A MULTI-DOMAIN ALE ALGORITHM FOR SIMULATING FLOWS INSIDE FREE-PISTON DRIVEN HYPERSONIC TEST FACILITIES

Transcription

1 A MULTI-DOMAIN ALE ALGORITHM FOR SIMULATING FLOWS INSIDE FREE-PISTON DRIVEN HYPERSONIC TEST FACILITIES Khalil Bensassi, and Herman Deconinck Von Karman Institute for Fluid Dynamics Aeronautics & Aerospace department for Space Vehicles

2 Outline Introduction 1 Introduction

3 Introduction Challenges The numerical simulation of internal flows inside ground hypersonic experimental facilities (e.g. shock and gun tunnels) is still a challenging task even with state-of-the-art computational tools It is mainly due to the complex unsteady multi-physics phenomena occurring during the extremely short testing times (of- ten in the order of milli-/micro-seconds). Objective The main objective behind this work is to develop and use numerical tools for simulating the flow in the driver-driven part of a free piston-type hypersonic facility in order to : provide a better understanding of the physical phenomena inside the facility assist the design of new components of the facility be a key component in fully specifying the facility s test flow conditions Computational codes L1d2 was written by P. A. Jacobs (University of Queensland, Australia) for simulating transient flow facilities such as gun shot wind tunnels. COOLFluiD is an open source platform designed for handling high-performance computing problems including re-entry aerothermodynamics, astrophysical and laboratory plasmas, multi-physics simulations. More at https ://github.com/andrealani/coolfluid/wiki.

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5 Experimental Data 4E E+6 3E+8 High Reynolds Medium Reynolds Low Reynolds 2 High Reynolds Medium Reynolds Low Reynolds 3E+6 High Reynolds Medium Reynolds Low Reynolds Po [Pa] 2E+8 Pt2 [Pa] 15 1 Qw [w/m2] 2E+6 1E+8 5 1E Time [ms] Time [ms] Time [ms] Measured Data The total pressure in the reservoir. The stagnation pressure in the chamber. The stagnation heat flux in the chamber. High Reynolds Medium Reynolds Low Reynolds E E E+6

6 What are we going to simulate?.5 Y[m] X[m] Schematic of VKI-Longhot driver-driven tube We are interested in simulating the flow in the driver-driven part of the facility, mainly the first 5 ms of the test. The dynamics of the piston inside the driven tube is computed by means of an Arbitrary Lagrangian Eulerian solver.

7 Numerical method Arbitrary Eulerian-Lagrangian formulation Eulerian-based finite volume formulation is used when the computational domain is constant in space/time. Lagrangian-based finite volume formulation is obtained by equating mesh motion to fluid motion. The formulation used for intermediate cases, in which the computational domain evolves and deforms independently, is called an Arbitrary Eulerian-Lagrangian formulation (ALE). Ω(t) ( Q Fx t dω + Ω(t) x + Fy ) dω = y where Q = (ρ, ρu, ρv, ρe) T is the vector of conserved variables, F are the convective fluxes. d QdΩ + dt ((u ug) n) Q + pn x Ω i (t) Γ i (t) pn y dγ i = (1) p(u n) The rate of change of the conservative variables Q in Ω(t) is due to convection through Γ i (t) with relative speed u r = u u g ; pressure surface terms.

8 Numerical method The convective flux are computed e computed using an ALE formulation of AUSM+ Crank-Nicholson is used for time integration. The discretized system of nonlinear equations can be written as : ( F U n+1) = (2) where the function F i (U n+1) reads : ( F U n+1) = T (Ω n+1 U n+1 Ω n U n) 1 t R(U n+1 ) + R(U n ) 2 whose unknowns are U n+1. The system can be solved using an iterative Newton procedure which is based on successive iterations, starting from an initial guess U and requiring the linearization of F : F(U) F(U n ) + J (U n ) ( U U n). (3) Herein, J is the Jacobian matrix of the nonlinear system. The resulting linear system and the update of the approximate solution can be expressed as : { J (U n ) U n = F(U n ) U n+1 = U n + U (4) At each time step, the system given by Eq. (4) is solved implicitly until convergence through a Newton iteration.

9 ALE Simulation : Multi-domain strategy Arbitrary Lagrangian-Eulerian simulations of the diver- driven part of the facility was performed using COOLFLuiD. A multi-domain strategy was developed : the numerical domain representing the driver-driven tubes is divided into two subdomains that are generated during the meshing process as a single block with moving internal boundaries. the driver and the driven parts of the facility were considered as a one tube with the same diameter. Coupling with the fluid The piston dynamics is coupled to the fluid by using moving boundaries for the two subdomains : a p = Ap m p 1 N fl N fl i=1 P i,l 1 N fr N fr j=1 P j,r, V n+1 p = V n p + ap t, δn+1 = V n+1 p t

10 ALE Simulation : Mesh adaptation During the displacement of the piston inside the driven tube, the subdomain I is stretched and subdomain II is compressed. Thus, mesh deformation technique has to be used in order to adapt the grid at each time step. mesh adaptation (I) X n+1 i = X n i + δn+1 Xn i X driver end X n+1 pb (II) X n+1 j = X n j + Xn δn+1 j X driven end X n+1 + L pb piston Computational grid and piston position at time of [ms] Computational grid and piston position at time of 5 [ms]

