Chongam Kim

Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows Part II: Multi-dimensional limiting process

Through the analysis of conventional TVD limiters, a new multi-dimensional limiting function is derived for an oscillation control in multi-dimensional flows. And, multi-dimensional limiting process (MLP) is developed with the multi-dimensional limiting function. The major advantage of MLP is to prevent oscillations across a multi-dimensional discontinuity, and it is readily compatible with more than third order spatial interpolation. Moreover, compared with other higher order interpolation schemes such as ENO type schemes, MLP shows a good convergence characteristic in a steady problem and it is very simple to be implemented. In the present paper, third and fifth order interpolation schemes with MLP, named MLP3 and MLP5, are developed and tested for several real applications. Through extensive numerous test cases including an oblique stationary contact discontinuity, an expansion fan, a vortex flow, a shock wave/vortex interaction and a viscous shock tube problem, it is verified that MLP combined with M-AUSMPW+ numerical flux substantially improves accuracy, efficiency and robustness both in continuous and discontinuous flows. By extending the current approach to three-dimensional flows, MLP is expected to reduce computational cost and enhance accuracy even further.

Cures for the shock instability: development of a shock-stable Roe scheme

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 Roes Flux Difference Splitting (FDS). Roes 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 shock-stable scheme, Roes 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 Mach number-based functions f and g are introduced to balance damping and feeding rates, which leads to a shock-stable Roe scheme. 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, I 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.

Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows

Abstract The present papers deal with numerical methods toward the accurate and efficient computations of multi-dimensional steady/unsteady compressible flows. In Part I, a new spatial discretization technique is introduced to reduce excessive numerical dissipation in a non-flow-aligned grid system. Through the analysis of TVD limiters, a criterion is proposed to predict cell-interface states accurately both in smooth region and in discontinuous region. According to the criterion, a new way of re-evaluating the cell-interface convective flux in AUSM-type methods is developed. The resultant flux reduces numerical dissipation remarkably in multi-dimensional flows. Also, the monotonicity of AUSM-type methods is achieved by modifying the pressure splitting function directly based on the governing equations and the detection of sonic transition point with respect to a cell-interface. It is noted that the newly formulated AUSM-type flux for Multi-dimensional flows, named M-AUSMPW+, possesses many improved properties in term of accuracy, computational efficiency, monotonicity and grid independency. Through numerous test cases from contact and shock discontinuities, vortex flow, shock wave/boundary-layer interaction to viscous shock tube problems, M-AUSMPW+ proves to be efficient and about twice more accurate than conventional upwind schemes. The three-dimensional implementation of M-AUSMPW+ is expected to provide accuracy and efficiency improvement furthermore.

Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids

In this paper, we derive the three-dimensional limiting condition and present three-dimensional limiting process for hyperbolic conservation laws. The basic idea of multidimensional limiting condition is that the multi-dimensionally interpolated values at a vertex point should be within the maximum and minimum cell-average values of neighboring cells for the monotonic distribution. By applying the MLP (Multi-dimensional Limiting Process) to the three dimensional Euler and Navier-Stokes equations, we can achieve monotonic characteristics, which results in the enhancement of solution accuracy, convergence behavior.

Multi-dimensional limiting process for three-dimensional flow physics analyses

The present paper deals with an efficient and accurate limiting strategy for multi-dimensional compressible flows. The multi-dimensional limiting process (MLP) which was successfully proposed in two-dimensional case [K.H. Kim, C. Kim, Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows. Part II: Multi-dimensional limiting process, J. Comput. Phys. 208 (2) (2005) 570-615] is modified and refined for three-dimensional application. For computational efficiency and easy implementation, the formulation of MLP is newly derived and extended to three-dimensional case without assuming local gradient. Through various test cases and comparisons, it is observed that the newly developed MLP is quite effective in controlling numerical oscillation in multi-dimensional flows including both continuous and discontinuous regions. In addition, compared to conventional TVD approach, MLP combined with improved flux functions does provide remarkable increase in accuracy, convergence and robustness in steady and unsteady three-dimensional compressible flows.

