An investigation of tip vortex formation and structure is an important problem of aerodynamics. The tip vortices are formed on outboard edges of wings and controls of flight vehicles (FV) due to the pressure difference. They exert influence on the aerodynamic characteristics of wings, the noise level, and the flight safety [1, 2]. The penetration of a FV or an element of its design into a tip vortex can lead to the loss of control or the breakdown. Different structural units, such as winglets, sharklets, and other types of wingtips [3], are used to reduce the tip vortices. The wingtips of subsonic FVs increase considerably the lift and reduce the drag. The possible application of the wingtips of the winglet type at supersonic velocities is investigated in [4] at M = 1.62. The authors of that study arrive at conclusion that in this flow regime the winglets are ineffective in reducing the wing drag.
One of the directions of perfecting the flight vehicle aerodynamics, which has been intensely developed most recently, is the use of structural units made of porous materials. The porous materials applied in flight vehicles can be produced basing on either various metals (nickel, bronze, etc.) or the materials based on heat-resisting porous carbon-based materials which possess high heat stability (2500 K) and a small specific density.
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There has been much research devoted to the use of porous materials for stabilizing boundary layers [5‐8], in bleed air-intake ducts [9], diffusers [10], nose cowls [11‐13], and other structural units of flight vehicles.
For this reason, both the effect of porous design elements and tip vortices are of interest. The studies concerning this subject are few in number; we can mention, for example, the description of the NASA patent [14]. Some new and helpful effects can be found in this field. This study is devoted to the formation of tip vortices in the conditions, when either the entire wing or its part are made of a porous material. By way of illustration, we consider the straight wing of simple geometry in supersonic flow.
MATHEMATICAL MODEL AND NUMERICAL ALGORITHM
The calculations presented in this study were performed using the S3D program package [15] developed and implemented into software in the Keldysh Institute of Applied Mathematics. This program complex is intended for solving three-dimensional aerodynamic problems. An implicit difference scheme applied in this complex uses the LU SGS method for solving the system of linear equations. The fluxes on the cell faces are calculated using the Godunov interpolation scheme.
To describe flows of a perfect viscous compressible gas the system of unsteady Reynolds-averaged Navier–Stokes (URANS) equations with the one-equation Spalart–Allmaras (SA) turbulence model for compressible flows is used. The equations are discretized using the finite volume method which, as distinct from the finite difference method, can be applied to any geometry, operates with different grids, and makes it possible to avoid the problems with metric singularities of generalized coordinates.
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The model of flow in a porous material is based on the restricted Baer–Nunziato model [16] which was primarily developed for describing the combustion and detonation processes in granular explosives. In this model, the medium is considered as a two-phase continuum that consists of the solid skeleton of unreacted explosive and the gas phase of combustion products. The model used in this study is a reduction of the Baer–Nunziato model based on the assumption that the solid skeleton remains fixed.
The instantaneous gas state is determined by its density ρ, the velocity vector u, and the pressure p. The gas is assumed to be a compressible, viscous, and heat-conducting medium. The skeleton is supposed to possess heat capacity and thermal conductivity, while the gas and skeleton temperatures are assumed to be the same.
It is assumed that the skeleton microstructure is isotropic and is characterized by the distribution of the skeleton volume fraction φ = φ(x). The quantity α = α(x) = 1 – φ(x) is the local distribution of pores (porosity).
The geometric shape of a continuous solid body can be preassigned in two ways, namely, either letting α = 0 or using the no-slip boundary conditions. In this study, the surfaces of continuous bodies are determined by the boundary conditions. Other-than-unity α values are used only in calculating the flows in porous regions of the wings.
