DSpace at VNU: Effect of the sliding friction on heat transfer in high-speed rarefied gas flow simulations in CFD

Lea,b,*, Ngoc Anh Vuc, Le Tan Locd a Divison of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam b Fac

Effect of the sliding friction on heat transfer in high-speed rare fied gas

flow simulations in CFD

Nam T.P Lea,b,*, Ngoc Anh Vuc, Le Tan Locd

a Divison of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

b Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

c Department of Aerospace Engineering, Ho Chi Minh City University of Technology, VNU-HCM, Viet Nam

d Faculty of Aviation Technology, Vietnam Aviation Academy (VAA), Ho Chi Minh City, Viet Nam

a r t i c l e i n f o

Article history:

Received 25 January 2016

Received in revised form

15 May 2016

Accepted 8 June 2016

Keywords:

Sliding friction

Heat transfer

Surface gas temperature

NACA0012 airfoil

Biconic

a b s t r a c t

Different nature of the heat transfer computation between the CFD and DSMC methods in rarefied gas flow simulations leads to the difference of their heat transfer results This paper will revisit a formulation

of heat transfer in CFD that includes the sliding friction part for a planar surface The sliding friction was first introduced by Maslen, and used to be omitted in calculating the CFD heat transfer This formulation will be extended to compute the heat transfer over the curved surfaces such as NACA0012 micro-airfoil and the sharp 25e55-deg biconic Two these configurations 1) NACA0012 micro-airfoil (M ¼ 2) with various Kn¼ 0.026, 0.1 and 0.26 with the angles-of-attack from 0-deg to 20-deg., and a sharp 25e55-deg biconic (M¼ 15.6) are simulated by the CFD method, which solves the Navier-Stokes equations within the OpenFOAM framework Moreover, the flat plat cases with various wall temperatures (Tw¼ 77 K and 294 K) are also undertaken to investigate the significance of the sliding friction on the heat transfer The CFD simulation results of the heat transfer involving the sliding friction give good agreements with the DSMC data, especially the NACA0012 micro-airfoil cases with high Knudsen numbers Kn¼ 0.1 and 0.26

© 2016 Elsevier Masson SAS All rights reserved

1 Introduction

The design of high speed vehicles requires accurate predictions

of the surface properties in theflight These quantities are typically

the surface temperature, the surface pressure, the shear stress, and

heat transfer The surface pressure and the skin friction forces

acting on the surface of the vehicles are integrated to calculate the

aerodynamic forces and the moments as lift, drag and pitching

moments The peak surface temperature or heatflux are integrated

over time and mapped over the vehicle surface as part of the

pro-cess to design the thermal protection system In this paper we will

focus on the wall heat transfer work, and try to reduce the gap

between the DSMC and CFD simulation results The typical

methods have been used for simulating rarefied gas flows as the

CFD and DSMC methods In the CFD method, in particular solving

the Navier-Stokes (NS) equations, the Fourier heat conduction law has been adopted for computing the wall heat transfer In the DSMC method the wall heat transfer has been calculated based on the velocities[1] The lack of the velocity term in CFD calculation al-ways results in the difference between the CFD and DSMC heat transfers In order to overcome this problem, Maslenfirst intro-duced the sliding friction part to compute the wall heat transfer in CFD for a planar surface[2]without any physical explanation The sliding friction (or shear work) had been employed in the energy equation in CFD for simulating the low-speed rarefied gas micro-flows, and gave good results[3e7] In high-speed rarefied gas flows, the wall heat transfer included the sliding friction was evaluated to compare with experimental data for rarefied gas flows over the flat plate[8], and agreed well with experiment In CFD rarefied gas simulations, we often use the slip velocity and temperature jump boundary conditions applied to the surface to improve the NS equations in the slip regime Since the slip velocity boundary condition is applied to the surfaces leading to the appearance of the sliding friction of gas molecules over the surface This slip will generate more heat transfer between gas molecules and the sur-face In the past, there were the work[8e12] using the sliding

* Corresponding author Divison of Computational Mathematics and Engineering,

Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City,

Viet Nam.

