THEORETICAL & APPLIED MECHANICS LETTERS 1, 022001 (2011)

Effects of mesh resolution on hypersonic heating predictionQuanhua Sun,a) Huiyu Zhu, Gang Wang, and Jing FanKey Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences,Beijing 100190, China

(Received 03 November 2010; accepted 26 January 2011; published online 10 March 2011)

Abstract Aeroheating prediction is a challenging and critical problem for the design and optimizationof hypersonic vehicles. One challenge is that the solution of the Navier-Stokes equations stronglydepends on the computational mesh. In this letter, the eect of mesh resolution on heat ux predic-tion is studied. It is found that mesh-independent solutions can be obtained using ne mesh, whoseaccuracy is conrmed by results from kinetic particle simulation. It is analyzed that mesh-inducednumerical error comes mainly from the ux calculation in the boundary layer whereas the temperaturegradient on the surface can be evaluated using a wall function. Numerical schemes having strongcapability of boundary layer capture are therefore recommended for hypersonic heating prediction.c 2011 The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1102201]Keywords hypersonic ow, aeroheating, CFD, mesh resolution

Hypersonic ight vehicles typically refer to aircraftsthat can y in the atmosphere with more than ve timesthe speed of sound. With the increase of the ightspeed, vehicles could suer terrible thermal loading dueto shock heating and skin friction from the ambient at-mosphere, which is based on the fact that the surfaceheat ux or aeroheating on an aircraft is proportionalto the cube of the ight speed, whereas the drag forceis proportional to the square of the speed. However,prediction of aeroheating is a hard task both exper-imentally and numerically. For instance, Bertin andCummings listed the aeroheating prediction as one ofthe most challenging problems in computational uiddynamics (CFD).1

In the literature, issues on aeroheating predictionhave been widely studied,2 including various physicalmodels, numerical scheme, and mesh resolution. Amongthese issues, physical models are probably the mostchallenging one, which addresses the ow physics suchas real gas eects, wall catalysis, radiation and cooling,transition and turbulence. Eects of physical models onaeroheating, however, are already qualitatively knownthanks to tremendous eorts.1 The mesh resolution, onthe other hand, is purely numerical. It is well-knownthat numerical results of aeroheating depend stronglyon the employed mesh.3{5 In CFD practices, mesh inde-pendence study is usually performed by rening meshesuntil numerical results agree with experimental or ightdata. The bad thing is that the solution becomes inac-curate when the mesh is further rened. In other words,an improper mesh could predict larger or smaller heat-ing for a hypersonic ow. Therefore, analysis of detailedeect of mesh resolution on heat ux is benecial to thedesign of eective mesh for hypersonic heating predic-tion.

To start the analysis, a simple case, supersonic owover a front step, is rst simulated. The governing equa-

a)Corresponding author. Email: qsun@imech.ac.cn.

tions are the Navier-Stokes equations. The simulatedgas is argon, which is preferred for kinetic particle sim-ulation that is adopted for the purpose of validation.The numerical scheme employed is the muscl type nitevolume method with Roe's FDS scheme and the min-mod limiter for the convective ux evaluation and thecentral dierence scheme for the viscous ux calcula-tion. Other specication of the case is: step height 2 h,wall temperature 1 000 K, free stream gas temperature200 K, free stream Mach number 10, free stream Knud-sen number (=h) 0.001, free stream Reynolds number13 000.

The mesh independence is studied by employing twosets of meshes. The rst set is equally-spaced meshwhose mesh size is set at 1/5, 1/10, 1/20, 1/40, 1/80 ofthe semi-height of the step, respectively. The second setemploys clustered mesh where the mesh near the sur-face is smoothly rened based on the equally-space meshwhose size is 1/80 h, so that the smallest mesh size inthe normal direction becomes 1/2, 1/4, 1/8, 1/16, 1/32,1/64, 1/128 of the base mesh size 1/80 h, respectively.Simulations show that the mesh-independent results canbe obtained.

