Numerical Study of Hypersonic Aerodynamics and Heating on a ...

AIAA 2017-2162 International Space Planes and Hypersonic Systems and Technologies Conferences 6-9 March 2017, Xiamen, China 21st AIAA International Space Planes and Hypersonics Technologies Conference

Numerical Study of Hypersonic Aerodynamics and Heating on a Cylinder at Mach 6

Downloaded by NATIONAL UNIVERSITY OF DEFENSE on September 20, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.2017-2162

Chen Li-li1, Guo Zheng2, Hou Zhong-xi3, and Wang Wen-kai4, College of Aerospace Science and Engineering , National University of Defense Technology Changsha, Hunan, 410073, P.R.China and Huang Chun-hua5 China Satellite Maritime Tracking and Control Department Jiangyin, Jiangsu, 214413, P.R.China Prediction of surface heating flux and pressure distribution is fundamental and critical to the design of the thermal protection system (TPS) for hypersonic aircraft. This paper studies the effect of different grid levels and different convective fluxes discretization. Roe and AUSM+ spatial discretization is analyzed and compared. The CFD-FASTRAN is very excellent software in computing hypersonic aerodynamic and heating. The typical 2-D cylinder is selected to complete this work on the FASTRAN flatform at Ma 6. The results show that pressure distribution is less susceptible to first layer grid scale and spatial discretization schemes. While grid Reynolds number is between 0.61 and 61, the pressure error is about 5% in the stagnation region and that is less 2% in other region compared with experiment. However, Heat flux has a close connection with first layer grid scale and spatial discretization schemes. Roe scheme is susceptible to grid level and AUSM+ shows feeble sensitivity to grid. AUSM+ can gain relative stable numerical data that fluctuate less than 8%, when Reynolds number is between 0.61 and 61. Calculated results of 1th order AUSM+ is approach to 2th order AUSM+ at different grid level. Calculated results of 1th order Roe shows evident discrepancy with 2th order Roe. While grid Reynolds number is less than 10, 1th order Roe scheme can gain values close to experiment data. Therefore, it is essential that grid Reynolds number should be less than 10 to gain numerical aerodynamic heating values, AUSM+ scheme is more suitable to gain heat flux tendency at extensive grid Reynolds number at Ma 6. Additionally, the effect of unit Reynolds number and freestream temperature is investigated, the results show that higher unit Reynolds number can generate severer heat flux and higher freestream temperature can lead to more serious aerodynamic heating.

Nomenclature CFD P T 



RANS FVS FDS Re Pr

= = = = = = = = = =

computational fluid dynamics pressure temperature dynamic viscosity gas density Reynolds Averaged Navier-Stokes flux vector splitting flux difference splitting Reynolds number Prandtl number

1

Postgraduate Student, College of Aerospace Science and Engineering, National University of Defense Technology, [email protected]. 2 Professor, College of Aerospace Science and Engineering, National University of Defense Technology, [email protected]. 3 Professor, College of Aerospace Science and Engineering, National University of Defense Technology, [email protected]. 4 Postgraduate Student, College of Aerospace Science and Engineering, National University of Defense Technology, [email protected]. 5 Assistant Engineer, China Satellite Maritime Tracking and Control Department, [email protected]. 1 American Institute of Aeronautics and Astronautics

Copyright © 2017 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Ma q_w y+ theta

= = = =

Mach number heating rate non-dimensional wall distance for wall-bounded viscous flows angle from cylinder forward stagnation point