E-mail address: letuanphuongnam@tdt.edu.vn (N.T.P Le).

Contents lists available atScienceDirect International Journal of Thermal Sciences

j o u r n a l h o m e p a g e : w w w e l s e v ie r c o m / l o c a t e / i j t s

http://dx.doi.org/10.1016/j.ijthermalsci.2016.06.020

1290-0729/© 2016 Elsevier Masson SAS All rights reserved.

International Journal of Thermal Sciences 109 (2016) 334e341

friction term proposed for planar surface to compute the wall heat

transfer However, many researchers have used the Fourier law of

heat conduction for calculating the heat transfer in their CFD high

speed rarefied gas flow simulations[13e22]that have omitted the

sliding friction part Recently Hong et al [9] pointed out the

physical reason of the inclusion of the sliding friction was

comprehensively explained in terms of both conservation law of

energy and the kinetic theory of gases Hadjiconstantinou [10]

derived an expression for fully developed slipflow and concluded

that the sliding friction (shear work) at the wall should be included

to calculate the heatflux from the wall when viscous heat

gener-ation needs to be considered Moreover, the boundary condition of

the heatflux of the higher moment method (R-13 equations)

pro-posed by Rana et al.[23]also included the slip velocity and stress

terms The simulation results in Refs.[9,10,23]gave good

agree-ment with DSMC and experiagree-mental data in internal rarefied gas

microflows

The heat transfer caused by the sliding friction part was

computed based on the slip velocity and the shear stress[2] Our

previous work[13]simulating rarefied gas flow of the NACA0012

airfoil the CFD heat transfers were calculated by the Fourier heat

conduction law, and did not match those of DSMC data They

showed that there were large differences between the DSMC and

CFD heat transfers since the gasflow velocity increases This can be

explained by the lack of the sliding friction part, which involves the

slip velocity and shear stress (e.g the velocity gradient) terms, in

the CFD heat transfer computation In the present work, we will

revisit a formulation of the wall heat transfer proposed by Maslen

[2]for a planar surface, and will extend this one to the curved

geometrical surfaces such as NACA0012 micro-airfoil and the sharp

25e55-deg biconic The NACA0012 micro-airfoil case with Mach

number, M¼ 2, is investigated with various angles-of-attack (AOA)

from 0-deg to 20-deg and Knudsen numbers from 0.026 to 0.26

The sharp 25e55-deg biconic case is simulated with M ¼ 15.6 The

flat plate cases [25](M ¼ 6.1) with different wall temperatures

Tw¼ 77 K and Tw ¼ 294 K are also simulated to investigate the

significance of the sliding friction on the wall heat transfer All

simulations are undertaken in the OpenFOAM framework[24] The

wall heat transfer involving the sliding friction is implemented in

OpenFOAM as a new post-processing utility The heat transfers

with and without the sliding friction in all CFD simulations will be

compared with those of the DSMC and experimental data

[13e15,26,27] We also investigate whether the heat transfer with

the sliding friction can capture that of DSMC at high Knudsen

numbers, Kn¼ 0.1 and 0.26 (i.e the transition regime)

2 CFD and DSMC heat transfers in rarefied gas flow

simulations

In CFD, the wall heat transfer is often calculated by the Fourier

heat conduction law that only depends on the temperature

gradient normal to the surface The DSMC wall heat transfer can be

obtained by sampling the difference of energyfluxes in which the

heat transfer is a function of molecular velocity, u[28]

qw DSMC¼tF$numDA

2

4XN

j¼1

 1

2msimu

2 j

i

XN

j¼1

 1

2msimu

2 j

r3

5 (1)

where Fnumis the number of real molecules each simulated particle

represents (weight factor); msim is the mass of the simulated

molecule in DSMC; The superscripts“i” and “r” denote the values of

before and after impacting the wall element, respectively; the

subscript“w” indicates the wall value; N is the total number of

molecules impacting the surface element during sampling time t;

andDA is the surface area

Due to different nature of the heatflux computation between two methods, there is always a gap between the CFD and DSMC wall heat transfer simulation results[13] We will therefore revisit the formulation of the heat transfer proposed by Maslen[2]that included the sliding friction part The heat transfer in the gas slip flow over the flat plate was first introduced by Maslen[2]including the Fourier heat conduction



kvT , and sliding friction



 usmvu vy

 parts,

q¼ kvTvy usmvuvy ; (2) where k is the thermal conductivity; T is the temperature; usis the slip velocity;mis the viscosity; and y is spatial coordinate in y di-rection (normal didi-rection) Maslen also stated that it was the cor-rect expression for heat transfer in slipflow without further proof

