The 20th Conference of echanical Engineering Network of Thailand 18-20 October 2006, Nakhon Ratchasima, Thailand luid Thermal nteraction of High peed Compressible Viscous low Past Uncooled and Cooled tructures by Adaptive esh Wiroj Limtrakarn Department of echanical Engineering, aculty of Engineering, Thammasat University, Bangkok 12120, Thailand Tel: 0-2564-3001-9, E-mail: limwiroj@engr.tu.ac.th Abstract nteraction behaviors of high speed compressible viscous flow and thermal structural response of uncooled and cooled structures are presented. The compressible viscous laminar flow behavior based on the Navier tokes equations is predicted by using an adaptive cellcentered finite-element method. The energy equation and the quasi-static structural equations for aerodynamically heated structures are solved by applying the Galerkin finite-element method. The mass transport element is used to predict temperature distribution of cooled structure. The finite-element formulation and computational procedure are described. The high-speed flow and structural heat phenomena of uncooled and cooled structures are studied by ach 10 flow past a flat plate to demonstrate their interaction. 1 ntroduction luid Thermal analysis methods have an important role in the design of high speed flight vehicles, such as hypersonic airbreathing vehicles [1], for predicting vehicles aerothermostructural performance. ignificant coupling occurs between high-speed flow phenomena, aerodynamic heating rates on structural surfaces, structural temperature and their gradients, as well as structural deformations and stresses, creating multidisciplinary interaction phenomena. These coupled effects indicate that the analysis of high-speed flowstructure interaction is an important consideration to highspeed vehicle design. uch coupled effects have been studied by a number of researchers recently. Computational fluid and structural dynamics commercial programs were combined together for predicting the flow and structure behaviors [2, 3]. An adaptive finite element method was proposed [4] to predict complex behavior of fluid thermal structure interaction. n the present paper, combination of mass transport and integrated flowthermal-structural analysis approach for predicting each disciplinary behavior and their interaction is presented. or high-speed compressible flows, the cell-centered finite-element method [5, 6] is combined with an adaptive meshing technique to solve the Navier-tokes equations. The Galerkin finite-element method is applied to solve the structural energy and mass transport equations for temperature distribution. The paper starts by explaining the theoretical formulation of high-speed compressible flow, structural heat transfer, and mass transport. The high speed flow, structural heat transfer, and mass transport behaviors are then studied by applying the present method to ach 10 flow past a uncooled and cooled flat plate to demonstrate their interdisciplinary coupling. 2. Theoretical ormulation and olution Procedure 2.1 Governing Equations The equations for the high-speed compressible flow, the structural heat transfer, mass transport, and the structural analysis in two dimensions are described below. High-speed compressible viscous flow The equations for high-speed compressible viscous laminar flow are represented by the conservation of mass, momentum, and energy. These equations are written in the conservation form [7] as { U } t x + { } + { } E y 0 (1) where the subscript denotes the fluid analysis. The vector { U } contains the fluid conservation variables defined by { U } ρu ρv ρε T ρ (2) where ρ is the fluid density, u and v are the velocity components in the x and y directions, respectively, and ε is the total energy. The vectors { E } and { } consist of the flux components in the x and y directions, respectively [6]. tructural heat transfer The thermal response of the structure is described by the energy equation in the conservation form as t U T + ET + T x y G T (3) where the subscript T denotes the structural heat transfer analysis. The vector U T contains the thermal conservation variable defined by U T ρ TcTTT (4)
