Circular Arc Bump Research Articles

Comparison of Robustness and Efficiency for SIMPLE and CLEAR Algorithms with 13 High-Resolution Convection Schemes in Compressible Flows

In this article, a comparison is made between the robustness and efficiency of the CLEAR algorithm and the SIMPLE algorithm on nonorthogonal curvilinear coordinates for compressible flows. Thirteen different high-order convection schemes are employed in the calculations. Subsonic flow, transsonic flow, and supersonic flow in a channel with a circular arc bump and compressible flow in a Laval nozzle are used as test cases. The CLEAR algorithm shows huge potential to compute the transsonic flow in the Laval nozzle and high-speed compressible flows. Results with the ADBQUICKEST scheme, the HLPA scheme, and the MUSCL scheme are stable for both the compressible SIMPLE and CLEAR algorithms for all the mentioned cases.

Finite difference gas-kinetic BGK scheme for compressible inviscid flow computation

In this paper, the gas-kinetic Bhatnagar–Gross–Krook (BGK) scheme is developed using the finite difference method and applied to simulate 2D compressible inviscid flow. The numerical scheme employed in this study, namely the BGK scheme, is based on the collisional Boltzmann model. The high-order resolution of this scheme is obtained by utilising the monotone upstream-centred schemes for conservation law-type primitive variable reconstruction. As for the time integration part of the scheme, a multistage Runge–Kutta method is employed. In order to assess the computational characteristics of the numerical scheme, namely accuracy and robustness, three flow problems are selected in this study. They are supersonic channel flow, supersonic wedge cascade and circular arc bump. The computed results of these test cases are validated with available exact solutions and numerical results from the central difference scheme with total variation diminishing (TVD) formulation. It shows that the BGK scheme is an accurate and robust scheme in comparison to the central difference scheme with TVD when used to simulate 2D compressible inviscid flows.

Higher‐order bounded differencing schemes for compressible and incompressible flows

AbstractIn recent years, three higher‐order (HO) bounded differencing schemes, namely AVLSMART, CUBISTA and HOAB that were derived by adopting the normalized variable formulation (NVF), have been proposed. In this paper, a comparative study is performed on these schemes to assess their numerical accuracy, computational cost as well as iterative convergence property. All the schemes are formulated on the basis of a new dual‐formulation in order to facilitate their implementations on unstructured meshes. Based on the proposed dual‐formulation, the net effective blending factor (NEBF) of a high‐resolution (HR) scheme can now be measured and its relevance on the accuracy and computational cost of a HR scheme is revealed on three test problems: (1) advection of a scalar step‐profile; (2) 2D transonic flow past a circular arc bump; and (3) 3D lid‐driven incompressible cavity flow. Both density‐based and pressure‐based methods are used for the computations of compressible and incompressible flow, respectively. Computed results show that all the schemes produce solutions which are nearly as accurate as the third‐order QUICK scheme; however, without the unphysical oscillations which are commonly inherited from the HO linear differencing scheme. Generally, it is shown that at higher value of NEBF, a HR scheme can attain better accuracy at the expense of computational cost. Copyright © 2006 John Wiley & Sons, Ltd.

B311 低温風洞へのLifetime Imaging法の適用可能性

The intensity-based method is a most common technique by which Pressure-Sensitive Paint images can be converted to the surface pressure. However, this method cannot be applied to a cryogenic wind tunnel because the thermal equilibrium condition cannot be achieved when stopping the tunnel to take the wind-off reference image. In the present experiment, we applied the lifetime-based method to a cryogenic wind tunnel to calculate surface pressure from two images obtained during the wind-on condition. In this experiment, the circular-arc bump model sprayed with PtTFPP/PTMSP paint was tested at the JAXA 0.1m Cryogenic Transonic Wind Tunnel using a UV-LED pulse illuminator and a multi-gated CCD camera. As a result, we could succeed in visualizing surface pressure distribution induced by a shock wave on the model by using lifetime PSP imaging method. This result has demonstrated a feasibility of using lifetime imaging method to a cryogenic wind tunnel.