[2] It should be noted that this relation can be derived with the kinetic theory for a nonuniform gas Its derivation had been briefly presented in Ref.[8]and is now re-summarized here

In the gas kinetic theory the three dimensional motion of indi-vidual molecules between collisions is governed by the equation of motion which is a set of 3 s order differential equations They are expressed in both physical space (x, y, z) and velocity space (U, V, W) The velocity components (u, v, w) of a molecule can be referred

to their average values by writing,

u¼ u þ U; v ¼ v þ V; w ¼ w þ W[29], where u; v; w are the compo-nents of the mass velocity of the gas, and U, V, W are the compocompo-nents

of velocity of a molecule referred to the mass velocity[29] The next energyflux to a planar surface can be expressed as follows[8,29]

E¼1

2r Z

þ∞

∞

dU

Zþ∞

∞

dV

Zþ∞

∞

Vf

u2þ V2þ W2

where f is the Maxwell distribution function of molecular velocity;

r is density; and the definition of thermal velocity,

C2¼ U2þ V2þ W2[29] Since we attempt to calculate all the macroscopic properties of a gas from the motion of its molecules it

is clear that the function f is fundamental importance The role played by the law of distribution of molecular velocities becomes apparent when we consider the formation of the mean values which define the state of a gas [29] So the average value of a quantity Q is a function of the velocity components So we integral equation(3), and making the use of the integral definition of an average quantity Q[8,29]:

Q≡

Zþ∞

∞

dU

Zþ∞

∞

dV

Zþ∞

∞

The energyflux over a planar surface (flat plate) is expressed as:

[8,29]

E¼1

where u is now the velocity parallel to the surface (i.e the slip velocity, usat near the surface) By definition in Ref.[29], the second term is the component of the stress tensor such as shear stress From the momentum-transfer equations derived by the kinetic theory of gas and the equation of motion of thefluid element in the

x, y, and z directions according to Newton’s second law described in Ref.[29]we have

N.T.P Le et al / International Journal of Thermal Sciences 109 (2016) 334e341 335

txy¼ rUV¼ mvvvx þvuvy; (6)

for the gasflow over a flat plate The gas flow over the flat plate

(planar surface) we then consider the flow in the horizontal

(streamwise) direction only So the termvvvxhas been neglected[8]

From the energy-transfer equations derived by the kinetic

the-ory of gas described in Ref.[29]we have

rVC2¼ 15

2mvyv C32

!

The definition of temperature in the kinetic theory of gases is

based on the assumption the molecules possess translational

en-ergy only in which the internal enen-ergy arises entirely from the

translational motion of the molecules[29]

So we have

rVC2¼ 15

2mRvT

where R is the gas constant, for a monatomic gas cv¼ 1.5 R the

coefficient of heat conduction k ¼ 2.5cvm[29], then1rVC2¼ kvT

So we have the heat transfer in slipflow over the flat plate is

presented[8]

q¼ kvTvy usmvuvy ¼ kvT

vy ustxy: (10) Derivation of heat transfer of the slip gasflow over the flat plate

figured out that there are two parts: 1) the usual contribution due

to conduction heatflux normal to the surfacekvT

, and 2) the second one due to sliding friction (or shear work), ustxy This

latter term is no zero if there is some slip at the surface[2] From

this one for gasflow over a flat plate we can express the general

formulation of the heat transfer of the slip gasflow over the curved

surface that we will consider the shear stresses and slip velocities in

three directions as follows:

q¼ kVnT us$S$ðn$PÞ (11)

where Vn≡n$V is the component of the gradient normal to

boundary surface; symbol“$” is inner product;Pis stress tensor;

tensor S¼ I¡nn, where n is the unit normal vector defined as

positive in the direction pointing out of the surface and I is identity

tensor, removes normal components of stresses Thefirst term on

the right hand side of equation(11)expresses the heatflux to the

gas, and the second term presents the sliding frictional work by the

slip over the curved surface Equation(11)is implemented as a new

post-processing utility in OpenFOAM to use with the solver

“rhoCentralFoam”