where c T is the specific heat of structure. The heat flux components E T and T are and E T TT kt and T x G T is the heat source. TT kt (5) y ass Transport The thermal reponse of the cooled structure is presnted by adding the mass transport equation in the form as Γ e ({ G } + { G }) nˆ dγ (10) where the flux vectors { G } and { V} and viscous flux vectors of { E + } and { V V } V G are the inviscid E +, respectively, and nˆ is the unit vector normal to the element boundary, Γ e. Equation (10) is evaluated by summing the normal fluxes from all sides, Γ e, of the element. The fluxes normal to the element sides are then approximated by the numerical inviscid and viscous. By applying an explicit time marching algorithm [7], Eq. (10) becomes fluxes, { } G and { G V} t U + E x + x 0 (6) Ae t { Un+ 1 U } δ ({ } { GV} ) n G + (11) where the subscript denotes the mass tranport analysis. The vector U contains the mass transport conservation variable defined by U ρ ct (7) where c is the specific heat of structure. The heat flux components E T and T are E ρ cu T and 2.2 inite-element ormulation T k (8) y The cell-centered finite-element method is applied to the Navier-tokes equations to derive the finiteelement equations. The Galerkin finite element approach is applied to the structural heat transfer equation, mass transport equation, and the equilibrium equations to derive the corresponding finite-element equations. The derivation procedures are briefly described below. inite-element flow equations The method of weighted residuals [8] is applied to Eq. (1) over the element domain, Ω, by using a unit interpolation function as t { } U dω x E dω { } y dω { } The Gauss divergence theorem is then applied to the flux integral terms of Eq. (9) to yield, x { } E dω y dω + { } (9) where U n 1 + and U n are the conservation variables at the time steps n+1 and n, respectively; A e is the element area; δ is the length of the element side. t is the allowable time step following the CL and viscous stability requirement [6]. G V is the average viscous flux and G is the average inviscid flux which is given by G [ GL GR + A* (UL U )] 1 R 2 + (12) where the superscripts L and R denote the left and right elements, respectively. The last term in Eq. (12) may be viewed as an artificial diffusion needed for the solution stability. This diffusion is represented by the product of the Jacobian matrix A * and the difference between the left and right element conservation variables [4]. U L and U R By substituting Eq. (12) into Eq. (11), then Eq. (11) becomes Ae t 1 2 { Un+ 1 U n } δ [{ L} + { R} + ({ L} { R G G A* U U })] { V} δ G (13) inite-element structural heat transfer equations The method of weighted residuals is applied to Eq. (3), over the element domain, Ω, by assuming a linear distribution of the conservation variable, U T, and the flux components E T and T. The finite-element equations can then be derived in the form:
[ ]{ } n + U 1 T { R T} n 1 + { T} n 2 R (14) where [ ] is the mass matrix, and { U T} n + 1 { U T} n+ 1 { U T} n at time n+1. The { R T} n 1 and { R T} n 2 vectors are associated with the thermal fluxes within each element and across the element boundary, respectively, and are given by, N x { } n R T 1 N dω{ En T } Γ e N y N dω{ T n } { R T} n 2 { }( En + n Tnx T n y ) + (15) N dγ (16) shock line from the sharp leading edge. The accuracy of the shock resolution and the shock angle strongly depends on the finite element mesh near the sharp leading edge. A total of 15,090 triangular elements are generated in the inviscid region and 1,230 quadrilateral elements inside the boundary layer. Ten graded layers of quadrilateral elements are used inside the boundary layer to capture steep temperature gradients for the accurate aerodynamic heating rate prediction. The predicted flow solution is shown by the density contours in igure 2(b). 0.3 m inite-element mass transport equations The galerkin method is applied to Eq. (6) in the same fashion as in the structural heat transfer analysis. The finite element equations can then be deived the form: [ ]{ } n + C U 1 { R } n 1 + { } n 2 R (17) where [ C ] is the mass matrix, and { } n + U 1 { } n + U 1 { U } n at time n+1. The { R } n 1 and { R } n 2 vectors are associated with the thermal fluxes within each element and across the element boundary, respectively, and are given by, { R } n 1 { R } n 2 { } N N n dω T x k x Ω (18) N n N u dω U x Ω (19) 10 Computational Domain hock Wave y x Convective coolant igure 1. ach 10 flow past a flat plate. 8,865 nodes 16,320 elements 4. Application A ach number 10 flow past a flat plate is presented to evaluate performance of the adaptive cell-centered finite-element method for high-speed flow analysis. The problem statement of a mach 10 flow past a flat plate as shown in igure 1 had the flow entering through the left boundary of the computational fluid domain. The shock wave is created from the leading edge as highlighted in the figure. The inlet flow conditions consist of specifying ρ 4.303E-05 kg/m 3, u 1,418.7 m/s, v 0, ε 1.043 J/kg, Re 5,580 with the wall temperature of 288.16 K. The combined method of the cell-centered finite-element analysis and the adaptive meshing technique is applied to solve the problem. igure 2(a) shows the final adaptive mesh that consists of small elements clustered along the (a) inal adaptive mesh (b) Density distribution (x10-5 kg/m 3 ) igure 2. inal adaptive mesh and corresponding density contours for mach 10 viscous flow past a flat plate.