Adaptive Unstructured Mesh Refinement of Supersonic Channel Flows

An isotropic local mesh refinement technique for unstructured meshes is presented. The present work pertains to inviscid, steady, supersonic internal flows contained in a channel configuration. A density-based mesh refinement sensor function is used to identify regions in the mesh for refinement. A face-cell finite-volume method, permitting dynamic changes to mesh connectivity, is employed in the mesh refinement strategy. The AUSM+ convection scheme is used to compute the interface numerical flux using Mach number and pressure splitting functions. The present method is capable of recursive multi-level mesh adaption. Local mesh refinement equivalent to 64 and 256 times the coarse mesh resolution can be obtained using four and five levels of refinement, respectively. Results for two popular supersonic channel test cases are presented: Mach 1.4 flow over a 4% thick circular arc bump, and Mach 2.0 flow over a 10° compression ramp. The effort associated with the mesh refinement of the bump case accounts for only 4.6% of the total simulation time. For the ramp case, a factor of 4.2 for memory storage requirements and 7.7 for simulation time is required to obtain an equivalent uniform fine mesh solution.

Large-eddy simulation of transonic turbulent flow over a bump

Transonic turbulent boundary-layer flow over a circular-arc bump has been computed by high-resolution large-eddy simulation of the compressible Navier–Stokes equations. The inflow turbulence was prescribed using a new technique, in which known dynamical features of the inner and outer part of the boundary-layer were exploited to produce a standard turbulent boundary-layer within a short distance of the inflow. This method was separately tested for a flat plate turbulent boundary-layer, for which results compared well with direct numerical simulation databases. Simulation of the bump flow was carried out using high-order methods, with the dynamic Smagorinsky model used for sub-grid terms in the momentum and energy equations. Simulations were carried out at a Reynolds number of 233,000 based on bump length and free-stream properties upstream of the bump. At a back pressure equal to 0.65 times the stagnation pressure, a normal shock was formed near the bump trailing-edge and a peak mean Mach number of 1.16 was observed. Turbulence fluctuations decayed in the favourable pressure gradient region of the flow, before being amplified due to the shock interaction and boundary-layer separation. The effect of Reynolds number on turbulence intensity upstream of the shock is discussed in connection with a laminarisation parameter. With reference to turbulence modelling, anisotropy levels are not unreasonably high in the shock interaction region and shock unsteadiness was not found to be an issue. Of more relevance to the perceived poor performance of models for this type of flow may be the extremely rapid rise and decay of turbulence levels in the separated shear layer immediately under the shock-wave.

Novel Pressure-Sensitive Paint for Cryogenic and Unsteady Wind-Tunnel Testing

A novel pressure-sensitive paint (PSP) formulation suitable for use in cryogenic and unsteady wind-tunnel testing has been developed. This new PSP uses poly [1-(trimethylsilyl)-1-propyne] [poly(TMSP)] as a binder. Poly(TMSP) is a glassy polymer having extremely high gas permeability. Unlike the conventional polymer-based paints, PSP based on poly(TMSP) maintains oxygen sensitivity even at cryogenic temperatures and exhibits fast response time. This paint is sprayable on any model surface including stainless steel and ceramics. These capabilities allow us to apply this PSP to a cryogenic wind tunnel and a short-duration shock tunnel. To demonstrate the capability of poly(TMSP)-PSP for pressure measurements in a cryogenic wind tunnel, a circular-arc bump model and a delta wing model were tested in the National Aerospace Laboratory 0.1-m Cryogenic Wind Tunnel at 100 K. It has been verified that poly(TMSP)-PSP can provide high SIN pressure distribution data at high O 2 concentration. A comparison of the measured intensity data with the pressure tap measurements has suggested that the in situ calibration can be used to obtain quantitative pressure data