3 Nonequilibrium boundary conditions

As we consider the sliding friction part in the computation of

wall heat transfer, so the slip velocity and temperature jump

boundary conditions must be applied to the surfaces Thefirst order

Maxwell/Smoluchowski boundary conditions are adopted for our

CFD simulations Thefirst-order conventional slip condition was

developed by Maxwell in 1890 for aflat plate, based on his earlier

work on the theory of viscosity in gases Although it is not perfect, it

is still the simplest and most useful description of the slip condition that we have It is designed to work with the continuum governing equations in the CFD method, such as the NS equations Thefirst order Maxwell slip condition is expressed as follows[30,31]:

uþ



2su

su



lðS$VnuÞ ¼ uw



2su

su



l

mS$ðn$PmcÞ

3 4

m

wherelis the Maxwellian mean free path;gis the specific heat ratio;Pmc¼mðVuÞT



2

 ItrðVuÞ



is the tensor; Pr is the Prandtl number;mis viscosity; p is the pressure and uwis the wall velocity The right-hand side of the equation(12)contains three terms that are associated with (in order): the surface velocity, the so-called curvature effect, and thermal creep The first-order slip condi-tions mentioned above, the accuracy of the simulation results is decided by a free parameter in the slip condition, namely the tangential momentum accommodation coefficient, which varies from 0 to 1 The tangential momentum accommodation coefficient,

su, determines the proportion of molecules reflected from the surface specularly (equal to 1su) or diffusely (equal tosu), and

0su 1

The temperature jump condition was an early development by Smoluchowski[32], and was driven by the heatflux to the surface

in the normal direction:

Tþ2sT

sT ðgþ 1ÞPr2g lVnT¼ Tw; (13) where Pr is the Prandtl number; Twis the wall temperature The temperature jump condition also depends on a free parameter,sT, namely the thermal accommodation coefficient, which varies from

0 to 1, and decides the accuracy of the simulation results The thermal accommodation coefficient is used to ascribe the temper-atures of the receding molecules Specularly reflected molecules recede from the surface with their original incident energy, and diffusely reflected molecules have their energies adjusted to those would arise in a mass of gas in equilibrium at the temperature of the surface Perfect energy exchange between the gas and the solid surface corresponds tosT¼ 1, and no energy exchange tosT¼ 0

If we make nondimensionalization with a reference length, ve-locity and temperature (such as free-stream veve-locity and temper-ature), these boundary conditions (Eqs (12) and (13)) can be expressed in the term of Kn as presented in Ref.[32] It is well known that the rarefaction effect depends on the Knudsen number

A rarefaction effect can be studied by solving the momentum and energy equations with the slip boundary conditions, in which the velocity slip, temperature jump, and shear stress work on the wall are taken into account[9]

The Maxwellian mean free path is defined as follows[33]:

l¼m r

ffiffiffiffiffiffiffiffiffi

p

2RT

r

where the viscositymis computed by the Maxwell model for two-dimensional gas[34]:

m¼

ffiffiffiffiffiffiffiffiffiffi

mkB

p

r

1

pd2T

where m is mass of a gas molecule, m¼ 46.5  1027kg for

nitro-gen, and 48.1 1027kg for air[1]; d is diameter of gas molecules, N.T.P Le et al / International Journal of Thermal Sciences 109 (2016) 334e341