I. Introduction


vehicles are rapidly developing in the aeronautical and astronautical realm. Nevertheless, because Hypersonic ground experimental facilities has costly expense and cannot simulate the real flight environment, it is necessary to employ computational fluid dynamic (CFD) method to gain some important aerodynamic values. In recent years, CFD method becomes more and more widespread in hypersonic aerodynamic and heating prediction and can gain satisfactory values. However, CFD heavily rely on grid level, algorithm and discretization schemes. For different configurations, it may need particular mesh and scheme, in the numerical calculation, grid is at the first place and vital, especially for aerodynamic heating. Scholars has conducted affluent research on grid effect in hypersonic flow simulation. Lee Joon Ho1 investigates the hypersonic viscous flow difference of Roe's FDS and AUSM+ schemes around a blunt body. Ref.2 argues that grid convergence is critical to the aerodynamic prediction on the surface of vehicles. Jed E. Marquart3 studies the aerodynamic effects of heating and cooling a circular cross-section cylinder surface,shows that the freestream air temperature significantly influences the drag coefficient. Menshov I S and Nakamura Y4 give the result that it is necessary that grid Reynolds number should be less than 3 to obtain accurate heat flux. Klaus Hannemann5 investigates the cylinder shock layer density profiles and heat flux though CFD and HEG shock tunnel at Mach 8.78 that involves high temperature chemical reaction. At hypersonic Mach numbers, shock waves and boundary layers are generally regions where the flow gradients are very high, solving the conservation equations in such regions can usually lead to numerically stiff problems and result in non-physical flow property calculations and instabilities6. To eliminate the stiffness of the computation and non-physical oscillations, some spatial discretization methodology is proposed to ensure high levels of accuracy, robustness and efficiency for computing hypersonic flows. Upwind-biased numerical schemes are popular in computing hypersonic flows. There are two methods of identifying the upwind directions, namely the flux vector splitting (FVS) approach and the flux difference splitting (FDS) approach. FDS formulations has the advantage of the solution of the local Riemann problem, which results in more accurate solutions. Roe’s Riemann solver is the most popular among FDS schemes due to its efficiency and accuracy. The AUSM scheme resolves shear layers much better than Roe’s scheme, and is more suited to be used in computing hypersonic flows. Keiichi Kitamura7 develops a new pressure flux for AUSM-Family Schemes for hypersonic heating computations. Commonly, it is regarded that the flow that Mach number is more than 6 is hypersonic, however, more research mainly in more than 10 Mach, even higher Mach, for example reentry problems. It is seldom found that aerodynamic heating is numerically studied at Mach 6. Wieting A.R. 8 investigated the shock wave interference heating on a cylindrical leading edge though plenty experiment at Ma 6 and 8 in his doctor’s dissertation. In this paper, the cylinder is selected to investigate the aerodynamic and heating at Mach 6, different grid level and spatial discretization schemes are exploited to compute the aerodynamic and heating, the numerical values are compared with experiment8.

II. Governing Equations The flow solver is a Reynolds Averaged Navier-Stokes (RANS) solver that utilizes finite volume approach. The governing equations are the compressible Navier-Stokes equations as follows,

U ( Ei  Ev ) ( Fi  Fv ) (Gi  Gv )    0 t x y z

(1)

Where, U is conservation scalar quantity, Ei , Fi , Gi represent the flux quantity at x, y and z direction, respectively. The subscript i represents inviscid flux vector and v is viscid flux vector.

U    , u,  v,  w,  e

T

2 American Institute of Aeronautics and Astronautics


u     2  u  p  Ei    uv  ,     uw   e  p u    t

v    u v    2  Fi   v  p  ,     vw   e  p v    t

  w     uw  Gi    vw      w2  p      et  p  w

      0 0 0        xy  xx  xz             F   yy Ev   xy  yz  Gv     v          yz  xz  zz       u yx  v yy  w yz  q y  u zx  v zy  w zz  qz  u xx  v xy  w xz  qx  ,

where，

 xx  2

u 2 v 2 w 2    V ,  yy  2    V ,  zz  2    V , x 3 y 3 z 3  u v   v w   u w    , zx   xz     ,  , yz   zy      z x   y x   z y 

 xy   yx    qx  k

T T T p 1 , qy  k , qz  k , et    u 2  v 2  w2  . x y z   1  2

Laminar viscous model is selected to solve the flowfield. Specific heat ratio is set as constant 1.4. Sutherland formula is selected to calculate viscosity and thermal conductivity is defined by Pr as follows,

T 1.5   1.4605  10 T  112 C p k Pr 6

(2) (3)

The governing equations are discretized and numerically integrated based on a finite-volume approach. Roe’ FDS and Liu’ AUSM are used as spatial discretization schemes, both Roe and AUSM schemes are first order spatially accurate, the spatial order of the accuracy can be increased to second or third order spatial accuracy by the added limiter. In addition, for Roe’ FDS first order scheme, the entropy fix is a requirement if supersonic expansions are expected within a flow. This fix can remove the presence of expansion shocks at the sonic point. It is introduced to eliminate the non-physical oscillation in the vicinity of shock wave and sonic point. Entropy fix is presented as,

i  max  i , max 

(4)

Where max  V  c , V is the local velocity magnitude, c is the local speed of sound. The entropy fix is set 0.2 both for the linear waves and non-linear waves.

