A new treatment of cell-interface convective fluxes in AUSM-type methods is introduced to substantially reduce the numerical dissipation in a smooth region, including a contact discontinuity, without compromising accuracy in shock regions. The core idea of the new scheme is the modification of the transferred property at a cell-interface under the consideration of physical and multi-dimensional phenomena. The transferred property of the M-AUSMPW+ is revealed in two aspects. One is that the newly defined cellinterface value is closer to the real physical value. The other is that it can eliminate numerical dissipation effectively in a non-flow aligned grid system.

An optimal shape design approach is presented for a subsonic S-shaped intake using aerodynamic sensitivity analysis. Two-equation turbulence model is employed to capture strong counter vortices in the S-shaped duct more precisely. Sensitivity analysis is performed for the three-dimensional Navier-Stokes equations coupled with two-equation turbulence models using a discrete adjoint method For code validation, the result of the flow solver is compared with experiment data and other computational results of bench marking test. To study the influence oj turbulence models and grid refinement on the duct flow analysis, the results from several turbulence models are compared with one another and the minimum number of grid points, which can yield an accurate solution is investigated The adjoint variable code is validated by comparing the complex step derivative results. To realize a sufficient and flexible design space, NURBS equations are introduced as a geometric representation and a new grid modification technique, Least Square NURBS Grid Approximation is applied With the verified flow solver, the sensitivity analysis code and the geometric modification technique, the optimization of S-shaped intake is carried out and the enhancement of overall intake performance is achieved The designed S-shaped duct is tested in several off-design conditions to confirm the robustness of the current design approach. As a result, the capability and the efficiency of the present design tools are successfully demonstrated in three-dimensional highly turbulent internal flow design and off-design conditions.

This paper describes an optimal flow control method for compressible laminar/turbulent flows utilizing discrete unsteady aerodynamic sensitivity analysis methods. Unsteady aerodynamic sensitivity codes are developed using a direct differentiation method and an adjoint method, respectively, from two-dimensional unsteady compressible Navier-Stokes equations. Optimal flow controls are conducted by minimizing an unsteady objective function defined at an instant instead of integrating a response for a period of time. Unsteady sensitivity derivatives of the objective function are calculated by the unsteady sensitivity codes, and optimization is conducted utilizing a linear line search method at every physical time step. Several flow control examples of academic interest including circular cylinder vortex shedding control and airfoil shock buffet control show satisfactory results. The present active flow control method utilizing the unsteady sensitivity analysis is robust and problem independent compared to conventional active control methods.

아크히터의 작동특성을 예측하고 아크히터 자체의 설계와 해석을 위해, 세그먼트 아크히터(segmented arc-heater)의 내부 유동을 계산하였다. 지금까지 몇몇 연구자들이 아크히터 내부 유동을 수학적으로 모델링하여 일부 아크히터의 내부 유동을 정확하게 계산하는데 성공하였지만, 다양한 아크히터의 여러 운용 조건을 모두 만족하는 수학적 모델을 완성하지 못했다. 본 연구에서는 수학적 모델링의 범용성 확보를 위해, 아크 히터 내부의 난류 유동에 중점을 두었다. 기존의 대수 난류 모델을 대신하여 세 개의 2방정식 난류를 사용하였으며, 계산 결과 $k-\varepsilon$ 난류 모델이 아크히터 내부의 유동을 모사하는데 적합하다는 것을 확인했으며, 난류가 아크히터 유동을 특성을 좌우하는 중요한 요소 중에 하나라는 것을 밝혔다. Flows in segmented arc-heaters have been calculated for prediction of experimental operating condition or for analysis and design of arc-heater itself. Some researchers succeeded in calculating accurately inner flows of a arc-heater, but could not made mathematical models which satisfy various operating conditions for many arc-heaters. this study is forced on turbulence for the generality of mathematical model. Instead of algebraic turbulence models which are frequently used for calculating inner flow of arc-heater, two equation turbulent models are used. Prediction results agree well with experiment data and it was confirmed that $k-\varepsilon$ two equation turbulence model is appropriate for a flow in an arc heater throughout extensive numerical testing.