336

d¼ 4.17  1010m for nitrogen, and 4.19 1010m for air[1]; and

kBis the Boltzmann constant

Our CFD simulations are undertaken in OpenFOAM[24] The

open source CFD software, OpenFOAM, uses the finite volume

numeric to solve systems of partial differential equations ascribed

on any three-dimensional unstructured mesh of polygonal cells

The CFD solver“rhoCentralFoam” in the OpenFOAM which solves

the NS equations with the Maxwell slip and Smoluchowski jump

boundary conditions withsu¼sT¼ 1 is selected for performing all

our CFD simulations In this solver, the NS equations are

numeri-cally solved with the high-resolution central scheme described in

Ref.[17] The accommodation coefficients of unity are equivalent to

the diffuse reflections simulated in the DSMC data[13e15,26]

4 Numerical setups

The profile of the NACA 0012 airfoil is computed by the

expression below[35]:

y¼ 0:6



0:2969x

c

0:5

 0:126x

c



 0:3537x

c

2

þ 0:2843x

c

3

 0:1015x

c

4

;

(16) where c is the chord length, c¼ 0.04 m; x is the running distance

along the chord (0 x  c); y is the half thickness of the airfoil The

dimensions of geometries of the airfoil and biconic cases are

pre-sented in Figs 1a and 2a, respectively Numerical setups of

computational domains for these two cases are presented inFigs 1a

and 2b The computational domain of the biconic case is similar to that given in Ref.[13] The structured C-type mesh is used for the airfoil case, as seen inFig 1b, and thefinal mesh is adapted from our previous work in Refs.[13,14]with 600 cells on the airfoil surface; the values L¼ 6c and H ¼ 3c This mesh had been validated with DSMC and experimental data in the previous work[13,14] For the two-dimensional axisymmetric biconic case, the geometry is specified as a wedge of one cell thickness running along the plane

of geometry The axisymmetric wedge planes must be specified as separated patches of type“wedge”, as seen inFig 2b Structured mesh of the biconic case is presented inFig 2c Freestream con-ditions (indicated by the symbol“∞”), and working gas of all airfoil cases are presented inTable 1 The freestream conditions of the biconic case[26]are: p∞¼ 2.23 Pa, M∞¼ 15.6, T∞¼ 42.6 K, and the wall temperature Tw¼ 297.2 K with working gas as nitrogen A typical mesh for a flat plate simulation is regular rectangular A mesh independence analysis was completed tofind the final mesh (Dx ¼Dy¼ 0.117  103 m) for the convergence solution The

freestream conditions of theflat plate case[25]are: p∞¼ 2.97 Pa,

M∞¼ 6.1, T∞¼ 83.4 K, the length of plate L ¼ 0.0875 m, and the wall temperatures Tw¼ 77 K and 294 K with working gas as nitrogen

5 Heat transfer simulation results and discussions 5.1 Sharp 25e55-deg biconic case

Fig 3compares the CFD heat transfers with those of DSMC[26]

and experimental[27]data In 0< x < 0.05 m, the heat transfer with the sliding friction is close to DSMC data[26] The CFD simulations underpredict the heat transfers along the fore-cone in comparing with DSMC and experimental data There are increases in heat transfer across the recirculation zone The heat transfers signi fi-cantly drop in the range 0.072 m x  0.095 m within the area of the recirculation zone, and then rapidly increase the peak values of 1) 103.59 kW/m2at the location x¼ 0.1037 m for the heat transfer without sliding friction, 2) 98.88 kW/m2 at the location

x¼ 0.1045 m for that with the sliding friction, 3) 124.40 kW/m2at the location x¼ 0.1027 m for DSMC data, and 4) 112.93 kW/m2at the location x¼ 0.1023 m for experimental data in the area of the shock-shock interaction It is rational that DSMC and NS solutions disagree on some points due to following reasons, i.e enhanced non-equilibrium effects that undermine the NS basic assumptions

as continuumflow field Past these peak locations, the heat trans-fers rapidly drop and oscillate along the second cone Again the CFD results are lower than DSMC and experimental data than along this cone surface (0.105 m  x  0.153 m) On the base surface, (0.153 m x  0.192 m), there is good agreement between all CFD, DSMC and experimental data Overall, the DSMC method predicts better heat transfer than the CFD method in comparing with the experimental data

