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UDC 551.511.61 DOI: 10.18698/1812-3368-2022-3-86-95
NUMERICAL INVESTIGATION OF A SUPERSONIC FLOW IN THE NEAR WAKE REGION OF A CYLINDRICAL AFTERBODY
A.M. Molchanov alexmol_2000@mail.ru
D.S. Yanyshev dyanishev@gmail.com
L.V. Bykov bykovlv@mai.ru
Moscow Aviation Institute (National Research University), Moscow, Russian Federation
Abstract
A computational study of a supersonic flow in the base region and the nearest wake of a cylindrical body moving at a supersonic speed have been carried out. A mathematical model of high-enthalpy flows is presented. In this case, the "prehistory" of the flow was taken into account, i.e., the configuration of the computational domain was as close as possible to the real one. The use of various turbulence models for calculating flow in the base region and the nearest wake was analyzed. The following turbulence models were considered: 1) the Spalart — Allmaras model; 2) SST model; 3) standard k-s model; 4) k-s model with compressibility correction; 5) k-s RNG (renormalized group) model; 6) k-s Realizable model; 7) standard Reynolds Stress (RS) model; 8) RS BSL (Reynolds stress baseline) model. Based on a comparison of the calculation results with experimental data, it is shown that: 1) when calculating the flow in the base region and in the wake of the vehicle, it is very important to take into account the "prehistory" of the flow, i.e., to calculate the flow around the entire vehicle; 2) the best match was obtained using Reynolds Stress models and the k-s RNG model
Keywords
Computational fluid dynamics, turbulence, external flow, supersonic flow
Received 26.01.2022 Accepted 14.02.2022 © Author(s), 2022
Introduction. The theoretical study of a flow in the base area and in the wake of a high-speed vehicle is important in many aerospace applications. The separated flow arising in the base area of such aerodynamic bodies can have a significant effect on various flight characteristics. Low bottom pressure causes drag that affects stability, trajectory and range. The reverse flow can significantly increase the heat flow to the structural elements of the vehicle.
It is generally accepted (see, e.g. [1-3]) that the theoretical approach based on the use of Reynolds-averaged Navier — Stokes equations is not very successful
in describing such flows due to several reasons, such as the instability of the recirculation zone and the difficulty of modeling turbulent characteristics in a compressible mixing layer.
It was shown in [4] that the "prehistory" of a flow has a very large effect on the distribution of parameters in the bottom area and in the wake, i.e., the flow along the cylindrical part of the vehicle and the nose section.
The purpose of this work is to analyze the use of various turbulence models for calculating the flow in the base area and in the wake of a high-speed vehicle. And the "prehistory" is taken into account, i.e., the configuration of the computational domain is as close as possible to the real one.
Supersonic, axisymmetric base flow. A schematic diagram of the mean flowfield structure in the near wake of a cylindrical afterbody aligned with a supersonic flow is shown in Fig. 1 [4].
5
Fig. 1. Flow diagram of base flow in the wake of a vehicle moving at supersonic speed:
1 — expansion waves; 2 — free shear layer; 3 — recompression region; 4 — trailing wake;
5 — rear stagnation point
The supersonic afterbody freestream flow undergoes a strong expansion centered at the base comer as the turbulent boundary layer separates geometrically from the body. A free shear layer is formed which separates the outer inviscid flow from a relatively large recirculation region immediately downstream of the base. The intense turbulent mixing and energy exchange that characterize the free shear layer are important in determining the flowfield properties throughout the near wake including the recirculation region. As the free shear layer approaches the axis of symmetry, a recompression process occurs which eventually realigns the flowfield with the axis. A rear stagnation
point, where the mean velocity vanishes, is located on the centerline and separates the recirculation region from the wake which develops downstream.
Governing equations. To describe the flow of a multicomponent gas mixture, the following system of equations written in vector form is used
f£ & *)-H,
with main vector
convective vector
U =
pu1 pu2 pu3 pET pCi
pC№-i pTi
pT Nt
FC =
pu1 puiu1 + 5l1 p pulu2 + ... + 5i2 p puiu3 + + 5'3 p pui (ET + p/p) puiC1
puCNc-i pu'T
pulTM
viscous vector
Ff =
0
X1'1 X1'2 X1'3
X uj xij - q j=1
- Ji
~ JNc-1
- gl
- gNt
source vector
H
Nc
Y
0,0,0,0, - X h°ws, wW 1,..., wNc-i, St,I, ..., ST,Nt
s = 1
and Kronecker delta in cartesian frame
i 10 0 ^
8ij =
0 1 0
V0 0 b
Fluxes are computed using the following formulae:
x11 =-» 3
2 (~^uuL-^uuL диъ ^
dx1 dx2 dx3
X12 =x21
f
du1 ^ du2 dx1 dx1
\
22
X22 = — u
3
2 _ÇuL
dx2 dx1 dx3
X13 =x31
23 32 X23 = X32 = ^
v L/л L/л J
' du2 du3 ^ +
v-x dx2 y
( Du1 | du3 ^ dx3 dx1
dx3 dx2
д dh
33
x33 = —u 3
q' =
Pr dx'
JS =-
2 ( „ du3 gu1 du2 } dx3 dx1 dx2
Д dCs
Sc dx1
Here p is the density of the gas mixture; ui are velocity components; p is pressure; xij is viscous stress tensor; ^ is effective coefficient of dynamic viscosity (takes into account both molecular and turbulent transport); Cs is mass fraction of species s; ws is the rate of formation of species s as a result of chemical reactions; Nc is the number of species of the gas mixture; JS is the diffusion flux of the species s; ET is the thermodynamic part of the total energy; qi is the density of the heat flux due to thermal conductivity and diffusion; Pr, Sc are the effective Prandtl and Schmidt numbers; h is enthalpy; h° is the enthalpy of formation of the species s.