Eu's GH(Genera1ized Hydrodynamic) equations are presented for analyzing a hypersonic flow over a double-cone geometry which shows various aerodynamic phenomena such as shock-shock interaction, shock-boundary layer interaction, etc. In order to analyze rarefied hypersonic flow, axisymrnetric generalized hydrodynamic equations are developed and validated by Rothe nozzle flow problem. Two kinds of solid surface boundary conditions nonslip and Langmuir's slip BC are examined, too. The hypersonic rarefied flow results acquired by GH equations are compared with experimental data and Navier-Stokes equations calculations with slip and nonslip boundary conditions. The calculations by GH equations show the more accurate flow predictions than those of NavierStokes equations, and GH equastion with some assumptions was able to be found that it is a useful tool to analyze rarefied hypersonic flows.

The Grid is a communication service that collaborates dispersed high performance computers so that those can be shared and worked together. It enables the analysis of huge-sized problem with the reduction of computation time by collaborating high-performanced computing resources in dispersed organizations. Thus, the present paper focuses on the efficient flow calculation in the Grid. As a large-scale computation, the separation motion of boosters attached on the multi-stage launch vehicle is analyzed by using 3-D compressible unsteady Navier-Stokes flow solver and rigid body dynamics. Simultaneously, a simple load balance algorithm for heterogeneous computing environment is proposed and applied to a 3-D problem that uses multiblock mesh.

The authors acknowledge the financial support provided by the Korea Science and Engineering Foundation (Grant R01-2002-000- 00329-0) and by the Brain Korea 21 Project.

A parallelized design optimization approach is presented for a subsonic S-shaped inlet using aerodynamic sensitivity analysis. Two-equation turbulence model is adopted to predict the strong counter vortices in the S-shaped duct more precisely. Sensitivity analysis is performed for the three-dimensional NavierStokes equations coupled with two-equation turbulence models using a discrete adjoint method. For code validation, the result of the flow solver is compared with experiment data and bench marking data of other computation researches. To study the influence of turbulence models and grid refinement in the duct flow analysis, the results using several turbulence models are compared with each other on various grid systems. The adjoint variable code is validated by comparison with the finite difference results. The capability and the efficiency of the present design tools are successfully demonstrated in three-dimensional subsonic inlet flow analysis and design optimization.

A feasibility study is carried out by investigating the effects of a usual assumption of constant turbulent eddy viscosity on the aerodynamic design using an adjoint variable method, one of the most efficient gradient-based optimization techniques. Accurate unsteady and steady flow analyses are followed by the aerodynamic sensitivity analysis for the Navier-Stokes equations coupled with two-equation turbulence models. A challengeable approach for high-lift design optimization at higher angles of attack is also proposed, which is based on unsteady sensitivity analysis using a dual time-stepping method and the chimera overset grid scheme. Through the comparison of the sensitivity gradients with respect to all of the design variables including angle of attack, it is observed that the constant turbulent eddy-viscosity assumption might provide inaccurate gradients in sensitivity analyses such as transonic airfoil with a strong shock and high-lift airfoil at a high angle of attack close to stall angle. Simultaneously, however, the final design results indicate that both approaches are acceptable in engineering applications. Both the single- and multi-element airfoil design optimizations using the constant eddy-viscosity assumption are carefully assessed in terms of design accuracy, computer memory overheads, and total design time in various design examples.

Numerical instability is studied based on numerics and mathematics. It is frequently observed in the region where velocity is zero. In that region, Euler equations have numerous solutions and, thus, it is impossible to determine an unique solution with only governing equations. The unique solution can be determined by additional outer flow conditions or outer numerical disturbances. This peculiarity of Euler equations can cause numerical instability in a shock discontinuity since information on the unique solution under undisturbed conditions is lost by disturbances. As a result, shock instability cannot be removed completely in the scheme which is consistent with Euler equations.

The new Multi-dimensional Higher order Interpolation Scheme named MHIS is developed. The major advantage of MHIS is the complete multi- dimensional monotonic characteristics. In addition, MHIS is more than the 3 rd order space accuracy, it has the same level of efficiency as the previous TVD MUSCL approach and it shows the similar convergence characteristics. MHIS is so simple but very accurate scheme compared with other higher order interpolation schemes such as ENO, WENO and so on. In this paper, the 3rd order and the 5th order MHIS are developed and tested for several real applications.