5.2 NACA0012 airfoil cases 5.2.1 M∞¼ 2, AOA ¼ 0-deg, Kn ¼ 0.026 Considering the case with small Kn¼ 0.026 and AOA ¼ 0-deg shown inFig 4, DSMC data[13]reach a peak value of 240 W/m2

at the airfoil tip Past the peak value they gradually decrease to the location x/c¼ 0.1, and oscillate along the surface The heat transfer with the sliding friction obtains the peak value of 200 W/m2, and thereafter reduces gradually to the location x/c¼ 0.13 It is a nearly constantfinite value in 0.13 < x/c < 0.9, and increases in the range 0.9 x/c  1 There is a good agreement between the DSMC data and the heat transfer with the sliding friction The heat transfer without the sliding friction disagrees with the DSMC data and is the lowest values along the surface

Fig 1 a): Geometry, dimensions and numerical setups of the NACA 0012 airfoil case

N.T.P Le et al / International Journal of Thermal Sciences 109 (2016) 334e341 337

5.2.2 M∞¼ 2, AOA ¼ 10-deg, Kn ¼ 0.026 Continuing the case Kn¼ 0.026 with the change of AOA ¼ 10-deg, the calculated heat transfers are shown in Fig 5 for the lower surface The heat transfer without the sliding friction dis-agrees with the DSMC data [13] and predicts the lowest heat transfer values along the surface The DSMC data reach a peak value

c

Fig 2 a): Geometry and dimensions of the sharp 25e55-deg biconic case [26] b): Numerical setups of the sharp 25e55-deg biconic case [13] c): Structured mesh of the sharp 25e55-deg biconic case, 256  256 (every fourth line presented) [13]

Table 1

Freestream conditions and working gas of the NACA0012 micro-airfoil [14]

Case p ∞ (Pa) M ∞ u ∞ (m/s) T ∞ (K) T w (K) Gas AOA (deg.) Kn

Airfoil 2.78443 2 509 161 290 Air 0, 10 0.026

Airfoil 0.72395 2 509 161 290 Air 10 0.1

Airfoil 0.27844 2 509 161 290 Air 20 0.26

Fig 3 Heat flux distribution along the biconic surface.

Fig 4 Heat flux distribution over the airfoil surface of the case M ¼ 2, AOA ¼ 0-deg,

Kn ¼ 0.026.

N.T.P Le et al / International Journal of Thermal Sciences 109 (2016) 334e341 338

of 290 W/m2 at the airfoil tip and then reduce gradually to the

location x/c¼ 0.1 Past this location they oscillate along the surface

The heat transfer with the sliding friction obtains the peak value of

125 W/m2 and thereafter decreases gradually to the location x/

c¼ 0.1 It is a nearly constant finite value in 0.1 < x/c < 0.9 before

slightly increasing in 0.9 x/c  1, and is close to the DSMC data

Fig 6presents the calculated heat transfer distributions over the

upper surface The DSMC data and the heat transfer with sliding

friction reach the peak values of 395 W/m2and 50 W/m2,

respec-tively at the airfoil tip Thereafter both of them rapidly decrease to

the location x/c¼ 0.4, and are nearly constant finite values along

the surface There is a good agreement between them The heat

transfer without the sliding friction obtains a negative value of

110 W/m2at the airfoil tip It rapidly reduces until the location x/

c¼ 0.5, and then increases gradually along the surface There is a

difference between the heat transfer without the sliding friction

and DSMC data

5.2.3 M∞¼ 2, AOA ¼ 0-deg, Kn ¼ 0.1

Fig 7presents the computed heat transfer distributions of the

case Kn¼ 0.1 with AOA ¼ 10-deg The heat transfer without the

sliding friction obtains the negative value at the airfoil tip, and then

rapidly increases to reach the peak value of 126.26 W/m2at the

location x/c¼ 0.012 Past the peak value, it gradually decreases along the surface In the range 0.05< x/c  1, its values are the lowest values, and have a gap with the DSMC data[14] The DSMC data and heat transfer with the sliding friction reach peak values of

