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Numerical Supersonic Aerodynamics of the Soyuz/ST Rocket Fairing

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Numerical Supersonic Aerodynamics of the Soyuz/ST Rocket Fairing Numerical Supersonic Aerodynamics of the Soyuz/ST Rocket Fairing Javier Urzay 2nd November 2004 Abstract The supersonic blunt-body problem is one of the most classical challenges in CFD of compressible flows. The change of mathematical behavior of the Euler equations across transonic flow zones made impossible its solution until a time-dependent ap- proach was first proposed by Moretti and Abbett [1]. The time-dependent approach is based on integrating the unsteady conservation equations until a steady solution is achieved. In this study, a similar method is employed to compute the solution of the full Navier-Stokes equations around a rocket fairing using a time-dependent approach with an explicit method that is second-order accurate in space and time. The grid reproduces the geometry of the rocket fairing and is generated using an ellip- tic transform, which requires integration of two non-linearly coupled elliptic equations. 1 The Soyuz Payload Fairing The type-ST fairing of the Soyuz rocket consists of a two-halves shell carbon-fiber rein- forced plastic structure [2]. The fairing hosts payloads such as satellites or other space instrumentation. The fairing structure is a cylindrical body attached to a blunt nose that enables a detached shock and prevents strong aerodynamic heating of the payload. It has a 4.110 m external diameter and provides the largest available volume for spacecraft accommodation in the Soyuz Launch Vehicles family. The Soyuz rocket is based on the original design of Sergei Korolev of the R-7A rocket that put the Sputnik Satellite into orbit in 1957. A complete family of rockets followed after that included Vostok, Molniya, Voskhod and finally the four-stage Soyuz, which is employed widely in manned and un- manned space missions by the European Space Agency (E.S.A.). The main manufacturers and partner organizations are the Russian Aviation and Space Agency (ROSAVIACOS- MOS), the European Aeronautics, Defense, and Space Company (E.A.D.S), the Samara Space Center (TsSKB-Progress) and ArianeSpace. 1 Figure 1: Schematic Soyuz/ST rocket. The spacecraft consists of four stages: Boosters, Core Stage, 3rd Stage, and Fregat Upper Stage. The payload fairing sits on top of the rocket hosting the upper stage and satellite payload. 2 2 Grid Generation: The Elliptic Transform The detailed geometry of the fairing is displayed schematically in figure 2. Figure 2: Dimensions (mm) of the payload fairing. Because of the relatively complex geometry, the most appropriate method for con- structing the grid is the elliptic transform. This method solves the coordinates fx; yg as a function of transformed coordinates variables fξ; ηg, whose space corresponds to a rectangular domain. To do so, the coupled elliptic and strongly non-linear equations @2x @2x @2x @x @y @x @y 2@x @x α − 2β + γ = − − P + Q @ξ2 @ξ@η @η2 @ξ @η @η @ξ @ξ @η @2y @2y @2y @x @y @x @y 2@y @y α − 2β + γ = − − P + Q @ξ2 @ξ@η @η2 @ξ @η @η @ξ @ξ @η need to be integrated, where the coefficients are given by @x2 @y 2 α = + @η @η @x @x @y @y β = + @ξ @η @ξ @η @x2 @y 2 γ = + . @ξ @ξ The functions P and Q force the grid points to be clustered around a point or a line, respectively. In the present configuration, the mesh is forced to be squeezed around the internal boundary corresponding to the fairing surface, with Q being given by −d·|η−η j Q(η) = c · sign(η − ηmin)e min : A Gauss-Seidel iteration and a 2nd order accurate discretization in space are employed to solve the set of elliptic differential equations. The resulting grid is shown in figure 3, while figure 4 shows the clustering of points near the neighborhood of the nose to capture the boundary layer. 3 Figure 3: Elliptic grid of the payload fairing (m). Figure 4: Zoom around the leading nose showing the clustered grid points. 