Aerodynamics heating rate from fluid analysis is applied along fluid structure interface surface. The bottom surface of flat plate is convectively cooled by hydrogen stream with the inlet temperature of 27.78 K. The hydrogen flow rate is 0.0471736 kg/s and the coolant film coefficient is 90.823 kw/m 2. The finite element model consists of 2,360 triangular elements and 118 mass transport element representing the hydrogen coolant flow. aterials of flat plate is copper. Temperature, K 360 350 340 330 0. 0.1 0.2 0.3 x, m (a) Uncooled structure Temperature, K 30 28 through the thickness. Copper and beryllium are the candidate materials. igure 3(b) compares temperature distribution of both materials along interface of cooled structure at time t 0.1 sec. 29.73 K and 29.92 K are the maximum temperature at leading edge of copper and beryllium, respectively. Beryllium has a little higher temperature at leading edge and after x 0.02 m. beryllium has temperature 1.2 K less than copper. 5. Concluding Remarks The multidisciplinary interaction behaviors of highspeed compressible flow, structural heat transfer, mass transport, and structural response were presented using the adaptive finite-element method. The finite-element method based on the cell-centered algorithm was used to predict the high-speed compressible flow behavior. The method was combined with the adaptive meshing technique to improve the flow accuracy. The ach 10 flow past a flat plate was then used to study the flowstructure interaction and to evaluate the performance of the proposed analysis procedure. The example highlights the interaction behavior between the high-speed flow, mass transport, and the thermal-structural response of uncooled and cooled structure. The high-speed compressible viscous flow behavior is well predicted by adaptive finite element method and the fluid structure interaction behavior by the proposed finite-element coupling procedure. 26 Copper Beryllium 0. 0.1 0.2 0.3 x, m Acknowledgments This research is partially supported by the Thailand Research und (TR) for the enior cholar Professor Pramote Dechaumphai. (b) Cooled tructure igure 3. Temperature distribution along x distance at t 0.1 sec. igure 3 (a) shows the temperature distribution at time t 0.1 sec. along fluid structure interface based on uncooled structure with insulated boundary conditions along left, right and bottom surface. The maximum temperature is 358.74 K on the leading edge. lat plate materials with high thermal conductivity are used to reduce the leading edge temperature, temperature gradient References [1] Glass, D. E., erski, N. R., Glass, C. E., 2002. Airframe Research and Technology for Hypersonic Airbreathing Vehicles, AAA Paper 2002-5137. [2] Baum, J. D., 2002. Development of a Coupled CD/CD ethodology using an Embedded CD Approach, Conference on Computational Physics. [3] Lohner, R. Baum, J. D. estreau, E. harov, D. Charman, C. Pelessone, D., 2003. Adaptive Embedded Unstructured Grid ethods, AAA Paper 03-1116. [4] Limtrakarn, W., and Dechaumphai, P., 2003. Computations of High peed Compressible lows
with Adaptive Cell Centered inite Element ethod, J Chin nst Eng, Vol. 26, pp. 553-563. [5] Gnoffo, P. A., 1986. Application of Program LAURA to Three-dimensional AOTV lowfields, AAA Paper 86-0565. [6] Dechaumphai, P., and Limtrakarn, W., 1999. Adaptive Cell-Centered inite Element Technique for Compressible lows, Journal of Energy, Heat and ass Transfer, Vol. 21, pp. 57-65. [7] Hirsch, C., 1988. Numerical Computation of nternal and External lows. Vol. 1, New York: Wiley. [8] Zienkiewicz, O. C., and Taylor, R. L., 2000. The inite Element ethod. ifth Ed., Woburn: Butterworth-Heinemann.