150 W/m2and 169 W/m2, respectively, at the airfoil tip Past these peak values, they gradually drop to the location x/c¼ 0.15, and are nearly constantfinite values along the surface for both of them, except the latter one increasing in the range 0.9< x/c  1.0 The heat transfer with the sliding friction is close to the DSMC data

Fig 8 shows the simulation results of heat transfer over the upper surface, and is similar the heat transfers over the lower surface The heat transfer without the sliding friction is negative at the airfoil tip whereas that with sliding friction and the DSMC data

[14]are positive The heat transfer without the sliding friction does not give good agreement with DSMC data That with the sliding friction and the DSMC data reach the peak values of 126 W/m2and

179 W/m2, respectively and are close together in 0< x/c < 0.92

5.2.4 M∞¼ 2, AOA ¼ 0-deg, Kn ¼ 0.26 The heat transfer distributions over the lower surface for the case with high Kn¼ 0.26 and AOA ¼ 20-deg are presented inFig 9 The DSMC data[14]have the peak value of 91 W/m2at the airfoil tip Past this peak value they gradually decrease to the location x/

Fig 5 Heat flux distribution over the airfoil lower surface of the case M ¼ 2, AOA ¼

10-deg, Kn ¼ 0.026.

Fig 6 Heat flux distribution over the airfoil upper surface of the case M ¼ 2,

¼ 10-deg, Kn ¼ 0.026.

Fig 7 Heat flux distribution over the airfoil lower surface of the case M ¼ 2, AOA ¼ 10-deg, Kn ¼ 0.1.

Fig 8 Heat flux distribution over the airfoil upper surface of the case M ¼ 2,

¼ 10-deg, Kn ¼ 0.1.

N.T.P Le et al / International Journal of Thermal Sciences 109 (2016) 334e341 339

c¼ 0.1 and are nearly constant a finite value along the lower

sur-face At the airfoil tip the heat transfer with the sliding friction has a

positive value while that without the sliding friction is a negative

value They increase to reach peak values of 34 W/m2at location x/

c¼ 0.01 for the heat transfer with the sliding friction, and 36 W/m2

at x/c¼ 0.04 for the other one Past the peak values that without the

sliding friction gradually decreases along the surface The heat

transfer with the sliding friction generally gives good agreement

with the DSMC data while that without the sliding friction does not

Fig 10shows the simulation heat transfers over the upper

sur-face Both of the DSMC data [14]and the heat transfer with the

sliding friction reach the peak values of 75 W/m2and 30 W/m2,

respectively, and they are generally close together along the

sur-face The heat transfer without the sliding friction does not give

agreement with DSMC data and there is gradually increase in the

range 0.84< x/c < 0.96 before decreasing rapidly to the location x/

c¼ 1.0

5.3 Flat plate cases, M∞¼ 6.1

All calculated heat transfers of theflat plate cases are presented

inFigs 11 and 12 The dsmcFoam solver[24]was used for DSMC

simulations on the same test cases, using the variable hard sphere

model The normal gradient of temperature, the slip velocity, and shear stress play a part in determining the wall heat transfer For the hot-wall case (Tw¼ 294 K), near the leading edge, the wall heat transfer without the sliding friction is negative while that is posi-tive even close to the leading edge of the cold-wall case (Tw¼ 77 K) The effect of the sliding friction on the wall heat transfers is

sig-nificant near the leading edge for both cases due to the large slip velocity and shear stress, which results in their positive values They obtain the peak values of 12.13 kW/m2and 10.33 kW/m2for the cold-wall and hot-wall cases, respectively, at near the leading edge Thereafter they gradually decrease along theflat plate In 0.005 m x  0.02 m the wall heat transfers with the sliding friction of two cases are close to DSMC data, seen inFigs 11 and 12

In 0.02 m x  0.0875 m all DSMC and CFD simulation results are close together This can be explained that the slip velocity and shear stress obtain the peak values near the leading edge, and thereafter they rapidly decrease to the almost low constant values along the flat plate in 0.02 m  x  0.0875 m So the sliding friction does not strongly affect the wall heat transfers The wall heat transfers of the cold-wall case are greater than those of the hot-wall case

5.4 Discussions The CFD wall heat transfers with the sliding friction over the

Fig 9 Heat flux distribution over the airfoil lower surface of the case M ¼ 2,

AOA ¼ 20-deg, Kn ¼ 0.26.