Figure 1 displays the temperature eld (T T1)=(Tshock T1) of the ow where a bow shock can beeasily observed. The gas temperature jumps across theshock and keeps increasing until it reaches the surfacethat is relatively cool. Simulations show that the oweld can be captured correctly even with a coarse meshhaving the size of 1/80 h. The heat ux on the sur-face, however, depends strongly on the mesh resolution.Figure 2 shows the heat ux coecient on surface lo-cated at 0.5 h height, where dx is the smallest meshsize. Clearly, the value of simulated heat ux increasesquickly with mesh renement at early stage. After itreaches its maximum, the heat ux converges graduallyto a constant that is the mesh-independent value. Forthis case, a mesh size around 1/1 000 of the semi stepheight is needed to reach the mesh-independent value.If experimental data are available and assumed to be

022001-2 Q.H. Sun, H.Y. Zhu, G. Wang and J. Fan Theor. Appl. Mech. Lett. 1, 022001 (2011)

Fig. 1. Temperature eld T = (T T1)=(Tshock T1) ofargon ow over a front step where step height=2 h, Tw =1 000 K, M1 = 10; T1 = 200 K, Kn1(= =h)=0.001,Re1 = 13 000:

consistent with the constant value, then a simulationcould give good prediction even with a relatively coarsemesh (mesh size of 1/40 h), but bad prediction will thenshow up with ner meshes, which explains the observa-tion in some studies.5

Fig. 2. Predicted heat ux coecient obtained from a set ofmeshes. The simulation is for the ow over a front step case.The mesh is denoted by the smallest mesh size dx. The uxis on the surface located at 0.5 h height.

It is necessary to validate whether the convergedheat ux is the physical solution of the ow. A to-tally dierent numerical technique, the direct simula-tion Monte Carlo (DSMC) method,6 is employed to sim-ulate the same ow. The DSMC method is a particleapproach that simulates a large number of microscopicmolecules by tracking their motions and collisions. Itis numerically expensive to simulate the current owwhere the global Knudsen number is only 0.001. Fig-ure 3 shows the heat ux coecient obtained from both

CFD and DSMC simulations where the CFD solutionis the mesh-independent data. The overall agreementbetween the two solutions is very good. Dierence isobserved only in a very small region near the step cor-ner, which can be attributed to two factors. One is thatit is hard for CFD to capture the ow gradients nearthe corner. Another is that the local Knudsen numbernear the corner is large and the Navier-Stokes equationsbecome invalid in this small region. Therefore, it canbe concluded that the mesh-independent result is thephysical solution of the Navier-Stokes equations.

Fig. 3. Comparison of heat ux coecient along the frontstep obtained using CFD and DSMC simulations for the owover a front step case.

The eect in Fig.2 of mesh size on the heating pre-diction is observed in many ow simulations. Figure4 presents CFD results for several problems including

ow over front-step, cylinder, and sphere under dierent

ow conditions. The simulated gas includes argon andair. In the plot, the mesh size is represented by the sur-face grid Reynolds number (Reg = cacy=c) wherethe length scale is the mesh size in the normal direc-tion and all values are at the near surface cell. Clearly,mesh-independent results are obtained for all the caseswhen Reg is less than 5, which agrees well with nd-ings in the Ref. 1. Of course, many factors will aectthe behavior of mesh dependence for aeroheating. Forinstance, dierent numerical scheme will have slight dif-ferent converging process of mesh resolution as shownin Fig. 5 where the ow over a cylinder is simulated.

It should be mentioned that the surface gridReynolds number is only one measure of the mesh size.A very important fact is that Reg species only the sizeof the surface mesh. The size of other part of meshwill also aect heat ux evaluation, which is often over-looked by some researchers. In practice, larger cells areused in domain away from the aircraft surface in or-der to save computational cost. Then the ow may notbe resolved locally with the large cells. In addition, thenon-uniform mesh will produce additional numerical er-ror. This may be the reason why meshes having verysmall surface cells may predict bad results sometime.

022001-3 Effects of mesh resolution on hypersonic heating prediction Theor. Appl. Mech. Lett. 1, 022001 (2011)

Fig. 4. Eects of surface grid Reynolds number on heat

ux predicted at the stagnant point where the heat ux isnormalized by the grid-independent value for several owcases.

Fig. 5. Heat ux coecient predicted by typical numericalschemes for ow over a cylinder. Schemes employed are:Roe's FDS scheme, Roe's scheme with entropy x, van Leer'sFVS scheme, AUSM scheme, AUSM+ scheme, and exactRiemann (Godunov) solver.