A feasibility study is carried out by investigating the effects of a usual assumption of constant turbulent eddy viscosity on the aerodynamic design using an adjoint variable method, one of most efficient gradient-based optimization techniques. Accurate steady and unsteady flow analyses are followed by the aerodynamic sensitivity analysis for the Navier-Stokes equations coupled with two-equation turbulence models. The capability of the present sensitivity analysis code to treat complex geometry using the chimera overlaid grid is also demonstrated by analyzing the flow over multielement airfoil. Like the mean flow equations, the turbulence model equations are also hand-differentiated to accurately calculate the sensitivity derivatives of flow quantities in turbulent viscous flows. With two-equation turbulence it is observed that the constant turbulent eddy viscosity assumption in the adjoint variable method could lead to inaccurate results in several test cases of transonic airfoil with strong shock and multielement airfoil at high angle of attack. Especially for the flow over high-lift airfoil close to stall, both flow analysis and sensitivity analysis are performed in an unsteady, time-accurate manner using the dual time stepping method. In addition, the effects of constant eddy viscosity assumption on single and multielement airfoil design optimization are carefully investigated in various design examples. Introduction As computational power advances, design optimization tools using computational fluid dynamics (CFD) have played a more important role in aerodynamic design. Among advanced optimization techniques, gradient-based optimization method has been widely used and even applied to multidisciplinary design optimization (MDO). In general, a gradient-based design optimization requires two steps. The first is to obtain the search direction that defines how the design variables will be changed for design improvement. The second is, so called one-dimensional search, to determine how far the design variables will move in this direction. This basic process is repeated until it approaches to the optimum. In one-dimensional search, an accurate and efficient flow solver is indispensable for the computation of pressure distribution and aerodynamic load coefficients such as lift, drag and pitching moment which are used in an objective function to be either minimized or maximized. Especially for the high-lift design optimization, it was * Postdoctorial Researcher, Institute of Advanced Machinery and Design, AIAA Member. T Assistant Professor, Dep't of Aerospace Engineering, AIAA Member. 1 Professor, Dep't of Aerospace Engineering, AIAA Senior Member. Copyright © 2002 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. previously noted that the unsteady, time-accurate computation is definitely required for the accurate computation of the flow over a high-lift device at a higher angle of attack close to stall angle where massive flow separation might occur. In determining the search direction, the gradients of design variables were traditionally calculated by the finitedifference method. It is, however, too expensive to compute the flow field iteratively with the incremented values of a design variable for complex two-dimensional or three-dimensional problems. In addition, this method is so sensitive to the step size of a design variable that it sometimes provides inaccurate signs or sensitivity derivatives. 4 Therefore, more robust techniques have been proposed using direct differentiation methods and adjoint variable methods. Compared with the direct differentiation methods, the adjoint variable methods are more advantageous for their capability to compute the gradients of the objective function and constraints when the number of design variables is much larger than that of the objective function and constraints. The adjoint variable methods adopt the formulation of the gradient in either a discrete or a continuous approach. In the discrete approach, which is used in the present work, the discretized governing equations are differentiated with respect to design variables whereas the adjoint equations are first differentiated and American Institute of Aeronautics and Astronautics Paper 2002-0262 (c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. then discretized in the continuous approach.' It is necessary to incorporate the effect of turbulence in differentiating the governing equations to treat high Reynolds number flows more accurately. It is, however, very difficult to fully hand-differentiate the governing equations including the viscous terms and turbulence terms. Thus, some software tools such as automatic differentiation' ' n are used for the Navier-Stokes equations with a turbulence model. However, this approach is generally less efficient, in terms of computing time and computer memory, than hand-differentiation codes.' In the present work, the Navier-Stokes equations coupled with two-equation turbulence models are fully differentiated by human hand. Among most popular two-equation turbulence models, the k-co SST model' 14 is mainly used and then compared with the original k — co model' 16 and the standard k-s model. Like the mean flow equations, the turbulence model equations are also hand-differentiated to accurately compute the sensitivity derivatives of flow quantities with respect to design variables in turbulent flows. A usual assumption of constant turbulent eddy viscosity (CTEV, hereafter) in the adjoint variable methods has been previously used in aerodynamic design optimizations '. Recently, the accuracy of this assumption in sensitivity analysis for turbulent flows was reported using two-equation turbulence models on chimera overlaid grids and a one-equation turbulence model on unstructured grids in Refs. 2 and 4, respectively. In the present study, the feasibility of constant eddy viscosity assumption is carefully studied by comparing the sensitivity gradients using the CSAAV code, and the effects of this assumption on single and multielement airfoil design optimization are further investigated in various design examples: subsonic and transonic designs for drag minimization and lift maximization, and high-lift designs for lift to drag ratio maximization and even C/max maximization. Numerical Background Flow Analysis The Compressible Flow Analysis Navier-Stokes (CFANS2D/3D) solver, which has been well verified in many applications'', is used for the computation of turbulent viscous flows over single and multielement airfoils. The governing equations are the twodimensional, unsteady, compressible Navier-Stokes equations coupled with two-equation turbulence models: the k-co SST model', the k-co model', and the standard k-s model. The governing equations are transformed in generalized coordinates and are solved with a finite-volume method. Using a backward Euler implicit method, the governing equations adopting the dual time stepping method are discretized in time and linearized in delta form as:

The separation motions of boosters attached on the various three-stage rockets are analysed by using 3-D compressible unsteady Navier-Stokes flow solver and rigid body dynamics. For the governing equations, six degree-of-freedom rigid body equations of motion are integrated into Navier-Stokes solution procedure to determine the aerodynamic-dynamic coupled motions. A Chimera overset grid technique is adopted for the calculation of the present configuration and grid around the core rocket is composed of 3 zones to represent fins in the core rocket. Flow solver is parallelized to reduce the calculation time, and an efficient parallelization algorithm for Chimera grid technique is proposed. AUSMPW+ scheme is used for the spatial discretization and LU-SGS for the time integration. Developed flow solver is validated by comparing the computed results with wind tunnel data around the Titan IV launch vehicle, and applied to the various three-stage rockets. From the analyses, booster trajectories are predicted and aerodynamic characteristics around the vehicle at each time interval are examined at various separation condition. In addition, additional jettisoning forces and moments needed for a safe separation are examined.

This paper deals with the development of an improved Roe scheme that is free from the shock instability and still preserves the accuracy and efficiency of the original Roe's Flux Difference Splitting (EDS). Roe's FDS is known to possess good accuracy but to suffer from the shock instability, such as the carbuncle phenomenon. As the first step towards a shockstable scheme, Roe's FDS is compared with the HLLE scheme to identify the source of the shock instability. Through a linear perturbation analysis on the odd-even decoupling problem, damping characteristic is examined and control functions / and g are introduced into Roe's FDS to cure the shock instability. In order to satisfy the conservation of total enthalpy, which is crucial in predicting surface heat transfer rate in high speed steady flows, an analysis of dissipation mechanism in the energy equation is carried out to find out the error source and to make the proposed scheme preserve total enthalpy. By modifying the maximum-minimum wave speed, the problem of expansion shock and numerical instability in the expansion region is also remedied without sacrificing the exact capturing of contact discontinuity. Various numerical tests concerned with the shock instability are performed to validate the robustness of the proposed scheme. Then, viscous flow test cases ranging from transonic to hypersonic regime are calculated to demonstrate the accuracy, robustness, and other essential features of the proposed scheme. ITRODUCTION It is essential that a numerical representation of inviscid fluxes, namely a numerical flux function, should guarantee the high level of accuracy, efficiency and robustness in computational fluid dynamics (CFD). In the last three decades, numerous numerical flux functions have been developed and much progress has been achieved [2,5,11,14,19]. The Flux Difference Splitting (FDS) framework is one of the most successful groups among the various approaches to design numerical schemes and is Graduate Student f Assistant Professor, Member AIAA, * Professor, Senior Member AIAA 55 Senior Researcher, Member AIAA Copyright © 2002 by the American Institute of Aeronautics and Astronautics Inc. All rights reserved widely used and studied. FDS schemes are generally based on the idea due to Godunov [1] and the Riemann problem is utilized locally. Godunov showed that after preparing piecewise constant initial data from cell-averaged flow values, numerical flux at a cell interface can be calculated through the exact solution of the Riemann problem. Although this strategy provides a way to obtain a good shock-capturing scheme, the Riemann problem is highly nonlinear and has no closed form solution. In order to overcome this deficiency, many have tried to simplify the step of numerical flux calculation, which leads to the family of Godunov-type schemes or approximate Riemann solvers, such as Roe's FDS [2], Osher's FDS [3] and HLLEM [5], and etc. These FDS schemes can capture contact discontinuity accurately and give good resolution for the boundary layer in viscous flow calculation. Despite these advantages and the good shock capturing property, some disastrous failings are also found in certain problems. This pathological behavior, usually represented as the 'carbuncle phenomenon', was first observed by Peery and Imlay [6] for blunt body computations with Roe's FDS. The carbuncle phenomenon refers to a protuberant shock profile obtained when a supersonic flow over a blunt body is calculated. Quirk [7] reported that approximate Riemann solvers generally suffer from such failings. After the carbuncle phenomenon was observed, many attempts were made to unveil the cause and to cure these failings. The attempts to cure the shock instability can be generally categorized into two groups. One is to use an alternative dissipative scheme in a hybrid manner and the other to employ an entropy fix. Quirk [7] noticed that some schemes possessing the property of the good