11 ALE Simulation : Flow field t = 25[ms] t = 25[ms] Log(P) [Pa]: M : t = 5[ms] t = 5[ms] Log(P) [Pa]: M : t = 55[ms] t = 55[ms] Log(P) [Pa]: M : t = 6[ms] t = 6[ms] Log(P) [Pa]: M : t = 64[ms] t = 64[ms] Log(P) [Pa]: M : Pressure flow field in logarithmic scale Mach number flow field

12 Testcase definition Numerical testcase Y[m] X[m] The test gas is Nitrogen T = 3 [K] End of the driven tube is a wall boundary condition p driver [Pa] p driven [Pa] Mass [kg] Length [m] The principal features of the one-dimensional flow code, L1d2, are Quasi-one-dimensional formulation for the gas-dynamics. There is only one spatial coordinate but gradual variation of duct area is allowed. The ability to simulate several independent (or interacting) slugs of gas. Also, several pistons and multiple diaphragms may be included. Coupling to the gas dynamics is via the boundary conditions of the gas slugs A Lagrangian discretization of the gas slugs. This is done by dividing each gas slug into a set of control-masses (or gas particles) and following the positions of these particles.

13 COOLFluiD-ALE vs L1d2 : pressures on the piston faces 8E Front face pressure [Pa] 6E+7 4E+7 2E+7 COOLFluiD_ALE L1d2 Front face pressure [Pa] COOLFluiD_ALE L1d Time [s] Piston back face pressure along time Time [s] Piston front face pressure a long time

14 COOLFluiD-ALE vs L1d2 : speed and position of the piston COOLFluiD_ALE L1d2 6 4 COOLFluiD_ALE L1d2 Position [m] Piston speed [m/s] Time [s] Position of the piston along time Time [s] speed of the piston a long time

15 2E+6 6E+6 1E+7 1.4E+7 1.8E+7 2.2E+7 2.6E+7 3E+7 COOLFluiD-ALE : Two dimensional effect of the chambrage During the previous computations, the geometry of the driver-driven tubes was simplified and the same diameter was used for both tubes. In the current section, we consider the real geometry of this part of the VKI-Longshot. The re-meshing procedure is applied on the driven tube, and on a small part of the tube located between the piston back face and the junction area between the two tubes. P [Pa] Log(P) [Pa]: ALE computation without chambrage ALE computation with chambrage

16 ALE Simulation : Two dimensional effect of the chambrage t = 25[ms] M: M: t = 5[ms] Piston speed [m/s] COOLFluiD_ALE L1d2 COOLFluiD_ALE D/d=1 L1d2 D/d=1 t = 55[ms] Time [s] M: COOLFluiD_ALE L1d2 COOLFluiD_ALE D/d=1 L1d2 D/d=1 M: t = 6[ms] Piston position [m] Mach number flow field Time [s]

17 Coupling with the nozzle In order to simulate the pressure rise in the reservoir during the transient part of the VKI-Longshot facility, the convergent part of the contoured nozzle was added at the end of the driven tube. 5E+8 I II III 4E+8 End of the driven tube and the convergent part of the nozzle The geometry of the contoured nozzle ends just after the throat where the flow is supersonic, thus a supersonic outlet boundary condition is applied. In order to keep a good mesh resolution inside the nozzle, no remeshing is applied on this part of the domain. P [Pa] 3E+8 2E+8 1E Time [ms]

18 COOLFluiD-ALE vs Experiment t = 68.6[ms] P [Pa] 3.E+6 2.9E+7 5.4E+7 8.E+7 Experiment COOLFluiD_ALE P [Pa] t = 68.7[ms] 3.E+6 2.9E+7 5.4E+7 8.E+7 4E+8 P [Pa] 2E+8 P [Pa] t = 68.8[ms] 3.E+6 2.9E+7 5.4E+7 8.E Time [s] t = 68.9[ms] P [Pa] 3.E+6 2.9E+7 5.4E+7 8.E+7 4E+8 Experiment COOLFluiD_ALE P [Pa] t = 69[ms] 2E+8 P [Pa] 3.E+6 2.9E+7 5.4E+7 8.E Time [s]

19 Conclusions A multi-domain Arbitrary Lagrangian Eulerian approach was developed for an efficient and accurate prediction of the dynamics of the piston and the related flow patterns in the facility. The results obtained with the new methodology were verified against those produced by a well-established 1D Lagrangian code (L1d2) showing an excellent agreement. The general form of the proposed method allows for accounting for multi-dimensional effects such as the chambrage and pressure rise in the reservoir. The numerical results including those effects were validated against the measured reservoir pressure and a good matching was obtained. Future work Simulate the full VKI-Longshot by coupling the driver-driven tubes and the complete contoured nozzle. Include viscous effects and thermal nonequilibrium for modelling the starting process in the nozzle. Compare simulations with the available experimental data at the exit of the countered nozzle. Acknowledgements Sebastien Paris (the VKI Longshot guru who kindly provided all experimental data) Financial support from FWO G N grant (from Flemish Scientific Foundation)