The pressure of the gas mixture is the sum of the partial pressures of all species:
p = TPs = X A = P M- T, (8)
s = 1 s = 1 Ms M
where T is the temperature; RU is the universal gas constant; Ms is the molar mass of species s; M^ is the apparent molecular weight of the gas mixture,
Mz =
( Nc C Л
s=i Ms
-1
(9)
The main system includes parameters characterizing turbulence T1, T2,.., TNt, the choice of which depends on the used turbulence model.
In this paper, the following turbulence models were considered (for the sake of brevity mathematical description of the models is omitted, for their thorough description address, e.g., to [5] and [6]): 1) the Spalart — Allmaras model; 2) SST (shear stress transport) model; 3) standard k-s model; 4) k-e model with compressibility correction; 5) k-s RNG (renormalized group) model; 6) k-s Realizable model; 7) standard Reynolds Stress (RS) model; 8) RS BSL (Reynolds stress baseline) model.
Results and discussion. The experimenta data from [4, 7, 8] was used for testing the calculation method. Free stream parameters: M = 2.5, T = 130 K, p = 30 141 Pa. Body diameter 63.5 mm.
The program Universe CFD [9-11], developed at the MAI by the Department of Thermal Engineering, was used for the calculations. We also used the methods and general approaches from [12-14].
The computational grid for flow around the vehicle, the flow in the base region and in the near wake is shown in Fig. 2. It was built as purely structured multiblock grid with quad elements and then converted to unstructured one to use within Universe CFD unstructured solver.
Fig. 2. Computational grid
Grid independence study was performed with different several meshes of different sizes. The one shown on Fig. 2 was found to be optimal to use for all case studies. Its overall size was 16 233 cells.
Figure 3 shows some results of calculating the velocity on axis in the base region and in the near wake using various turbulence models.
и/щ u!u\
0 1 2 3 4 5 6 7 x/R 0 1 2 3 4 5 6 7 x/R
с d
Fig. 3. Axial velocity distribution along centerline in the base region of a cylindrical model in a supersonic flow M = 2.5:
a — the Spalart — Allmaras model; b — k-e RNG model; c — model RS standard; d — model RS BSL; 1 — experiment [3], 2 — experiment [5]; 3, 4 — calculations (3 — no compression correction, 4 — with compression correction)
Figure 4 shows the dependence of the base pressure on the freestream Mach number. The standard RS model as well as RS BSL model are quite successful from this point of view.
pipi
1.5 2.0 2.5 3.0 3.5 4.0 M
Fig. 4. Base pressure ratio versus freestream Mach number:
circles — experimental data [6]; 1 — SST turbulence model; 2 — k-e RNG turbulence model; 3 — standard RS turbulence model; 4 — RS BSL turbulence model
In this view it is interesting also to look at Reynolds shear stress values obtained via RS BSL model (Fig. 5). As it can be seen from the figure, the obtained profiles of the Reynolds shear closely resemble experimental data.
r/R r/R
Fig. 5 (beginning). Radial distribution of dimensionless Reynolds shear stress
u'v' / u2) in different sections:
a — x/D = 1.26; b — x/D = 1.42; 1 — experimental data as per [8]; 2 — RS BSL turbulence model
r/R
1.5 1.0 0.5 0
-1.0 -0.5 0 0.5 1.0 -u'v'/uf c
Fig. 5 (end). Radial distribution of dimensionless Reynolds shear stress {-u'v' 7 u2)
in different sections: c — x/D =1.73; 1 — experimental data as per [8]; 2 — RS BSL turbulence model
Although, Reynolds stress models take much more computational effort (5 equations are being solved in 2d case and 7 in 3d case), as we can see, they are worth implementing for modeling flows with complex vortex structures.
Nevertheless, for the future work it might be prudent to investigate behavior of the eddy viscosity models with anisotropic corrections for the flows of the considered type.
Conclusion. Based on the analysis of the obtained data and their comparison with the experiments, the following conclusions can be drawn.
When calculating the flow in the base region and in the wake of the vehicle, it is very important to take into account the "prehistory" of the flow, i.e., to calculate the flow around the entire vehicle.
To better describe the velocity distribution in the base region, it is recommended to use models of RS type, which confirms the superiority of the models of this type over simpler eddy-viscosity models such as ks and SST, the main shortcoming of which, is their inability to account for history of flow upstream which influences motions of the larger scale eddies and contributes to the transfer of scalar quantities such as turbulent energy (see [15, 16]).
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Molchanov Aleksandr M. — Dr. Sc., Assoc. Professor, Department of Thermal Engineering, Moscow Aviation Institute (National Research University) (Volokolamskoe shosse 4, Moscow, 125993 Russian Federation).
Yanyshev Dmitry S. — Cand. Sc., Assoc. Professor, Department of Thermal Engineering, Moscow Aviation Institute (National Research University) (Volokolamskoe shosse 4, Moscow, 125993 Russian Federation).
Bykov Leonid V. — Cand. Sc., Assoc. Professor, Department of Thermal Engineering, Moscow Aviation Institute (National Research University) (Volokolamskoe shosse 4, Moscow, 125993 Russian Federation).
Please cite this article as:
Molchanov A.M., Yanyshev D.S., Bykov L.V. Numerical investigation of a supersonic flow in the near wake region of a cylindrical afterbody. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 3 (102), pp. 86-95. DOI: https://doi.org/10.18698/1812-3368-2022-3-86-95