III. Cylinder Geometry and Grid Generation Cylinder is a typical hypersonic shock verification test model, numerous experiment is carried to investigative the flow property at multiple Mach3, 8-10. Ref.8 gives the experimental pressure and heat transfer data with experiment “RUN 37”,“RUN 38”and “RUN 65”at Ma 6.5 from tests in the NASA Langley 8-Foot High Temperature Tunnel(8’ HTT) and “RUN 9”and “RUN 32” at Ma 6.3 in the Calspan 48-Inch Hypersonic Shock Tunnel. The “RUN 38” is chosen to verify the validation of numerical method in this paper, the model is a 3-in (76mm) diameter cylinder, the freestream total temperature is 1900K,unit Reynolds number Re  1.36  106 /m 3 American Institute of Aeronautics and Astronautics

The freestream condition is shown in table 1. Table 1. Freestream Condition

Ma

T





p

Tw

5

6.46 235.7 0.0103 650.86 294 1.52  10 A sort of structured grid is generated by CFD-GEOM, in order to study the effect of grid level on hypersonic flow, three grids are divided at different first cell height, the overall grid is 200  200 (the normal mesh number 200 nodes and along cylinder wall 200 nodes). The grids G1, G2 and G3 are distinguished with ymin (first layer cell height) 3.8  105 m , 3.8  106 m and 3.8  107 m , respectively. Here, grid cell Reynolds number is given by


Recell =

  u ymin 

(5)

Where, Recell is the first layer grid height Reynolds number. Table 2. Grid Description of different Grid Levels

(a)G1

Grid

Scale

ymin

Recell

G1 G3 G4

200  200 200  200 200  200

3.8  105 m 3.8  106 m

60.8 6.08 0.61

3.8  107 m

(a)G2 Figure 1. Grid of 2-D cylinder

(a)G3

The inlet boundary was assigned as a “far-field” boundary condition, the outlet as a “inflow/outflow” boundary condition, the cylinder surfaces (200 elements) were assigned as isothermal (294K) no-slip boundaries and the bottom boundary is “symmetry” condition.

IV. Results and Discussion A. Cases Description Different levels grid and spatial schemes are presented in this section, for spatial scheme, it is 1th order spatial accuracy, it becomes 2th order with flux limiter, here, Min-Mod (L) limiter is applied. There are 12 cases analyzed in the following. The concrete description are shown in table 3. Table 3. Cases Description Cases Case1 Case2 Case3

Grid G1 G1 G1

Spatial schemes Roe’ FDS Roe’ FDS Liu’ AUSM

Flux Limiter Min-Mod(L) -


Spatial Accuracy 1st order 2st order 1st order


Case4 Case5 Case6 Case7 Case8 Case9 Case10 Case11 Case12

G1 G2 G2 G2 G2 G3 G3 G3 G3

Liu’ AUSM Roe’ FDS Roe’ FDS Liu’ AUSM Liu’ AUSM Roe’ FDS Roe’ FDS Liu’ AUSM Liu’ AUSM

Min-Mod(L) Min-Mod(L) Min-Mod(L) Min-Mod(L) Min-Mod(L)

2st order 1st order 2st order 1st order 2st order 1st order 2st order 1st order 2st order

B. Parameters Analysis Based on different grid and spatial schemes, the calculated results are shown in three parts. Pressure, heat flux, density and y+ are extracted on the cylinder surface, respectively. Pressure and heat flux values are compared with experimental data8. At every grid level, above four spatial schemes results are plotted in a chart.

1. ymin = 3.8 105 m

(a)pressure distribution

(b)heat flux distribution

(c) gas density distribution (d)y+ distribution Figure 2. Calculated results comparison of cylinder at G1 grid For grid G1, Fig.2 gives the pressure, heat flux, gas density and y+ distribution. Roe and AUSM+ scheme can obtain consistent pressure values with experiment. However, heat flux presents conspicuous difference. Results of Roe 2ord scheme is close to experiment. The error of stagnation heat flux of Roe 1ord and Roe 2ord markedly reaches 100%, while error of that for AUSM+ 1ord and AUSM+ 2ord is only 2%. The density change can be neglected at different schemes, as they are highly identical. Fig. 2(d) shows that the wall y+ is less than 5.



2. ymin = 3.8 106 m



(c)gas density distribution (d)y+ distribution Figure 3. Calculated results comparison of cylinder at G2 grid As Fig.2, Fig.3 also gives the calculated corresponding values of grid G2, the pressure distribution and gas density does not obviously change due to grid or spatial scheme. At grid G2, the values of heat flux by Roe 1ord match well with experiment. The results by AUSM+ 1ord and AUSM+ 2ord are less than experiment, yet variance with spatial order is tiny.