As a result, we arrive at the following system of equations which describe in the continuum approximation compressible flows in the constrained environment of a porous permeable skeleton
$$\frac{{\partial \alpha \rho }}{{\partial t}} + \frac{{\partial \alpha \rho {{u}_{k}}}}{{\partial {{x}_{k}}}} = 0,$$
(1)
$$\frac{{\partial \alpha \rho {{u}_{i}}}}{{\partial t}} + \sum\limits_{k = 1}^3 {\frac{{\partial \alpha (\rho {{u}_{i}}{{u}_{k}} + p{{\delta }_{{ik}}})}}{{\partial {{x}_{k}}}}} = p\frac{{\partial \alpha }}{{\partial {{x}_{i}}}} + \sum\limits_{k = 1}^3 {\frac{{\partial \alpha {{\tau }_{{ik}}}}}{{\partial {{x}_{k}}}}} - {{g}_{i}},$$
(2)
$$\frac{{\partial \alpha \rho E}}{{\partial t}} + \frac{{\partial \left( {1 - \alpha } \right){{E}_{{\text{s}}}}}}{{\partial t}} + \sum\limits_{k = 1}^3 {\frac{{\partial \alpha \rho H{{u}_{k}}}}{{\partial {{x}_{k}}}}} = \sum\limits_{k - 1}^3 {\left( {\frac{{\partial \alpha {{\tau }_{{ik}}}{{u}_{i}}}}{{\partial {{x}_{k}}}} - \frac{{\partial {{\theta }_{k}}}}{{\partial {{x}_{k}}}}} \right)} .$$
(3)
Here, the notation is commonly accepted: i, k = 1, 2, 3 are the indices; ρ is the density, uk are the velocity vector components, p is the pressure, τik = 2μefeik – 2/3μefejjδik is the viscous stress tensor, eik = 0.5(∇iuk + ∇kui) is the strain rate tensor, μef = μmol + μturb is effective viscosity, H = E + p/ρ is the total enthalpy, E = Et + W is the total energy of the gas phase, \({{E}_{t}} = p{\text{/}}[\rho \left( {\gamma - 1} \right)]\) is the internal energy per unit mass (determined by the equation of state for an ideal, calorically perfect gas), W = 0.5\(\Sigma _{{k = 1}}^{3}u_{k}^{2}\) is the kinetic energy per unit mass, Es = CsT is the thermal energy of the skeleton, Cs is the heat capacity of the solid phase per unit volume, \({{\theta }_{k}} = - \lambda {{\nabla }_{k}}T\) is the molecular heat flux, λ is the thermal conductivity of the two-phase medium, Т is the temperature, and δik is the Kronecker tensor.
The molecular viscosity coefficient is assumed to be a function of the local air temperature Т given by the semi-empirical Sutherland formula. The gas thermal-conductivity coefficient λg is related with the dynamic viscosity coefficient μ by the Prandtl number Pr which is assumed to be constant: \(\Pr = 0.733\). The thermal conductivity coefficient of the solid phase λs is assume to be constant and determined by material properties. The coefficient of thermal conductivity of the two-phase medium λ = αλg + (1 – α) λs.
The body force \({{g}_{i}}\) in Eq. (2) is the viscous friction force that acts from the skeleton on the gas. It depends on the local velocity and the density of the gas and the skeleton microstructure (or the pore microstructure). Generally, the viscous friction force can be expressed in terms of the viscous drag coefficient in the Ergun form [17]where sp = Selem/Velem is the skeleton dispersity, that is, the ratio of the interphase surface area to the skeleton volume per unit physical volume, and Cd is the drag coefficient determined by the empiric Ergun formulas as a function of the Reynilds number and the porosity.
$${{g}_{i}} = \frac{{\left( {1 - \alpha } \right){{s}_{p}}{{C}_{d}}\rho \left| {\mathbf{u}} \right|}}{8},$$
A more detailed description of the numerical algorithms and mathematical models used in this study can be found in [18].
The parallel algorithms of the numerical solution were realized in the multiprocessor K-100 system of the Keldysh Institute of Applied Mathematics [19].
NUMERICAL CALCULATIONS
The vortex wake in supersonic flow was investigated with reference to the example of a straight wing with 0.2 mm-thick leading and trailing edges. The wing chord and semi-span were 30 and 47.5 mm long, respectively. The wing airfoils near the base and on the outer edge are shown in Fig. 1. The wing shape between the given airfoils is such, that all the surfaces are plane, namely, a plane quadrangle in the midsection and triangles near the leading and trailing edges. The wing schematics in plan are shown in v.
Fig.
1. Wing schematics: general view (a) and the airfoil near the edge (b) and the base (c).
The wing tip (cross-hatched in Fig. 1) is made of porous materials with different porosities, that is, volume fraction of the gas phase.
The porous material applied in the model is considered to be a continuous two-phase medium determined by two parameters, namely, porosity α, that is, the ratio of the gas phase volume to the overall volume, and the characteristic pore size interpreted as the mean diameter of the channels in the porous material.