Sensitivity Analysis for the Navier-Stokes Equations with Two-Equation Turbulence Models

Aerodynamic sensitivity analysis is performed for the Navier ‐Stokes equations, coupled with two-equation turbulence models using a discrete adjoint method and a direct differentiation method, respectively. Like the mean e ow equations, the turbulence model equations are also hand differentiated to calculate accurately the sensitivity derivatives of e ow quantities with respect to design variables in turbulent viscous e ows. Both the direct differentiation code and the adjoint variable code adopt the same time integration scheme with the e ow solver to solve the differentiated equations efe ciently. The sensitivity codes are then compared with the e ow solver in terms of solution accuracy, computing time, and computer memory requirements. The sensitivity derivatives obtained from the sensitivity codes with different turbulence models are compared with each other. Using two-equation turbulence models, it is observed that a usual assumption of constant turbulent eddy viscosity in adjoint methods may lead to inaccurate results in a case of turbulent e ows involving strong shocks. The capability of the present sensitivity codes to treat complex geometry is successfully demonstrated by analyzing the e ows over multielement airfoils on chimera overlaid grid systems.

Numerical Study on the Unsteady-Force-Generation Mechanism of Insect Flapping Motion

Detailed numerical simulations are conducted to investigate aerodynamic characteristics of unsteady-force generation by a two-dimensional flapping motion under a forward-flight condition. A realistic wing trajectory called the figure-eight motion is extracted from a blowflys tethered flight under freestream. Computed results show complex vortical flowfields that exhibit very interesting and distinctive unsteady characteristics. Lift is mainly generated during downstroke motion by a high effective angle of attack due to translational and lagging motion. On the other hand, a large amount of thrust is abruptly generated at the end of upstroke motion. Vortical structure in the wake and the pressure field shows that vortex-pairing and vortex-staying mechanisms can be presented as strong evidence for the abrupt large thrust generation, which is fundamentally different from the inverse Karman vortex that is used to explain thrust generation by sinusoidal oscillating airfoil. Additional numerical simulations are conducted to examine the effects of motion components of figure-eight motion. From numerical results and comparisons, it is observed that wing rotational motion at the end of upstroke is crucial in generating the pairing and staying of vortices, which eventually leads to the abrupt thrust generation.

Design of Flapping Airfoil for Optimal Aerodynamic Performance in Low-Reynolds Number Flows

Unsteady, viscous, incompressible flows over an airfoil under flapping motion are numerically investigated. Depending on key parameters such as Reynolds number, reduced frequency, and flapping amplitudes, a flapping airfoil could experience complex flow fields. The trailing-edge vortex plays an important role to induce inverse Karman-vortex street which is a jetlike flow on the downstream and then generates thrust. And, the leading-edge separation vortex is closely related on the propulsive efficiency. Through careful computations of several pitching, plunging, and plunging combined with pitching modes in terms of flow and/or geometry parameters, the key physical flow phenomenon dictating the aerodynamic characteristics of flapping airfoil is identified. Based on the analysis of thrust coefficient and propulsive efficiency a new airfoil shape for optimal aerodynamic performance is proposed. The improved performance of the new flapping airfoil is validated in terms of thrust coefficient and propulsive efficiency in various low-Reynolds number flow regimes.

Computations of Homogeneous-Equilibrium Two-Phase Flows with Accurate and Efficient Shock-Stable Schemes

For accurate and efficient computations of compressible gas-liquid two-phase mixture flows, the AUSMPW+ and RoeM schemes (for which the accuracy, efficiency, and robustness have been successfully demonstrated in gas dynamics) are extended to two-phase flows at all speeds. From the mixture equations of state, a new shock-discontinuity-sensing term suitably scaled for two-phase flows is derived and its performance is validated. In addition, several numerical difficulties appearing in the development of the two-phase AUSMPW+ and RoeM schemes are analyzed and successfully cured. The two-phase AUSMPW+ and RoeM schemes are then efficiently preconditioned for the simulation of all Mach number flows by employing the existing AUSM or Harten-Lax-van Leer with contact restoration types of preconditioning strategies. Various gas-liquid two-phase flows, from highly compressible to nearly incompressible flow conditions, are tested. The numerical results show the accurate and robust behavior of the proposed schemes for all speeds of two-phase flows.

Feasibility Study of Constant Eddy-Viscosity Assumption in Gradient-Based 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 efe cient gradient-based optimization techniques. Accurate unsteady and steady e ow analyses are followed by the aerodynamic sensitivity analysis for theNavier‐ 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 transonicairfoilwithastrongshockandhigh-liftairfoilata highangleofattackcloseto stallangle.Simultaneously, however, thee nal design resultsindicate that both approachesare acceptable in engineering applications. Both the single- and multi-element airfoil design optimizations using the constant eddy-viscosity assumption are carefully assessedintermsofdesignaccuracy,computermemoryoverheads,andtotaldesigntimeinvariousdesignexamples.