3. ymin = 3.8 107 m





(c)gas density distribution (d)y+ distribution Figure 4. Calculated results comparison of cylinder at G3 grid It can be found that Fig.3 and Fig.4 represent accordant tendency and distribution in pressure, heat flux and density. Similarly, heat flux is the focus for numerical values in this research. The heat flux of AUSM+ scheme still extremely steady not limited by grid and spatial order. Above all, it can be concluded that heat flux can gain relatively stable values while the grid Reynolds number is less than 10. Roe scheme appears intense susceptibility to grid level and spatial order. Comparatively, AUSM+ demonstrates favourable applicability to different grid and spatial order. Accordingly, AUSM+ is a more eligible alternative in computing heating for hypersonic vehicle. Simultaneously, because the heat flux results of AUSM+ 1ord and AUSM+ 2ord coincide well with each other, it is recommended that AUSM+ 1ord is selected to compute heat flux for 2-D blunt body at Mach 6 to reduce simulation time and save storage. C. Flowfield Analysis

(a) ymin = 3.8  105 m pressure contour

(b) ymin = 3.8  106 m pressure contour


(c) ymin = 3.8  107 m pressure contour


Figure 5. Pressure contour comparison of different grid levels and spatial schemes Fig.5 demonstrates pressure contour comparison at different grid and spatial scheme. Pressure contour illustrates little variance with grid level but visible difference with spatial scheme. 2ord scheme displays pressure oscillation near shock wave. Because 1order accuracy upwind scheme always has dissipation and can smooth the variance of flowfield, especially in the discontinuity of shock.

(a) ymin = 3.8  105 m Temperature contour

(b) ymin = 3.8  106 m Temperature contour

(c) ymin = 3.8  107 m Temperature contour Figure 6. Temperature contour comparison of different grid levels and spatial schemes Temperature contours can clearly depict the temperature distribution in the flowfield. Fig.6(a), Fig.6(b), Fig.6(c) give the temperature contour at grid G1, G2, G3, respectively. For same scheme, contours resemble at different grid, for same grid, contours are also approximate at different scheme, but 1order and 2 order scheme lead to evident distinction. 1 order scheme can obtain smooth flowfield distribution, the reason is similar to that 8 American Institute of Aeronautics and Astronautics

of pressure contour in Fig.5. Although temperature contours appear little discrepancy at different grid, the heat flux distributions vary obviously. Compared with heat flux distribution, it can conclude that proper grid and spatial scheme is specifically significant in aerodynamic heating calculation for hypersonic flow. D. Unit Reynolds Number Effect In this section, the effect of unit Reynolds number is investigated by changing the freestream pressure, the temperature is preserved as constant 235.7K, the unit Reynolds number varies from 0.51 106 /m to


2.03 106 /m . The concrete description is displayed in table 4. Table 4. Cases description of different unit Reynolds number Ma

T (K)



6.46 6.46 6.46 6.46 6.46

235.7 235.7 235.7 235.7 235.7

1.52  105 1.52  105 1.52  105 1.52  105 1.52  105

p (pa) 260 455 650 845 1040

Tw (K)

Re( 10 /m )

294 294 294 294 294

0.51 0.89 1.27 1.65 2.03

6

(a) heat flux distribution (b) pressure distribution Figure 7. Heat flux and pressure distribution versus unit Reynolds number Heat flux appears evident difference at different unit Reynolds number in Fig.7, at the same position, the heat rate increases with unit Reynolds number augmentation, however, the heat flux is not in linear proportion to that. In Fig.7(b), pressure change tendency resembles heat flux. Pressure increases with unit Reynolds number increasing. Therefore, it is certain that Reynolds number effects are not negligible in hypersonic flow. E. Freestream Temperature Effect In this section, the effect of freestream temperature is investigated, the unit Reynolds number is preserved as constant 1.36 106 /m . With the temperature changing, the viscosity, static pressure and freestream velocity also change to ensure the Mach number and unit Reynolds number invariable. The concrete description is displayed in table 5. Table 5. Cases description of different freestream temperature

u  m/s 



Tw (K) Re( 106 /m )

Ma

T (K)

6.46

100

0.689  105

1303

206.4

294

1.36

6.46

150

1.024  10

5

1596

375.7

294

1.36

6.46

200

1.324  105

1843

560.9

294

1.36

250

1.595  10

5

2060

755.4

294

1.36

1.842  10

5

2257

955.8

294

1.36

6.46 6.46

300

p (pa)



(a) heat flux distribution (b) pressure distribution Figure 8. Heat flux and pressure distribution versus freestream temperature It can be seen that freestream temperature has tremendous effect on heat flux of hypersonic flow in Fig.8(a). The maximum heat rate at 300K is 9 times as much as that at 100K, which reveals that temperature-decreasing can reduce the heat flux on the surface of hypersonic vehicles at same Mach and Reynolds number. Fig.8(b) illustrates the pressure distribution at different temperature, it can be concluded that higher freestream temperature would produce greater surface pressure for hypersonic flow at the same Mach and Reynolds number.