The checking calculations were performed for a continuous wing and then the calculations for five porosity values were carried out: α = 0.4, 0.6, 0.7, 0.8, and 0.9.
The characteristic pore size was taken to be 0.1 mm.
All the calculations were performed for the same oncoming flow with the Mach number M = 3, the Reynolds number based on 1 m Re = 8 803 209, and the angle of attack α = 10°.
For the purpose of checking the grid convergence the calculations were performed on two grids.
The block-regular grid 1 consists of 5 951 600 cells with refinement near the wing surface, the leading and trailing edge, and the side edge. The cell thickness near the wing surface is 0.04 mm. The size of the region behind the wing is equal to nine chord lengths and that on wing exterior equals to the wing half-span. The no-slip condition is preassigned on the surface of the continuous wing portion and the interface between the continuous and porous surfaces. The mirror symmetry condition is preassigned in the z = 0 plane, where the wing base is mounted, the supersonic entry is assumed at the forward boundary of the domain, and the extrapolation condition is fulfilled on the other outer boundaries.
Grid 2 is also block-regular, with the cell number 7 312 800; near the leading and trailing edges the cell size in the longitudinal x direction is reduced by a factor of 10 compared with grid 1, while that in the horizontal, transverse z direction is reduced by the factor of 1.5 near the side edge. The cell size in the y direction remains the same.
The origin of coordinates is in the plane of symmetry, near the leading edge of the wing. The x axis is directed downstream, the z axis is directed from the plane of symmetry to the wing edge, and the y axis is perpendicular to the flow and directed downward. The geometric dimensions are expressed in meters.
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RESULTS OF THE CALCULATIONS
An analysis of the results of the numerical simulations showed that for all porosity values considered a tip vortex is formed. It is analogous to the vortex on a continuous wing but it is located near the end of the continuous region rather than in the vicinity of the edge of the porous wing tip. A circulation zone arises above the porous region of the wing but in all the cases the vortex center lies above the continuous region of the wing.
Since the air flowing from the wing undersurface passes now through the porous material, its velocity decreases, the vortex structure becomes complicated and nonsymmetric, and in most cases the vortex intensity is reduced. At large porosity values the vortex intensity may increase (Fig. 7).
Figure 2 presents the vortex structure in the transverse plane passing through the trailing edge of the wing for different porosity values and in the case of the control calculation for the continuous wing. The vector lines of the crossflow velocity field (uz, uy) are presented. The porous region of the wing is encircled with a dash line.
Fig. 2.
Vector lines of the crossflow velocity field (uz, uy) in the plane of the x = 0.0293 section (trailing edge of the wing) for different values of the porosity α.
It is interesting to note that at α = 0.8 the vortex structure is qualitatively similar with that of the vortex on the continuous wing, only the vortex is displaced at a distance equal to the porous wingtip width. This means that the material with this porosity exerts a somewhat smaller effect on the flow structure and the wing is near-equivalent to a continuous wing of smaller span. However, some quantitative difference can be observable.
A variation in the vortex structure leads to a variation in the pressure distribution. In Fig. 3 the pressure coefficients are presented on the wing surface, in the z = 0.042 section for the continuous check-in wing and for α = 0.6. Clearly that the pressure decreases on the wing undersurface and increases on the upper side.
Fig. 3.
Pressure coefficient on the wing surface z = 0.042.
In Fig. 4 the results of calculations on two grids are presented. Here, the pressure coefficients for α = 0.6 calculated on grid 2, the same as in Fig. 3, are compared with the results of calculations on grid 1. Clearly that there is a difference near the leading edge, where a shock is formed, but on the greater portion of the wing surface the discrepancy is small.
Fig. 4.
Pressure coefficient calculated on the fine 1 and coarse 2 grids.
The pressure redistribution leads to a variation of the wing drag, lift, and lift-drag ratio. In Table 1 the aerodynamic coefficients (Cxs, Cys) are presented for the wing midsection and the wing edge, which is porous for α ≠ 0. The drag of the continuous wing midsection consists of two components, namely, the pressure drag and the friction drag on the material surface. When a porous material is in flow, a third component is added to the two preceding. At small values of α the air flows over the material, the flow within the material is only slight, and, therefore, the third component is small. At large α a considerable portion of the air passes through the material but the internal friction drag is small due to a great number of air voids in the material. The greatest drag must be observable at medium values of α, and we can see that at α = 0.6 it is actually higher than for the other values.