新色素を用いた感圧塗料の低温風洞への適用

A new type of pressure-sensitive paint for pressure measurement in a cryogenic wind tunnel is present. This paint is composed of Pyrene-based luminophores adsorbed onto anodized aluminum surface (AA-PSP). Various derivatives of Pyrene were calibrated by changing the mole fraction of oxygen in the 0.1 m Transonic Cryogenic Wind Tunnel (TCWT) at National Aerospace Laboratory (NAL). It has been verified that Pyrene-based AA-PSPs show high oxygen sensitivity in the 02 concentration range from 4 to 1000ppm. These AA-PSPs were applied to a 14%-thick circular arc-bump model and tested in TCWT at 100 Kelvin. Surface pressure distributions on the model were clearly imaged with high spatial resolution and S/N ratio. A comparison of the measured intensity data with the pressure tap measurements has suggested that the in situ calibration can be used to obtain quantitative pressure data.

Comparison of Relaxation Methods for the Euler Equations.

The performance of four relaxation methods for the compressible Euler equations is examined. The Euler equations are cast into the Euler implicit delta form. Four spatial discretization methods, the central difference (CD), total variation diminishing of Yee and Harten (TVD), flux difference splitting of Roe (FDS) and flux vector splitting of Van Leer (FVS), are applied to the spatial derivatives of the right-hand side. Flux splitting Gauss Seidel (FSGS), approximate flux splitting Gauss Seidel (AFSGS-I and AFSGS-II), and lower-upper symmetric Gauss Seidel (LU-SGS) methods are tested for the solution of the implicit equations. Numerical results are presented for a two-dimensional flow about a circular arc bump. Compatibility and efficiency of the numerical solutions are analyzed for all combinations between relaxation methods and spatial discretization methods. Steady-state solutions are independent of the relaxation methods. The AFSGS-II method gives the best efficiency for all spatial discretization methods.

An efficient shock capturing scheme for the pseudo-unsteady equations representing steady inviscid flow

An efficient algorithm is presented for the solution of the steady Euler equations of gas dynamics. The scheme is based on solving linearised Riemann problems approximately and in more than one dimension incorporates operator splitting. The scheme is applied to a standard test problem of flow down a channel containing a circular arc bump for three different mesh sizes.

Discontinuous Galerkin Finite Element Method for Euler and Navier-Stokes Equations

A finite element method for the Euler and Navier-Stokes equations has been developed. The spatial discretization involves the discontinuous Galerkin finite element method and Lax-Friedrichs flux method. The temporal discretizations used are the explicit Runge-Kutta time integrations. The scheme is formally second-order accurate in space and time. A dynamic mesh algorithm is included to simulate flows over moving bodies. The inviscid flows passing through a channel with circular arc bump, through the NACA 0012 airfoil, and the laminar flows passing over a flat plate with shock interaction are investigated to confirm the accuracy and convergence of the finite element method. Also the unsteady flow through a pitching NACA 0012 airfoil is performed to prove the capability of the present method.

An approximate riemann solver for the H-system in generalized coordinates

An approximate Riemann solver is presented for the solution of the pseudo-unsteady equations representing steady inviscid flow of an ideal gas, commonly known as the H-system. A simple modification enables nonideal gases to be treated. The scheme applies to the system in generalized coordinates enabling the use of body-fitted meshes. The scheme is applied to a standard test problem of flow down a channel containing a circular arc bump.

Adaptive finite volume upwind approach on mixed quadrilateral-triangular meshes

An adaptive cell-vertex finite volume upwind approach on unstructured mixed quadrilateral-triangular meshes, where the quadrilaterals are directionally stretched in the flow regions having one-dimensional features, has been developed to solve the unsteady Euler equations. In the present approach, the Runge-Kutta time integration, Roe's Riemann solver, MUSCL differencing with characteristic interpolation variables, a new treatment of the rotated extrapolation boundary condition, and a modified technique of the global/local regeneration for directionally stretched meshes are included. By using different combinations of interpolation variables, limiter functions, and numerical implementations of boundary conditions, a systematic study is made to understand the characteristics of the current approach. In this work, the isolated oblique shock problem, shock reflection at a wall, supersonic flow passing through a channel with a 4% circular arc bump, transonic flows around single- and two-element airfoils, as well as the shock propagation in a channel are investigated. It is concluded that the present solution procedure demonstrates good convergence performance and provides accurate and high-resolution results for the steady and unsteady inviscid flows.