Fig 10 Heat flux distribution over the airfoil upper surface of the case M ¼ 2,

¼ 20-deg, Kn ¼ 0.26.

Fig 11 Heat flux distribution over the flat plate, T w ¼ 77 K.

Fig 12 Heat flux distribution over the flat plate, T w ¼ 294 K.

N.T.P Le et al / International Journal of Thermal Sciences 109 (2016) 334e341 340

micro-airfoil and biconic surfaces of all simulation cases give good

agreements with DSMC data, even for the airfoil cases with high

Kn¼ 0.1 and 0.26, while those without the sliding friction do not

This can be explained by the different nature of the computation of

the heat transfers between DSMC (Eq.(1)) and the CFD method

without the sliding friction (i.e the Fourier law of conduction

q¼ kVnT) The contribution of the velocity terms in equation(1)

such as molecular velocity results in the heat transfer of DSMC to be

positive at near the tip of the airfoil, while those of CFD without the

sliding friction are negative due to the temperature gradient only

Once the slip velocity boundary applying to the surface, the gas

molecules will slip over the surface to generate more the heat

transfer that was computed based on the slip velocity and shear

stress terms

In the previous work[13,14] and our CFD simulations of the

NACA0012 micro-airfoil cases, the shear stresses reached the peak

values at the tip of the airfoil Past these values, they gradually

decrease along the airfoil surface Corresponding to the shear stress

distributions the slip velocities were slow due to viscous stress at

the tip airfoil, and gradually increase to trailing edge of the airfoil

Since the heat transfers over the surface including the sliding

friction, which are computed by the inner product of the shear

stresses and slip velocities, are greater than those without the

sliding friction This results in the values of heat transfers including

the sliding friction are positive at the tip of airfoil, and close to the

DSMC data along the micro-airfoil surface

For the biconic case the slip velocity and shear stress reached

the peak values at the tip of biconic and then gradually decrease in

x< 0.05 m The wall heat transfer involving the sliding friction

gives good agreement with DSMC data at the tip of biconic

(x< 0.05 m) Past this location, the heat transfer with and without

the sliding friction are close together because the slip velocity and

shear stress are small values in 0.05 m< x < 0.2 m presented in our

CFD simulations This results in the sliding friction term does not

significantly affect the heat transfer calculation in

0.05 m x  0.2 m

For theflat plate case the sliding friction strongly affects the wall

heat transfer near the leading edge due to the slip velocity and

shear stress By involving the sliding friction part, the CFD heat

transfers give good agreements with DSMC data for the cases

Kn> 0.1

6 Conclusions

The present work revisited the computations of the CFD heat

transfer with and without the sliding friction in high-speed rarefied

gasflow simulations The formulation of the heat transfer including

the sliding friction was re-evaluated for the flat plate case and

extended to the curved surfaces such as NACA0012 micro-airfoil

and the sharp 25e55-deg biconic From all simulation results

ob-tainedfigured out the importance of the sliding friction part in

calculating the wall heat transfer in CFD This represents the work

done directly over the surface by the shear and is nonzero only if

the slip velocity condition is applied to the surface The addition of

the sliding friction part in the computation of CFD heat transfers

leads to good agreement with DSMC for all cases considered,

especially for the cases with high Knudsen numbers, Kn¼ 0.1 and

0.26 (i.e in the transition regime)

Acknowledgements

The authors thank Dr Craig White at the University of Glasgow

for providing data from his DSMC simulations of theflat plate cases

This research is funded by Vietnam National Foundation for Science

and Technology Development (NAFOSTED) under grant number 107.03-2015.16

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