a)



5 10 15 200 (∘)

005

01

015

02

Cd

(b)




5

10

15

20

25

S ＡＨ

(WK

)

01 015 02005Cd


+ 119896eff1205832 120597120597119909119896 120597120597119909119896 (34)







２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(c) 120572 = 20∘









(35)




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

0

2

4

6

8

10

In(S

ＡＨ

(WK

))

(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

minus4

minus2

0

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(c) 120572 = 20∘




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)





generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739



NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018









２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06






(c) 120572 = 20∘









(35)




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

0

2

4

6

8

10

In(S

ＡＨ

(WK

))

(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

minus4

minus2

0

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(c) 120572 = 20∘




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)





generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739



NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018









２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

0

2

4

6

8

10

In(S

ＡＨ

(WK

))

(a) 120572 = 0∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(b) 120572 = 10∘

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

02 04 06 08 10xc

minus4

minus2

0

2

4

6

8

10

12

In(S

ＡＨ

(WK

))

(c) 120572 = 20∘




２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)





generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739



NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018









２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

0

5

10

15

20

25

S ＡＨ

(WK

)

(a)

２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

5 10 15 200 (∘)

005

01

015

Cd

(b)


２ = 205e + 06

２ = 240e + 06

２ = 288e + 06

２ = 342e + 06

２ = 377e + 06

005 01 0150Cd

0

5

10

15

20

25

S ＡＨ

(WK

)





generation119909119888 = 05 119909119888 = 10 119909119888 = 20 119909119888 = 400∘ 135 800 942 953 934 924 921 8915∘ 153 900 1127 1105 1092 1076 1071 101010∘ 223 1200 1778 1663 1645 1630 1624 147215∘ 415 1900 3463 3122 3115 3103 3096 2739



NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018










NACA0021NACA0024

5 10 15 200 (∘)

2

4

6

8

10

12S Ａ

Ｈ(W

K)

(a)


NACA0021NACA0024

5 10 15 200 (∘)

002

004

006

008

01

Cd

(b)



NACA0021NACA0024

002 004 006 008 010Cd

0

2

4

6

8

10

12

14

S ＡＨ

(WK

)









minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018










minus10

minus5

0

5

10

yc

(a)

(b) (c)









Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018










Medium-fine gird

02 04 06 08 10xc

100

101

102

Wal

l Y p

lus



Fine girdExperiment

02 04 06 08 10xc

1

08

06

04

02

0

minus02

minus04

minus06

Cp





5 10 150 (∘)

100

150

200

250

300

Cdtimes10

4




References




































Journal of












Journal of







Volume 2018






Volume 2018










































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

CFD Prediction of Airfoil Drag in Viscous Flow Using the Entropy Generation Method · 2019. 7. 30. · CFD Prediction of Airfoil Drag in Viscous Flow Using the Entropy Generation