Inviscid and Viscous Solutions for Airfoil/Cascade Flows Using a Locally Implicit Algorithm on Adaptive Meshes

A numerical solution procedure, which includes a locally implicit finite volume scheme and an adaptive mesh generation technique, has been developed to study airfoil and cascade flows. The Euler/Navier–Stokes, continuity, and energy equations, in conjunction with Baldwin-Lomax model for turbulent flow, are solved in the Cartesian coordinate system. To simulate physical phenomena efficiently and correctly, a mixed type of mesh, with unstructured triangular cells for the inviscid region and structured quadrilateral cells for the viscous, boundary layer, and wake regions, is introduced in this work. The inviscid flow passing through a channel with circular arc bump and the laminar flows over a flat plate with/without shock interaction are investigated to confirm the accuracy, convergence, and solution-adaptibility of the numerical approach. To prove the reliability and capability of the present solution procedure further, the inviscid/viscous results for flows over the NACA 0012 airfoil, NACA 65-(12)10 compressor, and one advanced transonic turbine cascade are compared to the numerical and experimental data given in related papers and reports.

Locally implicit total-variation-diminishing schemes on unstructuredtriangular meshes

A numerical solution procedure that includes locally implicit total-variation-diminishing schemes and adaptive mesh generation techniques has been developed in this work. In a Cartesian coordinate system, the Euler equations are solved by using a cell-centered finite volume algorithm. A new construction of symmetric total-variation-diminishing schemes on unstructured triangular meshes is presented. The validation of the present solution-adaptive methods is confirmed by comparison with related numerical results for inviscid flows around an isolated NACA 0012 airfoil and passing through a channel with a circular arc bump in transonic and supersonic flow regimes. To further prove the feasibility of this approach, a two-element airfoil flow is also investigated. Furthermore, one unsteady transonic channel flow is studied to demonstrate the reliability and capability of the present solution procedure for a time accurate calculation.

Uniformly second-order-accurate essentially nonoscillatory schemes for the Euler equations

Two time-level explicit and implicit finite-difference shock-capturin g schemes based on the characteristic flux difference splitting method and the modified flux approach with the essentially nonoscillatory (ENO) property of Harten and Osher have been developed for the two-dimensional Euler equations. The methods are conservative, uniformly second-order accurate in time and space, even at local extrema. General coordinate systems are used to treat complex geometries. Standard alternating direction implicit approximate factorization is used for constructing implicit schemes. Numerical results have been obtained for unsteady shock wave reflection around general two-dimensional blunt bodies and for steady transonic flows over a circular arc bump in a channel. Properties of ENO schemes as applied to two-dimensional flows with multiple embedded discontinuities are discussed. Comparisons of the performance between the present ENO schemes and our previous total variation diminishing schemes is also included.

Transonic, turbulent boundary-layer separation generated on an axisymmetric flow model

Experimental data describing the transonic, turbulent, separated flow generated by an axisymmetric flow model are presented. The model consisted of a circular-arc bump affixed to a straight, circular cylinder aligned with the flow direction. Measurements of the mean velocity, turbulence intensity, and Reynolds shear-stress profiles were made in the separated flow. These data revealed dramatic changes in the shear-stress levels as the flow passed through the interaction to reattachment. Behavior of the turbulence reaction to the imposed pressure gradients was examined in terms of the mixing length and the excursions of the turbulence from equilibrium.

Application of the Godunov method and its second-order extension to cascade flow modeling

The Godunov method and a new second-order accurate extension of the method are used for the solution of two-dimensional Euler equations. Both numerical schemes are described in detail. Their performances in the subsonic, transonic, and supersonic flow regimes are first tested on the problem of flow in a channel with a circular arc bump. The niethods are then applied to calculate the transonic flow through a supercritical com­ pressor cascade designed by J. Sanzo For this case, the solution with the second-order extension of the Godunov method gives verygood agreement with the design distribution of parameters given by Sanzo