V. Conclusion This paper researches the aerodynamic and heating on a 2-D cylinder for hypersonic flow at Mach 6, mainly focus on the effect of different gird and spatial schemes. Through three sorts of grid level at different first layer grid height and different schemes, 12 cases are numerically analyzed. Pressure distribution, heat flux, density and y+ are plotted and compared. For pressure prediction, different grid and spatial schemes gain highly identical distribution compared with experiment data. For heat flux prediction, the results show that Roe scheme is special sensitive to first layer grid height in heat flux calculation, while the grid Reynolds number is less than 10, the Roe scheme can gain relatively stable values. Meanwhile, Roe 1ord and Roe 2ord gain different heat flux distribution at same grid. The Roe 1ord can gain heat flux values more close to experiment at fine grid that grid Reynolds number is less than 10. Overall, Roe scheme requests sternly to grid, it is difficult to master and gain credible values in reality. However, for AUSM+, regardless of grid and spatial scheme, the heat flux distribution are consistent, so, the AUSM+ scheme is not sensitive to grid and has fine dependability at extensive grid. Aerodynamic heating computing is inherently difficult in hypersonic flow, the experiment values error generally reaches about 20%, so an accurate and reliable tendency prediction seems greatly important. Therefore, it is advisable to apply AUSM+ scheme to compute in hypersonic flow. Additionally, the effect of unit Reynolds number and freestream temperature are investigated, heat flux distribution has intimate link with unit Reynolds number and freestream temperature, higher Reynolds number and higher temperature both would bring about more serious aerodynamic heating.

References 1

Lee Joon Ho, Hyan Rho Oh. “Numerical analysis of hypersonic viscous flow around a blunt body using Roe's FDS and AUSM+ schemes,” AIAA 97-2054, 1997. 2 Bertin J J., M. Cummings R. “Critical Hypersonic Aerothermodynamic Phenomena,” Annual Review of Fluid Mechanics, Vol.38, No.1, 2006, pp.129-157. 3 Jed E. Marquart, Endicott Derick S. “CFD Investigation of the Drag Effects of Heating and Cooling Cylinders in Crossflow”, 6th AIAA Flow Control Conference, AIAA 2012-3045, New Orleans, Louisiana, 2012. 4 Menshov I S, Y Nakamura. “Numerical Simulation and Experimental Comparison for High-speed Non-equilibrium Air Flows”. Fluid Dynamics Research, Vol.27, No.5, 2000, pp.305-334. 5 Klaus Hannemann, Schramm Jan Martinez, Karl Sebastian, Beck Walter H. “Cylinder Shock Layer Density Profiles Measured in High Enthalpy Flows in HEG”. 22nd AIAA Aerodynamic Measurement Technology and Ground Testing Conference, AIAA 2002–2913, St. Louis, MO, June 24–28, 2002. 6 Susheel Kumar Sekhar. “Viscous Hypersonic Flow Physics Predictions using Unstructured Cartesian Grid Techniques,” Ph.D. Dissertation, Georgia Institute of Technology, 2012. 10 American Institute of Aeronautics and Astronautics


7 Keiichi Kitamura, Shima Eiji. “A New Pressure Flux for AUSM-Family Schemes for Hypersonic Heating Computations,” 20th AIAA Computational Fluid Dynamics Conference, AIAA 2011-3056, 2011. 8 Wieting A.R. “Experimental study of shock wave interference heating on a cylindrical leading edge,” Ph.D. Dissertation, Old Dominion University, 1987. 9 Francesco Grasso, Purpura Carlo, Chanetz Bruno, Delery Jean. “Type III and type IV shock/shock interferences: theoretical and experimental aspects,” Aerospace Science and Technology,Vol.7, 2003, pp, 93–106. 10 Schuelein Erich. “Effects of Laminar-Turbulent Transition on the Shock-Wave/Boundary-Layer Interaction,” 44th AIAA Fluid Dynamics Conference, AIAA 2014-3332, Atlanta, GA,16-20 June, 2014.