capturing of contact discontinuity show carbuncle phenomena while others free from carbuncle phenomena do not capture contact discontinuity accurately. Thus, it is suggested that a dissipative scheme, such as HLLE, should be used in shock region while a less dissipative scheme, such as Roe's FDS, should be used elsewhere. In order to flag the cell interface where a dissipative scheme is needed, a pressure gradient sensor is used. Wada and Liou [8], by the same philosophy, suggest a similar flagging procedure but they use a sonic point. For a less dissipative scheme, AUSMDV is used and Hanel's FVS is used for a dissipative scheme. This cure turns out to be very efficient, as shown by the American Institute of Aeronautics and Astronautics (c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. results reported in [7,8]. However, this approach always needs a proper counterpart which complements defects of the original flux function, and the selection of a proper numerical scheme itself is a critical problem. An inadequate counterpart may contaminate the accuracy of numerical solutions, especially in cases of high speed flows. An entropy fix to the linear wave field is a method to limit the minimum value of the wave speed, which is equivalent to the addition of extra numerical dissipation to damp out spurious oscillation. Peery and Imlay [6] propose an anisotropic function for an entropy fix, and Lin [9] designs an isotropic correction function using a pressure gradient sensor. Although this approach may successfully cure the carbuncle phenomenon, its performance always depends on the location and/or the amount of numerical dissipation added. Improper entropy fix may easily broaden shock wave profile and/or deteriorate boundary layer resolution. Thus, the development of a proper sensor which determines the location and the amount of numerical dissipation is crucial. The two approaches to cure the shock instability problem, i.e. the use of dissipative numerical schemes and the employment of an entropy fix, are fundamentally the same in the sense that extra numerical dissipation is added to the original scheme in a way or another, and both need a detection procedure which usually involves a tuning coefficient. So far, it is generally believed that a scheme that can capture contact discontinuity exactly, i.e. a scheme that has vanishing dissipation in stationary contact discontinuity, cannot avoid the shock instability, and the only way to prevent it is to add enough dissipation to damp out oscillation. However, Liou [10] observes that all the tested numerical functions that suffer from the shock instability have a term depending on pressure difference in the mass flux while those free from the shock instability are independent of pressure difference in the mass flux. Based on the numerical analysis and experiment, he suggests the following conjecture: 'The condition D ^ 0, VM , in the mass flux is necessary for a scheme to develop, as t increases, the shock instability as manifested by the odd-even decoupling and carbuncle phenomena. On the other hand, the condition D =0,VM, is sufficient for a scheme to prevent the shock instability from occurring. Here, D stands for the dissipation term depending on pressure difference. This result indicates that it is possible to devise a FDS flux function free from the shock instability with vanishing dissipation in capturing stationary contact discontinuity. Xu [11] explains the shock instability using the Bernoulli equation and a convergent-divergent nozzle concept. According to his explanation, vanishing dissipation in the direction parallel to the shock and the contribution of pressure fluctuation to the mass flux cause the shock instability. The analysis in Ref. [11] is in a qualitative agreement with Liou's conjecture in the sense that the pressure term in the mass flux triggers the shock instability. The present study aims at the design of a new Roebased FDS flux function that is free from shock instability. Following Liou's conjecture and Xu's explanation, we focus on the pressure term in the mass flux of Roe's FDS. In order to maintain the high level of robustness and accuracy, we impose that the newly developed flux function should satisfy the following criteria: • The new flux function should not have any tunable parameter. • The new flux function should capture contact discontinuity for the accurate resolution of boundary layer. • Total enthalpy should be conserved for the accurate prediction of surface heat transfer rate in high speed steady flows. • Robustness in the expansion region should be substantially improved and entropy-violating expansion shock should be removed. The present paper is organized as follows. After introduction, a list of the shock instability is briefly reviewed with some examples. Then, we present the analysis procedure of Roe's flux function and propose Roe with Mach number-based function (RoeM) schemes. And, we present extensive numerical results and discuss properties of the proposed schemes. In order to demonstrate various properties of the flux functions, we apply the proposed schemes to steady and unsteady problems. Finally, concluding remarks are given.