Table 1.
Porosity | 0 | 0.6 | 0.8 | 0.9 |
|---|---|---|---|---|
C
xs
| 0.3 | 0.292 | 0.294 | 0.295 |
C
xp
| 0.019 | 0.068 | 0.025 | 0.009 |
C
x
| 0.319 | 0.360 | 0.319 | 0.304 |
C
ys
| 1.263 | 1.22 | 1.228 | 1.229 |
C
yp
| 0.084 | 0.028 | 0.008 | 0.002 |
C
y
| 1.347 | 1.248 | 1.236 | 1.231 |
K | 4.22 | 3.47 | 3.87 | 4.05 |
The differences in the vortex structure and intensity are conserved in the process of its downstream evolution. In Fig. 5 the longitudinal vorticity Ωx = ∂uz/dy – ∂uy/dz and the Mach number at the center of the vortex are presented for the continuous wing and for α = 0.6.
Fig. 5.
Longitudinal vorticity (Ωx,103 s–1) and Mach number (M) at the vortex center for the porous 1 and continuous 2 wings.
The Mach number at the vortex axis is considerably smaller in the case of the porous wingtip. In this case, the difference in the gas velocity is slight, while the speed of sound and the pressure turn out to be greater.
In Fig. 6 we have plotted the isolines of longitudinal vorticity in the section x = 0.2348 (6.85 chord lengths from the trailing edge) behind the continuous wing (continuous curves) and the wing provided with a porous tip with the porosity α = 0.6 (dashed curves). For the sake of clarity the contours of the wing and its trailing edge are also presented in the figure. The porous wingtip is covered with gray. Clearly that the vorticity is considerably reduced (its maximum has become 23.9 rather than 29.7), while the center of the vortex has displaced from the edge toward the boundary between the continuous and porous parts of the wing.
Fig. 6.
Contours of the longitudinal vorticity (103 s–1) in the sections x = 0.0293 (left) and x = 0.2348 (right).
In Fig. 7 the flow parameters are presented, namely, the longitudinal vorticity Ωx and the tangential Mach number Myz along the line passing through the vortex center, perpendicular to its axis. The sections x = 0.1348 (at a distance of 3.5 chord lengths behind the edge) and x = 0.2348 (6.85 chord lengths) are presented. The curves are plotted for the continuous wing (α = 0) and three porosity values α = 0.4, 0.6, and 0.8.
Fig. 7.
Longitudinal vorticity Ωx and tangential Mach number Myz in the sections x = 0.1348 and 0.2348.
Despite the fact that the vortex centers on the continuous wing (α = 0) and the wing with a porous tip are at different points in the z coordinate, it can be seen in Fig. 7 that in all the cases the vortex center positions in the y coordinate are similar in value. It is interesting to note that the vortex structures on the continuous and α = 0.8 wings are very similar and differ from each other considerably less than on the wing with other porosity values. This indicates that the wingtip with a high porosity has only a small effect on the vortex and this wing is near-equivalent with a smooth wing having a smaller span.
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SUMMARY
The results of the numerical investigation of a tip vortex in supersonic М = 3 flow are presented for a wing with the wingtip made of a porous material at different porosity values. In the calculations the mathematical model of gas flow through a porous medium, the algorithms of turbulent flow modeling, and their parallel realization on multiprocessor computational systems were used.
The numerical results obtained show that a tip vortex formed on the wing with a porous tip is similar with that formed on a continuous wing. However, the vortex parameters, structure, and location differ considerably. In all the cases considered the vortex is formed near the edge of the continuous portion of the wing rather than near the boundary of the porous material.
When the porosity value is smaller than 0.8, the vortex becomes less intense; at the porosities 0.8 and 0.9 the vortex strength increases somewhat. The wing drag and lift decrease. Thus, there arises a possibility of controlling the vortex generation process.
The calculations are performed for the model wing with a simple geometry. They demonstrate the possibility in principle to control the tip vortex parameters using porous wingtips. In the case of other wing geometries the effects can be different. An analogous problem for a delta wing is of interest and can be subject of further studies.
CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest.
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Translated by M.Lebedev