4 3 The Navier-Stokes Equations in Generalized Coordinates 3.1 Formulation Given the coordinate transformation x = x(ξ; η) y = y(ξ; η) the Jacobian J(ξ; η) is defined as @x @y @(x; y) @ξ @ξ J ≡ ≡ @(ξ; η) @x @y : @η @η An appropriate change of partial differential operators must be made according to the change of variables @(') 1 h@y @(') @y @(')i = − @y J @η @ξ @ξ @η @(') 1 h@x@(') @x@(')i = − @x J @ξ @η @η @ξ for any fluid variable '. The gas is assumed to be ideal and calorically perfect, with a constant Prandtl number and temperature-dependent viscosity and thermal conductivity. Using the coordinate transformation above, the Navier-Stokes equations can be written as Continuity: @ρ 1 h@y @ @y @ @x @ @x @ i + (ρu) − (ρu) + (ρv) − (ρv) = 0 (1) @t J @η @ξ @ξ @η @ξ @η @η @ξ x-Momentum: @ 1 h@y @ @y @ (ρu) + (ρu2 + P − τ ) − (ρu2 + P − τ )+ @t J @η @ξ xx @ξ @η xx @x @ @x @ i + (ρuv − τ ) − (ρuv − τ ) = 0 (2) @ξ @η xy @η @ξ xy y-Momentum: @ 1 h@y @ @y @ (ρv) + (ρuv − τ ) − (ρuv − τ )+ @t J @η @ξ xy @ξ @η xy @x @ @x @ i + (ρv2 + P − τ ) − (ρv2 + P − τ ) = 0 (3) @ξ @η yy @η @ξ yy 5 Total Energy: @ jvj2 1 n@y @ h jvj2 P i ρ e + + ρ e + + u + q − uτ − vτ @t 2 J @η @ξ 2 ρ x xx xy @y @ h jvj2 P i − ρ e + + u + q − uτ − vτ + @ξ @η 2 ρ x xx xy @x @ h jvj2 P i + ρ e + + v + q − uτ − vτ @ξ @η 2 ρ y xy yy @x @ h jvj2 P io − ρ e + + v + q − uτ − vτ = 0 (4) @η @ξ 2 ρ y xy yy Equation of State: P = ρRgT (5) Local Thermodynamic Equilibrium: e = cvT (6) Fourier's Law: k(T )h@y @T @y @T i q = − − (7) x J @η @ξ @ξ @η k(T )h@x@T @x@T i q = − − (8) y J @ξ @η @η @ξ Poisson's Law: 2 µ(T )h @y @u @y @u @x@v @x@v i τ = 2 − 2 + − (9) xx 3 J @η @ξ @ξ @η @η @ξ @ξ @η 2 µ(T )h @x@v @x@v @y @u @y @v i τ = 2 − 2 + − (10) yy 3 J @ξ @η @η @ξ @ξ @η @η @ξ µ(T )h@x@u @x@u @y @v @y @v i τ = − + − (11) xy J @ξ @η @η @ξ @η @ξ @ξ @η where the second coefficient of viscosity is calculated from the Stokes' approximation 2 λ = − 3 µ. Additionally, the dynamic viscosity and thermal conductivity variations are given by the Sutherland's Law and the constant Prandtl-number assumption, T 3=2 Tref + 110 µ(T ) = µo (12) Tref T + 110 µ(T )c k(T ) = p : (13) P r 6 The system 1-13 represents the full set of conservation equations required to solve the problem. For numerical integration purposes, it is expedient to reformulate these equations in terms of the principal fluxes φ , E and F, namely [4] 8 ρ 9 > > <> ρu => φ = ρv (14) > 2 > > v > : ρ e + 2 ; 8 ρu 9 > 2 > <> ρu + P − τxx => E= (15) ρuv − τxy > 2 > > v P > : ρ e + 2 + ρ u + qx − uτxx − vτxy ; 8 ρv 9 > > <> ρuv − τxy => F = 2 : (16) ρv + P − τyy > 2 > > v P > : ρ e + 2 + ρ v + qy − uτxy − vτyy ; In these variables, the Navier-Stokes equations become @φ 1 h@y @E @y @Ei 1 h@x@F @x@Fi + − + − = 0: (17) @t J @η @ξ @ξ @η J @ξ @η @η @ξ Equation 17 represents the system of conservation equations written in strong conser- vative form, which facilitates the numerical treatment of the shock discontinuities. These consist of discontinuities in the primitive variables ρ, v, P and T , whose values are not defined at the shock. Conversely, the flux variables E and F are conserved through the shock as prescribed by the integral form of the conservation equations across. As a re- sult, the strong conservative form 17 enhances numerical stability despite the inherent discontinuities in the primitive variables. 3.2 Boundary Conditions The boundary conditions are shown schematically in figure 5. 3.3 Numerical Formulation A second-order accurate Mac'Cormack's explicit method is employed to transform (17) into a finite-differences form. The principal flux φ at time t + ∆t is given by t t+∆t t+∆t t 1h@φ @φc i φi;j = φi;j + + (18) 2 @t i;j @t i;j corresponding to a predictor-corrector set. The average derivative in time is composed by two terms, namely t+∆t @φt φb − φt 1 h@y Et − Et @y Et − Et i = i;j i;j = i+1;j i;j − i;j+1 i;j + @t i;j ∆t Ji;j @η i;j ∆ξ @ξ i;j ∆η 1 h@x Ft − Ft @x Ft − Ft i + i;j+1 i;j − i+1;j i;j (19) Ji;j @ξ i;j ∆η @η i;j ∆ξ which represents the linearization of the time derivative at stage t, where both E and F 7 Figure 5: Boundary conditions. The free stream is characterized by the velocity, pressure, temperature and two thermodynamic coefficients. A non-slip condition is used on the fairing surface. The outgoing boundaries employ interpolation from the internal flow field. t+∆t are known. The symbol φbi;j is the predicted flux at step t + ∆t. The above numerical derivative can be computed from the known step-t conditions.
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