Our simulations have indicated that the size ratio be-tween neighboring cells should be less than 1.2 to avoidobvious numerical error.

From numerical simulation, it is clear that the meshsize should be very small near the surface to predictcorrectly the heat ux. The main reason may be thatthe gradients of ow properties such as temperature arevery large near the surface or in the boundary layer.Figure 6 shows a typical temperature prole in theboundary layer where results from several meshes areplotted. It is found that the temperature distributionis nonlinear, thus numerical error could occur duringthe space discretization. Flux evaluation is then themain source for the error, especially the evaluation ofthe temperature gradient on the surface.

In fact, the temperature distribution near the sur-

Fig. 6. Typical temperature prole along the normal direc-tion in the boundary layer for the case of ow over a frontstep.

face can be estimated using wall function. Wall functionhas been employed for bounded turbulent ow simula-tions. For hypersonic laminar ow, we can also derivewall function for temperature and velocity. In hyper-sonic boundary layer, quasi one dimensional ow canbe assumed. Then the temperature along the surfacenormal direction can be approximated as

T (y) = Tw (cy + 1)1

!+1 ; (1)

when the viscosity employs the VHS model6 that /T!, where c = [(Tc=Tw)

!+1 1]=yc. The surface heattransfer is calculated as

q = w 1! + 1

(Tc/Tw)!+1 1

Tc/Tw 1 Tc Tw

yc: (2)

Expression (1) is quite accurate for hypersonic ows.Figure 7 shows the temperature gradients evaluated atdierent locations for the front-step case. The wall func-tion can give good results even when the mesh is large,which illustrates the benet of the wall function. How-ever, the mesh independence of heat ux evaluation isonly slightly improved using the wall function, whichindicates that the numerical error comes from the uxevaluation in the boundary layer.

To verify the error source, the ux on a mesh faceis calculated using dierent mesh sizes based on ana-lytic expressions of ow properties within the boundarylayer. The temperature is approximated using Eq. (1).The velocity prole can also be derived. For instance,the tangent velocity component in the boundary layercan be approximated as

u =a

c(! + 1)

T

Tw 1

+

b

c(! + 1)

T

Tw

264

TTw

!+1!+2!+1c

y

375b

(! + 2) c2; (3)

022001-4 Q.H. Sun, H.Y. Zhu, G. Wang and J. Fan Theor. Appl. Mech. Lett. 1, 022001 (2011)

Fig. 7. Temperature gradients evaluated at dierent surfacelocations using linear expression and wall function for thecase of ow over a front step.

where a=(@u=@y)0, b=dp=wdx, c= 1ych

TcTw

!+11i.

The pressure is assumed constant along the normal di-rection, thus the density distribution is also available.With the known ow properties, the ux at a locationof 1/50 h away from the surface is calculated using thecorresponding discrete values. It is found that the cal-culated uxes have good accuracy when the mesh sizeis very small. When the mesh size increases, the uxerror increases and depends on the numerical scheme.Figure 8 shows the energy ux calculated with dierentmesh size using typical schemes involving the minmodlimiter. It is noticed that the error appears when themesh size is about 5 microns (h/dx=1000) for the Roescheme with entropy x (Roe2), which agrees with theresults in Fig. 1. Therefore, the ux evaluation is themain source for numerical error when the mesh size islarge.

It can be concluded that hypersonic heating can becorrectly predicted as long as the governing equationsare valid. It turns out that the mesh resolution plays avery important role in numerical simulations. Very ne

Fig. 8. Energy ux (kg2/s3) through a mesh face calculatedusing numerical schemes when the discrete values on themesh are obtained from wall functions for the case of owover a front step. The mesh face is located at x = 0:0149 mand y = 0:5 h where the front step is at x = 0:015 m andh = 0:005 m.

mesh is usually required near the surface of hypersonicvehicles in order to resolve the nonlinear distribution of

ow properties. Numerical schemes having strong ca-pability in capturing boundary layer are recommendedfor hypersonic heating prediction.

This work was supported by the National Natural Sci-

ence Foundation of China (50836007, 90816012, 10621202.)

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